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\begin{document}
\title{Stabbing Pairwise Intersecting Disks by Five Points\footnote{
A preliminary version appeared as
S.~Har-Peled, H.~Kaplan, W.~Mulzer, L.~Roditty, P.~Seiferth,
M.~Sharir, and M.~Willert.
\emph{Stabbing Pairwise Intersecting Disks by Five Points.}
\begin{abstract}
Suppose we are given a set $\mathcal{D}$ of $n$ pairwise
intersecting disks in the plane. A planar point set $P$
\emph{stabs} $\mathcal{D}$ if and only if each disk in
$\mathcal{D}$ contains at least one point from $P$. We present a
deterministic algorithm that takes $O(n)$ time to find five points
that stab $\mathcal{D}$. Furthermore, we give a simple example of
13 pairwise intersecting disks that cannot be stabbed by three
points. Moreover, we present a simple argument showing that
eight disks can be stabbed by at most three points.
This provides a simple---albeit slightly weaker---algorithmic
version of a classical result by Danzer that such a set
$\mathcal{D}$ can always be stabbed by four points.
\end{abstract}
\section{Introduction}
The \emph{maximum clique problem} is a classic problem in combinatorial
optimization~\cite{Karp72}: given a simple graph $G = (V, E)$,
find a maximum-cardinality
set $C \subseteq V$ of vertices such that
any two distinct vertices in $C$ are adjacent.
In 1972, Karp proved that the maximum
clique problem is NP-hard~\cite{Karp72}.
Even worse, a subsequent line of research
showed that the maximum clique problem is hard to
approximate. In particular,
we now know that for any fixed ${\varepsilon} > 0$, if there is a polynomial-time
algorithm that approximates maximum clique in an $n$-vertex graph
up to a factor of $n^{1-{\varepsilon}}$, then
$\text{P} = \text{NP}$~\cite{zuckerman2006linear}
However, if the
input graph has additional structure, the problem
can become easier. For example, if the input is the intersection graph of
a set of disks in the plane, the maximum clique problem admits
efficient (approximation)
algorithms: for unit disk graphs,
it can be solved in polynomial time~\cite{clark1990unit},
while for general disk intersection graphs, there is a
randomized EPTAS~\cite{bonamy2018eptas}. Earlier, Amb\"uhl
and Wagner~\cite{ambuhl2005clique} presented a polynomial-time
algorithm that computes
a $\tau/2$-approximation for the maximum clique in a general disk
intersection graph, where $\tau$ is the minimum \emph{stabbing number} of
any arrangement of pairwise intersecting disks in the plane, i.e.,
the minimum number of points that are needed to stab every disk in
such an arrangement.
Motivated by this
application,
our goal here is to understand this stabbing number better.
Let $\D$ be a set of $n$ disks in the plane. If every \emph{three}
disks in $\D$ intersect, then Helly's theorem shows that the whole
intersection $\bigcap\D$ of $\D$ is
nonempty~\cite{Helly23,Helly30,Radon21}. In other words, there is a
single point $p$ that lies in all disks of $\D$, that is, $p$
\emph{stabs} $\D$. More generally, when we know only that every
\emph{pair} of disks in $\D$ intersect, there must be a point set $P$
of constant size such that each disk in $\D$ contains at least one
point in $P$ -- the minimum cardinality of $P$ is the \emph{stabbing number}
of $\D$. It is indeed not surprising that $\D$ can be stabbed by a
constant number of points,
but for some time, the exact bound remained elusive. Eventually,
in July 1956 at an Oberwolfach seminar, Danzer presented the answer:
four points are always sufficient and sometimes necessary to stab any
finite set of pairwise intersecting disks in the plane.
Danzer was not satisfied with his original
argument, so he never formally published it. In 1986, he presented a
new proof~\cite{Danzer86}. Previously, in 1981, Stach\'o\xspace had already
given an alternative proof~\cite{Stacho81}, building on a previous
construction of five stabbing points~\cite{Stacho65}. This line of
work was motivated by a result of Hadwiger and Debrunner, who showed
that three points suffice to stab any finite set of pairwise
intersecting \emph{unit} disks~\cite{HadwigerDe55}. In later work,
these results were significantly generalized and extended, culminating
in the celebrated $(p, q)$-theorem that was proven by Alon and
Kleitman in 1992~\cite{AlonKl92}.
See also a recent paper by
Dumitrescu and Jiang that studies generalizations of the stabbing
problem for translates and homothets of a convex
body~\cite{DumitrescuJi11}.
Danzer's published proof~\cite{Danzer86} is fairly involved.
It uses a compactness argument that does not seem to be
constructive, and one part of the argument relies on
an underspecified
verification by computer. Therefore, it is
quite challenging to check the correctness
of the argument, let alone to derive any intuition
from it.
There seems to be no obvious way to turn it into
an efficient algorithm for finding a stabbing set of size four. The
proof of Stach\'o\xspace~\cite{Stacho81} is simpler, but
it is obtained through a lengthy case analysis that requires
a very disciplined and focused reader. Here, we present a new
argument that yields five stabbing points. Our proof is constructive,
and it lets us find the stabbing set in deterministic linear time.
Following the conference version of this paper,
Carmi, Katz, and Morin published a manuscript
in which they present an algorithm that can find four
stabbing points
in linear time~\cite{CarmiKaMo18}.
As for lower bounds, Gr\"unbaum\xspace gave an example of 21 pairwise
intersecting disks that cannot be stabbed by three
points~\cite{Grunbaum59}. Later, Danzer reduced the number of disks to
ten~\cite{Danzer86}. This example is close to optimal, because every
set of eight disks can be stabbed by three points, as mentioned
by Stach\'o\xspace~\cite{Stacho65} and
formally proved in Section~\ref{sec:simple_bounds} below. However, it
is hard to verify Danzer's lower bound example---even with dynamic
geometry software, the positions of the disks cannot be visualized
easily.
We present a new and simple proof that shows
that the stabbing number of $\D$ is
upper bounded by $5$. Moreover, we obtain a linear time algorithm
that can find these $5$ stabbing points. Finally, we present a simple
construction of $13$ pairwise intersecting disks that cannot be stabbed
by $3$ points, and work out a proof of Stach\'o\xspace's eight-disk claim.
\section{The Geometry of Pairwise Intersecting Disks}
\label{sec:geometry}
Let $\D$ be a set of $n$ pairwise intersecting disks in the plane. A
disk $D_i \in \D$ is given by its center $c_i$ and its radius
$r_i$.
To simplify the analysis, we make the following assumptions:
(i) the radii of the disks are pairwise distinct; (ii) the
intersection of any two disks has a nonempty interior; and (iii) the
intersection of any three disks is either empty or has a nonempty
interior. A simple perturbation argument can then handle the
degenerate cases.
\begin{figure}\label{fig:3_lens}
\end{figure}
The \emph{lens} of two disks $D_i, D_j \in \D$ is the set
$L_{i, j} = D_i \cap D_j$. Let $u$ be any of the two intersection
points of the boundary of $D_i$ and the boundary of $D_j$.
The angle $\angle c_iuc_j$ is
called the \emph{lens angle} of $D_i$ and $D_j$. It is at most
$\pi$. A finite set $\mathcal{C}$ of disks is \emph{Helly} if their
common intersection $\bigcap \mathcal{C}$ is nonempty. Otherwise,
$\mathcal{C}$ is \emph{non-Helly}. We present some useful geometric
lemmas.
\begin{lemma}
\label{lem:triple}
Let $\{D_1, D_2, D_3\}$ be a set of three pairwise intersecting
disks that is non-Helly. Then, the set contains two disks with
lens angle larger than $2 \pi/3$.
\end{lemma}
\begin{proof}
Since $\{D_1, D_2, D_3\}$ is non-Helly, the lenses $L_{1, 2}$,
$L_{1, 3}$ and $L_{2, 3}$ are pairwise disjoint. Let $u$ be the
vertex of $L_{1, 2}$ nearer to $D_{3}$, and let $v$, $w$ be the
analogous vertices of $L_{1,3}$ and $L_{2,3}$ (see
\cref{fig:3_lens}, left). Consider the simple hexagon
$c_1 u c_2 w c_3v$, and write $\angle u$, $\angle v$, and
$\angle w$ for its interior angles at $u$, $v$, and $w$. The sum
of all interior angles is $4\pi$. Thus,
$\angle u + \angle v + \angle w < 4\pi$, so at least one angle is
less than $4\pi/3$. It follows that the corresponding lens angle,
which is the exterior angle at $u$, $v$, or $w$ must be larger
than $2\pi/3$.
\end{proof}
\begin{lemma}
\label{lem:subset}
Let $D_1$ and $D_2$ be two intersecting disks with $r_1 \geq r_2$
and lens angle at least $2 \pi / 3$. Let $E$ be the unique disk
with radius $r_1$ and center $c$, such that
\begin{enumerate}
\item[(i)] the centers $c_1$,
$c_2$, and $c$ are collinear and $c$ lies on the same side of
$c_1$ as $c_2$; and
\item[(ii)] the lens angle of $D_1$ and $E$ is
exactly $2\pi/3$ (see \cref{fig:3_lens}, right).
\end{enumerate}Then, if $c_2$
lies between $c_1$ and $c$, we have $D_2 \subseteq E$.
\end{lemma}
\begin{proof}
Let $x \in D_2$. Since $c_2$ lies between $c_1$ and $c$, the
triangle inequality gives
\begin{equation}
\label{eq:x:c}
|xc| \leq |xc_2| + |c_2c| = |xc_2| + |c_1c| - |c_1c_2|.
\end{equation}
Since $x \in D_2$, we get $|xc_2| \leq r_2$. Also, since $D_1$
and $E$ have radius $r_1$ each and lens angle $2\pi/3$, it follows
that $|c_1c| = \sqrt{3} \, r_1$. Finally,
$|c_1c_2| = \sqrt{r_1^2+r_2^2-2r_1r_2\cos\alpha}$, by the law of
cosines, where $\alpha$ is the lens angle of $D_1$ and $D_2$. As
$\alpha \geq 2\pi/3$ and $r_1 \geq r_2$, we get
\begin{math}
\cos\alpha
\leq
-1/2
= (\sqrt{3}-3/2)-\sqrt{3}+1
\leq
(\sqrt{3}-3/2){r_1}/{r_2}-\sqrt{3}+1,
\end{math}
As such, we have
\begin{align*}
|c_1 c_2|^2
&=
r_1^2 + r_2^2 - 2r_1r_2\cos\alpha
\geq
r_1^2 + r_2^2 - 2r_1r_2 {\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}igl( \bigl(\sqrt{3} - 3/2 \bigr)
\frac{r_1}{r_2} - \sqrt{3} + 1{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}igr)
\\&
=
r_1^2 - 2\bigl(\sqrt{3} - 3/2 \bigr)
r_1^2 +2 ( - \sqrt{3} + 1)r_1 r_2 + r_2^2
\\&
=
(1 - 2\sqrt{3} + 3 )r_1^2 +2( - \sqrt{3} + 1)r_1 r_2 + r_2^2
=
\bigl(r_1(\sqrt{3}-1)+r_2 \bigr)^2.
\end{align*}
Plugging this into Equation~\ref{eq:x:c} gives
$|xc| \leq r_2 +
\sqrt{3}r_1-(r_1\left(\sqrt{3}-1)+r_2\right)=r_1$, i.e.,
$x \in E$.
\end{proof}
\begin{lemma}
\label{lem:D:E}
Let $D_1$ and $D_2$ be two intersecting disks with equal radius
$r$ and lens angle $2\pi/3$. There is a set $P$ of four points so
that any disk $F$ of radius at least $r$ that intersects both
$D_1$ and $D_2$ contains a point of $P$.
\end{lemma}
\begin{proof}
Consider the two tangent lines of $D_1$ and $D_2$, and let $p$ and
$q$ be the midpoints on these lines between the respective two
tangency points. We set $P = \{ c_1, c_2, p, q\}$; see
\cref{fig:P:covers}.
\begin{figure}\label{fig:P:covers}
\end{figure}
Given the disk $F$ that intersects both $D_1$ and $D_2$, we shrink
its radius, keeping its center fixed, until either the radius
becomes $r$ or until $F$ is tangent to $D_1$ or $D_2$. Suppose the
latter case holds and $F$ is tangent to $D_1$. We move the center
of $F$ continuously along the line spanned by the center of $F$
and $c_1$ towards $c_1$, decreasing the radius of $F$ to maintain
the tangency. We stop when either the radius of $F$ reaches $r$ or
$F$ becomes tangent to $D_2$. We obtain a disk $G \subseteq F$
with center $c = (c_x, c_y)$ so that either: (i)
$\text{radius}(G) = r$ and $G$ intersects both $D_1$ and $D_2$; or
(ii) $\text{radius}(G) \geq r$ and $G$ is tangent to both $D_1$
and $D_2$. Since $G \subseteq F$, it suffices to show that
$G \cap P \neq \emptyset$.
We introduce a coordinate system, setting the origin $o$ midway
between $c_1$ and $c_2$, so that the $y$-axis passes through $p$
and $q$. Then, as in \cref{fig:P:covers}, we have
$c_1 = (-\sqrt{3}\,r / 2, 0)$, $c_2 = (\sqrt{3}\,r / 2, 0)$,
$q = (0, r)$, and $p = (0, -r)$.
For case (i), let $D_1^2$ be the disk of radius $2r$ centered at
$c_1$, and $D_2^2$ the disk of radius $2r$ centered at
$c_2$. Since $G$ has radius $r$ and intersects both $D_1$ and
$D_2$, its center $c$ has distance at most $2r$ from both $c_1$
and $c_2$, i.e., $c \in D_1^2 \cap D_2^2$. Let $D_p$ and $D_q$ be
the two disks of radius $r$ centered at $p$ and $q$. We will show
that $D_1^2 \cap D_2^2 \subseteq D_1 \cup D_2 \cup D_p \cup D_q$.
Then it is immediate that $G \cap P \neq \emptyset$. By symmetry,
it is enough to focus on the upper-right quadrant
$Q = \{(x,y) \mid x \geq 0, y \geq 0\}$. We show that all points
in $D_1^2 \cap Q$ are covered by $D_2 \cup D_q$. Without loss of
generality, we assume that $r = 1$. Then, the two intersection
points of $D_1^2$ and $D_q$ are
$t_1 = (\frac{5\sqrt{3} - 2\sqrt{87}}{28}, \frac{38 +
3\sqrt{29}}{28}) \approx (-0.36, 1.93)$ and
$t_2 = (\frac{5\sqrt{3} + 2\sqrt{87}}{28}, \frac{38 -
3\sqrt{29}}{28}) \approx (0.98, 0.78)$, and the two
intersection points of $D_1^2$ and $D_2$ are
$s_1 = (\frac{\sqrt{3}}{2}, 1) \approx (0.87, 1)$ and
$s_2 = (\frac{\sqrt{3}}{2}, -1) \approx (0.87, -1)$. Let $\gamma$
be the boundary curve of $D_1^2$ in $Q$. Since
$t_1, s_2 \not\in Q$ and since $t_2 \in D_2$ and $s_1 \in D_q$, it
follows that $\gamma$ does not intersect the boundary of
$D_2 \cup D_q$ and hence $\gamma \subset D_2 \cup D_q$.
Furthermore, the subsegment of the $y$-axis from $o$ to the start
point of $\gamma$ is contained in $D_q$, and the subsegment of the
$x$-axis from $o$ to the endpoint of $\gamma$ is contained in
$D_2$. Hence, the boundary of $D_1^2 \cap Q$ lies completely in
$D_2 \cup D_q$, and since $D_2 \cup D_q$ is simply connected, it
follows that $D_1^2 \cap Q \subseteq D_2 \cup D_q$, as desired.
For case (ii), since $G$ is tangent to $D_1$ and $D_2$, the center
$c$ of $G$ is on the perpendicular bisector of $c_1$ and $c_2$, so
the points $p$, $o$, $q$ and $c$ are collinear. Suppose without
loss of generality that $c_y\geq 0$. Then, it is easily checked
that $c$ lies above $q$, and
$\text{radius}(G) + r= |c_1c| \geq |oc|=r + |qc|$, so $q \in G$.
\end{proof}
\begin{figure}\label{fig:C:D:diff:size}
\end{figure}
\begin{lemma}
\label{lem:C:D:diff:size}
Consider two intersecting disks $D_1$ and $D_2$ with $r_1\geq r_2$
and lens angle at least $2\pi/3$. Then, there is a set $P$ of
four points such that any disk $F$ of radius at least $r_1$ that
intersects both $D_1$ and $D_2$ contains a point of $P$.
\end{lemma}
\begin{proof}
Let $\ell$ be the line through $c_1$ and $c_2$. Let $E$ be the
disk of radius $r_1$ and center $c \in \ell$ that satisfies the
conditions (i) and (ii) of \cref{lem:subset}. Let
$P = \{c_1, c, p, q\}$ as in the proof of \cref{lem:D:E}, with
respect to $D_1$ and $E$ (see \cref{fig:3_lens}, right). We claim
that
\[
D_1\cap F\neq\emptyset\ \wedge\ D_2\cap F\neq\emptyset\
\Rightarrow\ E\cap F\neq\emptyset. \tag{*}
\]
Once (*) is established, we are done by \cref{lem:D:E}. If
$D_2\subseteq E$, then (*) is immediate, so assume that
$D_2 \not\subseteq E$. By \cref{lem:subset}, $c$ lies between $c_1$
and $c_2$. Let $k$ be the line through $c$ perpendicular to
$\ell$, and let $k^+$ be the open halfplane bounded by $k$ with
$c_1 \in k^+$ and $k^-$ the open halfplane bounded by $k$ with
$c_1 \not\in k^-$. Since $|c_1c| = \sqrt{3}\,r_1 > r_1$, we have
$D_1 \subset k^+$; see \cref{fig:C:D:diff:size}. Recall that $F$
has radius at least $r_1$ and intersects $D_1$ and $D_2$. We
distinguish two cases: (i) there is no intersection of $F$ and
$D_2$ in $k^+$, and (ii) there is an intersection of $F$ and $D_2$
in $k^+$; see \cref{fig:C:D:diff:size} for the two cases.
For case (i), let $x$ be any point in $D_1 \cap F$. Since we know
that $D_1 \subset k^+$, we have $x\in k^+$. Moreover, let $y$ be
any point in $D_2 \cap F$. By assumption, $y$ is not in $k^+$,
but it must be in the infinite strip defined by the two tangents
of $D_1$ and $E$. Thus, the line segment $\overline{xy}$
intersects the diameter segment $k\cap E$. Since $F$ is convex,
the intersection of $\overline{xy}$ and $k\cap E$ is in $F$, so
$E \cap F \neq \emptyset$.
For case (ii), fix $x \in D_2\cap F \cap k^+$ arbitrarily.
Consider the triangle $\Delta xcc_2$. Since $x \in k^+$, the angle
at $c$ is at least $\pi/2$. Thus,
$|xc| \leq |xc_2|$. Also, since $x \in D_2$, we know that
$|xc_2| \leq r_2 \leq r_1$. Hence, $|xc| \leq r_1$, so $x \in E$
and (*) follows, as $x \in E \cap F$.
\end{proof}
\section{Existence of Five Stabbing Points}
With these tools we can now show that
there is a stabbing set with five points.
\begin{theorem}
\label{thm:existence}
Let $\D$ be a set of $n$ pairwise intersecting disks in the plane.
There is a set $P$ of five points such that each disk in $\D$
contains at least one point from $P$.
\end{theorem}
\begin{proof}
If $\D$ is Helly, there is a single point that lies in all disks
of $\D$. Thus, assume that $\D$ is non-Helly, and let
$D_1, D_2, \dots, D_n$ be the disks in $\D$ ordered by increasing
radius. Let $i^*$ be the smallest index with
$\bigcap_{i \leq i^*} D_i = \emptyset$. By Helly's
theorem~\cite{Helly23,Helly30,Radon21}, there are indices
$j, k < i^*$ such that $\{D_{i^*}, D_j, D_k\}$ is non-Helly. By
\cref{lem:triple}, two disks in $\{D_{i^*}, D_j, D_k\}$ have
lens angle at least $2\pi/3$. Applying
\cref{lem:C:D:diff:size} to these two disks, we obtain a set
$P'$ of four points so that every disk $D_i$ with $i \geq i^*$
contains at least one point from $P'$. Furthermore, by definition
of $i^*$, we have $\bigcap_{i < i^*} D_i \neq \emptyset$, so there
is a point $q$ that stabs every disk $D_i$ with $i < i^*$. Thus,
$P = P' \cup \{q\}$ is a set of five points that stabs every disk
in $\D$, as desired.
\end{proof}
\paragraph{Remark.} A weakness in our proof is that it combines two
different stages, one
of finding the point $q$ that stabs all the small disks, and one of
constructing the four points of \cref{lem:C:D:diff:size} that stab all
the larger disks. It is an intriguing challenge to merge the two
arguments so that altogether they only require four points. The proof of
Carmi et al.~\cite{CarmiKaMo18} uses a different approach.
\section{Algorithmic Considerations}
The proof of \cref{thm:existence} leads to a simple $O(n \log n)$ time
algorithm for finding a stabbing set of size five.
For this, we need an oracle that decides whether a given
set of disks is Helly. This has already been done by
L\"{o}ffler and van Kreveld~\cite{loffler2010largest}, in a
more general context:
\begin{lemma}[Theorem~6 in \cite{loffler2010largest}]
\label{lem:helly-oracle}
Given a set of $n$ disks, the problem of choosing a point in
each disk such that the smallest enclosing circle of the resulting
point set has minimum radius can be solved in $O(n)$ deterministic time.
\end{lemma}
Now, an $O(n\log n)$-time algorithm for finding the five
stabbing points is based on
the analysis in the proof of \cref{thm:existence}. It works as follows:
first, we sort the disks in $\D$ by increasing radius.
This takes $O(n\log n)$ time. Let
$\D = \langle D_1, \dots, D_n \rangle$ be the resulting order.
Next, we use binary search with the oracle
from \cref{lem:helly-oracle} to determine the smallest index $i^*$ such that
the prefix $\{D_1, \dots ,D_{i^*}\}$ is non-Helly.
This yields the disk $D_{i^*}$. We have to invoke
the oracle $O(\log n)$ times, which
gives a total time of $O(n\log n)$ for this step. After that,
we use another binary search with the oracle
from \cref{lem:helly-oracle} to determine
the smallest index $k < i^*$ such that
$\{D_{i^*}, D_1, \dots, D_k\}$ is non-Helly. This costs
$O(n\log n)$ time as well.
Then, we perform a linear search to find an index $j < k$ such that
$\{D_j,D_k,D_{i^*}\}$
is a non-Helly triple. This step works in $O(n)$ time. Finally, we use
\cref{lem:helly-oracle} to obtain in $O(n)$ time a stabbing point
$q$ for the Helly set
$\{D_1, \dots ,D_{i^* - 1}\}$ and
the method from the proof of \cref{thm:existence} to
extend $q$ to a stabbing set for the whole set $\D$. This
last step works in $O(1)$ time since
the result depends solely on $\{D_j, D_k, D_{i^*}\}$.
Hence, we can state our claimed theorem.
\begin{theorem}
Given a set $\D$ of $n$ pairwise intersecting disks in the plane,
we can find in $O(n\log n)$ time a set $P$ of five points such that
every disk of $\D$ contains at least one point of $P$.
\end{theorem}
The proof of \cref{lem:helly-oracle} uses the \emph{LP-type framework} by
Sharir and Welzl~\cite{Chazelle01,SharirWe92}. As we will see next, a
more sophisticated application of the framework directly leads to a
deterministic linear time algorithm to find a stabbing set with
five points.
\paragraph*{The LP-type framework.} An \emph{LP-type problem}
$(\H, w, \leq)$ is an abstract generalization of a low-dimensional
linear program. It consists of a finite set of \emph{constraints}
$\H$, a \emph{weight function} $w: 2^{\H} \rightarrow {\cal W}} \def\R{{\mathbb R}} \def\N{{\mathbb N}$, and a
\emph{total order} $({\cal W}} \def\R{{\mathbb R}} \def\N{{\mathbb N}, \leq)$ on the weights. The weight function
$w$ assigns a weight to each subset of constraints. It must fulfill
the following two axioms:
\begin{itemize}
\item \textbf{Monotonicity}: for any $\H' \subseteq \H$ and
$H \in \H$, we have $w\big(\H' \cup \{H\}\big) \leq w(\H')$;
\item \textbf{Locality:} for any ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y} \subseteq \H' \subseteq \H$
with $w({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}) = w(\H')$ and for any $H \in \H$, we have that if
$w\big({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y} \cup \{H\}\big) = w({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y})$, then also
$w\big(\H' \cup \{H\}\big) = w(\H')$.
\end{itemize}
Given a subset $\H' \subseteq \H$, a \emph{basis} for
$\H'$ is an inclusion-minimal set ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y} \subseteq \H'$ with
$w({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}) = w(\H')$. The \emph{combinatorial dimension} of $(\H,w,\leq)$
is the maximum size of any basis of any subset of $H$.
The goal in an LP-type problem is to determine
$w(\H)$ and a corresponding basis ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$ for $\H$.
Next, given a set ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y} \subseteq \H$ and
a constraint $H \in \H$, we say that $H$ \emph{violates} ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$ if
$w\big({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y} \cup \{H\}\big) < w({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y})$.
A generalization of Seidel's algorithm for low-dimensional linear
programming~\cite{Seidel91,SharirWe92} shows that we can solve an
LP-type problem
in $O(|\H|)$ expected time, provided that a constant time algorithm for the
following problem is available. Here and below, the constant factor in the
$O$-notation may depend on the combinatorial dimension.
\begin{itemize}
\item \textbf{Violation test:} Given a basis ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$ and a constraint $H\in\H$,
determine whether $H$ violates ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$
and return an error message if ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$ is not a basis for any
$\H'\subseteq\H$.\footnote{Here, we follow the presentation of
Chazelle and \Matousek~\cite{ChazelleMa96}. Sharir and
Welzl~\cite{SharirWe92}
use a violation test without the error message. Instead, they need an
additional \emph{basis computation} primitive: given a basis ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$
and a constraint $H \in \H$, find a basis for ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y} \cup
\{H\}$. If a violation test with error message exists and if the
combinatorial dimension
is a constant, a basis computation primitive can easily be
implemented by brute-force enumeration.}
\end{itemize}
For a deterministic solution, we need an additional computational
assumption. Let ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}\subseteq\H$ be a basis
of any subset $\H'\subseteq \H$, we use $\vio({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y})$ to denote the set of
all constraints
$H\in\H$ that violate ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$, i.e., that have $w({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}\cup\{H\})<w({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y})$. Consider
the \emph{range space}
$(\H, \Rr=\{\vio({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}) \mid {\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y} \text{ is a basis for some }\H' \subseteq \H\})$.
For a subset $\Y\subseteq\H$, we let $(\Y,\Rr_{\Y})$ be
the \emph{induced range space}, that is,
$\Rr_{\Y}=\{\Y \cap R \mid R\in\Rr\}$.
Chazelle and \Matousek~\cite{ChazelleMa96}
have shown that an LP-type problem can be solved in
$O(\vert\H\vert)$ \emph{deterministic} time
if there is a constant-time violation test as stated above
and the following computational assumption holds:
\begin{itemize}
\item \textbf{Oracle:} Given a subset $\Y\subseteq\H$, we can
compute some superset $\Rr'\supseteq\Rr_{\Y}$
in time $\vert\Y\vert^{O(1)}$.
\end{itemize}
During the following discussion, we will show that the problem of
finding a non-Helly triple as in
\cref{thm:existence} is LP-type and fulfills the four requirements for the
algorithm of Chazelle and \Matousek.
\begin{figure}\label{fig:lptype}
\end{figure}
\paragraph*{Remark.} L\"offler and van Kreveld provide
proofs that the underlying problem in \cref{lem:helly-oracle} is of
LP-type,
but they do not give arguments for the two computational assumptions,
see~\cite{loffler2010largest}. However, it is not difficult
to also verify the two missing statements.
\paragraph*{Geometric observations.} The \emph{distance} between
two closed sets $A, B \subseteq \R^2$ is defined as
$d(A, B) = \min\ \{\vert ab\vert \mid a \in A, b \in B\}$. From now on, we
assume that all points in $\bigcup\D$ have positive
$y$-coordinates. This can be ensured with linear overhead by an
appropriate translation of the input. We denote by $D_{\infty}$ the
closed halfplane below the $x$-axis. It is interpreted as a disk with
radius $\infty$ and center at $(0, -\infty)$. First, observe that for
any subsets ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_1 \subseteq {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2 \subseteq \D\cup\{D_\infty\}$,
we have that if ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_1$ is non-Helly, then
${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2$ is non-Helly. For any ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R} \subseteq \D \cup \{D_\infty\}$,
we say that a disk $D$ \emph{destroys}
${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ if ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R} \cup \{ D \}$ is non-Helly. Observe that $D_{\infty}$
destroys every non-empty subset of $\D$. Moreover, if ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ is
non-Helly, then every disk is a destroyer.
See \cref{fig:lptype} for an example. We can make the
following two observations.
\begin{lemma}
\label{lem:unique:v}
Let ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R} \subseteq \D$ be Helly and $D$ a destroyer of ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$.
Then, the point $v \in \bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ with minimum distance to
$D$ is unique.
\end{lemma}
\begin{proof}
Suppose there are two distinct points $v \neq w \in \bigcap {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$
with $d(v, D) = d\big(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}, D \big) = d(w, D)$. Since
$\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ is convex, the segment $\overline{vw}$ lies in
$\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$. Now, if $D \neq D_\infty$, then every point in the
relative interior of $\overline{vw}$ is strictly closer to $D$
than $v$ and $w$. If $D = D_\infty$, then all points in
$\overline{vw}$ have the same distance to $D$, but since
$\bigcap {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ is strictly convex, the relative interior of
$\overline{vw}$ lies in the interior of $\bigcap {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$, so there
must be a point in $\bigcap {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ that is closer to $D$ than $v$ and
$w$. In either case, we obtain a contradiction to the assumption
$v \neq w$ and $d(v, D) = d\big(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}, D\big) = d(w, D)$. The
claim follows.
\end{proof}
Let ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}\subseteq \D$ be Helly and $D$ a destroyer of ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$. The unique point
$v \in \bigcap {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ with minimum distance to $D$ is called the
\emph{extreme point} for ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ and $D$ (see \cref{fig:lptype}, right).
\begin{figure}\label{fig:destroyers}
\end{figure}
\begin{lemma}
\label{lem:destroyer:dist}
Let ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_1 \subseteq {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2 \subseteq\D$ be two Helly sets and
$D$ a destroyer of ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_1$ (and thus of ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2$). Let
$v \in \bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_1$ be the extreme point for ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_1$ and $D$. We
have $d\big(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_1, D\big) \leq d\big(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2, D\big)$.
In particular, if $v \in \bigcap {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2$, then
$d\big(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_1, D\big) = d\big(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2, D\big)$ and $v$ is
also the extreme point for ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2$ and $D$. If $v \not\in \bigcap {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2$,
then $d\big(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_1, D\big) < d\big(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2, D\big)$.
\end{lemma}
\begin{proof}
The first claim holds trivially: let $w \in \bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2$ be the
extreme point for ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2$ and $D$. Since ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_1 \subseteq {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2$, it
follows that $w \in \bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_1$, so
$d\big(\bigcap {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_1, D \big) \leq d(w, D) = d\big(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2,
D\big)$. If $v \in \bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2$, then
$d\big(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_1, D\big) \leq d\big(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2, D\big) \leq
d(v, D) = d\big(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_1, D\big)$, so $v = w$, by
\cref{lem:unique:v}. If $v \notin\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2$, then
$d\big(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_1, D\big) < d\big(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2, D\big)$, by
\cref{lem:unique:v} and the fact that ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_1 \subseteq {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_2$.
See \cref{fig:destroyers}.
\end{proof}
Let ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ be a subset of $\D$. For $0 < r \leq \infty$ we
define ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r}$ as the set of all disks in
${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ with radius smaller than $r$. Recall that we assume
that all the radii are pairwise distinct.
A disk $D$ with radius $r$, $0 < r \leq \infty$, is called
\emph{smallest destroyer} of ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ if (i) $D \in {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ or $D=D_{\infty}$,
(ii) $D$ destroys ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r}$, and (iii) there is no disk $D'\in{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r}$
that destroys ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r}$. Observe that Property~(iii)
is the same as saying that ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r}$ is Helly.
See \cref{fig:lptype} for an example.
Let ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ be a subset of $\D$ and $D$ the
smallest destroyer of ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$.
We write $\rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$ for the radius of $D$
and $\dist({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$ for
the distance between $D$ and the set $\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<\rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})}$, i.e.,
$\dist({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})= d\big(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<\rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})}, D\big)$. Now, if ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ is Helly,
then $D=D_{\infty}$ and thus $\rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}) = \infty$.
If ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ is non-Helly,
then $D \in {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ and thus $\rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})<\infty$.
In both cases, $\dist({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$ is the distance between $D$ and
the extreme point for ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<\rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})}$ and $D$. We define the
\emph{weight of ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$} as $w({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}) = (\rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}),-\dist({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}))$, and
we denote by $\leq$ the lexicographic order on $\R^2$. Chan
observed, in a slightly different context, that $(\D, w ,\leq)$ is
LP-type~\cite{Chan04}. However, Chan's paper does not contain a
detailed proof for this fact. Thus, in the following lemmas, we show
the two LP-type axioms, present a constant time violation test,
and a polynomial-time oracle. We start with the monotonicity axiom
followed by the locality axiom.
\begin{figure}\label{fig:axiom1}
\end{figure}
\begin{lemma}
\label{lem:LPtype:axiom1}
For any ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R} \subseteq \D$ and $E \in \D$, we have
$w\big({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R} \cup\{E\}\big) \leq w({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$.
\end{lemma}
\begin{proof}
Set ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^* = {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R} \cup \{E\}$. Let $D$ be the smallest destroyer of
${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$, and let $r = \rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$ be the radius of $D$.
Since $D$ destroys ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{< r}$, the set ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{< r}\cup\{D\}$ is
non-Helly. Moreover, since ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{< r}\cup\{D\}\subseteq {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*_{< r}\cup\{D\}$,
we know that ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*_{< r}\cup\{D\}$ is also non-Helly.
Therefore, $D$ destroys ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*_{< r}$ and we can derive
$\rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*) \leq \rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$. If we have
$\rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*) < \rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$, we are done. Hence, assume that
$\rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*) = \rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$. Then $D$ is the smallest destroyer of ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*$,
and \cref{lem:destroyer:dist} gives
$-\dist({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*) = -d\big(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*_{<r}, D\big) \leq -
d(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r}, D) = -\dist({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$. Hence,
$w\big({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*) \leq w({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$. See \cref{fig:axiom1} for an
illustration.
\end{proof}
\begin{lemma}
\label{lem:LPtype:axiom3}
Let ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y} \subseteq {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R} \subseteq\D$ with $w({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}) = w({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$ and let
$E \in \D$. Then, if $w \big({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y} \cup \{E\}\big) = w({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y})$, we also
have $w\big({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R} \cup \{E\}\big) = w({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$.
\end{lemma}
\begin{proof}
Set ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^* = {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R} \cup \{E\}$, ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}^* = {\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}\cup \{E\}$. Let
$r = \rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$ and $D$ be the smallest destroyer of ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$.
Since $w({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}) = w({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}) = w({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}^*)$, we have that $D$ is also the
smallest destroyer of ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$ and of ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}^*$.
If the radius of $E$ is larger than $r$, then $E$ cannot be the smallest
destroyer of ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*$, so $w\big({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*\big) = w({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$.
Thus, assume that $E$ has radius less than $r$. Let $v$ be the
extreme point of
${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r}$ and $D$. Since $w({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}^*) = w({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y})$, we know that
$d\big(\bigcap{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}_{<r}, D\big) = d\big(\bigcap{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}^*_{<r}, D\big) =
d(v, D)$. Now, \cref{lem:destroyer:dist} implies that $v \in E$,
since $E \in {\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}^*_{<r}$. Thus, the set
${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*_{<r} = {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r} \cup \{E\}$ is Helly and therefore, there is no
disk $D'\in {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*_{<r}$ that destroys ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*_{<r}$.
Furthermore, since $D$ destroys ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r}$ and
${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r} \subset {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*_{<r}$, the disk $D$ also destroys ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*_{<r}$.
Therefore, $D$ is also the smallest destroyer of ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*$, so
$\rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*) = r = \rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$. Finally, since
${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}^*_{<r} \subseteq {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*_{<r}$ we can use
\cref{lem:destroyer:dist} to derive
\[
d{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}ig(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r}, D{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}ig) = d{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}ig(\bigcap{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}^*_{<r}, D{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}ig)
\leq d{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}ig(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}^*_{<r}, D{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}ig) \leq d(v, D) =
d{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}ig(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r}, D{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}ig).
\]
The claim follows.
\end{proof}
Next, we are going to describe the violation test for $(\D, w, \leq)$: given
a basis
${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y} \subseteq \D$ and a disk $E \in \D$, check whether $E$
violates ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$, i.e., whether $w\big({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y} \cup \{E\}\big) < w({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y})$, and
return an error message
if ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$ is not a basis. But first, we show that the combinatorial dimension of
$(\D, w, \leq)$ is at most $3$.
\begin{figure}\label{fig:axiom2}
\end{figure}
\begin{lemma}
\label{lem:LPtype:axiom2}
For each ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R} \subseteq \D$, there is a set ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y} \subseteq {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ with
$|{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}| \leq 3$ and $w({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}) = w({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$.
\end{lemma}
\begin{proof}
Let $D$ be the smallest destroyer of ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$. Let
$r = \rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$ be the radius of $D$, and let
$v \in \bigcap {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r}$ be the extreme point for ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r}$ and
$D$.
First of all, we observe that $v$ cannot be in the
interior of $\bigcap {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r}$, since $v$ minimizes the distance to $D$.
Thus, there has to be a non-empty
subset $\A \subseteq {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r}$ such that $v$ lies on the boundary of
each disk of $\A$.
Let $\A$ be a minimal set such that $d(\bigcap \A, D) = d(v, D)$.
It follows that $|\A| \leq 2$.
See \cref{fig:axiom2} for an illustration.
First, assume that $\A = \{E\}$. Then, since
$d(E, D) = d(v,D)>0$, we know that $E\cap D=\emptyset$.
As the disks in ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ intersect pairwise,
we derive $D\notin {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ and hence $D=D_\infty$. Setting
${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}= \A$, we get $\rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})=\infty=\rad({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y})$ and
$\dist({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})=d(v,D)=d(E,D)=\dist({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y})$.
Thus, $|{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}|\leq 3$ and $w({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y})=w({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$.
Second, assume that $\A=\{E,F\}$.
Then, $v$ is one of the two vertices of the lens $L= E \cap F$.
Next, we show that $d(L, D)\geq d(v, D)$. Assume for the sake of
contradiction that there is a point $w \in L$ with $d(w, D) < d(v, D)$.
By general position and since $v$ is the intersection
of two disk boundaries, there is a relatively open neighborhood
$N$ around $v$ in $\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r}$ such that $N$ is also
relatively open in $L$. Since $L$ is convex, there
is a point $x \in N$ that also lies in the relative interior of
the line segment $\overline{wv}$. Then, $d(x, D) < d(v, D)$ and
$x \in \bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r}$. This yields a contradiction, as $v$ is the extreme
point for ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r}$ and $D$. Thus, we have $d(L, D)\geq d(v, D)$
which also shows hat $D \cap E \cap F= \emptyset$.
We set ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y} = \{E,F\}$, if ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ is Helly (i.e., $D=D_\infty$), and
${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y} = \{D, E, F\}$, if ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ is non-Helly (i.e., $D \in {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$). In both cases,
we have ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y} \subseteq {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$ and $|{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}| \leq 3$. Moreover, we can conclude
that $D$ destroys ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}_{<r} = \{E,F\}$, and since ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}_{<r}$ is Helly, $D$
is the smallest destroyer of ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$. Hence, we have $\rad({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}) = r = \rad({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y})$.
To obtain $\dist({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}) = \dist({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$, it remains to show
$d(\bigcap{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}_{<r},D)=d(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r},D)$. Since ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}_{<r} \subseteq {\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r}$,
we can use \cref{lem:destroyer:dist} as well as $d(L, D)\geq d(v, D)$ to
derive
\[
d{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}ig(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r},D{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}ig) \geq d{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}ig(\bigcap{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}_{<r},D{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}ig),
= d(L,D) \geq d(v,D) = d{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}ig(\bigcap{\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}_{<r},D{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}ig)
\]
as desired. We conclude that $w({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y})=w({\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R})$.
We remark that the set ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$ is actually a basis for ${\cal C}} \def{\varepsilon}{{\varepsilon}} \def\Rr{{\cal R}$: if ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$ is a non-Helly
triple, then removing any disk from ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$ creates a Helly set and
increases the radius of the smallest destroyer to $\infty$. If
$|{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}| \leq 2$, then $D_\infty$ is the smallest destroyer of ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$
and the minimality follows directly from the definition.
\end{proof}
Following the argument of the last proof, the violation test is now immediate.
We present pseudo-code in Algorithm~\ref{alg:violationTest}.
It obviously needs constant time. Finally, to apply
the algorithm of Chazelle and \Matousek, we still need to check
that there is a polynomial-time oracle that computes a superset of
$\Rr_{\Y}$ for a given
set of disks $\Y$.
\begin{algorithm}
\caption{The violation test.}
\begin{algorithmic}[1]
\Procedure {violates}{set ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}\subseteq\D$, disk $E\in\D$ with radius $r'$}
\If {$|{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}| > 3$ or $|{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}| = 3$ and ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$ is Helly}
\Return \textbf{``{\boldmath ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$} is not a basis.''}
\EndIf
\If {$|{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}| = 2$ and the $y$-minimum of $\bigcap {\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$ is also
the $y$-minimum of a single disk of ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$}
\State \Return \textbf{``{\boldmath ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$} is not a basis.''}
\EndIf
\If {${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}=\{D_1\}$}
\If {the $y$-minimum in $E \cap D_1$ differs from the $y$-minimum in $D_1$}
\State \Return \textbf{``{\boldmath $E$} violates {\boldmath ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$}.''}
\Else\ \Return \textbf{``{\boldmath $E$} does not violate {\boldmath ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$}.''}
\EndIf
\EndIf
\If {${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}=\{D_1,D_2\}$}
\State $v=\argmin\ \{w_y\mid w\in D_1\cap D_2\}$
\If {$v\notin E$}
\Return \textbf{``{\boldmath $E$} violates {\boldmath ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$}.''}
\Else\ \Return \textbf{``{\boldmath $E$} does not violate {\boldmath ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$}.''}
\EndIf
\Else
\algorithmiccomment{${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$ is of size $3$, non-Helly, and does not
contain $D_\infty$.}
\State $D=$ smallest destroyer of ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$
\State $\{D_1,D_2\}={\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}\setminus\{D\}$
\State $r=\rad({\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y})$
\If{$r'>r$}
\Return \textbf{``{\boldmath $E$} does not violate {\boldmath ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$}.''}
\Else
\State $v=\argmin\ \{d(w,E)\mid w\in D_1\cap D_2\}$
\If {$v\notin E$}
\Return \textbf{``{\boldmath $E$} violates {\boldmath ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$}.''}
\Else\ \Return \textbf{``{\boldmath $E$} does not violate {\boldmath ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$}.''}
\EndIf
\EndIf
\EndIf
\EndProcedure
\end{algorithmic}
\label{alg:violationTest}
\end{algorithm}
\begin{lemma}
\label{lem:oracle}
Given a set $\Y\subseteq\D$ of disks, we can compute a
superset of $\Rr_{\Y}$
in time $O(\vert\Y\vert^4)$.
\end{lemma}
\begin{proof}
Let $v\in\R^2$ and $r>0$.
First, we let $R_v=\{D\in\Y\mid v\notin D\}$ be the range of all disks
that do not contain
$v$. Second, let $R_{v,r}$ be the range of all disks of diameter
smaller than $r$
that do not contain the point $v$, i.e.,
$R_{v,r}=\{D\in\Y\mid v\notin D\text{ and } r_D<r\}$.
We define $\Rr'$ to be the set of
all ranges $R_v$ over all $v$ and subsequently, we let $\Rr''$ be the set
of all ranges $R_{v,r}$ over all $v$ and $r$, that is,
$\Rr''=\{R_{v,r}\mid v\in\R^2\text{ and } r>0\}$.
The discussion from the previous lemmas shows that for any basis ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$,
there is a
point $v_{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y} \in \R^2$ and a radius $r_{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}>0$ such that a disk $E \in \D$
with radius $r_E$ violates ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$ if
and only if $v_{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y} \not\in E$ and $r_E<r_{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$. Hence, we have
$\Rr''\supseteq\Rr_{\Y}$.
We show how to compute $\Rr''$ in polynomial time. For this, we
first construct $\Rr'$.
For the given set $\Y$ of disks, we compute the arrangement $A(\Y)$ and then
focus on the facets
of $A(\Y)$. Since the arrangement has $O(\vert\Y\vert^2)$
facets, we can compute $A(\Y)$ in time $O(\vert\Y\vert^3)$
using a simple brute-force approach (faster algorithms exist,
but are not needed here).
Clearly, for two points $v$ and $w$ of the same facet of $A(\Y)$,
we have $R_v=R_w$. Therefore, for a given facet $f$,
we pick an arbitrary point $v\in f$, and we compute $R_v$ by a linear scan
of $\Y$.
Summing over all facets, we can thus compute $\Rr'$ in time $O(\vert\Y\vert^3)$.
Finally, to compute $\Rr''$, we iterate over all $O(\vert\Y\vert^2)$
ranges in $\Rr'$. Given a range $R_v\in\Rr'$,
we get all $R_{v,r}$ for $r>0$ by first sorting $R_v$ by increasing radii
and then taking every prefix of the
sorted list of disks. For a fixed $v$, this can be done in time
$O(\vert\Y\vert^2)$. Hence, $\Rr''$ can be computed in
$O(\vert\Y\vert^4)$ time. The claim follows.
\end{proof}
The following lemma summarizes the discussion so far.
\begin{lemma}
\label{lem:solve:LPtype}
Given a set $\D$ of $n$ pairwise intersecting disks in the plane,
we can decide in $O(n)$ deterministic time whether $\D$ is Helly.
If so, we can compute a point in $\bigcap\D$ in $O(n)$
deterministic time. If not, we can compute the smallest destroyer
$D$ of $\D$ and two disks $E, F \in \D_{<r}$ that form a non-Helly
triple with $D$. Here, $r$ is the radius of $D$.
\end{lemma}
\begin{proof}
Since (i) $(\D, w, \leq)$ is LP-type, (ii) the violation test needs
constant
time, and (iii) the oracle needs polynomial time, we can apply
the deterministic algorithm of Chazelle and
\Matousek~\cite{ChazelleMa96} to compute
$w(\D) = (\rad(\D), -\dist(\D))$ and a corresponding basis ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$ in
$O(n)$ time. Then, $\D$ is Helly if and only if
$\rad(\D) = \infty$. If $\D$ is Helly, then $|{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}| \leq 2$. We
compute the unique point $v \in \bigcap{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$ with
$d(v, D_\infty) = d\big(\bigcap{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}, D_{\infty}\big)$. Since
${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y} \subseteq \D$ and
$d\big(\bigcap{\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y},D_{\infty}\big) = d\big(\bigcap\D,
D_{\infty}\big)$, we have $v \in \bigcap\D$ by
\cref{lem:destroyer:dist}. We output $v$. If $\D$ is non-Helly, we
simply output ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$, because ${\cal B}} \def\V{{\cal V}} \def\H{{\cal H}} \def\Y{{\cal Y}$ is a non-Helly triple with the
smallest destroyer $D$ of $\D$ and two disks $E, F \in\D_{<r}$,
where $r$ is the radius of $D$.
\end{proof}
\begin{theorem}
Given a set $\D$ of $n$ pairwise intersecting disks in the plane,
we can find in deterministic $O(n)$ time a set $P$ of five points such that
every disk of $\D$ contains at least one point of $P$.
\end{theorem}
\begin{proof}
Using the algorithm from \cref{lem:solve:LPtype}, we decide whether
$\D$ is Helly. If so, we return the extreme point computed by the
algorithm. Otherwise, the algorithm gives us a non-Helly triple
$\{D, E, F\}$, where $D$ is the smallest destroyer of $\D$ and
$E, F \in\D_{<r}$, with $r$ being the radius of $D$. Since
$\D_{<r}$ is Helly, we can obtain in $O(n)$ time a stabbing point
$q \in \bigcap\D_{<r}$ by using the algorithm from
\cref{lem:solve:LPtype} again. Next, by \cref{lem:triple}, there are
two disks in $\{D, E, F\}$ whose lens angle is at least
$2\pi/3$. Let $P'$ be the set of four points from the proof of
\cref{lem:C:D:diff:size}. Then, $P = P' \cup \{q\}$ is a set of
five points that stabs every disk in $\D$.
\end{proof}
\section{Simple Bounds}
\label{sec:simple_bounds}
We now provide some easy lower and upper bounds on the number of disks
for which a
certain number of stabbing points is necessary or sufficient.
\paragraph{Eight disks can be stabbed by three points.}
For the proof that any set of eight pair-wise intersecting disks can
be stabbed by at most three points, we show the following lemma.
\begin{lemma}
\label{lem:5-disks}
Let $\D$ be a set of at least $5$ pairwise intersecting disks.
Then, $\D$ contains a Helly-triple.
\end{lemma}
\begin{proof}
Let $\D$ be a set of exactly $5$ pairwise intersecting disks.
We assume that no three centers of the disks are on a line, since otherwise
these three disks are a Helly-triple.
Since the complete graph $K_5$ does not have a planar embedding, there have
to be four different disks $D_1,\dots,D_4\in\D$ with centers
$c_1,\dots, c_4$ and
radii $r_1,\dots,r_4$ such that
the line segments $c_1c_3$ and $c_2c_4$ intersect, see \cref{fig:4_disks}.
Let $x$ be the intersection point.
\begin{figure}\label{fig:4_disks}
\end{figure}
Moreover, let $\alpha$ (resp., $\beta$) be the intersection of
the lens $L_{1,3}$
(resp., $L_{2,4}$) and the line segment $c_1c_3$ (resp., $c_2c_4$). If $x$ is in
$\alpha$ or $\beta$, we are done. Otherwise, let $y$ be the point of $\alpha$
that is closest to $x$ and let $z$ be the point of $\beta$ closest to $x$.
We can assume without loss of generality that $\vert xy\vert\leq\vert xz\vert$
and $x\notin D_4$.
Using the triangle inequality, We can derive
\[
\vert c_2y\vert
\leq \vert c_2x\vert +\vert xy\vert
\leq \vert c_2x\vert+ \vert xz\vert \leq r_2
\]
to conclude that $y\in D_1\cap D_2\cap D_3$.
\end{proof}
Now consider a set $\D$ of $8$ pairwise intersecting disks. Using
\cref{lem:5-disks},
we can find a Helly-triple in $\D$. Among the remaining $5$ disks,
we find a second
Helly-triple. The remaining two disks can be stabbed by one point.
This reasoning
yields the following corollary, which was already mentioned by
Stach\'o\xspace~\cite{Stacho65}.
\begin{corollary}
Every set $\D$ of at most $8$ pairwise intersecting disks
can be stabbed by 3 points.
\end{corollary}
\paragraph{13 disks with 4 stabbing points.}
Danzer presented a set of $10$ pairwise intersecting pseudo-disks with
stabbing number
four \cite{Danzer86}. However,
it is not clear to us how these $10$ pseudo-disks can be realized as pairwise
intersecting Euclidean disks
achieving the same stabbing number. Moreover, it is another
open problem whether $9$
pairwise intersecting disks can be stabbed by three points.
Instead, we want to describe a
set of $13$ pairwise intersecting disks in the plane
such that no point set of size three can pierce all of them.
The
construction begins with an inner disk $A$ of radius $1$ and three
larger disks $D_1$, $D_2$, $D_3$ of equal radius, so that each pair
of disks in $\{A,D_1,D_2,D_3\}$ is tangent.
For $i = 1, 2, 3$, we denote the contact point of $A$ and
$D_i$ by $\xi_i$.
We add six more disks as follows. For $i=1,2,3$, we draw
the two common outer tangents to $A$ and $D_i$, and denote by
$T_i^-$ and $T_i^+$ the halfplanes that are bounded by these
tangents and are openly disjoint from $A$. The labels $T_i^-$ and
$T_i^+$ are chosen such that the points of tangency between $A$
and $T_i^-$, $D_i$, and $T_i^+$, appear along the boundary of $A$ in this
counterclockwise order. One can show that the nine points of
tangency between $A$ and the other disks and tangents are pairwise
distinct (see \cref{fig:lower}).
We regard the six halfplanes
$T_i^-$, $T_i^+$, for $i=1,2,3$, as (very large) disks; in the
end, we can apply a suitable inversion to turn the disks and
halfplanes into actual disks, if so desired.
\begin{figure}\label{fig:lower}
\end{figure}
Finally, we construct three additional disks $A_1$, $A_2$, $A_3$. To
construct $A_i$, we slightly expand $A$ into a disk $A'_i$ of radius
$1 + {\varepsilon}_1$, while keeping the tangency with $D_i$ at $\xi_i$. We
then roll $A'_i$ clockwise along $D_i$, by a tiny angle
${\varepsilon}_2 \ll {\varepsilon}_1$, to obtain $A_i$.
This gives a set of $13$ disks. For sufficiently small ${\varepsilon}_1$ and
${\varepsilon}_2$, we can ensure the following properties for each $A_i$: (i)
$A_i$ intersects all other $12$ disks; (ii) the nine intersection
regions $A_i \cap D_j$, $A_i \cap T_j^-$, $A_i \cap T_j^+$, for
$j = 1,2,3$, are pairwise disjoint; and (iii) $\xi_i\notin A_i$.
\begin{theorem}
\label{thm:lower:bound}
The construction yields a set of $13$ disks that cannot be stabbed
by $3$ points.
\end{theorem}
\begin{proof}
Consider any set $P$ of three points. Set
$A^* = A \cup A_1 \cup A_2 \cup A_3$. If
$P \cap A^* = \emptyset$, we have unstabbed disks, so suppose that
$P \cap A^* \neq \emptyset$. For $p \in P \cap A^*$, property~(ii)
implies that $p$ stabs at most one of the nine remaining disks
$D_j$, $T_j^+$ and $T_j^-$, for $j = 1, 2, 3$. Thus, if
$P \subset A^*$, we would have unstabbed disks, so we may assume
that $|P \cap A^*| \in \{1, 2\}$.
Suppose first that $|P \cap A^*| = 2$. As just argued, at most two
of the remaining disks are stabbed by $P \cap A^*$. The following
cases can then arise.
\begin{enumerate}[(a)]
\item None of $D_1$, $D_2$, $D_3$ is stabbed by $P \cap A^*$.
Since $\{D_1, D_2, D_3\}$ is non-Helly and a non-Helly set
must be stabbed by at least two points, at least one disk
remains unstabbed.
\item Two disks among $D_1$, $D_2$, $D_3$ are stabbed by
$P \cap A^*$. Then the six unstabbed halfplanes form many
non-Helly triples, e.g., $T_1^-$, $T_2^-$, and $T_3^-$, and
again, a disk remains unstabbed.
\item The set $P \cap A^*$ stabs one disk in
$\{D_1, D_2, D_3\}$ and one halfplane. Then, there is (at
least) one disk $D_i$ such that $D_i$ and its two tangent
halfplanes $T_i^-$, $T_i^+$ are all unstabbed by $P \cap A^*$.
Then, $\{ D_i, T_i^-, T_i^+ \}$ is non-Helly, and at least $2$
more points are needed to stab it.
\end{enumerate}
Suppose now that $|P \cap A^*| = 1$, and let
$P \cap A^* = \{p\}$. We may assume that $p$ stabs all four
disks $A$, $A_1$, $A_2$, $A_3$, since otherwise a disk would stay
unstabbed. By property (iii), we can derive $p \not\in \{\xi_1, \xi_2, \xi_3\}$.
Now, since $p\in A \setminus \{\xi_1, \xi_2, \xi_3\}$, the point $p$ does
not stab any of $D_1$, $D_2$, $D_3$. Moreover, by property (ii),
the point $p$ can only stab at most one of the remaining halfplanes.
Since $\{D_1, D_2, D_3\}$ is
non-Helly, it requires two stabbing points. Moreover, since
$|P \setminus \{p\}| = 2$, it must be the case that one point $q$ of
$P \setminus A^*$ is the point of tangency of two of these disks,
say $q = D_2 \cap D_3$. Then, $q$ stabs only two of the six
halfplanes, say, $T_1^-$ and $T_1^+$. But then,
$\{ D_1, T_2^+, T_3^- \}$ is non-Helly and does not contain any
point from $\{p, q\}$. At least one disk remains unstabbed.
\end{proof}
\section{Conclusion}
We gave a simple linear-time algorithm, based on techniques for solving
LP-type problems,
to find five stabbing points
for a set of pairwise intersecting disks in the plane.
The arXiv manuscript by Carmi, Katz, and Morin~\cite{CarmiKaMo18}
claims a similar linear-time algorithm for finding four stabbing points.
It would now be interesting to see whether these results,
the ones by Danzer, Stach\'o\xspace, and ours, could be used to find
new deterministic approximation algorithms for computing large cliques in
disk graphs;
refer to~\cite{ambuhl2005clique, bonamy2018eptas} for the known algorithms.
On the lower-bound side, it is still not known
whether nine disks can always be stabbed by three points or not. For eight
disks, we provided a proof that three points always suffice,
as already mentioned by Stach\'o\xspace~\cite{Stacho65}.
The lower bound construction of Danzer with ten
disks~\cite{Danzer86}
can easily be verified for pseudo-disks. However,
the example is not easy to draw, even with
the help of geometry processing
software. Until now, we were not able to check whether his
pseudo-disk arrangement can
be realized as a Euclidean disk arrangement.
\end{document} |
\begin{document}
\title[Annealed deviations of random walk in random
scenery]{\large Annealed deviations of random walk in random
scenery}
\author[Nina Gantert, Wolfgang
K{\"o}nig and Zhan Shi]{}
\maketitle
\thispagestyle{empty}
\centerline {{\scriptscriptstylec Nina Gantert$^1$\/,
Wolfgang K{\"o}nig$^{2}$\/} and {\scriptscriptstylec Zhan Shi$^{3}$\/}}
\centerline {\em $^1$ Fachbereich Mathematik und Informatik der Universit{\"a}t M{\"u}nster,}
\centerline {\em Einsteinstra\scriptscriptstyles e 62, D-48149 M{\"u}nster, Germany}
\centerline{\em
{\tt gantert@math.uni-muenster.de}}
\centerline{\em $^2$Mathematisches Institut, Universit{\"a}t Leipzig,}
\centerline{\em Augustusplatz 10/11, D-04109 Leipzig, Germany}
\centerline{\tt koenig@math.uni-leipzig.de}
\centerline{\em $^3$ Laboratoire de Probabilit{\'e}s et
Mod{\`e}les Al{\'e}atoires, Universit{\'e} Paris VI,}
\centerline{\em 4 place Jussieu, F-75252 Paris Cedex 05, France}
\centerline{\tt zhan@proba.jussieu.fr}
\centerline{\scriptscriptstylemall(18 November, 2005)}
\begin{quote}
{\scriptscriptstylemall {\bf Abstract:}} Let $(Z_n)_{n\in\N}$ be a
$d$-dimensional {\it random walk in random scenery}, i.e.,
$Z_n=\scriptscriptstyleum_{k=0}^{n-1}Y(S_k)$ with $(S_k)_{k\in\N_0}$ a
random walk in $\Z^d$ and $(Y(z))_{z\in\Z^d}$ an i.i.d.~scenery,
independent of the walk. The walker's steps have mean
zero and some finite exponential moments.
We identify the speed and the rate of the logarithmic decay of
$\P(\frac 1n Z_n>b_n)$ for various choices of sequences $(b_n)_n$ in $[1,\infty)$. Depending
on $(b_n)_n$ and the upper tails of the scenery, we identify different regimes for the speed
of decay and different variational formulas for the rate functions. In contrast to recent
work \cite{AC02} by A.~Asselah and F.~Castell, we consider sceneries {\it unbounded\/} to infinity.
It turns out that there are interesting connections to large deviation properties of
self-intersections of the walk, which have been studied recently by X.~Chen \cite{C03}.
\end{quote}
\begin{quote}
{\scriptscriptstylemall {\bf R\'esum\'e~:}} Soit $(Z_n)_{n\in\N}$ une marche
al\'eatoire en paysage al\'eatoire sur $\Z^d$~; il s'agit du
processus d\'efini par $Z_n=\scriptscriptstyleum_{k=0}^{n-1}Y(S_k)$, o\`u
$(S_k)_{k\in\N_0}$ est une marche al\'eatoire \`a valeurs dans $\Z^d$,
et
le paysage al\'eatoire $(Y(z))_{z\in\Z^d}$ est une famille de variables
al\'eatoires i.i.d.\ independante de la marche. On suppose que
$S_1$ est centr\'ee et admet certains moments exponentiels finis.
Nous identifions la vitesse et la fonction de taux de $\P(\frac 1n
Z_n>b_n)$, pour diverses suites $(b_n)_n$ \`a valeurs dans $[1,\infty[$.
Selon le comportement de $(b_n)_n$ et de la queue de distribution du
paysage al\'eatoire, nous d\'ecouvrons diff\'erents r\'egimes ainsi
que diff\'erentes formules variationnelles pour les fonctions de taux.
Contrairement au travail r\'ecent de A.~Asselah and
F.~Castell~\cite{AC02}, nous \'etudions le cas o\`u le paysage
al\'eatoire
n'est {\it pas born\'e\/}. Finalement, nous observons des liens
int\'eressants avec certaines propri\'et\'es d'auto-intersection de la
marche $(S_k)_{k\in\N_0}$, r\'ecemment \'etudi\'ees par X.~Chen
\cite{C03}.
\end{quote}
\noindent
{\it MSC 2000.} 60K37, 60F10, 60J55.
\noindent
{\it Keywords and phrases.} Random walk in random scenery, local time, large deviations, variational formulas.
\eject
\scriptscriptstyleetcounter{section}{0}
\scriptscriptstyleection{Introduction}
\label{Intro}
\scriptscriptstyleubsection{Model and motivation.}\label{sec-model}
\noindent Let $S=(S_n)_{n\in\N_0}$ be a random walk on $\Z^d$
starting at the origin. We denote by $\P$ the underlying
probability measure and by $\E$ the corresponding expectation.
We assume that $\E[S_1]=0$ and $\E[|S_1|^2]<\infty$.
Defined on the same probability space, let $Y=(Y(z))_{z\in\Z^d}$ be an
i.i.d.~sequence of random variables, independent of the
walk. We refer to $Y$ as the {\it random scenery.}
Then the process $(Z_n)_{n\in\N}$ defined by
$$
Z_n=\scriptscriptstyleum_{k=0}^{n-1} Y(S_k),\qquad n\in\N,
$$
where $\N=\{1, 2, \ldots\},$ is called a {\it random walk in random scenery}, sometimes
also referred to as the {\it Kesten-Spitzer random walk in random scenery}, see \cite{KS79}. An
interpretation is as follows. If a random walker has to pay
$Y(z)$ units at any time he/she visits the site $z$, then $Z_n$
is the total amount he/she pays by time $n-1$.
The random walk in random scenery has been introduced and
analyzed for dimension $d\not=2$ by H.~Kesten and F.~Spitzer \cite{KS79} and
by E.~Bolthausen \cite{B89} for $d= 2$. The case $d=1$ was treated independently by A. ~N. ~Borodin \cite{Bo79a}, \cite{Bo79b}. Under the assumption that
$Y(0)$ has expectation zero and variance $\scriptscriptstyleigma^2\in(0,\infty)$, their results imply that
\begin{equation}
\label{weakconv}
\frac 1n Z_n\approx a_n^{\scriptscriptstylesup{0}} \begin{cases}n^{-\frac 14}&\mbox{if }d=1,\\
(\frac n{\log n})^{-\frac 12}&\mbox{if }d=2,\\
n^{-\frac 12}&\mbox{if }d\ge 3.
\end{cases}
\end{equation}
More precisely,
$\frac 1{na_n^{\scriptscriptstylesup{0}}}Z_n$ converges in distribution towards some non-degenerate
random variable. The limit is Gaussian in $d\ge 2$ and a
convex combination of Gaussians (but not Gaussian) in $d=1$. This
can be roughly explained as follows.
In terms of the so-called {\it local times\/} of the walk,
\begin{equation}
\label{loctim}
\end{lemma}l_n(z)=\scriptscriptstyleum_{k=0}^{n-1} \1_{\{S_k=z\}},\qquad n\in\N, \;\;
z\in\Z^d,
\end{equation}
\noindent the random walk in random scenery may be identified as
\begin{equation}
\label{Znrepr}
Z_n=\scriptscriptstyleum_{z\in\Z^d}Y(z)\end{lemma}l_n(z).
\end{equation}
\noindent The number of effective summands in \eqref{Znrepr} is
equal to the {\it range\/} of the walk, i.e., the number of sites
visited by time $n-1$. Hence, conditional on the
random walk, $Z_n$ is, for dimension $d\ge 3$, a sum of
$\Ocal(n)$ independent copies of finite multiples of $Y(0)$,
and hence it is plausible that $Z_n/n^{1/2}$ converges to a
normal variable. The same assertion with logarithmic
corrections is also plausible in $d=2$. However, in $d=1$, $Z_n$ is
roughly a sum of
$\Ocal(n^{1/2})$ copies of independent variables with
variances of order $\Ocal( n)$, and this suggests the
normalization in \eqref{weakconv} as well as a non-normal limit.
In this paper, we analyse deviations $\{\frac 1n Z_n>b_n\}$ for
various choices of sequences $(b_n)_{n\in\N}$ in $[1,\infty)$. We
determine the speed and the rate of the logarithmic
asymptotics of the probability of this event as $n\to\infty$, and we
explain the typical behaviour of the random walk and the
random scenery on this event.
This problem has been addressed in recent work \cite{CP01}, \cite{AC02} and
\cite{Ca04} by F.~Castell in partial collaboration with F.~Pradeilles and
A.~Asselah for Brownian motion instead of random walk.
While \cite{CP01} and \cite{Ca04} treat the case of a continuous Gaussian scenery for $b_n=n^{1/2}$
and ${\operatorname {cst.}\,}\leq b_n\ll n^{1/2}$, respectively, the case of an arbitrary
bounded scenery (constant on the unit cubes) and $b_n={\operatorname {cst.}\,}$
is considered in \cite{AC02}.
See also \cite{AC02} for further references on this topic and \cite{AC05a} and \cite{GHK04}
for recent results on the random walk case.
The main novelty of the present paper is the study of arbitrary sceneries {\it unbounded\/}
to $+\infty$ and general scale functions $b_n \geq {\operatorname {cst.}\,}$ in the discrete setting.
On the technical side, in particular the proof of the upper bound is rather demanding
and requires new techniques. We solve this part of the problem by a careful analysis
of high integer moments, a technique which has been recently established in the study
of intersection properties of random motions.
A very rough, heuristic explanation of the interplay between the deviations of the
random walk in random scenery and the tails of the scenery at infinity and the dimension $d$ is as follows.
In order to realize the event $\{\frac 1n Z_n>b_n\}$, it is clear
that the scenery has to assume larger values on the range of the walk than usual.
In order to keep the probabilistic cost for this low, the random walker
has to keep its range small, i.e., it has to concentrate on less sites by time $n$ than usual.
The {\it optimal\/} joint strategy of the scenery and the walk is determined by
a balance between the respective costs. The optimal strategies in the cases considered
in the present paper are homogeneous. More precisely, the scenery and the walk
each approximate optimal (rescaled) profiles in a large, $n$-dependent box.
These optimal profiles are determined by a (deterministic) variational problem.
The topic of the present paper has deep connections to large deviation properties of
self-intersections of the walk. This is immediate in the important special case of a
standard Gaussian scenery $Y$. Indeed, the conditional distribution of $Z_n$ given
the random walk $S$ is a centered Gaussian with variance equal to
\begin{equation}\label{Lambdadef}
\Lambda_n=\scriptscriptstyleum_{z\in \Z^d}\end{lemma}l_n(z)^2=\|\end{lemma}l_n\|_2^2,
\end{equation}
which is often called the {\it self-intersection local time}. Hence, large deviations for the random walk in Gaussian scenery would be a consequence of an appropriate large deviation statement for self-intersection local times. However, the latter problem is notoriously difficult and is, up to the best of our knowledge, open in the precision we would need in the present paper. (However, compare to interesting and deep work on self-intersections and mutual intersections by X.~Chen \cite{C03}.) Recent results for self-intersection local times for random walks in dimension $d \geq 5$ and applications to random walk in random scenery are given in \cite{AC05b}.
The remainder of Section~\ref{Intro} is organized as follows. Our main results are in Section~\ref{results}, a heuristic derivation may be found in Section~\ref{sec-heur}, a partial result for Gaussian sceneries for dimension $d=2$ is in Section~\ref{Sec-Gauss}. The structure of the remainder of the paper is as follows. In Section~\ref{sec-varform} we analyse the variational formulas, in Section~\ref{sec-prep} we present the tools for our proofs of the main results, in Sections~\ref{s:ub} and \ref{s:lb} we give the proofs of the upper and the lower bounds, respectively, and finally in the appendix, Section~\ref{sec-proofLDP}, we provide the proof of a large deviation principle that is needed in the paper.
\scriptscriptstyleubsection{Results}\label{results}
\noindent Our precise assumptions on the random walk, $S$, are the following. The walker starts at $S_0=0$, and the steps have mean zero and some finite exponential moments, more precisely,
\begin{equation}
\label{emom}
\E[e^{t |S_1|}]<\infty\quad\mbox{for some }t>0.
\end{equation}
By $\Gamma\in\R^{d\times d}$ we denote the covariance matrix of the walk's step distribution. Hence,
$S$ lies in the domain of attraction of the Brownian motion with covariance matrix $\Gamma$. We assume that $\Gamma$ is a regular matrix. Furthermore, we assume that $S$ is strongly aperiodic, i.e., for any $z\in\Z^d$, the smallest subgroup of $\Z^d$ that contains $\{z+x\colon \P(S_1=x)>0\}$ is $\Z^d$ itself. Finally, to avoid technical difficulties, we also assume that the transition function of the walk is symmetric, i.e., $p(0,z)= p(0,-z)$ for $z\in\Z^d$, where $p(z,\widetilde z)$ denotes the walker's one-step probability from $z\in\Z^d$ to $\widetilde z\in\Z^d$.
Our assumptions on the scenery are the following. Let $Y=(Y(z))_{z\in\Z^d}$ be a family of i.i.d.~random variables, not necessarily having finite expectation, such that
\begin{equation}\label{assumption}
\E[e^{t Y(0)}]<\infty\quad\mbox{for every }t>0.
\end{equation}
In particular, the {\it cumulant generating function \/}
of $Y(0)$, is finite:
\begin{equation}\label{Hdef}
H(t)=\log\E[e^{tY(0)}]<\infty,\qquad t>0.
\end{equation}
In some of our results, we additionally suppose the following.
\noindent{\bf Assumption (Y).} {\it There are constants $D>0$ and $q>1$
such
that}
$$
\log\P(Y(0)>r)\scriptscriptstyleim -D r^q,\qquad r\to\infty.
$$
According to Kasahara's exponential Tauberian theorem (see \cite[Th.~4.12.7]{BGT87}), Assumption (Y) is equivalent to
\begin{equation}\label{cumgenfct}
H(t)\scriptscriptstyleim \widetilde D t^p,\quad \mbox{as }t\to\infty,
\qquad\mbox{where}\qquad
\widetilde D=(q-1)(D q^q)^{1/(1-q)}\qquad\mbox{and}\qquad\frac 1q+\frac 1p=1.
\end{equation}
In our first main result, we consider the case of sequences $(b_n)_n$
tending to infinity slower than $n^{\frac 1q}$. By $\nabla$ we denote the usual gradient acting on sufficiently regular functions
$\R^d\to\R$. By $H^1(\R^d)$ we denote the usual Sobolev space, and we write $\|\nabla \psi\|_2^2=\int_{\R^d}|\nabla\psi(x)|^2\,{\rm d} x$.
We use the notation $b_n\gg c_n$ if $\lim_{n\to\infty}b_n/c_n=\infty$.
\begin{theorem}[Very large deviations]\label{inter} Suppose that
Assumption (Y) holds with some $q > \frac d2$. Pick a sequence $(b_n)_{n\in\N}$ satisfying $1\ll b_n\ll
n^{\frac 1q}$. Then
\begin{equation}\label{interasy}
\lim_{n\to\infty}n^{-\frac d{d+2}}
b_n^{-\frac {2q}{d+2}}\log \P(\scriptscriptstylemfrac 1n Z_n>b_n)=- K_{D,q},
\end{equation}
where
\begin{equation}\label{kdq}
K_{D,q}\equiv \inf\Bigl\{
\frac 12\|\Gamma^{\frac 12}\nabla \psi\|_2^2+D\|\psi^2\|_{p}^{-q}\colon
\psi\in H^1(\R^d),\|\psi\|_2=1\Bigr\},
\end{equation}
(we recall that $\frac 1p+\frac 1q=1$), and $K_{D,q}$ is positive.
\end{theorem}
\begin{bem}\label{rem-inter} For $q \in(1,\frac d2)$, (\ref{interasy}) also holds true, but $K_{D,q} = 0$. Indeed, this follows from Proposition~\ref{constpos} below together with our proof of Theorem~\ref{inter}. One can also see this directly by giving an explicit lower bound for $\log\P(\scriptscriptstylemfrac 1n Z_n>b_n)$ which runs on a strictly smaller scale than $n^{\frac d{d+2}}
b_n^{\frac {2q}{d+2}}$.
It remains an open problem in this paper to determine the {\it precise\/} logarithmic rate of
$\P(\scriptscriptstylemfrac 1n Z_n>b_n)$ in the case $q\in(1,\frac d2)$.
The case $q=\frac d2$ seems even more delicate and is also left open in the present paper.
The case $q\in(0,1)$ has been studied in \cite{GHK04}.
$\Diamond$
\end{bem}
Note that the variational problem in \eqref{kdq} is of independent interest;
it also appeared in \cite[Theorem 1.1]{BAL91} in the context of heat
kernel asymptotics.
In Proposition~\ref{constpos} below it turns out that $K_{D,q}$ is positive if and only if $q\geq \frac d2$.
Our next result essentially extends \cite[Th.~2.2]{AC02} from the case of bounded sceneries to the case in \eqref{assumption}.
\begin{theorem}[Large deviations]\label{lin} Suppose that \eqref{assumption} holds.
Assume that $\E[Y(0)]=0$, and set $\overline p\equiv\limsup_{t\to\infty}\frac{\log H(t)}{\log t}$. Assume that $\overline p<\infty$ in $d\leq 2$ respectively $\overline p<\frac d{d-2}$ in $d\geq 3$. Then, for any $u>0$ satisfying $u\in \scriptscriptstyleupp(Y(0))^\circ$,
\begin{equation}\label{linasy}
\lim_{n\to\infty}n^{-\frac d{d+2}}\log\P(\scriptscriptstylemfrac 1n Z_n>u)= -K_{H}(u),
\end{equation}
where
\begin{equation} \label{K3def}
K_H(u)\equiv \inf\Bigl\{
\frac 12\|\Gamma^{\frac 12}\nabla \psi\|_2^2+\Phi_H(\psi^2,u)\colon \psi\in
H^1(\R^d),\|\psi\|_2=1\Bigr\},
\end{equation}
and
\begin{equation}\label{Phidef}
\Phi_H(\psi^2,u)=\scriptscriptstyleup_{\gamma\in(0,\infty)}\Bigl[\gamma
u-\int_{\R^d}H(\gamma \psi^2(y))\,{\rm d} y\Bigr].
\end{equation}
The constant $K_H(u)$ is positive.
\end{theorem}
Switching to the scenery $-Y$, one may, under appropriate conditions, use Theorem~\ref{lin} to obtain the \lq other half\rq\ of a full large deviation principle for $(\frac 1n Z_n)_n$. This was carried out in \cite{AC02} for bounded sceneries. For Brownian motion in a Gaussian scenery, a result analogous to Theorems~\ref{inter} and \ref{lin} is \cite[Th.~2]{Ca04}.
Note that the constant $K_H(u)$ depends on the entire scenery distribution, while $K_{D,q}$ in \eqref{kdq} only depends on its upper tails.
\begin{bem} A statement analogous to Remark~\ref{rem-inter} also applies here: for dimensions $d \geq 3$, when $ \liminf_{t\to\infty}\frac{\log H(t)}{\log t} > \frac d{d-2}$, (\ref{linasy}) also holds true, but $K_H(u) = 0$ for any $u>0$. It was shown recently in \cite{AC05a} that under assumption (Y) with $q\in(1, \frac{d}{2})$ and an additional symmetry assumption, $\log\P(\scriptscriptstylemfrac 1n Z_n>b_n)$ is of the order $n^{\frac q{q+2}}$. The case $q\in(0,1)$ has been studied in \cite{GHK04}.
$\Diamond$
\end{bem}
\begin{bem}[Large deviations and non-convexity]\label{rem-LDP} It is easy to see that, in the special case where $H(t)=\widetilde Dt^p$ (see \eqref{cumgenfct}), $K_H(u)=u^{\frac{2q}{d+2}} K_{D,q}$, for any $u>0$. (For asymptotic scaling relations see Lemma~\ref{constasy}.) In particular, $\frac 1n Z_n$ satisfies a large deviation principle on $(0,\infty)$ with speed $ n^{\frac d{d+2}}$ and rate function $u\mapsto u^{\frac{2q}{d+2}} K_{D,q}$. This function is strictly convex for $q>\frac d2 +1$ and strictly concave for $q<\frac d2 +1$. In the important special case of a centered Gaussian scenery, Theorem~\ref{lin} contains non-trivial information only in the case $d\in\{1,2,3\}$, in which the rate function is strictly convex, linear and strictly concave, respectively; see also \cite{CP01} and \cite{Ca04}.
The non-convexity around zero for bounded sceneries in $d\in\{3,4\}$ was found in \cite{AC02} by proving that $K_H(u)\geq C u^{\frac 4{d+2}}$ as $u\to0$ for some positive constant $C$.
$\Diamond$
\end{bem}
The upper bounds in Theorems~\ref{inter} and \ref{lin} are proved in Section~\ref{s:ub}, and the lower bounds in Section~\ref{s:lb}. We consider only sequences $b_n\ge 1$ there. The case $a_n^{\scriptscriptstylesup{0}}\ll b_n\ll 1$ seems subtle and is left open in the present paper; however see Section~\ref{Sec-Gauss} for a partial result.
Our next proposition gives almost sharp criteria for the positivity of the constants $K_{D,q}$ and $K_H(u)$ appearing in Theorems~\ref{inter} and \ref{lin}.
\begin{prop} [Positivity of the constants]\label{constpos} Fix $d\in\N$ and $p,q>1$ satisfying $\frac 1p+\frac 1q=1$.
\begin{enumerate}
\item[(i)] For any $D>0$,
\begin{equation}\label{Kdef}
K_{D,q}= (d+2)\Bigl(\frac D2\Bigr)^{\frac 2{d+2}}\Bigl(\frac{
\chi_{d,p}}d\Bigr)^{\frac d{d+2}},
\end{equation}
where
\begin{equation}\label{chiident}
\chi_{d,p}\inf\Bigl\{\frac 1{2}\|\Gamma^{\frac 12}\nabla \psi\|_2^2\colon \psi\in
H^1(\R^d)
\colon\|\psi\|_2=1=\|\psi\|_{2p}\Bigr\}.
\end{equation}
The constant $\chi_{d,p}$ is positive if and only if $d\leq \frac {2p}{p-1}=2q$.
Hence, $K_{D,q}$ is positive if and only if $d\leq \frac {2p}{p-1}=2q$.
\item[(ii)] The constant $K_H(u)$ is positive for any $u>\E[Y(0)]=0$ if
$$
\limsup_{t\to\infty}\frac{\log H(t)}{\log
t}<\begin{cases}\infty&\mbox{if }d\leq 2,\\
\frac d{d-2}&\mbox{if }d\geq3.
\end{cases}
$$
For $d\geq 3$, if $\liminf_{t\to\infty}\frac{\log H(t)}{\log t}>\frac d{d-2}$, then $K_H(u)=0$ for any $u>0$.
\end{enumerate}
\end{prop}
The proof of Proposition \ref{constpos} is in Section~\ref{sec-varform}. There we also clarify the relation between $\chi_{d,p}$ and the so-called {\it Gagliardo-Nirenberg\/} constant.
Now we formulate asymptotic relations between the rates obtained in Theorems~\ref{inter} and \ref{lin}.
\begin{lemma}[Asymptotic scaling relations]\label{constasy} Fix $D>0$ and $q>1$, and recall \eqref{cumgenfct}.
\begin{enumerate}
\item [(i)] Assume that $H(t)\scriptscriptstyleim\widetilde D t^p$ as $t\to\infty$, then
\begin{equation}\label{Kasy2}
K_H(u)\scriptscriptstyleim u^{\frac {2q}{d+2}} K_{D,q}\qquad\mbox{as }u\to\infty.
\end{equation}
\item [(ii)] Assume that $\E[Y(0)]=0$ and $\E[Y(0)^2]=1$, then
\begin{equation}\label{Kasy3}
K_H(u)\leq u^{\frac {4}{d+2}} \bigl[K_{\frac 12,2}+o(1)\bigr]\qquad\mbox{as }u{\rm d}ownarrow 0.
\end{equation}
\end{enumerate}
\end{lemma}
The proof of Lemma~\ref{constasy} is in Section~\ref{Sec-ScalLim}.
\begin{bem} We conjecture that the lower bound in \eqref{Kasy3} also holds under an appropriate upper bound on $H$. It is clear (see Remark~\ref{rem-LDP} and note the monotonicity of $K_H(u)$ in $H$) that $u^{-4/(d+2)}K_H(u)\geq K_{D,2}$ for every $u>0$ if $H(t)\leq \widetilde D t^2$ for every $t\geq 0$. The positivity of $\liminf_{u{\rm d}ownarrow 0} u^{-4/(d+2)}K_H(u)$ (for cumulant generating functions $H$ of {\it bounded\/} variables) is contained in \cite{AC02} as part of the proof for non-convexity of the rate function $K_H$ in $d\in\{3,4\}$. Since $K_{D,2}=0$ in $d> 4$, it is clear that this proof must fail in $d> 4$.
$\Diamond$
\end{bem}
Lemma~\ref{constasy}(i) is consistent with Theorems~\ref{inter} and \ref{lin}.
\scriptscriptstyleubsection{Heuristic derivation of Theorems~\ref{inter} and \ref{lin}}\label{sec-heur}
\noindent The asymptotics in \eqref{interasy} and \eqref{linasy}
are based on large deviation principles for scaled versions of the
walker's local times $\end{lemma}l_n$ and the scenery $Y$.
A short summary of the joint optimal strategy of the walker and the scenery is the following.
Let us first explain the exponential decay rate of the probabilities under consideration.
Assume that $1\ll b_n\ll n^{\frac 1q}$.
In order to contribute optimally to the event $\{\frac 1n Z_n>b_n\}$, the walker spreads out over a
region whose diameter is of order $\alpha_n$ (for a particular choice of
$\alpha_n$, depending on the sequence $(b_n)_n$). The cost for this behavior is
$e^{\Ocal(n\alpha_n^{-2})}$. The scenery assumes extremely
large values within that region, more precisely: values of the order $b_n$.
The cost for doing that is $\exp\{\Ocal(b_n^q\alpha_n^d)\}$, under Assumption (Y).
The choice of $\alpha_n$ is now determined by putting
\begin{equation}\label{alphachoice}
\frac n{\alpha_n^2}=\alpha_n^d b_n^q.
\end{equation}
A calculation shows that for this choice of $\alpha_n$ both sides of
\eqref{alphachoice} are equal to the logarithmic decay order of the probability
$\P(\frac 1n Z_n>b_n)$ in Theorem~\ref{inter}.
Next we give a more precise argument for the very large deviations (Theorem~\ref{inter}) which also explains the constants on the right hand side
of \eqref{interasy}. Introduce the scaled and normalized
version of the walker's local times,
\begin{equation}\label{Lndef}
L_n(x)=\frac{\alpha_n^d}n \end{lemma}l_n\bigl(\lfloor
x\alpha_n\rfloor\bigr),\qquad x\in \R^d.
\end{equation}
Then $L_n$ is a random element of the set
\begin{equation}
\Fcal=\Bigl\{\psi^2\in L^1(\R^d)\colon \|\psi\|_2=1\Bigr\}
\end{equation}
of all Lebesgue probability densities on $\R^d$.
Furthermore, introduce the scaled version of the field,
\begin{equation}\label{Yscaled}
\overline Y_n(x)=\frac 1{b_n}Y\bigl(\lfloor
x\alpha_n\rfloor\bigr),\qquad x\in \R^d.
\end{equation}
Then we have, writing $\langle\cdot,\cdot\rangle$ for the inner product on $L^2(\R^d)$,
\begin{equation}\label{Znscale}
\scriptscriptstylemfrac 1n Z_n=\scriptscriptstylemfrac 1n\scriptscriptstyleum_{z\in\Z^d}\frac n{\alpha_n^d}L_n\bigl({\textstyle{\frac z{\alpha_n}}}\bigr)
{b_n}\overline Y_n\bigl({\textstyle{\frac z{\alpha_n}}}\bigr)=b_n\langle L_n,\overline Y_n\rangle.
\end{equation}
Hence, the logarithmic asymptotics of the probability $\P(\scriptscriptstylemfrac 1n Z_n>b_n)
=\P(\langle L_n,\overline Y_n\rangle>1)$ will be
determined by a combination of large deviation principles for $L_n $ and $\overline Y_n$.
In the spirit of the celebrated large deviation
theorem of Donsker and Varadhan, the distributions of $L_n$
satisfy a weak large deviation principle in the weak $L^1$-topology on
$\Fcal$ with speed $n\alpha_n^{-2}$ and rate function $\Ical \colon
\Fcal\to[0,\infty]$ given by
\begin{equation}\label{Idef}
\Ical(\psi^2)=\begin{cases} \frac 1{2} \bigl\Vert\Gamma^{\frac
12}\nabla\psi \bigr\Vert_2^2
&\mbox{if } \psi\in H^1(\R^d),\\
\infty&\mbox{otherwise.}
\end{cases}
\end{equation}
Roughly speaking, this principle says that, for $\psi^2\in \Fcal$,
\begin{equation}\label{LDP1heur}
\P(L_n\approx \psi^2)\approx \exp\Big\{-\frac n{\alpha_n^2}\Ical(\psi^2)\Big\},\qquad n\to\infty.
\end{equation}
Using Assumption (Y), we see that the distributions of $\overline Y_n$
should satisfy, for any $R>0$, a weak large deviation principle
on some appropriate set of sufficiently regular functions $[-R,R]^d\to(0,\infty)$ with speed
$\alpha_n^d b_n^q$ and rate function
$$
\Phi_{D,q}(\varphi)=D\int_{[-R,R]^d} \varphi^q(x)\,{\rm d} x,
$$
as the following heuristic calculation suggests:
\begin{equation}\label{LDP2heur}
\begin{aligned}
\P(\overline Y_n\approx \varphi\mbox{ on }[-R,R]^d)&\approx
\P\Bigl(Y(z)>b_n \varphi\bigl({\textstyle{\frac z{\alpha_n}}}\bigr)
\mbox{ for }z\in [-R\alpha_n,R\alpha_n]^d\cap\Z^d\Bigr)\\
&\approx \prod_{z\in [-R\alpha_n,R\alpha_n]^d\cap\Z^d}
\exp\Bigl\{-D \bigl[b_n \varphi\bigl({\textstyle{\frac z{\alpha_n}}}\bigr)\bigr]^q\Bigr\}\\
&\approx \exp\Bigl\{ -D \alpha_n^d b_n^q\int_{[-R,R]^d} \varphi^q(x)\,{\rm d} x\Bigr\}.
\end{aligned}
\end{equation}
Note that the speeds of the two large deviation principles in
\eqref{LDP1heur} and \eqref{LDP2heur} are equal because of \eqref{alphachoice}.
Using the two large deviation principles and \eqref{Znscale}, we see that
$$
\P({\scriptscriptstylemfrac 1n} Z_n>b_n)\approx \exp\Bigl\{-\frac n{\alpha_n^2} \widetilde K_{D,q}\Bigr\},
$$
where
\begin{equation}\label{Ktildedef}
\widetilde K_{D,q}=\inf\{\Ical(\psi^2)+D\|\varphi\|_q^q\colon \psi^2\in\Fcal,
\varphi\in \Ccal_+(\R^d),\langle\psi^2,\varphi\rangle=1\Bigr\}.
\end{equation}
It is an elementary task to evaluate the infimum on $\varphi$ and to check that indeed
$K_{D,q}=\widetilde K_{D,q}$. This ends the heuristic explanation of Theorem~\ref{inter}.
The situation in the large deviation case, Theorem~\ref{lin},
is similar, when we put $b_n=1$. See \cite{AC02} for a heuristic argument in this case.
We distinguish the two cases of very large deviations (V)
and large deviations (L). The choices of $b_n$ and $\alpha_n$ in the respective cases are
the following.
\begin{equation}\label{bnalphachoice}
\begin{array}{lllrcl}
\mbox{case (V):}&\quad \mbox{Hypothesis of Theorem \ref{inter}}, &\quad 1\ll b_n\ll n^{\frac 1q},\qquad &\alpha_n&=&n^{\frac 1{d+2}}b_n^{-\frac q{d+2}},\\
\mbox{case (L):}&\quad \mbox{Hypothesis of Theorem \ref{lin}}, &\quad b_n=1,\qquad &\alpha_n&=&n^{\frac 1{d+2}}.
\end{array}
\end{equation}
\scriptscriptstyleubsection{Small deviations for Gaussian sceneries}\label{Sec-Gauss}
\noindent Theorems \ref{inter} and \ref{lin} do not handle sequences $(b_n)_n$ satisfying $a_n^{\scriptscriptstylesup{0}}\ll b_n\ll 1$, where we recall from \eqref{weakconv} that $a_n^{\scriptscriptstylesup{0}}$ is the scale of the convergence in distribution. In this regime, we present a partial result for Gaussian sceneries and simple random walk in $d=2$. This result is based on a deep result by Brydges and Slade \cite{BSl} about exponential moments of the renormalized self-intersection local time of simple random walk.
\begin{lemma}[Small deviations for Gaussian sceneries]\label{small} Assume that $Y(0)$ is a standard Gaussian random variable and that $(S_n)_n$ is the simple random walk, and assume that $d= 2$. Let
${n}^{-1/2}(\log n)^{1/2} = a_n^{\scriptscriptstylesup{0}} \ll b_n\ll a_n^{\scriptscriptstylesup{1}} \equiv n^{-1/2} \log n$, then
\begin{equation}\label{smallasy}
\lim_{n\to\infty}\frac {\log n}{b_n^2 n}\log\P({\scriptscriptstylemfrac 1n }Z_n>b_n)= -\frac \pi 4.
\end{equation}
\end{lemma}
\begin{Proof}{Proof}
As we mentioned in Section~\ref{sec-model}, the distribution of the random walk in random scenery, $Z_n$, is easily identified in terms of the walk's {self-intersection local time} $\Lambda_n$ defined in \eqref{Lambdadef}. More precisely, the conditional distribution of $Z_n$ given the walk $S$ is $\Ncal\times \scriptscriptstyleqrt{\Lambda_n}$, where $\Ncal$ is a standard normal variable, independent of the walk. The typical behavior of the self-intersection local time is as follows \cite{BSl}
\begin{equation}\label{typical}
\E\bigl[\Lambda_n\bigr]\scriptscriptstyleim\frac 2 \pi\bigl(na_n^{\scriptscriptstylesup{0}}\bigr)^2=\frac 2 \pi n\log n, \qquad n\to\infty.
\end{equation}
We prove now the upper bound in {\eqref{smallasy}}. Recall that $d=2$ and introduce the centered and normalized self-intersection local
time,
$$
\gamma_n=\frac 1n \Bigl(\Lambda_n-\E\bigl[\Lambda_n\bigr]\Bigr).
$$
\noindent Use Chebyshev's
inequality to obtain, for any $\theta>0$ and any $n\in\N$,
\begin{equation}\label{Chebychev}
\P(\scriptscriptstylemfrac 1n Z_n>b_n) \le \E\bigl[e^{\theta Z_n}\bigr] e^{-\theta b_n n}.
\end{equation}
Using the above characterization of the distribution of $Z_n$, we see that
\begin{equation}\label{Znident}
\E\bigl[e^{\theta Z_n}\bigr]=\E\bigl[\E\bigl[e^{\theta Z_n}\,\big|\,S\bigr]\bigr]=\E\bigl[\E\bigl[\exp\bigl\{\theta \Ncal \scriptscriptstyleqrt{\Lambda_n}\bigr\}\,\big|\, S\bigr]\bigr]=\E\bigl[e^{\frac 12\theta^2\Lambda_n}\bigr]=\E\bigl[e^{\frac 12\theta^2 n\gamma_n}\bigr]e^{\frac 12\theta^2 \E[\Lambda_n]}.
\end{equation}
According to Theorem~1.2 in \cite{BSl}, $\lim_{n\to\infty}
\E[e^{c\gamma_n}]$ exists and is finite for any $c<c_0$, where $c_0>0$ is some
positive constant. Now
pick $\theta=\theta_n= \pi\, b_n/(2\log n)$. Note that
$\theta_n^2\, n\to 0$ because of $b_n\ll n^{-1/2}\log n$, and
therefore the first factor on the right hand side of \eqref{Znident} is bounded,
according to the above mentioned result of Brydges and Slade. Use \eqref{typical} on the right hand side of \eqref{Znident} and substitute in \eqref{Chebychev} to obtain
$$
\log\P({\scriptscriptstylemfrac 1n} Z_n>b_n)\le - (1+o(1)) {\frac \pi 4} \,
\frac{b_n^2 n}{\log n}.
$$
\noindent This is the upper bound in (\ref{smallasy}).
Now we prove the lower bound in \eqref{smallasy}. Using the above characterization of the distribution of $Z_n$, we obtain, for any $\theta>0$,
\begin{equation}
\label{loweresti1}
\P(\scriptscriptstylemfrac 1n Z_n>b_n)\ge \P(\Ncal>\theta)\, \P\left( \Lambda_n>\frac{n^2b_n^2}{\theta^2}\right) .
\end{equation}
Fix an arbitrary $c\in (0, {\frac 2 \pi})$. We apply
\eqref{loweresti1} to
$\theta=b_n(\frac{n}{c\log n})^{1/2}$ and obtain
$$
\log \P(\scriptscriptstylemfrac 1n Z_n>b_n)\ge -\frac 1{2}\,b_n^2 \frac{n}{c \log n}(1+o(1))
+\log \P\left( \Lambda_n >c n\log n
\right),\qquad n\to\infty.
$$
By the Paley--Zygmund inequality (Kahane \cite{K85} p.~8) stating
that $\P (X> r \E [X]) \ge (1-r)^2 \E [X]^2/\E [X^2]$ for all
$r\in (0,1)$ and all square-integrable random variables $X$, we
obtain that
$$
\P\left( \Lambda_n>c
n\log n \right)\geq \bigl(1-\bigl({\scriptscriptstylemfrac{c\pi}2}\bigr)^2\bigr)\,\frac{\E[\Lambda_n]^2}{\E[\Lambda_n^2]}.
$$
Recall from (\ref{typical}) that
$\E[\Lambda_n] \scriptscriptstyleim {\frac 2 \pi} n\log n$ as $n\to
\infty$. On the other hand, Bolthausen \cite{B89} proved that
Var$[\Lambda_n] = \Ocal (n^2)$. Therefore, $\E[\Lambda_n^2]\scriptscriptstyleim \E[\Lambda_n]^2$, and, consequently,
$$
\liminf_{n\to \infty}\, \P\left( \Lambda_n>c
n\log n \right) >0.
$$
\noindent Therefore,
$$
\liminf_{n\to \infty}\, \frac {\log n} {b_n^2n} \log \P(\scriptscriptstylemfrac 1n Z_n>b_n) \ge
-{\frac 1 {2c}}.
$$
\noindent Letting $c\uparrow{\frac 2 \pi}$, this yields the lower bound in (\ref{smallasy}).
\end{Proof}
\qed
\scriptscriptstyleection{Variational formulas}\label{sec-varform}
In this section we prove Proposition~\ref{constpos} and Lemma~\ref{constasy}. In Section~\ref{subs:positivity} we prove a necessary and sufficient criterion for positivity of the constant $\chi_{d,p}$ defined in \eqref{chiident}. The relation to the Gagliardo-Nirenberg constant is discussed in Section~\ref{bem-GagNir}, and the relation to the constant $K_{D,q}$ defined in \eqref{kdq} is proved in Section~\ref{sec-Kchirel}, where we also finish the proof of Proposition~\ref{constpos}. Finally, Lemma~\ref{constasy} is proved in Section~\ref{Sec-ScalLim}.
\scriptscriptstyleubsection{Positivity of $\boldsymbol{\chi_{d,p}}$}
\label{subs:positivity}
\begin{lemma}\label{chipos}
The constant $\chi_{d,p}$ is positive if and only if $d\leq\frac{2p}
{p-1}$.
\end{lemma}
\begin{Proof}{Proof} Certainly, it suffices to do the proof only in
the case where $\frac 12\Gamma$ is the identity matrix.
See \cite[Sect.~2]{C03} for an alternate proof of the positivity of $\chi_{d,p}$ in the subcritical dimensions, $d<\frac{2p}{p-1}$, using the relation to the Gagliardo-Nirenberg constant, which we explain in Section~\ref{bem-GagNir}.
Let us recall standard Sobolev inequalities (see \cite[Theorems~8.3,
8.5]{LL97}). There are positive constants
$S_d$ for $d\geq 3$ and $S_{2,r}$ for $r>2$ such that
\begin{equation}\label{Sobolev}
\begin{array}{rcll}
S_d\|\psi \|_{2d/(d-2)}^2&\leq &\|\nabla \psi \|_2^2,\qquad&\mbox{for }
d\geq 3, \psi \in D^1(\R^d)\cap L^2(\R^d),\\
S_{2,r}\|\psi \|_{r}^2&\leq& \|\nabla \psi \|_2^2+\|\psi \|_2^2,
\qquad&\mbox{for }d=2, \psi \in H^1(\R^d),\; r>2.
\end{array}
\end{equation}
Here $D^1(\R^d)$ denotes the set of locally integrable functions $\R^d\to\R$
which vanish at infinity and possess a distributional derivative in $L^2(\R^d)$.
Let us first do the proof for the case $3\le d\leq\frac{2p}
{p-1}$.
For any $\psi\in H^1(\R^d)$ that satisfies
$\|\psi\|_2=1=\|\psi\|_{2p}$,
we may use the above Sobolev inequality and obtain that $\|\nabla
\psi\|_2^2
\geq {\operatorname {cst.}\,}\|\psi\|_{2d/(d-2)}^2$. We now rewrite
$$
\int_{\R^d} \psi^{\frac {2d}{d-2}}(t)\,{\rm d} t=\int_{\R^d}\bigl(\psi^{2p-2}(t)\bigr)^{\frac
2{(d-2)(p-1)}}\,\psi^2(t)\,{\rm d} t.
$$
Recall that $\psi^2$ is a probability density. Therefore, an application
of
Jensen's inequality to the convex map $x\mapsto x^{2/[(d-2)(p-1)]}$ yields
that
$\|\psi\|_{2d/(d-2)}$ satisfies a lower bound in terms of a power of $\|\psi\|_{2p}$,
which is equal to one. Hence, on the set of those $\psi\in H^1(\R^d)$
that satisfy $\|\psi\|_2=1=\|\psi\|_{2p}$, the map $\psi\mapsto
\|\nabla \psi\|_2^2$ is bounded away from zero. Now compare to
\eqref{chiident}
to see that this implies the assertion in the case $3\le d \le \frac{2p} {p-1}$.
Now we turn to $d=2$ with $p>1$ arbitrary. By a scaling $\psi_\beta= \beta^{\frac
d2}\psi(\cdot\,\beta)$, we can
find, for any ${\rm d}elta>0$, a $c({\rm d}elta)>0$ such that
\begin{equation}\label{chid=2}
\chi_{2,p}=c({\rm d}elta)\inf\Bigl\{\|\nabla \psi\|_2^2\colon \psi\in H^1(\R^d)
,\|\psi\|_2=1,\|\psi\|_{2p}={\rm d}elta\Bigr\}.
\end{equation}
Now we choose ${\rm d}elta$ such that $2{\rm d}elta^{-2}=S_{2,2p}$, the Sobolev
constant in \eqref{Sobolev}
for $d=2$ and $r=2p$. Then we have, for any $\psi$ in the set on the right
hand side of \eqref{chid=2},
$$
2=\frac 2{{\rm d}elta^2}\|\psi\|_{2p}^2=S_{2,2p}\|\psi\|_{2p}^2\leq \|\nabla \psi
\|_2^2+\|\psi \|_2^2=\|\nabla \psi \|_2^2+1,
$$
and hence it follows that $\chi_{2,p}\geq c({\rm d}elta)>0$.
Now we show that $\chi_{2d,p}\leq 2\chi_{d,p}$ for any $d\in\N$ and
$p\in (0,\infty)$. This simply
follows from the observation that, for any $\psi\in H^1(\R^d)$, the
function $\psi\otimes \psi\in H^1(\R^{2d})$
satisfies
$$
\| \nabla(\psi\otimes \psi)\|_2^2=2 \|\nabla\psi\|_2^2.
$$
Using this, the estimate $\chi_{2d,p}\leq 2\chi_{d,p}$ easily follows,
since $\|\psi\otimes\psi \|_2=\|\psi\|^2_2$ and $\|\psi\otimes \psi\|_{2p}=\|\psi\|^2_{2p}$.
In particular, this shows that $\chi_{1,p}>0$ for any $p>1$.
It remains to show that $\chi_{d,p}=0$ for $d>\frac{2p}{p-1}$. It is sufficient to
construct a sequence of sufficiently regular functions $\psi_n\colon\R^d\to[0,\infty)$
such that $\|\psi_n\|_2$ and $\|\psi_n\|_{2p}$ both converge towards some positive numbers,
but $\|\nabla\psi_n\|_2$ vanishes as $n\to\infty$. In order to do this, pick some
rotationally invariant function $\psi^2=f\circ|\cdot|\in\Fcal$ whose radial part
$f\colon(0,\infty)\to(0,\infty)$ satisfies
$$
f(r)=D\times\begin{cases}r^{-\gamma} &\mbox{if
}r\in(0,1),\\
1&\mbox{if }r\in[1,A],\\
A^{2d} r^{-2d}&\mbox{if }r>A,
\end{cases}
$$
where $A,D,\gamma>0$ are constants to be determined. Let $\omega_d$ denote the surface of the unit ball in $\R^d$.
The following statements can be easily verified by some tedious but elementary calculations:
\begin{eqnarray}
\gamma<d&\Longrightarrow&\|\psi\|_2^2=\frac{\omega_d}d D\Bigl[2A^d+\frac \gamma{d-\gamma}\Bigr]<\infty,\\
\gamma<\frac dp&\Longrightarrow&\|\psi\|_{2p}^{2p}={\omega_d} D^p\frac pd \Bigl[\frac\gamma{d-p\gamma}+A^d\frac 2{2p-1}\Bigr]<\infty,\\
\gamma<d-2&\Longrightarrow&\|\nabla \psi\|_2^2=\frac 14 \omega_d D\Bigl[\frac{\gamma^2}{d-\gamma-2}+A^{d-2}\frac{4d^2}{2+d}\Bigr]<\infty.
\end{eqnarray}
Since $p>1$ and $\frac dp<d-2$, we only have to assume that $\gamma<\frac dp$. Now we pick sequences $D_n$, $A_n$ and $\gamma_n$ such that all the following conditions are satisfied as $n\to\infty$:
$$
D_n\to 0,\qquad A_n\to\infty,\qquad \gamma_n\uparrow \frac dp,\qquad D_nA_n^d\to 1,\qquad\frac{D^p_n}{d-p\gamma_n}\to 1.
$$
Let $\psi_n$ be defined as the $\psi$ above with these parameters. Then we have, as $n\to\infty$,
$$
\|\psi_n\|_2^2\to 2\frac{\omega_d}d,\qquad \|\psi_n\|_{2p}^{2p}\to \omega_d,\qquad \|\nabla \psi_n\|_2^2\to 0.
$$
This ends the proof.
\end{Proof}
\qed
\scriptscriptstyleubsection{Relation to the Gagliardo-Nirenberg constant}\label{bem-GagNir}
\noindent Actually, for dimensions $d\geq 2$ in the special case that $\frac 12 \Gamma$ is the identity matrix, the constant $\chi_{d,p}$ in \eqref{chiident} can be identified in terms of the {\it Gagliardo-Nirenberg\/} constant, $\kappa_{d,p}$, as follows. Assume that $d\geq 2$ and $1<p<\frac d{d-2}$. Then $\kappa_{d,p}$ is defined as the smallest constant $C$ in the {\it Gagliardo-Nirenberg inequality\/}
\begin{equation}\label{GagNir}
\|\psi\|_{2p}\leq C \|\nabla\psi\|_2^{\frac{d(p-1)}{2p}}\|\psi\|_2^{1-\frac{d(p-1)}{2p}},\qquad \psi\in H^1(\R^d).
\end{equation}
This inequality received a lot of interest from physicists and analysts, and it has deep connections to Nash's inequality and logarithmic Sobolev inequalities. Furthermore, it also plays an important role in recent work of Chen \cite{C03} on self-intersections of random walks. See \cite[Sect.~2]{C03} for more on the Gagliardo-Nirenberg inequality.
It is clear that
\begin{equation}\label{kappadef}
\kappa_{d,p}=\scriptscriptstyleup_{\psi\in H^1(\R^d),\psi\not=0}\frac{\|\psi\|_{2p}}{\|\nabla\psi\|_2^{\frac{d(p-1)}{2p}}\|\psi\|_2^{1-\frac{d(p-1)}{2p}}}
=\Bigl(\inf_{\psi\in H^1(\R^d)\colon\|\psi\|_2=1}\|\psi\|_{2p}^{-\frac {4q}d}\|\nabla\psi\|_2^2\Bigr)^{-\frac d{4q}}.
\end{equation}
Clearly, the term over which the infimum is taken remains unchanged if $\psi$ is replaced by $\psi_\beta(\cdot)=\beta^{\frac d2}\psi(\cdot\,\beta)$ for any $\beta>0$. Hence, we can freely add the condition $\|\psi\|_{2p}=1$ and obtain that $\kappa_{d,p}=\chi_{d,p}^{-\frac d{4q}}$. In particular, the variational formulas for $\kappa_{d,p}$ in \eqref{kappadef} and for $\chi_{d,p}$ in \eqref{chiident} have the same maximizer(s) respectively minimizer(s). It is known that \eqref{kappadef} does possess a maximizer, and this is an infinitely smooth, positive and rotationally invariant function (see \cite{We83}). Uniqueness of the minimizer holds in $d\in\{2,3,4\}$ for any $p\in(1,\frac d{d-2})$, and in $d\in\{5,6,7\}$ for any $p\in(1,\frac 8d)$, see \cite{MS81}.
\scriptscriptstyleubsection{Relation between $\boldsymbol{K_{D,q}}$ and $\boldsymbol{\chi_{d,p}}$ (Proposition~\ref{constpos})}\label{sec-Kchirel}
Now we prove the remaining assertions of Proposition~\ref{constpos}.
(i) The relation \eqref{Kdef} is proved by an elementary scaling argument and optimization.
Indeed, replace $\psi$ by $\psi_\beta(\cdot)=\beta^{d/2}\psi(\cdot\,\beta)$ in \eqref{kdq} and
optimize explicitly on $\beta>0$. Afterwards the additional constraint $\|\psi\|_{2p}=1$
may freely be added. {From} \eqref{Kdef} and Lemma~\ref{chipos} the last assertion follows.
(ii) We only show the positivity of $K_H(u)$ for $d\geq 3$ and
${\overline p}<\frac d{d-2}$; the argument for $d\leq 2$ and any ${\overline p}>1$ is the same.
Since we assumed that $\E[Y(0)]=0$, we may pick some
${\rm d}elta>0$ such that $H(t)\leq u t/2$ for
$t\in[0,{\rm d}elta]$.
Pick $\varepsilon>0$ such that ${\overline p}+\varepsilon<
\frac d{d-2}$, then there is $c({\rm d}elta,\varepsilon)>0$
depending on
${\rm d}elta,\varepsilon$ and $H$ only, such that $H(t)\leq c({\rm d}elta,\varepsilon) t^{{\overline p}+\varepsilon}$ for any
$t\in[{\rm d}elta,
\infty)$. Then $H(t) \le \frac u2 \, t + c({\rm d}elta,\varepsilon) t^{{\overline p}+\varepsilon}$
for any $t\ge 0$, which implies that, for any $\psi\in H^1(\R^d)$ satisfying $\|\psi\|_2=1$,
$$
\begin{aligned}
\Phi_H(\psi^2,u)&\geq \scriptscriptstyleup_{\gamma>0}\Bigl[\gamma u-\int
\frac{u\gamma}2\, \psi^2(x)\,{\rm d} x-\int c({\rm d}elta,\varepsilon) (\gamma
\psi^2(x))^{{\overline p}+\varepsilon}\,{\rm d} x\Bigr]\\
&= \scriptscriptstyleup_{\gamma>0}\Bigl[ \, \frac u 2 \,
\gamma -c({\rm d}elta,\varepsilon)\gamma^{{\overline p}+\varepsilon}
\|\psi^2\|_{{\overline p}+\varepsilon}^{{\overline p}+\varepsilon}\Bigr].
\end{aligned}
$$
Now carry out the optimization over $\gamma$ to see that
$$
\Phi_H(\psi^2,u)\geq C \|\psi^2\|_{{\overline p}+\varepsilon}^{-q_\varepsilon},\qquad \mbox{where }\frac1{{\overline p}+\varepsilon}+\frac 1 {q_\varepsilon}=1,
$$
and $C>0$ depends on $u$, ${\overline p}+\varepsilon$ and $c({\rm d}elta,\varepsilon)$ only.
Hence, $K_H(u)\geq K_{C,q_\varepsilon}$. Since $d \le 2q_\varepsilon$, this is positive by assertion (i).
Now we show that $K_H(u)=0$ for any $u>0$ if ${\underline p}\equiv \liminf_{t\to\infty}
\frac{\log H(t)}{\log t}>\frac d{d-2}$. First we do this for a random variable $\widetilde Y(0)$ under the
assumption that $\E[\widetilde Y(0)]=1$. Pick $\varepsilon>0$ such that ${\underline p}-\varepsilon>\frac d{d-2}$. Since $H'(0)=1$,
there is $C>0$ such that $H(t)\geq C t^{{\underline p}-\varepsilon}$ for any $t\geq 0$. Hence, the above argument applies
and shows that $K_H(u)\leq K_{D,q_\varepsilon}$ for some $D>0$, where $q_\varepsilon$ is determined by
$\frac 1{{\underline p}-\varepsilon}+\frac 1{q_\varepsilon}=1$. Since ${\underline p}-\varepsilon>\frac d{d-2}$, the condition $d\leq \frac{2({\underline p}-\varepsilon)}{{\underline p}-\varepsilon-1}$ is violated.
Again assertion (i) implies that $K_H(u)=0$.
Let now $Y(0)$ have expectation $0$, then $\widetilde Y(0)=Y(0)+1$ has expectation 1. If $\widetilde H$ denotes the cumulant generating function of $\widetilde Y(0)$, then we have, according to the above, $K_{\widetilde H}(u)=0$ for any $u>0$. Since $K_{\widetilde H}(u)$ is well-defined, non-negative and non-decreasing for all $u\in\R$, we also have $K_{\widetilde H}(u)=0$ for any $u\in \R$. Obviously, $\widetilde H(t)=H(t)+t$ and $K_{\widetilde H}(u)=K_H(u-1)$ for any $u\in\R$, and this implies the statement.
\scriptscriptstyleubsection{Scaling relations (Lemma~\ref{constasy})}\label{Sec-ScalLim}
In this section, we prove Lemma~\ref{constasy}.
(i) Fix $\varepsilon>0$, then there is some $C>0$ such that
$$
-Ct+(\widetilde D-\varepsilon)t^p\leq H(t)\leq Ct+(\widetilde D+\varepsilon)t^p,\qquad t\geq 0.
$$
Using this in the definition of $\Phi_H(\psi^2,u)$, we obtain, for any $\psi\in H^1(\R^d)$,
$$
\scriptscriptstyleup_{\gamma>0}\Bigl\{\gamma (u-C)-\gamma^p(\widetilde D+\varepsilon)\|\psi^2\|_p^p\Bigr\}
\leq \Phi_H(\psi^2,u)\leq \scriptscriptstyleup_{\gamma>0}\Bigl\{\gamma (u+C)-\gamma^p
(\widetilde D-\varepsilon)\|\psi^2\|_p^p\Bigr\}.
$$
The suprema may easily be evaluated, and we obtain, for some $\eta_1,
\eta_2>0$, which vanish as $\varepsilon{\rm d}ownarrow 0$,
$$
(D-\eta_1)\|\psi^2\|_p^{-q} (u-C)^q\leq \Phi_H(\psi^2,u)\leq (D+\eta_2)\|\psi^2\|_p^{-q} (u+C)^q.
$$
Using this in the definition of $K_H(u)$ in \eqref{K3def}, we obtain
$$
K_{(D-\eta_1)(u-C)^q,q}\leq K_H(u)\leq K_{(D+\eta_2)(u+C)^q,q}.
$$
Now use Proposition~\ref{constpos}(i), in particular \eqref{Kdef}, and
use that $\eta_1,\eta_2\to 0$ as $\varepsilon{\rm d}ownarrow0$.
(ii) Substituting $\psi(\cdot)=u^{d/(d+2)}\psi_0(\cdot\, u^{2/(d+2)})$
and $\gamma= u^{(2-d)/(2+d)}\gamma_0$ yields that
\begin{equation}\label{KHscal}
u^{-\frac 4{d+2}}K_H(u)= \inf_{\|\psi_0\|_2=1}\Bigl\{\frac 12
\|\Gamma^{\frac 12}\nabla \psi_0\|_2^2 +\scriptscriptstyleup_{\gamma_0>0}\Bigl(
\gamma_0-\int u^{-2}H\bigl(u\gamma_0 \psi_0^2(x)\bigr)\,{\rm d} x\Bigr)\Bigr\}.
\end{equation}
It remains to show that the limit superior of the right
hand side as $u{\rm d}ownarrow 0$ is not larger than $K_{\frac 12,2}$.
This is shown as follows. Let $\psi_*\in H^1(\R^d)$ be an
$L^2$-normalized bounded minimizer in the variational formula in
\eqref{kdq} for $D=\frac 12$ and $q=2$. Its existence is proven in the same way as in
\cite{We83}, where the case $\Gamma={\rm Id}$ was considered.
Hence we have $\lim_{u{\rm d}ownarrow 0}\int u^{-2}H\bigl(u\gamma_0 \psi_*^2(x)\bigr)\,{\rm d} x
=\frac 12 \gamma_0^2 \|\psi_*^2\|_2^2$, uniformly in $\gamma_0$ on compacts of $[0,\infty)$.
Hence, the supremum on the right hand side of \eqref{KHscal} converges towards
$\scriptscriptstyleup_{\gamma_0>0}(\gamma_0-\frac 12 \gamma_0^2 \|\psi_*^2\|_2^2)=\frac 12 \|\psi_*^2\|_2^{-2}$.
Replacing on the right hand side of \eqref{KHscal} the infimum on $\psi_0$ by $\psi_*$,
we arrive at $\limsup_{u{\rm d}ownarrow 0}u^{-4/(d+2)} K_H(u)\leq K_{\frac 12,2}$, which is
\eqref{Kasy3}.
\scriptscriptstyleection{Proof of Theorems~\ref{inter} and \ref{lin}: Preparations}\label{sec-prep}
\noindent In this section we prepare for the proofs of our main
results, Theorems~\ref{inter} and \ref{lin}. Our proofs follow the
strategy of the proof of \cite[Theorem~2.2]{AC02}. That is, the proofs
of the lower bounds essentially follow the outline described in Section~\ref{sec-heur},
and the proofs of the upper bounds use an exponential Chebyshev inequality with a random
parameter. However, due to the unboundedness of the scenery in our case, we face a
serious additional difficulty, which we will overcome using a recently developed technique.
As we have already indicated in Section~\ref{sec-heur}, our main tools are large deviation
principles for the walker's local times and for the scenery. These principles are presented
in Sections~\ref{sec-LDP} and \ref{sec-LDPsc}, respectively.
However, for the application of these two principles, there are three main technical obstacles:
\begin{enumerate}
\item[(1)] the principles hold only on compact subsets of the space,
\item[(2)] the scaled scenery must be smoothed,
\item[(3)] the scaled scenery must be cut down to bounded size.
\end{enumerate}
The first obstacle will be handled later by making a connection to the
periodized version of the random walk, which is a standard recipe.
Hence, it will be necessary to approximate the variational formulas appearing in our main results
by finite-space versions, and this is carried out in Section~\ref{sec-appr}.
The necessity of the smoothing arises from the fact that the map $(\psi^2,\varphi)\mapsto
\langle \psi^2,\varphi\rangle$ is not continuous in the product of the topologies on which
the large deviation principles are based. This was already pointed out in \cite{AC02}. The remedy is a smoothing
procedure which was introduced in \cite{AC02} and will be adapted in Section~\ref{sec-smooth} below.
However, this procedure only works for {\it uniformly bounded\/} sceneries, and this
explains the necessity of a cutting argument for the scenery. This obstacle was not present
in \cite{AC02} and is the main technical challenge in the present paper, see Section~\ref{sec-cut}.
\scriptscriptstyleubsection{Large deviations for the local times}\label{sec-LDP}
\noindent In this section, we formulate one of our main tools: large deviation
principles for the normalized and scaled local times.
These principles are essentially standard and well-known, however, some of the principles
we use do not seem to have been proven in the literature, and therefore we shall provide
a proof for them in the appendix.
For the convenience of the reader, we recall the notion of a large deviation
principle. A sequence $(X_n)_{n\in\N}$ of random variables (or their distributions),
taking values in a topological space $\Xcal$,
satisfy a {\it large deviation principle\/} with {\it speed\/} $(\gamma_n)_{n\in\N}$ and
{\it rate function\/} $\Ical\colon \Xcal\to[0,\infty]$, if the following two statements hold:
\begin{eqnarray}
\limsup_{n\to\infty}\frac1{\gamma_n}\log P(X_n\in
F)&\leq& - \inf_{F} \Ical,\qquad F\scriptscriptstyleubset \Xcal\mbox{
closed},\label{upperbound}\\
\liminf_{n\to\infty}\frac1{\gamma_n}\log P(X_n\in
O)&\geq& -\inf_{O} \Ical,\qquad O\scriptscriptstyleubset \Xcal\mbox{
open}.\label{lowerbound}
\end{eqnarray}
This definition equally applies if the measure $P$ has not full mass, but happens to be
a subprobability measure only.
We shall need large deviation principles for a rescaled version of the local times of
our random walk. More precisely, we shall
need two slightly different principles:
one on never leaving a given cube in $\Z^d$ and $\R^d$, respectively, and another one for the
periodized version of the walk on that cube.
We recall
that we have listed our assumptions on the random walk at the beginning of Section~\ref{results}.
For $R>0$, we denote by $B_R=[-R,R]^d\cap\Z^d$ the centered box in
$\Z^d$ with radius $R$. By $S^{\scriptscriptstylemallsup{R}}=(S^{\scriptscriptstylemallsup{R}}_0,S^{\scriptscriptstylemallsup{R}}_1,{\rm d}ots)$
we denote the random walk on the torus $B_R$, i.e., the walk on $B_R$ (with the
opposite sides identified with each other) having transition kernel
\begin{equation}\label{kernelR}
p^{\scriptscriptstylemallsup{R}}(z,\widetilde z)=\scriptscriptstyleum_{k\in\Z^d}p(z,\widetilde z+2k\lfloor
R\rfloor),\qquad z,\widetilde z\in B_R,
\end{equation}
where $p(\cdot,\cdot)$ denotes the transition kernel of $S$. Note
that $p^{\scriptscriptstylemallsup{R}}$ is symmetric since $p$ is.
The local times of $S^{\scriptscriptstylemallsup{R}}$ are denoted by
\begin{equation}\label{perloctim}
\end{lemma}l_n^{\scriptscriptstylesup{R}}(z)=\scriptscriptstyleum_{k\in\Z^d}\end{lemma}l_n(z+2k\lfloor R\rfloor),\qquad z\in B_R.
\end{equation}
We consider rescaled versions of $\frac 1n\end{lemma}l_n$ and
$\frac 1n \end{lemma}l_n^{\scriptscriptstylemallsup{R}}$. Recall the normalized and rescaled version
$L_n$ of the local times $\end{lemma}l_n$ defined in \eqref{Lndef}.
By $\Fcal_R$ we denote the subset of those functions in $\Fcal$ whose
support lies in $Q_R=[-R,R]^d$.
Note that
\begin{equation}\label{supps}
\scriptscriptstyleupp(L_n)\scriptscriptstyleubset Q_R\qquad\Longleftrightarrow
\qquad\scriptscriptstyleupp(\end{lemma}l_n)\scriptscriptstyleubset B_{R\alpha_n}.
\end{equation}
Denote the scaled version of the torus-version of the local times, $\frac 1n \end{lemma}l_n^{\scriptscriptstylemallsup {R\alpha_n}}$, by
$L_n^{\scriptscriptstylemallsup{R}}\colon Q_R\to[0,\infty)$. Then
$L_n^{\scriptscriptstylemallsup{R}}$ is a random element of the set $\Fcal^{\scriptscriptstylemallsup{R}}$ of probability densities on the
torus $Q_R=[-R,R]^d$, whose opposite sides are identified with each other.
We define a rate function
$\Ical^{\scriptscriptstylemallsup{R}}\colon\Fcal^{\scriptscriptstylemallsup{R}}\to[0,\infty]$ by
\begin{equation}\label{IRdef}
\Ical^{\scriptscriptstylemallsup{R}}(\psi^2)=\frac 1{2}\int_{Q_R}\bigl|\Gamma^{\frac 12}\nabla_R
\psi(x)\big|^2\,{\rm d} x,
\end{equation}
if $\psi$ has an extension to an element of $H^1(\R^d)$,
and $\Ical^{\scriptscriptstylemallsup{R}}(\psi^2)=\infty$ otherwise. Here $\nabla_R$
denotes the gradient on the torus $Q_R$, i.e., with periodic boundary condition.
The topology used on the sets $\Fcal_R$
and on $\Fcal^{\scriptscriptstylemallsup{R}}$ are the weak topologies induced by the
test integrals against
the continuous bounded functions on
$Q_R$. If we identify any element of $\Fcal_R$ resp.~of
$\Fcal^{\scriptscriptstylemallsup{R}}$
with a probability measure, then this topology is just the usual weak
topology
on the set of probability measures on $Q_R$. In this
case, we extend the respective rate functions trivially by $\infty$ to
the set of measures not having a density.
\begin{lemma}[Large deviation principles for $L_n$]\label{LDP} Fix $R>0$.
Assume that $\alpha_n \to \infty$ and
$$
\alpha_n^d\ll \begin{cases} \scriptscriptstyleqrt n&\mbox{if }d=1,\\
\frac n{\log n}&\mbox{if }d=2,\\
n&\mbox{if }d\geq 3,
\end{cases}
$$
as $n\to\infty$. Then
the following two facts hold true.
\begin{enumerate}
\item[(i)] The distributions of $L_n$ under $\P(\,\cdot\,\cap\{\scriptscriptstyleupp(L_n)\scriptscriptstyleubset Q_R\})$
satisfy a large deviation principle on $\Fcal_R$
with speed $n\alpha_n^{-2}$ and rate function $\Ical_R$, the
restriction of $\Ical$ defined in
\eqref{Idef} to $\Fcal_R$.
\item[(ii)] The distributions of $L_n^{\scriptscriptstylemallsup{R}}$ under $\P$
satisfy a large deviation principle on $\Fcal^{\scriptscriptstylemallsup{R}}$ with
speed $n\alpha_n^{-2}$ and rate function $\Ical^{\scriptscriptstylemallsup{R}}$
given in \eqref{IRdef}.
\end{enumerate}
\end{lemma}
The upper bound \eqref{upperbound} of the principle in (i) for the
special case of
simple random walk and $\alpha_n=n^{\frac 1{d+2}}$ has been
proven by Donsker and Varadhan \cite{DV79}, Section~3.
We have deferred the proof of Lemma~\ref{LDP} to the
Appendix, Section~\ref{sec-proofLDP}. We feel that
the statement and its proof are standard and should be known
to the experts,
but we could not find a reference in the literature.
Our proof basically follows the route of \cite{Ga77}, which has
become standard by now. The strategy for the proof of (i) can be roughly summarized
as follows (the proof of (ii) is analogous). We shall identify the cumulant generating function
of $L_n$ (i.e., the logarithmic asymptotics of exponential moments
of test integrals against continuous and bounded functions $f$)
in terms of the Dirichlet eigenvalue of the operator
$\frac{1}{2}\nabla\cdot\Gamma\nabla + f$. In a second step, we prove
the large deviation principle via what is called now the
abstract {\em G\"artner-Ellis theorem\/} and identify the
rate function of the large deviation principle as the Legendre
transform of the eigenvalue.
\scriptscriptstyleubsection{Large deviations for the scenery}\label{sec-LDPsc}
\noindent In the proofs of the lower bounds in Theorems~\ref{inter} and \ref{lin}, we shall rely on
precise large deviation lower bounds for the scenery, tested against fixed
functions. The precise formulations are given for the respective cases here.
Recall from \eqref{bnalphachoice} the two cases (V) and (L), which correspond to Theorems~\ref{inter} and \ref{lin}, respectively.
\noindent We begin, in case (V ) with a large deviation
principle for the rescaled scenery $\overline Y_n$ defined in \eqref{Yscaled}.
\begin{lemma}\label{LDPfieldcont} Assume the case (V) in (\ref{bnalphachoice}), and
pick sequences $(b_n)_n$ and $(\alpha_n)_n$ as in \eqref{bnalphachoice}.
Fix $R>0$ and a continuous function
$\varphi\colon Q_R\to (0,\infty)$. Then
\begin{equation}
\liminf_{n\to\infty}\frac1{\alpha_n^d b_n^q}\log\P\bigl(\overline Y_n
\geq \varphi\mbox{ on }Q_R\bigr)\geq-D\|\varphi\|_q^q.
\end{equation}
\end{lemma}
\begin{Proof}{Proof} Fix some small $\varepsilon>0$.
It is easy to see that, for sufficiently large $n\in\N$,
$$
\begin{aligned}
\P\bigl(\overline Y_n
\geq \varphi\mbox{ on }Q_R\bigr)&=\prod_{z\in B_{R\alpha_n}}
\P\bigl(Y(z)\geq b_n \varphi(\scriptscriptstylemfrac z{\alpha_n})\bigr)\\
&\geq \exp\Bigl\{-(D-\varepsilon)b_n^q\scriptscriptstyleum_{z\in B_{R\alpha_n}}\varphi(\scriptscriptstylemfrac z{\alpha_n})^q\Bigr\}\\
&\geq\exp\Bigl\{-(D-2\varepsilon)\alpha_n^d b_n^q\|\varphi\|_q^q\Bigr\}.
\end{aligned}
$$
\end{Proof}
\qed
Let us now proceed with case (L).
\begin{lemma}\label{LDPfieldL} Assume the case (L) in (\ref{bnalphachoice}) and fix $R>0$, $M>0$ and a positive continuous function $\psi^2 \colon Q_R\to (0,\infty)$. Recall that $\alpha_n=n^{\frac 1{d+2}}$. Let $\widetilde H_M$ be the conditional cumulant generating function of $Y(0)$ given that $Y(0)\geq -M$. Then, for any $u>0$,
\begin{equation}
\liminf_{n\to\infty}\frac1{\alpha_n^d}\log\P\Bigl(\int_{Q_R}\overline Y_n(x) \psi^2 (x)\,{\rm d} x\geq u\,\Big|\,Y(z)\geq -M\quad \forall z\in B_{R\alpha_n}\Bigr)\geq \Phi_{\widetilde H_M}(\psi^2 ,u;R),
\end{equation}
where
\begin{equation}\label{PhiRdef}
\Phi_H(\psi^2,u;R)=\scriptscriptstyleup_{\gamma>0}\Bigl(\gamma u-\int_{Q_R}H(\gamma\psi^2(x))\,{\rm d} x\bigr)\Bigr)
\end{equation}
is the $Q_R$-version of $\Phi_H$ defined in \eqref{Phidef}.
\end{lemma}
\begin{Proof}{Proof}
For any $\gamma>0$, we have
$$
\begin{aligned}
\E\Bigl[&\exp\Bigl\{\gamma\alpha_n^d\int_{Q_R}\overline Y_n(x) \psi^2 (x)\,{\rm d} x\Bigr\}\,\Big|\,Y(z)\geq -M\quad \forall z\in B_{R\alpha_n}\Bigr]\\
&=\E\Bigl[\exp\Bigl\{\gamma\scriptscriptstyleum_{z\in B_{R\alpha_n}} Y(z)\alpha_n^d\int_{z/\alpha_n+[0,1/\alpha_n]^d}\psi^2(x)\,{\rm d} x\Bigr\}\,\Big|\,Y(z)\geq -M\quad \forall z\in B_{R\alpha_n}\Bigr]\\
&=\prod_{z\in B_{R\alpha_n}}e^{(1+o(1))\widetilde H_M\big(\gamma \psi^2 ({\scriptscriptstylemfrac {z}{\alpha_n}})\big)}\\
&=\exp\Bigl\{{\alpha_n^d}\int_{Q_R} \widetilde H_M(\gamma \psi^2 (x))\,{\rm d} x\,(1+o(1))\Bigr\}.
\end{aligned}
$$
According to a variant of the G\"artner-Ellis theorem, $\int_{Q_R}\overline Y_n(x) \psi^2 (x)\,{\rm d} x$ satisfies, under conditioning on $Y(z)\geq -M$ for all $z\in B_{R\alpha_n}$, a large deviation principle on $(0,\infty)$ with speed $\alpha_n^{d}$ and rate given by the Legendre transform of the map $\gamma\mapsto \int_{Q_R} \widetilde H_M(\gamma \psi^2 (x))\,{\rm d} x$. This transform is equal to the map $u\mapsto\Phi_{\widetilde H_M}(\psi^2 ,u;R)$.
\end{Proof}
\qed
\scriptscriptstyleubsection{The cutting argument}\label{sec-cut}
\noindent In this section we provide the cutting argument for the scenery in the cases (V) and (L).
Our method consists of a careful analysis of the $k$-th moments of the random walk in random
scenery, where $k=k_n$ is chosen in an appropriate dependence of $n$. Variants of this
method have recently been developed in the study of mutual intersections of random paths
in \cite{C03} and \cite{KM02}.
Fix sequences $(b_n)_n$ and $(\alpha_n)_n$ as in \eqref{bnalphachoice}
and consider the scaled normalized scenery $\overline Y_n$ as defined in \eqref{Yscaled}.
Fix $M>0$. We use the notation
\begin{equation}\label{ydeco}
y^{\scriptscriptstylesup{\leq M}}=(y\wedge M)\vee (-M)\qquad \mbox{and}\qquad
y^{\scriptscriptstylesup{> M}}=(y-M)_+,\qquad\mbox{for any }y\in \R.
\end{equation}
Later we shall estimate the scaled scenery $\overline Y_n$ by
$\overline Y_n\leq \overline Y_n^{\scriptscriptstylesup{\leq M}}
+\overline Y_n^{\scriptscriptstylesup{> M}}$. Here we show how we shall handle the second term.
\begin{prop}[Scenery cutting]\label{fieldcut} Assume one of the cases (V) or (L) in (\ref{bnalphachoice}).
Then, for any $\varepsilon>0$,
\begin{equation}
\lim_{M\to\infty}\limsup_{n\to\infty}\frac{\alpha_n^2}n\log
\P\bigl(\langle L_n,\overline Y_n^{\scriptscriptstylesup{> M}}\rangle >\varepsilon)=-\infty.
\end{equation}
\end{prop}
\scriptscriptstyleetcounter{step}{0}
\begin{Proof}{Proof}
\begin{step} It suffices to establish that there exists $C_M>0$ satisfying $\lim_{M\to\infty }C_M=0$ and
\begin{equation}\label{goal}
\E\bigl[\langle \end{lemma}l_n, Y^{\scriptscriptstylesup{>Mb_n}}\rangle^k\bigr]\leq n^k b_n^k C_M^k,\qquad n\in\N,\mbox{ where }k=\frac n{\alpha_n^2}.
\end{equation}
\end{step}
\begin{Proof}{Proof} Use the Markov inequality to estimate, for any $\varepsilon,M>0$ and $n,k\in\N$,
$$
\P\bigl(\langle L_n,\overline Y_n^{\scriptscriptstylesup{> M}}\rangle>\varepsilon)\leq \varepsilon^{-k}\E\bigl[\langle L_n,\overline Y_n^{\scriptscriptstylesup{>M}}\rangle^k\bigr]=\varepsilon^{-k}(nb_n)^{-k}\E\bigl[\langle \end{lemma}l_n,Y^{\scriptscriptstylesup{>Mb_n}}\rangle^k\bigr].
$$
Now put $k=n\alpha_n^{-2}$ and observe that the estimate in \eqref{goal} for some $C_M\to 0$ as $M\to\infty$ implies Proposition~\ref{fieldcut}.
\end{Proof}\qed
Our next step is a variant of the well-known periodization technique which projects the random walk in random scenery into a fixed box. Recall from \eqref{perloctim} the local times of the periodized random walk.
\begin{step}[Periodization]\label{step-per} For any $R,n,k\in\N$ and for any i.i.d.\ scenery $Y$ which is independent of the random walk,
\begin{equation}\label{writeoutform}
\E\bigl[\langle\end{lemma}l_n,Y\rangle^k\bigr]\leq \scriptscriptstyleum_{z_1,{\rm d}ots,z_k\in B_R}\E\Bigl[\prod_{i=1}^k \end{lemma}l_n^{\scriptscriptstylesup{R}}(z_i)\Bigr]\prod_{x\in B_R}\E\bigl[|Y(0)|^{\#\{i\colon z_i=x\}}\bigr].
\end{equation}
\end{step}
\begin{Proof}{Proof} We write out
\begin{equation}\label{writeout}
\E\bigl[\langle\end{lemma}l_n,Y\rangle^k\bigr]=\scriptscriptstyleum_{z_1,{\rm d}ots,z_k\in B_R}\scriptscriptstyleum_{m_1,{\rm d}ots,m_k\in\Z^d}\E\Bigl[\prod_{i=1}^k\end{lemma}l_n(z_i+2Rm_i)\Bigr]\E\Bigl[\prod_{i=1}^k Y(z_i+2Rm_i)\Bigr].
\end{equation}
We use that the scenery is i.i.d.~and derive, with the help of Jensen's inequality, the estimate
$$
\begin{aligned}
\E\Bigl[\prod_{i=1}^k Y(z_i+2Rm_i)\Bigr]&=\prod_{x\in B_R}\prod_{y\in \Z^d}\E\Bigl[Y(y)^{\#\{i\colon z_i=x,z_i+2Rm_i=y\}}\Bigr]\\
&\leq \prod_{x\in B_R}\prod_{y\in \Z^d}\E\Bigl[|Y(0)|^{\#\{i\colon z_i=x\}}\Bigr]^{\frac{\#\{i\colon z_i=x,z_i+2Rm_i=y\}}{\#\{i\colon z_i=x\}}}\\
&=\prod_{x\in B_R}\E\Bigl[|Y(0)|^{\#\{i\colon z_i=x\}}\Bigr].
\end{aligned}
$$
Use this in \eqref{writeout} and carry out the sum over $m_1,{\rm d}ots,m_k$ to finish.
\end{Proof}\qed
In the next step we estimate the term in \eqref{writeoutform} that involves the walker's local times. We denote by $S^{\scriptscriptstylesup{R}}$ the periodized version of the random walk in $B_R$ and by $p_s^{\scriptscriptstylesup{R}}(x,y)$ its transition probability from $x$ to $y$ in $s$ steps. By
\begin{equation}
G^{\scriptscriptstylesup{R}}_\lambda (x, y) = \scriptscriptstyleum\limits_{s=0}^\infty e^{-\lambda s}p_s^{\scriptscriptstylesup{R}}(x,y),
\end{equation}
we denote the Green's function associated with the periodized walk, geometrically stopped with parameter $\lambda>0$. $\mathfrak{S}_k$ denotes the set of permutations of $1,{\rm d}ots,k$.
\begin{step}\label{step-LocTim} Fix $R>0$, $\lambda>0$ and $k\in\N$.
Then, for any $n\in\N$, and for any $z_1,{\rm d}ots,z_k\in B_{R}$,
\begin{equation}\label{locestiGreen}
\E\Bigl[\prod_{i=1}^k \end{lemma}l_n^{\scriptscriptstylesup{R}}(z_i)\Bigr]\leq e^{\lambda n}\scriptscriptstyleum_{\scriptscriptstyleigma\in\mathfrak{S}_k}\prod_{i=1}^k G^{\scriptscriptstylesup{R}}_{\lambda}\bigl(z_{\scriptscriptstyleigma(i-1)},z_{\scriptscriptstyleigma(i)}\bigr).
\end{equation}
\end{step}
\begin{Proof}{Proof} Writing out the local times, we obtain
\begin{equation}\label{loctimesti1}
\begin{aligned}
\E\Bigl[\prod_{i=1}^k \end{lemma}l_n^{\scriptscriptstylesup{R}}(z_i)\Bigr]&\leq \scriptscriptstyleum_{t_1,{\rm d}ots,t_k=0}^n\P\Bigl(S^{\scriptscriptstylesup{R}}_{t_i}=z_i, \, i=1,{\rm d}ots,k\Big)\\
&\leq \scriptscriptstyleum_{0\leq t_1\leq t_2\leq {\rm d}ots\leq t_k\leq n}\scriptscriptstyleum_{\scriptscriptstyleigma\in\mathfrak{S}_k}\P\Bigl(S^{\scriptscriptstylesup{R}}_{t_{\scriptscriptstyleigma(i)}}=z_i, \, i=1,{\rm d}ots,k\Big)\\
&=\scriptscriptstyleum_{\scriptscriptstyleigma\in\mathfrak{S}_k}\scriptscriptstyleum_{s_1,{\rm d}ots,s_k\in\N_0}\1\Bigl\{\scriptscriptstyleum_{i=1}^ks_i\leq n\Bigr\}\prod_{i=1}^k p_{s_i}^{\scriptscriptstylesup{R}}\bigl(z_{\scriptscriptstyleigma(i-1)},z_{\scriptscriptstyleigma(i)}\bigr),
\end{aligned}
\end{equation}
where in the last line we substituted $s_i=t_i-t_{i-1}$ and wrote $\scriptscriptstyleigma^{-1}$ instead of $\scriptscriptstyleigma$. We put $\scriptscriptstyleigma(0)=0$ and $z_0=0$. Now we estimate the indicator by
$$
\1\Bigl\{\scriptscriptstyleum_{i=1}^ks_i\leq n\Bigr\}\leq e^{\lambda n}\prod_{i=1}^k e^{-s_i \lambda}.
$$
Using this in \eqref{loctimesti1} and carrying out the sums over $s_1,{\rm d}ots,s_k$, we arrive at the assertion.
\end{Proof}
\qed
In order to further estimate the Greenian term on the right of \eqref{locestiGreen}, we shall later need the following.
\begin{step}\label{step-Green}Fix $R>0$ and $p'\in(1,\frac d{d-2})$, if $d \geq 3$, or
$p'>1$ if $d \in \{1,2\}$. Then there is a constant $C>0$ such that, for any $n\in\N$ and any $x\in B_{R\alpha_n}$,
\begin{equation}\label{Greenesti}
\scriptscriptstyleum_{y\in B_{R\alpha_n}}G^{\scriptscriptstylesup{R\alpha_n}}_{\alpha_n^{-2}}(x,y)^{p'}\leq C\alpha_n^{d+(2-d)p'}.
\end{equation}
\end{step}
\begin{Proof}{Proof} For $d\leq 4$, we estimate, with the help of Jensen's inequality, and using that $p_s^{\scriptscriptstylesup{R\alpha_n}}(x,y)$ is not bigger than one and that its sum on $y\in B_{R\alpha_n}$ equals one,
$$
\begin{aligned}
\scriptscriptstyleum_{y\in B_{R\alpha_n}}G^{\scriptscriptstylesup{R\alpha_n}}_{\alpha_n^{-2}}(x,y)^{p'}&=\scriptscriptstyleum_{y\in B_{R\alpha_n}}\Big(\scriptscriptstyleum_{s=0}^\infty e^{-s\alpha_n^{-2}}p_s^{\scriptscriptstylesup{R\alpha_n}}(x,y)\Big)^{p'}\\
&\leq \big(1-e^{-\alpha_n^{-2}}\big)^{p'-1}\scriptscriptstyleum_{y\in B_{R\alpha_n}}\scriptscriptstyleum_{s=0}^\infty e^{-s\alpha_n^{-2}}p_s^{\scriptscriptstylesup{R\alpha_n}}(x,y)\\
&\leq \big(1-e^{-\alpha_n^{-2}}\big)^{p'-2}\scriptscriptstyleim\alpha_n^{4-2p'}.
\end{aligned}
$$
Now noting that $4-2p'\leq d+(2-d)p'$ for $d\leq 4$ finishes the proof of \eqref{Greenesti}.
For $d\geq 4$, we use another argument, which is based on the estimate \cite[Th.~2]{Uc98} $G(0,y)\leq C |y|^{2-d}$ for any $y\in\Z^d\scriptscriptstyleetminus\{0\}$, where $G$ is the Green's function for the free (i.e., non-stopped and non-periodized) random walk, and $C>0$ is constant. Certainly, it suffices to take $x=0$. We use $C>0$ and $c >0$ to denote generic
positive constants, not depending on $n$ or $y$, which may change their values from line to line. We estimate
\begin{equation}\label{Gsplit}
G^{\scriptscriptstylesup{R\alpha_n}}_{\alpha_n^{-2}}(0,y)\leq G(0,y)+\scriptscriptstyleum_{m\in\Z^d\scriptscriptstyleetminus\{0\}}\scriptscriptstyleum_{s\in\N_0}e^{-s\alpha_n^{-2}}p_s(0,y+2mR\alpha_n).
\end{equation}
For the first term, use the above mentioned result to see that $\scriptscriptstyleum_{y\in B_{R\alpha_n}}G(0,y)^{p'}\leq C\alpha_n^{d+(2-d)p'}$. With $\gamma>0$ a small auxiliary parameter, we split the sum on $s$ in the parts where $s\leq \gamma |m|\alpha_n$ and the remainder. Recall that the walker's steps have some exponential moments, see \eqref{emom}. Hence, we can estimate, if $\gamma$ is small enough ($\gamma<\frac R4/\log\E[e^{|S_1|}]$ suffices), for $|m|\geq 1$ and $s\leq \gamma |m|\alpha_n$, and all $y\in B_{R\alpha_n}$,
\begin{equation}\label{leqesti}
\begin{aligned}
p_s(0,y+2mR\alpha_n)&\leq \P(|S_s|\geq |y+2mR\alpha_n|)\leq \E[e^{|S_1|}]^s e^{-|y+2mR\alpha_n|}\leq \E[e^{|S_1|}]^se^{-R\alpha_n|m|}\\
&\leq e^{-c|m|\alpha_n}.
\end{aligned}
\end{equation}
This gives, for any $y\in B_{R\alpha_n}$,
\begin{equation}\label{firstsum}
\scriptscriptstyleum_{m\in\Z^d\scriptscriptstyleetminus\{0\}}\scriptscriptstyleum_{s\in\N_0\colon s\leq \gamma\alpha_n |m|}e^{-s\alpha_n^{-2}}p_s(0,y+2mR\alpha_n)\leq C \scriptscriptstyleum_{m\in\Z^d\scriptscriptstyleetminus\{0\}}e^{-c|m|\alpha_n}=o(\alpha_n^{2-d}).
\end{equation}
The remainder is estimated as follows. We use the local central limit theorem (see \cite[Ch.~VII, Thm.~13]{Pe75}) to deduce that there are $C>0$ and $c>0$ such that
\begin{equation}\label{LCLT}
p_s(0,x)\leq \frac C{s^{d/2}}e^{-c|x|^2/s}+C s^ {-d},\qquad s\in\N,\, x\in\Z^d.
\end{equation}
This gives, for any $y\in B_{R\alpha_n}$,
$$
\begin{aligned}
\scriptscriptstyleum_{m\in\Z^d\scriptscriptstyleetminus\{0\}}&\scriptscriptstyleum_{s\in\N_0\colon s\geq \gamma\alpha_n |m|}e^{-s\alpha_n^{-2}}p_s(0,y+2mR\alpha_n)\\
&\leq
C\scriptscriptstyleum_{ s\geq \gamma\alpha_n}e^{-s\alpha_n^{-2}}\Big[s^{-d/2}\scriptscriptstyleum_{0< |m|\leq s/(\gamma\alpha_n)}e^{-c|m|^2\alpha_n^2/s}+\Big(\frac s{\alpha_n}\Big)^d s^{-d}\Big],
\end{aligned}
$$
where we interchanged the sums on $s$ and $m$, and we also used that $|y+2mR\alpha_n|\geq |m|\alpha_n$ for $m\in\Z^d\scriptscriptstyleetminus\{0\}$. Using the substitution $w=|m|\alpha_n/\scriptscriptstyleqrt s$, the sum on $m$ is estimated by
$$
\scriptscriptstyleum_{0< |m|\leq s/(\gamma\alpha_n)}e^{-c|m|^2\alpha_n^2/s}
\leq C \Big(\frac s{\alpha_n^2}\Big)^{d/2}\int_{\alpha_n/\scriptscriptstyleqrt s}^{\scriptscriptstyleqrt s/\gamma}{\rm d} w\, w^{d-1}e^{-cw^2}\leq C \Big(\frac s{\alpha_n^2}\Big)^{d/2}.
$$
Since $\scriptscriptstyleum_{ s\in\N_0}e^{-s\alpha_n^{-2}}\leq C\alpha_n^2$, this implies that
\begin{equation}\label{geqesti}
\scriptscriptstyleum_{m\in\Z^d\scriptscriptstyleetminus\{0\}}\scriptscriptstyleum_{s\in\N_0\colon s\geq \gamma\alpha_n |m|}e^{-s\alpha_n^{-2}}p_s(0,y+2mR\alpha_n)\leq C\alpha_n^{2-d}.
\end{equation}
Use \eqref{firstsum} and \eqref{geqesti} in \eqref{Gsplit} to conclude.
\end{Proof}
\qed
The next step is a preparation for the estimate of the last term in \eqref{writeoutform}.
\begin{step}\label{step-Yesti} Let $Y$ be a random variable that satisfies
\begin{equation}\label{ttail}
\limsup_{r\to\infty}r^{-q}\log\P(Y>r)<0
\end{equation}
for some $q> 1$.
\begin{enumerate}
\item[(i)]
Fix $L>0$. Then there is $C_{M,L}>0$ such that $\lim_{M\to\infty}C_{M,L}=0$ such that, for every $n\in\N$ and $M>0$ and $b_n \geq 1$,
\begin{equation}
\E\bigl[(Y-M b_n)_+^{Lb_n^q}\bigr]^{\frac 1{Lb_n^q}}\leq b_n C_{M,L}.
\end{equation}
\item[(ii)] There is a constant $C>0$ such that, for any $\mu \in\N$,
\begin{equation}
\E[Y_+^\mu ]\leq \mu^{\frac 1q \mu }C^\mu.
\end{equation}
\end{enumerate}
\end{step}
\begin{Proof}{Proof} From our assumption on $Y$, we know that there are $C,D>0$ and $q> 1$ such that $\P(Y>s)\leq C e^{-Ds^q}$ for all $s>0$.
{\it Proof of (i).} We write $L$ instead of $Lb_n^q$ and have
\begin{equation}\label{expectY}
b_n^{-L}\E\bigl[(Y-M b_n)_+^{L}\bigr]=b_n^{-L}\int_0^\infty \P\bigl((Y-Mb_n)^L>t\bigr)\,{\rm d} t=L\int_0^\infty s^{L-1}\P(Y>(s+M)b_n)\,{\rm d} s.
\end{equation}
Now use the above estimate $\P(Y>(s+M)b_n)\leq C\exp\{-D(s+M)^q b_n^q\}$ for all $s>0$. Furthermore, use that $(s+M)^q\geq s^q+M^q$. This gives
\begin{equation}
\E\bigl[(Y-M b_n)_+^{L}\bigr] \leq b_n^L LC e^{-DM^qb_n^q}\int_0^\infty s^{L-1} e^{-D(sb_n)^q}\, {\rm d} s=LC e^{-DM^qb_n^q}\int_0^\infty s^{L-1} e^{-Ds^q}\,{\rm d} s.
\end{equation}
The change of variables $t=D s^q$ turns this into
\begin{equation}\label{momYesti}
\E\bigl[(Y-M b_n)_+^{L}\bigr] \leq L C e^{-DM^qb_n^q}D^{-L/q}q \Gamma(L/q),
\end{equation}
where $\Gamma$ denotes the Gamma-function.
Note that $\Gamma(x)\leq (C_1x)^x$ for some $C_1>0$ and all $x\geq 1$. Now we replace $L$ by $Lb_n^q$ and take the $(L b_n^q)$-th root to obtain
$$
\E\bigl[(Y-M b_n)_+^{Lb_n^q}\bigr]^{\frac 1{Lb_n^q}}
\leq \widetilde C (L b_n^q)^{\frac 1{Lb_n^q}}L^{\frac 1q} e^{-M^qD/L} b_n,
$$
where $\widetilde C$ does not depend on $L$ nor on $M$ or $n$. Since $b_n\geq 1$, the assertion is proved.
{\it Proof of (ii).} From \eqref{momYesti} with $M=0$ and $L=\mu$, we have $\E[Y_+^\mu]\leq \mu C \,D^{-\mu/q} \Gamma(\mu/q)$.
Recalling that $\Gamma(x)\leq (C_1x)^x$ for some $C_1>0$ and all $x\geq 1$, we arrive at the assertion.
\end{Proof}
\qed
\begin{step}\label{conclusion} Conclusion of the proof.
\end{step}
\begin{Proof}{Proof} Fix $R>0$ and let $B=B_{R\alpha_n}$ be the centered box in $\Z^d$ with radius $R\alpha_n$.
Note that in both cases (V) and (L), (\ref{ttail}) is satisfied with $q>\frac d2$.
Let $p$ be defined by $1=\frac 1p+\frac 1q$. Then, in both cases, $p \in (1, \frac{d}{d-2})$ if $d \geq 3$ and $p> 1$ if $d=2$. Put $k=n\alpha_n^{-2}$. Recall that $\alpha_n^{d+2}=nb_n^{-q}$. Recall that it suffices to prove \eqref{goal}. In the following, we shall use $C$ to denote a generic positive constant which depends on $R$, $q$ and $D$ only and may change its value from line to line.
Use Steps~\ref{step-per}--\ref{step-LocTim} for the scenery $Y$ replaced by $Y^{\scriptscriptstylesup{>Mb_n}}$ and $R$ replaced by $R\alpha_n$ and with $\lambda=\alpha_n^{-2}$ to obtain
\begin{equation}\label{cuttingesti2}
\E\bigl[\langle \end{lemma}l_n, Y^{\scriptscriptstylesup{>Mb_n}}\rangle^k\bigr]\leq e^{k}\scriptscriptstyleum_{\scriptscriptstyleigma\in\mathfrak{S}_k}\scriptscriptstyleum_{z_1,{\rm d}ots,z_k\in B} \prod_{i=1}^k G^{\scriptscriptstylesup{R\alpha_n}}_{\alpha_n^{-2}}\bigl(z_{\scriptscriptstyleigma(i-1)},z_{\scriptscriptstyleigma(i)}\bigr)\prod_{x\in B}\E\bigl[(Y(0)-Mb_n)_+^{\mu_x}\bigr],
\end{equation}
where we abbreviated $\mu_x=\#\{i\colon z_i=x\}$. Let us estimate the last term. We fix a parameter $L>0$ and split the product on $x\in B$ into the subproducts on $B_{\scriptscriptstylesup{L}}=\{x\in B\colon \mu_x\leq L b_n^q\}$ and $B_{\scriptscriptstylesup{L}}^{\rm c}=B\scriptscriptstyleetminus B_{\scriptscriptstylesup{L}}$. We estimate, with the help of Step~\ref{step-Yesti},
\begin{equation}
\begin{aligned}
\prod_{x\in B}\E\bigl[(Y(0)-Mb_n)_+^{\mu_x}\bigr]&\leq \prod_{x\in B_{\scriptscriptstylesup{L}}}\E\bigl[(Y(0)-Mb_n)_+^{Lb_n^q}\bigr]^{\frac {\mu_x}{Lb_n^q}}\prod_{x\in B_{\scriptscriptstylesup{L}}^{\rm c}}\E[Y(0)_+^{\mu_x}]\\
&\leq \prod_{x\in B_{\scriptscriptstylesup{L}}}(C_{M,L}b_n)^{\mu_x}\prod_{x\in B_{\scriptscriptstylesup{L}}^{\rm c}}(C\mu_x^{\frac 1q})^{\mu_x}.
\end{aligned}
\end{equation}
Let us abbreviate the term on the right hand side by $K(\mu)$ where $\mu=(\mu_x)_{x\in B}$. Now we pick numbers $p'>p$, $q'>1$ such that $\frac 1{p'}+\frac 1{q'}=1$ and, if $d \geq 3$, $p'<\frac d{d-2}$, and use H\"older's inequality in \eqref{cuttingesti2} to obtain
\begin{equation}\label{cuttingesti3}
\E\bigl[\langle \end{lemma}l_n, Y^{\scriptscriptstylesup{>Mb_n}}\rangle^k\bigr]\leq e^{k}\scriptscriptstyleum_{\scriptscriptstyleigma\in\mathfrak{S}_k}\Bigl(\scriptscriptstyleum_{z_1,{\rm d}ots,z_k\in B} \prod_{i=1}^k G^{\scriptscriptstylesup{R\alpha_n}}_{\alpha_n^{-2}}\bigl(z_{\scriptscriptstyleigma(i-1)},z_{\scriptscriptstyleigma(i)}\bigr)^{p'}\Bigr)^{\frac 1{p'}}\Bigl(\scriptscriptstyleum_{z_1,{\rm d}ots,z_k\in B}K(\mu)^{q'}\Bigr)^{\frac 1{q'}}.
\end{equation}
Using \eqref{Greenesti} in Step~\ref{step-Green}, the term in the first brackets may be estimated by
\begin{equation}\label{Gesti}
\Bigl(\scriptscriptstyleum_{z_1,{\rm d}ots,z_k\in B} \prod_{i=1}^k G^{\scriptscriptstylesup{R\alpha_n}}_{\alpha_n^{-2}}\bigl(z_{\scriptscriptstyleigma(i-1)},z_{\scriptscriptstyleigma(i)}\bigr)^{p'}\Bigr)^{\frac 1{p'}}\leq C^k \alpha_n^{2k} \alpha_n^{-\frac 1{q'}dk}.
\end{equation}
Now we estimate the last term in \eqref{cuttingesti3}. By $A_k$ we denote the set of maps $\mu\colon B\to\N_0$ such that $\scriptscriptstyleum_{x\in B}\mu_x=k$. Observe that, for any $\mu\in A_k$, we have
$$
\#\{(z_1,{\rm d}ots,z_k)\in B^k\colon \mu_x=\#\{i\colon z_i=x\}\quad \forall x\in B\}=\frac {k!}{\prod_{x\in B}\mu_x!}.
$$
Hence,
\begin{equation}\label{betaexpression}
\scriptscriptstyleum_{z_1,{\rm d}ots,z_k\in B}K(\mu)^{q'}\leq C^k k!\scriptscriptstyleum_{\mu\in A_k} \prod_{x\in B_{\scriptscriptstylesup{L}}}C_{M,L}^{q'\mu_x}\prod_{x\in B_{\scriptscriptstylesup{L}}}\Bigl(\frac{b_n^{q'}}{\mu_x}\Bigr)^{\mu_x}\prod_{x\in B_{\scriptscriptstylesup{L}}^{\rm c}}\mu_x^{-(1-\frac {q'}q)\mu_x}.
\end{equation}
Since $q'<q$, we have that $r\equiv 1-\frac {q'}q$ is positive. According to the definition of $B_{\scriptscriptstylesup{L}}$, the last term in \eqref{betaexpression} can be estimated by
\begin{equation}\label{lasttermesti}
\prod_{x\in B_{\scriptscriptstylesup{L}}^{\rm c}}\mu_x^{-(1-\frac {q'}q)\mu_x}\leq \prod_{x\in B_{\scriptscriptstylesup{L}}^{\rm c}}\bigl(L^{-r}b_n^{(q'-q)}\bigr)^{\mu_x}.
\end{equation}
The penultimate term in \eqref{betaexpression} can be estimated as
\begin{equation}\label{second-last}
\prod_{x\in B_{\scriptscriptstylesup{L}}}\Bigl(\frac{b_n^{q'}}{\mu_x}\Bigr)^{\mu_x}\leq C^k\prod_{x\in B_{\scriptscriptstylesup{L}}}b_n^{(q'-q)\mu_x},
\end{equation}
since we have, using also Jensen's inequality for the logarithm,
$$
\begin{aligned}
\prod_{x\in B_{\scriptscriptstylesup{L}}}\Bigl(\frac{b_n^{q'}}{\mu_x}\Bigr)^{\mu_x}
&=\exp\Bigl\{\Bigl(\scriptscriptstyleum_{y\in B_{\scriptscriptstylesup{L}}}\mu_y\Bigr)\scriptscriptstyleum_{x\in B_{\scriptscriptstylesup{L}}}\frac{\mu_x}{\scriptscriptstyleum_{y\in B_{\scriptscriptstylesup{L}}}\mu_y}\log\frac{b_n^{q'}}{\mu_x}\Bigr\}\\
&\leq\exp\Bigl\{\Bigl(\scriptscriptstyleum_{y\in B_{\scriptscriptstylesup{L}}}\mu_y\Bigr)\log\scriptscriptstyleum_{x\in B_{\scriptscriptstylesup{L}}}\frac{b_n^{q'}}{\scriptscriptstyleum_{y\in B_{\scriptscriptstylesup{L}}}\mu_y}\Bigr\}\\
&=\prod_{x\in B_{\scriptscriptstylesup{L}}}\Bigl(\frac{b_n^{q'}\#B_{\scriptscriptstylesup{L}}}{\scriptscriptstyleum_{y\in B_{\scriptscriptstylesup{L}}}\mu_y}\Bigr)^{\mu_x}.
\end{aligned}
$$
Now use that $\#B_{\scriptscriptstylesup{L}}\leq\#B\leq C\alpha_n^d=Ckb_n^{-q}$ and observe that there is a constant $C>0$ such that $(\frac kl)^l\leq C^k$, for any $l\in \{1,{\rm d}ots,k\}$, since the map $y\mapsto y\log y$ is bounded on $(0,1]$. Using \eqref{lasttermesti} and \eqref{second-last} in \eqref{betaexpression}, we obtain, for some constant $C_M>0$, satisfying $\lim_{M\to\infty} C_M=0$,
\begin{equation}\label{Kesti}
\begin{aligned}
\scriptscriptstyleum_{z_1,{\rm d}ots,z_k\in B}K(\mu)^{q'}&\leq C^k k! b_n^{(q'-q)k}\scriptscriptstyleum_{\mu\in A_k}\prod_{x\in B_{\scriptscriptstylesup{L}}}C_{M,L}^{q'\mu_x}\prod_{x\in B_{\scriptscriptstylesup{L}}^{\rm c}}L^{-r\mu_x}\\
&\leq C^k k! b_n^{(q'-q)k}\# A_k \Big(\max\{C_{M,L}^{q'}, L^{-r}\}\Big)^k\leq
C_M^{q' k}k! b_n^{(q'-q)k},
\end{aligned}
\end{equation}
where we choose $L$ in dependence on $M$ such that $\lim_{M\to\infty}\max\{C_{M,L}^{q'}, L^{-r}\}=0$, and we estimated $\# A_k =\binom {k+|B|}{|B|}\leq e^{o(k)}$ (recall that $k=n\alpha_n^2$).
Using \eqref{Kesti} and \eqref{Gesti} in \eqref{cuttingesti3}, we arrive at
\begin{equation}
\E\bigl[\langle \end{lemma}l_n, Y^{\scriptscriptstylesup{>Mb_n}}\rangle^k\bigr]\leq C_M^k k!\alpha_n^2k \alpha_n^{-\frac 1{q'}dk}\Bigl(k! b_n^{(q'-q)k}\Bigr)^{\frac 1{q'}}.
\end{equation}
Now recall that $b_n^q\alpha_n^d=k=n\alpha_n^{-2}$ and use Stirling's formula to see that the right hand side of this estimate is bounded from above by $C_M^k (nb_n)^k$ for some $C_M\to 0$ as $M\to\infty$. This ends the proof of Proposition~\ref{fieldcut}.
\end{Proof}
\qed
\end{Proof}
\qed
\scriptscriptstyleubsection{Smoothing the scenery}\label{sec-smooth}
\noindent In this section we provide the smoothing argument for the field. This
will be an adaptation of results of \cite[Sect.~3]{AC02}. Fix some smooth, rotationally invariant, and
$L^1$-normalized function $\kappa\colon\R^d\to[0,\infty)$ with $\scriptscriptstyleupp(\kappa)\scriptscriptstyleubset Q_1$, and put $\kappa_{\rm d}elta(\cdot)
={\rm d}elta^{-d}\kappa(\cdot/{\rm d}elta)$ for some small ${\rm d}elta>0$.
The convolution of two functions $f,g\colon
\R^d\to\R$ is denoted by $f*g$. Assume any of the cases (V) and (L) and
choose $(b_n)_n$ and $(\alpha_n)_n$
according to \eqref{bnalphachoice}.
We consider the rescaled and cut-down field
$\overline Y_n^{\scriptscriptstylesup{\leq M}}\colon\R^d\to[-M,M]$; see \eqref{ydeco}. Recall the scaled and normalized
local times $L_n$ from \eqref{Lndef}. By $\Bcal_M(\R^d)$ we denote the set of all measurable functions $\R^d\to[-M,M]$.
\begin{lemma}[Scenery smoothing]\label{FieSmoo} Fix $M>0$. Then, for any $\varepsilon>0$,
\begin{equation}\label{uniformsmoothing}
\lim_{{\rm d}elta{\rm d}ownarrow 0}\limsup_{n\to\infty}\frac{\alpha_n^2}n\log \scriptscriptstyleup_{f\in \Bcal_M(\R^d)}\P\bigl(|
\langle L_n,[f-f*\kappa_{\rm d}elta]\rangle|>\varepsilon)=-\infty.
\end{equation}
In particular,
\begin{equation}\label{fieldsmoothing}
\lim_{{\rm d}elta{\rm d}ownarrow 0}\limsup_{n\to\infty}\frac{\alpha_n^2}n\log \P\bigl(|
\langle L_n,[\overline Y_n^{\scriptscriptstylesup{\leq M}}-\overline Y_n^{\scriptscriptstylesup{\leq M}}*\kappa_{\rm d}elta]\rangle|>\varepsilon)=-\infty.
\end{equation}
\end{lemma}
\begin{Proof}{Proof} Certainly, it suffices to prove \eqref{uniformsmoothing} for $M=1$. We adapt the proof of \cite[Lemma~3.1]{AC02}, which is the same statement for $M=1$ and Brownian motion instead of random walk in Brownian scaling. We shall write $\Bcal$ instead of $\Bcal_1(\R^d)$.
Since all exponential moments of the steps are assumed finite, we have
$$
\lim_{R\to\infty}\limsup_{n\to\infty}\frac{\alpha_n^2}n\log \P\bigl(\scriptscriptstyleupp(\end{lemma}l_n)\not\scriptscriptstyleubset B_{R_n}\bigr)=-\infty,
$$
where $R_n=R n\alpha_n^{-1}$. Hence, it suffices to show, for every $R>0$,
\begin{equation}
\lim_{{\rm d}elta{\rm d}ownarrow 0}\limsup_{n\to\infty}\frac{\alpha_n^2}n\log \scriptscriptstyleup_{f\in \Bcal}\P\bigl(|
\langle L_n,f-f*\kappa_{\rm d}elta\rangle|>\varepsilon, \scriptscriptstyleupp(\end{lemma}l_n)\scriptscriptstyleubset B_{R_n})=-\infty.
\end{equation}
We prove this only without absolute value signs, since the complementary inequality is proved in the same way. Fix $f\in\Bcal$. Chebyshev's inequality yields, for any $a>0$,
\begin{equation}\label{aimsmooth}
\begin{aligned}
\P\bigl(
\langle &L_n,f-f*\kappa_{\rm d}elta\rangle>\varepsilon, \scriptscriptstyleupp(\end{lemma}l_n)\scriptscriptstyleubset B_{R_n})\\
&\leq \E\Bigl[\exp\Bigl\{a \frac n{\alpha_n^{2}}\langle L_n,f-f*\kappa_{\rm d}elta\rangle\Bigr\}\1\{\scriptscriptstyleupp(\end{lemma}l_n)\scriptscriptstyleubset B_{R_n}\}\Bigr]e^{-a\varepsilon n\alpha_n^{-2}}.
\end{aligned}
\end{equation}
Introduce a discrete version $\varphi_n\colon \Z^d\to\R$ of $f-f*\kappa_{\rm d}elta$ by
\begin{equation}
\varphi_n(z)=\alpha_n^{d}\int_{z\alpha_n^{-1}+[0,\alpha_n^{-1})^d}[f-f*\kappa_{\rm d}elta](x)\,{\rm d}
x,\qquad z\in\Z^d.
\end{equation}
Note that
\begin{equation}\label{phinrescale}
\begin{aligned}
\frac n{\alpha_n^2} \langle L_n,f-f*\kappa_{\rm d}elta\rangle&=\alpha_n^{d-2}\int[f-f*\kappa_{\rm d}elta](x)
\end{lemma}l_n\bigl(\lfloor x\alpha_n\rfloor\bigr)\,{\rm d} x
=\alpha_n^{-2}\scriptscriptstyleum_{z\in \Z^d}\end{lemma}l_n(z)\varphi_n(z)\\
&=\alpha_n^{-2}\scriptscriptstyleum_{k=0}^n\varphi_n(S_k).
\end{aligned}
\end{equation}
We first express the expectation on the right side of \eqref{aimsmooth} in terms of an
expansion with respect to an appropriate orthonormal system of eigenvalues
and eigenfunctions in $\R^{B_{R_n}}$. We write $\E_z$ for
expectation with respect to the random walk when started at $z\in\Z^d$, in
particular $\E=\E_0$. By \eqref{phinrescale}, for any $z,\widetilde
z\in B_{R_n}$,
\begin{equation}\label{rewriteexp}
\E_z\Bigl[\exp\Bigl\{a\frac n{\alpha_n^2}
\langle L_n,f-f*\kappa_{\rm d}elta\rangle\Bigr\}\1\{\scriptscriptstyleupp(\end{lemma}l_n)\scriptscriptstyleubset B_{R_n}\}\1\{S_n=\widetilde z\}\Bigr]= e^{\frac a2 \alpha_n^{-2}(\varphi_n(z)+\varphi_n(\widetilde
z))}A^n(z,\widetilde z),
\end{equation}
where $A^n$ is the $n$-th power of the symmetric matrix $A$
having components
\begin{equation}
A(z,\widetilde z)=e^{\frac a2
\alpha_n^{-2}\varphi_n(z)}p(z,\widetilde z)e^{\frac a2 \alpha_n^{-2}\varphi_n(\widetilde z)},\qquad
z,\widetilde z\in B_{R_n}.
\end{equation}
Using an expansion in terms of the eigenvalues $\lambda_{k}(n)$, $k\in\{1,{\rm d}ots,|B_{R_n}|\}$, of
$A$ and an orthonormal basis of $\R^{B_{R_n}}$ consisting
of corresponding eigenfunctions $v_{k,n}$ we obtain, for any $z,\widetilde z\in B_{R_n}$,
\begin{equation}\label{rewriteexp2}
A^n(z,\widetilde z)=\scriptscriptstyleum_{k=1}^{|B_{R_n}|}\lambda_{k}(n)^n v_{k,n}(z)v_{k,n}(\widetilde z).
\end{equation}
We assume that the
eigenvalues $\lambda_{k}(n)$ are in decreasing order, and the principal
eigenvector $v_{1,n}$ is positive in $B_{R_n}$.
Now we use this for the expectation on the right side of \eqref{aimsmooth}, which is equal to the
sum over $\widetilde z\in B_{R_n}$ of the left side of
\eqref{rewriteexp} at $z=0$. We obtain an upper bound by summing the right hand
side of \eqref{rewriteexp2} over $z,\widetilde z\in B_{R_n}$. Continuing the
upper bound with the help of Parseval's identity gives
\begin{equation}\label{lowupp}
\begin{aligned}
\E\Bigl[&\exp\Bigl\{a\frac n{\alpha_n^2}
\langle L_n, f-f*\kappa_{\rm d}elta\rangle\Bigr\}\1\{\scriptscriptstyleupp(\end{lemma}l_n)\scriptscriptstyleubset B_{R_n}\}\Bigr]\\
&\leq (1+o(1))\scriptscriptstyleum_{k=1}^{|B_{R_n}|}\lambda_{k}(n)^n \scriptscriptstyleum_{z,\widetilde
z\in B_{R_n}}v_{k,n}(z)v_{k,n}(\widetilde z)
\leq (1+o(1))\lambda_{1}(n)^n\scriptscriptstyleum_{k=1}^{|B_{R_n}|}\langle
v_{k,n},\1\rangle^2\\
&\leq (1+o(1))\lambda_{1}(n)^n|B_{R_n}|,
\end{aligned}
\end{equation}
where we denote by $\langle\cdot,\cdot\rangle$ and $\|\cdot\|_2$ the
inner product and Euclidean norm on $\R^{B_{R_n}}$.
Recall that $R_n=Rn\alpha_n^{-1}$. Our assumptions on $(\alpha_n)_n$ imply that
$|B_{R_n}|=e^{o(n\alpha_n^{-2})}$ as $n\to\infty$. Hence, as $n\to\infty$,
\begin{equation}\label{Evalueesti}
\frac{\alpha_n^2}n\log\E\Bigl[\exp\Bigl\{a\frac n{\alpha_n^2}
\langle L_n, f-f*\kappa_{\rm d}elta\rangle\Bigr\}\1\{\scriptscriptstyleupp(\end{lemma}l_n)\scriptscriptstyleubset B_{R_n}\}\Bigr]\leq o(1)+\alpha_n^2 \bigl[\lambda_{1}(n)-1\bigr].
\end{equation}
Recall the Rayleigh-Ritz principle, $\lambda_{1}(n)=\max_{\|g\|\leq 1}\langle A
g,g\rangle$, where the maximum runs over all $\end{lemma}l^2$-normalized vectors $g\colon
\Z^d\to (0,\infty)$ with support in $B_{R_n}$. Recall that $|\varphi_n|\leq 2$. Then, as
$n\to\infty$, we have, for any $\end{lemma}l^2$-normalized vector $g$,
\begin{equation}\label{eigenvesti1}
\begin{aligned}
\alpha_n^2 \bigl[\langle A g,g\rangle-1\bigr]&=\alpha_n^2
\Bigl(\scriptscriptstyleum_{z,\widetilde z}\bigl(e^{\frac
a2\alpha_n^{-2}[\varphi_n(z)+\varphi_n(\widetilde z)]}-1\bigr)p(z,\widetilde z)g(z)g(\widetilde
z)+\scriptscriptstyleum_{z,\widetilde z}\bigl(p(z,\widetilde z)-{\rm d}elta_{z,\widetilde
z}\bigr)g(z)g(\widetilde z)\Bigr)\\
&=a\langle\varphi_n,g^2\rangle+a\langle\varphi_n, g\,(pg-g)\rangle+\Ocal(\alpha_n^{-2})-\alpha_n^2 \Ical^{\scriptscriptstylesup {\rm d}}(g^2),
\end{aligned}
\end{equation}
where we recall that the walk is assumed symmetric, and we introduced its Dirichlet form,
\begin{equation}\label{Dirichlet}
\Ical^{\scriptscriptstylesup {\rm d}}(g^2) =\frac 12\scriptscriptstyleum_{z,\widetilde z\in\Z^d}p(z,\widetilde z)\bigl(g(z)-g(\widetilde z)\bigr)^2,\qquad g\in\end{lemma}l_2(\Z^d),
\end{equation}
and we wrote $pg(z)=\scriptscriptstyleum_{\widetilde z}p(z,\widetilde z)g(\widetilde z)$.
The second term on the right hand side of \eqref{eigenvesti1} is estimated as follows, using that $|\varphi_n|\leq 2$.
\begin{equation}\label{eigenvesti2}
\begin{aligned}
\langle\varphi_n, g\,(pg-g)\rangle&=\frac 12\scriptscriptstyleum_{z,\widetilde z}\varphi_n(z)p(z,\widetilde z)\Big[-\bigl(g(z)-g(\widetilde z)\big)^2+\big(g(\widetilde z)-g(z)\big)\big(g(z)+g(\widetilde z)\big)\Big]\\
&\leq 2\Ical^{\scriptscriptstylesup {\rm d}}(g^2)+\scriptscriptstyleqrt{2\Ical^{\scriptscriptstylesup {\rm d}}(g^2)}\scriptscriptstyleqrt{\frac12\scriptscriptstyleum_{z,\widetilde z}|\varphi_n(z)|p(z,\widetilde z)(g(z)+g(\widetilde z))^2}\\
&\leq 2\Ical^{\scriptscriptstylesup {\rm d}}(g^2)+\frac 8\varepsilon \Ical^{\scriptscriptstylesup {\rm d}}(g^2)+\frac\varepsilon4,
\end{aligned}
\end{equation}
where we used the inequality $\scriptscriptstyleqrt{2ab}\leq 8a/\varepsilon+\varepsilon b/16$ for $a,b,\varepsilon>0$ in the last step.
The first term on the right hand side of \eqref{eigenvesti1} is estimated as follows. We introduce $g_n(x)=g(\lfloor x\alpha_n\rfloor)$.
\begin{equation}\label{eigenvesti3}
\begin{aligned}
\langle\varphi_n,g^2\rangle&=\alpha_n^d\int{\rm d} x\,f(x) \Big(g_n^2(x)-\int{\rm d} y\,\kappa_{\rm d}elta(y) g_n^2(x+y)\Big)\\
&\leq \alpha_n^d\int{\rm d} x\, \int{\rm d} y\,\kappa_{\rm d}elta(y)\big|g_n^2(x)-g_n^2(x+y)\big|\\
&\leq \alpha_n^d\int{\rm d} x\, \scriptscriptstyleqrt{\int{\rm d} y\,\kappa_{\rm d}elta(y)(g_n(x)-g_n(x+y))^2}\scriptscriptstyleqrt {\int{\rm d} y\,\kappa_{\rm d}elta(y)(g_n(x)+g_n(x+y))^2}\\
&\leq \frac 4{\varepsilon}\alpha_n^d\int{\rm d} x\, \int{\rm d} y\,\kappa_{\rm d}elta(y)\big(g_n(x)-g_n(x+y)\big)^2+\frac\varepsilon8 \alpha_n^d\int{\rm d} x\,\Big(g_n^2(x)+\int{\rm d} y\,\kappa_{\rm d}elta(y)g_n^2(x+y)\Big)\\
&\leq \frac 4{\varepsilon}\alpha_n^d\int{\rm d} x\, \int{\rm d} y\,\kappa_{\rm d}elta(y)\big(g_n(x)-g_n(x+y)\big)^2+\frac \varepsilon4,
\end{aligned}
\end{equation}
where we used that $|f|\leq 1$ in the second step, H\"older's inequality in the third, and the inequality $\scriptscriptstyleqrt{2ab}\leq 4a/\varepsilon+\varepsilon b/8$ in the fourth step. Now pick some almost everywhere differentiable function $\psi_n\colon\R^d\to\R$ such that $\psi_n(z/\alpha_n)=\alpha_n^{d/2}g(z)$ for any $z\in\Z^d$, then a Taylor expansion gives that
\begin{equation}\label{eigenvesti4}
\begin{aligned}
\alpha_n^d\int{\rm d} x\, &\int{\rm d} y\,\kappa_{\rm d}elta(y)\big(g_n(x)-g_n(x+y)\big)^2\\
&=\alpha_n^{-d}\scriptscriptstyleum_{z,\widetilde z}\Big(\psi_n({\textstyle{\frac z{\alpha_n}}})-\psi_n({\textstyle{\frac {z+\widetilde z}{\alpha_n}}})\Big)^2\int_{\widetilde z/\alpha_n+[0,1/\alpha_n]^d}{\rm d} y\,\kappa_{\rm d}elta(y)\\
&=\alpha_n^{-d}\scriptscriptstyleum_{z,\widetilde z}\Big(\int_0^1{\rm d} t\, \frac{\widetilde z}{\alpha_n}\cdot\nabla\psi_n({\textstyle{\frac{z+t\widetilde z}{\alpha_n}}})\Big)^2\int_{\widetilde z/\alpha_n+[0,1/\alpha_n]^d}{\rm d} y\,\kappa_{\rm d}elta(y)\\
&\leq \alpha_n^{-d} \scriptscriptstyleum_{\widetilde z}\int_{\widetilde z/\alpha_n+[0,1/\alpha_n]^d}{\rm d} y\,\kappa_{\rm d}elta(y)|{\textstyle{\frac {\widetilde z}{\alpha_n}}}|^2\int_0^1{\rm d} t\,\scriptscriptstyleum_z\big|\nabla\psi_n({\textstyle{\frac{z+t\widetilde z}{\alpha_n}}})\big|^2\\
&\leq C{\rm d}elta^2\|\nabla\psi_n\|_2^2\leq C{\rm d}elta^2\|\Gamma^{\frac 12}\nabla\psi_n\|_2^2,
\end{aligned}
\end{equation}
where we remark that $\int{\rm d} y\,\kappa_{\rm d}elta(y)|y|^2\leq C{\rm d}elta^2$ for some $C>0$.
Now we specialize the choice of $\psi_n$ to
$$
\psi_n(x)=\alpha_n^{d/2}\Big[g_n(x)+\scriptscriptstyleum_{i=1}^d\big(\alpha_n x_i-\lfloor\alpha_n x_i\rfloor\big)\Big(g\big(\lfloor\alpha_n x\rfloor+{\rm e}_i\big)-g\big(\lfloor\alpha_n x\rfloor\big)\Big)\Big],
$$
where ${\rm e}_i$ denotes the $i$-th unit vector. Then $\psi_n$ is the linear interpolation of the rescaling of $g$, and $\partial_i\psi_n(x)=\alpha_n^{d/2+1}(g(\lfloor\alpha_n x\rfloor+{\rm e}_i)-g(\lfloor\alpha_n x\rfloor))$. Similarly to \eqref{eigenvesti4}, one derives
\begin{equation}\label{eigenvesti5}
\begin{aligned}
\alpha_n^2 \Ical^{\scriptscriptstylesup {\rm d}}(g^2)&=\int_0^1{\rm d} t\,\int_0^1{\rm d} s\,\scriptscriptstyleum_{z\in\Z^d}p(0,z)\scriptscriptstyleum_{i,j=1}^dz_iz_j\int{\rm d} x\,
\partial_i \psi_n\Big(\frac{\lfloor\alpha_n x\rfloor+tz}{\alpha_n}\Big)
\partial_j\psi_n\Big(\frac{\lfloor\alpha_n x\rfloor+sz}{\alpha_n}\Big)\\
&=\int_0^1{\rm d} t\,\int_0^1{\rm d} s\,\scriptscriptstyleum_{z}p(0,z)\scriptscriptstyleum_{i,j=1}^dz_iz_j\int{\rm d} x\,\partial_i \psi_n(x)\partial_j\psi_n(x)\\
&=\|\Gamma^{\frac 12}\nabla\psi_n\|_2^2.
\end{aligned}
\end{equation}
Now use \eqref{eigenvesti5} in \eqref{eigenvesti4} and this in \eqref{eigenvesti3}, and substitute \eqref{eigenvesti3} and \eqref{eigenvesti2} in \eqref{eigenvesti1} to obtain, for any $a>0$, for $n$ sufficiently large and all $\end{lemma}l^2$-normalized $g\in\end{lemma}l^2(\Z^d)$ with support in $B_{R_n}$,
$$
\alpha_n^2 \bigl[\langle A g,g\rangle-1\bigr]\leq \frac 12a\varepsilon -\alpha_n^2\Ical^{\scriptscriptstylesup {\rm d}}(g^2)\Big(1-C\frac {{\rm d}elta^2 a}\varepsilon\Big),
$$
for some $C>0$ which does not depend on $n$, $g$, $\varepsilon$ or on $a$.
Now we choose $a=\varepsilon/(2C{\rm d}elta^2)$ and obtain $\alpha_n^2 \bigl[\langle A g,g\rangle-1\bigr]\leq \frac 12a\varepsilon$. Taking the supremum over all $g$'s considered, we obtain that $\alpha_n^2[\lambda_{1}(n)-1]\leq \frac 12a\varepsilon$. Using this in \eqref{Evalueesti} and \eqref{Evalueesti} in \eqref{aimsmooth}, we obtain that
$$
\mbox{l.h.s.~of \eqref{fieldsmoothing}}\leq \limsup_{{\rm d}elta{\rm d}ownarrow 0}-\frac 12 a\varepsilon=-\lim_{{\rm d}elta{\rm d}ownarrow 0}\frac {\varepsilon^2}{4C{\rm d}elta^2}=-\infty,
$$
and the proof is finished.
\end{Proof}\qed
\scriptscriptstyleubsection{Various approximations}\label{sec-appr}
\noindent In the proofs of Theorems~\ref{inter} and \ref{lin} we shall need a couple of
approximations to the variational formulas in \eqref{K3def} and \eqref{kdq}.
In particular, we need to show that they may be approximated by finite-space approximations
and by smoothed versions of the functions involved in the variational formula.
As in Section~\ref{sec-smooth}, by $\kappa=\kappa_1\colon\R^d\to[0,\infty)$ we denote a smooth, rotationally invariant $L^1$-normalized function, and we put $\kappa_{\rm d}elta(x)={\rm d}elta^{-d}\kappa_1(x{\rm d}elta^{-1})$ for ${\rm d}elta>0$. Hence, $\kappa_{\rm d}elta$ is a smooth approximation of the Dirac measure at zero.
\begin{lemma}[Approximations of $K_{H}$]\label{KHappr}
For any $u>0$,
\begin{equation}
\limsup_{{\rm d}elta{\rm d}ownarrow 0}\limsup_{R\to\infty}K_{H}^{\scriptscriptstylesup{0}}(u;{\rm d}elta, R)\leq K_H(u)\leq\liminf_{{\rm d}elta{\rm d}ownarrow 0}\liminf_{R\to\infty}K_{H}^{\scriptscriptstylesup{\rm per}}(u;{\rm d}elta, R),
\end{equation}
where
\begin{eqnarray}
K_{H}^{\scriptscriptstylesup{0}}(u;{\rm d}elta, R)&=&\inf\Bigl\{
\frac 12\|\Gamma^{\frac 12}\nabla \psi\|_2^2+\Phi_H(\psi^2*\kappa_{\rm d}elta,u;R)\colon \psi\in
H^1(\R^d),\scriptscriptstyleupp(\psi)\scriptscriptstyleubset Q_R,\label{KHR0def}\\
&&\qquad\qquad\qquad\qquad\qquad\qquad \|\psi\|_2=1\Bigr\},\nonumber \\
K_{H}^{\scriptscriptstylesup{\rm per}}(u;{\rm d}elta, R)&=&\inf\Bigl\{
\frac 12\|\Gamma^{\frac 12}\nabla_{R} \psi\|_2^2+\Phi_H(\psi^2*\kappa_{\rm d}elta,u;R)\colon \psi\in
H^1(Q_R),\|\psi\|_2=1\Bigr\},\label{KHRperdef}
\end{eqnarray}
and $\Phi_H(\psi^2,u;R)$ is defined in \eqref{PhiRdef}.
In \eqref{KHRperdef}, $\nabla_R$ denotes the gradient on the torus $Q_R$, i.e., with periodic boundary condition.
\end{lemma}
\begin{Proof}{Proof}
Fix ${\rm d}elta>0$. In the first step, we carry out the limit as $R\to\infty$ on both sides to obtain
\begin{equation}\label{Rlimtwithdelta}
\limsup_{R\to\infty}K_{H}^{\scriptscriptstylesup{0}}(u;{\rm d}elta, R)\leq K_H(u;{\rm d}elta)\leq\liminf_{R\to\infty}K_{H}^{\scriptscriptstylesup{\rm per}}(u;{\rm d}elta, R),
\end{equation}
where $K_H(u;{\rm d}elta)$ is defined as $K_H(u)$ in \eqref{K3def} with $\Phi_H(\psi^2,u)$ replaced by $\Phi_H(\psi^2*\kappa_{\rm d}elta,u)$. The proof of \eqref{Rlimtwithdelta} follows standard patterns (see the proof of \cite[Lemma~3.7]{AC02}, e.g.) and we do not carry this out here. Hence, the only thing left to do is to show that $\lim_{{\rm d}elta{\rm d}ownarrow 0}K_H(u;{\rm d}elta)=K_H(u)$.
Using the convexity of $H$, it is easy to derive with the help of Jensen's inequality that, for any $\gamma>0$ and any $\psi$,
$$
\int H\bigl(\gamma \psi^2*\kappa_{\rm d}elta(y)\bigr)\,{\rm d} y\leq \int H\bigl(\gamma\psi^2(y)\bigr)\,{\rm d} y.
$$
As a consequence, we have $\Phi_H(\psi^2*\kappa_{\rm d}elta,u)\geq \Phi_H(\psi^2,u)$ and therefore $K_{H}(u;{\rm d}elta)\geq K_{H}(u)$ for any ${\rm d}elta>0$.
We argue now that $\limsup_{{\rm d}elta{\rm d}ownarrow 0}K_{H}(u;{\rm d}elta)\leq K_{H}(u)$. Indeed, fix some small $\varepsilon>0$ and pick some bounded approximative $\varepsilon$-minimizer for $K_{H}(u)$, i.e., a bounded function $\overline \psi\in H^1(\R^d)$ satisfying $\|\overline\psi\|_2=1$ and
$$
\frac 12\|\Gamma^{\frac 12}\nabla \overline\psi\|_2^2+\Phi_H(\overline\psi^2,u)\leq K_{H}(u) +\varepsilon.
$$
Using the mean-value theorem and the fact that $\|\overline\psi^2*\kappa_{\rm d}elta-\overline\psi^2\|_1\to0$ as ${\rm d}elta{\rm d}ownarrow 0$ (see \cite[Th.~2.16]{LL97}), it is elementary to show that we have $\int H\bigl(\gamma \overline\psi^2*\kappa_{\rm d}elta(y)\bigr)\,{\rm d} y \to \int H\bigl(\gamma \overline\psi^2(y)\bigr)\,{\rm d} y$ as ${\rm d}elta{\rm d}ownarrow 0$, uniformly in $\gamma$ on any compact subset of $[0,\infty)$. As a consequence, we have $\lim_{{\rm d}elta{\rm d}ownarrow 0}\Phi_H(\overline \psi^2*\kappa_{\rm d}elta,u)=\Phi_H(\overline \psi^2,u)$ and therefore
\begin{equation}\label{Kdeltaconv}
\begin{aligned}
\limsup_{{\rm d}elta{\rm d}ownarrow 0}K_{H}(u;{\rm d}elta)&\leq \frac 12\|\Gamma^{\frac 12}\nabla \overline\psi\|_2^2+\limsup_{{\rm d}elta{\rm d}ownarrow 0}\Phi_H(\overline \psi^2*\kappa_{\rm d}elta,u)=\frac 12\|\Gamma^{\frac 12}\nabla \overline\psi\|_2^2+\Phi_H(\overline\psi^2,u)\\
&\leq K_{H}(u) +\varepsilon.
\end{aligned}
\end{equation}
Now let $\varepsilon{\rm d}ownarrow 0$.
\end{Proof}\qed
Lemma~\ref{KHappr} implies the corresponding statement for the case (V):
\begin{cor}[Approximations of $K_{D,q}$]\label{KDqappr}
Fix $D>0$ and $q>1$ and recall that $\frac 1p+\frac 1q=1$. Then
\begin{equation}
\limsup_{R\to\infty}K_{D,q}^{\scriptscriptstylesup{0}}(R)\leq K_{D,q}\leq\liminf_{{\rm d}elta{\rm d}ownarrow 0}\liminf_{R\to\infty}K_{D,q}^{\scriptscriptstylesup{\rm per}}({\rm d}elta,R),
\end{equation}
where
\begin{eqnarray}
K_{D,q}^{\scriptscriptstylesup{0}}(R)&=&\inf\Bigl\{
\frac 12\|\Gamma^{\frac 12}\nabla \psi\|_2^2+D\|\psi^2\|_{p}^{-q}\colon\psi\in H^1(\R^d),\scriptscriptstyleupp(\psi)\scriptscriptstyleubset Q_R,\|\psi\|_2=1\Bigr\},\\
K_{D,q}^{\scriptscriptstylesup{\rm per}}({\rm d}elta,R)&=&\inf\Bigl\{
\frac 12\|\Gamma^{\frac 12}\nabla_R \psi\|_2^2+D\|\psi^2*\kappa_{\rm d}elta\|_{p}^{-q}\colon\psi\in H^1(Q_R),\|\psi\|_2=1\Bigr\},
\end{eqnarray}
and $\nabla_R$ is the gradient on the torus $Q_R$, i.e., with periodic boundary condition.
\end{cor}
\begin{Proof}{Proof}
We apply Lemma~\ref{KHappr} to the special choice $u=1$ and $H(t)=\widetilde D t^p$, where $p$ and $\widetilde D$ are as in \eqref{cumgenfct}. It is easy to see that for this choice of $H$, we have $\Phi_H(\psi^2,1)=D\|\psi^2\|_p^{-q}$.
\end{Proof}\qed
\scriptscriptstyleection{Proof of the upper bounds in Theorems~\ref{inter} and \ref{lin}}
\label{s:ub}
\noindent This section is devoted to the proof of the upper bounds in
Theorems~\ref{inter} and \ref{lin}. They are in Sections~\ref{subs:inter} and \ref{subs:ld}, respectively. Our proofs essentially follow the proof of \cite[Theorem~2.2]{AC02}.
\scriptscriptstyleubsection{Very-large deviation case (Theorem~\ref{inter})}
\label{subs:inter}
\noindent In this section we are under Assumption (Y) with $q>\frac d2$, and consider a sequence $(b_n)$ with $1\ll b_n \ll
n^{\frac 1q}$. We have to smoothen the scenery, as we have explained at the beginning of Section~\ref{sec-prep}. In order to do this, we have to cut down the scenery to bounded size. As soon as the smoothing argument has been carried out, we may relax the boundedness assumption.
Recall the scaled and normalized local times $L_n$ from \eqref{Lndef}
and the scaled normalized scenery $\overline Y_n$ from \eqref{Yscaled}.
Recall the notation $y^{\scriptscriptstylesup{\leq M}}=[y\wedge M]\vee(-M)$ from \eqref{ydeco}, and recall the
delta-approximation $\kappa_{\rm d}elta\colon\R^d\to[0,\infty)$ to the Dirac measure from the beginning
of Section~\ref{sec-smooth}.
Note that, for any $M,\varepsilon,{\rm d}elta>0$,
\begin{equation}
\begin{aligned}
\P(\scriptscriptstylemfrac 1n Z_n>b_n)&\leq \P\bigl(\langle \overline Y_n^{\scriptscriptstylesup{\leq M}}*\kappa_{\rm d}elta,
L_n\rangle >1-2\varepsilon\bigr)\\
&\qquad+\P\bigl(\langle |\overline Y_n^{\scriptscriptstylesup{\leq M}}*\kappa_{\rm d}elta
-\overline Y_n^{\scriptscriptstylesup{\leq M}}|,L_n\rangle>\varepsilon\bigr)
+\P(\langle \overline Y_n^{\scriptscriptstylesup{>M}},L_n\rangle>\varepsilon).
\end{aligned}
\end{equation}
Recall that, by our choice of $\alpha_n$, we have
\begin{equation}\label{alphachoiceI}
n^{\frac d{d+2}}b_n^{\frac {2q}{d+2}}=\frac n{\alpha_n^2}.
\end{equation}
Hence, by Proposition~\ref{fieldcut}, Lemma~\ref{FieSmoo} and Corollary~\ref{KDqappr}, it suffices to
prove, for any $M,{\rm d}elta>0$ and $R\in\N$,
\begin{equation}\label{upperaim}
\limsup_{\varepsilon{\rm d}ownarrow 0}\limsup_{n\to\infty}\frac {\alpha_n^2}n\log
\P\bigl(\langle \overline Y_n^{\scriptscriptstylesup{\leq M}}*\kappa_{\rm d}elta,L_n\rangle >1-2\varepsilon\bigr)\leq -K_{D,q}^{\scriptscriptstylesup{\rm per}}({\rm d}elta,R),
\end{equation}
where $K_{D,q}^{\scriptscriptstylesup{\rm per}}({\rm d}elta,R)$ is defined in Corollary~\ref{KDqappr}. Note that
$$
\begin{aligned}
\langle \overline Y_n^{\scriptscriptstylesup{\leq M}}*\kappa_{\rm d}elta,L_n\rangle
&=\frac 1{b_n\alpha_n^d}\scriptscriptstyleum_{z\in\Z^d}\bigl[\bigl(Y(z)\wedge(Mb_n)\bigr)\vee
(-Mb_n)\bigr]L_n*\kappa_{\rm d}elta\Bigl(\frac z{\alpha_n}\Bigr)\\
&\leq \frac 1{b_n\alpha_n^d}\scriptscriptstyleum_{z\in\Z^d}\bigl[Y(z)\vee
(-Mb_n)\bigr]L_n*\kappa_{\rm d}elta\Bigl(\frac z{\alpha_n}\Bigr).
\end{aligned}
$$
Introduce the cumulant generating function of $Y(0)\vee (-M)$,
$$
H_M(t)=\log \E[e^{t[Y(0)\vee (-M)]}].
$$
Using the exponential Chebyshev inequality and carrying out the expectation over the scenery, we obtain, for any $\gamma>0$, the upper bound
\begin{equation}\label{Iproof1}
\begin{aligned}
\P\bigl(\langle \overline Y_n^{\scriptscriptstylesup{\leq M}}*\kappa_{\rm d}elta,L_n\rangle >1-2\varepsilon\bigr)
&\leq\E\Bigl[ e^{-\gamma(1-2\varepsilon) nb_n} e^{\gamma n\alpha_n^{-d}\scriptscriptstyleum_z[Y(z)\vee (-Mb_n)]L_n*\kappa_{\rm d}elta\bigl(\frac z{\alpha_n}\bigr)}\Bigr]\\
&\leq \E\Bigl[e^{-\gamma (1-2\varepsilon) nb_n}\exp\Bigl\{\scriptscriptstyleum_{z\in\Z^d} H_{Mb_n}
\Bigl(\gamma n\alpha_n^{-d}L_n*\kappa_{\rm d}elta\Bigl(\frac z{\alpha_n}\Bigr)\Bigr)\Bigr\}\Bigr].
\end{aligned}
\end{equation}
Since $H_{Mb_n}$ is convex and satisfies $H_{Mb_n}(0)=0$, it is also
superadditive.
Hence,
for any $\gamma>0$ and any $x\in\Z^d$, we have
\begin{equation}\label{Hperesti}
\scriptscriptstyleum_{k\in \Z^d} H_{Mb_n}\bigl(\gamma \end{lemma}l_n(x+2k\lfloor R\rfloor)\bigr)\leq
H_{Mb_n}(\gamma\end{lemma}l_n^{\scriptscriptstylesup{R}}(x)),
\end{equation}
Therefore, the right hand in (\ref{Iproof1}) side does not become smaller if $L_n*\kappa_{\rm d}elta$ is replaced by its periodized version, $(L_n*\kappa_{\rm d}elta)^{\scriptscriptstylesup{R}}(x)=\scriptscriptstyleum_{k\in\Z^d}L_n*\kappa_{\rm d}elta(x+kR)$, for $x\in [-R,R]^d$. Furthermore, note that
$$
(L_n*\kappa_{\rm d}elta)^{\scriptscriptstylesup{R}}(x)=\scriptscriptstyleum_{k\in\Z^d}\int_{\R^d} L_n(y)\kappa_{\rm d}elta(x+kR-y)\,{\rm d} y=\int_{\R^d} L_n^{\scriptscriptstylesup{R}}(y)\kappa_{\rm d}elta(x-y)\,{\rm d} y=L_n^{\scriptscriptstylesup{R}}*\kappa_{\rm d}elta(x),
$$
for any $x\in[-R,R]^d$. Hence, we may replace $L_n$ on the right of \eqref{Iproof1} by its periodized version $L_n^{\scriptscriptstylesup{R}}$.
According to \eqref{cumgenfct}, for any $\varepsilon>0$, we may choose a $c(\varepsilon)>0$
such that
\begin{equation}\label{Hesti}
H(t)\leq c(\varepsilon) t+(1+\varepsilon)\widetilde D\, t^p,\qquad t\in[0,\infty).
\end{equation}
Since $e^{H_M(t)}\leq e^{H(t)}+1$, we also have the estimate in \eqref{Hesti} for $H_{Mb_n}$ instead of $H$. Hence, since $\kappa_{\rm d}elta$ and $L_n$ are $L^1$-normalized,
\begin{equation}\label{Intercalc1}
\P\bigl(\langle \overline Y_n^{\scriptscriptstylesup{\leq M}}*\kappa_{\rm d}elta,L_n\rangle >1-2\varepsilon\bigr)
\leq\E\Bigl[e^{-\gamma(1-2\varepsilon) nb_n}e^{c(\varepsilon)\gamma n}\exp\Bigl\{\gamma^p(\widetilde D+\varepsilon)\alpha_n^d (n\alpha_n^{-d})^p\| L_n^{\scriptscriptstylesup{R}}*\kappa_{\rm d}elta\|_p^p\Bigr\}\Bigr].
\end{equation}
We choose the value of $\gamma$ optimal for $\varepsilon=0$, which is
\begin{equation}\label{gammachoiceI}
\gamma=\frac{\alpha_n^d}n b_n^{\frac 1{p-1}}\Bigl(p\widetilde D\|L_n*\kappa_{\rm d}elta^p\|_p^p\Bigr)^{-\frac 1{p-1}}=\frac{\alpha_n^d}n b_n^{\frac 1{p-1}}Dq\bigl\|L_n^{\scriptscriptstylesup{R}}*\kappa_{\rm d}elta\bigr\|_p^{-q},
\end{equation}
where we recalled that $1=\frac 1p+\frac 1q$ and $\widetilde D=(q-1)(Dq^q)^{\frac 1{1-q}}$. Note that the map $\mu\mapsto \|\mu*\kappa_{\rm d}elta\|_p$ is bounded and continuous (in the weak $L^1$-topology) on the set of probability measures on $[-R,R]^d$. Indeed, the continuity is seen with the help of Lebesgue's theorem, and the boundedness follows from the following application of Jensen's inequality:
\begin{equation}\label{Jensentrick}
\begin{aligned}
\|\mu*\kappa_{\rm d}elta\|_p^p&=(2R)^d\int_{[-R,R]^d} \frac{{\rm d} x}{(2R)^d}\Big|\int_{\R^d}\mu({\rm d} y)\kappa_{\rm d}elta(x-y)\Big|^p\\
&\geq (2R)^d \Bigl(\int_{[-R,R]^d}\frac{{\rm d} x}{(2R)^d}\int_{\R^d}\mu({\rm d} y)\kappa_{\rm d}elta(x-y)\Bigr)^p\\
&=(2R)^{d(1-p)},
\end{aligned}
\end{equation}
since $\kappa_{\rm d}elta$ is $L^1$-normalized.
Recall that $b_n^q=n\alpha_n^{-(d+2)}$. For the choice of $\gamma$ in \eqref{gammachoiceI}, for large $n$, we can estimate the first two terms in the expectation on the right of \eqref{Intercalc1} by $e^{-\gamma(1-2\varepsilon) nb_n}e^{c(\varepsilon)\gamma n}\leq e^{-\gamma(1-3\varepsilon) nb_n}$, since we have in particular $\gamma\ll b_n$.
Substituting $\gamma$ in \eqref{Intercalc1}, we obtain
\begin{equation}\label{Interdev1}
\P\bigl(\langle \overline Y_n^{\scriptscriptstylesup{\leq M}}*\kappa_{\rm d}elta,L_n\rangle >1-2\varepsilon\bigr)
\leq\E\Bigl[\exp\Bigl\{-(D+\varepsilon C)\frac n{\alpha_n^2}\bigl\|L_n^{\scriptscriptstylesup{R}}*\kappa_{\rm d}elta\bigr\|_p^{-q}\Bigr\}\Bigr],
\end{equation}
where $C>0$ depends on $D, R$ and $q$ only. Now we can finally apply the large deviation principle in Lemma~\ref{LDP}(ii) to the right hand side of \eqref{Interdev1}. This yields the estimate in \eqref{upperaim} without $\limsup_{\varepsilon{\rm d}ownarrow 0}$ and with $D$ replaced by $D+\varepsilon C$. Letting $\varepsilon{\rm d}ownarrow 0$, we easily see that \eqref{upperaim} is satisfied, which ends the proof of the upper bound in Theorem~\ref{inter}.
\scriptscriptstyleubsection{Large-deviation case (Theorem~\ref{lin})}
\label{subs:ld}
\noindent In this section, we prove the upper bound in Theorem~\ref{lin}, i.e., in the case (L). The proof follows the pattern of the corresponding proof in \cite{AC02} and is analogous to the proof of Theorem~\ref{inter} in Section~\ref{subs:inter}, and hence we keep it short. Pick $b_n=1$ and $\alpha_n=n^{\frac 1{d+2}}$,
in accordance with \eqref{bnalphachoice}. Furthermore, fix $u>0$.
By Proposition~\ref{fieldcut} and Lemmas~\ref{FieSmoo} and \ref{KHappr}, it is sufficient to prove that, for any ${\rm d}elta>0$ and $R\in\N$,
\begin{equation}\label{upperaimlar}
\limsup_{\varepsilon{\rm d}ownarrow0}\limsup_{n\to\infty}n^{-\frac d{d+2}}\log
\P\bigl(\langle \overline Y_n^{\scriptscriptstylesup{\leq M}}*\kappa_{\rm d}elta,L_n\rangle >u-\varepsilon\bigr)\leq -K_{H}^{\scriptscriptstylesup{\rm per}}(u;{\rm d}elta, R),
\end{equation}
where $K_{H}^{\scriptscriptstylesup{\rm per}}(u;{\rm d}elta,R)$ is defined in Lemma~\ref{KHappr}. Fix a small $\varepsilon>0$. Analogously to \eqref{Iproof1}, we have the estimate
\begin{equation}
\P\bigl(\langle \overline Y_n^{\scriptscriptstylesup{\leq M}}*\kappa_{\rm d}elta,L_n\rangle >u-\varepsilon \bigr)\leq \E\Bigl[e^{-\gamma (u-2\varepsilon) n}\exp\Bigl\{\scriptscriptstyleum_{z\in\Z^d} H_{M}
\Bigl(\gamma n\alpha_n^{-d}L_n^{\scriptscriptstylesup{R}}*\kappa_{\rm d}elta\Bigl(\frac z{\alpha_n}\Bigr)\Bigr)\Bigr\}\Bigr],
\end{equation}
for any $\gamma>0$. Replacing $\gamma n\alpha_n^{-d}$ by $\gamma$, turning the sum into an integral, passing to the optimum over $\gamma$ and using the notation in \eqref{PhiRdef}, we obtain
\begin{equation}\label{Lcalc}
\P\bigl(\langle \overline Y_n^{\scriptscriptstylesup{\leq M}}*\kappa_{\rm d}elta,L_n\rangle >u\bigr)\leq\E\Bigl[\exp\Bigl\{-\frac n{\alpha_n^2}\Phi_{H_M}(L_n^{\scriptscriptstylesup{R}}*\kappa_{\rm d}elta,u-2\varepsilon;R)\Bigr\}\Bigr],
\end{equation}
where we also recall that $\alpha_n^d=n\alpha_n^{-2}$. Again, for fixed ${\rm d}elta>0$ and $R>0$, we can let $M\to\infty$ and $\varepsilon{\rm d}ownarrow 0$ to replace $\Phi_{H_M}(L_n^{\scriptscriptstylesup{R}}*\kappa_{\rm d}elta,u-2\varepsilon;R)$ by $\Phi_{H}(L_n^{\scriptscriptstylesup{R}}*\kappa_{\rm d}elta,u;R)$ on the right side of \eqref{Lcalc}. Analogously to \eqref{Jensentrick}, one shows that $\Phi_H(\psi^2,u)\leq |Q_R| \, \scriptscriptstyleup_{\gamma>0}\bigl(\gamma u -H(\gamma)\bigr)<\infty$ for any continuous $\psi\colon Q_R\to [0,\infty)$ satisfying $\int_{Q_R}\psi^2=1$. Hence, the map $\mu\mapsto \Phi_{H}(\mu*\kappa_{\rm d}elta,u;R)$ is bounded and continuous on the set of probability measures on $Q_R$, and we may apply the large deviation principle in Lemma~\ref{LDP}(ii). This, followed by $\varepsilon{\rm d}ownarrow 0$, implies that \eqref{upperaimlar} holds for any ${\rm d}elta>0$ and $R\in\N$. This finishes the proof of the upper bound in Theorem~\ref{lin}.
\scriptscriptstyleection{Proofs of the lower bounds in Theorems~\ref{inter} and \ref{lin}}
\label{s:lb}
In this section we prove the lower bounds in Theorems~\ref{inter} and \ref{lin}. Our proofs are variants of the analogous proofs in \cite{AC02}; they roughly follow the heuristics in Section~\ref{sec-heur}.
\scriptscriptstyleubsection{Very-large deviation case (Theorem~\ref{inter})}\label{sec-lowinter}
\noindent Suppose we are in the case (V) and pick sequences $(b_n)_n$ and $(\alpha_n)_n$
as in \eqref{bnalphachoice}. Fix $R>0$ and any continuous positive function $\varphi\colon Q_R\to(0,\infty)$.
Recall the scaled local times and scenery, $L_n$ and $\overline Y_n$, in \eqref{Lndef} and
\eqref{Yscaled}.
If $\overline Y_n\geq
\varphi$ on $
Q_R$ and $\scriptscriptstyleupp(L_n)\scriptscriptstyleubset Q_R$, then
\begin{equation}
Z_n=b_n n \langle L_n, \overline Y_n\rangle
\geq b_n n \langle L_n, \varphi\rangle.
\end{equation}
Hence, we obtain the lower bound, for any $n\in\N$,
\begin{equation}\label{interlow1}
\P\bigl(\scriptscriptstylemfrac 1n Z_n>b_n\bigr)\geq \P(\langle L_n,\varphi\rangle \geq
1, \scriptscriptstyleupp(L_n)\scriptscriptstyleubset Q_R)\,\P\bigl(\overline Y_n\geq
\varphi\mbox{ on }Q_R\bigr).
\end{equation}
With the help of the large deviation principle in Lemma~\ref{LDP}(i) it is easy to deduce that
\begin{equation}\label{interlow2}
\begin{aligned}
\lim_{n\to\infty}&\frac{\alpha_n^2}n\log \P\bigl(\langle L_n,\varphi\rangle
\geq 1, \scriptscriptstyleupp(L_n)\scriptscriptstyleubset Q_R\bigr)\\
&=-\inf\bigl\{\Ical_R(\psi^2)\colon \psi\in H^1(\R^d),\scriptscriptstyleupp(\psi)\scriptscriptstyleubset Q_R,\|\psi\|_2=1, \langle
\psi^2,\varphi\rangle\geq 1 \bigr\}.
\end{aligned}
\end{equation}
{From} Lemma~\ref{LDPfieldcont} we have, recalling that $n\alpha_n^{-2}=\alpha_n^d b_n^q$,
\begin{equation}\label{Ylowinter}
\liminf_{n\to\infty}\frac{\alpha_n^2}n\log \P\bigl(\overline Y_n\geq
\varphi\mbox{ on }Q_R\bigr)\geq -D\|\varphi\|_q^q.
\end{equation}
Using \eqref{interlow2} and \eqref{Ylowinter} in \eqref{interlow1} and optimizing on $\varphi$, we obtain the lower bound
\begin{equation}\label{lowPinter}
\liminf_{n\to\infty}\frac{\alpha_n^2}n\log
\P\bigl(\scriptscriptstylemfrac 1n Z_n>b_n\bigr)\geq -\widetilde K_{D,q}^{\scriptscriptstylesup{0}}(R),
\end{equation}
where
\begin{equation}
\widetilde K_{D,q}(R)=\inf_{\psi\in H^1(\R^d)\colon \|\psi\|_2=1,
\scriptscriptstyleupp(\psi)\scriptscriptstyleubset B_R}\Bigl(\Ical_R(\psi^2)+D
\inf_{\varphi\in\Ccal_+(Q_R)\colon \langle \psi^2,\varphi\rangle \geq 1}
\|\varphi\|_q^q\Bigr).
\end{equation}
It is easy to see that the inner infimum is equal to $\|\psi^2\|_p^{-q}$. Hence, $\widetilde K_{D,p}(R)=K_{D,p}^{\scriptscriptstylesup{0}}(R)$ as defined in Corollary~\ref{KDqappr}. Now Corollary~\ref{KDqappr} finishes the proof
of the lower bound in Theorem~\ref{inter}.
\scriptscriptstyleubsection{Large-deviation case (Theorem~\ref{lin})}
\noindent Recall from Section~\ref{sec-heur} that $\frac 1n Z_n=\langle L_n, \overline Y_n\rangle$. We want to apply the large deviation principles of Lemma~\ref{LDP}(i) for $L_n$ and Lemma~\ref{LDPfieldL} for $\overline Y_n$. However, as has been pointed out in \cite{AC02}, the map $(\mu,f)\mapsto \langle \mu,f\rangle$ is not continuous in the product of the weak topologies. Hence, we partially follow the strategy of \cite{AC02} and use Lemma~\ref{FieSmoo} to smoothen the field $\overline Y_n$. In order to apply Lemma~\ref{FieSmoo}, we first have to cut down the field to bounded size, which we do with the help of Proposition~\ref{fieldcut}. However, this works only for cutting the {\it large\/} values of the field, but not the small ones. In order to be able to use also a lower bound for the field, we intersect with the event that $Y(z)\geq -M$ for all $z$'s appearing, and use a large deviation principle for the conditional field.
Let us turn to the details. Let $u>0$ satisfying $u\in\scriptscriptstyleupp(Y(0))^\circ$. We fix small parameter $\varepsilon,{\rm d}elta>0$ such that $u+\varepsilon\in\scriptscriptstyleupp(Y(0))^\circ$ and large parameters $M$ and $R$. On the intersection of the events $\{\scriptscriptstyleupp(L_n)\scriptscriptstyleubset Q_R\}$ and $\{Y(z)\geq -M\, \forall z\in B_{R\alpha_n}\}$, we can estimate
$$
\frac 1n Z_n=\langle L_n, \overline Y_n\rangle \geq\langle L_n, \overline Y_n^{\scriptscriptstylesup{\leq M}}\rangle=\langle L_n*\kappa_{\rm d}elta, \overline Y_n^{\scriptscriptstylesup{\leq M}}\rangle+\langle L_n, \overline Y_n^{\scriptscriptstylesup{\leq M}}-\overline Y_n^{\scriptscriptstylesup{\leq M}}*\kappa_{\rm d}elta\rangle.
$$
We write $\P^{\scriptscriptstylesup{>-M}}$ for the conditional measure $\P(\,\cdot\,|\,Y(z)\geq -M\, \forall z\in\Z^d)$. Hence, we obtain the lower bound
\begin{equation}
\begin{aligned}
\P(\scriptscriptstylemfrac 1n Z_n>u)&\geq \P^{\scriptscriptstylesup{>-M}}\Bigl(\scriptscriptstyleupp(L_n)\scriptscriptstyleubset Q_R, \langle L_n*\kappa_{\rm d}elta, \overline Y_n^{\scriptscriptstylesup{\leq M}}\rangle>u+\varepsilon\Bigr)\P(Y(0)\geq -M)^{|B_{R\alpha_n}|}\\
&\qquad-\P(\langle L_n, \overline Y_n^{\scriptscriptstylesup{\leq M}}-\overline Y_n^{\scriptscriptstylesup{\leq M}}*\kappa_{\rm d}elta\rangle>\varepsilon).
\end{aligned}
\end{equation}
Using Lemma~\ref{FieSmoo} for the last term on the right hand side, and noting that $\P(Y(0)\geq -M)\to0$ as $M\to\infty$, it becomes clear that it suffices to estimate the first term on the right side. In order to do this, fix a positive continuous function $g\colon Q_R\to(0,\infty)$ satisfying $\int_{Q_R}g(x)\,{\rm d} x =1$ such that $g$ can be extended to an element of $H^1(\R^d)$. Let $B_\varepsilon(g)$ denote a weak $\varepsilon$-neighborhood of $g$. Then we have
$$
\begin{aligned}
\P^{\scriptscriptstylesup{>-M}}\Bigl(&\scriptscriptstyleupp(L_n)\scriptscriptstyleubset Q_R, \langle L_n*\kappa_{\rm d}elta, \overline Y_n^{\scriptscriptstylesup{\leq M}}\rangle>u+\varepsilon\Bigr)\\
&\geq \P(L_n\in B_\varepsilon(g), \scriptscriptstyleupp(L_n)\scriptscriptstyleubset Q_R)\P^{\scriptscriptstylesup{>-M}}\Bigl(\langle g*\kappa_{\rm d}elta, \overline Y_n^{\scriptscriptstylesup{\leq M}}\rangle>u+2\varepsilon\Bigr).
\end{aligned}
$$
According to Lemma~\ref{LDP}, the first term on the right is equal to $\exp\{-n\alpha_n^{-2} \inf_{\psi^2\in B_\varepsilon(g)}\Ical_R(\psi^2)(1+o(1))\}$, and according to Lemma~\ref{LDPfieldL}, the latter term is equal to $\exp\{-n\alpha_n^{-2} \Phi_{\widetilde H_M}(g*\kappa_{\rm d}elta,u-2\varepsilon,R)(1+o(1))\}$. Summarizing, we obtain, for any $R>0$ and any continuous positive function $g\colon Q_R\to(0,\infty)$, if $M$ is sufficiently large and ${\rm d}elta>0$ sufficiently small,
\begin{equation}
\liminf_{n\to\infty}\frac {\alpha_n^2}n\log \P(\scriptscriptstylemfrac 1n Z_n>u)\geq -\Bigl[\Ical_R(g)+\Phi_{\widetilde H_M}(g*\kappa_{\rm d}elta,u+2\varepsilon,R)\Bigr]+\eta_M,
\end{equation}
for some $\eta_M{\rm d}ownarrow 0$ as $M\to\infty$. Passing to the infimum over all $g$ and writing $\psi^2$ instead of $g$, we obtain
\begin{equation}
\liminf_{n\to\infty}\frac {\alpha_n^2}n\log \P(\scriptscriptstylemfrac 1n Z_n>u)\geq -\inf\limits_{\psi\in H^1(\R^d)\colon \scriptscriptstyleupp(\psi)\scriptscriptstyleubset Q_R}\Bigl[\Ical_R(\psi^2)+\Phi_{\widetilde H_M}(\psi^2*\kappa_{\rm d}elta,u+2\varepsilon,R)\Bigr]+\eta_M.
\end{equation}
Since $\psi^2*\kappa_{\rm d}elta$ is bounded uniformly in $\psi$, and since $\widetilde H_M(t)\to H(t)$ as $M\to\infty$, uniformly in $t$ on compacts, we can let $M\to\infty$. Furthermore, we also let $\varepsilon{\rm d}ownarrow0$ and obtain
\begin{equation}
\liminf_{n\to\infty}\frac {\alpha_n^2}n\log \P(\scriptscriptstylemfrac 1n Z_n>u)\geq -K_H(u;{\rm d}elta,R),
\end{equation}
for any ${\rm d}elta>0$ and $R>0$, where $K_H^{\scriptscriptstylesup{0}}(u;{\rm d}elta,R)$ is defined in Lemma~\ref{KHappr}. Now use Lemma~\ref{KHappr} to finish the proof of the lower bound in Theorem~\ref{lin}.
\scriptscriptstyleection{Appendix: Proof of the large deviation principle for the local times}\label{sec-proofLDP}
\noindent In this section, we prove the scaled large deviation principles in
Lemma~\ref{LDP}.
Although the statement should be familiar to experts and the proof is
fairly standard, we could not find it
in the literature. Therefore, we provide a proof. Let us mention that the lower bound of the following
Lemma~\ref{lem-cumul} (without the indicator on $\{\scriptscriptstyleupp(L_n)\scriptscriptstyleubset Q_R\}$, however) is contained in \cite{CL02}.
Fix $R>0$. For bounded and continuous functions $ f \colon Q_R\to\R$, we
denote by
\begin{equation}\label{Lapeigenv}
\lambda_R( f )=\max\Bigl\{\langle f ,\psi ^2\rangle-\frac{1}{2}\|\Gamma^{\frac
12}\nabla \psi \|_2^2\colon \psi \in H^1(\R^d),\scriptscriptstyleupp( \psi )\scriptscriptstyleubset Q_R,\|
\psi \|_2 = 1\Bigr\}
\end{equation}
the principal eigenvalue of the operator
$\frac{1}{2}\nabla\cdot\Gamma\nabla+ f $ in $Q_R$ with Dirichlet boundary
condition. (We denote the inner product and norm on $L^2(Q_R)$ by $\langle\cdot,\cdot\rangle$
and $\|\cdot\|_2$.)
The main step in the proof of Lemma~\ref{LDP}(i) is the
following.
\begin{lemma}\label{lem-cumul}
For any bounded and continuous function $ f \colon Q_R\to\R$, the
limit
\begin{equation}\label{cumul}
\lim_{n\to\infty}\frac {\alpha_n^2}n\log \E\Bigl[\exp\Bigl\{\frac
n{\alpha_n^2} \langle f ,L_n\rangle\Bigr\}\1\{\scriptscriptstyleupp(L_n)\scriptscriptstyleubset Q_R\}\Bigr]
\end{equation}
exists and is equal to $\lambda_R( f )$.
\end{lemma}
\begin{Proof}{Proof} In the following, we abreviate $B=B_{R\alpha_n}$.
Introduce a scaled version $ f _n\colon \Z^d\to\R$ of $ f $ by
\begin{equation}
f _n(z)=\alpha_n^{d}\int_{z\alpha_n^{-1}+[0,\alpha_n^{-1})^d} f (x)\,{\rm d}
x,\qquad z\in\Z^d.
\end{equation}
Note that $ f _n(\lfloor \cdot\,\alpha_n\rfloor)\to f $
uniformly on $Q_R$. Furthermore, note that
\begin{equation}
\frac n{\alpha_n^2} \langle f ,L_n\rangle=\alpha_n^{d-2}\int_{Q_R} f (x)
\end{lemma}l_n\bigl(\lfloor x\alpha_n\rfloor\bigr)\,{\rm d} x
=\alpha_n^{-2}\scriptscriptstyleum_{z\in B}\end{lemma}l_n(z) f _n(z)
=\scriptscriptstyleum_{k=0}^{n-1}\alpha_n^{-2} f _n(S_k).
\end{equation}
For notational convenience, we assume that $\alpha_n^2$ and $n\alpha_n^{-2}$ are integers. Using the Markov property, we split the expectation over the path $(S_0,{\rm d}ots,S_n)$ into $n\alpha_n^{-2}$ expectations over paths of length $\alpha_n^2$. By $\E_z$ we denote the expectation with respect to the random walk starting at $z\in\Z^d$, then we have
\begin{equation}\label{expansionexp}
\begin{aligned}
\E\Bigl[&\exp\Bigl\{\frac
n{\alpha_n^2} \langle f ,L_n\rangle\Bigr\}\1\{\scriptscriptstyleupp(L_n)\scriptscriptstyleubset Q_R\}\Bigr]\\
&=\E\Bigl[\exp\Bigl\{\frac 1{\alpha_n^2}\scriptscriptstyleum_{k=0}^{n-1} f _n(S_k)\Bigr\}\1\{\scriptscriptstyleupp(\end{lemma}l_n)\scriptscriptstyleubset B\}\Bigr]\\
&=\scriptscriptstyleum_{z_1,{\rm d}ots,z_{n\alpha_n^{-2}}\in B}\prod_{i=1}^{n\alpha_n^{-2}}\E_{z_{i-1}}\Bigl[\exp\Bigl\{\frac 1{\alpha_n^2}\scriptscriptstyleum_{k=0}^{\alpha_n^2-1} f _n(S_k)\Bigr\}\1\{\scriptscriptstyleupp(\end{lemma}l_{\alpha_n^2})\scriptscriptstyleubset B\}\1\{S_{\alpha_n^2}=z_i\}\Bigr]\\
&=\int_{Q_R^{n\alpha_n^{-2}}}{\rm d} x_1{\rm d}ots{\rm d} x_{n\alpha_n^{-2}}\,\prod_{i=1}^{n\alpha_n^{-2}}\Bigl[\alpha_n^d
\E_{\lfloor x_{i-1}\alpha_n\rfloor}\Bigl[\exp\Bigl\{\frac 1{\alpha_n^2}\scriptscriptstyleum_{k=0}^{\alpha_n^2-1} f _n(S_k)\Bigr\}\\
&\qquad\qquad\qquad\qquad\times\1\{\scriptscriptstyleupp(\end{lemma}l_{\alpha_n^2})\scriptscriptstyleubset B\}\1\{S_{\alpha_n^2}=\lfloor x_{i}\alpha_n\rfloor\}\Bigr)\Bigr].
\end{aligned}
\end{equation}
Let $(B_t)_{t\geq 0}$ be the Brownian motion on $\R^d$ with covariance matrix $\Gamma$, and let ${\tt E}_x$ denote the corresponding expectation, when $B_0=x\in\R^d$. Then $(\alpha_n^{-1}S_{\lfloor t\alpha_n^2\rfloor})_{t\geq 0}$ converges weakly towards $(B_t)_{t\geq 0}$ in distribution, and from a local central limit theorem (see \cite[P7.9, P7.10]{S76}) it follows that, uniformly in $x,y\in Q_R$,
\begin{equation}
\begin{aligned}
\lim_{n\to\infty}\alpha_n^d&\E_{\lfloor x\alpha_n\rfloor}\Bigl[\exp\Bigl\{\frac 1{\alpha_n^2}\scriptscriptstyleum_{k=0}^{\alpha_n^2-1} f _n(S_k)\Bigr\}\1\{\scriptscriptstyleupp(\end{lemma}l_{\alpha_n^2})\scriptscriptstyleubset B\}\1\{S_{\alpha_n^2}=\lfloor y\alpha_n\rfloor\}\Bigr]\\
&={\tt E}_x\Bigl(\exp\Bigl\{\int_0^1 f (B_s)\,{\rm d} s\Bigr\}\1\{B_{[0,1]}\scriptscriptstyleubset Q_R\};B_1\in {\rm d} y\Bigr)\Big/ {\rm d} y.
\end{aligned}
\end{equation}
Substituting this on the right hand side of \eqref{expansionexp} and again using the Markov property, we obtain that, as $n\to\infty$,
\begin{equation}\label{expansion2}
\E\Bigl[\exp\Bigl\{\frac
n{\alpha_n^2} \langle f ,L_n\rangle\Bigr\}\1\{\scriptscriptstyleupp(L_n)\scriptscriptstyleubset Q_R\}\Bigr]=e^{o(n\alpha_n^{-2})}{\tt E}_0\Bigl(\exp\Bigl\{\int_0^{n\alpha_n^{-2}} f (B_s)\,{\rm d} s\Bigr\}\1\{B_{[0,n\alpha_n^{-2}]}\scriptscriptstyleubset Q_R\}\Bigr).
\end{equation}
It is well-known that the expectation on the right hand side of \eqref{expansion2} is equal to $\exp\{\frac n{\alpha_n^2}[\lambda_R( f )+o(1)]\}$ as $n\to\infty$, and this ends the proof of Lemma~\ref{lem-cumul}.
\end{Proof}\qed
\begin{Proof}{Proof of Lemma~\ref{LDP}(i)} We shall
apply a version of the abstract G\"artner-Ellis theorem (see
\cite[Sect.~4.5]{DZ93}). (There is no problem in applying that result for {\em
sub\/}probability measures instead of probability measure.) More precisely,
we shall apply \cite[Cor.~4.5.27]{DZ93}, which implies the statement of
Lemma~\ref{LDP}(i) under the following two assumptions: (1) the
distributions of $L_n$ under $\P(\cdot\, ,\scriptscriptstyleupp(L_n)\scriptscriptstyleubset Q_R)$ form an exponentially tight
family, and (2) the limit in \eqref{cumul} exists and is a finite,
G\^ateau-differentiable and lower semicontinuous function of $ f $. These
two points are satisfied in our case. Indeed, (1) is trivially satisfied
since we consider subprobability measures on a compact set $Q_R$, and
(2) follows from Lemma~\ref{lem-cumul}, together with \cite{Ga77}, where
the G\^ateau-differentiability and lower semicontinuity of the map
$ f \mapsto \lambda_R( f )$ is shown. An application of
\cite[Cor.~4.5.27]{DZ93} therefore yields the validity of a large deviation
principle as stated in Lemma~\ref{LDP}(i).
It remains to identify the rate function obtained in
\cite[Cor.~4.5.27]{DZ93} with the rate function of Lemma~\ref{LDP}(i), $\Ical_R$.
The rate function appearing in \cite[Cor.~4.5.27]{DZ93}, $\widetilde
\Ical_R$, is the Legendre transform of $\lambda_R(\cdot)$:
\begin{equation}
\widetilde \Ical_R( \psi^2 )=\scriptscriptstyleup_{ f \in
\Ccal(Q_R)}\bigl[\langle \psi^2 , f \rangle-\lambda_R( \psi^2 )\bigr],\qquad \psi^2 \in\Fcal_R.
\end{equation}
It is obvious from \eqref{Lapeigenv} that $\lambda_R(\cdot)$ is itself
the Legendre transform of $\Ical_R$, since $\Ical_R$ is equal to
$\infty$ outside $\Fcal_R$. Because of the convexity inequality for gradients
(see \cite[Theorem~7.8]{LL97}), $\Ical_R$ is a convex function on $\Fcal_R$.
According to the Duality Lemma \cite[Lemma~4.5.8]{DZ93}, the Legendre
transform of $\lambda_R(\cdot)$ is equal to $\Ical_R$, i.e., we have that
$\widetilde \Ical_R= \Ical_R$. This finishes the proof of
Lemma~\ref{LDP}(i).
\end{Proof}\qed
\begin{Proof}{Proof of Lemma~\ref{LDP}(ii)}
This is a modification of the proof of part (i) above; we point out the
differences only. Recall that we identify the
box $B_R=\{\lfloor -R\rfloor+1,{\rm d}ots,\lfloor R\rfloor-1\}^d$ with the
torus $\{\lfloor -R\rfloor+1,{\rm d}ots,\lfloor R\rfloor\}^d$ where $\lfloor
R\rfloor$ is identified with $\lfloor -R\rfloor+1$. Analogously, we
conceive $Q_R=[-R,R]^d$ as the $d$-dimensional torus with the opposite
sides identified.
For a continuous bounded function $ f \colon Q_R\to\R$,
introduce the principal eigenvalue of the operator
$\frac{1}{2}\nabla\cdot\Gamma\nabla + f $ on $L^2(Q_R)$ with periodic boundary
condition:
\begin{equation}
\lambda^{\scriptscriptstylemallsup{R}}( f )=\max\Bigl\{\int_{Q_R} f (x)
\psi^2(x)\,{\rm d} x-\frac{1}{2}\int_{Q_R}\big|\Gamma^{\frac 12}\nabla_R \psi(x)\big|^2\,{\rm d} x\colon \psi\in\Ccal_1(Q_R),\int_{Q_R}\psi^2(x)\,{\rm d} x=1\Bigr\},
\end{equation}
where we recall that $\nabla_R$ is the gradient of the torus $Q_R$.
The main step in the proof of Lemma~\ref{LDP}(ii) is to show
that, for any continuous bounded function $ f \colon
Q_R\to\R$,
\begin{equation}\label{eigenasyR}
\lambda^{\scriptscriptstylemallsup{R}}( f )=\lim_{n\to\infty}\frac
{\alpha_n^2}n\log \E\Bigl[\exp\Bigl\{\frac
n{\alpha_n^2}\langle f ,L_n^{\scriptscriptstylemallsup{R\alpha_n}}\rangle\Bigr\}\Bigr].
\end{equation}
This is done in the same way as in the proof of Lemma~\ref{lem-cumul}, noting that
the process $(\alpha_n^{-1}S^{\scriptscriptstylemallsup{R\alpha_n}}_{t\alpha_n^2})_{t\geq 0}$ converges weakly in distribution towards $(B_t^{\scriptscriptstylemallsup{R}})_{t\ge 0}$, the Brownian motion with covariance matrix $\Gamma$, wrapped around the torus $Q_R$. Also using a local central limit theorem, we obtain, as $n\to\infty$,
\begin{equation}
\E\Bigl[\exp\Bigl\{\frac
n{\alpha_n^2}\langle f ,L_n^{\scriptscriptstylemallsup{R\alpha_n}}\rangle\Bigr\}\Bigr)=e^{o(n\alpha_n^{-2})}
{\tt E_0}\Bigl(\exp\Bigl\{\int_0^{n\alpha_n^{-2}} f (B_s^{\scriptscriptstylesup{R}})\,{\rm d} s\Bigr\}\Bigr].
\end{equation}
It is well-known that the expectation on the right side is equal to $\exp\{\frac n{\alpha_n^2}[\lambda^{\scriptscriptstylemallsup{R}}( f )+o(1)]\}$ as $n\to\infty$, and this shows that also \eqref{eigenasyR} holds. The remainder of the proof of Lemma~\ref{LDP}(ii) is the same as the proof of Lemma~\ref{LDP}(i).
\end{Proof}\qed
\noindent {\bf Acknowledgment. } This work was partially supported by DFG grant Ko 2205/1-1. W.~K. thanks the Deutsche Forschungsgemeinschaft for awarding a Heisenberg grant (realized in 2003/04). W.~K.\ and N.~G.\ thank the Laboratoire de Probabilit{\'e}s for its hospitality. All three authors thank Francis Comets for helpful discussions. We also thank the referee for
carefully reading the first version of the paper.
\end{document} |
\begin{document}
\title{Ultrabright and narrowband intra-fiber biphoton source at ultralow pump power}
\author{Alexander Bruns}
\affiliation{Institut für Angewandte Physik, Technische Universität Darmstadt, Hochschulstraße 6, 64289 Darmstadt, Germany}
\email{alexander.bruns@tu-darmstadt.de}
\author{Chia-Yu Hsu}
\affiliation{Institut für Angewandte Physik, Technische Universität Darmstadt, Hochschulstraße 6, 64289 Darmstadt, Germany}
\affiliation{Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan}
\author{Sergiy Stryzhenko}
\affiliation{Institut für Angewandte Physik, Technische Universität Darmstadt, Hochschulstraße 6, 64289 Darmstadt, Germany}
\affiliation{Institute of Physics, National Academy of Science of Ukraine, Nauky Avenue 46, Kyiv 03028, Ukraine}
\author{Enno Giese}
\affiliation{Institut für Angewandte Physik, Technische Universität Darmstadt, Hochschulstraße 6, 64289 Darmstadt, Germany}
\author{Leonid P. Yatsenko}
\affiliation{Institute of Physics, National Academy of Science of Ukraine, Nauky Avenue 46, Kyiv 03028, Ukraine}
\author{Ite A. Yu}
\affiliation{Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan}
\affiliation{Center for Quantum Technology, Hsinchu 30013, Taiwan}
\author{Thomas Halfmann}
\affiliation{Institut für Angewandte Physik, Technische Universität Darmstadt, Hochschulstraße 6, 64289 Darmstadt, Germany}
\author{Thorsten Peters}
\affiliation{Institut für Angewandte Physik, Technische Universität Darmstadt, Hochschulstraße 6, 64289 Darmstadt, Germany}
\begin{abstract}
Nonclassical photon sources of high brightness are key components of quantum communication technologies. We here demonstrate the generation of narrowband, nonclassical photon pairs by employing spontaneous four-wave mixing in an optically-dense ensemble of cold atoms within a hollow-core fiber. The brightness of our source approaches the limit of achievable generated spectral brightness at which successive photon pairs start to overlap in time. For a generated spectral brightness per pump power of up to \SI{2e9}{\pairs\per\second\per\mega\hertzMilli} we observe nonclassical correlations at pump powers below \SI{100}{\nano\watt} and a narrow bandwidth of $2\pi\times$ \SI{6.5}{\mega\hertz}.
In this regime we demonstrate that our source can be used as a heralded single-photon source.
By further increasing the brightness we enter the regime where successive photon pairs start to overlap in time and the cross-correlation approaches a limit corresponding to thermal statistics.
Our approach of combining the advantages of atomic ensembles and waveguide environments is an important step towards photonic quantum networks of ensemble-based elements.
\end{abstract}
\maketitle
\section{Introduction\label{sec:Intro}}
Nonclassical photon sources are of paramount importance for optical quantum communication \cite{sangouard_quantum_2011,slussarenko_photonic_2019}. Key requirements for these sources are high single-photon purity and high brightness, which are typically difficult to achieve simultaneously. If these photon sources are to be interfaced with atomic or solid-state quantum memories, spectrally narrow single photons in the \si{\mega\hertz} regime are additionally required to match the relevant atomic transitions \cite{eisaman_invited_2011}. In the context of quantum networks \cite{kimble_quantum_2008} and moving towards real-life applications, the integrability of the technology into optical waveguides becomes crucial \cite{wang_integrated_2021}. This, e.g., has the advantage of better scalability, lower pump powers, and improved efficiency due to better mode-matching as compared to free-space setups.
While deterministic sources include the timing information by design, probabilistic sources require an additional heralding mechanism. It is commonly realized by implementing a probabilistic source of correlated photon pairs and using one of the photons as a herald to obtain timing information about the second photon \cite{eisaman_invited_2011}. Such correlated photon pairs are typically generated by nonlinear optical processes such as spontaneous parametric down-conversion (SPDC) or spontaneous four-wave mixing (SFWM). Both of these processes can provide a high spectral biphoton brightness at narrow bandwidths either due to the inherent frequency filtering of SFWM or by employing an additional cavity. The highest generated spectral brightnesses (GSB) reported so far in excess of \SI{1e5}{\pairs\per\second\per\mega\hertz} were achieved using a variety of different experimental systems. These are waveguides combined with cavities \cite{luo_direct_2015,steiner_ultrabright_2021}, a bulk crystal inside a cavity \cite{tsai_ultrabright_2018}, and a room-temperature atomic ensemble \cite{chen_room-temperature_2022}.
The bandwidths of these biphoton sources ranged from sub-MHz \cite{chen_room-temperature_2022} to around \SI{100}{\mega\hertz} \cite{steiner_ultrabright_2021} and the generated spectral brightness per pump power (GSBP) ranged from \SI{3e4}{\pairs\per\second\per\mega\hertzMilli} \cite{luo_direct_2015} to \SI{2e8}{\pairs\per\second\per\mega\hertzMilli} \cite{steiner_ultrabright_2021}. While each system and technique has its own advantages and it is therefore difficult to compare the values achieved, they put our results into context.
In the following, we report on the first implementation of a biphoton source using SFWM in an ensemble of cold rubidium atoms loaded into a hollow-core fiber. To avoid collisions of the cold atoms with the fiber wall, thereby reducing the coherence time, the atoms are guided by an optical dipole trap. By using cold atoms we achieve an order of magnitude lower biphoton bandwidth as compared to fibers filled with warm gases \cite{cordier_raman-free_2020,lopez-huidobro_fiber-based_2021}. This was first shown by Corzo \textit{et al.}, who interfaced cold atoms with a nanofiber to generate narrowband single photons on-demand \cite{corzo_waveguide-coupled_2019}. However, using a hollow-core fiber instead of a nanofiber has the benefit that all light fields, including the pump, are guided in the same optical mode. Thus, we obtain intrinsically optimal mode-matching as well as strong light-atom coupling at orders of magnitude lower pump powers. This results in a GSBP of up to \SI{2e9}{\pairs\per\second\per\mega\hertzMilli}, which is a 10-fold increase over the previous record \cite{steiner_ultrabright_2021} at 100-fold reduced pump power and 10-fold lower bandwidth of $2\pi\times$ \SI{6.5}{\mega\hertz}, which is directly compatible with atomic quantum memories. Moreover, we show that by tuning the brightness of our source even higher, we reach the ultimate achievable limit of spectral brightness, at which successive photon pairs start to overlap in time. In this regime, the cross-correlation approaches a limit expected for thermal statistics.
To characterize our photon source, we present thorough measurements on the cross- and auto-correlations, bandwidth, and GSB of the photon pairs. Where possible, we compare our experimental results to theoretical simulations for a quantitative analysis. Due to the high GSB, narrow bandwidth, and waveguide-coupled single-mode emission, possible applications of our biphoton source include integration into photonic quantum networks \cite{kimble_quantum_2008} using atomic ensembles as building blocks \cite{sangouard_quantum_2011}, quantum key distribution via satellite links \cite{bedington_progress_2017}, and photonic quantum metrology \cite{polino_photonic_2020}.
\begin{figure*}
\caption{(a) Double-$\Lambda$ system for SFWM in ${}
\label{fig:Setup}
\end{figure*}
\section{Experimental Methods \label{sec:Setup}}
Our SFWM coupling scheme for photon-pair generation using ${}^{87}\text{Rb}$ atoms is shown in figure \ref{fig:Setup}(a).
As we do not intentionally lift the degeneracy of the Zeeman states, only the hyperfine states are shown. A pump beam (frequency $\omega_P$) with power $P$ on the D$_2$ line, blue-detuned by $\Delta=\SI{53}{\Gamma_{\mathrm{D}2}}$ from the transition $\ket{1}\leftrightarrow\ket{4}$ to minimize optical pumping, generates Stokes (S) photons (frequency $\omega_S$) via spontaneous Raman scattering. The collinear control beam (frequency $\omega_C$) with Rabi frequency $\Omega_C$ on the D$_1$ line is applied resonantly to the transition $\ket{2}\leftrightarrow\ket{3}$. This creates a narrow transparency window via electromagnetically induced transparency (EIT) \cite{fleischhauer_electromagnetically_2005} for the correlated anti-Stokes (AS) photons (frequency $\omega_{AS}$) in the SFWM process \cite{balic_generation_2005}.
The nonlinear optical medium in our experiment is an ensemble of up to \num{2.5e5} atoms at a temperature of about \SI{1}{\milli\kelvin}, a length of about $L=\SI{6}{\centi\meter}$ and a radial diameter of $d=\SI{3.4}{\micro\meter}$ that is located inside a hollow-core photonic bandgap fiber (HCPBGF) \cite{poletti_hollow-core_2013} (\textit{NKT Photonics}, HC800-02, \SI{14}{\centi\meter} long, numerical aperture $\sim\num{0.15}$, \SI{5.5}{\micro\meter} mode field $1/\mathrm{e}^2$ diameter of intensity). The process of preparing the cold atoms inside a magneto-optical trap (MOT) and guiding them into the HCPBGF\ using a far-off resonant trap (FORT) at a wavelength of \SI{820}{\nano\meter} was previously explained and characterized in detail \cite{blatt_one-dimensional_2014,peters_loading_2021}. For simplicity, we therefore omit here all details regarding the preparation of the atomic ensemble inside the HCPBGF.
The optical setup of the SFWM experiment is schematically shown in figure \ref{fig:Setup}(b). All laser beams are generated using home-built diode laser systems with linewidths of several hundred kHz (except of the FORT \cite{peters_loading_2021}). The timing and frequency of the laser beams are controlled by acousto-optic modulators with switching times of about \SI{100}{\nano\second}. Because all beams of SFWM are guided in a single optical mode of the HCPBGF, we achieve Rabi frequencies of the order of $\Gamma$ with nanowatts of optical power only. The complete experimental setup requires an optical power of about \SI{300}{\milli\watt} to operate the MOT and the FORT. Pump and control fields are orthogonally linearly polarized and combined in front of the HCPBGF\ using a polarizing beamsplitter (PBS). As the involved wavelengths span \SI{40}{\nano\meter} we use achromatic half-wave plates to align the polarization axes of the linearly polarized laser fields to the optical axis of the HCPBGF\ at the input. This allows for achieving the highest degree of polarization at the output and thus optimum polarization filtering as required for operation at the photon level \cite{peters_single-photon-level_2020}. As confirmed by numerical simulations (see appendix \ref{sec:appendixPolSim}) of the SFWM process including the Zeeman structure and laser field polarizations, the generated S (AS) field is orthogonally polarized to the pump (control) field. Thus, the two generated fields can be separated by a PBS which also serves as a first filter stage. The optical noise originating from residual pump (control) light in the S (AS) channel separated by \SI{6.8}{\giga\hertz} is attenuated by up to \SI{40}{\decibel}. After polarization filtering the photons are coupled into single-mode fibers which serve as spatial filters to isolate the HCPBGF 's optical mode. As a second filtering stage we use temperature-tuned monolithic etalons \cite{palittapongarnpim_note_2012,ahlrichs_monolithic_2013} with an attenuation of up to \SI{45}{\decibel} to further attenuate the optical noise in the S (AS) channel that originates from the pump (control) beam (separated by \SI{6.8}{\giga\hertz}) and the control (pump) beam (separated by \SI{15}{\nano\meter}). Finally, we employ narrowband optical filters (blocking optical depth (OD) $>4$) to further attenuate the \SI{15}{\nano\meter} separated optical noise and broadband noise originating from stray light. The laser beams not directly involved in the SFWM process, i.e., preparation beam and FORT, are aligned counter-propagating to the pump and control beams to protect the photon counting equipment and to minimize optical noise. The residual optical noise (measured without loading atoms into the HCPBGF) is about \SI{2800}{\per\second} (\SI{1200}{\per\second}) for the S (AS) channel.
To detect and characterize the generated photon pairs, we implement Hanbury Brown \& Twiss setups for both photonic channels by splitting the signals with 50:50 fiber beamsplitters before guiding them to single-photon counting modules (SPCM, \textit{Excelitas}, SPCM-AQRH-13). The overall detection efficiency in both channels is about \SI{8}{\percent}, including the transmission through all elements after the HCPBGF\ and the intrinsic quantum efficiencies of our detectors. We acquire the timings of all detected photons with time-tagging electronics (\textit{Swabian Instruments}, Timetagger 20) for further analysis. The combined timing jitter of the detection system is specified as $<$\SI{400}{\pico\second}, while the temporal dynamics of the biphoton waveform are on a timescale of a few \SI{10}{\nano\second} (see Figure 2). Hence, we neglect this jitter in the analysis of our data.
Figure \ref{fig:Setup}(c) shows the temporal sequence of the experiment. After the atoms are captured in the MOT and subsequently loaded into the HCPBGF, we start the experimental SFWM phase. To avoid any influence of the strong FORT on the SFWM process, we periodically switch off the FORT for periods of \SI{2.4}{\micro\second} before switching it on again for \SI{1.6}{\micro\second} to recapture the atoms and prevent collisions with the fiber wall. During each period where the FORT is off, the atoms are first prepared in the ground state $\ket{1}$ by optical pumping with the SFWM control beam and an additional orthogonally-polarized preparation beam on the D$_2$ line to address all magnetic sublevels of ground state $\ket{2}$. Note that we do not prepare the atoms in a single Zeeman state. Subsequently, while keeping the control beam on, we simultaneously apply the pump beam to drive the SFWM process for a duration of \SI{1.2}{\micro\second}. We acquire data from the SPCMs only during this SFWM phase. Due to the finite temperature of the atomic ensemble, the periodic switching of the FORT leads to a non-negligible loss of atoms over time. To characterize these changing experimental conditions, we measured time-resolved absorption spectra by probing the AS transition with a weak laser pulse. We observed a peak OD of \num{155} on the $\ket{1}\leftrightarrow\ket{3}$ transition that decays exponentially with a time constant of \SI{3.2}{\milli\second}. Therefore, we choose to repeat the experimental window \num{3500} times for a total duration of \SI{12}{\milli\second}, to include the full range of ODs available to us. After this time, basically all atoms are lost and we start with a new MOT loading cycle. Because the OD is changing over two orders of magnitude during the measurements, we also modulate the control power in a similar way to maintain similar EIT conditions over a larger range of measurement windows. We can then post-select the range of experimental windows (or ODs) in the evaluation. Narrow ranges lead to sharply defined experimental conditions, while wider ranges increase the detection rates at the cost of averaging over changing conditions. The duty cycle, i.e., the ratio of SFWM data acquisition phases to the total duration of the experiment depends on the post-selected parameter ranges and is typically around 1:2000. The generation rates given throughout this paper are corrected for the respective duty cycle.
\section{Experimental Results \label{sec:Results}}
By implementing the experiment as described in the previous section, we generate correlated S-AS photon pairs for a wide range of the experimentally controllable parameters OD and pump power. This allows for tuning the spectral brightness and cross-correlation of our photon-pair source, which we will discuss later in more detail.
\begin{figure*}
\caption{(a) Exemplary temporal biphoton waveform. Cross-correlation \gTwo{S,AS}
\label{fig:ResultBiphotonQuality}
\end{figure*}
First, however, we discuss the properties of the observed photon pairs at fixed experimental conditions yielding a good signal-to-noise ratio at a GSBP of about \SI{300}{\pairs\per\second\per\mega\hertzNano} (see discussion of figure \ref{fig:ResultVsRate}) which is comparable to the record GSBP reported in \cite{steiner_ultrabright_2021}. Figure \ref{fig:ResultBiphotonQuality} (a) shows exemplary data (blue circles) of the cross-correlation $g^{(2)}_{S,AS}(\tau)$ of the photon pairs as a function of the time delay $\tau$ between S and subsequent AS detection events. We obtained the data by normalizing the coincidences to the background of accidental coincidences. We acquired this background by correlating events from different measurement windows, hence removing any physical correlation from the data.
Clearly, the cross-correlation exceeds the classical limit of $g^{(2)}_{S,AS}(\tau)=2$ (red dotted line) for a range of delays. The small damped oscillations visible in the waveform indicate that we work in the transition region between the damped Rabi oscillation and the group delay regimes, as defined in \cite{du_narrowband_2008,kolchin_electromagnetically-induced-transparency-based_2007}. We verified experimentally that the individual S and AS fields exhibit thermal statistics, i.e., $g^{(2)}_{S,S}(0)=g^{(2)}_{AS,AS}(0)\approx2$. To obtain sufficiently good statistics for these measurements within a feasible integration time, we had to raise the generation rate by increasing the pump power to $\gtrsim\SI{100}{\nano\watt}$.
To quantify the violation of the Cauchy-Schwarz inequality \cite{clauser_experimental_1974} we calculated $\mathcal{R}=({g^{(2)}_{S,AS}})^2/(g^{(2)}_{S,S}\,g^{(2)}_{AS,AS})$. The peak value is $\mathcal{R}=\num{97(24)}$, which clearly violates the classical limit of $\mathcal{R}\leq 1$ by 4 standard deviations and demonstrates the nonclassical nature of the photon pairs.
To obtain the bandwidth of the photon-pair source, we compare the experimental data shown in figure \ref{fig:ResultBiphotonQuality}(a) to a simulated waveform using the theory presented in \cite{du_narrowband_2008}. The relevant equations used are summarized in appendix \ref{sec:Theory}. First, we evaluated equation \eqref{eqBiphotonWaveform} starting with parameters determined from other measurements and subsequently optimized these parameters for the best agreement between simulation and experimental data. The features of the experimental data (blue circles) can be reproduced by the simulation (orange solid line). The discrepancy between the parameter sets of experiment and simulation might be explained by deviations of our experiment from the assumptions made in \cite{du_narrowband_2008}. There, homogeneous atomic density and pump/control intensities are assumed, which is clearly not the case inside the HCPBGF. Moreover, the simulation considers only population of a single Zeeman level, whereas in our case the population is initially distributed among the Zeeman levels of $\ket{F=1}$. We use the simulated waveform to extract the biphoton spectrum as shown in the inset of figure \ref{fig:ResultBiphotonQuality}(a). Here, $\delta$ is the single-photon detuning of the generated AS photon frequency components from their central frequency $\omega_{AS}$. The spectral bandwidth (FWHM) is $2\pi\times\SI{6.5}{\mega\hertz}\approx\SI{1.1}{\Gamma_{\mathrm{D}1}}$ demonstrating compatibility of our source with other rubidium-based experiments. The characteristic timescale of the biphotons is thus \SI{24}{\nano\second}.
We now turn to interpreting the photon-pair source as a heralded single-photon source. Instead of using the S for heralding the AS detection events, as it is typically done, we use the AS to herald the S events for the following reason: As the Raman gain is not negligible compared to the FWM gain for our current parameters, uncorrelated photons can be generated which act as additional noise. This process is mainly relevant for the S channel as the pump is coupled to the populated state $\ket{1}$, whereas the control is coupled to the unpopulated state $\ket{2}$. Using the AS detection events as heralding events for the creation of single photons in the S channel, we evade the influence of Raman noise on the heralding itself. As the AS photon is delayed by the slow-light effect of EIT \cite{fleischhauer_electromagnetically_2005} and thus exits the medium after the S photon, choosing the AS as heralding photon is somewhat counter-intuitive. If necessary, however, the temporal order of S and AS detection events could be easily changed by sending the S photons through a fiber optical delay line.
In figure \ref{fig:ResultBiphotonQuality}(b) we plot the auto-correlation of the S photons conditioned on the detection of an AS photon, \gTwo{S,S|AS}. To obtain data of sufficient quality within a reasonable acquisition time, we use the following method \cite{fasel_high-quality_2004,seri_laser-written_2018}: Any heralded S1(S2) detection event (see figure \ref{fig:Setup}) is used as a start trigger. For any subsequent S2(S1) detection event, we record the number of additional AS events $n$ between the two S events. The number of S1-S2(S2-S1) pairs with $n$ AS events in between gives the amplitude of the $n$-th ($-n$-th) bin in the resulting histogram. The zero bin thus corresponds to more than one S photon being heralded by the same AS photon. From this data we can now determine \gTwo{S,S|AS}$(0)$ as the amplitude of the zero bin normalized to the expected value that we get from a linear fit (red dashed line) to the remaining bins \cite{fasel_high-quality_2004}. We observe anti-bunching of the heralded S photons with \gTwo{S,S|AS}$(0)=\num{0.08(1)}$, clearly violating the classical bound of \gTwo{S,S|AS}$(0)\ge 1$ \cite{grangier_experimental_1986} and confirming a non-zero projection onto the single-photon Fock state for \gTwo{S,S|AS}$(0)\le 0.5$ \cite{grunwald_effective_2019}. We calculate the heralding efficiency as the ratio of the pair generation rate and the heralding (AS) rate corrected for transmission losses and find a value of \SI{43}{\percent}. The optimal mode matching in the HCPBGF\ facilitates such efficient heralding. Similar values were reported using a nanofiber \cite{corzo_waveguide-coupled_2019}.
\begin{figure}
\caption{Photon pair properties versus pump power. The experimental parameters are $OD=40-80$ and the control Rabi frequency $\Omega_C=\SI{2.8}
\label{fig:ResultVsRate}
\end{figure}
Next, we analyze the performance of our biphoton source when varying the pump power $P$ over three orders of magnitude. Figure \ref{fig:ResultVsRate}(a) shows the measured peak cross-correlation \gTwo{S,AS} (blue circles) as a function of $P$. With increasing pump power the cross-correlation, i.e., the purity of the photon pairs, reduces following the expected $1/P$ dependency. Nevertheless, we observe nonclassical correlations over the full range of pump powers. We use the model described in appendix \ref{sec:NoiseModel} to calculate the peak cross-correlation that includes the measured unconditional as well as pump power-dependent optical noise in the S and AS channel (dashed line). The model is described in more detail in the discussion of figure \ref{fig:Result2D}(c). The same parameter set determined in figure \ref{fig:Result2D}(c) is also used for the simulation shown in figure \ref{fig:ResultVsRate}(a) without any additional fitting parameter. We use a linear dependence between the generated brightness and the pump power as shown in \ref{fig:ResultVsRate}(b) (see next paragraph). Experiment and theory agree well over the whole range of pump powers.
In figure \ref{fig:ResultVsRate}(b) we show the corresponding GSB (blue circles, left hand side axis). We obtain this rate from the detected rate by correcting it for optical background, transmission losses, detection efficiencies, and the duty cycle of the experiment. The dashed blue line is a linear fit of type $\textrm{GSB}=\textrm{GSBP}\cdot P$. The fit confirms the expected linear dependence for pump powers up to \SI{200}{\nano\watt}. For higher pump powers, the spectral generation rate increases slower. We suspect that in this regime population redistribution due to optical pumping is no longer negligible. When we normalize the spectral generation rate with regard to the pump power, we obtain the GSBP. The experimental values (orange triangles, right hand side axis) range from \SIrange{200}{410}{\per\s\per\mega\hertz\per\nano\watt}. The orange dashed line represents the fitted value $\textrm{GSBP}=\SI{312(24)}{\pairs\per\second\per\mega\hertzNano}\approx \SI{3e8}{\pairs\per\second\per\mega\hertzMilli}$. In this intermediate parameter regime the GSBP of our source is comparable to the highest reported value of \SI{2e8}{\pairs\per\second\per\mega\hertzMilli} using a waveguide coupled to an on-chip microring cavity \cite{steiner_ultrabright_2021}, however, at a 10-fold reduced bandwidth and a 100-fold reduced pump power. This very efficient conversion of pump power into narrowband photon pairs is enabled by the intrinsically large overlap between the cold atomic ensemble and the light fields in the HCPBGF\ as well as the optimal mode matching between the four involved fields due to all of them being guided in a single optical mode.
\begin{figure}
\caption{(a) Peak cross-correlation and (b) GSBP vs. pump power and OD. The control Rabi frequency for $OD\gtrsim 55$ is \SI{2.8}
\label{fig:Result2D}
\end{figure}
Next, we investigate the dependence of the peak cross-correlation \gTwo{S,AS} and the GSBP on the pump power as well as the OD. The results are shown in figure \ref{fig:Result2D}. We note that our system allows for an easy tuning of the OD over an order of magnitude. This, combined with a tuning of the pump power results in a tunability of the GSBP of our biphoton source by two orders of magnitude. The regimes of high generation rate but low-quality photon pairs (upper right corner at high OD and high pump power) and low generation rate with high-quality photon pairs (lower left corner at low OD and low pump power) can be clearly identified in figure \ref{fig:Result2D}(a) and (b). The parameter range of the data presented in figures \ref{fig:ResultBiphotonQuality} and \ref{fig:ResultVsRate} lies in between these two regimes and is marked by the blue and green rectangles. The highest values for \gTwo{S,AS} are observed in the low OD and low pump power regime. When we restrict ourselves to parameters with \gTwo{S,AS}$\geq 3$, the highest observed GSBP is \SI{2e9}{\pairs\per\second\per\mega\hertzMilli}. For values \gTwo{S,AS}$\gtrsim50$, the relative uncertainties become larger than \SI{30}{\percent}, due to the low number of counts during the finite measurement duration. This measurement demonstrates the versatile nature of our photon-pair source. Depending on the application requirements, the parameter regime can be easily adjusted.
Finally, we investigate the limit of the GSB achieved with our system. To this end, we plot in figure \ref{fig:Result2D}(c) the GSB vs. the peak cross-correlation \gTwo{S,AS}. The data are taken from figures (a) and (b). Note that the grey datapoints correspond to a count rate where the saturation of the SPCMs can not be neglected anymore. As we can see, for higher spectral brightnesses the peak cross-correlation decreases and reaches a lower limit near two. We fitted the data using the model described in appendix \ref{sec:NoiseModel} that is based on a single-mode description and thermal statistics. We start from equation \eqref{eq:NoiseModel} and write the pair number as $n=\alpha\cdot\mathrm{GSB}$ with a scaling factor $\alpha$. The orange line thus corresponds to a fitted function of the form
\begin{equation}
\label{eqGSB}
g^{(2)}_\text{S,AS} = \frac{ 2 +\frac{1}{\alpha\mathrm{GSB}} + \frac{\mathcal{N}_{S}\mathcal{N}_{AS}}{T_{S}T_{AS} (\alpha\mathrm{GSB})^2}}{\left[1 + \frac{\mathcal{N}_{S}}{T_{S} \alpha\mathrm{GSB}} \right]\left[1 + \frac{\mathcal{N}_{AS}}{T_{AS} \alpha\mathrm{GSB}} \right]}
\end{equation}
with noise contributions $\mathcal{N}_j$ and detection efficiencies $T_j$, where we used measured values as described in appendix \ref{sec:NoiseModel}. The scaling factor is the only free parameter for our fit and was determined to be $\alpha=103$. As the dataset includes points obtained with different pump powers, the pump-dependent noise contributions also vary. This is visualized by the orange-shaded area.
Nonclassical correlations with \gTwo{S,AS}$>3$ are maintained up to a GSB of around $\SI{2e5}{\pairs\per\second\per\mega\hertz}$.
Beyond this brightness we approach the ultimate obtainable limit of $\mathrm{GSB}=\pi/2\times\SI{e6}{\pairs\per\second\per\mega\hertz}$, where successive photon pairs start to overlap in time \cite{chen_room-temperature_2022}. The transition from temporally separated to overlapping pairs is visualized by the blue color gradient in figure \ref{fig:Result2D}(c). In this range, the peak value of $g^{(2)}_{S,AS} \to 2$ indicates that thermal statistics apply while the S and AS fields are still correlated and can be described by a single mode. A detailed discussion of this regime is beyond the scope of this work and will be addressed in future work. Nonetheless, our data clearly demonstrates that our biphoton source can be operated near the ultimate limit of generated spectral brightness with a high tunability over two orders of magnitude, narrow bandwidth, and at ultralow pump powers.
\section{Conclusion and Outlook\label{sec:Conclusion}}
We demonstrated the first nonclassical photon-pair source based on SFWM in an ensemble of cold Rubidium atoms interfaced with a HCPBGF. The strong confinement of atoms and light fields within the HCPBGF\ leads to enhanced optical nonlinearities as compared to free space setups. It results in a GSBP of up to \SI{2e9}{\pairs\per\second\per\mega\hertzMilli} at pump powers in the regime up to \SI{100}{\nano\watt} and for a cross-correlation \num{\geq3}. We determined the biphoton linewidth as $2\pi\times$\SI{6.5}{\mega\hertz} which is thus compatible with atomic quantum memories and similar to the linewidth achievable with free-space setups. This represents a 10-fold increase of the GSBP compared to the previous record using a microring cavity, at a 10-fold reduced bandwidth and a 100-fold reduced pump power. Furthermore, the photon pairs exhibit orthogonal polarizations, thereby allowing for a relatively simple separation without the need for elaborate filtering and separation techniques \cite{caltzidis_atomic_2021}.
We verified the nonclassical nature of the photon pairs by measuring their cross-correlation with a violation of the Cauchy-Schwarz inequality by a factor of \num{97(24)} as well as a heralded auto-correlation \num{\ll0.5}.
Moreover, by increasing the generated spectral brightness even further, we reached the regime where different photon pairs start to overlap in time. Here, we demonstrated that the cross-correlation approaches a limit corresponding to thermal statistics, i.e., S and AS photons exhibit bunching.
Our hollow-core fiber-based biphoton source therefore combines the advantages of free-space and waveguide photon sources.
Examples of possible applications include easy integration into photonic networks \cite{kimble_quantum_2008} based on atomic ensembles \cite{sangouard_quantum_2011} due to the narrowband emission into a single transverse mode, employment in quantum key distribution satellite links \cite{bedington_progress_2017} due to suitable transmission band wavelengths of \SI{780}{\nano\meter} and \SI{795}{\nano\meter}, and as light source for photonic quantum metrology \cite{polino_photonic_2020}. Additionally, the ultralow pump power required to generate biphotons combined with integrating the HCPBGF\ with an atom chip \cite{keil_fifteen_2016} holds promise for a significant miniaturization of the setup. Finally, the high GSB could be applied, e.g., to increase the brightness of temporally-multiplexed photon sources \cite{meyer-scott_single-photon_2020}.
To further improve and extend our HCPBGF-based biphoton source one could envision the following options: First, using a ladder-type scheme as in \cite{lee_highly_2016,davidson_bright_2021} the currently dominating Raman noise could be avoided.
Second, replacing our currently pulsed loading of the HCPBGF\ by a continuous loading scheme, the duty cycle could be increased substantially. Combined with in-fiber magic-wavelength trapping \cite{hilton_dual-color_2019,yoon_laser-cooled_2019} photon pairs could be generated continuously without the need to periodically modulate the guiding potential. We estimate that this could enhance the detected pair rates by up to three orders of magnitude. Third, by implementing intra-fiber laser cooling \cite{wang_enhancing_2022} the ensemble temperature could be reduced thereby prolonging the coherence time, i.e., further reducing the bandwidth.
Moreover, we note that as the generation of biphotons only requires atoms loaded \textit{into} instead of \textit{through} the fiber (note that in our case only the first 6~cm of the 14~cm long fiber are filled), the output side of the HCPBGF\ could be spliced directly to a (polarization-maintaining) fiber \cite{thapa_splicing_2006,kristensen_low-loss_2008}, thereby enabling direct connection to a photonic network, possibly after wavelength conversion to the telecom band in a waveguide \cite{albrecht_waveguide_2014}.
Finally, a detailed investigation of the photon statistics for the highest spectral brightness, where successive photon pairs overlap in time, may permit the generation of multi-photon states.
\section*{Acknowledgments}
The authors thank the group of T. Walther for providing us with their home-built ultra-low noise laser diode driver with high modulation bandwidth. The project received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 765075, as well as by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project number 410249930.
\appendix
\section{Theoretical treatment of SFWM \label{sec:Theory}}
\subsection{Biphoton waveform \label{sec:TheoryBiphoton}}
We here summarize the basic theory to simulate the biphoton waveform and spectrum based on the work of Du \textit{et al.} \cite{du_narrowband_2008}. We focus only on the relevant equations that enable comparison with our experimental results in figure \ref{fig:ResultBiphotonQuality}.
The double $\Lambda$-type system is introduced in figure \ref{fig:Setup}(a) of the main text. We additionally define the single-photon detuning of the AS frequency components $\omega$ as $\delta=\omega-\omega_{AS}$. Due to the collinear setup the phase matching condition, written in terms of the wavenumbers $k_i$, reads $\Delta k=k_{AS}+k_{S}-k_{C}-k_{P}=0$. We assume that all atoms are initially prepared in ground state $\ket{1}$.
We are interested in temporal correlations of the created photon pairs as a function of the time delay $\tau$ between the paired $S$ and $AS$ photons. They are determined by the relative biphoton wavefunction
\begin{equation}
\label{eqBiphotonWaveform}
|\psi(\tau)|^2 = \left|
\frac{L}{2\pi} \int \kappa(\delta) \Phi(\delta) \text{e}^{-\text{i}\delta\tau} \mathrm{d}\delta
\right|^2
\end{equation}
with the longitudinal detuning function
\begin{equation}
\Phi(\delta)=\text{sinc}\left(\frac{\Delta k L}{2}\right)\text{e}^{\text{i}(k_{AS}+k_{S})L/2}
\end{equation}
and the nonlinear parametric coupling coefficient
\begin{equation}
\kappa(\delta)=-\text{i}\frac{\sqrt{\omega_{AS}\omega_{S}}}{2c}\chi^{(3)}(\delta)E_P E_C,
\end{equation}
where $E_i$ are the electric fields of the respective transitions. The third-order nonlinear susceptibility of the AS field is given by
\begin{equation}
\begin{aligned}
\chi^{(3)}(\delta) &= \frac{
\mathcal{N} \mu_{13} \mu_{32} \mu_{24} \mu_{41} / (\epsilon_0 \hbar^3)
}{
(\Delta + \text{i} \frac{\Gamma_{\mathrm{D}2}}{2})
\left[
|\Omega_C|^2 - 4(
\delta + \text{i} \frac{\Gamma_{\mathrm{D}1}}{2}
) (\delta+ \text{i} \gamma_{12})
\right].
}
\end{aligned}
\end{equation}
We introduced here the transition dipole matrix elements $\mu_{ij}$, the atomic density $\mathcal{N}=N/(L \pi (d/2)^2)$ and the control Rabi frequency $\Omega_C=\mu_{23}E_C/\hbar$.
To evaluate the detuning function $\Phi(\delta)$, we calculate the complex wave numbers of the created fields as
$k_i(\delta)=\frac{\omega_i}{c}\sqrt{1+\chi_i(\delta)}$ with the linear susceptibilities given by
\begin{align}
\chi_S(\delta) &= \frac{
\mathcal{N} |\mu_{24}|^2 (\delta-\text{i}\frac{\Gamma_{\mathrm{D}1}}{2}) / (\epsilon_0\hbar)
}{
|\Omega_C|^2 - 4(\delta - \text{i}\frac{\Gamma_{\mathrm{D}1}}{2}) (\delta-\text{i}\gamma_{12})
} \times \frac{
|\Omega_P|^2
}{
\Delta^2+\left(\frac{\Gamma_{\mathrm{D}2}}{2}\right)^2
}, \\
\chi_{AS}(\delta)&=\frac{4\mathcal{N}|\mu_{13}|^2(\delta+\text{i}\gamma_{12})/(\epsilon_0\hbar)}{|\Omega_C|^2-4(\delta+\text{i}\frac{\Gamma_{\mathrm{D}1}}{2})(\delta+\text{i}\gamma_{12})}.
\end{align}
Here, $\Omega_P=\mu_{14}E_P/\hbar$ is the pump Rabi frequency.
Using these equations allows for calculating the expected waveform $|\psi(\tau)|^2$ and the biphoton spectrum $|\kappa(\delta)\Phi(\delta)|^2$.
\subsection{Polarization-resolved spatio-temporal simulation of SFWM\label{sec:appendixPolSim}}
In this section we describe the procedure for numerical simulation of the SFWM process taking into account a total of 16 magnetic sublevels of the involved hyperfine states as well as light field polarizations. This simulation served to confirm the polarizations of the generated S and AS fields for varying polarization configurations ($\parallel$ and $\perp$) of the pump and control fields. This was necessary as we are currently unable to prepare the population in a single Zeeman state and use arbitrary polarizations configurations as in free-space setups due to the HCPBGF s birefringence, which requires the use of linear polarizations.
We assume classical fields, use a plane-wave approximation and describe the atom-field interactions using a density matrix approach in rotating wave approximation.
We simulate the process of SFWM by solving a system of partial differential equations
which represent light propagation and time evolution of the density matrix operator $\hat{\rho}$ with appropriate random initial conditions.
Denoting the set of magnetic sublevels of the ground states $\ket{1}$ and $\ket{2}$ by the index $g$ and a similar set of magnetic sublevels of the excited states by the index $e$ we can write the density matrix equations in the short matrix form.
In this notation all quantum operators are spelled as block matrices:
\begin{equation}
\hat{O} = \begin{bmatrix}
O_{gg} & O_{ge} \\
O_{eg} & O_{ee} \\
\end{bmatrix}.
\end{equation}
Then the propagation equations read
\begin{equation}
\frac{\partial E_m(z, t)}{\partial z}
= -4\pi i k_m \mathcal{N} \operatorname{tr}[ \hat{\mu}^m_{ge} \cdot \rho_{eg}(z, t) ].
\label{eq:sim_propagation}
\end{equation}
Here $E_m$ is the complex amplitude of the electric field in mode $m$, $k_m$
is its wavenumber, and $\hat{\mu}^m$ is the mode's dipole moment operator
where the matrix elements are taken from \cite{steck_rubidium_2021}.
The mode index $m$ runs across all fields we take into account
(pump, control, Stokes, and anti-Stokes), as well as their respective polarizations,
where we use $\sigma^\pm$ as a polarization basis.
The time evolution matrix equations for the density matrix read
(here and below we assume
$\Gamma_\mathrm{D1} \simeq \Gamma_\mathrm{D2} \simeq \Gamma$)
\begin{widetext}
\begin{equation}
\begin{aligned}
\frac{\partial \rho_{gg}}{\partial t} &=
i [\rho_{gg}, \Delta_g] + \frac{i}{2} (
\rho_{ge} \cdot \Omega_{eg} - \Omega_{ge} \cdot \rho_{eg}
) + \Gamma \, R_g \circ C_{ge} \cdot \rho_{ee} \cdot C_{eg}
- \gamma \circ \rho_{gg}, \\
\frac{\partial \rho_{eg}}{\partial t} &= i (
\rho_{eg} \cdot \Delta_g - \Delta_e \cdot \rho_{eg}
) + \frac{i}{2} (
\rho_{ee} \cdot \Omega_{eg} - \Omega_{eg} \cdot \rho_{gg}
) - \frac{\Gamma}{2} (C_{eg} \cdot C_{ge}) \cdot \rho_{eg},\\
\frac{\partial \rho_{ee}}{\partial t} &=
i [\rho_{ee}, \Delta_e] + \frac{i}{2} (
\rho_{eg} \cdot \Omega_{ge} - \Omega_{eg} \cdot \rho_{ge}
) - \frac{\Gamma}{2} \{ C_{eg} \cdot C_{ge}, \rho_{ee} \}. \\
\end{aligned}
\label{eq:sim_evolution}
\end{equation}
\end{widetext}
Here, $\Delta_g$ and $\Delta_e$ are diagonal matrices with the summed detunings
of the considered modes on the diagonal.
$\Omega_{ge} = \Omega_{eg}^\dag$ are submatrices of the Rabi frequency operator defined by
\begin{equation}
\hbar \hat{\Omega} = \sum_m E_m \hat{\mu}^m.
\end{equation}
The matrices $C_{ge} = C_{eg}^\dag$ contain the Clebsch-Gordan coefficients as
follows:
\begin{equation}
\Bra{\begin{matrix}
F_e \\
M_e \\
\end{matrix}} C_{eg} \Ket{\begin{matrix}
F_g \\
M_g \\
\end{matrix}}
= \braket{F_e, M_e | F_g, M_g; 1, M_e-M_g}.
\end{equation}
$R_g$ is the Kronecker delta of the $F$ quantum number:
\begin{equation}
\Bra{\begin{matrix}
F_1 \\
M_1 \\
\end{matrix}} R_g \Ket{\begin{matrix}
F_2 \\
M_2 \\
\end{matrix}} = \delta_{F_1 F_2}
\end{equation}
and the operation $\circ$ stands for the element-wise multiplication of matrices.
This way, $R_g$ prevents coherence between $\ket{1}$ and $\ket{2}$ being generated
when population is transferred from the excited to the ground levels.
The matrix $\gamma$ describes decoherence effects in the ground states:
\begin{equation}
\Bra{\begin{matrix}
F_1 \\
M_1 \\
\end{matrix}}\gamma \Ket{\begin{matrix}
F_2 \\
M_2 \\
\end{matrix}} = \begin{cases}
0 \text{ if } F_1 = F_2 \text{ and } M_1 = M_2, \\
\gamma_{12} \text{ otherwise.} \\
\end{cases}
\end{equation}
To imitate the vacuum fluctuations we set the Stokes field at the
beginning of the fiber to a random value with correlation time
$1/\Gamma$ in the following way:
\begin{equation}
\begin{aligned}
E_{S\pm}(z=0, t=0) &= 0, \\
E_{S\pm}(z=0, t + \delta t)
&= E_{S\pm}(z=0, t) e^{-\frac{\Gamma_\mathrm{D2}}{2} \, \delta t} \\
&\quad + \nu_{\pm} E_\mathrm{vacuum} \sqrt{1 - e^{-\Gamma_\mathrm{D2} \, \delta t}},
\end{aligned}
\label{initial}
\end{equation}
where $\delta t \ll 1/\Gamma$ is the time step,
$\nu_{\pm}$ is a random complex number from the normalized
complex Gaussian distribution, and the field
$E_\mathrm{vacuum}$ corresponds to the Rabi frequency of
$\sim 10^{-5} \Gamma$.
The sign $\pm$ in the index points out that we use these
initial conditions both for the left and the right
polarization components.
The time evolution equation (\ref{eq:sim_evolution}) with initial conditions (\ref{initial}) were solved numerically
using the Kutta-Merson method.
On each Kutta-Merson step we solved the propagation equation
(\ref{eq:sim_propagation}) by integrating its right-hand side using
Simpson's rule\footnote{L. V. Blake, \textit{A Modified Simpson’s Rule and Fortran Subroutine for Cumulative Numerical Integration of a Function Defined by Data Points}, Tech. Rep. (Defense Technical Information Center, Fort Belvoir, VA, 1971)}.
The results of these simulations confirmed that the generated S and AS fields exhibit orthogonal linear polarizations with respect to the injected pump and control fields of also linear orthogonal polarizations. Therefore it was experimentally possible to use polarization filtering (see figure \ref{fig:Setup}(b)) in addition to frequency filtering and obtain a sufficiently high extinction ratio for the strong collinear pump and control fields.
\section{
Model for the detected cross-correlation
\label{sec:NoiseModel}
}
In this section we present a simple single-mode model~\cite{rohde_modelling_2006} that incorporates loss and detector inefficiencies for the fit of the cross-correlation function $g^{(2)}_{S,AS}$.
For that, we introduce the bosonic annihilation operator
\begin{equation}
\label{eq.det_photons}
\hat{b}_j= t_j \hat{a}_j + r_j \hat{\mu}_j
\end{equation}
of photons detected in channel $j={S,AS}$.
Here, $\hat{a}_j$ denotes the (bosonic) annihilation operator of Stokes or anti-Stokes photons generated during SFWM and $\hat{\mu}_j$ the (bosonic) annihilation operator of noise photons in channel $j$.
Imperfect detection is introduced in equation~\eqref{eq.det_photons} though a beam splitter transformation with $T_j+R_j = |t_j|^2 + |r_j|^2=1$.
Hence, $T_j$ is the efficiency of the detector including all loss in channel $j$.
If we assume no correlation between the noise and the generated photons, i.\,e., $\braket{\hat{a}_j^\dagger \hat{\mu}_j}=0$, we can connect the detected photon numbers
$N_j= \braket{\hat{b}_j^\dagger \hat{b}_j}=T_j n_j + \mathcal{N}_j$
to the number of generated Stokes and anti-Stokes photons $n_j=\braket{\hat{a}_j^\dagger \hat{a}_j}$.
We furthermore introduced the noise detected in channel $j$ through $\mathcal{N}_j = R_j \braket{\hat{\mu}_j^\dagger \hat{\mu}_j}$.
The detected cross-correlation function $g^{(2)}_{S,AS}=\braket{\hat{b}_{S}^\dagger\hat{b}_{AS}^\dagger \hat{b}_{S}\hat{b}_{AS}}/(N_{S} N_{AS}) $ takes the form
\begin{equation}
g^{(2)}_{S,AS} = \frac{\braket{\hat{a}_{S}^\dagger\hat{a}_{AS}^\dagger \hat{a}_{S}\hat{a}_{AS}}}{n_{S} n_{AS}} \frac{T_{S}n_{S} T_{AS}n_{AS}}{N_{S} N_{AS}} + \frac{\mathcal{N}_{S}\mathcal{N}_{AS}}{N_{S} N_{AS}}.
\end{equation}
Here, we have again assumed no correlation between the noise and the generated photons, i.\,e., $\braket{\hat{a}_{S}^\dagger\hat{a}_{AS}^\dagger \hat{\mu}_{S}\hat{\mu}_{AS}}=0$
and uncorrelated noise in both channels, i.\,e., $R_{S}R_{AS} \braket{\hat{\mu}_{S}^\dagger\hat{\mu}_{AS}^\dagger \hat{\mu}_{S}\hat{\mu}_{AS}}= \mathcal{N}_{S}\mathcal{N}_{AS}$.
If there is perfect correlation between Stokes and anti-Stokes fields, the number of photons generated in each channel corresponds to the number of generated pairs $n=n_{S}=n_{AS}$.
We have verified that both the Stokes and anti-Stokes fields exhibit close to thermal statistics, which directly implies $\braket{\hat{a}_{S}^\dagger\hat{a}_{AS}^\dagger \hat{a}_{S}\hat{a}_{AS}}/(n_{S} n_{AS}) = 2 + 1/n$ for perfectly correlated fields.
In this case, the cross-correlation function takes the form
\begin{equation}
\label{eq:NoiseModel}
g^{(2)}_\text{S,AS} = \frac{ 2 +\frac{1}{n} + \frac{\mathcal{N}_{S}\mathcal{N}_{AS}}{T_{S}T_{AS} n^2}}{\left[1 + \frac{\mathcal{N}_{S}}{T_{S} n} \right]\left[1 + \frac{\mathcal{N}_{AS}}{T_{AS} n} \right]}
\end{equation}
and reduces to the ideal case for negligible noise. For dominant noise, however, the cross-correlation approaches the limit of $1+1/(n+\mathcal{N})$.
Note that in a multi-mode model the number $2$ has to be replaced by $1+1/M$, where $M$ is the number of detected modes~\cite{ivanova_multiphoton_2006}.
To determine the noise for the fits in main body of the paper, we make the ansatz
$\mathcal{N}_j= [r_j^0+r_j^0(P)+r_j(P)]\tau_c$, where $r_i^0$ is the noise rate with no atoms loaded in the HCPBGF\ and no pump beam present, i.e., optical noise from stray light, residual control light and the dark counts of the SPCMs.
The additional optical noise caused by the pump beam itself, but still measured without atoms present in the HCPBGF, is denoted by $ r_j^0(P)$.
The remaining contribution $r_j(P)$ accounts for the noise originating from the cold atoms and it is dominated by Raman noise and residual optical pumping, but is neglected in the remainder of our analysis.
We measured the contributions without atoms to be $r_S^0=\SI{2800}{\per\second}$, $r_{AS}^0=\SI{1200}{\per\second}$, $r_S^0(P)=\SI{84}{\per\second\per\nano\watt}\cdot P$ and $r_{AS}^0(P)=\SI{7.6}{\per\second\per\nano\watt}\cdot P$.
The characteristic time scale $\tau_c = \SI{24}{ns}$ is assumed to be of the order of the temporal duration of a biphoton, obtained from the data in figure \ref{fig:ResultBiphotonQuality}(a).
Moreover, we use $T_{S}= T_{AS}= \num{0.08}$ in accordance with the measured transmission losses and specified detector efficiencies.
\end{document} |
\begin{document}
\title[Minimal accessible categories]
{Minimal accessible categories}
\author[J. Rosick\'{y}]
{J. Rosick\'{y}}
\thanks{Supported by the Grant Agency of the Czech Republic under the grant
19-00902S}
\address{
\newline J. Rosick\'{y}\newline
Department of Mathematics and Statistics\newline
Masaryk University, Faculty of Sciences\newline
Kotl\'{a}\v{r}sk\'{a} 2, 611 37 Brno, Czech Republic\newline
rosicky@math.muni.cz
}
\begin{abstract}
We give a purely category-theoretic proof of the result of Makkai and Par\'e saying that the category $\Lin$ of linearly ordered
sets and order preserving injective mappings is a minimal finitely accessible category. We also discuss the existence of a minimal $\aleph_1$-accessible category.
\end{abstract}
\keywords{}
\subjclass{}
\maketitle
\section{Introduction}
One of striking results of \cite{MP} is that the category $\Lin$ of linearly ordered sets and order preserving injective mappings
is a minimal finitely accessible category. This means that for every large finitely accessible category $\ck$ there is a faithful functor $\Lin\to\ck$ preserving directed colimits. \cite{MP} does not contain a proof of this result -- Makkai and Par\'e just
say that it essentially follows from the work of Morley \cite{M}. Since there are many applications of this result (see, e.g., \cite{LR}), it might be useful to give an explicit proof of it. We do it by transferring the standard model-theoretic argument to the language of accessible categories. Another, more model-theoretic proof, of the theorem of Makkai and Par\'e was recently given by Boney \cite{Bo}.
The minimality of $\Lin$ among finitely accessible categories implies its minimality among ($\infty,\omega$)-elementary categories (see \cite{MP} 3.4.1) and, even, among accessible categories
with directed colimits whose morphisms are monomorphisms (\cite{LR} 2.5).
One cannot expect that $\Lin$ is a minimal accessible category because there is no faithful functor from $\Lin$ to the
$\aleph_1$-accessible category of well ordered sets and order preserving injective mappings. The reason is that any well ordered set $A$ is iso-rigid, it means that every isomorphism $A\to A$ is the identity. Using \cite{HPT}, we give an example
of a $\aleph_1$-accessible category $\ck$ having every object $K$ rigid, i.e., every morphism $K\to K$ is the identity.
This yields a candidate for a minimal $\aleph_1$-accessible category. Similarly, one gets a candidate for a minimal
$\aleph_\alpha$-accesible category.
\vskip 1mm
\noindent
{\bf Acknowledgement.} We are grateful to T. Beke for useful discussions concerning this paper.
\section{Skolem cover}
Let $\ck$ be a finitely accessible category and $\ca$ its representative small full subcategory of finitely presentable objects
(i.e., any finitely presentable object of $\ck$ is isomorphic to some $A\in \ca$). Let
$$
E:\ck\to\Set^{\ca^{\op}}
$$
be the canonical embedding that takes each $K\in\ck$ to the contravariant functor $\ck(-,K):\ca\to\Set$. We note that, by Proposition 2.8 in \cite{AR},
this functor is fully faithful and preserves directed colimits and finitely presentable objects. Following Theorem 4.17 in \cite{AR}, $\ck$
is equivalent to a finitary-cone-injectivity class $\Inj(T)$ in $\Set^{\ca^{\op}}$; this means that there is a set $T$ of cones $a=(a_i:X\to EA_i)_{i\in I}$
where $X$ is finitely presentable in $\Set^{\ca^{\op}}$ and $A_i\in \ca$, $i\in I$ such that $\Inj(T)$ consists of functors $F$ injective to each cone
$a\in T$. The latter means that for any morphism $f:X\to F$ there is $i\in I$ and $g:EA_i\to F$ with $ga_i=f$.
Let $S(\ck)$ be the category whose objects are $(F,a_F)_{a\in T}$ consisting of $F:\ca^{\op}\to\Set$ with $a_F$ assigning to a cone $a$ and
$f:X\to F$ a morphism $a_F(f):EA_i\to F$ for some $i\in I$ such that $a_F(f)a_i=f$. Morphisms $(F,a_F)\to (F',a_{F'})$ are
natural transformations $\varphi:F\to F'$ such that $a_{F'}(\varphi f)=\varphi a_F(f)$. The forgetful functor $G:S(\ck)\to\Set^{\ca^{\op}}$
is faithful and has values in $\Inj(T)$. Its codomain restriction $S(\ck)\to\Inj(T)$ is surjective on objects. Since $E:\ck\to\Inj(T)$ is an equivalence,
we get a faithful functor $H:S(\ck)\to\ck$ which is essentially surjective on objects, i.e., any $K\in\ck$ is isomorphic to some $H(F,\tilde{a})$.
\begin{lemma}\label{le2.1}
The category $S(\ck)$ is finitely accessible and $H:S(\ck)\to\ck$ preserves directed colimits.
\end{lemma}
\begin{proof}
Let $D:\cd\to S(\ck)$ be a directed diagram and consider the colimit $\delta: GD\to F$ in $\Set^{\ca^{\op}}$. Then $\colim D =(F,a_F)$ where
$a_F(f)=\delta_d a_{Dd}(g)$ where $f=\delta_d g$. Since $X$ is finitely presentable, the description is correct. Thus $S(\ck)$ has directed
colimits and $G$ preserves them. Hence $H$ preserves them too.
If $F$ is finitely presentable in $\Set^{\ca^{\op}}$ then any $(F,a_F)$ is finitely presentable in $S(\ck)$. In order to show that any $(F,a_F)$
is a directed colimit of finitely presentable objects in $S(\ck)$ it suffices to express $F$ as a directed colimit of finitely presentable objects
$F_d$ in $\Set^{\ca^{\op}}$ and complete them to $(F_d,a_{F_d})$ using finite presentability of $X$ again. Then $(F,a_F)$ is a directed colimit of
$(F_d,a_{F_d})$. Thus $S(\ck)$ is finitely accessible.
\end{proof}
In fact, we have shown that
$$
S(\ck)=S(\Ind\ca)=\Ind S(\ca)
$$
$S(\ck)$ will be called a \textit{Skolem cover} of $\ck$ because it is a skolemization of the $L_{\infty,\omega}$-theory corresponding to $T$.
Let $U:\Set^{\ca^{\op}}\to\Set$ assign to $F$ the set $\coprod_{A\in\ca}FA$. The functor $U$ is faithful and preserves directed colimits. Thus
$(\ck,UE)$ and $(S(\ck),UG)$ are concrete finitely accessible categories with concrete directed colimits and $H:S(\ck)\to \ck$ is a concrete
functor.
\begin{lemma}\label{le2.2}
Let $(F,a_F)\in S(\ck)$ and $Z\subseteq UG(F,a_F)$. Then there is the smallest subobject $(F_Z,a_{F_Z})$ of $(F,a_F)$ such that $Z\subseteq UGF_Z$.
\end{lemma}
\begin{proof}
Let $F_0$ be the smallest subfunctor of $F$ such that $Z\subseteq UF_0$; let $\sigma:F_0\to F$ denote the inclusion. Consider a cone $a:X\to EA_i$
in $T$ and a morphism $f:X\to F_0$. Then the composition $\sigma f$ factorizes through some $a_i$. Let $F_1$ be a colimit in $\Set^{\ca^{\op}}$
of the diagram
$$
\xymatrix{
F_0 \\
&&\\
X\ar[uu]^f \ar@{.}[ur] \ar [rr]_{a_i} && EA_i
}
$$
consisting of all spans $(f,a_i)$ above. We iterate this construction by replacing $F_0$ with $F_1$, etc. In this way, we get the chain
$F_0\to F_1\to\dots F_n\to\dots$. Then $F_Z=\colim F_n$ carries the desired smallest subobject of $(F,a_F)$.
\end{proof}
This is the virtue of the skolemization and reflects the fact that the skolemized theory is universal. We skolemized cone-injectivity while
algebraic factorization systems (see \cite{GT}) skolemize injectivity. J. Bourke \cite{B} came to the same point from a different motivation.
\begin{rem}\label{re2.3}
{
\em
For any $Z$, there is only a set of non-isomorphic $(F_Z,a_{F_Z})$, $F:\ca^{\op}\to\Set$.
}
\end{rem}
\section{Minimal finitely accessible categories}
\begin{theo}\label{th3.1}
For any large finitely accessible category $\ck$ there is a faithful functor $\Lin\to\ck$ preserving directed colimits.
\end{theo}
\begin{proof}
Following \ref{le2.2}, we can assume that $\ck$ is equipped with a faithful functor $U:\ck\to\Set$ preserving directed colimits and such that
for any subset $Z\subseteq UK$ there is the smallest subobject $K_Z$ of $K$ such that $Z\subseteq UK_Z$. Let $\cl$ be the category with objects
$(K,X)$ where $K\in\ck$ and $X\subseteq UK$ is linearly ordered. Morphisms $(K_1,X_1)\to (K_2,X_2)$ are morphisms $f:K_1\to K_2$ such that
$Uf$ induces the order preserving mapping $X_1\to X_2$. The category $\cl$ has directed colimits given as
$$
\colim (K_i,X_i)=(\colim K_i,\colim X_i)
$$
and any $(K,X)$ with $K$ finitely presentable in $\ck$ and $X$ finite is finitely presentable in $\cl$. Thus $\cl$ is finitely accessible
and the forgetful functor $\cl\to\ck$ preserves directed colimits.
For a $\cl$-object $(K,X)$, let $\rho_{(K,X)}$ be the greatest ordinal $0<\rho_{(K,X)}\leq\omega,|X|$ such that for any $n<\rho_{(K,X)}$
and any $a_1< a_2<\dots <a_n$ and $b_1< b_2<\dots < b_n$ in $X$ there is an isomorphism $s:K_{\{a_1,\dots,a_n\}}\to K_{\{b_1,\dots,b_n\}}$ such
that $Us(a_i)=b_i$ for $i=1,\dots,n$. Assume that there is $(K,X)\in\cl$ with $\rho_{(K,X)}=\omega$. Then $X$ is infinite and for any $n<\omega$
there is a chain $a_{n1}< a_{n2}<\dots < a_{nn}$ in $X$. We will construct a functor $F:\Lin\to\ck$ as follows. Finitely presentable objects in $\Lin$
are finite chains $C_n$ with elements $1< 2<\dots <n$. Put $F_0(C_n)=K_{a_{n1},\dots,a_{nn}}$. Given an injective order preserving mapping $h:C_m\to C_n$, let
$Fh$ be the composition
$$
K_{a_{m1},\dots,a_{mm}}\to K_{a_{nh(1)},\dots,a_{nh(m)}}\to K_{a_{n1},\dots,a_{nn}}
$$
where the first morphism is the isomorphism $s$ above and the second morphism is the inclusion. Given $h_1:C_k\to C_m$ and $h_2:C_m\to C_n$ then
it is easy to see that $F_0(h_2h_1)=F_0(h_2)F_0(h_1)$. In fact, we always get the isomorphism
$$
K_{a_{k1},\dots,a_{kk}}\to K_{a_{nh_2h_1(1)},\dots a_{nh_2h_1(k)}}
$$
followed by the inclusion $K_{a_{nh_2h_1(1)},\dots a_{nh_2h_1(k)}}\to K_{a_{n1},\dots,a_{nn}}$. Thus we get the functor $F_0:\FinLin\to\ck$
defined on finite linear orderings. Since $\Lin=\Ind\FinLin$, $F_0$ extends to a functor $F:\Lin\to\ck$ preserving directed colimits.
Since $F_0$ is faithful, $F$ is faithful too.
Assume that $\rho_{(K,X)}<\omega$ for any $(K,X)\in\cl$. We put $(K_1,X_1)<(K_2,X_2)$ provided that $\rho_{(K_2,X_2)}<\rho_{(K_1,X_1)}$ and
$(K_1)_{\{a_1,\dots,a_{\rho_{(K_2,X_2)}}\}}\cong (K_2)_{\{b_1,\dots b_{\rho_{(K_2,X_2)}}\}}$ for any $a_1< \dots < a_{\rho_{(K_2,X_2)}}$ in $X_1$
and any $b_1,\dots <b_{\rho_{(K_2,X_2)}}$ in $X_2$. Then $<$ partially orders objects of $\cl$ and this order is well-founded in the sense
that there is no decreasing chain
$$
\dots <(K_n,X_n) < (K_{n-1},X_{n-1}) <\dots < (K_1,X_1).
$$
Such chain would yield a diagram
$$
(K_1)_{\{a_{11}\}}\to (K_2)_{\{a_{21},a_{22}\}}\to (K_n)_{\{a_{n1},\dots,a_{nn}\}}
$$
whose colimit $(K,X)$ in $\cl$ has $\rho_{(K,X)}=\omega$. Thus we can assign an ordinal $\alpha(K,X)$ to each $(K,X)\in\cl$ in such a way
that
$$
\alpha(K,X)=\sup_{(K',X')<(K,X)}\alpha(K',X')+1.
$$
Following \ref{re2.3}, there is an infinite cardinal $\mu$ greater or equal to the number of non-isomorphic objects $K_X$ for $X$ finite
and $K$ arbitrary. For $(K,X)\in\cl$, choose $a_1<\dots <a_{\rho_{(K,X)-1}}$ in $X$ and put
$$
(K,X)^\ast= (K_{\{a_1,\dots,a_{\rho_{(K,X)-1}}\}},X\cap UK_{\{a_1,\dots,a_{\rho_{(K,X)-1}}\}}).
$$
We will prove that
$$
|X|<\exp_{\omega(\alpha(K,X)^\ast+1})(\mu)
$$
for any $(K,X)\in\cl$. Recall that $\exp_0(\mu)=\mu$, $\exp_{\xi+1}(\mu)=2^{\exp_\xi(\mu)}$ and $\exp_\eta(\mu)=sup_{\xi<\eta}\exp_\xi(\mu)$.
Since $(K,UK)\in\cl$ for any $K$ in $\ck$, this inequality implies that $\ck$ is small.
The proof will use the recursion on $\alpha(K,X)^\ast$.
Let $\alpha(K,X)^\ast=0$ and assume that $|X|\geq\exp_\omega(\mu)$. The set $X^n$ is decomposed into $\leq\mu$ parts following isomorphisms types
of $K_{\{a_1,\dots,a_n\}}$. Following the Erd\"os-Rado partition theorem (see \cite{J}, Exercise 29.1), there is $X_0\subseteq X$ such that $X_0>\mu$
and $K_{\{a_1,\dots,a_n\}}\cong K_{\{b_1,\dots,b_n\}}$ for any $a_1<\dots<a_n$ and $b_1<\dots<b_n$ in $X_0$. Thus
$$\label{claim}
(K,X_0)^\ast=(K_{\{a_1,\dots,a_n\}},X_0\cap UK_{\{a_1,\dots,a_n\}})<(K,X)^\ast,
$$
which is impossible because $\alpha(K,X)^\ast=0$.
Assume that the claim holds for any $(K,X)\in\cl$ with $\alpha(K,X)^\ast<\beta$ and consider $(L,Y)\in\cl$ with $\alpha(L,Y)^\ast=\beta$. Assume that
$|Y|\geq\exp_{\omega(\alpha(L,Y)^\ast+1)}(\mu)$ and let $n=\rho_{(L,Y)}$. We have
$$
|Y|\geq\exp_{\omega(\beta+1)}(\mu)>\exp_{\omega\beta+n-1}(\mu)=\exp_{n-1}(\exp_{\omega\beta}(\mu).
$$
Following the Erd\"os-Rado partition theorem, there is $Y_0\subseteq Y$ such that $|Y_0|>\exp_{\omega\beta}(\mu)$ and
$L_{\{b_1,\dots,b_n\}}\cong L_{\{c_1,\dots,c_n\}}$ for each $b_1<\dots <b_n$ and $c_1<\dots <c_n$ in $Y_0$. Then $\rho_{(L,Y_0)}> n$ and
$(L,Y_0)<(L,Y)$. Thus $(L,Y_0)^\ast < (L,Y)^\ast$. Hence $\alpha(L,Y_0)^\ast<\beta$ and thus
$$
|Y_0|<\exp_{\omega(\alpha(L,Y_0)^\ast+1}(\mu)\leq\exp_{\omega\beta}(\mu),
$$
which is a contradiction.
\end{proof}
\section{Towards minimal $\lambda$-accessible categories}
\begin{exam}\label{re4,2}
{
\em
The category $\cw$ of well-ordered sets is $\aleph_1$-accessible and any its object $K$ is iso-rigid in the sense that the only isomorphism
$K\to K$ is the identity. Thus there is no faithful functor $\Lin\to\cw$ and a prospective minimal $\aleph_1$-accessible category is iso-rigid.
}
\end{exam}
\begin{exam}\label{hpt}
{
\em
There is an $\aleph_1$-accessible category $\cl$ having all objects $K$ rigid in the sense that the only morphism $K\to K$ is the identity. Thus there is no faithful functor $\cw\to\cl$.
The construction of $\cl$ is motivated by \cite{HPT}, II.3. Let $\ck$ be the category of structures $(A,<,R,S,\sup,s)$ where $<$ is a well-ordering,
$R$ is a unary relation, $S$ is an $\omega$-ary relation, $\sup$ is the countable join and $s$ is the unary operation of taking the successor.
Let $T$ be the following set of axioms:
\begin{enumerate}
\item $(\forall x_0,x_1,y_1,\dots,x_n,y_n,\dots)(S(x_0,x_1,\dots,x_n,\dots)\wedge S(x_0,y_1,\dots,y_n,\dots)\to \newline\bigwedge_{0<n}x_n=y_n)$
\item $(\forall x_0,x_1,\dots,x_n,\dots)(S(x_0,x_1\dots,x_n\dots)\to(\bigwedge_{0<n}x_n<x_{n+1})\wedge x_0=\sup x_n)$
\item $(\forall x)(\exists(y_1,\dots,y_n,\dots)(\bigwedge_{0<n}(y_n<y_{n+1})\wedge x=\sup y_n)\newline \to(\exists x_1,\dots,x_n,\dots)S(x,x_1,\dots,x_n,\dots)$
\item $(\forall x)(R(x)\leftrightarrow \neg(\exists y)(x=s(y))\wedge \neg(\exists x_1,\dots,x_n,\dots)S(x,x_1,\dots,x_n,\dots)$
\end{enumerate}
Let $A_2$ be the set of isolated elements of $A$, $A_0$ be the set of all limit elements of $a\in A$ such that $S(a,a_1,\dots,a_n,\dots)$ for some
$a_1,\dots,a_n,\dots\in A$ and $A_1=A\setminus (A_0\cup A_2)$. All the sets $A_2$, $A_0$ and $A_1$ are preserved by homomorphisms $f:A\to B$
(due to $s$, $S$ and $R$ resp.).
This category clearly has $\aleph_1$-directed colimits. Objects $A$ of $\cl$ generated by $0$ are ordinals $\omega_1$ with a choice of $S$ for every $a\in A_0$. Thus there is $\aleph_0^{\aleph_1}=2^{\aleph_1}$ such objects. These objects are
$\aleph_1$-presentable and the same is true for objects $\omega_1\cdot\alpha$ where $\alpha<\omega_1$. Clearly, every object
of $\cl$ is an $\aleph_1$-directed colimit of these objects $\omega_1\cdot\alpha$, $\alpha<\omega_1$. Thus $\cl$ is
$\aleph_1$-accessible.
Assume that there exists a morphism $f:A\to A$ in $\cl$ which is not the identity. Let $a$ be the least element in $A$ such that $f(a)\neq a$.
Since A is a well-ordered set and $f$ is injective, $a<f(a)$. Hence
$$
a<f(a)<f^2(a)<\dots <f^n(a)<\dots
$$
Let $b=\sup f^n(a)$. There are $b_1<b_2<\dots <b_n<\dots$ such that $S(b,b_1,\dots,b_n,\dots)$. Since $S(f(b),f(b_1),\dots,f(b_n),\dots)$,
we have $f(b)=\sup f(b_n)$. For each $n$ there is $k$ such that $b_n<f^k(a)$. Hence $f(b_n)<f^{k+1}(a)$ and thus $f(b)=b$. Therefore $f(b_n)=b_n$
for each $n$. Since $a<b_m$ for some $m$, $f^n(a)<b_m$ for each $n$. Hence $b\leq b_m$, which is a contradiction.
}
\end{exam}
\begin{rem}
{
\em
(1) Let $\cl_1$ be a full subcategory of $\cl$ where we choose $S$ for every $a\in A_0$ in every object generated by $0$. This category does not depend of the choices of $S$ and is also $\aleph_1$-accessible. In fact, it is $\Ind_{\aleph_1}(\cc_1)$ where
$\cc_1$ is the category of ordinals $\omega_1\cdot\alpha$, $\alpha<\omega_1$ with non-identity morphisms
$$
\omega_1\cdot f:\omega\cdot\alpha\to\omega_1\cdot\beta
$$
where $f:\alpha\to\beta$ is an order preserving injective mapping with $\alpha<\beta$. The category $\cc_1$ is, in fact,
the category of ordinals $\alpha<\omega_1$ where non-identity morphisms are order preserving injective mappings $\alpha\to\beta$
for $\alpha<\beta<\omega_1$.
(2) The category $\FinLin$ is the category $\cc_0$ of ordinals $\alpha<\omega$ where non-identity morphisms are order preserving
injective mapping $\alpha\to\beta$ for $\alpha<\beta<\omega$. Observe that $\FinLin$ is rigid, i.e., the only morphisms
$\alpha\to\alpha$ are the identities. Hence $\cl_1=\Ind_{\omega_1} \cc_1$ is $\aleph_1$-modification of a minimal $\aleph_0$-accessible category $\Lin$.
(3) Let $\cc_\gamma$ be the category of ordinals $\alpha<\omega_\gamma$ where non-identity morphisms are order preserving
injective mapping $\alpha\to\beta$ for $\alpha<\beta<\omega_\gamma$. Then $\cl_\gamma = \Ind_{\aleph_\gamma}\cc_\gamma$ is
an $\aleph_\gamma$-accessible category.
}
\end{rem}
\begin{pb}
{
\em
Is $\cl_1$ a minimal $\aleph_1$-accessible category? This means that for every large $\aleph_1$-accessible category $\ck$ there is a faithful functor $\cl_1\to\ck$ preserving $\aleph_1$-directed colimits.
Similarly, is $\cl_\gamma$ a minimal $\aleph_\gamma$-accessible category
for $0<\gamma$?
}
\end{pb}
\end{document} |
\begin{document}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}[section]
\theoremstyle{remark}
\newtheorem{rmk}{Remark}[section]
\newtheorem{exa}{Example}[section]
\def\square{
${\vcenter{\vbox{\hrule height.4pt \hbox{\vrule width.4pt
height7pt \kern7pt \vrule width.4pt} \hrule height.4pt}}}$}
\def\mathbb R{\mathbb R}
\def\mathbb Z{\mathbb Z}
\def\mathbb C{\mathbb C}
\def\mathbb H{\mathbb H}
\def\mathbb CP{\mathbb{CP}}
\newenvironment{pf}{\noindent {\it Proof:}\quad}{\square \vskip 12pt}
\title{Napoleon in isolation}
\author{Danny Calegari}
\address{Department of Mathematics \\ UC Berkeley \\ Berkeley, CA 94720}
\email{dannyc@math.berkeley.edu}
\maketitle
\begin{abstract}
Napoleon's theorem in elementary geometry describes how certain linear
operations on plane polygons of arbitrary shape always produce regular
polygons. More generally,
certain triangulations of a polygon that tiles
$\mathbb R^2$ admit deformations which keep fixed the symmetry group of
the tiling. This gives rise to
{\em isolation phenomena} in cusped hyperbolic $3$-manifolds, where
hyperbolic Dehn surgeries on some collection of cusps leaves the
geometric structure at some other collection of cusps unchanged.
\end{abstract}
\section{Geometric isolation}
\subsection{Definition}
Let $M$ be a complete, finite volume hyperbolic $3$-manifold with $n$
torus cusps, which we denote $c_1, \dots c_n$. The following definition is
found in \cite{wNaR93}:
\begin{defn}
A collection of cusps $c_{j_1},\dots c_{j_m}$ is {\em geometrically
isolated} from a collection $c_{i_1},\dots c_{i_n}$ if any hyperbolic
Dehn surgery on any collection of the $c_{i_k}$ leaves the geometric
structure on all the $c_{j_l}$ invariant.
\end{defn}
Note that this definition is {\em not} symmetric in the collections
$c_{i_k}$ and $c_{j_l}$, and in fact there are examples which show that
a symmetrized definition is strictly stronger (see \cite{wNaR93}).
More generally, we can ask for some prescribed set of fillings on the
$c_{i_k}$ which leave the $c_{j_l}$ invariant. Generalized (non-integral)
hyperbolic surgeries on a cusp are a holomorphic parameter for the
space of all (not necessarily complete) hyperbolic structures on $M$
with a particular kind of allowable singularities (i.e. generalized
cone structures) in a neighborhood of the complete structure. Moreover, the
complex dimension of the space of geometric shapes on a complete cusp
is $1$. Consequently, for dimension reasons whenever $n>m$ there will be
families of generalized surgeries leaving the geometric structures on the
$c_{j_l}$ invariant. There is no particular reason to expect, however, that
any of these points will correspond to an {\em integral} surgery on the
$c_{i_k}$. When there is a $1$ complex dimensional holomorphic family
of isolated generalized surgeries which contains infinitely many integral
surgeries, we say that we have an example of an {\em isolation phenomenon}.
Neumann and Reid describe other qualities of isolation in \cite{wNaR93}
including the following:
\begin{defn}
A collection of cusps $c_{j_1} \dots c_{j_m}$ is {\em strongly isolated}
from a collection $c_{i_1} \dots c_{i_m}$ if after any hyperbolic Dehn
surgeries on any collection of the $c_{j_l}$, a further surgery on any
collection of the $c_{i_k}$ leaves the geometry of the (possibly filled)
cusps $c_{j_l}$ invariant.
A collection of cusps $c_{j_1} \dots c_{j_m}$ is {\em first-order isolated}
from a collection $c_{i_1} \dots c_{i_m}$ if the derivative of the
deformation map from generalized surgeries on the $c_{i_k}$ to the
space of cusp shapes on the $c_{j_l}$ vanishes at the complete structure.
\end{defn}
By using the structure of the $\Phi$ function defined in \cite{wNdZ85},
Neumann and Reid show that strong isolation and first order isolation
{\em are} symmetric relations. First order isolation can be restated
in terms of group cohomology, and is studied in some papers of Kapovich,
notably \cite{mK92}.
In this paper we produce new constructions of isolation phenomena of
various qualities, both by extending or modifying known constructions,
and by introducing a conceptually original construction based on
Napoleon's theorem in plane geometry. The sheer wealth of examples
that these techniques can produce, used in combination, strongly suggests
that instances of isolation phenomena are not isolated phenomena.
\subsection{Holomorphic rigidity}
Suppose we want to show that a cusp $d$ is isolated from a cusp $c$. Then
since the shape of the complete cusp $d$ depends holomorphically on the
generalized surgery on $c$, it suffices to show that infinitely many
surgeries on $c$ keep the structure of $d$ fixed, since these can only
accumulate at the complete (unfilled) structure, where we know the
function relating the structure on $d$ to the surgery on $c$ is regular.
We describe this holomorphic structure in more detail. Choose a
meridian $m$ of a cusp $c$. Let $M_{(p,q)}$ be obtained by doing $(p,q)$
surgery on $c$. The hyperbolic structure on $M_{(p,q)}$ determines
a representation $\rho_{p,q}:\pi_1(M) \to PSL(2,\mathbb C)$.
The image of $[m]$ under this representation has a well--defined {\em complex
length} $u$ in $\mathbb C/2\pi i \mathbb Z$, which is the logarithm of the ratio of the
eigenvalues of $\rho_{p,q}(m)$. We may choose a branch of the logarithm
so that the value $0$ corresponds to the complete structure. Then small
deformations of the hyperbolic structure on $M$ corresponding to generalized
surgeries on $c$ are parameterized in a
$2:1$ way by $u$. The set of Euclidean structures on a complete cusp
$d$ are parameterized by the complex orbifold
$\mathcal{M}_1 = \mathbb H^2 / PSL(2,\mathbb Z)$, and
it is well--known that the map from $u \to \mathcal{M}_1$
is analytic, and regular at $0$. For more detail, see \cite{wNdZ85}.
\subsection{Rigid orbifolds}
A systematic study of isolation was initiated in \cite{wNaR93}.
Most of the examples constructed in \cite{wNaR93} of pairs of cusps $c_1,c_2$
which are geometrically isolated have the property that there is a totally
geodesic rigid triangle orbifold separating the two cusps. Such a separating
surface splits the manifold $M$ into two pieces $M_1,M_2$ where $c_i$ sits
as a cusp in $M_i$. Then a surgery on $c_i$ deforms only the piece $M_i$,
keeping the geometry of the splitting orbifold unchanged, and $M_i$ can then
be glued to $M_{i+1}$ to produce a complete hyperbolic structure on $M$.
One sees that the geometry of the entire piece containing the unfilled cusp
is unchanged by this operation, and therefore that each of the two cusps
is isolated from the other cusp.
If $M$ covers an orbifold $N$ containing a totally geodesic triangle orbifold
which separates cusps $c$ and $d$, then in $M$ any surgeries on lifts of
$c$ which descend to $N$ will leave unaffected the structures on lifts of $d$.
\subsection{Rigid cusps}
A refinement of the construction above comes when the rigid orbifold is
boundary parallel. The square torus $\mathbb R^2/\mathbb Z\oplus i\mathbb Z$ and the hexagonal
torus $\mathbb R^2/\mathbb Z\oplus \frac {1+\sqrt{3}i} 2 \mathbb Z$ have rotational symmetries
of order $4$ and $6$ respectively. If these symmetries about a cusp $c$
can be made to extend over the entire manifold $M$, then any surgery preserving
these symmetries will keep the cusp $c$ square or hexagonal respectively.
Note that isolation phenomena produced by this method tend to be one--way.
\begin{exa}
Let $M=T^2 \times I - K$ where $K$ is a knot which has $4$--fold rotational
symmetry, as seen from either of the $T^2$ ends of $T^2 \times I$. Then
$M$ has $4$--fold rotational symmetry which is preserved after $(p,q)$
filling on the knot $K$. This symmetry keeps the shape of the two
ends of $T^2 \times I$ square after every surgery on $K$. In general,
surgery on either of the ends of the $T^2$ will disrupt the symmetry and
not exhibit any isolation.
\end{exa}
\subsection{Mutation}
An incompressible surface in a hyperbolic manifold does not need to be
rigid for certain topological symmetries to be realized as geometric
symmetries. In a finite volume hyperbolic $3$--manifold, an
incompressible, $\partial$--incompressible
surface $S$ without accidental parabolics
which is not the fiber of a fibration over $S^1$ is quasifuchsian, and
therefore corresponds to a unique point in $Teich(S) \times Teich(\bar{S})$.
For surfaces $S$ of low genus, the tautological curve over the
Teichm\" uller space of $S$ has certain symmetries which restrict to a
symmetry of each fiber --- that is, of each Riemann surface topologically
equivalent to $S$. There is a corresponding symmetry of the universal
curve over $Teich(\bar{S})$ and therefore of a quasifuchsian representation
of $S$. Geometrically, this means that one can cut along a minimal surface
representing the class of $S$ and reglue it by an automorphism which
preserves the geometric structure to give a complete
nonsingular hyperbolic structure on a new manifold. Actually, one does
not need to know there is an equivariant minimal surface along which one
can cut --- one can perform the cutting and gluing at the level of limit
sets by using Maskit's combinations theorems (\cite{bM87}). This operation is
called {\em mutation} and is studied extensively by Ruberman, for instance
in \cite{dR87}.
If $S$ is a sphere with $4$ punctures, we can think of $S$ as a Riemann
sphere with $4$ points deleted. After a M\" obius transformation, we can
assume $3$ of these points are at $0,1,\infty$ and the $4$th is at the
complex number $z$. The full symmetry group $\mathsf{S}_4$ does not act
holomorphically on $S$ except in very special cases, but the subgroup
consisting of $(12)(34)$ and its conjugates {\em does}
act holomorphically. This group is known as the {\em Klein $4$--group}.
Geometrically, where we find an appropriate
$4$--punctured sphere in a hyperbolic manifold $M$, we can cut along the
sphere and reglue after permuting the punctures. This operation leaves
invariant the geometric structure on those pieces of the manifold that
do not meet the sphere. If $S$ is a spherical orbifold or cone manifold
with $4$ equivalent cone points, we can similarly cut and reglue.
The sphere along which the mutation is performed is called a
{\em Conway sphere}. The observation that mutation can be performed for
spherical orbifolds is standard: one can always find a finite manifold
cover of any hyperbolic orbifold, by Selberg's lemma. The spherical
orbifold lifts to an incompressible surface in this cover, and one can
cut and reglue equivariantly in the cover. In fact, local rigidity for
small cone angles developed by Hodgson and Kerckhoff in \cite{cHsK98} suggests
that this can be done just as easily for the cone manifold case, but this
is superfluous for our applications.
\begin{exa}
Let $M$ be a manifold with a single cusp $c$ which admits a $\mathbb Z/3\mathbb Z$ symmetry
that acts on $c$ as a rotation. This symmetry forces $c$ to be hexagonal.
Let $p$ be a point in $M$ not fixed by $\mathbb Z/3\mathbb Z$, and let $M'$ be obtained
by equivariantly removing $3$ balls centered at the orbit of $p$. Glue another
manifold with $\mathbb Z/3\mathbb Z$ symmetry to $M'$ along its spherical boundaries
$S_1,S_2,S_3$ so that the symmetries on either side are compatible,
to make $M''$. Let $K$ be a $\mathbb Z/3\mathbb Z$-invariant knot in $M''$ which intersects
each of the spheres $S_i$ in $4$ points and which is sufficiently complicated
that its complement in $M''$ is atoroidal and the $4$--punctured spheres
$S_i$ are incompressible and $\partial$--incompressible. Perform a
mutation on $S_1$ which destroys the $\mathbb Z/3\mathbb Z$--symmetry to get a manifold
$N$ with two cusps which we refer to as $c$ and $d$, where $d$ corresponds
to the core of $K$. Then $c$ is hexagonal, since mutation does not
affect the geometric structure away from the splitting surface. Moreover,
for large integers $r$,
any generalized $(r,0)$ surgery on $d$ will preserve the
fact that the $S_i$ are incompressible, $\partial$--incompressible
spheres with $4$ cone points of order $r$, and therefore we can undo the
mutation on $S_1$ to see that these surgeries do not affect the hexagonal
structure on $c$. But if a real $1$--parameter family of surgeries on
$d$ do not affect the structure on $c$, then {\em every} surgery on $d$
keeps the structure on $c$ fixed, so $c$ is isolated from $d$.
Note that for general $(p,q)$ surgeries
on $d$, the spheres $S_i$ will be destroyed and the surgered manifold
will not be mutation--equivalent to a $\mathbb Z/3\mathbb Z$ symmetric manifold. Moreover,
for sufficiently generic $K$, there will be no rigid triangle orbifolds
in $N$ separating $c$ from $d$. As far as we know, this is the first
example of geometric isolation to be constructed that is not forced by
a rigid separating or boundary parallel surface.
\end{exa}
We note in passing that mutation followed by surgery often preserves other
analytic invariants of hyperbolic cusped manifolds, such as volume
and (up to a constant), Chern--Simons invariants. Ruberman observed
in \cite{dR87} that if $K$ and a Conway sphere $S$ are {\em unlinked},
then a mutation which preserves this unlinking can actually be
achieved by mutation along the genus $2$ surface obtained by tubing
together the sphere $S$ with a tubular neighborhood of $K$. This
mutation corresponds to the hyperelliptic involution of a genus
$2$ surface. Since this surface is present and incompressible for
all but finitely many surgeries on $K$ and its mutant, the surgered
mutants are mutants themselves. If $S$ and $K$ are not unlinked, no
such hyperelliptic surface can be found, and in fact in this case
there is no clear relationship between invariants of the manifold obtained
after surgeries on the original
knot and on the mutant. Perhaps this makes the persistence of isolation
under mutation more interesting, since it shows that some of the effects of
mutation are persistent under surgeries where other effects are destroyed.
\section{Napoleon's theorem}
\subsection{Triangulations and hyperbolic surgery}
A finite volume complete but not compact hyperbolic $3$--manifold $M$
can be decomposed into a finite union of (possibly degenerate)
ideal tetrahedra glued together along their faces. An ideal tetrahedron
determines and is determined by its fourtuple of endpoints on the
sphere at infinity of $\mathbb H^3$. Identifying this sphere with $\mathbb CP^1$,
we can think of the ideal tetrahedron as a $4$--tuple of complex
numbers. The isometry type of the tetrahedron is determined by the
{\em cross--ratio} of these $4$ points; equivalently, if we move three
of the points to $0,1,\infty$ by a hyperbolic isometry, the isometry type
of the tetrahedron is determined by the location of the $4$th point,
that is by a value in $\mathbb C -\lbrace 0,1 \rbrace$. This value is referred
to as the {\em simplex parameter} of the ideal tetrahedron. A combinatorial
complex $\Sigma$
of simplices glued together can be realized geometrically
as an ideal triangulation of a finite volume hyperbolic manifold (after
removing the vertices of $\Sigma$) if
certain equations in the simplex parameters are satisfied. These equations
can be given explicitly by examining the links of vertices of the
triangulation.
For a finite volume non--compact hyperbolic manifold, all the links of
vertices of $\Sigma$ should be tori $T_j$. There are induced triangulations
$\tau_j$ of these tori by the small triangles obtained by cutting off the tips
of the tetrahedra in $\Sigma$. Let $\Sigma = \bigcup_i \Delta_i$ and let
$z_i$ be a hypothetical assignment of simplex parameters to the tetrahedra
$\Delta_i$. Then the horoball sections of a hypothetical hyperbolic
structure on $\Sigma' = \Sigma - \text{vertices}$ are Euclidean tori
triangulated by Euclidean triangles, which are the horoball sections of
the $\Delta_i$. If the simplex parameter of an ideal tetrahedron is
$z$, the Euclidean triangles obtained as sections of the horoballs centered
at its vertices have the similarity type of the triangle in $\mathbb C$ with
vertices at $0,1,z$. A path in the dual $1$--skeleton to $\tau_j$ determines
a developing map of $\tau_j$ in $\mathbb C$: given a choice of the initial triangle,
there is a unique choice of each subsequent triangle within its fixed
Euclidean similarity type such that the combinatorial edge it shares
with its predecessor is made to geometrically agree with it.
The {\em holonomy} of a closed path in this dual skeleton is the Euclidean
similarity taking the initial triangle to the final triangle.
There are two necessary conditions to be met in order for
the hyperbolic structures on the $\Delta_i$ to glue up to give a hyperbolic
structure on $\Sigma'$. These conditions are actually sufficient in a
small neighborhood of the complete structure as described in \cite{wNdZ85}
and made rigorous in \cite{cPjP00}.
\begin{itemize}
\item{The {\em edge equations}: the holonomy around a vertex of
$\tau_j$ should be trivial.}
\item{The {\em cusp equations}: the holonomy around the meridian and
longitudes of the $T_j$ should be {\em translations}.}
\end{itemize}
These ``equations'' can be restated as identities of the form
$$\sum_i c_{ij} \ln(z_i) + d_{ij} \ln(1-z_i) = \pi i e_j$$
for some collection of integers $c_{ij},d_{ij},e_j$
and some appropriate choices
of branches of the logarithms of the $z_i$.
A $(p,q)$ hyperbolic dehn surgery translates in this context to replacing the
cusp equations for some cusp by the condition that the holonomy around the
meridian and longitude give Euclidean similarities $h_m,h_l$ such that
$h_m^ph_l^q = \text{id}$. The analytic co--ordinates on Dehn surgery space
determined by the analytic parameters $z_i$ are holomorphically related to
the $u$ co--ordinates alluded to earlier. The geometry of a complete cusp
is determined by the translations corresponding to the holonomy of the
meridian and longitude.
This is all described in great detail in \cite{wNdZ85} and \cite{cPjP00}.
\subsection{Tessellations with forced symmetry}
In this section, we produce examples of isolation phenomena which do
not come from rigidity or mutation, but rather from the following theorem,
known as ``Napoleon's Theorem'' (\cite{hCsG67}):
\begin{thm}[``Napoleon's Theorem'']
Let $T$ be a triangle in $\mathbb R^2$. Let $E_1,E_2,E_3$ be three equilateral
triangles constructed on the sides of $T$. Then the centers of the $E_i$
form an equilateral triangle.
\end{thm}
\begin{pf}
Let the vertices of $T$ be $0,1,z \in \mathbb C$. The three centers of the $E_i$
are of the form $a_iz + b_i$ for certain complex numbers $a_i,b_i$
so the shape of the resulting triangle is
a holomorphic function of $z$. Let $\omega = (1+\frac i {\sqrt{3}})/2$.
If $z$ is real and between $0$ and $1$ then
the center of $E_1$ splits the line between $0$ and $\omega$ in the
ratio $z,1-z$, the center of $E_2$ splits the line between $\omega$ and $1$
in the ratio $z,1-z$, and the center of $E_3$ is at $1-\omega$. But a
clockwise rotation through $\pi/3$ about $1-\omega$ takes the line between
$0$ and $\omega$ to the line between $\omega$ and $1$; i.e. it takes the
center of $E_1$ to the center of $E_2$. Thus the theorem is true for real
$z$ and by holomorphicity, it is true for all $z$.
\end{pf}
\begin{figure}
\caption{Three isometric triangles and three equilateral triangles make up
a hexagon which tiles the plane with symmetry group $\mathsf{S}
\end{figure}
Napoleon's theorem gives rise to an interesting phenomenon in plane
geometry: fix a triangle $T$. Then three triangles isometric to $T$
and three equilateral triangles with side length equal to the sides of
$T$ can be glued together to make a hexagon which tiles the plane with
symmetry group $\mathsf{S}(3,3,3)$ as in figure 1. The edge lengths of
the three equilateral triangles are equal to the three edge lengths of
$T$, and around each vertex the angles are the three interior angles of
$T$ together with three angles equal to $\frac {2\pi} 6$. It follows that
the hexagon in question exists and tiles the plane; to see that it has
the purported symmetry group, observe that the combinatorics of the
triangulation have a $3$--fold symmetry about the centers of the three
equilateral triangles bounding some fixed triangle of type $T$. These
three centers are the vertices of an equilateral triangle, by Napoleon's
theorem; it follows that the symmetry group of the tiling contains the
group generated by three rotations of order $3$ with centers at the
vertices of an equilateral triangle --- that is to say, the symmetry
group $\mathsf{S}(3,3,3)$.
If we imagine that these triangles
are the asymptotic horoball sections of ideal tetrahedra going out a
cusp of a hyperbolic manifold, we see that appropriate deformations of
the tetrahedral parameters change the triangulation {\em but not the
geometry} of the cusp. For $T$ a horoball section of an ideal tetrahedron
with simplex parameter $z$, the holonomy around vertices is just
$$z \cdot \omega \cdot \frac {z-1} z \cdot \omega \cdot \frac 1 {1-z} \cdot
\omega = 1$$
where $\omega = \frac {1 + i\sqrt{3}} 2$ is the similarity type of an
equilateral triangle, and the holonomy around the meridian and longitude
are both translations by complex numbers $z_1,z_2$ whose ratio is
$\frac {1 + i\sqrt{3}} 2$.
\begin{figure}
\caption{Two cusps with triangle parameters related as indicated in the
diagram can be deformed independently through (incomplete) affine structures.
We follow the conventions of \cite{wNdZ85}
\end{figure}
\begin{exa}
A scheme to parlay this theorem into isolation phenomena is given by
the following setup: we have two hexagons $H_1,H_2$ each tiled by six
equilateral triangles. We divide these twelve triangles into two
similarity types, corresponding to the triangles in $\mathbb C$ with vertices at
$\lbrace 0,1,z \rbrace$ and $\lbrace 0,1,w \rbrace$,
and we choose a preferred vertex for each triangle corresponding to $0$,
in the manner indicated in figure 2. For an arbitrary
choice of complex numbers $z$ and $w$,
the hexagons can be realized geometrically to give
affine structures on the tori obtained by gluing opposite sides of $H_1$
together and similarly for $H_2$. Initially set
$z = w = \frac {1 + i\sqrt{3}} 2$. Deforming $z$ but keeping $w$ fixed
changes the affine structure on the first torus but leaves the structure
on the second torus unchanged. For, the combinatorial triangulation of
the second torus is exactly the triangulation of a fundamental domain of
the tessellation in figure 1. It follows that for $w = \frac {1 + i\sqrt{3}} 2$
and $z$ arbitrary, the universal cover of the second torus is tiled
by a tessellation with symmetry group $\mathsf{S}(3,3,3)$ and the torus is
therefore hexagonal.
Similarly, deforming $w$ and keeping $z = \frac {1 + i\sqrt{3}} 2$
changes the affine structure on the second torus but leaves the
structure on the first torus unchanged, since now we can identify the
combinatorial triangulation of the {\em first} torus with the triangulation
of a fundamental domain of the tessellation in figure 1.
This configuration of ideal tetrahedra with horoball sections equal to
the two cusps in this figure can be realized geometrically
by arranging six regular ideal tetrahedra in the upper--half space in a
hexagonal pattern with the common edge of the tetrahedra going from $0$ to
$\infty$. The pattern seen from infinity looking down is $H_1$, and the
pattern seen from $0$ looking up is $H_2$. The pictures are aligned so
that the real line has its usual orientation. Glue the twelve free faces of
the tetrahedra in such a way as to make $H_1$ and $H_2$ torus cusps. This
gives an orbifold $N$ whose underlying manifold is $T^2 \times I$,
with orbifold locus three arcs of cone angle
$2\pi/3$ each running between two $(3,3,3)$ triangle cusps arranged in the
obvious symmetrical manner.
Unfortunately, under surgeries of the cusps of $N$, the simplices do not
deform in the manner required by Napoleon's theorem. However, if we pass
to a $3$-fold cover, this problem can be corrected.
\begin{figure}
\caption{``Napoleon's $3$-manifold'' has a decomposition into ideal
tetrahedra such that surgery on a ``dark'' cusp deforms the triangulation
of the other ``dark'' cusps but keeps the shape of the cusp constant, as
guaranteed by Napoleon's theorem. A similar relation holds for the ``light''
cusps.}
\end{figure}
Let $L$ be the link depicted in figure $3$, and let $M = S^3 - L$. Then
$M$ admits a complete hyperbolic structure which can be decomposed into
$18$ regular ideal tetrahedra. It follows that $M$ is commensurable with
the figure $8$ knot complement. In fact, $M$ is the $3$-fold cover of $N$
promised above.
Geometrically, arrange $18$ regular ideal tetrahedra
in the upper half-space with a common vertex at infinity so that a
horoball section intersects the collection the pattern depicted in figure
$4$. If we glue the $12$ external vertical sides of this collection of
tetrahedra in the indicated manner, it gives a horoball section of one of the
cusps of $M$. It remains to glue up the $18$ faces of the ideal tetrahedra
with all vertices on $\mathbb C$. Figure $4$ has an obvious decomposition into
$3$ regular hexagons, each composed of $6$ equilateral triangles. The
$3$ edges of the complex associated to the centers of these three hexagons
have a common endpoint at $\infty$, and intersect $\mathbb C$ at three points
$p_1,p_2,p_3$. Six triangles come together at each of the points $p_i$, and
a horoball centered at each of these points intersects $6$ tetrahedra in a
hexagonal pattern. Glue opposite faces of these hexagons to produce $3$
cusps centered at $p_1,p_2,p_3$. The result of all this gluing produces
the manifold $M$. By construction, this particular choice of triangulation
produces $6$ hexagonal cusps, $3$ made up of $6$ equilateral triangles and
$3$ made up of $18$ equilateral triangles.
\begin{figure}
\caption{A horoball section of the cusp $d_1$ of Napoleon's $3$--manifold.
When Dehn surgery is performed on $c_1$, the triangles making up $d_1$
deform in the manner indicated. The triangles marked with a circle stay
equilateral under the deformation.}
\end{figure}
The components of $L$ fall into two
sets of $3$ links, depicted in the figure as the darker and the lighter
links, which we denote by $c_1,c_2,c_3$ and $d_1,d_2,d_3$
respectively. The group of symmetries of $M$ permutes the cusps by the
group $(\mathsf{S}_3 \times \mathsf{S}_3) \rtimes \mathbb Z/2\mathbb Z$ where the
conjugation action of the generator of $\mathbb Z/2\mathbb Z$ takes $(\sigma_i,\sigma_j)$
to $(\sigma_j,\sigma_i)$. If $G$ is the entire group of symmetries of
$M$, then there is a short exact sequence
$$ 0 \longrightarrow \mathbb Z/3\mathbb Z \longrightarrow G \longrightarrow
(\mathsf{S}_3 \times \mathsf{S}_3) \rtimes \mathbb Z/2\mathbb Z \longrightarrow 0 $$
\vskip 12pt
so the order of $G$ is $216$.
These links have the property that a surgery on $c_i$ keeps
$c_{i+1}$ and $c_{i+2}$ hexagonal, but distorts the structure at the
$d_j$, and vice versa. On the other hand, a surgery on {\em both} $c_i$ and
$c_{i+1}$ {\em does} distort the structure at $c_{i+2}$. In terms of the
picture already described, the configuration of $18$ triangles in
figure 4 decomposes into 3 hexagons of 6 triangles. These three hexagons
glue up in the obvious way to give hexagonal triangulations of the $c_i$.
Under surgery on $c_i$, the tetrahedra intersecting $c_i$ deform to
satisfy the new modified cusp equations. Under this deformation, the other
triangle types must deform to keep the other cusps complete. It can be
easily checked that the triangulations of the cusps $c_{i+1},c_{i+2}$ lift
to the symmetric tessellations of $\mathbb R^2$ depicted in figure 1, and therefore
the similarity types of these cusps stay hexagonal. However, under
surgery on {\em both}
$c_i$ and $c_{i+1}$, the similarity types of triangles making
up cusp $c_{i+2}$ are not related in any immediately apparent way to
the picture in figure 1. The proof that $c_2$ and $c_3$ are
isolated from $c_1$
is essentially just a calculation that under surgery on $c_1$ say,
the simplex parameters solving the relevant edge and cusp equations
for the combinatorial triangulation vary as indicated in figure 4.
One may check experimentally that the similarity
type of $c_3$ is not constant under fillings on both
$c_1$ and $c_2$, using for example, Jeff Weeks' program
{\tt snappea}, available from \cite{jWsn}, for finding hyperbolic
structures on $3$--manifolds, or Andrew Casson's provably accurate
program {\tt cusp}
(\cite{aCcu}).
\begin{defn}
Say that $3$ cusps are in {\em Brunnian isolation} when a surgery on one
of them leaves invariant the structure at the other $2$, but a surgery on
two of the cusps can change the structure of the third.
\end{defn}
With this definition, we observe that Napoleon's $3$--manifold has two sets
of cusps in Brunnian isolation.
One can see that there is an automorphism of $S^3$ of order $2$ fixing
two components of $L$ and permuting the other $4$ components in pairs.
The quotient by this automorphism is an orbifold $N$ which
has two regular cusps and two pillow cusps. We call the pillow cusps
$c_p, d_p$, and the regular cusps $c_r,d_r$ where $c_p$ is a quotient
of $c_1$, $c_r$ is covered by $c_2 \cup c_3$, and similarly for the $d_i$.
The cusp $c_p$ is first-order isolated but not isolated from $c_r$.
Similarly, $d_p$ is first-order isolated but not isolated from $d_r$.
This can be easily observed by noting that $c_r$ {\em is} isolated from
$c_p$, and therefore it is first-order isolated from it (by the
corresponding properties in the cover). It follows
that $c_p$ is first-order isolated from $c_r$, by \cite{wNaR93}. To see
that $c_p$ is {\em not} isolated from $c_r$, it suffices to pass to the
cover and perform an equivariant surgery there. Alternatively, one can
easily check by hand using {\tt snappea} that the geometry of the
cusp $c_p$ changes when one performs surgery on $c_r$.
\end{exa}
\begin{exa}
The $2$--cusped orbifold $A$ first described in \cite{wNaR93} and studied in
\cite{dC96} displays Napoleonic tendencies, where the version of
Napoleon's theorem we use now concerns right triangles. It is obtained by
$(2,0)$ surgery on the light cusps of the link complement portrayed in
figure 5. Coincidentally, it is $2$--fold covered by that very link
complement. Let $T$ be the right triangle with side lengths
$\lbrace 1,1,\sqrt{2} \rbrace$, and let $S$ be the unit square in $\mathbb C$.
Pick a point $p \in S$ and construct four triangles $T_i$ all similar to
$T$ with one vertex of the diagonal at $p$ and the other vertex of the
diagonal at a vertex of $S$, such that the triangle is clockwise of the
diagonal, seen from $p$. Then the $8$ vertices of these triangles away
from $p$ are the vertices of an octagon which tiles the plane with
quotient space the square torus. This corresponds exactly to the
triangulation of a horoball section of a complete cusp in the orbifold $A$
after a deformation of the other cusp. One observes that the link
complement in figure 3 is obtained from the link complement in figure 5
by drilling out two curves. Are the isolation phenomena associated with
the two links related?
\end{exa}
\begin{figure}
\caption{A link in $S^3$ whose complement displays Napoleonic tendencies}
\end{figure}
\end{document} |
\begin{document}
\title{\Large Many-to-many Correspondences between Partitions:\\Introducing a Cut-based Approach\thanks{This work is partially supported by DFG grant FINCA (ME-3619/3-1) within the SPP 1736 Algorithms for Big Data.}
\begin{abstract}
\small
Let $\mathcal{P}$ and $\mathcal{P}'$ be finite partitions of the
set $V$. Finding good correspondences between the parts of $\mathcal{P}$
and those of $\mathcal{P}'$ is helpful in classification,
pattern recognition, and network analysis.
Unlike common similarity measures for partitions that yield only a single value,
we provide specifics on how $\mathcal{P}$ and $\mathcal{P'}$
correspond to each other.
To this end, we first define natural collections of best
correspondences under three constraints \textsc{$C_{\vee}$}\xspace, \textsc{$C_{\mathcal{P}}$}\xspace, and \textsc{$C_{\wedge}$}\xspace.
In case of \textsc{$C_{\vee}$}\xspace, the best correspondences form a
minimum cut basis of a certain bipartite graph, whereas the other two
lead to minimum cut bases of $\mathcal{P}$
w.\,r.\,t.\xspace $\mathcal{P}'$.
We also introduce a constraint, \textsc{$C_{m}$}\xspace, which tightens \textsc{$C_{\wedge}$}\xspace;
both are useful for finding consensus partitions.
We then develop branch-and-bound algorithms for finding minimum $P_s$-$P_t$
cuts of $\mathcal{P}$ and thus $\vert \mathcal{P} \vert -1$ best
correspondences under \textsc{$C_{\mathcal{P}}$}\xspace, \textsc{$C_{\wedge}$}\xspace, and \textsc{$C_{m}$}\xspace, respectively.
In a case study, we use the correspondences to gain insight into a
community detection algorithm. The results suggest, among others, that only very minor losses in the
quality of the correspondences occur if the branch-and-bound
algorithm is restricted to its greedy core. Thus, even for graphs with
more than half a million nodes and hundreds of communities, we can
find hundreds of best or almost best correspondences in less than a minute.\\[0.25ex]
\textbf{Keywords:} Many-to-many correspondences, similarities of partitions,
minimum cut basis, (graph) clustering
\end{abstract}
\section{Introduction}
\label{sec:intro}
Objective and quantitative methods to help humans with the task of grouping
objects in a meaningful way are the subject of cluster
analysis~\cite{Everitt2011a}.
We con\-si\-der the case in which
the parts are non-overlapping and form a partition of data points into
parts/clusters/groups/regions/communities.
Even small changes in the data can provoke a clustering algorithm to
split or merge clusters and thus produce different local levels of
detail -- as an example, imagine a (dynamic) clustering algorithm working on data
changing over time. Comparisons of partitions resulting in a single number
expressing total (dis)similarity do not provide specifics
on how the clusters have split or merged, and a comparison restricted
to one-to-one correspondences may be in\-sufficient.
This paper is about a new approach for comparing two partitions
$\mathcal{P} = \{P_1, \dots, P_{\vert \mathcal{P} \vert}\}$ and
$\mathcal{P}' = \{P'_1, \dots, P'_{\vert \mathcal{P}' \vert}\}$ of the
same set.\footnote{If $\mathcal{P}$ and $\mathcal{P}'$ are partitions of
sets that are different but have a large intersection, $W$, one can
turn $\mathcal{P}$ and $\mathcal{P}'$ into the two related
partitions $\{P_1 \cap W, \dots, P_{\vert \mathcal{P} \vert} \cap
W\}$ and $\{P'_1 \cap W, \dots, P'_{\vert \mathcal{P}' \vert} \cap
W\}$ of (the same set) $W$. If $W$ is large enough, the
correspondences between the two new partitions will still reveal
specifics on similarities between $\mathcal{P}$ and $\mathcal{P}'$.}
In Section~\ref{sec:app:soa} we describe some related work and specify properties that
our approach shares with standard similarity measures for
partitions~\cite{Meila2007a,Wagner2007a}.
The crucial difference is that we provide specifics on how $\mathcal{P}$ and
$\mathcal{P'}$ \emph{correspond} to each other, as opposed to just a single
number.
More specifically, a good many-to-many correspondence, short:
\emph{correspondence}, is a pair $(\mathcal{S}, \mathcal{S}')$ with
$\mathcal{S} \subseteq \mathcal{P}$, $\mathcal{S}' \subseteq
\mathcal{P}'$ and a low value of
\begin{equation}
\label{eq:firstMin1}
\phi(\mathcal{S}, \mathcal{S}') := \vert U_{\mathcal{S}} \triangle
U_{\mathcal{S}'} \vert,
\end{equation}
\noindent where $U_{\mathcal{S}}$ denotes the union of all sets in
$\mathcal{S}$, $\triangle$ denotes the symmetric difference, and
$\vert \cdot \vert$ denotes cardinality [total weight] if the elements
of $V$ are unweighted [weighted]. Thus, minimizing $\phi(\cdot,
\cdot)$ means finding similarities between $\mathcal{P}$ and
$\mathcal{P}'$ modulo unions of parts.
\begin{figure}
\caption{Top: Two segmentations (by hand) of the same image. Poor
match between individual regions left and right. Bottom: A good
correspondence $(\mathcal{S}
\label{fig:sheep}
\end{figure}
Among others, correspondences between partitions may be used to describe changes of
ground truth, discrepancies between a model and ground truth, or to
compare different solutions from (variations of) a (possibly
non-deterministic) algorithm.
To illustrate correspondences further, we turn to applications in which
$\mathcal{P}$ and $\mathcal{P}'$ are segmentations, i.\,e.,\xspace partitions of
a set of pixels/voxels into regions.
We assume for simplicity that $\mathcal{P}$ and $\mathcal{P}'$ are based on the same
image. Ideally, a region
corresponds to a real-world object; see Figure~\ref{fig:sheep} for an example
of correspondences between different segmentations.
Finding such regions is hindered by noise, under-segmentation, over-segmentation or
occlusion. More scenarios motivating a comparison of $\mathcal{P}$
and $\mathcal{P}'$ using correspondences
are described in Appendix~\ref{subsec:app:image analysis}.
vs.\ pace{0.5ex}
\paragraph*{Contributions and outline.}
\label{subsec:contrib}
In Section~\ref{sec:problem-statement}, we define the problem and investigate
the connection between correspondences and cuts. We go on by introducing four constraints
(\textsc{$C_{\vee}$}\xspace, \textsc{$C_{\mathcal{P}}$}\xspace, \textsc{$C_{\wedge}$}\xspace, and \textsc{$C_{m}$}\xspace) on correspondences, ordered from weak to strong.
Our main objective is to develop methods for finding good correspondences between two partitions of the
same set w.\,r.\,t.\xspace all four constraints.
(Due to space constraints, we focus on \textsc{$C_{\mathcal{P}}$}\xspace-correspondences.)
This includes (i) an analytic objective function for finding optimal non-trivial
\textsc{$C_{\mathcal{P}}$}\xspace-correspondences and a characterization of the problem in terms
of symmetric submodular minimization (see
Section~\ref{subsec:C2_submod}), (ii) a description of a natural
collection of $\vert \mathcal{P} \vert - 1$ good \textsc{$C_{\mathcal{P}}$}\xspace-correspondences
(see Section~\ref{subsec:minST}) and (iii) asymptotic time
complexities for finding natural collections of \textsc{$C_{\mathcal{P}}$}\xspace-correspondences
(see Section~\ref{subsec:goodC2_running_times}).
To \emph{compute} good correspondences between two partitions
$\mathcal{P}$ and $\mathcal{P}'$ in practice, we develop branch-and-bound
algorithms for finding minimum $P_s$-$P_t$ cuts of $\mathcal{P}$
under the constraints \textsc{$C_{\mathcal{P}}$}\xspace, \textsc{$C_{\wedge}$}\xspace and \textsc{$C_{m}$}\xspace, respectively, see
Section~\ref{sec:BB}. The algorithms are built around a greedy
algorithm each, and the restriction to these greedy cores provides
an alternative for calculating not always optimal but
typically good \textsc{$C_{\mathcal{P}}$}\xspace-, \textsc{$C_{\wedge}$}\xspace- and \textsc{$C_{m}$}\xspace-correspondences quickly.
In Section~\ref{sec:real-sim} we use one of many possible
applications to evaluate the correspondence concept and our algorithms
for computing them.
We investigate the effect that (i) a refinement option and
(ii) non-determinism has on the output of a community
detection (= graph clustering) algorithm. It turns out that these two effects can indeed
be characterized in terms of correspondences:
refinement does not change the general cluster assignment
significantly, whereas non-determinism leads to more
drastic changes. Also, from an algorithmic point of view, only
minor losses in the solution quality
are observed if the branch-and-bound algorithm is restricted to its greedy
core. Thus, even for graphs with millions of edges
and hundreds of communities, we can find hundreds of best or
almost best correspondences in less than a minute.
\section{Correspondences, cuts, and constraints}
\label{sec:problem-statement}
This section lays the notational ground for
computing good meaningful correspondences.
\subsection{Correspondences, cuts and optimal partners.}
\label{subsec:cuts}
The element $\mathcal{S}$ of a correspondence $(\mathcal{S}, \mathcal{S}')$
with $\mathcal{S} \notin \{\emptyset, \mathcal{P}\}$ gives rise to a cut $(\mathcal{S},
\mathcal{P} \setminus \mathcal{S})$ of $\mathcal{P}$. We measure the
size (weight) of such a cut by
\begin{equation}
\label{eq:phi}
\phi_{\mathcal{P}'}(\mathcal{S}) := \min_{\mathcal{S}' \subseteq
\mathcal{P}'}\phi(\mathcal{S}, \mathcal{S}').
\end{equation}
\noindent Given $\mathcal{S} \subseteq \mathcal{P}$, \emph{one} way
to minimize $\phi(\mathcal{S}, \mathcal{S}')$ is to let $\mathcal{S}'$ be
\begin{equation}
\label{eq:partner}
\mathcal{S_{\downarrow}}' := \{P' \in \mathcal{P}' : \vert U_{\mathcal{S}}
\cap P' \vert > \frac{\vert P'\vert}{2}\}
\end{equation}
\noindent We call \emph{any} $\mathcal{S}' \in \mathcal{P}'$ with
$\phi(\mathcal{S}, \mathcal{S}') = \phi(\mathcal{S},
\mathcal{S_{\downarrow}}')$ an \emph{optimal partner} of
$\mathcal{S}$. In contrast to $S_{\downarrow}'$, an optimal
partner of $\mathcal{S}$ may contain $P'$ with $\vert
U_{\mathcal{S}} \cap P' \vert = \vert P'\vert / 2$. A small cut
$(\mathcal{S}, \mathcal{P} \setminus \mathcal{S})$ gives rise to a
good correspondence $(\mathcal{S}, \mathcal{S}')$, where
$\mathcal{S}'$ is an optimal partner of $\mathcal{S}$. In this paper,
we frequently switch between correspondences and cuts.
\subsection{Examples of cuts and correspondences.}
\label{subsec:app:examples-cuts-corres}
\begin{figure}
\caption{Upper [lower] row of six [five] disks depicts partition
$\mathcal{P}
\label{fig:example}
\end{figure}
Figure~\ref{fig:example} depicts partitions $\mathcal{P}$,
$\mathcal{P}'$ of a set $V$ with 31 elements. The elements of $V$ are
represented by symbols indicating membership to the parts of
$\mathcal{P}$. The two subsets $\mathcal{S}$ of $\mathcal{P}$ giving
rise to the smallest cuts $(\mathcal{S}, \mathcal{P} \setminus
\mathcal{S})$ (size is 2) are the sets $\{P_1, P_2\}$ and $\{P_5,
P_6\}$. The optimal partners of these subsets are the subsets
$\{P'_1\}$ and $\{P'_4, P'_5\}$ of $\mathcal{P}'$, giving rise to the
correspondences $(\{P_1, P_2\}, \{P'_1\})$ and $(\{P_5, P_6\}, \{P'_4,
P'_5\})$, respectively. The optimal partner of $\{P_4\}$ is
$\{P'_3\}$.
If we reverse the roles of $\mathcal{P}$ and $\mathcal{P}'$ (see
Figure~\ref{fig:revExample} in Appendix~\ref{sub:exmp-cuts-corres}), the counterparts of the best
correspondences from before are the new best correspondences, e.\,g.,\xspace
$(\{P'_1\}, \{P_1, P_2\})$, and $(\{P'_4, P'_5\}, \{P_5, P_6\})$. The
counterpart of $(\{P_4\},\{P'_3\})$, however, is
gone. Indeed, the optimal partner of $\mathcal{S}' = \{P'_3\}$ is not
$\{P_4\}$ but $\emptyset$.
Thus, one must be aware that
swapping the roles of $\mathcal{P}$ and $\mathcal{P}'$ cannot always
be compensated by swapping $\mathcal{S}$ and $\mathcal{S}'$ in a
correspondence.
\subsection{Constraints on correspondences.}
\label{subsec:constraints}
We now define four constraints on correspondences
$(\mathcal{S}, \mathcal{S}')$, ordered from weak to strong, and
suggest cases in which they can be used.
Our first and weakest constraint, called \textsc{$C_{\vee}$}\xspace, just excludes correspondences that are trivial or
very bad:
\textsc{$C_{\vee}$}\xspace: $\mathcal{S} \notin \{\emptyset, \mathcal{P}\} \vee
\mathcal{S}' \notin \{\emptyset, \mathcal{P}'\}$. For more on
\textsc{$C_{\vee}$}\xspace-correspondences see Appendix~\ref{sec:app:goodC1}.
A more specific constraint that makes sense is \textsc{$C_{\mathcal{P}}$}\xspace: $\mathcal{S}
\notin \{\emptyset, \mathcal{P}\}$, i.\,e.,\xspace that $(\mathcal{S},
\mathcal{P} \setminus \mathcal{S})$ is a cut of
$\mathcal{P}$. \textsc{$C_{\mathcal{P}}$}\xspace-correspondences are useful if one wants to
understand the formation of $\mathcal{P}$ in terms of
$\mathcal{P}'$. Exchanging the roles of $\mathcal{P}$ and
$\mathcal{P}'$ yields an analogous asymmetric constraint.
If one wants a correspondence to cut $\mathcal{P}$ \emph{and}
$\mathcal{P}'$, one can require \textsc{$C_{\wedge}$}\xspace: $\mathcal{S} \notin
\{\emptyset, \mathcal{P}\} \wedge \mathcal{S}' \notin \{\emptyset,
\mathcal{P}'\}$. In particular, a good \textsc{$C_{\wedge}$}\xspace-correspondence gives
rise to two similar cuts $(U_{\mathcal{S}}, U_{\mathcal{P} \setminus
\mathcal{S}})$ and $(U_{\mathcal{S}'}, U_{\mathcal{P}' \setminus
\mathcal{S}'})$ of $V$. For such a pair of similar cuts one can find
a cut $(U_{\mathcal{S}^*}, U_{\mathcal{P} \setminus \mathcal{S}^*})$
that mediates between $(U_{\mathcal{S}}, U_{\mathcal{P} \setminus
\mathcal{S}})$ and $(U_{\mathcal{S}'}, U_{\mathcal{P}' \setminus
\mathcal{S}'})$ in that $\max\{\vert \mathcal{S}^* \triangle
\mathcal{S} \vert, \vert \mathcal{S}^* \triangle \mathcal{S}' \vert\}$
is minimum. The overlay of $k$ such medial cuts then results in a
consensus partition $\mathcal{P}_c$ between $\mathcal{P}$ and
$\mathcal{P}'$ with $k - 1 \leq \vert \mathcal{P}_c \vert \leq 2^k$
(the medial cuts may or may not cross).
\textsc{$C_{\mathcal{P}}$}\xspace-correspondences that do not fulfill \textsc{$C_{\wedge}$}\xspace, however, can
provide useful information if one wants to detect erratic differences
between $\mathcal{P}$ and $\mathcal{P}'$. As an example, assume that
$\mathcal{P}$ and $\mathcal{P}'$ consist of communities in a
network at times $t$ and $t' > t$, re\-spec\-tively. Moreover, let $P$
be a community in $\mathcal{P}$. If $(\{P\}, \mathcal{P} \setminus
\{P\})$ is in the minimum cut basis of $\mathcal{P}$ and if
$\emptyset$ is an optimal partner of $\{P\}$, this tells us that $P$
has disintegrated over time in a way that cannot be explained by a
good correspondence between $\mathcal{S}$ and $\mathcal{S'}$
(more on \textsc{$C_{\wedge}$}\xspace-correspondences in Appendix~\ref{sec:app:C3C4}).
A correspondence $(\mathcal{S}, \emptyset)$, however, is good, i.\,e.,\xspace
$\phi(\mathcal{S}, \emptyset)$ is low, whenever $U_{\mathcal{S}}$ is
small (note that $\mathcal{S} \neq \emptyset$ and $\mathcal{S}' =
\emptyset$ fulfill $\vert U_{\mathcal{S}} \triangle U_{\mathcal{S}'}
\vert = \vert U_{\mathcal{S}} \vert$). Even a good
\textsc{$C_{\wedge}$}\xspace-correspondence $(\mathcal{S}, \mathcal{S}')$ can be awkward,
e.\,g.,\xspace if $\vert U_{\mathcal{S}} \vert$ and $\vert U_{\mathcal{S}'}
\vert$ are small and $U_{\mathcal{S}} \cap U_{\mathcal{S}'} =
\emptyset$. Indeed, this means that $\vert U_{\mathcal{S}} \triangle
U_{\mathcal{S}'} \vert$ is still small, i.\,e.,\xspace the correspondence
$(\mathcal{S}, \mathcal{S}')$ is good, while $\mathcal{S}$ and
$\mathcal{S}'$ ``have nothing in common''. The purpose of
Definition~\ref{def:mutual} is to exclude these correspondences, i.\,e.,\xspace
to ensure that $\mathcal{S}$ and $\mathcal{S}'$ ``have a lot in
common'':
\begin{definition}
A correspondence
$(\mathcal{S}, \mathcal{S}')$ is called \emph{mutual} if all of the
following holds.
\begin{enumerate}
\setlength\itemsep{1pt}
\item $\vert P \cap U_{\mathcal{S}'} \vert \geq \frac{\vert P
\vert}{2}$ for all $P \in \mathcal{S}$,
\item $\vert P \cap U_{\mathcal{S}'} \vert \leq \frac{\vert P
\vert}{2}$ for all $P \in \mathcal{P} \setminus \mathcal{S}$,
\item $\vert P' \cap U_{\mathcal{S}} \vert \geq \frac{\vert P'
\vert}{2}$ for all $P' \in \mathcal{S}'$ and
\item $\vert P' \cap U_{\mathcal{S}} \vert \leq \frac{\vert P'
\vert}{2}$ for all $P' \in \mathcal{P}' \setminus \mathcal{S}'$.
\end{enumerate}
\label{def:mutual}
\end{definition}
If $(\mathcal{S}, \mathcal{S}')$ with $\mathcal{S} \notin \{\emptyset,
\mathcal{P}\}$ is mutual, then $\mathcal{S}' \notin \{\emptyset,
\mathcal{P}\}$ too. Thus, a new meaningful constraint on
correspondences $(\mathcal{S}, \mathcal{S}')$ that is stronger than
\textsc{$C_{\wedge}$}\xspace is:
\begin{equation*}
\textsc{$C_{m}$}\xspace: \mathcal{S} \notin \{\emptyset, \mathcal{P}\} \wedge
\mbox{$(\mathcal{S}, \mathcal{S}')$ is mutual}.
\end{equation*}
For more on \textsc{$C_{m}$}\xspace-correspondences see Appendix~\ref{sec:app:C3C4}.
In Appendix~\ref{subsec:app:mutual} we show that an optimal
\textsc{$C_{\mathcal{P}}$}\xspace-correspondence or \textsc{$C_{\wedge}$}\xspace-correspondence is either mutual or
fulfills $\vert \mathcal{S} \vert \in \{1, \vert
\mathcal{P} \vert - 1\} \vee \vert \mathcal{S}' \vert \in \{1, \vert
\mathcal{P}' \vert - 1\}$.
\section{\textsc{$C_{\mathcal{P}}$}\xspace-correspondences}
\label{sec:goodC2}
In this section we
reduce the problem of finding a good
\textsc{$C_{\mathcal{P}}$}\xspace-correspondence $(\mathcal{S}, \mathcal{S}')$ to the problem of
finding a small cut $(\mathcal{S}, \mathcal{P} \setminus \mathcal{S})$
of $\mathcal{P}$.
We then show that $\phi_{\mathcal{P}'}(\cdot)$ is a symmetric
submodular function on $2^{\mathcal{P}}$. Symmetry of
$\phi_{\mathcal{P}'}(\cdot)$ implies that one can find a minimum cut
basis of $\mathcal{P}$ by computing $\vert \mathcal{P} \vert - 1$
minimum $P_s$-$P_t$ cuts of $\mathcal{S}$.
Finally, we discuss asymptotic
running times for finding good \textsc{$C_{\mathcal{P}}$}\xspace-correspondences.
\subsection{\textsc{$C_{\mathcal{P}}$}\xspace-correspondences and submodularity.}
\label{subsec:C2_submod}
The constraint \textsc{$C_{\mathcal{P}}$}\xspace: $\mathcal{S} \notin \{\emptyset, \mathcal{P}\}$
on a correspondence $(\mathcal{S}, \mathcal{S}')$ does not constrain
$\mathcal{S}'$. The search for a ``good'' \textsc{$C_{\mathcal{P}}$}\xspace-correspondence thus
basically amounts to finding $\emptyset \neq \mathcal{S} \subsetneq
\mathcal{P}$ such that $\phi_{\mathcal{P}'}(\mathcal{S}) :=
\min_{\mathcal{S}' \in \mathcal{P}'}\phi(\mathcal{S}, \mathcal{S}')$
(Eq.~(\ref{eq:phi})) is ``low''. Once we have $\mathcal{S}$,
we can find an optimal partner $\mathcal{S}'$ of $\mathcal{S}$ via
Eq.~(\ref{eq:partner}). The problem with this approach is that,
in its present form, $\phi_{\mathcal{P}'}(\mathcal{S})$ depends on
$\mathcal{S}'$. Proposition~\ref{prop:eq:firstMin1} provides an
analytic expression for $\phi_{\mathcal{P}'}(\cdot)$ that does not
contain $\mathcal{S}'$:
\begin{prop}[Proof in Appendix~\ref{proof:eq:firstMin1}]
\label{prop:eq:firstMin1}
\begin{align}
&\phi_{\mathcal{P}'}(\mathcal{S}) =\sum_{P' \in
\mathcal{P}'} \vert P' \vert \operatorname{peak}(\frac{\vert U_{\mathcal{S}}
\cap P' \vert}{\vert P' \vert}), \quad \label{eq:Frac}
\mbox{where} \\ &\operatorname{peak}(x) := \begin{cases} x, &
\mbox{if } x \leq 1/2\\ 1 - x, & \mbox{if } x >
1/2 \end{cases} \label{eq:peak}
\end{align}
\end{prop}
The function $\operatorname{peak}(\cdot)$ in Eq.~(\ref{eq:peak}), called
(classification) error in~\cite{Tan2005a}, is an example of a
\emph{generator} as defined in~\cite{Simovici2002a}: a function $f :
[0,1] \mapsto \mathbb{R}$ is a generator if it is concave and $f(0) =
f(1) = 0$ (hence $f(\cdot)$ is also subadditive). In addition,
$\operatorname{peak}(\cdot)$ is symmetric, i.\,e.,\xspace $\operatorname{peak}(p) = \operatorname{peak}(1-p)$ for all
$p \in [0,1]$. Other examples of symmetric generators are the
binary entropy function
$H(\cdot)$~\cite{MacKay2003a,Tan2005a,Simovici2002a} and the Gini
impurity measure $G(\cdot)$~\cite{Tan2005a,Simovici2002a}. We
could have chosen $\operatorname{H}(\cdot)$, $\operatorname{G}(\cdot)$ or any other
nontrivial symmetric generator (computable in constant time)
instead of $\operatorname{peak}(\cdot)$. The minimization of
Eq.~(\ref{eq:Frac}) would then have the same asymptotic time
complexity (see Section~\ref{subsec:goodC2_running_times}).
\begin{definition}[Symmetric, (sub)modular]
\label{def:submodular}
Let $\mathcal{P}$ be a set. A function $\Pi : 2^{\mathcal{P}} \mapsto
\mathbb{R} $ is called \emph{symmetric} if $\Pi(\mathcal{S}) =
\Pi(\mathcal{P} \setminus \mathcal{S})$ for all $\mathcal{S} \subseteq
\mathcal{P}$. Furthermore, $\Pi(\cdot)$ is called \emph{submodular} if
$\Pi(\mathcal{S}_1 \cup \mathcal{S}_2) \leq \Pi(\mathcal{S}_1) +
\Pi(\mathcal{S}_2) - \Pi(\mathcal{S}_1 \cap \mathcal{S}_2)$ for all
$\mathcal{S}_1, \mathcal{S}_2 \subseteq \mathcal{P}$. If $\Pi(\cdot)$
fulfills the above with ``$=$'' instead of ``$\leq$'', then
$\Pi(\cdot)$ is called \emph{modular}.
\end{definition}
\begin{prop}[Proof in Appendix~\ref{proof:prop:submodular}]
\label{prop:submodular}
$\phi_{\mathcal{P}'}(\cdot)$ in Eq.~(\ref{eq:Frac}) is symmetric
and submodular.
\end{prop}
\subsection{Minimum $P_s$-$P_t$ cuts.}
\label{subsec:minST}
We are actually interested in a larger set of
\emph{good} correspondences between $\mathcal{P}$ and $\mathcal{P}'$
(rather than a single one)
or, equivalently, in a larger set of small
cuts of $\mathcal{P}$. A natural set of small cuts is formed by
minimum $P_s$-$P_t$ cuts:
\begin{definition}
\label{def:s-t-cuts}
Let $P_s \neq P_t \in \mathcal{P}$. Any pair $(\mathcal{S}_s,
\mathcal{S}_t)$ with $P_s \in \mathcal{S}_s$, $P_t \in \mathcal{S}_t$
and $\mathcal{S}_t = \mathcal{P} \setminus \mathcal{S}_s$ is called a
\emph{$P_s$-$P_t$ cut} of $\mathcal{P}$. A $P_s$-$P_t$ cut $(\mathcal{S}_s,
\mathcal{S}_t)$ is minimum if $\phi_{\mathcal{P}'}(\mathcal{S}_s)$,
and thus $\phi_{\mathcal{P}'}(\mathcal{S}_t)$, is minimum
w.\,r.\,t.\xspace all $P_s$-$P_t$ cuts.
\end{definition}
Analogous to graphs, there exists a minimum cut basis of $\mathcal{P}$
w.\,r.\,t.\xspace $\phi_{\mathcal{P}'}(\cdot)$ made up of $\vert
\mathcal{P} \vert - 1$ minimum $P_s$-$P_t$ cuts. This follows from
$\phi_{\mathcal{P}'}(\cdot)$ being
symmetric~\cite{Cheng1992a}. Moreover, the cuts in the minimum basis
are non-crossing (two cuts are non-crossing if their cut sides are
pairwise nested or disjoint~\cite{HartmannW12cut}). This is a consequence of
$\phi_{\mathcal{P}'}(\cdot)$ being
submodular~\cite{Gomory1961a,Queyranne98a}. A minimum
basis of cuts of $\mathcal{P}$ can be represented concisely by a Gomory-Hu tree~\cite{Gomory1961a}.
\subsection{Asymptotic time for minimum cut basis of \textsc{$C_{\mathcal{P}}$}\xspace-correspondences.}
\label{subsec:goodC2_running_times}
To compute a minimum basis of cuts of $\mathcal{P}$ under the
constraint \textsc{$C_{\mathcal{P}}$}\xspace, we have to compute $\vert \mathcal{P} \vert - 1$
minimum $P_s$-$P_t$
cuts~\cite{Gomory1961a,Gusfield90a}. Unfortunately, computing such a
cut in the setting of general symmetric submodular minimization is as
hard as minimizing a general non-symmetric submodular
function~\cite{Queyranne98a}; $\mathcal{O}(\vert \mathcal{P}
\vert^7) \log \vert \mathcal{P} \vert$ evaluations of
$\phi_{\mathcal{P}'}(\cdot)$ would be needed to find a single
minimum $P_s$-$P_t$ cut~\cite[Theorem
4.3]{Iwata2001a}. Fortunately, finding $P_s$-$P_t$ cuts is
easier in our case (proof in
Appendix~\ref{proof:prop:run-time-ctwo}):
\begin{prop}
\label{prop:running-nontrivial:P}
A minimum cut basis of $\mathcal{P}$ w.\,r.\,t.\xspace
$\phi_{\mathcal{P}'}(\cdot)$ can be computed in time
\begin{equation}
\mathcal{O}(\vert V \vert + \vert \mathcal{P} \vert^3 \vert \mathcal{P}'
\vert + \vert \mathcal{P} \vert^2 \vert \mathcal{P}'
\vert^2)
\label{firstBigO}
\end{equation}
vs.\ pace{-0.25cm}\centering \text{or in}vs.\ pace{-0.25cm} \\
\begin{equation}
\label{secondBigO}
\begin{cases}
\mathcal{O}(\vert V \vert + \vert \mathcal{P} \vert^3 \vert \mathcal{P}' \vert \log (2 +
\vert \mathcal{P} \vert^2 / \vert \mathcal{P}' \vert)), & \text{if~}
\vert \mathcal{P} \vert \leq \vert \mathcal{P}' \vert\\
\mathcal{O}(\vert V \vert + \vert \mathcal{P} \vert^2 \vert \mathcal{P}' \vert^2 \log (2 +
\vert \mathcal{P}' \vert^2 / \vert \mathcal{P} \vert)), &
\text{otherwise}
\end{cases}
\end{equation}
\end{prop}
The new notation in Definition~\ref{def:distribs} below helps
to prove Propositions~\ref{prop:evaluateFrac} and~\ref{prop:complexity}.
\begin{definition}[Distributions $d_{P'}\lbrack \cdot \rbrack$]
\label{def:distribs}
Let $P' \in \mathcal{P}'$. The distribution of $P'$ w.\,r.\,t.\xspace $\mathcal{P}$
is the vector $d_{P'}[\cdot]$ of length $\vert \mathcal{P} \vert$
defined by $d_{P'}[i] := \vert P_i \cap P' \vert \mbox{~for~} 1 \leq i
\leq \vert \mathcal{P} \vert$.
\end{definition}
The computation of all distributions (necessary to compute $\vert
U_{\mathcal{S}} \cap P' \vert$ in Eqs.~(\ref{eq:partner})
and~(\ref{eq:Frac})), i.\,e.,\xspace the contingency table~\cite{Wagner2007a}, takes
time $\mathcal{O}(\vert V \vert + \vert \mathcal{P} \vert \vert \mathcal{P}'
\vert)$, see Appendix~\ref{subsec:goodC1_running_times}.
The next result, Proposition~\ref{prop:evaluateFrac},
follows directly from the fact that, due to
$\vert U_{\mathcal{S}} \cap P' \vert = \sum_{i: P_i \in \mathcal{S}}
d_{P'}[i]$, the term $\vert U_{\mathcal{S}} \cap P' \vert$ can be
computed in $\mathcal{O}(\vert \mathcal{P} \vert)$ for any $P' \in
\mathcal{P}'$. It allows to derive Proposition~\ref{prop:complexity} afterwards.
\begin{prop}
\label{prop:evaluateFrac}
Given all distributions and $\mathcal{S} \subseteq \mathcal{P}$, the
calculation of $\mathcal{S}'$, as defined in
Eq.~(\ref{eq:partner}), and the evaluation of
$\phi_{\mathcal{P}'}(\mathcal{S})$, as defined in
Eq.~(\ref{eq:Frac}), can both be done in time $\mathcal{O}(\vert
\mathcal{P} \vert \vert \mathcal{P}' \vert)$.
\end{prop}
\begin{prop}
\label{prop:complexity}
Finding an optimal \textsc{$C_{\mathcal{P}}$}\xspace-correspondence takes time $\mathcal{O}(\vert V
\vert + \vert \mathcal{P} \vert^4 \vert \mathcal{P}' \vert)$ if one
first minimizes $\mathcal{S}$ in Eq.~(\ref{eq:Frac}) through
general symmetric submodular minimization and then determines the
optimal partner $\mathcal{S}'$ of $\mathcal{S}$ using
Eq.~(\ref{eq:partner}). For a proof see
Appendix~\ref{proof:prop:complexity}.
\end{prop}
Interestingly, the asymptotic time for computing the minimum cut basis
of $\mathcal{P}$ is lower than that for computing just one optimal
\textsc{$C_{\mathcal{P}}$}\xspace-correspondence using symmetric submodular minimization.\footnote{The only
way the latter time could be lower than or as low as $\mathcal{O}(\vert V
\vert + \vert \mathcal{P} \vert^3 \vert \mathcal{P}' \vert + \vert
\mathcal{P} \vert^2 \vert \mathcal{P}' \vert^2)$ (see
Eq.~(\ref{firstBigO})) would entail $\vert \mathcal{P} \vert^4
\vert \mathcal{P}' \vert \leq \vert \mathcal{P} \vert^2 \vert
\mathcal{P}' \vert^2$.
Eq.~(\ref{secondBigO}) then yields that the
minimum cut basis can be computed in time $\mathcal{O}(\vert V \vert + \vert
\mathcal{P} \vert^3 \vert \mathcal{P}' \vert)$.}
\section{Computing minimum $P_s$-$P_t$ cuts}
\label{sec:BB}
With the intent of improving the running time of the results
in the previous section for practical purposes, we continue with branch-and-bound (B\&B)
and greedy techniques.
\subsection{Basic branch-and-bound algorithm.}
\label{subsec:BB_basicAlgo}
Let $P_s \neq P_t \in \mathcal{P}$. Our goal is to find a minimum
$P_s$-$P_t$ cut $(\mathcal{S}_s, \mathcal{S}_t)$ of $\mathcal{P}$ w.\,r.\,t.\xspace
$\phi_{\mathcal{P}'}(\cdot)$.
The idea be\-hind our algorithm is to first set $\mathcal{S}_s :=
\{P_s\}$, $\mathcal{S}_t := \{P_t\}$ and then let $\mathcal{S}_s$ and
$\mathcal{S}_t$ compete for the remaining parts in $\mathcal{P}$ until
$\mathcal{S}_s \cup \mathcal{S}_t = \mathcal{P}$. To curtail the
exponentially growing number of possibilities that arise when
assigning new parts, i.\,e.,\xspace parts in $\mathcal{P} \setminus
(\mathcal{S}_s \cup \mathcal{S}_t)$, to either $\mathcal{S}_s$ or
$\mathcal{S}_t$, we need a lower bound $b(\mathcal{S}_s \cup
\mathcal{S}_t)$ on how low $\phi_{\mathcal{P}'}(\mathcal{S})$ can
possibly get for $\mathcal{S}$ with $\mathcal{S}_s \subseteq
\mathcal{S}$ and $\mathcal{S} \cap \mathcal{S}_t =
\emptyset$. Proposition~\ref{prop:bound} below guarantees that the
bound defined next is admissible.
\begin{definition}
\label{def:bound}
Let $\mathcal{S}_s, \mathcal{S}_t \subseteq \mathcal{P}$ with $P_s \in
\mathcal{S}_s$, $P_t \in \mathcal{S}_t$ and $\mathcal{S}_s \cap
\mathcal{S}_t = \emptyset$. We set $b(\mathcal{S}_s, \mathcal{S}_t)
:=\sum_{P' \in \mathcal{P}'} \min\{\vert U_{\mathcal{S}_s} \cap P'
\vert, \vert U_{\mathcal{S}_t} \cap P' \vert\}$.
\end{definition}
\begin{prop}[Proof in Appendix~\ref{proof:bound}]
\label{prop:bound}
Let $\mathcal{S}_s, \mathcal{S}_t \subseteq \mathcal{P}$ with $P_s \in
\mathcal{S}_s$, $P_t \in \mathcal{S}_t$ and $\mathcal{S}_s \cap
\mathcal{S}_t = \emptyset$. Moreover, let $\mathcal{S} \supseteq
\mathcal{S}_s$, $\mathcal{S} \cap \mathcal{S}_t = \emptyset$. Then,
$b(\mathcal{S}_s, \mathcal{S}_t) \leq
\phi_{\mathcal{P}'}(\mathcal{S})$.
\end{prop}
\noindent We still have to make decisions on (i) the choice of
the next part $P$ from $\mathcal{P} \setminus (\mathcal{S}_s \cup
\mathcal{S}_t)$ that we use to extend either $\mathcal{S}_s$ or
$\mathcal{S}_t$ and (ii) whether we assign $P$ to $\mathcal{S}_s$ or
$\mathcal{S}_t$.
Our strategy for (ii) is to assign $P$ to $\mathcal{S}_s$ or
$\mathcal{S}_t$ according to the (optimistic) prospect
$b(\mathcal{S}_s, \mathcal{S}_t)$, i.\,e.,\xspace $P$ is assigned such that the
new value $b(\mathcal{S}_s, \mathcal{S}_t)$ is minimum. We prefer
minimizing $b(\mathcal{S}_s, \mathcal{S}_t)$ over minimizing
$\phi_{\mathcal{P}'}(\mathcal{S}_s)$ and/or
$\phi_{\mathcal{P}'}(\mathcal{S}_t)$ because the latter two numbers
can both be high although the prospect for finding a good cut of
$\mathcal{P}$ is still good. Due to the definition of $b(\cdot,
\cdot)$ and the symmetry of $\phi_{\mathcal{P}'}(\cdot)$, however, the
objectives of minimizing $b(\mathcal{S}_s, \mathcal{S}_t)$,
$\phi_{\mathcal{P}'}(\mathcal{S}_s)$ and
$\phi_{\mathcal{P}'}(\mathcal{S}_t)$ will have converged by the time
when $\mathcal{S}_s$ and $\mathcal{S}_t$ are fully grown, i.\,e.,\xspace
$\mathcal{S}_s \cup \mathcal{S}_t = \mathcal{P}$.
Our strategy for (i), the choice of $P$, aims at shifting
the backtracking phases of our B\&B algorithm to scenarios
in which $\mathcal{S}_s$ and $\mathcal{S}_t$ are already large and
where chances are that we are close to a new minimum of
$\phi_{\mathcal{P}'}(\cdot)$. To this end, we pick $P$ from
$\mathcal{P} \setminus (\mathcal{S}_s \cup \mathcal{S}_t)$ such that
the alternative between putting $P$ into $\mathcal{S}_s$ or into
$\mathcal{S}_t$ matters the most in terms of $b(\cdot,
\cdot)$. Formally,
\begin{equation}
\small
\label{eq:nextP}
P = \operatorname{argmax}\limits_{P \in \mathcal{P}
\setminus (\mathcal{S}_s \cup \mathcal{S}_t)} \vert b(\mathcal{S}_s \cup \{P\},
\mathcal{S}_t) -
b(\mathcal{S}_s, \mathcal{S}_t \cup \{P\}) \vert
\end{equation}
After initializing $\mathcal{S}_s$ and $\mathcal{S}_t$, our
B\&B algorithm calls $\textsc{greedy}\xspace(\mathcal{S}_s, \mathcal{S}_t,
\infty)$, which is shown as Algorithm~\ref{algo:BB} in Section~\ref{subsec:greedy}. In later calls of
$\textsc{greedy}\xspace(\mathcal{S}_s, \mathcal{S}_t, bestSoFar)$, we always have
$(\mathcal{S}_s \supsetneq \{P_s\} \vee \mathcal{S}_t \supsetneq
\{P_t\}) \wedge \mathcal{S}_s \cap \mathcal{S}_t = \emptyset$, and
$bestSoFar$ amounts to the minimum weight ($\phi_{\mathcal{P}'}$
value) of the $P_s$-$P_t$ cuts found so far (see lines 5-10 of
Algorithm~\ref{algo:BB} in Appendix~\ref{subsec:app:bb}). Crucial questions \emph{after} any call
of $\textsc{greedy}\xspace$ are
\begin{itemize}
\item[1)] whether $\textsc{greedy}\xspace(\cdot, \cdot, \cdot)$ needs to be invoked
again and, \emph{if so},
\item[2a)] which of the most recent assignments of parts (to
$\mathcal{S}_s$ or $\mathcal{S}_t$) should be undone when
backtracking and
\item[2b)] which alternative line for searching a minimum $P_s$-$P_t$
cut is taken after backtracking, i.\,e.,\xspace what is the input for the next
$\textsc{greedy}\xspace(\cdot, \cdot, \cdot)$ call.
\end{itemize}
The answer to 1) is ``as long as $\mathcal{S}_s \supsetneq \{P_s\}$ or
$\mathcal{S}_t \supsetneq \{P_t\}$''. In other words, we stop when
$(\mathcal{S}_s, \mathcal{S}_t)$ has shrunk to its initialization
$(\{P_s\}, \{P_t\})$ (see lines 3 and 23 of Algorithm~\ref{algo:BB}).
The answer to 2a) and 2b), in turn, as well as more details on our B\&B algorithm (such as pseudocode),
can be found in Appendix~\ref{subsec:app:bb}.
How to extend the B\&B algorithm from \textsc{$C_{\mathcal{P}}$}\xspace to \textsc{$C_{\wedge}$}\xspace and \textsc{$C_{m}$}\xspace
is described in Appendix~\ref{subsubsec:extensions}.
\subsection{Speeding up the B\&B algorithm.}
\label{subsec:BB_extensions}
As before, let $\mathcal{S}_s$ and $\mathcal{S}_t$ consist of the parts
of $\mathcal{P}$ that have already been assigned to the $P_s$-side and
the $P_t$-side of the cut, respectively. To tighten the bound at the
current state of assembling $\mathcal{S}_s$ and $\mathcal{S}_t$, we
take a closer look at parts $P' \in \mathcal{P}'$ that overlap well
with $\mathcal{S}_s$ or $\mathcal{S}_t$ already. Specifically, let $P'
\in \mathcal{P}'$ such that $\vert P' \cap U_{\mathcal{S}_t} \vert
\geq \vert P' \vert / 2$. Then, if no backtracking behind the current
state occurs, a new assignment of some $P \in \mathcal{P}$ to the
$P_s$-side will increase the value of $\phi_{\mathcal{P}'}(\cdot)$ by
at least $I_s(P, P') := \vert P \cap P' \vert$. Exchanging the roles
of $s$ and $t$ may yield alternative increases $I_t(P, P')$ (based on
other $P'$). Thus,
\begin{equation*}
I_s(P) := \sum(I(P, P') \mbox{~:~} P' \mbox{ fulfills } \vert P' \cap U_{\mathcal{S}_t}
\vert \geq \vert P' \vert / 2)
\end{equation*}
\noindent and analogously defined $I_t(P)$ are the increases of
$\phi_{\mathcal{P}'}(\cdot)$ if $P$ is assigned to the $s$-side or
$t$-side, respectively. Hence, summing up the terms $\min\{I_s(P),
I_t(P)\}$ over all $P$ not yet assigned to any side yields a lower
bound on the future increase of the objective function.
Apart from improving the bound, a second way to curtail the search is
to interpret current $\mathcal{S}_s$ and $\mathcal{S}_t$ as two
\textsc{$C_{\mathcal{P}}$}\xspace-correspondences $(\mathcal{S}_s, \mathcal{S}_s')$ and
$(\mathcal{S}_t, \mathcal{S}_t')$, where $\mathcal{S}_s'$ and
$\mathcal{S}_t'$ are optimal partners of $\mathcal{S}_s$ and
$\mathcal{S}_t$, respectively.
\subsection{Greedy heuristic.}
\label{subsec:greedy}
Algorithm~\ref{algo:greedy}, $\textsc{greedy}\xspace$ $(\mathcal{S}_s, \mathcal{S}_t, bestSoFar)$, is at the heart of our B\&B
algorithm. It greedily extends a pair $(\mathcal{S}_s,
\mathcal{S}_t)$ and terminates prematurely, i.\,e.,\xspace with $\mathcal{S}_s
\cup \mathcal{S}_t \neq \mathcal{P}$, if there is no chance to find
$\mathcal{S}$ with $\phi_{\mathcal{P}'}(\mathcal{S}) < bestSoFar$.
In the first call of $\textsc{greedy}\xspace(\mathcal{S}_s, \mathcal{S}_t, bestSoFar)$,
we have $\mathcal{S}_s = \{P_s\}$, $\mathcal{S}_s = \{P_s\}$ and
$bestSoFar = \infty$. In particular, $\textsc{greedy}\xspace(\{P_s\}, \{P_t\},
\infty)$ does \emph{not} end prematurely, i.\,e.,\xspace it delivers a
$P_s$-$P_t$ cut $(\mathcal{S}, \mathcal{P} \setminus \mathcal{S})$.
While $\textsc{greedy}\xspace$ does not guarantee optimality, it will be interesting
if its quality is acceptable in practice.
\begin{algorithm}[tb]
\caption{Algorithm $\textsc{greedy}\xspace(\mathcal{S}_s, \mathcal{S}_t, bestSoFar)$
for extending a pair $(\mathcal{S}_s, \mathcal{S}_t)$ with
$\mathcal{S}_s \cap \mathcal{S}_t = \emptyset$ towards a pair
$(\mathcal{S}, \mathcal{P} \setminus \mathcal{S})$ with
$\mathcal{S} \supseteq \mathcal{S}_s$ and $\mathcal{S} \cap
\mathcal{S}_t = \emptyset$ as long as there is a chance that
$\phi_{\mathcal{P}'}(\mathcal{S}) < bestSoFar$}
\label{algo:greedy}
\begin{algorithmic}[1]
\While{$(\mathcal{S}_s \cup \mathcal{S}_t \neq \mathcal{P})
\wedge b((\mathcal{S}_s, \mathcal{S}_t) < bestSoFar)$}
\State Find $P \in \mathcal{P} \setminus (\mathcal{S}_s \cup
\mathcal{S}_t)$ that fulfills Eq.~(\ref{eq:nextP})
\If{$b(\mathcal{S}_s \cup \{P\}, \mathcal{S}_t) <
b(\mathcal{S}_s, \mathcal{S}_t \cup \{P\})$}
\State $\mathcal{S}_s \gets \mathcal{S}_s \cup \{P\}$
\Else
\State $\mathcal{S}_t \gets \mathcal{S}_t \cup \{P\}$
\EndIf
\EndWhile
\end{algorithmic}
\end{algorithm}
\section{Correspondences in community detection}
\label{sec:real-sim}
In the experiments of this section we
not only evaluate the performance of our B\&B and $\textsc{greedy}\xspace$ algorithms,
but also gain insight into different variants of the Louvain method, a
community detection algorithm. Community detection is a graph
clustering problem well-known in (social) network
analysis~\cite{Fortunato201075}, resulting in a partition of the
graph's node set. Finding correspondences between communities is
challenging when the nodes come without attributes that could help
with the task (like colored pixels in images), which is the case here.
\subsection{Louvain method (LM) and variants.}
\label{subsec:LM}
LM~\cite{Blondel:2008uq} is a locally greedy, bottom-up multilevel
algorithm. It is very popular for community detection by maximizing
the objective function modularity (this
modularity~\cite{girvan2002community} should not be confused with the
one in Definition~\ref{def:submodular}). On each hierarchy level, LM
assigns nodes to communities iteratively, while
maximizing modularity greedily.
The communities on each level are contracted into single
nodes, giving rise to the graph on the next level. The solution of
the coarsest graph is then successively expanded to the next finer
level, respectively.
In~\cite{DBLP:journals/tpds/StaudtM16} a shared-memory parallelization
of LM, called PLM, is provided.
PLM is not deterministic since the outcome depends on the order of the threads. We
denote the single-threaded (sequential) version of PLM by SLM. Both
versions have been extended by an optional \emph{refinement} phase:
after each expansion, nodes are again moved for modularity gain. SLM
with refinement is denoted by SLMR.
\subsection{Research questions and approach.}
Let now $\mathcal{P}$ and $\mathcal{P}'$ be from SLM and SLMR,
respectively. We first want to know whether the transition from
$\mathcal{P}$ to $\mathcal{P}'$ is best described as (a) communities
merely exchanging elements with each other, but otherwise remaining
as they are or (b) involving unions and break-ups of communities. An
analogous question arises when $\mathcal{P}$ and $\mathcal{P}'$ are
the partitions returned by different (non-deterministic) runs of
PLM.
Second, we want to test if the choice between \textsc{$C_{\mathcal{P}}$}\xspace- and
\textsc{$C_{\wedge}$}\xspace-correspondences matters when comparing communities.
Third, we want to evaluate the tradeoff between quality and running time
for both the B\&B algorithm and the heuristic $\textsc{greedy}\xspace$.
To answer the first question, we use the $\vert \mathcal{P} \vert - 1$
best \textsc{$C_{\mathcal{P}}$}\xspace-correspondences: if the
communities merely exchange elements with each other, most
\textsc{$C_{\mathcal{P}}$}\xspace-correspondences $(\mathcal{C}, \mathcal{C}')$ should be such
that $\vert \mathcal{C} \vert = \vert \mathcal{C}' \vert$. Conversely,
unions and break-ups of communities should result in many instances
with $\vert \mathcal{C} \vert \neq \vert \mathcal{C}' \vert$.
To answer the second and the third question, we compute the
$\vert \mathcal{P} \vert - 1$ best \textsc{$C_{\mathcal{P}}$}\xspace- and the $\vert \mathcal{P}
\vert - 1$ best \textsc{$C_{\wedge}$}\xspace-correspondences with the
B\&B algorithm and the heuristic $\textsc{greedy}\xspace$, respectively. For each of these
four scenarios, we then aggregate the $\vert \mathcal{P} \vert - 1$
best correspondences by calculating what can be called \emph{total
dissimilarity}, i.\,e.,\xspace the sum of the $\phi_{\mathcal{P}'}(\cdot,
\cdot)$ values of the $\vert \mathcal{P} \vert - 1$ best
correspondences, divided by the total number of vertices.
In experiments involving (non-deterministic) PLM, we smooth total dissimilarity by averaging over 10 runs.
We compare the results from the four scenarios as follows. Let
$d_{\mathcal{P}}$, $d_{\wedge}$, $d^h_{\mathcal{P}}$ and
$d^h_{\wedge}$ be the total dissimilarity from the scenarios (i)
\textsc{$C_{\mathcal{P}}$}\xspace, B\&B, (ii) \textsc{$C_{\wedge}$}\xspace, B\&B, (iii)
\textsc{$C_{\mathcal{P}}$}\xspace, heuristic, and (iv) \textsc{$C_{\wedge}$}\xspace, heuristic, respectively. We form
the ratios $r_1 = d_{\mathcal{P}}/d_{\wedge}$, $r_2 =
d_{\mathcal{P}}/d^h_{\mathcal{P}}$ and $r_3 =
d_{\mathcal{P}}/d^h_{\wedge}$. In case of experiments
involving PLM, different scenarios come with different sets of ten PLM-generated partitions each.
As input for PLM, SLM, and SLMR we choose a collection of 15 diverse
and widely used complex networks from two popular
archives~\cite{Bader2017a,Leskovec2014a}. These networks are listed
in Table~\ref{tab:extra_social}. For each network and
each comparison, i.\,e.,\xspace one run of PLM vs.\ another run of PLM or SLM vs.\
SLMR, we get a pair of partitions $\mathcal{P}$ and $\mathcal{P}'$.
Note that we are not aware of comparable many-to-many correspondences approaches,
so that a comparison to existing methods has to be omitted.
Our sequential code implementing the algorithms presented in
Section~\ref{sec:BB} is written in C++; it uses the LM implementations
of NetworKit~\cite{Staudt2014}.
vs.\ pace{-2.5ex}
\begin{center}
\begin{table*}[ht]
\caption{Complex networks used for comparing partitions. The column
{\small \emph{\# communities}} indicates the average number of communities
generated by PLM (average over 20 runs).}
\label{tab:extra_social}
\begin{center}
\begin{small}
\scalebox{0.78}{
\begin{tabular}{ l | l | r | r | r | r }
Graph ID & Name & \#vertices & \#edges & \# communities & Network type\\ \hline \hline
1 & \textsc{p2p-Gnutella} & \numprint{6405} & \numprint{29215} & {12.7} & filesharing network\\\hline
2 & \textsc{PGPgiantcompo} & \numprint{10680} & \numprint{24316} & {95.7} & network of PGP users\\\hline
3 & \textsc{email-EuAll} & \numprint{16805} & \numprint{60260} & {48.4} & network of connections via email\\\hline
4 & \textsc{as-22july06} & \numprint{22963} & \numprint{48436} & {26.1} & autonomous systems in the internet\\\hline
5 & \textsc{soc-Slashdot0902} & \numprint{28550} & \numprint{379445} & {144.4} & news network\\\hline
6 & \textsc{loc-brightkite\_edges} & \numprint{56739} & \numprint{212945} & {264.7} & location-based friendship network\\\hline
7 & \textsc{loc-gowalla\_edges} & \numprint{196591} & \numprint{950327} & {509.7} & location-based friendship network\\\hline
8 & \textsc{coAuthorsCiteseer} & \numprint{227320} & \numprint{814134} & {181.5} & citation network\\\hline
9 & \textsc{wiki-Talk} & \numprint{232314} & \numprint{1458806} & {632.3} & user interactions through edits\\\hline
10 & \textsc{citationCiteseer} & \numprint{268495} & \numprint{1156647} & {124.8} & citation network\\\hline
11 & \textsc{coAuthorsDBLP} & \numprint{299067} & \numprint{977676} & {181.7} & citation network\\\hline
12 & \textsc{web-Google} & \numprint{356648} & \numprint{2093324} & {159.0} & hyperlink network of web pages\\\hline
13 & \textsc{coPapersCiteseer} & \numprint{434102} & \numprint{16036720} & {266.9} & citation network\\\hline
14 & \textsc{coPapersDBLP} & \numprint{540486} & \numprint{15245729} & {146.2} & citation network\\\hline
15 & \textsc{as-skitter} & \numprint{554930} &\numprint{5797663} & {226.8} & network of internet service providers\\\hline
\end{tabular}}
\end{small}
\end{center}
\end{table*}
\end{center}
\begin{figure*}
\caption{A data point $(n, n')$ indicates that there are
\textsc{$C_{\mathcal{P}
\label{fig:web}
\end{figure*}
\subsection{Results.}
For any network in Table~\ref{tab:extra_social}, all $\vert \mathcal{P} \vert - 1$ best
\textsc{$C_{\mathcal{P}}$}\xspace-cor\-res\-pon\-den\-ces $(\mathcal{S}, \mathcal{S}')$ between
partitions from runs of SLM and SLMR (both deterministic)
fulfill $\vert \mathcal{S} \vert = \vert \mathcal{S}' \vert$.
(Figure~\ref{fig:web}(a) shows the corresponding result for \textsc{web-Google}.)
This indicates that, between SLM and SLMR, the communities merely exchange
elements with each other and that there are no unions and no
break-ups of communities. Calculations of $r_1$, $r_2$ and $r_3$
for SLM vs.\ SLMR and all networks yield values between $1.0$ and
$1.024$. Thus, none of the choices, i.\,e.,\xspace B\&B vs.\ heuristic
and \textsc{$C_{\mathcal{P}}$}\xspace vs.\ \textsc{$C_{\wedge}$}\xspace, has considerable impact on quality.
Figure~\ref{fig:web}(b) shows the results of analogous experiments with
PLM vs.\ PLM instead of SLM vs.\ SLMR. In contrast to Figure~\ref{fig:web}(a), numerous
\textsc{$C_{\mathcal{P}}$}\xspace-correspondences $(\mathcal{S}, \mathcal{S}')$ are unbalanced in
that $\vert \mathcal{S} \vert$ and $\vert \mathcal{S}' \vert$ differ
considerably. This indicates that the non-determinism of PLM causes
unions and break-ups of communities.
Figure~\ref{fig:web}(c) shows that the fluctuations of total
dissimilarity (after some averaging) do not follow any trend in
terms of the four scenarios. Since the B\&B algorithm
cannot perform worse than the corresponding heuristic on a given
partition, this indicates that the fluctuations are due to the
non-determinism of PLM, and that the heuristics are as good as the
corresponding B\&B algorithm.
Running times of our B\&B algorithm fluctuate considerably, e.\,g.,\xspace
between 38 and 9555 seconds in ten runs for the graph
\textsc{wiki-Talk}. Minimum, maximum and mean running times for
graphs in Table~\ref{tab:extra_social} are shown in
Table~\ref{tab:running_BB} (Appendix~\ref{sec:app:run}). Recall that running times refer to
computing the best $\vert \mathcal{P} \vert -1$ correspondences. Not
surprisingly, running times tend to increase enormously with
increasing numbers of communities, despite the strong fluctuations.
In the vast majority of cases, however, the B\&B algorithm terminates
within a few minutes, even for the larger instances.
The analogue running times of our greedy heuristic are much more
stable, never exceeding 40 seconds; for details see
Table~\ref{tab:running_greed} (Appendix~\ref{sec:app:run}). As expected, due to the absence of
backtracking, the trend toward higher running times for increasing
numbers of communities is less pronounced than for the B\&B algorithm.
Nonetheless, as mentioned above, the aggregated quality ($r_i \in
[1.0; 1.024]$) shows that $\textsc{greedy}\xspace$ yields very good results already.
To summarize, non-de\-ter\-mi\-nism of PLM disrupts the communities in
a more fundamental way (frequent unions or break-ups of communities)
than the refinement phase.
Also, the choices (i) B\&B vs.\
heuristic and (ii) \textsc{$C_{\mathcal{P}}$}\xspace vs.\ \textsc{$C_{\wedge}$}\xspace have a minor impact on the
quality of the correspondences. Most of the time the B\&B algorithm is
fast (less than one minute), but outliers with running times of a few
hours do exist. In the context of community detection, however, it
suffices to run $\textsc{greedy}\xspace$, which yields very good correspondences
quickly in all cases.
Another option would be to terminate the
B\&B algorithm after a certain amount of time, taking the best result
found.
\section{Related work}
\label{sec:app:soa}
\subsection{Similarity measures for partitions.}
Wagner and Wagner~\cite{Wagner2007a} provide a comprehensive
collection of similarity measures for partitions $\mathcal{P}$ and $\mathcal{P}'$ of the same set $V$. They can
all be derived from the contingency table of $\mathcal{P}$ and
$\mathcal{P}'$.
Ref.~\cite{Wagner2007a} groups the
similarity measures into three groups:
\begin{enumerate}
\item Measures based on considering all unordered pairs $\{v, w\}$ of $V$ and counting the 4 cases arising from the
distinction as to whether $v$ and $w$ belong to the same part or to
different parts of $\mathcal{P}$ and the analogous distinction with
$\mathcal{P}'$ instead of $\mathcal{P}$. Examples of such measures
are the Rand index~\cite{Rand1971a} and the adjusted Rand
index~\cite{Hubert1985a}.
\item Measures that involve a sum over maximum $P_i$, $P'_j$ overlaps,
where the sum is over the $P_i$, the maximum is over the $P'_j$, and
the overlaps are defined in various ways. One example is the
$\mathcal{F}$-measure~\cite{Larsen1999a,Fung2008a}. Typically,
theses measures yield different results if the roles of
$\mathcal{P}$ and $\mathcal{P}'$ are exchanged. The set function
$\phi_{\mathcal{P}'}(\cdot)$ defined in this paper, see
Eqs.~(\ref{eq:Frac}) and~(\ref{eq:peak}), has similar
properties in that it (i) aggregates $P_i$, $P'_j$ overlaps over
certain $P_i$ in a nonlinear way, and (ii) may vary if the roles of
$\mathcal{P}$ and $\mathcal{P}'$ are exchanged.
\item Measures that involve mutual information, e.\,g.,\xspace Normalized Mutual
Information~\cite{Strehl2003a}. Here, the common ground with our
approach to defining correspondences is that we can replace the
function $\operatorname{peak}(\cdot)$, see Eqs.~(\ref{eq:Frac})
and~(\ref{eq:peak}), by the binary entropy function without altering
the nature of our optimization problem.
\end{enumerate}
\subsection{Impurity measures.}
The value $\phi_{\mathcal{P}'}(\mathcal{S})$ indicates
how well the parts of $\mathcal{P}'$ fit into $U_{\mathcal{S}}$ or $V
\setminus U_{\mathcal{S}}$, see Eqs.~(\ref{eq:Frac})
and~(\ref{eq:peak}). Impurity measures, as defined
in~\cite{Tan2005a,Simovici2002a}, seem to be based on a similar
idea. Using our setting and notation, Simovici
et al.\xspace~\cite{Simovici2002a} define the impurity of a subset $L$ of the
ground set $V$ relative to $\mathcal{P}$ and generated by
$\operatorname{peak}(\cdot)$ as
$\operatorname{IMP}^{\operatorname{peak}}_{\mathcal{P}'}(L) = \vert L \vert \sum_{P' \in \mathcal{P}'} \operatorname{peak}(\frac{\vert L \cap P' \vert}{\vert L \vert})$.
We can turn $\operatorname{IMP}^{\operatorname{peak}}_{\mathcal{P}'}(U_{\mathcal{S}})$ into
$\phi_{\mathcal{P}'}(\mathcal{S})$ by (i) pulling
$U_{\mathcal{S}}$ under the sum (mathematically correct) and (ii)
exchanging the roles of $U_{\mathcal{S}}$ and $P'$ under the sum
(mathematically incorrect). For us it is important to have the roles
as they are in $\phi_{\mathcal{P}'}(\mathcal{S})$ because
this is what makes $\phi_{\mathcal{P}'}(\cdot)$ a
submodular and symmetric function. These properties, in turn, make it
possible to find the best nontrivial $\mathcal{S}$ in polytime.
Despite this mismatch between $\operatorname{IMP}^{\operatorname{peak}}_{\mathcal{P}'}(\cdot)$ and
$\phi_{\mathcal{P}'}(\cdot)$, studying
$\operatorname{IMP}^{\operatorname{peak}}_{\mathcal{P}'}(\cdot)$ helped to develop the intuition
behind our approach.
The main properties of $\operatorname{IMP}^{\operatorname{peak}}_{\mathcal{P}'}(\cdot)$ and
related measures, as formulated and proven in~\cite{Simovici2002a},
are preserved if $\operatorname{peak}(\cdot)$ is replaced by another
\emph{generator}, as defined in~\cite{Simovici2002a}, i.\,e.,\xspace another
concave and subadditive function $f : [0,1] \mapsto \mathbb{R}$ with $f(0) =
f(1) = 0$. Likewise, if we replace $\operatorname{peak}(\cdot)$ in
Eq.~(\ref{eq:Frac}) by another generator, we will arrive at
similar definitions of correspondences. This also does not change the
kind and asymptotic complexity of the optimization problems posed by
our approach.
\section{Conclusions and outlook}
Recall that small data changes can lead clustering methods to split
or merge clusters. By computing many-to-many correspondences,
one can recover the most crucial split and merge operations.
Here, \textsc{$C_{\mathcal{P}}$}\xspace-correspondences are ideal
in that the many-to-many correspondences and the associated split and
merge operations make up a hierarchy.
For \textsc{$C_{\mathcal{P}}$}\xspace-correspondences there exists a minimum basis of
\emph{non-cros\-sing} $P_s$-$P_t$ cuts of $\mathcal{P}$ w.\,r.\,t.\xspace $\mathcal{P}'$ that, in turn, yield a hierarchy of the $\vert
\mathcal{P} \vert - 1$ best correspondences between $\mathcal{P}$ and
$\mathcal{P}'$ via optimal partners.
Under \textsc{$C_{\wedge}$}\xspace,
the cuts in a minimum cut basis
are crossing in general.
On the upside and in contrast to
\textsc{$C_{\mathcal{P}}$}\xspace-correspondences,
a good \textsc{$C_{\wedge}$}\xspace-correspondence gives rise to two
similar cuts $(U_{\mathcal{S}}, U_{\mathcal{P} \setminus
\mathcal{S}})$ and $(U_{\mathcal{S}'}, U_{\mathcal{P}' \setminus
\mathcal{S}'})$ of $V$. For such a pair of similar cuts it is
easy to find a cut that mediates between them. The overlay
of $k$ such medial cuts then results in a consensus partition.
In our B\&B algorithm, one has to choose the next candidates for
extension of either $\mathcal{S}_s$ or $\mathcal{S}_t$. This choice
may also involve application-specific criteria such as color or shape
in image analysis. Such additional information may help our B\&B
algorithm to stay in the lane, and is expected to accelerate it.
We see our B\&B algorithm as a starting point for
fast heuristics to find high-quality correspondences. In the
experiments of Section~\ref{sec:real-sim} we have seen that turning
off the backtracking in our B\&B algorithm (i.\,e.,\xspace running $\textsc{greedy}\xspace$ only once)
has only a negligible effect on the quality of the results.
\begin{small}
vs.\ pace{1.5ex}
\textbf{Acknowledgements.}
We thank Christian Staudt for helpful discussions and the anonymous
reviewers for helping to improve the paper in various respects.
\end{small}
\appendix
\section{Appendix}
\label{sec:app}
\subsection{Correspondences in image analysis.}
\label{subsec:app:image analysis}
\pagenumbering{roman}
\setcounter{page}{1}
Scenarios in which it makes sense to compare
$\mathcal{P}$ and $\mathcal{P}'$ using correspondences can be as follows:
(i) $\mathcal{P}$ is the result of a segmentation algorithm and
$\mathcal{P}'$ describes ground truth, e.\,g.,\xspace if $\mathcal{P}$ is a
segmented satellite image and if $\mathcal{P}'$ describes land use
that has been determined in the field by experts. Here, the aim of a
comparison might be to identify areas where $\mathcal{P}$ suffers
from over-segmentation (unions of regions of $\mathcal{P}$ that
correspond well to single regions of $\mathcal{P}'$), from
under-segmentation (single regions of $\mathcal{P}$ that
correspond well to unions of regions of $\mathcal{P}'$) or
more intricate combinations of over-segmentation and
under-segmentations.
(ii) $\mathcal{P}$ and $\mathcal{P}'$ describe ground truth at
different times. Sticking to land use, a good
correspondence $(\mathcal{S}, \mathcal{S}')$ with $\vert \mathcal{S}
\vert, \vert \mathcal{S}' \vert > 1$ may indicate crop rotation.
(iii) $\mathcal{P}$ and $\mathcal{P}'$ are results of different
segmentation algorithms applied to the same image, and/or the two
segmentations are based on different physical measurements, e.\,g.,\xspace
channels in Satellite Imagery or CT vs.\ MRI in medical
imaging. Then, a good correspondence $(\mathcal{S}, \mathcal{S}')$
provides strong evi\-dence that the feature described by $\mathcal{S}$
is not an artifact. For an example of correspondences between
different segmentations see Figure~\ref{fig:sheep}.
\FloatBarrier
\subsection{Examples of cuts and correspondences.}
\label{sub:exmp-cuts-corres}
Illustration for example from Section~\ref{sec:problem-statement}:
\begin{figure}
\caption{Same scenario as in Figure~\ref{fig:example}
\label{fig:revExample}
\end{figure}
vs.\ pace{-4ex}
\subsection{\textsc{$C_{\vee}$}\xspace-correspondences.}
\label{sec:app:goodC1}
We show how a natural collection of $\vert \mathcal{P} \vert + \vert
\mathcal{P}' \vert - 1$ best \textsc{$C_{\vee}$}\xspace-correspondences emerges from a
minimum cut basis of an edge-weighted bipartite graph $G=(\mathcal{P}
\sqcup \mathcal{P}', E, \omega(\cdot))$, where the edge set $E$
consists of all $\{P, P'\}$ with $P \in \mathcal{P}$, $P' \in
\mathcal{P}'$ and $P \cap P' \neq \emptyset$. The edge weights are
given by $\omega(\{P, P'\}) = \vert P \cap P' \vert$. The cuts in the
basis may be chosen such that they are non-crossing, see
Section~\ref{subsec:bipartite}. Time complexities for
finding good \textsc{$C_{\vee}$}\xspace-cor\-res\-pon\-den\-ces are discussed in
Section~\ref{subsec:goodC1_running_times}.
\subsubsection{\textsc{$C_{\vee}$}\xspace-correspondences from cuts of
a bipartite graph}
\label{subsec:bipartite}
Recall that \textsc{$C_{\vee}$}\xspace: $\mathcal{S} \notin \{\emptyset, \mathcal{P}\} \vee
\mathcal{S}' \notin \{\emptyset, \mathcal{P}'\}$ is our weakest
constraint. It merely excludes trivial and very bad correspondences.
Finding good correspondences through finding small cuts of certain
bipartite graphs has already been proposed in the context of mutual
document and word clustering~\cite{Zha2001a}. Here, we start by
rewriting $\phi(\mathcal{S}, \mathcal{S}')$.
\begin{align*}
\phi(\mathcal{S}, \mathcal{S}') &= \vert
U_{\mathcal{S}} \setminus U_{\mathcal{S}'} \vert + \vert
U_{\mathcal{S}'} \setminus U_{\mathcal{S}} \vert \nonumber\\ &= \vert
U_{\mathcal{S}} \cap (V \setminus U_{\mathcal{S}'})\vert + \vert
U_{\mathcal{S}'} \cap (V \setminus U_{\mathcal{S}})\vert
\nonumber\\ &= \sum_{P' \notin \mathcal{S}'} \vert U_{\mathcal{S}}
\cap P'\vert + \sum_{P \notin \mathcal{S}} \vert U_{\mathcal{S}'} \cap
P \vert \nonumber\\ &= \sum_{P' \notin
\mathcal{S}'}(\sum_{P \in \mathcal{S}} \vert P \cap P' \vert) + \sum_{P \notin
\mathcal{S}}(\sum_{P' \in \mathcal{S}'} \vert P \cap P' \vert).
\label{eq:bipart}
\end{align*}
Let $G=(W, E, \omega(\cdot))$ with $\omega:E \mapsto \mathbb{R}_{\geq 0}$ be
the edge-weighted bipartite graph defined by (i) $W := \mathcal{P}
\sqcup \mathcal{P}'$, where $\sqcup$ denotes the disjoint union, (ii)
$E := \{\{P, P'\} \mbox{~with~} P \in \mathcal{P}, P' \in \mathcal{P}'
\mbox{~and~} P \cap P' \neq \emptyset\}$ and (iii) $E := \{\{P, P'\}
\mbox{~with~} P \in \mathcal{P}, P' \in \mathcal{P}' \mbox{~and~} P
\cap P' \neq \emptyset\}$ and $\omega(\{P, P'\}) := \vert P \cap P'
\vert$,
where we distinguish between $P \in \mathcal{P}$ and $P' \in
\mathcal{P}'$, even if $P = P'$. Then, $\phi(\mathcal{S},
\mathcal{S}')$ equals the total weight of the cut $(\mathcal{S} \sqcup
\mathcal{S}', (\mathcal{P} \setminus \mathcal{S}) \sqcup (\mathcal{P}'
\setminus \mathcal{S}'))$.
Thus, a minimum cut basis of $G$ gives rise to a minimum basis of
\textsc{$C_{\vee}$}\xspace-correspondences. A minimum cut basis of $G$, in turn, can be
chosen such that the cuts are non-crossing~\cite{Gomory1961a}. Hence,
a minimum basis of \textsc{$C_{\vee}$}\xspace-correspondences can be represented by a
Gomory-Hu tree with vertex set $W$.
\subsubsection{Asymptotic time for minimum cut basis of \textsc{$C_{\vee}$}\xspace-correspondences}
\label{subsec:goodC1_running_times}
To build a minimum cut basis of \textsc{$C_{\vee}$}\xspace-correspondences, we first
generate the bipartite graph $G$. To this end, we compute the
contingency table of $\mathcal{P}$ and $\mathcal{P}'$ (weighted
adjacency matrix of $G$), i.\,e.,\xspace the matrix whose entry at $(i,j)$ equals
$\vert P_i \cap P'_j \vert$. Initializing the contingency table to
zero entries takes time $\mathcal{O}(\vert \mathcal{P} \vert \vert
\mathcal{P}' \vert)$. The contingency table can then be filled in one
traversal of $V$, provided that deciding on the membership of any $v
\in V$ to a part in $\mathcal{P}$ and $\mathcal{P}'$ takes constant
time. Asymptotic time for computing the contingency table \emph{and}
building $G$ is the same as for just computing the contingency table,
i.\,e.,\xspace $\mathcal{O}(\vert V \vert + \vert \mathcal{P} \vert \vert \mathcal{P}'
\vert)$.
Then, based on $G$ and using the algorithms by Gomory or
Gusfield~\cite{Gomory1961a,Gusfield90a}, one can compute a minimum cut
basis of $G$. The asymptotic time of both algorithms amounts to that
of $\vert W \vert - 1$ calculations of minimum $Q$-$R$ cuts, $Q, R \in
W$. Given $Q, R \in W$, a minimum $Q$-$R$ cut can be found in
$\mathcal{O}(\vert W \vert \vert E \vert)$ time using an algorithm
in~\cite{Orlin2013a}, which also works for general $G$. Alternatively,
one can use an algorithm in~\cite{Ahuja1994a} which finds a minimum
$Q$-$R$ cut of $G$ in time $\mathcal{O}(\mu \vert E \vert \log(2 + \mu^2 /
\vert E \vert))$, where $\mu = \min\{\vert \mathcal{P} \vert, \vert
\mathcal{P}' \vert\}$. The latter algorithm makes use of $G$ being
bipartite and can yield a lower asymptotic time than the former if (i)
$(\vert \mathcal{P} \vert \ll \vert \mathcal{P}' \vert) \vee (\vert
\mathcal{P}' \vert \ll \vert \mathcal{P} \vert)$ and (ii) $G$ is
sparse. Proposition~\ref{prop:running-nontrivial:G} summarizes the
running times of the two algorithms and expresses them in our terms,
i.\,e.,\xspace $V$, $\mathcal{P}$ and $\mathcal{P}'$.
\begin{prop}
\label{prop:running-nontrivial:G}
A minimum cut basis of $G$ can be computed in $\mathcal{O}(\vert V \vert +
\vert \mathcal{P} \vert^3 \vert \mathcal{P}' \vert + \vert \mathcal{P}
\vert \vert \mathcal{P}' \vert^3)$ or in
\begin{equation*}
\mathcal{O}(\vert V \vert + \vert \mathcal{P} \vert^2 \vert \mathcal{P}'
\vert^2 \log (2 + \frac{(\min\{\vert \mathcal{P} \vert, \vert
\mathcal{P}' \vert\})^2}{\max\{\vert \mathcal{P} \vert, \vert
\mathcal{P}' \vert\}})).
\end{equation*}
\end{prop}
\begin{proof}
Recall that generating $G$ takes time $\mathcal{O}(\vert V \vert + \vert
\mathcal{P} \vert \vert \mathcal{P}' \vert)$. Total time is the sum of
the latter and time for $\vert W \vert - 1$ calculations of minimum
$Q$-$R$ cuts.
Using the algorithm in~\cite{Orlin2013a}, $\vert W \vert - 1$
calculations of minimum $Q$-$R$ cuts take time $\mathcal{O}(\vert W \vert^2
\vert E \vert) = \vert \mathcal{P} \vert^3 \vert \mathcal{P}' \vert +
\vert \mathcal{P} \vert^2 \vert \mathcal{P}' \vert^2 + \vert
\mathcal{P} \vert \vert \mathcal{P}' \vert^3 = \vert \mathcal{P}
\vert^3 \vert \mathcal{P}' \vert + \vert \mathcal{P} \vert \vert
\mathcal{P}' \vert^3$. This yields the first asymptotic time.
To see that the second asymptotic time is valid, note that (i) $\vert
E \vert \leq \vert \mathcal{P} \vert \vert \mathcal{P}' \vert$ and
(ii) $\vert E \vert \geq \max\{\vert \mathcal{P} \vert, \vert
\mathcal{P}' \vert\}$. The remainder of the proof is
straightforward.
\end{proof}
\subsection{Proofs.}
\subsubsection{Proof of Proposition~\ref{prop:eq:firstMin1}.}
\label{proof:eq:firstMin1}
\begin{proof}
\noindent Starting with Eq.~(\ref{eq:firstMin1}), we get
\begin{align}
\phi(\mathcal{S}, \mathcal{S}') &= \vert U_{\mathcal{S}} \setminus
U_{\mathcal{S}'} \vert + \vert U_{\mathcal{S}'} \setminus
U_{\mathcal{S}} \vert\nonumber\\ &= \vert U_{\mathcal{S}} \cap (V
\setminus U_{\mathcal{S}'})\vert + \sum_{P' \in \mathcal{S}'} \vert P'
\setminus U_{\mathcal{S}}\vert\nonumber\\ &= \sum_{P' \notin
\mathcal{S}'} \vert U_{\mathcal{S}} \cap P'\vert + \sum_{P' \in
\mathcal{S}'}(\vert P' \vert - \vert U_{\mathcal{S}} \cap P'
\vert)\label{eq:secondMin}.
\end{align}
\noindent By letting $\mathcal{S}'$ be an optimal partner of
$\mathcal{S}$, e.\,g.,\xspace by calculating $\mathcal{S}'$ using
Eq.~(\ref{eq:partner}), we minimize the contribution (damage) of
each $P' \in \mathcal{P}'$ to the right hand side of
Eq.~(\ref{eq:secondMin}), and thus minimize $\phi(\mathcal{S},
\cdot)$. Insertion of $\mathcal{S}'$ from Eq.~(\ref{eq:partner})
then yields
\begin{small}
\begin{eqnarray}
\min_{\mathcal{S}' \subseteq \mathcal{P}'}
\phi(\mathcal{S},\mathcal{S}') =& \sum_{P' \in \mathcal{P}'}
\min\{\vert U_{\mathcal{S}} \cap P'\vert, \vert P' \vert - \vert
U_{\mathcal{S}} \cap P' \vert\}\nonumber\\ =& \sum_{P' \in
\mathcal{P}'} \vert P' \vert \min\{\frac{\vert U_{\mathcal{S}} \cap
P'\vert}{\vert P' \vert}, 1 - \frac{\vert U_{\mathcal{S}} \cap P'
\vert}{\vert P' \vert}\}\nonumber\\ =& \sum_{P' \in \mathcal{P}'}
\vert P' \vert \operatorname{peak}(\frac{\vert U_{\mathcal{S}} \cap P' \vert}{\vert
P' \vert}).
\label{eq:towardPeak}
\end{eqnarray}
\end{small}
\end{proof}
\subsubsection{Proof of Proposition~\ref{prop:submodular}.}
\label{proof:prop:submodular}
\begin{proof}
The symmetry of $\phi_{\mathcal{P}'}(\cdot)$ follows from
that of $\operatorname{peak}(\cdot)$. Sums and multiples of submodular functions are
submodular~\cite{Schrijver2003a}. Thus, to show that
$\phi_{\mathcal{P}'}(\cdot)$ is submodular, it suffices to
show that $\phi_i(\mathcal{S}) := \operatorname{peak}(\frac{\vert U_{\mathcal{S}}
\cap P'_i \vert}{\vert P'_i \vert})$ in Eq.~(\ref{eq:Frac}) is
submodular for all $i$.
Indeed, the $\phi_i(\cdot)$ are of the form $c(m(\cdot))$, where
$c(\cdot)$ is concave and $m(\cdot)$ is non-negative modular. Any
function of this form is
submodular~\cite{Stobbe2010b,Bilmes2012a}.
\end{proof}
\subsubsection{Proof of Proposition~\ref{prop:running-nontrivial:P}.}
\label{proof:prop:run-time-ctwo}
\begin{proof}
By means of the bipartite graph $G$ in
Section~\ref{subsec:bipartite}, finding a mi\-ni\-mum $P_s$-$P_t$
cut $(\mathcal{S}, \mathcal{P} \setminus \mathcal{S})$ of
$\mathcal{P}$ can be achieved through (i) finding a minimum
$P_s$-$P_t$ cut $(\mathcal{S} \sqcup \mathcal{S}', (\mathcal{P}
\setminus \mathcal{S}) \sqcup (\mathcal{P}' \setminus
\mathcal{S}'))$ of $G$ and (ii) extracting $(\mathcal{S},
\mathcal{P} \setminus \mathcal{S})$. Step (i) is analogous to the
proof of Proposition~\ref{prop:running-nontrivial:G}.
\end{proof}
\subsubsection{Proof of Proposition~\ref{prop:complexity}.}
\label{proof:prop:complexity}
\begin{proof}
Below, we refer to Algorithm OPTIMAL-SET
from~\cite{Queyranne98a}, where a symmetric submodular function
$f(\cdot)$ is minimized by building the set minimizing $f(\cdot)$ from
scratch. OPTIMAL-SET consists of $\mathcal{O}(\vert \mathcal{P} \vert^3)$
evaluations of $\phi_{\mathcal{P}'}(\cdot)$~\cite[Theorem
3]{Queyranne98a}.
Due to (i) the fact that all distributions can be computed in
$\mathcal{O}(\vert V \vert + \vert \mathcal{P} \vert \vert \mathcal{P}'
\vert)$, (ii) Proposition~\ref{prop:submodular} of this paper, (iii)
Theorem~3 in~\cite{Queyranne98a} (which uses OPTIMAL-SET) and (iv)
Proposition~\ref{prop:evaluateFrac} of this paper, the asymptotic
running time for mi\-ni\-mi\-zing Eq.~(\ref{eq:Frac}) under the
constraint $\mathcal{S} \notin \{\emptyset, \mathcal{P}\}$ amounts to
$\mathcal{O}(\vert V \vert + \vert \mathcal{P} \vert^3 \vert \mathcal{P}
\vert \vert \mathcal{P}'\vert) = \mathcal{O}(\vert V \vert + \vert
\mathcal{P} \vert^4 \vert \mathcal{P}' \vert)$.
\end{proof}
\subsubsection{Proof of Proposition~\ref{prop:bound}.}
\label{proof:bound}
\begin{proof}
\begin{align}
\phi_{\mathcal{P}'}(\mathcal{S}) &=\sum_{P' \in
\mathcal{P}'} \vert P' \vert \operatorname{peak}(\frac{\vert U_{\mathcal{S}} \cap
P' \vert}{\vert P' \vert})\nonumber\\ &= \sum_{P' \in \mathcal{P}'}
\vert P' \vert \min\{\frac{\vert U_{\mathcal{S}} \cap P' \vert}{\vert
P' \vert}, \frac{\vert U_{\mathcal{P} \setminus \mathcal{S}} \cap P'
\vert}{\vert P' \vert}\}\nonumber\\ &= \sum_{P' \in \mathcal{P}'}
\min\{\vert U_{\mathcal{S}} \cap P' \vert, \vert U_{\mathcal{P}
\setminus \mathcal{S}} \cap P' \vert\}\\ &\geq b(\mathcal{S}_s,
\mathcal{S}_t).
\end{align}
\end{proof}
\subsection{\textsc{$C_{\wedge}$}\xspace-correspondences and \textsc{$C_{m}$}\xspace-correspondences.}
\label{sec:app:C3C4}
We first show that finding an optimal \textsc{$C_{\wedge}$}\xspace-correspondence between
$\mathcal{P}$ and $\mathcal{P}'$ amounts to finding a nontrivial
minimum $\mathcal{S}$ of a symmetric and non-submodular function
$\phi^*: 2^{\mathcal{P}} \mapsto \mathbb{R}_{\geq 0}$. Alternatively,
$\mathcal{S}$ can be found through minimizing $\vert \mathcal{P}
\vert^2$ non-symmetric submodular functions. As for \textsc{$C_{\mathcal{P}}$}\xspace, a good
$\textsc{$C_{\wedge}$}\xspace$-correspondence is essentially a small cut of
$\mathcal{P}$. Since $\phi^*$ is symmetric, there exists a minimum
cut basis containing $\vert \mathcal{P} \vert - 1$ minimum $P_s$-$P_t$
cuts of $\mathcal{P}$. These cuts give rise to a natural collection of
$\vert \mathcal{P} \vert - 1$ best \textsc{$C_{\wedge}$}\xspace-correspondences.
The rest of the section is on \textsc{$C_{m}$}\xspace-correspondences. We derive a
property of \textsc{$C_{m}$}\xspace-correspondences which suggests that finding good
\textsc{$C_{m}$}\xspace-correspondences is more difficult than submodular
minimization. Nevertheless, the techniques that we developed for
finding good correspondences under the constraints $\textsc{$C_{\mathcal{P}}$}\xspace$ and
$\textsc{$C_{\wedge}$}\xspace$ may be useful for finding at least a subset of good
\textsc{$C_{m}$}\xspace-correspondence in a real-world application.
\paragraph*{\textsc{$C_{\wedge}$}\xspace-correspondences.}
The constraint \textsc{$C_{\wedge}$}\xspace: $\mathcal{S} \notin \{\emptyset, \mathcal{P}\}
\wedge \mathcal{S}' \notin \{\emptyset, \mathcal{P}'\}$ ensures that
$(U_{\mathcal{S}}, U_{\mathcal{P} \setminus \mathcal{S}})$ and
$(U_{\mathcal{S}'}, U_{\mathcal{P}' \setminus \mathcal{S}'})$ are cuts
of $V$. This is a prerequisite for finding a consensus partition via
good correspondences, see Section~\ref{subsec:constraints}.
Finding an optimal \textsc{$C_{\wedge}$}\xspace-correspondence $(\mathcal{S},
\mathcal{S}')$ amounts to finding $\emptyset \neq \mathcal{S}
\subsetneq \mathcal{P}$ with a minimum value of
$\phi^*~:~2^{\mathcal{P}} \mapsto
\mathbb{R}_{\geq 0}$ defined as
\begin{equation*}
\label{eq:phistar}
\phi^*(\mathcal{S}) := \left\{
\begin{array}{ll}
0 & \mbox{if } \mathcal{S} \in \{\emptyset, \mathcal{P}\},\\
\min_{\emptyset \neq \mathcal{S}' \subsetneq \mathcal{P}'} \vert
U_{\mathcal{S}} \triangle U_{\mathcal{S}'}\vert & \mbox{otherwise.}
\end{array}
\right.
\end{equation*}
\begin{prop} $\phi^*(\cdot)$
is symmetric and not submodular.
\label{prop:submodular-nondeg}
\end{prop}
\begin{proof}
If $\mathcal{S} \in \{\emptyset, \mathcal{P}\}$, then
$\phi^*(\mathcal{P} \setminus \mathcal{S}) = \phi^*(\mathcal{S}) =
0$. Otherwise,
\begin{align*}
\phi^*(\mathcal{P} \setminus \mathcal{S})
&= \min_{\emptyset \neq \mathcal{S}' \subsetneq \mathcal{P}'} \vert
U_{\mathcal{P} \setminus \mathcal{S}} \triangle
U_{\mathcal{S}'}\vert\\ &= \min_{\emptyset \neq \mathcal{S}'
\subsetneq \mathcal{P}'} \vert U_{\mathcal{P} \setminus \mathcal{S}}
\triangle U_{\mathcal{P}' \setminus \mathcal{S}'}\vert\\ &=
\min_{\emptyset \neq \mathcal{S}' \subsetneq \mathcal{P}'} \vert
U_{\mathcal{S}} \triangle U_{\mathcal{S}'}\vert =
\phi^*(\mathcal{S}).
\end{align*}
\noindent A counterexample to submodularity of
$\phi^*(\cdot)$ is provided in
Figure~\ref{fig:exampleG}.
\end{proof}
\begin{figure}
\caption{\textsc{Counterexample to submodularity of}
\label{fig:exampleG}
\end{figure}
Analogous to \textsc{$C_{\mathcal{P}}$}\xspace-correspondences, we reformulate the problem of
finding an optimal \textsc{$C_{\wedge}$}\xspace-correspondence under the constraint $P_i'
\in \mathcal{S}'$ and $P_j' \in \mathcal{P}' \setminus
\mathcal{S}'$. Let $\mathcal{S} \subseteq \mathcal{P}$ and
$\mathcal{S}' \subseteq \mathcal{P}'$. Then,
Eq.~(\ref{eq:secondMin}) and the constraint imply
\begin{align}
\vert U_{\mathcal{S}} \triangle U_{\mathcal{S}'} \vert = \vert
U_{\mathcal{S}} \cap P_j' \vert + \vert P_i' \vert - \vert
U_{\mathcal{S}} \cap P_i' \vert + \nonumber\\
\sum_{\substack{P'
\notin \mathcal{S}'\\P' \notin \{P_i', P_j'\}}} \vert
U_{\mathcal{S}} \cap P'\vert + \sum_{\substack{P' \in \mathcal{S}'\\P'
\notin \{P_i', P_j'\}}}(\vert P' \vert - \vert U_{\mathcal{S}}
\cap P' \vert)
\label{eq:thirdMin}
\end{align}
Analogous to Eq.~(\ref{eq:towardPeak}) we set
\begin{equation}
\label{eq:partner1Copy}
\mathcal{S}' := \{P_i'\} \cup \{P' \in \mathcal{P}' \setminus
\{P_i', P_j'\} : \vert U_{\mathcal{S}} \cap P' \vert > \frac{\vert
P'\vert}{2}\},
\end{equation}
and thus minimize the contribution (damage) of each $P' \in
\mathcal{P}'$ in the sums of Eq.~(\ref{eq:thirdMin}). In
particular, the following holds for any $\emptyset \subseteq
\mathcal{S} \subseteq \mathcal{P}$:
\begin{align*}
\min_{\mathcal{S}' \subseteq \mathcal{P}'} \vert U_{\mathcal{S}}
\triangle U_{\mathcal{S}'} \vert =
\vert U_{\mathcal{S}} \cap P_j' \vert + \vert P_i' \vert - \vert U_{\mathcal{S}} \cap P_i' \vert \\
+ \nonumber \sum_{\substack{P' \in \mathcal{P}'\\P' \notin \{P_i',
P_j'\}}} \min\{\vert U_{\mathcal{S}} \cap P'\vert, \vert P' \vert
- \vert U_{\mathcal{S}} \cap P' \vert\} \nonumber\\
= \vert U_{\mathcal{S}} \cap P_j' \vert + \vert P_i' \vert - \vert
U_{\mathcal{S}} \cap P_i' \vert + \nonumber\\
\sum_{\substack{P' \in \mathcal{P}'\\P' \notin \{P_i', P_j'\}}} \vert P' \vert
\min\{\frac{\vert U_{\mathcal{S}} \cap P'\vert}{\vert P' \vert}, 1 -
\frac{\vert U_{\mathcal{S}} \cap P' \vert}{\vert P' \vert}\}
\nonumber\\
= \vert U_{\mathcal{S}} \cap P_j' \vert + \vert P_i'
\vert - \vert U_{\mathcal{S}} \cap P_i' \vert +
\nonumber\\
\sum_{\substack{P' \in \mathcal{P}'\\P' \notin \{P_i',
P_j'\}}} \vert P' \vert \operatorname{peak}(\frac{\vert U_{\mathcal{S}} \cap P'
\vert}{\vert P' \vert}).
\end{align*}
\noindent Proposition~\ref{prop:constrained} below summarizes our
findings.
\begin{prop}
An optimal \textsc{$C_{\wedge}$}\xspace-correspondence
$(\mathcal{S}, \mathcal{S}')$ under the constraint $P_i' \in
\mathcal{S}'$ and $P_j' \in \mathcal{P}' \setminus \mathcal{S}'$ can
be computed by first finding $\emptyset \neq \mathcal{S} \subsetneq
\mathcal{S}$ that minimizes the term
\begin{align*}
\phi^*_{i',j'}(\mathcal{S}) := \vert U_{\mathcal{S}} \cap P_j' \vert +
\vert P_i' \vert - \vert U_{\mathcal{S}} \cap P_i' \vert + \\
\sum_{\substack{P' \in \mathcal{P}'\\P' \notin \{P_i', P_j'\}}} \vert
P' \vert \operatorname{peak}(\frac{\vert U_{\mathcal{S}} \cap P' \vert}{\vert P'
\vert})
\end{align*}
\noindent and then setting $\mathcal{S}'$ as in
Eq.~(\ref{eq:partner1Copy}).
\label{prop:constrained}
\end{prop}
\begin{prop}$\phi^*_{i',j'}(\cdot)$ is submodular and not
symmetric.
\label{prop:submodular_constrained}
\end{prop}
\begin{proof}
Sums and positive multiples of submodular functions are
submodular~\cite{Schrijver2003a}. Thus, since $f(\mathcal{S}) := \vert
U_{\mathcal{S}} \cap P_j' \vert$ and $g(\mathcal{S}) := \vert P_i'
\vert - \vert U_{\mathcal{S}} \cap P_i' \vert$ are modular functions,
submodularity of $\phi^*_{i',j'}(\cdot)$ follows from
$h(\mathcal{S}) := \operatorname{peak}(\frac{\vert U_{\mathcal{S}} \cap P'
\vert}{\vert P' \vert})$ being submodular for all $P' \in
\mathcal{P}'$. The latter was shown in the proof of
Proposition~\ref{prop:submodular}. To see that
$\phi^*_{i',j'}(\cdot)$ is not symmetric, first note that
$\phi^*_{i',j'}(\mathcal{S}) = \phi^*_{j',i'}(\mathcal{P \setminus
\mathcal{S}})$. Thus, symmetry of $\phi^*_{i',j'}(\cdot)$ would
imply $\phi^*_{i',j'}(\mathcal{S}) = \phi^*_{j',i'}(\mathcal{S})$
for all $\emptyset \neq \mathcal{S} \subsetneq \mathcal{P}$. For a
counterexample see Figure~\ref{fig:exampleG}.
\end{proof}
The function $\phi^*(\cdot)$ is symmetric, see
Proposition~\ref{prop:submodular-nondeg}. Thus, we can compute a
minimum cut basis of $\mathcal{P}$ w.\,r.\,t.\xspace $\phi^*(\cdot)$ by finding a
certain collection of $\vert \mathcal{P} \vert - 1$ minimum
$P_s$-$P_t$ cuts of $\mathcal{P}$~\cite{Cheng1992a}. Minimum
$P_s$-$P_t$ cuts are defined as in Definition~\ref{def:s-t-cuts} with
the exception that ``minimum'' now is w.\,r.\,t.\xspace $\phi^*(\cdot)$. In
contrast to the $\vert \mathcal{P} \vert - 1$ cuts in the minimum
basis w.\,r.\,t.\xspace $\phi_{\mathcal{P}'}(\cdot)$, the cuts in the minimum basis
w.\,r.\,t.\xspace $\phi^*(\cdot)$ are crossing cuts, in general.
\paragraph*{\textsc{$C_{m}$}\xspace-correspondences.}
These correspondences raise two major difficulties. First, the sets
$\mathcal{S}$ in mutual correspondences $(\mathcal{S}, \mathcal{S}')$
do not form a lattice family~\cite{Goemans95a}. The latter is a family
$\mathcal{L}$ of subsets of a set $V$ such that $A, B \in \mathcal{L}$
implies $A \cap B, A \cup B \in \mathcal{L}$. For an example of mutual
correspondences causing a lattice family conflict see
Figure~\ref{fig:exampleG}. Second, it can occur that there are no
\textsc{$C_{m}$}\xspace-correspondences at all, which raises a serious problem to any
B\&B algorithm for finding
\textsc{$C_{m}$}\xspace-correspondences.
\subsection{Optimal \textsc{$C_{\mathcal{P}}$}\xspace-correspondences and mutual correspondences.}
\label{subsec:app:mutual}
Propositions~\ref{prop:char_nontriv} and~\ref{prop:char_nondeg} below
tells us that an optimal \textsc{$C_{\mathcal{P}}$}\xspace-correspondence or
\textsc{$C_{\wedge}$}\xspace-cor\-res\-pon\-dence is either mutual or simple. Here, ``simple''
means that Eq.~(\ref{eq:simple}) is fulfilled.
\begin{prop}
If an optimal \textsc{$C_{\mathcal{P}}$}\xspace-correspondence $(\mathcal{S}, \mathcal{S}')$
is not mutual, then
\begin{equation}
\label{eq:simple}
\vert \mathcal{S} \vert \in \{1, \vert \mathcal{P} \vert - 1\} \vee \vert \mathcal{S}' \vert \in \{1, \vert \mathcal{P}' \vert - 1\}.
\end{equation}
\label{prop:char_nontriv}
\end{prop}
\begin{proof}
Let $(\mathcal{S}, \mathcal{S}')$ be an optimal \textsc{$C_{\mathcal{P}}$}\xspace-correspondence
that is not mutual. First assume that $(\mathcal{S}, \mathcal{S}')$
does not fulfill item $1.$ in Definition~\ref{def:mutual}. Then there
exists $P \in \mathcal{S}$ with $\vert U_{\mathcal{S} \setminus \{P\}}
\triangle U_{\mathcal{S}'} \vert < \vert U_{\mathcal{S}} \triangle
U_{\mathcal{S}'} \vert$. Since $(\mathcal{S}, \mathcal{S}')$ is an
optimal \textsc{$C_{\mathcal{P}}$}\xspace-correspondence, the correspondence $(\mathcal{S}
\setminus \{P\}, \mathcal{S}')$ cannot fulfill \textsc{$C_{\mathcal{P}}$}\xspace, i.\,e.,\xspace $\vert
\mathcal{S} \vert = 1$. Likewise, items $2.$, $3.$ and $4.$ imply
$\vert \mathcal{S} \vert = \vert \mathcal{P} \vert - 1$, $\vert
\mathcal{S}' \vert = 1$ and $\vert \mathcal{S}' \vert = \vert
\mathcal{P}' \vert - 1$, respectively.
\end{proof}
\noindent An analogous proof leads to an analogous characterization
of \textsc{$C_{\wedge}$}\xspace-cor\-res\-pond\-en\-ces.
\begin{prop}
If an optimal \textsc{$C_{\wedge}$}\xspace-cor\-res\-pon\-dence $(\mathcal{S}, \mathcal{S}')$ is
not mutual, then Eq.~(\ref{eq:simple}) holds.
\label{prop:char_nondeg}
\end{prop}
\subsection{Details on B\&B algorithm.}
\label{subsec:app:bb}
The following notation will make it easier to formulate our
B\&B algorithm presented as Algorithm~\ref{algo:BB}.
\begin{notation}
\label{notation:indices}
W.\,l.\,o.\,g.\ the parts in $\mathcal{S}_s \cup \mathcal{S}_t \setminus
\{P_s, P_t\}$ are denoted by $\hat{P}_1, \dots, \hat{P}_{\vert
\mathcal{S}_s \cup \mathcal{S}_t \vert - 2}$, and the indices
reflect the order in which the parts were added to $\mathcal{S}_s
\setminus \{P_s\}$ or $\mathcal{S}_t \setminus \{P_t\}$ (the larger
an index, the later the part was added).
\end{notation}
\begin{algorithm}[tbh]
\caption{B\&B algorithm for finding a minimum
$P_s$-$P_t$ cut.}
\label{algo:BB}
\begin{algorithmic}[1]
\State $\mathcal{S}_s \gets \{P_s\}$, $\mathcal{S}_t \gets
\{P_t\}$
\State $bestSoFar \gets \infty$
\Do
\State $\textsc{greedy}\xspace(\mathcal{S}_s, \mathcal{S}_t, bestSoFar)$
\If{$\mathcal{S}_s \cup \mathcal{S}_t = \mathcal{P}$}
\Comment{i.\,e.,\xspace we have found a $P_s$-$P_t$ cut}
\If{$b(\mathcal{S}_s, \mathcal{S}_t) < bestSoFar$}
\Comment{$b(\mathcal{S}_s, \mathcal{S}_t) = \phi_{\mathcal{P}'}(\mathcal{S}_s) = \phi_{\mathcal{P}'} (\mathcal{S}_t)$}
\State $\mathcal{S} \gets \mathcal{S}_s$
\State $bestSoFar \gets b(\mathcal{S}_s, \mathcal{S}_t)$
\EndIf
\EndIf
\State $i \gets 0$ \Comment{Beginning of undo}
\Do
\If{$\hat{P}_{\vert \mathcal{S}_s \cup \mathcal{S}_t \vert-2-i}
\in \mathcal{S}_s$}
\State $\mathcal{S}_s \gets \mathcal{S}_s \setminus \hat{P}_{\vert \mathcal{S}_s \cup \mathcal{S}_t \vert-2-i}$
\Else \Comment{i.\,e.,\xspace $\hat{P}_{\vert \mathcal{S}_s \cup \mathcal{S}_t \vert-2-i}
\in \mathcal{S}_t$}
\State $\mathcal{S}_t \gets \mathcal{S}_t \setminus \hat{P}_{\vert \mathcal{S}_s \cup \mathcal{S}_t \vert-2-i}$
\EndIf
\State $i \gets i + 1$
\doWhile{$(\mathcal{S}_s, \mathcal{S}_t) \neq (\{P_s\}, \{P_t\})
\wedge \textsc{dejaVu}(A(\mathcal{S}_s, \mathcal{S}_t))$} \Comment{End
of undo}
\If{$(\mathcal{S}_s, \mathcal{S}_t) \neq (\{P_s\}, \{P_t\})$}
\State $(\mathcal{S}_s, \mathcal{S}_t) \gets A(\mathcal{S}_s, \mathcal{S}_t)$
\EndIf
\doWhile{$(\mathcal{S}_s, \mathcal{S}_t) \neq (\{P_s\}, \{P_t\})$}
\State {\bf return} $(\mathcal{S}, \mathcal{P} \setminus \mathcal{S})$
\end{algorithmic}
\end{algorithm}
After initializing $\mathcal{S}_s$ and $\mathcal{S}_t$, our
B\&B algorithm calls $\textsc{greedy}\xspace(\mathcal{S}_s, \mathcal{S}_t,
\infty)$, see Algorithm~\ref{algo:BB} in Section~\ref{subsec:greedy}. In later calls of
$\textsc{greedy}\xspace(\mathcal{S}_s, \mathcal{S}_t, bestSoFar)$, we always have
$(\mathcal{S}_s \supsetneq \{P_s\} \vee \mathcal{S}_t \supsetneq
\{P_t\}) \wedge \mathcal{S}_s \cap \mathcal{S}_t = \emptyset$, and
$bestSoFar$ amounts to the minimum weight ($\phi_{\mathcal{P}'}$
value) of the $P_s$-$P_t$ cuts found so far (see lines 5-10 of
Algorithm~\ref{algo:BB}).
The following definition will make it easier to address the remaining
questions whose answer was left open in Section~\ref{subsec:BB_basicAlgo}.
\begin{definition}
If $\hat{P}_{\vert \mathcal{S}_s \cup \mathcal{S}_t \vert - 2}$ is
contained in $\mathcal{S}_s$, the \emph{alternative} to $(\mathcal{S}_s,
\mathcal{S}_t)$ called $A(\mathcal{S}_s, \mathcal{S}_t)$
is $(\mathcal{S}_s \setminus \{\hat{P}_{\vert
\mathcal{S}_s \cup \mathcal{S}_t \vert - 2}\}, \mathcal{S}_t
\cup \{\hat{P}_{\vert \mathcal{S}_s \cup \mathcal{S}_t \vert -
2}\})$. If $\hat{P}_{\vert \mathcal{S}_s \cup \mathcal{S}_t
\vert - 2}$ is contained in $\mathcal{S}_t$, the \emph{alternative} to
$(\mathcal{S}_s, \mathcal{S}_t)$ is $A(\mathcal{S}_s, \mathcal{S}_t) :=
(\mathcal{S}_s \cup
\{\hat{P}_{\vert \mathcal{S}_s \cup \mathcal{S}_t \vert - 2}\},
\mathcal{S}_t \setminus \{\hat{P}_{\vert \mathcal{S}_s \cup
\mathcal{S}_t \vert - 2}\})$.
\end{definition}
The answer to 2a) now is ``Undo the assignment of $\hat{P}_{\vert
\mathcal{S}_s \cup \mathcal{S}_t \vert - 2}$. Keep undoing the
latest assignments until some $\hat{P}_{\vert \mathcal{S}_s \cup
\mathcal{S}_t \vert - 2 - i}$, $i \geq 1$, is reached such that
$\textsc{greedy}\xspace(\cdot, \cdot, \cdot)$ has \emph{not} yet been called with the
first two arguments given by $A(\mathcal{S}_s, \mathcal{S}_t)$.'' In
the pseudocode of Algorithm~\ref{algo:BB}, a boolean function called
$\textsc{dejaVu}(\cdot, \cdot)$ is used to express whether $A(\mathcal{S}_s,
\mathcal{S}_t)$ has entered the call of $\textsc{greedy}\xspace(\cdot, \cdot, \cdot)$
before, see line 19 of Algorithm~\ref{algo:BB}. This line guarantees
termination of our B\&B algorithmB\&B algorithm. The answer to 2b) then
is ``call $\textsc{greedy}\xspace(\cdot, \cdot, \cdot)$ with $A(\mathcal{S}_s,
\mathcal{S}_t)$ and the current value of $bestSoFar$'' (see lines 21
and 4 of Algorithm~\ref{algo:BB}).
\if #0
\noindent We conclude this section with two implementation details.
\begin{itemize}
\item We implement the boolean function $\textsc{dejaVu}(\cdot, \cdot)$ as a
boolean vector called $doneWith[\cdot]$. The two arguments
$\mathcal{S}_s$, $\mathcal{S}_t$ of $\textsc{dejaVu}(\cdot, \cdot)$
correspond to a single index $i$ of $doneWith$. Specifically, $i$ is
the cardinality of $\mathcal{S}_s \cup \mathcal{S}_t \setminus
\{P_s, P_t\}$ or, equivalently, the highest index, $\vert
\mathcal{S}_s \cup \mathcal{S}_t \vert - 2$, in
Notation~\ref{notation:indices}. The vector $doneWith$ has length
$\vert \mathcal{P} \vert - 2$, and its entries are initialized to
false. Anytime $\textsc{greedy}\xspace(\cdot, \cdot, \cdot)$ is called with
$(\mathcal{S}_s, \mathcal{S}_t) = A(\hat{\mathcal{S}}_s,
\hat{\mathcal{S}}_t)$ for some $\hat{\mathcal{S}}_s,
\hat{\mathcal{S}}_t \subset \mathcal{P}$, $doneWith[i]$ is set to
true. Furthermore, if backtracking goes back behind an
index $j$, $doneWith[j]$ is set to false.
\item For efficient updates of $b(\mathcal{S}_s, \mathcal{S}_t)$ in
Algorithm~\ref{algo:BB}, i.\,e.,\xspace when $\mathcal{S}_s$ or $\mathcal{S}_t$
grows or shrinks by one part, we use distributions as in
Section~\ref{subsec:goodC2_running_times} --- this time the \emph{rows} of the
contingency table of $\mathcal{P}$ and $\mathcal{P}'$. Specifically,
let $P \in \mathcal{P}$. Then $d_{P}[\cdot]$ is a vector of length
$\vert \mathcal{P}' \vert$ defined by $d_{P}[j] := \vert P \cap P'_j
\vert \mbox{~for~} 1 \leq j \leq \vert \mathcal{P}' \vert$. We
define $d_{\mathcal{S}_s} := \sum_{P \in \mathcal{S}_s} d_{P}$. Now,
adding or removing a part $P$ from $\mathcal{S}_s$ corresponds to
the command $d_{\mathcal{S}_s} \gets d_{\mathcal{S}_s} + d_{P}$ and
$d_{\mathcal{S}_s} \gets d_{\mathcal{S}_s} - d_{P}$,
respectively. By carrying along $d_{\mathcal{S}_s}$ and
$d_{\mathcal{S}_t}$, we can compute the bounds $b(\cdot, \cdot)$ as
follows.
\begin{equation*}
\label{eq:bound2}
b(\mathcal{S}_s, \mathcal{S}_t) = \sum^{\vert \mathcal{P}' \vert}_{j=1}
\min\{d_{\mathcal{S}_s}[j], d_{\mathcal{S}_t}[j]\}.
\end{equation*}
Since the distributions $d_{P'}\lbrack \cdot \rbrack$ are basically
long vectors, our implementation is well-suited for parallel
processing.
\end{itemize}
\fi
\subsection{Extensions of B\&B from \textsc{$C_{\mathcal{P}}$}\xspace to \textsc{$C_{\wedge}$}\xspace and \textsc{$C_{m}$}\xspace.}
\label{subsubsec:extensions}
The extension from \textsc{$C_{\mathcal{P}}$}\xspace to \textsc{$C_{\wedge}$}\xspace needs two adaptations. First, an
early exit $(\mathcal{S}_s, \mathcal{S}_s')$ or $(\mathcal{S}_t,
\mathcal{S}_t')$ must fulfill $\mathcal{S}_s' \neq \emptyset$ and
$\mathcal{S}_t' \neq \emptyset$, respectively. Second, assume that our
B\&B algorithm has reached a point where all $P \in
\mathcal{P}$ have been assigned to the $s$-side or to the $t$-side. If
$\mathcal{S}'$ is still in $\{\emptyset, \mathcal{P}\}$, we modify it
such that it is not in $\{\emptyset, \mathcal{P}\}$ anymore and such
that the damage to $\phi_{\mathcal{P}'}(\cdot)$ is minimum.
If $\textsc{$C_{m}$}\xspace$-correspondences are to be found, the search can be
interrupted whenever there exists $P_t \in \mathcal{S}_t$ such that
$\vert P_t \cap U_{\mathcal{S}_s'} \vert > \vert P_t \vert / 2$. A second
analogous criterion for interrupting the search arises from exchanging
the roles of $s$ and $t$. Moreover, early exits $(\mathcal{S}_s,
\mathcal{S}_s')$ [$(\mathcal{S}_t, \mathcal{S}_t')$] have to be
checked for mutuality of $\mathcal{S}_s$ and $\mathcal{S}_s'$
[$\mathcal{S}_t$ and $\mathcal{S}_t'$]. Analogously, at any point
where all $P \in \mathcal{P}$ have been assigned to the $s$-side or to
the $t$-side, the current correspondence $(\mathcal{S}, \mathcal{S}')$
must be checked for mutuality of $\mathcal{S}$ and $\mathcal{S}'$.
\subsection{Running times.}
\label{sec:app:run}
The detailed running times of the algorithms under consideration are given in
Tables~\ref{tab:running_BB} and~\ref{tab:running_greed} below.
\begin{table}[h!]
\caption{Running times (in seconds) for calculating the $\vert
\mathcal{P} \vert - 1$ best correspondences using the B\&B algorithm
from Section~\ref{sec:BB}. Minima, mean values and maxima are over
10 runs (the community detection algorithm is non-deterministic).}
\label{tab:running_BB}
\begin{center}
\begin{small}
\scalebox{0.78}{
\begin{tabular}{ l | l | r | r | r | r }
Graph ID & Name & Min & Mean & Max\\ \hline \hline
1 & \textsc{p2p-Gnutella} & {0.052} & {0.060} & {0.070}\\\hline
2 & \textsc{PGPgiantcompo} & {0.256} & {0.313} & {0.379}\\\hline
3 & \textsc{email-EuAll} & {0.255} & {0.370} & {0.574}\\\hline
4 & \textsc{as-22july06} & {0.292} & {0.329} & {0.386}\\\hline
5 & \textsc{soc-Slashdot0902} & {1.048} & {1.557} & {2.855}\\\hline
6 & \textsc{loc-brightkite\_edges} & {4.309} & {5.520} & {11.210}\\\hline
7 & \textsc{loc-gowalla\_edges} & {28.013} & {50.265} & {240.330}\\\hline
8 & \textsc{coAuthorsCiteseer} & {19.871} & {29.574} & {56.117}\\\hline
9 & \textsc{wiki-Talk} & {38.403} & {1230.200} & {9554.700}\\\hline
10 & \textsc{citationCiteseer} & {12.297} & {13.353} & {15.488}\\\hline
11 & \textsc{coAuthorsDBLP} & {27.791} & {23.892} & {26.606}\\\hline
12 & \textsc{web-Google} & {20.414} & {22.309} & {25.432}\\\hline
13 & \textsc{coPapersCiteseer} & {77.912} & {356.130} & {1961.300}\\\hline
14 & \textsc{coPapersDBLP} & {38.824} & {36.438} & {39.386}\\\hline
\end{tabular}}
\end{small}
\end{center}
\end{table}
\begin{table}[h!]
\caption{Running times (in seconds) for calculating $\vert \mathcal{P}
\vert - 1$ correspondences using the algorithm \textsc{greedy}\xspace from
Section~\ref{sec:BB}. Minima, mean values and maxima are over 10
runs (the community detection algorithm is non-deterministic).}
\label{tab:running_greed}
\begin{center}
\begin{small}
\scalebox{0.78}{
\begin{tabular}{ l | l | r | r | r | r }
Graph ID & Name & Min & Mean & Max\\ \hline \hline
1 & \textsc{p2p-Gnutella} & {0.034} & {0.037} & {0.041}\\\hline
2 & \textsc{PGPgiantcompo} & {0.119} & {0.136} & {0.150}\\\hline
3 & \textsc{email-EuAll} & {0.134} & {0.166} & {0.205}\\\hline
4 & \textsc{as-22july06} & {0.125} & {0.148} & {0.189}\\\hline
5 & \textsc{soc-Slashdot0902} & {0.435} & {0.691} & {0.958}\\\hline
6 & \textsc{loc-brightkite\_edges} & {2.102} & {2.173} & {2.304}\\\hline
7 & \textsc{loc-gowalla\_edges} & {12.980} & {14.411} & {15.479}\\\hline
8 & \textsc{coAuthorsCiteseer} & {7.852} & {8.340} & {9.086}\\\hline
9 & \textsc{wiki-Talk} & {23.290} & {28.893} & {36.077}\\\hline
10 & \textsc{citationCiteseer} & {6.198} & {6.801} & {7.144}\\\hline
11 & \textsc{coAuthorsDBLP} & {10.620} & {11.194} & {11.683}\\\hline
12 & \textsc{web-Google} & {8.815} & {9.575} & {10.243}\\\hline
13 & \textsc{coPapersCiteseer} & {18.849} & {20.035} & {22.214}\\\hline
14 & \textsc{coPapersDBLP} & {15.583} & {16.974} & {19.239}\\\hline
15 & \textsc{as-skitter} & {21.423} & {22.466} & {24.134}\\\hline
\end{tabular}}
\end{small}
\end{center}
\end{table}
\end{document} |
\begin{document}
\begin{frontmatter}
\title{$P_k$ and $C_k$-structure and substructure connectivity of hypercubes}
\author[]{Yihan Chen\corref{cor1}}
\ead{2016750518@smail.xtu.edu.cn}
\author[]{Bicheng Zhang\corref{cor1}}
\ead{zhangbicheng@xtu.edu.cn}
\address{School of Mathematics and Computational Science, Xiangtan Univerisity, Xiangtan, Hunan, 411105, PR China}
\cortext[cor1]{Corresponding author}
\begin{abstract}
Hypercube is one of the most important networks to interconnect processors in multiprocessor computer systems. Different kinds of connectivities are important parameters to measure the fault tolerability of networks. Lin et al.\cite{LinStructure} introduced the concept of $H$-structure connectivity $\kappa(Q_n;H)$ (resp. $H$-substructure connectivity $\kappa^s(Q_n;H)$) as the minimum cardinality of $F=\{H_1,\dots,H_m\}$ such that $H_i (i=1,\dots,m)$ is isomorphic to $H$ (resp. $F=\{H'_1,\dots,H'_m\}$ such that $H'_i (i=1,\dots,m)$ is isomorphic to connected subgraphs of $H$) such that $Q_n-V(F)$ is disconnected or trivial. In this paper, we discuss $\kappa(Q_n;H)$ and $\kappa^s(Q_n;H)$ for hypercubes $Q_n$ with $n\geq 3$ and $H\in \{P_k,C_k|3\leq k\leq 2^{n-1}\}$. As a by-product, we solve the problem mentioned in \cite{ManeStructure}.
\end{abstract}
\begin{keyword}
Structure connectivity;\;Substructure connectivity;\;Hypercubes;\;Path;\;Cycle
\end{keyword}
\end{frontmatter}
\section{Introduction}
In the design of multiprocessor computer systems, the analysis of topological properties of interconnection networks is very important. Usually, an efficient interconnection topology can offer higher speed and better quality of information transmission between processors, most importantly, it can also provide high fault tolerability.
Many regular networks have been proposed as the model to interconnect processors, such as hypercube, mesh and twiced cube etc. In the past, people used vertex connectivity to measure the fault tolerability of interconnect networks, their works focuse on the effect of individual processors becoming fault. People soon discovered that there were some shortcomings in the using of vertex connectivity, in order to make up for these shortcomings, Harary defined the conditional connectivity. Later, many people generalized vertex connectivity in various ways and got many benefits, for example, restricted connectivity, $g$-extra connectivity and $R_g$-connectivity etc.
Lin et al.\cite{LinStructure} introduced the concept of $H$-structure connectivity $\kappa(Q_n;H)$ and the $H$-substructure connectivity $\kappa^s(Q_n;H)$ and determined them for $H\in\{K_1,K_{1,1},K_{1,2},K_{1,3},C_4\}$. Li et al.\cite{LI2019169} determined $\kappa(H_n;T)$ and $\kappa^s(H_n;T)$ with $H_n$ the twisted hypercube and $T\in\{K_{1,r},P_k|r=3,4 \text{ and } 1\leq k\leq n\}$. Mane \cite{ManeStructure} proved that if $n\geq 4$, then for each $2\leq m \leq n-2$, $\kappa(Q_n;C_{2^m})\leq n-m$ and gave an open problem: For $n\geq 4$ and for each $2\leq m\leq n-2$, can we prove $\kappa(Q_n;C_{2^m})= n-m$?
In this paper, we discuss $\kappa(Q_n;H)$ and $\kappa^s(Q_n;H)$ for hypercubes $Q_n$ with $n\geq 3$ and $H\in \{P_k,C_k|3\leq k\leq 2^{n-1}\}$. As a by-product, we solve the problem above.
The rest of this paper is organized as follows. In \emph{Section 2} we give the fundamental definitions and theorems. In \emph{Section 3} we discuss $\kappa(Q_n;P_k)$ and $\kappa^s(Q_n;P_k)$ for hypercubes $Q_n$ with $n\geq 3$ and $3\leq k\leq 2^{n-1}$. In \emph{Section 4} we discuss $\kappa(Q_n;C_k)$ and $\kappa^s(Q_n;C_k)$ for hypercubes $Q_n$ with $n\geq 3$ and $4\leq k\leq 2^{n-1}$, $k$ is even and solve the open problem above.
\section{Definitions and preliminaries}
We are consistent with \cite{diestel2012graph} in terms of notations and definitions. Let $G(V,E)$ be an undirected simple graph with vertex set $V$ and edge set $E$, two vertices $u$ and $v$ are \emph{adjacent} if and only if they are two endpoints of some edge $uv$ in $E$. Let $S$ be a subset of $V$, we denote $N_G(S)$ the set of neighborhoods of $S$, particularly $N_G(\{u\})=N_G(u)$ is the set of neighborhoods of $u$. In graph theory, a \emph{path} of length $k-1$ is a sequence of distinct vertices $P_k=(v_0,v_1,\dots,v_{k-1})$ in which $v_iv_{i+1}\in E$ for $i=0,1,\dots,k-2$. A \emph{cycle} of length $l$ is a sequence of vertices $C_l=(v_0,v_1,\dots,v_l)$ in which $v_0=v_l$ and $v_i$ are distinct with $v_iv_{i+1}\in E$ for $i=0,1,\dots,l-1$. An undirected graph $G(V,E)$ is said to be \emph{connected} if and only if it has at least one vertex and there is a path between every pair of its vertices.\par
A \emph{$n$-dimensional hypercube} $Q_n$ is an undirected simple graph with vertex set $V(Q_n)=\{v=x_v^0x_v^1\dots x_v^{n-1}\mid x_v^i=0\;or\;1, i=0,\dots,n-1 \}$ and edge set $E(Q_n)$. Two vertices are adjacent if and only if the $n$-bit binary strings corresponding to them are differ in exactly one bit position. Let $v$ be a vertex of $Q_n$, we denote $(v)^i$ the neighborhood of $v$ whose $n$-bit binary string is differ from $v$'s in exactly the $i^{th}$ bit position for $i=0,1,\dots,n-1$. We notice that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges, and it can be divided into two $n-1$-dimentional hypercubes. We set $Q_n^i$ the induced subgraph of $Q_n$ with vertex set $V(Q_n^i)=\{v\mid v\in V(Q_n), x_v^{n-1}=i \}$ for $i=0,1$. It is easy to see that $Q_n^i$ is isomorphic to $Q_{n-1}$.\par
In the following, let's introduce some necessary definitions about the \emph{structure and substructure connectivity}, let $H$ and $G$ be undirected simple graph, $H$ is connected.
\begin{dfn}
A \emph{$H$-structure cut (resp. $H$-substructure cut)} of $G$ is a set $F$ of connected subgraphs of $G$ whose elements are isomorphic to $H$ \emph{(resp. connected subgraph of $H$)} and $G-V(F)$ is trivial or disconnected.
\end{dfn}
See {\bf Fig 1} and {\bf Fig 2} for an illustration.
\begin{center}
\textbf{Fig 1.} A $C_4$-structure cut
\end{center}
\begin{center}
\textbf{Fig 2.} A $C_4$-substructure cut
\end{center}
\begin{dfn}
The \emph{$H$-structure connectivity (resp. $H$-substructure connectivity)} denote by $\kappa(G;H)$ (\emph{resp.} $\kappa^s(G;H)$) is the minimum cardinality of $H$-structure cuts (\emph{resp.} $H$-substructure cuts).
\end{dfn}
\begin{dfn}(See {\cite{ZHOU2017208}})
Given a non-negative integer $g$, \emph{$g$-extra connectivity} of $G$ denote by $\kappa_g(G)$ is the minimum cardinality of sets of vertices in $G$ whose deletion disconnects $G$ and leaves each components with at least $g+1$ vertices.
\end{dfn}
\begin{lem}\label{dist2nodes}
Any two vertices in $V(Q_n)$ with a distance of $2$ have exactly $2$ common neighborhoods.
\end{lem}
\begin{proof}
According to the symmetry of the hypercube $Q_n$, without loss of generalicity we assume that the two vertices are $u=000\dots0$ and $v=110\dots0$, obviously their common neighborhoods are $u'=100\dots0$ and $v'=010\dots0$.
\end{proof}
\begin{thm}\label{gextra}(See {\cite{YangExtraconnectivity}})
If $n\geq 4$, then $\kappa_g(Q_n)=(g+1)n-2g-\binom{g}{2}$ for $0\leq g\leq n-4$, and $\kappa_g(Q_n)=\frac{n(n-1)}{2}$ for $n-3\leq g\leq n$.
\end{thm}
\begin{thm}\label{linsresult}(See {\cite{LinStructure}})
For $n\geq 4$
\begin{equation}
\label{res1}
\kappa(Q_n;H)=
\begin{cases}
n & \text{$H=K_1$,} \\
n-1 & \text{$H=K_{1,1}$,} \\
\lceil \frac{n}{2} \rceil & \text{$H\in \{K_{1,2},K_{1,3}\}$,} \\
n-2 & \text{$H=C_4$.}
\end{cases}
\end{equation}
\begin{equation}
\label{res2}
\kappa^s(Q_n;H)=
\begin{cases}
n & \text{$H=K_1$,} \\
n-1 & \text{$H=K_{1,1}$,} \\
\lceil \frac{n}{2} \rceil & \text{$H\in \{K_{1,2},K_{1,3},C_4\}$.}
\end{cases}
\end{equation}
\end{thm}
\begin{thm}\label{Manesresult}(See \cite{ManeStructure})
if $n\geq 4$, then for each $2\leq m \leq n-2$, $\kappa(Q_n;C_{2^m})\leq n-m$.
\end{thm}
\begin{lem}\label{TAMI}(See \cite{TAMIZHCHELVAM2019})
For an integer $n\geq 4$, $\kappa(Q_n;C_6)\geq \lceil \frac{n}{3}\rceil$.
\end{lem}
\begin{thm}\label{hamilton}(See {\cite{SaadTopological}})
$Q_n$ is Hamiltonian.
\end{thm}
\section{$\kappa(Q_n;P_k)$ and $\kappa^s(Q_n;P_k)$ for $n\geq 3$ and $3\leq k \leq 2^{n-1}$}
In this section we discuss $H$-structure connectivity and $H$-substructure connectivity of $Q_n$ with $H$ the path of length $k-1$.
\begin{lem}\label{pathleq}
If $n\geq 3$ and $3\leq k \leq 2^{n-1}$, then
\begin{equation} \kappa^s(Q_n;P_k)\leq \kappa(Q_n;P_k)\leq
\begin{cases}
\lceil \frac{2n}{k+1} \rceil & \text{if $k$ is odd,} \\
\lceil \frac{2n}{k} \rceil & \text{if $k$ is even.}
\end{cases}
\end{equation}
\end{lem}
\begin{proof}
We set $v=00\dots0$ being a vertex in $Q_n$.
\vskip .2cm
{\bf Case 1.} $k$ is odd.
\vskip .2cm
{\bf Subcase 1.1.} $k\geq 2n-1$.
We notice that there is a hamiltonian cycle in $Q_n^1$ with ${({(v)}^{n-2})}^{n-1}(v)^{n-1}$ as one of its edges by Theorem \ref{hamilton} and the symmetry of $Q_n^1$. For each $i=1,2,\dots,2^{n-1}-2$, set $u_i$ the $i^{th}$ vertex after ${\left({\left(v\right)}^{n-2}\right)}^{n-1}$ and $(v)^{n-1}$ along the hamiltonian cycle. Then $F=\left\{\left((v)^0,{\left(\left(v\right)^0\right)}^1,(v)^1,\dots,{(v)}^{n-2},{\left({\left(v\right)}^{n-2}\right)}^{n-1},(v)^{n-1},u_1,\dots,u_{k-(2n-1)}\right)\right\}$ is a $P_k$-structure cut of $Q_n$ leaves at least two connected components in $Q_n-V(F)$ and $\{v\}$ is one of them. Therefore $\kappa(Q_n;P_k)\leq 1=\lceil \frac{2n}{k+1} \rceil$.
\begin{center}
\end{center}
\begin{center}
\textbf{Fig 3.} $k\geq 2n-1$
\end{center}
\vskip .2cm
{\bf Subcase 1.2.} $1\leq k\leq 2n-1$.
\vskip .2cm
{\bf Subcase 1.2.1.}$\frac{k+1}{2}$ divides $n$.
We set $F=\{P_k^i|i=0,1,\dots,\frac{2n}{k+1}-1\}$ with \\ $P_k^i=\left((v)^{i\frac{k+1}{2}},{\left(\left(v\right)^{i\frac{k+1}{2}}\right)}^{i\frac{k+1}{2}+1},(v)^{i\frac{k+1}{2}+1},\dots,(v)^{(i+1)\frac{k+1}{2}-1}\right)$. Then $F$ is obviously a $P_k$-structure cut of $Q_n$, implies $\kappa(Q_n;P_k)\leq \frac{2n}{k+1}=\lceil \frac{2n}{k+1} \rceil$.
\vskip .2cm
{\bf Subcase 1.2.2.}$\frac{k+1}{2}$ does not divide $n$.
We set $F=\{P_k^i|i=0,1,\dots,\lceil \frac{2n}{k+1} \rceil-1\}$ with\\ $P_k^i=\left((v)^{i\frac{k+1}{2}},{\left(\left(v\right)^{i\frac{k+1}{2}}\right)}^{i\frac{k+1}{2}+1},(v)^{i\frac{k+1}{2}+1},\dots,(v)^{(i+1)\frac{k+1}{2}-1}\right)$ for $i=0,\dots,\lceil \frac{2n}{k+1} \rceil-2$ and $P_k^{\lceil \frac{2n}{k+1} \rceil-1}=\left((v)^{n-\frac{k+1}{2}},{\left(\left(v\right)^{n-\frac{k+1}{2}}\right)}^{n-\frac{k+1}{2}+1},(v)^{n-\frac{k+1}{2}+1},\dots,(v)^{n-1}\right)$. Then $F$ is obviously a $P_k$-structure cut of $Q_n$, implies $\kappa(Q_n;P_k)\leq \lceil \frac{2n}{k+1} \rceil$.\\
Let's have $n=5$ and $k=3$ as an example:
\begin{center}
\textbf{Fig 4.} $n=5$ and $k=3$
\end{center}
\vskip .2cm
{\bf Case 2.} $k$ is even.
\vskip .2cm
{\bf Subcase 2.1.} $k\geq 2n$. The proof is as same as {\bf Subcase 1.1.}
\vskip .2cm
{\bf Subcase 2.2.} $2\leq k\leq 2n-2$.
\vskip .2cm
{\bf Subcase 2.2.1.} $\frac{k}{2}$ divides $n$.
We set $F=\{P_k^i|i=0,1,\dots,\frac{2n}{k}-1\}$ with \\ $P_k^i=\left((v)^{i\frac{k}{2}},{\left(\left(v\right)^{i\frac{k}{2}}\right)}^{i\frac{k}{2}+1},(v)^{i\frac{k}{2}+1},\dots,(v)^{(i+1)\frac{k}{2}-1},{\left(\left(v\right)^{(i+1)\frac{k}{2}-1}\right)}^{(i+1)\frac{k}{2}}\right)$ for $i=0,\dots, \frac{2n}{k}-2$ and $P_k^{\frac{2n}{k}-1}=\left((v)^{\left(\frac{2n}{k}-1\right)\frac{k}{2}},{\left(\left(v\right)^{\left(\frac{2n}{k}-1\right)\frac{k}{2}}\right)}^{\left(\frac{2n}{k}-1\right)\frac{k}{2}+1},(v)^{\left(\frac{2n}{k}-1\right)\frac{k}{2}+1},\dots,(v)^{n-1},{\left(\left(v\right)^{n-1}\right)}^{0}\right)$. Then $F$ is obviously a $P_k$-structure cut of $Q_n$, implies $\kappa(Q_n;P_k)\leq \frac{2n}{k}=\lceil \frac{2n}{k} \rceil$.
\vskip .2cm
{\bf Subcase 2.2.2.} $\frac{k}{2}$ does not divide $n$.
We set $F=\{P_k^i|i=0,1,\dots,\lceil \frac{2n}{k} \rceil-1\}$ with \\ $P_k^i=\left((v)^{i\frac{k}{2}},{\left(\left(v\right)^{i\frac{k}{2}}\right)}^{i\frac{k}{2}+1},(v)^{i\frac{k}{2}+1},\dots,(v)^{(i+1)\frac{k}{2}-1},{\left(\left(v\right)^{(i+1)\frac{k}{2}-1}\right)}^{(i+1)\frac{k}{2}}\right)$ for $i=0,\dots,\lceil \frac{2n}{k} \rceil-2$ and $P_k^{\lceil \frac{2n}{k} \rceil-1}=\left((v)^{n-\frac{k}{2}},{\left(\left(v\right)^{n-\frac{k}{2}}\right)}^{n-\frac{k}{2}+1},(v)^{n-\frac{k}{2}+1},\dots,(v)^{n-1},{\left(\left(v\right)^{n-1}\right)}^0\right)$. Then $F$ is obviously a $P_k$-structure cut of $Q_n$, implies $\kappa(Q_n;P_k)\leq \lceil \frac{2n}{k} \rceil$.
\end{proof}
\begin{lem}\label{forleqk-1}
For $n\geq 3$ and $3\leq k \leq 2^{n-1}$, let $k=3q_k+r_k$ with non-negative integers $q_k$ and $r_k$, $0\leq r_k\leq 2$. If $u$ and $v$ are adjacent in $Q_n-V(P_k)$, then $|N_{Q_n}(\{u,v\})\cap V(P_k)|\leq 2q_k+r_k$. In particular, $|N_{Q_n}(\{u,v\})\cap V(P_k)|\leq k-1$.
\end{lem}
\begin{proof}
Set the vertices in $V(P_k)$ in turn as $v_i$ for $i=1,2,\dots,k$. First of all, we prove $|N_{Q_n}(\{u,v\})\cap \{v_{i-1},v_i,v_{i+1}\}|\leq 2$ for $i=2,3,\dots,k-1$ by contradiction. Suppose $|N_{Q_n}(\{u,v\})\cap \{v_{i'-1},v_{i'},v_{i'+1}\}|=3$ for some integer $2\leq i'\leq k-1$, without loss of generality, we assume $u$ is adjacent to $v_{i'-1}$ and $v_{i'+1}$ and $v$ is adjacent to $v_{i'}$. Then $d(u,v_{i'})=2$ but $u$ and $v_{i'}$ have $3$ common neighborhoods: $v$,$v_{i'-1}$,$v_{i'+1}$, leads a contradiction by Lemma \ref{dist2nodes}.
Then we prove the lemma by induction on $k$. For $k=3$, $|N_{Q_n}(\{u,v\})\cap \{v_{1},v_2,v_{3}\}|\leq 2$ by the statement above. Assume that $|N_{Q_n}(\{u,v\})\cap V(P_k)|\leq 2q_k+r_k$ is true for $3\leq k \leq l$ with integer $3\leq l \leq 2^{n-1}-1$. For $k=l+1$, $|N_{Q_n}(\{u,v\})\cap V(P_{l+1})|\leq |N_{Q_n}(\{u,v\})\cap V(P_{l})|+1\leq 2q_l+r_l+1$ by inductive hypothesis.
\vskip .2cm
{\bf Case 1.} $r_{l+1}=1$ and $2$. Notice that in this case $r_{l+1}=r_{l}+1$ and $q_{l+1}=q_{l}$, so we have $|N_{Q_n}(\{u,v\})\cap V(P_{l+1})|\leq 2q_l+r_l+1=2q_{l+1}+r_{l+1}$.
\vskip .2cm
{\bf Case 2.} $r_{l+1}=0$ and $l\geq 5$. Since $|N_{Q_n}(\{u,v\})\cap \{v_{i-1},v_i,v_{i+1}\}|\leq 2 (i=2,3,\dots,k-1)$ we have $|N_{Q_n}(\{u,v\})\cap V(P_{l+1})|\leq |N_{Q_n}(\{u,v\})\cap V(P_{l-2})|+2\leq 2q_{l-2}+2\leq 2q_{l+1}$.\\
In particular, $|N_{Q_n}(\{u,v\})\cap V(P_k)|\leq 2q_k+r_k=k-q_k\leq k-1$.
\end{proof}
\begin{lem}\label{pathgeq}
For $n\geq 3$ and $3\leq k\leq 2^{n-1}$,
\begin{equation} \kappa^s(Q_n;P_k)\geq
\begin{cases}
\lceil \frac{2n}{k+1} \rceil & \text{if $k$ is odd,} \\
\lceil \frac{2n}{k} \rceil & \text{if $k$ is even.}
\end{cases}
\end{equation}
\end{lem}
\begin{proof}
Let $F=\{P_1^{1},P_1^{2},\dots,P_1^{a_1},P_2^{1},\dots,P_2^{a_2},\dots,P_k^{1},\dots,P_k^{a_k}\}$ be a set of paths in $Q_n$ of length less than $k-1$.
For $n=3$, $k$ must be $3$ or $4$. Suppose to the contrary that $Q_3-V(F)$ is disconnected with $|F|\leq \lceil \frac{6}{k} \rceil -1=1$ $(k=3,4)$. If $k=3$, then $Q_3-V(F)$ is connected by Theorem \ref{linsresult}. If $k=4$ and $a_4=0$, then $Q_3-V(F)$ is connected by Theorem \ref{linsresult}. If $a_4=1$, let $F=\{P_4\}$ and all possible cases of $P_4$ are shown in {\bf Fig 5} by the symmetry of $Q_3$, and we can see $Q_3-V(F)$ is connected.
\begin{center}
\begin{center}
\textbf{Fig 5.}
\end{center}
\end{center}
For $n\geq 4$, we have the following cases.
{\bf Case 1.} $k$ is odd.
We prove that for $|F|\leq \lceil \frac{2n}{k+1} \rceil-1$, $Q_n-V(F)$ is connected by contradiction. Suppose $Q_n-V(F)$ is disconnected, let $C$ be one of the smallest components of it, that means components who have the smallest number of vertices.
\vskip .2cm
{\bf Subcase 1.1.} $|V(C)|=1$. Let $V(C)=\{u\}$, each path in $F$ contains at most $\frac{k+1}{2}$ neighbors of $u$ since there is no $3$-cycle in $Q_n$. Therefore $\frac{k+1}{2}|F|\geq n$ i.e. $\frac{k+1}{2}(\lceil \frac{2n}{k+1} \rceil-1)\geq n$, a contradiction with $\frac{k+1}{2}(\lceil \frac{2n}{k+1} \rceil-1)< n$.
\vskip .2cm
{\bf Subcase 1.2.} $|V(C)|\geq 2$.
For $n=4$, we have $|V(F)|\leq k(\lceil \frac{8}{k+1} \rceil-1)<6$, a contradiction with $\kappa_1(Q_4)=\frac{4(4-1)}{2}=6$ by Theorem \ref{gextra}. For $n\geq 5$, we have
$\kappa_1(Q_n)=2n-2$ by Theorem \ref{gextra}, the number of deleted vertices should be at least $2n-2$ i.e. $|V(F)|\geq 2n-2$. But $|V(F)|\leq k(\lceil \frac{2n}{k+1} \rceil-1)\leq k(\frac{2n+k-1}{k+1}-1)=\frac{k}{k+1}(2n-2)<2n-2$, a contradiction.
\vskip .2cm
{\bf Case 2.} $k$ is even.
We prove that for $|F|\leq \lceil \frac{2n}{k} \rceil-1$, $Q_n-V(F)$ is connected by contradiction. Suppose $Q_n-V(F)$ is disconnected, let $C$ be one of the smallest components of it.
\vskip .2cm
{\bf Subcase 2.1.} $|V(C)|=1$.
Let $V(C)=\{u\}$, each path in $F$ contains at most $\frac{k}{2}$ neighbors of $u$. Therefore $\frac{k}{2}|F|\geq n$ i.e. $\frac{k}{2}(\lceil \frac{2n}{k} \rceil-1)\geq n$, a contradiction with $\frac{k}{2}(\lceil \frac{2n}{k} \rceil-1)< n$.
\vskip .2cm
{\bf Subcase 2.2.} $|V(C)|=2$. Let $V(C)=\{u,v\}$, it is obvious that $|N_{Q_n}(\{u,v\})|=2n-2$. For each path $P \in F$, we have $|N_{Q_n}(\{u,v\})\cap V(P)|\leq k-1$ by Lemma \ref{forleqk-1}. Therefore $F$ must has at least $\lceil \frac{2n-2}{k-1} \rceil$ elements, but $|F|\leq \lceil \frac{2n}{k} \rceil -1\leq \frac{2n+k-2}{k}-1=\frac{2n-2}{k}<\lceil \frac{2n-2}{k-1} \rceil$, a contradiction.
\vskip .2cm
{\bf Subcase 2.3.} $|V(C)|\geq 3$.
\vskip .2cm
{\bf Subcase 2.3.1.} For $n\geq 6$, we have $|V(F)|\leq k(\lceil \frac{2n}{k} \rceil-1)<3n-5=\kappa_2(Q_n)$, a contradiction.
\vskip .2cm
{\bf Subcase 2.3.2.}
For $n=5$, we have $|V(F)|\leq k(\lceil \frac{10}{k} \rceil-1)<\frac{5(5-1)}{2}=10=\kappa_2(Q_5)$, a contradiction.
\vskip .2cm
{\bf Subcase 2.3.3.}
For $n=4$ and $k=4,8$, we have $|V(F)|\leq k(\lceil \frac{8}{k} \rceil-1)<\frac{4(4-1)}{2}=6=\kappa_2(Q_4)$, a contradiction.
\begin{table}[H]
\centering
\begin{tabular}{|c|c|c|cccc}
\cline{1-3} \cline{5-7}
$k$ &
$k(\lceil \frac{10}{k} \rceil-1)$ &
$\kappa_2(Q_5)$ &
\multicolumn{1}{c|}{} &
\multicolumn{1}{c|}{$k$} &
\multicolumn{1}{c|}{$k(\lceil \frac{8}{k} \rceil-1)$} &
\multicolumn{1}{c|}{$\kappa_2(Q_4)$} \\ \cline{1-3} \cline{5-7}
$4$ &
$8$ &
\multirow{4}{*}{$10$} &
\multicolumn{1}{c|}{} &
\multicolumn{1}{c|}{$4$} &
\multicolumn{1}{c|}{$4$} &
\multicolumn{1}{c|}{\multirow{2}{*}{$6$}} \\ \cline{1-2} \cline{5-6}
$6$ & $6$ & & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$8$} & \multicolumn{1}{c|}{$0$} & \multicolumn{1}{c|}{} \\ \cline{1-2} \cline{5-7}
$8$ & $8$ & & & & & \\ \cline{1-2}
$\geq 10$ & $0$ & & & & & \\ \cline{1-3}
\end{tabular}
\end{table}
For $k=6$, we have $\lceil \frac{8}{6} \rceil-1=1$. If $a_6=0$, we can think of $F$ as a $P_5$-substructure cut and $Q_4-V(F)$ is connected by {\bf Case 1}, a contradiction.
If $a_6=1$, let $F=\{P_6\}$, without loss of generality, we assume that the first $3$ vertices of $P_6$ are $u=0000$, $v=1000$ and $w=1100$ by the symmetry of $Q_4$. If $|V(P_6)\cap V(Q_4^1)|=0$, then $Q_4^1-V(F)$ is connected, therefore $Q_4-V(F)$ is connected since there is a perfect matching between $Q_4^0$ and $Q_4^1$, a contradiction. If $|V(P_6)\cap V(Q_4^1)|=1$, let $V(P_6)\cap V(Q_4^1)=\{x\}$, then $(x)^3\in V(P_6)$ and obviously $Q_4^1-\{x\}$ is connected, therefore $Q_4-V(F)$ is connected by the perfect matching, a contradiction. If $|V(P_6)\cap V(Q_4^1)|=2$, all possible cases of $P_6$ are shown in {\bf Fig 6}, and we can see $Q_4-V(F)$ is connected, a contradiction.
\begin{center}
\begin{center}
\textbf{Fig 6.} Different types of lines represent different ways to construct $P_6$ with startpoint u.
\end{center}
\end{center}
If $|V(P_6)\cap V(Q_4^1)|=3$, by Theorem \ref{linsresult}, $Q_4^i-V(F)$ is connected for $i=0,1$. Suppose $Q_4-V(F)$ is disconnected, then we have $(y)^3 \in V(F)$ for each $y \in Q_4^0-V(F)$ implies $2^3-3\leq 3$, a contradiction. Therefore $Q_4-V(F)$ is connected, a contradiction.
\end{proof}
By Lemma \ref{pathleq} and Lemma \ref{pathgeq}, we have the following theorem.
\begin{thm}\label{paththm}
For $n\geq 3$ and $3\leq k\leq 2^{n-1}$
\begin{equation} \kappa^s(Q_n;P_k)= \kappa(Q_n;P_k)=
\begin{cases}
\lceil \frac{2n}{k+1} \rceil & \text{if $k$ is odd,} \\
\lceil \frac{2n}{k} \rceil & \text{if $k$ is even.}
\end{cases}
\end{equation}
\end{thm}
\section{$\kappa(Q_n;C_k)$ and $\kappa^s(Q_n;C_k)$ for $n\geq 3$ and $4\leq k \leq 2^{n-1}$ and $k$ is even}
In this section we discuss $H$-structure connectivity and $H$-substructure connectivity of $Q_n$ with $H$ the cycle of length $k$.
\begin{lem}\label{cycleoddks}
For $n\geq 3$ and odd integer $3\leq k\leq 2^{n-1}$, $\kappa^s(Q_n;C_k)=\lceil \frac{2n}{k+1} \rceil$.
\end{lem}
\begin{proof}
$\kappa^s(Q_n;C_k)=\kappa^s(Q_n;P_k)=\lceil \frac{2n}{k+1} \rceil$ by Theorem \ref{paththm} since there is no odd cycle in $Q_n$.
\end{proof}
\begin{lem}\label{cycleevenks}
For $n\geq 3$ and even integer $3\leq k\leq 2^{n-1}$, $\kappa^s(Q_n;C_k)\leq\lceil \frac{2n}{k} \rceil$.
\end{lem}
\begin{proof}
$\kappa^s(Q_n;C_k)\leq \kappa^s(Q_n;P_k)=\lceil \frac{2n}{k} \rceil$ by Theorem \ref{paththm}.
\end{proof}
\begin{lem}\label{cycleleqk-1}
For $n\geq 3$ and $3\leq k\leq 2^{n-1}$, if vertices $u$ and $v$ are adjacent in $Q_n-V(C_k)$, then $|N_{Q_n}(\{u,v\})\cap V(C_k)|\leq k-1$.
\end{lem}
\begin{proof}
We prove this lemma by contradiction. Suppose that $|N_{Q_n}(\{u,v\})\cap V(C_k)|=k$ i.e. each vertex in $V(C_k)$ is adjacent to either $u$ or $v$. Take vertices $v_i\in V(C_k)$ for $i=1,2,3$, such that $v_1v_2\in E(Q_n)$ and $v_2,v_3\in E(Q_n)$. Without loss of generality, we assume that $v_1,v_3\in N_{Q_n}(u)$ and $v_2\in N_{Q_n}(v)$, then $d(u,v_2)=2$ but $N_{Q_n}(u)\cap N_{Q_n}(v_2)=\{v,v_1,v_3\}$, a contradiction.
\end{proof}
\begin{lem}\label{cyclesubstr}
For $n\geq 3$ and even integer $4\leq k \leq 2^{n-1}$, $\kappa^s(Q_n;C_k)\geq \lceil\frac{2n}{k} \rceil$.
\end{lem}
\begin{proof}
Let $F=\{P_1^{1},P_1^{2},\dots,P_1^{a_1},P_2^{1},\dots,P_2^{a_2},\dots,P_k^{1},\dots,P_k^{a_k},C_k^{1},\dots,C_k^{b}\}$ be a set of paths of length less than $k-1$ and cycles of length $k$. We prove that $Q_n-V(F)$ is connected for $|F|\leq \lceil\frac{2n}{k} \rceil-1$ .
\vskip .2cm
{\bf Case 1.} $b=0$. In this case, we can think of $F$ a $P_k$-substructure cut, $Q_n-V(F)$ is connected by Theorem \ref{paththm}.
\vskip .2cm
{\bf Case 2.} $b>0$.
\vskip .2cm
{\bf Subcase 2.1.} $n=3$. $k$ must be $4$ and $|F|\leq \lceil\frac{6}{4} \rceil-1=1$. Let $F=\{C_4\}$ and $V(C_4)=\{v_0=000,v_1=100,v_2=110,v_3=010\}$ by the symmetry of $Q_3$, it is clear that $Q_3-V(F)$ is connected.
\vskip .2cm
{\bf Subcase 2.2.} $n\geq 4$. We prove that $Q_n-V(F)$ is connected by contradiction. Suppose $Q_n-V(F)$ is disconnected, let $C$ be one of the smallest components of it.
\vskip .2cm
{\bf Subcase 2.2.1.} $|V(C)|=1$.
Let $V(C)=\{u\}$, each element in $F$ contains at most $\frac{k}{2}$ neighbors of $u$. Therefore $\frac{k}{2}|F|\geq n$ i.e. $\frac{k}{2}(\lceil \frac{2n}{k} \rceil-1)\geq n$, a contradiction.
\vskip .2cm
{\bf Subcase 2.2.2.} $|V(C)|=2$.
Let $V(C)=\{u,v\}$, it is obvious that $|N_{Q_n}(\{u,v\})|=2n-2$. For each element $G \in F$, we have $|N_{Q_n}(\{u,v\})\cap V(G)|\leq k-1$ by Lemma \ref{forleqk-1} and Lemma \ref{cycleleqk-1}. Therefore $F$ must has at least $\lceil \frac{2n-2}{k-1} \rceil$ elements, but $|F|\leq \lceil \frac{2n}{k} \rceil -1\leq \frac{2n+k-2}{k}-1=\frac{2n-2}{k}<\lceil \frac{2n-2}{k-1} \rceil$, a contradiction.
\vskip .2cm
{\bf Subcase 2.2.3.} $|V(C)|\geq 3$.
\vskip .2cm
{\bf Subcase 2.2.3.1.} For $n\geq 5$ and $n=4, k=4,8$ the discussion is the same as Lemma \ref{pathgeq}.
\vskip .2cm
{\bf Subcase 2.2.3.2.} For $n=4, k=6$, $|F|\leq \lceil\frac{8}{6} \rceil-1=1$. Let $F=\{C_6\}$, $F$ is a $C_6$-structure cut and $2=\lceil \frac{4}{3} \rceil\leq\kappa(Q_4,C_6)\leq 1$ by Lemma \ref{TAMI}, a contradiction.
\end{proof}
By Lemma \ref{cycleevenks} and Lemma \ref{cyclesubstr}, we have the following theorem.
\begin{thm}\label{cycleksandkgeq}
For $n\geq 3$ and even integer $3\leq k\leq 2^{n-1}$, $\kappa(Q_n;C_k)\geq \kappa^s(Q_n;C_k)=\lceil \frac{2n}{k}\rceil$.
\end{thm}
\begin{lem}\label{n=3}
For $n=3$, it is obvious that $\kappa(Q_3;C_4)=2$.
\end{lem}
\begin{lem}\label{cubecycle}(See \cite{SaadTopological})
A cycle of length $l$ can be mapped into $Q_n$ when $l$ is even and $4\leq l \leq 2^n$.
\end{lem}
\begin{lem}\label{oddpathcycle}
For $n\geq 2$ and odd integer $1\leq q \leq 2^n-1$, each pair of adjacent vertices $u$ and $v$ in $Q_n$ have a path of length $q$ between them.
\end{lem}
\begin{proof}
The lemma is true when $q=1$ since $u,v$ are adjacent. When $q\geq 3$, there is a cycle $C_{q+1}$ of length $q+1$ with $uv\in E(C_{q+1})$, $C_{q+1}-\{uv\}$ is the path required.
\end{proof}
\begin{lem}\label{cycleleq}
For $n\geq 4$ and $k$ even,
\begin{equation}
\kappa(Q_n;C_k)
\begin{cases}
=n-2 & \text{k=4,} \\
\leq \lceil \frac{2n}{k} \rceil & \text{$n\geq 5$ and $6\leq k\leq 2^{n-2}$.}
\end{cases}
\end{equation}
\end{lem}
\begin{proof}
For $k=4$, we have $\kappa(Q_n;C_4)=n-2$ by Theorem \ref{linsresult}.
For $n\geq 5$ and $6\leq k\leq 2^{n-2}$,
we set $u=00\dots0$ being a vertex in $Q_n$.
\vskip .2cm
{\bf Case 1.} $\frac{k}{2} \leq n$.
\vskip .2cm
{\bf Subcase 1.1.} $n$ is divided by $\frac{k}{2}$. Let $F = \{ C^i_k | i = 0,
2, \ldots, \lceil \frac{2n}{k} \rceil - 1 = \frac{2n}{k} - 1 \}$ with $C^i_k
= \left( (u)^{i \frac{k}{2}}, \left( (u)^{i \frac{k}{2}} \right)^{i
\frac{k}{2} + 1}, (u)^{i \frac{k}{2} + 1}, \ldots, (u)^{(i + 1)
\frac{k}{2} - 1}, \left( (u)^{(i + 1) \frac{k}{2} - 1} \right)^{i
\frac{k}{2}}, (u)^{i \frac{k}{2}} \right)$. Then $F$ is a $C_k$-structure cut of $Q_n$.
\begin{center}
\begin{center}
\textbf{Fig 7.} An example for $n=6$ and $k=6$.
\end{center}
\end{center}
\vskip .2cm
{\bf Subcase 1.2.} $n$ is not divided by $\frac{k}{2}$. Let $F = \{ C^i_k | i =
0, 2, \ldots, \lceil \frac{2n}{k} \rceil - 1 \}$ with $C^i_k = \left(
(u)^{i \frac{k}{2}}, \left( (u)^{i \frac{k}{2}} \right)^{i \frac{k}{2}
+ 1}, (u)^{i \frac{k}{2} + 1}, \ldots, (u)^{(i + 1) \frac{k}{2} - 1},
\left( (u)^{(i + 1) \frac{k}{2} - 1} \right)^{i \frac{k}{2}}, (u)^{i
\frac{k}{2}} \right)$ for $i = 0, \ldots, \lceil \frac{2n}{k} \rceil - 2$ and
$$C^{\lceil \frac{2n}{k} \rceil - 1}_k = \left( (u)^{n - \frac{k}{2}},
\left( (u)^{n - \frac{k}{2}} \right)^{n - \frac{k}{2} + 1}, (u)^{n -
\frac{k}{2} + 1}, \ldots, (u)^{n - 1}, \left(\left(u\right)^{n - 1}\right)^{n -
\frac{k}{2}}, (u)^{n - \frac{k}{2}} \right) .$$ Then $F$ is a $C_k$-structure cut of $Q_n$.
\begin{center}
\begin{center}
\textbf{Fig 8.} An example for $n=5$ and $k=6$.
\end{center}
\end{center}
\vskip .2cm
{\bf Case 2.} $\frac{k}{2} \geq n+1$. In this case $\lceil\frac{2n}{k}\rceil=1$ and $k\geq 2n+2$. Let $Q$ be the induced subgraph of $Q_n$ with $V(Q)=\{v=x_0x_1\dots x_{n-3}01|x_i=0\text{ or }1,i=0,1,\dots,n-3 \}$, obviously $\left((u)^0\right)^{n-1}$ and $(u)^{n-1}$ are belong to $V(Q)$, by Lemma \ref{oddpathcycle}, there is a path $P$ of length $k-(2n-1)$ between $\left((u)^0\right)^{n-1}$ and $(u)^{n-1}$ in $Q$. Let $F=\{C_k\}$ with $$C_k = \left( (u)^0, \left(\left(u\right)^0\right)^1,
\ldots, \left(\left(u\right)^{n - 2}\right)^{n - 1}, (u)^{n - 1}, P, \left(\left(u\right)^0\right)^{n - 1}, (u)^0\right)$$Then $F$ is a $C_k$-structure cut of $Q_n$.
\begin{center}
\begin{center}
\textbf{Fig 9.}
\end{center}
\end{center}
\end{proof}
By Lemma \ref{cycleoddks}, Lemma \ref{cycleksandkgeq}, Lemma \ref{n=3} and Lemma \ref{cycleleq} we can obtain the following theorem.
\begin{thm}\label{cycleres}
For $n\geq 3$
\begin{equation}
\kappa^s(Q_n;C_k)=
\begin{cases}
\lceil\frac{2n}{k+1}\rceil & \text{$k$ is odd and $3\leq k\leq 2^{n-1}$,} \\
\lceil\frac{2n}{k}\rceil & \text{$k$ is even and $4\leq k\leq 2^{n-1}$.}
\end{cases}
\end{equation}
\begin{equation}
\kappa(Q_n;C_k)
\begin{cases}
=2 & \text{$n=3,k=4$,} \\
=n-2 & \text{$n\geq 4$ and $k=4$,}\\
=\lceil\frac{2n}{k}\rceil & \text{$n\geq 5$, $k$ is even and $6\leq k\leq 2^{n-2}$,}\\
\geq \lceil\frac{2n}{k}\rceil & \text{$n\geq 4$, $k$ is even and $2^{n-2}+2\leq k\leq 2^{n-1}$.}
\end{cases}
\end{equation}
\end{thm}
\begin{lem}\label{kks2^m}
For $n=4,5$ and $2\leq m\leq n-2$, $\kappa(Q_n;C_{2^m})=n-m$.
\end{lem}
\begin{proof}
We have $\kappa(Q_n;C_{2^m})\geq \lceil\frac{n}{2^{m-1}}\rceil$ by Lemma \ref{cycleksandkgeq}. We also have $\kappa(Q_n;C_{2^m})\leq n-m$ by Lemma \ref{Manesresult}.
\begin{center}
\begin{table}[H]
\centering
\begin{tabular}{|c|c|c|}
\hline
& $\lceil\frac{n}{2^{m-1}}\rceil$ & $n-m$ \\ \hline
$n=4,m=2$ & 2 & 2 \\ \hline
$n=5,m=2$ & 3 & 3 \\ \hline
$n=5,m=3$ & 2 & 2 \\ \hline
\end{tabular}
\end{table}
\end{center}
Therefore the statement is proved.
\end{proof}
\begin{lem}\label{kks2mcy}
For $n\geq 6$ and $3\leq m\leq n-2$, $\kappa(Q_n;C_{2^m})=\kappa^s(Q_n;C_{2^m})=\lceil\frac{n}{2^{m-1}}\rceil$
\end{lem}
\begin{proof}
By Theorem \ref{cycleres}.
\end{proof}
\begin{lem}\label{budengs}
For $n\geq 6$ and $3\leq m\leq n-2$, $\lceil\frac{n}{2^{m-1}}\rceil<n-m$.
\end{lem}
\begin{proof}
For a given integer $n\geq 6$, let $f (m) =\frac{n}{2^{m-1}} + 1 - (n- m)$. Derivate $f(m)$ with respect to $m$, $f' (m) = 1 - 2^{1 - m} n \ln 2$, again derivate $f'(m)$, we have $f'' (m) = 2^{1 -m} n (\ln 2)^2 $. Obviously, $f'' (m) \geq 0$ which implies $f'(m)$ monotonically increasing respect to $m$.
After simple calcutions we obtain
\begin{equation}
\begin{cases}
f' (3) = 1 - \frac{n \ln 2}{4} \\
f' (n - 2) =
1 - \frac{n \ln 2}{2^{n - 3}}
\end{cases}
\end{equation}
$f' (3)<0$ when $n\geq 6$. Let $g (n) =f' (n - 2)= 1
- \frac{n \ln 2}{2^{n - 3}}$, derivate $g(n)$ with respect to $n$, $g' (n) = - 2^{3 - n} \ln 2 (1
- nln2) >0$ when $n\geq 6$, therefore $g(n)$ monotonically increasing respect to $n$ and $\min\limits_{n\geq 6} g(n)=g(6)=1 - \frac{6 \ln
2}{8} > 0$, this implies $f' (n - 2)=g (n)>0$ and $f(m)$ firstly monotonically decreasing and then monotonically increasing respect to $m$. Therefore $\max\limits_{m} f (m) =\max\limits_{m}\{ f (3), f (n - 2) \}$.
By simple calculations we obtain
\begin{equation}
\begin{cases}
f (3) = 4 - \frac{3 n}{4} < 0 \\
f (n - 2) = \frac{n}{2^{n - 3}} - 1
\end{cases}
\end{equation}
Let $h (n) = \frac{n}{2^{n - 3}} - 1$, derivate $h(n)$ with respect to $n$, $h' (n) = 2^{3 - n} (1- nln2) <0$ when $n\geq 6$. Therefore $h(n)$ monotonically decreasing respect to $n$ and $\max\limits_{n\geq 6}h(n)=h(6)=\frac{6}{8} - 1 <
0$, therefore $f (n - 2) < 0$ implies $f (m) = \frac{n}{2^{m-1}} + 1 - (n - m) < 0$ i.e. $\lceil \frac{n}{2^{m-1}} \rceil \leq \frac{n}{2^{m-1}} + 1 < n - m$.
\end{proof}
By Theorem \ref{cycleres}, Lemma \ref{kks2^m}, Lemma \ref{kks2mcy} and Lemma \ref{budengs}, we obtain the following theorem as the solution of the open problem in \cite{ManeStructure}.
\begin{thm}
For $n\geq 4$ and for each $2\leq m \leq n-2$
\begin{equation}
\kappa(Q_n;C_{2^m})=
\begin{cases}
=n-m& \text{$n=4,5$ or $n\geq 4,m=2$,} \\
=\lceil\frac{n}{2^{m-1}}\rceil<n-m& \text{$n\geq 6$ and $3\leq m\leq n-2$.}
\end{cases}
\end{equation}
\end{thm}
In Theorem \ref{cycleres} we have $\kappa(Q_n;C_k)\geq \lceil\frac{2n}{k}\rceil$ for $n\geq 4$, $k$ even and $2^{n-2}+2\leq k\leq 2^{n-1}$, but in this case, whether $\kappa(Q_n;C_k)$ equals $\lceil\frac{2n}{k}\rceil$ is still a question.
\end{document} |
\begin{document}
\begin{abstract}
This is the first of two papers in which we introduce and study two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields.
One of these zeta functions encodes the numbers of isomorphism classes of irreducible complex representations of finite dimensions of
congruence quotients of the associated group and the other one encodes the numbers of conjugacy classes of each size of such quotients.
In this paper, we show that these zeta functions satisfy Euler factorisations and almost all of their Euler factors are rational and satisfy functional equations.
Moreover, we show that such bivariate zeta functions specialize to (univariate) class number zeta functions.
In case of nilpotency class~$2$, bivariate representation zeta functions also specialize to (univariate) twist representation zeta functions.
\end{abstract}
\mathbf{m}aketitle
\thispagestyle{empty}
\section{Introduction and statement of main results}
\subsection{Introduction}\label{intro}
Let~$G$ be a group and, for $n \in \mathbf{m}athds{N}$, write
\begin{align*}
r_n(G)&=|\{\text{isomorphism classes of $n$-dimensional irreducible complex}\\
&\mathfrak{p}hantom{r(G)}\text{representations of } G\}|,\\
c_n(G)&=|\{\text{conjugacy classes of $G$ of cardinality }n\}|.
\end{align*}
If~$G$ is a topological group, we only consider continuous representations.
We study the bivariate zeta functions of groups associated to unipotent group schemes encoding either the numbers $r_n(Q)$ or the numbers $c_n(Q)$ of certain finite quotients~$Q$ of the infinite groups considered.
We first recall the definitions of (univariate) representation and conjugacy class zeta functions.
\begin{dfn}\label{univa} Let~$G$ be a group and $s$ a complex variable.
\begin{enumerate}
\item If all~$r_n(G)$ are finite, then the \emph{representation zeta function} of~$G$ is
\[\zeta^\textup{irr}_{G}(s)=\sum_{n=1}^{\infty}r_n(G)n^{-s}. \]
\item If all~$c_n(G)$ are finite, then the \emph{conjugacy class zeta function} of~$G$ is
\[\zeta^\textup{cc}_{G}(s)=\sum_{n=1}^{\infty}{c}_n(G)n^{-s}.\]
\end{enumerate}
\end{dfn}
The groups considered in the present paper are groups associated to unipotent group schemes which are obtained from nilpotent Lie lattices; see Section~\text{Re}f{groups}.
From here on, let~$K$ denote a number field and~$\mathcal{O}$ its ring of integers.
Let~$\mathbf{m}athbf{G}$ be a unipotent group scheme over~$\mathcal{O}$. The group~$\mathbf{m}athbf{G}(\mathcal{O})$ is a finitely generated, torsion-free nilpotent group ($\mathbf{m}athcal{T}$-group for short);
see~\cite[Section~2.1.1]{StVo14}.
We observe that for a $\mathbf{m}athcal{T}$-group~$G$ the numbers $r_n(G)$ and $c_n(G)$ are not all finite. For this reason, one cannot define representation and conjugacy class zeta functions of~$G$ as in Definition~\text{Re}f{univa}.
In the representation case, many authors have overcome this by considering zeta functions encoding the $n$-dimensional irreducible complex representations of such groups up to tensoring by one-dimensional representations; see Section~\text{Re}f{app2}.
Our idea is to investigate zeta functions encoding the relevant data $r_n(Q)$ or $c_n(Q)$ of principal congruence quotients~$Q$ of~$\mathbf{m}athbf{G}(\mathcal{O})$. We define bivariate complex functions: one variable concerns either the dimensions of the representations considered or the cardinalities of the conjugacy classes considered, and the other variable concerns the principal congruence subgroups.
\begin{dfn}\label{biva} The \emph{bivariate representation} and the \emph{bivariate conjugacy class zeta functions} of $\mathbf{m}athbf{G}(\mathcal{O})$ are
\begin{align*}
\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathcal{O})}(s_1,s_2)&=\sum_{(0) \neq I \unlhd \mathcal{O} }\zeta^\textup{irr}_{\mathbf{m}athbf{G}(\mathcal{O}/I)}(s_1)|\mathcal{O}:I|^{-s_2}\text{ and}\\
\mathcal{Z}^\textup{cc}_{\mathbf{m}athbf{G}(\mathcal{O})}(s_1,s_2)&=\sum_{(0) \neq I \unlhd \mathcal{O} }\zeta^\textup{cc}_{\mathbf{m}athbf{G}(\mathcal{O}/I)}(s_1)|\mathcal{O}:I|^{-s_2},
\end{align*}
respectively, where $s_1$ and $s_2$ are complex variables.
\end{dfn}
These series converge for $s_1$ and $s_2$ with sufficiently large real parts; see Section~\text{Re}f{conv}.
We remark that the zeta functions defined above depend not only on the group of $\mathcal{O}$-points $\mathbf{m}athbf{G}(\mathcal{O})$, but implicitly on the ring~$\mathcal{O}$ and the
$\mathcal{O}$-scheme $\mathbf{m}athbf{G}$ which give rise to the finite quotients $\mathbf{m}athbf{G}(\mathcal{O}/I)$, modulo ideals~$I$ of~$\mathcal{O}$. It is convenient and customary to use the short notation $\mathbf{m}athbf{G}(\mathcal{O})$ in the index to indicate this more complex dependency.
In Proposition~\text{Re}f{factor}, we establish the Euler decompositions
\[\lvlzf^{\ast}_{\mathbf{m}athbf{G}(\mathcal{O})}(s_1,s_2)=\mathfrak{p}rod_{\mathfrak{p} \in \textup{Spec}p} \lvlzf^{\ast}_{\mathbf{m}athbf{G}(\mathcal{O}c)}(s_1,s_2),\]
where $\ast\in\{\textup{irr},{c}up\}$ and~$\mathcal{O}c$ is the completion of~$\mathcal{O}$ at the nonzero prime ideal~$\mathfrak{p}$. When considering a fixed prime ideal~$\mathfrak{p}$, we write simply $\mathfrak{o}=\mathcal{O}c$
and $\mathbf{m}athbf{G}_N:=\mathbf{m}athbf{G}(\mathfrak{o}p)$. With this notation, the local factor at~$\mathfrak{p}$ is given by
\begin{align}\label{localfactors}\lvlzf^{\ast}_{\mathbf{m}athbf{G}(\mathcal{O}c)}(s_1,s_2)=\lvlzf^{\ast}_{\mathbf{m}athbf{G}(\mathfrak{o})}(s_1,s_2)= \sum_{N=0}^{\infty}\zeta^{\ast}_{\mathbf{m}athbf{G}_N}(s_1)|\mathfrak{o}:\mathfrak{p}|^{-Ns_2}.\end{align}
\begin{exm} Let~$\mathbf{m}athbf{G}(\mathcal{O})$ be the free Abelian torsion-free group~$\mathcal{O}^m$, and let~$\mathfrak{p}$ be a nonzero prime ideal of~$\mathcal{O}$ with $q=|\mathcal{O}:\mathfrak{p}|$. Then, for $N \in \mathbf{m}athds{N}_0=\mathbf{m}athds{N}\cup \{0\}$, we have
\[r_{q^i}(\mathbf{m}athbf{G}_N)={c}_{q^i}(\mathbf{m}athbf{G}_N)=\begin{cases}
q^{mN},&\text{ if }i=0,\\
0,&\text{ otherwise.}
\end{cases}
\]
Therefore, for $\ast \in \{\textup{irr},{c}up\}$,
\[\lvlzf^{\ast}_{\mathbf{m}athbf{G}(\mathfrak{o})}(s_1,s_2)=\lvlzf^{\ast}_{\mathfrak{o}^{m}}(s_1,s_2)=\sum_{N=0}^{\infty}q^{N(m-s_2)}=\mathfrak{f}rac{1}{1-q^{m-s_2}}.\]
Consequently $\lvlzf^{\ast}_{\mathcal{O}^m}(s_1,s_2)=\zeta_{{K}}(s_2-m)$ with $\zeta_{{K}}(s)$ denoting the Dedekind zeta function of the number field~$K$.
Moreover, the local factor at~$\mathfrak{p}$ satisfies the functional equation
\[\lvlzf^{\ast}_{\mathfrak{o}^{m}}(s_1,s_2)\mathbf{m}id_{q\to q^{-1}}=-q^{m-s_2}\lvlzf^{\ast}_{\mathfrak{o}^{m}}(s_1,s_2).\qedhere\]
\end{exm}
Certain zeta functions of groups related to representations---or the local factors of such functions---are known to be rational functions satisfying functional equations; for instance, representation zeta functions of certain pro-$p$ groups~\cite[Theorem~A]{AKOV13}, and local factors of twist representation zeta functions---see Section~\text{Re}f{app2}---of groups of the form $\mathbf{m}athbf{G}(\mathcal{O})$~\cite[Theorem~A]{StVo14}, where $\mathbf{m}athbf{G}$ is a unipotent group scheme obtained from a nilpotent $\mathcal{O}$-Lie lattice.
As for zeta functions related to conjugacy classes, the so-called class number zeta functions---see Section~\text{Re}f{app1}---of certain groups are rational; for instance the local factors of class number zeta functions of Chevalley groups $G(\mathfrak{o})$~\cite[Theorem~C]{BDOP}, where $\mathfrak{o}$ is the valuation ring of a non-Archimedean local field of any (sufficiently large) characteristic, and class number zeta functions of compact $p$-adic analytic groups~\cite[Theorem~1.2]{duSau05}.
This motivates our main result, which concerns the above mentioned features for the local factors of bivariate representation and conjugacy class zeta functions of groups of the form~$\mathbf{m}athbf{G}(\mathcal{O})$ obtained from nilpotent Lie lattices; see Section~\text{Re}f{groups}.
\begin{thm}\label{thmA} Let $\mathcal{O}$ be the ring of integers of a number field~$K$, and let $\mathbf{m}athbf{G}$ be a unipotent group scheme obtained from a nilpotent $\mathcal{O}$-Lie lattice~$\Lambda$. For each $\ast \in\{\textup{irr},\textup{cc}\}$, there exist a positive integer $t^{\ast}$ and a rational function
\[R^{\ast}(X_1, \dots, X_{t^{\ast}}, Y_1,Y_2)\text{ in } \mathbf{m}athds{Q}(X_1, \dots, X_{t^{\ast}},Y_1,Y_2)\]
such that, for all but finitely many nonzero prime ideals $\mathfrak{p}$ of $\mathcal{O}$, there exist algebraic integers $\lambda_{1}^{\ast}(\mathfrak{p})$, $\dots$,
$\lambda_{t^{\ast}}^{\ast}(\mathfrak{p})$ for which the following holds. For any finite extension $\mathfrak{O}$ of $\mathfrak{o}:=\mathcal{O}c$ with relative degree of inertia $f=f(\mathfrak{O},\mathfrak{o})$,
\[\mathcal{Z}_{\mathbf{m}athbf{G}(\mathfrak{O})}^{\ast}(s_1,s_2)=R^{\ast}(\lambda_{1}^{\ast}(\mathfrak{p})^f, \dots, \lambda_{t^{\ast}}^{\ast}(\mathfrak{p})^f,q^{-fs_1},q^{-fs_2}),\]
where $q=|\mathcal{O}:\mathfrak{p}|$.
Moreover, inverting parameters in the rational function $R^{\ast}$ yields that these local factors satisfy the functional equation
\[\lvlzf^{\ast}_{\mathbf{m}athbf{G}(\mathfrak{O})}(s_1,s_2)\mathbf{m}id_{\substack{q \mathcal{O}ghtarrow q^{-1}\\ \lambda_{j}^{\ast}(\mathfrak{p}) \mathcal{O}ghtarrow \lambda_{j}^{\ast}(\mathfrak{p})^{-1}}}=-q^{f(h-s_2)}\lvlzf^{\ast}_{\mathbf{m}athbf{G}(\mathfrak{O})}(s_1,s_2),\]
where $h=\dim_K(\Lambda \otimes K)$.
\end{thm}
The statement of Theorem~\text{Re}f{thmA} is analogous to~\cite[Theorem~A]{StVo14}, and its proof heavily relies on the techniques of~\cite{AKOV13, StVo14}; see Section~\text{Re}f{pthmA}. The main tools used in the proof of Theorem~\text{Re}f{thmA} are the Kirillov orbit method, the Lazard correspondence, and $\mathfrak{p}$-adic integration.
Since the rational functions~$R^{\textup{irr}}$ and $R^{{c}up}$ of Theorem~\text{Re}f{thmA} only depend on the $\mathcal{O}$-scheme $\mathbf{m}athbf{G}$, it follows that almost all local factors $\lvlzf^{\ast}_{\mathbf{m}athbf{G}(\mathcal{O}c)}(s_1,s_2)$ only depend on~$\mathbf{m}athbf{G}$ and the chosen prime ideal~$\mathfrak{p}$ of~$\mathcal{O}$, not on the group of $\mathcal{O}c$-points $\mathbf{m}athbf{G}(\mathcal{O}c)$. We keep the notation $\lvlzf^{\ast}_{\mathbf{m}athbf{G}(\mathcal{O}c)}(s_1,s_2)$ to make clear which ring of integers and prime ideal are being considered.
\begin{rmk}\label{rmk:HrMa}
Local multivariate zeta functions counting the number of equivalence classes in some uniformly definable family of equivalence relations are known to be rational functions; see~\cite[Theorem~1.3]{HrMa}. As an application of this theorem, the authors prove in~\cite[Section~8]{HrMa} rationality for all local factors of twist representation zeta functions of~$\mathbf{m}athcal{T}$-groups, partially extending~\cite[Theorem~A]{StVo14}. Nevertheless, the techniques of~\cite{HrMa} do not assure that these local factors satisfy functional equations, as in~\cite[Theorem~A]{StVo14} or in Theorem~\text{Re}f{thmA}. We hope that the methods of~\cite{HrMa} can be applied to the bivariate zeta functions defined here to show rationality of all of their local factors. We point out that if $\mathbf{m}athbf{G}(\mathcal{O})$ is a $\mathbf{m}athcal{T}$-group as in Theorem~\text{Re}f{thmA} which has nilpotency class~$2$, then all local factors of $\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathcal{O})}(s_1,s_2)$ are rational functions; see Remark~\text{Re}f{condition2} and Proposition~\text{Re}f{intpadic}.
\end{rmk}
\begin{rmk}
The author is not aware whether the results of Theorem~\text{Re}f{thmA} remain true for groups associated to unipotent group schemes in positive characteristic or for arithmetic groups associated to non-unipotent group schemes, since the techniques used here to prove Theorem~\text{Re}f{thmA} do not apply in such cases.
However one could obtain a positive answer for the former question if the techniques of~\cite{HrMa} do apply for these bivariate zeta functions, in which case~\cite[Corollary~6.8]{HrMa} would assure that almost all local factors of the bivariate zeta functions of~$\mathbf{m}athbf{G}(\mathbf{m}athds{F}_p[t])$ are also rational, where $\mathbf{m}athds{F}_p$ is the field with~$p$ elements.
\end{rmk}
Some applications of bivariate zeta functions are given below in Sections~\text{Re}f{app1} and~\text{Re}f{app2}. Specifically, we obtain results on previously studied (univariate) zeta functions by specializing the bivariate zeta functions introduced here.
It would be interesting to understand which other kinds of information one can extract from bivariate zeta functions.
In~\cite{PL18II}, we explicitly compute bivariate zeta functions of three infinite families of nilpotent groups. As a consequence, we obtain explicit formulae for two (univariate) zeta functions of these groups.
We also provide an application in combinatorics: the formulae for bivariate representation zeta functions of these groups are shown to be related to statistics of certain Weyl groups, leading to formulae for joint distributions of three statistics; see~\cite[Propositions~5.5 and 5.6]{PL18II}.
\subsection{Application 1: class number zeta functions}\label{app1}
An advantage of the study of the bivariate zeta functions of Definition~\text{Re}f{biva} is that they can be used to investigate (univariate) class number zeta functions, which encode the class numbers of principal congruence quotients of the groups considered.
Recall that the \emph{class number}~$\class(G)$ of a finite group $G$ is the number of its conjugacy classes or, equivalently, the number of its irreducible complex characters. In particular, $\class(G)=\zeta^\textup{cc}_G(0)=\zeta^\textup{irr}_G(0)$.
\begin{dfn} The \emph{class number zeta function} of the $\mathbf{m}athcal{T}$-group $\mathbf{m}athbf{G}(\mathcal{O})$ is
\[\zeta^\textup{k}_{\mathbf{m}athbf{G}(\mathcal{O})}(s)=\sum_{(0) \neq I \unlhd \mathcal{O}}\class(\mathbf{m}athbf{G}(\mathcal{O}/I))|\mathcal{O}:I|^{-s},\]
where $s$ is a complex variable.
\end{dfn}
As for the bivariate zeta functions of Definition~\text{Re}f{biva}, the class number zeta function defined above depend not only
on the group of $\mathcal{O}$-points $\mathbf{m}athbf{G}(\mathcal{O})$, but also on~$\mathcal{O}$ and the $\mathcal{O}$-scheme $\mathbf{m}athbf{G}$. We adopt the notation $\mathbf{m}athbf{G}(\mathcal{O})$ in the index to indicate this more complex dependency.
The term `conjugacy class zeta function' is sometimes used for what we call `class number zeta function'; see, for instance,~\cite{BDOP,Ro17,ask2, duSau05}.
Clearly,
\begin{align}\label{specialization}
\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathcal{O})}(0,s)=\mathcal{Z}^\textup{cc}_{\mathbf{m}athbf{G}(\mathcal{O})}(0,s)=
\zeta^\textup{k}_{\mathbf{m}athbf{G}(\mathcal{O})}(s).
\end{align}
A consequence of Theorem~\text{Re}f{thmA} is that almost all local factors of the class number zeta function of $\mathbf{m}athbf{G}(\mathcal{O})$ are rational in $\lambda_i(\mathfrak{p})$, $q$, and $q^{-s}$ and
behave uniformly under base extension. Moreover, for a finite extension $\mathfrak{O}$ of $\mathfrak{o}$ with relative degree of inertia $f=f(\mathfrak{O},\mathfrak{o})$, the local factors satisfy
the functional equation
\[\zeta^\textup{k}_{\mathbf{m}athbf{G}(\mathfrak{O})}(s)\mathbf{m}id_{\substack{q \mathcal{O}ghtarrow q^{-1}\\ \lambda_{j}^{\ast}(\mathfrak{p}) \mathcal{O}ghtarrow \lambda_{j}^{\ast}(\mathfrak{p})^{-1}}}=-q^{f(h-s)}\zeta^\textup{k}_{\mathbf{m}athbf{G}(\mathfrak{O})}(s). \]
Rossmann showed independently in~\cite{Ro17} that class number zeta functions of certain nilpotent groups $G \leq \mathbf{m}athbf{G}l_d(\mathcal{O}c)$ are rational functions and satisfy functional equations. This is a consequence of~\cite[Theorems~4.10 and~4.18]{Ro17} together with
the specialization of ask zeta functions to class number zeta functions given in~\cite[Theorem~1.7]{Ro17}.
\subsection{Application 2: twist representation zeta functions}\label{app2}
A $\mathbf{m}athcal{T}$-group of nilpotency class~$c=2$ is called a $\mathbf{m}athcal{T}_2$-group.
The bivariate representation zeta function of a $\mathbf{m}athcal{T}_2$-group $\mathbf{m}athbf{G}(\mathcal{O})$ specializes to its twist representation zeta function, whose definition we now recall.
Nontrivial $\mathbf{m}athcal{T}$-groups have infinitely many one-dimensional irreducible complex representations.
For this reason, one cannot define the representation zeta function of a $\mathbf{m}athcal{T}$-group $G$ as in Definition~\text{Re}f{univa}.
Instead, for a $\mathbf{m}athcal{T}$-group~$G$, one considers equivalence classes on the set of its irreducible complex representations:
two representations $\rho$, $\sigma$ of~$G$ are called~\emph{twist-equivalent} if there exists a one-dimensional
representation $\chi$ of $G$ such that $\rho \cong \chi \otimes \sigma$. This is an equivalence relation on the set of irreducible complex representations of~$G$ whose equivalence classes are called \emph{twist-isoclasses}.
Let $\widetilde{r}_n(G)$ be the number of twist-isoclasses of $n$-dimensional irreducible complex representations of~$G$. If~$G$ is a topological group, we only consider continuous representations.
The $\widetilde{r}_n(G)$ are all finite, see~\cite[Theorem~6.6]{LuMa85}.
\begin{dfn} The \emph{twist representation zeta function} of a $\mathbf{m}athcal{T}$-group $G$ is
\[\zeta^{\widetilde{\textup{irr}}}_G(s)=\sum_{n=1}^{\infty}\widetilde{r}_n(G)n^{-s},\] where $s$ is a complex variable.
\end{dfn}
Twist representation zeta functions of $\mathbf{m}athcal{T}$-groups have been previously investigated, for instance, in~\cite{DuVo14, HrMa, Ro17comp, StVo14, Vo10}.
Explicit examples of (local factors of) twist representation zeta functions of $\mathbf{m}athcal{T}$-groups can be found in~\cite{Ezzphd,Ro17comp,Snophd,StVo14, StVo17}.
Let $\mathbf{m}athbf{G}(\mathcal{O})$ be a $\mathbf{m}athcal{T}_2$-group obtained from a unipotent $\mathcal{O}$-Lie lattice~$\Lambda$ as explained in Section~\text{Re}f{groups}.
In Section~\text{Re}f{irr}, we show that the bivariate representation zeta function of $\mathbf{m}athbf{G}(\mathcal{O})$ specializes to its twist representation zeta function as follows.
For a fixed nonzero prime ideal~$\mathfrak{p}$ of~$\mathcal{O}$, let $\mathfrak{g}=\Lambda\otimes_{\mathcal{O}}\mathcal{O}c$ and let~$\mathfrak{g}'$ be the derived Lie sublattice of~$\mathfrak{g}$.
Denote by~$r$ the torsion-free rank of $\mathfrak{g}/\mathfrak{g}'$. Then Proposition~\text{Re}f{repzeta} states
\begin{equation}\label{specializationirr}\mathfrak{p}rod_{\mathfrak{p} \in \textup{Spec}p}\left((1-q^{r-s_2})\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathcal{O}c)}(s_1,s_2)\mathbf{m}id_{\substack{s_1\to s-2\\ s_2\to r\mathfrak{p}hantom{-2}}}\mathcal{O}ght)=
\zeta^{\widetilde{\textup{irr}}}_{\mathbf{m}athbf{G}(\mathcal{O})}(s),\end{equation}
provided both the left-hand side and the right-hand side converge.
No specialization of the form~\eqref{specializationirr} is expected to exist in case of nilpotency class~$c>2$; see~\cite[Section~3.3]{PLphd} for details.
In Section~\text{Re}f{irr}, we exhibit a $\mathbf{m}athcal{T}$-group of nilpotency class~$3$ whose bivariate representation zeta function does not specialize to its twist representation zeta function.
We conclude this section with an example which illustrates Theorem~\text{Re}f{thmA} and specializations~\eqref{specialization} and~\eqref{specializationirr}.
\begin{exm}\label{exintro} Let~$\mathbf{m}athbf{H}(\mathcal{O})$ denote the Heisenberg group of upper uni-triangular $3\times 3$-matrices over~$\mathcal{O}$.
In Example~\text{Re}f{Heis}, we show that, for a given nonzero prime ideal~$\mathfrak{p}$ of~$\mathcal{O}$ with $|\mathcal{O}:\mathfrak{p}|=q$, the bivariate zeta functions of~$\mathbf{m}athbf{H}(\mathfrak{o})$ are given by
\begin{align}
\label{Heisirr}\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{H}(\mathfrak{o})}(s_1,s_2) &=\mathfrak{f}rac{1-q^{-s_1-s_2}}{(1-q^{1-s_1-s_2})(1-q^{2-s_2})} ~\text{ and}&\\
\label{Heiscc}\mathcal{Z}^\textup{cc}_{\mathbf{m}athbf{H}(\mathfrak{o})}(s_1,s_2) &=\mathfrak{f}rac{1-q^{-s_1-s_2}}{(1-q^{1-s_2})(1-q^{2-s_1-s_2})}. &
\end{align}
In particular, these are rational functions in~$q$, $q^{-s_1}$, and~$q^{-s_2}$, and
\[\lvlzf^{\ast}_{\mathbf{m}athbf{H}(\mathfrak{o})}(s_1,s_2)\mathbf{m}id_{q\to q^{-1}}=-q^{3-s_2}\lvlzf^{\ast}_{\mathbf{m}athbf{H}(\mathfrak{o})}(s_1,s_2),\]
for each $\ast \in \{ \textup{irr},\textup{cc}\}$.
Specializations~\eqref{specialization} and~\eqref{specializationirr} yield
\[\zeta^\textup{k}_{\mathbf{m}athbf{H}(\mathfrak{o})}(s)=\mathfrak{f}rac{1-q^{-s}}{(1-q^{1-s})(1-q^{2-s})}\hspace{0.3cm}\text{ and }\hspace{0.3cm} \zeta^{\widetilde{\textup{irr}}}_{\mathbf{m}athbf{H}(\mathfrak{o})}(s)=\mathfrak{f}rac{1-q^{-s}}{1-q^{1-s}}.\]
The expression of the class number zeta function agrees with the formula given in~\cite[Section~9.3, Table~1]{Ro17}. This example also occurs in~\cite[Section~8.2]{BDOP}, corrected by a sign mistake.
The expression of the twist representation zeta function accords with~\cite[Theorem~B]{StVo14}.
We further note that
\[\zeta^\textup{k}_{\mathbf{m}athbf{H}(\mathfrak{o})}(s)\mathbf{m}id_{q \to q^{-1}}=-q^{3-s}\zeta^\textup{k}_{\mathbf{m}athbf{H}(\mathfrak{o})}(s).\qedhere\]
\end{exm}
\section{General properties}
\subsection{Convergence}\label{conv}
It is well known that, if a complex sequence $(a_n)_{n\in\mathbf{m}athds{N}}$ grows at most polynomially, the Dirichlet series $D((a_n)_{n\in\mathbf{m}athds{N}},s):=\sum_{n=1}^{\infty}a_nn^{-s}$ converges for
$s \in \mathbf{m}athds{C}$ with sufficiently
large real part. We now show that an analogous result holds for double Dirichlet series.
For simplicity, we write $(a_{n,m}):=(a_{n,m})_{n,m\in\mathbf{m}athds{N}}$.
\begin{dfn} A double sequence $(a_{n,m})$ of complex numbers is said to have \emph{polynomial growth} if there exist positive integers
$\alpha_1$ and $\alpha_2$ and a constant $C>0$ such that
$|a_{n,m}|<Cn^{\alpha_1}m^{\alpha_2}$ for all $n,m \in \mathbf{m}athds{N}$.
\end{dfn}
\begin{pps}\label{double}
If the double sequence $(a_{n,m})$ has polynomial growth, then there exist $\alpha_1$, $\alpha_2\in \mathbf{m}athds{R}$ such that the double Dirichlet series
\[D((a_{n,m}), s_1, s_2):=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}a_{n,m}n^{-s_1}m^{-s_2}\]
converges absolutely for $(s_1,s_2)\in \mathbf{m}athds{C}^2$ satisfying $\text{Re}(s_1)>\alpha_1$ and $\text{Re}(s_2)>\alpha_2$.
\end{pps}
\begin{proof}
Let $\beta_1$, $\beta_2 \in \mathbf{m}athds{N}$ and $C>0$ be such that $|a_{n,m}|<Cn^{\beta_1}m^{\beta_2}$, for all $n,m\in \mathbf{m}athds{N}$. Then
\[\sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \left|\mathfrak{f}rac{a_{n,m}}{n^{s_1}m^{s_2}}\mathcal{O}ght|\leq C\sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \mathfrak{f}rac{1}{n^{\text{Re}(s_1)-\beta_1}m^{\text{Re}(s_2)-\beta_2}}.\]
The relevant statement of Proposition~\text{Re}f{double} then follows from the fact that, for $p,q \in \mathbf{m}athds{R}$, the harmonic double series
\[\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}\mathfrak{f}rac{1}{k^{p}l^{q}}\]
converges if and only if $p>1$ and $q>1$; see \cite[Example~7.10(iii)]{GhLi}.
\end{proof}
For a unipotent $\mathcal{O}$-group scheme~$\mathbf{m}athbf{G}$ and positive integers~$m$ and $n$, write
\[r_{n,m}(\mathbf{m}athbf{G}(\mathcal{O}))=\sum_{\substack{I \unlhd \mathcal{O}\\ |\mathcal{O}:I|=m}}r_n(\mathbf{m}athbf{G}(\mathcal{O}/I))
\text{ and }c_{n,m}(\mathbf{m}athbf{G}(\mathcal{O}))=\sum_{\substack{I \unlhd \mathcal{O}\\ |\mathcal{O}:I|=m}}c_n(\mathbf{m}athbf{G}(\mathcal{O}/I)).\]
The bivariate representation and the bivariate conjugacy class zeta functions of $\mathbf{m}athbf{G}(\mathcal{O})$ are given by the following double Dirichlet series with nonnegative coefficients:
\begin{align*}
\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathcal{O})}(s_1,s_2)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}r_{n,m}(\mathbf{m}athbf{G}(\mathcal{O}))n^{-s_1}m^{-s_2},\\
\mathcal{Z}^\textup{cc}_{\mathbf{m}athbf{G}(\mathcal{O})}(s_1,s_2)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}c_{n,m}(\mathbf{m}athbf{G}(\mathcal{O}))n^{-s_1}m^{-s_2}.
\end{align*}
\begin{pps}
The bivariate zeta functions $\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathcal{O})}(s_1,s_2)$ and $\mathcal{Z}^\textup{cc}_{\mathbf{m}athbf{G}(\mathcal{O})}(s_1,s_2)$ converge (at least) on some open domain of the form
\begin{equation}\label{DAB}\mathbf{m}athscr{D}_{\alpha_1,\alpha_2}:=\{(s_1,s_2)\in \mathbf{m}athds{C}^2\mathbf{m}id \text{Re}(s_1)>\alpha_1,~\text{Re}(s_2)> \alpha_2\},
\end{equation}
for some real constants $\alpha_1$ and $\alpha_2$.
\end{pps}
\begin{proof}
Set $\mathfrak{g}amma_m(\mathcal{O}):=|\{I \unlhd \mathcal{O} \mathbf{m}id |\mathcal{O}:I|=m\}|$.
The Dedekind zeta function of the number field $K$ is given by $\zeta_{{K}}(s)=\sum_{m=1}^{\infty}\mathfrak{g}amma_m(\mathcal{O})m^{-s}$,
and is known to converge for $\text{Re}(s)>1$. In particular, there exists a positive real constant~$C$ such that, for each $M \in \mathbf{m}athds{N}$, it holds that $\sum_{m=1}^{M}\mathfrak{g}amma_m(\mathcal{O})<CM$.
Given $I \unlhd \mathcal{O}$, the finite group $\mathbf{m}athbf{G}(\mathcal{O}/I)$ is a congruence quotient of a torsion-free nilpotent and finitely generated group.
Then there exists $\mathbf{m}athscr{P}(X)\in \mathbf{m}athds{Z}[X]$ such that, for all $I \unlhd \mathcal{O}$, the cardinality of $\mathbf{m}athbf{G}(\mathcal{O}/I)$ is bounded by $\mathbf{m}athscr{P}(m)$, where $m=|\mathcal{O}:I|$.
Given $I \unlhd \mathcal{O}$, the finite group $\mathbf{m}athbf{G}(\mathcal{O}/I)$ has at most $|\mathbf{m}athbf{G}(\mathcal{O}/I)|$ conjugacy classes.
Consequently, for each $(n,m)\in \mathbf{m}athds{N}^2$,
\[c_{n,m}(\mathbf{m}athbf{G}(\mathcal{O}))=\sum_{\substack{ I \unlhd \mathcal{O} \\ |\mathcal{O}:I|=m}}c_n(\mathbf{m}athbf{G}(\mathcal{O}/I)) < C m\mathbf{m}athscr{P}(m).\]
Analogously, $r_{n,m}(\mathbf{m}athbf{G}(\mathcal{O})) < C m\mathbf{m}athscr{P}(m)$, since $r_n(\mathbf{m}athbf{G}(\mathcal{O}/I)) \leq |\mathbf{m}athbf{G}(\mathcal{O}/I)|$.
\end{proof}
When finite, the abscissa of convergence of a Dirichlet series $\sum_{n=1}^{\infty}a_nn^{-s}$ gives the precise degree of polynomial growth of the sequence $(\sum_{i=1}^{n}a_i)_{n\in \mathbf{m}athds{N}}$.
However, for double Dirichlet series $\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}a_{n,m}n^{-s_1}m^{-s_2}$, an analogue of an abscissa of convergence might not be unique.
As mentioned in Example~\text{Re}f{exintro}, the bivariate representation zeta function of the Heisenberg group $\mathbf{m}athbf{H}(\mathcal{O})$ is given by
\begin{align*} \mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{H}(\mathfrak{o})}(s_1,s_2) =\mathfrak{f}rac{1-q^{-s_1-s_2}}{(1-q^{1-s_1-s_2})(1-q^{2-s_2})}.\end{align*}
The maximal domain of convergence of $\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{H}(\mathfrak{o})}(s_1,s_2)$ is
\[\mathbf{m}athscr{D}_{\mathbf{m}athbf{H}}:=\{ (s_1,s_2)\in \mathbf{m}athds{C}^2 \mathbf{m}id \text{Re}(s_1+s_2)> 2 \text{ and }\text{Re}(s_2)>3\}.\]
In contrast with the one variable case, there is more than one choice of constants $(\alpha_1,\alpha_2) \in \mathbf{m}athds{R}^2$ such that~$\mathbf{m}athscr{D}_{\alpha_1,\alpha_2}$ is a maximal domain of the form~\eqref{DAB} with the property that $\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{H}(\mathcal{O})}(s_1,s_2)$ converges on it.
For instance, $\mathbf{m}athscr{D}_{-1,3}$ and $\mathbf{m}athscr{D}_{-2,4}$ are two such domains.
However, no choice of $(\alpha_1,\alpha_2)$ is such that~$\mathbf{m}athscr{D}_{\alpha_1,\alpha_2}$ coincides with the maximal domain of convergence~$\mathbf{m}athscr{D}_{\mathbf{m}athbf{H}}$ of $\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{H}(\mathfrak{o})}(s_1,s_2)$.
\subsection{Euler products}
Our main results concern properties of local factors of bivariate representation and bivariate conjugacy class zeta functions.
In this section, we show that the corresponding global zeta functions can be written as products of such local terms, allowing us to relate local results to the global zeta
functions. Here, $\mathbf{m}athbf{G}$~denotes a unipotent $\mathcal{O}$-group scheme.
\begin{pps}\label{factor}
For each $\ast \in \{\textup{irr},\textup{cc}\}$ and for $s_1$ and $s_2$ with sufficiently large real parts, the following Euler decomposition holds.
\[ \lvlzf^{\ast}_{\mathbf{m}athbf{G}(\mathcal{O})}(s_1,s_2)=
\mathfrak{p}rod_{\mathfrak{p} \in \textup{Spec}p} \lvlzf^{\ast}_{\mathbf{m}athbf{G}(\mathcal{O}c)}(s_1,s_2).\]
\end{pps}
\begin{proof}
It suffices to show that, for any ideal $I$ of finite index in $\mathcal{O}$ with prime decomposition $I=\mathfrak{p}_{1}^{e_1}\cdots \mathfrak{p}_{r}^{e_r}$, with $\mathfrak{p}_i \neq \mathfrak{p}_j$ if $i \neq j$,
the following holds
\[\zeta^{\ast}_{\mathbf{m}athbf{G}(\mathcal{O}/I)}(s)=\mathfrak{p}rod_{i=1}^{r}\zeta^{\ast}_{\mathbf{m}athbf{G}(\mathcal{O}/\mathfrak{p}_{i}^{e_i})}(s).\]
Unipotent groups satisfy the strong approximation property; see~\cite[Lemma~5.5]{PlRa94}. This gives an isomorphism
\begin{equation}\label{sap}\mathbf{m}athbf{G}(\mathcal{O}/I) \cong \mathbf{m}athbf{G}(\mathcal{O}/\mathfrak{p}_{1}^{e_1})\times \dots \times \mathbf{m}athbf{G}(\mathcal{O}/\mathfrak{p}_{r}^{e_r}).\end{equation}
We first show the relevant statement of Proposition~\text{Re}f{factor} for the representation case.
Given a positive integer~$n$, write $[n]=\{1,\dots, n\}$.
For a group~$G$, denote by $\textup{Irr}(G)$ the set of its irreducible characters. A consequence of~\eqref{sap} is
\[\textup{Irr}(\mathbf{m}athbf{G}(\mathcal{O}/I)) \cong \textup{Irr}(\mathbf{m}athbf{G}(\mathcal{O}/\mathfrak{p}_{1}^{e_1}))\times \dots \times \textup{Irr}(\mathbf{m}athbf{G}(\mathcal{O}/\mathfrak{p}_{r}^{e_r})).\]
Write $\textup{Irr}_i=\textup{Irr}(\mathbf{m}athbf{G}(\mathcal{O}/\mathfrak{p}_{i}^{e_i}))$.
Since $r_n(\mathbf{m}athbf{G}(\mathcal{O}/I))=|\{\chi \in \textup{Irr}(\mathbf{m}athbf{G}(\mathcal{O}/I)): \chi(1)=n\}|$, it follows that
\begin{align*}
\zeta^\textup{irr}_{\mathbf{m}athbf{G}(\mathcal{O}/I)}(s)=\sum_{\chi \in \textup{Irr}(\mathbf{m}athbf{G}(\mathcal{O}/I))}\chi(1)^{-s} &=\sum_{(\chi_1, \dots, \chi_r) \in \textup{Irr}_1\times \dots \times \textup{Irr}_{r}} \chi_1(1)^{-s}\cdots\chi_r(1)^{-s}\\
=\mathfrak{p}rod_{i=1}^{r}\sum_{\chi_i \in \textup{Irr}_i}\chi_i(1)^{-s} \hspace{0.35cm}
&=\mathfrak{p}rod_{i=1}^{r}\zeta^\textup{irr}_{\mathbf{m}athbf{G}(\mathcal{O}/\mathfrak{p}_{i}^{e_i})}(s).
\end{align*}
For the conjugacy class zeta function, we use the fact that each conjugacy class~$C$ of $\mathbf{m}athbf{G}(\mathcal{O}/\mathfrak{p}_{1}^{e_1})\times \dots \times \mathbf{m}athbf{G}(\mathcal{O}/\mathfrak{p}_{r}^{e_r})$ is of the form
$C=C_1 \times \dots \times C_r$, where~$C_i$ is a conjugacy class of $\mathbf{m}athbf{G}(\mathcal{O}/\mathfrak{p}_{i}^{e_i})$, for each $i\in [r]$. Thus
\[{c}_n(\mathbf{m}athbf{G}(\mathcal{O}/I))=\sum_{\substack{n_1,\dots,n_r\in \mathbf{m}athds{N}_0\\n_1 \cdots n_r=n}}{c}_{n_1}(\mathbf{m}athbf{G}(\mathcal{O}/\mathfrak{p}_{1}^{e_1}))\cdots{c}_{n_r}(\mathbf{m}athbf{G}(\mathcal{O}/\mathfrak{p}_{r}^{e_r})).\]
Set $q_i=|\mathcal{O}:\mathfrak{p}_i|$. In Section~\text{Re}f{chcc}, we show that all conjugacy classes of $\mathbf{m}athbf{G}(\mathcal{O}/\mathfrak{p}_{i}^{e_i})$ have size a power of~$q_i$. Consequently
\begin{align*}
\zeta^\textup{cc}_{\mathbf{m}athbf{G}(\mathcal{O}/I)}(s) &=\sum_{n=1}^{\infty}\sum_{\substack{n_1,\dots,n_r\in \mathbf{m}athds{N}_0\\q_{1}^{n_1} \dots q_{r}^{n_r}=n}}{c}_{q_{1}^{n_1}}(\mathbf{m}athbf{G}(\mathcal{O}/\mathfrak{p}_{1}^{e_1})) \cdots
{c}_{q_{r}^{n_r}}(\mathbf{m}athbf{G}(\mathcal{O}/\mathfrak{p}_{r}^{e_r}))(q_{1}^{n_1}\dots q_{r}^{n_r})^{-s}&\\
&=\mathfrak{p}rod_{k=1}^{r}\left(\sum_{n_k=0}^{\infty}{c}_{q_{k}^{n_k}}(\mathbf{m}athbf{G}(\mathcal{O}/\mathfrak{p}_{k}^{e_k}))q_{k}^{-n_ks}\mathcal{O}ght)=
\mathfrak{p}rod_{k=1}^{r}\zeta^\textup{cc}_{\mathbf{m}athbf{G}(\mathcal{O}/\mathfrak{p}_{k}^{e_k})}(s).\qedhere
\end{align*}
\end{proof}
\section{Group schemes obtained from nilpotent Lie lattices}\label{groups}
Here, we explain briefly how the unipotent groups schemes $\mathbf{m}athbf{G}$ of interest in this work are constructed; for more details see~\cite[Section~2.1.2]{StVo14}.
An $\mathcal{O}$-Lie lattice is a free and finitely generated $\mathcal{O}$-module~$\Lambda$ together with an antisymmetric $\mathcal{O}$-bilinear form $[\cdot,\cdot]$ which satisfies the Jacobi identity.
Let $(\Lambda, [\cdot,\cdot])$ be a nilpotent $\mathcal{O}$-Lie lattice of $\mathcal{O}$-rank~$h$. Let $\mathbf{m}athscr{B}=(x_1, \dots, x_h)$ be an $\mathcal{O}$-basis for~$\Lambda$.
For each $\mathcal{O}$-algebra $R$, set $\Lambda(R)=\Lambda \otimes_{\mathcal{O}} R$. Then
\[\mathbf{m}athscr{B}_R=(x_1 \otimes 1_R, \dots, x_h\otimes 1_R)\] is an $R$-basis of~$\Lambda(R)$.
If~$\Lambda$ has nilpotency class~$c$ and satisfies $\Lambda' \subseteq c!\Lambda$, where $\Lambda'=[\Lambda,\Lambda]$ is its derived Lie sublattice, we may define a group operation~$\ast$ on~$\Lambda(\mathcal{O})$ by means of the Hausdorff series.
The assumption $\Lambda' \subseteq c!\Lambda$ assures that all denominators in the Hausdorff series will cancel out, so that the coordinates of the operation~$\ast$ in terms of~$\mathbf{m}athscr{B}$ are given by polynomials $f_1$, $\dots$, $f_h: \mathcal{O}^{2h} \to \mathcal{O}$, say, with coefficients in~$\mathcal{O}$. That is, given $a=\sum_{i=1}^{h}a_ix_i$, $b=\sum_{i=1}^{h}b_ix_i \in \Lambda(\mathcal{O})$, we have
\[a \ast b =\sum_{i=1}^{h}f_i(\underline{a},\underline{b})x_i,\]
where $\underline{a}=(a_1, \dots, a_h)$, $\underline{b}=(b_1, \dots, b_h) \in \mathcal{O}^h$.
For each $\mathcal{O}$-algebra~$R$, one may give $\Lambda(R)$ a group structure by defining the group operation~$\ast$ on $\Lambda(R)$ as follows.
\[\text{ for all } a,b \in \Lambda(R): a \ast b = \sum_{i=1}^{f}f_i(\underline{a},\underline{b}) (x_i\otimes 1_R),\]
where \[a=\sum_{i=1}^{h}a_i(x_i\otimes 1_R),~ b=\sum_{i=1}^{h}b_i(x_i\otimes 1_R) \in \Lambda(R),\]
and $\underline{a}=(a_1, \dots, a_h)$, $\underline{b}=(b_1, \dots, b_h) \in \mathcal{O}^h$.
This defines a unipotent $\mathcal{O}$-group scheme $\mathbf{m}athbf{G}=\mathbf{m}athbf{G}_{\Lambda}$ which is isomorphic as a scheme to affine $h$-space over~$\mathcal{O}$ which represents the group functor $R \mathbf{m}apsto (\Lambda(R),\ast)$.
The $\mathcal{O}$-group scheme~$\mathbf{m}athbf{G}$ is such that $\mathbf{m}athbf{G}(\mathcal{O})$ is a $\mathbf{m}athcal{T}$-group of nilpotency class~$c$.
If~$R$ is an $\mathcal{O}$-algebra whose underlying additive group is a finitely generated pro-$p$ group, for instance $R=\mathcal{O}c$, then $\mathbf{m}athbf{G}(R)$ is a finitely generated pro-$p$ group of nilpotency class~$c$.
For Lie lattices~$\Lambda$ of nilpotency class~$2$, a different construction of such unipotent group schemes is given in~\cite[Section~2.4.1]{StVo14}, in which case the hypothesis $\Lambda' \subseteq 2\Lambda$ is not needed. However, if this condition is satisfied, the unipotent group schemes obtained via such construction coincide with the latter ones.
\begin{rmk}
Definition~\text{Re}f{biva} of bivariate zeta functions of groups~$\mathbf{m}athbf{G}(\mathcal{O})$ might be extended to $\mathbf{m}athcal{T}$-groups which are not necessarily of the form~$\mathbf{m}athbf{G}(\mathcal{O})$.
As explained in~\cite[Section~5]{DuVo14} every $\mathbf{m}athcal{T}$-group~$G$ is virtually of the form~$\mathbf{m}athbf{G}(\mathbf{m}athds{Z})$ for some unipotent $\mathbf{m}athds{Z}$-group scheme~$\mathbf{m}athbf{G}$ obtained from a nilpotent $\mathbf{m}athds{Z}$-Lie lattice. We might then define $\lvlzf^{\ast}_G(s_1,s_2)$ as the corresponding bivariate zeta function of a finite index subgroup~$H$ of~$G$ which is of the form~$\mathbf{m}athbf{G}(\mathbf{m}athds{Z})$.
That is,
\[\lvlzf^{\ast}_{G}(s_1,s_2)=\lvlzf^{\ast}_{G,H}(s_1,s_2):=\lvlzf^{\ast}_{\mathbf{m}athbf{G}(\mathbf{m}athds{Z})}(s_1,s_2),~\ast\in\{\textup{irr}, {c}up\}.\]
If~$G$ has two subgroups $H_1=\mathbf{m}athbf{G}_1(\mathbf{m}athds{Z})$ and $H_2=\mathbf{m}athbf{G}_2(\mathbf{m}athds{Z})$ of finite index, then $H_1$ and $H_2$ are commensurable and, therefore, they have the same pro-$p$ completion for all but finitely many prime integers~$p$; see~\cite[Lemma~1.8]{Pi71}.
In particular, $\lvlzf^{\ast}_{\mathbf{m}athbf{G}_1(\mathbf{m}athds{Z}_p)}(s_1,s_2)=\lvlzf^{\ast}_{\mathbf{m}athbf{G}_2(\mathbf{m}athds{Z}_p)}(s_1,s_2)$, for all but finitely many primes~$p$, that is, although $\lvlzf^{\ast}_{G,H_1}(s_1,s_2)$ and $\lvlzf^{\ast}_{G,H_2}(s_1,s_2)$ may not coincide, they are almost the same in the sense that they coincide except for finitely many local factors.
In this point of view, bivariate zeta functions also yield invariants for $\mathbf{m}athcal{T}$-groups, namely domains of convergence and of meromorphy.
More precisely, we prove in~\cite[Theorem~1]{PL18III} that the maximal domains of convergence of the bivariate zeta functions of groups of the form $\mathbf{m}athbf{G}(\mathcal{O})$---when finitely many local factors are disregarded---are independent of the ring of integers~$\mathcal{O}$ and admit meromorphic continuations to domains which are also independent of~$\mathcal{O}$.
\end{rmk}
From now on, we assume that~$\mathbf{m}athbf{G}$ is a unipotent $\mathcal{O}$-group scheme obtained from a nilpotent $\mathcal{O}$-Lie lattice~$\Lambda$.
\section{Bivariate zeta functions and \texorpdfstring{$\mathfrak{p}$}{p}-adic integrals}\label{padicintegral}
Our results rely on the fact that local bivariate representation and local bivariate conjugacy class zeta functions of groups associated to unipotent group schemes can be written in terms of $\mathfrak{p}$-adic integrals.
In Section~\text{Re}f{spadic}, we show how to write most of the local factors of these zeta functions in terms of $\mathfrak{p}$-adic integrals using the methods of~\cite[Section~2.2]{Vo10}. A consequence is that these local factors are given by rational functions as stated in Theorem~\text{Re}f{thmA}.
In Section~\text{Re}f{pthmA}, we use the results of~\cite[Section~2.1]{Vo10} and~\cite[Section~4.1]{AKOV13} to show that these local factors satisfy functional equations and that they are uniform under base extension, concluding the proof of Theorem~\text{Re}f{thmA}.
The methods of~\cite[Section~2]{Vo10} were also applied in~\cite{Ro17} to show functional equations for the ask zeta functions of modules of matrices over compact discrete valuation rings~$\mathfrak{O}$ in characteristic zero. This result provides functional equations for the class number zeta functions of certain nilpotent groups $G \leq \mathbf{m}athbf{G}l_d(\mathfrak{O})$.
Furthermore, these methods were applied to show results similar to Theorem~1.4 in~\cite{AKOV13}, for the representation zeta functions of certain $p$-adic analytic groups, and in~\cite{StVo14}, for the local factors of twist representation zeta functions of groups of the form~$\mathbf{m}athbf{G}(\mathcal{O})$.
We now recall the methods of~\cite[Section~2.2]{Vo10}.
For the rest of this section, let $\mathfrak{p}$~be a fixed nonzero prime ideal of~$\mathcal{O}$ and $\mathfrak{o}=\mathcal{O}c$.
Denote by~$q$ the cardinality of~$\mathcal{O}/\mathfrak{p}$ and by~$p$ its characteristic.
Recall that given an element $z \in \mathfrak{o}$, the ideal $(z) \lhd \mathcal{O}$ has prime factorisation $(z)=\mathfrak{p}^{e}\mathfrak{p}_{1}^{e_1}\cdots\mathfrak{p}_{r}^{e_r}$ such that $\mathfrak{p}_i\neq \mathfrak{p}$,
for all $i\in[r]$. The $\mathfrak{p}$-\emph{adic valuation} of $z$ is $v_{\mathfrak{p}}(z)=e$, and its $\mathfrak{p}$-\emph{adic norm} is $|z|_{\mathfrak{p}}=q^{-v_{\mathfrak{p}}(z)}$. Equivalently, $v_{\mathfrak{p}}(z)=e$ and $|z|_{\mathfrak{p}}=q^{-e}$ if $z \in \mathfrak{p}^e\setminus \mathfrak{p}^{e+1}$.
For each $j \in \mathbf{m}athds{N}$, denote by $\|\cdot\|_{\mathfrak{p}}$ the maximum norm of $\mathfrak{o}^j$ with respect to $|\cdot|_{\mathfrak{p}}$; that is, for $\mathbf{m}athbf{z}=(z_1, \dots , z_j)\in \mathfrak{o}^j$, let $\|\mathbf{m}athbf{z}\|_{\mathfrak{p}}=\mathbf{m}ax\{|z_k|_{\mathfrak{p}}\}_{k=1}^{j}$.
For $N \in \mathbf{m}athds{N}$, we also denote by $v_{\mathfrak{p}}$ the function on $\mathfrak{o}/\mathfrak{p}^N$ given as follows: let~$\overline{z}$ be the image of $z \in \mathfrak{o}$ under $\mathfrak{o} \to \mathfrak{o}/\mathfrak{p}^N$ and assume $z \in \mathfrak{p}^e \setminus \mathfrak{p}^{e+1}$. Then $v_{\mathfrak{p}}(\overline{z})=e$ if $0 \leq e <N$, and $v_{\mathfrak{p}}(\overline{z})=+\infty$, otherwise.
We make the following distinction: $\mathfrak{p}^m$ denotes the $m$th ideal power $\mathfrak{p}\cdots\mathfrak{p}$, whilst $\mathfrak{p}^{(m)}$ denotes the $m$-fold Cartesian power $\mathfrak{p} \times \dots \times \mathfrak{p}$.
For $k$, $ N\in \mathbf{m}athds{N}$, set
\begin{align*}W_{k}(\mathfrak{o}p)&:=((\mathfrak{o}p)^k)^{*}=\{\mathbf{m}athbf{x}\in(\mathfrak{o}p)^k\mathbf{m}id v_{\mathfrak{p}}(\mathbf{m}athbf{x})=0\},\\
W^{\lri}_{k}&:=(\mathfrak{o}^k)^{*}\mathfrak{p}hantom{(/p^N)}=\{\mathbf{m}athbf{x}\in\mathfrak{o}^k\mathbf{m}id v_{\mathfrak{p}}(\mathbf{m}athbf{x})=0\},
\end{align*}
and let $W_{k}((0))=(0)^k$ for each $k \in \mathbf{m}athds{N}$.
Let $\mathfrak{p}i\in \mathfrak{o}$ be a uniformizer of $\mathfrak{o}$. Given a matrix $M\in \Mat_{m \times n}(\mathfrak{o}/\mathfrak{p}^N)$, we write $\nu(M)=(m_1, \dots, m_{\epsilon})$ to indicate the elementary divisor type of~$M$, where $0 \leq \epsilon \leq \mathbf{m}in\{m,n\}$.
In the following, denote by~$\text{Frac}(\mathfrak{o})$ the field of fractions of~$\mathfrak{o}$.
Let $n \in \mathbf{m}athds{N}$ and let $\mathbf{m}athcal{R}(\underline{Y})=\mathbf{m}athcal{R}(Y_1, \dots, Y_n)$ be a matrix of polynomials $\mathbf{m}athcal{R}(\underline{Y})_{ij}\in\mathfrak{o}[\underline{Y}]$ with
\[u_\mathbf{m}athcal{R}=\mathbf{m}ax\{\textup{rk}_{\text{Frac}(\mathfrak{o})}\mathbf{m}athcal{R}(\mathbf{m}athbf{z}) \mathbf{m}id \mathbf{m}athbf{z}\in \mathfrak{o}^n\}.\] For each $\mathbf{m}athbf{m}\in \mathbf{m}athds{N}_{0}^{u_\mathbf{m}athcal{R}}$, write
\begin{align*}
\mathbf{m}athfrak{N}_{\mathbf{m}}^{\mathbf{m}athcal{R}}(\mathfrak{o}p)&:=\{\mathbf{m}athbf{y} \in W_{n}(\mathfrak{o}p) \mathbf{m}id \nu(\mathcal{R}(\mathbf{y}))= \mathbf{m}athbf{m}\}\text{ and }\\
\mathbf{m}athds{N}r_{\mathbf{m}}(\mathfrak{o}p)&:=|\mathbf{m}athfrak{N}_{\mathbf{m}}^{\mathbf{m}athcal{R}}(\mathfrak{o}p)|.\nonumber
\end{align*}
The number $\mathbf{m}athds{N}r_{\mathbf{m}}(\mathfrak{o}p)$ is zero unless $\mathbf{m}athbf{m}=(m_1, \dots, m_{u_{\mathcal{R}}})$ satisfies \[0 = m_1 \leq \dots \leq m_{u_{\mathcal{R}}} \leq N.\]
Let $\underline{r}=(r_1, \dots, r_{u_{\mathcal{R}}})$ be a vector of variables. Consider the Poincaré series
\begin{equation}\label{poincare}\mathcal{P}^{\lri}_{\mathbf{m}athcal{R}}(\underline{r},t)=\sum_{\substack{N \in \mathbf{m}athds{N}\\ \mathbf{m} \in \mathbf{m}athds{N}_{0}^{u_{\mathcal{R}}}}}
\mathbf{m}athds{N}r_{\mathbf{m}}(\mathfrak{o}p)q^{-tN-\sum\limits_{i=1}^{u_{\mathcal{R}}}r_im_i}.
\end{equation}
In~\cite[Section 2.2]{Vo10} it is shown how to describe the series~\eqref{poincare} in terms of the $\mathfrak{p}$-adic integral
\begin{equation}\label{Vo10}\mathfrak{p}int_{\mathbf{m}athcal{R}}(\underline{r},t)=\mathfrak{f}rac{1}{1-q^{-1}}\int_{(x,\underline{y}) \in \mathfrak{p}\times W^{\lri}_{n}}|x|_{\mathfrak{p}}^{t}\mathfrak{p}rod_{k=1}^{u_{\mathcal{R}}}
\mathfrak{f}rac{\|F_{k}(\mathbf{m}athcal{R}(\underline{y})) \cup xF_{k-1}(\mathbf{m}athcal{R}(\underline{y}))\|_{\mathfrak{p}}^{r_k}}{\|F_{k-1}(\mathbf{m}athcal{R}(\underline{y}))\|_{\mathfrak{p}}^{r_{k}}}d\mathbf{m}u,
\end{equation}
where $\mathbf{m}u$ is the additive Haar measure normalised so that $\mathbf{m}u(\mathfrak{o}^{n+1})=1$;
$F_j(\mathbf{m}athcal{R}(\underline{y}))$ is the set of nonzero $j \times j$-minors of $\mathbf{m}athcal{R}(\underline{y})$.
More precisely, in~\cite[Section 2.2]{Vo10} it is shown that~\eqref{poincare} satisfies
\begin{equation}\label{Vo14gen}
\mathcal{P}^{\lri}_{\mathbf{m}athcal{R}}(\underline{r},t)=\mathfrak{p}int_{\mathbf{m}athcal{R}}(\underline{r},t-n-1).
\end{equation}
Suppose now that $M\in \Mat_{n\times n}(\mathfrak{o}/\mathfrak{p}^N)$ is an antisymmetric matrix. Then, for some $\xi \in [n]_0:=\{0,1,\dots,n\}$, the elementary divisor type of~$M$ is of the form
\[\nu(M)=(m_1, m_1, m_2, m_2, \dots, m_{\xi},m_{\xi}).\]
If $M$ is antisymmetric, we write $\tilde{\nu}(M)=(m_1, m_2, \dots, m_{\xi})$ for its reduced elementary divisor type, that is, to indicate~$\nu(M)=(m_1, m_1, m_2, m_2, \dots, m_{\xi},m_{\xi})$.
Assume now that $\mathbf{m}athcal{R}(\underline{Y})$ is antisymmetric, in which case $u_{\mathcal{R}}$ is even. For each $\mathbf{m} \in \mathbf{m}athds{N}_{0}^{u_{\mathcal{R}}/2}$, write
\begin{align*}\widetilde{\mathbf{m}athfrak{N}}_{\mathbf{m}}^{\mathbf{m}athcal{R}}(\mathfrak{o}p)&=\{\mathbf{m}athbf{y} \in W_{n}(\mathfrak{o}p) \mathbf{m}id \tilde{\nu}(\mathbf{m}athcal{R}(\textbf{y}))= \mathbf{m}athbf{m}\} \text{ and } \\
\mathbf{m}athds{N}r_{\mathbf{m}}(\mathfrak{o}p)&=|\widetilde{\mathbf{m}athfrak{N}}_{\mathbf{m}}^{\mathbf{m}athcal{R}}(\mathfrak{o}p)|.
\end{align*}
For $\mathbf{m}athcal{R}(\underline{Y})$ antisymmetric, we assume that the vector of variables $\underline{r}$ is of the form $\underline{r}=\left(r_1,r_1,\dots,r_{u_{\mathcal{R}}/2},r_{u_{\mathcal{R}}/2}\mathcal{O}ght)$ so that
\begin{equation}\label{poincareas}\mathcal{P}^{\lri}_{\mathbf{m}athcal{R}}(\underline{r},t)=\sum_{N \in \mathbf{m}athds{N},~\mathbf{m} \in \mathbf{m}athds{N}_{0}^{u_{\mathcal{R}}/2}}
\mathbf{m}athds{N}r_{\mathbf{m}}(\mathfrak{o}p)q^{-tN-2\sum\limits_{i=1}^{u_{\mathcal{R}}/2}r_im_i}.
\end{equation}
Recall the notation $[n]=\{1,\dots, n\}$, for $n \in \mathbf{m}athds{N}$. Given $x\in \mathfrak{o}$ with $v_{\mathfrak{p}}(x)=N$, $\mathbf{m}athbf{y}\in\mathfrak{o}^{n}$, and $k\in[u_{\mathcal{R}}]$, we obtain from \cite[Lemma~4.6(i) and~(ii)]{Ro17} the following for the antisymmetric
matrix $\mathbf{m}athcal{R}(\mathbf{m}athbf{y})$ with $\tilde{\nu}(\mathbf{m}athcal{R}(\mathbf{m}athbf{y}))=(m_1, \dots, m_{u_{\mathcal{R}}})$:
\[\mathfrak{f}rac{\|F_{2k}(\mathbf{m}athcal{R}(\mathbf{m}athbf{y})) \cup x F_{2k-1}(\mathbf{m}athcal{R}(\mathbf{m}athbf{y}))\|_{\mathfrak{p}}}{\|F_{2k-1}(\mathbf{m}athcal{R}(\mathbf{m}athbf{y}))\|_{\mathfrak{p}}}=
\mathfrak{f}rac{\|F_{2k-1}(\mathbf{m}athcal{R}(\mathbf{m}athbf{y})) \cup x F_{2(k-1)}(\mathbf{m}athcal{R}(\mathbf{m}athbf{y}))\|_{\mathfrak{p}}}{\|F_{2(k-1)}(\mathbf{m}athcal{R}(\mathbf{m}athbf{y}))\|_{\mathfrak{p}}}=q^{-\mathbf{m}in(m_k,N)},\]
and
\[\mathfrak{f}rac{\|F_{2k}(\mathbf{m}athcal{R}(\mathbf{m}athbf{y})) \cup x^2 F_{2(k-1)}(\mathbf{m}athcal{R}(\mathbf{m}athbf{y}))\|_{\mathfrak{p}}}{\|F_{2(k-1)}(\mathbf{m}athcal{R}(\mathbf{m}athbf{y}))\|_{\mathfrak{p}}}=q^{-2\mathbf{m}in(m_k,N)}.\]
Therefore, if $\mathbf{m}athcal{R}(\underline{Y})$ is an antisymmetric matrix, the series~\eqref{poincareas} can be described by the $\mathfrak{p}$-adic integral
\begin{align}&\mathcal{P}^{\lri}_{\mathbf{m}athcal{R}}(\underline{r},t)=
\mathfrak{p}int_{\mathbf{m}athcal{R}}(\underline{r},t-n-1)=\nonumber\\
&\label{Vo10as}\mathfrak{f}rac{1}{1-q^{-1}}\int_{(x,\underline{y}) \in \mathfrak{p}\times W^{\lri}_{n}}|x|_{\mathfrak{p}}^{t-n-1}
\mathfrak{p}rod_{k=1}^{u_{\mathcal{R}}/2}\mathfrak{f}rac{\|F_{2k}(\mathbf{m}athcal{R}(\underline{y})) \cup x^2F_{2(k-1)}(\mathbf{m}athcal{R}(\underline{y}))\|_{\mathfrak{p}}^{r_k}}
{\|F_{2(k-1)}(\mathbf{m}athcal{R}(\underline{y}))\|_{\mathfrak{p}}^{r_{k}}}d\mathbf{m}u.
\end{align}
\subsection{The numbers \texorpdfstring{$r_n(\mathbf{m}athbf{G}_N)$}{rn(GN)} and \texorpdfstring{${c}_n(\mathbf{m}athbf{G}_N)$}{cn(GN)}}\label{chcc}
Recall the notation $\mathbf{m}athbf{G}_N=\mathbf{m}athbf{G}(\mathfrak{o}p)$. We now write the local bivariate zeta functions at $\mathfrak{p}$ in terms of sums encoding the elementary divisor types of certain matrices
associated to $\Lambda$.
This is done by rewriting the numbers $r_n(\mathbf{m}athbf{G}_N)$ and $c_n(\mathbf{m}athbf{G}_N)$, for $n\in \mathbf{m}athds{N}$ and $N \in \mathbf{m}athds{N}_0$, in terms of numbers~$\mathbf{m}athds{N}r_{\mathbf{m}}(\mathfrak{o}p)$ defined at the beginning of Section~\text{Re}f{padicintegral}. In each case, $\mathbf{m}athcal{R}$ is one of the two commutator matrices of
$\Lambda$ which we now define.
Set $\mathfrak{g}=\Lambda(\mathfrak{o})=\Lambda\otimes_{\mathcal{O}} \mathfrak{o}$. Let $\mathfrak{g}'$ be the derived Lie sublattice of $\mathfrak{g}$, and let $\mathfrak{z}$ be its center. Consider the torsion-free $\mathcal{O}$-ranks
\begin{align*} & h=\textup{rk}(\mathfrak{g}), & & a=\textup{rk}(\mathfrak{g}/\mathfrak{z}), & & b=\textup{rk}(\mathfrak{g}'), & & r= \textup{rk}(\mathfrak{g}/\mathfrak{g}'), & & z=\textup{rk}(\mathfrak{z}) .\end{align*}
For~$R$ either~$\mathcal{O}$ or~$\mathfrak{o}$, let~$M$ be a finitely generated $R$-module with a submodule~$N$. The \emph{isolator}~$\iota(N)$ of~$N$ in~$M$ is the smallest submodule~$L$ of~$M$ containing~$N$ such that~$M/L$ is torsion free. In particular $\mathfrak{z}=\iota(\mathfrak{z})$; see~\cite[Lemma~2.5]{StVo14}.
Set $k=\textup{rk}(\iota(\mathfrak{g}')/\iota(\mathfrak{g}'\cap\mathfrak{z}))=\textup{rk}(\iota(\mathfrak{g}'+\mathfrak{z})/\mathfrak{z})$.
The commutator matrices are defined with respect to a fixed $\mathfrak{o}$-basis
\[\mathbf{m}athscr{B}=(e_1, \dots, e_h)\] of the $\mathfrak{o}$-Lie lattice $\mathfrak{g}$, satisfying the conditions
\begin{align*}
(e_{a-k+1}, \dots, e_{a}) & \text{ is an $\mathfrak{o}$-basis for }\iota(\mathfrak{g}'+\mathfrak{z}),\\
(e_{a+1}, \dots, e_{a-k+b}) & \text{ is an $\mathfrak{o}$-basis for }\iota(\mathfrak{g}' \cap \mathfrak{z}),\text{ and}\\
(e_{a+1}, \dots, e_{h}) &\text{ is an $\mathfrak{o}$-basis for }\mathfrak{z}.
\end{align*}
Denote by $\overline{\mathfrak{p}hantom{X}}$ the natural surjection $\mathfrak{g} \to \mathfrak{g}/\mathfrak{z}$.
Let $\mathbf{m}athbf{e}=(e_1, \dots, e_a)$. Then
\[\overline{\mathbf{m}athbf{e}}=(\overline{e_1}, \dots, \overline{e_a})\] is an $\mathfrak{o}$-basis of $\mathfrak{g}/\mathfrak{z}$.
The $e_i$ can be chosen so that there are nonnegative integers $c_1,\dots, c_{b}$ with the property that
\begin{align*}
(\overline{\mathfrak{p}i^{c_1}e_{a-k+1}}, \dots, \overline{\mathfrak{p}i^{c_k}e_{a}}) &\text{ is an $\mathfrak{o}$-basis of }\overline{\mathfrak{g}'+\mathfrak{z}}\text{ and}\\
(\mathfrak{p}i^{c_{k+1}}e_{a+1}, \dots, \mathfrak{p}i^{c_b}e_{a-k+b}) &\text{ is an $\mathfrak{o}$-basis of }\mathfrak{g}'\cap\mathfrak{z},
\end{align*}
by the elementary divisor theorem.
Fix an $\mathfrak{o}$-basis $\mathbf{m}athbf{f}=(f_1, \dots, f_b)$ for $\mathfrak{g}'$ satisfying
\begin{align*}
(\overline{f_1}, \dots, \overline{f_k})&=(\overline{\mathfrak{p}i^{c_1}e_{a-k+1}}, \dots, \overline{\mathfrak{p}i^{c_k}e_{a}}) \text{ is an $\mathfrak{o}$-basis of }\overline{\mathfrak{g}'+\mathfrak{z}}\text{ and}\\
(f_{k+1}, \dots, f_{b})&=(\mathfrak{p}i^{c_{k+1}}e_{a+1}, \dots, \mathfrak{p}i^{c_b}e_{a-k+b}) \text{ is an $\mathfrak{o}$-basis of }\mathfrak{g}'\cap\mathfrak{z}.
\end{align*}
For $i,j \in [a]$ and $k \in [b]$, let $\lambda_{ij}^{k}\in \mathfrak{o}$ be the structure constants satisfying
\[[e_i,e_j]=\sum_{k=1}^{b}\lambda_{ij}^{k}f_k.\]
The following matrices were previously defined in~\cite[Definition~2.1]{ObVo15}.
\begin{dfn}\label{commutator} The $A$\emph{-commutator} and the $B$\emph{-commutator matrices} of $\mathfrak{g}$ with respect to $\mathbf{m}athbf{e}$ and $\mathbf{m}athbf{f}$ are
\begin{align*}
A(X_1, \dots, X_a)&=\left( \sum_{j=1}^{a}\lambda_{ij}^{k}X_j\mathcal{O}ght)_{ik}\in \Mat_{a \times b}(\mathfrak{o}[\underline{X}]),\text{ and} \\
B(Y_1,\dots, Y_b)&=\left( \sum_{k=1}^{b}\lambda_{ij}^{k}Y_k\mathcal{O}ght)_{ij}\in \Mat_{a \times a}(\mathfrak{o}[\underline{Y}]),
\end{align*}
respectively, where $\underline{X}=(X_1, \dots, X_a)$ and $\underline{Y}=(Y_1, \dots, Y_b)$ are independent variables.
\end{dfn}
For each $\mathbf{m}athbf{y}\in \mathfrak{o}^b$, the matrix $B(\mathbf{m}athbf{y})$ is antisymmetric.
Fix $N \in \mathbf{m}athds{N}$. The congruence quotient $\mathbf{m}athbf{G}_N$ is a finite $p$-group of nilpotency class~$c$.
Set $\mathfrak{g}n:=\Lambda \otimes_{\mathfrak{o}} \mathfrak{o}p$ and $\mathfrak{z}n=\mathfrak{z}\otimes_{\mathfrak{o}} \mathfrak{o}p$, and let $\mathfrak{g}nc=\mathfrak{g}\otimes_\mathfrak{o} \mathfrak{o}p$.
Tensoring~$\mathbf{m}athbf{e}$ and $\mathbf{m}athbf{f}$ with $\mathfrak{o}p$ yields ordered sets
\[\mathbf{m}athbf{e}_N=(e_{1}^{N}, \dots, e_{a}^{N})\text{ and }\mathbf{m}athbf{f}_N=(f_{1}^{N},\dots,f_{b}^{N})\] such that
$\overline{\mathbf{m}athbf{e}}=(\overline{e_{1}^{N}}, \dots,\overline{e_{a}^{N}})$ is an $\mathfrak{o}p$-basis for $\mathfrak{g}n/\mathfrak{z}n$ and $\mathbf{m}athbf{f}_N$ is an $\mathfrak{o}p$-basis for $\mathfrak{g}nc$ as $\mathfrak{o}p$-modules, where $\overline{~\cdot~}$ is the natural surjection $\mathfrak{g}n \to \mathfrak{g}n/\mathfrak{z}n$.
Given an element $\omega$ of $\widehat{\mathfrak{g}n}=\text{Hom}_{\mathfrak{o}}(\mathfrak{g}n,\mathbf{m}athds{C}^{\times})$, set
\[ B_{\omega}^{N}: \mathfrak{g}n \times \mathfrak{g}n ~\mathcal{O}ghtarrow ~ \mathbf{m}athds{C}^{\times} ,~ (u,v) \mathbf{m}apsto \omega([u,v]).\]
The \emph{radical} of $B_{\omega}^{N}$ is
\[\textup{Rad}(B_{\omega}^{N})=\{u \in \mathfrak{g}n\mathbf{m}id \text{ for all } v \in \mathfrak{g}n :~B_{\omega}^{N}(u,v)=1 \}.\]
Observe that $B_{\omega}$ depends only on the restriction of~$\omega$ to $\mathfrak{g}nc$.
For this reason, given $\widetilde{\omega} \in \widehat{\mathfrak{g}nc}$ and any extension $\omega$ of~$\widetilde{\omega}$ to $\mathfrak{g}n$, we write~$B_{\widetilde{\omega}}$ for~$B_{\omega}$.
For $x \in \mathfrak{g}n/\mathfrak{z}n$, following~\cite[Section~3.1]{ObVo15}, we define
\begin{alignat*}{5} \textup{ad}_x:\mathfrak{g}n/\mathfrak{z}n & \to \mathfrak{g}nc &\hspace{0.5cm}&\text{and}\hspace{0.5cm} & \textup{ad}_{x}^{\star}: \widehat{\mathfrak{g}nc} &\to \widehat{\mathfrak{g}n/\mathfrak{z}n}\\
y & \mathbf{m}apsto [y,x] &\hspace{0.5cm}&& ~\omega & \mathbf{m}apsto \omega \circ \textup{ad}_{x}.
\end{alignat*}
Observe that in the definition of the $\mathfrak{o}p$-module homomorphism $\textup{ad}_x$ we identify~$\mathfrak{g}n/\mathfrak{z}n$ with the $\mathfrak{o}p$-submodule of~$\mathfrak{g}n$ generated by~$\textbf{e}_N$.
The dimensions of the irreducible complex representations and the sizes of conjugacy classes of~$\mathbf{m}athbf{G}_N$ are powers of~$p$ and, according to~\cite[Section~3]{ObVo15},
for $c<p$, the numbers $r_{p^i}(\mathbf{m}athbf{G}_N)$ and $c_{p^i}(\mathbf{m}athbf{G}_N)$ are given by
\begin{align}
\label{chi} r_{p^i}(\mathbf{m}athbf{G}_N)&=\left|\left\{\omega \in \widehat{\mathfrak{g}nc} \bigm| |\textup{Rad}(B_{\omega}^{N}):\mathfrak{z}n|=p^{-2i}|\mathfrak{g}n/\mathfrak{z}n|\mathcal{O}ght\}\mathcal{O}ght|\hspace{0.05cm}|\mathfrak{g}n/\mathfrak{g}nc|~p^{-2i}, \\[0.1cm]
\label{cci}{c}_{p^i}(\mathbf{m}athbf{G}_N)&=\left|\left\{x \in \mathfrak{g}n/\mathfrak{z}n \bigm| |\textup{Ker}(\textup{ad}_{x}^{\star})|=p^{-i}|\widehat{\mathfrak{g}nc}|\mathcal{O}ght\}\mathcal{O}ght|\hspace{0.05cm}|\mathfrak{z}n|~p^{-i}.
\end{align}
The first formula is a consequence of the Kirillov orbit method, which reduces the problem of enumerating the characters of $\mathbf{m}athbf{G}_N$ to the problem of determining the indices
in $\mathfrak{g}n$ of $\textup{Rad}(B_{\omega}^{N})$ for $\omega \in \widehat{\mathfrak{g}nc}$; see~\cite[Theorem 3.1]{ObVo15}.
The second formula reflects the fact that the Lazard correspondence induces an order-preserving correspondence between subgroups of $\mathbf{m}athbf{G}_N$ and sublattices of $\mathfrak{g}n$, and maps
normal subgroups to ideals. Moreover, centralizers of elements in~$\mathbf{m}athbf{G}_N$ correspond to centralizers of elements in~$\mathfrak{g}n$ under the Lazard correspondence.
The cardinalities of $\mathfrak{g}n$ and $\mathfrak{g}n/\mathfrak{z}n$ are powers of $q$, and hence so are the cardinalities of $\textup{Rad}(B_{\omega}^{N})/\mathfrak{z}$ and $\textup{Ker}(\textup{ad}_{x}^{\star})$.
It follows that $r_{n}(\mathbf{m}athbf{G}_N)$ and $c_n(\mathbf{m}athbf{G}_N)$ can only be nonzero if $n$ is a power of $q$.
The next step is to relate~\eqref{chi} and~\eqref{cci} to the commutator matrices of~$\mathfrak{g}n$ with respect to~$\mathbf{m}athbf{e}_N$ and $\mathbf{m}athbf{f}_N$.
Using arguments analogous to the ones of~\cite[Section~2]{ObVo15}, we define the coordinate systems
\begin{align*} \mathfrak{p}hi_N: \mathfrak{g}n/\mathfrak{z}n &\to (\mathfrak{o}p)^a, && x=\sum_{j=1}^{a}x_je_{j}^{N} \mathbf{m}apsto \mathbf{m}athbf{x}=(x_1, \dots, x_a),&\\
\mathfrak{p}si_N:\widehat{\mathfrak{g}nc} &\to (\mathfrak{o}p)^b, && \omega=\sum_{j=1}^{b}y_jf_{j}^{N\vee} \mathbf{m}apsto \mathbf{m}athbf{y}=(y_1, \dots, y_b),&
\end{align*}
where, for $N \in \mathbf{m}athds{N}_0$, $\mathbf{m}athbf{f}_{N}^{\vee}=(f_{1}^{N\vee}, \dots, f_{b}^{N\vee})$ is the dual $\mathfrak{o}/\mathfrak{p}$-basis for \[\widehat{\mathfrak{g}nc}=\text{Hom}_{\mathfrak{o}}(\mathfrak{g}nc,\mathbf{m}athds{C}^{\times}).\]
We notice that $\mathfrak{g}_1/\mathfrak{z}_1$ and $\mathfrak{g}'_1$ are regarded as $\mathfrak{o}/\mathfrak{p}$-vector spaces in the construction of~\cite[Section~2]{ObVo15}.
In the coordinate systems above, we regard $\mathfrak{g}n/\mathfrak{z}n$ and $\mathfrak{g}nc$ as $\mathfrak{o}/\mathfrak{p}^N$-modules for all $N\in \mathbf{m}athds{N}$.
\begin{lem}\label{ObVo3.3} Given $x \in \mathfrak{g}n/\mathfrak{z}n$ with $\mathfrak{p}hi_N(x)=\mathbf{m}athbf{x}$, and $\omega \in \widehat{\mathfrak{g}nc}$ with $\mathfrak{p}si_N(\omega)=\mathbf{m}athbf{y}$, the following holds.
\begin{align*}
x \in \textup{Rad}(B_{\omega}^{N})/\mathfrak{z}n \text{ if and only if } B(\mathbf{m}athbf{y})\mathbf{m}athbf{x}^{\textup{tr}}=0,\\
\omega \in \textup{Ker}(\textup{ad}_{x}^{\star}) \text{ if and only if } A(\mathbf{m}athbf{x})\mathbf{m}athbf{y}^{\textup{tr}}=0.
\end{align*}
\end{lem}
\begin{proof}
An element $x \in \mathfrak{g}n/\mathfrak{z}n$ belongs to $\textup{Rad}(B_{\omega}^{N})/\mathfrak{z}$ exactly when $\omega[v,x]=1$, for all $v \in \mathfrak{g}n/\mathfrak{z}n$, whilst an element $\omega\in \widehat{\mathfrak{g}nc}$ belongs to
$\textup{Ker}(\textup{ad}_{x}^{\star})$ exactly when $\omega[v,x]=1$ for all $v \in \mathfrak{g}n/\mathfrak{z}n$. Expressing these conditions in coordinates, we see that both expressions hold.
We prove the second claim in detail.
Fix $x \in \mathfrak{g}n/\mathfrak{z}n$ with \[\mathfrak{p}hi_N(\overline{x})=\mathbf{m}athbf{x}=(x_1, \dots, x_a).\]
It holds that
\begin{equation}\label{eix}
\left[e_{i}^{N},x\mathcal{O}ght]=\sum_{j=1}^{a}x_j[e_{i}^{N},e_{j}^{N}]=\sum_{j=1}^{a}\sum_{l=1}^{b}\lambda_{ij}^{l}x_jf_{l}^{N}.
\end{equation}
We want to determine which elements $\omega\in \widehat{\mathfrak{g}nc}$ satisfy $\omega([v,x])=1$, for all $v\in\mathfrak{g}n/\mathfrak{z}n$.
Consider $\mathfrak{p}si_N(\omega)=\mathbf{m}athbf{y}=(y_1, \dots , y_b)$, i.e., $\omega=\mathfrak{p}rod_{k=1}^{b}(f_{k}^{N\vee})^{y_k}$. Because of~\eqref{eix}, for each $i\in[a]$,
\[\omega([e_{i}^{N},x])=\mathfrak{p}rod_{k=1}^{b}\left(f_{k}^{N\vee}\left(\sum_{j=1}^{a}\sum_{l=1}^{b}\lambda_{ij}^{l}x_jf_{l}^{N}\mathcal{O}ght)\mathcal{O}ght)^{y_k}=
\mathfrak{p}rod_{k=1}^{b}\left(f_{k}^{N\vee}\left(f_{k}^{N}\mathcal{O}ght)\mathcal{O}ght)^{y_k\sum_{j=1}^{a}\lambda_{ij}^{k}x_j}.\]
This expression equals $1$ exactly when $\sum_{k=1}^{b}y_k\sum_{j=1}^{a}\lambda_{ij}^{k}x_j=0$. Now, by definition, $\sum_{j=1}^{a}\lambda_{ij}^{k}x_j=A(\mathbf{m}athbf{x})_{ik}$, where
$A(\mathbf{m}athbf{x})$ is the $A$-commutator matrix of Definition~\text{Re}f{commutator} evaluated at $\mathbf{m}athbf{x}$.
Consequently, $\omega \in \textup{Ker}(\textup{ad}_{x}^{\star})$ if and only if
\[\sum_{k=1}^{b}A(\mathbf{m}athbf{x})_{ik}y_k=0,\text{ for all }i\in [a],\]
that is, $A(\mathbf{m}athbf{x})\mathbf{m}athbf{y}^{\textup{tr}}=0$.
\end{proof}
Applying Lemma~\text{Re}f{ObVo3.3} to~\eqref{chi}, we rewrite the numbers $r_{q^i}(\mathbf{m}athbf{G}_N)$ in terms of solutions of the system
$B(\mathbf{m}athbf{y})\mathbf{m}athbf{x}^{\textup{tr}}=0$ and, applying Lemma~\text{Re}f{ObVo3.3} to~\eqref{cci}, we rewrite the numbers ${c}_{q^i}(\mathbf{m}athbf{G}_N)$ in terms of solutions of the system $A(\mathbf{m}athbf{x})\mathbf{m}athbf{y}^{\textup{tr}}=0$.
In each case, we consider the elementary divisor type of the corresponding matrix.
Fix an elementary divisor type $\tilde{\nu}(B(\mathbf{m}athbf{y}))=(m_1, \dots, m_{u_{B}})$, where
\[2u_{B}=\mathbf{m}ax\{\textup{rk}_{\text{Frac}(\mathfrak{o})}B(\mathbf{m}athbf{z}) \mathbf{m}id \mathbf{m}athbf{z}\in \mathfrak{o}^b\}.\]
Since $B(\mathbf{m}athbf{y})$ is similar to the matrix $\textup{Diag}(\mathfrak{p}i^{m_1},\mathfrak{p}i^{m_1}, \dots,\mathfrak{p}i^{m_{u_{B}}},\mathfrak{p}i^{m_{u_{B}}},\mathbf{m}athbf{0}_{a-2u_{B}})$, where $\mathbf{m}athbf{0}_{a-2u_{B}}=(0,\dots,0)\in \mathbf{m}athds{Z}^{a-2u_{B}}$,
the system $B(\mathbf{m}athbf{y})\mathbf{m}athbf{x}^{\textup{tr}}=0$ in $\mathfrak{o}/\mathfrak{p}^N$ is
equivalent to
\[\begin{cases} x_1\mathfrak{p}hantom{2u_b-} \equiv x_2\mathfrak{p}hantom{ub} \equiv &0 ~\bmod \mathfrak{p}^{N-m_1},\\
x_3\mathfrak{p}hantom{2u_b-} \equiv x_4\mathfrak{p}hantom{ub} \equiv &0 ~\bmod \mathfrak{p}^{N-m_2},\\
&\vdots \\
x_{2u_{B}-1} \equiv x_{2u_{B}} \equiv &0 ~\bmod \mathfrak{p}^{N-m_{u_{B}}}.
\end{cases}\]
For $2u_{B} <a$, the elements $x_{2u_{B}+1}, \dots , x_a$ are arbitrary elements of $\mathfrak{o}/\mathfrak{p}^N$, and
\[|\{x \in \mathfrak{o}/\mathfrak{p}^N \mathbf{m}id x \equiv 0 \bmod \mathfrak{p}^{N-m_j}\}|=q^{m_j}.\]
Hence, the number of solutions of $B(\mathbf{m}athbf{y})\mathbf{m}athbf{x}^{\textup{tr}}=0$ in $\mathfrak{o}/\mathfrak{p}^N$ is
\[q^{2(m_1+ \dots +m_{u_{B}})+(a-2u_{B})N}.\]
In other words, $\tilde{\nu}(B(\mathbf{m}athbf{y}))=(m_1, \dots, m_{u_{B}})$ implies \[|\textup{Rad}(B_{\omega}^{N}) / \mathfrak{z}n|=q^{2(m_1+ \dots + m_{u_{B}})+(a-2u_{B})N}.\]
Lemma~\text{Re}f{ObVo3.3} then assures that $|\textup{Rad}(B_{\omega}^{N}) / \mathfrak{z}n|=q^{-2i}|\mathfrak{g}n/\mathfrak{z}n|=q^{aN-2i}$, whenever $B(\mathbf{m}athbf{y})$ has elementary divisor type $(m_1, \dots, m_{u_{B}})$ satisfying
\[\sum_{j=1}^{u_{B}}m_j=u_{B} N-i.\]
Consequently, for $r=\textup{rk}(\mathfrak{g}/\mathfrak{g}')=h-b$, expression~\eqref{chi} can be rewritten as
\begin{equation}\label{chin} r_{q^i}(\mathbf{m}athbf{G}_N)=\sum_{\mathbf{m} \in \mathcal{D}_{B}^{N}}
|\{\mathbf{m}athbf{y} \in (\mathfrak{o}/\mathfrak{p}^N)^b \mathbf{m}id \tilde{\nu}(B(\mathbf{m}athbf{y}))=\mathbf{m}\}|q^{rN-2i},
\end{equation}
where
\[\mathcal{D}_{B}^{N}:=\left\{\mathbf{m}=(m_1, \dots, m_{u_{B}}) \in \mathbf{m}athds{N}_{0}^{u_{B}} \bigm| m_1 \leq \dots \leq m_{u_{B}} \leq N,~\textstyle\sum\limits_{i=1}^{u_{B}}m_i=u_{B} N-i\mathcal{O}ght\}.\]
Analogously, if $\nu(A(\mathbf{m}athbf{x}))=(m_1, \dots, m_{u_{A}})$, where
\[u_{A}:=\mathbf{m}ax\{\textup{rk}_{\text{Frac}(\mathfrak{o})}A(\mathbf{m}athbf{z}) \mathbf{m}id \mathbf{m}athbf{z}\in \mathfrak{o}^a\},\]
the equality $A(\mathbf{m}athbf{x})\mathbf{m}athbf{y}^{\textup{tr}}=0$ has $q^{m_1+m_2 +\dots +m_{u_{A}}+(b-u_{A})N}$ solutions in $\mathfrak{o}/\mathfrak{p}^N$. For $z=\textup{rk}(\mathfrak{z})=h-a$, this yields
\begin{equation}\label{ccin} c_{q^i}(\mathbf{m}athbf{G}_N)=
\sum_{\mathbf{m} \in \mathcal{D}_{A}^{N}}
|\{\mathbf{m}athbf{x} \in (\mathfrak{o}/\mathfrak{p}^N)^a \mathbf{m}id {\nu}(A(\mathbf{m}athbf{x}))=\mathbf{m}\}|q^{zN-i},
\end{equation}
where
\[\mathcal{D}_{A}^{N}:=\left\{\mathbf{m}=(m_1, \dots, m_{u_{A}}) \in \mathbf{m}athds{N}_{0}^{u_{A}} \bigm| m_1 \leq \dots \leq m_{u_{A}} \leq N,~\textstyle\sum\limits_{i=1}^{u_{A}}m_i=u_{A} N-i\mathcal{O}ght\}.\]
For a matrix $\mathbf{m}athcal{R}(\underline{Y})=\mathbf{m}athcal{R}(Y_1, \dots, Y_n)$ of polynomials as the one at the beginning of Section~\text{Re}f{padicintegral} and for
$\mathbf{m}athbf{m}=(m_1, \dots, m_{u_{\mathcal{R}}})\in \mathbf{m}athds{N}_{0}^{u_{\mathcal{R}}}$, define
\[ \mathbf{m}athfrak{W}_{\mathbf{m}}^{\mathbf{m}athcal{R}}(\mathfrak{o}p):=\{\mathbf{m}athbf{y} \in (\mathfrak{o}/\mathfrak{p}^N)^n \mathbf{m}id \nu(\mathcal{R}(\mathbf{y}))=\mathbf{m}athbf{m}\}.\]
Expressions~\eqref{chin} and~\eqref{ccin} are written in terms of cardinalities of such sets, which are related to the numbers $\mathbf{m}athds{N}r_{\mathbf{m}}(\mathfrak{o}p)$ as follows.
Write
\[\mathbf{m}athbf{m}-m=(m_1-m, \dots, m_{u_{\mathcal{R}}}-m)\text{ for all }m \in \mathbf{m}athds{N}_{0}.\]
If $\mathbf{m}athcal{R}(\mathbf{m}athbf{y})$ is such that $v_\mathfrak{p}(\mathbf{m}athbf{y})=v_{\mathfrak{p}}(\mathbf{m}athcal{R}(\mathbf{m}athbf{y}))$ for all $\mathbf{m}athbf{y} \in \mathfrak{o}^n$, then
\begin{equation}\label{wm} |\mathbf{m}athfrak{W}_{\mathbf{m}}^{\mathbf{m}athcal{R}}(\mathfrak{o}p)|= \mathbf{m}athds{N}r_{\mathbf{m}-m_1}(\mathfrak{o}/\mathfrak{p}^{N-m_1}).\end{equation}
Indeed, the map $\mathbf{m}athfrak{N}_{\mathbf{m}-m_1}^{\mathbf{m}athcal{R}}(\mathfrak{o}/\mathfrak{p}^{N-m_1}) \to \mathbf{m}athfrak{W}_{\mathbf{m}}^{\mathbf{m}athcal{R}}(\mathfrak{o}p)$ given by $\mathbf{m}athbf{y}\mathbf{m}apsto \mathfrak{p}i^{m_1}\mathbf{m}athbf{y}$ is a bijection.
Equality~\eqref{wm} provides the following reformulations of~\eqref{chin} and~\eqref{ccin}.
\hspace{-0.5cm}
\begin{lem}\label{chin2}
For each $i \in \mathbf{m}athds{N}_0$ and $N\in \mathbf{m}athds{N}_0$,
\begin{align}
\label{lemch} r_{q^i}(\mathbf{m}athbf{G}_N)&=\sum_{\mathbf{m} \in \mathbf{m}athcal{D}_{B}^{N}}\mathbf{m}athds{N}b_{\mathbf{m}-m_1}(\mathfrak{o}/\mathfrak{p}^{N-m_1})q^{rN-2i},\\
\label{lemcc} c_{q^i}(\mathbf{m}athbf{G}_N)&=\sum_{\mathbf{m} \in \mathbf{m}athcal{D}_{A}^{N}}\mathbf{m}athds{N}a_{\mathbf{m}-m_1}(\mathfrak{o}/\mathfrak{p}^{N-m_1})q^{zN-i}.
\end{align}
\end{lem}
\begin{rmk}\label{condition2}
As explained in Section~\text{Re}f{groups}, for $\mathcal{O}$-Lie lattices $\Lambda$ of nilpotency class~$2$, a different construction for~$\mathbf{m}athbf{G}=\mathbf{m}athbf{G}_{\Lambda}$ is given in~\cite[Section~2.4.1]{StVo14} which does not require the assumption $\Lambda' \subseteq 2\Lambda$.
For groups associated to such schemes a Kirillov orbit method formalism was formulated which is valid for all primes; see~\cite[Section~2.4.2]{StVo14}.
Consequently, \cite[Lemma~2.13]{StVo14} assures that~\eqref{lemch} holds for all primes~$\mathfrak{p}$ if~$\Lambda$ has nilpotency class~$2$.
\end{rmk}
\subsection{\texorpdfstring{$\mathfrak{p}$}{p}-adic integrals}\label{spadic}
We now write the local factors of the bivariate zeta functions of $\mathbf{m}athbf{G}(\mathcal{O})$ in terms of Poincaré series such as~\eqref{poincare}.
Recall from Section~\text{Re}f{chcc} that the dimensions of irreducible complex representations as well as the sizes of the conjugacy classes of
$\mathbf{m}athbf{G}(\mathfrak{o})$ are powers of $q$, allowing us to write the representation and the conjugacy class zeta functions of the congruence quotient $\mathbf{m}athbf{G}_N=\mathbf{m}athbf{G}(\mathfrak{o}p)$ as
\[\zeta^\textup{irr}_{\mathbf{m}athbf{G}_N}(s)=\sum_{i=0}^{\infty}r_{q^i}(\mathbf{m}athbf{G}_N)q^{-is}~\text{ and }~\zeta^\textup{cc}_{\mathbf{m}athbf{G}_N}(s)=\sum_{i=0}^{\infty}c_{q^i}(\mathbf{m}athbf{G}_N)q^{-is}.\]
These sums are finite, since $\mathbf{m}athbf{G}_N$ is a finite group.
Applying this to~\eqref{localfactors}, the definition of the local factors of the bivariate zeta functions, one obtains
\begin{align*}
\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathfrak{o})}(s_1,s_2)&=\sum_{N=0}^{\infty}\sum_{i=0}^{\infty}r_{q^i}(\mathbf{m}athbf{G}_N)q^{-is_1-Ns_2} \text{ and }\\
\mathcal{Z}^\textup{cc}_{\mathbf{m}athbf{G}(\mathfrak{o})}(s_1,s_2)&=\sum_{N=0}^{\infty}\sum_{i=0}^{\infty}{c}_{q^i}(\mathbf{m}athbf{G}_N)q^{-is_1-Ns_2}.
\end{align*}
Recall that $z=\textup{rk}(\mathfrak{z})=h-a$ and $r=\textup{rk}(\mathfrak{g}/\mathfrak{g}')=h-b$. If $c=2$ or $p>c>2$, then~\eqref{lemch} yields
\begin{equation}\label{rlocal}\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathfrak{o})}(s_1,s_2)=\sum_{N=0}^{\infty}\sum_{i=0}^{\infty}\sum_{\mathbf{m}\in \mathbf{m}athcal{D}_{B}^{N}}
\mathbf{m}athds{N}b_{\mathbf{m}-m_1}(\mathfrak{o}/\mathfrak{p}^{N-m_1})q^{-(s_2-r)N-(2+s_1)i}.
\end{equation}
If $p>c$, then~\eqref{lemcc} yields
\begin{equation}\label{cclocal}\mathcal{Z}^\textup{cc}_{\mathbf{m}athbf{G}(\mathfrak{o})}(s_1,s_2)=
\sum_{N=0}^{\infty}\sum_{i=0}^{\infty}\sum_{\mathbf{m} \in \mathbf{m}athcal{D}_{A}^{N}}
\mathbf{m}athds{N}a_{\mathbf{m}-m_1}(\mathfrak{o}/\mathfrak{p}^{N-m_1})q^{-(s_2-z)N-(1+s_1)i}.
\end{equation}
We now show how to rewrite these sums as Poincaré series of the form~\eqref{poincare}. In preparation for this, we need two lemmata.
\begin{lem}\label{terms} Let $s$ be a complex variable, $(a_m)_{m\in \mathbf{m}athds{N}_0}$ a sequence of real numbers, and $q\in \mathbf{m}athds{Z}_{\mathfrak{g}eq 2}$. Provided both series converge, the following holds:
\[\sum_{N=1}^{\infty}\sum_{m=0}^{N-1}a_mq^{-sN}= \mathfrak{f}rac{q^{-s}}{1-q^{-s}}\left(\sum_{N=0}^{\infty}a_Nq^{-sN}\mathcal{O}ght).\]
\end{lem}
\begin{proof} This is due to a short manipulation of geometric series; see~\cite[Lemma~3.2.9]{PLphd}.
\end{proof}
\begin{lem}\label{rewr} Let $s$ and $t$ be complex variables. Let $\mathbf{m}athcal{R}(\underline{Y})=\mathbf{m}athcal{R}(Y_1, \dots, Y_n)$ be a matrix of polynomials
$\mathbf{m}athcal{R}(\underline{Y})_{ij}\in\mathfrak{o}[\underline{Y}]$. If $\mathbf{m}athcal{R}$ is not antisymmetric, set $u=\mathbf{m}ax\{\textup{rk}_{\mathbf{m}athds{F}rac(\mathfrak{o})}\mathbf{m}athcal{R}(\mathbf{m}athbf{z}) \mathbf{m}id \mathbf{m}athbf{z}\in \mathfrak{o}^n\}$. Otherwise, set $u=\tfrac{1}{2}\mathbf{m}ax\{\textup{rk}_{\mathbf{m}athds{F}rac(\mathfrak{o})}\mathbf{m}athcal{R}(\mathbf{m}athbf{z}) \mathbf{m}id \mathbf{m}athbf{z}\in \mathfrak{o}^n\}$. Moreover, let $q=|\mathfrak{o}/\mathfrak{p}|$. Provided both series converge, the following holds:
\begin{align}\label{sumlema}&\sum_{N=0}^{\infty}\sum_{i=0}^{\infty}\sum_{\mathbf{m} \in \mathcal{D}^N(uN-i)}
\mathbf{m}athds{N}r_{\mathbf{m}-m_1}(\mathfrak{o}/\mathfrak{p}^{N-m_1})q^{-sN-ti}\\
&=\mathfrak{f}rac{1}{1-q^{-s}}\left(1+\sum_{N=1}^{\infty}\sum_{\mathbf{m} \in \mathbf{m}athds{N}_{0}^{u}}\hspace{-0.2cm}\mathbf{m}athds{N}r_{\mathbf{m}}(\mathfrak{o}p)q^{-(s+ut)N+t \sum\limits_{j=1}^{u}m_j}\mathcal{O}ght),\nonumber
\end{align}
where for each $c \in \mathbf{m}athds{N}_0$,
\[\mathcal{D}^N(c):=\left\{\mathbf{m}=(m_1,\dots,m_u)\in \mathbf{m}athds{N}_{0}^{u} \mathbf{m}id m_1\leq \dots \leq m_u \leq N,\textstyle\sum\limits_{i=1}^{u}m_i=c\mathcal{O}ght\}.\]
\end{lem}
\begin{proof}
Set $\mathbf{m}athbf{m}=(m_1, \dots, m_u)$ and recall the notation
\[\mathbf{m}athbf{m}-m=(m_1-m,\dots, m_u-m)\text{ for }m \in \mathbf{m}athds{N}_0.\]
As $\mathbf{m}athds{N}r_{\mathbf{m}}(\mathfrak{o}p)=0$ unless $0=m_1 \leq m_2 \leq \dots \leq m_{u} \leq N$, in which case \
\[0\leq \textstyle\sum_{j=1}^{u}m_j \leq uN,\]
the condition $\textstyle\sum_{j=1}^{u}m_j=uN-i$ implies that the only values of~$i$ which are relevant for the sum~\eqref{sumlema} are $0 \leq i \leq uN$.
Hence, the expression on the left-hand side of~\eqref{sumlema} can be rewritten as
\begin{equation}\label{rewr2} 1+\sum_{N=1}^{\infty}\sum_{i=0}^{uN}\sum_{\mathbf{m}\in\mathcal{D}^N(uN-i)}
\mathbf{m}athds{N}r_{\mathbf{m}-m_1}(\mathfrak{o}/\mathfrak{p}^{N-m_1})q^{-sN-ti}.
\end{equation}
In the following, we make use of the notation \[\mathcal{D}_{\leq}^N(c):=\left\{\mathbf{m}=(m_1,\dots,m_u)\in \mathbf{m}athds{N}_{0}^{u} \mathbf{m}id m_1\leq \dots \leq m_u \leq N,\textstyle\sum\limits_{i=1}^{u}m_i\leq c\mathcal{O}ght\}.\]
Restricting the summation in~\eqref{rewr2} to $m_1=0$ yields
\[\sum_{N=1}^{\infty}\sum_{\substack{\mathbf{m} \in \mathcal{D}_{\leq}^N((u-1)N)\\ m_1=0}}
\mathbf{m}athds{N}r_{\mathbf{m}}(\mathfrak{o}p)q^{-sN-t(uN-\sum\limits_{j=2}^{u}m_j)}.\]
Since $\mathbf{m}athds{N}r_{\mathbf{m}}(\mathfrak{o}p)=0$ unless $0=m_1 \leq m_2 \leq \dots \leq m_{u} \leq N$, we may rewrite this sum as
\begin{equation*} \sum_{N=1}^{\infty}\sum_{\mathbf{m} \in \mathbf{m}athds{N}_{0}^{u} }\mathbf{m}athds{N}r_{\mathbf{m}}(\mathfrak{o}p)q^{-(s+ut)N+t\sum\limits_{j=1}^{u}m_j}=:\mathbf{m}athcal{S}(s,t).
\end{equation*}
Our goal now is to write the part of the summation in~\eqref{rewr2} with $m_1> 0$ in terms of $\mathbf{m}athcal{S}(s,t)$.
Restricting the summation in~\eqref{rewr2} to $m_1>0$ yields
\begin{align}
& \sum_{N=1}^{\infty}\sum_{i=0}^{u N}\sum_{\substack{\mathbf{m} \in \mathcal{D}^N(uN-i)\\ m_1> 0}}
\mathbf{m}athds{N}r_{\mathbf{m}-m_1}(\mathfrak{o}/\mathfrak{p}^{N-m_1})q^{-sN-ti}\nonumber \\
\label{change}&= \sum_{N=1}^{\infty}\sum_{m=1}^{N}\sum_{\substack{\mathbf{m} \in \mathcal{D}_{\leq}^N(uN)\\ m_1=m}}
\mathbf{m}athds{N}r_{\mathbf{m}-m_1}(\mathfrak{o}/\mathfrak{p}^{N-m_1})q^{-sN-t(uN-\sum\limits_{j=1}^{u}m_j)}.
\end{align}
By writing $m'_j=m_j-m_1$, we obtain $\textstyle\sum_{j=1}^{u}m_j=um_1+\sum_{j=2}^{u}m'_j$. Moreover, for each $m \in [N]$,
\[\{\mathbf{m}-m \mathbf{m}id \mathbf{m} \in \mathcal{D}_{\leq}^N(uN),~ m_1=m\}=\{\mathbf{m} \in \mathcal{D}_{\leq}^{N-m}(u(N-m)) \mathbf{m}id m_1=0\}.\] Then, we may rewrite~\eqref{change} as
\begin{align}
&\sum_{N=1}^{\infty}\sum_{m=1}^{N}\sum_{\substack{\mathbf{m} \in \mathcal{D}_{\leq}^{N-m}(u(N-m))\\ m_1=0}}
\mathbf{m}athds{N}r_{\mathbf{m}}(\mathfrak{o}/\mathfrak{p}^{N-m})q^{-sN-t(u(N-m)-\sum\limits_{j=2}^{u}m_{j})}\nonumber\\
&=\sum_{N=1}^{\infty}q^{-sN}\sum_{m=0}^{N-1}\sum_{\substack{\mathbf{m} \in \mathcal{D}_{\leq}^{m}(um)\\ m_1=0}}
\mathbf{m}athds{N}r_{\mathbf{m}}(\mathfrak{o}/\mathfrak{p}^m)q^{t\sum\limits_{j=2}^{u}m_{j}-tum} \nonumber\\
&=\label{notzero}\sum_{N=1}^{\infty}q^{-sN}\sum_{m=0}^{N-1}\sum_{\mathbf{m} \in \mathbf{m}athds{N}_{0}^{u}}
\mathbf{m}athds{N}r_{\mathbf{m}}(\mathfrak{o}/\mathfrak{p}^m)q^{t\sum\limits_{j=1}^{u}m_j-tum}.
\end{align}
Apply Lemma~\text{Re}f{terms} to~\eqref{notzero} by setting
\[a_m:=\sum_{\mathbf{m} \in \mathbf{m}athds{N}_{0}^{u}}\mathbf{m}athds{N}r_{\mathbf{m}}(\mathfrak{o}/\mathfrak{p}^m)q^{t\sum_{j=1}^{u}m_j-tum}.\]
This gives that~\eqref{notzero} equals
\begin{align*}
&\mathfrak{f}rac{q^{-s}}{1-q^{-s}}\left(1+\sum_{N=1}^{\infty}\sum_{\mathbf{m} \in \mathbf{m}athds{N}_{0}^{u}}\mathbf{m}athds{N}r_{\mathbf{m}}(\mathfrak{o}p)q^{-(s+ut)N+t\sum\limits_{j=1}^{u}m_j}\mathcal{O}ght)
&=\mathfrak{f}rac{q^{-s}}{1-q^{-s}}\left(1+\mathbf{m}athcal{S}(s,t)\mathcal{O}ght).
\end{align*}
Combining the expressions for the parts of the sum with $m_1=0$ and $m_1> 0$ yields
\begin{align*}&\sum_{N=0}^{\infty}\sum_{i=0}^{\infty}\sum_{\mathbf{m} \in \mathcal{D}^N(uN-i)}
\mathbf{m}athds{N}r_{\mathbf{m}}(\mathfrak{o}p)q^{-sN-ti}\\
&=1+\mathbf{m}athcal{S}(s,t)+\mathfrak{f}rac{q^{-s}}{1-q^{-s}}\left( 1+\mathbf{m}athcal{S}(s,t)\mathcal{O}ght)=\mathfrak{f}rac{1}{1-q^{-s}}\left(1+\mathbf{m}athcal{S}(s,t)\mathcal{O}ght).\qedhere
\end{align*}
\end{proof}
\begin{pps} If either $c=2$ or $p>c>2$, then
\begin{align}\label{rint} &\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathfrak{o})}(s_1,s_2)=\\ \mathfrak{f}rac{1}{1-q^{r-s_2}}
&\left(1+\sum_{N=1}^{\infty}\sum_{\mathbf{m}\in\mathbf{m}athds{N}_{0}^{u_{B}}} \mathbf{m}athds{N}b_{\mathbf{m}}(\mathfrak{o}p)q^{-N(u_{B} s_1+s_2+2u_{B}-r)-2\sum\limits_{j=1}^{u_{B}}m_j\mathfrak{f}rac{(-s_1-2)}{2}}\mathcal{O}ght).\nonumber
\end{align}
Moreover, if $p>c$, then
\begin{align}
\label{cint} &\mathcal{Z}^\textup{cc}_{\mathbf{m}athbf{G}(\mathfrak{o})}(s_1,s_2)= \\\mathfrak{f}rac{1}{1-q^{z-s_2}}
&\left(1+\sum_{N=1}^{\infty}\sum_{\mathbf{m}\in\mathbf{m}athds{N}_{0}^{u_{A}}}\mathbf{m}athds{N}a_{\mathbf{m}}(\mathfrak{o}p)q^{-N(u_{A} s_1+s_2+u_{A}-z)-\sum\limits_{j=1}^{u_{A}}m_j(-s_1-1)}\mathcal{O}ght).\nonumber
\end{align}
\end{pps}
\begin{proof}
By setting $s=s_2-r$ and $t=2+s_1$, and considering $\mathbf{m}athcal{R}$ to be the $B$-commutator matrix of $\Lambda$ on the left-hand side of~\eqref{sumlema}, we obtain~\eqref{rlocal}.
Under these substitutions, Lemma~\text{Re}f{rewr} shows~\eqref{rint}.
Analogously, by setting $s=s_2-z$ and $t=1+s_1$, and considering $\mathbf{m}athcal{R}$ to be the $A$-commutator matrix of $\Lambda$, the left-hand side of~\eqref{sumlema} equals~\eqref{cclocal}, so that, under these substitutions, Lemma~\text{Re}f{rewr} shows~\eqref{cint}.
\end{proof}
Expression~\eqref{rint} is of the form~\eqref{poincareas} with
\[t=u_{B} s_1+s_2+2u_{B}-r\text{ and }r_k=\mathfrak{f}rac{-s_1-2}{2}\text{ for each }k \in [u_{B}],\]
whilst~\eqref{cint} is~\eqref{poincare} with
\[t=u_{A} s_1+s_2+u_{A}-z\text{ and }r_k=-s_1-1\text{ for each }k \in [u_{A}].\]
Therefore these choices of $t$ and $\underline{r}$ applied to~\eqref{Vo10as} and to~\eqref{Vo14gen} yield the following.
Recall $a+z=\textup{rk}(\mathfrak{g}/\mathfrak{z})+\textup{rk}(\mathfrak{z})=\textup{rk}(\mathfrak{g})=h$ and $b+r=\textup{rk}(\mathfrak{g}')+\textup{rk}(\mathfrak{g}/\mathfrak{g}')=h$. For $k \in \mathbf{m}athds{N}$ write $\mathbf{m}athbf{1}_{k}=(1, \dots , 1) \in \mathbf{m}athds{Z}^k$.
\begin{pps}\label{intpadic} If either $c=2$ or $p>c>2$, then
\begin{align}
\label{rpadic} &\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathfrak{o})}(s_1,s_2)=
&\mathfrak{f}rac{1}{1-q^{r-s_2}}\left(1+\mathfrak{p}int_{B}\left(\left(\tfrac{-s_1-2}{2}\mathcal{O}ght)\mathbf{m}athbf{1}_{u_{B}},u_{B} s_1+s_2+2u_{B}-h-1\mathcal{O}ght)\mathcal{O}ght).
\end{align}
Moreover, if $p>c$, then
\begin{align}
\label{ccpadic} &\mathcal{Z}^\textup{cc}_{\mathbf{m}athbf{G}(\mathfrak{o})}(s_1,s_2)=
\mathfrak{f}rac{1}{1-q^{z-s_2}}\left(1+\mathfrak{p}int_{A}\left((-s_1-1)\mathbf{m}athbf{1}_{u_{A}},u_{A} s_1+s_2+u_{A}-h-1\mathcal{O}ght)\mathcal{O}ght).
\end{align}
\end{pps}
Specialization~\eqref{specialization} applied to~\eqref{rpadic} and to~\eqref{ccpadic} yields
\begin{align}\label{cctok}\zeta^\textup{k}_{\mathbf{m}athbf{G}(\mathfrak{o})}(s)&=
\mathfrak{f}rac{1}{1-q^{z-s}}\left(1+\mathfrak{p}int_{A}\left(-\mathbf{m}athbf{1}_{u_{A}},s+u_{A}-h-1\mathcal{O}ght)\mathcal{O}ght),\\
&\label{rtok}=\mathfrak{f}rac{1}{1-q^{r-s}}\left(1+\mathfrak{p}int_{B}\left(-\mathbf{m}athbf{1}_{u_{B}},s+2u_{B}-h-1\mathcal{O}ght)\mathcal{O}ght).
\end{align}
\begin{rmk}
Formula~\eqref{cctok} coincides with the $\mathfrak{p}$-adic integral obtained from the $\mathfrak{p}$-adic integral~\cite[formula~(4.3)]{Ro17} together with the
specialization given in \cite[Theorem~1.7]{Ro17}.
In fact, for each $x \in \mathfrak{g}$, let $\textup{ad}_x:\mathfrak{g} \to \mathfrak{g}'$ be the adjoint homomorphism
\[\textup{ad}_{x}(z)=[z,x],\text{ for all }z \in \mathfrak{g}.\]
As in Section~\text{Re}f{chcc}, let $\mathbf{m}athscr{B}=(e_1, \dots, e_h)$ be a basis of $\mathfrak{g}$ with the properties described there; we use the notation that was set up in this context.
For each $x \in \mathfrak{g}$, we can write $x=\sum_{i=1}^{h}x_ie_i$, for some $x_i \in \mathfrak{o}$. Let $\mathbf{m}athbf{x}=(x_1, \dots, x_h) \in \mathfrak{o}^h$.
The $b \times h$-matrix representing the linear transformation $\textup{ad}_x$ is such that its submatrix composed of its first $a$ columns is the transpose $A(\mathbf{m}athbf{x})^{\textup{tr}}$ of the $A$-commutator matrix of~$\Lambda$, and the remaining columns have only zero entries.
We observe that the above mentioned integrals of~\cite{Ro17} are taken over $\mathfrak{o} \times \mathfrak{o}^a$ instead of $\mathfrak{p} \times W^{\lri}_a$ as in~\eqref{cctok}. Formula~\eqref{cctok} coincides with the above mentioned $\mathfrak{p}$-adic integral due to~\cite[Lemma~2.2.4]{PLphd}.
\end{rmk}
\begin{exm}\label{Heis}
Let $\mathbf{m}athbf{H}(\mathcal{O})$ be the Heisenberg group over~$\mathcal{O}$ considered in Example~\text{Re}f{exintro}.
The unipotent group scheme~$\mathbf{m}athbf{H}$ is obtained from the $\mathbf{m}athds{Z}$-Lie lattice \[\Lambda=\langle x_1,x_2, y\mathbf{m}id [x_1,x_2]-y\rangle.\]
The commutator matrices of $\mathfrak{g}=\Lambda(\mathfrak{o})$ with respect to the ordered sets $\mathbf{m}athbf{e}=(x_1,x_2)$ and $\mathbf{m}athbf{f}=(y)$ are
\[ A(X_1, X_2)=\left[ \begin {array}{c} X_2\\ \noalign{\mathbf{m}edskip} -X_1\end {array} \mathcal{O}ght] ~\text{ and }~ B(Y)=\left[ \begin {array}{cc} 0&Y\\ \noalign{\mathbf{m}edskip}-Y&0\end {array}
\mathcal{O}ght].\]
The $A$-commutator matrix has rank~$1$ and the $B$-commutator matrix has rank~$2$ over the respective fields of rational functions, that is, $u_{A}=u_{B}=1$.
Moreover, $h=\textup{rk}(\mathfrak{g})=3$, and
\[F_{1}(A(X_1,X_2))=\{-X_1,X_2\}, \hspace{0.5cm} F_2(B(Y))=\{Y^2\}.\]
In particular, if $(x_1,x_2)\in W^{\lri}_{2}$, i.e., $v_{\mathfrak{p}}(x_1,x_2)=0$, then $\|F_1(A(x_1,x_2))\|_{\mathfrak{p}}=1$.
Also, if $y \in W^{\lri}_1$, then, in particular, $v_{\mathfrak{p}}(y^2)=0$, which gives $\|F_2(B(y))\|_{\mathfrak{p}}=1$.
It follows from Proposition~\text{Re}f{intpadic} that
\begin{align*}
\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{H}(\mathfrak{o})}(s_1,s_2) &=\mathfrak{f}rac{1}{1-q^{2-s_2}}\left(1+(1-q^{-1})^{-1}\int_{(w, y) \in \mathfrak{p} \times W^{\lri}_{1}}|w|_{\mathfrak{p}}^{s_1+s_2-2}d\mathbf{m}u\mathcal{O}ght),\\
\mathcal{Z}^\textup{cc}_{\mathbf{m}athbf{H}(\mathfrak{o})}(s_1,s_2) &=\mathfrak{f}rac{1}{1-q^{1-s_2}}\left(1+(1-q^{-1})^{-1}\int_{(w, x_1,x_2) \in \mathfrak{p} \times W^{\lri}_{2}}|w|_{\mathfrak{p}}^{s_1+s_2-3}d\mathbf{m}u\mathcal{O}ght).\\
\end{align*}
Expressions~\eqref{Heisirr} and~\eqref{Heiscc} for $\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{H}(\mathfrak{o})}(s_1,s_2)$ and $\mathcal{Z}^\textup{cc}_{\mathbf{m}athbf{H}(\mathfrak{o})}(s_1,s_2)$ given in Example~\text{Re}f{exintro} are then consequence of the following well-known fact: for $k \in \mathbf{m}athds{N}$ and $t \in \mathbf{m}athds{C}$,
\begin{equation}\label{integral1}
\int_{w\in\mathfrak{p}^k}|w|_{\mathfrak{p}}^{t}d\mathbf{m}u=\mathfrak{f}rac{q^{-k(t+1)}(1-q^{-1})}{1-q^{-k(t+1)}},
\end{equation}
provided the $\mathfrak{p}$-adic integral on the left-hand side converges.
\end{exm}
\subsection{Twist representation zeta functions}\label{irr}
In this section, we assume that~$\mathbf{m}athbf{G}$ is the unipotent group scheme associated to a nilpotent $\mathcal{O}$-Lie lattice~$\Lambda$ of nilpotency class~$2$ without the assumption $\Lambda'\subseteq 2\Lambda$, constructed as in~~\cite[Section~2.4.1]{StVo14}.
We provide a univariate specialization of the bivariate representation zeta function of $\mathbf{m}athbf{G}(\mathfrak{o})$ which results in the twist representation zeta function of this group.
According to~\cite[Corollary~2.11]{StVo14}, the twist representation zeta function of $\mathbf{m}athbf{G}(\mathfrak{o})$ is given by
\[\zeta^{\widetilde{\textup{irr}}}_{\mathbf{m}athbf{G}(\mathfrak{o})}(s)=1+\mathfrak{p}int_{B}(-\tfrac{s}{2}\mathbf{m}athbf{1}_{u_{B}}, u_Bs-b-1),\]
where $b=\textup{rk}(\mathfrak{g}')$, $2u_B=\mathbf{m}ax\{\textup{rk}_{\text{Frac}(\mathfrak{o})}B(\mathbf{m}athbf{z}) \mathbf{m}id \mathbf{m}athbf{z}\in \mathfrak{o}^b\}$, $\mathbf{m}athbf{1}_{u_{B}}=(1, \dots, 1) \in \mathbf{m}athds{Z}^{u_{B}}$, and $\mathfrak{p}int_{B}(\underline{r},t)$ is the integral
$\mathfrak{p}int_{\mathbf{m}athcal{R}}(\underline{r},t)$ given in~\eqref{Vo10as} with $\mathbf{m}athcal{R}(\underline{Y})$ being regarded as the $B$-commutator matrix $B(\underline{Y})$ of~$\mathfrak{g}$.
Recall that $r=\textup{rk}(\mathfrak{g}/\mathfrak{g}')=h-b$. Proposition~\text{Re}f{intpadic} states
\[(1-q^{r-s_2})\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathfrak{o})}(s_1,s_2)=1+\mathfrak{p}int_{B}\left(\tfrac{-2-s_1}{2}\mathbf{m}athbf{1}_{u_{B}},u_{B} s_1+s_2+2u_{B}-h-1\mathcal{O}ght).\]
Comparing the expressions for $\zeta^{\widetilde{\textup{irr}}}_{\mathbf{m}athbf{G}(\mathfrak{o})}(s)$ and $(1-q^{r-s_2})\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathfrak{o})}(s_1,s_2)$, we obtain the desired specialization.
\begin{pps}\label{repzeta} If $\mathbf{m}athbf{G}(\mathfrak{o})$ has nilpotency class~$2$, then
\begin{equation*}
(1-q^{r-s_2})\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathfrak{o})}(s_1,s_2)\mathbf{m}id_{\substack{s_1\to s-2\\ s_2\to r\mathfrak{p}hantom{-2}}}=\zeta^{\widetilde{\textup{irr}}}_{\mathbf{m}athbf{G}(\mathfrak{o})}(s),
\end{equation*}
provided both the left-hand side and the right-hand side converge.
\end{pps}
In the following example, we exhibit a $\mathbf{m}athcal{T}$-group of nilpotency class~$3$ whose bivariate representation zeta function does not specialize to its twist representation zeta function.
\begin{exm}\label{irr3}
Consider the following free nilpotent $\mathbf{m}athds{Z}$-Lie lattice on~$2$ generators of class~$3$
\[\mathfrak{f}_{3,2}=\langle x_1,x_2,y,z_1,z_2 \mathbf{m}id i,j\in \{1,2\}: ~[x_1,x_2]-y,~[y,x_i]-z_i, ~[z_i,x_j], ~[z_i,y], ~[z_1,z_2] \rangle.\]
Let~$\mathbf{m}athfrak{F}_{3,2}$ denote the unipotent group scheme obtained from~$\mathfrak{f}_{3,2}$, and denote by~$\mathfrak{z}_{3,2}$ and by~$\mathfrak{f}'_{3,2}$ the center and
the derived Lie lattice of~$\mathfrak{f}_{3,2}$, respectively.
The $B$-commutator matrix of~$\mathfrak{f}_{3,2}$ with respect to $\mathbf{m}athbf{e}=(y,x_1,x_2)$ and $\mathbf{m}athbf{f}=(z_1,z_2,y)$ is
\[
B(Y_1,Y_2,Y_3)=\left[ \begin {array}{ccc} 0&Y_1&Y_2\\ \noalign{\mathbf{m}edskip}-Y_1&0&Y_3
\\ \noalign{\mathbf{m}edskip}-Y_2&-Y_3&0\end {array} \mathcal{O}ght] .
\]
Thus, $u_{B}=1$, $F_0(B(\underline{Y}))=\{1\}$, and $F_2(B(\underline{Y}))\supseteq \{Y_{1}^{2},Y_{2}^{2},Y_{3}^{2}\}$.
It follows from Proposition~\text{Re}f{intpadic} and~\eqref{integral1} that
\begin{align}
\mathcal{Z}^\textup{irr}_{\mathbf{m}athfrak{F}_{2,3}(\mathfrak{o})}(s_1,s_2)&=\mathfrak{f}rac{1}{1-q^{2-s_2}}\left(1+(1-q^{-1})^{-1}\int_{(w,y_1,y_2,y_3)\in\mathfrak{p}\times W^{\lri}_{3}}|w|_{\mathfrak{p}}^{s_1+s_2-4}d\mathbf{m}u\mathcal{O}ght)\nonumber \\
\label{lvlf23}&=\mathfrak{f}rac{1-q^{-s_1-s_2}}{(1-q^{2-s_2})(1-q^{3-s_1-s_2})}. \hspace{0.5cm}
\end{align}
By implementing his methods in \textup{Zeta}~\cite{RoZeta}, Rossmann provides in~\cite[Table~1]{Ro17comp} the following formula for the twist representation zeta function of $\mathbf{m}athfrak{f}_{3,2}$---denoted by $L_{5,9}$ in~\cite{Ro17comp}---,~provided $q$ is sufficiently large
\begin{equation}\label{rf23}\zeta^{\widetilde{\textup{irr}}}_{\mathbf{m}athfrak{F}_{3,2}(\mathfrak{o})}(s)=\mathfrak{f}rac{(1-q^{-s})^2}{(1-q^{1-s})(1-q^{2-s})}.
\end{equation}
Comparing~\eqref{lvlf23} and~\eqref{rf23}, we see that there is no specialization of form~\eqref{specializationirr} for the bivariate representation zeta function of~$\mathbf{m}athfrak{F}_{3,2}(\mathfrak{o})$ that leads to its twist representation zeta function.
For the sake of completeness, we now calculate the bivariate conjugacy class and the class number zeta functions of~$\mathbf{m}athfrak{F}_{3,2}(\mathfrak{o})$.
The $A$-commutator matrix of~$\mathfrak{f}_{3,2}$ with respect to~$\mathbf{m}athbf{e}$ and~$\mathbf{m}athbf{f}$ is
\[A(X_1,X_2,X_3)= \left[ \begin {array}{ccc} X_2&X_3&0\\ \noalign{\mathbf{m}edskip}-X_1&0&X_3
\\ \noalign{\mathbf{m}edskip}0&-X_1&-X_2\end {array} \mathcal{O}ght].\]
Thus, $u_{A}=2$, $F_0(A(\underline{X}))=\{1\}$, $F_1(A(\underline{X}))=\{-X_1,\mathfrak{p}m X_2,X_3\}$, and $F_2(A(\underline{X}))\supseteq \{X_{1}^{2},-X_{2}^{2},X_{3}^{2}\}$.
Hence
\begin{align*}
\mathcal{Z}^\textup{cc}_{\mathbf{m}athfrak{F}_{2,3}(\mathfrak{o})}(s_1,s_2)&=\mathfrak{f}rac{1}{1-q^{2-s_2}}\left(1+(1-q^{-1})^{-1}\int_{(w,x_1,x_2,x_3)\in\mathfrak{p}\times W^{\lri}_{3}}|w|_{\mathfrak{p}}^{2s_1+s_2-4}d\mathbf{m}u\mathcal{O}ght)\\
&=\mathfrak{f}rac{1-q^{-2s_1-s_2}}{(1-q^{2-s_2})(1-q^{3-2s_1-s_2})}.
\end{align*}
Specialization~\eqref{specialization} yields
\[\zeta^\textup{k}_{\mathbf{m}athfrak{F}_{2,3}(\mathfrak{o})}(s)=\mathfrak{f}rac{1-q^{-s}}{(1-q^{2-s})(1-q^{3-s})}.\]
This formula agrees with the one given in~\cite[Section~9.3, Table~1]{Ro17}.
\end{exm}
\subsection{Local functional equations---proof of Theorem~\text{Re}f{thmA}}\label{pthmA}
Proposition~\text{Re}f{intpadic} assures that, for each $\ast \in \{\textup{irr}, {c}up\}$, almost all local factors of~$\lvlzf^{\ast}_{\mathbf{m}athbf{G}(\mathcal{O})}$ are given by a rational function~$R^{\ast}$ in certain parameters, as stated in Theorem~\text{Re}f{thmA}.
In this section, we conclude the proof of Theorem~\text{Re}f{thmA} by showing that the integrals given in Proposition~\text{Re}f{intpadic} behave uniformly under base extension and that they satisfy functional equations.
Fix a nonzero prime ideal~$\mathfrak{p}$ satisfying the conditions of Proposition~\text{Re}f{intpadic}.
Let~$L$ be a finite extension of $K=\mathbf{m}athds{F}rac(\mathcal{O})$ with ring of integers~$\mathcal{O}_L$. For a fixed prime ideal~$\mathfrak{P}$ of~$\mathcal{O}_L$ dividing~$\mathfrak{p}$, write~$\mathfrak{O}$ for the localisation~$\mathcal{O}_{L,\mathfrak{P}}$.
Denote the relative degree of inertia by $f=f(\mathfrak{O},\mathfrak{o})$, and hence $|\mathfrak{O} / \mathfrak{P}|=q^{f}$.
Set~$\mathfrak{g}_{L}=\Lambda(\mathfrak{O})$, and let~$\mathfrak{z}_{L}$ and~$\mathfrak{g}'_{L}$ be the center and the derived Lie sublattice of~$\mathfrak{g}_{L}$, respectively.
Since~$\mathcal{O}_L$ is a ring of integers of a number field~$L$, we can choose ordered sets~$\mathbf{m}athbf{e}$ and~$\mathbf{m}athbf{f}$ as the ones of Section~\text{Re}f{chcc} such that~$\overline{\mathbf{m}athbf{e}}$ and~$\mathbf{m}athbf{f}$ are bases of~$\mathfrak{g}_{L}/\mathfrak{z}_{L}$ and~$\mathfrak{g}_{L}'$,
respectively. Let $A(\underline{X})$ and $B(\underline{Y})$ be the commutator matrices of~$\mathfrak{g}_{L}$ with respect to~$\mathbf{m}athbf{e}$ and~$\mathbf{m}athbf{f}$; see Definition~\text{Re}f{commutator}.
Consider the functions.
\begin{align*}\widetilde{\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathfrak{O})}}(s_1,s_2)&:=1+\mathscr{Z}^{\Lri}_{B}\left((-s_1-2)/2,u_{B} s_1+s_2+2u_{B}-h-1\mathcal{O}ght),\\
\widetilde{\mathcal{Z}^\textup{cc}_{\mathbf{m}athbf{G}(\mathfrak{O})}}(s_1,s_2)&:=1+\mathscr{Z}^{\Lri}_{A}\left(-s_1-1,u_{A} s_1+s_2+u_{A}-h-1\mathcal{O}ght),
\end{align*}
where $\mathscr{Z}^{\Lri}_{B}(\underline{r},t)$ and $\mathscr{Z}^{\Lri}_{A}(\underline{r},t)$ are the integrals given in~\eqref{Vo10as} and~\eqref{Vo10}, respectively.
We have shown in Proposition~\text{Re}f{intpadic} that
\begin{align*}
\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathfrak{O})}(s_1,s_2)&=\mathfrak{f}rac{1}{1-q^{f(r-s_2)}}\widetilde{\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathfrak{O})}}(s_1,s_2)\text{ and }&\\
\mathcal{Z}^\textup{cc}_{\mathbf{m}athbf{G}(\mathfrak{O})}(s_1,s_2)&=\mathfrak{f}rac{1}{1-q^{f(z-s_2)}}\widetilde{\mathcal{Z}^\textup{cc}_{\mathbf{m}athbf{G}(\mathfrak{O})}}(s_1,s_2).&
\end{align*}
It is clear that the terms $(1-q^{f(r-s_2)})^{-1}$ and $(1-q^{f(r-s_2)})^{-1}$ are given by rational functions in~$q^f$ and~$q^{-fs_2}$, and that they satisfy functional equations under inversion of~$q^{f}$. Therefore, to prove Theorem~\text{Re}f{thmA}, it suffices to show that $\widetilde{\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathfrak{O})}}(s_1,s_2)$ and $\widetilde{\mathcal{Z}^\textup{cc}_{\mathbf{m}athbf{G}(\mathfrak{O})}}(s_1,s_2)$ behave uniformly under base extension and satisfy functional equations.
We first show that the integrands of $\widetilde{\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathfrak{O})}}(s_1,s_2)$ and $\widetilde{\mathcal{Z}^\textup{cc}_{\mathbf{m}athbf{G}(\mathfrak{O})}}(s_1,s_2)$ are defined over~$\mathcal{O}$, that is, that only their domains of integration vary with the ring~$\mathfrak{O}$.
The $\mathfrak{O}$-bases $\overline{\mathbf{m}athbf{e}}$ and $\mathbf{m}athbf{f}$ are only defined locally, and hence so are the matrices $A(\underline{X})$ and $B(\underline{Y})$.
We must assure that there exist $\mathfrak{O}$-bases~$\overline{\mathbf{m}athbf{e}}$ and~$\mathbf{m}athbf{f}$ as the ones of Section~\text{Re}f{chcc} such that the commutator matrices $A(\underline{X})$ and $B(\underline{Y})$, defined with respect to~$\mathbf{m}athbf{e}$ and~$\mathbf{m}athbf{f}$ are defined over~$\mathcal{O}$, and hence so are the sets of polynomials $F_j(A(\underline{X}))$ and $F_{2j}(B(\underline{Y}))$.
Since the matrix $B(\underline{Y})$ is the same as the one appearing in the integrands of~\cite[formula~(2.8)]{StVo14} and $A(\underline{X})$ is obtained in an analogous way,
the argument of~\cite[Section~2.3]{StVo14} also holds in this case. Namely, we choose an $\mathcal{O}$-basis~$\mathbf{m}athbf{f}$ for a free finite-index $\mathcal{O}$-submodule of the isolator $i(\Lambda')$ of the derived $\mathcal{O}$-Lie sublattice of~$\Lambda$; see Section~\text{Re}f{chcc}.
By~\cite[Lemma~2.5]{StVo14}, $\mathbf{m}athbf{f}$~can be extended to an $\mathcal{O}$-basis~$\mathbf{m}athbf{e}$ for a free $\mathcal{O}$-submodule~$M$ of finite index of~$\Lambda$.
If the residue characteristic~$p$ of~$\mathfrak{p}$ does not divide $|\Lambda:M|$ or $|i(\Lambda'): \Lambda'|$, this basis~$\mathbf{m}athbf{e}$ may be used to
obtain an $\mathfrak{O}$-basis for~$\Lambda(\mathfrak{O})$, by tensoring the elements of~$\mathbf{m}athbf{e}$ with~$\mathfrak{O}$.
\begin{rmk}
The condition ``$p$ does not divide $|i(\Lambda'): \Lambda'|$'' is missing in~\cite{StVo14}, but this omission does not affect the proof of~\cite[Theorem~A]{StVo14}, since this
condition only excludes a finite number of prime ideals~$\mathfrak{p}$.
This was first pointed out in~\cite[Section~3.3]{DuVo14}.
\end{rmk}
We now recall the general integrals given in~\cite[Section~2.1]{Vo10} and show that the integrals~$\widetilde{\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathfrak{O})}}(s_1,s_2)$ and~$\widetilde{\mathcal{Z}^\textup{cc}_{\mathbf{m}athbf{G}(\mathfrak{O})}}(s_1,s_2)$ are special cases of such integrals, so that the arguments given in~\cite[Section~4]{AKOV13} assure that they satisfy functional equations. Recall $[u]=\{1, \dots, u\}$ for $u \in \mathbf{m}athds{N}$.
Fix $l,m,n \in \mathbf{m}athds{N}$. For each $k \in [l]$, let $J_k$ be a finite index set. Fix $I \subseteq [n-1]$. Further, fix non-negative integers~$e_{ikj}$ and finite sets~$F_{kj}(\underline{Y})$ of polynomials over~$\mathfrak{o}$, for $k \in [l], j\in J_k$ and $i \in I$.
Also, let $\mathbf{m}athcal{W}(\mathfrak{o})\subseteq \mathfrak{o}^m$ be a union of cosets modulo~$\mathfrak{p}^{(m)}$. Define
\begin{equation}\label{Vogen}
\mathfrak{p}ints_{\mathbf{m}athcal{W}(\mathfrak{o}),I}(\underline{s})=\int_{\mathfrak{p}^{(|I|)}\times \mathbf{m}athcal{W}(\mathfrak{o})}
\mathfrak{p}rod_{k=1}^{l}\left|\left|\bigcup_{j \in J_k} \left( \mathfrak{p}rod_{i \in I}x_{i}^{e_{ikj}}\mathcal{O}ght)F_{kj}(\underline{y})\mathcal{O}ght|\mathcal{O}ght|_{\mathfrak{p}}^{s_k}d\mathbf{m}u,
\end{equation}
where $\underline{s}=(s_1, \dots, s_l)$ is a vector of complex variables and $\underline{x}=(x_i)_{i \in I}$ and $\underline{y}=(y_1, \dots, y_m)$ are independent integration variables.
In~\cite[Corollary~2.4]{Vo10}, by studying the transformation of the integral~\eqref{Vogen} under a principalisation~$(Y,h)$ of the ideal $\mathfrak{p}rod_{k,j}(F_{kj}(\underline{Y}))$, Voll proved a functional equation for~\eqref{Vogen} under inversion of the parameter~$q$ under certain invariance and regularity conditions. In particular, it is required that the principalisation~$(Y,h)$ has good reduction modulo~$\mathfrak{p}$, a condition that is satisfied for almost all prime ideals~$\mathfrak{p}$.
We now relate the integrals of Proposition~\text{Re}f{intpadic} with the general integral~\eqref{Vogen}.
Set $I=\{1\}$ and write $x_1=x$. Set~$n=b$, $m=b^2$, $l=2u_{B}+1$, and $J_k=\{1,2\}$, if $k \in [u_{B}]$ and $J_{k}=\{1\}$ if $u_{B}<k\leq 2u_{B}+1$.
We also set $\mathbf{m}athcal{W}(\mathfrak{O})=\mathbf{m}athbf{G}l_b(\mathfrak{O})$, and
\begin{table*}[h]
\centering
\begin{tabular}{ c | c | c | c }
$k$ & $j$ & $F_{kj}$ & $e_{1kj}$ \\
\hline
$\leq u_{B}$ & $1$ & $F_{2k}(B(\underline{y}))$ & $0$ \\
$u_{B}<k\leq 2u_{B}$ & $1$ & $F_{2(k-1-u_{B})}(B(\underline{y}))$ & $0$ \\
$2u_{B}+1$ & $1$ & $\{1\}$ & $1$ \\
$\leq u_{B}$ & $2$ & $F_{2(k-1)}(B(\underline{y}))$ & 2
\end{tabular}.
\end{table*}
We see that, with this set-up, the integral~\eqref{Vogen} is equal to
\begin{align}
\label{intb2}\mathfrak{p}ints_{\mathbf{m}athbf{G}l_b(\mathfrak{O}),\{1\}}(\underline{s})=\int_{\mathfrak{P}\times \mathbf{m}athbf{G}l_b(\mathfrak{O})} &\|x\|_{\mathfrak{P}}^{s_{2u_{B}+1}}\cdot\\
\mathfrak{p}rod_{k=1}^{u_{B}}\|F_{2k}(B(\underline{Y}))\cup x^2F_{2(k-1)}(B(\underline{Y}))\|_{\mathfrak{P}}^{s_k}
&\mathfrak{p}rod_{k=u_{B}+1}^{2u_{B}}\|F_{2(k-1-u_{B})}(B(\underline{Y}))\|_{\mathfrak{P}}^{s_{k}}d\mathbf{m}u.\nonumber
\end{align}
Set
\[\mathbf{m}athbf{a}_{1}^{\textup{irr}}=(-\mathfrak{f}rac{1}{2}\mathbf{m}athbf{1}_{u_{B}},\mathfrak{f}rac{1}{2}\mathbf{m}athbf{1}_{u_{B}},u_{B}), \hspace{0.3cm} \mathbf{m}athbf{a}_{2}^{\textup{irr}}=(\mathbf{m}athbf{0}_{u_{B}}, \mathbf{m}athbf{0}_{u_{B}},1),\]
\[\mathbf{m}athbf{b}^{\textup{irr}}=(-\mathbf{m}athbf{1}_{u_{B}}, \mathbf{m}athbf{1}_{u_{B}}, 2u_{B}-h-1),\]
where $\mathbf{m}athbf{1}_{u_{B}}=(1, \dots, 1)\in \mathbf{m}athds{Z}^{u_{B}}$ and $\mathbf{m}athbf{0}_{u_{B}}=(0,\dots,0)\in\mathbf{m}athds{Z}^{u_{B}}$.
Although the domain of integration of the integral~\eqref{intb2} involves $\mathbf{m}athbf{G}l_b(\mathfrak{O})$, the integrand only depends on the entries of the first column, say, as explained in~\cite[Section~4.1.3]{AKOV13}.
It follows that
\[\widetilde{\mathcal{Z}^\textup{irr}_{\mathbf{m}athbf{G}(\mathfrak{O})}}(s_1,s_2)=1+\mathfrak{f}rac{1}{1-q^{-f}}\left(\mathfrak{p}rod_{k=1}^{b-1}(1-q^{-fk})\mathcal{O}ght)^{-1}\hspace{-0.2cm}
\mathfrak{p}ints_{\mathbf{m}athbf{G}l_b(\mathfrak{O}),\{1\}}\left(\mathbf{m}athbf{a}_{1}^{\textup{irr}}s_1+\mathbf{m}athbf{a}_{2}^{\textup{irr}}s_2+\mathbf{m}athbf{b}^{\textup{irr}}\mathcal{O}ght).\]
Analogously, for~$n=a$, $m=a^2$, one can find appropriate data $l\in \mathbf{m}athds{N}$, $J_k$, $e_{1jk}$, and $F_{kj}(\underline{X})$ such that
\[\widetilde{\mathcal{Z}^\textup{cc}_{\mathbf{m}athbf{G}(\mathfrak{O})}}(s_1,s_2)=1+\mathfrak{f}rac{1}{1-q^{-f}}\left(\mathfrak{p}rod_{k=1}^{a-1}(1-q^{-fk})\mathcal{O}ght)^{-1}\hspace{-0.2cm}
\mathfrak{p}ints_{\mathbf{m}athbf{G}l_a(\mathfrak{O}),\{1\}}(\mathbf{m}athbf{a}_{1}^{{c}up}s_1+\mathbf{m}athbf{a}_{2}^{{c}up}s_2+\mathbf{m}athbf{b}^{{c}up}),\]
for $\mathbf{m}athbf{a}_{1}^{{c}up}=(-\mathbf{m}athbf{1}_{u_{A}},\mathbf{m}athbf{1}_{u_{A}},u_{A})$, $\mathbf{m}athbf{a}_{2}^{{c}up}=(\mathbf{m}athbf{0}_{u_{A}}, \mathbf{m}athbf{0}_{u_{A}},1)$,
$\mathbf{m}athbf{b}^{{c}up}=(-\mathbf{m}athbf{1}_{u_{A}}, \mathbf{m}athbf{1}_{u_{A}}, u_{A}-h-1)$.
Theorem~\text{Re}f{thmA} then follows by the arguments given in~\cite[Section~4]{AKOV13}.
\end{document} |
\begin{document}
\sectionhead{Contributed research article}
\volume{XX}
\volnumber{YY}
\year{20ZZ}
\month{AAAA}
\begin{article}
\title{Accessible Computation of Tight Symbolic Bounds on Causal Effects
using an Intuitive Graphical Interface}
\author{by Gustav Jonzon, Michael C Sachs, and Erin E Gabriel}
\maketitle
\abstract{
Strong untestable assumptions are almost universal in causal point
estimation. In particular settings, bounds can be derived to narrow the
possible range of a causal effect. Symbolic bounds apply to all settings
that can be depicted using the same directed acyclic graph (DAG) and for
the same effect of interest. Although the core of the methodology for
deriving symbolic bounds has been previously developed, the means of
implementation and computation have been lacking. Our \texttt{R}-package
\CRANpkg{causaloptim} \citep{causaloptim} aims to solve this usability
problem by implementing the method of \citep{generalcausalbounds} and
providing the user with a graphical interface through \CRANpkg{shiny}
that allows for input in a way that most researchers with an interest in
causal inference will be familiar; a DAG (via a point-and-click
experience) and specifying a causal effect of interest using familiar
counterfactual notation.
}
\hypertarget{introduction-and-background}{
\section{Introduction and
Background}\label{introduction-and-background}}
A common goal in many different areas of scientific research is to
determine causal relationships between one or more exposure variables
and an outcome. Prior to any computation or inference, we must clearly
state all assumptions made, i.e., all subject matter knowledge
available, regarding the causal relationships between the involved
variables as well as any additional variables, called confounders, that
may not be measured but influence at least two other variables of
interest. These assumptions are usually encoded in a causal directed
acyclic graph (DAG), with directed edges encoding direct causal
influences, which conveniently depicts all relevant information and has
become a familiar tool in applied research \citep{greenland1999causal}.
Such a DAG not only clearly states the assumptions made by the
researcher, but also comes with a sound methodology for causal
inference, in the form of identification results as well as derivation
of causal estimators \citep{pearl2009causality}.
Unfortunately, point identification of a desired causal effect typically
requires an assumption of no unmeasured confounders, in some form. When
there are unmeasured confounders, it is sometimes still possible to
derive bounds on the effect, i.e., a range of possible values for the
causal effect in terms of the observed data distribution. Symbolic
bounds are algebraic expressions for the bounds on the causal effect
written in terms of probabilities that can be estimated using observed
data. Alexander Balke and Judea Pearl first used linear programming to
derive tight symbolic bounds in a simple binary instrumental variable
setting \citep{balke1997bounds}. Balke wrote a program in \texttt{C++}
to take a linear programming problem as text file input, perform
variable reduction, conversion of equality constraints into inequality
constraints, and perform the vertex enumeration algorithm of
\citep{mattheiss1973algorithm}. This program has been used by
researchers in the field of causal inference
\citep{balke1997bounds, cai2008bounds, sjolander2009bounds, sjolander2014bounds}
but it is not particularly accessible because of the technical challenge
of translating the DAG plus causal query into the constrained
optimization problem and to determine whether it is linear. Moreover,
the program is not optimized and hence does not scale well to more
complex problems. Since they only cover a simple instrumental variable
setting, it has also not been clear to what extent their techniques
extend to more general settings, nor how to apply them to more complex
queries. Thus, applications of this approach have been limited to a
small number of settings and few attempts to generalize the method to
more widely applicable settings have been made.
Recent developments have expanded the applicability by generalizing the
techniques and the causal DAGs and effects to which they apply
\citep{generalcausalbounds}. These new methods have been applied in
novel observational and experimental settings
\citep{gabriel2020causal, gabriel2021nonparametric, gabriel2022sharp}.
Moreover, through the \texttt{R} package \CRANpkg{causaloptim}
\citep{causaloptim}, these computations are now accessible. With
\CRANpkg{causaloptim}, the user needs only to give input in a way they
would usually express their causal assumptions and state their target
causal estimand; through a DAG and counterfactual expression. Providing
DAGs through textual input is an awkward experience for most users, as
DAGs are generally communicated pictorially. Our package
\CRANpkg{causaloptim} provides a user-friendly graphical interface
through a web browser, where the user can draw their DAG in a way that
is familiar to them. The methodology that underpins
\CRANpkg{causaloptim} is not universal however; some restrictions on the
DAG and query are imposed. These are validated and communicated to the
user through the graphical interface, which guides the user through
providing the DAG and query, adding any extra conditions beyond those
encoded in the DAG, computing, interpreting and exporting the bounds for
various further analyses.
There exist few other \texttt{R}-packages related to causal bounds and
none to our knowledge for computation of symbolic bounds.
\CRANpkg{bpbounds} \citep{bpbounds-package} provides a text-based
interface to compute numeric bounds for the original single instrumental
variable example of Balke and Pearl and extends this by being able to
compute bounds given different types of data input including a ternary
rather than binary instrument. There is also a standalone program
written in \texttt{Java} by the \texttt{TETRAD\ Project}
(\url{https://github.com/cmu-phil/tetrad}) that includes a graphical
user interface and has a wrapper for \texttt{R}. Its focus, however, is
on causal discovery in a given sample data set, and although it can also
compute bounds, it can do so only numerically for the given data set.
In this paper we describe our \texttt{R} package \CRANpkg{causaloptim},
first focusing on the graphical and programmatic user interfaces in the
next 2 sections. Then we highlight some of our interesting functions and
data structures that may be useful in other contexts. We provide a
summary of the theoretical background and methods, while referring to
the companion paper \citep{generalcausalbounds} for the details. We
illustrate the use of the package with some numeric examples and close
with a discussion and summary.
\hypertarget{graphical-user-interface}{
\section{Graphical User Interface}\label{graphical-user-interface}}
In the following, we will work through the binary instrumental variable
example, where we have 3 observed binary variables \(X\), \(Y\), \(Z\),
and we want to determine the average causal effect of \(X\) on \(Y\)
given by total causal risk difference, in the presence of unmeasured
confounding by \(U_R\) and an instrumental variable \(Z\). Our causal
DAG is given by \(Z\to X\to Y\) and \(X\leftarrow U_R\to Y\) and our
causal query is \(P(Y(X=1)=1)-P(Y(X=0)=1)\), where we use \(Y(X = x)\)
to denote the potential outcome for \(Y\) if \(X\) were intervened upon
to have value \(x\).
\CRANpkg{causaloptim} includes a graphical user interface using
\CRANpkg{shiny} \citep{shiny}. The interface is launched in the user's
default web browser by calling \texttt{specify\_graph()}. Once the
\CRANpkg{shiny} app is launched, the user is presented with an
interactive display as shown in Figure \ref{fig:InterfaceStart}, in
which they can draw their causal DAG. This display is divided into a
left side \(\mathcal{L}\) and right side \(\mathcal{R}\) to classify the
vertices according to the class of DAGs that the method covers. In
particular, the existence of unmeasured confounders is assumed within
each of these sides, but not between them, and any causal influence
between the two sides must originate in \(\mathcal{L}\). Thus, for the
example, we would want to put the instrumental variable on the left
side, but the exposure and outcome on the right side. In the web version
of this article an interactive version of this interface is shown at the
end of this section.
\begin{Schunk}
\begin{figure}\label{fig:InterfaceStart}
\end{figure}
\end{Schunk}
\hypertarget{specifying-the-setting-by-drawing-a-causal-diagram-and-adding-attributes}{
\subsection{Specifying the setting by drawing a causal diagram and
adding
attributes}\label{specifying-the-setting-by-drawing-a-causal-diagram-and-adding-attributes}}
\begin{Schunk}
\begin{figure}\label{fig:DAG-1}
\label{fig:DAG-2}
\label{fig:DAG}
\end{figure}
\end{Schunk}
The DAG is drawn using a point-and-click device (e.g., a mouse) to add
vertices representing variables (by Shift-click) and name them (using
any valid variable name in \texttt{R}), and to draw edges representing
direct causal influences (Shift+drag) between them. The vertices may
also be moved around, renamed and deleted (as can the edges) as also
described in an instruction text preceding the DAG interface. As shown
in Figure \ref{fig:DAG}, for the example we add a vertex \(Z\) on the
left side, and vertices \(X\) and \(Y\) on the right side. Then the
\(Z \to X\) and \(X \to Y\) edges are added by Shift+clicking on a
parent vertex and dragging to the child vertex. There is no need to add
the unmeasured confounder variable \(U_R\) as it is assumed and added
automatically.
Importantly, the nodes may be selected and assigned additional
information. In \(\mathcal{R}\) a variable may be assigned as unobserved
(click+`u'). All observed variables are assumed categorical and their
cardinality (i.e., number of levels) may be set (click+`c' brings up a
prompt for this this number; alternatively a short-cut click+`any digit'
is provided), with the default being binary. Although the causal query
(i.e., the causal effect of interest) is entered subsequently, the DAG
interface provides a convenient short-cut; a node \(X\) may be assigned
as an exposure (click+`e') and another \(Y\) as outcome (click+`y'),
whereupon the default query is the total causal risk difference
\(P(Y(X=1)=1)-P(Y(X=0)=1)\). Finally, an edge may be assigned as
representing an assumed monotonic influence (click+`m'). The nodes and
edges change appearance according to their assigned characteristics
(Figure \ref{fig:Cardinality}) and violations to the restrictions
characterizing the class of DAGs are detected and communicated to the
user.
\begin{Schunk}
\begin{figure}\label{fig:Cardinality-1}
\label{fig:Cardinality-2}
\label{fig:Cardinality-3}
\label{fig:Cardinality}
\end{figure}
\end{Schunk}
Once the DAG has been drawn, the user may click the button ``Analyze the
graph'', upon which the DAG is interpreted and converted into an
annotated \CRANpkg{igraph}-object \citep{igraph} as described in the
implementation details below, and the results are displayed in graphical
form to the user (Figure \ref{fig:causalDAG}). The addition of \(U_R\),
the common unmeasured cause of \(X\) and \(Y\), is added and displayed
in this static plot.
\begin{Schunk}
\begin{figure}\label{fig:causalDAG-1}
\label{fig:causalDAG-2}
\label{fig:causalDAG}
\end{figure}
\end{Schunk}
\hypertarget{specifying-the-causal-query}{
\subsection{Specifying the causal
query}\label{specifying-the-causal-query}}
Next, the user is asked to specify the causal query, i.e., causal effect
of interest. If no outcome variable has been assigned in the DAG then
the input field for the causal query is left blank and a query needs to
be specified. In our example, since we have assigned an exposure and
outcome using the DAG interface, the total causal risk difference
\(P(Y(X=1)=1)-P(Y(X=0)=1)\) is suggested.
\hypertarget{specifying-optional-additional-constraints}{
\subsection{Specifying optional additional
constraints}\label{specifying-optional-additional-constraints}}
Finally, the user is given the option to provide any additional
constraints besides those imposed by the DAG. This may be considered an
optional advanced feature where, e.g., monotonicity of a certain direct
influence of \(Z\) on \(X\) may be assumed by entering
\(X(Z=1)\ge X(Z=0)\), with any such extra constraints separated by line
breaks. If this feature is used, the input is followed by clicking the
button ``Parse'', which identifies and fixes them.
\hypertarget{computing-the-symbolic-tight-bounds-on-the-query-under-the-given-constraints}{
\subsection{Computing the symbolic tight bounds on the query under the
given
constraints}\label{computing-the-symbolic-tight-bounds-on-the-query-under-the-given-constraints}}
As the final step, the button ``Compute the bounds'' is clicked,
whereupon the constraints and objective are compiled into an
optimization problem which is then solved for tight causal bounds on the
query symbolically in terms of observational quantities (conditional
probabilities of the observed variables in the DAG) and the expressions
are displayed alongside information on how the parameters are to be
interpreted in terms of the given variable names (Figure
\ref{fig:causalDAG}). During computation, a progress indicator is shown,
and the user should be aware that complex and/or high-dimensional
problems may take significant time. The interface also provides a
feature to subsequently convert the bounds to \LaTeX-code using standard
probabilistic notation for publication purposes.
Once done, clicking ``Exit and return objects to R'' stops the
\CRANpkg{shiny} app and returns all information about the DAG, query and
computed bounds to the \texttt{R}-session. This information is bundled
in a list containing the graph, query, parameters and their
interpretation, the symbolic tight bounds as expressions as well as
implementations as \texttt{R}-functions and further log information
about the formulation and optimization procedures.
\hypertarget{programmatic-user-interface}{
\section{Programmatic user
interface}\label{programmatic-user-interface}}
Interaction may also be done entirely programmatically as we illustrate
with the same binary instrumental variable example. First we create the
\texttt{igraph} object using the \texttt{graph\_from\_literal} function.
Once the basic graph is created, the necessary vertex and edge
attributes are added. The risk difference is defined as a character
object. The \texttt{analyze\_graph} function is the workhorse of
\CRANpkg{causaloptim}; it translates the causal graph, constraints, and
causal effect of interest into a linear programming problem. This linear
programming object, stored in \texttt{obj} in the code below, gets
passed to \texttt{optimize\_effect\_2} which performs vertex enumeration
to obtain the bounds as symbolic expressions in terms of observable
probabilities.
\begin{Schunk}
\begin{Sinput}
graph <- igraph::graph_from_literal(Z -+ X, X -+ Y,
Ul -+ Z, Ur -+ X, Ur -+ Y)
V(graph)$leftside <- c(1, 0, 0, 1, 0)
V(graph)$latent <- c(0, 0, 0, 1, 1)
V(graph)$nvals <- c(2, 2, 2, 2, 2)
E(graph)$rlconnect <- c(0, 0, 0, 0, 0)
E(graph)$edge.monotone <- c(0, 0, 0, 0, 0)
riskdiff <- "p{Y(X = 1) = 1} - p{Y(X = 0) = 1}"
obj <- analyze_graph(graph, constraints = NULL, effectt = riskdiff)
bounds <- optimize_effect_2(obj)
bounds
\end{Sinput}
\begin{Soutput}
#> lower bound =
#> MAX {
#> p00_0 - p00_1 - p10_1 - p01_1,
#> p00_0 - p00_1 - p10_0 - p10_1 - p01_0,
#> p00_0 - p00_1 + p10_0 - 2p10_1 - 2p01_1,
#> -p10_1 - p01_1,
#> -p10_0 - p01_0,
#> -p00_0 + p00_1 - 2p10_0 + p10_1 - 2p01_0,
#> -p00_0 + p00_1 - p10_0 - p10_1 - p01_1,
#> -p00_0 + p00_1 - p10_0 - p01_0
#> }
#> ----------------------------------------
#> upper bound =
#> MIN {
#> 1 - p10_1 - p01_0,
#> 1 + p00_0 + p10_0 - 2p10_1 - p01_1,
#> 2 - p00_1 - p10_0 - p10_1 - 2p01_0,
#> 1 - p10_1 - p01_1,
#> 1 - p10_0 - p01_0,
#> 1 + p00_1 - 2p10_0 + p10_1 - p01_0,
#> 2 - p00_0 - p10_0 - p10_1 - 2p01_1,
#> 1 - p10_0 - p01_1
#> }
\end{Soutput}
\end{Schunk}
The resulting bounds object contains character strings representing the
bounds and logs containing detailed information from the vertex
enumeration algorithm. The bounds are printed to the console but more
features are available to facilitate their use. The
\texttt{interpret\_bounds} function takes the bounds and parameter names
as input and returns an \texttt{R} function implementing vectorized
forms of the symbolic expressions for the bounds.
\begin{Schunk}
\begin{Sinput}
bounds_function <- interpret_bounds(bounds$bounds, obj$parameters)
str(bounds_function)
\end{Sinput}
\begin{Soutput}
#> function (p00_0 = NULL, p00_1 = NULL, p10_0 = NULL, p10_1 = NULL, p01_0 = NULL,
#> p01_1 = NULL, p11_0 = NULL, p11_1 = NULL)
\end{Soutput}
\end{Schunk}
The results can also be used for numerical simulation using
\texttt{simulate\_bounds}. This function randomly generates
counterfactuals and probability distributions that satisfy the
constraints implied by the DAG and optional constraints. It then
computes and returns the bounds as well as the true causal effect.
If one wants to bound a different effect using the same causal graph,
the \texttt{update\_effect} function can be used to save some
computation time. It takes the object returned by
\texttt{analyze\_graph} and the new effect string then returns an object
of class \texttt{linearcausalproblem} that can be optimized:
\texttt{obj2\ \textless{}-\ update\_effect(obj,\ "p\{Y(X\ =\ 1)\ =\ 1\}")}.
Finally, \LaTeX-code may also be generated using the function
\texttt{latex\_bounds} as in
\texttt{latex\_bounds(bounds\$bounds,\ obj\$parameters)} yielding \tiny
\begin{align*}
\mbox{Lower bound} &= \mbox{max} \left. \begin{cases} P(X = 0, Y = 0 | Z = 0) - P(X = 0, Y = 0 | Z = 1) - P(X = 1, Y = 0 | Z = 1) - P(X = 0, Y = 1 | Z = 1),\\
P(X = 0, Y = 0 | Z = 0) - P(X = 0, Y = 0 | Z = 1) - P(X = 1, Y = 0 | Z = 0) - P(X = 1, Y = 0 | Z = 1) - P(X = 0, Y = 1 | Z = 0),\\
P(X = 0, Y = 0 | Z = 0) - P(X = 0, Y = 0 | Z = 1) + P(X = 1, Y = 0 | Z = 0) - 2P(X = 1, Y = 0 | Z = 1) - 2P(X = 0, Y = 1 | Z = 1),\\
-P(X = 1, Y = 0 | Z = 1) - P(X = 0, Y = 1 | Z = 1),\\
-P(X = 1, Y = 0 | Z = 0) - P(X = 0, Y = 1 | Z = 0),\\
-P(X = 0, Y = 0 | Z = 0) + P(X = 0, Y = 0 | Z = 1) - 2P(X = 1, Y = 0 | Z = 0) + P(X = 1, Y = 0 | Z = 1) - 2P(X = 0, Y = 1 | Z = 0),\\
-P(X = 0, Y = 0 | Z = 0) + P(X = 0, Y = 0 | Z = 1) - P(X = 1, Y = 0 | Z = 0) - P(X = 1, Y = 0 | Z = 1) - P(X = 0, Y = 1 | Z = 1),\\
-P(X = 0, Y = 0 | Z = 0) + P(X = 0, Y = 0 | Z = 1) - P(X = 1, Y = 0 | Z = 0) - P(X = 0, Y = 1 | Z = 0) \end{cases} \right\} \\
\mbox{Upper bound} &= \mbox{min} \left. \begin{cases} 1 - P(X = 1, Y = 0 | Z = 1) - P(X = 0, Y = 1 | Z = 0),\\
1 + P(X = 0, Y = 0 | Z = 0) + P(X = 1, Y = 0 | Z = 0) - 2P(X = 1, Y = 0 | Z = 1) - P(X = 0, Y = 1 | Z = 1),\\
2 - P(X = 0, Y = 0 | Z = 1) - P(X = 1, Y = 0 | Z = 0) - P(X = 1, Y = 0 | Z = 1) - 2P(X = 0, Y = 1 | Z = 0),\\
1 - P(X = 1, Y = 0 | Z = 1) - P(X = 0, Y = 1 | Z = 1),\\
1 - P(X = 1, Y = 0 | Z = 0) - P(X = 0, Y = 1 | Z = 0),\\
1 + P(X = 0, Y = 0 | Z = 1) - 2P(X = 1, Y = 0 | Z = 0) + P(X = 1, Y = 0 | Z = 1) - P(X = 0, Y = 1 | Z = 0),\\
2 - P(X = 0, Y = 0 | Z = 0) - P(X = 1, Y = 0 | Z = 0) - P(X = 1, Y = 0 | Z = 1) - 2P(X = 0, Y = 1 | Z = 1),\\
1 - P(X = 1, Y = 0 | Z = 0) - P(X = 0, Y = 1 | Z = 1) \end{cases} \right\}.
\end{align*} \normalsize
\hypertarget{implementation-and-program-overview}{
\section{Implementation and Program
Overview}\label{implementation-and-program-overview}}
An overview of the main functions and their relations is depicted as a
flow chart in Figure \ref{fig:Overview}. All functions may be called
individually by the user at the \texttt{R}-console and all input, output
and interaction available through the \CRANpkg{shiny} app has
corresponding availability at the \texttt{R}-console as well.
\citep{generalcausalbounds} define the following class of problems for
which the query in general is not identifiable, but for which a
methodology to derive symbolic tight bounds on the query is provided.
The causal DAG consists of a finite set
\(\mathcal{W}=\{W_1,\dots,W_n\}=\mathcal{W}_\mathcal{L}\cup\mathcal{W}_\mathcal{R}\)
of categorical variables with
\(\mathcal{W}_\mathcal{L}\cap\mathcal{W}_\mathcal{R}=\varnothing\), no
edges going from \(\mathcal{W}_\mathcal{R}\) to
\(\mathcal{W}_\mathcal{L}\) and no external common parent between
\(\mathcal{W}_\mathcal{L}\) and \(\mathcal{W}_\mathcal{R}\), but
\emph{importantly} external common parents \(\mathcal{U}_\mathcal{L}\)
and \(\mathcal{U}_\mathcal{R}\) of variables within
\(\mathcal{W}_\mathcal{R}\) and \(\mathcal{W}_\mathcal{R}\) may not be
ruled out. Nothing is assumed about any characteristics of these
confounding variables \(\mathcal{U}_\mathcal{L}\) and
\(\mathcal{U}_\mathcal{R}\).
The causal query may be any linear combination of joint probabilities of
factual and counterfactual outcomes expressed in terms of the variables
in \(\mathcal{W}\) and may always be expressed as a sum of probabilities
of response function variables of the DAG. It is subject to the
restriction that each outcome variable is in \(\mathcal{W}_\mathcal{R}\)
and if \(\mathcal{W}_\mathcal{L}\ne\varnothing\) it is also subject to a
few regularity conditions as detailed in \citep{generalcausalbounds}.
Tight bounds on the query may then be derived symbolically in terms of
conditional probabilities of the observable variables \(\mathcal{W}\).
Algorithms 1 and 2 in \citep{generalcausalbounds} construct the
constraint space and causal query in terms of the joint probabilities of
the response function variables and in \CRANpkg{causaloptim} are
implemented in the functions \texttt{create\_R\_matrix} and
\texttt{create\_effect\_vector} respectively. Both are called as
sub-procedures of the function \texttt{analyze\_graph} to translate the
causal problem to that of optimizing an objective function over a
constraint space. The implementation of Algorithm 1 involves
constructing the response functions themselves as actual
\texttt{R}-functions. Evaluating these correspond to evaluating the
structural equations of the causal DAG.
The conditions on the DAG suffice to ensure that the causal query will
depend only on the response functions corresponding to the variables in
\(\mathcal{W}_\mathcal{R}\) and that the exhaustive set of constraints
of their probabilities are linear in a subset of conditional
probabilities of observable variables (Proposition 2 in
\citep{generalcausalbounds}), and the conditions on the query in turn
ensure that it may be expressed as a linear combination of joint
probabilities of the response functions of the variables in
\(\mathcal{W}_\mathcal{R}\) (Proposition 3 in
\citep{generalcausalbounds}).
Once this formulation of the causal problem as a linear program has been
set up, a vertex enumeration method is employed to compute the extrema
symbolically in terms of conditional probabilities of the observable
variables.
The main and interesting functions will be described in some detail
below. We begin however with an overview of how they are tied together
by the \CRANpkg{shiny} app.
\begin{Schunk}
\begin{figure}\label{fig:Overview}
\end{figure}
\end{Schunk}
\hypertarget{specify_graph}{
\subsubsection{\texorpdfstring{\texttt{specify\_graph}}{specify\_graph}}\label{specify_graph}}
The graphical interface is launched by \texttt{specify\_graph()}, or
preferably \texttt{results\ \textless{}-\ specify\_graph()}. Once the
\CRANpkg{shiny} app is stopped, the input, output and other useful
information is returned by the function, so we recommend assigning it to
a variable so they are saved in the \texttt{R}-session and may easily be
further analyzed and processed. All further function calls will take
place automatically as the user interacts with the web interface. Thus,
from a basic user perspective, \texttt{specify\_graph} is the main
function. The core functionality however is implemented in the functions
\texttt{analyze\_graph}, \texttt{optimize\_effect\_2} and their
subroutines.
The \texttt{JavaScript} that handles the communication between the
\CRANpkg{shiny} server and the input as the user draws a DAG through the
web interface uses on the project \texttt{directed-graph-creator}, an
interactive tool for creating directed graphs, created using
\texttt{d3.js} and hosted at
\url{https://github.com/cjrd/directed-graph-creator}, which has been
modified for the purpose of causal diagrams. The modification binds the
user inputs as they interact with the graph to \CRANpkg{shiny} so that
the directed graph and its attributes set by the user are reactively
converted into an \texttt{igraph}-object for further processing. Since
directed graphs are common in many computational and statistical
problems, this \CRANpkg{shiny} interface may also be valuable to many
other \texttt{R}-package authors and maintainers who may wish to provide
their users with an accessible and intuitive way to interact with their
software.
The server listens to a reactive function that, as the user draws the
DAG, collects information about the current edges, collects and
annotates vertices, adds left- and right-side confounding, and returns
an annotated \texttt{igraph}-object, comprising information about the
connectivity along with some additional attributes; for each variable,
its name, cardinality, latency-indicator and side-indicator, and for
each edge, a monotonicity-indicator and (to detect and communicate
violations on direction) a right-to-left-indicator. The server meanwhile
also monitors the DAG for any violation of the restriction that each
edge between \(\mathcal{L}\) and \(\mathcal{R}\) must go \emph{from}
\(\mathcal{L}\) \emph{to} \(\mathcal{R}\), and if detected directly
communicates this to the user through a text message in the
\CRANpkg{shiny} app.
\hypertarget{analyze_graph}{
\subsubsection{\texorpdfstring{\texttt{analyze\_graph}}{analyze\_graph}}\label{analyze_graph}}
The function \texttt{analyze\_graph} takes a DAG (in the form of an
\texttt{igraph} object), optional constraints, and a string representing
the causal effect of interest and proceeds to construct and return a
linear optimization problem (a \texttt{linearcausalproblem}-object) from
these inputs.
First, some basic data structures are created to keep track of the
observed variables, their possible values, the latent variables, and
whether they are in \(\mathcal{L}\) or \(\mathcal{R}\). Once these basic
data-structures have been created, the first task of the algorithm is to
create the response function variables (for each variable, observed or
not, except \(U_\mathcal{L}\) and \(U_\mathcal{R}\)). Probabilities of
these will be the entities \(\mathbf{q}\) in which the objective
function (representing the target causal effect) is expressed and will
constitute the points in the space it is optimized over, where this
space itself is constrained by the the relationships between them and
observed conditional probabilities \(\mathbf{p}\).
\hypertarget{create_response_function}{
\subsubsection{\texorpdfstring{\texttt{create\_response\_function}}{create\_response\_function}}\label{create_response_function}}
The function \texttt{create\_response\_function} returns a list
\texttt{respvars} that has a named entry for each observed variable,
containing its response function variable and response function. If
\(X\) is an observed variable with \(n\) response functions, then they
are enumerated by \(\{0,\dots,n-1\}\). Its entry \texttt{respvars\$X}
contains the response function variable \(R_X\) of \(X\), and is a list
with two entries. The first, \texttt{respvars\$X\$index}, is a vector
containing all the possible values of \(R_X\), i.e., the integers
\((0,\dots,n-1)\). The second, \texttt{respvars\$X\$values} is itself a
list with \(n\) entries; each containing the particular response
function of \(X\) corresponding to its index. Each such response
function is an actual \texttt{R}-function and may be evaluated by
passing it any possible values of the parents of \(X\) as arguments.
Next, the response function variables are used in the creation of a
matrix of unobserved probabilities. Specifically the joint probabilities
\(P(\mathbf{R}_\mathcal{R}=\mathbf{r}_\mathcal{R})\) for each possible
value-combination \(\mathbf{r}_\mathcal{R}\) of the response function
variables \(\mathbf{R}_\mathcal{R}\) of the right-side-variables
\(\mathbf{W}_\mathcal{R}\). In \citep{generalcausalbounds}, the possible
value-combinations \(\mathbf{r}_\mathcal{R}\) are enumerated by
\(\gamma\in\{1,\dots,\aleph_\mathcal{R}\}\) with corresponding
probabilities \(q_\gamma:=P(\mathbf{R}_\mathcal{R}=\mathbf{r}_\gamma)\)
being components of the vector
\(\mathbf{q}\in[0,1]^{\aleph_\mathcal{R}}\).
\hypertarget{create_r_matrix}{
\subsubsection{\texorpdfstring{\texttt{create\_R\_matrix}}{create\_R\_matrix}}\label{create_r_matrix}}
The constraints that the DAG and observed conditional probabilities
\(\mathbf{p}\) (in \texttt{p.vals}) impose on the unobserved
probabilities \(\mathbf{q}\) (represented by \texttt{variables}) are
linear. Specifically, there exists a matrix whose entries are the
coefficients relating \texttt{p.vals} to \texttt{variables}. This matrix
is called \(P\) in \citep{generalcausalbounds}, where its existence is
guaranteed by Proposition 2 and its construction is detailed in
Algorithm 1, which is implemented in the function
\texttt{create\_R\_matrix}. This function returns back a list with two
entries; a vector of strings representing the linear constraints on the
unobserved \(\mathbf{q}\in[0,1]^{\aleph_\mathcal{R}}\) imposed by and in
terms of the observed \(\mathbf{p}\in[0,1]^B\) and the numeric matrix
\(R\in\{0,1\}^{(B+1)\times\aleph_\mathcal{R}}\) of coefficients
corresponding to these constraints as well as the probabilistic ones and
given by \(R=\begin{pmatrix}\mathbf{1}\\P\end{pmatrix}\) where
\(P\in\{0,1\}^{B\times\aleph_\mathcal{R}}:\mathbf{p}=P\mathbf{q}\), so
\(R\mathbf{q}=\begin{pmatrix}1\\\mathbf{p}\end{pmatrix}\).
This determines the constraint space as a compact convex polytope in
\(\mathbf{q}\)-space, i.e., in \(\mathbb{R}^{\aleph_\mathcal{R}}\). To
create the matrix, we define a recursive function \texttt{gee\_r} that
takes two arguments; a positive integer \texttt{i} being the index
\(i\in\{1,\dots,n\}\) of a variable \(W_i\in\mathcal{W}\) (i.e.~the
\(i^{th}\) component of \(\mathbf{W}\) or, equivalently, the
\texttt{i}th entry of \texttt{obsvars}) and a vector \texttt{r} being a
value \(\mathbf{r}\in\nu(\mathbf{R})\) in the set \(\nu(\mathbf{R})\) of
all possible value-vectors of the joint response function variable
\(\mathbf{R}\). This recursive function is called for each variable in
\texttt{obsvars} and for each possible value of the response function
variable vector. The base case is reached if the variable has no
parents, in which case the list corresponding to the response function
variable \(R_{W_i}\) of \(W_i\) is extracted from \texttt{respvars}.
From this list, the entry whose index matches the \texttt{i}th index of
\texttt{r} (i.e.~the one corresponding to the response function variable
value \(r_i=\)\texttt{r{[}i{]}}) is extracted and finally its value,
i.e., the corresponding response function itself, is extracted and is
evaluated on an empty list of arguments, since it is a constant function
and determined only by the value \(r_i\).
The recursive case is encountered when \texttt{parents} is non-empty. If
so, then for each parent in \texttt{parents}, its index in
\texttt{obsvars} is determined and \texttt{gee\_r} is recursively called
with the same vector \texttt{r} as first argument but now with this
particular index (i.e.~that of the current parent) as second argument.
The numeric values returned by these recursive calls are then
sequentially stored in a vector \texttt{lookin}, whose entries are named
by those in \texttt{parents}. Just as in the base case, the response
function corresponding to the particular value \(r_i\) of the response
function variable \(R_{W_i}\) (i.e.~the response function of the
variable \texttt{obsvars{[}i{]}} that has the index \texttt{r{[}i{]}})
is extracted from \texttt{respvars} and is now evaluated with arguments
given by the list \texttt{lookin}. Note that \texttt{gee\_r(r,\ i)}
corresponds to the value \(w_i=g^*_{W_i}(\mathbf{r})\) in
\citep{generalcausalbounds}.
Then the values that match the observed probabilities are recorded, the
corresponding entries in the current row of the matrix \texttt{R} are
set to 1 and a string representing the corresponding equation is
constructed and added to the vector of constraints.
\hypertarget{parse_effect}{
\subsubsection{\texorpdfstring{\texttt{parse\_effect}}{parse\_effect}}\label{parse_effect}}
Now that the constraint space has been determined, the objective
function representing the causal query needs to be specified as a linear
function of the components of \(\mathbf{q}\), i.e., \texttt{variables}.
First the causal query that has been provided by the user as a
text-string in \texttt{effectt} is passed to the function
\texttt{parse\_effect}, which identifies its components including nested
counterfactuals and creates a data structure representing the causal
query. This structure includes nested lists which represent all
interventional paths to each outcome variable in the query.
Once the nested list \texttt{effect} is returned back to
\texttt{analyze\_graph}, it checks that the requirements (see
Proposition 3 in \citep{generalcausalbounds}) on the query are fulfilled
before creating the linear objective function. Despite these regularity
conditions, a large set of possible queries may be entered using
standard counterfactual notation, using syntax described in the
accompanying instruction text along with examples such as
\(P(Y(M(X = 0), X = 1) = 1) - P(Y(M(X = 0), X = 0) = 1)\); the natural
direct effect \citep{pearl2001direct} of a binary exposure \(X\) at
level \(M=0\) on a binary outcome \(Y\) \emph{not} going through the
mediator \(M\), in the presence of unmeasured confounding between \(M\)
and \(Y\) \citep{sjolander2009bounds}.
\hypertarget{create_effect_vector}{
\subsubsection{\texorpdfstring{\texttt{create\_effect\_vector}}{create\_effect\_vector}}\label{create_effect_vector}}
Now that the required characteristics of the query have been
established, the corresponding objective function will be constructed by
the function \texttt{create\_effect\_vector} which returns a list
\texttt{var.eff} of string-vectors; one for each term in the query. Each
such vector contains the names (strings in \texttt{variables}) of the
response function variables of the right-side (i.e.~the components of
\(\mathbf{q}\)) whose sum corresponds the that particular term. The
function \texttt{create\_effect\_vector} implements Algorithm 2 of
\citep{generalcausalbounds} with the additional feature that if the user
has entered a query that is incomplete in the sense that there are
omitted mediating variables on paths from base/intervention variables to
the outcome variable, then this is interpreted as the user intending the
effects of the base/intervention variables to be propagated through the
mediators, so that they are set to their ``natural'' values under this
intervention. These mediators are detected and their values are set
accordingly.
We define a recursive function \texttt{gee\_rA} that takes three
arguments; a positive integer \texttt{i} (the index \(i\) of a variable
\(W_i\in\mathcal{W}=\)\texttt{obsvars}), a vector \texttt{r} (a value
\(\mathbf{r}\in\nu(\mathbf{R})\) in the set \(\nu(\mathbf{R})\) of all
possible value-vectors of the joint response function variable
\(\mathbf{R}\)) and a string \texttt{path} that represents an
interventional path and is of the form ``X -\textgreater{} \ldots{}
-\textgreater{} Y'' if not \texttt{NULL}. The base case is reached
either if \texttt{path} is non-\texttt{NULL} and corresponds to a path
to the intervention set or if \texttt{parents} is empty. In the former
case, the corresponding numeric intervention-value is returned, and in
the latter case, the value of the corresponding response function called
on the empty list of arguments is returned just as in the base case of
\texttt{gee\_r}. The recursive case is encountered when \texttt{path} is
\texttt{NULL} and \texttt{parents} is non-empty. This recursion proceeds
just as in \texttt{gee\_r}, but now rather with a recursive call to
\texttt{gee\_rA}, whose third argument is now
\texttt{path\ =\ paste(gu,\ "-\textgreater{}",\ path)} where the string
in \texttt{gu} is the name of the parent variable in \texttt{parents}
whose index \texttt{i} in \texttt{obsvars} is the second argument of
this recursive call. This construction traces the full path taken from
the outcome of interest to the variable being intervened upon. Note that
\texttt{gee\_rA(r,\ i,\ path)} corresponds to the value
\(w_i=h^{A_i}_{W_i}(\mathbf{r},W_i)\) in \citep{generalcausalbounds}. A
matrix is now created just as in the observational case, but this time
using \texttt{gee\_rA} instead of \texttt{gee\_r} .
\hypertarget{optimize_effect_2}{
\subsubsection{\texorpdfstring{\texttt{optimize\_effect\_2}}{optimize\_effect\_2}}\label{optimize_effect_2}}
Once the constraints on \(\mathbf{q}\) as well as the effect of interest
in terms of \(\mathbf{q}\) have been established, it remains only to
optimize this expression over the constraint space. Here, \(\mathbf{c}\)
denotes the constant gradient vector of the linear objective function
and \(P\) denotes the coefficient matrix of the linear restrictions on
\(\mathbf{q}\) in terms of \(\mathbf{q}\) imposed by the causal DAG. By
adding the probabilistic constraints on \(\mathbf{q}\) we have arrived
at e.g.~the following linear program giving a tight lower bound on the
average causal effect
\(\theta_\mathbf{q} = P\{Y(X = 1) = 1\} - P\{Y(X = 0) = 1\}\) in the
simple instrumental variable problem of the introductory section:
\begin{align*}
\min_\mathbf{q} \theta_\mathbf{q}
&=\min\{\mathbf{c}^\top\mathbf{q}\mid \mathbf{q}\in\mathbb{R}^{16},\mathbf{q}\geq\mathbf{0}_{16\times1},\mathbf{1}_{1\times 16} \mathbf{q}=1,P\mathbf{q}=\mathbf{q}\}\\
&=\max\{\begin{pmatrix}1&\mathbf{q}^\top\end{pmatrix}\mathbf{y}\mid\mathbf{y}\in\mathbb{R}^{9},\mathbf{y}\geq \mathbf{0}_{9\times 1},\begin{pmatrix}\mathbf{1}_{16\times 1}&P^\top\end{pmatrix}\leq\mathbf{c}\}\\
&=\max\{\begin{pmatrix}1&\mathbf{q}^\top\end{pmatrix}\bar{\mathbf{y}}\mid\bar{\mathbf{y}}\text{ is a vertex of }\{\mathbf{y}\in\mathbb{R}^{9}\mid \mathbf{y}\geq\mathbf{0}_{9\times 1},R^\top\leq\mathbf{c}\}\}
\end{align*}
Since we allow the user to provide additional linear inequality
constraints (e.g.~it may be quite reasonable to assume the proportion of
``defiers'' in the study population of our example to be quite low), the
actual primal and dual linear programs may look slightly more
complicated, but this small example still captures the essentials.\\
In general, given the matrix of linear constraints on the observable
probabilities implied by the DAG and an optional user-provided matrix
inequality, we construct the coefficient matrix and right hand side
vector of the dual polytope.
The optimization via vertex enumeration step in \CRANpkg{causaloptim} is
implemented in the function \texttt{optimize\_effect\_2} which uses the
double description method for vertex enumeration, as implemented in the
\CRANpkg{rcdd} package \citep{rcdd}. This step of vertex enumeration has
previously been the major computational bottleneck. The approach is now
based on \texttt{cddlib}
(\url{https://people.inf.ethz.ch/fukudak/cdd_home/}), which has an
implementation of the Double Description Method (dd). Any convex
polytope can be dually described as either an intersection of
half-planes (which is the form we get our dual constraint space in) or
as a minimal set of vertices of which it is the convex hull (which is
the form we want it in) and the dd algorithm efficiently converts
between these two descriptions. \texttt{cddlib} also uses exact rational
arithmetic, so there is no need to worry about any numerical instability
issues. The vertices of the dual polytope are obtained and stored as
rows of a matrix with \texttt{hrep\ \textless{}-\ rcdd::makeH(a1,\ b1);}
\texttt{vrep\ \textless{}-\ rcdd::scdd(hrep);}
\texttt{vertices\ \textless{}-\ vrep\$output{[}vrep\$output{[},\ 1{]}\ ==\ 0\ \&\ vrep\$output{[},\ 2{]}\ ==\ 1,\ -c(1,\ 2),\ drop=FALSE{]}}.
The rest is simply a matter of plugging them into the dual objective
function, evaluating the expression and presenting the results. The
first part of this is done by
\texttt{apply(vertices,\ 1,\ function(y)\ evaluate\_objective(c1\_num,\ p,\ y))}
(here
\texttt{(c1\_num,p)}\(=(\begin{pmatrix}b_{\ell}^\top&1\end{pmatrix},p)\)
separates the dual objective gradient into its numeric and symbolic
parts).
\CRANpkg{causaloptim} also contains a precursor to to
\texttt{optimize\_effect\_2}, called \texttt{optimize\_effect}. This
legacy function uses the original optimization procedure written in
\texttt{C++} by Alexander Balke and involves linear program formulation
followed by the vertex enumeration algorithm of
\citep{mattheiss1973algorithm}. This has worked well for very simple
settings but has struggled severely with even remotely complex ones and
thus been insufficient for the ambitions of \CRANpkg{causaloptim}. The
efficiency gains of \texttt{optimize\_effect\_2} over the legacy code
have reduced the computation time for several setting from hours to
milliseconds.
\hypertarget{numeric-examples}{
\section{Numeric Examples}\label{numeric-examples}}
\hypertarget{a-mediation-analysis}{
\subsection{A Mediation Analysis}\label{a-mediation-analysis}}
In \citep{sjolandernaturaldirecteffects}, the author derives bounds on
natural direct effects in the presence of confounded intermediate
variables and applies them to data from the Lipid Research Clinics
Coronary Primary Prevention Trial \citep{freedmandata}, where subjects
were randomized to cholestyramine treatment and presence of coronary
heart disease events as well as levels of cholesterol were recorded
after a 1-year follow-up period. We let \(X\) be a binary treatment
indicator, with \(X=0\) indicating actual cholestyramine treatment and
\(X=1\) indicating placebo. We further let \(Y\) be an indicator of the
occurrence of coronary heart disease events within follow-up, with
\(Y=0\) indicating event-free follow-up and \(Y=1\) indicating an event.
We finally let \(M\) be a dichotomized (cut-off at \(280\ mg/dl\))
cholesterol level indicator, with \(M=0\) indicating levels
\(<280\ mg/dl\) and \(M=1\) indicating levels \(\ge280\ mg/dl\). The
causal assumptions are summarized in the DAG shown in Figure
\ref{fig:mediation-fig}, where \(U_l\) and \(U_r\) are unmeasured and
the latter confounds the effect of \(M\) on \(Y\).
\begin{Schunk}
\begin{figure}\label{fig:mediation-fig}
\end{figure}
\end{Schunk}
\begin{Schunk}
\begin{Sinput}
b <- igraph::graph_from_literal(X -+ Y, X -+ M, M -+ Y,
Ul -+ X, Ur -+ Y, Ur -+ M)
V(b)$leftside <- c(1, 0, 0, 1, 0)
V(b)$latent <- c(0, 0, 0, 1, 1)
V(b)$nvals <- c(2, 2, 2, 2, 2)
E(b)$rlconnect <- c(0, 0, 0, 0, 0, 0)
E(b)$edge.monotone <- c(0, 0, 0, 0, 0, 0)
\end{Sinput}
\end{Schunk}
Using the data from Table IV of \citep{sjolandernaturaldirecteffects},
we compute the observed conditional probabilities.
\begin{Schunk}
\begin{Sinput}
# parameters of the form pab_c, which represents
# the probability P(Y = a, M = b | X = c)
p00_0 <- 1426/1888 # P(Y=0,M=0|X=0)
p10_0 <- 97/1888 # P(Y=1,M=0|X=0)
p01_0 <- 332/1888 # P(Y=0,M=1|X=0)
p11_0 <- 33/1888 # P(Y=1,M=1|X=0)
p00_1 <- 1081/1918 # P(Y=0,M=0|X=1)
p10_1 <- 86/1918 # P(Y=1,M=0|X=1)
p01_1 <- 669/1918 # P(Y=0,M=1|X=1)
p11_1 <- 82/1918 # P(Y=1,M=1|X=1)
\end{Sinput}
\end{Schunk}
We proceed to compute bounds on the controlled direct effect
\(CDE(0) = P(Y(M = 0, X = 1) = 1) - P(Y(M = 0, X = 0) = 1)\) of \(X\) on
\(Y\) not passing through \(M\) at level \(M=0\), the controlled direct
effect \(CDE(1) = P(Y(M = 1, X = 1) = 1) - P(Y(M = 1, X = 0) = 1)\) at
level \(M=1\), the natural direct effect
\(NDE(0) = P(Y(M(X = 0), X = 1) = 1) - P(Y(M(X = 0), X = 0) = 1)\) of
\(X\) on \(Y\) at level \(X=0\) and the natural direct effect
\(NDE(1) = P(Y(M(X = 1), X = 1) = 1) - P(Y(M(X = 1), X = 0) = 1)\) at
level \(X=1\).
\begin{Schunk}
\begin{Sinput}
CDE0_query <- "p{Y(M = 0, X = 1) = 1} - p{Y(M = 0, X = 0) = 1}"
CDE0_obj <- analyze_graph(b, constraints = NULL, effectt = CDE0_query)
CDE0_bounds <- optimize_effect_2(CDE0_obj)
CDE0_boundsfunction <- interpret_bounds(bounds = CDE0_bounds$bounds,
parameters = CDE0_obj$parameters)
CDE0_numericbounds <- CDE0_boundsfunction(p00_0 = p00_0, p00_1 = p00_1,
p10_0 = p10_0, p10_1 = p10_1,
p01_0 = p01_0, p01_1 = p01_1,
p11_0 = p11_0, p11_1 = p11_1)
CDE1_query <- "p{Y(M = 1, X = 1) = 1} - p{Y(M = 1, X = 0) = 1}"
CDE1_obj <- update_effect(CDE0_obj, effectt = CDE1_query)
CDE1_bounds <- optimize_effect_2(CDE1_obj)
CDE1_boundsfunction <- interpret_bounds(bounds = CDE1_bounds$bounds,
parameters = CDE1_obj$parameters)
CDE1_numericbounds <- CDE1_boundsfunction(p00_0 = p00_0, p00_1 = p00_1,
p10_0 = p10_0, p10_1 = p10_1,
p01_0 = p01_0, p01_1 = p01_1,
p11_0 = p11_0, p11_1 = p11_1)
NDE0_query <- "p{Y(M(X = 0), X = 1) = 1} - p{Y(M(X = 0), X = 0) = 1}"
NDE0_obj <- update_effect(CDE0_obj, effectt = NDE0_query)
NDE0_bounds <- optimize_effect_2(NDE0_obj)
NDE0_boundsfunction <- interpret_bounds(bounds = NDE0_bounds$bounds,
parameters = NDE0_obj$parameters)
NDE0_numericbounds <- NDE0_boundsfunction(p00_0 = p00_0, p00_1 = p00_1,
p10_0 = p10_0, p10_1 = p10_1,
p01_0 = p01_0, p01_1 = p01_1,
p11_0 = p11_0, p11_1 = p11_1)
NDE1_query <- "p{Y(M(X = 1), X = 1) = 1} - p{Y(M(X = 1), X = 0) = 1}"
NDE1_obj <- update_effect(CDE0_obj, effectt = NDE1_query)
NDE1_bounds <- optimize_effect_2(NDE1_obj)
NDE1_boundsfunction <- interpret_bounds(bounds = NDE1_bounds$bounds,
parameters = NDE1_obj$parameters)
NDE1_numericbounds <- NDE1_boundsfunction(p00_0 = p00_0, p00_1 = p00_1,
p10_0 = p10_0, p10_1 = p10_1,
p01_0 = p01_0, p01_1 = p01_1,
p11_0 = p11_0, p11_1 = p11_1)
\end{Sinput}
\end{Schunk}
We obtain the same symbolic bounds as
\citep{sjolandernaturaldirecteffects} and the resulting numeric bounds
are given in Table \ref{tab:mediation-bounds} which of course agree with
those of Table V in \citep{sjolandernaturaldirecteffects}.
\begin{Schunk}
\begin{table}
\caption{\label{tab:mediation-bounds}Bounds on the controlled and natural direct effects.}
\centering
\begin{tabular}[t]{l|r|r}
\hline
& lower & upper\\
\hline
CDE(0) & -0.20 & 0.39\\
\hline
CDE(1) & -0.78 & 0.63\\
\hline
NDE(0) & -0.07 & 0.56\\
\hline
NDE(1) & -0.55 & 0.09\\
\hline
\end{tabular}
\end{table}
\end{Schunk}
\hypertarget{a-mendelian-randomization-study-of-the-effect-of-homocysteine-on-cardiovascular-disease}{
\subsection{A Mendelian Randomization Study of the Effect of
Homocysteine on Cardiovascular
Disease}\label{a-mendelian-randomization-study-of-the-effect-of-homocysteine-on-cardiovascular-disease}}
Mendelian randomization \citep{mendelian} assumes certain genotypes may
serve as suitable instrumental variables for investigating the causal
effect of an associated phenotype on some disease outcome.
In \citep{st0232}, the authors investigate the effect of homocysteine on
cardiovascular disease using the 677CT polymorphism (rs1801133) in the
Methylenetetrahydrofolate Reductase gene as an instrument. They use
observational data from \citep{meleady_thermolabile_2003} in which the
outcome is binary, the treatment has been made binary by a suitably
chosen cut-off at \(15\mu mol/L\), and the instrument is ternary (this
polymorphism can take three possible genotype values).
With \(X\) denoting the treatment, \(Y\) the outcome and \(Z\) the
instrument, the conditional probabilities are given as follows.
\begin{Schunk}
\begin{Sinput}
params <- list(p00_0 = 0.83, p00_1 = 0.88, p00_2 = 0.72,
p10_0 = 0.11, p10_1 = 0.05, p10_2 = 0.20,
p01_0 = 0.05, p01_1 = 0.06, p01_2 = 0.05,
p11_0 = 0.01, p11_1 = 0.01, p11_2 = 0.03)
\end{Sinput}
\end{Schunk}
The computation using \CRANpkg{causaloptim} is done using the following
code.
\begin{Schunk}
\begin{Sinput}
# Input causal DAG
b <- graph_from_literal(Z -+ X, Ul -+ Z, X -+ Y, Ur -+ X, Ur -+ Y)
V(b)$leftside <- c(1, 0, 1, 0, 0)
V(b)$latent <- c(0, 0, 1, 0, 1)
V(b)$nvals <- c(3, 2, 2, 2, 2)
E(b)$rlconnect <- c(0, 0, 0, 0, 0)
E(b)$edge.monotone <- c(0, 0, 0, 0, 0)
# Construct causal problem
obj <- analyze_graph(b, constraints = NULL,
effectt = "p{Y(X = 1) = 1} - p{Y(X = 0) = 1}")
# Compute bounds on query
bounds <- optimize_effect_2(obj)
# Construct bounds as function of parameters
boundsfunction <- interpret_bounds(bounds = bounds$bounds,
parameters = obj$parameters)
# Insert observed conditional probabilities
numericbounds <- do.call(boundsfunction, as.list(params))
round(numericbounds, 2)
\end{Sinput}
\begin{Soutput}
#> lower upper
#> 1 -0.09 0.74
\end{Soutput}
\end{Schunk}
Our computed bounds agree with those computed using \CRANpkg{bpbounds}
as well as those estimated using Theorem 2 of \citep{richardson2014ace},
who independently derived expressions for tight bounds that are
applicable to this setting.
\hypertarget{summary-and-discussion}{
\section{Summary and Discussion}\label{summary-and-discussion}}
The methods and algorithms described in \citep{generalcausalbounds} to
compute symbolic expressions for bounds on non-identifiable causal
effects are implemented in the package \CRANpkg{causaloptim}. Our aim
was to provide a user-friendly interface to these methods with a
graphical interface to draw DAGs, specification of causal effects using
standard notation for potential outcomes, and an efficient
implementation of vertex enumeration to reduce computation times. These
methods are applicable to a wide variety of causal inference problems
which appear in biomedical research, economics, social sciences and
more. Aside from the graphical interface, programming with the package
is encouraged to promote reproducibility and advanced use. Our package
includes automated unit tests and also tests for correctness by
comparing the symbolic bounds derived using our program to independently
derived bounds in particular settings.
Our implementation uses a novel approach to draw DAGs using
\texttt{JavaScript} in a web browser that can then be passed to
\texttt{R} using \CRANpkg{shiny}. This graphical approach can be adapted
and used in other settings where graphs need to be specified and
computed on, such as other causal inference settings, networks, and
multi-state models. Other algorithms and data structures that could be
more broadly useful include the representation of structural equations
as \texttt{R} functions, recursive evaluation of response functions, and
parsing of string equations for causal effects and constraints.
\address{
Gustav Jonzon\\
Department of Medical Epidemiology and Biostatistics, Karolinska
Institutet\\%
\\
\url{https://ki.se/meb}\\%
\href{mailto:gustav.jonzon@ki.se}{\nolinkurl{gustav.jonzon@ki.se}}
}
\address{
Michael C Sachs\\
Department of Public Health, University of Copenhagen\\%
\\
\url{https://biostat.ku.dk/}\\%
\textit{ORCiD: \href{https://orcid.org/0000-0002-1279-8676}{0000-0002-1279-8676}}\\%
\href{mailto:michael.sachs@sund.ku.dk}{\nolinkurl{michael.sachs@sund.ku.dk}}
}
\address{
Erin E Gabriel\\
Department of Public Health, University of Copenhagen\\%
\\
\url{https://biostat.ku.dk/}\\%
\textit{ORCiD: \href{https://orcid.org/0000-0002-0504-8404}{0000-0002-0504-8404}}\\%
\href{mailto:erin.gabriel@sund.ku.dk}{\nolinkurl{erin.gabriel@sund.ku.dk}}
}
\end{article}
\end{document} |
\begin{document}
\title{A mixed complementarity problem approach for steady-state voltage and frequency stability analysis\\
\thanks{This material was based upon work supported by the U.S. Department of Energy, Office of Science, under Contract No. DE-AC02-06CH11357.}
}
\author{\IEEEauthorblockN{Youngdae Kim}
\IEEEauthorblockA{\textit{Mathematics and Computer Science Division} \\
\textit{Argonne National Laboratory}\\
Lemont, IL, USA\\
youngdae@anl.gov}
\and
\IEEEauthorblockN{Kibaek Kim}
\IEEEauthorblockA{\textit{Mathematics and Computer Science} \\
\textit{Argonne National Laboratory}\\
Lemont, IL, USA\\
kimk@anl.gov}
}
\maketitle
\begin{abstract}
We present a mixed complementarity problem (MCP) approach for a steady-state stability analysis of voltage and frequency of electrical grids.
We perform a theoretical analysis providing conditions for the global convergence and local quadratic convergence of our solution procedure, enabling fast computation time.
Moreover, algebraic equations for power flow, voltage control, and frequency control are compactly and incrementally formulated as a single MCP that subsequently is solved by a highly efficient and robust solution method.
Experimental results over large grids demonstrate that our approach is as fast as the existing Newton method with heuristics while showing more robust performance.
\end{abstract}
\begin{IEEEkeywords}
power flow, voltage and frequency regulation, mixed complementarity
\end{IEEEkeywords}
\section{Introduction}
\label{sec:intro}
The growing penetration of renewable energy resources into the electrical grid and the increasing interdependency between the natural gas network and the grid draw attention to the impact of these components on grid stability.
Unlike traditional generators, the intermittency and lack of reactive power generation capability of renewable energy resources can significantly degrade grid stability. Natural gas as the largest source of U.S. electricity generation~\cite{EIA21} also can have a huge impact on stable operations of the grid, as we have seen in the Texas power outage in the winter storm of early 2021.
One of the key measures for assessing grid stability is a steady-state power flow analysis with voltage regulation and frequency control.
It basically computes a solution to alternating current power flows of the grid formulated as a nonlinear system of equations that also satisfies several complicated conditions modeling the interactions between grid components to implement those regulations.
An example of such conditions is voltage set point control via reactive power, which regulates voltage magnitude to stay at its set point by controlling its corresponding reactive power.
Computationally, these additional conditions make it difficult to directly apply the the Newton--Raphson (NR) method~\cite{Tinney-Hart67}, which has been the norm for conventional power flow analysis without regulations.
Heuristics have been developed~\cite{Stott74,Zhao08} that fix some variable values and switch the bus type (e.g., PV-PQ switching) at each iteration of the NR method so that it can still be used to solve the reduced system of equations.
However, they can potentially cause numerical issues~\cite{Zhao08} leading to divergence.
The lack of theoretical analysis of conditions that guarantee the convergence of these heuristics further complicates the issue to avoid such divergence.
Recent works~\cite{Sundaresh14,Murray15,Tinoco16} introduce complementarity constraints
$0 \le a \perp b \ge 0$ (which denotes $ab=0,a,b \ge 0$)
to directly incorporate regulation conditions in the model.
They then transform the problem into a computationally more tractable form via equation reformulation using the Fischer--Burmeister function defined by $\phi(a,b)=\sqrt{a^2+b^2}-a-b$ so that $0 \le a \perp b \ge 0 \Leftrightarrow \phi(a,b)=0$.
In particular,~\cite{Murray15} showed global convergence results that lay grounds for a numerically more robust performance than that of the NR method.
The computation time of this approach, however, is usually much slower than that of the NR method with heuristics, as also shown in~\cite{Murray15,Tinoco16}.
The reason is partly that the solution procedure does not exploit complementarity structure since it is obscured by reformulations into equations, resulting in no special treatment possible for it.
In this paper we show that such problems can be cast as a mixed complementarity problem (\text{MCP}{}) with a proper formulation, as will be described in Section~\ref{sec:formulations}.
In particular, we can leverage the convergence theory and solution methods for \text{MCP}{}s to efficiently tackle our problem.
We show sufficient conditions for obtaining global convergence as well as local quadratic convergence that enable fast computation time.
Such fast convergence is made through the solution method for \text{MCP}{}~\cite{Dirkse95,Ferris99-2} that specifically utilizes the complementarity structure.
Similar to the NR method, at each iteration it linearizes the problem at the current iterate; however, instead of solving a system of equations, it solves a linear complementarity problem (LCP), which provides a first-order approximation to the original \text{MCP}{}.
Numerical experiments in Section~\ref{sec:exp} demonstrate that the computational time of our \text{MCP}{} approach is as fast as that of the NR method, while showing a more robust performance.
The rest of the paper is organized as follows.
In Section~\ref{sec:background} we briefly introduce \text{MCP}{} formulations, strong regularity, and their solution method.
Section~\ref{sec:formulations} introduces our \text{MCP}{} formulations for a steady-state power flow analysis with voltage and frequency regulations.
Numerical results are presented in Section~\ref{sec:exp}, and we conclude the paper in Section~\ref{sec:conclusion}.
\section{Background of mixed complementarity problems}
\label{sec:background}
In this section we briefly introduce \text{MCP}{}s, their solution method based on the generalized equation~\cite{Robinson79}, and conditions that guarantee local superlinear or quadratic convergence.
An $\text{MCP}{}(B,F)$ is defined by a vector-valued continuous function $F:\mathbf{R}^n \rightarrow \mathbf{R}^n$ and a box constraint $B=[l,u]=\prod_{i=1}^n[l_i,u_i]$ with $l_i \le u_i$ and $l_i, u_i \in \mathbf{R} \cup \{-\infty,+\infty\}$ for $i=1,\dots,n$.
We note that $F$ is a square system of equations; that is, it has the same number of variables and equations.
We say that $x$ is a solution to the $\text{MCP}{}(B,F)$ if it satisfies one of the following three conditions for each $i=1,\dots,n$:
\begin{equation}
\begin{aligned}
x_i = l_i \quad \& \quad F_i(x) \ge 0\\
l_i \le x_i \le u_i \quad \& \quad F_i(x)=0\\
x_i=u_i \quad \& \quad F_i(x) \le 0 .
\end{aligned}
\label{eq:mcp}
\end{equation}
We use a complementarity notation $l_i \le x_i \le u_i \perp F_i(x)$ to denote~\eqref{eq:mcp}.
Its vector form $l \le x \le u \perp F(x)$ implies that~\eqref{eq:mcp} holds componentwise.\footnote{As a variation, we may use the notation $l_i \le x_i \perp F_i(x) \ge 0$ when $u_i=\infty$. This emphasizes that $F_i(x)$ should be nonnegative at a solution since the third condition of~\eqref{eq:mcp} cannot hold in this case. Similarly, we may denote $x_i \le u_i \perp F_i(x) \le 0$ when $l_i=-\infty$. If $l_i=-\infty$ and $u_i=\infty$, we may omit both bounds and simply denote $x_i \perp F_i(x)$.}
MCPs subsume many different problem classes.
When $B=\mathbf{R}^n$, the $\text{MCP}(B,F)$ becomes a square system of nonlinear equations seeking $x \in \mathbf{R}^n$ that satisfies $F(x)=0$.
The Karush--Kuhn--Tucker conditions of an optimization problem $\min_{l \le x \le u}\; f(x) \; \text{s.t.}\; c(x)=0$ can be formulated as an $\text{MCP}(B,F)$ with $B=[l,u] \times \mathbf{R}^m$ and $F(x,\lambda)=((\nabla f(x)+\nabla c(x)\lambda)^T, (c(x))^T)^T \in \mathbf{R}^{n+m}$, where $c(x),\lambda \in \mathbf{R}^m$.
Many other different problem classes, including generalized Nash equilibrium problems and quasi-variational inequalities, also can be formulated as MCPs.
We refer to~\cite{Kim18,Kim19-MPC} for details.
The generalized equation (GE) provides a machinery to solve MCPs~\cite{Dirkse95,Josephy79}.
It is defined by
\begin{equation}
0 \in F(x) + N_B(x),
\label{eq:ge}
\end{equation}
where $N_B(x)$ is a normal map to $B$ at $x \in B$ defined by $N_B(x):=\{v \mid \langle v, y-x \rangle \le 0, \forall y \in B\}$ and $N_B(x):=\emptyset$ when $x \notin B$.
One can easily verify that $x$ satisfies~\eqref{eq:ge} if and only if it satisfies~\eqref{eq:mcp}.
A Newton's method~\cite{Dirkse95,Josephy79} finds a solution to the GE by iteratively linearizing and solving the linearized GE, where it becomes an LCP in this case with $F$ being an affine function.
This method is similar to the NR method for a nonlinear system of equations. However, it takes into account complementarity at its linearization, and its complementary pivoting solution procedure effectively exploits the complementarity structure in finding a solution.
We will see its fast and robust computational performance in Section~\ref{sec:exp}.
A key condition to define the domain of attraction for local convergence to a solution $x^*$ of~\eqref{eq:ge} is strong regularity, defined below:
\begin{definition}[Strong regularity~\cite{Robinson80}]
Let $x^*$ be a solution to~\eqref{eq:ge}.
Let $T:=LF_{x^*} + N_B$, where $LF_{x^*}:=F(x^*)+\nabla F(x^*)(x-x^*)$.
We say that $x^*$ is strongly regular if and only if there exist neighborhoods $U$ of the origin and $V$ of $x^*$ such that the restriction to $U$ of $T^{-1}\cap V$ is a single-valued Lipschitzian function from $U$ to $V$.
\label{def:strong-regularity}
\end{definition}
We note that for a system of nonlinear equations $F(x)=0$ strong regularity reduces to the condition that $\nabla F(x^*)$ has a continuous linear inverse map.
This condition is a typical assumption for the local convergence of the NR method.
In Section~\ref{sec:formulations} we present conditions for local convergence of our power flow analysis with voltage and frequency regulations via MCPs.
The following shows an asymptotic convergence rate of the Newton's method~\cite{Dirkse95,Josephy79} for the GE under strong regularity.
\begin{theorem}[Local $Q$-quadratic convergence~\cite{Josephy79}]
For a given $\text{MCP}(B,F)$ with $B \neq \emptyset$, suppose that $x^*$ is a strongly regular solution of it and that $\nabla F$ is locally Lipschitz continuous.
Then for $x^k$ near $x^*$, the sequence generated by solving LCPs is locally Q-quadratically convergent to $x^*$.
\label{thm:local-convergence}
\end{theorem}
\section{Mixed complementarity formulations for a unified power flow analysis}
\label{sec:formulations}
This section presents \text{MCP}{} formulations that encapsulate power flow equations, voltage regulation, and frequency control in an incremental fashion.
Starting with power flow equations in Section~\ref{subsec:power-flow}, we incrementally build \text{MCP}{} formulations in Sections~\ref{subsec:voltage-control}--\ref{subsec:frequency-control}, each of which models a specific regulation and control feature.
At the end of this section, we describe the global and local convergence results.
We note that complementarity constraints for voltage and frequency regulations are known in the literature, but our work is the first showing that these also can be compactly formulated as a single \text{MCP}{}, enabling us to utilize its convergence theory and solution method.
\subsection{Modeling power flow equations}
\label{subsec:power-flow}
\begin{table}[t]
\caption{Variables fixed/free at each bus $i$ according to its type}
\label{tbl:fix-vars}
\centering
\begin{tabular}{|c|c|c|}
\hline
Bus type & Variables fixed & Variables free \\\hline
Slack & $v_i,\delta_i$ & $p_{g_i},q_{g_i}$\\
PQ bus & $p_{g_i},q_{g_i}$ & $v_i,\delta_i$\\
PV bus & $p_{g_i},v_i$ & $\delta_i,q_{g_i}$\\
\hline
\end{tabular}
\end{table}
In conventional power flow analysis, we are interested in finding voltage magnitudes and angles of the grid for a given set of parameters such as load, real power generation, voltage magnitudes at regulated buses, and network coefficients.
To achieve this, we solve the following system of nonlinear equations defined at each bus $i=1,\dots,N$:
\begin{equation}
\begin{aligned}
P_i(x) &= p^{\text{inj}}_i - \sum_{k=1}^N v_iv_k(G_{ik}\cos(\delta_{ik})+B_{ik}\sin(\delta_{ik}))\\
Q_i(x) &= q^{\text{inj}}_i - \sum_{k=1}^N v_iv_k(G_{ik}\sin(\delta_{ik})-B_{ik}\cos(\delta_{ik})),
\end{aligned}
\label{eq:pf-def}
\end{equation}
where $p^{\text{inj}}_i := p_{g_i} - P_{d_i}$ and $q^{\text{inj}}_i := q_{g_i} - Q_{d_i}$ are the net real and reactive power injections at bus $i$ with $p_{g_i}$ and $q_{g_i}$ being real and reactive power generated at the bus and $P_{d_i}$ and $Q_{d_i}$ denoting its real and reactive loads.
Voltage magnitude and angle at bus $i$ are denoted by $v_i$ and $\delta_i$, respectively, and the angle difference is defined by $\delta_{ik} := \delta_i - \delta_k$.
We denote by $x:=([p_{g_i},q_{g_i},v_i,\delta_i]_{i=1}^N)$ the encapsulation of all variables.
$G_{ik}$ and $B_{ik}$ are network coefficients computed from a nodal admittance matrix and bus shunt values.
Later in Section~\ref{subsec:transfomer-switched-shunt} they may be changed to variables for voltage regulation.
When solving~\eqref{eq:pf-def}, some components of variable $x$ are fixed depending on the bus type as described in Table~\ref{tbl:fix-vars}.
Since we are interested mainly in voltage magnitudes and angles, we solve for equations that involve these voltage values as variables.
These equations correspond to the real and reactive power flow equations at PQ buses and real power flow equations at PV buses.
Therefore the conventional power flow analysis solves the following square system of nonlinear equations:
\begin{equation}
\begin{bmatrix}
P_{\text{PQ}}(x)\\
Q_{\text{PQ}}(x)\\
P_{\text{PV}}(x)
\end{bmatrix} = 0,
\label{eq:pf-eq}
\end{equation}
where we use the notation $\text{PQ}$ or $\text{PV}$ in the subscript to indicate the set of bus indices belonging to the indicated bus type.
We note that variables in~\eqref{eq:pf-eq} are $\delta_{\text{PQ}}, v_{\text{PQ}}$, and $\delta_{\text{PV}}$.
Using the fact that variables are set to be free variables in~\eqref{eq:pf-eq}, we can formulate~\eqref{eq:pf-eq} as an \text{MCP}{} as follows:
\begin{equation}
\begin{aligned}
\delta_{\text{PQ}} &&\perp&& P_{\text{PQ}}(x)\\
v_{\text{PQ}} &&\perp&& Q_{\text{PQ}}(x)\\
\delta_{\text{PV}} &&\perp&& P_{\text{PV}}(x)\\.
\end{aligned}
\label{eq:mcp-pf}
\end{equation}
From~\eqref{eq:mcp}, one can easily verify that $x$ solves~\eqref{eq:pf-eq} if and only if it solves~\eqref{eq:mcp-pf}.
\subsection{Modeling generator voltage control}
\label{subsec:voltage-control}
Generator voltage control can be compactly formulated by using the following MCP formulation:
\begin{equation}
\begin{aligned}
q^{\min}_{g_{\text{PV}}} \le q_{g_{\text{PV}}} \le q^{\max}_{g_{\text{PV}}} &&\perp&& v_{\text{PV}} - v^{\text{sp}}_{\text{PV}},
\end{aligned}
\label{eq:mcp-gen-vol}
\end{equation}
where $v^{\text{sp}}_i$ denotes the set point of the voltage magnitude for bus $i \in \text{PV}$.
From~\eqref{eq:mcp}, we see that $v_i=v^{\text{sp}}_i$ when $q^{\min}_{g_i} < q_{g_i} < q^{\max}_{g_i}$, $v_i \ge v^{\text{sp}}_i$ for $q_{g_i}=q^{\min}_{g_i}$, and $v_i \le v^{\text{sp}}_i$ for $q_{g_i}=q^{\max}_{g_i}$.
Therefore~\eqref{eq:mcp-gen-vol} formulates the desired behavior for generator voltage control.
In contrast to the conventional power flow analysis, we note that $v_i, i \in \text{PV}$ in~\eqref{eq:mcp-gen-vol} is not fixed to its set point.
Since both $v_i$ and $q_{g_i}$ are variables for $i \in \text{PV}$, the reactive power flow equations at PV buses should be added in addition to the complementarity constraints~\eqref{eq:mcp-gen-vol}.
Together with~\eqref{eq:mcp-pf}, the following incrementally formulates a power flow analysis with generator voltage control as an \text{MCP}{}:
\begin{equation}
\begin{aligned}
v_{\text{PV}} &&\perp&& Q_{\text{PV}}(x)\\
q^{\min}_{g_{\text{PV}}} \le q_{g_{\text{PV}}} \le q^{\max}_{g_{\text{PV}}} &&\perp&& v_{\text{PV}} - v^{\text{sp}}_{\text{PV}}.
\end{aligned}
\label{eq:mcp-gen-vol-full}
\end{equation}
\subsection{Modeling tap-charging transformer and switched shunt voltage control}
\label{subsec:transfomer-switched-shunt}
These additional controllers are used to regulate the bounds on voltage magnitudes as defined below using \text{MCP}{} formulations.
We note that a similar formulation has been given in~\cite{Tinney-Hart67}.
\begin{equation}
\begin{aligned}
0 \le u^+_{\text{PV}} &&\perp&& v_{\text{PV}} + v^-_{\text{PV}} - v^{\min}_{\text{PV}} \ge 0\\
0 \le u^-_{\text{PV}} &&\perp&& v^{\max}_{\text{PV}} - v_{\text{PV}} + v^+_{\text{PV}} \ge 0\\
0 \le v^+_{\text{PV}} &&\perp&& u_{\text{PV}} - u^{\min}_{\text{PV}} \ge 0\\
0 \le v^-_{\text{PV}} &&\perp&& u^{\max}_{\text{PV}} - u_{\text{PV}} \ge 0\\
u_{\text{PV}} &&\perp&& u_{\text{PV}} - u^{\text{sp}}_{\text{PV}} - u^+_{\text{PV}} + u^-_{\text{PV}}
\end{aligned}
\label{eq:mcp-tf-ss-vol}
\end{equation}
The first two conditions allow an increase (or a decrease) of a tap ratio $u_{\text{PV}}$ from its set point $u^{\text{sp}}_{\text{PV}}$, when a voltage magnitude reaches its limit.
The third and fourth conditions together with the first two conditions capture a further drift from voltage magnitude limits when the corresponding tap ratio reaches its limit.
For switched shunt devices, we can employ similar \text{MCP}{} formulations.
We note that tap ratios and susceptances in general take discrete values—choosing among finite values—but we instead treat them as continuous variables in~\eqref{eq:mcp-tf-ss-vol}.
This is a common approach as in \cite{Stott74} and still provides useful values for finding a discrete solution~\cite{Tinoco16,Stott74,Chang88}.
In addition to the generator voltage control by~\eqref{eq:mcp-gen-vol-full}, the formulation~\eqref{eq:mcp-tf-ss-vol} can be easily concatenated to~\eqref{eq:mcp-gen-vol-full} to include in the model the effect of tap-changing transformer and switched shunt devices for voltage control as well.
This provides a great flexibility in modeling the impact of various regulation components in power flow analysis via MCPs.
In Section 4.3, we present a modeling example of incrementally including voltage regulating components to regulate the voltage magnitude.
\subsection{Modeling primary frequency control}
\label{subsec:frequency-control}
Primary frequency control is the governor's reaction to real power imbalance and can be formulated by using an \text{MCP}{} formulation as described in~\eqref{eq:mcp-freq}.
When $p^{\min}_{g_i} < p_{g_i} < p^{\max}_{g_i}$, we have $p_{g_i} = p^{\text{sp}}_{g_i} + \nu_{g_i}\Delta f$ following the linear generation characteristic.
If $p_{g_i}=p^{\max}_{g_i}$, then it stays at its limit; but the frequency could further drop, resulting in $p_{g_i} < p^{\text{sp}}_{g_i} + \nu_{g_i}\Delta f$.
Similarly, we could have $p_{g_i} > p^{\text{sp}}_{g_i} + \nu_{g_i}\Delta f$ when $p_{g_i}=p^{\min}_{g_i}$.
In~\eqref{eq:mcp-freq}, we note that the real power flow equation of the slack bus is added to the formulation for the synchronized generation.
\begin{equation}
\begin{aligned}
\Delta f &&\perp&& P_{\text{slack}}(x)\\
p^{\min}_{g_i} \le p_{g_i} \le p^{\max}_{g_i} &&\perp&& p_{g_i} - (p^{\text{sp}}_{g_i} + \nu_{g_i} \Delta f)\\
&&&&\forall i \in \text{PV} \cup \text{Slack}
\end{aligned}
\label{eq:mcp-freq}
\end{equation}
By combining~\eqref{eq:mcp-freq} with voltage regulation formulations in Sections~\ref{subsec:voltage-control} and~\ref{subsec:transfomer-switched-shunt}, we can perform a power flow analysis with voltage and frequency regulations simultaneously using a single \text{MCP}{} model.
\subsection{Convergence results}
The following proposition shows sufficient conditions for strong regularity of a solution to an \text{MCP}{}.
The conditions can be applied to any \text{MCP}{}s defined in the preceding sections.
By applying Theorem~\ref{thm:local-convergence} with strong regularity, we obtain a local $Q$-quadratic convergence result.
\begin{proposition}[Theorem 3.1 in~\cite{Robinson80}]
Let $x^*$ be a solution to an $\text{MCP}(B,F)$.
Let $\alpha:=\{i \mid x^{\min}_i < x^*_i < x^{\max}_i, F_i(x^*)=0\}, \beta:=\{i \mid x^*_i \in \{x^{\min}_i,x^{\max}_i\}, F_i(x^*)=0\}$, and $\gamma := \{i \mid x^*_i \in \{x^{\min}_i,x^{\max}_i\}, F_i(x^*)\neq 0\}$.
If the submatrix $\nabla F(x^*)_{\alpha,\alpha}$ is nonsingular and the Schur complement $\nabla F(x^*)_{\alpha\cup\beta,\alpha\cup\beta}/\nabla F(x^*)_{\alpha,\alpha}:=\nabla F(x^*)_{\beta,\beta} - \nabla F(x^*)_{\beta,\alpha}\nabla F(x^*)_{\alpha,\alpha}^{-1}\nabla F(x^*)_{\alpha,\beta}$ has positive principal minors (a $P$-matrix), then $x^*$ is a strongly regular solution.
\label{prop:local-conv}
\end{proposition}
We note that when $\beta = \emptyset$ in Proposition~\ref{prop:local-conv}, the conditions correspond to having a continuous linear inverse mapping for the reduced space defined by $\alpha$.
Global convergence results are obtained by assuming a sequence having an accumulation point that converges to a strongly regular solution.
See~\cite[Theorem 5]{Ferris99-2} for details.
\section{Experimental results}
\label{sec:exp}
In this section we present numerical results of our \text{MCP}{} approach and compare its performance with those of the NR method with switching heuristics and the FB-based optimization approach~\cite{Murray15}.
Experiments were performed over large grids included in the MATPOWER package~\cite{Zimmerman11} on a Mac machine having Intel 6-Core i7@2.6 GHz and 32 GB of memory.
The PATH and Ipopt were used for solving \text{MCP}{} and the FB-based problem, respectively.
\subsection{Generator voltage control}
\label{subsec:exp-gen-vol}
In Table~\ref{tbl:exp-gen-vol} we demonstrate the computational performance of each method for voltage control via reactive power.
For \text{MCP}{} we solve the \text{MCP}{} defined by~\eqref{eq:mcp-pf} and~\eqref{eq:mcp-gen-vol-full} in this case.
Our \text{MCP}{} method showed the most robust performance as solving all of the problems, whereas the NR method failed the convergence on 3120sp, and the FB method could not find a solution for ACTIVSg70k.
Also, the \text{MCP}{} method demonstrated computation time as fast as that of the NR method, while the FB method showed a much slower computation time.
\begin{table*}[!t]
\centering
\caption{Generator voltage control}
\label{tbl:exp-gen}
\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|}
\hline
\multicolumn{1}{|c|}{\multirow{3}{*}{Data}} & \multicolumn{3}{c|}{NR method} & \multicolumn{3}{c|}{\text{MCP}{}} & \multicolumn{3}{c|}{FB method}\\\cline{2-10}
& \multicolumn{1}{c|}{\multirow{2}{*}{Iter}} & \multicolumn{1}{c|}{Time} & \multicolumn{1}{c|}{$\max$}
& \multicolumn{1}{c|}{\multirow{2}{*}{Iter}} & \multicolumn{1}{c|}{Time} & \multicolumn{1}{c|}{$\max$}
& \multicolumn{1}{c|}{\multirow{2}{*}{Iter}} & \multicolumn{1}{c|}{Time} & \multicolumn{1}{c|}{$\max$}\\
& & (secs) & $|v-v^{\text{sp}}|$ & & (secs) & $|v-v^{\text{sp}}|$ & & (secs) & $|v-v^{\text{sp}}|$\\\hline
1354pegase & 4 & 0.06 & 2.64e-02 & 4 & 0.05 & 2.64e-02 & 22 & 0.82 & 2.64e-02\\
2869pegase & 6 & 0.18 & 1.82e-02 & 6 & 0.29 & 1.64e-02 & 24 & 2.21 & 1.65e-02\\
3120sp & f & n/a & n/a & 6 & 0.15 & 6.95e-02 & 925 & 82.15 & 6.95e-02\\
6468rte & 5 & 0.26 & 2.79e-02 & 4 & 0.28 & 2.78e-02 & 301 & 66.83 & 2.79e-02\\
9241pegase & 6 & 0.52 & 2.47e-02 & 7 & 0.92 & 2.46e-02 & 74 & 36.95 & 2.47e-02\\
13659pegase & 5 & 0.63 & 5.66e-02 & 5 & 0.93 & 5.02e-03 & 35 & 23.59 & 5.03e-03\\
ACTIVSg10k & 5 & 0.36 & 4.07e-05 & 3 & 1.40 & 4.06e-05 & 37 & 10.95 & 1.67e-04\\
ACTIVSg25k & 4 & 0.80 & 5.83e-04 & 5 & 1.37 & 5.82e-04 & 253 & 263.05 & 7.31e-04\\
ACTIVSg70k & 5 & 3.12 & 1.04e-03 & 5 & 3.55 & 1.03e-03 & f & n/a & n/a\\\hline
\end{tabular}
\label{tbl:exp-gen-vol}
\end{table*}
We found for the 3120sp data that the solutions from the FB method and our \text{MCP}{} method violate some of the voltage magnitude bounds.
Since reactive power cannot control such violations once it reaches its limit, we need another controller such as tap ratios and switched shunt devices, as described in Section~\ref{subsec:transfomer-switched-shunt}, in order to guarantee the feasibility.
This will be addressed in the following section.
\subsection{Voltage control using transformer tap ratios and switched shunt devices}
\label{subsec:exp-tt-ss-vol}
To control the problematic voltage magnitudes of the 3120sp data to be within their bounds, we incorporated~\eqref{eq:mcp-tf-ss-vol} into our \text{MCP}{} problem (so we solved~\eqref{eq:mcp-pf}+\eqref{eq:mcp-gen-vol-full}+\eqref{eq:mcp-tf-ss-vol}) by allowing some buses to change their tap ratios and switched shunt devices.
Figure~\ref{fig:exp-tr-ss-vol} demonstrates how voltage bound violations were decreasing and eventually removed as we increased the allowed bounds on the tap ratios and switched shunt devices.
These results demonstrate the capability of our method to easily simulate the effect of several voltage controllers in an incremental fashion.
\begin{figure}
\caption{Using transformer tap ratios only}
\label{subfig:exp-tap}
\caption{Using both tap ratios and switched shunts}
\label{subfig:exp-sshunt}
\caption{Voltage range regulation using transformers and shunt devices over 3120sp}
\label{fig:exp-tr-ss-vol}
\end{figure}
\subsection{Frequency and voltage regulations.}
\label{subsec:exp-freq-vol}
In Table~\ref{tbl:exp-freq}, we report the numerical results from our method for the MCP problem \eqref{eq:mcp-pf}+\eqref{eq:mcp-gen-vol-full}+\eqref{eq:mcp-freq} with frequency and voltage controls on ACTIVSg25k data.
For frequency control we gradually increased real power generation loss by turning off up to 5 generators that had the most real power generation among other generators (i.e., IDs 1557, 2345, 2346, 235, and 2015 in the order of removed).
Those generators were chosen from the solution of generator voltage control we performed in Section~\ref{subsec:exp-gen-vol}, and we warm-started from that solution to solve the \text{MCP}{}.
To the best of our knowledge, only one recent paper~\cite{sanchez2020integrated} uses NR-based heuristics for solving the problem without any convergence guarantee. However, we do not consider NR method because of the lack of convergence guarantee.
In all cases, the \text{MCP}{} was able to quickly find a solution containing frequency changes while controlling the voltage magnitudes to stay at their set points as much as possible and be within their bounds as well.
\begin{table}[tbhp]
\centering
\caption{Frequency and voltage regulations over ACTIVSg25k}
\label{tbl:exp-freq}
\begin{tabular}{|c|c|c|c|}
\hline
Generation loss & Frequency & $\max |v-v^{\text{sp}}|$ & Time (secs)\\\hline
1,299 MW & 59.92 Hz & 3.41e-03 & 1.27\\
2,597 MW & 59.84 Hz & 9.50e-03 & 1.26\\
3,895 MW & 59.76 Hz & 1.96e-02 & 1.28\\
5,185 MW & 59.68 Hz & 2.80e-02 & 1.56\\
6,455 MW & 59.62 Hz & 2.90e-02 & 1.77\\\hline
\end{tabular}
\label{tbl:exp-freq}
\end{table}
\section{Conclusion}
\label{sec:conclusion}
We presented a mixed complementarity problem approach for a steady-state power flow analysis with voltage regulation and frequency control. The key advantage of our approach is leveraging the existing algorithms with the theoretical support for the global convergence with quadratic local convergence rate, which guarantees numerical stability and fast computation.
We demonstrated the greater computational performance of our approach, as compared with the existing NR method and FB method, by using large MATPOWER test instances.
\iffalse
\section*{Acknowledgment}
The preferred spelling of the word ``acknowledgment'' in America is without
an ``e'' after the ``g''. Avoid the stilted expression ``one of us (R. B.
G.) thanks $\ldots$''. Instead, try ``R. B. G. thanks$\ldots$''. Put sponsor
acknowledgments in the unnumbered footnote on the first page.
\fi
{\footnotesize
\noindent\fbox{\parbox{0.47\textwidth}{
The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory (``Argonne''). Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE-AC02-06CH11357. The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).}
}
}
\end{document} |
\begin{document}
\title{Linear chord diagrams with long chords}
\begin{abstract}
A linear chord diagram of size $n$ is a partition of the set $\{1,2,\cdots,2n\}$ into sets of size two, called chords.
From a table showing the number of linear chord diagrams of degree $n$ such that every chord has length at least $k$,
we observe that if we proceed far enough along the diagonals, they are given by a geometric sequence.
We prove that this holds for all diagonals, and identify when the effect starts.
\end{abstract}
\section{Introduction}
A linear chord diagram is a matching of $\{1,2,\cdots,2n\}$.
Chord diagrams arise in many different contexts, from the study of RNA ~\cite{Reidys11} to knot theory ~\cite{Chmutov12}. In combinatorics, chord diagrams show up in the m\'{e}nage problem ~\cite{Lucas91}, partitions ~\cite{Hsieh73}, and interval orders ~\cite{Winkler90}.
This paper will address diagrams where there is a specified minimum length for each chord.
From a table counting the number of such diagrams for $n$ and $k$, we observe that if we proceed far enough along the diagonals, they are given by a geometric sequence.
We prove that this holds for all diagonals, and identify when the effect starts.
\section{Statement of Result}
A \textit{linear chord diagram} of \textit{size} $n$ is a partition of the set $\{1,2,\cdots,2n\}$ into parts of size 2.
We can draw linear chord diagrams with arcs connecting the partition blocks.
\begin{center}
\begin{tikzpicture}[scale=.45]
\drawDiagram{6}{1/3,2/6,4/5}
\end{tikzpicture}
\end{center}
If $c = \{s_{c},e_{c}\}$ where $s_{c} < e_{c}$ is a block of a linear chord diagram.
We say that $s_{c}$ is the \textit{start point} of $c$ and $e_{c}$ is the \textit{end point}.
Then \textit{length} of $c$ is $e_{c} - s_{c}$.
We say that a chord $c$ \textit{covers} $i$ if $s_{c} < i < e_{c}$.
We say that a chord $c$ \textit{covers} a chord $d$ if it covers $s_{d}$ and $e_{d}$.
\begin{definition}
Let $D_{n}$ denote the set of all linear chord diagrams with $n$ chords.
Let $\minimalClass{k}{}$ denote the class of all linear chord diagrams such that every chord has length at least k.
Let $\minimalClass{k}{n}$ denote the set of all linear chord diagrams with $n$ such that every chord has length at least k.
\end{definition}
Table \ref{tab:data1} shows the sizes of $\minimalClass{k}{n}$ for various $n$ and $k$.
If $k$ is fixed, $\minimalClass{n}{k}$ can be computed using on the order of $2^{k}n^{2}$ arithmetic operations.
$a_{n} = \abs{\minimalClass{n}{2}}$ and $b_{n} = \abs{\minimalClass{n}{3}}$ can be computed using linear recurrences:
\begin{align*}
a_{n} & = (2n-1)a_{n-1} + a_{n-2} \\
b_{n} & = (2n+2)b_{n-1} - (6n-10)b_{n-2} + (6n-16)b_{n-3} - (2n-8)b_{n-4} - b_{n-5}.
\end{align*}
The recurrence for $\abs{\minimalClass{n}{2}}$, can be found in ~\cite{Hazewinkel95}; the recurrence for $\abs{\minimalClass{n}{3}}$ is new.
Conjecturally, there are linear recurrences for every $\minimalClass{n}{k}$ where $k$ is fixed: We will address these matters elsewhere.
{
\begin{table}
\caption {Counting chord diagram with long chords} \label{tab:data1}
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$n$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ \hline
$\abs{\minimalClass{n}{1}}$ & 1 & 3 & 15 & 105 & 945 & 10395 & 135135 & 2027025 & 34459425 & 654729075 & 13749310575 \\ \hline
$\abs{\minimalClass{n}{2}}$ & 0 & 1 & \cellcolor{black!20!white} 5 & 36 & 329 & 3655 & 47844 & 721315 & 12310199 & 234615096 & 4939227215 \\ \hline
$\abs{\minimalClass{n}{3}}$ & 0 & 0 & 1 & \cellcolor{black!20!white} 10 & 99 & 1146 & 15422 & 237135 & 4106680 & 79154927 & 1681383864 \\ \hline
$\abs{\minimalClass{n}{4}}$ & 0 & 0 & 0 & 1 & \cellcolor{black!20!white} 20 & \cellcolor{black!20!white} 292 & 4317 & 69862 & 1251584 & 24728326 & 535333713 \\ \hline
$\abs{\minimalClass{n}{5}}$ & 0 & 0 & 0 & 0 & 1 & \cellcolor{black!20!white} 40 & \cellcolor{black!20!white} 876 & 16924 & 332507 & 6944594 & 156127796 \\ \hline
$\abs{\minimalClass{n}{6}}$ & 0 & 0 & 0 & 0 & 0 & 1 & \cellcolor{black!20!white} 80 & \cellcolor{black!20!white} 2628 & \cellcolor{black!20!white} 67404 & 1627252 & 39892549 \\ \hline
$\abs{\minimalClass{n}{7}}$ & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \cellcolor{black!20!white} 160 & \cellcolor{black!20!white} 7884 & \cellcolor{black!20!white} 269616 & 8075052 \\ \hline
$\abs{\minimalClass{n}{8}}$ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \cellcolor{black!20!white} 320 & \cellcolor{black!20!white} 23652 & \cellcolor{black!20!white} 1078464 \\ \hline
$\abs{\minimalClass{n}{9}}$ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \cellcolor{black!20!white} 640 & \cellcolor{black!20!white} 70956 \\ \hline
$\abs{\minimalClass{n}{10}}$ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \cellcolor{black!20!white} 1280 \\ \hline
$\abs{\minimalClass{n}{11}}$ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \hline
\end{tabular}
}
The first four rows can be found in the OEIS under the identification numbers A001147, A000806, A190823, and A190824, respectively.
\end{table}
}
Here we address the diagonals of the table. The shaded squares highlight a pattern.
The number in the square one below and one to the right, is exactly $(n-k+1)$ our current square.
This pattern holds for all such squares,
\begin{theorem}\label{thm:main}
Let $n$ and $k$ be positive integers such that $n \geq 3(n - k)$ and $n \geq k$.
Then $\abs{\minimalClass{k+1}{n+1}} = (n-k+1)\abs{\minimalClass{k}{n}}$.
\end{theorem}
\section{Outline of the proof}
We consider each diagonal separately.
We refer to the $i^{\text{th}}$ diagonal as all the entries such that $(n - k + 1) = i$.
For any entry $\minimalClass{k}{n}$ the $i^{th}$ diagonal we create $(n - k + 1)$ functions $\alpha_{n,k,j}$ ($j \in \{0,\cdots,n-k\}$) which are injective into $\minimalClass{k+1}{n+1}$.
We show that the images of these functions are disjoint and cover $\minimalClass{k+1}{n+1}$.
And so there are $(n-k+1)$-times as many elements in $\minimalClass{k+1}{n+1}$ as there are in $\minimalClass{k}{n}$.
To create the bijection $\alpha_{n,k,j}$ we consider the middle $2(n-k)$ indices.
Here is an example from an element of $\minimalClass{4}{6}$
\begin{center}
\begin{tikzpicture}[scale=.45]
\draw (0,0) -- (11,0);
\draw(0,0) -- (2,2) -- (4,0);
\draw(1,0) -- (5,4) -- (9,0);
\draw(2,0) -- (5,3) -- (8,0);
\draw(3,0) -- (5,2) -- (7,0);
\draw(5,0) -- (7.5,2.5) -- (10,0);
\draw(6,0) -- (8.5,2.5) -- (11,0);
\node [below] at (0,0) {1};
\node [below] at (1,0) {2};
\node [below] at (2,0) {3};
\node [below] at (3,0) {4};
\node [below] at (4,0) {5};
\node [below] at (5,0) {6};
\node [below] at (6,0) {7};
\node [below] at (7,0) {8};
\node [below] at (8,0) {9};
\node [below] at (9,0) {10};
\node [below] at (10,0) {11};
\node [below] at (11,0) {12};
\draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0pt,yshift=0pt]
(3.8,-1) -- (7.2,-1) node [black,midway,yshift=-.5cm]
{};
\end{tikzpicture}
\end{center}
Any chords starting or ending in the middle indices are highlighted
\begin{center}
\begin{tikzpicture}[scale=.45]
\draw (0,0) -- (11,0);
\draw[very thick](0,0) -- (2,2) -- (4,0);
\draw(1,0) -- (5,4) -- (9,0);
\draw(2,0) -- (5,3) -- (8,0);
\draw[very thick](3,0) -- (5,2) -- (7,0);
\draw[very thick](5,0) -- (7.5,2.5) -- (10,0);
\draw[very thick](6,0) -- (8.5,2.5) -- (11,0);
\node [below] at (0,0) {1};
\node [below] at (1,0) {2};
\node [below] at (2,0) {3};
\node [below] at (3,0) {4};
\node [below] at (4,0) {5};
\node [below] at (5,0) {6};
\node [below] at (6,0) {7};
\node [below] at (7,0) {8};
\node [below] at (8,0) {9};
\node [below] at (9,0) {10};
\node [below] at (10,0) {11};
\node [below] at (11,0) {12};
\draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0pt,yshift=0pt]
(3.8,-1) -- (7.2,-1) node [black,midway,yshift=-.5cm]
{};
\end{tikzpicture}
\end{center}
A new chord is inserted covering only the indices in the middle
\begin{center}
\begin{tikzpicture}[scale=.45]
\draw (0,0) -- (13,0);
\draw[very thick](0,0) -- (2.5,2.5) -- (5,0);
\draw(1,0) -- (6,5) -- (11,0);
\draw(2,0) -- (6,4) -- (10,0);
\draw[very thick](3,0) -- (5.5,2.5) -- (8,0);
\draw[gray, dashed](4,0) -- (6.5,2.5) -- (9,0);
\draw[very thick](6,0) -- (9,3) -- (12,0);
\draw[very thick](7,0) -- (10,3) -- (13,0);
\node [below] at (0,0) {1};
\node [below] at (1,0) {2};
\node [below] at (2,0) {3};
\node [below] at (3,0) {4};
\node [below] at (4,0) {5};
\node [below] at (5,0) {6};
\node [below] at (6,0) {7};
\node [below] at (7,0) {8};
\node [below] at (8,0) {9};
\node [below] at (9,0) {10};
\node [below] at (10,0) {11};
\node [below] at (11,0) {12};
\node [below] at (12,0) {13};
\node [below] at (13,0) {14};
\draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0pt,yshift=0pt]
(3.8,-1) -- (9.2,-1) node [black,midway,yshift=-.5cm]
{};
\end{tikzpicture}
\end{center}
The new chord then has its start point iteratively swapped with the starting points of the unbolded cords, starting with the one that started last and stopping when there are $j$ unswapped unbolded chords.
\begin{center}
\begin{tikzpicture}[scale=0.45]
\begin{scope}[shift={(0,-0.5)}]
\draw (0,0) -- (11,0);
\draw[very thick](0,0) -- (2,2) -- (4,0);
\draw(1,0) -- (5,4) -- (9,0);
\draw(2,0) -- (5,3) -- (8,0);
\draw[very thick](3,0) -- (5,2) -- (7,0);
\draw[very thick](5,0) -- (7.5,2.5) -- (10,0);
\draw[very thick](6,0) -- (8.5,2.5) -- (11,0);
\node [below] at (0,0) {1};
\node [below] at (1,0) {2};
\node [below] at (2,0) {3};
\node [below] at (3,0) {4};
\node [below] at (4,0) {5};
\node [below] at (5,0) {6};
\node [below] at (6,0) {7};
\node [below] at (7,0) {8};
\node [below] at (8,0) {9};
\node [below] at (9,0) {10};
\node [below] at (10,0) {11};
\node [below] at (11,0) {12};
\node at (5.5,-2) {$D$};
\draw[->] (11.5,1) -- (12.5,1);
\end{scope}
\begin{scope}[shift={(13,-0.5)}]
\draw (0,0) -- (13,0);
\draw[very thick](0,0) -- (2.5,2.5) -- (5,0);
\draw(1,0) -- (6,5) -- (11,0);
\draw(2,0) -- (6,4) -- (10,0);
\draw[very thick](3,0) -- (5.5,2.5) -- (8,0);
\draw[gray, dashed](4,0) -- (6.5,2.5) -- (9,0);
\draw[very thick](6,0) -- (9,3) -- (12,0);
\draw[very thick](7,0) -- (10,3) -- (13,0);
\node [below] at (0,0) {1};
\node [below] at (1,0) {2};
\node [below] at (2,0) {3};
\node [below] at (3,0) {4};
\node [below] at (4,0) {5};
\node [below] at (5,0) {6};
\node [below] at (6,0) {7};
\node [below] at (7,0) {8};
\node [below] at (8,0) {9};
\node [below] at (9,0) {10};
\node [below] at (10,0) {11};
\node [below] at (11,0) {12};
\node [below] at (12,0) {13};
\node [below] at (13,0) {14};
\node at (6.5,-2) {$\alpha_{6,4,2}(D)$};
\draw[->] (13.5,1) -- (14.5,1);
\end{scope}
\begin{scope}[shift={(0,-9.5)}]
\draw (0,0) -- (13,0);
\draw[very thick](0,0) -- (2.5,2.5) -- (5,0);
\draw(1,0) -- (6,5) -- (11,0);
\draw[gray, dashed](2,0) -- (5.5,3.5) -- (9,0);
\draw[very thick](3,0) -- (5.5,2.5) -- (8,0);
\draw(4,0) -- (7,3) -- (10,0);
\draw[very thick](6,0) -- (9,3) -- (12,0);
\draw[very thick](7,0) -- (10,3) -- (13,0);
\node [below] at (0,0) {1};
\node [below] at (1,0) {2};
\node [below] at (2,0) {3};
\node [below] at (3,0) {4};
\node [below] at (4,0) {5};
\node [below] at (5,0) {6};
\node [below] at (6,0) {7};
\node [below] at (7,0) {8};
\node [below] at (8,0) {9};
\node [below] at (9,0) {10};
\node [below] at (10,0) {11};
\node [below] at (11,0) {12};
\node [below] at (12,0) {13};
\node [below] at (13,0) {14};
\node at (6.5,-2) {$\alpha_{6,4,1}(D)$};
\draw[->] (13.5,1) -- (14.5,1);
\end{scope}
\begin{scope}[shift={(15,-9.5)}]
\draw (0,0) -- (13,0);
\draw[very thick](0,0) -- (2.5,2.5) -- (5,0);
\draw[gray, dashed](1,0) -- (5,4) -- (9,0);
\draw(2,0) -- (6.5,4.5) -- (11,0);
\draw[very thick](3,0) -- (5.5,2.5) -- (8,0);
\draw(4,0) -- (7,3) -- (10,0);
\draw[very thick](6,0) -- (9,3) -- (12,0);
\draw[very thick](7,0) -- (10,3) -- (13,0);
\node [below] at (0,0) {1};
\node [below] at (1,0) {2};
\node [below] at (2,0) {3};
\node [below] at (3,0) {4};
\node [below] at (4,0) {5};
\node [below] at (5,0) {6};
\node [below] at (6,0) {7};
\node [below] at (7,0) {8};
\node [below] at (8,0) {9};
\node [below] at (9,0) {10};
\node [below] at (10,0) {11};
\node [below] at (11,0) {12};
\node [below] at (12,0) {13};
\node [below] at (13,0) {14};
\node at (6.5,-2) {$\alpha_{6,4,0}(D)$};
\end{scope}
\end{tikzpicture}
\end{center}
\section{Details of the proof}
\begin{definition}
Let $C$ be a linear chord diagram, then we define $L_{n,k} = \{1,2,\cdots,k\}$, $M_{n,k} = \{k+1,k+3,\cdots,2n-k\}$, and $R_{n,k} = \{2n-k+1,2n-k+1,\cdots,2n\}$.
Let $C_{n,k}$ denote the set of all chords $c \in C$ such that $s_{c} \in M_{n,k}$ or $e_{c} \in M_{n,k}$, and $S_{C}$ denote the set of all chords $c \in C$ such that $c \notin C_{n,k}$.
\end{definition}
\begin{lemma}\label{lem:noChord}
Given any linear chord diagram in $\minimalClass{k}{n}$ such that $n \geq 3(n-k)$ and $n \geq k$, there is no chord $c$ such that $s_{c},e_{c} \in M_{n,k}$.
\end{lemma}
\begin{proof}
If a chord has both its start point and end point inside $M_{n,k}$, then the largest length it could have, is when it starts at $k+1$ and ends at $2n-k$.
So the maximum length any such chord could have is $2n-2k-1$.
But $n \geq 3(n-k)$ which is equivalent to $3k \geq 2n$.
Thus the maximum length any such chord could have is $2n - 2k -1 \leq 3k - 2k - 1 = k-1$.
But every chord must have length at least $k$.
Thus there is no chord such that its indices of the start point and end point lie inside $M_{n,k}$
\end{proof}
\begin{lemma}\label{lem:sameNumber}
Given any linear chord diagram in $\minimalClass{k}{n}$ such that $n \geq 3(n-k)$ and $n \geq k$, $C_{n,k}$ contains exactly $n-k$ chords that start in $M_{n,k}$ and $n-k$ chords that end in $M_{n,k}$.
\end{lemma}
\begin{proof}
We first observe that no chord has its end index in $L_{n,k}$, since it if did, its maximum length would be $k-1$.
Similarly, no chord has its start index in $R_{n,k}$ since it if did, its maximum length would be $2n-(2n-k+1) = k-1$.
Thus every index in $L_{n,k}$ is a start index, and every index in $R_{n,k}$ is an end index.
We also observe that $\abs{L_{n,k}} = \abs{R_{n,k}}$.
Consider all chords in $S_{c}$.
Since they neither start nor end in $M_{n,k}$, they must start in $L_{n,k}$ and end in $R_{n,k}$.
Thus $\abs{L_{n,k}} - \abs{S_{C}}$ chords start in $L_{n,k}$ and end in $M_{n,k}$, and $\abs{R_{n,k}} - \abs{S_{C}}$ chords end in $R_{n,k}$ and start in $M_{n,k}$.
By Lemma \ref{lem:noChord}, every Chord in $M$ either start in $L_{n,k}$ or ends in $R_{n,k}$.
Thus $M_{n,k}$ has the same number of start indices as end indices, and that number is $n-k$.
\end{proof}
\begin{lemma}\label{lem:left}
Given any linear chord diagram $C \in \minimalClass{k+1}{n+1}$ such that $n \geq 3(n-k)$ and $n \geq k$, let $a$ be the chord whose end index is $2n-k+2$ (i.e. the smallest element in $R_{n+1,k+1}$).
Let $m$ be the number of chords $b \in S_{C}$ such that $s_{b} < s_{a}$.
Then $m < n - k + 1$.
\end{lemma}
\begin{proof}
Let $M^{\ast}$ by the ordered set of all chords $c \in C_{n+1,k+1}$ in such that $e_{c} \in M$.
We say $k < c$ for $k,c \in M^{\ast}$ if $e_{k} < e_{c}$.
Observe that $M^{\ast}$ is completely ordered.
By Lemma \ref{lem:sameNumber}, we have $\abs{M^{\ast}} = n-k$
We may relabel the chords is $M^{\ast}$ to be $\{c_{1},c_{2},\cdots, c_{n-k}\}$.
Observe that by Lemma \ref{lem:sameNumber}, $e_{c_{i}} \leq (k+1) + (n-k) + i = n+i+1$.
Since $\ell_{c_{i}} = e_{c_{i}} - s_{c_{i}} \geq k+1$ we have $s_{c_{i}} \leq n + i + 1 - (k+1) = n - k + i$.
Let $m_{i}$ be the number of chords $a \in S_{C}$ such that $s_{a} < s_{c_{i}}$.
Then $m_{1} < n-k + 1$.
The largest number of start indices to the left of $s_{c_{2}}$ is $n-k + 1$, but if it were that large, one of them must be the start of $c_{1}$.
Thus $m_{2} < n-k + 1$.
By induction we have $m_{i} < n-k + 1$ for all $i$.
Now suppose $m \geq n-k + 1$, then $s_{c_{i}} < s_{a}$ for all $i$ since otherwise $m_{i} \geq n-k + 1$.
Thus $s_{a} \geq (n-k+1) + (n-k) + 1 = 2n-2k + 2$
Thus $\ell_{a}$ is bounded above by $2n-k+2 - (2n-2k + 2) = k < k+1$.
Thus $m < n-k + 1$.
\end{proof}
\begin{definition}
We define $\alpha_{n,k,i}$ for $i \in \{0,\cdots,n-k\}$, $n \geq k$, and $n \geq 3(n-k)$ to be a map from $\minimalClass{k}{n}$ to $D_{n}$ as follows.
Given a diagram $C$, we insert a new chord $c$ with start point right before $M_{n,k}$ and end point right after $M_{n,k}$ to get diagram $C^{\ast}$.
We then swap the start index of the new chord with the closest start index of a chord in $S_{C}$ to its left.
We continue to swap until there are $i$ start indices of chords in $S_{C}$ to its left.
Observe that since $n \geq 3(n-k)$, that the number of chords in $S$ is at least $n - (2n-2k) \geq 3(n-k) - 2(n-k) = n-k$
Thus every $\alpha$ exists and is well defined.
\end{definition}
\begin{example}
Obtaining $C^{\ast}$ from $C$ is shown below
\begin{center}
\begin{tikzpicture}[scale=.45]
\begin{scope}[shift={(0,-0.5)}]
\draw (0,0) -- (11,0);
\draw[very thick](0,0) -- (2,2) -- (4,0);
\draw(1,0) -- (5,4) -- (9,0);
\draw(2,0) -- (5,3) -- (8,0);
\draw[very thick](3,0) -- (5,2) -- (7,0);
\draw[very thick](5,0) -- (7.5,2.5) -- (10,0);
\draw[very thick](6,0) -- (8.5,2.5) -- (11,0);
\node [below] at (0,0) {1};
\node [below] at (1,0) {2};
\node [below] at (2,0) {3};
\node [below] at (3,0) {4};
\node [below] at (4,0) {5};
\node [below] at (5,0) {6};
\node [below] at (6,0) {7};
\node [below] at (7,0) {8};
\node [below] at (8,0) {9};
\node [below] at (9,0) {10};
\node [below] at (10,0) {11};
\node [below] at (11,0) {12};
\node at (6,-2) {$C$};
\draw[->] (11.5,1) -- (12.5,1);
\end{scope}
\begin{scope}[shift={(13,-0.5)}]
\draw (0,0) -- (13,0);
\draw[very thick](0,0) -- (2.5,2.5) -- (5,0);
\draw(1,0) -- (6,5) -- (11,0);
\draw(2,0) -- (6,4) -- (10,0);
\draw[very thick](3,0) -- (5.5,2.5) -- (8,0);
\draw[gray, dashed](4,0) -- (6.5,2.5) -- (9,0);
\draw[very thick](6,0) -- (9,3) -- (12,0);
\draw[very thick](7,0) -- (10,3) -- (13,0);
\node [below] at (0,0) {1};
\node [below] at (1,0) {2};
\node [below] at (2,0) {3};
\node [below] at (3,0) {4};
\node [below] at (4,0) {5};
\node [below] at (5,0) {6};
\node [below] at (6,0) {7};
\node [below] at (7,0) {8};
\node [below] at (8,0) {9};
\node [below] at (9,0) {10};
\node [below] at (10,0) {11};
\node [below] at (11,0) {12};
\node [below] at (12,0) {13};
\node [below] at (13,0) {14};
\node at (7,-2) {$C^{\ast}$};
\end{scope}
\end{tikzpicture}
\end{center}
Here is $\alpha_{3,2,0}$ applied to an element of $\minimalClass{2}{3}$
\begin{center}
\begin{tikzpicture}[scale=.45]
\begin{scope}[shift={(0,-.5)}]
\draw (0,0) -- (5,0);
\draw[very thick] (0,0) -- (1,1) -- (2,0);
\draw (1,0) -- (2.5,1.5) -- (4,0);
\draw[very thick] (3,0) -- (4,1) -- (5,0);
\node [below] at (0,0) {1};
\node [below] at (1,0) {2};
\node [below] at (2,0) {3};
\node [below] at (3,0) {4};
\node [below] at (4,0) {5};
\node [below] at (5,0) {6};
\draw[->] (5.5,1) -- (7.5,1);
\end{scope}
\begin{scope}[shift={(8,-.5)}]
\draw (0,0) -- (7,0);
\draw[very thick] (0,0) -- (1.5,1.5) -- (3,0);
\draw (1,0) -- (3.5,2.5) -- (6,0);
\draw[gray, dashed] (2,0) -- (3.5,1.5) -- (5,0);
\draw[very thick] (4,0) -- (5.5,1.5) -- (7,0);
\node [below] at (0,0) {1};
\node [below] at (1,0) {2};
\node [below] at (2,0) {3};
\node [below] at (3,0) {4};
\node [below] at (4,0) {5};
\node [below] at (5,0) {6};
\node [below] at (6,0) {7};
\node [below] at (7,0) {8};
\draw[->] (7.5,1) -- (9.5,1);
\end{scope}
\begin{scope}[shift={(18,-.5)}]
\draw (0,0) -- (7,0);
\draw[very thick] (0,0) -- (1.5,1.5) -- (3,0);
\draw (2,0) -- (4,2) -- (6,0);
\draw[gray, dashed] (1,0) -- (3,2) -- (5,0);
\draw[very thick] (4,0) -- (5.5,1.5) -- (7,0);
\node [below] at (0,0) {1};
\node [below] at (1,0) {2};
\node [below] at (2,0) {3};
\node [below] at (3,0) {4};
\node [below] at (4,0) {5};
\node [below] at (5,0) {6};
\node [below] at (6,0) {7};
\node [below] at (7,0) {8};
\end{scope}
\end{tikzpicture}
\end{center}
Here is $\alpha_{4,3,1}$ applied to an element of $\minimalClass{3}{4}$.
\begin{center}
\begin{tikzpicture}[scale=.45]
\begin{scope}[shift={(0,-.5)}]
\draw (0,0) -- (7,0);
\draw[very thick] (0,0) -- (1.5,1.5) -- (3,0);
\draw (1,0) -- (3.5,2.5) -- (6,0);
\draw (2,0) -- (3.5,1.5) -- (5,0);
\draw[very thick] (4,0) -- (5.5,1.5) -- (7,0);
\node [below] at (0,0) {1};
\node [below] at (1,0) {2};
\node [below] at (2,0) {3};
\node [below] at (3,0) {4};
\node [below] at (4,0) {5};
\node [below] at (5,0) {6};
\node [below] at (6,0) {7};
\node [below] at (7,0) {8};
\draw[->] (7.5,1) -- (9.5,1);
\end{scope}
\begin{scope}[shift={(10,-.5)}]
\draw (0,0) -- (9,0);
\draw[very thick] (0,0) -- (2,2) -- (4,0);
\draw (1,0) -- (4.5,3.5) -- (8,0);
\draw (2,0) -- (4.5,2.5) -- (7,0);
\draw[gray, dashed] (3,0) -- (4.5,1.5) -- (6,0);
\draw[very thick] (5,0) -- (7,2) -- (9,0);
\node [below] at (0,0) {1};
\node [below] at (1,0) {2};
\node [below] at (2,0) {3};
\node [below] at (3,0) {4};
\node [below] at (4,0) {5};
\node [below] at (5,0) {6};
\node [below] at (6,0) {7};
\node [below] at (7,0) {8};
\node [below] at (8,0) {9};
\node [below] at (9,0) {10};
\draw[->] (9.5,1) -- (11.5,1);
\end{scope}
\begin{scope}[shift={(22,-.5)}]
\draw (0,0) -- (9,0);
\draw[very thick] (0,0) -- (2,2) -- (4,0);
\draw (1,0) -- (4.5,3.5) -- (8,0);
\draw[gray, dashed] (2,0) -- (4,2) -- (6,0);
\draw (3,0) -- (5,2) -- (7,0);
\draw[very thick] (5,0) -- (7,2) -- (9,0);
\node [below] at (0,0) {1};
\node [below] at (1,0) {2};
\node [below] at (2,0) {3};
\node [below] at (3,0) {4};
\node [below] at (4,0) {5};
\node [below] at (5,0) {6};
\node [below] at (6,0) {7};
\node [below] at (7,0) {8};
\node [below] at (8,0) {9};
\node [below] at (9,0) {10};
\end{scope}
\end{tikzpicture}
\end{center}
where the thick lines are chords in $C_{n,k}$, the thin chords are in $S_{C}$ and the greyed dashed chord is the new inserted one.
\end{example}
\begin{definition}
We define $\beta_{n,k}$ for $n \geq k$, and $n > 3(n-k)$ to be a map from $\minimalClass{k}{n}$ to $D_{n-1}$ as follows.
Given a diagram $C$, we denote $c$ to be the cord with end point right after $M_{n,k}$.
We then swap the start index of the new chord with the closest start index of a chord in $S_{C}$ to its right.
We continue to swap until there are no more start indices of chords in $S_{C}$ to its right.
We then remove chord $c$.
\end{definition}
\begin{lemma}\label{lem:alpha}
$\alpha_{n,k,i}\para{\minimalClass{k}{n}} \subseteq \minimalClass{k+1}{n+1}$.
\end{lemma}
\begin{proof}
We see that the result will have $n+1$ chords, so it suffices to show that every chord has length at least $k+1$.
Consider a chord $c$ in $C_{n,k}$, it either has $s_{c} \in M_{n,k}$ and $e_{c} \in R_{n,k}$, in which case it length is increased by 1, since we inserted a index between $_{n,k}M$ and $R_{n,k}$.
Or $e_{c} \in M_{n,k}$ and $s_{c} \in L_{n,k}$, in which case it length is increased by 1, since we inserted a index between $M_{n,k}$ and $L_{n,k}$.
Since the length of such a chord had to be at least $k$ to begin with, it must have at least length $k+1$ after applying $\alpha$.
Consider the chord we just inserted.
It will cover all the indices in $M_{n,k}$, and every time we swap, another index will be covered.
Since there are a total of $n$ chords before inserting, of which $M_{n,k}$ contains $2n-2k$ of them, and it swaps until there are $i$ chord to its left in $S$, it swapped with at least $n-(2n-2k) - i$.
Recall that the length of the chord will be the number of indices it covers plus 1.
Thus its length is at least $1 +(2n-2k) + (n-(2n-2k)) - i = 1 + n - i \geq 1 + n - (n-k) = k+1$.
As desired.
Now consider chords in $S_{C}$.
There are two cases, either it has its start index swapped at some point or it didn't.
If it didn't, then it covers the new chord $c$, and has length greater then $c$'s length.
Thus the chord has length at least $k+1$ as desired.
If it did swap, then either its starting index increased by 1 or more.
Suppose that its starting index increased by 1.
Then the number of indices that lie in between its endpoints has increased by 1.
When we inserted $c$, it was increased by 2, but then we moved the starting index forward by 1, causing it to lose 1.
Thus its length increased by exactly 1.
Since it must of have length $k$ to begin, with, in now has length at least $k+1$.
Suppose that its starting index increased by more then 1.
Let $a$ be its original starting index after inserting $c$ and $b$ be its starting index after inserting and swapping $c$.
Then the index $b-1$ is the starting index of some point in $M_{n+1,k+1}$, since $b-a > 1$ and otherwise $b$ would have occurred sooner.
Thus the the chord with starting index $b-1$ has length at least $k+1$.
Since the ending index of our chord lies in $R$ which is at least 1 more then the ending index of the chord at $b-1$, the length of our chord after swapping is at least $k+1$.
Thus $\alpha_{n,k,i}\para{\minimalClass{k}{n}} \subseteq \minimalClass{k+1}{n+1}$ as desired.
\end{proof}
\begin{lemma}\label{lem:beta}
$\beta_{n,k}\para{\minimalClass{k}{n}} \subseteq \minimalClass{k-1}{n-1}$.
\end{lemma}
\begin{proof}
We see that the result will have $n-1$ chords, so it suffices to show that every chord has length at least $k-1$.
Consider a chord $r$ in $C_{n,k}$, it either has $s_{r} \in M_{n,k}$ and $e_{r} \in R_{n,k}$, in which case it length is decreased by 1, since we removed the first index in $R_{n,k}$.
Or $e_{r} \in M$ and $s_{r} \in L$, in which case it length is decreased by 1, since we removed the last index from $L_{n,k}$.
Since the length of such a chord had to be at least $k$ to begin with, it must have at least length $k-1$ after applying $\beta_{n,k}$.
Consider a chord $r$ in $S_{C}$
We break it into two cases:
Case 1: $s_{r}$ was swapped with $s_{c}$ at some point.
Then $s_{r}$ has decreased by at least 1, which means that $\ell_{r}$ increased by at least 1.
But when we remove $s_{c}$ a the end, $\ell_{r}$ is deceased by 2.
Thus $\ell_{r}$ never deceases by more then 1.
Since $\ell_{r} = k$, the length of $r$ must be at least length $k-1$ after applying $\beta_{n,k}$.
Case 2: $s_{r}$ did not swap with $s_{c}$ at some point.
Then $s_{r} < s_{c}$, which means that, $\ell_{r}$ is at least $2 + \ell_{c} = k+2$ since $\ell_{c}$ has length at least $k$.
When we remove $s_{c}$ a the end, $\ell_{r}$ is deceased by 2.
Thus $\ell_{r}$ never deceases by more then 2.
Since $\ell_{r} \geq k+2$, the length of $r$ must be at least length $k$ after applying $\beta_{n,k}$.
Thus $\beta_{n,k}\para{\minimalClass{k}{n}} \subseteq \minimalClass{k-1}{n-1}$ as desired.
\end{proof}
\begin{proof}[Proof (of theorem \ref{thm:main})]
We shall proceed by constructing $(n-k+1)$ injective function from $\minimalClass{k}{n}$ to $\minimalClass{k+1}{n+}$ such that their images partition $\minimalClass{k+1}{n+1}$.
Let $C \in \minimalClass{k}{n}$
Let $E_{n,k,i}$ be the set of all linear chord diagrams in $\minimalClass{k}{n}$ such that the chord $c$ with $e_{s} = 2n-k+1$ (i.e. the first index after $M_{n,k}$) has $i$ start points of chords in $S_{C}$ to its left.
Then by lemma \ref{lem:left} the collection $\{E_{n+1,k+1,0},\cdots,E_{n+1,k+1,n-k-1}\}$ partitions $\minimalClass{k+1}{n+1}$.
By construction we see that $\text{Im}(\alpha_{n,k,i}) \subseteq E_{n+1,k+1,i}$.
We also see that both $\restricto{\beta_{n+1,k+1}}{E_{n+1,k+1,i}} \circ \alpha_{n,k,i}$ and $\alpha_{n,k,i} \circ \restricto{\beta_{n+1,k+1}}{E_{n+1,k+1,i}}$ are the identity map.
Thus there is a bijection between $\minimalClass{k}{n}$ and $E_{n,k,i}$ for every $i$.
Thus
\begin{displaymath}
\abs{\minimalClass{k+1}{n+1}} = \sumover{i=0}{n-k}{\abs{\alpha_{n,k,i}\para{\minimalClass{k}{n}}}} = (n-k+1)\abs{\minimalClass{k}{n}}
\end{displaymath}
As desired.
\end{proof}
\end{document} |
\begin{document}
\title{Convex order for convolution polynomials of~Borel measures}
\author{Andrzej Komisarski}
\address{Andrzej Komisarski, Department of Probability Theory and Statistics, Faculty of Mathematics and Computer Science,
University of \L\'od\'z, ul. Banacha 22, 90-238 \L\'od\'z, Poland}
\email{andkom@math.uni.lodz.pl}
\author{Teresa Rajba}
\address{Teresa Rajba, University of Bielsko-Biala, Department of Mathematics, ul. Willowa 2, 43-309 Bielsko-Bia\l{}a, Poland}
\email{trajba@ath.bielsko.pl}
\keywords{Bernstein polynomials, stochastic order, stochastic convex order, convex functions,
functional inequalities related to convexity, Muirhead inequality}
\subjclass[2010]{Primary 26D15; Secondary 60E15, 39B62}
\maketitle
\begin{abstract}
We give necessary and sufficient conditions for Borel measures
to satisfy the inequality introduced by Komisarski, Rajba (2018).
This inequality is a generalization of the convex order inequality for binomial distributions, which was proved
by Mrowiec, Rajba, W\k{a}sowicz (2017), as a probabilistic version of the inequality for convex functions,
that was conjectured as an old open problem by I.~Ra\c{s}a.
We present also further generalizations using convex order inequalities between convolution polynomials of finite Borel measures.
We generalize recent results obtained by B.~Gavrea (2018) in the discrete case to general case. We give solutions to his open problems
and also formulate new problems.
\end{abstract}
\section{Introduction}
Let $\mu$ and $\nu$ be two finite Borel measures (e.g. probability distributions) on $\mathbb R$
with finite first moments (i.e. $\int|x|\ \mu(dx)<\infty$ and the same for $\nu$). We say that
$\mu$ is \emph{smaller than $\nu$ in the convex order} (denoted as $\mu\leqslantslantq_{\text{\rm cx}}\nu$) if $$\int_{\mathbb R}\varphi(x)\mu(dx)\leqslantslantq\int_{\mathbb R}\varphi(x)\nu(dx) \quad \text{for all convex functions }\ \varphi\colon\mathbb R\to\mathbb R$$
Note that both integrals always exist (finite or infinite).
Let $P$ and $Q$ be two real polynomials of $m$ variables.
They can be treated as convolution polynomials of finite Borel measures $\mu_1,\dots,\mu_m$ (product of variables corresponds to convolution of measures).
We are interested, when the relation $P(\mu_1,\dots,\mu_m)\leqslantslantq_{\text{\rm cx}} Q(\mu_1,\dots,\mu_m)$ holds.
Our investigation is motivated by the recent result of J.\ Mrowiec, T.\ Rajba and S.\ W\k{a}sowicz \cite{MRW2017}
who proved the following convex ordering relation for convolutions of binomial distributions $B(n,x)$ and $B(n,y)$ ($n\in\mathbb{N}$, $x,y\in[0,1]$):
\begin{equation}\label{eq:mainv3}
B(n,x)*B(n,y)\leqslantslantq_{\text{\rm cx}}\tfrac12(B(n,x)*B(n,x)+B(n,y)*B(n,y)),
\end{equation}
which is a probabilistic version of the inequality involving Bernstein polynomials and convex functions, that was conjectured as an open problem by
I.\ Ra\c{s}a \cite{Rasa2014b} (see also \cite{Abel2016}, \cite{AbelRasa2017}, \cite{KomRaj2018bis}, \cite{Gav2018} for further results on the I.\ Ra\c{s}a problem).
In \cite{KomRaj2018}, we gave a generalization of the inequality \eqref{eq:mainv3}. We introduced and studied the following convex ordering relation
\begin{equation}\label{eq:mainv2}
\mu*\nu\leqslantslantq_{\text{\rm cx}}\tfrac12(\mu*\mu+\nu*\nu),
\end{equation}
where $\mu$ and $\nu$ are two probability distributions on $\mathbb R$. The inequality \eqref{eq:mainv2} can be regarded as the Ra\c{s}a type inequality.
In \cite{KomRaj2018}, we proved Theorem 2.3 providing a~very useful sufficient condition for verification that $\mu$ and $\nu$ satisfy \eqref{eq:mainv2}.
We applied Theorem 2.3 for $\mu$ and $\nu$ from various families of probability distributions.
In particular, we obtained a new proof for binomial distributions, which is significantly simpler and shorter than that given in \cite{MRW2017}.
By \eqref{eq:mainv2}, we can also obtain inequalities related to some approximation operators associated with $\mu$ and $\nu$.
(such as Bernstein-Schnabl operators, Mirakyan-Sz\'asz operators, Baskakov operators and others, cf.~\cite{KomRaj2018}).
In \cite{KomRaj2018}, we considered also a generalization of \eqref{eq:mainv2}, taking a finite sequence of probability distributions in place of two probability distributions $\mu$ and $\nu$. We proved the Muirhead type inequality for convex orders for convolution polynomials, and we gave a strong generalization of Theorem 2.3.
If $\mu$ and $\nu$ are discrete probability distributions concentrated on the set of non-negative integers $\{0,1,2, \ldots \}$ with $a_k=\mu (\{ k\})$ and $b_k=\nu (\{ k\})$ ($k=0,1,2, \ldots $), then the inequality \eqref{eq:mainv2} is equivalent to the following inequality
\begin{equation}\label{eq:Rasatres}
\sum_{i=0}^n \sum_{j=0}^n \leqslantslantft(a_i \:a_j+b_i \:b_j\right)\varphi\leqslantslantft(i+j \right)\geqslantslantq
\sum_{i=0}^n \sum_{j=0}^n 2 \:a_i \:b_j \:\varphi\leqslantslantft(i+j \right)
\end{equation}
\noindent for all convex functions $\varphi: \mathbb R \to \mathbb R$.
B.~Gavrea \cite{Gav2018} studied the inequality \eqref{eq:Rasatres} with a convex function $\varphi:\mathbb R \to \mathbb R$
and non-negative sequences $(a_k)$, $(b_k)$ such that $\sum_{k}a_k=\sum_{k}b_k=1$ and $\sum_{k}a_kt^k<\infty$, $\sum_{k}b_kt^k<\infty$ for some $t>1$.
He gave necessary and sufficient conditions for the sequences $(a_k)$, $(b_k)$ to satisfy \eqref{eq:Rasatres}.
He did not use probabilistic methods. Instead, he used complex analysis.
In Section 2, we give necessary and sufficient conditions for \eqref{eq:mainv2}. We do not limit ourselves to the discrete case.
In our considerations, $\mu$ and $\nu$ are finite Borel measures on $\mathbb R$.
In the particular case of discrete probability distributions,
our assumptions on the sequences $(a_k)$ and $(b_k)$
are weaker then those given in \cite{Gav2018}.
In Section 3, we consider a generalization of \eqref{eq:mainv2} for more than two measures.
As a generalization of results from \cite{KomRaj2018}, we present the Ra\c{s}a type inequalities
for convex orders for convolution polynomials of finite Borel measures on $\mathbb R$.
In Section 4, we give solutions to B.~Gavrea's problems (presented in \cite{Gav2018}) and list some new problems.
\section{The basic case of two measures}
In the sequel we adapt some notation from theory of probability and stochastic orders (see \cite{Shaked2007}).
Let $\mu$ be a finite Borel measure (e.g. a~probability distribution) on $\mathbb R$.
For $x\in\mathbb R$ the delta symbol $\delta_x$ denotes the one-point probability distribution satisfying $\delta_x(\{x\})=1$.
Function $F(x)=F_\mu(x)=\mu((-\infty,x])$ ($x\in\mathbb R$) is the cumulative distribution function of $\mu$ (simply the distribution function).
The complementary cumulative distribution function, or simply the tail distribution, is defined as $\overline F(x)=\mu(\mathbb R)-F(x)=\mu((x,\infty))$.
If $\mu$ and $\nu$ are finite Borel measures such that $\mu(\mathbb R)=\nu(\mathbb R)$
and $F_\mu(x)\geqslantslantq F_\nu(x)$ for all $x\in\mathbb R$,
then $\mu$ is said to be \emph{smaller than $\nu$ in the~usual stochastic order} (denoted by $\mu\leqslantslantq_{\text{\rm st}}\nu$).
An important characterization of the~usual stochastic order for probability distributions is given in the following theorem.
\begin{theorem}[\cite{Shaked2007}, p. 5]\label{th:1a1}
Two probability distributions $\mu$ and $\nu$ satisfy $\mu\leqslantslantq_{\text{\rm st}}\nu$ if, and only if, there exist two random variables $X$ and $Y$
defined on the same probability space, such that the distribution of $X$ is $\mu$, the distribution of $Y$ is $\nu$
and $P(X\leqslantslantq Y)=1$.
\end{theorem}
In \cite{KomRaj2018}, we gave a very useful sufficient condition, that can be used for the verification of the inequality \eqref{eq:mainv2}.
\begin{theorem}[\cite{KomRaj2018}]\label{th:condition}
Let $\mu$ and $\nu$ be two probability distributions with finite first moments, such that $\mu\leqslantslantq_{\text{\rm st}}\nu$ or $\nu\leqslantslantq_{\text{\rm st}}\mu$. Then
\begin{equation}\label{eq:main}
\mu*\nu\leqslantslantq_{\text{\rm cx}}\tfrac12(\mu*\mu+\nu*\nu).
\end{equation}
\end{theorem}
As an application of Theorem~\ref{th:condition}, we obtain that \eqref{eq:main} holds for $\mu$ and $\nu$ from various families of probability distributions: binomial, Poisson, negative binomial, beta, gamma and Gaussian distributions.
The condition presented in Theorem~\ref{th:condition} is sufficient but it is not necessary.
In the following theorem we give a necessary and sufficient condition for finite Borel measures $\mu$ and $\nu$ to satisfy the inequality \eqref{eq:main}.
\begin{theorem}\label{th:necsuf}
Let $\mu$ and $\nu$ be two finite Borel measures on $\mathbb R$ with finite first moments. Let $F$ and $G$
be the distribution functions corresponding to $\mu$ and $\nu$, respectively. Then the following conditions are equivalent:
\begin{itemize}
\item[(1)] $\mu(\mathbb R)=\nu(\mathbb R)$ and $(F-G)*(F-G)\geqslantslantq0$,
\item[(2)] $\mu*\nu\leqslantslantq_{\text{\rm cx}}\frac12(\mu*\mu+\nu*\nu)$.
\end{itemize}
\end{theorem}
\begin{proof}
First we show that (2) implies $\mu(\mathbb R)=\nu(\mathbb R)$.
For the convex function $\varphi(x)=1$ ($x\in \mathbb R$) we have:
$$2\mu(\mathbb R)\nu(\mathbb R)=\int_{-\infty}^\infty 1\ (2\mu*\nu)(dx)\leqslantslantq\int_{-\infty}^\infty 1\ (\mu*\mu+\nu*\nu)(dx)=(\mu(\mathbb R))^2+(\nu(\mathbb R))^2.$$
In turn, taking the convex function $\varphi(x)=-1$ ($x\in \mathbb R$) we obtain:
$$-2\mu(\mathbb R)\nu(\mathbb R)=\int_{-\infty}^\infty (-1)\ (2\mu*\nu)(dx)\leqslantslantq\int_{-\infty}^\infty(-1)\ (\mu*\mu+\nu*\nu)(dx)=-(\mu(\mathbb R))^2-(\nu(\mathbb R))^2.$$
Consequently, we have $2\mu(\mathbb R)\nu(\mathbb R)=(\mu(\mathbb R))^2+(\nu(\mathbb R))^2$, which implies $(\mu(\mathbb R)-\nu(\mathbb R))^2=0$.
It follows that $\mu(\mathbb R)=\nu(\mathbb R)$. It remains to show that if $\mu(\mathbb R)=\nu(\mathbb R)$,
then (2) is equivalent to $(F-G)*(F-G)\geqslantslantq0$.
The relation (2) is equivalent to fulfilling the following inequality
\begin{equation}\label{rasa_v4}
\int_{-\infty}^\infty \varphi(x)(\mu*\mu+\nu*\nu)(dx)\geqslantslantq\int_{-\infty}^\infty \varphi(x)(2\mu*\nu)(dx)
\end{equation}
for all convex functions $\varphi:\mathbb R \to\mathbb R$.
Note that every convex function $\varphi$ is a pointwise limit
of an~increasing sequence $(\varphi_n)$ of convex, piecewise linear functions.
Therefore (due to monotone convergence theorem for integrals)
\eqref{rasa_v4} is valid for convex functions if, and only if, it is valid
for convex, piecewise linear functions.
On the other hand, every convex, piecewise linear function is a linear combination with non-negative coefficients
of a linear (affine) function and functions of the form \eqref{rasa_v5} (see below).
If $\varphi(x)=ax+b$ is a~linear function, then we have equality in \eqref{rasa_v4}
(both sides of~\eqref{rasa_v4} are equal to $2b(\mu(\mathbb R))^2+2a\mu(\mathbb R)(\int x\mu(dx)+\int x\nu(dx))$).
It follows that \eqref{rasa_v4} is valid for convex functions if, and only if, it is valid for functions of the form
\begin{equation}\label{rasa_v5}
\varphi(x)=(x-A)_+=\max(x-A,0),
\end{equation}
where $A\in\mathbb R$.
In the following computation symbols $\overline F$ and $\overline G$ stand
for the tail distributions of $\mu$ and $\nu$, respectively.
We use the Fubini Theorem several times.
We assume that all the definite integrals are integrals on open intervals. Let $\lambda$ denote the Lebesgue measure on the real line $\mathbb{R}$.
Besides the positive measures $\mu$ and $\nu$, we also study the signed measure $\mu-\nu$.
Integrability of all considered functions and applicability of the Fubini Theorem
follows from our assumption that $\mu$ and $\nu$ have finite first moments
(e.g., $\mu(\mathbb R)=\nu(\mathbb R)$ implies $F-G=\overline G-\overline F$,
hence
$\int_{-\infty}^\infty|F(t)-G(t)|\ \lambda(dt)\leqslantslantq
\int_{-\infty}^0(F(t)+G(t))\ \lambda(dt)
+\int_0^{\infty}(\overline F(t)+\overline G(t))\ \lambda(dt)=
\int_{-\infty}^\infty|x|\ \mu(dx)+\int_{-\infty}^\infty|x|\ \nu(dx)<\infty$).
Let $A\in\mathbb R$. Then we have
\begin{multline*}
\int_{-\infty}^\infty (x-A)_+(\mu*\mu+\nu*\nu-2\mu*\nu)(dx)=\int_A^\infty\int_A^x1\ \lambda(dz)(\mu-\nu)^{*2}(dx)=\bigg[A<z<x\bigg]=\\
\int_A^\infty\int_z^\infty1\ (\mu-\nu)^{*2}(dx)\lambda(dz)=
\int_A^\infty\int_{-\infty}^\infty\int_{z-v}^\infty1\ (\mu-\nu)(du)(\mu-\nu)(dv)\lambda(dz)=\\
\bigg[A<z<u+v\bigg]=\int_{-\infty}^\infty\int_{A-v}^\infty\int_A^{u+v}1\ \lambda(dz)(\mu-\nu)(du)(\mu-\nu)(dv)=\\
\bigg[\text{we substitute }t=z-u\bigg]=
\int_{-\infty}^\infty\int_{A-v}^\infty\int_{A-u}^v1\ \lambda(dt)(\mu-\nu)(du)(\mu-\nu)(dv)=\bigg[A<u+t<u+v\bigg]=\\
\int_{-\infty}^\infty\int_t^\infty\int_{A-t}^\infty1\ (\mu-\nu)(du)(\mu-\nu)(dv)\lambda(dt)=
\int_{-\infty}^\infty(\overline F(t)-\overline G(t))(\overline F(A-t)-\overline G(A-t))\ \lambda(dt)=\\
((\overline F-\overline G)*(\overline F-\overline G))(A)=((F-G)*(F-G))(A).
\end{multline*}
The above identity completes the proof of the theorem.
\end{proof}
\begin{remark}
Theorem \ref{th:necsuf} is a generalization of Theorem \ref{th:condition}. Indeed, if $\mu\leqslantslantq_{\text{\rm st}}\nu$ or $\nu\leqslantslantq_{\text{\rm st}}\mu$, then obviously the condition (1) in Theorem \ref{th:necsuf} is satisfied.
\end{remark}
In the following proposition we give more precise estimation of the difference of integrals given in \eqref{rasa_v4}.
\begin{proposition}
Let $\mu$ and $\nu$ be two finite Borel measures on $\mathbb R$ with finite first and second moments.
Let $\overline\mu=\int_{\mathbb R}x\ \mu(dx)$ and $\overline\nu=\int_{\mathbb R}x\ \nu(dx)$.
Assume, that $\mu*\nu\leqslantslantq_{\text{\rm cx}}\frac12(\mu*\mu+\nu*\nu)$ and $\varphi$ is a~twice differentiable convex function.
If both sides of \eqref{rasa_v4} are finite, then
$$\inf_x\varphi''(x)\cdot(\overline\mu-\overline\nu)^2\leqslantslantq\int_{\mathbb R}\varphi(x)(\mu*\mu+\nu*\nu-2\mu*\nu)(dx)\leqslantslantq\sup_x\varphi''(x)\cdot(\overline\mu-\overline\nu)^2$$
(we set $\infty\cdot0=\infty$ and $-\infty\cdot0=-\infty$).
\end{proposition}
\begin{proof}
The proofs of both inequalities are similar, therefore we will prove only the left one.
If $\inf_x\varphi''(x)=-\infty$, then the inequality is obvious (the left side is $-\infty$). Assume that
$m:=\inf_x\varphi''(x)$ is finite. Then $x\mapsto\varphi(x)-\frac m2\cdot x^2$ is a convex function. We have
$$\int_{\mathbb R}\leqslantslantft(\varphi(x)-\tfrac m2\cdot x^2\right)(\mu*\mu+\nu*\nu-2\mu*\nu)(dx)\geqslantslantq0,$$
which implies
$$\int_{\mathbb R}\varphi(x)(\mu*\mu+\nu*\nu-2\mu*\nu)(dx)\geqslantslantq
\tfrac m2\cdot\int_{\mathbb R}x^2(\mu*\mu+\nu*\nu-2\mu*\nu)(dx)=m\cdot(\overline\mu-\overline\nu)^2.$$
We need to justify the last equality. If $\mu=\nu=0$, then there is nothing to do.
If $a:=\mu(\mathbb R)=\nu(\mathbb R)>0$, then $\frac\mu a$ and $\frac\nu a$ are probability distributions.
We consider independent random variables $X_1,X_2,Y_1,Y_2$ such that the distribution
of $X_1$ and $X_2$ is $\frac\mu a$ and the distribution of $Y_1$ and $Y_2$ is $\frac\nu a$.
Then we have
\begin{multline*}
\int_{\mathbb R}x^2(\mu*\mu+\nu*\nu-2\mu*\nu)(dx)=
a^2\cdot\int_{\mathbb R}x^2\leqslantslantft(\tfrac\mu a*\tfrac\mu a+\tfrac\nu a*\tfrac\nu a-2\tfrac\mu a*\tfrac\nu a\right)(dx)=\\
a^2\leqslantslantft(\mathbb E(X_1+X_2)^2+\mathbb E(Y_1+Y_2)^2-2\mathbb E(X_1+Y_1)^2\right)=\\
a^2\leqslantslantft(\mathbb EX_1^2+\mathbb EX_2^2+2\mathbb EX_1\mathbb EX_2+\mathbb EY_1^2+\mathbb EY_2^2+2\mathbb EY_1\mathbb EY_2-2\mathbb EX_1^2-2\mathbb EY_1^2-4\mathbb EX_1\mathbb EY_1\right)=\\
2a^2(\mathbb EX_1-\mathbb EY_1)^2=2(\overline\mu-\overline\nu)^2.
\end{multline*}
In the proof of the right inequality we use the convexity of the function $x\mapsto\frac M2\cdot x^2-\varphi(x)$,
where $M=\sup_x\varphi''(x)$.
\end{proof}
Consider now the discrete probability distribution $\mu$ concentrated on the set of non-negative integers $\{0,1,2,\ldots\}$, with $a_k=\mu (\{ k\})$ ($k=0,1,2, \ldots $). Then the probability generating function corresponding to $\mu$ is given by the formula
$$f(z)=\sum_{k=0}^\infty a_kz^k.$$
\begin{theorem}\label{th:discr}
Let $\mu$ and $\nu$ be discrete probability distributions concentrated on the set of non-negative integers $\{0,1,2,\ldots\}$, with $a_k=\mu (\{ k\})$ and $b_k=\nu (\{ k\})$ ($k=0,1,2, \ldots $).
Assume that $\mu$ and $\nu$ have finite first moments. Let $F$, $f$ and $G$, $g$ be the distribution functions and the generating functions corresponding to $\mu$ and $\nu$, respectively. Then the following conditions are equivalent:
\begin{itemize}
\item[(1)] For all convex functions $\varphi: \mathbb R \to \mathbb R$
\begin{equation}\label{eq:Rasatres_v4}
\sum_{i=0}^\infty\sum_{j=0}^\infty\leqslantslantft(a_i \:a_j+b_i \:b_j\right)\varphi(i+j)\geqslantslantq
\sum_{i=0}^\infty\sum_{j=0}^\infty2a_i \:b_j \:\varphi(i+j).
\end{equation}
\item[(2)] $(F-G)*(F-G)\geqslantslantq0$.
\item[(3)] $\frac{d^k}{dz^k}\leqslantslantft(\frac{f(z)-g(z)}{z-1}\right)^2\bigg|_{z=0}\geqslantslantq0$ \quad \text{for}\ $k=0,1,\dots$.
\end{itemize}
\end{theorem}
\begin{proof}
Since \eqref{eq:Rasatres_v4} is equivalent to the relation $\mu*\nu\leqslantslantq_{\text{\rm cx}}\frac12(\mu*\mu+\nu*\nu)$,
the equivalence of (1) and (2) clearly follows from Theorem \ref{th:necsuf}. It suffices to prove the equivalence of (2) and (3).
In the following calculations, we use the existence and finiteness of the first moments of the probability distributions $\mu$ and $\nu$,
which implies that all the following series are absolutely convergent for $z\in[-1,1]$ and we can change the summation order.
By the equality $\sum_{k=0}^\infty a_k=1$, we have
$$\frac{f(z)-1}{z-1}=\sum_{k=0}^\infty a_k\frac{z^k-1}{z-1}=\sum_{k=1}^\infty\sum_{i=0}^{k-1}a_kz^i=
\sum_{i=0}^\infty\sum_{k=i+1}^\infty a_kz^i=\sum_{i=0}^\infty\overline F(i)z^i$$
for every $z\in[-1,1)$, where $\overline F$ is the tail distribution of $\mu$.
Note that $\sum_{i=0}^\infty\overline F(i)=\int_{\mathbb R}x\ \mu(dx)<\infty$.
Similarly, we obtain
$$\frac{g(z)-1}{z-1}=\sum_{i=0}^\infty\overline G(i)z^i,$$
where $\overline G$ is the tail distribution of $\nu$. Therefore for every $z\in[-1,1)$ we have
$$\leqslantslantft(\frac{f(z)-g(z)}{z-1}\right)^2=\leqslantslantft(\sum_{i=0}^\infty(\overline F-\overline G)(i)z^i\right)^2
=\leqslantslantft(\sum_{i=0}^\infty(G-F)(i)z^i\right)^2
=\sum_{i=0}^\infty((G-F)\hat{\ast}(G-F))(i)z^i.$$
Here $\hat{\ast}$ denotes the discrete convolution (the Euler product) of sequences.
Condition (3) is equivalent to the non-negativity of all the coefficients in the above series.
Note that if $i\in\mathbb Z$ and $x\in[i,i+1]$, then $((G-F)*(G-F))(x)=(x-i)((G-F)\hat{\ast}(G-F))(i)+(i+1-x)((G-F)\hat{\ast}(G-F))(i-1)$
(here we put $((G-F)\hat{\ast}(G-F))(i)=0$ for $i<0$). Therefore the non-negativity of all the terms $((G-F)\hat{\ast}(G-F))(i)$ is equivalent to (2).
The theorem is proved.
\end{proof}
\begin{remark}
B.~Gavrea \cite{Gav2018} also gave the condition (3) as necessary and sufficient to satisfy the condition (1), but assuming that the radii
of convergence of functions $f$ and $g$ are greater then 1 (in particular, there exist all the moments of $\mu$ and $\nu$).
The assumption in Theorem \ref{th:discr} is weaker, we assume only the existence of the first moments of $\mu$ and $\nu$.
Furthermore, in Theorem \ref{th:necsuf}, we give necessary and sufficient condition for all distributions, not just for discrete.
\end{remark}
\section{The case of $m$ measures}
In this section we consider $m$ finite Borel measures on $\mathbb R$, $\mu_1,\dots,\mu_m$, with finite first moments.
We denote $F_i(x)=\mu_i((-\infty,x])$ and $\overline F_i(x)=\mu_i((x,\infty))$ for $i=1,\dots,m$
and $x\in\mathbb R$.
Let $P$ and $Q$ be two polynomials of $m$ variables.
They can be treated as convolution polynomials of the measures $\mu_1,\dots,\mu_m$ (product of variables corresponds to convolution of measures).
We are interested, when the relation $P(\mu_1,\dots,\mu_m)\leqslantslantq_{\text{\rm cx}} Q(\mu_1,\dots,\mu_m)$ holds.
Since $\leqslantslantq_{\text{\rm cx}}$ is defined for non-negative measures, we generally assume that
the polynomials have non-negative coefficients, although in the proofs we consider also differences of such polynomials.
\begin{proposition}
Let $\mu_1,\dots,\mu_m$ be finite Borel measures on $\mathbb R$ with finite first moments.
For $i=1,\dots,m$, we denote $a_i=\mu_i(\mathbb R)$ and $b_i=\int x\ \mu_i(dx)$.
Let $P,Q$ be polynomials of $m$ variables with non-negative coefficients, such that $P(\mu_1,\dots,\mu_m)\leqslantslantq_{\text{\rm cx}} Q(\mu_1,\dots,\mu_m)$.
Then $P(a_1,\dots,a_m)=Q(a_1,\dots,a_m)$ and $\frac{\partial P}{\partial(b_1,\dots,b_m)}(a_1,\dots,a_m)
=\frac{\partial Q}{\partial(b_1,\dots,b_m)}(a_1,\dots,a_m)$ (the directional derivatives
along the vector $(b_1,\dots,b_m)$ at the point $(a_1,\dots,a_m)$).
\end{proposition}
\begin{proof}
Let $P=\sum_jp_j\prod_{i=1}^mx_i^{k_{j,i}}$ and $Q=\sum_jq_j\prod_{i=1}^mx_i^{l_{j,i}}$.
Considering the convex functions $\varphi(x)=1$ and $\varphi(x)=-1$ we obtain
$\int1\ P(\mu_1,\dots,\mu_m)=\int1\ Q(\mu_1,\dots,\mu_m)$, which implies $P(a_1,\dots,a_m)=Q(a_1,\dots,a_m)$.
Taking the convex functions $\varphi(x)=x$ and $\varphi(x)=-x$ we get
$\int x\ P(\mu_1,\dots,\mu_m)=\int x\ Q(\mu_1,\dots,\mu_m)$,
which is equivalent to $\sum_jp_j\sum_{i=1}^mk_{j,i}\frac{b_i}{a_i}\prod_{i=1}^ma_i^{k_{j,i}}
=\sum_jq_j\sum_{i=1}^ml_{j,i}\frac{b_i}{a_i}\prod_{i=1}^ma_i^{l_{j,i}}$.
Consequently we obtain $\frac{\partial P}{\partial(b_1,\dots,b_m)}(a_1,\dots,a_m)
=\frac{\partial Q}{\partial(b_1,\dots,b_m)}(a_1,\dots,a_m)$.
\end{proof}
\begin{theorem}\label{th:muir5}
Let $\mu_1,\dots,\mu_m$ be finite Borel measures on $\mathbb R$ with finite first moments.
We assume that $\mu_1(\mathbb R)=\dots=\mu_m(\mathbb R)$ and $(F_i-F_j)*(F_i-F_j)\geqslantslantq0$ for all $i,j=1,\dots,m$.
Let $P$ and $Q$ be polynomials of variables $x_1,\dots,x_m$ with non-negative coefficients, such that $(Q-P)(x_1,\dots,x_m)=\sum_{i\neq j}R_{i,j}(x_1,\dots,x_m)(x_i-x_j)^2$,
where $R_{i,j}$ are polynomials of variables $x_1,\dots,x_m$ with non-negative coefficients.
Then $P(\mu_1,\dots,\mu_m)\leqslantslantq_{\text{\rm cx}} Q(\mu_1,\dots,\mu_m)$.
\end{theorem}
\begin{proof}
Since $R_{i,j}$ are polynomials with non-negative coefficients,
it follows that $R_{i,j}(\mu_1,\dots,\mu_m)$ are finite non-negative measures.
Let $\varphi$ be a function which is affine or of form $\varphi(x)=(x-A)_+$, where $A\in\mathbb R$.
Then $\varphi$ is integrable with respect to every polynomial of measures $\mu_1,\dots,\mu_m$ and we have
\begin{multline*}
\int\varphi(x)\ (Q-P)(\mu_1,\dots,\mu_m)(dx)=
\sum_{i\neq j}\int\varphi(x)\ (R_{i,j}(\mu_1,\dots,\mu_m)*(\mu_i-\mu_j)^{*2})(dx)=\\
\sum_{i\neq j}\int\int\varphi(u+v)\ (\mu_i-\mu_j)^{*2}(du)R_{i,j}(\mu_1,\dots,\mu_m)(dv)\geqslantslantq0,
\end{multline*}
because, by Theorem \ref{th:necsuf}, the internal integral in each component of the above sum is non-negative.
Since other convex functions are limits of convex combinations with non-negative coefficients of functions considered above, we obtain $P(\mu_1,\dots,\mu_m)\leqslantslantq_{\text{\rm cx}} Q(\mu_1,\dots,\mu_m)$.
\end{proof}
\begin{remark}
Let $\mu_1,\dots,\mu_k$ be finite Borel measures on $\mathbb R$ with finite first moments.
If $\mu_1,\dots,\mu_k$ are pairwise comparable in the usual stochastic order
(for each $1\leqslantslantq i,j\leqslantslantq k$ we have $\mu_i\leqslantslantq_{\text{\rm st}}\mu_j$ or $\mu_i\geqslantslantq_{\text{\rm st}}\mu_j$),
then the inequalities $(F_i-F_j)*(F_i-F_j)\geqslantslantq0$ are satisfied for all $i,j=1,\dots,m$.
\end{remark}
Before we state the next theorem, we need to present two definitions.
In the set of all the $m$-tuples $\mathbf p=(p_1,\dots,p_m)$ of non-negative integers we consider the following quasiorder.
\begin{definition}
We say that $\mathbf q$ majorizes $\mathbf p$ (denoted by $\mathbf p\prec\mathbf q$ or $\mathbf q\succ\mathbf p$) if
\begin{enumerate}
\item $\sum_{l=1}^m\widehat p_l=\sum_{l=1}^m\widehat q_l$,
\item $\sum_{l=1}^k\widehat p_l\leqslantslantq\sum_{l=1}^k\widehat q_l$ for $k=1,\dots,m$,
\end{enumerate}
where $\mathbf{\widehat p}=(\widehat p_1,\dots,\widehat p_m)$ and $\mathbf{\widehat q}=(\widehat q_1,\dots,\widehat q_m)$
are nonincreasing permutations of $\mathbf p$ and $\mathbf q$, respectively
($\widehat p_1\geqslantslantq\dots\geqslantslantq\widehat p_m$ and $\widehat q_1\geqslantslantq\dots\geqslantslantq\widehat q_m$).
\end{definition}
The majorization has been studied in \cite{Hardy1952}, \cite{MarshallOlkin2011}, and many other sources.
The following condition ($S$) plays an important role:
We say that a pair $\mathbf p\prec\mathbf q$ satisfies the condition ($S$), if there exist $1\leqslantslantq l_1<l_2\leqslantslantq m$
such that $\widehat q_{l_1}=\widehat p_{l_1}+1$, $\widehat q_{l_2}=\widehat p_{l_2}-1$ and $\widehat q_l=\widehat p_l$ for $l\notin\{l_1,l_2\}$.
In \cite{KomRaj2018} we proved the following lemma.
\begin{lemma}\label{lm:muirh}
If $\mathbf p\prec\mathbf q$, then $\mathbf{\widehat p}=\mathbf{\widehat q}$
or there exist $\mathbf p=\mathbf p^0\prec\mathbf p^1\prec\dots\prec\mathbf p^I=\mathbf q$
such that $\mathbf p^{i-1}\prec\mathbf p^i$ satisfies ($S$) for $i=1,\dots,I$.
\end{lemma}
The main theorem of this section concerns polynomials defined as follows.
\begin{definition}
Let $m\in\mathbb N$ and let $\Pi$ be the set of all permutations of the set $\{1,\dots,m\}$.
For every $m$-tuple $\mathbf p=(p_1,\dots,p_m)$ of non-negative integers, we define the following polynomial:
$$W^{\mathbf p}(x_1,\dots,x_m):=\tfrac1{m!}\sum_{\pi\in\Pi}\prod_{l=1}^mx_{\pi(l)}^{p_l}.$$
\end{definition}
Clearly $W^{\mathbf p}$ is a symmetric polynomial with non-negative coefficients.
If $\mathbf q$ is a permutation of $\mathbf p$, then $W^{\mathbf q}=W^{\mathbf p}$.
In particular $W^{\mathbf p}=W^{\mathbf{\widehat p}}$.
\begin{theorem}\label{th:Muir}
Let $m\in\mathbb N$ and let $\mu_1,\dots,\mu_m$ be finite Borel measures on $\mathbb R$
satisfying $\mu_1(\mathbb R)=\dots=\mu_m(\mathbb R)$ and $\int|x|\ \mu_l(dx)<\infty$
for $l=1,\dots,m$.
Let $F_i(x)=\mu_i((-\infty,x])$ for $x\in\mathbb R$.
Assume that $(F_i-F_j)*(F_i-F_j)\geqslantslantq0$ for all $i$ and $j$.
If $\mathbf p\prec\mathbf q$,
then $W^{\mathbf p}(\mu_1,\dots,\mu_m)\leqslantslantq_{\text{\rm cx}} W^{\mathbf q}(\mu_1,\dots,\mu_m)$.
\end{theorem}
\begin{proof}
In view of Lemma \ref{lm:muirh} and transitivity of $\leqslantslantq_{\text{\rm cx}}$, it is enough
to consider the case when the pair $\mathbf p\prec\mathbf q$ satisfies condition ($S$).
Let $l_1<l_2$ be the indices given in condition ($S$).
For every $\pi\in\Pi$ we define $\pi'\in\Pi$ by $\pi'(l_1)=\pi(l_2)$, $\pi'(l_2)=\pi(l_1)$
and $\pi'(l)=\pi(l)$ for $l\notin\{l_1,l_2\}$. We have
\begin{multline*}
(W^{\mathbf q}-W^{\mathbf p})(x_1,\dots,x_m)=
\tfrac1{m!}\sum_{\pi\in\Pi}\leqslantslantft(\prod_{l=1}^mx_{\pi(l)}^{\widehat q_l}-\prod_{l=1}^mx_{\pi(l)}^{\widehat p_l}\right)=\\
\tfrac1{m!}\sum_{1\leqslantslantq u<v\leqslantslantq m}\sum_{\substack{\pi\in\Pi\\\pi(l_1)=u\\\pi(l_2)=v}}
\leqslantslantft(\prod_{l=1}^mx_{\pi(l)}^{\widehat q_l}+\prod_{l=1}^mx_{\pi'(l)}^{\widehat q_l}
-\prod_{l=1}^mx_{\pi(l)}^{\widehat p_l}-\prod_{l=1}^mx_{\pi'(l)}^{\widehat p_l}\right)=\\
\tfrac1{m!}\sum_{1\leqslantslantq u<v\leqslantslantq m}\sum_{\substack{\pi\in\Pi\\\pi(l_1)=u\\\pi(l_2)=v}}\prod_{l\neq l_1,l_2}x_{\pi(l)}^{\widehat p_l}
\leqslantslantft(x_u^{\widehat p_{l_1}+1}x_v^{\widehat q_{l_2}}+x_v^{\widehat p_{l_1}+1}x_u^{\widehat q_{l_2}}
-x_u^{\widehat p_{l_1}}x_v^{\widehat q_{l_2}+1}-x_v^{\widehat p_{l_1}}x_u^{\widehat q_{l_2}+1}\right)
\end{multline*}
Note that
\begin{multline*}
x_u^{\widehat p_{l_1}+1}x_v^{\widehat q_{l_2}}+x_v^{\widehat p_{l_1}+1}x_u^{\widehat q_{l_2}}
-x_u^{\widehat p_{l_1}}x_v^{\widehat q_{l_2}+1}-x_v^{\widehat p_{l_1}}x_u^{\widehat q_{l_2}+1}=\\
(x_u-x_v)\leqslantslantft(x_u^{\widehat p_{l_1}}x_v^{\widehat q_{l_2}}-x_v^{\widehat p_{l_1}}x_u^{\widehat q_{l_2}}\right)=
(x_u-x_v)^2\sum_{j=\widehat q_{l_2}}^{\widehat p_{l_1}-1}x_u^jx_v^{\widehat p_{l_1}+\widehat q_{l_2}-1-j}.
\end{multline*}
It follows that
$$(W^{\mathbf q}-W^{\mathbf p})(x_1,\dots,x_m)=
\sum_{1\leqslantslantq u<v\leqslantslantq m}(x_u-x_v)^2\sum_{\substack{\pi\in\Pi\\\pi(l_1)=u\\\pi(l_2)=v}}
\sum_{j=\widehat q_{l_2}}^{\widehat p_{l_1}-1}\frac{x_u^jx_v^{\widehat p_{l_1}+\widehat q_{l_2}-1-j}}{m!}\prod_{l\neq l_1,l_2}x_{\pi(l)}^{\widehat p_l}.
$$
By Theorem \ref{th:muir5}, we obtain $W^{\mathbf p}(\mu_1,\dots,\mu_m)\leqslantslantq_{\text{\rm cx}} W^{\mathbf q}(\mu_1,\dots,\mu_m)$.
\end{proof}
\begin{remark}
Theorem \ref{th:Muir} is an analogue of Muirhead's inequality (see \cite{Hardy1952}, Theorem 45 or \cite{MarshallOlkin2011}, Section 3G)
with positive numbers replaced by Borel measures on $\mathbb R$, multiplication replaced by convolution, and $\leqslantslantq$ replaced by $\leqslantslantq_{\text{\rm cx}}$.
Moreover, if $x_1,\dots,x_k>0$, then applying Theorem \ref{th:Muir} with $\mu_l=\delta_{\ln x_l}$ (for $l=1,\dots,k$)
and the convex function $\varphi(x)=e^x$, we obtain the classical Muirhead inequality with integer exponents:
If $\mathbf p\prec\mathbf q$ and $x_1,\dots,x_m>0$, then $W^{\mathbf p}(x_1,\dots,x_m)\leqslantslantq W^{\mathbf q}(x_1,\dots,x_m)$.
\end{remark}
If we apply Theorem \ref{th:Muir} with $(p)=(1,\dots,1)$ and $(q)=(m,0,\dots,0)$, we get
the following corollary, which generalizes
Ra\c{s}a type inequalities proved in \cite{MRW2017}, \cite{KomRaj2018} and \cite{Gav2018}.
\begin{corollary}
If $\mu_1,\dots,\mu_m$ are finite Borel measures on $\mathbb R$
satisfying assumptions of Theorem \ref{th:Muir}, then
$$\mu_1*\dots*\mu_m\leqslantslantq_{\text{\rm cx}}\tfrac1m\leqslantslantft[(\mu_1)^{*m}+\dots+(\mu_m)^{*m}\right].$$
In particular
$$\sum_{i_1,\dots,i_m=0}^n\bigl(b_{n,i_1}(x_1)\cdots b_{n,i_m}(x_1)+\dots+b_{n,i_1}(x_m)\dots b_{n,i_m}(x_m)
-mb_{n,i_1}(x_1)\dots b_{n,i_m}(x_m)\bigr)\varphi\leqslantslantft(\tfrac{i_1+\dots+i_m}{mn}\right)\geqslantslantq0,$$
in the case of $\mu_i=B(n,x_i)$ $(x_i\in[0,1])$ for $i=1,\dots,m.$
\end{corollary}
One might expect that every polynomial inequality valid for non-negative real numbers has its counterpart
for finite Borel measures and convex orders. The following example shows that it is very far from true.
\begin{example}
Let $P(x,y)=\frac12x^3y+\frac12xy^3$ and $Q(x,y)=\frac18x^4+\frac34x^2y^2+\frac18y^4$.
The polynomials $P$ and $Q$ are symmetric and homogeneous polynomials of degree $4$.
We have $Q(x,y)-P(x,y)=\frac18(x-y)^4\geqslantslantq0$ for every $x,y\in\mathbb R$.
Both $P$ and $Q$ have non-negative coefficients and $P(1,1)=Q(1,1)=1$.
It follows that $P(\mu,\nu)$ and $Q(\mu,\nu)$ are probability distributions
whenever $\mu$ and $\nu$ are probability distributions.
If the expected values (means) $\mathbb E\mu$ and $\mathbb E\nu$ are finite,
then $\mathbb E P(\mu,\nu)=2(\mathbb E\mu+\mathbb E\nu)=\mathbb E Q(\mu,\nu)$.
Despite all this regularity the inequality $P(\mu,\nu)\leqslantslantq_{\text{\rm cx}} Q(\mu,\nu)$
is not valid for $\mu=\delta_0$ and $\nu=\frac12\delta_0+\frac12\delta_1$
(note that $F-G\geqslantslantq 0$, hence $(F-G)*(F-G)\geqslantslantq 0$, cf.\ assumptions of Theorem 1 and Theorem \ref{th:Muir}).
Indeed, $P(\mu,\nu)=\frac5{16}\delta_0+\frac7{16}\delta_1+\frac3{16}\delta_2+\frac1{16}\delta_3$
and $Q(\mu,\nu)=\frac{41}{128}\delta_0+\frac{52}{128}\delta_1+\frac{30}{128}\delta_2+\frac4{128}\delta_3+\frac1{128}\delta_4$
and for the convex function $\varphi(x)=\max(0,x-2)$ we have $\int\varphi(x)P(\mu,\nu)(dx)=\frac1{16}>\frac6{128}=\int\varphi(x)Q(\mu,\nu)(dx)$,
hence $P(\mu,\nu)\not\leqslantslantq_{\text{\rm cx}} Q(\mu,\nu)$.
\end{example}
\section{Open problems}
For $n\in\mathbb{N}$
the classical Bernstein operators $B_n:\mathcal{C}([0,1])\to\mathcal{C}([0,1])$, defined by
$$\leqslantslantft(B_n f\right)(x)=\sum_{i=0}^n b_{n,i}(x)f\leqslantslantft(\tfrac in\right)\quad\text{for } x\in[0,1],$$
with the Bernstein basic polynomials
\[
b_{n,i}(x)=\binom{n}{i}x^i(1-x)^{n-i} \quad \text{for } \ i=0,1,\dots,n, \ x\in[0,1],
\]
are the most prominent positive linear approximation operators (see \cite{Lorentz1953}).
The inequality \eqref{eq:mainv3} is the probabilistic version of the following
inequality involving Bernstein polynomials and convex functions, that was conjectured as an open problem by
I.~Ra\c{s}a in~\cite{Rasa2014b}
\begin{equation}\label{MRW_eq:Rasa}
\sum_{i,j=0}^n\bigl(b_{n,i}(x)b_{n,j}(x)+b_{n,i}(y)b_{n,j}(y)-2b_{n,i}(x)b_{n,j}(y)\bigr)f\leqslantslantft(\frac{i+j}{2n}\right)\geqslantslant 0
\end{equation}
\par
\noindent for each convex function $f\in\mathcal{C}\bigl([0,1]\bigr)$ and for all $x,y\in[0,1]$.
Ra\c{s}a \cite{Rasa2017b} remarked, that \eqref{MRW_eq:Rasa} is equivalent to
\begin{equation}\label{Rasa_eq:Rasabis}
\leqslantslantft(B_{2n}f\right)(x) + \leqslantslantft(B_{2n}f\right)(y)
\geqslantslantq 2\;\sum_{i=0}^n \sum_{j=0}^n b_{n,i}(x)b_{n,j}(y)\ f\leqslantslantft(\frac{i+j}{2n}\right).
\end{equation}
B. Gavrea \cite{Gav2018} presented the following generalization of the problem of I.\ Ra\c{s}a \cite{Rasa2014b}.
\par
\textbf{Problem 1. \cite{Gav2018}} Let $D=[0,1]\times [0,1]$, $g\in C(D)$ and $n\in\mathbb{N}$. The Bernstein operator is then defined by
$$\leqslantslantft(B_{n,n}g\right)(x,y)=\sum_{i=0}^n \sum_{j=0}^n b_{n,i}(x)b_{n,j}(y)g\leqslantslantft(\frac{ i}{n},\frac{ j}{n}\right)\quad\text{for } \ (x,y)\in D.$$
Give a characterization of the class of of convex functions $g$ defined on $D$, satisfying
\begin{equation}\label{eq:Gavv1}
\leqslantslantft(B_{n,n}g\right)(x,x)+\leqslantslantft(B_{n,n}g\right)(y,y)-2\leqslantslantft(B_{n,n}g\right)(x,y)\geqslantslantq 0
\end{equation}
for all $(x,y)\in D$.
\begin{remark}[\cite{Gav2018}]
We note that, if $\varphi\in C([0,1])$ is a convex function, and
$$
g(x,y)=\varphi\leqslantslantft( \frac{x+y}{2} \right) \quad \text{for} \ (x,y)\in D,
$$
then \eqref{eq:Gavv1} coincides with the Ra\c{s}a inequality \eqref{MRW_eq:Rasa}.
\end{remark}
\begin{remark}
Note that if \eqref{eq:Gavv1} is satisfied for all $(x,y)\in D$, then also
\begin{equation}\label{eq:Gavv10}
\leqslantslantft(B_{n,n}g\right)(x,x)+\leqslantslantft(B_{n,n}g\right)(y,y)-2\leqslantslantft(B_{n,n}g\right)(y,x)\geqslantslantq 0
\end{equation}
for all $(x,y)\in D$. Adding inequalities \eqref{eq:Gavv1} and \eqref{eq:Gavv10}, we obtain
\begin{equation}\label{eq:Gavv20}
\leqslantslantft(B_{n,n}g\right)(x,x)+\leqslantslantft(B_{n,n}g\right)(y,y)-\leqslantslantft(B_{n,n}g\right)(x,y)-\leqslantslantft(B_{n,n}g\right)(y,x)\geqslantslantq 0.
\end{equation}
\end{remark}
Taking into account \eqref{eq:Gavv20}, we consider a modification of Problem 1.
\par
\textbf{Problem 1'.} Let $D=[0,1]\times [0,1]$, $g\in C(D)$ and $n\in\mathbb{N}$.
Give a characterization of the class of convex functions $g$ defined on $D$, satisfying
\begin{equation}\label{eq:Gavv2}
\leqslantslantft(B_{n,n}g\right)(x,x)+\leqslantslantft(B_{n,n}g\right)(y,y)-\leqslantslantft(B_{n,n}g\right)(x,y)-\leqslantslantft(B_{n,n}g\right)(y,x)\geqslantslantq 0
\end{equation}
for all $(x,y)\in D$.
\par
\begin{remark}\label{rem:prob}
The inequality \eqref{eq:Gavv2} has the probabilistic interpretation. It is equivalent to the following inequality
\begin{equation}\label{eq:Gavv3}
\mathbb{E}\, g\leqslantslantft(\frac{X}{n}, \frac{Y}{n}\right)+\mathbb{E}\, g\leqslantslantft(\frac{Y}{n}, \frac{X}{n}\right) \leqslantslantq \mathbb{E}\, g\leqslantslantft(\frac{X_1}{n}, \frac{X_2}{n}\right)+\mathbb{E}\, g\leqslantslantft(\frac{Y_1}{n}, \frac{Y_2}{n}\right),
\end{equation}
where $(X,Y)$, $(X_1,X_2)$, $(Y_1,Y_2)$ are pairs of independent random variables such that $X,X_1,X_2\sim B(n,x) $ and $Y,Y_1,Y_2\sim B(n,y)$.
\end{remark}
\par
We use the following notation: $X\sim\mu$ means that $\mu$ is the probability distribution of a random variable $X$.
The inequality \eqref{eq:Gavv3} is not satisfied for all convex functions $g\in C(D)$. Let us take $g(x,y)=|x-y|$. Then $g$ is convex,
$g(0,0)= g(1,1)=0$ and $g(0,1)= g(1,0)=1$. Let $X,X_1,X_2\sim B(n,0)=\delta_0$, $Y,Y_1,Y_2\sim B(n,1)=\delta_n$
be independent random variables. We obtain
$$\mathbb{E}\, g\leqslantslantft(\frac{X}{n},\frac{Y}{n}\right)+ \mathbb{E}\, g\leqslantslantft(\frac{Y}{n},\frac{X}{n}\right)=1+1>0+0=\mathbb{E}\, g\leqslantslantft(\frac{X_1}{n},\frac{X_2}{n}\right)+ \mathbb{E}\, g\leqslantslantft(\frac{Y_1}{n},\frac{Y_2}{n}\right),$$
consequently the inequality \eqref{eq:Gavv3} is not fulfilled.
\par
Let $g: \mathbb R^2 \to \mathbb R$ be a convex function. We consider the Jensen gap corresponding to $g$ that is given by
\begin{equation}\label{eq:cond2}
\mathcal{J}(g;(x_1, x_2), (y_1, y_2))=\frac{g(x_1,x_2)+g(y_1,y_2)}{2}-g\leqslantslantft( \frac{(x_1,x_2)+(y_1,y_2)}{2} \right)
\end{equation}
\noindent for all $x_1, x_2, y_1, y_2 \in \mathbb R.$
Since $g$ is convex,
\begin{equation*}\label{eq:cond1}
\mathcal{J}(g;(x_1, x_2), (y_1, y_2))\geqslantslantq 0\quad \text{and} \quad \mathcal{J}(g;(x_1, y_2), (y_1, x_2))\geqslantslantq 0
\end{equation*}
for all $x_1, x_2, y_1, y_2 \in \mathbb R$.
We will consider convex functions $g$ satisfying the inequality
\begin{equation}\label{eq:cond}
\mathcal{J}(g;(x_1, x_2), (y_1, y_2))\geqslantslantq \mathcal{J}(g;(x_1, y_2), (y_1, x_2))
\end{equation}
for all $x_1, x_2, y_1, y_2 \in \mathbb R$ such that
\begin{equation}(y_1-x_1)(y_2-x_2)>0.\end{equation}
By \eqref{eq:cond2}, we can write \eqref{eq:cond} in the form
\begin{equation*}
\frac{g(x_1,x_2)+g(y_1,y_2)}{2}-g\leqslantslantft( \frac{(x_1,x_2)+(y_1,y_2)}{2} \right) \geqslantslantq \frac{g(x_1,y_2)+g(y_1,x_2)}{2}-g\leqslantslantft( \frac{(x_1,y_2)+(y_1,x_2)}{2} \right),
\end{equation*}
or equivalently
\begin{equation}\label{eq:cond4}
g(x_1,x_2)+g(y_1,y_2) \geqslantslantq g(x_1,y_2)+g(y_1,x_2).
\end{equation}
The following theorem says, that if the convex function $g$ satisfies the inequality \eqref{eq:cond4}, which is equivalent to the inequality \eqref{eq:cond}, and the random variables $X$ and $Y$ (which are not necessary binomially distributed) are chosen to satisfy some sufficient condition, then
the inequality \eqref{eq:Gavv3} is satisfied (up to natural number $n$).
\begin{theorem}\label{th:cond}
Let $X$ and $Y$ be two independent random variables with finite first moments, such that
\begin{equation}\label{eq:cond5}
X\leqslantslantq_{\text{\rm st}} Y \quad \text{or} \quad Y\leqslantslantq_{\text{\rm st}} X.
\end{equation}
Let $g: \mathbb R^2 \to \mathbb R$ be a convex function satisfying
\begin{equation}\label{eq:cond6}
g(x_1,x_2)+g(y_1,y_2) \geqslantslantq g(x_1,y_2)+g(y_1,x_2)
\end{equation}
for all $x_1, x_2, y_1, y_2 \in \mathbb R$ such that
\begin{equation}(y_1-x_1)(y_2-x_2)>0.\end{equation}
Then
\begin{equation}\label{eq:cond7}
\mathbb{E}\, g(X_1,X_2)+\mathbb{E}\, g(Y_1,Y_2) \geqslantslantq \mathbb{E}\, g(X,Y)+\mathbb{E}\, g(Y,X),
\end{equation}
where $X_1,X_2$ and $Y_1,Y_2$ are independent random variables such that $X_1,X_2\sim X$ and $Y_1,Y_2\sim Y$.
\end{theorem}
\begin{proof}
Without loss of generality we may assume that $X\leqslantslantq_{\text{\rm st}} Y$.
By Theorem \ref{th:1a1}, there exist two independent random vectors $(X_1,Y_1)$ and $(X_2,Y_2)$ such that
\begin{equation}\label{eq:suf3}
X_1,X_2\sim X, \quad Y_1,Y_2\sim Y, \quad P(X_1\leqslantslantq Y_1)=1 \quad \text{and} \quad P(X_2\leqslantslantq Y_2)=1.
\end{equation}
By \eqref{eq:suf3} and \eqref{eq:cond6} we obtain
$$
P\leqslantslantft( g(X_1,X_2)+g(Y_1,Y_2) \geqslantslantq g(X_1,Y_2)+ g(Y_1,X_2) \right) =1
$$
which implies
\begin{equation}\label{eq:suf4}
\mathbb{E}\, g(X_1,X_2)+\mathbb{E}\, g(Y_1,Y_2) \geqslantslantq \mathbb{E}\, g(X_1,Y_2)+ \mathbb{E}\, g(Y_1,X_2).
\end{equation}
Since \eqref{eq:suf4} is equivalent to \eqref{eq:cond7}, the theorem is proved.
\end{proof}
Note that binomially distributed random variables $X\sim B(n,x)$ and $Y\sim B(n,y)$ satisfy the condition \eqref{eq:cond5} (see \cite{KomRaj2018}).
Therefore we obtain (from Theorem \ref{th:cond} and Remark \ref{rem:prob}) the following
sufficient condition for functions $g$ that appear in Problem 1'.
\begin{corollary}
Let $g: [0,1]^2 \to \mathbb R$ be a convex function satisfying
\begin{equation*}\label{eq:cond8}
g(x_1,x_2)+g(y_1,y_2) \geqslantslantq g(x_1,y_2)+g(y_1,x_2)
\end{equation*}
for all $x_1, x_2, y_1, y_2 \in \mathbb R$ such that $(y_1-x_1)(y_2-x_2)>0$.
Then
\begin{equation*}\label{eq:Gavv30}
\leqslantslantft(B_{n,n}g\right)(x,x)+\leqslantslantft(B_{n,n}g\right)(y,y)-\leqslantslantft(B_{n,n}g\right)(x,y)-\leqslantslantft(B_{n,n}g\right)(y,x)\geqslantslantq 0.
\end{equation*}
\end{corollary}
\par
We present a new open problem, which is a generalization of Problem 1 \cite{Gav2018}.
\par
\textbf{Problem 3.} Let $k\in\mathbb{N}$, $n_i \in \mathbb{N}$ for $i=1,\ldots k$, $\sum_{i=1}^k n_i=m$, $D=[0,1]^k$ and $g\in C(D)$. The Bernstein type operator $B_{n_1,\ldots,n_k}$ is defined by
$$\leqslantslantft(B_{n_1,\ldots,n_k}g\right)(x_1,\ldots,x_k)=\sum_{i_1=0}^{n_1}\ldots \sum_{i_k=0}^{n_k}b_{n_1,i_1}(x_1)\ldots b_{n_k,i_k}(x_k) g\leqslantslantft(\frac{ i_1}{n_1},\ldots,\frac{ i_k}{n_k}\right)$$
for $(x_1,\ldots,x_k)\in D$.
Give a characterization of the class of convex functions $g$ defined on $D$ and satisfying
\begin{equation}\label{eq:Gavv6}
\leqslantslantft(B_{n_1,\ldots,n_k}g\right)(x_1,\ldots,x_k)\leqslantslantq
\sum_{i=1}^k\frac{n_i}m\;\leqslantslantft(B_{n_1,\ldots,n_k}g\right)\leqslantslantft(x_i,\ldots,x_i\right)
\end{equation}
for all $(x_1,\ldots,x_k)\in D$.
\par
In \cite{KomRaj2018bis}, we proved the following generalization of the Ra\c{s}a inequality \eqref{Rasa_eq:Rasabis}.
\begin{theorem} [\cite{KomRaj2018bis}, Theorem 2.2, (2.5)]\label{th:2.1}
Let $k\in\mathbb{N}$, $n_i \in \mathbb{N}$ for $i=1,\ldots k$ and $\sum_{i=1}^k n_i=m$. Then
\par
\begin{equation}\label{eq:6prim}
\sum_{i_1=0}^{n_1}\ldots \sum_{i_k=0}^{n_k}b_{n_1,i_1}(x_1)\ldots b_{n_k,i_k}(x_k) \ \varphi\leqslantslantft(\frac{i_1+\ldots+i_k}m\right)\leqslantslantq
\sum_{i=1}^k\frac{n_i}m\;\leqslantslantft(B_{m}\varphi\right)\leqslantslantft(x_i\right)
\end{equation}
\par
for all convex functions $\varphi\in\mathcal{C}([0,1])$ and $x_1, \ldots, x_k \in [0,1]$, .
\end{theorem}
\begin{remark}
We note that, if $\varphi\in C([0,1])$ is a convex function, and
$$
g(x_1,\ldots,x_k)=\varphi\leqslantslantft( \frac{n_1}{m}x_1+ \ldots+ \frac{n_k}{m}x_k\right) \quad \text{for}\ (x_1,\ldots,x_k)\in D,
$$
then the inequality \eqref{eq:Gavv6} coincides with \eqref{eq:6prim}, which was proved in \cite{KomRaj2018bis}.
\end{remark}
\begin{remark}
If the inequality \eqref{eq:Gavv6} is satisfied for all $(x_1,\ldots,x_k)\in D$, then
\begin{multline}\label{rm:gav13}
\leqslantslantft(B_{n_1,\ldots,n_k}g\right)(x_1,\ldots,x_k)+\leqslantslantft(B_{n_1,\ldots,n_k}g\right)(x_2,\ldots,x_k,x_1)+\ldots +\leqslantslantft(B_{n_1,\ldots,n_k}g\right)(x_k,x_1,\ldots,x_{k-1}) \\
\leqslantslantq
\sum_{i=1}^k\;\leqslantslantft(B_{n_1,\ldots,n_k}g\right)\leqslantslantft(x_i,\ldots,x_i\right)
\end{multline}
for all $(x_1,\ldots,x_k)\in D.$
\end{remark}
\textbf{Problem 3'.} With assumptions such as in Problem 3., give a characterization of the class of convex functions $g$ defined on $D$ and satisfying \eqref{rm:gav13} for all $(x_1,\ldots,x_k)\in D$.
\par
B. Gavrea \cite{Gav2018} presented also the following open problem.
\par
\textbf{Problem 4. (\cite{Gav2018}, Problem 2.)}
If $a_{n,k}(x)=\binom{n+k}{k}\:(1-x)^{n+1}x^k$ and if $\varphi$ is a convex continuous function on $[0,1]$, prove or disprove the following inequality:
\begin{equation}\label{eq:Gavtres}
\sum_{i=0}^{\infty} \sum_{j=0}^{\infty} \leqslantslantft(a_{n,i}(x) \:a_{n,j}(x)+a_{n,i}(y) \:a_{n,j}(y)-2a_{n,i}(x) \:a_{n,j}(y)\right)\varphi\leqslantslantft(\frac{i+j }{2n+i+j}\right)\geqslantslantq0.
\end{equation}
\par
We will show that \eqref{eq:Gavtres} is not valid (in general).
Let $\varphi(u)=u$ and let $x\neq y$.
Let $\mu$ be the negative binomial probability distribution with parameters $n+1$ and $x$.
Then $\mu$ is concentrated on the set of non-negative integers and it satisfies
$\mu(\{k\})=\binom{n+k}{k}\:(1-x)^{n+1}x^k=a_{n,k}(x)$ for $k=0,1,\dots$.
Similarly, let $\nu$ be the negative binomial probability distribution with parameters $n+1$ and $y$.
Let $F$ and $G$ be the cumulative distribution functions of $\mu$ and $\nu$, respectively.
If $x<y$, then $F(u)<G(u)$ for $u\geqslantslantq0$. If $x>y$, then $F(u)>G(u)$ for $u\geqslantslantq0$ (see proof of Lemma~2.5.c in~\cite{KomRaj2018}).
In both cases we have $F(u)=G(u)=0$ for $u<0$, thus $(F-G)*(F-G)(u)>0$ for $u>0$.
Observe that for every $u\geqslantslantq0$ we have
$$\tfrac u{2n+u}=\tfrac u{2n}-\int_0^\infty\tfrac{4n}{(2n+y)^3}(u-y)_+\lambda(dy).$$
Consequently,
\begin{multline*}
\sum_{i=0}^\infty\sum_{j=0}^\infty\leqslantslantft(a_{n,i}(x) \:a_{n,j}(x)+a_{n,i}(y) \:a_{n,j}(y)-2a_{n,i}(x) \:a_{n,j}(y)\right)\tfrac{i+j}{2n+i+j}=
\int_{-\infty}^\infty\tfrac u{2n+u}(\mu*\mu+\nu*\nu-2\mu*\nu)(du)=\\
\int_{-\infty}^\infty\tfrac u{2n}(\mu*\mu+\nu*\nu-2\mu*\nu)(du)
-\int_{-\infty}^\infty\int_0^\infty\tfrac{4n}{(2n+y)^3}(u-y)_+\lambda(dy)(\mu*\mu+\nu*\nu-2\mu*\nu)(du)=\\
0-\int_0^\infty\tfrac{4n}{(2n+y)^3}\int_{-\infty}^\infty(u-y)_+(\mu*\mu+\nu*\nu-2\mu*\nu)(du)\lambda(dy)=\\
-\int_0^\infty\tfrac{4n}{(2n+y)^3}((F-G)*(F-G))(y)\lambda(dy)<0.
\end{multline*}
It follows that the inequality \eqref{eq:Gavtres} is not valid for $\varphi(u)=u$.
However, if $u\mapsto \varphi\leqslantslantft(\frac{u}{n+u}\right)$ is convex on $[0,\infty)$
(e.g. if $\varphi$ is convex and decreasing),
then inequality \eqref{eq:Gavtres} is valid (cf. Theorem~\ref{th:necsuf} above and Theorem~2.6.c in~\cite{KomRaj2018}).
Using the same method it can be shown that if $u\mapsto \varphi\leqslantslantft(\frac{u}{n+u}\right)$ is concave on $[0,\infty)$ but it is not linear
(e.g. if $\varphi$ is concave and strictly increasing), then \eqref{eq:Gavtres} is not valid.
\end{document} |
\begin{document}
\begin{abstract}
We consider stochastic differential equation involving pathwise integral with respect to fractional Brownian motion. The estimates for the Hurst parameter are constructed according to first- and second-order quadratic variations of observed values of the solution. The rate of convergence of these estimates to the true value of a parameter is established.
\end{abstract}
\begin{keyword}
Fractional Brownian motion; stochastic differential equation; first- and second-order quadratic variations; estimates of Hurst parameter; rate of convergence.
\MSC 60G22, 60H10
\end{keyword}
\title{The rate of convergence of estimate for Hurst index of fractional Brownian motion involved into stochastic differential equation noteref{t1}
\section{Introduction}
Consider stochastic differential equation
\begin{equation}\label{SIE}
X_t=\xi + \int_0^t f(X_s)\,d s + \int_0^t g(X_s)\,d B^H_s,\quad t\in[0,T],\ T>0,
\end{equation}
where $f$ and $g$ are measurable functions, $B^H$ is a fractional Brownian motion (fBm) with Hurst index $1/2<H<1$, $\xi$ is a random variable. It is well-known that almost all sample paths of $B^H$ have bounded
$p$-variations for $p>1/H.$ Therefore it is natural to define the integral with respect to fractional Brownian motion as pathwise Riemann-Stieltjes integral (see, e.g., \cite{y} for the original definition and \cite{dn1} for the advanced results).
A solution of stochastic differential equation \eqref{SIE} on a given filtered
probability space $(\Omega,{\mathcal F},{\bf P}, \mathbb{F}=\{\mathcal F_t\}, t\in[0,T])$, with respect to the
fixed fBm $(B^H,\mathbb{F}),$ $1/2<H<1$ and with $\mathcal F_0$-measurable initial condition $\xi$ is an adapted to
the filtration $\mathbb{F}$ continuous process $X=\{X_t\colon\ 0\ls t\ls T\}$ such
that $X_0=\xi$ a.s.,
\[
{\bf P}\bigg(\int_0^t\vert f(X_s)\vert\,ds+\big\vert\int_0^t
g(X_s)\,dB^H_s\big\vert<\infty\bigg)=1\qquad\mbox{for every}\ 0\ls t\ls T,
\]
and its almost all sample paths satisfy (\ref{SIE}).
For $0<\alpha\ls 1$, $\mathcal{C}^{1+\a}(\mathbb{R})$ denotes the set of all
$\mathcal{C}^{1}$-functions $g$: $\mathbb{R}\to\mathbb{R}$ such that
\[
\sup_{x}\vert g^\hbox{\bf P}ime(x) \vert + \sup_{x\neq y}\frac{\vert g^\hbox{\bf P}ime(x) - g^\hbox{\bf P}ime(y) \vert}{\vert x
- y \vert^\a}<\infty.
\]
Let $f$ be a Lipschitz function and let
$g\in\mathcal{C}^{1+\a}(\mathbb{R})$, $\frac1H-1<\alpha\ls 1$. Then there
exists a unique solution of equation (\ref{SIE}) with almost all sample paths in
the class of all continuous functions defined on $[0,T]$ with bounded $p$-variation for any $p>\frac1H$
(see \cite{du},
\cite{ly}, \cite{ly1} and \cite{kk}). Different (but similar in many features) approach to the integration with respect to fractional Brownian motion based on the integration in Besov spaces and corresponding stochastic differential equations were studied in \cite{nr}, see also \cite{biagini} and \cite{mishura} where the different approaches to stochastic integration and to stochastic differential equations involving fractional Brownian motion are summarized.
The main goal of the present paper is to establish the rate of convergence of two estimates of Hurst parameter to the true value of a parameter. The estimates are based on the two types of the quadratic variations of the observed solution to stochastic differential equation involving the integral with respect to fractional Brownian motion and considered on the fixed interval $[0,T]$. The paper is organized as follows: Section 2 contains some preliminary information. More precisely, subsection 2.1 describes the properties of $p$-variations and of the integrals with respect to the functions of bounded $p$-variations while subsection 2.2 contains the results on the asymptotic behavior of the normalized first- and second-order quadratic variations of fractional Brownian motion. Section 3 describes the rate of convergence of the first- and second-order quadratic variations of the solution to stochastic differential equation involving fBm. Section 4 contains the main result concerning the rate of convergence of the constructed estimates of Hurst index to its true value when the diameter of partitions of the interval $[0,T]$ tends to zero. Section 5 contains simulation results.
\section{Preliminaries}
\subsection{The functions of bounded p-variation}
First, we mention some information concerning $p$-variation and the functions of bounded $p$-variation. It is containing, e.g., in \cite{dn1} and \cite{y}.
Let interval $[a,b]\subset \mathbb{R}.$ Consider the following class of functions:
\[
\mathcal{W}_p\big([a,b]\big):=\big\{f: [a,b]\to\mathbb{R}{:}\ v_p\big(f;
[a,b]\big)<\infty\big\},
\]
where
\[
v_p\big(f; [a,b]\big)=\sup_{\pi}\sum_{k=1}^n \big\vert
f(x_k)-f(x_{k-1})\big\vert^p.
\]
Here $\pi=\{x_i{:}\ i=0,\ldots,n\}$ stands for any finite partition of $[a,b]$ such that
$a=x_0<x_i<\cdots<x_n=b$. Denote $\Pi([a,b])$ the class of such partitions. We say that function $f$ has
bounded $p$-variation on $[a;b]$ if $v_p(f;[a,b])< \infty$.
Let $V_p(f):=V_p(f;[a,b])=v_p^{1/p}(f; [a,b])$. Then for any fixed $f$ we have that $V_p(f)$ is a non-increasing
function of $p$. It means that for any $0<q<p$ the relation $V_p(f)\ls V_q(f)$ holds.
Let $a<c<b$ and let $f\in{\mathcal W}_p([a,b])$ for some $p\in(0,\infty).$ Then
\begin{gather*}
v_p\big(f;[a,c]\big)+v_p\big(f;[c,b]\big)\ls
v_p\big(f;[a,b]\big),\\[4pt]
V_p\big(f;[a,b]\big)\ls V_p\big(f;[a,c]\big)+V_p\big(f;[c,b]\big).
\end{gather*}
Let $f\in {\cal W}_q([a,b])$ and $h\in {\cal W}_p([a,b])$, where $p^{-1}+q^{-1}>1$. Then the well-known Love-Young inequality states that
\begin{equation}\label{1.2}
\Bigg\vert\int_a^bf\,d h-f(y)\big[ h(b)-h(a) \big]\Bigg\vert \ls C_{p,q}
V_q\big(f;[a,b]\big)V_p\big(h;[a,b]\big),
\end{equation}
whence
\begin{equation}\label{1.3}
V_p\Bigg(\int_a^{\hbox{$\bf \cdot$}} f\,d h;[a,b]\Bigg)\ls C_{p,q}
V_{q,\infty}\big(f;[a,b]\big) V_p\big(h;[a,b]\big).
\end{equation}
Here $V_{q,\infty}(f;[a,b])=V_q(f;[a,b])+\sup_{a\ls x\ls b}\vert f(x)\vert$,
$C_{p,q}=\zeta(p^{-1}+q^{-1})$ and $\zeta(s)=\sum_{n\gs 1} n^{-s}$ is the Riemann zeta function.
Further, for any $y\in[a,b]$
\begin{align}\label{1.3a}
V_p\Bigg(\int_a^{\hbox{$\bf \cdot$}} [f(x)-f(y)]\,d h(x);[a,b]\Bigg)\ls& C_{p,q}
\big[V_q\big(f;[a,b]\big)\nonumber\\+\sup_{a\ls x\ls b}\vert f(x)-f(y)\vert\big] V_p\big(h;[a,b]\big)
\ls& 2C_{p,q}V_q\big(f;[a,b]\big) V_p\big(h;[a,b]\big).
\end{align}
Denote
$$|A|_\infty=\sup_{x\in \mathbb{R}}|A(x)|,\;\;|A|_\alpha=\sup_{x,y\in \mathbb{R}}\frac{|A(x)-A(y)|}{|x-y|^\alpha}.
$$
Let $F$ be a Lipschitz function and let $G\in\mathcal{C}^{1+\a}(\mathbb{R})$ with $0<\alpha\ls 1$ and $1\ls p< 1+\a$.
Then for any $h\in\mathcal{W}_p([a,b])$
\begin{align}
V_{p,\infty}\big( F(h);[a,b]\big)\ls& L V_p\big( h;[a,b]\big) +\sup_{a\ls x\ls b}\vert F(h(x))-F(h(a))\vert+\vert F(h(a))\vert\nonumber\\
\ls& 2L V_p\big( h;[a,b]\big)+\vert F(h(a))\vert \label{1.3b},\\
V_{p/\alpha,\infty}\big( G(h);[a,b]\big)\ls& V_{p,\infty}\big( G(h);[a,b]\big)\ls 2\vert G^\hbox{\bf P}ime\vert_\infty V_p\big( h;[a,b]\big)+\vert G(h(a))\vert\label{1.3c},
\end{align}
and
\begin{align}
&V_{p/\alpha,\infty}\big( G^\hbox{\bf P}ime(h);[a,b]\big)\nonumber\\
&\quad\ls\vert G^\hbox{\bf P}ime\vert_\alpha V_p^\alpha\big(h;[a,b]\big) +\sup_{a\ls x\ls b}\vert G^\hbox{\bf P}ime(h(x))-G^\hbox{\bf P}ime(h(a))\vert+\vert G^\hbox{\bf P}ime(h(a))\vert\nonumber\\
&\quad\ls 2\vert G^\hbox{\bf P}ime\vert_\alpha V^\alpha_p\big( h;[a,b]\big)+\vert G^\hbox{\bf P}ime(h(a))\vert\,\label{1.3d}.
\end{align}
Let $f \in\mathcal{W}_p([a,b])$ and $p_1>p>0.$ Then
\begin{equation} \label{1.4}
V_{p_1}\bigl( f;[a,b] \bigr)\ls \hbox{\rm Osc}(f;[a,b])^{(p_1-p)/p_1}
V_p^{p/p_1}\bigl( f;[a,b] \bigr),
\end{equation}
where $\hbox{\rm Osc}(f;[a,b])=\sup\{\vert f(x)-f(y)\vert \colon\ x,y\in [a,b]\}$.
Take functions $f_1,f_2\in {\cal W}_p([a,b])$, $0<p<\infty$. Then $f_1f_2\in {\cal W}_p([a,b])$ and
\begin{equation}\label{1.4a}
V_{p}\big(f_1f_2;[a,b]\big)\leq V_{p,\infty}\big(f_1f_2;[a,b]\big)\ls C_p V_{p,\infty}\big(f_1;[a,b]\big)
V_{p,\infty}\big(f_2;[a,b]\big).
\end{equation}
Let $f_1 \in{\mathcal W}_q([a,b])$ and $f_2\in{\mathcal W}_p([a,b]).$ Then it follows from Young's version of
H\"older's inequality that for any partition
$\pi\in\Pi([a,b])$ and for any $p^{-1}+q^{-1}\gs 1$
\begin{equation}\label{1.5}
\sum_i V_{q}\big( f_1;[x_{i-1},x_i] \big)V_{p}\big( f_2;[x_{i-1},x_i] \big) \ls V_q\big(
f_1;[a,b] \big)V_p\big( f_2;[a,b] \big).
\end{equation}
Second, we state some facts from the theory of Riemann-Stieltjes integration. Let $f\in {\cal W}_q([a,b])$ and $h\in {\cal W}_p([a,b])$ with $0<p<\infty$, $q>0,$ $1/p+\allowbreak 1/q>1.$ Let symbol (R) stands for the Riemann integration, and (RS) stands for Riemann-Stieltjes integration. Then
integral $(RS)\int_a^b f\,\mathrm{d}h$ exists under the additional assumption that $f$ and $h$ have no common discontinuities.
\begin{prop}\label{2.1} Let $f: [a, b] \to \mathbb{R}$ be such function that for some $1 \ls p < 2$ $f\in \mathcal{CW}_p([a,b])$. Also, let $F: \mathbb{R} \to \mathbb{R}$ be a differentiable function with locally Lipschitz derivative $F^\hbox{\bf P}ime$. Then composition $F^\hbox{\bf P}ime(f)$ is Riemann-Stieltjes integrable with respect to $f$ and
\[
F(f(b))-F(f(a))=(RS)\int_a^b F^\hbox{\bf P}ime(f(x))\,df(x).
\]
\end{prop}
Furthermore, the following substitution rule holds.
\begin{prop}\label{2.2} Let $f_1,f_2$ and $f_3$ be functions from $\mathcal{CW}_p([a, b])$,
$1 \ls p < 2$. Then
\[
(RS)\int_a^b f_1(x)\,d\bigg((RS)\int_a^x f_2(y)\,df_3(y)\bigg)=(RS)\int_a^b f_1(x)f_2(x)\,df_3(x).
\]
\end{prop}
Finally, assume that
\[
F_1(x)=(R)\int_a^x f_1(y)\,dy\qquad\mbox{and}\qquad F_2(x)=(RS)\int_a^x f_2(y)\,df_3(y),
\]
where $f_1$ is continuous function, $f_2, f_3\in\mathcal{CW}_p([a,b])$ for some $1 \ls p < 2$, and $Q$ is a differentiable function with locally Lipschitz derivative $q$. It follows from Propositions \ref{2.1} and \ref{2.2} that
\begin{align}\label{2.3}
&Q(F_1(x)+F_2(x))-Q(0)=\int_a^x q(F_1(y)+F_2(y))\,d(F_1(y)+F_2(y))\nonumber\\
&\quad=\int_a^x q(F_1(y)+F_2(y))f_1(y)\,dy+\int_a^x q(F_1(y)+F_2(y))f_2(y)\,d f_3(y).
\end{align}
\subsection{Asymptotic property of the first- and second-order quadratic variations of fractional Brownian motion}
Consider the fractional Brownian motion (fBm) $B^H=\{B^H_t, t\in[0,T]\}$ with Hurst index $H\in(\frac12,1)$.
Its sample paths are almost all locally H\" older
up to order $H$. Moreover, for any $0<\gamma<H$ we have that $L^{H,\gamma}_T:= \sup_{\substack{s\ne t\\s,t\ls T}}{\frac{\vert B^H_t-B^H_s\vert}{\vert
t-s\vert^\gamma}}$ is finite a.s. and even more, ${\bf E}\big(L^{H,\gamma}_T\big)^k<\infty$
for any $ k\gs 1.$ The following estimate for the $p$-variation of fBm is evident:
\begin{equation}\label{1.6}
V_p\big(B^H;[s,t]\big)\ls L^{H,1/p}_T\, (t-s)^{1/p},
\end{equation}
where $s<t\ls T,$ $p>1/H$.
Let $\pi_n=\{0=t^n_0<t^n_1<\cdots<t^n_n=T\}$, $T>0$, be a sequence of
uniform partitions of interval $[0,T]$ with $t^n_k=\frac{kT}{n}$ for all $n\in\mathbb{N}$ and all $k\in\{0,\ldots,n\}$, and let $X$ be some real-valued stochastic process defined on the interval $[0,T]$.
\begin{dfn} The normalized first- and second-order quadratic variations of $X$
taking along the partitions $(\pi_n)_{n\in\mathbb{N}}$ and corresponding to the value $1/2<H<1$ are defined
as
\[
V_n^{(1)}(X,2)=n^{2H-1}\sum_{k=1}^n\big(\D^{(1)}_{k,n} X\big)^2,\quad
\D^{(1)}_{k,n} X=X(t^n_{k})-X(t^n_{k-1}),
\]
and
\[
V^{(2)}_n(X,2)=n^{2H-1}\sum_{k=1}^{n-1}\big(\D^{(2)}_{k,n} X\big)^2,\quad
\D^{(2)}_{k,n} X=X\big(t^n_{k+1}\big)-2X\big(t^n_{k}\big) +X\big(t^n_{k-1}\big).
\]
\end{dfn}
For simplicity, we shall omit index $n$ for points $t_k^n$ of partitions $\pi_n$.
It is known (see, e.g., Gladyshev \cite{glad}) that $V_n^{(1)}(B^H,2)\to T$ a.s. as $n\to\infty$. Also, it was proved in Benasi et al. \cite{bcij} and Istas et al. \cite{IL} that $V_n^{(2)}(B^H,2)\to (4-2^{2H})T$ a.s. as $n\to\infty$.
Denote
\begin{align*}
V^{(1)}_n(B^H,2)_t=&n^{2H-1}\sum_{k=1}^{r(t)}(\D^{(1)}_{k,n} B^H)^2\,,\\ V^{(2)}_n(B^H,2)_t=&n^{2H-1}\sum_{k=1}^{r(t)-1}(\D^{(2)}_{k,n} B^H)^2,\quad t\in[0,T],
\end{align*}
where $r(t)=\max\{k\colon\ t_k\leq t\}=[\frac{tn}{T}]$. It is evident that
\begin{equation}\label{lyg}
\hbox{\bf E} V^{(1)}_n(B^H,2)_t=\rho(t)\quad\mbox{and}\quad \hbox{\bf E} V^{(2)}_n(B^H,2)_t=(4-2^{2H})\rho(t),
\end{equation}
where $\rho(t)=\max\{t_k\colon\ t_k\ls t\}$.
The following classical result will be used in the proof of Theorem \ref{konv1.}.
\begin{lem}\label{LevyOct}{(L\'evy-Octaviani inequality)}
Let $X_1,\ldots,X_n$ be independent random variables. Then for any fixed $t,s\gs 0$
\[
\hbox{\bf P}\bigg(\max_{1\ls i\ls n}\bigg\vert\sum_{j=1}^i X_j\bigg\vert>t+s\bigg)\ls \frac{\hbox{\bf P}\big(\big\vert\sum_{j=1}^n X_j\big\vert>t\big)}{1-\max_{1\ls i\ls n} \hbox{\bf P}\big(\big\vert\sum_{j=i}^{n} X_j\big\vert>s\big)}\,.\qquad\Box
\]
\end{lem}
\begin{thm}\label{konv1.} The following asymptotic property holds for the first- and second-order quadratic variations of fractional Brownian motion:
\begin{equation}\label{mars4}
\sup_{t\ls T}\big\vert V_n^{(i)}(B^H,2)_t-\hbox{\bf E} V_n^{(i)}(B^H,2)_t\big\vert=\mathcal{O}\big(n^{-1/2}\ln^{1/2} n\big)\quad\mbox{a.s.},\; i=1,2.
\end{equation}
\end{thm}
\begin{rem}\label{rem7} It follows from \eqref{lyg} and \eqref{mars4} that $\sup_{t\ls T}\big\vert V_n^{(i)}(B^H,2)_t\big\vert=\mathcal{O}(1)$ a.s., $i=1,2.$.
\end{rem}
\begin{proof}
We can consider
$V_n^{(i)}\big(B^H,2\big)$ as the square of the Euclidean norm of the $n$-dimensional Gaussian vector $X_n$ with the components
\[
n^{H-1/2}\Delta^{(i)}_{k,n} B^H, \qquad 1\ls k\ls n-(i-1).
\]
Obviously, one can get a new $n$-dimensional Gaussian vector $\widetilde{X}_n$ with independent components applying the linear transformation to $X_n$. It means that there exist nonnegative real numbers $(\lambda^{(i)}_{1,n}, \ldots, \lambda^{(i)}_{n-(i-1),n})$ and such $n-(i-1)$-dimensional Gaussian vector $Y_n$ with independent Gaussian $\mathcal{N}(0,1)$-components that
\[
V_n^{(i)}\big(B^H,2\big)=\sum_{j=1}^{n-(i-1)}\lambda_{j,n}\big(Y^{(j)}_n\big)^2.
\]
The numbers $(\lambda^{(i)}_{1,n}, \ldots, \lambda^{(i)}_{n-(i-1),n})$ are the eigenvalues of the symmetric $n-(i-1)\times n-(i-1)$-matrix
\[
\Big(n^{2H-1}\hbox{\bf E}\big[\Delta^{(i)}_{j,n} B^H\Delta^{(i)}_{k,n} B^H\big]\Big)_{1\ls j,k\ls n-(i-1)}.
\]
Now we can apply the Hanson and Wright's inequality (see Hanson et al. \cite{HW} or Begyn \cite{begyn1}), and it yields that for $\eps>0$
\begin{align}\label{nelyg4}
\hbox{\bf P}\bigg(\bigg\vert\sum_{j=k}^{n-(i-1)}\lambda^{(i)}_{j,n}\big[\big(Y^{(j)}_n\big)^2-1\big]\bigg\vert\geq \eps\bigg) \ls& 2\exp\bigg({-}\min\bigg[\frac{C_1 \eps}{\l_{k,n}^{*(i)}}\,,\frac{C_2 \eps^2}{\sum_{j=k}^{n-(i-1)}(\l^{(i)}_{j,n})^2}\bigg]\bigg)\nonumber\\
\ls& 2\exp\bigg({-}\min\bigg[\frac{C_1 \eps}{\l_n^{*(i)}}\,,\frac{C_2 \eps^2}{\sum_{j=1}^{n-(i-1)}(\l^{(i)}_{j,n})^2}\bigg]\bigg),
\end{align}
where $C_1$, $C_2$ are nonnegative constants, $\l_{k,n}^{*(i)}=\max_{k\ls j\ls n-(i-1)}\l^{(i)}_{j,n}$, $\l_n^{*(i)}=\max_{1\ls j\ls n-(i-1)}\l^{(i)}_{j,n}$.
The evident equality holds:
\[
\sum_{j=1}^{n-(i-1)}\l^{(i)}_{j,n}=\hbox{\bf E} V_n^{(i)}\big(B^H,2\big).
\]
Furthermore, it follows from (\ref{lyg}) that the sequence $\hbox{\bf E} V_n^{(i)}\big(B^H,2\big),\;n\geq 1$ is bounded. So, the sums $\sum_{j=1}^{n-(i-1)}\l^{(i)}_{j,n}$ are bounded as well.
It is easy to check that
\[
\sum_{j=1}^{n-(i-1)}(\l^{(i)}_{j,n})^2\ls \l_n^{*(i)}\sum_{j=1}^{n-(i-1)}\l^{(i)}_{j,n}.
\]
Therefore for any $0<\eps\ls 1$ the inequality (\ref{nelyg4}) can be rewritten as
\begin{equation}\label{ivert}
\hbox{\bf P}\bigg(\bigg\vert\sum_{j=i}^{n-(i-1)}\lambda_{j,n}\big[\big(Y^{(j)}_n\big)^2-1\big]\bigg\vert\geq \eps\bigg)
\ls 2\exp\bigg({-}\frac{K\eps^2}{\l_n^{*(i)}}\bigg),
\end{equation}
where $K$ is a positive constant.
Now we use L\'evy-Octaviani inequality (see Lemma \ref{LevyOct}) and evident inequality
\[
\frac{x}{1-x}\ls 2x \qquad \mbox{for}\quad 0< x\ls 1/2
\]
to obtain the bound
\begin{gather*}
\hbox{\bf P}\bigg(\max_{1\ls k\ls n-(i-1)}\bigg\vert\sum_{j=1}^k\lambda^{(i)}_{j,n}\big[\big(Y^{(j)}_n\big)^2-1\big]\bigg\vert> 2\eps\bigg)\\
\ls \frac{2\exp\Big({-}\frac{K \eps^2}{\l_n^{*(i)}}\Big)}{1- 2\exp\Big({-}\frac{K \eps^2}{\l_n^{*(i)}}\Big)}\ls 4\exp\bigg({-}\frac{K \eps^2}{\l_n^{*(i)}}\bigg),
\end{gather*}
assuming that
\[
\exp\bigg({-}\frac{K \eps^2}{\l_n^{*(i)}}\bigg)\ls 1/4\quad\mbox{and}\quad 0<\varepsilon\leq 1.
\]
So, for the values of parameters mentioned above,
\[
\hbox{\bf P}\Bigg(n^{2H-1}\max_{1\ls k\ls n-(i-1)}\bigg\vert \sum_{j=1}^k (\D^{(i)}_{j,n} B^H)^2-\sum_{j=1}^k \hbox{\bf E}(\D^{(i)}_{j,n} B^H )^2 \bigg \vert> 2\eps\Bigg)\ls 4\exp\bigg({-}\frac{K \eps^2}{\l_n^{*(i)}}\bigg).
\]
Furthermore,
\[
\l_n^{*(i)}\ls K n^{2H-1}\max_{1\ls k\ls n-(i-1)}\sum_{j=1}^{n-(i-1)} \vert d_{jkn}^{(i)}\vert\,,
\]
where $d^{(i)}_{jkn}=\hbox{\bf E}\D^{(i)}_{j,n}B^H\D^{(i)}_{k,n}B^H$. From Gladyshev \cite{glad} and Begyn \cite{begyn1} we get
\begin{equation}\label{ivert1}
\l_n^{*(i)}\ls C n^{-1}.
\end{equation}
Now we set
\[
\eps_n^2=\frac{2C}{K}\,n^{-1}\ln n
\]
and conclude that
\begin{gather*}
\hbox{\bf P}\Bigg(n^{2H-1}\max_{1\ls k\ls n-(i-1)}\bigg\vert \sum_{j=1}^k (\D^{(i)}_{j,n} B^H)^2-\sum_{j=1}^k \hbox{\bf E}(\D^{(i)}_{k,n} B^H )^2 \bigg \vert> 2\eps_n\Bigg)\\
\ls 4\exp\bigg({-}2\ln n\bigg)=\frac{4}{n^2}\,.
\end{gather*}
It means that
\[
\sum_{n=0}^\infty \hbox{\bf P}\Bigg(n^{2H-1}\max_{1\ls k\ls n-(i-1)}\bigg\vert \sum_{j=1}^k (\D^{(i)}_{j,n} B^H)^2-\sum_{j=1}^k \hbox{\bf E}(\D^{(i)}_{j,n} B^H)^2 \bigg \vert> 2\eps_n\Bigg)<\infty.
\]
Finally, we get the statement of the present theorem from the Borel-Cantelli lemma and the evident equality
\begin{align*}
&\sup_{t\ls T}\big\vert V_n^{(i)}(B^H,2)_t-\hbox{\bf E} V_n^{(i)}(B^H,2)_t\big\vert\\
&\quad=n^{2H-1}\max_{1\ls k\ls n-(i-1)}\bigg\vert \sum_{j=1}^k (\D^{(i)}_{j,n} B^H)^2-\sum_{j=1}^k \hbox{\bf E}(\D^{(i)}_{j,n} B^H)^2 \bigg \vert\,.
\end{align*}
\end{proof}
\section{The rate of convergence of the first- and second-order quadratic variations of the solution of stochastic differential equation}
First, we formulate the following result from \cite{kubmel3} about convergence of first- and second-order quadratic variation.
\begin{thm}\label{var1}
Consider stochastic differential equation ~\eqref{SIE}, where function $f$ is Lipschitz and $g\in\mathcal{C}^{1+\a}$ for some $0<\a<1$. Let $X$ be its solution. Then
\begin{equation}\label{rezult}
V_n^{(i)}(X,2)\to c^{(i)}\int_0^T g^2\big(X(t)\big)\,d t\,\
\text{a.s.} \quad\text{as}\ n\to \infty,
\end{equation}
where
\[
c^{(i)}=\begin{cases}1& \mbox{for $i=1$},\\
(4-2^{2H})& \mbox{for $i=2$}.
\end{cases}
\]
\end{thm}
Second, we prove the following auxiliary result.
\begin{lem}\label{disc} Let $X$ be a solution of stochastic differential equation ~\eqref{SIE}. Define a step-wise process $X^{\pi}$ that is a discretization of process $X$:
\[
X^{\pi}_t=\begin{cases}X(t_k)& \mbox{for}\quad t\in[t_k,t_{k+1}),\ k=0,1,\ldots,n-2,\\
X(t_{n-1})& \mbox{for}\quad t\in[t_{n-1},t_n].
\end{cases}
\]
Then for any $p>\frac1H$ we have that
\begin{gather*}
\sup_{t\ls T}\vert X^{\pi}_t-X_t\vert= \mathcal{O}\big(n^{-1/p}\big).
\end{gather*}
\end{lem}
\begin{proof} Consider $t\in [t_k,t_{k+1})$. We get immediately from the Love-Young inequality (\ref{1.2}) that
\begin{align*}
\vert X^{\pi}_t-X_t\vert=&\bigg\vert\int_{\rho^n(t)}^t f(X_s)\,ds+\int_{\rho^n(t)}^t g(X_s)\,dB^{H}_s\bigg\vert\\
\ls& T n^{-1}\sup_{t_k\ls t\ls t_{k+1}}\vert f(X_t)\vert
+C_{p,p} V_{p,\infty}(g(X);[t_k,t_{k+1}]) V_p(B^H;[t_k,t_{k+1}]).
\end{align*}
Further,
\begin{equation}\label{mars1}
\sup_{t_k\ls t\ls t_{k+1}}\vert f(X_t)\vert\ls \sup_{t\ls T} \vert f(X_t)-f(\xi)\vert+\vert f(\xi)\vert\ls LV_p(X;[0,t])+\vert f(\xi)\vert,
\end{equation}
where $L$ is a Lipschitz constant for $f$, and
\begin{align}\label{mars2}
V_{p,\infty}(g(X);[t_k,t_{k+1}])\ls& \vert g^\hbox{\bf P}ime\vert_\infty V_p(X;[t_k,t_{k+1}])+\sup_{t\ls T} \vert g(X_t)-g(\xi)\vert+\vert g(\xi)\vert\nonumber\\
\ls& 2\vert g^\hbox{\bf P}ime\vert_\infty V_p(X;[0,T])+\vert g(\xi)\vert.
\end{align}
We get the statement of the lemma from \eqref{mars1}, \eqref{mars2} and inequality (\ref{1.6}) .\end{proof}
Now we prove the main result of this section which specifies the rate of convergence in Theorem \ref{var1}.
\begin{thm}\label{var2} Let the conditions of Theorem \ref{var1} hold and, in addition, $\alpha>\frac1H-1$. Then
\begin{equation}\label{mars3}
V_n^{(i)}(X,2)-c^{(i)}\int_0^T g^2\big(X(t)\big)\,d t=\mathcal{O}\big(n^{-1/4}\ln^{1/4} n\big).
\end{equation}
\end{thm}
\begin{proof} Decompose the left-hand side of \eqref{mars3} into three parts:
\begin{align*}
I_n^{(i)}:=&V_n^{(i)}(X,2)-c^{(i)}\int_0^T g^2\big(X(t)\big)\,d t = I_n^{(1,i)}+I_n^{(2,i)}+I_n^{(3,i)},
\end{align*}
where
\begin{align*} I_n^{(1,i)}=&n^{2H-1}\sum_{k=1}^{n-(i-1)} \big(\D_{k,n}^{(i)}X\big)^2-\sum_{k=1}^{n-(i-1)} g^2(X_{k-1+(i-1)})\big(\D_{k,n}^{(i)}B^H\big)^2,\\
I_n^{(2,i)}=&n^{2H-1}\sum_{k=1}^{n-(i-1)} g^2(X_{k-1+(i-1)})\big(\D_{k,n}^{(i)}B^H\big)^2\nonumber\\
&- \sum_{k=1}^{n-(i-1)} g^2(X_{k-1+(i-1)})\hbox{\bf E}\big(\D_{k,n}^{(i)} B^H\big)^2,\\
I_n^{(3,i)}=& n^{2H-1} \sum_{k=1}^{n-(i-1)} g^2(X_{k-1+(i-1)})\hbox{\bf E}\big(\D_{k,n}^{(i)}B^H\big)^2- c^{(i)}\int_0^T g^2(X_s)ds,
\end{align*}
and $X_k=X(t_k)$. We start with the most simple term $I_n^{(3,i)}$ and get immediately, similarly to bounds contained in \eqref{1.3c}, that for any $p>\frac1H$
\begin{align*}
\vert I_n^{(3,i)}\vert\ls& c^{(i)}\sum_{k=1-(i-1)}^{n-(i-1)}\int^{t_{k+(i-1)}}_{t_{k-1+(i-1)}} \big\vert g^2(X(t_{k-1+(i-1)}))-g^2(X_s) \big\vert\,ds\\
\ls& 2c^{(i)}T \vert g^\hbox{\bf P}ime\vert_\infty \sup_{t\ls T}\vert X^{\pi}_t-X_t\vert\cdot \big[\vert g^\hbox{\bf P}ime\vert_\infty V_p(X;[0,T])+\vert g(\xi)\vert\big]\,.
\end{align*}
In order to estimate $I_n^{(2,i)}$, denote
\[
S^{(i)}_t=n^{2H-1}\sum_{k=1}^{r(t)-({i-1})}\big(\Delta^{(i)}_{k,n} B^H\big)^2\,,\quad t\in[0,T],\quad i=1,2.
\]
Then
\begin{eqnarray*}
n^{2H-1}\sum_{k=1}^{n-({i-1})}g^2(X_{k-1+(i-1)})\big(\Delta^{(i)}_{k,n} B^H\big)^2=\int_0^T g^2(X_t)\,dS^{(i)}_t
\end{eqnarray*}
and
\begin{align*}
&n^{2H-1}\sum_{k=1}^{n-({i-1})}g^2(X_{k-1+(i-1)})\big[\big(\Delta^{(i)}_{k,n} B^H\big)^2-\hbox{\bf E} \big(\Delta^{(i)}_{k,n} B^H\big)^2\big]\\
&\quad= \int_0^T g^2(X_t)\,d\big[S^{(i)}_t-\hbox{\bf E} S^{(i)}_t\big].
\end{align*}
Note that $1/p+1/2>1$ for $\frac1H<p<2$. Therefore, we obtain from the Love-Young inequality and from (\ref{1.4})-(\ref{1.4a}) that
\begin{align*}
|I^{(2,i)}_n|=&\bigg\vert \int_0^T g^2(X_t)\,d\big[S^{(i)}_t-\hbox{\bf E} S^{(i)}_t\big]\bigg\vert\\
\ls& C_{p,2}V_{p,\infty}\big(g^2(X) ;[0,T]\big) V_2\big(S^{(i)}-\hbox{\bf E} S^{(i)} ;[0,T]\big)\\
\ls& C_{p,2}\big\{\hbox{\rm Osc}\big( S^{(i)}-\hbox{\bf E} S^{(i)} ;[0,T]\big)\big\}^{1/2} V_{p,\infty}\big(g^2(X);[0,T]\big)\\
&\times V_1^{1/2}\big(S^{(i)}-\hbox{\bf E} S^{(i)} ;[0,T]\big)\\
\ls& 2C_{p,2}\Big(\sup_{t\ls T}\big\vert S^{(i)}_t-\hbox{\bf E} S^{(i)}_t\big\vert\Big)^{1/2} V^2_{p,\infty}\big(g(X);[0,T]\big)\\
&\times \bigg[n^{2H-1}\sum_{k=1}^{n-(i-1)}\big(\Delta^{(i)}_{k,n} B^H\big)^2+c^{(i)}T\bigg]^{1/2}.
\end{align*}
It follows from Theorem \ref{konv1.}, Remark \ref{rem7}, and (\ref{1.3c}) that the rate of convergence of $I^{(2,i)}_n$ is $O(n^{-1/4}\ln^{1/4} n)$.
It still remains to estimate $I^{(1,i)}_n$. Consider only $i=2$, the proof for $i=1$ is similar.
Denote
\begin{align*}
J_{k}^1=&\int_{t_k}^{t_{k+1}}[f(X_s)-f(X_k)]\,ds-\int_{t_{k-1}}^{t_{k}}[f(X_s)-f(X_k)]\,ds,\\
J_{k}^2=&\int_{t_{k-1}}^{t_{k}} \bigg(g(X_k)- g(X_s)
-\int_{s}^{t_k} g^\hbox{\bf P}ime(X_k)f(X_k)\,du-\int_{s}^{t_k} g^\hbox{\bf P}ime(X_k)g(X_k)\,dB^H_u\bigg)dB^H_s,\\
J_{k}^3=&\int_{t_k}^{t_{k+1}} \bigg( g(X_s)-g(X_k)
-\int_{t_k}^s g^\hbox{\bf P}ime(X_k)f(X_k)\,du-\int_{t_k}^s g^\hbox{\bf P}ime(X_k)g(X_k)\,dB^H_u\bigg)dB^H_s, \\
J_{k}^4=&g^\hbox{\bf P}ime(X_k)f(X_k)\Big(\int_{t_k}^{t_{k+1}}(s-t_k)\,dB^H_s+\int_{t_{k-1}}^{t_{k}}(t_k-s)\,dB^H_s\Big),\\
J_{k}^5=&\frac{1}{2}\,g^\hbox{\bf P}ime(X_k)g(X_k)\Big( \big(\Delta_{k,n}^{(1)} B^H\big)^2+\big(\Delta_{k+1,n}^{(1)} B^H\big)^2\Big),\qquad
J_{k}^6=g(X_k)\Delta^{(2)}_{k,n} B^H.
\end{align*}
Equalities
\begin{align*}
\int_{t_{k-1}}^{t_{k}} \bigg(\int_{s}^{t_k} g^\hbox{\bf P}ime(X_k)g(X_k)\,dB^H_u\bigg)dB^H_s =& \frac{1}{2}\,g^\hbox{\bf P}ime(X_k)g(X_k) \big(\Delta_k^{(1)} B^H\big)^2,\\
\int_{t_k}^{t_{k+1}}\bigg(\int_{t_k}^s g^\hbox{\bf P}ime(X_k)g(X_k)\,dB^H_u\bigg)dB^H_s =&\frac{1}{2}\,g^\hbox{\bf P}ime(X_k)g(X_k) \big(\Delta_{k+1,n}^{(1)} B^H\big)^2.
\end{align*}
(see Proposition \ref{2.1}) imply
\[
\Delta^{(2)}_{k,n} X=\sum_{l=1}^{6}J_{k}^l.
\]
Taking into account Lipschitz property of $f$ and Lemma \ref{disc}, we can conclude that
\[
\vert f(X_t)-f(X_k)\vert\ls L \sup_{t\ls T}\vert X^{\pi}_t-X_t\vert=\mathcal{O}\big(n^{-1/p}\big).
\]
Therefore
\begin{align*}
\sum_{k=1}^{n-1}(J_{k}^1)^2\ls& 2Tn^{-1} \sum_{k=1}^{n-1}\int_{t_k}^{t_{k+1}} [f(X_s)-f(X_k)]^2\,ds\\
&+2Tn^{-1} \sum_{k=1}^{n-1}\int_{t_{k-1}}^{t_{k}} [f(X_s)-f(X_k)]^2\,ds\\
\ls& 4Tn^{-1} L^2\Big(\sup_{t\ls T}\vert X^{\pi}_t-X_t\vert\Big)^2 \\
=&\mathcal{O}\big(n^{-1-2/p}\big).
\end{align*}
Consider $J_{k}^2$. It follows from equality (\ref{2.3}) that for any fixed $t\in[t_{k-1},t_k]$
\begin{equation}\label{Ito1}
g(X_{k})-g(X_t)=\int^{t_k}_t g^\hbox{\bf P}ime(X_s)f(X_s)\,ds+\int^{t_k}_t g^\hbox{\bf P}ime(X_s)g(X_s)\,dB^H_s.
\end{equation}
Substituting equality (\ref{Ito1}) into $J_{k}^2$ we get
\begin{align}\label{mars5}
\vert J_{k}^2\vert\ls&\bigg\vert \int_{t_{k-1}}^{t_k}\int^{t_k}_s \big[ g^\hbox{\bf P}ime(X_u)f(X_u)
- g^\hbox{\bf P}ime(X_k)f(X_k)\big]du \,dB^H_s\bigg\vert\nonumber\\
&+\bigg\vert \int_{t_{k-1}}^{t_k}\int^{t_k}_s \big[ g^\hbox{\bf P}ime(X_u)g(X_u)- g^\hbox{\bf P}ime(X_k)g(X_k)\big]dB^H_u \,dB^H_s\bigg\vert\,.
\end{align}
Transforming identically the first term in the right-hand side of \eqref{mars5} and applying to it Love-Young inequality (\ref{1.2}), we conclude that for any $p>\frac1H$
\begin{align}\label{vert1}
&\bigg\vert\int_{t_{k-1}}^{t_k}\int^{t_k}_s \big[ g^\hbox{\bf P}ime(X_u)f(X_u)- g^\hbox{\bf P}ime(X_k)f(X_k)\big]du \,dB^H_s\bigg\vert \nonumber\\
&\quad\ls C_{p,1} V_1\bigg(\int_\cdot^{t_{k}} \big[ g^\hbox{\bf P}ime(X_u)f(X_u)
- g^\hbox{\bf P}ime(X_k)f(X_k)\big]du;[t_{k-1},t_k]\bigg)V_p\big( B^H;[t_{k-1},t_k]\big)\nonumber\\
&\quad\ls C_{p,1}V_p\big( B^H;[t_{k-1},t_k]\big)\int^{t_k}_{t_{k-1}} \big\vert g^\hbox{\bf P}ime(X_u)f(X_u)- g^\hbox{\bf P}ime(X_k)f(X_k)\big\vert\,du.
\end{align}
Henceforth we consider the following interval of the values of $p$: $\frac1H<p< 1+\a$. Then it follows from inequality (\ref{1.3a}) that the second term in the right-hand side of \eqref{mars5} admits the bound:
\begin{align}\label{mars6}
&\bigg\vert \int_{t_{k-1}}^{t_k}\int^{t_k}_s \big[ g^\hbox{\bf P}ime(X_u)g(X_u)- g^\hbox{\bf P}ime(X_k)g(X_k)\big]dB^H_u \,dB^H_s\bigg\vert\nonumber\\
&\quad\ls C_{p,p/\alpha} V_{p/\alpha}\bigg(\int^{t_{k}}_\cdot \big[ g^\hbox{\bf P}ime(X_u)g(X_u)
- g^\hbox{\bf P}ime(X_k)g(X_k)\big]dB^H_u;[t_{k-1},t_k]\bigg)\nonumber\\
&\qquad\times V_p\big( B^H;[t_{k-1},t_k]\big)\nonumber\\
&\quad\ls 2C^2_{p,p/\alpha}V_{p/\alpha}\big( g^\hbox{\bf P}ime(X_u)g(X_u)
;[t_{k-1},t_k]\big) V^2_p\big( B^H;[t_{k-1},t_k]\big).
\end{align}
We conclude from \eqref{mars5}--\eqref{mars6} that
\begin{align*}
\vert J^2_k\vert
\ls& Tn^{-1} C_{p,1} V_{p/\alpha}\big( g^\hbox{\bf P}ime(X)f(X);[t_k,t_{k+1}]\big)V_p\big( B^H;[t_k,t_{k+1}]\big)\\
&+ 2C^2_{p,p/\alpha} V_{p/\alpha}\big( g^\hbox{\bf P}ime(X)g(X);[t_k,t_{k+1}]\big)V^2_p\big( B^H;[t_k,t_{k+1}]\big).
\end{align*}
Applying inequalities (\ref{1.4a}) and (\ref{1.5}) we immediately obtain that
\begin{align*}
\sum_{k=1}^n \big( J^2_k\big)^2\ls& 2T^2 C^2_{p,1} n^{-2}\max_{0\ls k\ls n-1}\big[ V_{p/\alpha}\big( g^\hbox{\bf P}ime(X)f(X);[t_k,t_{k+1}]\big)
V_p\big( B^H;[t_k,t_{k+1}]\big)\big]\\
&\times V_{p/\alpha}\big( g^\hbox{\bf P}ime(X)f(X);[0,T]\big)V_p\big( B^H;[0,T]\big)\\
&+ 4C^4_{p,p/\alpha}\max_{0\ls k\ls n-1}\big[ V_{p/\alpha}\big( g^\hbox{\bf P}ime(X)g(X);[t_k,t_{k+1}]\big)V^3_p\big( B^H;[t_k,t_{k+1}]\big)\big]\\
&\times V_{p/\alpha}\big( g^\hbox{\bf P}ime(X)g(X);[0,T]\big)V_p\big( B^H;[0,T]\big)\\
\ls& 2 T^2 C^2_{p,1} n^{-2}\max_{0\ls k\ls n-1}\big[V_p\big( B^H;[t_k,t_{k+1}]\big)\big]
V^2_{p/\alpha,\infty}\big( g^\hbox{\bf P}ime(X);[0,T]\big)\\
&\times V^2_{p,\infty}\big( f(X);[0,T]\big)V_p\big( B^H;[0,T]\big)\\
&+ 4C^4_{p,p/\alpha}\max_{0\ls k\ls n-1}\big[V^3_p\big( B^H;[t_k,t_{k+1}]\big)\big]\\
&\times V^2_{p/\alpha,\infty}\big( g^\hbox{\bf P}ime(X);[0,T]\big) V^2_{p/\alpha,\infty}\big( g(X);[0,T]\big)V_p\big( B^H;[0,T]\big).
\end{align*}
It follows from the inequalities (\ref{1.3b})--(\ref{1.3d}) that the values of the variations
\[
V_{p,\infty}\big( f(X);[0,T]\big), \qquad V_{p/\alpha,\infty}\big( g(X);[0,T]\big), \qquad\text{and}\qquad V_{p/\alpha,\infty}\big( g^\hbox{\bf P}ime(X);[0,T]\big)
\]
are finite. Therefore we get from (\ref{1.6}) that
\[
\sum_{k=1}^n \big( J^2_k\big)^2= \mathcal{O}(n^{-2-1/p})+ \mathcal{O}(n^{-3/p})=\mathcal{O}(n^{-3/p}).
\]
The similar reasonings lead to the similar bound for $ J^3_k$, and we conclude that
\[
\sum_{k=0}^{n-1} [J^2_k+J^3_k]^2=\mathcal{O}\big(n^{-3/p}\big).
\]
Consider $J_k^4$. It consists of two terms that can be estimated in a similar way. Applying inequalities \eqref{1.2} and (\ref{1.6}), we obtain the following bound for the first term:
\begin{align*}
& \sum_{k=1}^{n-1} \big[g^\hbox{\bf P}ime(X_k)f(X_k)\big]^2\bigg(\int_{t_k}^{t_{k+1}}(s-t_k)\,dB^H_s\bigg)^2\\
&\quad\ls C^2_{p,1}\sum_{k=1}^{n-1} \big[g^\hbox{\bf P}ime(X_k)f(X_k)\big]^2 (t_{k+1}-t_k)^2V^2_p\big( B^H;[t_k,t_{k+1}]\big)\\
&\quad\ls n^{-1}T^2C^2_{p,1}\max_{1\ls k\ls n-1}\big[g^\hbox{\bf P}ime(X_k)f(X_k) V_p\big( B^H;[t_k,t_{k+1}]\big)\big]^2
=\mathcal{O}\big(n^{-1-2/p}\big).
\end{align*}
As a consequence,
\[
\sum_{k=1}^{n-1} \big(J^4_k\big)^2=\mathcal{O}\big(n^{-1-2/p}\big).
\]
Furthermore, note that under our assumptions $\sup_{s\in[0,T]}|g(X_s)|<\infty$ and $\sup_{s\in[0,T]}|g^\hbox{\bf P}ime(X_s)|<\infty$ a.s. Therefore we have for the first term in $J_k^5$ that
\begin{align*}
\sum_{k=1}^{n-1}\Big[ g^\hbox{\bf P}ime(X_k)g(X_k)
\big(\Delta_{k+1,n}^{(1)} B^H\big)^2\big]^2
\ls& \max_{1\ls k\ls n-1}\big[g^\hbox{\bf P}ime(X_k)g(X_k)\big]^2
\sum_{k=0}^{n-1}\big(\Delta_{k+1,n}^{(1)} B^H\big)^4\\
=&\mathcal{O}\big(n^{1-4/p}\big).
\end{align*}
The second term is bounded in a similar way, and we conclude that
\[
\sum_{k=1}^{n-1} \big(J^5_k\big)^2=\mathcal{O}\big(n^{1-4/p}\big).
\]
Thus
\[
\bigg(\sum_{l=1}^{5}J_{k}^l\bigg)^2=\mathcal{O}\big(n^{-1-2/p}\lor n^{-3/p}\lor n^{1-4/p}\big)=\mathcal{O}\big(n^{1-4/p}\big).
\]
At last,
\begin{align}\label{konv}
n^{2H-1}\sum_{k=1}^{n-1}\big[\Delta^{(2)}_{k,n}
X-g(X_k)\Delta^{(2)}_{k,n} B^H\big]^2=&\mathcal{O}\big(n^{1-4/p+2H-1}\big)\nonumber\\
=&\mathcal{O}\big(n^{-4/p+2H}\big)
\end{align}
for any $\frac1H< p< 1+\a$. Set $1/p=H-\eps$ for $\eps<(H/2-1/16)\wedge(H-\frac{1}{1+\alpha})$. Then
\begin{align}\label{konv1}
&V_n^{(2)}(X,2)-c^{(2)}\int_0^T g^2(X_s)\,ds\nonumber\\
&\quad=\mathcal{O}\big(n^{-4/p+2H}\big)+\mathcal{O}\big(n^{-1/4}\ln^{1/4} n\big)+\mathcal{O}\big(n^{-1/p}\big)\nonumber\\
&\quad=\mathcal{O}\big(n^{-2H+4\eps}\big)+\mathcal{O}\big(n^{-1/4}\ln^{1/4} n\big)+\mathcal{O}\big(n^{-H+\eps}\big)\nonumber\\
&\quad=\mathcal{O}\big(n^{-1/4}\ln^{1/4} n\big).
\end{align}
\end{proof}
\section{The rate of convergence of estimators of Hurst index}
Consider the following statistics: $$R_n^{(i)}=\frac{\sum_{k=1}^{2n-(i-1)}(\Delta^{(i)}_{k,2n}X)^2}{\sum_{k=1}^{n-(i-1)}(\Delta^{(i)}_{k,n}X)^2}$$ and construct the following estimate of Hurst index $H$:
\[
\widehat{H}^{(i)}_n=\bigg(\frac{1}{2} - \frac{1}{2\ln2}\ln R_n^{(i)}\bigg){\bf 1}_{\widetilde{C}_n},
\]
where
\[
\widetilde{C}_n=\bigg\{2^{-1}\big(1-2 n^{-1/4}(\ln n)^{1/4+\b}\big)\ls R_n^{(i)}\ls 1+2 n^{-1/4}(\ln n)^{1/4+\b} \bigg\},\qquad\beta>0.
\]
Further, introduce the following notation: $g^{(i)}(T)=c^{(i)}\int_0^T g^2(X_s)ds$.
\begin{thm}\label{mainthm} Let conditions of Theorem \ref{var1} hold with $\alpha>\frac1H-1.$ Also, let $X$ be a solution of (\ref{SIE}) and assume that random variable $g^{(i)}(T)$ is separated from zero: there exists a constant $c_0>0$ such that $g^{(i)}(T)\gs c_0$ a.s. Then $\widehat{H}^{(i)}_n$ is a strongly consistent estimator of the Hurst index $H$ and the following rate of convergence holds\emph{:}
\[
\vert \widehat H^{(i)}_n-H\vert=\mathcal{O}\big(n^{-1/4}(\ln n)^{1/4+\b}\big)\quad\mbox{a.s.},
\]
for any $\b>0$.
\end{thm}
\begin{proof} Consider a sequence $1>\d_n\downarrow 0$ as $n\rightarrow\infty$. It will be specified later on. Introduce the events
\[
C_n=\bigg\{\frac12(1-\d_n)\ls R_n^{(i)}\ls 1+\d_n \bigg\}.
\]
Also, introduce the notations
\[
A^{(i)}_n=V_{2n}^{(i)}(X,2)\quad \mbox{and}\quad B^{(i)}_n=V_n^{(i)}(X,2)
\]
and note that $2^{2H-1}R_n^{(i)}=\frac{A_n^{(i)}}{B_n^{(i)}}.$
Then $$ C_n=\bigg\{2^{2H-2}(1-\d_n)\ls\frac{A_n^{(i)}}{B_n^{(i)}}\ls 2^{2H-1}(1+\d_n )\bigg\},$$
and estimate $\widehat{H}^{(i)}_n$ has a form
\[
\widehat{H}^{(i)}_n=\bigg(\frac{1}{2} - \frac{1}{2\ln2}\ln R_n^{(i)}\bigg){\bf 1}_{{C}_n}.
\]
It is easy to see that $\overline C_n:=\Omega\backslash C_n$ has a form
\begin{align}
\overline C_n
=&\bigg\{\frac{A^{(i)}_n}{B^{(i)}_n}< 2^{2H-2}(1-\d_n)\bigg\}\bigcup \bigg\{\frac{A^{(i)}_n}{B^{(i)}_n}> 2^{2H-1}(1+\d_n)\bigg\}\nonumber\\
\subset& \bigg\{\bigg\vert\frac{A^{(i)}_n}{B^{(i)}_n}-1\bigg\vert>\d_n\bigg\}.
\end{align}
Then
\begin{align*}
\widehat{H}^{(i)}_n=&H{\bf 1}_{C_n}-\frac{1}{2\ln2}\ln \frac{(2n)^{2H-1}V_{2n}^{(i)}(X,2)}{ n^{2H-1}V_n^{(i)}(X,2)}\,{\bf 1}_{C_n}\\
=&H{\bf 1}_{C_n}-\frac{1}{2\ln2}\ln \frac{A^{(i)}_n}{B^{(i)}_n}\,{\bf 1}_{C_n}.
\end{align*}
The latter representation implies that
\begin{align}\label{1.7}
\big\vert\widehat{H}^{(i)}_n-H\big\vert\ls& H{\bf 1}_{\big\{\big\vert\frac{A^{(i)}_n}{B^{(i)}_n}-1\big\vert>\d_n\big\}}+\frac{1}{2\ln2}\bigg\vert\ln \frac{A^{(i)}_n}{ B^{(i)}_n}\bigg\vert\,{\bf 1}_{\big\{1-\d_n\ls\frac{A^{(i)}_n}{B^{(i)}_n}\ls 1+\d_n\big\}}\nonumber\\
&-\bigg(\frac{1}{2\ln2}\,\ln \frac{A^{(i)}_n}{ B^{(i)}_n}\bigg){\bf 1}_{\big\{2^{2H-2}(1-\d_n)\ls\frac{A^{(i)}_n}{B^{(i)}_n}< 1-\d_n \big\}}\nonumber\\
&+\bigg(\frac{1}{2\ln2}\,\ln \frac{A^{(i)}_n}{ B^{(i)}_n}\bigg){\bf 1}_{\big\{1+\d_n\ls\frac{A^{(i)}_n}{B^{(i)}_n}< 2^{2H-1}(1+\d_n ) \big\}}
:=\sum_{l=1}^4 L_n^l.
\end{align}
In what follows we need an elementary inequalities: $-\ln(1-x)\leq 2\ln(1+x)\ls 2x$ provided that $0\ls x\ls 1/2$.
Consider $L_n^2$. We divide it in two parts. As to the first part, it is obvious that
\[
\bigg(\ln \frac{A^{(i)}_n}{ B^{(i)}_n}\bigg){\bf 1}_{\big\{1-\d_n\ls \frac{A^{(i)}_n}{B^{(i)}_n}<1\big\}}=\bigg(\ln\bigg[1-\bigg(1-\frac{A^{(i)}_n}{ B^{(i)}_n}\bigg)\bigg]\bigg){\bf 1}_{\big\{1-\d_n\ls \frac{A^{(i)}_n}{B^{(i)}_n}<1\big\}},
\]
and
\[
1-\d_n\ls \frac{A^{(i)}_n}{B^{(i)}_n}<1\quad\text{implies that}\quad 0<1- \frac{A^{(i)}_n}{ B^{(i)}_n}\ls \d_n.
\]
Applying inequality $-\ln(1-x)\ls 2x$, $0\ls x\ls 1/2$, we deduce that for $\d_n\ls 1/2$
\[
\bigg(-\ln \frac{A^{(i)}_n}{ B^{(i)}_n}\bigg){\bf 1}_{\big\{1-\d_n\ls \frac{A^{(i)}_n}{B^{(i)}_n}<1\big\}}\ls 2\bigg(1-\frac{A^{(i)}_n}{ B^{(i)}_n}\bigg){\bf 1}_{\big\{1-\d_n\ls \frac{A^{(i)}_n}{B^{(i)}_n}<1\big\}}\ls 2\d_n{\bf 1}_{\big\{1-\d_n\ls \frac{A^{(i)}_n}{B^{(i)}_n}<1\big\}}.
\]
As to the second part,
\begin{align*}
\bigg(\ln \frac{A^{(i)}_n}{ B^{(i)}_n}\bigg){\bf 1}_{\big\{1\ls\frac{A^{(i)}_n}{B^{(i)}_n}\ls 1+\d_n\big\}}=&\bigg(\ln\bigg[1+\bigg(\frac{A^{(i)}_n}{ B^{(i)}_n}-1\bigg)\bigg]\bigg){\bf 1}_{\big\{1\ls\frac{A^{(i)}_n}{B^{(i)}_n}\ls 1+\d_n\big\}}\\
\ls&\bigg(\frac{A^{(i)}_n}{ B^{(i)}_n}-1\bigg){\bf 1}_{\big\{1\ls\frac{A^{(i)}_n}{B^{(i)}_n}\ls 1+\d_n\big\}}
\ls \d_n{\bf 1}_{\big\{1\ls\frac{A^{(i)}_n}{B^{(i)}_n}\ls 1+\d_n\big\}}.
\end{align*}
Consider $L_n^3$. From here we easy deduce that
\begin{align*}&-\bigg(\frac{1}{2\ln2}\,\ln \frac{A^{(i)}_n}{ B^{(i)}_n}\bigg){\bf 1}_{\big\{2^{2H-2}(1-\d_n)\ls\frac{A^{(i)}_n}{B^{(i)}_n}< 1-\d_n \big\}}\\
&\quad\ls -\frac{1}{2\ln2}\big[\ln \big(2^{2H-2}(1-\d_n)\big)\big]{\bf 1}_{\big\{2^{2H-2}(1-\d_n)\ls\frac{A^{(i)}_n}{B^{(i)}_n}< 1-\d_n \big\}}\\
&\quad\ls \bigg((1-H)-\frac{\ln (1-\d_n)}{2\ln2}\bigg){\bf 1}_{\big\{2^{2H-2}(1-\d_n)\ls\frac{A^{(i)}_n}{B^{(i)}_n}< 1-\d_n \big\}}\\
&\quad\ls \bigg(1-H+\frac{\d_n}{\ln2}\bigg){\bf 1}_{\big\{2^{2H-2}(1-\d_n)\ls\frac{A^{(i)}_n}{B^{(i)}_n}< 1-\d_n\big\}}\\
&\quad\ls (1+2\d_n) {\bf 1}_{\big\{\frac{A^{(i)}_n}{B^{(i)}_n}< 1-\d_n \big\}}.
\end{align*}
The term $L_n^4$ is estimated similarly as the second part of $L_n^2$. Thus we get
\begin{align*}
\bigg(\ln \frac{A^{(i)}_n}{ B^{(i)}_n}\bigg){\bf 1}_{\big\{1+\d_n\ls\frac{A^{(i)}_n}{B^{(i)}_n}\ls 2^{2H-1}(1+\d_n)\big\}}\ls&
\bigg(\frac{A^{(i)}_n}{ B^{(i)}_n}-1\bigg){\bf 1}_{\big\{1+\d_n\ls\frac{A^{(i)}_n}{B^{(i)}_n}\ls 2^{2H-1}(1+\d_n)\big\}}\\
\ls& (1+2\d_n){\bf 1}_{\big\{\frac{A^{(i)}_n}{B^{(i)}_n}> 1+\d_n\big\}}.
\end{align*}
Summarizing, we conclude that
\begin{align*}
\vert \widehat{H}^{(i)}_n-H\vert\ls&(1+2\d_n){\bf 1}_{\big\{\big\vert\frac{A^{(i)}_n}{B^{(i)}_n}-1\big\vert>\d_n\big\}}+2\d_n{\bf 1}_{\big\{1-\d_n\ls \frac{A^{(i)}_n}{B^{(i)}_n}\ls 1+\d_n\big\}}\\
\ls&(1+2\d_n){\bf 1}_{\big\{\big\vert\frac{A^{(i)}_n}{B^{(i)}_n}-1\big\vert>\d_n\big\}}+2\d_n.
\end{align*}
Now, let $\b>0$. Note that
\begin{align*}
&\bigg\{\bigg\vert\frac{A^{(i)}_n}{B^{(i)}_n}-1\bigg\vert>\d_n\bigg\}\\
&\quad\subset\bigg\{\bigg\vert\frac{A^{(i)}_n}{B^{(i)}_n}-1\bigg\vert>\d_n, B^{(i)}_n\gs (\ln n)^{-\b} \bigg\}\bigcup\big\{B^{(i)}_n< (\ln n)^{-\b} \big\}\\
&\quad=\big\{\vert A^{(i)}_n-B^{(i)}_n\vert> \d_n B^{(i)}_n, B^{(i)}_n\gs (\ln n)^{-\b}\big\}\cup\big\{B^{(i)}_n< (\ln n)^{-\b} \big\}\\
&\quad\subset \big\{\big\vert A^{(i)}_n-B^{(i)}_n\big\vert> \d_n (\ln n)^{-\b}\big\}\cup\big\{B^{(i)}_n< (\ln n)^{-\b} \big\}.
\end{align*}
Therefore
\[
\vert \widehat{H}^{(i)}_n-H\vert\ls(1+2\d_n){\bf 1}_{\{\vert A^{(i)}_n-B^{(i)}_n\vert> \d_n (\ln n)^{-\b}\}\cup\{B^{(i)}_n< (\ln n)^{-\b}\}}+2\d_n.
\]
It follows from (\ref{konv1}) that
\[
\vert A^{(i)}_n-B^{(i)}_n\vert=\mathrm{O}\big(n^{-1/4}\ln^{1/4} n\big)
\]
and
\[
\bigg\vert B^{(i)}_n-c^{(i)}\int_0^T g^2(X_s)ds\bigg\vert=\mathrm{O}\big(n^{-1/4}\ln^{1/4} n\big).
\]
Obviously, for any $n>\exp\{\big(\frac{2}{c_0}\big)^{\frac{1}{\beta}}\}$ we have that $g^{(i)}(T)\gs c_0\geq 2(\ln n)^{-\b}$ a.s. and
\[
\big\{B^{(i)}_n< (\ln n)^{-\b} \big\}=\big\{B^{(i)}_n< (\ln n)^{-\b},g^{(i)}(T)\gs 2(\ln n)^{-\b} \big\}.
\]
Now, let $\d_n<(\ln n)^{-\b}$. Then it is not hard to deduce that
\begin{align*}
&\big\{B^{(i)}_n< (\ln n)^{-\b},g^{(i)}(T)\gs 2(\ln n)^{-\b} \big\}\\
&\quad=\big\{B^{(i)}_n< (\ln n)^{-\b},g^{(i)}(T)\gs 2(\ln n)^{-\b},B^{(i)}_n<g^{(i)}(T)-\d_n \big\}\\
&\quad\subset\big\{\vert B^{(i)}_n-g^{(i)}(T)\vert>\d_n \big\}.
\end{align*}
Therefore,
\[
\big\{B^{(i)}_n< (\ln n)^{-\b} \big\}\subset\big\{\vert B^{(i)}_n-g^{(i)}(T)\vert>\d_n \big\}
\]
if $n>\exp\{\big(\frac{2}{c_0}\big)^{\frac{1}{\beta}}\}$.
Finally, specify $\d_n$. More precisely, set $\d_n=n^{-1/4}(\ln n)^{1/4+2\b}$, $\b>0$. Note that $\d_n<(\ln n)^{-\b}$ for sufficiently large $n$ and, moreover,
\[
\frac{\mathcal{O}\big(n^{-1/4}\ln^{1/4} n\big)}{\d_n(\ln n)^{-\b}}=\frac{\mathcal{O}\big(n^{-1/4}\ln^{1/4} n\big)}{n^{-1/4}(\ln n)^{1/4+\b}} \longrightarrow 0\qquad\mbox{a.s.}\quad\mbox{as}\ n\to\infty.
\]
The latter relation together with Theorem \ref{var2} imply that for any $\omega\in\Omega^{\hbox{\bf P}ime}$ with $P(\Omega^{\hbox{\bf P}ime})=1$ there exists $n_0=n_0(\omega)$ such that for any $n>n_0$
\[
{\bf 1}_{\big\{\vert A^{(i)}_n-B^{(i)}_n\vert> \d_n (\ln n)^{-\b}\big\}\cup\big\{B^{(i)}_n< (\ln n)^{-\b}\big\}}=0\quad\mbox{a.s.},
\]
and we obtain the proof.
\end{proof}
\section{Simulation results}
Consider fractional Ornstein-Uhlenbeck process that is the solution of the linear stochastic differential equation
\[
dX_t= - X_tdt + dB^H_t,\qquad X_0=0.
\] with the step $0.05$ and for increasing (in the logarithmic scale) number $n$
of points from $n=10^2$ to $n=10^6$.
Table 1 presents the values of the difference $\vert \widehat H^{(1)}_n-H\vert$ for the values of $H$ from $0.55$ to $0.95$.
We can conclude that the difference $\vert \widehat H^{(1)}_n-H\vert$ decreases rapidly in $n$ and for fixed value of $n$ increases in $H$.
Table 2 demonstrates that the rate of convergence agrees with Theorem \ref{mainthm}, at least, for $\beta=0.05$.
Moreover, we can see from Table 3 that in the case of the linear equation the rate of convergence for $H\in(0.5, 0.7)$ can be estimated by $n^{-1/2}(\ln n)^{1/2}$.
\begin{table}[h]
\caption{ $\vert\widehat H^{(1)}_n-H\vert$ }\label{t1}
\centering
{\footnotesize{
\begin{tabular}{llllllllll}
\hline
&\multicolumn{9}{c}{$n$ points}
\\
\cline{2-10}
\noalign{
}
\textbf{H} &\multicolumn{1}{c}{$100$} &\multicolumn{1}{c}{$250$} &\multicolumn{1}{c}{$1000$} &\multicolumn{1}{c}{$2500$} &\multicolumn{1}{c}{$10^4$} &\multicolumn{1}{c}{$2.5\cdot10^4$} & \multicolumn{1}{c}{$10^5$} & \multicolumn{1}{c}{$2.5\cdot10^5$} & \multicolumn{1}{c}{$10^6$} \\
\hline \noalign{
} \textbf{0.55} & 0,08401& 0,05488& 0,02124& 0,01467& 0,00777& 0,00549& 0,00195& 0,00160& 0,00079\\
\textbf{0.6} & 0,07216& 0,04145& 0,02213& 0,01286& 0,00683& 0,00466& 0,00214& 0,00137& 0,00069\\
\textbf{0.65} & 0,07761& 0,04811& 0,01972& 0,01296& 0,00626& 0,00414& 0,00210& 0,00144&0,00066\\
\textbf{0.7}& 0,05364& 0,03403& 0,02023& 0,01219& 0,00608& 0,00341& 0,00183& 0,00125&0,00065\\
\textbf{0.75} & 0,06485& 0,03798& 0,02187& 0,01147& 0,00707& 0,00424& 0,00211&0,00140& 0,00083\\
\textbf{0.8 }& 0,05938& 0,03884& 0,02040& 0,01307& 0,00791& 0,00528& 0,00303& 0,00227&0,00120\\
\textbf{0.85} & 0,04666& 0,03577& 0,02105& 0,01684& 0,01011& 0,00753& 0,00511& 0,00384&0,00249\\
\textbf{0.9} & 0,06311& 0,04642& 0,03037& 0,02338& 0,01667& 0,01352& 0,00984& 0,00801&0,00599\\
\textbf{0.95} & 0,06219& 0,04763& 0,03488& 0,02907& 0,02295& 0,02018& 0,01640& 0,01448&0,01213\\
\hline
\end{tabular}
}}
\end{table}
\begin{table}[h!]
\caption{ $\vert\widehat H^{(1)}_n-H\vert\cdot n^{0.25}(\ln n)^{-0.3}$ }\label{t1}
\centering
{\footnotesize{
\begin{tabular}{llllllllll}
\hline
&\multicolumn{9}{c}{$n$ points}
\\
\cline{2-10}
\noalign{
}
\textbf{H} &\multicolumn{1}{c}{$100$} &\multicolumn{1}{c}{$250$} &\multicolumn{1}{c}{$1000$} &\multicolumn{1}{c}{$2500$} &\multicolumn{1}{c}{$10^4$} &\multicolumn{1}{c}{$2.5\cdot10^4$} & \multicolumn{1}{c}{$10^5$} & \multicolumn{1}{c}{$2.5\cdot10^5$} & \multicolumn{1}{c}{$10^6$} \\
\hline
\noalign{
}
\textbf{0.55} & 0.16802& 0.13070& 0.06689& 0.05596& 0.03991& 0.03449& 0.01663& 0.01681& 0.01136 \\
\textbf{0.6} & 0.14433& 0.09873& 0.06969& 0.04906& 0.03506& 0.02923& 0.01828& 0.01435& 0.00994\\
\textbf{0.65} & 0.15522& 0.11458& 0.06212& 0.04942& 0.03215& 0.02602& 0.01793& 0.01510& 0.00954\\
\textbf{0.7}& 0.10727& 0.08104& 0.06370& 0.04652& 0.03125& 0.02142& 0.01565& 0.01311& 0.00931\\
\textbf{0.75}& 0.12970& 0.09044& 0.06888& 0.04376& 0.03631& 0.02664& 0.01804& 0.01471& 0.01189\\
\textbf{0.8 }& 0.11877& 0.09250& 0.06426& 0.04987& 0.04066& 0.03313& 0.02590& 0.02382& 0.01723\\
\textbf{0.85} & 0.09331& 0.08519& 0.06628& 0.06425& 0.05193& 0.04726& 0.04363& 0.04031& 0.03580\\
\textbf{0.9} & 0.12622& 0.11055& 0.09563& 0.08920& 0.08563& 0.08485& 0.08410& 0.08413& 0.08612\\
\textbf{0.95} & 0.12438& 0.11344& 0.10984& 0.11089& 0.11789& 0.12669& 0.14009& 0.15206& 0.17450\\
\hline
\end{tabular}
}}
\end{table}
\begin{table}[h!]
\caption{ $\vert\widehat H^{(1)}_n-H\vert\cdot n^{0.5}(\ln n)^{-0.5}$ }\label{t1}
\centering
{\footnotesize{
\begin{tabular}{llllllllll}
\hline
&\multicolumn{9}{c}{$n$ points}
\\
\cline{2-10}
\noalign{
}
\textbf{H} &\multicolumn{1}{c}{$100$} &\multicolumn{1}{c}{$250$} &\multicolumn{1}{c}{$1000$} &\multicolumn{1}{c}{$2500$} &\multicolumn{1}{c}{$10^4$} &\multicolumn{1}{c}{$2.5\cdot10^4$} & \multicolumn{1}{c}{$10^5$} & \multicolumn{1}{c}{$2.5\cdot10^5$} & \multicolumn{1}{c}{$10^6$} \\
\hline
\noalign{
}
\textbf{0.55} & 0.59403 & 0.56033 & 0.38778 & 0.39788 & 0.38848 & 0.41418 & 0.27528 & 0.34462 & 0.32254 \\
\textbf{0.6} & 0.51028 & 0.42327 & 0.40402 & 0.34882 & 0.34127 & 0.35107 & 0.30255 & 0.29419 & 0.28218 \\
\textbf{0.65} & 0.54879 & 0.49122 & 0.36011 & 0.35142 & 0.31289 & 0.31248 & 0.29686 & 0.30960 & 0.27064 \\
\textbf{0.7} & 0.37926 & 0.34744 & 0.36932 & 0.33076 & 0.30417 & 0.25725 & 0.25904 & 0.26879 & 0.26421 \\
\textbf{0.75} & 0.45857 & 0.38776 & 0.39935 & 0.31116 & 0.35343 & 0.31994 & 0.29864 & 0.30156 & 0.33742 \\
\textbf{0.8} & 0.41990 & 0.39658 & 0.37253 & 0.35462 & 0.39573 & 0.39783 & 0.42871 & 0.48825 & 0.48905 \\
\textbf{0.85} & 0.32990 & 0.36523 & 0.38428 & 0.45683 & 0.50541 & 0.56751 & 0.72214 & 0.82624 & 1.01618 \\
\textbf{0.9} & 0.44627 & 0.47396 & 0.55444 & 0.63424 & 0.83341 & 1.01898 & 1.39207 & 1.72453 & 2.44410 \\
\textbf{0.95} & 0.43976 & 0.48637 & 0.63677 & 0.78848 & 1.14744 & 1.52138 & 2.31880 & 3.11678 & 4.95257 \\
\hline
\end{tabular}
}}
\end{table}
\end{document} |
\begin{document}
\title{The generic representation theory of the Juyumaya algebra of braids and ties}
\begin{abstract}
In this paper we determine the complex generic representation theory of the Juyumaya algebra. We show that a certain specialization of this algebra is isomorphic to the small ramified partition algebra, introduced by P. Martin.
\end{abstract}
\section{Introduction}
The main result of this paper is a determination of the complex generic representation theory of a family of finite dimensional algebras $\{\mathcal{E}_n(x) \colon n \in \mathbb{N}, \; x \in \mathbb{C}\}.$ These algebras were introduced by Juyumaya in \cite{juyumaya99} and studied further by Aicardi and Juyumaya \cite{aicardi} and by Ryom-Hansen \cite{ryom2010}.
The Juyumaya algebras $\mathcal{E}_n(x)$ are a generalisation of the Iwahori-Hecke algebras \cite{mathas}. The complex generic representation theory of the Iwahori-Hecke algebras is reasonably well known (see e.g. \cite{mathas} for a review). Like the Iwahori-Hecke algebras it turns out, as we shall show, that the Juyumaya algebras are generically semisimple. In contrast to the Iwahori-Hecke case however, the generic representation theory of the Juyumaya algebras over the field of complex numbers was only known for the cases $n = 1,2,3$ \cite{aicardi}, \cite{ryom2010}. Here we determine the result for all $n.$
Our method is to establish, for each $n,$ an isomorphism of $\mathcal{E}_n(1)$ (over $\mathbb{C}$) with an algebra, the small ramified partition algebra $P_n^{\ltimes}$ \cite{martin}, of known complex representation theory and then use general arguments of Cline, Parshall and Scott \cite{cline1999}.
The paper is organized as follows: we start (in section \ref{ramPart}) by reviewing the small ramified partition algebra $P_n^{\ltimes}$. In section \ref{braidTie}, we recall the definition of the algebras $\mathcal{E}_n(x)$. In section \ref{mainRslt}, we prove that, for each $n$, the algebra $\mathcal{E}_n(1)$ and $P_n^{\ltimes}$ are isomorphic as $\mathbb{C}$-algebras. We use this result as well as other results including arguments in \cite{cline1999} to show that $\mathcal{E}_n(x)$ is semisimple (of given structure) over $\mathbb{C}$ for generic choices of $x$ in section \ref{repthry}.
\section{The Small Ramified Partition Algebras} \label{ramPart}
In order to define the \emph{small} ramified partition algebra, it will be helpful to recall the definition of the ramified partition algebra, given in \cite{martin2004}, from which this algebra can be constructed. We assume familiarity with the \emph{ordinary} partition algebra \cite{martin1994}.
\subsection{Some definitions and notation}
For $n \in \mathbb{N},$ we define $\underline{n} = \{1,2, \ldots,n\}$ and $\underline{n}' = \{1',2', \ldots, n'\}.$ Let $S_n$ denote the symmetric group on $\underline{n}$ and $\sigma_{i,i+1}$ the transposition $(i,i+1) \in S_n.$ When we write $\underline{d}$ for a \emph{poset}, we mean $\{1,2, \ldots, d\}$ equipped with the natural partial order $\leq$ (although we will often concentrate on the case $\underline{2} = (\{1,2\}, \; 1 \leq 2)$ in this paper). For a set $X$, we write $\mathcal{P}_X$ for the set of partitions of $X.$
\noindent For example,
\begin{eqnarray*}
\mathcal{P}_{\underline{2} \cup \underline{2}'} &=& \{\{\{1\},\{2\},\{1'\},\{2'\}\}, \{\{1,2,1',2'\}\}, \{\{1,2,1'\},\{2'\}\},\\
& & \{\{1,2,2'\},\{1'\}\}, \{\{1,1',2'\},\{2\}\}, \{\{2,1',2'\},\{1\}\}, \{\{1,2\},\{1',2'\}\},\\
& & \{\{1,1'\},\{2,2'\}\}, \{\{1,2'\},\{1',2\}\}, \{\{1,2\},\{1'\},\{2'\}\},\\
& & \{\{1,1'\},\{2\},\{2'\}\}, \{\{1,2'\},\{1'\},\{2\}\}, \{\{1',2\},\{1\},\{2'\}\},\\
& & \{\{2,2'\},\{1\},\{1'\}\}, \{\{1',2'\},\{1\},\{2\}\}\}.
\end{eqnarray*}
In an element of $\mathcal{P}_{\underline{n} \cup \underline{n}'}$ we call the individual subsets of the set of objects \emph{parts}. For instance, $\{1,2\}$ is a part of the partition $\{\{1,2\},\{1'\},\{2'\}\}$ in $\mathcal{P}_{\underline{2} \cup \underline{2}'}$. For $X' \subset X$ and $c \in P_X$ we define $c|_{X'}$ as the collection of the sets of the form $c_i \cap X'$ where $c_i$ are elements of the partition $c$.
\begin{defn}
For a set $X$, we define the \emph{refinement} partial order on $\mathcal{P}_X$ as follows. For $p, \;q \in \mathcal{P}_X,$ we say $p$ is a refinement of $q,$ denoted $p \leq q,$ if each part of $q$ is a union of one or more parts of $p.$
\end{defn}
\begin{prop}[See {\citep[Prop. 1]{martin2004}}]
Let $p, \; q \in \mathcal{P}_X$ and $Y \subseteq X.$ Then $p \leq q$ implies $p|_Y \leq q|_Y.$ \qed
\end{prop}
\begin{remark}
For $F$ a field, $n \in \mathbb{N}, \; \delta' \in F,$ the set $\mathcal{P}_{\underline{n} \cup \underline{n}'}$ is a basis for the partition algebra \cite{martin1994} which we denote by $P_n(\delta').$ The dimension of $P_n(\delta')$ is therefore the Bell number $B_{2n}$ (see \cite{cameron}). The group algebra $FS_n$ of the symmetric group $S_n$ is embedded in $P_n(\delta')$ as the span of the partitions with every part having exactly two elements, one primed and the other unprimed, of $\underline{n} \cup \underline{n}'.$
\end{remark}
\subsection{Ramified partition algebra}
The ramified partition algebra was introduced by Martin \cite{martin2004} as a generalisation of the ordinary partition algebra $P_n(\delta').$
\begin{defn}
Let $(T, \leq)$ be a finite poset. For a set $X,$ we define $\mathbf{P}_X^{T}$ to be the subset of the Cartesian product $\prod_T\mathcal{P}_X$ consisting of those elements $q = (q_i \colon i \in T)$ such that $q_i \leq q_j$ whenever $i \leq j.$ Any such element $q \in \mathbf{P}_X^{T}$ will be referred to as a \emph{T-ramified partition}.
\end{defn}
\noindent For example, some elements of $\mathbf{P}_{\underline{2} \cup \underline{2}'}^{\underline{2}}$ are listed below:
\begin{eqnarray*}
\pi_1 &=& (\{\{1\},\{2\},\{1'\},\{2'\}\}, \{\{1\},\{2\},\{1'\},\{2'\}\}) \\
\pi_2 &=& (\{\{1,2\},\{1'\},\{2'\}\}, \{\{1,2,1'\},\{2'\}\}) \\
\pi_3 &=& (\{\{1,2'\},\{2,1'\}\}, \{\{1,2,1',2'\}\}),
\end{eqnarray*}
and so on.
We now recall from \cite{martin2004} the diagrammatic realization of an element of $\mathbf{P}_{\underline{n} \cup \underline{n}'}^{T}.$ We shall only need the case $T = \underline{2}$ in this paper. We first look at an example therein. The diagram
\[
\includegraphics[scale=0.4]{ramfdiagram}
\]
\noindent represents $(\{\{1,2,3\},\{1',2'\},\{3'\},\{4,5'\},\{5,4'\}\},\{\{1,2,3,1',2'\},\{3'\},\{4,5'\},$\\
$\{5,4'\}\}).$ For $T = \underline{1}$ case, the diagram coincides with a partition algebra diagram (cf. \cite{halverson2005}, \cite{martin1996}, or \cite{martin2004}). Thus, a diagram of a $\underline{2}$-ramified partition can be thought of as an \emph{enhanced} partition algebra diagram in which every connected component lies inside an ``island'', and the union of components in the island is the less refined part. Note that islands can cross (as illustrated in the diagram above), but it is not hard to draw them unambiguously.
A diagram representing a T-ramified partition is not unique. We say two diagrams are equivalent if they give rise to the same T-ramified partition.
The term \emph{ramified partition diagram} (or sometimes \emph{ramified $2n$-partition diagram} to indicate the number of vertices) will be used to mean the equivalence class of the given diagram.
We refer to the interior line-segments in the underlying partition algebra diagram of a ramified partition diagram as \emph{bones}.
The composition of ramified $2n$-partition diagrams is as follows. First identify the bottom of one ramified $2n$-partition diagram with the top of the other. Then replace bone (resp. island) connected components that are isolated from the boundaries in composition by a factor $\delta_1$ (resp. $\delta_2$) as shown in Figure \ref{compRam}.
\begin{figure}
\caption{The composition of diagrams in $P_4^{(\underline{2}
\label{compRam}
\end{figure}
Throughout this paper, we shall identify a ramified partition with its ramified partition diagram and speak of them interchangeably.
\begin{prop}[See {\citep[Prop. 2]{martin2004}}]
For any $d$-tuple $\delta = (\delta_1, \ldots, \delta_d) \in F^d,$ the set $\mathbf{P}_{\underline{n} \cup \underline{n}'}^{T}$ forms a basis for a subalgebra of $\bigotimes_{t \in T}P_n(\delta_t).$
\end{prop}
\begin{proof}
The proof can be found in \cite{martin2004}, but is in essence the observation that bones continue to lie within islands under composition.
\end{proof}
For $\delta = (\delta_1, \delta_2, \ldots, \delta_d)$ $\in F^d,$ we define the \emph{T-ramified partition algebra $P_n^{(T)}(\delta)$} over $F$ as the finite dimensional algebra with basis $\mathbf{P}_{\underline{n} \cup \underline{n}'}^{T}$ and the above composition.
A bone connecting a vertex in the northern boundary to a vertex in the southern boundary of the frame will be called a \emph{propagating line}.
\noindent The complex generic representation theory of $P_n^{(T)}(\delta)$ has been determined in the case $T = \underline{2}$ in \cite{martin2004}. It was shown that there are many choices of $\delta$ such that $P_n^{(\underline{2})}(\delta)$ is not semisimple for sufficiently large $n,$ but that it is generically semisimple for all $n.$
\subsection{Small Ramified Partition Algebra $P_n^{\ltimes}$} \label{sRamPart}
In this section we recall the definition of the small ramified partition algebra. To define this algebra we require the following definitions.
\begin{defn}
We define $\text{diag}-\mathcal{P}_n$ to be the subset of $\mathcal{P}_{\underline{n} \cup \underline{n}'}$ such that $i,i'$ are in the same part for all $i \in \mathbb{N}.$
\end{defn}
For example, recall from \cite{martin1994} the special elements in $\mathcal{P}_{\underline{n} \cup \underline{n}'}$ as follows.
\begin{eqnarray*}
1 &=& \{\{1,1'\},\; \{2,2'\}, \; \ldots \{i,i'\}, \; \ldots \{n,n'\}\}\\
A^{i,j} &=& \{\{1,1'\},\; \{2,2'\}, \; \ldots \{i,i',j,j'\}, \; \ldots \{n,n'\}\}\\
\sigma_{i,j} &=& \{\{1,1'\},\; \{2,2'\}, \; \ldots \{i,j'\},\{j, i'\} \; \ldots \{n,n'\}\} \\
e_i &=& \{\{1,1'\},\; \{2,2'\}, \; \ldots \{i\},\{i'\}, \; \ldots \{n,n'\}\}.
\end{eqnarray*}
Here, $1$ and $A^{i,j}$ are in $\text{diag}-\mathcal{P}_n$.
\begin{defn}\label{Delta}
For any $\delta' \in F,$ we define $\Delta_n$ as the subalgebra of $P_n(\delta')$ generated by $$\langle S_n, A^{i,j} (i,j = 1,2, \ldots, n) \rangle.$$
\end{defn}
\begin{prop} \label{prop:injMap}
The map $$\ltimes \colon S_n \times \text{diag}-\mathcal{P}_n \to S_n \times \mathcal{P}_{\underline{n} \cup \underline{n}'}$$ given by $$(a,b) \mapsto (a,ba)$$ defines an injective map.
\end{prop}
\begin{proof}
The well-definedness of $\ltimes$ is clear. To prove that $\ltimes$ is an injective map, it suffices to show that if $(a,ba)$ is equal to $(c,dc)$ in $S_n \times \mathcal{P}_{\underline{n} \cup \underline{n}'}$, then $(a,b)$ is equal to $(c,d)$ in $S_n \times \text{diag}-\mathcal{P}_n.$
Assume that $(a,ba) = (c,dc).$ Since $a = c,$ then $bc = dc.$ But $c$ is invertible, thus, $b=d.$
\end{proof}
\noindent Note that $\ltimes$ is not a surjective map.
\begin{defn}
We define $\mathbb{P}_{\underline{n} \cup \underline{n}'}$ to be the subset of the Cartesian product $S_n \times \mathcal{P}_{\underline{n} \cup \underline{n}'}$ given by the elements $q = (q_1,q_2)$ such that $q_1$ is a refinement of $q_2$.
\end{defn}
\begin{coro}
The set of $\ltimes(S_n \times \text{diag}-\mathcal{P}_n)$ lies in $\mathbb{P}_{\underline{n} \cup \underline{n}'}$ and forms a basis for a subalgebra of $FS_n \otimes_F \Delta_n.$ \qed
\end{coro}
\begin{defn}
The associative algebra $P_n^{\ltimes}$ over $F$ is the free $F$-module with the set of $\ltimes(S_n \times \text{diag}-\mathcal{P}_n)$ as basis and multiplication inherited from the multiplication on $P_n^{(\underline{2})}(\delta).$ This is the \emph{small ramified partition algebra} \cite{martin}.
\end{defn}
\noindent It is easy to check that
\begin{lemma}
The multiplication on $P_n^{\ltimes}$ is well-defined up to equivalence. \qed
\end{lemma}
There is a diagram representation of the set of $\ltimes(S_n \times \text{diag}-\mathcal{P}_n)$ since its elements are $\underline{2}$-ramified partitions (see \cite{martin}).
\begin{example} \label{explRamified}
The map defined in Proposition \ref{prop:injMap} is illustrated by the following pictures.
\includegraphics[scale=0.4]{ramfDiag}
\noindent In particular, these pictures describe the diagrammatic realization of some basis elements in $P_4^{\ltimes}.$
\end{example}
\begin{coro}[See {\citep[\S 3.4]{martin}}] \label{coro:dimSRP}
The dimension of $P_n^{\ltimes}$ is given by $n!B_n,$ where $B_n$ is the Bell number. \qed
\end{coro}
\begin{remark}
Notice that, $P_n^{\ltimes}$ is spanned by diagrams with $n$ propagating lines (See Example \ref{explRamified}). This means that, unlike the ramified partition algebras, the small ramified partition algebras do not depend on parameter $\delta.$
\end{remark}
\begin{defn}
For any $\delta' \in F,$ we define $\Gamma_n$ as the subalgebra of $P_n(\delta')$ generated by $$\langle A^{i,j} (i,j = 1,2, \ldots, n) \rangle .$$
\end{defn}
Note that the natural injection of $\Gamma_n$ into $P_n^{\ltimes}$ is given by $$A^{i,j} \mapsto (1,A^{i,j})$$ and there exists a natural injection of $FS_n$ into $P_n^{\ltimes}$ given by $$\sigma_{i,i+1} \mapsto (\sigma_{i,i+1},\sigma_{i,i+1}).$$
\begin{prop}[See {\citep[Prop. 2]{martin}}] \label{genSmll}
The algebra $P_n^{\ltimes}$ is generated by $(1,A^{i,i+1})$ and $(\sigma_{i,i+1}, \sigma_{i,i+1})$ ($i = 1,2, \ldots, n-1$). \qed
\end{prop}
\section{The Juyumaya algebra of braids and ties} \label{braidTie}
Following \cite{ryom2010}, we recall the Juyumaya algebra over the ring $\mathbb{C}[u,u^{-1}].$
\begin{defn}[See {\citep[\S 2]{ryom2010}}]
Let $u$ be an indeterminate over $\mathbb{C}$ and $\mathcal{A}$ be the principal ideal domain $\mathbb{C}[u,u^{-1}]$. The algebra $\mathcal{E}_n^{\mathcal{A}}(u)$ over $\mathcal{A}$ is the unital associative $\mathcal{A}$-algebra generated by the elements $T_1,~T_2, \ldots, T_{n-1}$ and $E_1,~E_2, \ldots, E_{n-1},$ which satisfy the defining relations
\begin{eqnarray*}
(A1) &T_iT_j = T_jT_i & \hbox{ if } ~|i-j| > 1\\
(A2) &E_iE_j = E_jE_i & \forall \; i,j\\
(A3) &E_i^2 = E_i \\
(A4) &E_iT_i = T_iE_i\\
(A5) &E_iT_j = T_jE_i & \hbox{ if } ~|i-j|>1\\
(A6) &T_iT_jT_i = T_jT_iT_j & \hbox{ if } ~|i-j| = 1\\
(A7) &E_jT_iT_j = T_iT_jE_i & \hbox{ if } ~|i-j| = 1\\
(A8) &E_iE_jT_j = E_iT_jE_i = T_jE_iE_j & \hbox{ if } ~|i-j| = 1\\
(A9) &T_i^2 = 1+(u-1)E_i(1-T_i)
\end{eqnarray*}
\end{defn}
Let $\mathbb{C}(u)$ be the field of rational function. We define $\mathcal{E}^0_n(u)$ as $$\mathcal{E}^0_n(u) := \mathcal{E}_n^{\mathcal{A}}(u) \otimes_{\mathcal{A}} \mathbb{C}(u)$$ where $\mathbb{C}(u)$ is made into an ${\mathcal{A}}$-module through inclusion.
\begin{coro}[See {\citep[Corollary 3]{ryom2010}}] \label{coro:dimAJ}
The dimension of $\mathcal{E}^0_n(u)$ is given by $n!B_n,$ where $B_n$ is the Bell number. \qed
\end{coro}
The Bell number making appearance in Corollary \ref{coro:dimAJ} indicates that there might be a connection between the Juyumaya algebra and the (small ramified) partition algebra. In section \ref{mainRslt} we present this connection.
From the presentation of $\mathcal{E}_n^{\mathcal{A}}(u),$ relations $(A1)$, $(A6)$, $(A9)$ form a deformation of the Coxeter relations (see \citep[\S 1]{mathas}) of the symmetric group $S_n$. It is straightforward to verify the following result.
\begin{prop}
There exists a homomorphism from $\mathcal{E}_n^{\mathcal{A}}(u)$ to the group ring $\mathbb{\mathcal{A}}S_n$ of the symmetric group given by
\begin{eqnarray*}
X \colon \mathcal{E}_n^{\mathcal{A}}(u) & \to & \mathcal{A}S_n \\
T_i & \mapsto & \sigma_{i,i+1} \\
E_i & \mapsto & 0.
\end{eqnarray*} \qed
\end{prop}
\noindent In particular, $\mathcal{A}S_n$ is isomorphic to a quotient of $\mathcal{E}_n^{\mathcal{A}}(u)$ when $u = 1.$
\section{Relationship of the Juyumaya algebra to $P_n^{\ltimes}$} \label{mainRslt}
In response to a remark of Ryom-Hansen in \cite{ryom2010}, we present new results that establish a connection between the Juyumaya algebra and the partition algebra, via the small ramified partition algebra.
Let $\mathbb{C}$ be the field of complex numbers which is a $\mathbb{C}[u,u^{-1}]$-algebra (that is, with $u$ specified to a complex number $x$). Denote the $\mathbb{C}$-algebra $\mathcal{E}_n^{\mathcal{A}}(u) \otimes_{\mathcal{A}} \mathbb{C}$ by $\mathcal{E}_n(x).$ Here, we shall only need the case $x = 1.$
\begin{prop} \label{prop:ajAlgHomo}
The map $\rho \colon \mathcal{E}_n(1) \to \mathbb{C}S_n \otimes_{\mathbb{C}} \Delta_n$ given by
\begin{eqnarray*}
E_i & \mapsto & (1, A^{i,i+1})\\
T_i & \mapsto & (\sigma_{i,i+1}, \sigma_{i,i+1})
\end{eqnarray*}
defines a $\mathbb{C}$-algebra homomorphism.
\end{prop}
\begin{proof}
First, it is easy to see that the tensor product $\mathbb{C}S_n \otimes_{\mathbb{C}} \Delta_n$ makes sense. To show that this map is an algebra homomorphism we check that the relations (A1)--(A9) hold when $(1, A^{i,i+1})$ is put in place of $E_i$ and $(\sigma_{i,i+1}, \sigma_{i,i+1})$ is put in place of $T_i$ as follows.
\begin{itemize}
\item[(A1)] $\rho(T_iT_j) = (\sigma_{i,i+1}, \sigma_{i,i+1})\; (\sigma_{j,j+1},\sigma_{j,j+1})$ $= (\sigma_{i,i+1}\sigma_{j,j+1}, \; \sigma_{i,i+1}\sigma_{j,j+1})$ and \\
$\rho(T_jT_i) = (\sigma_{j,j+1}, \sigma_{j,j+1})$ $(\sigma_{i,i+1},\sigma_{i,i+1}) = (\sigma_{j,j+1}\sigma_{i,i+1},$ $\sigma_{j,j+1}\sigma_{i,i+1}).$\\
Since $|i-j|>1,$ $\sigma_{i,i+1}\sigma_{j,j+1} = \sigma_{j,j+1}\sigma_{i,i+1}.$ Thus, \\
$(\sigma_{i,i+1}\sigma_{j,j+1},$ $\sigma_{i,i+1}\sigma_{j,j+1}) = (\sigma_{j,j+1}\sigma_{i,i+1},$ $\sigma_{j,j+1}\sigma_{i,i+1})$ as required.
Diagrammatically, this may be represented as follows.
\includegraphics[scale=0.4]{compRamf1}
\item[(A2)] $\rho(E_iE_j) = (1, A^{i,i+1})\; (1, A^{j,j+1}) = (1, \; A^{i,i+1}A^{j,j+1}) = (1, \; A^{j,j+1}A^{i,i+1})$\\
$ = (1, A^{j,j+1})\; (1, A^{i,i+1}) = \rho(E_jE_i)$.
The second equality follows from the definition of the tensor product of algebras, the third equality is a consequence of commutativity of $A^{k,k+1}$s and the fourth equality follows again from the definition of the tensor product of algebras.
\item[(A3)] $\rho(E_i^2) = (1, A^{i,i+1})\; (1, A^{i,i+1}) = (1, \; A^{i,i+1}A^{i,i+1}) = (1, \; A^{i,i+1}) = \rho(E_i).$
\item[(A4)] Similar to the proof of relation (A2), $\rho(E_iT_i) = (1, A^{i,i+1})\; (\sigma_{i,i+1}, \; \sigma_{i,i+1})$ \\ $= (1 \sigma_{i,i+1}, \; A^{i,i+1} \sigma_{i,i+1}) = (\sigma_{i,i+1} 1, \; \sigma_{i,i+1} A^{i,i+1})$\\ $= (\sigma_{i,i+1}, \;\sigma_{i,i+1}) \; (1, A^{i,i+1}) = \rho(T_iE_i).$
\item[(A5)] $\rho(E_iT_j) = (1, A^{i,i+1})\; (\sigma_{j,j+1}, \; \sigma_{j,j+1}) = (1 \sigma_{j,j+1}, \; A^{i,i+1} \sigma_{j,j+1})$ and \\
$\rho(T_jE_i) = (\sigma_{j,j+1}, \;\sigma_{j,j+1}) \; (1, A^{i,i+1}) = (\sigma_{j,j+1} 1, \; \sigma_{j,j+1} A^{i,i+1}).$\\
Since $|i-j|>1, \; A^{i,i+1} \sigma_{j,j+1} = \sigma_{j,j+1} A^{i,i+1}.$ Thus, \\
$(\sigma_{j,j+1}, \; A^{i,i+1} \sigma_{j,j+1}) = (\sigma_{j,j+1}, \; \sigma_{j,j+1} A^{i,i+1})$ as required.
\item[(A6)] $T_iT_jT_i$ corresponds to $(\sigma_{i,i+1}, \sigma_{i,i+1})\; (\sigma_{j,j+1}, \; \sigma_{j,j+1}) \; (\sigma_{i,i+1}, \sigma_{i,i+1}) = (\sigma_{i,i+1}\sigma_{j,j+1}\sigma_{i,i+1}, \sigma_{i,i+1}\sigma_{j,j+1}\sigma_{i,i+1})$ and \\
$T_jT_iT_j$ corresponds to $(\sigma_{j,j+1}, \sigma_{j,j+1})\; (\sigma_{i,i+1}, \; \sigma_{i,i+1}) \; (\sigma_{j,j+1}, \sigma_{j,j+1}) = (\sigma_{j,j+1}\sigma_{i,i+1}\sigma_{j,j+1}, \sigma_{j,j+1}\sigma_{i,i+1}\sigma_{j,j+1}).$\\
Since $|i-j|=1, \; \sigma_{i,i+1}\sigma_{j,j+1}\sigma_{i,i+1} = \sigma_{j,j+1}\sigma_{i,i+1}\sigma_{j,j+1}.$\\
Thus, $(\sigma_{i,i+1}\sigma_{j,j+1}\sigma_{i,i+1}, \sigma_{i,i+1}\sigma_{j,j+1}\sigma_{i,i+1}) = \\ (\sigma_{j,j+1}\sigma_{i,i+1}\sigma_{j,j+1}, \sigma_{j,j+1}\sigma_{i,i+1}\sigma_{j,j+1})$ as required.
\item[(A7)] The element $E_jT_iT_j$ is mapped to $(1, A^{j,j+1}) (\sigma_{i,i+1},\sigma_{i,i+1}) (\sigma_{j,j+1},\sigma_{j,j+1})$ \\
$ = (1 \sigma_{i,i+1}\sigma_{j,j+1}, \; A^{j,j+1}\sigma_{i,i+1}\sigma_{j,j+1})$ and \\
the element $T_iT_jE_i$ is mapped to $(\sigma_{i,i+1},\sigma_{i,i+1}) (\sigma_{j,j+1},\sigma_{j,j+1}) (1, A^{i,i+1}) \\ = (\sigma_{i,i+1}\sigma_{j,j+1} 1, \; \sigma_{i,i+1}\sigma_{j,j+1} A^{i,i+1}).$\\
Since $|i-j|=1, \; A^{j,j+1}\sigma_{i,i+1}\sigma_{j,j+1}= \sigma_{i,i+1}\sigma_{j,j+1}A^{i,i+1}$ and the result follows.
Proving relation (A7) using diagrams:
\includegraphics[scale=0.4]{compA7}
\item[(A8)]The relation $E_iE_jT_j$ corresponds to $(1,A^{i,i+1})(1,A^{j,j+1})(\sigma_{j,j+1},\sigma_{j,j+1})$ $= (11\sigma_{j,j+1}, A^{i,i+1}A^{j,j+1}\sigma_{j,j+1}),$\\
the relation $E_iT_jE_i$ corresponds to $(1,A^{i,i+1})(\sigma_{j,j+1},\sigma_{j,j+1})(1,A^{i,i+1})= (1\sigma_{j,j+1}1, A^{i,i+1}\sigma_{j,j+1}A^{i,i+1}),$ and \\
the relation $T_jE_iE_j$ corresponds to $(\sigma_{j,j+1},\sigma_{j,j+1})(1,A^{i,i+1})(1,A^{j,j+1})= (\sigma_{j,j+1}11, \sigma_{j,j+1}A^{i,i+1}A^{j,j+1}).$\\
$A^{i,i+1}A^{j,j+1}\sigma_{j,j+1} = A^{i,i+1}\sigma_{j,j+1}A^{i,i+1} = \sigma_{j,j+1}A^{i,i+1}A^{j,j+1}$ since $|i-j| = 1$ as required.
Relation (A8) may be described using diagrams as follows.
\includegraphics[scale=0.4]{compA8}
\item[(A9)] Since $u$ is fixed to $1,$ it follows that $T_i^2 = 1$ and the relation corresponds to $(\sigma_{i,i+1}, \sigma_{i,i+1}) \; (\sigma_{i,i+1}, \sigma_{i,i+1}) = (\sigma_{i,i+1}\sigma_{i,i+1}, \sigma_{i,i+1}\sigma_{i,i+1}) = (1,1)$ as required.
\end{itemize}
\end{proof}
We leave it as an exercise to use diagrams to check relations (A2) -- (A6) and (A9). Next we show that
\begin{theo} \label{isoMap}
The map $\phi \colon \mathcal{E}_n(1) \to P_n^{\ltimes}$ given by
\begin{eqnarray*}
E_i & \mapsto & (1, A^{i,i+1})\\
T_i & \mapsto & (\sigma_{i,i+1}, \sigma_{i,i+1})
\end{eqnarray*}
defines a $\mathbb{C}$-algebra isomorphism.
\end{theo}
\begin{proof}
The map $\phi$ is well-defined since by Proposition \ref{genSmll} $(1, A^{i,i+1})$ and $(\sigma_{i,i+1}, \sigma_{i,i+1})$ generates precisely $P_n^{\ltimes}$.
In order to check that $\phi$ is an algebra homomorphism, we need to verify that the defining relations of $\mathcal{E}_n(1)$ are satisfied in $P_n^{\ltimes}$ and this has already been shown in Proposition \ref{prop:ajAlgHomo}. All that remains is to show that the map is an isomorphism.
By Corollary \ref{coro:dimAJ} and by Corollary \ref{coro:dimSRP}, the dimensions of $\mathcal{E}_n(1)$ and $P_n^{\ltimes}$ are equal. Moreover, the map $\phi$ is surjective since the images of the generators generate $P_n^{\ltimes}$.
Thus, the preceding facts together imply that $\phi$ is an isomorphism.
\end{proof}
\section{Representation theory} \label{repthry}
Generic irreducible representations of the Juyumaya algebra are constructed for the cases $n = 2,3$ in \cite{aicardi}, \cite{ryom2010}. Here we do all other cases, by reference to \cite{martin}.
In the previous section we established, for each $n \in \mathbb{N},$ an isomorphism between the algebras $\mathcal{E}_n(1)$ and $P_n^{\ltimes}.$ With this result we have implicitly determined the complex representation theory of $\mathcal{E}_n(1)$ since the representation theory of $P_n^{\ltimes}$ over $\mathbb{C}$ is already known (cf. \cite{martin}) and
\begin{theo}[See {\citep[Theorem 4]{martin}}] \label{srpSplit}
For all $n,$ the algebra $P_n^{\ltimes}$ over $\mathbb{C}$ is split semisimple. \qed
\end{theo}
We can now prove that
\begin{theo}
For all $n,$ the algebra $\mathcal{E}_n(x)$ is generically semisimple.
\end{theo}
(By \emph{generically} we mean in a Zariski open subset of the complex space)
\begin{proof}
By Theorem \ref{isoMap}, $\mathcal{E}_n(1)$ is isomorphic to the algebra $P_n^{\ltimes}$ and by Theorem \ref{srpSplit} $P_n^{\ltimes}$ is split semisimple over $\mathbb{C}$. This implies that $\mathcal{E}_n(1)$ is split semisimple over $\mathbb{C}$. Split semisimplicity is a generic property according to Parshall, Cline and Scott \citep[\S 1]{cline1999}. That is, the split semisimplicity property holds (in our case) on a Zariski open subset (see \cite{smith2000}) of complex space. Therefore, $\mathcal{E}_n(x)$ is split semisimple for generic choices of $x \in \mathbb{C}.$ But $\mathcal{E}_n(x)$ is semisimple if and only if it is split semisimple since we are working over an algebraically closed field of characteristic zero.
\end{proof}
\noindent {\bf Acknowledgements.} The results of this paper form part of the author's doctoral thesis. The author would like to express her gratitude to Paul Martin for introducing her to this field and for helpful discussions and encouragement, and to Andrew Reeves, Alison Parker, Maud De Visscher, for useful comments on the earlier version of the paper. She would also like to thank City University London (research studentship) for financial support. She is grateful to the University of Leeds for hospitality.
\end{document} |
\begin{document}
\footnotetext{Support of the research of all authors by the Austrian Science Fund (FWF), project I~4579-N, and the Czech Science Foundation (GA\v CR), project 20-09869L, entitled ``The many facets of orthomodularity'', is gratefully acknowledged.}
\title{Representability of Kleene posets and Kleene lattices}
\begin{abstract}
A Kleene lattice is a distributive lattice equipped with an antitone involution and satisfying the so-called normality condition. These lattices were introduced by J.~A.~Kalman. We extended this concept also for posets with an antitone involution. In our recent paper \cite{CLP}, we showed how to construct such Kleene lattices or Kleene posets from a given distributive lattice or poset and a fixed element of this lattice or poset by using the so-called twist product construction, respectively. We extend this construction of Kleene lattices and Kleene posets by considering a fixed subset instead of a fixed element.
Moreover, we show that in some cases, this generating poset can be embedded into the resulting Kleene poset.
We investigate the question when a Kleene poset can be represented by a Kleene poset obtained by the
mentioned construction. We show that a direct product of representable Kleene posets is again representable and hence a direct product of finite chains is representable. This does not hold in general for subdirect products, but we show some examples where it holds. We present large classes of representable and non-representable Kleene posets. Finally, we investigate two kinds of extensions of a distributive poset $\mathbf A$, namely its Dedekind-MacNeille completion $\BDM(\mathbf A)$ and a completion $G(\mathbf A)$ which coincides with $\BDM(\mathbf A)$ provided $\mathbf A$ is finite. In particular we prove that if $\mathbf A$ is a Kleene poset then its extension $G(\mathbf A)$ is also a Kleene lattice. If the subset $X$ of principal order ideals of $\mathbf A$ is involution-closed and doubly dense in $G(\mathbf A)$ then it generates $G(\mathbf A)$ and it is isomorphic to $\mathbf A$ itself.
\end{abstract}
{\bf AMS Subject Classification:} 06D30, 06A11, 06B23, 06D10, 03G25
{\bf Keywords:} Kleene lattice, normality condition, Kleene poset, pseudo-Kleene poset, representable Kleene lattice, embedding, twist-product, Dedekind-MacNeille completion
\section{Introduction}
Kleene lattices serve as algebraic semantics for a specific De Morgan logic. The latter is a logic equipped with a negation $'$ satisfying the double negation law $x''=x$. Here the unary operation $'$ is assumed to be antitone with respect to the induced order, but it need not be a complementation. In order to enrich the properties of such a negation, it is natural to ask the so-called {\em normality condition}, i.e.,\ the inequality
\[
x\wedge x'\leq y\vee y'.
\]
Of course, if $'$ is a complementation, then this inequality is satisfied automatically. But often, the negation in a De Morgan logic has this property, and hence such a negation turns out to be close to complementation. Distributive lattices with an antitone involution satisfying the normality condition are called {\em Kleene lattices} and were introduced by J.~A.~Kalman \cite K (under a different name). To emphasize the importance of this concept, let us note that every MV-algebra, i.e.,\ the algebraic semantics of \L ukasiewicz's many-valued logic, is a Kleene lattice. Moreover, MV-algebras are also crucial in the logic of quantum events because every so-called lattice effect algebra is composed of blocks, which are MV-algebras. Due to this, the question how to construct Kleene lattices is of some interest and importance.
If instead of lattices, only posets are considered, one obtains so-called {\em Kleene posets}. If we also forget distributivity, we get {\em pseudo-Kleene posets}. We will introduce both notions later.
Our previous paper \cite{CLP} showed how to construct a Kleene lattice $\mathbf K$ from a given distributive lattice $\mathbf L=(L,\vee,\wedge)$ employing the so-called twist product and its reduction using a non-empty subset $S$ of $L$. In such a case, we say that $\mathbf K$ is representable. However, a lot of problems mentioned in \cite{CLP} remain open. Among them, we would like to try to solve the following ones:
\begin{itemize}
\item Determine classes of representable Kleene lattices as well as classes of not representable Kleene lattices.
\item Can these constructions be extended to Kleene posets?
\item Can every poset be embedded into a Kleene poset obtained by such a construction?
\item Is the Dedekind-MacNeille completion of a representable poset a representable Kleene lattice?
\end{itemize}
The present paper aims to get at least partial answers to the mentioned questions. We show that direct products of chains can be considered as representable Kleene lattices and study certain ordinal sums of distributive lattices. We prove that if a pseudo-Kleene poset $\mathbf K$ of odd cardinality can be represented by a distributive poset $\mathbf A$ and a non-empty subset $S$ of $A$, then $S$ must be a singleton, i.e.,\ $S=\{a\}$, and $\mathbf A$ can be embedded into $\mathbf K$ in such a way that $a$ is mapped onto the unique fixed point of\, {}$'$. We prove further results on representable Kleene posets and Kleene lattices.
First, we recall or introduce several concepts on ordered sets (posets).
Let $(A,\leq)$ be a poset, $b,c\in A$ and $B,C\subseteq A$. We say
\[
B\leq C\text{ if }x\leq y\text{ for all }x\in B\text{ and }y\in C.
\]
Instead of $B\leq\{c\}$, $\{b\}\leq C$ and $\{b\}\leq\{c\}$ we simply write $B\leq c$, $b\leq C$ and $b\leq c$, respectively. Further, we define
\begin{align*}
L(B) & :=\{x\in A\mid x\leq B\}, \\
U(B) & :=\{x\in A\mid B\leq x\}.
\end{align*}
Instead of $L(B\cup C)$, $L(B\cup\{c\})$, $L(\{b\}\cup C)$, $L(\{b,c\})$ and $L\big(U(B)\big)$ we simply write $L(B,C)$, $L(B,c)$, $L(b,C)$, $L(b,c)$ and $LU(B)$, respectively. Analogously we proceed in similar cases.
A subset $I$ of $A$ is said to be a {\em Frink ideal} if $LU(M) \subseteq I$
for each finite subset $M \subseteq I$. Similarly, a
subset $F$ of $A$ is said to be a {\em Frink filter} if $UL(N) \subseteq F$
for each finite subset $N \subseteq F$.
An order-preserving map $f$ between posets $\mathbf A$ and
$\mathbf B$ is said to be
\it an \hbox{{$L$}{$U$}}-morphism \rm\ if
\begin{equation}\label{LUmor}
\hbox{{$ L$}}\big(f(X)\big)=\hbox{{$ L$}}\big(f(\hbox{{$U$}}\hbox{{$L$}}(X)\big)\quad \hbox{and}\quad
\hbox{{$U$}}\big(f(Y)\big)=\hbox{{$U$}}\big(f(\hbox{{$L$}}\hbox{{$ U$}}(Y)\big)
\end{equation}
\noindent
for all non-empty finite subsets $X, Y\subseteq A$.
We say that an \hbox{{$L$}{$U$}}-morphism $f$ is {\em an \hbox{{$L$}{$U$}}-embedding}
({\em \hbox{{$L$}{$U$}}-isomorphism})\
if $f$ is {\em order reflecting} (the inverse map to $f$ is an
\hbox{{$L$}{$U$}}-morphism).
An {\em antitone involution} on $\mathbf A$ is a unary operation $'$ on $A$, satisfying
\begin{align*}
& x\leq y\text{ implies }y'\leq x', \\
& x''=x
\end{align*}
($x,y\in A$).
An element $y \in A$ is said to be a {\em complement} of $x \in A$
if $L(x, y) =L(A)$ and $U(x, y) = U(A)$. $\mathbf A$ is said to be
{\em complemented} if each element of $A$ has a complement in $\mathbf A$.
$\mathbf A$ is said to be {\em Boolean} if it is distributive and complemented.
If $\mathbf A$ has a greatest element $1$ and a smallest element $0$,
then an
antitone involution $'$ on $A$ is called an {\em orthocomplementation} if
$L(x,x')=\{0\}$ and $U(x,x')=\{1\}$.
\section{Constructions of Kleene posets}
Let $\mathbf A=(A,\leq)$ be a poset. Then the {\em twist product} of $\mathbf A$ is defined as $(A^2,\sqsubseteq)$ where
\[
(x,y)\sqsubseteq(z,u)\text{ if }x\leq z\text{ and }u\leq y
\]
($(x,y),(z,u)\in A^2$). It is easy to see that the twist product of $\mathbf A$ is a poset again. This construction was successfully applied in the study of the so-called Nelson-type algebras \cite{BC}.
Now we define the central concept of our paper, which is used to represent Kleene lattices and Kleene posets within twist products. Let $S$ be a non-empty subset of $A$. Define
\[
P_S(\mathbf A):=\{(x,y)\in A^2\mid L(x,y)\leq S\leq U(x,y)\}.
\]
Instead of $P_{\{a\}}(\mathbf A)$, we simply write $P_a(\mathbf A)$.
The ordered triple $(A,\leq,{}')$ is called a {\em pseudo-Kleene poset} if $'$ is an antitone involution on $\mathbf A$ and the {\em normality condition}
\begin{equation}\label{equ1}
L(x,x')\leq U(y,y')\text{ for all }x,y\in A
\end{equation}
holds. The {\em poset} $\mathbf A$ is called {\em distributive} if one of the following equivalent LU-identities is satisfied:
\begin{align*}
L\big(U(x,y),z\big) & \approx LU\big(L(x,z),L(y,z)\big), \\
U\big(L(x,z),L(y,z)\big) & \approx UL\big(U(x,y),z\big), \\
U\big(L(x,y),z\big) & \approx UL\big(U(x,z),U(y,z)\big), \\
L\big(U(x,z),U(y,z)\big) & \approx LU\big(L(x,y),z\big), \\
L\big(U(x_1,x_2, \dots,x_n),z\big) & \approx LU\big(L(x_1,z),L(x_2,z), \dots, L(x_n,z)\big),\\
U\big(L(x_1,x_2, \dots,x_n),z\big) & \approx UL\big(U(x_1,z),U(x_2,z), \dots, U(x_n,z)\big).
\end{align*}
A {\em Kleene poset} is a distributive pseudo-Kleene poset.
\begin{remark}\rm
Recall that
D. Zhu in {\cite{Zhu}} introduced the notion of a Kleene poset as an
ordered triple $(A,\leq,{}')$ such that $'$ is an antitone involution on $\mathbf A$ and the
{\em Zhu condition}
\begin{equation}\label{equzhu}
x\leq x' \text{ and } y'\leq y\text{ implies }
x\leq y \text{ for all }x,y\in A
\end{equation}
holds. In fact, his concept is precisely the pseudo-Kleene poset in our sense because he does not assume distributivity of $(A,\leq)$.
\end{remark}
\begin{lemma}
Let $\mathbf A=(A,\leq,{}')$ be a poset with an antitone involution $'$. Then
the {normality condition} {\rm (\ref{equ1})} is equivalent to
the {Zhu condition} {\rm (\ref{equzhu})}.
\end{lemma}
\begin{proof}
Assume first that the {normality condition} holds. Let $x,y\in A, x\leq x',$ and $y'\leq y$.
Then $L(x)=L(x,x')\leq U(y,y')=U(y)$. Hence $x\leq y$.
Now assume that the {Zhu condition} holds. Let $x,y\in A$, $u\in L(x,x')$, and
$v\in U(y,y')$. Since $u\leq x$ we obtain that $x'\leq u'$. It follows that
$u\leq u'$. Similarly, $v'\leq v$. From the {Zhu condition} we conclude $u\leq v$,
i.e., $L(x,x')\leq U(y,y')$.
\end{proof}
Since Kleene algebras are distributive lattices and Zhu does not assume any
distributivity condition in his definition, we will use the notion of a Kleene poset in our sense.
For any poset $\mathbf A=(A,\leq)$ and any non-empty subset $S$ of $A$ we define $(x,y)':=(y,x)$ for all $(x,y)\in A^2$ and $\mathbf P_S(\mathbf A):=\big(P_S(\mathbf A),\sqsubseteq,{}')$. Instead of $\mathbf P_{\{a\}}(\mathbf A)$ we simply write $\mathbf P_a(\mathbf A)$.
Let $p_1$ and $p_2$ denote the first and second projection from $P_S(\mathbf A)$ to $A$, respectively.
The following lemma shows how to produce pseudo-Kleene posets from an arbitrarily given poset.
\begin{lemma}
Let $\mathbf A=(A,\leq)$ be a poset and $S$ a non-empty subset of $A$. Then $\mathbf P_S(\mathbf A)$ is a pseudo-Kleene poset and
\begin{align*}
L(X) & =\big(L(p_1(X))\times U(p_2(X))\big)\cap P_S(\mathbf A), \\
U(X) & =\big(U(p_1(X))\times L(p_2(X))\big)\cap P_S(\mathbf A)
\end{align*}
for all $X\subseteq P_S(\mathbf A)$.
\end{lemma}
\begin{proof}
Clearly, $\mathbf P_S(\mathbf A)$ is a poset with an antitone involution. Let
$(a,b),(c,d)\in P_S(\mathbf A)$, $(e,f)\in L\big((a,b),(b,a)\big)$ and $(g,h)\in U\big((c,d),(d,c)\big)$. Then $e\in L(a,b)$,
$f\in U(a,b)$, $g\in U(c,d)$, $h\in L(c,d)$, $L(a,b)\leq S\leq U(a,b)$ and $L(c,d)\leq S\leq U(c,d)$ and hence $e\leq S\leq f$ and $h\leq S\leq g$ which implies $e\leq S\leq g$ and $h\leq S\leq f$, i.e.\ $(e,f)\sqsubseteq(g,h)$ showing $L\big((a,b),(b,a)\big)\leq U\big((c,d),(d,c)\big)$.
Let $X\subseteq P_S(\mathbf A)$. Then the following are equivalent:
\begin{align*}
&(a,b)\in P_S(\mathbf A), (a,b) \sqsubseteq X,\\
&(a,b)\in P_S(\mathbf A), a \leq p_1(X), b \geq p_2(X),\\
&(a,b)\in P_S(\mathbf A), a \in L(p_1(X)), b \in U(p_2(X)),\\
&(a,b)\in \big(L(p_1(X))\times U(p_2(X))\big)\cap P_S(\mathbf A).
\end{align*}
Hence $L(X) =\big(L(p_1(X))\times U(p_2(X))\big)\cap P_S(\mathbf A)$. Similarly, we obtain
that $U(X) =\big(U(p_1(X))\times L(p_2(X))\big)\cap P_S(\mathbf A)$.
\end{proof}
Let $\mathbf A=(A,\leq)$ be a poset and $B\subseteq A$. Recall that $\mathbf A$ is said to satisfy the {\em Ascending Chain Condition} {\rm(ACC)} or the {\em Descending Chain Condition} {\rm(DCC)} if in $\mathbf A$, every strictly ascending chain or every strictly descending chain, respectively, is finite. Let $\Max B$ and $\Min B$ denote the set of all maximal and minimal elements of $B$, respectively.
Hence if $\mathbf A=(A,\leq)$ satisfies {\rm(ACC)} or {\rm(DCC)} then every
$\emptyset\not = B\subseteq A$ contains maximal or minimal elements, respectively.
\begin{lemma}
Let $\mathbf A=(A,\leq)$ be a poset and $S$ a non-empty subset of $A$.
\begin{enumerate}[{\rm(i)}]
\item If $(S,\leq)$ satisfies the {\rm ACC} and the {\rm DCC}, then $P_S(\mathbf A)=P_{(\Max S)\cup(\Min S)}(\mathbf A)$,
\item if $\bigwedge S$ and $\bigvee S$ exist, then $P_S(\mathbf A)=P_{\{\bigwedge S,\bigvee S\}}(\mathbf A)$.
\end{enumerate}
\end{lemma}
\begin{proof}
\
\begin{enumerate}[(i)]
\item If $(S,\leq)$ satisfies the ACC, then every element of $S$ lies under some maximal element of $S$, and if $(S,\leq)$ meets the DCC, then every element of $S$ lies over some minimal element of $S$.
\item If $\bigwedge S$ and $\bigvee S$ exist then we have
\begin{align*}
P_S(\mathbf A) & =\{(x,y)\in A^2\mid L(x,y)\leq S\leq U(x,y)\}= \\
& =\{(x,y)\in A^2\mid L(x,y)\leq\bigwedge S\leq\bigvee S\leq U(x,y)\}
=P_{\{\bigwedge S,\bigvee S\}}(\mathbf A).
\end{align*}
\end{enumerate}
\end{proof}
A {\em subset} $B$ of a poset $(A,\leq)$ is called {\em convex} if
\[
x,z\in B, y\in A\text{ and }x\leq y\leq z\text{ imply }y\in B.
\]
Let $S\subseteq A$. We put $co(S):=LU(S)\cap UL(S)$.
\begin{lemma}
Let $\mathbf A=(A,\leq)$ be a poset and $S\subseteq A$. Then $co(S)$ is a convex set
containing $S$,
$L(S)=L(co(S))$, $U(S)=U(co(S))$, and $P_S(\mathbf A)=P_{co(S)}(\mathbf A)$.
\end{lemma}
\begin{proof}
Let $x,z\in co(S), y\in A$ and $x\leq y\leq z$. Then $\{x, z\}\leq U(S)$. Since $y\leq z$, we obtain that $\{y\}\leq U(S)$, i.e., $y\in LU(S)$. Similarly, $y\in UL(S)$, i.e.,
$y\in co(S)$. Clearly, $S\subseteq LU(S)\cap UL(S)=co(S)$, $L(co(S))\subseteq L(S)$
and $U(co(S))\subseteq U(S)$. We have $co(S)\subseteq UL(S)$ and $co(S)\subseteq LU(S)$.
We conclude $L(S)=LUL(S)\subseteq L(co(S))$ and $U(S)=ULU(S)\subseteq U(co(S))$,
i.e., $L(S)=L(co(S))$ and $U(S)=U(co(S))$.
Finally, since $S\subseteq co(S)$ we obtain
$P_{co(S)}(\mathbf A) \subseteq P_S(\mathbf A)$. Assume now that
$(x,y)\in P_S(\mathbf A)$. Then $L(x,y)\leq S\leq U(x,y)$.
Hence $L(x,y)\subseteq L(S)=L(co(S))$ and $U(x,y)\subseteq U(S)=U(co(S))$.
We conclude $L(x,y)\leq co(S)\leq U(x,y)$, i.e., $(x,y)\in P_{co(S)}(\mathbf A)$.
\end{proof}
We are going to show that for a given poset $\mathbf A=(A,\leq)$ and an element $a$ of $A$, the constructed pseudo-Kleene poset $\mathbf P_a(\mathbf A)$ contains $A$ as a convex subset.
\begin{lemma}\label{lem2}
Let $\mathbf A=(A,\leq)$ be a poset and $a\in A$ and let $f$ denote the mapping from $A$ to $P_a(\mathbf A)$ defined by
\[
f(x):=(x,a)\text{ for all }x\in A.
\]
Then $f(A)$ is a convex subset of $\big(P_a(\mathbf A),\sqsubseteq\big)$,
$\mathbf A$ can be \hbox{{$L$}{$U$}}-embedded into $\big(P_{a}(\mathbf A),\sqsubseteq\big)$, and $f$ is an order isomorphism from $(A,\leq)$ to $\big(f(A),\sqsubseteq\big)$.
\end{lemma}
\begin{proof}
It is clear that $f$ is a mapping from $A$ to $P_a(\mathbf A)$. If $b,c\in A$, $(d,e)\in P_a(\mathbf A)$ and $(b,a)\sqsubseteq(d,e)\sqsubseteq(c,a)$, then $a\leq e\leq a$ and hence $e=a$, which implies $(d,e)\in f(A)$. This shows that $f(A)$ is a convex subset of $\big(f(A),\sqsubseteq\big)$.
Assume now that $X\subseteq A$ is finite and non-empty.
We compute:
$$\begin{aligned}
U(f(X))&=\big(U(X)\times L(a)\big) \cap P_{a}(\mathbf A)
=\big(ULU(X)\times L(a)\big) \cap P_{a}(\mathbf A)=U(f(LU(X))),\\
L(f(X))&=\big(L(X)\times U(a)\big) \cap P_{a}(\mathbf A)
=\big(LUL(X)\times U(a)\big) \cap P_{a}(\mathbf A)=L(f(UL(X))).
\end{aligned}
$$
Finally, for any $x,y\in A$, $x\leq y$ and $(x,a)\sqsubseteq(y,a)$ are equivalent. We conclude that $f$ is an order isomorphism from $(A,\leq)$ to $\big(f(A),\sqsubseteq\big)$.
\end{proof}
\section{Embeddings}
In Lemma~\ref{lem2}, we showed that for a poset $\mathbf A=(A,\leq)$ and an element $a$ of $A$, there is an embedding of $\mathbf A$ into $\big(P_a(\mathbf A),\sqsubseteq\big)$. A similar result can be shown for a distributive lattice $\mathbf L=(L,\vee,\wedge)$ and element $a$ of $L$. However, if the non-empty subset $S$ of $A$ is not a singleton, it is not so easy to find an embedding of $\mathbf L$ into $\big(P_S(\mathbf L),\sqcup,\sqcap\big)$. The following theorem provides a solution to this problem in a particular case.
\begin{theorem}\label{th3}
Let $\mathbf A=(A,\leq)$ be a distributive poset and $a,b\in A$ with $a\leq b$ and assume that there exists an orthocomplementation
${}'$ on $([a,b],\leq)$. Further, assume that for every $x\in A$ satisfying $L(a)\subset L(U(x,a),b)\subset L(b)$ we have $L(U(x,a),b)= L(x)$. Then $\mathbf A$
can be \hbox{{$L$}{$U$}}-embedded into
$\big(P_{\{a,b\}}(\mathbf A),\sqsubseteq\big)$
and the poset $([a,b],\leq, {}')$ is Boolean.
\end{theorem}
\begin{proof}
Put
\begin{align*}
I & :=\{x\in A\mid L(x, b)\leq a\}, \\
F & :=\{x\in A\mid b\leq U(x, a)\}.
\end{align*}
It is easy to see $a\in I$ and $b\in F$. Let us show that $I$ is a Frink ideal and
$F$ is a Frink filter. Assume that $X\subseteq I$, $X$ finite. Let $X=\emptyset$. Then
either $LU(X)=\emptyset$ or $LU(X)=\{0\}$ where $0$ is the smallest element of $\mathbf A$. In both
cases, $LU(X)\subseteq I$. Suppose now that $X\not=\emptyset$, $X=\{x_1, \dots, x_n\}$ and $x\in LU(X)$.
Then $L(x,b)\subseteq L(U(X), b)=LU\big(L(x_1,b),L(x_2,b), \dots, L(x_n,b)\big)\subseteq
LU\big(L(a)\big)=L(a)$. Hence $L(x, b)\leq a$. Similarly, $F$ is a Frink filter.
Let $'$ be an orthocomplementation on $([a,b],\leq)$ and define $f\colon L\rightarrow P_{\{a,b\}}(\mathbf L)$ as follows:
\[
f(x):=\left\{
\begin{array}{ll}
(x,b) & \text{if }x\in I, \\
(x,a) & \text{if }x\in F, \\
(x,x') & \text{otherwise}
\end{array}
\right.
\]
($x\in L$). Of course, $a'=b$ and $b'=a$.
If $x\in I$, then $L(x, b)\leq a\leq b\leq U(x,b)$ and $(x,b)\in P_{\{a,b\}}(\mathbf A)$. If
$x\in F$, then $L(x, a)\leq a\leq b\leq U(x,a)$ and $(x,a)\in P_{\{a,b\}}(\mathbf A)$.
Assume first that $I\cap F\not=\emptyset$. Let $z\in I\cap F$.
Then $L(z,b)\leq a$ and $b\leq U(z,a)$. We conclude
$$b\in LU(z,a)\cap L(b)=LU\big(L(z,b), L(a,b)\big)\subseteq LU(L(a))=L(a).$$
Hence $a=b$ and $I=F=A$. Evidently, $f$ is an \hbox{{$L$}{$U$}}-embedding by Lemma~\ref{lem2}.
From now on, we will assume that $I\cap F=\emptyset$. We have that $a<b$.
Let $c\in A$. If $c\notin I\cup F$ then
$L(c, b)\not\subseteq L(a)$ and $U(c, a)\not\subseteq U(b)$, i.e., $ L(b)=LU(b)\not\subseteq LU(c, a)$.
$LU(c,a)\wedge L(b)=L\big(U(c,a),b\big)=LU\big(L(c,b),L(a,b)\big)
=LU\big(L(c,b),a\big)=L(c,b)\vee L(a)$ implies
\[
L(a)\subset L(U(c,a),b)\subset L(b)
\]
and hence $L(U(c,a),b)= L(c)$. This shows $a< c < b$. Therefore
\[
A\setminus(I\cup F)\subseteq [a,b]\setminus \{a,b\}.
\]
If $c\in I\cap[a,b]$, then $a\leq c\in L(c,b)\leq a$, which implies $c=a$. This implies
\[
I\cap[a,b]=\{a\}.
\]
Dually, we obtain
\[
F\cap[a,b]=\{b\}.
\]
Therefore, we have a partition
$$
A=I\cupdot F \cupdot ([a,b]\setminus \{a,b\}).
$$
We conclude
\[
f(x)=(x,x')\text{ for all }x\in[a,b].
\]
Since $'$ is an orthocomplementation on $([a,b],\leq)$ we obtain that
$L(x,x')= L(a) \subset L(b) = LU(x,x')$, i.e.,
$(x,x')\in P_{\{a,b\}}(\mathbf A)$.
We have proved that $f$ is well-deï¬ned.
Let us verify that $f$ is an \hbox{{$L$}{$U$}}-embedding. We first show that
$f$ is an \hbox{{$L$}{$U$}}-morphism. Clearly, $f$ is order preserving.
Let us put $Z_I=Z\cap I$, $Z_{(a,b)}=Z\cap ([a,b]\setminus \{a,b\})$, and
$Z_F=Z\cap F$ for any subset $Z\subseteq A$.
Assume now that $X\subseteq A$ is finite and non-empty.
Suppose first that $X_F\not=\emptyset$. Then
$LU(X)\cap F\not=\emptyset$ and $U(X)\subseteq F$.
We compute:
$$
U(f(X))=\big(U(X)\times L(a)\big) \cap P_{\{a,b\}}(\mathbf A)
=\big(ULU(X)\times L(a)\big) \cap P_{\{a,b\}}(\mathbf A)=U(f(LU(X))).
$$
From now on, we will assume that $X_F=\emptyset$.
Case~1. Let $X_{(a,b)}=\emptyset$.
Then $X_I=X$ and $X\subseteq LU(X)\subseteq I$. We compute:
$$
\begin{aligned}
&U\big(f(X)\big)=U(X\times \{b\})=\big(U(X)\times L(b)\big) \cap P_{\{a,b\}}(\mathbf A)\
\text{and}\ U\big(f(LU(X))\big)=\\
&U\big(LU(X)\times \{b\}\big)=\big(ULU(X)\times L(b)\big) \cap P_{\{a,b\}}(\mathbf A)=
\big(U(X)\times L(b)\big) \cap P_{\{a,b\}}(\mathbf A).
\end{aligned}
$$
Case~2. Let $X_{(a,b)}\not=\emptyset$.
Then $f(X)=X_I\times \{b\}\cup \{(x,x')\mid x\in X_{(a,b)}\}$,
$X_{(a,b)}'\subseteq [a,b]\setminus \{a,b\}$
and
$L(a)\subseteq L(X_{(a,b)}')\subseteq L(b)$. We compute:
that
$$
\begin{aligned}
U\big(f(X)\big)&=\big(U(X)\times L(X_{(a,b)}')\big)\cap P_{\{a,b\}}(\mathbf A)
\supseteq U(f(LU(X))), \\
U\big(f(LU(X))\big)&=\begin{cases}\big(ULU(X)\times L(a)\big)\cap P_{\{a,b\}}(\mathbf A),&
\text{if } LU(X)\cap F\not=\emptyset,\\
\big(ULU(X)\times L((LU(X)_{(a,b)})')\big)\cap P_{\{a,b\}}(\mathbf A),&\text{otherwise.}
\end{cases}\\
&=\begin{cases}\big(U(X)\times L(a)\big)\cap P_{\{a,b\}}(\mathbf A),&
\text{if } LU(X)\cap F\not=\emptyset,\\
\big(U(X)\times L((LU(X)_{(a,b)})')\big)\cap P_{\{a,b\}}(\mathbf A),\phantom{UL}&\text{otherwise.}
\end{cases}
\end{aligned}
$$
Suppose first that $LU(X)\cap F\not=\emptyset$. Assume that $y\in L(X_{(a,b)}')\setminus L(a)$. We have
$$
L(a)\subset L(U(y,a),b)=L(U(y,a))\subseteq L(x')\subset L(b)
$$
for every $x\in X_{(a,b)}$. Since $X_{(a,b)}$ is non-empty,
we therefore obtain
$$
L(a)\subset L(U(y,a),b)=L(y)\subset L(b),
$$
i.e., $a<y <b$ and $a<y' <b$. Moreover, $y\in L(X_{(a,b)}')$ yields $y'\in U(X_{(a,b)})$, i.e.,
$\emptyset\not=LU(X)\cap F \subseteq LU(X_I,y')\cap F$. Hence there is an element $z\in F$ such that $z\in LU(X_I,y')$
and $b\leq U(z,a)$. Let $X_I=\{x_{i_1}, \dots, x_{i_k}\}$ and
$X_{(a,b)}=\{x_{j_1}, \dots, x_{j_l}\}$.
We compute:
$$
\begin{aligned}
b\in LU(z,a)\cap L(b)&\subseteq LU\big(LU(X_I, y'),a\big)\cap L(b)=
LU(X_I, y')\cap L(b)\\
&=L\big(U(x_{i_1},\dots, x_{i_k}, y'),b\big)=
LU\big(L(x_{i_1},b), \dots, L(x_{i_k},b), L(y',b)\big)\\
&\subseteq LU\big(L(a), \dots, L(a), L(y')\big)=L(y'),
\end{aligned}
$$
i.e., $b\leq y' < b$, a contradiction. Hence $L(a)= L(X_{(a,b)}')$ and
$U\big(f(LU(X))\big)=U\big(f(X)\big)$.
Suppose now that $LU(X)\cap F=\emptyset$.
Let us check that $L(X_{(a,b)}')=L\big(\{u'\mid u\in LU(X)\cap ([a,b]\setminus \{a,b\})\}\big)$.
Clearly, $L(a)\subseteq L\big(\{u'\mid u\in LU(X)\cap ([a,b]\setminus \{a,b\})\}\big)
\subseteq L(X_{(a,b)}')\subseteq L(b)$. The first inclusion holds
since $a < u < b$ and $a < u' < b$. The second one follows from the fact that $X_{(a,b)}\subseteq LU(X)$ and $X_{(a,b)}\subseteq ([a,b]\setminus \{a,b\})$. Hence also $X_{(a,b)}\subseteq LU(X)\cap ([a,b]\setminus \{a,b\})\subseteq [a,b]$, i.e., $X_{(a,b)}'\subseteq \big(LU(X)\cap ([a,b]\setminus \{a,b\})\big)'$.
Assume that $y\in L(X_{(a,b)}')\setminus L(a)$. We have
$$
L(a)\subset L\big(U(y,a),b\big)=L\big(U(y,a)\big)\subseteq L(x')\subset L(b)
$$
for every $x\in X_{(a,b)}$. Since $X_{(a,b)}$ is non-empty (i.e.,
at least one $x\in X_{(a,b)}$ exists) we obtain
$$
L(a)\subset L\big(U(y,a),b\big)=L(y)\subset L(b),
$$
i.e., $a<y <b$ and $a<y' <b$. Moreover, $y\in L\big(X_{(a,b)}'\big)$ yields $y'\in U\big(X_{(a,b)}\big)$, i.e., $LU\big(X_{(a,b)}\big)\leq y'$.
Let $u\in LU(X)\cap ([a,b]\setminus \{a,b\})$. We compute:
$$
\begin{aligned}
u\in LU(X)\cap L(b)&=LU\big(L(x_{i_1}, b), \dots, L(x_{i_k}, b), L(x_{j_1}, b), \dots, L(x_{j_l}, b)\big)\\
&\subseteq LU\big(L(a), \dots, L(a), L(x_{j_1}), \dots, L(x_{j_l}\big))=LU\big(X_{(a,b)}\big)\leq y'.
\end{aligned}
$$
This implies $u\leq y'$, i.e.,
$y\leq u'$ and $LU(X)\cap L(b)=LU\big(X_{(a,b)}\big)$. Therefore, $y\in L\big(\{u'\mid u\in LU(X)\cap ([a,b]\setminus \{a,b\})\}\big)$.
We compute:
$$
\begin{aligned}
U\big(f(LU(X))\big)&=U\big(\big(LU(X)\cap I\big)\times \{b\}\cup
\{(u,u')\mid u\in LU(X)\cap ([a,b]\setminus \{a,b\})\}\big)\\
&=\big(ULU(X)\times L\big(\{u'\mid u\in LU(X)\cap ([a,b]\setminus \{a,b\})\}
\big)\cap P_{\{a,b\}}(\mathbf A)\\
&= \big(ULU(X)\times L(X_{(a,b)}')\big)\cap P_{\{a,b\}}(\mathbf A)
= \big(U(X)\times L(X_{(a,b)}')\big)\cap P_{\{a,b\}}(\mathbf A)\\
&=U\big(f(X)\big).
\end{aligned}
$$
Similarly, we obtain that $L\big(f(UL(X))\big)=L\big(f(X)\big)$ for every
finite and non-empty $X\subseteq A$.
Assume now that $c, d\in A$, $f(c)=(c,u)\sqsubseteq (d,v)=f(d)$. Then $c\leq d$ and $f$ is order-reflecting.
It remains to check that $([a,b],\leq, {}')$ is a Boolean poset. Clearly, it is
complemented. We have to verify that it is distributive.
Assume that $x, y, z\in [a,b]$. If $\{x, y,z\}\cap \{a,b\}\not=\emptyset$
then evidently $L_{[a,b]}\big(U_{[a,b]}(x,y),z\big) =
L_{[a,b]}U_{[a,b]}\big(L_{[a,b]}(x,z),L_{[a,b]}(y,z)\big)$.
Hence we may assume that $a< x, y, z < b$.
Let
$d\in U_{[a,b]}\big(L_{[a,b]}(x, z), L_{[a,b]}(y, z)\big)$ and
$e \in L_{[a,b]}\big(U_{[a,b]}(x, y),z\big)$.
We will show that $d\in U\big(L(x, z), L(y, z)\big)$ and
$e\in L\big(U(x, y),z\big)$.
Then
$d \geq L(x, z)$. Namely, if $d \not\geq L(x, z)$ then
there is $h\in A\setminus [a,b]$
such that $h\not \leq d$, $h< x$ and $h<z$. Hence also $h< b$ and
$h\not\leq a$.
Clearly,
$L(a)\subseteq L\big(U(h,a),b\big)=LU(h,a)\subseteq L(b)$. Assume first $L(a)= LU(h,a)$. Then
$h\in LU(h,a)=L(a)$, a contradiction. We have
$ LU(h,a)\subseteq L(x)\subset L(b)$. Hence
$L(a)\subset L\big(U(h,a),b\big)=L\big(U(h,a)\big)\subset L(b)$, i.e., $a< h <b$, a contradiction again.
Therefore really $d \geq L(x, z)$ and similarly $d \geq L(y, z)$. Hence
$d\in U\big(L(x, z), L(y, z)\big)$.
Further, $e\in L(z)$ since $e\in L_{[a,b]}(z)$. Let us check that $e\in LU(x, y)$. Assume that
$e\not\in LU(x, y)$. Since $e \in L_{[a,b]}\big(U_{[a,b]}(x, y),z\big)$ there exists
$g\in U(x,y)\setminus U_{[a,b]}(x, y)$ such that $e\not\leq g$. We have
$a < x \leq g\not=b$. Hence
$L(a)\subseteq L\big(U(g,a),b\big)=L(g,b)\subseteq L(b)$.
Suppose first that
$L(g,b)=L(b)$. Then $e\leq b < g$, a contradiction with $e\not\leq g$. Also,
$L(a)\subset L(x)\subseteq L(g,b)=L\big(U(g,a),b\big) \subset L(b)$. We conclude $a< g < b$,
which is impossible since $g\notin [a,b]$.
Therefore $e\in LU(x, y)$ and we obtain
$e\in LU(x, y)\cap L(z)=L\big(U(x, y),z\big)$.
Since $\mathbf A$ is distributive we see that $e\leq d$.
Consequently $U_{[a,b]}L_{[a,b]}\big(U_{[a,b]}(x,y),z\big)\supseteq
U_{[a,b]}\big(L_{[a,b]}(x, z), L_{[a,b]}(y, z)\big)$
which yields that $([a,b],\leq)$ is distributive.
\end{proof}
\begin{remark} \rm Note that Theorem \ref{th3} generalizes
\cite[Lemma 6]{CLP} formulated for distributive lattices. We also prefer to use orthocomplementation instead of antitone complementation as in
\cite[Lemma 6]{CLP}.
\end{remark}
In the sequel, we use the following notation: If $\mathbf A_1$ and $\mathbf A_2$ are posets with top and bottom elements, respectively, then by $\mathbf A_1+_a\mathbf A_2$ we denote
the ordinal sum of $\mathbf A_1$ and $\mathbf A_2$ where the top element $a$ of $\mathbf A_1$ is identified with the bottom element of $\mathbf A_2$. If $\mathbf A_1$, $\mathbf A_2$, and $\mathbf A_3$
are posets with top element, bottom and top element, and bottom element, respectively, then by
$\mathbf A_1+_a\mathbf A_2+_b\mathbf A_3$ we denote the ordinal sum of $\mathbf A_1$, $\mathbf A_2$, and $\mathbf A_3$ where the top element $a$ of $\mathbf A_1$ is identified with the bottom element of $\mathbf A_2$ and the top element $b$ of $\mathbf A_2$ is identified with the bottom element of $\mathbf A_3$. For every poset $\mathbf A$, let $\mathbf A^d$ denote its dual.
\begin{lemma}\label{osumdist}
Let $\mathbf A_1=(A_1,\leq)$ and $\mathbf A_2=(A_2,\leq)$ be distributive posets with top element $a$ and bottom element $a$, respectively. Then $\mathbf A_1+_a\mathbf A_2$ is a distributive poset.
\end{lemma}
\begin{proof}
Let $x\in A_1$ and $y\in A_2$. Then $L_{\mathbf A_1+_a\mathbf A_2}(x)=L_{\mathbf A_1}(x)$,
$L_{\mathbf A_1+_a\mathbf A_2}(y)=L_{\mathbf A_2}(y)\cup A_1$,
$U_{\mathbf A_1+_a\mathbf A_2}(y)=U_{\mathbf A_2}(y)$,
$U_{\mathbf A_1+_a\mathbf A_2}(x)=U_{\mathbf A_1}(x)\cup A_2$.
Assume that $x, y, z\in \mathbf A_1+_a\mathbf A_2$. If some pair of elements
$x, y, z$ is comparable, then evidently
\begin{align*}
L_{\mathbf A_1+_a\mathbf A_2}\big(U_{\mathbf A_1+_a\mathbf A_2}(x,y),z\big)
& = L_{\mathbf A_1+_a\mathbf A_2}U_{\mathbf A_1+_a\mathbf A_2}
\big(L_{\mathbf A_1+_a\mathbf A_2}(x,z),L_{\mathbf A_1+_a\mathbf A_2}(y,z)\big).
\end{align*}
Suppose now that there is no pair of comparable elements from
$\{x, y, z\}$. Then either $\{x, y, z\}\subseteq A_1$ or
$\{x, y, z\}\subseteq A_2$. Assume first that $\{x, y, z\}\subseteq A_1$.
We compute:
\begin{align*}
L_{\mathbf A_1+_a\mathbf A_2}\big(U_{\mathbf A_1+_a\mathbf A_2}(x,y),z\big)
& = L_{\mathbf A_1+_a\mathbf A_2}\big(U_{\mathbf A_1}(x,y)\cup A_2,z\big) \\
& = L_{\mathbf A_1}\big(U_{\mathbf A_1}(x,y),z\big) =
L_{\mathbf A_1}U_{\mathbf A_1}\big(L_{\mathbf A_1}(x,z),L_{\mathbf A_1}(y,z)\big)\\
& =
L_{\mathbf A_1}U_{\mathbf A_1}\big(L_{\mathbf A_1+_a\mathbf A_2}(x,z),L_{\mathbf A_1+_a\mathbf A_2}(y,z)\big)\\
& = L_{\mathbf A_1+_a\mathbf A_2}U_{\mathbf A_1+_a\mathbf A_2}
\big(L_{\mathbf A_1+_a\mathbf A_2}(x,z),L_{\mathbf A_1+_a\mathbf A_2}(y,z)\big).
\end{align*}
The case $\{x, y, z\}\subseteq A_2$ can be verified by the same procedure.
\end{proof}
The following proposition enables us to determine a broad class of representable Kleene posets by using ordinal sums of distributive posets.
\begin{proposition}\label{prop2}
Let $\mathbf A_1=(A_1,\leq)$ and $\mathbf A_2=(A_2,\leq)$ be distributive posets with top element $a$ and bottom element $b$, respectively, and $\mathbf B=(B,\leq,{}')$ a Boolean poset with bottom element $a$ and top element $b$. If $a\not=b$ or $a$ is join-irreducible in $\mathbf A_1$ and meet-irreducible in $\mathbf A_2$ then $\big(P_{ab}(\mathbf A_1+_a\mathbf B+_b\mathbf A_2),\sqsubseteq,{}'\big)$ is
a Kleene poset and
\[
\big(P_{ab}(\mathbf A_1+_a\mathbf B+_b\mathbf A_2),\sqsubseteq\big)\cong(\mathbf A_1\times\mathbf A_2^d)+_{(a,b)}\mathbf B+_{(b,a)}(\mathbf A_2\times\mathbf A_1^d).
\]
\end{proposition}
\begin{proof}
We have
\[
P_{ab}(\mathbf A_1+_a\mathbf B+_b\mathbf A_2)=(A_1\times A_2)\cup\{(x,x')\mid x\in B\}\cup(A_2\times A_1)
\]
and the order relations on both sides coincide. The equality follows from the following facts:
\begin{enumerate}
\item if $x\in B\setminus \{a,b\}$ then the only element $y\in A_1\cup B\cup A_2$ satisfying
$L_{\mathbf A_1+_a\mathbf B+_b\mathbf A_2}(x,y)\leq a$ and
$U_{\mathbf A_1+_a\mathbf B+_b\mathbf A_2}(x,y)\geq b$ is the element $x'\in B$,
\item if $x\in A_1$ and $(x,y)\in P_{ab}(\mathbf A_1+_a\mathbf B+_b\mathbf A_2)$ then
$y\in A_2$,
\item if $x\in A_2$ and $(x,y)\in P_{ab}(\mathbf A_1+_a\mathbf B+_b\mathbf A_2)$ then
$y\in A_1$,
\item $(A_1\times A_2)\cup\{(x,x')\mid x\in B\}\cup(A_2\times A_1)\subseteq
P_{ab}(\mathbf A_1+_a\mathbf B+_b\mathbf A_2)$.
\end{enumerate}
Since the dual of a distributive poset is again
distributive and the cartesian product of distributive posets is distributive, we obtain by Lemma
\ref{osumdist} that the ordinal sum $(\mathbf A_1\times\mathbf A_2^d)+_{(a,b)}\mathbf B+_{(b,a)}(\mathbf A_2\times\mathbf A_1^d)$ is also distributive. Hence
$\big(P_{ab}(\mathbf A_1+_a\mathbf B+_b\mathbf A_2),\sqsubseteq,{}'\big)$ is
a Kleene poset.
\end{proof}
\begin{corollary}\label{cor3}
Let $\mathbf A_1=(A_1,\leq)$ and $\mathbf A_2=(A_2,\leq)$ be distributive posets with top element $a$ and bottom element $b$, respectively, and $\mathbf B=(B,\leq,{}')$ a Boolean poset with bottom element $a$ and top element $b$. If $a\not=b$ or $a$ is join-irreducible in $\mathbf A_1$ and meet-irreducible in $\mathbf A_2$ then $\big(P_{ab}(\mathbf A_1+_a\mathbf B),\sqsubseteq,{}'\big)$ and
$\big(P_{ab}(\mathbf B+_b\mathbf A_2),\sqsubseteq,{}'\big)$ are Kleene posets and
\begin{align*}
\big(P_{ab}(\mathbf A_1+_a\mathbf B),\sqsubseteq\big) & \cong\mathbf A_1+_{(a,b)}\mathbf B+_{(b,a)}\mathbf A_1^d, \\
\big(P_{ab}(\mathbf B+_b\mathbf A_2),\sqsubseteq\big) & \cong\mathbf A_2^d+_{(a,b)}\mathbf B+_{(b,a)}\mathbf A_2.
\end{align*}
\end{corollary}
\begin{proof}
It is enough to put $\mathbf A_2:=\mathbf 1$ or $\mathbf A_1:=\mathbf 1$ and use Proposition~\ref{prop2}.
\end{proof}
\begin{corollary}\label{cor4}
Let $\mathbf A_1=(A_1,\leq)$ and $\mathbf A_2=(A_2,\leq)$ be distributive posets with top and bottom element $a$, respectively. If $a$ is join-irreducible in $\mathbf A_1$ and meet-irreducible in $\mathbf A_2$ then
\begin{align*}
\big(P_a(\mathbf A_1+_a\mathbf A_2),\sqsubseteq\big) & \cong(\mathbf A_1\times\mathbf A_2^d)+_{(a,a)}(\mathbf A_2\times\mathbf A_1^d), \\
\big(P_a ({\mathbf A}_1),\sqsubseteq\big) & \cong\mathbf A_1+_{(a,a)}\mathbf A_1^d, \\
\big(P_a(\mathbf A_2),\sqsubseteq\big) & \cong\mathbf A_2^d+_{(a,a)}\mathbf A_2.
\end{align*}
\end{corollary}
\begin{proof}
It is enough to put $\mathbf B:=\mathbf 1$ and use Proposition~\ref{prop2} and Corollary~\ref{cor3}.
\end{proof}
For some of the Kleene posets described in Proposition~\ref{prop2}, we can construct the embeddings as follows:
\begin{corollary}
Let $\mathbf A_1=(A_1,\leq)$ and $\mathbf A_2=(A_2,\leq)$ be distributive posets with top element $a$ and bottom element $b$, respectively, and $\mathbf B=(B,\leq,{}',a,b)$ a non-trivial bounded Boolean poset, put $\mathbf A:=\mathbf A_1+_a\mathbf B+_b\mathbf A_2$ and define
$f\colon A\rightarrow P_{ab}(\mathbf L)$ as follows:
\[
f(x):=\left\{
\begin{array}{ll}
(x,b) & \text{if }x\leq a, \\
(x,x') & \text{if }a\leq x\leq b, \\
(x,a) & \text{if }b\leq x
\end{array}
\right.
\]
{\rm(}$x,y\in A${\rm)}. Then $f$ is an embedding from $\mathbf A$ into $\big(P_{ab}(\mathbf A),\sqsubseteq\big)$, and $f(A)$ is a convex subset of $\big(P_{ab}(\mathbf L),\sqsubseteq\big)$.
\end{corollary}
\begin{proof}
The first assertion is a special case of Theorem~\ref{th3}. We have
\begin{align*}
\mathbf A & =\mathbf A_1+_a\mathbf A+_2\mathbf A_2, \\
P_{ab}(\mathbf A) & =(A_1\times A_2)\cup\{(x,x')\mid x\in B\}\cup(A_2\times A_1), \\
f(A) & =(A_1\times\{b\})\cup\{(x,x')\mid x\in B\}\cup(A_2\times\{a\}).
\end{align*}
Now assume $(c,d),(h,i)\in f(A)$, $(e,g)\in P_{ab}(\mathbf A)$ and $(c,d)\sqsubseteq(e,g)\sqsubseteq(h,i)$. Then $c\leq e\leq h$ and $a\leq i\leq g\leq d\leq b$, and hence $a\leq g\leq b$.
If $e\in B$ then $f(e)=(e,e')=(e,g)$. Assume now that $e< a$. Then $g\geq b$, i.e., $g=b$.
We conclude that $f(e)=(e,b)=(e,g)$. Finally, suppose that $b< e$. Hence $g\leq a$, i.e., $g=a$
and $f(e)=(e,a)=(e,g)$.
Summing up, $f(A)$ is a convex subset of $\big(P_{ab}(\mathbf A),\sqsubseteq\big)$.
\end{proof}
\section{Representable Kleene posets}
The following result shows how to construct representable Kleene posets
using the direct product of known representable Kleene posets.
\begin{theorem}\label{th4}
Let $\mathbf A_i=(A_i,\leq)$ be a poset and $S_i$ a non-empty subset of $A_i$ for every $i\in I$. Put
\[
\mathbf A:=\prod_{i\in I}\mathbf A_i\text{ and }S:=\prod_{i\in I}S_i.
\]
Then
\[
\mathbf P_S(\mathbf A)\cong\prod_{i\in I}\mathbf P_{S_i}(\mathbf A_i).
\]
Moreover, if $\mathbf A_i$ is a distributive poset for every $i\in I$ then $\mathbf A$ is a distributive poset.
\end{theorem}
\begin{proof}Recall that, for a non-empty subset $X\subseteq \prod_{i\in I} A_i$,
$L_{\mathbf A}(X)=\prod_{i\in I} L_{{\mathbf A}_i}\big(p_i(X)\big)$ and
$U_{\mathbf A}(X)=\prod_{i\in I} U_{{\mathbf A}_i}\big(p_i(X)\big)$.
Let us show that the mapping $f$ from $\prod\limits_{i\in I}P_{S_i}(\mathbf A_i)$ to $P_S(\mathbf A)$ defined by
\[
f\big((x_i,y_i)_{i\in I}\big):=\big((x_i)_{i\in I},(y_i)_{i\in I}\big)
\]
for all $(x_i,y_i)_{i\in I}\in\prod\limits_{i\in I}P_{S_i}(\mathbf A_i)$ is an isomorphism from $\prod\limits_{i\in I}\mathbf P_{S_i}(\mathbf A_i)$ to $\mathbf P_S(\mathbf A)$. \\
First, we have to check that $\big((x_i)_{i\in I},(y_i)_{i\in I}\big)\in P_S(\mathbf A)$. \\
Evidently, $S\not=\emptyset$. Suppose $a=(a_i)_{i\in I}\in S$. Then $a_i\in S_i$ for every $i\in I$. Since $(x_i,y_i)\in P_{S_i}(\mathbf A_i)$ we have that
$L_{\mathbf A_i}(x_i, y_i)\leq a_i\leq U_{\mathbf A_i}(x_i, y_i)$. Hence
$L_{\mathbf A}\big((x_i)_{i\in I}, (y_i)_{i\in I}\big)\leq a\leq U_{\mathbf A}((x_i)_{i\in I}, (y_i)_{i\in I})$. \\
Second, let us check that $f$ is an order embedding. Assume $(x_i,y_i)_{i\in I},(u_i,v_i)_{i\in I}\in\prod\limits_{i\in I}P_{S_i}(\mathbf A_i)$. Then the following are equivalent: $(x_i,y_i)_{i\in I}\sqsubseteq(u_i,v_i)_{i\in I}$ in $\prod\limits_{i\in I}\mathbf P_{S_i}(\mathbf A_i)$; $x_i\leq u_i$ and $v_i\leq y_i$ for every $i\in I$; $(x_i)_{i\in I}\leq(u_i)_{i\in I}$ and $(v_i)_{i\in I}\leq(y_i)_{i\in I}$ in $\mathbf A$; $\big((x_i)_{i\in I},(y_i)_{i\in I}\big)\sqsubseteq\big((u_i)_{i\in I},(v_i)_{i\in I}\big)$ in $\mathbf P_S(\mathbf A)$. \\
Third, we have to verify that $f$ is surjective. Let $z\in P_S(\mathbf A)$.
Then $z=(x,y)\in A^2$ and $L_{\mathbf A}(x, y)\leq a\leq U_{\mathbf A}(x, y)$
for all $a\in S$. We have $x=(x_i)_{i\in I}$ and $y=(y_i)_{i\in I}$ with
$x_i,y_i\in A_i$ for every $i\in I$. Let us check that $(x_j,y_j)\in P_{S_j}(\mathbf A_j)$ for every $j\in I$. Let $a_j\in S_j$ (this is possible since all $S_i$ are non-empty) and extend this to some $a:=(a_i)_{i\in I}\in S$. Then we have
\[
L_{\mathbf A}\big((x_i)_{i\in I}, (y_i)_{i\in I}\big)\leq a\leq U_{\mathbf A}\big((x_i)_{i\in I}, (y_i)_{i\in I}\big).
\]
Therefore $L_{\mathbf A_i}(x_i, y_i)\leq a_i\leq U_{\mathbf A_i}(x_i, y_i)$ for all $i\in I$, in particular for $i=j$. \\
Fourth, let us show that $f$ preserves $'$. Assume that $(x_i,y_i)_{i\in I}\in\prod\limits_{i\in I} P_{S_i}(\mathbf A_i)$. Then $(y_i,x_i)_{i\in I}\in\prod\limits_{i\in I} P_{S_i}(\mathbf A_i)$ and $\big((x_i)_{i\in I},(y_i)_{i\in I}\big),\big((y_i)_{i\in I},(x_i)_{i\in I}\big)\in P_S(\mathbf A)$. We compute
\begin{align*}
f\Big(\big((x_i,y_i)_{i\in I}\big)'\Big) & =f\big((y_i,x_i)_{i\in I}\big)=\big((y_i)_{i\in I},(x_i)_{i\in I}\big)=\big((x_i)_{i\in I},(y_i)_{i\in I}\big)'= \\
& =\Big(f\big((x_i,y_i)_{i\in I}\big)\Big)'.
\end{align*}
Suppose now that $\mathbf A_i$ is a distributive poset for every $i\in I$.
Since the distributive law for
$\mathbf A$ can be checked componentwise, $\mathbf A$ is a distributive poset.
\end{proof}
\begin{lemma}\label{chaininv}
Let ${\mathbf C}$ be a bounded chain with involution and $a\in C$ such that $a\leq a'$
and $x\in [a,a']$ implies $x\in \{a,a'\}$. Then ${\mathbf C}$ is a
representable Kleene poset and ${\mathbf C}\cong {\mathbf P}_{[a,a']}([a, 1])$.
\end{lemma}
\begin{proof}
Assume first that $a=a'$. From \cite[Lemma 18]{CLP} we know that
$P_a(\mathbf [a, 1])=\big(\{a\}\times[a,1]\big)\cup\big([a,1]\times\{a\}\big)
\cong \mathbf C$. The isomorphism $f$ from $\mathbf C$ to ${\mathbf P}_a(\mathbf [a, 1])$
is given by
\begin{align*}
f(x)=\begin{cases}(a,x')&\text{ if } x\leq a,\\
(x, a)&\text{ if } a<x.
\end{cases}
\end{align*}
Moreover $f(a)=(a,a)$.
Suppose now that $a<a'$. Let us show that
${\mathbf C}\cong {\mathbf P}_{\{a,a'\}}([a, 1])$. We define an isomorphism $g$
from $\mathbf C$ to ${\mathbf P}_{\{a,a'\}}([a, 1])$ as follows:
\begin{align*}
g(x)=\begin{cases}(a,x')&\text{ if } x\leq a,\\
(x, a)&\text{ if } a<x.
\end{cases}
\end{align*}
Moreover $g(a)=(a,a')$ and $g(a')=(a',a)$.
\end{proof}
\begin{remark} \rm Let us denote by ${\mathcal R}{\mathcal C}$ the class of
bounded chains with involution satisfying the assumption of Lemma
\ref{chaininv}. Clearly, all finite chains are in
${\mathcal R}{\mathcal C}$, which was proved already in
\cite[Corollary 21]{CLP}. Hence due to Theorem \ref{th4} and
Lemma \ref{chaininv} direct products of chains from ${\mathcal R}{\mathcal C}$
form a class of representable Kleene lattices.
\end{remark}
By Theorem \ref{th4}, a direct product of representable Kleene
posets ${\ \mathbf K}_i$ is
again representable, and the set $S$ for this product is just the
direct product of the sets $S_i$ for ${\ \mathbf K}_i$. The natural question arises
if a similar result also holds for a subdirect product of representable Kleene
lattices. We can show that, in particular cases, this is true. Let us consider the following example.
\begin{example} Let ${\ \mathbf K}_1$ be the Kleene lattice depicted in Fig.~1 and
${\ \mathbf K}_2$ the two-element chain considered as a Kleene lattice:
\vspace*{3mm}
\begin{center}
\setlength{\unitlength}{8mm}
\begin{tabular}{c c c}
\begin{picture}(2,4)
\put(1,0){\circle*{.3}}
\put(1,1){\circle*{.3}}
\put(0,2){\circle*{.3}}
\put(2,2){\circle*{.3}}
\put(1,3){\circle*{.3}}
\put(1,4){\circle*{.3}}
\put(1,1){\line(-1,1)1}
\put(1,1){\line(0,-1)1}
\put(1,1){\line(1,1)1}
\put(1,3){\line(-1,-1)1}
\put(1,3){\line(1,-1)1}
\put(1,3){\line(0,1)1}
\put(1.35,-.3){$0_{{\mathbf K}_1}$}
\put(1.35,.8){$y$}
\put(-.6,1.8){$x$}
\put(-2.399,1.8){${\ \mathbf K}_1=\ $}
\put(2.35,1.8){$x'$}
\put(1.35,2.8){$y'$}
\put(1.35,4.1){$1_{{\mathbf K}_1}$}
\end{picture}&\phantom{xxxxxccccxxxxxxxxx}&
\begin{picture}(2,3)
\put(1,2){\circle*{.3}}
\put(1,3){\circle*{.3}}
\put(1,2){\line(0,1)1}
\put(1.35,1.8){$0_{{\mathbf K}_2}$}
\put(1.35,3.3){$1_{{\mathbf K}_2}$}
\put(-1.1399,1.8){${\ \mathbf K}_2=\ $}
\end{picture}\\
&&\\
&{\rm Fig.~1}&
\end{tabular}
\end{center}
Then ${\ \mathbf K}_1$ is representable by means of ${\ \mathbf L}_1$ and
${S}_1$ and, similarly, ${\ \mathbf K}_2$ is representable by means
of ${\ \mathbf L}_2$ and ${S}_2$ as shown in Fig.~2.
\vspace*{3mm}
\begin{center}
\setlength{\unitlength}{8mm}
\begin{tabular}{c c c}
\begin{picture}(2,4)
\put(1,1){\circle*{.3}}
\put(0,2){\circle*{.3}}
\put(2,2){\circle*{.3}}
\put(1,3){\circle*{.3}}
\put(1,4){\circle*{.3}}
\put(1,1){\line(-1,1)1}
\put(1,1){\line(1,1)1}
\put(1,3){\line(-1,-1)1}
\put(1,3){\line(1,-1)1}
\put(1,3){\line(0,1)1}
\put(1.35,.8){$b$}
\put(-.6,1.8){$a$}
\put(-2.399,1.8){${\ \mathbf L}_1=\ $}
\put(2.35,1.8){$a'$}
\put(1.35,2.8){$b'$}
\put(1.35,4.1){$1_{{\mathbf L}_1}$}
\end{picture}&\phantom{xxcccxxx}&
\begin{picture}(2,3)
\put(1,2){\circle*{.3}}
\put(1,3){\circle*{.3}}
\put(1,2){\line(0,1)1}
\put(1.35,1.8){$0_{{\mathbf L}_2}$}
\put(1.35,3.3){$1_{{\mathbf L}_2}$}
\put(-1.1399,1.8){${\ \mathbf L}_2=\ $}
\end{picture}\\
$\begin{array}{r l}
\multicolumn{2}{c}{S_1=\{a, a', b, b'\}}\\[0.2cm]
1_{{\mathbf K}_1} \mapsto (1_{\mathbf L_1},b),& y'\mapsto (b',b) \\
x \mapsto (a,a'),& x'\mapsto (a',a) \\
0_{{\mathbf K}_1} \mapsto (b,1_{{\mathbf L}_1})
,& y\mapsto (b,b') \\
\end{array}$&&
$\begin{array}{r l}
\multicolumn{2}{c}{S_2=\{0_{{\mathbf L}_2}, 1_{{\mathbf L}_2}\}}\\[0.2cm]
1_{{\mathbf K}_2} \mapsto (1_{{\mathbf L}_2},0_{{\mathbf L}_2}),
& 0_{{\mathbf K}_2} \mapsto (0_{{\mathbf L}_2},1_{{\mathbf L}_2})\\
& \\
&\\
\end{array}$\\
&{\rm Fig.~2}&
\end{tabular}
\end{center}
Hence ${{\mathbf K}_1}\cong \mathbf P_{S_1}(\mathbf L_1)$ and
${{\mathbf K}_2}\cong \mathbf P_{S_2}(\mathbf L_2)$.
Consider now the Kleene lattice ${\mathbf K}={{\mathbf K}_1}\times {{\mathbf K}_2}$.
By Theorem \ref{th4} it is representable by means of
${\ \mathbf L}={\ \mathbf L}_1\times {{\mathbf L}_2}$ and
$S={S}_1\times {{S}_2}$, see Fig.~3.
Consider now two subdirect products of ${\mathbf K}$ which are Kleene lattices.
\begin{enumerate}[{\rm1.)}]
\item We start with ${{\mathbf K}^s}$, see Fig.~4.
Take $L^{s}:=(L_1\times L_2)\cap K^{s}$ and $S':=S\cap L^{s}$, see
Fig.~5. Then
$L^{s}=(L_1\times L_2) \setminus \{(1_{{\mathbf L}_1},0_{{\mathbf L}_2})\}$ and $S'=S$.
It is elementary to show that $\mathbf K^s\cong\mathbf P_{S'}(\mathbf L^s)$.
Thus it is representable.
\item Now, let us consider the subdirect product ${{\mathbf K}^0}$ and put
$L^{0}:=(L_1\times L_2)\cap K^{0}$ and $S^{0}:=S\cap L^{0}$ as depicted in Fig.~6.
\vskip0.3cm
Then $\mathbf K^0\cong\mathbf P_{S^0}(\mathbf L^0)$, and
${{\mathbf K}^0}$ is representable.
\end{enumerate}
\end{example}
Note that not every subdirect product of two representable Kleene posets need be representable. On the one hand, if $\mathbf K=(K,\vee,\wedge,{}')$ is a Kleene poset
that is isomorphic to some $\mathbf P_S(\mathbf A)$, then $\mathbf A$ may not be embeddable into $\mathbf K$ and thus $S$ may not be considered
as a subset of $K$. On the other hand, every finite distributive lattice is a subdirect product of finite chains and every finite chain is representable, but non-representable Kleene lattices exist (see Theorem~\ref{th1}).
\begin{lemma}\label{lem1}
Let $\mathbf A=(A,\leq)$ be a finite poset and $S$ a non-empty subset of $A$. Then $|P_S(\mathbf A)|$ is odd if and only if $|S|=1$.
\end{lemma}
\begin{proof}
Let $a,b,c\in A$. Then $(b,c)\in P_S(\mathbf A)$ if and only if $(c,b)\in P_S(\mathbf A)$. If $|S|=1$, say $S=\{a\}$, then $(b,b)\in P_S(\mathbf A)$ if and only if $b=a$. Otherwise, $(b,b)\notin P_S(\mathbf A)$.
\end{proof}
The following three lemmas will be helpful to determine a large class of representable, respectively non-representable, Kleene posets.
\begin{lemma}\label{lem3}
Let $\mathbf A=(A,\leq)$ be a poset and $a\in A$ and assume $\big(P_a(\mathbf A),\sqsubseteq)$ to have no three-element antichain. Then $a$ is comparable with every element of $A$ and join- and meet-irreducible.
\end{lemma}
\begin{proof}
If there would exist some element $b$ of $A$ with $b\parallel a$ then $\{(a,a),(a,b),(b,a)\}$ would be a three-element antichain of $\big(P_a(\mathbf A),\sqsubseteq)$. If $a$ would not be join-irreducible then there would exist $c,d\in A\setminus\{a\}$ with $c\vee d=a$ and $\{(a,a),(c,d),(d,c)\}$ would be a three-element antichain of $\big(P_a(\mathbf A),\sqsubseteq)$. If, finally, $a$ would not be meet-irreducible then there would exist $e,f\in A\setminus\{a\}$ with $e\wedge f=a$ and $\{(a,a),(e,f),(f,e)\}$ would be a three-element antichain of $\big(P_a(\mathbf A),\sqsubseteq)$.
\end{proof}
\begin{lemma}\label{lem5}
Let $\mathbf K=(K,\leq,{}')$ be a finite pseudo-Kleene poset, $\mathbf A=(A,\leq)$ a poset and $S$ a non-empty subset of $A$. Then
\begin{enumerate}[{\rm(i)}]
\item the antitone involution $'$ on $K$ has at most one fixed point,
\item the antitone involution $'$ on $K$ has a fixed point if and only if $|K|$ is odd,
\item the antitone involution $'$ on $P_S(\mathbf A)$ has a fixed point if and only if $|S|=1$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $a,b\in K$ and $c,d,e\in A$.
\begin{enumerate}[(i)]
\item If $a'=a$ and $b'=b$ then $L(a) = L(a,a')\leq U(b,b')=U(b)$
and $ L(b)= L(b,b')\leq U(a,a')= U(a)$ and hence $a=b$.
\item The set
\[
\bigcup_{x\in K}\{x,x'\}^2
\]
is an equivalence relation on $K$ having only two-element classes if $'$ on $K$ has no fixed point and having only two-element classes, but precisely
one one-element class otherwise.
\item We have $(c,c)\in P_c(\mathbf A)$ and $(c,c)'=(c,c)$ in $P_c(\mathbf A)$. Now assume $|S|>1$. If $(d,e)$ would be a fixed point of $'$ in $P_S(\mathbf A)$ then $d=e$ and we would have $L(d)=L(d,d)\leq S\leq U(d,d)= U(d)$ and hence $d\leq s\leq d$, i.e., $d=s$ for all $s\in S$ contradicting $|S|>1$. Hence $'$ has no fixed point in $P_S(\mathbf A)$.
\end{enumerate}
\end{proof}
In the following lemma, we derive an upper bound for $|A|$ provided $P_a(\mathbf A)$ is finite. This result will be used in the following theorem describing representable Kleene posets of odd cardinality.
\begin{lemma}\label{lem4}
Let $\mathbf A=(A,\leq)$ be a poset and $a\in A$ and assume
$P_a(\mathbf A)$ to be finite. Then $A$ is finite and $|A|<|P_a(\mathbf A)|/2+1$.
\end{lemma}
\begin{proof}
Since $P_a(\mathbf A)\supseteq(\{a\}\times A)\cup(A\times\{a\})$ we have that $A$
is finite and $|P_a(\mathbf A)|\geq2|A|-1$ whence $|A|\leq(|P_a(\mathbf A)|+1)/2<|P_a(\mathbf A)|/2+1$.
\end{proof}
\begin{theorem}\label{th2}
Let $\mathbf K=(K,\leq,{}')$ be a finite representable Kleene poset with an odd number of elements. Then $'$ has exactly one fixed point $a$, and there exists some subposet $\mathbf A$ of $(K,\leq)$ of cardinality less than $|K|/2+1$ containing $a$ such that $\mathbf P_a(\mathbf A)\cong\mathbf K$.
\end{theorem}
\begin{proof}
Since $\mathbf K$ is representable, there exists some poset $\mathbf A^*=(A^*,\leq)$ and some non-empty subset $S$ of $A^*$ such that $\mathbf P_S(\mathbf A^*)\cong\mathbf K$. According to Lemma~\ref{lem5}, $'$ has a fixed point in $K$ and hence $|S|=1$, say $S=\{b\}$, again because of Lemma~\ref{lem5}. According to Lemma~\ref{lem4}, $A^*$ is finite, and $|A^*|<|K|/2+1$. Let $f$ denote an isomorphism from $P_b(\mathbf A^*)$ to $\mathbf K$. Obviously, $(b,b)$ is the unique fixed point of $'$ in $P_b(\mathbf A^*)$, and hence $f(b,b)=a$. Let $g$ denote the embedding $x\mapsto(x,b)$ of $\mathbf A^*$ into $(P_b(\mathbf A^*),\sqsubseteq)$. Then $f\circ g$ is an embedding of $\mathbf A^*$ into $(K,\leq)$ mapping $b$ onto $a$. Hence, if $A:=f\big(g(A^*)\big)$ then $\mathbf A:=(A,\leq)$ is a subposet of $(K,\leq)$ isomorphic to $\mathbf A^*$ and $\mathbf P_a(\mathbf A)\cong \mathbf P_b(\mathbf A^*)\cong\mathbf K$.
\end{proof}
The following theorem shows a class of non-representable Kleene lattices.
\begin{theorem}\label{th1}
Let $\mathbf C$ be a finite chain containing more than one element and $\mathbf B$ the four-element Boolean algebra and let $\mathbf K=(K,\vee,\wedge,{}')$ denote the Kleene lattice $\mathbf C+_b\mathbf B+_c\mathbf B+_d\mathbf C$. Then $\mathbf K$ is not representable.
\end{theorem}
\begin{proof}
Using the method of indirect proof, let us suppose $\mathbf K$ to be representable. According to Theorem~\ref{th2}, there exists some subposet $\mathbf A=(A,\leq)$ of $(K,\vee,\wedge)$ containing the element $e\in A$ such that $\mathbf P_e(\mathbf A)\cong\mathbf K$ and
$(e,e)\mapsto c$. Moreover, $e$ is both meet-irreducible and join-irreducible.
Corollary~\ref{cor4} shows that $\mathbf A$ cannot be a chain. Hence $A$ must contain
two non-comparable elements $u$ and $v$ such that $u$ and $v$ are
comparable with $e$. Hence either
$u,v\in U(e)$ or $u,v\in L(e)$.
In the first case $(u,e)$ and $(v,e)$ cover $(e,e)$ and $(u,e)\sqcap (v,e)=(e,e)$, i.e., $u\wedge v=e$ in
$\mathbf A$, a contradiction with meet-irreducibility of $e$.
In the second case $(u,e)$ and $(v,e)$ are covered by $(e,e)$ and $(u,e)\sqcup (v,e)=(e,e)$, i.e., $u\vee v=e$ in
$\mathbf A$, a contradiction with join-irreducibility of $e$. This shows that $\mathbf K$ is not representable.
\end{proof}
Note that since the proof of Theorem~\ref{th1} uses Lemma~\ref{lem3} and since in Lemma~\ref{lem3} we have the assumption that $\big(P_a(\mathbf A),\sqsubseteq)$ does not have a three-element antichain, one cannot replace the four-element Boolean algebra $\mathbf B$ in Theorem~\ref{th1} by a larger Boolean algebra.
\section{Dedekind-MacNeille completion of Kleene posets}
Recall that the {\em Dedekind-MacNeille completion} $\BDM(\mathbf A)$ of a poset $\mathbf A=(A,\leq)$ is the complete lattice $\big(\DM(\mathbf A),\subseteq\big)$ where
\[
\DM(\mathbf A):=\{L(B)\mid B\subseteq A\}=\{C\subseteq A\mid LU(C)=C\}.
\]
For Kleene posets, we can show the following:
\begin{example}
Consider the Kleene poset $\mathbf A$ depicted in Figure~7 such that
$a'=d$ and $b'=c$:
\vspace*{2mm}
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\begin{tabular}{@{}c c c c c@{}}
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\put(5,3){\line(-1,1)4}
\put(5,3){\line(0,1)4}
\put(3,9){\line(-1,-1)2}
\put(3,9){\line(1,-1)2}
\put(2.85,.25){$0$}
\put(.25,2.85){$a$}
\put(5.4,2.85){$b$}
\put(.25,6.85){$c$}
\put(5.4,6.85){$d$}
\put(2.85,9.4){$1$}
\put(2.2,-1.75){{\rm Fig.~7}}
\end{picture}&\phantom{xxxxx} &
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\put(-.8,6.85){$(a,b)$}
\put(-.8,10.85){$(d,0)$}
\put(5.3,6.85){$(b,a)$}
\put(5.3,10.85){$(c,0)$}
\put(2.35,13.4){$(1,0)$}
\put(2.2,-1.75){{\rm Fig.~8}}
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\put(5,11){\line(-1,1)2}
\put(1.9,.25){$L\big((0,1)\big)$}
\put(-1.9,2.85){$L\big((0,c)\big)$}
\put(5.3,2.85){$L\big((0,d)\big)$}
\put(3.3,4.85){$L\big((a,b),(b,a))\big)$}
\put(-1.9,6.85){$L\big((a,b)\big)$}
\put(-1.9,10.85){$L\big((d,0)\big)$}
\put(5.3,6.85){$L\big((b,a)\big)$}
\put(3.3,8.85){$L\big((d,0),(c,0)\big)$}
\put(5.3,10.85){$L\big((c,0)\big)$}
\put(1.9,13.4){$L\big((1,0)\big)$}
\put(2.2,-1.75){{\rm Fig.~9}}
\end{picture}
\end{tabular}
\vspace*{12mm}
Put $S:=\{a,b\}$. Then the poset $\big(P_S(\mathbf A),\sqsubseteq\big)$ is visualized in Figure~8.
The Dedekind-MacNeille completion $\BDM\big(P_S(\mathbf A),\sqsubseteq\big)$ of this poset is depicted in Figure~9, where
\begin{align*}
L\big((0,1)\big) & =\{(0,1)\}, \\
L\big((0,c)\big) & =\{(0,1),(0,c)\}, \\
L\big((0,d)\big) & =\{(0,1),(0,d)\}, \\
L\big((a,b),(b,a)\big) & =\{(0,1),(0,c),(0,d)\}, \\
L\big((a,b)\big) & =\{(0,1),(0,c),(0,d),(a,b)\}, \\
L\big((b,a)\big) & =\{(0,1),(0,c),(0,d),(b,a)\}, \\
L\big((d,0),(c,0)\big) & =\{(0,1),(0,c),(0,d),(a,b),(b,a)\}, \\
L\big((d,0)\big) & =\{(0,1),(0,c),(0,d),(a,b),(b,a),(d,0)\}, \\
L\big((c,0)\big) & =\{(0,1),(0,c),(0,d),(a,b),(b,a),(c,0)\}, \\
L\big((1,0)\big) & =\{(0,1),(0,c),(0,d),(a,b),(b,a),(d,0),(c,0),(1,0)\}.
\end{align*}
The Dedekind-MacNeille completion $\BDM(\mathbf A)$ of the given poset $\mathbf A$ is visualized in Figure~10; here the involution is given by
$L(a)'=L(d)$, $L(b)'=L(c)$:
\vspace*{2mm}
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\put(5,3){\line(-1,1)4}
\put(3,9){\line(-1,-1)2}
\put(3,9){\line(1,-1)2}
\put(2.4,.25){$L(0)$}
\put(-.7,2.85){$L(a)$}
\put(5.3,2.85){$L(b)$}
\put(3.3,4.85){$L(c,d)$}
\put(-.7,6.85){$L(c)$}
\put(5.3,6.85){$L(d)$}
\put(2.4,9.4){$L(1)$}
\put(2.2,-1.75){{\rm Fig.~10}}
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\put(1,11){\line(1,1)2}
\put(5,11){\line(-1,1)2}
\put(1.5,.25){$\big(L(0),L(1)\big)$}
\put(-3.05,2.85){$\big(L(0),L(c)\big)$}
\put(5.3,2.85){$\big(L(0),L(d)\big)$}
\put(3.3,4.85){$\big(L(0),L(c,d)\big)$}
\put(-3.05,6.85){$\big(L(a),L(b)\big)$}
\put(-3.05,10.85){$\big(L(d),L(0)\big)$}
\put(5.3,6.85){$\big(L(b),L(a)\big)$}
\put(3.3,8.85){$\big(L(c,d),L(0)\big)$}
\put(5.3,10.85){$\big(L(c),L(0)\big)$}
\put(1.5,13.4){$\big(L(1),L(0)\big)$}
\put(2.2,-1.75){{\rm Fig.~11}}
\end{picture} &
\end{tabular}
\vspace*{12mm}
Finally, the lattice $\Big(P_{\{L(s)\mid s\in S\}}\big(\BDM(\mathbf A)\big),\sqcup,\sqcap\Big)$ is depicted in Figure~11, where
\begin{align*}
\big(L(0),L(1)\big) & =(\{0\},\{0,a,b,c,d,1\}), \\
\big(L(0),L(c)\big) & =(\{0\},\{0,a,b,c\}), \\
\big(L(0),L(d)\big) & =(\{0\},\{0,a,b,d\}), \\
\big(L(0),L(c,d)\big) & =(\{0\},\{0,a,b\}), \\
\big(L(a),L(b)\big) & =(\{0,a\},\{0,b\}), \\
\big(L(b),L(a)\big) & =(\{0,b\},\{0,a\}), \\
\big(L(c,d),L(0)\big) & =(\{0,a,b\},\{0\}), \\
\big(L(d),L(0)\big) & =(\{0,a,b,d\},\{0\}), \\
\big(L(c),L(0)\big) & =(\{0,a,b,c\},\{0\}), \\
\big(L(1),L(0)\big) & =(\{0,a,b,c,d,1\},\{0\}).
\end{align*}
Hence, in this case, the lattices $\Big(P_{\{L(s)\mid s\in S\}}\big(\BDM(\mathbf A)\big),\sqcup,\sqcap\Big)$ and $\BDM\big(P_S(\mathbf A),\sqsubseteq\big)$ are isomorphic.
\end{example}
The lattices mentioned above need not be isomorphic for distributive posets
$\mathbf A$, which are not Kleene posets.
\begin{example}\label{counterdm}
Consider the distributive poset $\mathbf A$ which is not a Kleene poset depicted in Figure~12:
\vspace*{8mm}
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\put(1,3){\line(1,-1)1}
\put(1,3){\line(0,1)1}
\put(-.6,1.8){$c$}
\put(2.35,1.8){$d$}
\put(1.35,2.8){$b$}
\put(.85,4.3){$1$}
\put(.4,1.0){{\rm Fig.~12}}
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\put(.4,1.0){{\rm Fig.~13}}
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\put(1,4){\line(1,-1)1}
\put(1.35,1.8){$\emptyset=LU(\emptyset)$}
\put(-1.45096,2.8){$L(c,d)$}
\put(2.35,2.8){$L(d,c)$}
\put(-0.385,4.3){$LU\big((c,d), (d,c)\big)$}
\put(.4,1.0){{\rm Fig.~14}}
\end{picture}
\end{tabular}
\vspace*{-2mm}
Put $S:=\{c,d\}$. Then the poset $\big(P_S(\mathbf A),\sqsubseteq\big)$ is visualized in Figure~13.
The Dedekind-MacNeille completion $\BDM\big(P_S(\mathbf A),\sqsubseteq\big)$ of this poset is depicted in Figure~14 and the Dedekind-MacNeille completion $\BDM(\mathbf A)$ of the given poset $\mathbf A$ is visualized in Figure~15:
\vspace*{12mm}
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\put(1,1){\circle*{.3}}
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\put(1,3){\line(0,1)1}
\put(1.35,.8){$\emptyset=L(c,d)$}
\put(-1.098,1.8){$L(c)$}
\put(2.35,1.8){$L(d)$}
\put(1.35,2.8){$L(b)$}
\put(.85,4.3){$L(1)$}
\put(.4,-1.4){{\rm Fig.~15}}
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\put(1.5,2.8){$(L(b), \emptyset)$}
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\end{picture}&
\end{tabular}
\vspace*{12mm}
Finally, the lattice $\Big(P_{\{L(s)\mid s\in S\}}\big(\BDM(\mathbf A)\big),\sqcup,\sqcap\Big)$ is depicted in Figure~16.
Hence, in this case, the lattices $\Big(P_{\{L(s)\mid s\in S\}}\big(\BDM(\mathbf A)\big),\sqcup,\sqcap\Big)$ and $\BDM\big(P_S(\mathbf A),\sqsubseteq\big)$
are not isomorphic.
\end{example}
Let $\mathbf A=(A,\leq)$ be a distributive poset and let $Fin(A)$ denote the set of all finite subsets of $A$. We put (see \cite{Niederle})
$$
G(\mathbf A) := \{L\big(U(A_1),...,U(A_n)\big)\mid n \in \mathbb N_{+} \&
\forall i, 1\leq i\leq n, \emptyset\not= A_i \in Fin(\mathbf A)\}.
$$
Note that $G(\mathbf A)$ is a subset of $\BDM(\mathbf A)$,
containing all principal ideals. It is worth noticing that $G(\mathbf A)=\BDM(\mathbf A)$ provided $A$ is finite. Moreover, from (\cite[Proposition~31]{Niederle}) we immediately
obtain that any element of $G(\mathbf A)$ is also of the form $LU\big(L(B_1),...,L(B_n)\big)$ where
$B_i$ are finite non-empty subsets of $A$.
The following definition and theorem are motivated by a similar result of Niederle for Boolean posets (\cite[Theorem~17]{Niederle}).
\begin{definition}\label{doubly}
\begin{enumerate}
\item Let $\mathbf A=(A,\leq)$ be a poset. A subset $X$ of $A$ is called {\em doubly dense in $\mathbf A$} if $a=\bigvee_{\mathbf A}\big(L(a)\cap X\big)=\bigwedge_{\mathbf A}\big(U(a)\cap X\big)$ for all $a\in A$.
\item Let $\mathbf A=(A,\leq,{}')$ be a poset with an antitone involution $'$. A subset $X$ of $A$ is called {\em involution-closed and doubly dense in $\mathbf A$} if $X'\subseteq X$ and $X$ is doubly dense in $\mathbf A$.
\end{enumerate}
\end{definition}
We will need the following
\begin{proposition}[\cite{Niederle}, Proposition 33]\label{niedga}
Let $\mathbf A=(A,\leq)$ be a distributive poset. Then
$\big(G(\mathbf A),\subseteq\big)$
is a distributive lattice and $X= \{L(a) \mid a \in A\}$ is doubly dense in
$G(\mathbf A)$,
generates $G(\mathbf A)$ and $(X, \subseteq)$ is isomorphic to $\mathbf A$.
\end{proposition}
In what follows, if $\mathbf A=(A,\leq,{}')$ is a
poset with an antitone involution $'$ and $X\subseteq A$, we define:
\begin{itemize}
\item $X':=\{x'\in A\mid x\in X\}$,
\item $X^\bot:=\{a\in A\mid a\leq x'\text{ for all }x\in X\}=L(X')$.
\end{itemize}
\begin{remark}\label{ddcd}\em
Recall that any involution-closed and doubly dense subset $X$ in $\mathbf A$ is a poset with induced order and involution. Moreover, if $\mathbf A=(A,\leq,{}')$ is a poset with an antitone involution $'$ then $A$ is an involution-closed and doubly dense subset in its Dedekind-MacNeille completion $\BDM(\mathbf A)$
with involution ${}^\bot$. This can be shown by the same arguments as in {\rm(\cite[Theorem 16]{Niederle})}, so we omit it.
\end{remark}
By the preceding remark, Proposition \ref{niedga} and
\cite[Theorem 34]{Niederle} we have the following
\begin{corollary}\label{dstinv}
Let $\mathbf A=(A,\leq,\, {}')$ be a distributive poset with an antitone involution $'$.
Then $\big(G(\mathbf A),\subseteq, {}^{\bot}\big)$
is a distributive lattice with an antitone involution
${}^{\bot}$ and $X= \{L(a) \mid a \in A\}$ is involution-closed and doubly dense in
$G(\mathbf A)$,
generates $G(\mathbf A)$ and $(X, \subseteq, {}^{\bot})$ is isomorphic to $\mathbf A$.
\end{corollary}
\begin{corollary}\label{embeddstinv}
{\bfseries Embedding theorem for distributive posets with an antitone involution.} The following
conditions are equivalent for a poset $\mathbf A$:
\begin{enumerate}[{\rm(i)}]
\item $\mathbf A$ is a distributive poset with an antitone involution;
\item $\mathbf A$ is an involution-closed and doubly dense subset of
a distributive lattice with an antitone involution.
\end{enumerate}
\end{corollary}
But we can prove more.
\begin{proposition}\label{dstkleene}
Let $\mathbf A=(A,\leq,\, {}')$ be a Kleene poset.
Then $(G\big(\mathbf A),\subseteq, {}^{\bot}\big)$
is a Kleene lattice and $X= \{L(a) \mid a \in A\}$ is involution-closed and doubly dense in
$G(\mathbf A)$,
generates $G(\mathbf A)$ and $(X, \subseteq, {}^{\bot})$ is isomorphic to $\mathbf A$.
\end{proposition}
\begin{proof} It is enough to check that for all $C, D\in G(\mathbf A)$, we have
$$
C\cap {C}^{\bot} \subseteq L\big(U(D\cup D^{\bot})\big).
$$
Assume first that $C=LU(E)$ and $D=LU(F)$ where $E, F$ are non-empty finite subsets of $A$.
Then $C=\bigvee \{L(e)\mid e\in E\}$ and $D=\bigvee \{L(f)\mid f\in F\}$. We compute:
\begin{align*}
&\big(\bigvee \{L(e)\mid e\in E\}\big)\wedge
\big(\bigwedge \{L(g)^{\bot}\mid g\in E\}\big)=
\bigvee_{e\in E} \Big(L(e)\wedge
\big(\bigwedge \{L(g)^{\bot}\mid g\in E\}\big)\Big)\\
&\leq \bigvee_{e\in E} \big(L(e)\wedge L(e)^{\bot}\big)=
\bigvee_{e\in E} \big(L(e)\wedge L(e')\big)
=\bigvee_{e\in E} L(e,e')\leq \bigwedge_{h\in F} LU(h,h')\\
&=\bigwedge_{h\in F} \big(L(h)\vee L(h')\big)\leq\bigwedge_{h\in F}
\Big(\big(\bigvee \{L(f)\mid f\in F\}\big)\vee L(h')\Big)\\[0.2cm]
&=\big(\bigvee \{L(f)\mid f\in F\}\big)\wedge
\big(\bigwedge \{L(h)^{\bot}\mid h\in F\}\big).
\end{align*}
Now, assume that $C=\bigwedge_{i=1}^{n} C_i$, $D=\bigwedge_{j=1}^{m} D_i$ where
$C_i=LU(E_i)$, $D_j=LU(F_j)$, $E_i$ and $F_j$ are non-empty finite subsets
of $A$, $1\leq i\leq n$
and $1\leq j\leq m$. We compute:
$$
\begin{array}{@{}r c l}
(\bigwedge_{i=1}^{n} C_i)\wedge (\bigvee_{k=1}^{n} C_k^{\bot})&=&
\bigvee_{k=1}^{n} \big(C_k^{\bot} \wedge (\bigwedge_{i=1}^{n} C_i)\big)\leq
\bigvee_{k=1}^{n} (C_k^{\bot} \wedge C_k)\leq
\bigwedge_{l=1}^{m} (D_l^{\bot} \vee D_l)\\[0.2cm]
&\leq&\bigwedge_{l=1}^{m} \big((\bigvee_{j=1}^{m} D_j^{\bot}) \vee D_l\big)=
(\bigvee_{j=1}^{m} D_j^{\bot})\vee (\bigwedge_{l=1}^{m} D_l).
\end{array}
$$
\end{proof}
\begin{theorem}\label{embeddstkleene}
{\bfseries Embedding theorem for Kleene posets.} The following
conditions are equi\-va\-lent for a poset $\mathbf A$:
\begin{enumerate}[{\rm(i)}]
\item $\mathbf A$ is a Kleene poset;
\item $\mathbf A$ is an involution-closed and doubly dense subset of
a Kleene lattice.
\end{enumerate}
\end{theorem}
\begin{proof} (i)$\,\,\Rightarrow$ (ii) has been proved in Proposition \ref{dstkleene}.
(ii)$\,\,\Rightarrow$ (i): From Corollary \ref{embeddstinv}, we know that $\mathbf A$ is a distributive
poset with an antitone involution ${}'$. But the involution reflects the Kleene condition.
Namely, let $x, y\in A$. Assume that $a\in L(x,x')$ and $b\in U(y,y')$. Then
$a\leq x\wedge x'\leq y\vee y'\leq b$ in the Kleene lattice. Hence $a\leq b$ in $\mathbf A$, i.e.,
$\mathbf A$ is a Kleene poset.
\end{proof}
\section*{Data availability statement}
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
Authors' addresses:
Ivan Chajda \\
Palack\'y University Olomouc \\
Faculty of Science \\
Department of Algebra and Geometry \\
17.\ listopadu 12 \\
771 46 Olomouc \\
Czech Republic \\
ivan.chajda@upol.cz
Helmut L\"anger \\
TU Wien \\
Faculty of Mathematics and Geoinformation \\
Institute of Discrete Mathematics and Geometry \\
Wiedner Hauptstra\ss e 8-10 \\
1040 Vienna \\
Austria, and \\
Palack\'y University Olomouc \\
Faculty of Science \\
Department of Algebra and Geometry \\
17.\ listopadu 12 \\
771 46 Olomouc \\
Czech Republic \\
helmut.laenger@tuwien.ac.at
Jan Paseka \\
Masaryk University Brno \\
Faculty of Science \\
Department of Mathematics and Statistics \\
Kotl\'a\v rsk\'a 2 \\
611 37 Brno \\
Czech Republic \\
paseka@math.muni.cz
\end{document} |
\begin{document}
\title{f Relations Between Quantum Maps and Quantum States}
\begin{abstract}
The relation between completely positive maps and compound
states is investigated in terms of the notion of quantum conditional
probability.
\end{abstract}
\section{Introduction}
In quantum information two problems play a relevant role, the
first one concerns the study of the dynamical change of states of a system
by means of completely positive maps, commonly
called channels, the second one is to describe correlations
between the initial and final states; such correlations are
described by compound states.
The connection between the two concepts is based on a very general
principle. Indeed, in any Hilbert space ${\mathcal H}$ there is an one-to-one correspondence
between the set ${\mathcal P}^p_q$ of $p$-contravariant $q$-covariant tensors and
the set ${\mathcal P}_{p+q}$ of $p+q$ covariant tensors. The equivalence is due to
the identification of ${\mathcal H}$ and its dual space ${\mathcal H}^\ast$ by means of the
hermitian product. Consequently, endomorphisms of ${\mathcal H}$ which are in ${\mathcal P}^1_1$ are in
one-to-one correspondence with $2$-covariant tensors in ${\mathcal P}_2$. In
particular, if we consider the Hilbert spaces of Hilbert-Schmidt operators on
${\mathcal H_1}$ and ${\mathcal H_2}$, any map of the Hilbert-Schmidt operators on ${\mathcal H_1}$
into those on ${\mathcal H_2}$ is associated to a state on the tensor product of
the spaces of Hilbert-Schmidt operators on ${\mathcal H_1}$ and ${\mathcal H_2}$.
The correspondence can also be formulated at $C^*$-algebraic
level. However, only the finite-dimensional case will be
considered here.
Let us consider two systems described by $(M_n,\, S(M_n))$ and
$(M_m,\, S (M_m))$. The first one describes an initial ({\it input})
system and the second one a final ({\it output}) system. The symbol
$M_n$ stands for the algebra of $n \times n$ complex matrices and
the symbol $S(M_n)$ stands for the set of all states on $M_n$, i.e.\
the set of all density matrices. Moreover, $I_n$ denotes the
identity matrix in $M_n$. Let us consider a map $\varphi^* \colon S
(M_n) \to S (M_m)$, such that its dual map $\varphi \,\colon M_m
\to M_n$ is completely positive and normalized, i.e.\ $\varphi (I_m)
= I_n$.
For an initial state $\rho \in M_n$ and final state $\varphi^*
(\rho) \in M_m$, a composite state $\omega \in S (M_n \otimes
M_m)$ should satisfy the following two conditions:
\begin{itemize}
\item[i)] \quad $ \omega (a \otimes I_m) = \rho (a)$, for all $a \in M_n$
\item[ii)] \quad $ \omega (I_n \otimes b) = \varphi^\ast(\rho) (b)$ for
all $b \in M_m$.
\end{itemize}
It is well known that joint probability measures do not
generally exist for quantum systems, therefore it is difficult to
define a compound state $\omega$ satisfying the above conditions.
The first construction of a compound state $\omega$ satisfying the
above two conditions has been given by Ohya [1,\,2]. Let $\rho \in
M_n$, then $\rho$ has the following spectral decomposition
\begin{equation}
\rho \;=\; \sum_k \lambda_k m_k \rho_k\,,
\end{equation}
where
\begin{equation} \rho_k \;=\;
\frac{1}{m_k} p_k\, , \qquad m_k \;=\; \mbox{Tr}\, p_k\, ,
\end{equation}
$\lambda_k$
are the eigenvalues of $\rho$, and $p_k$ are eigenprojectors of
$\rho$, respectively.
Then for any $\varphi^* \colon S (M_n) \to S (M_m)$ the compound
state $\omega_\varphi \in S (M_n \otimes M_m)$ has the form \begin{equation}
\omega_\varphi \;=\; \sum_k \lambda_k m_k\, \rho_k \otimes
\varphi^* (\rho_k)\,. \end{equation} Let us observe that $\omega_\varphi$ is a
separable state on $M_n \otimes M_m$, $\varphi$ is a positive
normalized map $\varphi \,\colon M_m \to M_n$, and the construction
of $\omega_\varphi$ is non-linear with respect to $\rho$.
We notice that the Ohya compound state can be constructed in the
case of general $C^*$-algebraic setting.
The construction of compound states which will be studied in the
present paper can be described as follows.
Let $\sigma \in S (M_n \otimes M_m)$, and suppose that \begin{equation}
\mbox{tr}_{\mathbb{C}^m}\, \sigma \;=\; \sigma_1 \;>\;0\,. \end{equation}
Then, one can define the following operator
\begin{equation}n
& \pi(\sigma) \, \colon \, \mathbb{C}^n
\otimes \mathbb{C}^m \;\longrightarrow\; \mathbb{C}^n \otimes
\mathbb{C}^m & \nonumber \\
& \pi(\sigma) \;=\; (\sigma_1^{-1/2} \otimes I_m)\, \sigma\, (\sigma_1^{-1/2}
\otimes I_m)\,,& \end{equation}n
which has the properties
\begin{equation}n
& \pi(\sigma) \;\geq\; 0 &\\
& \mbox{tr}_{\mathbb{C}^m}\, \pi(\sigma) \;=\; I_m \,. & \end{equation}n
It follows from (1.4) and (1.5) that the operator $\pi$ is the quantum analogue of
classical conditional probability.
Another definition of quantum conditional probability has been given in \cite{cerf}.
\begin{defn}\rm
A map
\begin{equation}
\pi \, \colon \, \mathbb{C}^n
\otimes \mathbb{C}^m \;\longrightarrow\; \mathbb{C}^n \otimes
\mathbb{C}^m
\end{equation}
is a quantum conditional probability (QCPO) iff
satisfies (1.6) and (1.7).
\end{defn}
For a given $\pi$ and any $\rho \in S (M_n)$ one can define a
compound state
\begin{equation} \omega \;=\; (\rho^{1/2} \otimes I_m)\, \pi\,
(\rho^{1/2} \otimes I_m)\,,
\end{equation}
which has the following properties
\begin{equation}
\mbox{tr}_{{\mathbb{C}^m}}\, \omega \;=\;\rho
\end{equation}
and
\begin{equation}
\mbox{tr}_{\mathbb{C}^n}\, \omega \;=\; \mbox{tr}_{\mathbb{C}^n}\, \pi
(\rho \otimes I_m) \;=\; \varphi^* (\rho)\,,
\end{equation}
where
\[
\varphi^* \, \colon \, S (M_n) \,\longrightarrow\, S (M_m)\, .
\]
The study of the relation between $\varphi^*$ and $\pi$ is based
on duality between quantum maps and composite states which has
been investigated in detail in [4,11] (see also references therein).
\section{Classification of Composite States and Positive Maps}
\setcounter{equation}{0}
It has been shown above that the construction of composite states is
based on the notion of quantum conditional probability operator
$(QCPO)\,$ $ \pi \,\colon \mathbb{C}^n \otimes \mathbb{C}^m \to
\mathbb{C}^n \otimes \mathbb{C}^m$. In what follows the case $m=n$
will be considered for simplicity.
Let ${\mathcal H}$ be a Hilbert space and ${\mathcal H}_1 =
{\mathcal H}_2 = {\mathcal H}$, the following order in the tensor
product ${\mathcal H} \otimes {\mathcal H} = {\mathcal H}_2
\otimes {\mathcal H}_1$ will be used, and the partial trace with
respect to the Hilbert space ${\mathcal H}_a$ will be denoted by
$\mbox{tr}_a$.
Let $\{ e_1, \ldots , e_n\}$ be a fixed orthonormal basis in
$\mathbb{C}^n$ and $e_{kl} = e_k (e_l, \,\cdot\,)$ be the
corresponding basis in $M_n$, then $a \in M_n$ can be written in the
form \begin{equation} a \;=\; \sum_{i,j=1}^n (e_i, a\,e_j) e_{ij} \;=\;
\sum_{i,j=1}^n{\rm tr}\, (a\, e_{ij}^*) e_{ij}\, , \end{equation} and the transpose
map $\mathcal{T} \,\colon M_n \to M_n$ (with respect to the basis
$\{e_1, \ldots , e_n\})$ has the form \begin{equation} \mathcal{T} (a) \;=\;
\sum_{i,j=1}^n (e_j, ae_i) e_{ij} \;=\; \sum_{i,j =1}^n e_{ij}\, a\,
e_{ij} \, . \end{equation} A generic element $x \in \mathbb{C}^n \otimes
\mathbb{C}^n$ can be written in the form \begin{equation} x \;=\; \sum_{i=1}^n
x_i \otimes e_i \;=\; \sum_{i=1}^n (a\, e_i) \otimes e_i\,, \end{equation}
where $x_1, x_2, \ldots , x_n \in \mathbb{C}^n$ and $a \in M_n$.
With every $a \in M_n$, such that $\mbox{tr} \,(a^*\,a)=1$, one can
associate one-dimensional projections $p_a \,\colon
\mathbb{C}^n\otimes \mathbb{C}^n \to \mathbb{C}^n\otimes
\mathbb{C}^n$ \begin{equation} p_a \;=\; \sum_{i,j=1}^n a\, e_{ij}\, a^* \otimes
e_{ij} \,. \end{equation} Moreover, two projections $p_a$ and $p_b$ are
orthogonal provided that $\mbox{tr} \,(a^* b)=0$. As a consequence
of (2.4) any positive operator $\hat{A} \,\colon \mathbb{C}^n
\otimes \mathbb{C}^n \to \mathbb{C}^n \otimes \mathbb{C}^n$ has the
form \begin{equation} \widehat{A} \;=\; \sum_{i,j=1}^n \sigma_{ij} \otimes
e_{ij} \;=\; \sum_{i,j=1}^n \varphi (e_{ij}) \otimes e_{ij}\,, \end{equation}
where \begin{equation} \varphi (e_{ij}) \;=\; \sum_{\alpha=1}^{n^2}
\lambda_\alpha\, a_\alpha \,e_{ij}\, a_\alpha^* \end{equation} and
\[
\lambda_\alpha \geq 0\, ,
\qquad\mbox{tr}\, (a_\alpha a_\beta^*) = \delta_{\alpha \beta}\, ,
\]
i.e.~$\{a_1, a_2, \ldots , a_{n^2}\}$ is an
orthonormal basis in $M_n$.
Relation (2.5) can also be rewritten in the form \begin{equation} \hat{\sigma}
\;=\; (\varphi \otimes \mbox{id}) \sum_{i,j=1}^n e_{ij} \otimes
e_{ij} \,, \end{equation} which gives the relation between elements of $(M_n
\otimes M_n)^+$ and completely positive maps in $M_n$. In order to
classify the states on $M_n \otimes M_n$ it is convenient to
introduce the following cones in $M_n \otimes M_n$: \begin{equation} V_s \;=\;
\mbox{conv}\, \Big\{ \sum_{i,j=1}^n a\, e_{ij}\, a^* \otimes
e_{ij}\, \colon \; a \in M_n\,, \,\,\, \mbox{rank}\, a \leq s \Big\}
\end{equation} where conv$\,X$ means convex (not normalized) set generated by
elements of $X$, and \begin{equation} V^s \;=\; (\mbox{id} \otimes \mathcal{T})
V_s\,, \end{equation} where $\mathcal{T}$ is the transpose map on $M_n$, i.e.
\begin{equation} V^s \;=\; \mbox{conv}\, \Big\{ \sum_{i,j}^n a\, e_{ij}\, a^*
\otimes e_{ji}\, \colon\; a \in M_n\, , \,\,\,\mbox{rank}\, a \leq
s\Big\} \, . \end{equation} It follows from (2.8) and (2.9) that the following
chains of inclusions \begin{equation}n & V_1 \subset V_2 \subset \ldots \subset
V_n , \qquad V^1 \subset V^2 \subset \ldots \subset
V^n\, ,& \nonumber \\
& V_1 \cap V^1 \subset V_2 \cap V^2 \subset \ldots \subset V_n
\cap V^n & \end{equation}n
hold true.
It is clear that $V_n$ coincides with the cone $(M_n \otimes
M_n)^+$ of all positive semidefinite elements of $M_n \otimes
M_n$, $V_1 =V^1$ is the cone generated by elements $a \otimes b$,
where $a,b$ are positive elements of $M_n$, i.e.\ $V_1$ coincides
with separable (not normalized) states on $M_n \otimes M_n$, while
$V_n \cap V^n$ is the set of all (not normalized) PPT states on
$M_n \otimes M_n$ (by definition).
Using the results of [4,\,5,\,6] the above cones can be used to
classify positive maps. Let $P_s$, $P^s$ and $P_s \cup P^r$ be the
cones of $s$-positive, $s$-copositive maps and sums of $s$-positive and
$r$-copositive ones, respectively. One can verify
that
\begin{equation}n
\varphi \in P_s &\Longleftrightarrow & (\varphi \otimes
\mbox{id}) V_s \in (M_n \otimes M_n)^+ \nonumber \\
\varphi \in P^s &\Longleftrightarrow & (\varphi \otimes \mbox{id})
V^s \in (M_n \otimes M_n)^+
\end{equation}n
and
\[
\varphi \in P_s \cup P^r \;\Longleftrightarrow\; (\varphi \otimes
\mbox{id}) V_s \cap V^r \in (M_n \otimes M_n)^+ \,.
\]
Relations (2.12) can be considered as an extension of the Horodecki
theorem [7] which gives the characterization of the cone $V_1$ in
terms of positive maps. It should be pointed out that our
knowledge concerning the above mentioned cones is rather poor. In
fact only the structure of cones $V_n$ and $V^n$ is known.
In the $n$-dimensional case an example of an element $V_2 \cap
V^2$ which is not separable has been given in [8].
The cones $V_r$ and $V^r$ and $V_r \cap V^s$ can be used for classification of
completely positive maps.
\begin{defn}\rm
A completely positive map $\varphi\,\colon M_n\longrightarrow M_n$
is said to be $s$-{\it comple\-tely positive} if \begin{equation}
\sum^n_{i,j=1}\varphi(e_{ij})\otimes e_{ij}\;\in\; V_s \end{equation} and
$(r,s)$-{\it completely positive} if
\[
\sum^n_{i,j=1}\varphi(e_{ij})\otimes
e_{ij}\;\in\; V_r\otimes V^s\,.
\]
\end{defn}
Let us observe that the set $\mathcal{P}_{rs}$ of all $(r,s)$-completely
positive maps is a subset of $P_n\cap P^n$, the subset of maps which are
completely positive and completely copositive. On the other hand,
$(r,s)$-completely positive maps generate PPT-states since the inclusion
$V_r \cap V^s\subseteq V_n \cap V^n$ holds.
It is also convenient to consider $s$-completely positive maps which
are $k$-coposi\-tive, i.e.~elements of the set $P_n\cap P^k$
($k<n$) which generate NPT states.
The construction of composite states in terms of QCPO will
be used to find out some classes of PPT and NPT states.
Indeed, taking into account (2.5), one finds out that the general
form of $\pi$ is given by
\begin{equation}
\pi \;=\; \sum_{i,j}^n \varphi (e_{ij}
)\otimes e_{ij}\,,
\end{equation}
where $\varphi$ is a completely positive
normalized map in $M_n$, i.e., $\varphi(I_n)=I_n$. It follows from (2.14) and (1.8) that the
composite state can be written in the form
\begin{equation}
\omega \;=\;
\sum_{i,j=1}^n \rho^{1/2} \, \varphi (e_{ij})\,\rho^{1/2} \, \otimes e_{ij}
\end{equation}
and
the relations
\begin{equation}n
\mbox{tr}_1 \, \omega &=& \rho\,, \nonumber \\
\mbox{tr}_2 \, \omega &=& (\mathcal{T} \circ \varphi^*)(\rho) \;=\;
\psi^* (\rho)\, , \end{equation}n
hold true.
The dual map
$\psi$ can be written as
\begin{equation}
\psi(e_{ij}) \;=\; \varphi (e_{ji}) \;=\;
(\varphi\circ {\mathcal{T}})(e_{ij})
\quad {\rm or}\quad
\psi^\ast\;=\;{\mathcal{T}}\circ\varphi^\ast \,.
\end{equation}
The properties of the composite state $\omega$ can be summarized as follows.
\begin{cor}\rm
The composite state $\omega$ is a PPT iff $\varphi$ is $(r,s)$-completely positive.
\end{cor}
\begin{cor}\rm
The composite state $\omega$ is NPT iff $\varphi$ is $r$-completely positive
and $k$-copositive $(k<n)$ provided
rank$\,\rho=n$.
\end{cor}
The above results imply that the
construction of PPT-states is reduced to normalized completely
positive and completely copositive maps, while, entangled but not
PPT-states are induced by normalized completely positive and
$k$-completely copositive maps.
Next we will analyze some examples of normalized completely positive
and completely copositive maps.
\begin{exmp}\rm
Let us consider the following QCPO:
\begin{equation}
\pi_\lambda \;=\; \frac{1-
\lambda }{n} \, I_n \otimes I_n + \lambda \sum_{i,j=1}^n e_{ij}
\otimes e_{ij}\,,
\end{equation}
where
\begin{equation}
- \frac{1}{n^2 - 1} \;\leq\; \lambda
\;\leq\; \frac{1}{n+1} \, .
\end{equation}
The above $\pi$ is, up to
normalization, the Horodecki state [8], and for $\lambda$
satisfying (2.19) $\pi$ is separable, i.e., $\pi_\lambda\in V^1\cap V_1$.
Let $\psi \,\colon M_n \to M_n$ be a normalized positive map, then
\begin{equation} \sigma_\lambda \;=\; (\psi \otimes \mbox{id})\, \pi_\lambda
\end{equation} is a QCPO which is separable, and the relation \begin{equation} \sum_{ij}
\varphi_\lambda(e_{ij}) \otimes e_{ij} \;=\; (\psi \otimes
\mbox{id}) \, \pi_\lambda \end{equation} defines completely positive and
completely copositive maps of the form \begin{equation} \varphi_\lambda (e_{ij})
\;=\; \frac{1 - \lambda}{n}\, I_n\, \delta_{ij} + \lambda \,\psi
(e_{ij})\,, \end{equation} which is $(1,1)$-completely positive.
\end{exmp}
\begin{exmp}\rm
Let us consider the following QPCO: \begin{equation} \pi_\gamma \;=\;
N_\gamma^{-1} \Bigl( n\, I_n \otimes I_n + \sum_{i,j=1}^n a_{ij}
\otimes e_{ij} \Bigr)\,, \end{equation} where \begin{equation}n
a_{ij} &=& n\, e_{ij}\,,\qquad i \neq j\,, \\
a_{ij} &=& \Bigl( 1 - \frac{1}{\gamma^2} \Bigr) (\gamma^2
e_{i+1, i+1} - e_{n+i-1, n+i-1} )\,\, (\mbox{mod}\,n)\,, \\
N_\gamma &=& n^2 + \Bigl( 1 - \frac{1}{\gamma^2} \Bigr) (\gamma^2 -
1)\,, \qquad \gamma^2 >0\, . \end{equation}n It has been shown in [4] that
$\pi_\gamma \in V_2 \cap V^2$ but it is not separable.
The relation
\begin{equation}
\sum_{i,j =1}^n \varphi_\gamma (e_{ij}) \otimes e_{ij}
\;=\; \pi_\gamma
\end{equation}
defines a normalized $(2,2)$-completely positive
map $\varphi$ which has the form
\begin{equation}
\varphi_\gamma (e_{ij}) \;=\; \frac{1}{N_\gamma} (n\, I_n\, \delta_{ij} +
a_{ij})\, .
\end{equation}
\end{exmp}
\begin{exmp}\rm
The map $\varphi$ given by (2.28) has the following form
\begin{equation}
\varphi (a) \;=\; \sum_{i,j}^n c_{ij} \, e_{ij}\, a\,e_{ij}^* + \mu a \,,
\end{equation}
where
\begin{equation}
\varphi(I_n)\;=\;\sum_{i=1}^n e_{ii}\Big(\sum^n_{j=1}c_{ij}+\mu\Big)\,.
\end{equation}
Taking into account
(2.2) one finds that the relation
\begin{equation}
\varphi (\mathcal{T} (a)) \;=\;
\mathcal{T}(\varphi (a))
\end{equation}
holds.
\end{exmp}
\begin{thm}
The map $\varphi$, as in (2.29), is completely positive iff the
following conditions \begin{equation} c_{ij} \;\geq\; 0\,,\qquad i \neq j \end{equation}
and \begin{equation} \bigl[ c_{ii} \delta_{ij} + \mu \bigr] \;\geq\; 0 \end{equation} are
satisfied.
\end{thm}
{\it Proof}.\quad
The map $\varphi$ can be written in the form \begin{equation} \varphi (a) \;=\;
\sum_{i,j=1}^n (\delta_{ij} c_{ii} + \mu) e_{ii}\, a\, e_{jj} +
\sum_{i\neq j} c_{jj}\, e_{ij}\, a\, e_{ij}^* \end{equation} on the other hand
the map $\psi$ is completely positive iff it has the form \begin{equation}
\psi(a) \;=\; \sum_{\alpha, \beta =1}^{n^2} \lambda_{\alpha \beta}
f_\alpha a f_\beta^*\,, \end{equation} where \begin{equation} \mbox{tr} \, (f_\alpha
f_\beta^*) \;=\; \delta_{\alpha \beta} \end{equation} and \begin{equation}
\bigl[\lambda_{\alpha \beta} \bigr] \;\geq\; 0\, . \end{equation} Taking into
account (2.35)--(2.37) and (2.34), one finds conditions (2.33) and
(2.34).
\begin{thm}
The map $\varphi$ is completely copositive iff the following
conditions
\begin{equation}n
c_{ii} + \mu &\geq& 0\,, \\
c_{ij} + c_{ji} &\geq& 2 \mu\,,\qquad i \neq j\,, \\
c_{ij} + c_{ji} &\geq& - 2 \mu\,,\,\quad i \neq j\,, \\
c_{ij} c_{ji} &\geq& \mu^2\,,\qquad i \neq j
\end{equation}n
are
satisfied.
\end{thm}
{\it Proof.}\quad
The map $\varphi$ is completely copositive iff the map $\mathcal{T}
\varphi$ is completely positive.
Taking into account (2.2) and (2.29) one finds \begin{equation}
(\mathcal{T}\circ \varphi) (a) \;=\; \sum_{i=1}^n (c_{ii} + \mu)
e_{ii}\, a\, e_{ii} + \sum_{i \neq j} (c_{ij}\, e_{ij}\, a \,
e_{ij}^* + \mu e_{ij}\, a\, e_{ij} )\,. \end{equation} From (2.42) one finds
the condition (2.38). Let us introduce trace orthonormal operators
\begin{equation}n f_{ij} &=&
\frac{1}{\sqrt{2}} (e_{ij} + e_{ji})\,, \nonumber \\
g_{ij} &=& \frac{-i}{\sqrt{2}} (e_{ij} - e_{ji})\,,\qquad i <j\,.
\end{equation}n Then the equality \begin{equation}n & & \hspace*{-20mm} \sum_{i \neq j}
(c_{ij}\, e_{ij}\, a\, e_{ij}^* + \mu e_{ij}\, a\, e_{ij}) \;=\;
\sum_{k < l} \Big\{ \Big( \frac{1}{2} (c_{kl} + c_{lk}) + \mu \Big)
f_{kl}\, a\, f_{kl}
\nonumber \\
& & \qquad\qquad +\; \Big( \frac{1}{2}(c_{kl} + c_{lk}) - \mu \Big)
g_{kl}\, a\, g_{kl} - \frac{i}{2} (c_{kl} - c_{lk}) f_{kl}\, a\,
g_{kl}
\nonumber \\
& & \qquad\qquad\qquad\qquad +\; \frac{i}{2} (c_{kl} - c_{lk}) g_{kl}\, a\, f_{kl}
\Big\}
\end{equation}n
holds true.
Taking into account (2.35) and (2.37) one finds that $\mathcal{T}
\circ \varphi$ is completely positive iff the matrices \begin{equation}
\Delta_{kl} \;=\; \left[ \begin{array}{ll} {\displaystyle \frac{1}{2} (C_{kl}
+ C_{lk}) + \mu}~~~~~~~ & {\displaystyle - \frac{i}{2} (C_{kl} - C_{lk})} \\
& \\
{\displaystyle \frac{i}{2} (C_{kl} - C_{lk})}~~~~~~~ & {\displaystyle \frac{1}{2}
(C_{kl} + C_{lk}) - \mu} \end{array} \right] \end{equation} are semipositive
definite. The conditions (2.39)--(2.41) are equivalent to
$\Delta_{kl} \geq 0$.
\begin{cor}\rm
The map
\begin{equation}
\varphi(a)\;=\;\sum_{i,j=1}^n c_{ij}\, e_{ij}\, a\,
e_{ij}^* + \mu a
\end{equation}
is completely positive and completely copositive, i.e. $\varphi\in P_n\cap P^n$ iff the
following conditions
\begin{equation}n
c_{ij} &>& 0\,,\\
c_{ij} c_{ji} &\geq& \mu^2\,,\qquad i \neq j\,, \\
c_{ii} + \mu &\geq& 0\,, \\
\bigl[c_{ii}\delta_{ij} + \mu\bigr] &\geq& 0 \end{equation}n are satisfied.
The $\varphi$ is not normalized, but since $\varphi(I_n)>0$, the map
\begin{equation}
\psi(a)\;=\;\varphi(I_n)^{-1/2}\, \varphi(a)\,\varphi(I_n)^{-1/2}
\end{equation}
is normalized, and the state
\begin{equation}
\sum_{i,j=1}^n \rho^{1/2} \psi(e_{ij})\rho^{1/2} \otimes e_{ij}
\end{equation}
is a PPT state.
\end{cor}
\begin{cor}\rm
It follows from Theorems 1 and 2 that choosing $c_{ij}=k$, $1\leq k<
n$, and $\mu=-1$ the map \begin{equation}n \varphi_k(a) &=& (k-1)\Big(
\sum_{i=1}^n e_{ii}\, a\, e_{ii} + \sum_{i<j} f_{ij}\, a\,
f_{ij}\Big) + (k+1) \sum_{i<j} g_{ij}\, a\, g_{ij} \end{equation}n is
completely positive, while the map \begin{equation} (\tau \circ
\varphi_k)(a)\;=\;k I_n \, ({\rm tr}\, a) -a \end{equation} is known to be
$k$-positive but not $(k+1)$-positive; i.e., $\varphi_k$ is
completely positive and $k$-copositive, and consequently the state
\begin{equation}
\frac{1}{kn-1}
\sum_{i,j=1}^n \rho^{1/2} \, \varphi_k (e_{ij})\,\rho^{1/2} \, \otimes e_{ij}
\end{equation}
is NPT (by definition) provided rank$\rho=n$.
\end{cor}
Using Theorems 1 and 2 specialized for the case $c_{ii}=c$ and $\mu=-1$.
\begin{cor}\rm
The map
\begin{equation}n \varphi(a) &= & \sum_{i=1}^n (c-1) e_{ii}\, a\, e_{ii}+
\sum_{i < j} \Big\{ \Big( \frac{1}{2} (c_{ij} + c_{ji}) -1 \Big)
f_{ij} a f_{ij}
\\
&+ & \Big( \frac{1}{2}(c_{ij} + c_{ji}) +1 \Big) g_{ij}\, a\, g_{ij}
- \frac{i}{2} (c_{ij} - c_{ji}) f_{ij}\, a\, g_{ij}
+\frac{i}{2} (c_{ij} - c_{ji}) g_{ij}\, a\, f_{ij}
\Big\} \nonumber
\end{equation}n
is completely positive iff $c\geq 1$ and $c_{ij} c_{ji}\geq 1\quad i\neq j$.
On the other hand
\begin{equation}n
(\tau \circ \varphi)(a)\;=\;
c \sum_{i=1}^{n-1} h_i\, a\, h_i + \sum_{i\neq j}c_{ij}\, e_{ij}\, a\, e_{ij}^\ast -\frac{n-c}{n}a\,,
\end{equation}n
where
$h_1,h_2,\ldots h_{n-1}\in M_n$, $h_i=h_i^\ast$, ${\rm tr}\, h_i=0$,
${\rm tr}\, h_i h_j=\delta_{ij}$
and $ {\rm tr}\, h_i f_{kl}={\rm tr}\, h_i g_{kl}=0$.
Using the result of [9] one finds that $(\tau \circ \varphi )$ is $k$-positive but not
$(k+1)$-positive provided the following conditions are satisfied \cite{tt}
\begin{equation}n
& 1\;\leq\; k \;\leq\; c \;<\; k+1 \;<\; n\,, & \\
&\displaystyle\frac{k}{n-k} \;\leq\; \frac{c_{ij}}{n-c}\;<\;\frac{k+1}{n-k-1}\,.&
\end{equation}n
In this case the map $\varphi$ generates NPT states.
\end{cor}
\section*{Acknowledgments}
The work of M.\,A. and G.\,M. has been partially supported by a cooperation grant INFN-CICYT.
The work of M.\,A. has also been partially supported by the Spanish MCyT grant
FPA2000-1252. A.\,K. has been supported by the Grant PBZ-MIN-008/P03/2003.
\end{document} |
\betaegin{document}
\betaegin{center}
\centerline{\lambdarge\betaf \vspace*{6pt} Harish-Chandra modules over the high rank}
\centerline{\lambdarge\betaf $W$-algebra $W(2,2)$}
\centerline{ Haibo Chen}
\end{center}
{\small
\parskip .005 truein
\betaaselineskip 3pt \lineskip 3pt
\noindent{{\betaf Abstract:}
In this paper, using the theory of $\mathcal{A}$-cover developed in \cite{B1,BF1}, we completely classify all simple Harish-Chandra modules over the high rank $W$-algebra $W(2,2)$. As a byproduct, we obtain the classification of simple Harish-Chandra
modules over the classical $W$-algebra $W(2,2)$ studied in \cite{LGZ,CLW,GLZ1}.
\vspace*{5pt}
\noindent{\betaf Key words:}
high rank
$W$-algebra $W(2,2)$, Harish-Chandra module, weight module.}
\noindent{\it Mathematics Subject Classification (2020):} 17B10, 17B65, 17B68.}
\parskip .001 truein\betaaselineskip 6pt \lineskip 6pt
\section{Introduction}
In the representation theory of infinite-dimensional Lie algebras, there is a very important class of weight modules called Harish-Chandra modules (namely, weight modules with finite-dimensional weight spaces). The classification of simple Harish-Chandra modules over the Virasoro algebra (also called $N=0$ superconformal algebra), conjectured by Kac (see \cite{K}), was given in \cite{M2}.
Combined \cite{MP} with \cite{S0}, a new method was presented to obtain this classification.
After that, a lot of general versions of the Virasoro algebra
have been investigated by some authors. Those include, but are not limited to,
the generalized Virasoro algebra (see, e.g., \cite{LZ2,Ma3,S1,S2,S,SZ,PS,GLZ}), the
generalized Heisenberg-Virasoro algebra (see \cite{GLZ0,LG,LZ1}), the $W$-algebra $W(2,2)$ (see \cite{LGZ,CLW,GLZ1}), the loop-Virasoro algebra (see \cite{GLZ1}),
and
so on.
To classify all simple Harish-Chandra modules over the
Lie algebra $W_n$ of vector fields on $n$-dimensional torus, Billig and Futorny developed a new technique called $\mathcal{A}$-cover theory in \cite{B1,BF1}. The result gained here was a generalized version of Mathieu's classification theorem for the Virasoro algebra. From then on, the $\mathcal{A}$-cover theory was used in some other Lie (super)algebras (see, e.g., \cite{BF2,CLW,XL,BFIK}).
The $W$-algebra $W(2,2)$ was introduced in \cite{ZD} by Zhang and Dong for investigated the classification of simple vertex
operator algebras generated by two weight $2$ vectors.
The centerless $W$-algebra $W(2,2)$ $\overline{\mathcal{W}}[\mathbb{Z}]$ can be obtained from the point of view of non-relativistic analogues of the conformal field theory. By using the ``non-relativistic limit'' on a pair of commuting
algebras $\mathfrak{vect}(S^1)\oplus \mathfrak{vect}(S^1)$ (see \cite{RU}) via a group contraction, one has the following generators
\betaegin{eqnarray*}
&&L_m=-t^{m+1}\frac{d}{dt}-(m+1)t^my\frac{d}{dy}-(m+1)\sigma t^m-m(m+1)\eta t^{m-1}y,
\\&&W_m=-t^{m+1}\frac{d}{dy}-(m+1)\eta t^m,
\end{eqnarray*}
where $m\in\mathbb{Z}$, $\sigma$ and $\eta$ are respectively the scaling dimension and a free parameter.
Then {\em the centerless $W$-algebra $W(2,2)$} is a Lie algebra with the basis $\{L_m,W_m\mid m\in\mathbb{Z}\}$ and the non-vanishing commutators as follows $$[L_m,L_{m^\prime}]=(m^\prime-m)L_{m+m^\prime},\ [L_m,W_{m^\prime}]=(m^\prime-m)W_{m+m^\prime}$$
for $m,m^\prime\in\mathbb{Z}$.
It is an infinite-dimensional extension
of an algebra called either non-relativistic or conformal Galilei algebra $\mathrm{CGA}(1)\cong\lambdangle L_{\pm1,0},W_{\pm1,0}\mathfrak{g}ammaangle$ (see \cite{G}).
Basically, the relationship with conformal algebras makes it widely studied
in string theory (see \cite{BS}).
Furthermore, $\overline{\mathcal{W}}[\mathbb{Z}]$ can be realized by the semidirect product of the Witt algebra $\overline{\mathcal{V}}[\mathbb{Z}]$
and the $\overline{\mathcal{V}}[\mathbb{Z}]$-module $A_{0,-1}$ of the intermediate series in \cite{KS}, that is, $\overline{\mathcal{W}}[\mathbb{Z}]\cong\overline{\mathcal{V}}[\mathbb{Z}]\ltimes A_{0,-1}$. What we concern most is the approach of realizing $\overline{\mathcal{W}}[\mathbb{Z}]$ from a truncated loop-Witt algebra (see, e.g.,
\cite{GLZ,GL}). The detailed description on it will be shown in Section \mathfrak{g}ammaef{115eqw}, which is closely associated with the usage of $\mathcal{A}$-cover theory. The aim of this paper is to present a completely classification of simple Harish-Chandra modules over the
high rank $W$-algebra $W(2,2)$, which reobtains the
classification result of classical $W$-algebra $W(2,2)$ (when $k=1$) studied in \cite{LGZ,CLW,GLZ1}.
The paper is organized as follows. In Section $2$, we introduce some notations and definitions related to the high rank $W$-algebra $W(2,2)$ and Harish-Chandra modules. We also recall some known classification theorems over several related Lie algebras for later use.
In Section $3$,
we give a classification of simple cuspidal modules over
the higher rank $W$-algebra $W(2,2)$ in Theorem \mathfrak{g}ammaef{the513}.
In Section $4$, we present a classification of simple Harish-Chandra modules over the higher rank $W$-algebra $W(2,2)$ in Theorem \mathfrak{g}ammaef{the622}.
Throughout the present article, we denote by $\mathbb{C}$, $\mathbb{R}$, $\mathbb{Z}$, $\mathbb{N}$ and $\mathbb{Z}_+$ the sets of complex numbers, real numbers, integers, nonnegative integers and positive integers, respectively.
All vector spaces and Lie algebras are over $\mathbb{C}$. All simple modules are considered to be non-trivial.
For a Lie algebra $\mathfrak{L}$, we use $U(\mathfrak{L})$ to denote
the universal enveloping algebra.
\section{Preliminaries}
\subsection{The high rank $W$-algebra $W(2,2)$ and its cuspidal module}\lambdabel{sec22}
The {\em high rank $W$-algebra $W(2,2)$} is an infinite dimensional Lie algebra
$$\mathcal{W}[\mathbb{Z}^k]=\betaigoplus_{\alphalpha\in\mathbb{Z}^k}\mathbb{C} L_\alphalpha \oplus \betaigoplus_{\alphalpha\in\mathbb{Z}^k}\mathbb{C} W_{\alphalpha} \oplus\mathbb{C} C,$$
which satisfies the following Lie brackets
\betaegin{equation}\lambdabel{def1.1}
\alphaligned
&[L_\alphalpha,L_\betaeta]= (\betaeta-\alphalpha)L_{\alphalpha+\betaeta}+\deltalta_{\alphalpha+\betaeta,0}\frac{\alphalpha^{3}-\alphalpha}{12}C,\\&
[L_\alphalpha,W_\betaeta]=(\betaeta-\alphalpha)W_{\alphalpha+\betaeta}+\deltalta_{\alphalpha+\betaeta,0}\frac{\alphalpha^{3}-\alphalpha}{12}C,
\\&
[W_\alphalpha,W_\betaeta]=[\mathcal{W}[\mathbb{Z}^k],C]=0,
\endaligned
\end{equation}
where $\alphalpha,\betaeta\in \mathbb{Z}^k,k\in\mathbb{Z}_+$.
Clearly, $\mathcal{W}[\mathbb{Z}^k]$ has an infinite-dimensional Lie subalgebra $\mathcal{V}[\mathbb{Z}^k]:=\mathrm{span}\{L_\alphalpha,C\mid \alphalpha\in\mathbb{Z}^k\}$, which is called {\em high rank Virasoro algebra}. Note that $\mathbb{C} C$ is the center of $\mathcal{W}[\mathbb{Z}^k]$. The quotient algebras $\overline{\mathcal{W}}[\mathbb{Z}^k]=\mathcal{W}[\mathbb{Z}^k]/\mathbb{C} C$ and $\overline{\mathcal{V}}[\mathbb{Z}^k]=\mathcal{V}[\mathbb{Z}^k]/\mathbb{C} C$ are respectively called {\em centerless high rank $W$-algebra $W(2,2)$}
and {\em high rank Witt algebra}.
When $k=1$, we say that
$\mathcal{V}[\mathbb{Z}]$ and $\mathcal{W}[\mathbb{Z}]$ are respectively {\em classical Virasoro algebra} and {\em classical $W$-algebra $W(2,2)$}.
For any $\alphalpha\in\mathbb{Z}^k\setminus\{0\}$, we know that $\mathcal{V}[\mathbb{Z}\alphalpha]$ and $\mathcal{W}[\mathbb{Z}\alphalpha]$ are respectively isomorphic
to the classical Virasoro algebra and classical $W$-algebra $W(2,2)$.
Now we recall some definitions related to the weight module. Consider a non-trivial module $M$ over $\mathcal{V}[\mathbb{Z}^k]$ or $\mathcal{W}[\mathbb{Z}^k]$. We set that the action of the central element $C$ is a scalar $c$.
The module $M$ is said to be {\em trivial} if the action of whole algebra on $M$ is trivial.
Denote
$M_\lambdambda=\{v\in M \mid L_0v=\lambdambda v\}$, which is called a {\em weight space} of weight $\lambdambda\in\mathbb{Z}^k$. We call that $M$ is
a {\em weight module} if $M=\betaigoplus_{\lambdambda\in\mathbb{Z}^k}M_{\lambdambda}$. Set
$\mathrm{Supp}(M)=\{\lambdambda\mid M_{\lambdambda}\neq0\}$,
which is called the {\em support} (or called the {\em weight set}) of $M$.
The indecomposable weight module $M$ with all
weight spaces one-dimensional is called the {\em intermediate series module}.
\betaegin{defi}\mathfrak{g}ammam
Let $M$ be a weight module over $\mathcal{W}[\mathbb{Z}^k]$.
\betaegin{itemize}
\item[{\mathfrak{g}ammam (1)}]
If
$\mathrm{dim} (M_\lambdambda)<+\infty$ for all $\lambdambda\in\mathrm{Supp}(M)$, then $M$ is called {\em Harish-Chandra module}.
\item[{\mathfrak{g}ammam (2)}]
If there exists some $K\in\mathbb{Z}_+$ such that $\mathrm{dim}(M_\lambdambda)<K$ for all $\lambdambda \in \mathrm{Supp}(M)$, then $M$ is called {\em cuspidal} (or {\em uniformly bounded}).
\end{itemize}
\end{defi}
We define a class of cuspidal $\mathcal{W}[\mathbb{Z}^k]$-modules as follows, which are exactly intermediate series modules for $\mathcal{W}[\mathbb{Z}^k]$.
\betaegin{defi}\mathfrak{g}ammam\lambdabel{def22}
For any $g,h \in\mathbb{C}$, the $\mathcal{W}[\mathbb{Z}^k]$-module $M(g,h;\mathbb{Z}^k)$ has a basis
$\{v_\betaeta \mid \betaeta\in\mathbb{Z}^k\}$ and the $\mathcal{W}[\mathbb{Z}^k]$-action:
\betaegin{eqnarray*}
&&L_\alphalpha v_\betaeta=(g+\betaeta+h\alphalpha)v_{\alphalpha+\betaeta},
\ W_\alphalpha v_\betaeta=C v_\betaeta=0.
\end{eqnarray*}
\end{defi}
It is clear that the modules $M(g,h;\mathbb{Z}^k)$ are isomorphic to the intermediate series modules of $\mathcal{V}[\mathbb{Z}^k]$.
By \cite{SZ}, we see that the modules $M(g,h;\mathbb{Z}^k)$ are reducible if and only if $g\in\mathbb{Z}^k$ and $h\in\{0,1\}$.
We use
$\overline{M}(g,h;\mathbb{Z}^k)$ to denote the unique non-trivial simple subquotient of $M(g,h;\mathbb{Z}^k)$. Then
$\mathrm{Supp}(\overline{M}(g,h;\mathbb{Z}^k))=g+\mathbb{Z}^k$ or $\mathrm{Supp}(\overline{M}(g,h;\mathbb{Z}^k))=\mathbb{Z}^k\setminus\{0\}$. We also define $\overline{M}(g,h;\mathbb{Z}^k)$ as
intermediate series modules of $\mathcal{W}[\mathbb{Z}^k]$.
\subsection{Generalized highest weight modules}\lambdabel{www9998}
In this section,
a general class of Lie algebras are considered. Assume that $\mathcal{H}=\sum_{\alphalpha\in\mathbb{Z}^k}\mathcal{H}_\alphalpha$ is a $\mathbb{Z}^k$-graded Lie algebra
such that $\mathcal{H}_0$ is abelian. And the gradation of $\mathcal{H}$ is the root space decomposition
with respect to $\mathcal{H}_0$.
Let $\mathfrak{g}$ be a subgroup of $\mathbb{Z}^k$ such that $\mathbb{Z}^k=\mathfrak{g}\oplus\mathbb{Z}\mu$ for
some $\mu\in\mathbb{Z}^k$. We define the subalgebra of $\mathcal{H}$ as follows
$$\mathcal{H}_{\mathfrak{g}}=\betaigoplus_{\alphalpha\in \mathfrak{g}}\mathcal{H}_\alphalpha,\ \mathcal{H}_{\mathfrak{g}}^+=\betaigoplus_{\alphalpha\in \mathfrak{g},m\in\mathbb{Z}_+}\mathcal{H}_{\alphalpha+m\mu},
\ \mathcal{H}_{\mathfrak{g}}^-=\betaigoplus_{\alphalpha\in \mathfrak{g},m\in\mathbb{Z}_+}\mathcal{H}_{\alphalpha-m\mu}.$$
Let $\mathcal{K}$ be a simple $\mathcal{H}_{\mathfrak{g}}$-module. Then $\mathcal{K}$ can be extended to an $(\mathcal{H}_{\mathfrak{g}}+\mathcal{H}_{\mathfrak{g}}^+)$-module
by defining $\mathcal{H}_{\mathfrak{g}}^+\mathcal{K}=0$. Now we can define the {\em generalized Verma module} $V_{\mathfrak{g},\mu,\mathcal{K}}$
for $\mathcal{H}$ as
$$V_{\mathfrak{g},\mu,\mathcal{K}}=\mathrm{Ind}_{\mathcal{H}_{\mathfrak{g}}^++\mathcal{H}_{\mathfrak{g}}}^{\mathcal{H}}\mathcal{K}
=U(\mathcal{H})\betaigotimes_{U(\mathcal{H}_{\mathfrak{g}}+\mathcal{H}_{\mathfrak{g}}^+)}\mathcal{K}.$$
It is easy to know that $V_{\mathfrak{g},\mu,\mathcal{K}}$ has a unique simple quotient module for $\mathcal{H}$ and we write it as $P_{\mathfrak{g},\mu,\mathcal{K}}^{\mathcal{H}}$. Then $P_{\mathfrak{g},\mu,\mathcal{K}}^{\mathcal{H}}$ is called a {\em simple
highest weight module}. As far as we know,
the generalized Verma module (or generalized highest weight module) was introduced and investigated in some other
references (see, e.g., \cite{BZ,F,Ma2}).
Fix a basis of $\mathbb{Z}^k$.
Assume that $M$ is a weight module for $\mathcal{H}$.
Then $M$ is called {\em dense} if $\mathrm{Supp}(M)=\lambdambda+\mathbb{Z}^k$ for some $\lambdambda\in \mathcal{H}_0^*$. On the other hand,
if there exist $\lambdambda\in\mathrm{Supp}(M), \tau\in \mathbb{R}^k\setminus \{0\}$ and $\betaeta\in\mathbb{Z}^k$ such that
$$\mathrm{Supp}(M)\subseteq \lambdambda+\betaeta+{\mathbb{Z}^k}_{\leq0}^{(\tau)},$$
where ${\mathbb{Z}^k}_{\leq0}^{(\tau)}=\{\alphalpha\in{\mathbb{Z}^k}\mid(\tau|\alphalpha)\leq0\}$
and $(\tau|\alphalpha)$ is the usual inner product in $\mathbb{R}^k$, then $M$ is called {\em cut}. Obviously, the modules
$P_{\mathfrak{g},\mu,\mathcal{K}}^\mathcal{H}$ defined above are cut modules. If there exist a $\mathbb{Z}$-basis
$\{\epsilon_1,\ldots,\epsilon_k\}$ of ${\mathbb{Z}^k}$ and $K\in \mathbb{Z}_+$ such that $\mathcal{H}_\alphalpha v=0$ for all
$\alphalpha=\mathcal{S}igma_{i=1}^k\alphalpha_i\epsilon_i$
with $\alphalpha_i>K, i\in\{1,\ldots,k\}$, then the element $v \in M$ is called a
{\em generalized highest weight vector}.
The following general result of cut
$\mathcal{H}$-modules appeared in Theorem $4.1$ of \cite{MZ}.
\betaegin{theo}\lambdabel{the311}
Let $[\mathcal{H}_\alphalpha,\mathcal{H}_\betaeta]=\mathcal{H}_{\alphalpha+\betaeta}$ for all $\alphalpha,\betaeta\in\mathbb{Z}^k, k\in\mathbb{Z}_+$ with $\alphalpha \neq\betaeta$. Assume that
$M$ is a simple weight module over $\mathcal{H}$, which is neither dense nor trivial. If $M$
contains a generalized highest weight vector, then $M$ is a cut module.
\end{theo}
\subsection{The know results}
The classification theorems of simple Harish-Chandra
modules over the classical $W$-algebra $W(2,2)$ and high rank Virasoro algebra will be recalled in
this section.
The following result for the high rank Virasoro algebra appeared in \cite{LZ2}.
\betaegin{theo}\lambdabel{411}
Let $k>1$. Any non-trivial simple Harish-Chandra module
for $\mathcal{V}[\mathbb{Z}^k]$ is either a module of intermediate series or isomorphic
to $P_{\mathfrak{g},\mu,\mathcal{K}}^{\mathcal{V}[\mathbb{Z}^k]}$ for some $\mu\in\mathbb{Z}^k\setminus\{0\}$, a subgroup $\mathfrak{g}$ of $\mathbb{Z}^k$ with
$\mathbb{Z}^k=\mathfrak{g}\oplus\mathbb{Z} \mu$ and a non-trivial simple intermediate series
$\mathcal{V}[\mathfrak{g}]$-module $\mathcal{K}$.
\end{theo}
The classification of Harish-Chandra modules over the classical $W$-algebra $W(2,2)$
was given in \cite{LGZ}, which was reobtained in\cite{GLZ1,CLW} by some new ideas.
\betaegin{theo}\lambdabel{the44378}
Any non-trivial simple Harish-Chandra module over
$\mathcal{W}[\mathbb{Z}]$ is either a module of intermediate series, or a highest/lowest
weight module.
\end{theo}
\section{Cuspidal module}\lambdabel{115eqw}
In this section, we determine the simple cuspidal module for
the higher rank $W$-algebra $W(2,2)$.
Let $\mathbb{Z}^k=\betaigoplus_{i=1}^{k}\mathbb{Z} \epsilon_i$, where $\epsilon_1,\epsilon_2,\ldots,\epsilon_k$ is a
$\mathbb{Z}$-basis of $\mathbb{Z}^k\subseteq \mathbb{C}$. Given $\alphalpha\in \mathbb{Z}^k$, we set
$\alphalpha=\sum_{i=1}^k\alphalpha_i\epsilon_i$ for $\alphalpha_i \in \mathbb{Z}$.
For any $\alphalpha,\betaeta\in\mathbb{Z}^k$ with $\alphalpha_i,\betaeta_i\in\mathbb{N}, i\in\{1,\ldots,k\}$, we denote
$$\alphalpha^\betaeta=\alphalpha_1^{\betaeta_1}\cdots\alphalpha_k^{\betaeta_k}\quad \mathrm{and}\quad \betaeta!=\betaeta_1!\cdots\betaeta_k!.$$
Conveniently, we denote $\partial:=\frac{d}{dt}$. The high rank $W$-algebra $W(2,2)$ can be realized from truncated high rank loop-Witt algebra $\overline{\mathcal{V}}[\mathbb{Z}^k]\otimes \betaig(\mathbb{C}[x]/\lambdangle x^2\mathfrak{g}ammaangle\betaig)$ (see, e.g., \cite{LGZ,GLZ1}), namely,
\betaegin{eqnarray*}
&&L_\alphalpha=t^{\alphalpha+1}\partial\otimes1,
\ W_{\alphalpha}=t^{\alphalpha+1}\partial\otimes x,
\end{eqnarray*}
where $\alphalpha\in\mathbb{Z}^k$.
Denote $\mathcal{A}=\mathrm{span}\{t^\alphalpha\otimes 1 \mid \alphalpha\in\mathbb{Z}^k\}$, which is a unital associative algebra with multiplication $(t^\alphalpha\otimes1) (t^\betaeta\otimes1)=t^{\alphalpha+\betaeta}\otimes1$ for $\alphalpha,\betaeta\in \mathbb{Z}^k$. For convenience, we write $t^{\alphalpha+1}\partial=t^{\alphalpha+1}\partial\otimes1$ and $t^{\alphalpha}=t^{\alphalpha}\otimes1$ for $\alphalpha\in\mathbb{Z}^k$.
\subsection{$\mathcal{A}\overline{\mathcal{W}}[\mathbb{Z}^k]$-module}
We describe the structure of cuspidal $\overline{\mathcal{W}}[\mathbb{Z}^k]$-modules that admit a compatible action of the commutative unital algebra $\mathcal{A}$.
\betaegin{defi}\mathfrak{g}ammam (see \cite{BF1})
A module $M$ is called an {\em $\mathcal{A}\overline{\mathcal{W}}[\mathbb{Z}^k]$-module} if it is a module for both $\overline{\mathcal{W}}[\mathbb{Z}^k]$ and the commutative unital algebra $\mathcal{A}=\mathbb{C}[t^{\pm1}]\otimes1$
with these two structures being compatible:
\betaegin{eqnarray}\lambdabel{511}
y(fv)=(yf)v + f(yv) \quad \mathrm{for}\ f\in \mathcal{A}, y \in \overline{\mathcal{W}}[\mathbb{Z}^k], v\in M.
\end{eqnarray}
\end{defi}
Let $M$ be a weight module over $\mathcal{A}\overline{\mathcal{W}}[\mathbb{Z}^k]$. From \eqref{511}, we see that the action
of $\mathcal{A}$ is compatible with the weight grading of $M$:
$$\mathcal{A}_\alphalpha M_\lambdambda\subset M_{\alphalpha+\lambdambda} \quad \mathrm{for}\ \alphalpha, \lambdambda\in\mathbb{Z}^k.$$
We suppose that $\mathcal{A}\overline{\mathcal{W}}[\mathbb{Z}^k]$-module $M$ has a weight space decomposition, and one of the weight
spaces is finite-dimensional. According to all non-zero homogeneous elements of $\mathcal{A}$
are invertible, we know that all weight spaces of $M$ have the same dimension. Then
$M$ is also a free $\mathcal{A}$-module of a finite rank. It is clear that $\mathcal{A}\overline{\mathcal{W}}[\mathbb{Z}^k]$-module $M$ is
cuspidal (as $\overline{\mathcal{W}}[\mathbb{Z}^k]$-modules).
Assume that $M$ is a cuspidal $\mathcal{A}\overline{\mathcal{W}}[\mathbb{Z}^k]$-module. Let $W=M_g$ for $g\in\mathbb{Z}^k$ and $\mathrm{dim}(W)<\infty$.
From $M$ is a free $\mathcal{A}$-module, we can write
$$M\cong\mathcal{A}\otimes W.$$
\betaegin{lemm}\lambdabel{qas3.3}
Let $M$ defined as above.
For any $\alphalpha,n\in\mathbb{Z}^k$, we have $W_\alphalpha(t^nv)=t^n(W_\alphalpha v)$ for $v\in M$.
\end{lemm}
\betaegin{proof}
For any $\betaeta,n\in\mathbb{Z}^k$, by \eqref{511}, we have
\betaegin{eqnarray}\lambdabel{4rnlk}
W_\betaeta(t^nv)=(W_\betaeta t^n)v+t^n(W_\betaeta v).
\end{eqnarray}
Note that $W_\betaeta t^{n}=nt^{n+\betaeta}\otimes x=n(t^{n+\betaeta}\otimes1)(1\otimes x)$.
It is clear that $[1\otimes x,\overline{\mathcal{W}}[\mathbb{Z}^k]\oplus \mathcal{A}]=0$.
Then there is a homomorphism of algebras $\chi: 1\otimes x \mathfrak{g}ammaightarrow \mathbb{C}$ such
that $1\otimes x $ acts on $M$ as $\chi(1\otimes x )\in\mathbb{C}$. So the action of $W_\betaeta t^n$ on $M$ can be written as $n\mu t^{n+\betaeta}$, where $\mu\in\mathbb{C}$.
Now from
\betaegin{eqnarray*}
&&0
=[W_{\alphalpha},W_\betaeta](t^nv)
=n\mu^2(\betaeta-\alphalpha)t^{\alphalpha+\betaeta+n}v,
\end{eqnarray*}
we get $\mu=0$ by taking $n\neq0,\alphalpha\neq\betaeta$, namely, $(W_\betaeta t^n)v=0$.
Putting this into \eqref{4rnlk}, one has $W_\alphalpha(t^nv)=t^n(W_\alphalpha v)$ for $v\in M$. The lemma has been proved.
\end{proof}
\betaegin{remark}\lambdabel{rem322}
We note that the $\mathcal{A}\overline{\mathcal{W}}[\mathbb{Z}^k]$-module is a module for the semidirect product Lie
algebras $\overline{\mathcal{W}}[\mathbb{Z}^k]\ltimes\mathcal{A}$ (the action of $\mathcal{A}$ as a unital commutative associative algebra). The Lie
brackets between $\overline{\mathcal{W}}[\mathbb{Z}^k]$ and $\mathcal{A}$ are given by $[L_m, t^n]=nt^n, [W_m, t^n]=0$ for $m,n\in\mathbb{Z}^k$.
\end{remark}
For $m \in\mathbb{Z}^k$, we consider the following operator
$$\mathfrak{D} (m):W \mathfrak{g}ammaightarrow W.$$
It can be defined as the restriction to $W$ of the composition $t^{-m}\circ(t^{m+1}\partial)$ regarded also as an
operator on $M$. Note that $\mathfrak{D}(0)=g \mathrm{Id}$.
According to \eqref{511}, Lemma \mathfrak{g}ammaef{qas3.3} and the finite-dimensional operator $\mathfrak{D} (m)$, we get the action on $M$ as follows
\betaegin{eqnarray}\lambdabel{5533}
&&L_m(t^nv)=(t^{m+1}\partial)(t^nv)=nt^{m+n}v+t^{m+n}\mathfrak{D} (m)v,
\ W_m(t^nv)=t^n(W_mv),
\end{eqnarray}
where $m,n\in\mathbb{Z}^k,v\in W$.
Based on
\eqref{def1.1} and \eqref{5533},
it is easy to derive the Lie
bracket (also see Lemma $3.2$ in \cite{B1}):
\betaegin{eqnarray}\lambdabel{556677hjj}
[\mathfrak{D} (s),\mathfrak{D} (m)]=(m-s)\mathfrak{D} (s+m)-m\mathfrak{D} (m)+s\mathfrak{D} (s).
\end{eqnarray}
Next, we show that $\mathfrak{D} (m)$ can be expressed as a polynomial in
$m=(m_1,\ldots,m_k)$.
\betaegin{theo}\lambdabel{533rree}
Assume that $M$ is a cuspidal $\mathcal{A}\overline{\mathcal{W}}[\mathbb{Z}^k]$-module, $M= \mathcal{A}\otimes W$, where $W=M_g,g\in\mathbb{Z}^k$.
Then the action of $\overline{\mathcal{W}}[\mathbb{Z}^k]$ on $M$ is presented as
\betaegin{eqnarray*}
&&L_m(t^nv)=nt^{m+n}v+t^{m+n}\mathfrak{D} (m)v,
\ W_m(t^nv)=t^n(W_mv),
\end{eqnarray*}
$m,n\in\mathbb{Z}^k,v\in W$,
where the family of operators $\mathfrak{D} (m): W\mathfrak{g}ammaightarrow W$ can be shown as an $\mathrm{End}(W)$-valued polynomial in $m=(m_1,\ldots,m_k)$ with the constant term $\mathfrak{D} (0)=g \mathrm{Id}$, and $\mathrm{Id}$ is the identification endomorphism of $W$.
\end{theo}
\betaegin{proof} By $m\in\mathbb{Z}^k$, we can write $m=\sum_{{i}=1}^km_{{i}}\epsilon_{{i}}$, where $m_{{i}}\in\mathbb{Z}$, $\epsilon_1,\epsilon_2,\ldots,\epsilon_k$ is a
$\mathbb{Z}$-basis of $\mathbb{Z}^k$.
According to Theorem $2.2$ in \cite{B1}, we obtain that $\mathfrak{D} (m_{{i}}\epsilon_{{i}})$ is a polynomial in $m_i\in \mathbb{Z}$ with
coefficients in $\mathrm{End}(W)$ for all ${{i}}\in \{1,\ldots,k\}$. Now suppose that $\mathfrak{D} (\sum_{{{i}}=1}^{{{j}}-1}m_{{i}}\epsilon_{{i}})$
is a polynomial in $\alphalpha_1,\ldots,\alphalpha_{{{j}}-1}$ for some $1<{{j}}\leq k$. For $m_{{j}}\in\mathbb{Z}$, it follows from \eqref{556677hjj} that
\betaegin{eqnarray*}
&&(\sum_{{{i}}=1}^{{{j}}-1}m_{{i}}
\epsilon_{{i}}-m_{{j}}\epsilon_{{j}})\mathfrak{D} (\sum_{{{i}}=1}^{{{j}}}m_{{i}}\epsilon_{{i}})
\\&=&[\mathfrak{D} (m_{{j}}\epsilon_{{j}}),
\mathfrak{D}(\sum_{{{i}}=1}^{{{j}}-1}m_{{i}}\epsilon_{{i}})]
+\sum_{{{i}}=1}^{{{j}}-1}(m_{{i}}\epsilon_{{i}})\mathfrak{D} (\sum_{{{i}}=1}^{{{j}}-1}m_{{i}}\epsilon_{{i}})
-(m_{{j}}\epsilon_{{j}})\mathfrak{D} (\alphalpha_{{j}}\epsilon_{{j}}).
\end{eqnarray*}
Consider $m_{{j}}\neq0$. Then from the linearly independence of $\epsilon_1,\ldots,\epsilon_{{j}}$, one has $\sum_{{{i}}=1}^{{{j}}-1}m_{{i}}\epsilon_{{i}}
-m_{{j}}\epsilon_{{j}}\neq0$. By the induction assumption, we conclude that $\mathfrak{D} (\sum_{{{i}}=1}^{{{j}}}m_{{i}}\epsilon_{{i}})$
is a polynomial in $m_1,\ldots,m_{{{j}}}$, where $1<{{j}}\leq k$. Choosing ${{j}}=k$, one can see that
$\mathfrak{D} (m)$ is a polynomial in $m_1,\ldots,m_k$. By the definition of operator $\mathfrak{D}(m)$, one has $\mathfrak{D} (0)=g \mathrm{Id}$ for $g\in\mathbb{C}$. We complete the proof.
\end{proof}
We can write $\mathfrak{D} (m)$ in the form (also see \cite{B1,BF2,LG})
\betaegin{eqnarray}\lambdabel{dd555}
\sum_{{\widetilde{i}}\in\mathbb{N}^k}\frac{m^{\widetilde{i}}}{{\widetilde{i}}!}D^{({\widetilde{i}})},
\end{eqnarray}
where ${\widetilde{i}}!=\prod_{{\widetilde{j}}=1}^k{\widetilde{i}}_{\widetilde{j}}!$ and only has a finite number of the nonzero operators $D^{({\widetilde{i}})}\in \mathrm{End}(W)$. Note that $\mathfrak{D}(0)=D^{(\widetilde{0})}$.
For $m,s\in\mathbb{Z}^k$, by \eqref{556677hjj}, we have
\betaegin{eqnarray}\lambdabel{eq55}
&&\nonumber \sum_{{\widetilde{i}},{\widetilde{j}}\in\mathbb{N}^k}\frac{s^{\widetilde{i}}m^{\widetilde{j}}}{{\widetilde{i}}!{\widetilde{j}}!}[D^{({\widetilde{i}})},D^{({\widetilde{j}})}]
\\&=&\nonumber (\sum_{{\widetilde{i}}\in\mathbb{N}^k}\frac{s^{\widetilde{i}}}{{\widetilde{i}}!}D^{({\widetilde{i}})})(\sum_{{\widetilde{j}}\in\mathbb{N}^k}\frac{m^{\widetilde{j}}}{{\widetilde{j}}!}D^{({\widetilde{j}})})
-(\sum_{{\widetilde{j}}\in\mathbb{N}^k}\frac{m^{\widetilde{j}}}{{\widetilde{j}}!}D^{({\widetilde{j}})})(\sum_{{\widetilde{i}}\in\mathbb{N}^k}\frac{s^{\widetilde{i}}}{{\widetilde{i}}!}D^{({\widetilde{i}})})
\\&=&\nonumber [\mathfrak{D} (s),\mathfrak{D} (m)]=(s-m)\mathfrak{D} (s+m)-s\mathfrak{D} (s)+m\mathfrak{D} (m)
\\&=& \sum_{l=1}^k\betaig(\sum_{{\widetilde{i}},{\widetilde{j}}\in\mathbb{N}^k}\frac{s^{\widetilde{i}}m^{\widetilde{j}}}{{\widetilde{i}}!({\widetilde{j}}-\epsilon_l)!}D^{({\widetilde{i}}+{\widetilde{j}}-\epsilon_l)}\betaig)
-\sum_{l=1}^k\betaig(\sum_{{\widetilde{i}},{\widetilde{j}}\in\mathbb{N}^k}\frac{s^{\widetilde{i}}m^{\widetilde{j}}}{({\widetilde{i}}-\epsilon_l)!{\widetilde{j}}!}D^{({\widetilde{i}}+{\widetilde{j}}-\epsilon_l)}\betaig).
\end{eqnarray}
Comparing the coefficients of $\frac{s^{\widetilde{i}}m^{\widetilde{j}}}{{\widetilde{i}}!{\widetilde{j}}!}$ in \eqref{eq55}, we check that
\betaegin{eqnarray}\lambdabel{22231lk}
[D^{({\widetilde{i}})},D^{({\widetilde{j}})}]=\left\{\betaegin{array}{llll}
\sum_{l=1}^k({\widetilde{j}}_l-{\widetilde{i}}_l)\epsilon_lD^{({\widetilde{i}}+{\widetilde{j}}-\epsilon_l)} &\mbox{if}\
{\widetilde{i}},{\widetilde{j}}\in\mathbb{N}^k\betaackslash \{0\},\\[4pt]
0 &\mbox{if}\
{\widetilde{i}}=0\ \mathrm{or}\ {\widetilde{j}}=0.
\end{array}\mathfrak{g}ammaight.
\end{eqnarray}
From \eqref{22231lk},
the operators $\mathcal{G}=\mathrm{span}\betaig\{D^{(\widetilde{i})}\mid \widetilde{i} \in \mathbb{N}^k\betaackslash \{0\}\betaig\}$ yield a Lie algebra.
For $\widetilde{i}\in\mathbb{N}^k$, let $|\widetilde{i}|=\widetilde{i}_1+\widetilde{i}_2+\cdots+\widetilde{i}_k$.
Denote $$\mathcal{G}_p=\mathrm{span}\{D^{(\widetilde{i})}\mid \widetilde{i}\in\mathbb{N}^k,|\widetilde{i}|-1=p\}\quad \mathrm{for}\ p\in\mathbb{N}.$$
Then $\mathcal{G}=\betaigoplus_{p\in\mathbb{N}}\mathcal{G}_p$ is a $\mathbb{Z}$-graded Lie algebra (also see \cite{LG,BF2,B1}).
From \eqref{22231lk}, we know that $\mathcal{G}_0=\mathrm{span}\{D^{(\epsilon_{i})}\mid {i}=1,\ldots,k\}$ is a subalgebra of $\mathcal{G}$, whose Lie algebra structure is presented as
$$[D^{(\epsilon_{i})},D^{(\epsilon_{j})}]
=\epsilon_{j}D^{(\epsilon_{i})}-\epsilon_{i}D^{(\epsilon_{j})},$$
where ${i},{j}\in\{1,2,\ldots,k\}$.
It is easy to get that $[\mathcal{G}_0,\mathcal{G}_0]$ is nilpotent. Hence, $\mathcal{G}_0$ is a solvable Lie
algebra. Now we define the following one-dimensional $\mathcal{G}$-module $V(h)=\mathbb{C} v\neq0$ for
any $h\in\mathbb{C}$:
\betaegin{eqnarray}
D^{(\epsilon_{i})}v=h \epsilon_{i}v \quad \mathrm{for}\ {i}\in\{1,\ldots,k\}.
\end{eqnarray}
The following lemma was proved in \cite{LG}.
\betaegin{lemm}\lambdabel{555rree}
Assume that $T$ and $W$ are finite-dimensional simple modules over $\mathcal{G}_0$ and $\mathcal{G}$, respectively. Then
\betaegin{itemize}
\item[\mathfrak{g}ammam(a)] $T\cong V(h)$ for $h\in\mathbb{C}$.
\item[\mathfrak{g}ammam(b)] $D^{(\widetilde{i})}W=0$ for any $\widetilde{i}\in\mathbb{N}^k$ with $|\widetilde{i}|$ sufficiently large;
\item[\mathfrak{g}ammam(c)] $\mathcal{G}_pW=0$ for all $p\in\mathbb{Z}_+$ and $W\cong V(h)$ as a $\mathcal{G}_0$-module for some $h\in\mathbb{C}$.
\end{itemize}
\end{lemm}
\betaegin{theo}\lambdabel{the566}
Any simple cuspidal $\mathcal{A}\overline{\mathcal{W}}[\mathbb{Z}^k]$-module is isomorphic to a
module of intermediate series $\overline{M}(g,h;\mathbb{Z}^k)$ for some $g,h\in\mathbb{C}$.
\end{theo}
\betaegin{proof}
Let $M$ be a simple cuspidal $\mathcal{A}\overline{\mathcal{W}}[\mathbb{Z}^k]$-module, $M= \mathcal{A}\otimes W$, where $W=M_g,g\in\mathbb{Z}^k$. For $n,\alphalpha\in\mathbb{Z}^k,v\in W$, based on Theorem \mathfrak{g}ammaef{533rree}, Lemma \mathfrak{g}ammaef{555rree} and \eqref{dd555}, we check that
\betaegin{eqnarray}\lambdabel{qiuy789}
L_\alphalpha (t^nv)&=&\nonumber nt^{\alphalpha+n} v+t^{\alphalpha+n}(\mathfrak{D}(\alphalpha) v)
\\&=&\nonumber t^{\alphalpha+n}\betaig((n+\mathfrak{D}(0)+\sum_{{\widetilde{i}}\in\mathbb{N}^k\setminus\{0\}}\frac{\alphalpha^{\widetilde{i}}}
{{\widetilde{i}}!}D^{({\widetilde{i}})})v\betaig)
\\&=&\nonumber t^{\alphalpha+n}\betaig((n+\mathfrak{D}(0)+\sum_{i=1}^k\alphalpha_iD^{({\epsilon_i})})v\betaig)
\\&=&\nonumber t^{\alphalpha+n}\betaig((n+g\mathrm{Id}+\sum_{i=1}^kh\alphalpha_i\epsilon_i)v\betaig)
\\&=&(n+g+h\alphalpha)(t^{\alphalpha+n}v),
\end{eqnarray}
where $g,h\in\mathbb{C}$.
Using \eqref{qiuy789} and $(\betaeta-\alphalpha)W_{\alphalpha+\betaeta}(t^nv)=(L_\alphalpha W_\betaeta-W_\betaeta L_\alphalpha)(t^nv)$, we obtain
\betaegin{eqnarray}\lambdabel{564rf}
(\betaeta-\alphalpha)t^n(W_{\alphalpha+\betaeta}v)=\betaeta t^{n+\alphalpha}(W_\betaeta v).
\end{eqnarray}
Taking $\betaeta=0$ in \eqref{564rf}, we immediately get $t^n(W_\alphalpha v)=0$ for $\alphalpha\neq0$.
Considering $\alphalpha=-\betaeta\neq0$ in \eqref{564rf} again, one checks $t^n(W_0v)=0$.
Then we conclude $t^n(W_\alphalpha v)=0$ for $\alphalpha,n\in\mathbb{Z}^k$, that is to say, $W_\alphalpha(t^nv)=0$.
This completes the proof.
\end{proof}
\subsection{$\mathcal{A}$-cover of a cuspidal $\overline{\mathcal{W}}[\mathbb{Z}^k]$-module}
We first recall the definitions of coinduced module and $\mathcal{A}$-cover (see \cite{BF1}).
\betaegin{defi}\mathfrak{g}ammam
A module {\em coinduced} from a $\overline{\mathcal{W}}[\mathbb{Z}^k]$-module $M$ is the space $\mathrm{Hom}(\mathcal{A},M)$ with the actions of $\overline{\mathcal{W}}[\mathbb{Z}^k]$ and $\mathcal{A}$ as follows
\betaegin{eqnarray*}
&&(\mathfrak{a}\varphi)(f)=\mathfrak{a}(\varphi(f))-\varphi(\mathfrak{a}(f)),
\ (y\varphi)(f)=\varphi(yf),
\end{eqnarray*}
where $\varphi\in\mathrm{Hom}(\mathcal{A},M), \mathfrak{a}\in \overline{\mathcal{W}}[\mathbb{Z}^k],f,y\in \mathcal{A}$.
\end{defi}
\betaegin{defi}\mathfrak{g}ammam\lambdabel{def5777}
An {\em $\mathcal{A}$-cover} of a cuspidal module $M$ over $\overline{\mathcal{W}}[\mathbb{Z}^k]$ is an $\mathcal{A}\overline{\mathcal{W}}[\mathbb{Z}^k]$-submodule $$\widehat{M}=\mathrm{span}\{\phi(\mathfrak{a},w)\mid \mathfrak{a}\in\overline{\mathcal{W}}[\mathbb{Z}^k],w\in M\}\subset\mathrm{Hom}(\mathcal{A},M),$$
where $\phi(\mathfrak{a},w):\mathcal{A}\mathfrak{g}ammaightarrow M$ is defined as
$$\phi(\mathfrak{a},w)(f)=(f\mathfrak{a})(w).$$
\end{defi}
The action of $\mathcal{A}\overline{\mathcal{W}}[\mathbb{Z}^k]$ on $\widehat{M}$ is given by
\betaegin{eqnarray*}
&&\mathfrak{b}\phi(\mathfrak{a},w)=\phi([\mathfrak{b},\mathfrak{a}],w)+\phi(\mathfrak{a},\mathfrak{b}w),
\\&&f\phi(\mathfrak{a},w)=\phi(f\mathfrak{a},w)\quad \mathrm{for}\ \mathfrak{a},\mathfrak{b}\in \overline{\mathcal{W}}[\mathbb{Z}^k],w\in M,f\in \mathcal{A}.
\end{eqnarray*}
Let $$\mathfrak{K}(M)=\left\{\sum_{\alphalpha\in\mathbb{Z}^k}\mathfrak{a}_\alphalpha\otimes w_{\alphalpha}\in\overline{\mathcal{W}}[\mathbb{Z}^k]\otimes M
\ \betaigg|\ \sum_{\alphalpha\in\mathbb{Z}^k}(f\mathfrak{a}_\alphalpha) w_{\alphalpha}=0,\quad\forall f\in\mathcal{A} \mathfrak{g}ammaight\}.$$ Then $\mathfrak{K}(M)$ is an
$\mathcal{A}\overline{\mathcal{W}}[\mathbb{Z}^k]$-submodule of $\overline{\mathcal{W}}[\mathbb{Z}^k]\otimes M$. The $\mathcal{A}$-cover $\widehat{M}$ can also be constructed as a quotient $\mathcal{A}\overline{\mathcal{W}}[\mathbb{Z}^k]$-module
$$(\overline{\mathcal{W}}[\mathbb{Z}^k]\otimes M)/\mathfrak{K}(M),$$
where $\overline{\mathcal{W}}[\mathbb{Z}^k]M=M$. Clearly, the following linear map
\betaegin{eqnarray*}
\Theta:\quad & \widehat{M}&\longrightarrow \overline{\mathcal{W}}[\mathbb{Z}^k]M
\\ &\mathfrak{a}\otimes w+\mathfrak{K}(M)&\longmapsto \mathfrak{a}w
\end{eqnarray*} is a $\overline{\mathcal{W}}[\mathbb{Z}^k]$-module epimorphism.
\betaegin{lemm}{\mathfrak{g}ammam (see \cite{BF1})}\lambdabel{lemm6765}
Let $M$ be a cuspidal module for $\overline{\mathcal{W}}[\mathbb{Z}^k]$. Then there exists $l\in \mathbb{Z}_+$ such that for all
$\alphalpha,\betaeta,\mathfrak{g}amma\in\mathbb{Z}^k$ the operator $\Omega_{\alphalpha,\betaeta}^{(l,\mathfrak{g}amma)}=\sum_{{i}=0}^l(-1)^{i}{l\choose {i}}L_{\alphalpha-{i}\mathfrak{g}amma}L_{\betaeta+{i}\mathfrak{g}amma}$ annihilates $M$.
\end{lemm}
\betaegin{lemm}\lambdabel{lemm676225}
Let $M$ be a cuspidal $\overline{\mathcal{W}}[\mathbb{Z}^k]$-module. Then there exists $r\in \mathbb{Z}_+$ such that for all
$\alphalpha,\betaeta,\mathfrak{g}amma\in\mathbb{Z}^k$ the following two operators $$\Omega_{\alphalpha,\betaeta}^{(r,\mathfrak{g}amma)}=\sum_{{i}=0}^r(-1)^{i}{r\choose {i}}L_{\alphalpha-{i}\mathfrak{g}amma}L_{\betaeta+{i}\mathfrak{g}amma}\quad \mathrm{and} \quad \widetilde{\Omega}_{\alphalpha,\betaeta}^{(r,\mathfrak{g}amma)}=\sum_{{i}=0}^r(-1)^{i}{r\choose {i}}W_{\alphalpha-{i}\mathfrak{g}amma}L_{\betaeta+{i}\mathfrak{g}amma}$$ annihilate $M$.
\end{lemm}
\betaegin{proof}
Note that $M$ is also a cuspidal module for $\mathcal{V}[\mathbb{Z}^k]$. According to Lemma \mathfrak{g}ammaef{lemm6765}, there exists
$l\in\mathbb{Z}_+$ such that $\Omega_{\alphalpha,\betaeta}^{(l,\mathfrak{g}amma)}M=0$ for all $\alphalpha,\betaeta,\mathfrak{g}amma\in\mathbb{Z}^k$. It
follows from this that
\betaegin{eqnarray}\nonumber
0&=&\Big(\sum_{{i}=0}^l(-1)^{i}{l\choose {i}}\betaig(L_{\alphalpha-({i}-1)\mathfrak{g}amma}L_{\betaeta+({i}-1)\mathfrak{g}amma}
-2L_{\alphalpha-{i}\mathfrak{g}amma}L_{\betaeta+{i}\mathfrak{g}amma}
+L_{\alphalpha-({i}+1)\mathfrak{g}amma}L_{\betaeta+({i}+1)\mathfrak{g}amma}\betaig)\Big)M
\\&=&\lambdabel{57uyt}\Big(\sum_{{i}=0}^{l+2}(-1)^{i}{l+2\choose {i}}L_{\alphalpha-({i}-1)\mathfrak{g}amma}L_{\betaeta+({i}-1)\mathfrak{g}amma}\Big)M.
\end{eqnarray}
Setting $r=l+2$ in \eqref{57uyt}, one gets $\Omega_{\alphalpha,\betaeta}^{(r,\mathfrak{g}amma)}M=0$ for all $\alphalpha,\betaeta,\mathfrak{g}amma\in\mathbb{Z}^k$.
For any $s\in\mathbb{Z}^k$, from Lemma \mathfrak{g}ammaef{lemm6765}, we immediately get
$[\Omega_{\alphalpha,\betaeta}^{(l,\mathfrak{g}amma)},W_s]M=0$ for all $\alphalpha,\betaeta,\mathfrak{g}amma\in\mathbb{Z}^k$.
Now for any $\alphalpha,\betaeta,s\in\mathbb{Z}^k$ and $\mathfrak{g}amma\in\mathbb{Z}^k\setminus\{0\}$, we can compute that
\betaegin{eqnarray*}
0&=&\Big([\Omega_{\alphalpha,\betaeta-\mathfrak{g}amma}^{(l,\mathfrak{g}amma)},W_{s+\mathfrak{g}amma}]
-2[\Omega_{\alphalpha,\betaeta}^{(l,\mathfrak{g}amma)},W_{s}]
+[\Omega_{\alphalpha,\betaeta+\mathfrak{g}amma}^{(l,\mathfrak{g}amma)},W_{s-\mathfrak{g}amma}]
\\&&-[\Omega_{\alphalpha+\mathfrak{g}amma,\betaeta-\mathfrak{g}amma}^{(l,\mathfrak{g}amma)},W_{s}]
+2[\Omega_{\alphalpha+\mathfrak{g}amma,\betaeta}^{(l,\mathfrak{g}amma)},W_{s-\mathfrak{g}amma}]-[\Omega_{\alphalpha+\mathfrak{g}amma,\betaeta+\mathfrak{g}amma}^{(l,\mathfrak{g}amma)},
W_{s-2\mathfrak{g}amma}]\Big)M
\\&=&\Big([\sum_{{i}=0}^l(-1)^{i}{l\choose {i}}L_{\alphalpha-{i}\mathfrak{g}amma}L_{\betaeta-\mathfrak{g}amma+{i}\mathfrak{g}amma},W_{s+\mathfrak{g}amma}]-2[\sum_{{i}=0}^l(-1)^{i}{l\choose {i}}L_{\alphalpha-{i}\mathfrak{g}amma}L_{\betaeta+{i}\mathfrak{g}amma},W_{s}]
\\&&+[\sum_{{i}=0}^l(-1)^{i}{l\choose {i}}L_{\alphalpha-{i}\mathfrak{g}amma}L_{\betaeta+\mathfrak{g}amma+{i}\mathfrak{g}amma},W_{s-\mathfrak{g}amma}]
-[\sum_{{i}=0}^l(-1)^{i}{l\choose {i}}L_{\alphalpha+\mathfrak{g}amma-{i}\mathfrak{g}amma}L_{\betaeta-\mathfrak{g}amma+{i}\mathfrak{g}amma},W_{s}]
\\&&+2[\sum_{{i}=0}^l(-1)^{i}{l\choose {i}}L_{\alphalpha+\mathfrak{g}amma-{i}\mathfrak{g}amma}L_{\betaeta+{i}\mathfrak{g}amma},W_{s-\mathfrak{g}amma}]
-[\sum_{{i}=0}^l(-1)^{i}{l\choose {i}}L_{\alphalpha+\mathfrak{g}amma-{i}\mathfrak{g}amma}L_{\betaeta+\mathfrak{g}amma+{i}\mathfrak{g}amma},W_{s-2\mathfrak{g}amma}]\Big)M
\\&=&\Big(\sum_{{i}=0}^l(-1)^{i}{l\choose {i}}\betaig(
(s+(2-{i})\mathfrak{g}amma-\betaeta)L_{\alphalpha-{i}\mathfrak{g}amma}W_{\betaeta+s+{i}\mathfrak{g}amma}
\\&&+(s+({i}+1)\mathfrak{g}amma-\alphalpha)W_{\alphalpha+s+(1-{i})\mathfrak{g}amma}L_{\betaeta+({i}-1)\mathfrak{g}amma}
\\&&-2\betaig((s-\betaeta-{i}\mathfrak{g}amma)L_{\alphalpha-{i}\mathfrak{g}amma}W_{\betaeta+s+{i}\mathfrak{g}amma}+(s-\alphalpha+{i}\mathfrak{g}amma)
W_{\alphalpha+s-{i}\mathfrak{g}amma}L_{\betaeta+{i}\mathfrak{g}amma}\betaig)
\\&&
+(s-\betaeta-({i}+2)\mathfrak{g}amma)L_{\alphalpha-{i}\mathfrak{g}amma}W_{\betaeta+s+{i}\mathfrak{g}amma}
+(s-\alphalpha+({i}-1)\mathfrak{g}amma)W_{\alphalpha+s-({i}+1)\mathfrak{g}amma}L_{\betaeta+({i}+1)\mathfrak{g}amma}
\\&&
-(s-\betaeta-({i}-1)\mathfrak{g}amma)L_{\alphalpha+(1-{i})\mathfrak{g}amma}W_{\betaeta+s+({i}-1)\mathfrak{g}amma}
-(s-\alphalpha+({i}-1)\mathfrak{g}amma)W_{\alphalpha+s+(1-{i})\mathfrak{g}amma}L_{\betaeta+({i}-1)\mathfrak{g}amma}
\\&&+2\betaig(
(s-\betaeta-({i}+1)\mathfrak{g}amma)L_{\alphalpha+(1-{i})\mathfrak{g}amma}W_{\betaeta+s+({i}-1)\mathfrak{g}amma}
+(s-\alphalpha+({i}-2)\mathfrak{g}amma)W_{\alphalpha+s-{i}\mathfrak{g}amma}L_{\betaeta+{i}\mathfrak{g}amma}\betaig)
\\&&
-(s-\betaeta-({i}+3)\mathfrak{g}amma)L_{\alphalpha+(1-{i})\mathfrak{g}amma}W_{\betaeta+s+({i}-1)\mathfrak{g}amma}
\\&&-(s-\alphalpha+({i}-3)\mathfrak{g}amma)W_{\alphalpha+s-({i}+1)\mathfrak{g}amma}L_{\betaeta+({i}+1)\mathfrak{g}amma}\betaig)\Big)M
\\&=&\Big(2\mathfrak{g}amma\sum_{{i}=0}^l(-1)^{i}{l\choose {i}}\betaig(W_{\alphalpha+s-({i}-1)\mathfrak{g}amma}L_{\betaeta+({i}-1)\mathfrak{g}amma}
-2W_{\alphalpha+s-{i}\mathfrak{g}amma}L_{\betaeta+{i}\mathfrak{g}amma}
+W_{\alphalpha+s-({i}+1)\mathfrak{g}amma}L_{\betaeta+({i}+1)\mathfrak{g}amma}\betaig)\Big)M
\\&=&\Big(2\mathfrak{g}amma\sum_{{i}=0}^{l+2}(-1)^{i}{l+2\choose {i}}W_{\alphalpha+s-({i}-1)\mathfrak{g}amma}L_{\betaeta+({i}-1)\mathfrak{g}amma}\Big)M.
\end{eqnarray*}
Moreover, the module $M$ can be annihilated by the operator $\widetilde{\Omega}_{\alphalpha,\betaeta}^{(l+2,0)}$.
Then we conclude that $\widetilde{\Omega}_{\alphalpha,\betaeta}^{(r,\mathfrak{g}amma)}M=\betaig(\sum_{{i}=0}^r(-1)^{i}{r\choose {i}}W_{\alphalpha-{i}\mathfrak{g}amma}L_{\betaeta+{i}\mathfrak{g}amma}\betaig)M=0$, where $r=l+2$ and all $\alphalpha,\betaeta,\mathfrak{g}amma\in\mathbb{Z}^k$. The lemma holds.
\end{proof}
\betaegin{prop}\lambdabel{pro510}
Let $M$ be a cuspidal module over $\overline{\mathcal{W}}[\mathbb{Z}^k]$. Then the $\mathcal{A}$-cover $\widehat{M}$ of $M$ is also a cuspidal $\mathcal{A}\overline{\mathcal{W}}[\mathbb{Z}^k]$-module.
\end{prop}
\betaegin{proof}
Let $M_\lambdambda$ be a weight space with weight $\lambdambda\in\mathbb{Z}^k$.
For $\alphalpha\in\mathbb{Z}^k$, we denote
$$\phi(L_\alphalpha\cup W_\alphalpha,M_\lambdambda)=\left\{\phi(L_\alphalpha,w),\phi(W_\alphalpha,w)\ \betaigg|\ w\in M_\lambdambda\mathfrak{g}ammaight\}\subset\widehat{M}.$$
By considering the weight spaces of $M$, the space
$\phi(L_\alphalpha\cup W_\alphalpha,M_\lambdambda)$ is finite-dimensional.
Since $\widehat{M}$ is an $\mathcal{A}$-module, we see that one of its weight spaces is finite-dimensional.
For a fixed weight $\betaeta\in\mathbb{Z}^k$, we will show that $\widehat{M}_\betaeta$
is finite-dimensional. Obviously, the space $\widehat{M}_\betaeta$
is spanned by the
set $$\Big(\betaigcup_{\mathfrak{g}amma\in\mathbb{Z}^k}\phi(L_{\betaeta-\mathfrak{g}amma},M_\mathfrak{g}amma)\Big)\cup
\Big(\betaigcup_{\mathfrak{g}amma\in\mathbb{Z}^k}\phi(W_{\betaeta-\mathfrak{g}amma},M_\mathfrak{g}amma)\Big).$$
Define a norm on $\mathbb{Z}^k$ as follows
$$\|\alphalpha\|=\sum_{{i}=1}^k|\alphalpha_{i}|,$$
where $\alphalpha=\sum_{{i}=1}^k\alphalpha_{i}\epsilon_{i}$.
By Lemma \mathfrak{g}ammaef{lemm676225}, there exists $r\in\mathbb{N}$ such that for all $\alphalpha,\betaeta,\mathfrak{g}amma\in\mathbb{Z}^k$ the following two operators
$$\Omega_{\alphalpha,\betaeta}^{(r,\mathfrak{g}amma)}=\sum_{{i}=0}^r(-1)^{i}{r\choose {i}}L_{\alphalpha-{i}\mathfrak{g}amma}L_{\betaeta+{i}\mathfrak{g}amma}\quad \mathrm{and}\quad \widetilde{\Omega}_{\alphalpha,\betaeta}^{(r,\mathfrak{g}amma)}=\sum_{{i}=0}^r(-1)^{i}{r\choose {i}}W_{\alphalpha-{i}\mathfrak{g}amma}L_{\betaeta+{i}\mathfrak{g}amma}$$
annihilate $M$, namely, $\Omega_{\alphalpha,\betaeta}^{(r,\mathfrak{g}amma)}v= \widetilde{\Omega}_{\alphalpha,\betaeta}^{(r,\mathfrak{g}amma)}v=0$ for all $\alphalpha,\betaeta,\mathfrak{g}amma\in\mathbb{Z}^k,v\in M$.
Hence, $\Omega_{\alphalpha,\betaeta}^{(r,\mathfrak{g}amma)}v$ and $\widetilde{\Omega}_{\alphalpha,\betaeta}^{(r,\mathfrak{g}amma)}v$ are both in $\mathfrak{K}(M)$.
\betaegin{clai}\lambdabel{clai11}
For any $\alphalpha,\betaeta\in\mathbb{Z}^k$, $\widehat{M}_{\alphalpha+\betaeta}$ is equal to
$$\mathrm{span} \left\{\phi(L_{\alphalpha+\betaeta}\cup W_{\alphalpha+\betaeta},M_0),
\phi(L_{\alphalpha-\mathfrak{g}amma},M_{\betaeta+\mathfrak{g}amma}),
\phi(W_{\alphalpha-\mathfrak{g}amma},M_{\betaeta+\mathfrak{g}amma})\ \betaigg|\ \mathfrak{g}amma\neq-\betaeta,\|\mathfrak{g}amma\|\leq\frac{kr}{2}\mathfrak{g}ammaight\}.
$$
\end{clai}
For all $q\in\mathbb{Z}^k$ and $w\in M_{\betaeta+q}$, we have $\phi(L_{\alphalpha-q},w)$ and $\phi(W_{\alphalpha-q},w)$ in
$\widehat{M}_{\alphalpha+\betaeta}$.
Now we prove this claim by induction on $\|q\|$.
If $|q_{i}|\leq\frac{r}{2}$ for all ${i}\in\{1,\ldots,k\}$,
the result clears. On the other hand, suppose $|q_{i}|>\frac{r}{2}$ for some ${i}\in\{1,\ldots,k\}$.
We may assume $q_{i}<-\frac{r}{2}$, and the other case $q_{i}>-\frac{r}{2}$ can be proved by the similar method.
It is easy to see that $\|q+{j}\epsilon_{i}\|<\|q\|$ for all ${j}\in\{1,\ldots,r\}$. We only need to give the proof
for $\betaeta+q\neq0$. From the action of $L_0$ on $M_{\betaeta+q}$ is a nonzero scalar, we can write
$w=L_0v$, where $v\in M_{\betaeta+q}$.
We will verify that
$$\sum_{{j}=0}^r(-1)^{j}{r\choose {j}}\phi(L_{\alphalpha-q-{j}\epsilon_{i}},L_{{j}\epsilon_{i}}v)= \sum_{{j}=0}^r(-1)^{j}{r\choose {j}}\phi(W_{\alphalpha-q-{j}\epsilon_{i}},L_{{j}\epsilon_{i}}v)=0$$ in $\widehat{M}$.
Based on Definition \mathfrak{g}ammaef{def5777} and Lemma \mathfrak{g}ammaef{lemm676225},
for any $m\in\mathbb{Z}^k$ we deduce
\betaegin{eqnarray*}
&&\sum_{{j}=0}^r(-1)^{j}{r\choose {j}}\phi(L_{\alphalpha-q-{j}\epsilon_{i}},L_{{j}\epsilon_{i}}v)(t^m)
\\&=&\sum_{{j}=0}^r(-1)^{j}{r\choose {j}}L_{\alphalpha+m-q-{j}\epsilon_{i}}L_{{j}\epsilon_{i}}v=\Omega_{\alphalpha+m-q,0}^{(r,\epsilon_{i})}v=0
\end{eqnarray*}
and
\betaegin{eqnarray*}
&&\sum_{{j}=0}^r(-1)^{j}{r\choose {j}}\phi(W_{\alphalpha-q-{j}\epsilon_{i}},L_{{j}\epsilon_{i}}v)(t^m)
\\&=&\sum_{{j}=0}^r(-1)^{j}{r\choose {j}}W_{\alphalpha+m-q-{j}\epsilon_{i}}L_{{j}\epsilon_{i}}v
=\widetilde{\Omega}_{\alphalpha+m-q,0}^{(r,\epsilon_{i})}v=0.
\end{eqnarray*}
Thus, one has
\betaegin{eqnarray}
&&\lambdabel{eq588}\phi(L_{\alphalpha-q},w)=-\sum_{{j}=1}^r(-1)^{j}{r\choose {j}}\phi(L_{\alphalpha-q-{j}\epsilon_{i}},L_{{j}\epsilon_{i}}v),
\\&&\lambdabel{eq599}
\phi(W_{\alphalpha-q},w)=-\sum_{{j}=1}^r(-1)^{j}{r\choose {j}}\phi(W_{\alphalpha-q-{j}\epsilon_{i}},L_{{j}\epsilon_{i}}v).
\end{eqnarray}
By induction assumption the right hand sides of \eqref{eq588} and \eqref{eq599} belong to $\widehat{M}_{\alphalpha+\betaeta}$, and so do
$\phi(L_{\alphalpha-q},w)$, $\phi(W_{\alphalpha-q},w)$. This proves the claim.
Therefore, $\widehat{M}_{\alphalpha+\betaeta}$ is finite-dimensional. The proposition follows.
\end{proof}
The Claim \mathfrak{g}ammaef{clai11} can also be described as follows.
\betaegin{remark}
For $\alphalpha,\betaeta,q\in\mathbb{Z}^k$ and $\betaeta+q\neq0,w\in M_{\betaeta+q}$, we get
\betaegin{eqnarray*}
&&\phi(L_{\alphalpha-q},w)\in \sum_{\|\mathfrak{g}amma\|\leq\frac{kr}{2}}\phi(L_{\alphalpha-\mathfrak{g}amma},M_{\betaeta+\mathfrak{g}amma})+\mathfrak{K}(M),
\\&&
\phi(W_{\alphalpha-q},w)\in\sum_{\|\mathfrak{g}amma\|\leq\frac{kr}{2}}\phi(W_{\alphalpha-\mathfrak{g}amma},M_{\betaeta+\mathfrak{g}amma})+\mathfrak{K}(M).
\end{eqnarray*}
\end{remark}
Now we give a classification for all simple cuspidal $\overline{\mathcal{W}}[\mathbb{Z}^k]$-modules.
\betaegin{theo}\lambdabel{the511}
Any simple cuspidal $\overline{\mathcal{W}}[\mathbb{Z}^k]$-module is isomorphic to a module of intermediate series $\overline{M}(g,h;\mathbb{Z}^k)$ for some $g,h\in\mathbb{C}$.
\end{theo}
\betaegin{proof}
Assume that $M$ is a simple cuspidal $\overline{\mathcal{W}}[\mathbb{Z}^k]$-module. It is clear that $\overline{\mathcal{W}}[\mathbb{Z}^k]M=M$. Then there exist an $\mathcal{A}$-cover $\widehat{M}$ of $M$ with a surjective homomorphism $\Theta:\widehat{M}\mathfrak{g}ammaightarrow M$.
It follows from Proposition \mathfrak{g}ammaef{pro510} that $\widehat{M}$ is a cuspidal $\mathcal{A}\overline{\mathcal{W}}[\mathbb{Z}^k]$-module. Hence, we can consider the composition series
$$0=\widehat{M}_0\subset\widehat{M}_1\subset\cdots\subset\widehat{M}_{\widehat{c}}=\widehat{M}$$
with the quotients $\widehat{M}_{\widehat{i}+1}/\widehat{M}_{\widehat{i}}$ being simple $\mathcal{A}\overline{\mathcal{W}}[\mathbb{Z}^k]$-modules. Let $d$ be the smallest
integer such that $\Theta(\widehat{M}_d)\neq0$. Then by the simplicity of $M$, we obtain
$\Theta(\widehat{M}_d)=M$ and $\Theta(\widehat{M}_{d-1})=0$.
So we get an $\mathcal{A}\overline{\mathcal{W}}[\mathbb{Z}^k]$-epimorphism
$$\overline{\Theta}:\widehat{M}_{d}/\widehat{M}_{d-1}\mathfrak{g}ammaightarrow M.$$
Now from Theorem
\mathfrak{g}ammaef{the566}, $\widehat{M}_{d}/\widehat{M}_{d-1}$ is isomorphic to a module of intermediate series $\overline{M}(g,h;\mathbb{Z}^k)$ for some $g,h\in\mathbb{C}$.
We complete the proof.
\end{proof}
Based on the representation
theory of $\mathcal{V}[\mathbb{Z}^k]$ studied in \cite{S1,S2}, we see that the action of $C$
on any simple cuspidal $\mathcal{W}[\mathbb{Z}^k]$-modules is trivial. Therefore, the category
of simple cuspidal modules over $\mathcal{W}[\mathbb{Z}^k]$ is equivalent to the category of simple cuspidal modules over $\overline{\mathcal{W}}[\mathbb{Z}^k]$. Then Theorem \mathfrak{g}ammaef{the511} can be described as follows.
\betaegin{theo}\lambdabel{the513}
Let $M$ ba a simple cuspidal module over $\mathcal{W}[\mathbb{Z}^k]$. Then $M$ is isomorphic to a simple quotient of intermediate
series module $M(g,h;\mathbb{Z}^k)$ for some $g,h\in\mathbb{C}$.
\end{theo}
\section{Non-cuspidal modules}
In this section, we determine the simple weight modules with finite-dimensional weight spaces
which are not cuspidal for the
higher rank $W$-algebras $W(2,2)$. These modules are called generalized highest weight modules and defined in Section \mathfrak{g}ammaef{www9998}.
The result of high rank Virasoro algebras of Theorem \mathfrak{g}ammaef{411} plays a key role in the following proof.
\betaegin{theo}\lambdabel{the611}
Let $M$ be a simple Harish-Chandra module over $\mathcal{W}[\mathbb{Z}^k]$. Then $M$ is either
a cuspidal module, or isomorphic to some $P_{\mathfrak{g},\mu,\mathcal{K}}^{\mathcal{W}[\mathbb{Z}^k]}$,
where $\mu\in\mathbb{Z}^k\setminus\{0\}$, $\mathfrak{g}$ is a subgroup of $\mathbb{Z}^k$ such that $\mathbb{Z}^k=\mathfrak{g}\oplus\mathbb{Z}\mu$ and $\mathcal{K}$ is a
non-trivial simple cuspidal $\mathcal{W}[\mathbb{Z}^k]_{\mathfrak{g}}$-module.
\end{theo}
\betaegin{proof}
Suppose that $M$ is not a cuspidal module over $\mathcal{W}[\mathbb{Z}^k]$. Let us recall that $M=\betaigoplus_{\alphalpha\in \mathbb{Z}^k}M_{\alphalpha}$ where $M_{\alphalpha}=\{w\in M \mid L_0w=(g+\alphalpha)w\}$. By Theorem \mathfrak{g}ammaef{the44378}, we see that the statement holds for any $\mathbb{Z}^k$ of
$k=1$.
Now suppose $k\mathfrak{g}eq2$. View $M$ as a $\mathcal{V}[\mathbb{Z}^k]$-module. Then based on Theorem \mathfrak{g}ammaef{411},
we obtain that the action of the central element $C$ on $M$ is trivial. Hence $M$ can be seen
as a $\overline{\mathcal{W}}[\mathbb{Z}^k]$-module. We fix a $\mathbb{Z}$-basis of $\mathbb{Z}^k$, which is also suitable for $\mathbb{Z}^k$. For any $\sigma\in \mathbb{R}^k$ and $g \in\mathbb{Z}^k$, we have the inner product $(\sigma|g)$. It follows from $[\overline{\mathcal{W}}[\mathbb{Z}^k]_\alphalpha, \overline{\mathcal{W}}[\mathbb{Z}^k]_\betaeta] = \overline{\mathcal{W}}[\mathbb{Z}^k]_{\alphalpha+\betaeta}$ that Theorem \mathfrak{g}ammaef{the311} can be applied
to $\mathbb{Z}^k$.
Since $M$ is not cuspidal, we can find some rank $k-1$ direct summand
$\widetilde{\mathbb{Z}^k}$ of $\mathbb{Z}^k$ such that $M_{\widetilde{\mathbb{Z}^k}}$ is not cuspidal. Without loss of generality,
we may assume that $\widetilde{\mathbb{Z}^k}$ is spanned by
$\{\epsilon_1, \epsilon_{2},\ldots,\epsilon_{k}\}\setminus\{\epsilon_{j}\}$, where
$j\in\{1,2,\ldots,k\}$. Then there exists some
$\widetilde{\alphalpha}\in \widetilde{\mathbb{Z}^k}$ such that
\betaegin{eqnarray}\lambdabel{ga611}
\mathrm{dim}(M_{-\widetilde{\alphalpha}})> 2k\betaig(\mathrm{dim}(M_{\epsilon_{j}})+
\sum_{{i}=1,{i}\neq j}^{k}\mathrm{dim}(M_{\epsilon_{j}+\epsilon_{{i}}})\betaig).
\end{eqnarray}
For simplicity, we denote $\xi_{j}=\widetilde{\alphalpha}+\epsilon_{j}$ and $\xi_{{i}}=\widetilde{\alphalpha}+\epsilon_{j}+\epsilon_{{i}}$ for any ${{i}}\in\{1,2,\ldots,k\}\setminus \{j\}$.
Then it is easy to check that the linear transformation sending each $\epsilon_{{i}}$ to $\xi_{{i}}$
for any ${{i}}\in\{1,\ldots,k\}$, has determinant $1$ and therefore $\{\xi_{{i}}\mid {i}=1,\ldots,k\}$ is also a $\mathbb{Z}$-basis of $\mathbb{Z}^k$. According to \eqref{ga611}, there exists some
nonzero element $w\in M_{-\widetilde{\alphalpha}}$ such that $L_{\xi_{{i}}}w=W_{\xi_{{i}}}w=0$ for all ${{i}}\in\{1,\ldots,k\}$. Thus,
$w$ is a generalized highest weight vector associated with the $\mathbb{Z}$-basis $\{\xi_{{i}} \mid {{i}}=1,\ldots,k\}$.
It is clear that $M$ is neither dense nor trivial. From Theorem \mathfrak{g}ammaef{the311}, there exist
some $\betaeta\in\mathbb{Z}^k$ and $\tau \in \mathbb{R}^k\setminus \{0\}$ such that $\mathrm{Supp}(M) \subseteq g+\betaeta+ {\mathbb{Z}^k}^{(\tau)}_{\leq0}$. Consider $M$ as a $\mathcal{V}[\mathbb{Z}^k]$-module. Then $M$
has a simple non-trivial $\mathcal{V}[\mathbb{Z}^k]$-subquotient, and we denote it by $\overline{M}^{\mathcal{V}}$, which is not cuspidal. By Theorem \mathfrak{g}ammaef{411}, we know that $\overline{M}^{\mathcal{V}}\cong P^{\mathcal{V}[\mathbb{Z}^k]}_{\mathfrak{g},\mu,\mathcal{K}}$ for some
nonzero $\mu\in\mathbb{Z}^k$, subgroup $\mathfrak{g}$ of $\mathbb{Z}^k$ with $\mathbb{Z}^k=\mathfrak{g}\oplus\mathbb{Z}\mu$ and $\mathcal{K}$ being a simple
intermediate series module over $\mathcal{V}[\mathfrak{g}]$. Write
$\mathbb{Z}^k_\tau=\{\alphalpha \in \mathbb{Z}^k\mid (\tau|\alphalpha)=0\}$. In particular, we have
$$g -\widetilde{c}\mu+\mathfrak{g} \subseteq \mathrm{Supp}(\overline{M}^{\mathcal{V}}) \subseteq \mathrm{Supp}(M) \subseteq g+\betaeta+{\mathbb{Z}^k}^{(\tau)}_{\leq0}$$
for sufficiently large $\widetilde{c}\in \mathbb{Z}_+$. This gives $\mathfrak{g}=\mathbb{Z}^k_\tau$ and $(\tau|\mu)>0$.
We set that $\widetilde{c}_0\in\mathbb{Z}$ is the maximal number such that $\mathcal{K}=M_{g+\widetilde{c}_0\mu+\mathfrak{g}}\neq0$. Hence,
$\mathcal{W}[\mathbb{Z}^k]^+\mathcal{K}=0$. Then it follows from the simplicity of $\mathcal{W}[\mathbb{Z}^k]$-module $M$ that the simple $\mathcal{W}[\mathfrak{g}]$-module $\mathcal{K}$ and $M=P^{\mathcal{W}[\mathbb{Z}^k]}_{\mathfrak{g},\mu,\mathcal{K}}$. At last, note that
$\mathcal{K}$ is non-trivial and cuspidal. This proves the theorem.
\end{proof}
Based on Theorems \mathfrak{g}ammaef{the513} and \mathfrak{g}ammaef{the611}, we give a classification of simple Harish-Chandra modules over the higher rank $W$-algebra $W(2,2)$.
\betaegin{theo}\lambdabel{the622}
Assume that $M$ is a non-trivial simple Harish-Chandra module over $\mathcal{W}[\mathbb{Z}^k]$ for some
$k\in\mathbb{Z}_+$.
\betaegin{itemize}
\item[\mathfrak{g}ammam(1)] If $M$ is cuspidal, then $M$ is isomorphic to some $\overline{M}(g,h;\mathbb{Z}^k)$
for some $g,h\in \mathbb{C}$;
\item[\mathfrak{g}ammam(2)] If $M$ is not cuspidal, then $M$ is isomorphic to
$P^{\mathcal{W}[\mathbb{Z}^k]}_{\mathfrak{g},\mu,\mathcal{K}}$ for some $\mu\in\mathbb{Z}^k\setminus\{0\}$, a subgroup $\mathfrak{g}$ of $\mathbb{Z}^k$ with $\mathbb{Z}^k=\mathfrak{g}\oplus\mathbb{Z}{\mu}$ and a non-trivial simple intermediate series $\mathcal{W}[\mathfrak{g}]$-module $\mathcal{K}$.
\end{itemize}
\end{theo}
Note that Theorem \mathfrak{g}ammaef{the44378} is a special case of Theorem \mathfrak{g}ammaef{the622} for $k=1$.
\end{document} |
\circ} \def\a{\bulletegin{document}
\title{Krawtchouk matrices from classical and quantum random walks}
\author{
Philip Feinsilver and Jerzy Kocik\\[.1in]
\small
Department of Mathematics\\
\small
Southern Illinois University\\
\small
Carbondale, IL 62901\\
\small
pfeinsil@math.siu.edu, jkocik@siu.edu
}
\maketitle
\thispagestyle{empty}
\circ} \def\a{\bulletegin{abstract}
Krawtchouk's polynomials occur classically as orthogonal polynomials
with respect to the binomial distribution.
They may be also expressed in the form of matrices,
that emerge as arrays of the values that the polynomials take.
The algebraic properties of these matrices provide a very interesting
and accessible example in the approach to probability theory
known as {\em quantum probability.}
First it is noted how the Krawtchouk matrices are connected to the
classical symmetric Bernoulli random walk.
And we show how to derive Krawtchouk matrices in the quantum probability
context via tensor powers of the elementary Ha\-da\-mard matrix.
Then connections with the classical situation are shown by calculating
expectation values in the quantum case.
\end{abstract}
\section{Introduction}
\label{sec:intro}
Some very basic algebraic rules can be expressed using matrices.
Take, for example,
$$
\circ} \def\a{\bulletegin{array}{lcl}
(a+b)^2 &=& a^2+2ab+b^2 \\
(a+b)(a-b) &=& a^2\phantom{+2ab\,\,\,} - b^2 \\
(a-b)^2 &=& a^2-2ab+b^2
\end{array}
\qquad\Rightarrow\qquad
K^{(2)} = \left[\circ} \def\a{\bulletegin{array}{rrr}
1 & 1 & 1 \cr
2 & 0 & -2 \cr
1 & -1 & 1 \cr \end{array}\right]
$$
(the expansion coefficients make up the columns of the matrix).
In general, we make the definition:
\circ} \def\a{\bulletegin{definition}\rm
The $N^{\mathrm{th}}$-order Krawtchouk matrix $\FFF N{}$ is an
$(N+1)\times(N+1)$ matrix,
the entries of which are determined by the expansion:
\circ} \def\a{\bulletegin{equation}
\label{eq:genkraw}
(1+v)^{N-j} \; (1-v)^j = \sum_{i=0}^{N} \ v^i K^{(N)}_{ij} \,.
\end{equation}
The left-hand-side
$
G(v)= (1+v)^{N-j}\; (1-v)^j
$
is thus the {\it generating function} for the
row entries of the $j^{\mathrm{th}}$ column of $K^{(N)}$.
Expanding gives an explicit expression:
$$
K^{(N)}_{ij}= \sum_{k} (-1)^k {j \choose k}
{N-j \choose i-k} \,.
$$
\end{definition}
Here are the Krawtchouk matrices of order zero and one:
\circ} \def\a{\bulletegin{equation}
\label{eq:krav2}
K^{(0)}=\left[\circ} \def\a{\bulletegin{array}{rr}{ 1 }\end{array}\right] \qquad
K^{(1)}=\left[
\circ} \def\a{\bulletegin{array}{rr}
1 & 1 \cr
1 & -1 \cr
\end{array}\right] \,.
\end{equation}
More examples can be found in Table 1 of Appendix 1.
In the remaining of the text, matrix indices run from $0$ to $N$.
\\
One may view the columns of Krawtchouk matrices
as \emph{generalized binomial coefficients}.
The rows define Krawtchouk \emph{polynomials}: for a fixed order
$N$, the $i$-th \emph{Krawtchouk polynomial} is the function
$$
K_i(j,N) = K^{(N)}_{ij}
$$
that takes its corresponding values from the $i$-th row.
One can easily show that $K_i(j,N)$ is indeed a
polynomial of degree $i$ in the variable $j$.
\\
Historically, Krawtchouk's polynomials were introduced and studied
by Mikhail Krawtchouk in the late 20's \cite{Kra, Kra2}.
Since then, they have appeared in many areas of mathematics and applications.
As orthogonal polynomials, they occur in the classic work by
Sz\"ego \cite{Sze}.
They have been studied from the point of view of
harmonic analysis and special functions,
e.g., in work of Dunkl \cite{Dun, DR}.
In statistical considerations, they arose in work of
Eagleson \cite{Eag}
and later Vere-Jones \cite{V-J}.
They play various roles in coding theory and combinatorics, for example, in
MacWilliams' theorem on weight enumerators \cite{MS, Lev},
and in association schemes \cite{Del1,Del2,Del3}.
\\
A classical probabilistic interpretation has been given in \cite{FS}.
In the context of the classical symmetric random walk,
it is recognized that Krawtchouk's polynomials
are elementary symmetric functions in variables taking values
$\pm1$. Specifically, if $\xi_i$ are independent Bernoulli random
variables taking values $\pm1$ with probability $\frac12$, then if
$j$ of the $\xi_i$ are equal to $-1$, the $i^{\rm th}$ elementary
symmetric function in the $\xi_i$ is equal to $K^{(N)}_{ij}$.
It turns out that the generating function (\ref{eq:genkraw}) is a
martingale in the parameter $N$. Details are in Section
\ref{sec:classical} below.
\\
As matrices, they appeared in the 1985 work of N. Bose \cite{B}
on digital filtering, in the context of the Cayley
transform on the complex plane.
The symmetric version of the Krawtchouk matrices has been considered
in \cite{FF}.
\\
Despite this wide research, the full potential, meaning and significance
of Krawtchouk polynomials is far from being complete.
In this paper we look at Krawtchouk matrices as \emph{operators} and
propose two new ways in which Krawtchouk matrices arise:
via classical and quantum random walks.
Especially the latter is of current interest.
The starting idea is to represent the second Krawtchouk matrix
(coinciding with the basic Hadamard matrix) as a sum of two operators
$$
\left[\circ} \def\a{\bulletegin{array}{rr}
1 & 1 \cr
1 & -1 \cr
\end{array}\right]
=
\left[\circ} \def\a{\bulletegin{array}{cc}
0 & 1\cr
1 & 0\cr
\end{array}\right]
+
\left[\circ} \def\a{\bulletegin{array}{rr}
1 & 0\cr
0 &-1\cr
\end{array}\right] .
$$
Via the technique of tensor products of underlying spaces we obtain
a relationship between Krawtchouk matrices and Sylvester-Hadamard
matrices.
\\
The reader should consult Parthasarathy's \cite{Par} for
material on quantum probability. It contains the operator theory
needed for the subject as well as showing the connections with
classical probability theory.\\
For information on Hadamard matrices as they appear here, we recommend
Yarlagadda and Hershey's work \cite{YH} which provides
an overview of the subject of Sylvester-Hadamard matrices,
indicating many interesting applications.
For statisticians, they point out that in Yates' factorial analysis,
the Hadamard transform provides a useful nonparametric test for association.
\\
Yet another area of significance of this research
lies in the quantum computing program \cite{Lom, Par}.
Details on this connection will appear in an independent work.
\\
This paper is organized as follows.
In the next section, we review basic properties
of Kraw\-tchouk matrices. The identities presented, although
basic, seem to be new and do not appear in the references cited.
Section \ref{sec:classical} presents the classical
probability interpretation. It may be viewed as a warm-up
leading to the \emph{quantum random walk}
introduced and studied in Section \ref{sec:quantum},
and to the relationship between Krawtchouk matrices
and Sylvester-Hadamard matrices.
The generating function techniques used there are original with the
present authors. In the last subsection,
calculating expectation values in the quantum case shows how the
quantum model is related to the classical random walk.
Appendix~1 has examples of Krawtchouk and
symmetric Krawtchouk matrices so that the reader may see concretely the
subject(s) of our discussion.
Appendices~2 (tensor products) and 3 (symmetric tensor spaces)
are included to aid the reader as well as to clarify the notation.
\\
\section{Basic properties of Krawtchouk matrices}
\label{sec:ident}
\noindent
(1)
The square of a Krawtchouk matrix is proportional to the identity matrix.
$$
(\FFF N{})^2 = 2^N\,I \,.
$$
This remarkable property allows one to define
a Fourier-like {\em Krawtchouk transform} on integer vectors.
\\
\\
(2)
The top row is all 1's. The bottom row has $\pm 1$'s with
alternating signs, starting with $+1$.
The leftmost entries are just binomial coefficients,
$\FFF N{i0} = {N\choose i}\phantom{\circ} \def\a{\bulletiggm|}$.
The rightmost entries are binomial
coefficients with alternating signs,
$\FFF N{iN} = (-1)^i{N\choose i}\phantom{\circ} \def\a{\bulletiggm|}$.
\\
\\
(3)
There is a four-fold symmetry:
$
|\FFF{N}{i\; j}| = |\FFF{N}{N-i\;j}| =
|\FFF{N}{i\;N-j} | = |\FFF{N}{N-i\;N-j}|
$.\\
Krawtchouk matrices generalize Pascal's triangle in the following sense:
Visualize a stack of Krawtchouk matrices, the order $N$ increasing
downwards.
Pascal's triangle is formed by the leftmost columns.
It turns out that Pascal's identity
holds for the other columns as well.
Less obvious is another identity --- call it dual Pascal.
\circ} \def\a{\bulletegin{proposition}
\label{prop:ids}
Set
$ a= \FFF N {i-1\;j},\,
b= \FFF N {i \;j}$,
$ A= \FFF {N+1} {i\;j},\,
B= \FFF {N+1} {i\;j+1}$ . \\
1. {\circ} \def\a{\bulletf (Cross identities)}
The following mutually inverse relations (Pascal and dual Pascal) hold:
\circ} \def\a{\bulletegin{displaymath}
\circ} \def\a{\bulletegin{array}{cc}
a+b=A \\
b-a=B \\
\end{array}
\qquad\hbox{and}\qquad
\circ} \def\a{\bulletegin{array}{cc}
A+B=2b \\
A-B=2a \,.
\end{array}
\end{displaymath}
2. {\circ} \def\a{\bulletf (Square identity)}
In a square of any four adjacent entries in a Krawtchouk matrix,
the entry in the left-bottom corner is the sum of the other three,
i.e.,
$$
\hbox{for } ~~
K =
\left[\circ} \def\a{\bulletegin{array}{cccc}
~ &\vdots&\vdots &\cr
\cdots & a & c &\cdots \cr
\cdots & b & d &\cdots\cr
~ &\vdots&\vdots &\cr
\end{array}\right]
\qquad\hbox{one has}\qquad
b=a+c+d.
$$
\end{proposition}
\circ} \def\a{\bulletegin{proof}
For $a+b$, consider
$$(1+v)^{N+1-j}(1-v)^j=(1+v)\, (1+v)^{N-j}(1-v)^j \,.$$
For $b-a$, consider
$$(1+v)^{N-j}(1-v)^{j+1}= (1-v) \, (1+v)^{N-j}(1-v)^j \,. $$
The inverse relations are immediate.
The square identity follows from the observation
$(a+b)+(c+d) = A+B=2b$, hence $a+c+d=b$.
\end{proof}
The square identity is useful in producing the entries of a Krawtchouk
matrix: fill the top row with 1's, the right-most column with
sign-alternating binomial coefficients. Then, apply the square identity to
reproduce the matrix.
\\
In summary, the identities considered above can be written as follows:
\\
{\circ} \def\a{\bulletf Cross identities:}
$$
\circ} \def\a{\bulletegin{array}{ccccc}
(i)& \FFF N{i-1\;j} + \FFF N{i \;j} = \FFF {N+1}{i\;j} &\quad&
(ii)& \FFF N{i \;j} + \FFF N{i \;j+1}= 2\FFF {N-1}{i\;j} \cr\mathstrut\cr
(iii)& \FFF N{i \;j} - \FFF N{i-1\;j} = \FFF {N+1}{i\;j+1} &\quad&
(iv)& \FFF N{i \;j} - \FFF N{i \;j+1}= 2\FFF {N-1}{i-1\;j}\,.\cr
\end{array}
$$
\\
{\circ} \def\a{\bulletf Square identity:}
$$
\FFF N {ij} = \FFF N{i-1\;j} + \FFF N{i-1\; j+1}+ \FFF N{i\;j+1} \,.
$$
\\
If each column of the matrix is multiplied by the corresponding binomial
coefficient, the matrix becomes symmetric.
Let $B^{(N)}$ denote the $(N+1)\times (N+1)$ diagonal matrix
with binomial coefficients
\circ} \def\a{\bulletegin{equation}
\label{eq:diagB}
B^{(N)}_{ii}={N\choose i}
\end{equation}
as its non-zero entries.
Then, for each $N\ge0$,
one defines the {\circ} \def\a{\bulletf symmetric Krawtchouk matrix} as
$$
S^{(N)}=\FFF N{}B^{(N)} \,.
$$
{\circ} \def\a{\bulletf Example:} For $N=3$, we have
$$
S^{(3)} =
\left[\circ} \def\a{\bulletegin{array}{rrrr}
1 & 1 & 1 & 1 \cr
3 & 1 & -1 & -3 \cr
3 & -1 & -1 & 3 \cr
1 & -1 & 1 & -1 \cr \end{array}\right]
\left[\circ} \def\a{\bulletegin{array}{rrrr}
1 & 0 & 0 & 0 \cr
0 & 3 & 0 & 0 \cr
0 & 0 & 3 & 0 \cr
0 & 0 & 0 & 1 \cr \end{array}\right]
=
\left[\circ} \def\a{\bulletegin{array}{rrrr}
1 & 3 & 3 & 1 \cr
3 & 3 & -3 & -3 \cr
3 & -3 & -3 & 3 \cr
1 & -3 & 3 & -1 \cr \end{array}\right] \,.
$$
Some symmetric Krawtchouk matrices are displayed in Table 2 of
Appendix 1.
\\
\section{Krawtchouk matrices and classical random walk}
\label{sec:classical}
In this section we will give a probabilistic meaning to
the Krawtchouk matrices and some of their properties.
\\
Let $\xi_i$ be independent symmetric Bernoulli random variables taking
values $\pm1$. Let $X_N=\xi_1+\cdots+\xi_N$ be the associated random
walk starting from $0$. Now observe that the generating function of
the elementary symmetric functions in the $\xi_i$ is a martingale, in
fact a discrete exponential martingale:
$$
M_N = \prod_{i=1}^N(1+v\xi_i)=\sum_k v^k
\alpha_k(\xi_1,\ldots,\xi_N) \,,
$$
where $\alpha_k$ denotes the $k^{\mathrm{th}}$ elementary symmetric
function.
The martingale property is immediate since each $\xi_i$ has mean $0$.
Suppose that at time $N$, the number of the $\xi_i$ that are equal to $-1$
is $j_N$, with the rest equal to $+1$. Then $j_N= (N-X_N)/2$
and $M_N$ can be expressed solely in terms of $N$ and $X_N$, or,
equivalently, of $N$ and $j_N$
$$
M_N = (1+v)^{N-j_N}(1-v)^{j_N}
= (1+v)^{(N+X_N)/2}(1-v)^{(N-X_N)/2} \,.
$$
From the generating function for the Krawtchouk
matrices, (\ref{eq:genkraw}), follows
$$
M_N = \sum_iv^iK^{(N)}_{i,j_N} \,,
$$
so that as functions on the Bernoulli space, each sequence of random
variables $K^{(N)}_{i,j_N}$ is a martingale.
\\
Now we can interpret two basic recurrences of Proposition \ref{prop:ids}.
For a fixed column of $K^{(N)}$, the corresponding column in
$K^{(N+1)}$ satisfies the Pascal triangle recurrence:
$$
\FFF N{i-1\;j} + \FFF N{i \;j} = \FFF {N+1}{i\;j} \,.
$$
To see this in the probabilistic setting, write
$M_{N+1}=(1+v\xi_N)M_N$. Observe that for $j_N$ to remain constant,
$\xi_N$ must take the value $+1$ and expanding
$(1+v)M_N$ yields the Pascal recurrence as in the proof of Proposition
\ref{prop:ids}.
It is interesting how the martingale property comes into play.
We have
$$
\FFF N{ij_N}= E(\FFF {N+1}{ij_{N+1}}|\xi_1,\ldots,\xi_N)
=\frac12\,\left(\FFF{N+1}{i\,j_N+1}+\FFF{N+1}{ij_N}\right)\,,
$$
since half the time $\xi_{N+1}$ is $-1$, increasing $j_N$ by 1, and
half the time $j_N$ is unchanged. Thus, writing $j$ for $j_N$,
$$
\FFF N{ij}=\frac12\,\left(\FFF{N+1}{i\,j+1}+\FFF{N+1}{ij}\right) \,.
$$
Many further properties of Krawtchouk polynomials
may be derived from their interpretation as elementary symmetric
functions on the Bernoulli space with scope for probabilistic methods
as well.
\\
\section{Krawtchouk matrices and quantum random walk}
\label{sec:quantum}
In quantum probability, random variables are modeled by self-adjoint
operators on Hilbert spaces and independence by tensor products.
We can model a symmetric Bernoulli random walk as follows.
Consider a 2-dimensional Hilbert space $V={\circ} \def\a{\bulletf R}^2$
and two special $2\times2$ operators,
$$
F=\left[\circ} \def\a{\bulletegin{array}{cc}0&1\cr 1&0\cr\end{array}\right]\qquad\hbox{and}\qquad
G=\left[\circ} \def\a{\bulletegin{array}{rr}1&0\cr 0&-1\cr\end{array}\right]\,,
$$
satisfying $F^2=G^2=I$ (the $2\times2$ identity).
The fundamental Ha\-da\-mard matrix $H$ coincides with the second
Kraw\-tchouk matrix. Now we shall view it as a sum of the above operators
$$
H=F+G = \left[\circ} \def\a{\bulletegin{array}{rr} 1 & 1 \cr
1 & -1 \cr\end{array}\right] \,.
$$
One can readily check that
\circ} \def\a{\bulletegin{equation}
\label{eq:H}
FH =F(F+G) =(F+G)G =HG
\end{equation}
(use $F^2=G^2=I$). This, of course, can be viewed as
the spectral decomposition of $F$ and we can interpret the Hadamard
matrix as a matrix reducing $F$ to diagonal form.
\circ} \def\a{\bulletegin{remark}\rm \label{rem:exval}
Note that the exponentiated operator
$$
\exp(zF) = \left[\circ} \def\a{\bulletegin{array}{cc}\cosh z&\sinh z\cr \sinh z&\cosh
z\cr\end{array}\right]
$$
has the expectation value in the state $e_0$ equal to
\circ} \def\a{\bulletegin{equation}\label{eq:exval}
\langle e_0,\exp(zF)e_0\rangle = \cosh z\,,
\end{equation}
where $e_0$ denotes the transpose of $[1,0]$.
This coincides with the moment generating function for the symmetric
Ber\-noul\-li random variable taking values $\pm 1$,
showing that indeed we are dealing with the (quantum)
generalization of the classical model.
\end{remark}
The Hilbert space of states is represented
by the $N$-th tensor product of the original space $V$,
that is, by the $2^N$-dimensional Hilbert space $V^{\otimes N}$ \
(see Appendix~2 for notation).
Define the following linear operators, $N$ in all, in $V^{\otimes N}$
\circ} \def\a{\bulletegin{eqnarray*}
f_1 &=& F\otimes I \otimes\cdots\otimes I \\
f_2 &=& I\otimes F\otimes I \otimes\cdots\otimes I \\
\vdots &=& \vdots \\
f_N &=& I\otimes I \otimes\cdots\otimes F \,,
\end{eqnarray*}
each $f_i$ describing a flip at the $i$-th position.
These are the quantum equivalents of the random walk variables
from Section \ref{sec:classical}.
We shall consider the superposition of these independent
actions, setting
$$
X_F=f_1+\cdots+f_N \,.
$$
{\circ} \def\a{\bulletf Notation:}
For notational clarity,
since $N$ is fixed throughout the discussion, we drop the index $N$ from
the $X$'s.
\\
Analogously, we define:
\circ} \def\a{\bulletegin{eqnarray*}
g_1 &=& G\otimes I \otimes\cdots\otimes I\\
g_2 &=& I\otimes G\otimes I \otimes\cdots\otimes I\\
\vdots &=& \vdots \\
g_N &=& I\otimes I \otimes\cdots\otimes G \,,
\end{eqnarray*}
with $X_G=g_1+\cdots+g_N$.
Finally, let us extend $H$ to the $N$-fold tensor product,
setting $H_N=H^{\otimes N}$. These are the well-known Sylvester-Hadamard
matrices with the first few given here:
\def\circ} \def\a{\bullet{\circ} \def\a{\circ} \def\a{\bulletullet}
$$
H_{1}=\left[\circ} \def\a{\bulletegin{array}{cc}
\a &\a \cr
\a &\circ} \def\a{\bullet \cr
\end{array}\right]
\quad
H_{2}=\left[\circ} \def\a{\bulletegin{array}{cccc}
\a &\a &\a &\a \cr
\a &\circ} \def\a{\bullet &\a &\circ} \def\a{\bullet \cr
\a &\a &\circ} \def\a{\bullet &\circ} \def\a{\bullet \cr
\a &\circ} \def\a{\bullet &\circ} \def\a{\bullet &\a \cr
\end{array}\right]
\quad
H_{3}=\left[\circ} \def\a{\bulletegin{array}{cccccccc}
\a &\a &\a &\a &\a &\a &\a &\a \cr
\a &\circ} \def\a{\bullet &\a &\circ} \def\a{\bullet &\a &\circ} \def\a{\bullet &\a &\circ} \def\a{\bullet \cr
\a &\a &\circ} \def\a{\bullet &\circ} \def\a{\bullet &\a &\a &\circ} \def\a{\bullet &\circ} \def\a{\bullet \cr
\a &\circ} \def\a{\bullet &\circ} \def\a{\bullet &\a &\a &\circ} \def\a{\bullet &\circ} \def\a{\bullet &\a \cr
\a &\a &\a &\a &\circ} \def\a{\bullet &\circ} \def\a{\bullet &\circ} \def\a{\bullet &\circ} \def\a{\bullet \cr
\a &\circ} \def\a{\bullet &\a &\circ} \def\a{\bullet &\circ} \def\a{\bullet &\a &\circ} \def\a{\bullet &\a \cr
\a &\a &\circ} \def\a{\bullet &\circ} \def\a{\bullet &\circ} \def\a{\bullet &\circ} \def\a{\bullet &\a &\a \cr
\a &\circ} \def\a{\bullet &\circ} \def\a{\bullet &\a &\circ} \def\a{\bullet &\a &\a &\circ} \def\a{\bullet \cr
\end{array}\right] \,,
$$
etc.,
where, for typographical reasons, we use
$\circ} \def\a{\bulletullet$ for $1$ and $\circ$ for $-1$.
\\
It turns out that our $X$-operators intertwine
the Sylvester-Hadamard matrices.
For illustration, consider a calculation for $N=3$:
\circ} \def\a{\bulletegin{eqnarray*}
f_1H_3 &=& (F\otimes I\otimes I)(H\otimes H\otimes H)\\
&=& (H\otimes H\otimes H)(G\otimes I\otimes I)
= H_3g_1 \,,
\end{eqnarray*}
where the relation $FH=HG$ is used.
This clearly generalizes to $f_kH_N=H_Ng_k$ and,
by summing over $k$, yields an important relation:
$$
X_F H_N = H_N X_G \,.
$$
Now, we shall consider the symmetrized versions of the operators
(the reader is referred to Appendix 3 for the theory and methods used here).
Since products are preserved in the process of reduction to
the symmetric tensor space, we get
$$
\overline{X}_F\overline{H}_N = \overline{H}_N \overline{X}_G \,,
$$
the bars indicating the corresponding induced maps.
We know how to calculate $\overline{H}_N$ from the action of $H$ on
polynomials in degree $N$. For symmetric tensors the components in
degree $N$ are
$$
x_0^{N-k}x_1^k \,,
$$
where $0\le k\le N$.
\circ} \def\a{\bulletegin{proposition}
For each $N>0$, symmetric reduction of the $N^{\rm th}$ Hadamard
matrix results in the transposed $N^{\rm th}$ Krawtchouk matrix:
$$
(\overline{H}_N)_{ij} = \FFF{N}{ji} \,.
$$
\end{proposition}
\circ} \def\a{\bulletegin{proof}
Writing $(x,y)$ for $(x_0,x_1)$, we have in degree $N$ for the
$k^{\mathrm{th}}$ component:
$$
(x+y)^{N-k}(x-y)^k=\sum_l \overline{H}_{kl} \,\dir l. \,.
$$
Scaling out $x^N$ and replacing $v=y/x$ yields
the generating function for the Krawtchouk matrices with the
coefficient of $v^l$ equal to $\FFF{N}{lk}$. Thus the result.
\end{proof}
Now consider the generating function for the elementary symmetric
functions in the quantum variables $f_j$. This is the $N$-fold
tensor power
$$
{\mathcal F}_N (t) = (I+tF)^{\otimes N}
= I^{\otimes N}+t\,X_F+\cdots \,,
$$
noting that the coefficient of $t$ is $X_F$.
Similarly, define
$$
{\mathcal G}_N(t) = (I+tG)^{\otimes N}
= I^{\otimes N}+tX_G+\cdots \,.
$$
From $(I+tF)H=H(I+tG)$ we have
$$
{\mathcal F}_NH_N = H_N{\mathcal G}_N
\qquad\hbox{and}\qquad
\overline{\mathcal F}_N \overline{H}_N
= \overline{H}_N \overline{\mathcal G}_N \,.
$$
The difficulty is to calculate the action on the symmetric tensors for
operators, such as $X_F$, that are not pure tensor powers. However,
from ${\mathcal F}_N(t)$ and ${\mathcal G}_N(t)$ we can
recover $X_F$ and $X_G$ via
$$
X_F=\frac{d}{dt}\circ} \def\a{\bulletiggm|_{t=0}(I+tF)^{\otimes N},
\qquad
X_G=\frac{d}{dt}\circ} \def\a{\bulletiggm|_{t=0}(I+tG)^{\otimes N}
$$
with corresponding relations for the barred operators.
Calculating on polynomials yields the desired results as follows.
$$
I+tF = \left[\circ} \def\a{\bulletegin{array}{cc}1&t\cr t&1\cr\end{array}\right],\qquad
I+tG = \left[\circ} \def\a{\bulletegin{array}{cc}1+t&0\cr 0&1-t\cr\end{array}\right] \,.
$$
In degree $N$, using $x$ and $y$ as variables, we get the
$k^{\mathrm{th}}$ component for $\overline{X}_F$ and $\overline{X}_G$
via
\circ} \def\a{\bulletegin{eqnarray*}
\frac{d}{dt}\circ} \def\a{\bulletiggm|_{t=0}(x+ty)^{N-k}(tx+y)^k
&=& (N-k)\,\dirs k+1. +k\,\dirs k-1. \,,
\end{eqnarray*}
and since $I+tG$ is diagonal,
\circ} \def\a{\bulletegin{eqnarray*}
\frac{d}{dt}\circ} \def\a{\bulletiggm|_{t=0}(1+t)^{N-k}(1-t)^k\,\dir k.
&=& (N-2k)\,\dir k. \,.
\end{eqnarray*}
For example, calculations for $N=4$ result in
\circ} \def\a{\bulletegin{eqnarray*}
\overline{X}_F = \left[
\circ} \def\a{\bulletegin{array}{ccccc}0&4&0&0&0\cr
1&0&3&0&0\cr
0&2&0&2&0\cr
0&0&3&0&1\cr
0&0&0&4&0\cr\end{array}\right],&\qquad&
\overline{H}_4 = \left[\circ} \def\a{\bulletegin{array}{rrrrr}
1&4&6&4&1\cr
1&2&0&-2&-1\cr
1&0&-2&0&1\cr
1&-2&0&2&-1\cr
1&-4&6&-4&1\cr \end{array} \right]\,,\\
\mathstrut\\
\overline{X}_G = \left[\circ} \def\a{\bulletegin{array}{rrrrr}4&0&0&0&0\cr
0&2&0&0&0\cr
0&0&0&0&0\cr
0&0&0&-2&0\cr
0&0&0&0&-4\cr\end{array}\right] \,.&&
\end{eqnarray*}
Since $\overline{X}_G$ is the result of diagonalizing
$\overline{X}_F$, we observe that
\circ} \def\a{\bulletegin{corollary}
The spectrum of $\overline{X}_F$ is
$N,N-2,\ldots,2-N,-N$, coinciding with the support
of the classical random walk.
\end{corollary}
\circ} \def\a{\bulletigskip
\subsection{Expectation values}
To find the probability distributions associated to our $X_F$ operators,
we must calculate expectation values, cf. Remark \ref{rem:exval}. In the present context,
expectation values in two particular states are especially interesting.
Namely, in the state $e_0$ and in the normalized trace, which is the uniform
distribution on the spectrum. In the $N$-fold tensor product,
we want to consider expectation values in the ground state
$|\,000\ldots0\,\rangle$ and normalized traces.
Then we can go to the symmetric tensors.
\\
The scalar product on the tensor product space factors,
corresponding to independence in classical probability. Thus,
from (\ref{eq:exval}) one obtains the expectation value of $\exp(zX_F)$
in the ground state $|\,000\ldots0\,\rangle$ to be $(\cosh z)^N$.
Similarly, the trace of the tensor product of operators is the product
of their traces. So, for the trace, $\mathrm{tr}\, \exp(zF)=2\cosh z$ implies
$\mathrm{tr}\,\exp(zX_F)=2^N(\cosh z)^N$ and, after normalizing, this yields
$(\cosh z)^N$.
\\
For the barred operators, we consider the symmetric trace. Here we use
the {\circ} \def\a{\bulletf symmetric trace theorem}, detailed in Appendix 3. It tells us
that the generating function for the symmetric traces of any operator
$A$ in the various degrees is \\ $\det(I-tA)^{-1}$. Taking
$A=\exp(zF)$, we have
\circ} \def\a{\bulletegin{eqnarray*}
\det(I-te^{zF})^{-1} &=& [(1-te^z)(1-te^{-z})]^{-1}\\
&=& (1-2t\cosh z+t^2)^{-1} \,.
\end{eqnarray*}
The latter is the generating function for Chebyshev polynomials of the
second kind, $U_N$, so that the normalized symmetric trace is
$$
(N+1)^{-1}\mathrm{tr}\,_\mathrm{Sym}^N \exp(zF) = U_N(\cosh z)/(N+1) \,,
$$
which equals as well
$$
\frac{e^{z(N+1)} - e^{-z(N+1)}}{(e^z-e^{-z})(N+1)}
= \frac{\sinh (N+1)z}{(N+1)\,\sinh z} \,.
$$
This corresponds to a uniform distribution on the support of the
random walk at time $N$, namely, $-N,2-N,\ldots,N-2,N$.
\vskip.3in
{\circ} \def\a{\bulletf Acknowledgment.} We would like to thank Marlos Viana for inviting us
to participate in the special session and we extend our appreciation
for all the hard work involved in organizing the session as well as
related activities.
\\
\section*{Appendix 1: Krawtchouk matrices}
\circ} \def\a{\bulletigskip
\hrule
\circ} \def\a{\bulletigskip
\circ} \def\a{\bulletegin{eqnarray*}
K^{(0)}&=&\left[\circ} \def\a{\bulletegin{array}{r} 1 \end{array}\right]
\\
\\
K^{(1)}&=&\left[\circ} \def\a{\bulletegin{array}{rr} 1 & 1 \cr
1 & -1 \cr\end{array}\right]
\\
\\
K^{(2)}&=& \left[\circ} \def\a{\bulletegin{array}{rrr} 1 & 1 & 1 \cr
2 & 0 & -2 \cr
1 & -1 & 1 \cr \end{array}\right]
\\
\\
K^{(3)}&=&
\left[\circ} \def\a{\bulletegin{array}{rrrr}
1 & 1 & 1 & 1 \cr
3 & 1 & -1 & -3 \cr
3 & -1 & -1 & 3 \cr
1 & -1 & 1 & -1 \cr
\end{array}\right]
\\
\\
K^{(4)}&=&
\left[\circ} \def\a{\bulletegin{array}{rrrrr} 1 & 1 & 1 & 1 & 1 \cr
4 & 2 & 0 & -2 & -4 \cr
6 & 0 & -2 & 0 & 6 \cr
4 & -2 & 0 & 2 & -4 \cr
1 & -1 & 1 & -1 & 1 \cr \end{array}\right]
\\
\\
K^{(5)}&=&
\left[\circ} \def\a{\bulletegin{array}{rrrrrr} 1 & 1 & 1 & 1 & 1 & 1 \cr
5 & 3 & 1 & -1 & -3 & -5 \cr
10 & 2 & -2 & -2 & 2 & 10 \cr
10 & -2 & -2 & 2 & 2 & -10 \cr
5 & -3 & 1 & 1 & -3 & 5 \cr
1 & -1 & 1 & -1 & 1 & -1 \cr \end{array}\right]
\\
\\
K^{(6)}&=&
\left[\circ} \def\a{\bulletegin{array}{rrrrrrr}
1 & 1 & 1 & 1 & 1 & 1 & 1 \cr
6 & 4 & 2 & 0 & -2 & -4 & -6 \cr
15 & 5 & -1 & -3 & -1 & 5 & 15 \cr
20 & 0 & -4 & 0 & 4 & 0 & -20 \cr
15 & -5 & -1 & 3 & -1 & -5 & 15 \cr
6 & -4 & 2 & 0 & -2 & 4 & -6 \cr
1 & -1 & 1 & -1 & 1 & -1 & 1 \cr
\end{array}\right]
\end{eqnarray*}
\circ} \def\a{\bulletigskip
\hrule
\circ} \def\a{\bulletigskip
\centerline{{\circ} \def\a{\bulletf Table 1:} Krawtchouk matrices}
~
\hrule
\circ} \def\a{\bulletigskip
\circ} \def\a{\bulletegin{eqnarray*}
S^{(0)}&=&\left[\circ} \def\a{\bulletegin{array}{r} 1 \end{array}\right]
\\
\\
S^{(1)}&=&\left[\circ} \def\a{\bulletegin{array}{rr} 1 & 1 \cr
1 & -1 \cr\end{array}\right]
\\
\\
S^{(2)}&=& \left[\circ} \def\a{\bulletegin{array}{rrr} 1 & 2 & 1 \cr
2 & 0 & -2 \cr
1 & -2 & 1 \cr \end{array}\right]
\\
\\
S^{(3)}&=&
\left[\circ} \def\a{\bulletegin{array}{rrrr} 1 & 3 & 3 & 1 \cr
3 & 3 & -3 & -3 \cr
3 & -3 & -3 & 3 \cr
1 & -3 & 3 & -1 \cr \end{array}\right]
\\
\\
S^{(4)}&=&
\left[\circ} \def\a{\bulletegin{array}{rrrrr} 1 & 4 & 6 & 4 & 1 \cr
4 & 8 & 0 & -8 & -4 \cr
6 & 0 &-12 & 0 & 6 \cr
4 & -8 & 0 & 8 & -4 \cr
1 & -4 & 6 & -4 & 1 \cr \end{array}\right]
\\
\\
S^{(5)}&=&
\left[\circ} \def\a{\bulletegin{array}{rrrrrr} 1 & 5 & 10 & 10 & 5 & 1 \cr
5 & 15 & 10 & -10 & -15 & -5 \cr
10 & 10 &-20 & -20 & 10 & 10 \cr
10 &-10 &-20 & 20 & 10 & -10 \cr
5 &-15 & 10 & 10 & -15 & 5 \cr
1 & -5 & 10 & -10 & 5 & -1 \cr \end{array}\right]
\\
\\
S^{(6)}&=&
\left[\circ} \def\a{\bulletegin{array}{rrrrrrr} 1 & 6 & 15 & 20 & 15 & 6 & 1
\cr
6 & 24 & 30 & 0 & -30 & -24 & -6 \cr
15 & 30 & -15 & -60 & -15 & 30 & 15 \cr
20 & 0 & -60 & 0 & 60 & 0 & -20 \cr
15 & -30 & -15 & 60 & -15 & -30 & 15 \cr
6 & -24 & 30 & 0 & -30 & 24 & -6 \cr
1 & -6 & 15 & -20 & 15 & -6 & 1 \cr
\end{array}\right]
\end{eqnarray*}
\circ} \def\a{\bulletigskip
\hrule
\circ} \def\a{\bulletigskip
\centerline{{\circ} \def\a{\bulletf Table 2:} Symmetric Krawtchouk matrices}
\section*{Appendix 2: Tensor products}
Fulton and Harris \cite{FH} is a useful reference for
this section and the next. Also Parthasarathy \cite{Par}, Chapter II, is an
excellent reference.
\\
Let $V$ be a $d$-dimensional vector space over ${\circ} \def\a{\bulletf R}$.
We fix an orthonormal basis $\{e_0,\ldots,e_\delta\}$
with $d=1+\delta$. Denote tensor powers of $V$ by
$V^{\otimes N}$, so that $V^{\otimes 2}=V\otimes V$, etc.
A basis for $V^{\otimes N}$ is given by all $N$-fold tensor products
of the basis vectors $e_i\,$,
$$
|\,n_1n_2\ldots n_N\,\rangle =
e_{n_1}\otimes e_{n_2}\otimes \cdots\otimes e_{n_N} \,.
$$
Note that we can label these $d^N$ basis elements by all
numbers $0$ to $d^N-1$ and recover the tensor products
by expressing these numbers in base $d$,
putting leading zeros so that all extended labels are of length $N$.
\\
Now let $\{A_i:\; 1\le i\le N\}$ be a set of $N$ linear operators on $V$.
On $V^{\otimes N}$, the linear operator
$A=A_1\otimes A_2\otimes\cdots\otimes A_N$ acts on a basis vector
$|\,n_1n_2\ldots n_N\,\rangle$ by
$$
A |\,n_1n_2\ldots n_N\,\rangle =
A_1e_{n_1}\otimes\cdots\otimes A_Ne_{n_N} \,.
$$
This needs to be expanded and terms regrouped using bilinearity.
\\
If $A$ and $B$ are two $d\times d$ matrices, the matrix corresponding
to the operator $A\otimes B$ is the Kronecker product, a
$d^2\times d^2$ matrix having the block form:
$$\left[\circ} \def\a{\bulletegin{array}{ccc}
a_{00}B&\ldots&a_{0\delta}B\cr
a_{10}B&\ldots&a_{1\delta}B\cr
\vdots&\vdots&\vdots\cr
a_{\delta0}B&\ldots&a_{\delta\delta}B\cr\end{array}\right] \,.
$$
This, iteratively, is valid for higher-order tensor
products (associating from the left by convention).
The rows and columns of the matrix of a linear operator acting on
$V^{\otimes N}$ are conveniently labeled by associating to each
basic tensor
$|\,n_1n_2\ldots n_N\,\rangle$ the corresponding integer label
$\sum\limits_{k=1}^N n_kd^{N-k}$, which thus provides a canonical
ordering.
\\
\section*{Appendix 3: Symmetric tensor spaces}
Here we review symmetric tensor spaces
as spaces of polynomials in commuting variables.
This material is presented with a view to the infinite-dimensional
case in \cite{Par}, pp. 105ff.,
however we focus on the finite-dimensional context and include as well
an important observation contained in Theorem \ref{thm:symmtr}.
\\
The space $V^{\otimes N}$ can be mapped onto the space of symmetric
tensors, $V^{\otimes_S N}$ by identifying basis vectors
(in $V^{\otimes N}$) that are equivalent under all permutations.
Alternatively, one can identify the basic tensor
$|\,n_1n_2\ldots n_N\,\rangle$ with the monomial $x_{n_1}x_{n_2}\cdots
x_{n_N}$
in the commuting variables $x_0,\ldots,x_\delta$.
Hence we have a linear map
from tensor space into the space of polynomials, itself
isomorphic to the space of symmetric tensors:
$$
\overline{\phantom{W}}:\quad \circ} \def\a{\bulletigcup_{N\ge0}V^{\otimes N}
\ \longrightarrow \
{\circ} \def\a{\bulletf R}[x_0,\ldots,x_\delta]
\cong \circ} \def\a{\bulletigcup_{N\ge0} V^{\otimes_S N} \,.
$$
In the symmetric tensor space, tensor labels
need to count only occupancy, that is, the number of times a
basis vector of $V$ occurs in a given basic tensor of $V^{\otimes N}$.
We indicate occupancy by a multi-index which is the exponent of the
corresponding monomial. The dimension of $V^{\otimes_S N}$ is thus
$$
\hbox{dim}\; V^{\otimes_S N} = {N+d-1\choose d-1} \,,
$$
that is, the number of monomials homogeneous of degree $N$.
\\
Given an operator $A$ on $V$, let $A_N=A^{\otimes N}$. Then $A_N$
induces an operator $\overline{A}_N$ on $V^{\otimes_S N}$ from the action of
$A$ on polynomials, which we call the {\circ} \def\a{\bulletf symmetric representation
of $A$ in degree $N$}. For convenience we work dually with the tensor
components rather with the action on the basis vectors.
Denote the matrix elements of the action of
$\overline{A}_N$ by $\overline{A}_{mn}$. If $A$ has matrix entries
$A_{ij}$, let
$$
y_i=\sum_j A_{ij}x_j \,.
$$
Then the matrix elements of the symmetric representation
are defined by the relation (expansion):
$$
y_0^{m_0}\cdots y_\delta^{m_\delta}=
\sum_n \overline{A}_{mn}
x_0^{n_0}\cdots x_\delta^{n_\delta}
$$
with multi-indices $m$ and $n$.
\\
Composition of $A_1$ with $A_2$ shows that mapping
to the symmetric representation is an algebra homomorphism, i.e.,
$$
\overline{A_1A_2}=\overline{A}_1\overline{A}_2 \,.
$$
Explicitly, in basis notation
$$
\overline{(A_1A_2)}_{mn}=
\sum_r (\overline{A_1})_{mr}
(\overline{A_2})_{rn} \,.
$$
Define the {\circ} \def\a{\bulletf symmetric trace} in degree $N$ of $A$ as the trace of
the matrix elements of $\overline{A}_N$, i.e., the sum of the diagonal
matrix elements:
$$
\mathrm{tr}\,_{\rm Sym}^N A=\sum_{|m|=N}\overline{A}_{mm}
$$
with $|m|$ denoting, as usual, the sum of the components of $m$.
Observe that if $A$ is upper-triangular,
with eigenvalues $\lambda_1,\ldots,\lambda_d$, then the trace of
this action on the space of polynomials homogeneous of degree $N$ is
exactly $h_N(\lambda_1,\ldots,\lambda_d)$, the $N^{\mathrm{th}}$
homogeneous symmetric function in the $\lambda$'s.
\\
We recall a useful theorem on calculating the symmetric trace.
Since the mapping from $A$ to $\overline{A}_N$ is a homomorphism, a
similarity transformation on $A$ extends to one on $\overline{A}_N$ thus
preserving traces. Now, any matrix is similar to an upper-triangular
one with the same eigenvalues, thus follows \cite{Spr}:
\circ} \def\a{\bulletegin{thm}{\circ} \def\a{\bulletf Symmetric trace theorem}
\label{thm:symmtr}
Denoting by $\mathrm{tr}_{\mathrm{Sym}}^N$ the trace of the symmetric
representation on polynomials homogeneous of degree $N$,
$$
\frac{1}{\det(I-tA)}=\sum_{N=0}^\infty t^N\mathrm{tr}\,_{\mathrm{Sym}}^N A \,.
$$
\end{thm}
\circ} \def\a{\bulletegin{proof}
With $\{\lambda_i\}$ denoting the eigenvalues of $A$,
\circ} \def\a{\bulletegin{eqnarray*}
\frac{1}{\det(I-tA)}
&=&\prod_i\frac{1}{1-t\lambda_i}
= \sum_{N=0}^\infty t^Nh_N(\lambda_1,\ldots,\lambda_d)\cr
&=&\sum_{N=0}^\infty t^N\mathrm{tr}\,_{\mathrm{Sym}}^N A \,,\cr
\end{eqnarray*}
as stated above.
\end{proof}
\circ} \def\a{\bulletegin{remark}\rm Note that this result is equivalent to
{\sl MacMahon's Master Theorem} in combinatorics \cite{Mac}.
\end{remark}
\circ} \def\a{\bulletegin{remark}\rm
From another point of view, Chen and Louck \cite{CL}, considering
powers of the bilinear form $\sum_{i,j} x_iA_{ij}y_j$ rather than
just the linear form as done here, study representation functions that
are analogs of our symmetric Krawtchouk matrices.
Their $L_{\alpha,\circ} \def\a{\bulleteta}$ is our symmetric representation
scaled by multinomial factors.
In addition they suggest further interesting generalizations beyond
symmetric tensors.
\end{remark}
\circ} \def\a{\bulletegin{thebibliography}{10}
\circ} \def\a{\bulletibitem{B}
N. Bose,
\emph{Digital filters: theory and applications},
North-Holland, 1985.
\circ} \def\a{\bulletibitem{CL}
W.Y.C. Chen and J.D. Louck,
\emph{The combinatorics of a class of representation functions,}
Adv. in Math., {\circ} \def\a{\bulletf 140} (1998), 207--236.
\circ} \def\a{\bulletibitem{Del1}
P. Delsarte,
\emph{Bounds for restricted codes, by linear programming,}
Philips Res. Reports, {\circ} \def\a{\bulletf 27} (1972) 272--289.
\circ} \def\a{\bulletibitem{Del2}
P. Delsarte,
\emph{Four fundamental parameters of a code and their combinatorial
significance,}
{Info. \&\ Control}, {\circ} \def\a{\bulletf 23} (1973) 407--438.
\circ} \def\a{\bulletibitem{Del3}
P. Delsarte,
\emph{An algebraic approach to the association schemes of coding theory,}
{Philips Research Reports Supplements}, No. 10, 1973.
\circ} \def\a{\bulletibitem{Dun}
C.F. Dunkl,
\emph{A Krawtchouk polynomial addition theorem and wreath products of
symmetric
groups,}
{Indiana Univ. Math. J.}, {\circ} \def\a{\bulletf 25} (1976) 335--358.
\circ} \def\a{\bulletibitem{DR}
C.F. Dunkl and D.F. Ramirez,
\emph{Krawtchouk polynomials and the symmetrization of hypergraphs,}
{SIAM J. Math. Anal.}, {\circ} \def\a{\bulletf 5} (1974) 351--366.
\circ} \def\a{\bulletibitem{Eag}
G.K. Eagelson,
\emph{A characterization theorem for positive definite sequences of the
Krawtchouk polynomials,}
{Australian J. Stat}, {\circ} \def\a{\bulletf 11} (1969) 29--38.
\circ} \def\a{\bulletibitem{FF}
P. Feinsilver and R. Fitzgerald,
\emph{The spectrum of symmetric Krawtchouk matrices,}
{Lin. Alg. \& Appl.}, {\circ} \def\a{\bulletf 235} (1996) 121--139.
\circ} \def\a{\bulletibitem{FS}
P. Feinsilver and R. Schott,
\emph{Krawtchouk polynomials and finite probability theory,}
{Probability Measures on Groups X}, Plenum, 1991, pp. 129--135.
\circ} \def\a{\bulletibitem{FH}
W. Fulton and J. Harris,
\emph{Representation theory, a first course,}
Graduate texts in mathematics, {\circ} \def\a{\bulletf 129},
Springer-Verlag, 1991.
\circ} \def\a{\bulletibitem{Kra}
M. Krawtchouk,
\emph{Sur une generalisation des polynomes d'Hermite,}
{Comptes Rendus}, {\circ} \def\a{\bulletf 189} (1929) 620--622.
\circ} \def\a{\bulletibitem{Kra2}
M. Krawtchouk,
\emph{Sur la distribution des racines des polynomes orthogonaux,}
{Comptes Rendus}, {\circ} \def\a{\bulletf 196} (1933) 739--741.
\circ} \def\a{\bulletibitem{Lev}
V.I. Levenstein,
\emph{Krawtchouk polynomials and universal bounds for codes and design in
Hamming spaces,}
{IEEE Transactions on Information Theory}, {\circ} \def\a{\bulletf 41} 5 (1995)
1303--1321.
\circ} \def\a{\bulletibitem{Lom}
S.J. Lomonaco, Jr.,
\emph{A Rosetta Stone for quantum mechanics with an introduction
to quantum computation,}
http://www.arXiv.org/abs/quant-ph/0007045
\circ} \def\a{\bulletibitem{Mac}
P.A. MacMahon,
\emph{Combinatory analysis,}
Chelsea, New York, 1960.
\circ} \def\a{\bulletibitem{MS}
F.J. MacWilliams and N.J.A. Sloane,
\emph{The theory of Error-Correcting Codes,}
The Netherlands, North Holland, 1977.
\circ} \def\a{\bulletibitem{Par}
K.R. Parthasarathy,
\emph{An introduction to quantum stochastic calculus,}
Birkh\"auser, 1992.
\circ} \def\a{\bulletibitem{Spr}
T.A. Springer,
\emph{Invariant theory,}
Lecture notes in mathematics, {\circ} \def\a{\bulletf 585},
Springer-Verlag, 1977.
\circ} \def\a{\bulletibitem{Sze}
G. Szeg\"o,
{\sl Orthogonal Polynomials},
Colloquium Publications, Vol. 23, New York, AMS, revised eddition 1959,
35--37.
\circ} \def\a{\bulletibitem{V-J}
D. Vere-Jones,
\emph{Finite bivariate distributions and semi-groups of nonnegative matrices,}
{Q. J. Math. Oxford}, {\circ} \def\a{\bulletf 22} 2 (1971)
247--270.
\circ} \def\a{\bulletibitem{YH} R.K. Rao Yarlagadda and J.E. Hershey,
\emph{Hadamard matrix analysis and synthesis: with applications
to communications and signal/image processing,}
Kluwer Academic Publishers, 1997.
\end{thebibliography}
\end{document} |
\begin{document}
\title{On resumming periodic orbits in the spectra of
integrable systems}
\author{Alfredo M.~Ozorio de Almeida\dag\ddag, Caio H.~Lewenkopf\S\
and Steven Tomsovic \dag$\|$}
\address{\dag\ Max-Planck-Institut f\"ur Physik komplexer Systeme,
N\"othnitzer Str.~38, 01187 Dresden, Germany}
\address{\ddag\ Centro Brasileiro de Pesquisas F\'{\i}sicas, R. Xavier
Sigaud 150, 22290-180 Rio de Janeiro, Brazil}
\address{\S\ Instituto de F\'{\i}sica, Universidade do Estado do Rio de
Janeiro, R. S\~ao Francisco Xavier 524,
20559-900 Rio de Janeiro, Brazil}
\address{$\|$\ Department of Physics, Washington State University,
Pullman, WA99164-2814, USA}
\date{\today}
\begin{abstract}
Spectral determinants have proven to be valuable tools for resumming the
periodic orbits in the Gutzwiller trace formula of chaotic systems. We
investigate these tools in the context of integrable systems to which
these techniques have not been previously applied. Our specific model is a
stroboscopic map of an integrable Hamiltonian system with quadratic action
dependence, for which each stage of the semiclassical approximation can be
controlled. It is found that large errors occur in the semiclassical traces
due to edge corrections which may be neglected if the eigenvalues are obtained
by Fourier transformation over the long time dynamics. However, these errors
cause serious harm to the spectral approximations of an integrable system
obtained via the spectral determinants. The symmetry property of the spectral
determinant does not generally alleviate the error, since it sometimes
sheds a pair of eigenvalues from the unit circle.
By taking into account the leading order asymptotics of the edge corrections,
the spectral determinant method makes a significant recovery.
\end{abstract}
\pacs{ 03.65.Sq, 02.30.Ik, 02.30.Lt}
\maketitle
\section{Introduction}
\label{intro}
It has been generally accepted that spectral determinants, or
zeta-functions, provide optimal semiclassical estimates of the individual
energy levels of classically chaotic systems~\cite{cv,Cvitanovic89,Eckhardt89}.
These resummations of the periodic orbits in the Gutzwiller trace
formula~\cite{Gutzwiller71,Balian72}, originally developed for time-independent
systems, may be obtained in a variety of ways. The formulation of Berry and
Keating~\cite{Berry90JPA} relies explicitly on the instability of the periodic
orbits in such a way as to give an expression that does not depend on a sharp
cut-off of the orbit periods.
In contrast, Bogomolny's approach~\cite{Bogomolny90,Bogomolny92} reduces the
problem to the quantization of a map over a Poincare\'e surface of section,
without making any assumption about the nature of the classical motion. However,
two sharp boundaries are introduced. On the one hand, the Poincare\'e section
itself is bounded if the constant energy surface is compact. The limited area
of the section, which corresponds to a finite dimension of the Hilbert space,
then leads to a sharp cut-off in the period of the orbits.
The recent paper of Eckhart and Smilansky~\cite{Eckhardt01} also works with a
quantum map in a bounded region, but this is obtained stroboscopically instead
of with a Poincar\'e section.
The classical motion of integrable systems is restricted to invariant tori, which
determine closed invariant curves (also tori) of the Poincar\'e mapping.
For integrable systems, Berry and Tabor~\cite{bt} established the
equivalence of the Gutzwiller trace formula to the general forms of
Bohr-Sommerfeld quantization. The latter method is evidently the most
efficient for the calculation of individual levels.
Nevertheless the exercise of showing the equivalence of both methods
had the merit of clarifying the role of periodic orbits (forming continuous
tori for higher dimensional systems) in the density of states.
Indeed it raises an important question:
The Berry-Tabor equivalence involves the complete set of periodic orbits,
so how might one obtain correct energy levels from the resummation of a
finite selection of short orbits?
Our interest in this paper is resummation approximations for the spectra
of simple integrable systems. Particularly, we will focus on the effect
of introducing boundaries that do not interfere with the local classical
tori. For simplicity, consider such a system with a single degree of freedom.
For the classical Hamiltonian $H(I)$, where $(I,\theta)$ are the action-angle
variables~\cite{text}, the Bohr-Sommerfeld levels are
\begin{equation}
\label{bs}
E_n=H\left(\hbar\left(n+{1 \over 2}\right)\right) \ .
\end{equation}
Actually, it suffices to consider a case for which the action variables
are those of the harmonic oscillator
\begin{equation}
\label{ho}
I={1\over 2}(p^2+q^2) \ ,
\end{equation}
rendering the semiclassical approximation of \ (\ref{bs}) exact. It
is important to point out that large-$n$ states are well approximated by
\ (\ref{bs}) even for a general nonquadratic dependence of the
Hamiltonian on the phase space variables. The main question
is how well a spectral determinant method converges to the
eigenvalues $E_n$ in this case.
A conceptually clean approach to answering this question follows in the
spirit of Creagh~\cite{Creagh95}, who applied an extension of resummation
methods~\cite{Saracero92,Smilansky93JPA} to the perturbed cat maps. He found
worsening errors
in the semiclassical trace formula as the perturbation brought the map
out of the hyperbolic regime, but was unable to continue the perturbation
all the way to an integrable regime. Here, since $I$ is a constant of the
classical evolution, our consideration can be limited to a ring
${\cal I}_-<I<{\cal I}_+$, thus obtaining a finite phase space and hence a
finite Hilbert space of dimension $N=\{({\cal I}_+-{\cal I}_-)/\hbar \}$,
where $\{\ \cdot \ \}$ denotes the integer part. A stroboscopic map of
period $\tau$ can be defined and quantized in such a way that the dynamics
is determined by the evolution operator
\begin{equation}
\label{un}
U_N = \sum_{n=N_-}^{N_+} \exp\!\left(-{i\over \hbar}E_n\tau\right)
|n\rangle\langle n | \ .
\end{equation}
The quantity $\hbar N_-$ ($\hbar N_+$) is the smallest (largest) quantized action
greater (smaller) than ${\cal I}_-$ (${\cal I}_+$) and $|n\rangle$ are the eigenstates.
The definition of the
spectral determinant as $\det\left(1-zU_N\right)$ gives an $N^{th}$ order
polynomial in the variable $z$ and the roots of the characteristic
equation
\begin{equation}
\label{char}
P_N(z)\equiv \det\left(1-zU_N\right) =0
\end{equation}
are just
\begin{equation}
\label{ev}
z_n=\exp\left({i\over \hbar}E_n\tau\right) \ .
\end{equation}
Thus, knowledge of the phases $\phi_n$ of the zeros of the spectral
determinant, or zeta-function, determines the eigenvalues of the
stroboscopic map within the ring. These coincide with a subset of the
eigenvalues of the original system possessing an infinite Hilbert space.
The eigenenergies of the original system can be inferred from
\begin{equation}
\label{infer}
E_n={\hbar\over \tau}(\phi_n +2k\pi) \ .
\end{equation}
The determination is unique if, for the energy interval of the ring
$(E_-,E_+) = (E({\cal I}_-),E({\cal I}_+))$, the constraint
\begin{equation}
\label{constraint}
\left(E_+-E_-\right){\tau\over \hbar} \le 2\pi
\end{equation}
is respected. This scheme for locating eigenvalues of the original
Hamiltonian resembles that of~\cite{Eckhardt01}, but our objective of comparing
semiclassical methods can be achieved also in the context of the
continuous dynamical system confined to the ring, or even the discrete
map defined by (\ref{un}).
In the semiclassical limit, $\mbox{Tr}(U_N^l)$ is expressed as a sum over
periodic orbits of period $l$ for the classical stroboscopic map
generated by the Hamiltonian $H(I)$ in the time $\tau$ within the ring.
These orbits, with action $I_{m,l}$, such that
\begin{equation}
\label{deriv}
\tau {{\rm d}H(I)\over {\rm d}I} = {2\pi m\over l} \ ,
\end{equation}
define continuous curves, unless $I=0$. To simplify the treatment, we
consider ${\cal I}_->0$, which excludes the isolated periodic orbit at the
origin.
Note that the periodic orbits of the stroboscopic map comprise only a
small subset of those of the original system with continuous time.
Indeed, all the orbits of the latter are periodic whereas only those full
orbits with periods that are rationally related to the stroboscopic time,
$\tau$, are periodic in the map. The map resembles a Poincar\'e
map of an integrable system with two degrees of freedom in the sense that
all orbits lie on invariant curves, but the curves made up of periodic
orbits form a dense set of zero measure.
The semiclassical form for the spectral determinant is obtained from
the expression
\begin{equation}
\label{specdet}
\det\left(1-zU_N\right)=\exp \left[ - \sum_{l=1}^\infty {z^l\over l}
\ \mbox{Tr}(U_N^l) \right] \ .
\end{equation}
Even though the series only converges for $|z|<1$, it is possible to
follow~\cite{Saracero92} in noting that the Taylor expansion of the
exponential in Eq.\ (\ref{specdet}) can be identified with the finite
expansion of the spectral determinant (\ref{char}):
\begin{equation}
\label{expan}
P_N(z)= 1 + c_1 z + c_2 z^2 ... + c_N z^N \ ,
\end{equation}
where the coefficients are given by the following recurrence relation
\begin{equation}
\label{eq:recurrence}
c_k = - \frac{1}{k} \sum_{l=1}^k c_{k-l} \mbox{Tr}\,(U_N^l) \ .
\end{equation}
In this way the periodic orbits up to period $l=N$ determine, in
principle, all the eigenvalues of $U_N$.
A further reduction of the period of the orbits used in the semiclassical
spectral determinant results from the symmetry relation for the
coefficients~\cite{Bogomolny90}
\begin{equation}
\label{symm}
c_k=(-1)^N\det(U_N) c_{N-k} \ .
\end{equation}
The coefficients containing traces for long iteration times, $l>N/2$, can be obtained from
the coefficients with $l<N/2$. Hence, only the orbits of period $l<N/2$ are needed.
This is a particularly simple example of ``bootstrapping''~\cite{Berry85}
resulting from the finiteness of the Hilbert space. For chaotic systems,
the symmetry (\ref{symm}) is a fundamental tool because the number
of periodic orbits increases exponentially with $l$. This is not so for
integrable systems.
In Section~\ref{st}, we derive the semiclassical formula for $\mbox{Tr}(U^l)$ of
the integrable map defined in \ (\ref{un}). It will then become clear that
edge corrections to the periodic orbit sum cannot be neglected in this case.
Indeed these are the only corrections for the particular model that we study
numerically in Section~\ref{model}.
Though the absence of edge corrections severely affects the spectral determinant,
which has a natural cut-off in time, they cancel out in the Berry-Tabor formula
as discussed in Section~\ref{equiv}.
\section{The semiclassical trace}
\label{st}
Relation~(\ref{bs}) for the eigenvalues becomes exact in the case of the
harmonic oscillator actions (\ref{ho}). Likewise, the propagator (\ref{un}),
extended to all $n$, would give the full evolution operator for any given
function of the projection operators $|n\rangle \langle n|$.
However, the initial semiclassical approximation is to consider the action-angle
variables $(I,\theta)$ as appropriate conjugate variables for quantization.
From the action representation of the operator $U_N^l$, the $l^{th}$ power of
$U_N$ in \ (\ref{un}), the matrix elements in the angle representation turn
out to be
\begin{equation}
\label{me}
\langle \theta |U_N^l |\theta^\prime\rangle = {1 \over
2 \pi} \sum_{n=N_-}^{N_+}
\exp\left\{i\left[\left(n+{1\over 2}\right)(\theta-\theta^\prime)-l{\tau
E_n\over
\hbar}\right]\right\} \ ,
\end{equation}
where we used $\langle\theta | n\rangle = e^{-i\left(n+{1\over 2}\right)
\theta}/\sqrt{2\pi}$.
The Poisson transformation is now applied, changing to the continuous action variable
$I$, so that $E$ is interpolated by $H(I)$ as in \ (\ref{bs}).
This results in
\begin{equation}
\label{meI}
\langle \theta | U_N^l | \theta^\prime\rangle = {1\over 2\pi \hbar}
\sum_{m={-\infty}}^\infty (-1)^m \int_{{\cal I}_-}^{{\cal I}_+}
{\rm d}I \exp\left\{{i\over \hbar} \Big[ I (\theta-\theta^\prime) +
2\pi m - l\tau H(I) \Big] \right\} \ ,
\end{equation}
which is exactly equivalent to (\ref{me}). Tracing over the angle
variables gives
\begin{equation}
\label{trace}
\mbox{Tr}(U_N^l) = {1\over \hbar} \sum_{m={-\infty}}^\infty (-1)^m
\int_{{\cal I}_-}^{{\cal I}_+} {\rm d}I \exp\left\{{i\over \hbar} \Big[
2\pi mI - l\tau H(I) \Big] \right\} \ .
\end{equation}
The exponential in the integrand of (\ref{trace}) oscillates rapidly
in the semiclassical limit, and the largest contributions to the trace,
of order $\hbar^{1/2}$, come from the regions near the stationary phase
points at
\begin{equation}
\label{points}
l\tau {{\rm d}H(I)\over {\rm d} I} -2 \pi m = 0 \ ,
\end{equation}
which is the same condition as the one given by (\ref{deriv}) for the
periodic orbits of the classical stroboscopic map. The stationary phase
evaluation of the trace is therefore tied to the periodic orbits with actions $I_{m,l}$.
The periodic orbits can now be ordered by increasing period, $l$.
Equation (\ref{points}) also sets the range of possible repetitions
$(M_{l-}, M_{l+})$ of the period $l$ orbit. The frequencies,
$\omega={{\rm d}H(I)/ {\rm d} I}$ in
the limited range of action $({\cal I}_-,{\cal I}_+)$ are also bounded.
Thus, the number of periodic orbits in this range increases linearly with
$l$. The full stationary phase evaluation of (\ref{trace}) becomes
\begin{equation}
\label{eval}
\mbox{Tr}(U_N^l) = \sum_{m=M_{l-}}^{M_{l+}}
(-1)^m \left({2\pi \over \hbar l \tau \left|{{\rm
d^2}H(I)\over {\rm d} I^2}\right|_{I_{m,l}}} \right)^{1/2}\!\!\!\exp\left\{
{i\over \hbar} \left[ 2\pi mI_{m,l} - l\tau H(I_{m,l}) - \hbar{\pi\over
4}\right] \right\} \ .
\end{equation}
Not only are the stationary points of the integration specified by the
action variables of the periodic orbits, but the phase of each
contribution is given by the full action in units of $\hbar$,
\begin{equation}
\label{action}
{\cal S}_{m,l} = \int_0^{l\tau} {\rm d}t\ \left[ p \dot q - H\right ],
\end{equation}
except for the geometric ``Maslov term'', $\pi/4$.
Typically, the contributions to the trace from the endpoints ${\cal
I}_\pm$ to the integrals in (\ref{trace}) are only of order $\hbar$.
However, note that they cannot be separated from the stationary phase term
corresponding to a periodic orbit which lies very close to the boundary.
In fact, by varying the parameters of the system and $l$, essentially
all the stationary phase points pass close to the boundaries. The edge
corrections can be obtained as a uniform approximation~\cite{Eckhardt01}, but
this erases the simplicity of the periodic orbit expression for the
trace. It is shown in the following section that the limitation of the
allowed phase space to the ring $({\cal I}_-,{\cal I}_+)$ in no way
hinders the Berry-Tabor equivalence.
\section{The Berry-Tabor equivalence}
\label{equiv}
A version of the Gutzwiller trace formula that is appropriate to an integrable
map is simply obtained by Fourier analyzing the traces of $U_N^l$ in the discrete
time $l$:
\begin{equation}
\label{bt}
\sum_{l=0}^\infty e^{il\theta} \mbox{Tr}(U_N^l) = \mbox{Tr}{1\over 1-
e^{i\theta}U_N} = \sum_n {1 \over 1- \exp [i(\theta-l\tau E_n)]} \ .
\end{equation}
Here, the eigenvalues of the map are considered as poles of the
resolvent $(1-zU_N)^{-1}$ instead of zeros of $\det(1-zU_N)$.
Though the infinite series of traces in (\ref{bt}) appears to be a
cumbersome alternative to evaluating the finite determinant, Poisson
transformation to the periodic orbit expression can be performed for
the semiclassical resolvent,
\begin{eqnarray}
\label{transform}
\mbox{Tr}{1\over 1-e^{i\theta}U_N} &=& \sum_{m,l} (-1)^m \left({2\pi \over\hbar
l \tau \left| {{\rm d^2}H(I)\over {\rm d} I^2}\right|_{I_{m,l}}}
\right)^{1/2} \nonumber\\
&& \times\exp \left( {i\over\hbar}\left[ \hbar l \theta + 2\pi m
I_{m,l} - l \tau H(I_{m,l}) -\hbar {\pi\over 4}\right]\right) \ .
\end{eqnarray}
Interpolating the discrete time $l$ again by the continuous time
$s$ gives
\begin{eqnarray}
\label{back}
\mbox{Tr}{1\over 1-e^{i\theta}U_N} &=& \sum_k (-1)^m \left({2\pi \over\hbar \tau l
\left| {{\rm d^2}H(I)\over {\rm d} I^2}\right|_{I_{m,s}} }\right)^{1/2}
\nonumber\\
&&
\times\int_0^\infty {\rm d}s\ \exp\left\{{i\over\hbar}\left[ \hbar s \theta +
2\pi (m I_{m,s} + \hbar ks) - s\tau H(I_{m,s}) -\hbar {\pi\over
4}\right]\right\} \ .
\end{eqnarray}
The stationary phase points, $s_{m,k}(\theta)$, of the integrals in
Eq.~(\ref{back}) are given by
\begin{equation}
\label{statph2}
{{\rm d} {\cal S}_{m,s}\over {\rm d} s} + \hbar(\theta + 2 \pi k) = 0
\end{equation}
where ${\cal S}_{m,s}$ is the full action of the orbit singled out by
condition (\ref{points}). It is important to note that, while in (\ref{points}),
the stationary phase condition defined a periodic orbit of integer period
$l$ for the $\tau$-stroboscopic map, now $l$ has been replaced by the
continuous variable $s$. This condition can be reinterpreted in two
ways: (i) orbits are picked that are no longer closed, or (ii) the
selected orbit is still periodic, but for a different
$\tau(\theta)$. In either case,
\begin{equation}
\label{condit}
{{\rm d} {\cal S}_{m,s}\over {\rm d} s} = \tau {{\rm d} {\cal
S}_{m,s}\over {\rm d} t} = \tau E_{m,s} \ ,
\end{equation}
where the energy of the selected orbit appears in the last equality.
Thus, the resulting phase of each integral is given by the
reduced action for the orbits of energy $\hbar(\theta +2\pi k)/\tau$. The
stationary evaluation of the resolvent is
\begin{equation}
\label{resolv}
\mbox{Tr}{1\over 1-e^{i\theta}U_N} \approx \sum_{k,m} s_{1,k}(\theta) \>\exp \left[
{i\over \hbar} 2\pi mI\left({\hbar\over \tau}(\theta+2\pi k)\right) + i\pi
m\right] \ ,
\end{equation}
where $I(E)$ is the local inverse of $H(I)$. Thus, if $\hbar/\tau$ is
chosen such that the variation of $\tau H(I)/\hbar$ does not exceed $2\pi$
for $I$ in $({\cal I}_-,{\cal I}_+)$, there are at most a few orbits
contributing to the semiclassical resolvent for each value of
$\theta$. Since the repetitions must still be summed, the interpretation
(ii) for (\ref{statph2}) as a periodic orbit of another stroboscopic map
is perhaps the more appealing.
The condition for the resolvent to be singular is that all the $m$
repetitions are exactly in phase,
\begin{equation}
\label{bs2}
I\!\left({\hbar\over \tau}(\theta + 2\pi k)\right) = \left(j-{1\over 2}\right)\pi \ ,
\end{equation}
which is just the initial Bohr-Sommerfeld quantization. Thus, the
periodic orbit evaluation of the map resolvent retrieves the full
Berry-Tabor equivalence without any need to consider boundary corrections
to the trace. The reason why this works is that the singularities of the
resolvent only manifest themselves by summing the periodic contributions
for very long times, or multiple repetitions. For these contributions
there is an effective increase of the large parameter of the stationary
phase evaluation of $\mbox{Tr}(U_N^l)$ given by (\ref{eval}). The integrals
are then dominated by a very narrow region near each periodic orbit,
so that the boundary can be ignored for the high repetition of an orbit
even if it lies almost at the edge.
The evaluation of the spectral determinant may be regarded as a
resummation of the semiclassical sum for the poles of the resolvent, such
that the (discrete) time of the contributing periodic orbits is cut-off
by the dimension of the finite Hilbert space. We show in the next
section that the error in the traces of the propagator due to the edge corrections
can be magnified by the spectral determinant.
\section{Model Hamiltonian}
\label{model}
To illustrate the calculation of the spectral determinant, we choose the
simplest Hamiltonian that is a nonlinear function of the action, namely
\begin{equation}
\label{ham}
H(I) = {1\over 2} (I-{\cal I})^2 \qquad (I\ge 0) \ .
\end{equation}
The periodic orbit condition (\ref{deriv}) then reduces to
\begin{equation}
\label{simple}
I_{m,l} = {\cal I} + 2\pi {m\over l \tau} \
\end{equation}
and the stationary phase evaluation of the trace assumes the explicit form
\begin{equation}
\label{spsimp}
\mbox{Tr}(U_N^l) \approx \left({2 \pi \over \hbar l\tau}\right)^{1/2}
e^{-i\pi/4} \sum_m (-1)^m \exp \left[{i\over \hbar} 2\pi m\left( {\cal I}
+ {\pi m \over l \tau}\right)\right] \ .
\end{equation}
For the case where the parameter ${\cal I} = \hbar (n+1/2)$ (i.e. one of
the quantized actions), $\mbox{Tr}(U_N^l)$ reduces to a ``curlicue''. These
recursively spiraling patterns in the complex plane were shown by Berry
and Goldberg~\cite{Berry88} to have quite diverse characteristics depending
on $\tau/\hbar$, in the limit $N\rightarrow \infty$. This may also be
the case for non-quantized ${\cal I}$.
Two essentially different regions of the eigenspectrum can be studied.
In case (i), ${\cal I}>0$ and ${\cal I}_\pm={\cal I}\pm \sqrt{2\cal E} < 2
{\cal I}$. The energy levels there originate on two different branches of
$H(I)$, so there may be near degeneracies. This is a similar situation to
that expected for an integrable system with more degrees of freedom
(see e.g.~\cite{text}).
Otherwise, in case (ii), both ${\cal I}_+$ and ${\cal I}_-$ $>$ $2{\cal I}$,
so that a single branch of the $H(I)$ curve is sampled. The energy levels
are then quite regularly spaced with approximate separation of
Planck's constant times the average ${\rm d}H/{\rm d}I$ for this energy interval.
By choosing $E_\pm$ to satisfy condition (\ref{constraint}),
we also guarantee the regularity of the spectrum of $U_N$ itself.
\subsection{Case (i)}
\begin{figure}
\caption{Comparison between the exact (dotted lines) and the semiclassical values
(solid lines) of Re[Tr($U_N)$] as a function of $t/t_{\mbox{\scriptsize H}
\label{one}
\end{figure}
Let us first examine the semiclassical approximations to $\mbox{Tr}(U_N)$ for a single
period. Notice that the stationary phase approximation to the integrals in
(\ref{trace}) would be exact for the quadratic model if the limits
could be extended to $\pm \infty$. In other words, the edge corrections
are the only deviation of the periodic orbit approximation with respect to
the exact trace. These corrections are certainly not negligible for stroboscopic
parameters such that a periodic orbit is close to one of the edges.
The simplest asymptotic edge corrections, as deduced in~\cite{Eckhardt01}, are presented
in \ref{sec:appA}. The computed error due to
the edges is displayed in
Figure \ref{one}. Here we let the time $t$ vary
and study Tr($U_N$) as a function of $t/t_{\mbox{\scriptsize H}}$, where
$t_{\mbox{\scriptsize H}}=2\pi\Delta N/E$ is the Heisenberg time.
As the time is increased the integrand at the l.h.s.~of Equation (\ref{trace}) has more
stationary phase points.
The sudden jump in the semiclassical trace in Figure \ref{one}a is due to the following.
For small values of $t/t_{\mbox{\scriptsize H}}$ there is just a single stationary phase
point with $m=0$. Going beyond $t/t_{\mbox{\scriptsize H}} \approx 0.25$
all $|m|\le 1$ become stationary phase points. (It is near such transition points that the
semiclassical approximation is worse.) This situation repeats itself regularly
with increasing $t/t_{\mbox{\scriptsize H}}$ adding more stationary points to
the sum (\ref{spsimp}).
As seen in Figure~\ref{one}a, the semiclassical trace as given by (\ref{spsimp}) is not
an accurate approximation to the exact Tr$(U_N)$.
The suppression of the edge correction by means of a cosine weighted trace,
namely
\begin{equation}
\mbox{Tr}(\widetilde{U_N}) = \mbox{Tr} \left[U_N \cos\left(\frac{\pi
H(I)}{2E}\right)\right],
\end{equation}
improves the agreement enormously, as shown in Figure \ref{one}b.
Our calculations have been performed for case $\hbar= 0.01, E = 30.4571694,$
and ${\cal I} = 20.3489573$ giving $N=1~521$.
\begin{figure}
\caption{Comparison between the exact (white histograms) and the semiclassical
(grey histograms) values of (a) Im[Tr$(U_N^l$)] and (b) Re[Tr$(U_N^l$)] as a
function of $l$ for $N=48$.}
\label{fig:tracesN=48}
\end{figure}
We now return to the original programme of obtaining the eigenvalues of $U_N$,
defined for a fixed $\tau$, by resumming periodic orbits.
The first step is to calculate the traces of $U_N^l$. The coefficient of the
characteristic polynomial are then easily computed by (\ref{eq:recurrence}).
The roots of the resulting polynomial are obtained numerically.
The precision in finding the roots, limits this method to polynomials
with $N \le 50$.
We chose the model Hamiltonian parameters accordingly.
To illustrate case (i), we take the parameters $E$ and ${\cal I}$ to be the same
as above and $\hbar = 0.35$, which gives $N=48$.
Figure~\ref{fig:tracesN=48} displays the real and imaginary parts of
$\mbox{Tr}(U_N^l)$ for $l\le N$ using the semiclassical expression
(\ref{spsimp}) contrasted with their exact values.
As expected, the semiclassical approximation works reasonably well for the lowest
values of $l$ and gives very poor results for the largest ones.
It is worth noticing that similar calculations with the addition of an imaginary
part to the time in the trace (\ref{eval}) show a dramatic improvement in the
agreement between the semiclassical and the exact traces.
This is further evidence that the differences are due to edge corrections
and not to numerical inaccuracies.
Next, we calculate the coefficients $c_k$ of the characteristic polynomial
(\ref{char}) using the recursion relation (\ref{eq:recurrence}).
The symmetry relation (\ref{symm}) allows us to use only traces corresponding
to times shorter than $\tau_{\mbox{\scriptsize H}}/2$.
In Figure \ref{fig:coefN=28} we compare the exact coefficients $c_k$ with
those resulting from the semiclassical traces, with and without
symmetrization. To better illustrate the whole range of $c_k$ we show a
situation where $N=28$, corresponding to $\hbar= 1.0, E = 100.1234,$
and ${\cal I} = 41.2345$.
As before, the semiclassical approximation works well for the low $k$ coefficients
($k < 5$) and, obviously, also for the corresponding higher $k$'s if we impose the
symmetry (\ref{symm}).
For the remaining coefficients, the agreement with the exact $c_k$ is poor.
\begin{figure}
\caption{Comparison between the exact (gray), the semiclassical (white) and the
symmetrized semiclassical (black) values of (a) Im($c_k$) and (b) Re($c_k$) as a
function of $k$ for $N=28$.}
\label{fig:coefN=28}
\end{figure}
Finally we arrive at the eigenvalues of $U_N^l$.
Figure \ref{fig:rootsN=48} shows the results for $\hbar= 0.35, E = 30.4571694,$
and ${\cal I} = 20.3489573$, which is a rather typical situation.
The exact roots of the characteristic polynomial lie on the unit circle, as they
should. The standard semiclassical approximation destroys unitarity and the roots of
the corresponding characteristic polynomial are no longer restricted to the unit circle.
The enforcement of the symmetry (\ref{symm}) makes $|\det(U)| = 1$.
As a consequence the roots either lie exactly on the unit circle, or appear in pairs,
one inside and one outside the circle.
More precisely, the self-inversive symmetry of the characteristic polynomial renders
a symmetry in its zeros: if $z_k$ is a root then $1/z^*_k$ is also a root.
Indeed, it has been shown by Bogomolny et al.\cite{BBL96} that on average only about
$57\%$ of the roots of a random symmetrical polynomial lie on the unit circle.
Unfortunately, upon symmetrization individual roots do not necessarily come closer
to the exact ones, as compared with the standard procedure. Figure 4 shows that they can
even be pushed out, unless the standard semiclassical roots are
already close to the exact ones, in which case symmetrization tends to improve the
accuracy. Unfortunately this behaviour is not very systematic.
\begin{figure}
\caption{Comparison between the exact (asterisks), the semiclassical (open circles)
and the symmetrized semiclassical (open squares) roots $z_i$ of the characteristic
polynomial for $N=48$.}
\label{fig:rootsN=48}
\end{figure}
\subsection{Case (ii)}
We switch now to case (ii) and consider the situation where the energy
$E>{\cal I}^2/2$, or equivalently, both ${\cal I}_+$ and ${\cal I}_->2{\cal I}$.
As mentioned before, in this situation the levels are almost equally spaced with
separation of $\hbar {\rm d}H/{\rm d}I$.
From the semiclassical point of view, the most important distinction to case (i)
is the absence of the $m=0$ orbit.
Actually, for our simple model there will be no tori with orbits of period $l<N/2$
in case (ii). Indeed, the fraction, $f$, of zero traces in the semiclassical
approximation is easily seen to be:
\begin{equation}
f=\frac{1}{2}\frac{\sqrt{E_+}+\sqrt{E_-}}{\sqrt{E_+}-\sqrt{E_-}}.
\label{fraction}
\end{equation}
The optimum of $f\approx N/2$ is approached for $E_-\approx 0$, which implies
${\cal I}\leq 0$,
so we choose $\hbar= 1.0, {\cal I} = -1.03489573$ with $E_ -= 0.265678$
and $E_+=1250.0$, which corresponds to $N=48$.
Keeping $\tau$ as defined by (\ref{constraint}) gives Tr$(U_N^l)=0$ for
$l \le 24$.
The results are summarized by Figure \ref{fig:rootscusp}.
As in case (i) the simple semiclassical approximation shows roots outside
the unit circle. The standard semiclassical improvement by symmetrization
(``bottom up symmetrization")
erases all system information, since there are only nonzero traces for long
times. Indeed, the spectral determinant (\ref{expan}) reduces to
\begin{equation}
P_N (z)=1+(-1)^N \det(U_N)\>z_N.
\label{expan1}
\end{equation}
Instead, one can maximize the semiclassical information by symmetrizing the
lower coefficients in $P_N(z)$ from the higher $c_k$'s , containing the
traces with long orbits (``top down symmetrization"). However, the final
result is no better than the one without symmetrizing at all.
By shifting the parameters in case (ii) so that $H(I)$ becomes almost linear,
within the interval $({\cal I}_-,{\cal I}_+)$, the agreement
between the different approaches becomes much more reasonable
than in Figure \ref{fig:rootscusp}.
However, if one recalls that the mean level spacing is known and the levels
are almost equally spaced, some caution is then required.
The exact traces lead to a characteristic polynomial with self-inversive symmetry.
For small values of $l$, where there is no stationary phase and the semiclassical
traces are zero, the (modulus of) exact traces are small compared to unit.
On the other hand, we observe that, for large $l$ values, the semiclassical approximation
fails to reproduce the exact traces with the same precision.
Thus, the best semiclassical agreement corresponds surprisingly to the case where all
traces are taken as zero (bottom up).
As shown in Fig \ref{fig:rootscusp}, the top down symmetrization is superior to the
standard procedure only where the eigenvalues are sufficiently accurate.
It then brings most of the roots back to the unit circle. However, since
it employs some inaccurate large $l$ traces, (top down) symmetrization also produces
pairs of roots lying
at opposite sides of the unit circle, like in case (i).
\begin{figure}
\caption{Comparison between the exact (crosses), the semiclassical (open circles),
the ``bottom" symmetrized semiclassical (open squares) and the ``top" symmetrized
semiclassical (open triangles) roots $z_i$ of the characteristic polynomial for
$N=48$. Panel (a) displays all roots, while (b) is a scale blow up of the same
data showing that the nice agreement is only apparent.}
\label{fig:rootscusp}
\end{figure}
\section{Conclusions}
The main assets of using spectral determinants to resum the periodic orbits
in the Gutzwiller trace formula for chaotic quantum maps are:
(a) the requirement of unitarity is partially incorporated,
(b) the necessity of handling very long (and exponentially many) periodic orbits
with periods larger than $\tau_H/2$ is eliminated and
(c) the method produces isolated levels rather than a smoothed density of states.
Nevertheless, the method remains problematic in general, because of the necessity
of accounting for a formidable number of periodic orbits in the semiclassical
limit, which is a daunting task from the classical point of view.
Even though semiclassical spectra of integrable systems are obtained most efficiently
by generalizations of the Bohr-Sommerfeld rules, one could
expect that spectral determinants would still be more successful
tools for integrable spectra than the trace formula, since they take some account of
unitarity and here the classical periodic orbit structure is much
simpler than for chaotic systems.
Our study shows that this is not the case.
We find large errors in the semiclassical traces, which could, in principle,
be completely fixed by accounting for edge corrections. Indeed, this has been verified
in the case of the first order corrections in the Appendix. However, to follow such a path
would be at odds with the spirit of the present work. Rather than solving a trivial model,
our purpose was to assess the efficacy of the spectral determinant, symmetrized or not,
as a tool of reducing the effect of inaccuracies of the semiclassical
traces in the calculation of eigenvalues. The spectrum of our model is exactly known
and it allows complete control over each stage of the approximation.
Thus, we have worked solely within a framework that is based on the usual amplitudes
of periodic orbits in the trace formula. Even though our map has been obtained
by looking stroboscopically at a simple system with continuous time and its boundaries
are entirely arbitrary, it is arguable that our results may resemble those for a quantum map
that results by taking a ``Bogomolny section" of a Hamiltonian system
\cite{Bogomolny90, Bogomolny92}. The corresponding classical map must also have a boundary
and it will also be integrable,
if this is a property of the original Hamiltonian system.
The errors in the semiclassical traces, discussed in Section \ref{model},
yield inaccurate coefficients for the characteristic polynomial,
rendering poor approximations for the spectra of integrable systems.
The symmetry property of the spectral determinant does not necessarily
lead to better approximations. Actually in most of the cases studied
there is only improvement where the unsymmetrized eigenvalues were already
reasonably accurate. We can now understand this to account for the encouraging results
obtained by Creagh \cite{Creagh95} for the perturbed cat map. Even though the exact eigenvalues
are not determined semiclassically as in our case, his system lies very close to the
linear map, where the periodic orbit traces are exact, so that the symmetrization always
fixes the eigenvalues on the unit circle. In contrast, if one perturbs the traces in the present
integrable system continuously from their exact values to their semiclassical approximation,
the eigenvalues may collide on the unit circle and be knocked off as a pair,
of which we see many examples in our model. We have also tried fitting the spectrum by
treating $\det U_N$ as a free parameter, without any essential qualitative improvement
of the results.
It is important to mention that the edge correction may be neglected
if the eigenvalues are obtained by Fourier transformation over long
times, so that the Berry-Tabor equivalence does not need to be ``dressed"
because of the boundary. Of course, this is not much help if the system
is not integrable. Indeed, an important point concerning the Bogomolny
approach is that no assumption is made about characteristics of the
dynamics. Recently, this method has also been extended successfully to
describe the eigenstates themselves of a chaotic system \cite{SS00}.
The present study of the integrable limit, though not obtained by a
surface of section, suggests that caution may be required in any attempt
to extrapolate the general section-method to nearly integrable,
or mixed systems.
\section*{Acknowledgments}
We gratefully acknowledge discussions with M.~Saraceno. AMOA and CHL
acknowledge support from CNPq, and ST acknowledges support by the US
National Science Foundation under Grant No. PHY-0098027, and the
hospitality of CBPF and UERJ, Brazil.
\appendix
\section{Edge corrections}
\label{sec:appA}
The asymptotic form for the edge corrections follows by basic application
of the expansion
\begin{equation}
\label{edgeasym}
C(z) + i S(z) = \int_0^z {\rm d}t e^{i{\pi \over 2} t^2} \sim {e^{i{\pi
\over 4}} \over \sqrt{2}} {\rm sign}(z) - {i e^{i{\pi \over 2} z^2} \over
\pi z} + {\rm O}\left( |z|^{-3} \right) \ .
\end{equation}
where the first term gives the stationary phase contribution; actually,
it is the complex conjugate with $z$ real which turns out to be needed.
The second term is responsible for the edge corrections. Expanding the
argument of the exponential in (\ref{trace}) to quadratic order in
$I^\prime = I - I_{m,l}$ for the Hamiltonian in (\ref{ham}), gives
\begin{equation}
\label{argument}
Arg\ = {i \over \hbar}\left( 2\pi m \varphi + {2\pi^2 m^2 \over
l \tau } - { l\tau \over 2 } {I^\prime}^2\right)
\end{equation}
Rescaling the action variable by $\sqrt{\pi\hbar /l\tau}$ matches the
argument of (\ref{edgeasym}) with that of (\ref{argument}). After a
little algebra, the edge corrections $\epsilon$ take the form
\begin{equation}
\label{corre}
\epsilon \sim {i\over l \tau} \sum_m \exp\!\!\left\{{i\over \hbar} \left(2\pi m
\varphi + {2 \pi^2 m^2 \over l\tau} \right)\right\} \left[ {1\over {\cal I}_+}
\exp\!\!\left(-i{l\tau \over 2\hbar} {\cal I}_+^2\right) - {1\over {\cal I}_-}
\exp\left(-i{l\tau \over 2\hbar} {\cal I}_-^2\right) \right]
\end{equation}
For a discussion of edge corrections for the general case where the phase
in the integral (\ref{trace}) is not quadratic, see~\cite{Eckhardt01}.
\end{document} |
\begin{document}
\APAmaketitle
\section{Introduction}\label{Sec:Introduction}
Big data analysis has gained interest in recent years through providing new insights and unlocking hidden knowledge in different fields of study \parencite{karmakar2018statistical} including medicine \parencite{rehman2021leveraging}, fraud detection \parencite{vaughan2020efficient}, the oil and gas industry \parencite{nguyen2020systematic} and astronomy \parencite{zhang2015astronomy}.
However, the analysis of Big data can be challenging for traditional statistical methods and standard computing environments \parencite{wang2016statistical}.
\textcite{martinez2020modeling} discuss storage and modelling solutions when handling such a large amount of data.
In general, modelling solutions can be grouped into three broad categories: 1) \emph{sub-sampling methods}, where the analysis is performed on an informative sub-sample obtained from the Big data \parencite{kleiner2014scalable,ma2015statistical,ma2015leveraging,drovandi2017principles,wang2018logistic,wang2019linear,yao2019softmax,ai2020quantile,cheng2020IBOSSlogistic,lee2021fast,ai2021optimal,yao2021review}; 2) \emph{divide and recombine methods}, where the Big data set is divided into smaller blocks, and then the intended statistical analysis is performed on each block and subsequently recombined for inference \parencite{lin2011aggregated,guha2012large,cleveland2014divide,chang2017divide,li2020sequential}; 3) \emph{online updating of streamed data}, where statistical inference is updated as new data arrive sequentially \parencite{schifano2016online,xue2020online}.
In recent years, compared to divide and recombine methods, sub-sampling has been applied to a variety of regression problems, while online updating is typically only used for streaming data.
In addition, in cases where a large data set is not needed to answer a specific question with sufficient confidence, sub-sampling seems preferable as the analysis of the data can often be undertaken with standard methods.
Moreover, the computational efficiency of sub-sampling over analysing large data sets has been observed for parameter estimation in linear \parencite{wang2019linear} and logistic \parencite{wang2018logistic} regression models.
For these reasons, we focus on sub-sampling methods in this article.
The key challenge for sub-sampling methods is how to obtain an informative sub-sample that can be used to efficiently answer specific analysis questions and provide results that align with the analysis of the whole Big data set.
Two approaches for this exist in the literature: 1) randomly sample from the Big data with sub-sampling probabilities that are found based on a specific statistical model and objective (e.g., prediction, parameter estimation) \parencite{wang2018logistic,yao2019softmax,wang2019linear,ai2020quantile,cheng2020IBOSSlogistic,ai2021optimal,lee2021fast,yao2021review}; 2) select sub-samples based on an experimental design \parencite{drovandi2017principles,Laura2022Optimal}.
Randomly sampling with certain probabilities (based upon the definitions of $A$- or $L$- optimality criteria, see \textcite{atkinson2007optimum}) is the focus of this article, and has been applied for parameter estimation in a wide range of regression problems including softmax \parencite{yao2019softmax} and quantile regression \parencite{ai2020quantile}, and generalised linear models \parencite{wang2018logistic,wang2019linear,cheng2020IBOSSlogistic,ai2021optimal,yao2021review}.
In contrast, the approach based on an experimental design has only been applied for: 1) parameter estimation in logistic fixed and mixed effects regression models \parencite{drovandi2017principles}; 2) parameter estimation and prediction accuracy in linear and logistic regression models \parencite{Laura2022Optimal}.
A key feature of both of the current sub-sampling approaches is that they rely on a statistical model that is assumed to appropriately describe the Big data.
Given this is a potentially limiting assumption, \textcite{yu2022subdata} proposed to select the best candidate model from a pool of models based on the Bayesian Information Criterion (BIC) \parencite{schwarz1978estimating}.
This was applied to linear models, and resulted in sub-sampling probabilities that were more appropriate than those based on considering a single model.
Similarly, for linear regression, \textcite{shi2021model,meng2021lowcon} explored using space filling and orthogonal Latin hypercube experimental design techniques to allow for potential model misspecification.
In this paper, we propose that, instead of selecting a single best candidate model for the Big data, a set of models is considered and a model averaging approach is used for determining the sub-sampling probabilities.
Through adopting such an approach, it is thought that the analysis goal (e.g., efficient parameter estimation) should be achieved regardless of the preferred model for the data.
To implement this model robust approach, we consider sub-sampling based on $A$- and $L$- optimality criteria within the Generalised Linear Modelling framework, and provide theoretical support for using a model averaged approach based on each of these criteria.
Given we consider Generalised Linear Models (GLMs), our approach should be generally applicable across many areas of science and technology where a variety of data types are observed.
This is demonstrated through applying our proposed methods within a simulation study, and for the analysis of two real-world Big data problems.
The remainder of the article is structured as follows.
Section~\ref{Sec:Background} introduces GLMs and the existing probability-based sub-sampling approach of \textcite{ai2021optimal} and \textcite{yao2021review}.
Our proposed model robust sub-sampling approach is introduced in Section~\ref{Sec:ModRobOptSubMethod}, which is embedded with a GLM framework.
A simulation study is then used to assess the performance of our model robust approach in Section~\ref{Sec:SimulationAndRealWorldSetup}, and two real world applications are presented.
Section~\ref{Sec:Discussion} concludes the article with a discussion of the results and some suggestions for future research.
\section{Background}\label{Sec:Background}
There are variety of ways Big data can be sub-sampled.
In this section, we focus on the approach where sub-sampling probabilities are determined for each data point, and the Big data is sub-sampled (at random) based on these probabilities.
Such an approach was first proposed by \textcite{wang2018logistic} for logistic regression problems in Big data settings, and has been extended to a wide range of regression problems (e.g., \textcite{yao2019softmax,wang2019linear,ai2020quantile,cheng2020IBOSSlogistic,ai2021optimal,yao2021review}).
In this section, we describe such a sub-sampling approach as applied to GLMs based on the work of \textcite{ai2021optimal}.
\subsection{Generalised Linear Models}\label{Sec:GLMs}
Let a Big data set be denoted as $F_N=(\bm{X}_0,\bm{y})$, where $\bm{X}_0=({\bm{x}_0}_1,\ldots,{\bm{x}_0}_N)^T \in R^{N \times p}$ represents a data matrix based on the Big data set with $p$ covariates, $\bm{y}=(y_1,\ldots,y_N)^T$ represents the response vector and $N$ is the total number of data points.
To fit a GLM, consider the model matrix $\bm{X}=h(\bm{X}_0) \in R^{N \times (p+q)}$ where $h(.)$ is some function of $\bm{X}_0$ which creates an additional $q$ columns representing, for example, an intercept and/or higher-order terms.
A GLM can then be defined via three components: 1) distribution of response $\bm{y}$, which is from the exponential family (e.g., Normal, Binomial or Poisson); 2) linear predictor $\bm{\eta}=\bm{X}\bm{\theta}$, where $\bm{\theta}=(\theta_1,\ldots,\theta_{p+q})^T$ is the parameter vector; and 3) link function $g(.)$, which links the mean of the response to the linear predictor \parencite{nelder1972generalized}.
Throughout this article, the inverse link function $g^{-1}(.)$ is denoted by $u(.)$.
A common exponential form for the probability density or mass function of $y$ can be written as:
\begin{equation}\label{Eq:ypdf}
f(y;\omega,\gamma)=\exp{\Big(\frac{y\omega - \psi(\omega)}{a(\gamma)} + b(y,\gamma) \Big)},
\end{equation}
where $\psi(.), a(.)$ and $b(.)$ are some functions, $\omega$ is known as the natural parameter and $\gamma$ the dispersion parameter.
Based on Equation \eqref{Eq:ypdf} the link function $g(.)$ can then be defined as $g(\bm{\mu}) = \bm{\eta}$, where $\mu = E[y|\bm{X},\bm{\theta}] = \mbox{d} \psi(\omega)/\mbox{d}\omega$.
A general linear model, or linear regression model, is a special case of a GLM where $\bm{y} \sim N(\bm{\mu},\bm{\Sigma})$ and $g(.)$ is the identity link function $g(\bm{\mu})=\bm{\mu}$, such that $\bm{\mu}=\bm{X}\bm{\theta}$.
For logistic regression, $\bm{y} \sim \mbox{Bin}(n,\bm{\pi})$ and $g(.)$ is the logit link function $g(\bm{\pi})=\log(\bm{\pi}/(1-\bm{\pi}))$ such that $g(\bm{\pi})=\bm{X}\bm{\theta}$. Similarly, for Poisson regression, $\bm{y} \sim \mbox{Poisson}(\bm{\lambda})$ and $g(.)$ is the log link function $g(\bm{\lambda})=\log(\bm{\lambda})$ such that $g(\bm{\lambda})=\bm{X}\bm{\theta}$.
Note that the dispersion parameter $\gamma=1$ for the logistic and Poisson regression models.
\subsection{A general sub-sampling algorithm for GLMs}\label{Sec:GenSamAlgoGLMs}
As described by \textcite{ai2021optimal}, consider a general sub-sampling approach to estimate parameters $\bm{\theta}$ through a weighted log-likelihood function for GLMs (weights are the inverse of the sub-sampling probabilities).
A weighted likelihood function is considered, since an unweighted likelihood leads to biased estimates of model parameters for logistic regression, see \textcite{wang2019more}.
Define $\phi_i$ as the probability that row $i$ of $F_N$ is randomly selected, for $i=1,\ldots,N$, where $\sum_{i=1}^{N} \phi_i=1$ and $\phi_i \in (0,1)$.
A sub-sample $S$ of size $r$ is then drawn with replacement from $F_N$ based on $\bm{\phi}=(\phi_1,\ldots,\phi_N)$.
The selected responses, covariates and sub-sampling probabilities are then used to estimate the model parameters.
Pseudo-code for this general sub-sampling approach is provided in Algorithm~\ref{Algo:GenSam}.
\begin{algorithm}[htbp!]
\SetAlgoLined
\textbf{Sampling:} Assign $\phi_i,i=1,...,N,$ for $F_N$. \\ \nonl
Based on $\bm{\phi}$, draw an $r$ size sub-sample with replacement from $F_N$ to yield $S = \{\bm{x}_l^*,y_l^*,\phi_l^*\}_{l=1}^r = (\bm{X}^*,\bm{y}^*,\bm{\phi}^*)$. \\
\textbf{Estimation:} Based on $S$, find:
\begin{equation*}
\begin{aligned}
\tilde{\bm{\theta}} = \argmaxA_{\bm{\theta}} ~ \log{L(\bm{\theta}|\bm{X}^*,\bm{y}^*,\bm{\phi}^*)} \equiv \argmaxA_{\bm{\theta}}~ \frac{1}{r}\sum_{l=1}^{r} \frac{y^*_l u(\bm{\theta}^T\bm{x}^*_l) - \psi(u(\bm{\theta}^T\bm{x}^*_l))}{\phi^*_l}.
\end{aligned}
\end{equation*} \\
\textbf{Output:} $\tilde{\bm{\theta}}$ and $S$.
\caption{General sub-sampling algorithm \parencite{ai2021optimal}} \label{Algo:GenSam}
\end{algorithm}
From Algorithm~\ref{Algo:GenSam}, the first step is to assign sub-sampling probabilities $\phi_i$ to the rows of $F_N$.
The simplest approach is to assign each data point an equal probability of being selected.
These probabilities could also depend on the composition of $\bm{y}$, e.g., for binary data, one could sample proportional to the inverse of the number of successes and failures.
Based on these probabilities, a sub-sample $S$ of size $r$ is then drawn completely at random (with replacement) from $F_N$ to yield $\{\bm{x}^*_l,y^*_l,\phi^*_l\}_{l=1}^r$.
Based on this sub-sample, model parameters $\tilde{\bm{\theta}}$ are estimated via maximising a weighted log-likelihood function.
The estimates $\tilde{\bm{\theta}}$ can then be considered as estimates of what would be obtained if the whole Big data set were analysed.
The asymptotic properties of $\tilde{\bm{\theta}}$ based on the general sub-sampling approach given in Algorithm~\ref{Algo:GenSam} were derived by \textcite{ai2021optimal}.
These properties are outlined below as they form the basis for our extensions to model robust sub-sampling detailed in Section~\ref{Sec:ModRobOptSubMethod}.
\subsection[Asymptotic properties of \texorpdfstring{$\tilde{\bm{\theta}}$} from the general sub-sampling algorithm]{Asymptotic properties of $\tilde{\bm{\theta}}$ from the general sub-sampling algorithm} \label{Sec:Asymptotic}
To explore the asymptotic properties of the estimates obtained from the general sub-sampling algorithm, \textcite{ai2021optimal} made a number of assumptions which are outlined below.
To follow these, note that $\dot{\psi}(u(\bm{\theta}^T\bm{x}_i))$ and $\dot{u}(\bm{\theta}^T\bm{x}_i)$ denote the first-order derivatives of $\psi(u(\bm{\theta}^T\bm{x}_i))$ and $u(\bm{\theta}^T\bm{x}_i)$ (with respect to $\bm{\theta}$), respectively, with two dots similarly denoting the second-order derivatives.
In addition, the Euclidean norm of a vector $\bm{a}$ will be denoted as $||\bm{a}||=(\bm{a}^T\bm{a})^{1/2}$.
\begin{assumption}\label{Ass:1}
Assume that $\bm{X}\bm{\theta}$ lies in the interior of a compact set $K \in \Omega$ almost surely.
\end{assumption}
\begin{assumption}\label{Ass:2}
The regression coefficient $\bm{\theta}$ is an inner point of the compact domain $\Lambda_B=\{\bm{\theta} \in R^p : ||\bm{\theta}|| \leq B \}$ for some constant $B$.
\end{assumption}
\begin{assumption}\label{Ass:3}
Central moments condition: $N^{-1}\sum_{i=1}^{N}|y_i-\dot{\psi}(u(\bm{\theta}^T\bm{x}_i))|^4 = O_P(1)$ for all $\bm{\theta} \in \Lambda_B$.
\end{assumption}
\begin{assumption}\label{Ass:4}
As $N \rightarrow \infty$, the observed information matrix
$$ \bm{J_X} := \frac{1}{N} \sum_{i=1}^{N} \{\ddot{u}(\hat{\bm{\theta}}^T_{MLE}\bm{x}_i) \bm{x}_i \bm{x}^T_i[\dot{\psi}(u(\hat{\bm{\theta}}^T_{MLE}\bm{x}_i)) - y_i] + \ddot{\psi}(u(\hat{\bm{\theta}}^T_{MLE}\bm{x}_i))\dot{u}^2(\hat{\bm{\theta}}^T_{MLE}\bm{x}_i)\bm{x}_i \bm{x}^T_i\}$$
goes to a positive definite matrix in probability.
\end{assumption}
\begin{assumption}\label{Ass:5}
Require that the full sample covariates have finite $6$-th order moments, \,i.e., $E||\bm{x}_1||^6 \leq \infty$.
\end{assumption}
\begin{assumption}\label{Ass:6}
Assume $N^{-2}\sum_{i=1}^{N} ||\bm{x}_i||^s/ \phi_i = O_P(1)$ for $s=2,4$.
\end{assumption}
\begin{assumption}\label{Ass:7}
For $\delta=0$ and some $\delta > 0$, assume
$$\frac{1}{N^{2+\delta}} \sum_{i=1}^{N} \frac{|y_i - \dot{\psi}_i(u(\hat{\bm{\theta}}^T_{MLE}\bm{x}_i))|^{2+\delta} ||\dot{u}(\hat{\bm{\theta}}^T_{MLE}\bm{x}_i)\bm{x}_i||^{2+\delta}}{\phi^{1+\delta}_i} = O_P(1).$$
\end{assumption}
Assumptions H\ref{Ass:1} and H\ref{Ass:2} ensure that $\mbox{E}[y_i|\bm{x}_i] < \infty$ for all $i$, and was used by \textcite{clemenccon2014scaling} when investigating the impact of survey sampling with unequal inclusion probabilities on (stochastic) gradient descent-based-estimation methods in Big data problems.
H\ref{Ass:2} defines an admissible set which is required for ensuring consistency of estimates for model parameters for GLMs, see \textcite{fahrmeir1985consistency}.
Assumption H\ref{Ass:4} ensures (asymptotically) that $\bm{J_X}$, the observed Fisher information, is defined for the given model, and that it is positive definite and therefore non-singular.
To obtain a Bahadur representation, that is often useful in determining the asymptotic properties of statistical estimators of the sub-sampled estimator, Assumptions H\ref{Ass:3} and H\ref{Ass:5} are needed.
H\ref{Ass:6} and H\ref{Ass:7} are moment conditions on covariates and sub-sampling probabilities.
Assumption H\ref{Ass:7} is required by the Lindeberg-Feller central limit theorem, see \textcite{van2000asymptotic}.
Based on the above assumptions, the following theorems were proved by \textcite{ai2021optimal}.
The first theorem proves that the estimator from the sub-sampling algorithm is consistent where $||\tilde{\bm{\theta}} - \hat{\bm{\theta}}_{MLE}||$ can be made small with a large sub-sample size $r$, and $\hat{\bm{\theta}}_{MLE}$ is the maximum likelihood estimator of $\bm{\theta}$ based on $F_N$.
The second theorem proves that the approximation error, $\tilde{\bm{\theta}} - \hat{\bm{\theta}}_{MLE}$, given $F_N$ is approximately asymptotically Normally distributed with mean zero and variance $\bm{V}$.
\begin{theorem}\label{The:1}
If H\ref{Ass:1} to H\ref{Ass:7} hold then, as $N \rightarrow \infty$ and $r \rightarrow \infty$, $\tilde{\bm{\theta}}$ is consistent to $\hat{\bm{\theta}}_{MLE}$ in conditional probability given $F_N$. Moreover, the rate of convergence is $r^{-\frac{1}{2}}$. That is, with probability approaching one, for any $\epsilon > 0$, there exist finite $\Delta_{\epsilon}$ and $r_{\epsilon}$ such that
\begin{equation*}
P(||\tilde{\bm{\theta}} - \hat{\bm{\theta}}_{MLE}|| \ge r^{-\frac{1}{2}} \Delta_{\epsilon} | F_N ) < \epsilon, \forall r > r_{\epsilon}.
\end{equation*}
\end{theorem}
\begin{theorem}\label{The:2}
If H\ref{Ass:1} to H\ref{Ass:7} hold, then as $N \rightarrow \infty$ and $r \rightarrow \infty$, conditional on $F_N$ in probability,
\begin{equation*}
\bm{V}^{-1/2}(\tilde{\bm{\theta}}-\hat{\bm{\theta}}_{MLE}) \rightarrow N(\bm{0},\bm{I}),
\end{equation*}
in distribution, where $\bm{V}= \bm{J}^{-1}_{\bm{X}} \bm{V}_c \bm{J}^{-1}_{\bm{X}}= O_P(r^{-1})$ and
\begin{equation*}
\bm{V}_c = \frac{1}{rN^2} \sum_{i=1}^{N} \frac{\{y_i - \dot{\psi}(u(\hat{\bm{\theta}}^T_{MLE}\bm{x}_i))\}^2 \dot{u}^2(\hat{\bm{\theta}}^T_{MLE}\bm{x}_i)\bm{x}_i \bm{x}^T_i}{\phi_i}.
\end{equation*}
\end{theorem}
When applying the general sub-sampling algorithm, it may not be clear how to appropriately choose $\bm{\phi}$ depending upon the goal of the analysis (e.g., parameter estimation, response prediction, etc).
To address this, \textcite{ai2021optimal} proposed determining $\bm{\phi}$ based on an optimality criterion from experimental design (specifically $A$-optimality and $L$-optimality), and this led to the proposal of Theorems~\ref{The:3} and \ref{The:4} (below).
Such theorems provide the optimal choice for the sub-sampling probabilities to minimise the asymptotic mean squared error of $\tilde{\bm{\theta}}$ (or $\mbox{tr}(\bm{V})$) and $\bm{J_X} \tilde{\bm{\theta}}$ (or $\mbox{tr}(\bm{V}_c)$), denoted as $\bm{\phi}^{mMSE}$ and $\bm{\phi}^{mV_c}$, respectively.
We note that such sub-sampling probabilities are conditional on the assumption of a model being appropriate to describe the Big data.
\begin{theorem}\label{The:3}
The sub-sampling strategy is $A$-optimal if the sub-sampling probability is chosen such that
\begin{equation}
\phi^{mMSE}_i = \frac{|y_i - \dot{\psi}(u(\hat{\bm{\theta}}^T_{MLE}\bm{x}_i))|\, ||\bm{J}^{-1}_{\bm{X}} \dot{u}(\hat{\bm{\theta}}^T_{MLE}\bm{x}_i)\bm{x}_i ||}{\sum_{j=1}^{N} |y_j - \dot{\psi}(u(\hat{\bm{\theta}}^T_{MLE}\bm{x}_j))|\, ||\bm{J}^{-1}_{\bm{X}} \dot{u}(\hat{\bm{\theta}}^T_{MLE}\bm{x}_j)\bm{x}_j ||}, i=1,\ldots,N.
\end{equation}
\end{theorem}
Optimal sub-sampling probabilities from Theorem~\ref{The:3} can be computationally expensive as they require the calculation of $\bm{J}^{-1}_{\bm{X}}$.
Hence, the linear- or $L$- optimality criterion was proposed by \textcite{ai2021optimal}, which minimises the asymptotic mean squared error of $\bm{J_X}\tilde{\bm{\theta}}$ or $\mbox{tr}(\bm{J_X} \bm{V} \bm{J_X})$ which corresponds to minimising the variance of a linear combination of the parameters \parencite{atkinson2007optimum}.
This led to the following theorem.
\begin{theorem}\label{The:4}
The sub-sampling strategy is $L$-optimal if the sub-sampling probability is chosen such that
\begin{equation}
\phi^{mV_c}_i = \frac{|y_i - \dot{\psi}(u(\hat{\bm{\theta}}^T_{MLE}\bm{x}_i))|\, || \dot{u}(\hat{\bm{\theta}}^T_{MLE}\bm{x}_i)\bm{x}_i||}{
\sum_{j=1}^{N} |y_j - \dot{\psi}(u(\hat{\bm{\theta}}^T_{MLE}\bm{x}_j))|\, ||\dot{u}(\hat{\bm{\theta}}^T_{MLE}\bm{x}_j)\bm{x}_j ||}, i=1,\ldots,N.
\end{equation}
\end{theorem}
\textcite{wang2018logistic} applied Theorems~\ref{The:3} and \ref{The:4} to obtain the following optimal sub-sampling probabilities for logistic regression:
\begin{align}
\phi^{mMSE}_i = \frac{|y_i - \pi_i|\,|| \bm{J}^{-1}_{\bm{X}} \bm{x}_i ||}{\sum_{j=1}^{N} |y_j - \pi_j|\,|| \bm{J}^{-1}_{\bm{X}} \bm{x}_j||}, & \quad \phi^{mV_c}_i = \frac{|y_i - \pi_i|\,||\bm{x}_i||}{\sum_{j=1}^{N} |y_j - \pi_j|\,||\bm{x}_j||}
\end{align}
with $y_i \in \{0, 1\}$, $\pi_i = \exp{(\hat{\bm{\theta}}^T_{MLE}\bm{x}_i)}/(1+\exp{(\hat{\bm{\theta}}^T_{MLE}\bm{x}_i)})$, $\bm{J_X}=N^{-1}\sum_{h=1}^{N} \pi_h(1-\pi_h)\bm{x}_h {\bm{x}_h}^T$ and $i=1,\ldots,N$.
\textcite{ai2021optimal} applied Theorems~\ref{The:3} and \ref{The:4} to obtain the following optimal sub-sampling probabilities for Poisson regression:
\begin{align}
\phi^{mMSE}_i = \frac{|y_i - \lambda_i|\,|| \bm{J}^{-1}_{\bm{X}} \bm{x}_i ||}{\sum_{j=1}^{N} |y_j - \lambda_j|\,|| \bm{J}^{-1}_{\bm{X}} \bm{x}_j||}, & \quad \phi^{mV_c}_i = \frac{|y_i - \lambda_i|\,||\bm{x}_i||}{\sum_{j=1}^{N} |y_j - \lambda_j|\,||\bm{x}_j||}
\end{align}
with $y_i \in N_0$ or non-negative integers, $\lambda_i = \exp{(\hat{\bm{\theta}}^T_{MLE}\bm{x}_i)}$, $\bm{J_X}=N^{-1}\sum_{h=1}^{N} \exp{(\hat{\bm{\theta}}^T_{MLE}\bm{x}_h)}\bm{x}_h {\bm{x}_h}^T$ and $i=1,\ldots,N$.
Unfortunately, in practice, the optimal sub-sampling probabilities $\bm{\phi}^{mMSE}$ and $\bm{\phi}^{mV_c}$ cannot be determined as they depend on $\hat{\bm{\theta}}_{MLE}$.
To address this, based on Theorems~\ref{The:1} and \ref{The:2}, \textcite{ai2021optimal} proposed a two stage sub-sampling strategy where an initial random sample of the Big data is used to estimate $\hat{\bm{\theta}}_{MLE}$; an estimate which is then used with the results from Theorems~\ref{The:3} and \ref{The:4} to provide estimates of optimal sub-sampling probabilities.
Such an approach is thus termed a two stage sub-sampling approach, and is outlined in the next section.
\subsection{Optimal sub-sampling algorithm for GLMs}\label{Sec:OptSubAlgGLMs}
Theorems~\ref{The:1} to \ref{The:4} provide a theoretical basis for the optimal sub-sampling algorithm of \textcite{ai2021optimal}.
In this section, we describe this approach for GLMs which is outlined in Algorithm~\ref{Algo:OSGLMAC}.
This is a two stage algorithm where the general sub-sampling algorithm is initially applied.
For this, the sub-sampling probabilities could be $1/N$ for all observations or may be based on a stratified sampling technique.
Based on this initial sub-sample, estimates $\tilde{\bm{\theta}}$ are obtained.
Then, optimal sub-sampling probabilities are estimated using results from Theorems~\ref{The:3} and \ref{The:4}, where $\hat{\bm{\theta}}_{MLE}$ is approximated by $\tilde{\bm{\theta}}$.
\begin{algorithm}[H]
\SetAlgoLined
\nonl \textbf{Stage 1} \\
\textbf{Random Sampling :} Assign $\bm{\phi} = (\phi_1,\ldots,\phi_N)$ for $F_N$. For example, in logistic regression $\phi_i=\phi^{prop}$ represents proportional sub-sampling probabilities that are based on the response composition, and $\phi_i=N^{-1}$ which is uniform sub-sampling probabilities for Poisson regression.\\ \nonl
According to $\bm{\phi}$ draw a random sub-sample of size $r_0$, $S_{r_0}=\{\bm{x}^{r_0}_l,y^{r_0}_l,\phi^{r_0}_l\}_{l=1}^{r_0} = (\bm{X}^{r_0},\bm{y}^{r_0},\bm{\phi}^{r_0})$. \\
\textbf{Estimation:} Based on $S_{r_0}$, find:
\begin{equation*}
\begin{aligned}
\tilde{\bm{\theta}}^{r_0} = \argmaxA_{\bm{\theta}} ~ \log{L(\bm{\theta}|\bm{X}^{r_0},\bm{y}^{r_0},\bm{\phi}^{r_0})} \equiv \argmaxA_{\bm{\theta}} ~ \frac{1}{r_0}\sum_{l=1}^{r_0} \Big[\frac{y^{r_0}_l u(\bm{\theta}^T\bm{x}^{r_0}_l) - \psi(u(\bm{\theta}^T\bm{x}^{r_0}_l))}{\phi^{r_0}_l} \Big].
\end{aligned}
\end{equation*} \nonl \\
\textbf{Stage 2} \\
\textbf{Optimal sub-sampling probability :} Estimate $\bm{\phi}^{mMSE}$ or $\bm{\phi}^{mV_c}$ from Theorems~\ref{The:3} and \ref{The:4} using $\tilde{\bm{\theta}}^{r_0}$ in place of $\hat{\bm{\theta}}_{MLE}$. \\
\textbf{Optimal sub-sampling and estimation:} Based on the estimated sub-sampling probabilities, draw a sub-sample $S_r$ completely at random (with replacement) of size $r$ from $F_N$, such that $S_{r}=\{\bm{x}^r_l,y^r_l,\phi^r_l\}_{l=1}^{r} = (\bm{X}^{r},\bm{y}^{r},\bm{\phi}^{r})$. \\ \nonl
Form $S_{r_0+r}$ by combining $S_{r_0}$ and $S_{r}$, and obtain:
\begin{align*}
\tilde{\bm{\theta}} = & \argmaxA_{\bm{\theta}} ~ \log{L(\bm{\theta}|\bm{X}^{r_0},\bm{y}^{r_0},\bm{\phi}^{r_0},\bm{X}^{r},\bm{y}^{r},\bm{\phi}^{r})} \notag \\
\equiv & \argmaxA_{\bm{\theta}} ~ \frac{1}{r_0+r} \Bigg[ \sum_{k \in \{r_0,r\}} \sum_{l=1}^{k} \frac{y^{k}_l u(\bm{\theta}^T\bm{x}^{k}_l) - \psi(u(\bm{\theta}^T \bm{x}^{k}_l))}{\phi^{k}_l} \Bigg] .
\end{align*} \\
\textbf{Output :} $\tilde{\bm{\theta}}$ and $S_{r_0+r}$.
\caption{Two stage optimal sub-sampling algorithm for GLMs.}\label{Algo:OSGLMAC}
\end{algorithm}
The first stage of the algorithm is as shown in Algorithm~\ref{Algo:GenSam}.
In the second stage, $r \ge r_0$ data points are sampled with replacement from the Big data through the optimal sub-sampling probabilities (estimated based on the model parameters obtained from stage one).
These two sub-sampled data sets are then combined to yield a single data set, and this data set is used to estimate the model parameters.
Once these estimates have been obtained, $\tilde{\bm{V}}$, the variance-covariance matrix of $\tilde{\bm{\theta}}$, can be estimated via $\tilde{\bm{V}}=\tilde{\bm{J}}^{-1}_{\bm{X}} \tilde{\bm{V}}_c \tilde{\bm{J}}^{-1}_{\bm{X}}$, where
\begin{align}\label{Eq:V_theta}
\tilde{\bm{J_X}} = & \frac{1}{N(r_0+r)} \Bigg[ \sum_{k \in \{r_0,r\}} \sum_{l=1}^{k} \frac{\ddot{u}(\tilde{\bm{\theta}}^T\bm{x}^k_l) \bm{x}^k_l (\bm{x}^k_l)^T [\ddot{\psi}(u(\tilde{\bm{\theta}}^T\bm{x}^k_l)) - y^k_l] + \ddot{\psi}(u(\tilde{\bm{\theta}}^T\bm{x}^k_l)) \dot{u}^2(\tilde{\bm{\theta}}^T\bm{x}^k_l) \bm{x}^k_l (\bm{x}^k_l)^T}{\phi^k_l} \Bigg] \notag \\
\mbox{and} & \quad \tilde{\bm{V}_c} = \frac{1}{N^2(r_0+r)^2} \Bigg[\sum_{k \in \{r_0,r\}} \sum_{l=1}^{k} \frac{[y^k_l -\dot{\psi}(u(\tilde{\bm{\theta}}^T\bm{x}^k_l))]^2 \dot{u}^2(\tilde{\bm{\theta}}^T\bm{x}^k_l) \bm{x}^k_l (\bm{x}^k_l)^T}{ (\phi^k_l)^2} \Bigg].
\end{align}
Specific versions of the above algorithm under the logistic and Poisson regression models are given in Appendix~\ref{Appendix:Algorithms} in Algorithms~\ref{Algo:OSMAC} and \ref{Algo:OSPAC}, respectively.
\subsection{Limitations of the optimal sub-sampling algorithm}
The above optimal sub-sampling approach has a number of limitations.
One such limitation is the computational expense involved in obtaining the optimal sub-sampling probabilities, as these need to be found for each data point in the Big data set.
To address this, \textcite{lee2021fast} introduced a faster two stage sub-sampling procedure for GLMs using the Johnson-Lindenstrauss Transform (JLT) and sub-sampled Randomised Hadamard Transform (SRHT), which are techniques to downsize matrix volume.
Another limitation is that, as the approximated optimal sub-sampling probabilities are proportional to $|y_i -\dot{\psi}(u(\tilde{\bm{\theta}}^T\bm{x}_i))|$, an observation with $y_i \approx \dot{\psi}(u(\tilde{\bm{\theta}}^T\bm{x}_i))$ has a near zero probability of being selected and data-points with $y_i = \dot{\psi}(u(\tilde{\bm{\theta}}^T\bm{x}_i))$ will be never sub-sampled.
\textcite{ai2021optimal} proposed to resolve this by introducing $\epsilon$ (a small positive value, e.g., $10^{-6}$) to constrain the optimal sub-sampling probabilities by replacing $[y_i -\dot{\psi}(u(\tilde{\bm{\theta}}^T\bm{x}_i))]$ with $\max{(|y_i -\dot{\psi}(u(\tilde{\bm{\theta}}^T\bm{x}_i))|,\epsilon)}$ which ensures such data points have a non-zero probability of being selected.
Lastly, one of the major limitations of the approach that has not been addressed previously is the inherent assumption that the Big data can be appropriately described by a given model.
That is, the sub-sampling probabilities are evaluated based on an assumed model, and they are only optimal for this model.
We suggest that this is a substantial limitation as specifying such a model in practice can be difficult.
This motivates the development of methods that yield sub-sampling probabilities that are robust to the choice of model, and our proposed approach for this is outlined next.
\section{Model robust optimal sub-sampling method}\label{Sec:ModRobOptSubMethod}
In order to apply the above two stage sub-sampling approach, optimal sub-sampling probabilities need to be evaluated, and these are based on a model that is assumed to appropriately describe the data.
In practice, determining such a model may be difficult, and there could be a variety of models that appropriately describe the data.
Hence, a sub-sampling approach that provides robustness to the choice of model is desirable.
For this, we propose to consider a set of $Q$ models which can be constructed to encapsulate a variety of scenarios that may be observed within the data.
For each model in this set, define model probabilities $\alpha_q$ for $q=1,\ldots,Q$ such that $\sum_{q=1}^Q\alpha_q=1$, which represents our {\it a priori} belief about the appropriateness of each model.
Denote the model matrix for the $q$-th model as $\bm{X}_q= h_q(\bm{X}_0)$, i.e., some function of the data matrix $\bm{X}_0$.
To apply a sub-sampling approach for this model set, sub-sampling probabilities are needed, and they should be constructed such that the resulting data is expected to address the analysis aim, regardless of which model is actually preferred for the Big data.
For this purpose, we propose to form these sub-sampling probabilities by taking a weighted average (based on $\alpha_q)$ of the sub-sampling probabilities that would be obtained for each model (singularly).
This is the basic approach of our model robust optimal sub-sampling algorithm, for which further details are provided below, including a theoretical basis for constructing the sub-sampling probabilities in this way.
\subsection{Properties of model robust optimal sub-sampling algorithm}\label{Sec:ProforGLMModRob}
Updating the notation of the model matrix from $\bm{X}$ to $\bm{X}_q$ for the $q$-th model subsequently leads to analogous definitions for $\bm{x}_{qi}$, $\bm{\theta}_{q}$, $\hat{\bm{\theta}}_{qMLE}$, $\tilde{\bm{\theta}}_{q}$, ${\bm{J_X}}_q$, $\bm{V}_q$ and ${\bm{V}_q}_c$.
Assuming H\ref{Ass:1} to H\ref{Ass:7} hold for each of the $q$ models, Theorems~\ref{The:1} and \ref{The:2} apply straightforwardly to each of the models.
Extending the ideas from Section~\ref{Sec:Asymptotic}, optimal sub-sampling probabilities can be selected based on certain optimality criteria to, for example, ensure efficient estimates of parameters across the $Q$ models.
This leads to the following theorems.
\begin{theorem}\label{The:5}
For a set of $Q$ models with model probability $\alpha_q$ for the $q$-th model, $q=1,\ldots Q$, if the sub-sampling probabilities are selected as follows:
\begin{equation}
\phi^{mMSE}_i = \sum_{q=1}^{Q} \alpha_q \frac{|y_i - \dot{\psi}(u(\hat{\bm{\theta}_q}_{MLE}^T\bm{x}_{qi}))|\, ||\bm{J}^{-1}_{\bm{X}_q} \dot{u}(\hat{\bm{\theta}_q}_{MLE}^T\bm{x}_{qi})\bm{x}_{qi} ||}{\sum_{j=1}^{N} |y_j - \dot{\psi}(u(\hat{\bm{\theta}_q}_{MLE}^T\bm{x}_{qj}))|\, ||\bm{J}^{-1}_{\bm{X}_q} \dot{u}(\hat{\bm{\theta}_q}_{MLE}^T\bm{x}_{qj})\bm{x}_{qj} ||},
\end{equation}
$i=1,\ldots,N$, and $\sum_{q=1}^{Q} \alpha_q = 1$, then $\sum_{q=1}^{Q} \alpha_q \mbox{tr}(\bm{V}_q)$ attains its minimum.
\end{theorem}
The proof of Theorem~\ref{The:5} is available in Appendix~\ref{Appendix:Theorem}, which is an extension of the proof of Theorem~\ref{The:3} in \textcite{ai2021optimal}.
\begin{theorem}\label{The:6}
For a set of $Q$ models with model probability $\alpha_q$ for the $q$-th model, $q=1,\ldots Q$, if the sub-sampling probabilities are selected as follows:
\begin{equation}
{\phi}_i^{mV_c} = \sum_{q=1}^{Q}\alpha_q \frac{|y_i - \dot{\psi}(u(\hat{\bm{\theta}_q}_{MLE}^T\bm{x}_{qi}))|\, || \dot{u}(\hat{\bm{\theta}_q}_{MLE}^T\bm{x}_{qi}) \bm{x}_{qi} ||}{\sum_{j=1}^{N} |y_j - \dot{\psi}(u(\hat{\bm{\theta}_q}_{MLE}^T\bm{x}_{qj}))|\, ||\dot{u}(\hat{\bm{\theta}_q}_{MLE}^T\bm{x}_{qj}) \bm{x}_{qj} ||},
\end{equation}
$i=1,\ldots,N$, and $\sum_{q=1}^{Q} \alpha_q = 1$, then $\sum_{q=1}^Q \alpha_q \mbox{tr}({\bm{V}_q}_c)$ attains its minimum.
\end{theorem}
The proof of Theorem \ref{The:6} follows straightforwardly from the proof of Theorem~\ref{The:5}.
The above theorems can be applied to obtain optimal sub-sampling probabilities under the logistic regression model as follows:
\begin{align}
{\phi}^{mMSE}_i = \sum_{q=1}^{Q} \alpha_q \frac{|y_i - {\pi_q}_i| \, || \bm{J}^{-1}_{\bm{X}_q} \bm{x}_{qi} ||}{\sum_{j=1}^{N} |y_j - {\pi_q}_j| \, || \bm{J}^{-1}_{\bm{X}_q} \bm{x}_{qj} ||}, & \quad {\phi}^{mV_c}_i = \sum_{q=1}^{Q} \alpha_q \frac{|y_i - {\pi_q}_i| \, || \bm{x}_{qi}||}{\sum_{j=1}^{N} |y_j - {\pi_q}_j| \, || \bm{x}_{qj} ||}
\end{align}
with $y_i \in \{0,1\}$, ${\pi_q}_i = \exp{(\hat{\bm{\theta}}_{qMLE}^T\bm{x}_{qi})}/(1+\exp{(\hat{\bm{\theta}}_{qMLE}^T\bm{x}_{qi} )}),$ $\bm{J}_{\bm{X}_q}=N^{-1}\sum_{h=1}^{N} {\pi_q}_h(1-{\pi_q}_h) \bm{x}_{qh} (\bm{x}_{qh})^T$ and $i=1,\ldots,N$.
Similarly, optimal sub-sampling probabilities under the Poisson regression model can be obtained as follows:
\begin{align}
{\phi}^{mMSE}_i = \sum_{q=1}^{Q} \alpha_q \frac{|y_i - {\lambda_q}_i|\,|| \bm{J}^{-1}_{\bm{X}_q} \bm{x}_{qi} ||}{\sum_{j=1}^{N} |y_j - {\lambda_q}_j|\,|| \bm{J}^{-1}_{\bm{X}_q} \bm{x}_{qj}||}, & \quad \phi^{mV_c}_i = \sum_{q=1}^{Q} \alpha_q \frac{|y_i - {\lambda_q}_i|\,|| \bm{x}_{qi} || }{\sum_{j=1}^{N} |y_j - {\lambda_q}_j|\,|| \bm{x}_{qj} ||}
\end{align}
with $y_i \in N_0$ or non-negative integers, ${\lambda_q}_i = \exp{(\hat{\bm{\theta}}_{qMLE}^T\bm{x}_{qi})},$ $\bm{J}_{\bm{X}_q}=N^{-1}\sum_{h=1}^{N} {\lambda_q}_h \bm{x}_{qh} (\bm{x}_{qh})^T$ and $i=1,\ldots,N$.
As noted in Section~\ref{Sec:Asymptotic} and by \textcite{ai2021optimal}, these model robust optimal sub-sampling probabilities are based on the maximum likelihood estimator found by considering the whole Big data set.
Hence, a two stage procedure similar to Algorithm~\ref{Algo:OSGLMAC} is proposed for model robust sub-sampling, and this is outlined next.
\subsection{Model robust optimal sub-sampling algorithm for GLMs}\label{Sec:ModRobOptSubAlgforGLMs}
The two stage model robust optimal sub-sampling algorithm for GLMs is presented in Algorithm~\ref{Algo:OSGLMACMR}, where the estimates of sub-sampling probabilities based on the results of Theorems~\ref{The:5} and \ref{The:6} are used.
The specific algorithms for logistic and Poisson regression are available in Appendix~\ref{Appendix:Algorithms} as Algorithms~\ref{Algo:OSMACMR} and \ref{Algo:OSPACMR}, respectively.
\begin{algorithm}[H]
\SetAlgoLined
\nonl \textbf{Stage 1} \\
\textbf{Random Sampling :} Assign $\bm{\phi} = (\phi_1,\ldots,\phi_N)$ for $F_{N}$. For example, in logistic regression $\phi_i=\phi^{prop}$ represents proportional sub-sampling probabilities that are based on the response composition, and $\phi_i=N^{-1}$ which is uniform sub-sampling probabilities for Poisson regression.\\ \nonl
According to $\bm{\phi}$ draw random sub-samples of size $r_0$, such that $S_{r_0}=\{h_q({\bm{x}_0}^{r_0}_{l}),y^{r_0}_l,\phi^{r_0}_l\}_{l=1}^{r_0} = (h_q(\bm{X}_0^{r_0}),\bm{y}^{r_0},\bm{\phi}^{r_0})$ for $q=1,\ldots,Q$. \\
\textbf{Estimation:} For $q=1,\ldots,Q$ and $S_{r_0}$, find:
\begin{equation*}
\begin{aligned}
\tilde{\bm{\theta}^{r_0}_{q}} = \argmaxA_{\bm{\theta}_q} ~ \log{L(\bm{\theta}_q| \bm{X}^{r_0}_{q},\bm{y}^{r_0},\bm{\phi}^{r_0})} \equiv \argmaxA_{\bm{\theta}_q} ~ \frac{1}{r_0}\sum_{l=1}^{r_0} \Big[\frac{y^{r_0}_l u(\bm{\theta}_q^T \bm{x}^{r_0}_{ql}) - \psi(u(\bm{\theta}_q^T\bm{x}^{r_0}_{ql})}{\phi^{r_0}_l} \Big].
\end{aligned}
\end{equation*} \nonl \\
\textbf{Stage 2} \\
\textbf{Optimal sub-sampling probability:} Estimate optimal sub-sampling probabilities $\bm{\phi}^{mMSE}$ or $\bm{\phi}^{mV_c}$ using the results from Theorem~\ref{The:5} or \ref{The:6}, with $\hat{\bm{\theta}}_{qMLE}$ replaced by $\tilde{\bm{\theta}}^{r_0}_{q}$.\\
\textbf{Optimal sub-sampling and estimation:} Based on $\bm{\phi}^{mMSE}$ or $\bm{\phi}^{mV_c}$, draw sub-sample of size $r$ from $F_{N}$, such that $S_{r}=\{h_q({\bm{x}_0}^r_{l}),\bm{y}^r_l,\bm{\phi}^r_l\}_{l=1}^{r} = (h_q(\bm{X}_0^{r}),\bm{y}^{r},\bm{\phi}^{r})$ for $q=1,\ldots,Q$. \\ \nonl
Combine $S_{r_0}$ and $S_{r}$ to form $S_{(r_0+r)}$, and obtain :
\begin{align*}
\tilde{\bm{\theta}_{q}} = & \argmaxA_{\bm{\theta}_q} ~ \log{\big(L(\bm{\theta}_q| \bm{X}^{r_0}_q,\bm{y}^{r_0},\bm{\phi}^{r_0}, \bm{X}^{r}_q,\bm{y}^{r},\bm{\phi}^{r})\big)} \notag \\
\equiv & \argmaxA_{\bm{\theta}_q} ~ \frac{1}{r_0+r} \Bigg[\sum_{k \in \{ r_0,r\}} \sum_{l=1}^{k} \frac{y^{k}_l u(\bm{\theta}^T_q\bm{x}^{k}_{ql}) - \psi(u(\bm{\theta}^T_q\bm{x}^{k}_{ql}))}{\phi^{k}_l} \Bigg].
\end{align*} \\
\textbf{Output:} $\tilde{\bm{\theta}}_q$ and $S_{(r_0+r)}$ for $q=1,\ldots,Q$.
\caption{Two stage model robust optimal sub-sampling algorithm for GLMs.}\label{Algo:OSGLMACMR}
\end{algorithm}
Similar to Algorithm~\ref{Algo:OSGLMAC}, the first stage entails randomly sub-sampling $F_{N}$ (with replacement), and estimating model parameters for each of the $Q$ models.
Based on these estimated parameters, model specific sub-sampling probabilities are obtained, and these are combined based on $\alpha_q$, forming model robust optimal sub-sampling probabilities.
Subsequently, $r\ge r_0$ data points are sampled from $F_{N}$.
The two sub-samples are then combined, and each of the $Q$ models are fitted (separately) based on the weighted log-likelihood function, which should yield efficient estimates of parameters across each of the models.
Similar to estimating the variance-covariance matrix of a single model (Equation \eqref{Eq:V_theta}), $\tilde{\bm{V}}_q$, the variance-covariance matrix of $\tilde{\bm{\theta}_q}$ can be estimated, for $q=1,\ldots,Q$.
In the following section, the proposed model robust optimal sub-sampling method and the current optimal sub-sampling method are assessed through a simulation study and real world scenarios.
\section{Applications of optimal sub-sampling algorithms}\label{Sec:SimulationAndRealWorldSetup}
In this section, a simulation study and two real world applications are used to assess the performance of the proposed model robust optimal sub-sampling algorithm (Algorithm~\ref{Algo:OSGLMACMR}) compared to the optimal sub-sampling algorithm (Algorithm~\ref{Algo:OSGLMAC}), and random sampling.
The main results are presented in this section with some results presented in the Appendix.
The simulation study and real world applications were coded in the R statistical programming language with the help of Rstudio IDE, and our scripts are available through Github.
Appendix~\ref{Appendix:Code} provides specific Github hyperlinks to the code repositories for the simulation study and real world applications.
\subsection{Simulation study design}\label{Sec:Simulation}
To explore the performance of our model robust sub-sampling approach, a simulation study was constructed based on the logistic and Poisson regression models.
For each case, a set of $Q=4$ models were assumed based on \textcite{shi2021model}, and this set is summarised in Table~\ref{Tab:1}.
For each model, $F_{N}$ was constructed, by assuming a distribution for the covariates and corresponding response.
The performance of the three sampling methods were then compared for each $F_{N}$ through evaluating six scenarios: 1) random sampling to estimate the parameters of the data generating model; 2) optimal sub-sampling under the data generating model --
this simulates the case where an appropriate model was assumed for describing the Big data; 3)-5) optimal sub-sampling under alternative models i.e., optimal sub-sampling to estimate the parameters of the data generating model but where samples were obtained based on an alternative model -- this simulates undertaking optimal sub-sampling based on a `wrong' model; 6) model robust optimal sub-sampling, with $\alpha_q=1/4$ used to estimate the parameters of the data generating model, for $q=1,\ldots,4$.
To quantitatively compare each approach, the simulated Mean Squared Error (SMSE) was used. This was evaluated as follows:
\begin{equation}\label{Eq:SMSE}
SMSE(\bm{\theta}) = \frac{1}{M}{\sum_{m=1}^{M} \sum_{n=1}^{p+q} (\tilde{\theta}_{nm} - \theta_n)^2 },
\end{equation}
where $M$ is the number of simulations, $p+q$ is the number of parameters, $\theta_n$ is the $n$-th underlying parameter of the data generating model and $\tilde{\theta}_{nm}$ is the estimate of this parameter from the $m$-th simulation.
In addition, the determinant of the observed Fisher information matrix (i.e., the inverse of the variance-covariance matrix $\bm{V}$) was also compared across sub-sampling approaches, with the largest of such values being preferred.
For each simulation, $N=10000$, $r_0=100$, $r=100,200,\ldots,1400$ and $M=1000$.
\begin{table}[H]
\centering
\caption{Model set assumed for simulation study.}\label{Tab:1}
\begin{tabular}{l} \hline
\textbf{Model set} \\ \hline
$\theta_0 + \theta_1 \bm{x}_1 + \theta_2 \bm{x}_2$ \\
$\theta_0 + \theta_1 \bm{x}_1 + \theta_2 \bm{x}_2 + \theta_3 \bm{x}^2_1$ \\ $\theta_0 + \theta_1 \bm{x}_1 + \theta_2 \bm{x}_2 + \theta_3 \bm{x}^2_2$ \\ $\theta_0 + \theta_1 \bm{x}_1 + \theta_2 \bm{x}_2 + \theta_3 \bm{x}^2_1 + \theta_4 \bm{x}^2_2$ \\ \hline
\end{tabular}
\end{table}
Additionally, we only consider the $A$-optimal ($mMSE$) sub-sampling strategy in these simulations, as \textcite{wang2018logistic,ai2021optimal} indicated that this approach generally outperformed the $L$-optimal ($mV_c$) strategy.
We explore both optimality criteria in the real-world applications.
\subsubsection{Logistic regression}\label{Sec:LogisticRegression}
Following \textcite{wang2018logistic}, covariate data for the logistic regression model were simulated based on two distributions: Exponential ($\lambda$) and Multivariate Normal ($\bm{\mu},\bm{\Sigma}$).
The values of ($\lambda,\bm{\mu},\bm{\Sigma}$) and $\bm{\theta}$ are given in Table~\ref{Tab:3} for each data generating model.
For all models, the first element of $\bm{\theta}$ is the value of the parameter for the intercept.
While $\lambda$, $\bm{\mu}$ and $\bm{\Sigma}$ were selected arbitrarily, $\bm{\theta}$ was selected such that for $N=10000$, the data generating model was preferred over the alternative models based on the Akaike Information Criterion \parencite{akaike1974new}.
\begin{table}[H]
\centering
\caption{$\lambda,\bm{\mu},\bm{\Sigma}$ and $\bm{\theta}$ values are given to generate $x_1,x_2$ through Exponential and Multivariate Normal distributions and form $F_{N}$ for each data generating logistic regression model.}\label{Tab:3}
\begin{tabular}{lll} \hline
\multirow{3}{*}{\textbf{Covariates}} & \multicolumn{2}{c}{\textbf{Distributions}} \\ \cline{2-3}
& \textbf{Exponential} ($\lambda=\sqrt{3}$) & \textbf{Normal} $\Big(\bm{\mu}=[0,0]$,$\bm{\Sigma}= \begin{bmatrix} 1.5 & 0 \\ 0 & 1.5 \end{bmatrix}\Big)$ \\ \hline
$\bm{x}_1,\bm{x}_2$ & $\bm{\theta}=[-2,\hphantom{-}1.5,\hphantom{-}0.3]$ & $\bm{\theta}=[-1,\hphantom{-}0.5,\hphantom{-}0.1]$\\
$\bm{x}_1, \bm{x}_2, \bm{x}^2_1$ & $\bm{\theta}=[-2,\hphantom{-}1.7,-1.2,\hphantom{-}0.2]$ & $\bm{\theta}=[-1,\hphantom{-}0.5,\hphantom{-}0.5,\hphantom{-}0.7]$ \\
$\bm{x}_1, \bm{x}_2, \bm{x}^2_2$ & $\bm{\theta}=[-2,-1.3,\hphantom{-}1.9,\hphantom{-}0.9]$ & $\bm{\theta}=[-1,\hphantom{-}0.5,\hphantom{-}0.5,\hphantom{-}0.3]$ \\
$\bm{x}_1, \bm{x}_2, \bm{x}^2_1, \bm{x}^2_2$ & $\bm{\theta}=[-2,\hphantom{-}1.9,\hphantom{-}1.9,\hphantom{-}0.9,\hphantom{-}0.7]$ & $\bm{\theta}=[-1,\hphantom{-}0.5,\hphantom{-}0.5,\hphantom{-}0.5,\hphantom{-}0.5]$ \\ \hline
\end{tabular}
\end{table}
\begin{figure}
\caption{(a) Model information (larger is better) and (b) SMSE (smaller is better) for the sub-sampling methods for the logistic regression model under $mMSE$. Covariate data were generated from the Multivariate Normal distribution.}
\label{Fig:RWA_LR_TV_Nor}
\end{figure}
Figure~\ref{Fig:RWA_LR_TV_Nor} provides summaries of the SMSE and average model information over all models when the covariate data is generated through a Multivariate Normal distribution.
The SMSE and average model information indicate that, under optimal sub-sampling for $mMSE$, the data generating model is typically preferred within the model set.
This is expected as it is the case where the appropriate data generating model was correctly assumed to describe the Big data.
Of note, the proposed model robust approach performs similarly to the optimal sub-sampling approach.
Notable increases in the SMSE and decreases in the model information are observed when the incorrect model is considered for optimal sub-sampling compared to the data generating model.
Overall, random sampling tends to have the worst performance.
Similar results were obtained when the covariate data were generated through an Exponential distribution (Appendix~\ref{Appendix:Figures}).
\subsubsection{Poisson regression}\label{Sec:PoissonRegression}
The simulation study based on Poisson regression was constructed similar to the logistic regression case.
In terms of generating covariate values, Uniform and Multivariate Normal ($\bm{\mu}, \bm{\Sigma}$) distributions were used, see Table~\ref{Tab:5} and \textcite{ai2021optimal}.
Values for $\bm{\theta}$ were selected as described above, and are given in this table.
\begin{table}[htbp!]
\centering
\caption{$\bm{\mu},\bm{\Sigma}$ and $\bm{\theta}$ values are given to generate $x_1,x_2$ through uniform and Multivariate Normal distributions and form $F_{N}$ for each data generating Poisson regression model.}
\label{Tab:5}
\begin{tabular}{lll} \hline
\multirow{3}{*}{\textbf{Covariates}} & \multicolumn{2}{c}{\textbf{Distributions}} \\ \cline{2-3}
& \textbf{Uniform} & \textbf{Normal} $\Big(\bm{\mu}=[0,0]$,$\bm{\Sigma}= \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\Big)$ \\ \hline
$\bm{x}_1,\bm{x}_2$ & $\bm{\theta}=[1,\hphantom{-}0.3,\hphantom{-}0.3]$ & $\bm{\theta}=[1,\hphantom{-}0.5,\hphantom{-}0.1]$\\
$\bm{x}_1, \bm{x}_2, \bm{x}^2_1$ & $\bm{\theta}=[1,\hphantom{-}0.7,-0.5,\hphantom{-}0.4]$ & $\bm{\theta}=[1,-0.3,\hphantom{-}0.3,\hphantom{-}0.1]$ \\
$\bm{x}_1, \bm{x}_2, \bm{x}^2_2$ & $\bm{\theta}=[1,-0.4,\hphantom{-}0.5,\hphantom{-}0.3]$ & $\bm{\theta}=[1,\hphantom{-}0.3,-0.3,\hphantom{-}0.1]$ \\
$\bm{x}_1, \bm{x}_2, \bm{x}^2_1, \bm{x}^2_2$ & $\bm{\theta}=[1,\hphantom{-}0.5,\hphantom{-}0.5,-0.3,\hphantom{-}0.5]$ & $\bm{\theta}=[1,\hphantom{-}0.3,\hphantom{-}0.3,-0.3,\hphantom{-}0.1]$ \\ \hline
\end{tabular}
\end{table}
SMSE and average model information over all models when the covariate data was generated from a Multivariate Normal distribution and the response from generated form a Poisson regression model are shown in Figure~\ref{Fig:RWA_PR_TV_Nor}.
Generally, random sampling performs worst, while the proposed model robust approach and the optimal sub-sampling method based on the data generating model perform the best, and have similar SMSE and average model information values.
Again, the use of the optimal sub-sampling algorithm can lead to notable increases in SMSE when the assumed model is incorrect.
Similar results were obtained when the covariate data was generated from a uniform distribution (see Appendix~\ref{Appendix:Figures}).
\begin{figure}
\caption{(a) Model information (larger is better) and (b) SMSE (smaller is better) for the sub-sampling methods for the Poisson regression model under $mMSE$. Covariate data were generated from the standardised Multivariate Normal distribution.}
\label{Fig:RWA_PR_TV_Nor}
\end{figure}
\subsection{Real world applications}\label{Sec:RealWorldApplications}
The three sub-sampling methods are applied to analyse the ``Skin segmentation" and ``New York City taxi fare" data under logistic and Poisson regression, respectively.
In the simulation study, the parameters were specified for the data generating model.
However, in real world applications these are unknown.
In this situation the sub-sampling methods cannot be compared as in Section \ref{Sec:Simulation}.
Instead, for every model, the simulated mean squared error (SMSE) under each sub-sampling method is evaluated for various $r$ sub-sample sizes and $M$ simulations.
This can be evaluated as follows:
\begin{equation}
SSMSE(\hat{\bm{\theta}}_{MLE}) = \sum_{q=1}^{Q} SMSE_q(\hat{\bm{\theta}}_{MLE}),
\end{equation}
where $SMSE_q(\hat{\bm{\theta}}_{MLE})$ is the simulated mean squared error in Equation \eqref{Eq:SMSE} with $\bm{\theta}$ replaced by $\hat{\bm{\theta}}_{MLE}$.
In the following real world examples, the set of $Q$ models includes the main effects model, with intercept, and all possible combinations of quadratic terms for continuous covariates.
Again, these were constructed based on the work of \textcite{shi2021model}.
\subsubsection{Identifying skin from colours in images} \label{Sec:RealLogisticRegression}
\textcite{Skin_Data} considered the problem of identifying skin-like regions in images as part of the complex task of performing facial recognition.
For this purpose, \textcite{Skin_Data} collated the ``Skin segmentation" data set, which consists of RGB (R-red, G-green, B-blue) values of randomly sampled pixels from $N=245,057$ face images (out of which $50,859$ are skin samples and $194,198$ are non-skin samples) from various age groups, race groups and genders.
\textcite{bhatt2009efficient,binias2018pixel} applied multiple supervised machine learning algorithms to classify if images are skin or not based on the RGB colour data.
In addition, \textcite{abbas2019skin} conducted the same classification task for two different colour spaces, HSV (H-hue, S-saturation, V-value) and YCbCr (Y-luma component, Cb-blue difference chroma component, Cr-red difference chroma component), by transforming the RGB colour space.
We consider the same classification problem but use a logistic regression model.
Skin presence is denoted as one and skin absence is denoted as zero.
Each colour vector is scaled to have a mean of zero and a variance of one (initial range was between $0-255$).
To compare sub-sampling methods, we set $r_0=200,r=200,300,\ldots,1800$ for the sub-samples, and construct a set of $Q$ models by considering an intercept with main effects model with all covariates (scaled colors red,green and blue) as the base model, and form all alternative models by including different combinations of quadratic terms of all covariates.
This leads to $Q=8$.
Each of these models is considered equally likely {\it a priori}.
Figure~\ref{Fig:RWS_SkinData} shows the SSMSE values over the $Q$ models for various sub-sample sizes obtained by applying logistic regression to the ``Skin segmentation" data.
The proposed model robust approach performs similarly to the optimal sub-sampling method for $r=200$ and $r=300$.
However, when the sample size increases the optimal sub-sampling method performs poorly, and after $r=1300$ random sampling actually has lower SSMSE values than optimal sub-sampling.
The same is not true for our proposed model robust approach which has the lowest values of SSMSE throughout the selected sample sizes under $mMSE$ and $mV_c$ optimal sub-sampling criteria.
It is interesting that the optimal sub-sampling approach based on $mMSE$ performed worse than random sampling in comparison to the simulation study results.
Upon investigating this, it was found that it could potentially be explained by one of the models being particularly poor (subsequent higher SMSE with increasing $r$ values) for describing the data, and therefore led to inflated SSMSE values.
Despite this, we note that the model robust approach appears to perform well in general.
\begin{figure}
\caption{Summed SMSE over the available models for logistic regression applied on the ``Skin segmentation" data.}
\label{Fig:RWS_SkinData}
\end{figure}
\subsubsection{New York City taxi cab usage}\label{Sec:RealPoissonRegression}
New York City (NYC) taxi trip and fare information from 2009 onward, consisting of over $170$ million records each year, are publicly available courtesy of the New York City Taxi and Limousine Commission.
Some analyses of interest of these NYC taxi data include: a taxi driver's decision process to pick up a fair or cruise for customers which was modelled via a logistic regression model \parencite{yazici2013big}; taxi demand and how it is impacted by location, time, demographic information, socioeconomic status and employment status which was modelled via a multiple linear regression model \parencite{yang2014modeling}; and the dependence of taxi supply and demand on location and time considered via the Poisson regression model \parencite{yang2017modeling}.
In our application, we are interested in how taxi usage varies with day of the week (weekday/weekend), season (winter/spring/summer/autumn), fare amount, and cash payment.
The data used in our application is the ``New York City taxi fare'' data for the year 2013, hosted by the University of Illinois Urbana Champaign \parencite{illinoisdatabankIDB}.
Each data point includes the number of rides recorded against the medallion number (a license number provided for a motor vehicle to operate as a taxi) ($y$), weekday or not ($\bm{x_1}$), winter or not ($\bm{x_2}$), spring or not ($\bm{x_3}$), summer or not ($\bm{x_4}$), summed fare amount in dollars ($\bm{x_5}$), and the ratio of cash payment trips in terms of all trips ($\bm{x_6}$).
The continuous covariate $\bm{x_5}$ was scaled to have a range of zero and one.
Poisson regression was used to model the relationship between the number of rides per medallion and these covariates.
\begin{figure}
\caption{Summed SMSE over the available models for Poisson regression applied on the 2013 NYC taxi usage data.}
\label{Fig:RWS_NYCTaxiFareData}
\end{figure}
For our study, three sub-sampling methods are compared for the analysis of the taxi fare data, assigning $r_0=100$ and $r=200,\ldots,1900$ for sub-samples.
The model set consists of the main effects model($\bm{x_1},\bm{x_2},\bm{x_3},\bm{x_4},\bm{x_5},\bm{x_6}$) and all possible combinations of the quadratic terms of the continuous covariates ($\bm{x^2_5},\bm{x^2_6}$) which leads to $Q=4$.
Each of these models were considered equally likely {\it a priori}.
The SSMSE over the four models is shown in Figure~\ref{Fig:RWS_NYCTaxiFareData}.
Our proposed model robust approach outperforms the optimal sub-sampling method for almost all sample sizes for both $mMSE$ and $mV_c$ strategy.
Under $mV_c$, the model robust approach actually initially performs worse than the optimal sub-sampling approach but this is quickly reserved as $r$ is increased.
Random sampling performs the worst, suggesting that there is benefit in using targeted sampling approaches (as proposed here) over random selection.
\section{Discussion}\label{Sec:Discussion}
In this article, we proposed a model robust optimal sub-sampling approach for GLMs.
This new approach extends the current optimal sub-sampling approach by considering a set of models (rather than a single model) when determining the sub-sampling probabilities.
The exact formulation of these probabilities is derived using Theorems~\ref{The:5} and \ref{The:6}.
The robustness properties of this proposed approach were demonstrated in a simulation study, and in two real-world analysis problems, where it was shown to outperform optimal sub-sampling and random sampling.
Accordingly, we suggest that such an approach be considered in future Big data analysis problems.
The main limitation of the proposed approach that could be addressed in future research is extending the specification of the model set to a flexible class of models.
This could be, for example, through a generalised additive model \parencite{hastie1986generalized} or the inclusion of a discrepancy term in the linear predictor \parencite{krishna2021robust}.
Another avenue of interest that could also be explored is reducing the model set after stage one of the two-stage algorithm where models that clearly do not appear to be appropriate for the data could be dropped.
Both of these extensions are planned for future research.
\printbibliography[title={References},heading=bibnumbered]
\begin{appendices}\label{Appendix}
\section{Proof of Theorem~\ref{The:5}}\label{Appendix:Theorem}
\begin{proof}
\begin{align}
& \sum_{q=1}^{Q} \alpha_q \mbox{tr}{(\bm{V}_q)} = \sum_{q=1}^{Q} \alpha_q \mbox{tr}\Big( \bm{J}^{-1}_{\bm{X}_q} {\bm{V}_q}_c \bm{J}^{-1}_{\bm{X}_q} \Big) \notag \\
= & \frac{1}{N^2 r} \sum_{q=1}^{Q} \alpha_q \sum_{i=1}^{N} \mbox{tr}\Big[\frac{1}{{\phi_q}_i} \{y_i - \dot{\psi}(u(\hat{\bm{\theta}_q}^T_{MLE}\bm{x}_{qi}))\}^2 \bm{J}^{-1}_{\bm{X}_q} \notag \\ & \hspace{3cm} \dot{u}( \hat{\bm{\theta}_q}^T_{MLE} \bm{x}_{qi}) \bm{x}_{qi} [\dot{u}(\hat{\bm{\theta}_q}_{MLE}^T \bm{x}_{qi})\bm{x}_{qi}]^T \bm{J}^{-1}_{\bm{X}_q} \Big] \notag \\
= & \frac{1}{N^2 r} \sum_{q=1}^{Q} \alpha_q \sum_{i=1}^{N} \Big[\frac{1}{{\phi_q}_i} \{y_i - \dot{\psi}(u(\hat{\bm{\theta}_q}^T_{MLE} \bm{x}_{qi}))\}^2 || \bm{J}^{-1}_{\bm{X}_q} \dot{u}(\hat{\bm{\theta}_q}^T_{MLE} \bm{x}_{qi}) \bm{x}_{qi} ||^2 \Big] \notag \\
= & \frac{1}{N^2 r} \sum_{q=1}^{Q} \alpha_q \Big(\sum_{i=1}^{N} {\phi_q}_i \Big) \sum_{i=1}^{N} \Big[ \phi^{-1}_{qi} \{y_i - \dot{\psi} (u(\hat{\bm{\theta}_q}^T_{MLE} \bm{x}_{qi}))\}^2 || \bm{J}^{-1}_{\bm{X}_q} \dot{u}( \hat{\bm{\theta}_q}^T_{MLE} \bm{x}_{qi}) \bm{x}_{qi} ||^2 \Big] \notag \\
= & \frac{\alpha_1}{N^2 r} \Big(\sum_{i=1}^{N} {\phi_1}_i \Big) \sum_{i=1}^{N} \Big[ \phi^{-1}_{1i} \{y_i - \dot{\psi} (u(\hat{\bm{\theta}_1}^T_{MLE} \bm{x}_{1i}))\}^2 || \bm{J}^{-1}_{\bm{X}_1} \dot{u}( \hat{\bm{\theta}_1}^T_{MLE} \bm{x}_{1i}) \bm{x}_{1i} ||^2 \Big] + \ldots + \notag \\ & \frac{\alpha_Q}{N^2 r} \Big(\sum_{i=1}^{N} {\phi_Q}_i \Big) \sum_{i=1}^{N} \Big[ \phi^{-1}_{Qi} \{y_i - \dot{\psi} (u( \hat{\bm{\theta}_Q}^T_{MLE} \bm{x}_{Qi}))\}^2 || \bm{J}^{-1}_{\bm{X}_Q} \dot{u}(\hat{\bm{\theta}_Q}^T_{MLE} \bm{x}_{Qi}) \bm{x}_{Qi}||^2 \Big]\notag \\
\ge & \frac{\alpha_1}{N^2 r} \sum_{i=1}^{N} \{y_i - \dot{\psi} (u(\hat{\bm{\theta}_1}^T_{MLE} \bm{x}_{1i}))\}^2 || \bm{J}^{-1}_{\bm{X}_1} \dot{u}(\hat{\bm{\theta}_1}^T_{MLE} \bm{x}_{1i}) \bm{x}_{1i} ||^2 \Big] + \ldots + \notag \\ & \frac{\alpha_Q}{N^2 r} \sum_{i=1}^{N} \{y_i - \dot{\psi} (u(\hat{\bm{\theta}_Q}^T_{MLE} \bm{x}_{Qi}))\}^2 || \bm{J}^{-1}_{\bm{X}_Q} \dot{u}(\hat{\bm{\theta}_Q}^T_{MLE} \bm{x}_{Qi}) \bm{x}_{Qi} ||^2 \Big] \notag \\
= & \frac{1}{N^2 r} \sum_{q=1}^{Q} \alpha_q \sum_{i=1}^{N} \{y_i - \dot{\psi} (u(\hat{\bm{\theta}_q}^T_{MLE} \bm{x}_{qi}))\}^2 || \bm{J}^{-1}_{\bm{X}_q} \dot{u}( \hat{\bm{\theta}_q}^T_{MLE} \bm{x}_{qi}) \bm{x}_{qi} ||^2 \Big]
\end{align}
where the last inequality follows from the Cauchy-Schwarz inequality, and the equality in it holds if and only if
$${\phi_q}_i \propto |y_i - \dot{\psi} (u(\hat{\bm{\theta}_q}^T_{MLE} \bm{x}_{qi}))| \, || \bm{J}^{-1}_{\bm{X}_q} \dot{u}( \hat{\bm{\theta}_q}^T_{MLE} \bm{x}_{qi}) \bm{x}_{qi} ||.$$
Here we define $0/0=0$, and this equivalent to removing data points with $|y_i - \dot{\psi} (u(\hat{\bm{\theta}_q}^T_{MLE} \bm{x}_{qi}))| = 0$ in the expression of ${\bm{V}_q}_c$.
For the $q$-th model sub-sampling probabilities would be
\begin{equation}
{\phi_q}^{mMSE}_i = \frac{|y_i - \dot{\psi}(u( \hat{\bm{\theta}_q}^T_{MLE} \bm{x}_{qi}))|\, || \bm{J}^{-1}_{\bm{X}_q} \dot{u}(\hat{\bm{\theta}_q}^T_{MLE} \bm{x}_{qi}) \bm{x}_{qi} ||}{\sum_{j=1}^{N} |y_j - \dot{\psi}(u(\hat{\bm{\theta}_q}^T_{MLE} \bm{x}_{qj}))|\, || \bm{J}^{-1}_{\bm{X}_q} \dot{u}( \hat{\bm{\theta}_q}^T_{MLE} \bm{x}_{qj}) \bm{x}_{qj} ||},
\end{equation}
$i=1,\ldots,N$ and $\phi_q = \sum_{i=1}^{N} {\phi_q}_i^{mMSE} = 1$.
If so, using the {\it a priori} probabilities $\alpha_q$ the model average optimal sub-sampling probabilities is chosen such that
\begin{equation}
{\phi_i}^{mMSE} = \sum_{q=1}^{Q} \alpha_q {\phi_q}^{mMSE}_i,
\end{equation}
$i=1,\ldots,N$, $\sum_{q=1}^{Q} \alpha_q = 1$ then $\sum_{q=1}^{Q} \alpha_q \mbox{tr}(\bm{V}_q)$ attains its minimum. \\
Similarly optimal sub-sampling probabilities can be obtained for $L$-optimality.
\end{proof}
\section{Algorithms}\label{Appendix:Algorithms}
\begin{algorithm}[H]
\SetAlgoLined
\nonl \textbf{Stage 1} \\
\textbf{Random Sampling:} Assign $\bm{\phi} = (\phi_1,\ldots,\phi_N)$ for $F_N$. Here, $\bm{\phi}=\phi^{prop}$, where $\phi^{prop}_i=(2N_0)^{-1}$ if the $i$-th response element is zero, otherwise $\phi^{prop}_i=(2N_1)^{-1}$, and $N_0$ and $N_1$ are the number of elements in $\bm{y}$ when $y_i \in 0$ and $y_i \in 1$, respectively.\\ \nonl
According to $\bm{\phi}$ draw a random sub-sample of size $r_0$, $S_{r_0}=\{\bm{x}^{r_0}_l,y^{r_0}_l,\phi^{r_0}_l\}_{l=1}^{r_0}$. \\
\textbf{Estimation:} Based on $S_{r_0}$, using the Newton Raphson method find $\tilde{\bm{\theta}}^{r_0}$ until $\tilde{\bm{\theta}}^{t+1}$ and $\tilde{\bm{\theta}}^t$ are `close' enough ($\tilde{\bm{\theta}}^{t+1} - \tilde{\bm{\theta}}^t < 10^{-4}$),
\begin{equation}
\begin{aligned}
\tilde{\bm{\theta}}^{t+1} = \tilde{\bm{\theta}}^t + \Bigg(\sum_{l=1}^{r_0} \frac{\pi_l(\tilde{\bm{\theta}}^t)(1-\pi_l(\tilde{\bm{\theta}}^t))\bm{x}^{r_0}_l (\bm{x}^{r_0}_l)^T}{\phi^{r_0}_l}\Bigg)^{-1} \Bigg(\sum_{l=1}^{r_0} \frac{(y^{r_0}_l-\pi_l(\tilde{\bm{\theta}}^t))\bm{x}^{r_0}_l}{\phi^{r_0}_l} \Bigg), \notag
\end{aligned}
\end{equation}
where $\mbox{logit} ~ \pi_l(\bm{\theta})=\bm{\theta}^T\bm{x}_l$. \nonl \\
\textbf{Stage 2} \\
\textbf{Optimal sub-sampling probability:} Using $\tilde{\bm{\theta}}^{r_0}$ estimate optimal sub-sampling probabilities,
\begin{equation}
\begin{aligned}
\phi^{mMSE}_i= \frac{|y_i-\pi_i(\tilde{\bm{\theta}}^{r_0})|\,||
\tilde{\bm{J}}^{-1}_{\bm{X}} \bm{x}_i||}{ \sum_{j=1}^N |y_j-\pi_j(\tilde{\bm{\theta}}^{r_0})|\,|| \tilde{\bm{J}}^{-1}_{\bm{X}} \bm{x}_j||} \quad \mbox{or} \quad \phi^{mV_c}_i= \frac{|y_i-\pi_i(\tilde{\bm{\theta}}^{r_0})|\,||\bm{x}_i||}{ \sum_{j=1}^N |y_j-\pi_j(\tilde{\bm{\theta}}^{r_0})|\,||\bm{x}_j||}, \notag
\end{aligned}
\end{equation}
where $\tilde{\bm{J_X}}={(Nr_0)}^{-1} \sum_{l=1}^{r_0} (\phi_l^{r_0})^{-1} \pi_l(\tilde{\bm{\theta}}^{r_0})(1-\pi_l(\tilde{\bm{\theta}}^{r_0})) {\bm{x}_l}^{r_0} ({\bm{x}_l}^{r_0})^T$ and $i=1,\ldots,N$.\\
\textbf{Optimal sub-sampling and estimation:} Using estimated $\bm{\phi}^{mMSE}$ or $\bm{\phi}^{mV_c}$ draw a sub-sample $S_r$ completely at random (with replacement) of size $r$ from $F_N$, such that $S_r=\{\bm{x}^r_l,y^r_l,\phi^r_l\}_{l=1}^r$. \\ \nonl
Form $S_{r_0+r}$ by combining $S_{r_0}$, $S_r$ and obtain $\tilde{\bm{\theta}}$ until $\tilde{\bm{\theta}}^{t+1}$ and $\tilde{\bm{\theta}}^t$ are 'close' enough ($\tilde{\bm{\theta}}^{t+1} - \tilde{\bm{\theta}}^t < 10^{-4}$):
\begin{equation}
\begin{aligned}
\tilde{\bm{\theta}}^{t+1} = \tilde{\bm{\theta}}^t + \Bigg(\sum_{k \in \{r_0,r\}} \sum_{l=1}^{k} \frac{\pi_l(\tilde{\bm{\theta}}^t)(1-\pi_l(\tilde{\bm{\theta}}^t))\bm{x}^k_l (\bm{x}^k_l)^T}{\phi^k_l}\Bigg)^{-1} \Bigg(\sum_{k \in \{r_0,r\}} \sum_{l=1}^{k} \frac{(y^k_l-\pi_l(\tilde{\bm{\theta}}^t))\bm{x}^k_l}{\phi^k_l} \Bigg). \notag
\end{aligned}
\end{equation} \\ \nonl
Estimate the variance-covariance matrix of $\tilde{\bm{\theta}}$, $\tilde{\bm{V}}$, by $\tilde{\bm{J}}^{-1}_{\bm{X}} \tilde{\bm{V}}_c\tilde{\bm{J}}^{-1}_{\bm{X}}$, where,
\begin{align}
\tilde{\bm{J_X}} = & \frac{1}{N(r_0+r)} \sum_{k \in \{r_0,r\}} \sum_{l=1}^{k} \frac{\pi_l(\tilde{\bm{\theta}})(1-\pi_l(\tilde{\bm{\theta}})) {\bm{x}^k_l} ({\bm{x}^k_l})^T}{\phi^k_l} \notag \\ \mbox{and} &\quad \tilde{\bm{V_c}} = \frac{1}{N^2(r_0+r)^2} \sum_{k \in \{r_0,r\}} \sum_{l=1}^{k} \frac{(y^k_l-{\pi_l}(\tilde{\bm{\theta}}))^2 {\bm{x}^k_l} ({\bm{x}^k_l})^T}{ ({\phi^k_l})^2}. \notag
\end{align} \\
\textbf{Output :} $\tilde{\bm{\theta}}, S_{r_0+r}$ and $\tilde{\bm{V}}$.
\caption{Two stage optimal sub-sampling algorithm for logistic regression.} \label{Algo:OSMAC}
\end{algorithm}
\begin{algorithm}[H]
\SetAlgoLined
\nonl \textbf{Stage 1} \\
\textbf{Random Sampling:} Assign $\bm{\phi} = (\phi_1,\ldots,\phi_N)$ for $F_N$. Here, $\phi_i = {N^{-1}}$.\\ \nonl
According to $\bm{\phi}$ draw a random sub-sample of size $r_0$, $S_{r_0}=\{\bm{x}^{r_0}_l,y^{r_0}_l,\phi^{r_0}_l\}_{l=1}^{r_0} = (\bm{X}^{r_0},\bm{y}^{r_0},\bm{\phi}^{r_0})$.\\
\textbf{Estimation:} Based on $S_{r_0}$, find:
\begin{equation}
\begin{aligned}
\tilde{\bm{\theta}}^{r_0} = \argmaxA_{\bm{\theta}} ~ \log{ L(\bm{\theta}|\bm{X}^{r_0},\bm{y}^{r_0},\bm{\phi}^{r_0})} \equiv \argmaxA_{\bm{\theta}} ~ \frac{1}{r_0} \sum_{l=1}^{r_0} \Big[\frac{y^{r_0}_l \bm{\theta}^T \bm{x}^{r_0}_l - \lambda_l(\bm{\theta}) - \log{(y^{r_0}_l!)}}{\phi^{r_0}_l} \Big], \notag
\end{aligned}
\end{equation}
where $\log~{\lambda_l(\bm{\theta})} = \bm{\theta}^T\bm{x}_l.$ \nonl \\
\textbf{Stage 2} \\
\textbf{Optimal sub-sampling probability:} Using $\tilde{\bm{\theta}}^{r_0}$ estimate optimal sub-sampling probabilities,
\begin{equation}
\begin{aligned}
\phi^{mMSE}_i= \frac{|y_i-\lambda_i(\tilde{\bm{\theta}}^{r_0})| \, || \tilde{\bm{J}}^{-1}_{\bm{X}} \bm{x}_i||}{ \sum_{j=1}^N |y_j-\lambda_j(\tilde{\bm{\theta}}^{r_0})| \, || \tilde{\bm{J}}^{-1}_{\bm{X}} \bm{x}_j||} \quad \mbox{or} \quad \phi^{mV_c}_i= \frac{|y_i-\lambda_i(\tilde{\bm{\theta}}^{r_0})| \, ||\bm{x}_i||}{ \sum_{j=1}^N |y_j-\lambda_j(\tilde{\bm{\theta}}^{r_0})| \, || \bm{x}_j||}, \notag
\end{aligned}
\end{equation}
where $\tilde{\bm{J_X}}={(Nr_0)}^{-1} \sum_{l=1}^{r_0} (\phi_l^{r_0})^{-1} \lambda_l(\tilde{\bm{\theta}}^{r_0}) {\bm{x}_l}^{r_0} ({\bm{x}_l}^{r_0})^T$ and $i=1,\ldots,N$.\\
\textbf{Optimal sub-sampling and estimation:} Using estimated $\bm{\phi}^{mMSE}$ or $\bm{\phi}^{mV_c}$ draw a sub-sample $S_r$ completely at random (with replacement) of size $r$ from $F_N$, such that $S_r=\{\bm{x}^r_l,y^r_l,\phi^r_l\}_{l=1}^r = (\bm{X}^r,\bm{y}^r,\bm{\phi}^r)$. \\ \nonl
Form $S_{r_0+r}$ by combining $S_{r_0}$, $S_r$ and obtain:
\begin{align}
\tilde{\bm{\theta}} = & \argmaxA_{\bm{\theta}} ~ \log{ L(\bm{\theta}|\bm{X}^{r_0},\bm{y}^{r_0},\bm{\phi}^{r_0},\bm{X}^r,\bm{y}^r,\bm{\phi}^r)} \notag \\ \equiv & \argmaxA_{\bm{\theta}} ~\frac{1}{r_0+r} \sum_{k \in \{r_0,r\}} \sum_{l=1}^{k} \Big[\frac{y^k_l \bm{\theta}^T\bm{x}^k_l - \lambda_l(\bm{\theta}) - \log{(y^k_l!)}}{\phi^k_l} \Big]. \notag
\end{align} \\ \nonl
Estimate the variance-covariance matrix of $\tilde{\bm{\theta}}$, $\tilde{\bm{V}}$, by $\tilde{\bm{J}}^{-1}_{\bm{X}} \tilde{\bm{V}}_c \tilde{\bm{J}}^{-1}_{\bm{X}}$, where,
\begin{align}
\tilde{\bm{J_X}} = & \frac{1}{N(r_0+r)} \sum_{k \in \{r_0,r\}} \sum_{l=1}^{k} \frac{\lambda_l(\tilde{\bm{\theta}}) {\bm{x}^k_l} ({\bm{x}^k_l})^T}{ \phi^k_l} \notag \\ \mbox{and} & \quad \tilde{\bm{V_c}} = \frac{1}{N^2(r_0+r)^2} \sum_{k \in \{r_0,r\}} \sum_{l=1}^{k} \frac{(y^k_l-{\lambda_l}(\tilde{\bm{\theta}}))^2 {\bm{x}^k_l} ({\bm{x}^k_l})^T}{ ({\phi^k_l})^2}. \notag
\end{align} \\
\textbf{Output :} $\tilde{\bm{\theta}}, S_{r_0+r}$ and $\tilde{\bm{V}}$.
\caption{Two stage optimal sub-sampling algorithm for Poisson regression.}\label{Algo:OSPAC}
\end{algorithm}
\begin{algorithm}[H]
\SetAlgoLined
\nonl \textbf{Stage 1} \\
\textbf{Random Sampling:} Assign $\bm{\phi} = (\phi_1,\ldots,\phi_N)$ for $F_{N}$. Here, $\bm{\phi} =\phi^{prop}$, where $\phi^{prop}_i=(2N_0)^{-1}$ if the $i$-th response element is $0$, otherwise $\phi^{prop}_i=(2N_1)^{-1}$, and $N_0$ and $N_1$ are the number of elements in $\bm{y}$ when $y_i \in 0$ and $y_i \in 1$, respectively.\\ \nonl
According to $\bm{\phi}$ draw random sub-samples of size $r_0$, such that $S_{r_0}=\{h_q(\bm{{x_0}}^{r_0}_{l}),y^{r_0}_l,\phi^{r_0}_l\}_{l=1}^{r_0}$ for $q=1,\ldots,Q$. \\
\textbf{Estimation:} For $q=1,\ldots,Q$ and $S_{r_0}$, using the Newton Raphson method find $\tilde{\bm{\theta}_q}^{r_0}$ until $\tilde{\bm{\theta}_q}^{t+1}$ and $\tilde{\bm{\theta}_q}^t$ are `close' enough ($\tilde{\bm{\theta}_q}^{t+1} - \tilde{\bm{\theta}_q}^t < 10^{-4}$),
\begin{equation}
\begin{aligned}
\tilde{\bm{\theta}_q}^{t+1} = \tilde{\bm{\theta}_q}^t + \Bigg(\sum_{l=1}^{r_0} \frac{\pi_l(\tilde{\bm{\theta}_q}^t)(1-\pi_l(\tilde{\bm{\theta}_q}^t)) \bm{x}_{ql} (\bm{x}_{ql})^T }{ \phi_l} \Bigg)^{-1} \Bigg(\sum_{l=1}^{r_0} \frac{(y_l - \pi_l(\tilde{\bm{\theta}_q}^t)) \bm{x}_{ql}}{\phi_l} \Bigg), \notag
\end{aligned}
\end{equation}
where $\mbox{logit} ~ \pi_l(\bm{\theta}_q)= \bm{\theta}^T_q\bm{x}_{ql}$. \nonl \\
\textbf{Stage 2} \\
\textbf{Optimal sub-sampling probability:} Based on $\tilde{\bm{\theta}_q}^{r_0}$ and $\alpha_q$ (for $q=1,\ldots,Q$) estimate optimal sub-sampling probabilities, $$\phi^{mMSE}_i= \sum_{q=1}^{Q} \frac{\alpha_q |y_i-\pi_i(\tilde{\bm{\theta}_q}^{r_0})| \, || \tilde{\bm{J}}^{-1}_{\bm{X}_q} \bm{x}_{qi} ||}{ \sum_{j=1}^N |y_j-\pi_j(\tilde{\bm{\theta}_q}^{r_0})| \, || \tilde{\bm{J}}^{-1}_{\bm{X}_q} \bm{x}_{qj} ||} \quad \mbox{or} \quad \phi^{mV_c}_i= \sum_{q=1}^{Q} \frac{\alpha_q |y_i-\pi_i(\tilde{\bm{\theta}_q}^{r_0})| \, || \bm{x}_{qi} ||}{ \sum_{j=1}^N |y_j-\pi_j(\tilde{\bm{\theta}_q}^{r_0})| \, || \bm{x}_{qj}||},$$ where $\tilde{\bm{J}}_{\bm{X}_q}={(Nr_0)}^{-1} \sum_{l=1}^{r_0} (\phi_l^{r_0})^{-1} \pi_l (\tilde{\bm{\theta}_q}^{r_0})(1-\pi_l(\tilde{\bm{\theta}_q}^{r_0})) {\bm{x}^{r_0}_{ql}} (\bm{x}^{r_0}_{ql})^T$ and $i=1,\ldots,N$.\\
\textbf{Optimal sub-sampling and estimation:} Based on $\bm{\phi}^{mMSE}$ or $\bm{\phi}^{mV_c}$ draw a sub-sample of size $r$ from $F_{N}$, such that $S_r=\{h_q(\bm{{x_0}}^r_{l}),y^r_l,\phi^r_l\}_{l=1}^r$ for $q=1,\ldots,Q$. \\ \nonl
Combine $S_{r_0},S_r$ and form $S_{(r_0+r)}$, and obtain $\tilde{\bm{\theta}_q}$ until $\tilde{\bm{\theta}_q}^{t+1}$ and $\tilde{\bm{\theta}_q}^t$ are 'close' enough ($\tilde{\bm{\theta}_q}^{t+1} - \tilde{\bm{\theta}_q}^t < 10^{-4}$):
\begin{equation}
\begin{aligned}
& \tilde{\bm{\theta}_q}^{t+1} = \tilde{\bm{\theta}_q}^t + \Bigg[\sum_{k \in \{r_0,r\}} \sum_{l=1}^{k} \frac{\pi_l(\tilde{\bm{\theta}_q}^t)(1-\pi_l(\tilde{\bm{\theta}_q}^t)) \bm{x}^{k}_{ql} (\bm{x}^{k}_{ql})^T}{\phi^k_l}\Bigg]^{-1} \Bigg[\sum_{k \in \{r_0,r\}}\sum_{l=1}^{k} \frac{(y^k_l-\pi_l(\tilde{\bm{\theta}_q}^t)) \bm{x}^{k}_{ql} }{\phi^k_l} \Bigg]. \notag
\end{aligned}
\end{equation} \\ \nonl
Estimate the variance-covariance matrix of $\tilde{\bm{\theta}}_q$, $\tilde{\bm{V}}_q$, by $\tilde{\bm{J}}^{-1}_{\bm{X}_q} \tilde{\bm{V}}_{q_c} \tilde{\bm{J}}^{-1}_{\bm{X}_q}$, where,
\begin{align}
\tilde{\bm{J}}_{\bm{X}_q} = & \frac{1}{N(r_0+r)} \sum_{k\in \{r_0,r\}} \sum_{l=1}^{k} \frac{\pi_l(\tilde{\bm{\theta}_q})(1-\pi_l(\tilde{\bm{\theta}_q})) \bm{x}^{k}_{ql} (\bm{x}^{k}_{ql})^T}{ \phi^k_l} \notag \\ \mbox{and} & \quad \tilde{\bm{V}}_{q_c} = \frac{1}{N^2(r_0+r)^2} \sum_{k \in \{r_0,r\}} \sum_{l=1}^{k} \frac{(y^k_l-{\pi_l}(\tilde{\bm{\theta}_q}))^2 \bm{x}^{k}_{ql} (\bm{x}^{k}_{ql})^T}{ ({\phi^k_l})^2}. \notag
\end{align} \\
\textbf{Output :} $\tilde{\bm{\theta}}_q, S_{(r_0+r)}$ and $\tilde{\bm{V}}_q$ for $q=1,\ldots,Q$.
\caption{Two stage model robust optimal sub-sampling algorithm for logistic regression.}\label{Algo:OSMACMR}
\end{algorithm}
\begin{algorithm}[H]
\SetAlgoLined
\nonl \textbf{Stage 1} \\
\textbf{Random Sampling :} Assign $\bm{\phi} = (\phi_1,\ldots,\phi_N)$ for $F_{N}$. Here, $\phi_i = {N^{-1}}$.\\ \nonl
According to $\bm{\phi}$ draw random sub-samples of size $r_0$, such that $S_{r_0}=\{h_q({\bm{x}_0}^{r_0}_{l}),y^{r_0}_l,\phi^{r_0}_l\}_{l=1}^{r_0} = (h_q(\bm{X}_0^{r_0}),\bm{y}^{r_0},\bm{\phi}^{r_0})$ for $q=1,\ldots,Q$. \\
\textbf{Estimation:} For $q=1,\ldots,Q$ and $S_{r_0}$ find:
\begin{equation}
\begin{aligned}
\tilde{\bm{\theta}_q}^{r_0} = \argmaxA_{\bm{\theta}_q} ~ \log{L(\bm{\theta}_q| \bm{X}^{r_0}_{q},\bm{y}^{r_0},\bm{\phi}^{r_0})} \equiv \argmaxA_{\bm{\theta}_q} ~ \frac{1}{r_0} \sum_{l=1}^{r_0} \Big[\frac{y^{r_0}_l \bm{\theta}^T_q \bm{x}^{r_0}_{ql} - \lambda_l(\bm{\theta}_q) - \log{(y^{r_0}_l!)}}{\phi^{r_0}_l} \Big], \notag
\end{aligned}
\end{equation}
where $\log{\lambda_l(\bm{\theta}_q)} = \bm{\theta}^T_q \bm{x}_{ql}.$ \nonl \\
\textbf{Stage 2} \\
\textbf{Optimal sub-sampling probability:} For $q=1,\ldots,Q$, using $\tilde{\bm{\theta}_{q}^{r_0}}$ estimate optimal sub-sampling probabilities,
\begin{equation}
\begin{aligned}
\phi^{mMSE}_i= \sum_{q=1}^{Q}\frac{\alpha_q|y_i-\lambda_i(\tilde{\bm{\theta}_{q}}^{r_0})| \, || \tilde{\bm{J}}^{-1}_{\bm{X}_q} \bm{x}_{qi} ||}{ \sum_{j=1}^N |y_j-\lambda_j(\tilde{\bm{\theta}_{q}}^{r_0})| \, || \tilde{\bm{J}}^{-1}_{\bm{X}_q} \bm{x}_{qj} ||} \quad \mbox{or} \quad \phi^{mV_c}_i= \sum_{q=1}^{Q} \frac{\alpha_q|y_i-\lambda_i(\tilde{\bm{\theta}_{q}}^{r_0})| \, || \bm{x}_{qi} ||}{ \sum_{j=1}^N |y_j-\lambda_j(\tilde{\bm{\theta}_{q}}^{r_0})| \, || \bm{x}_{qj} ||}, \notag
\end{aligned}
\end{equation}
where $\tilde{\bm{J}}_{\bm{X}_q}={(Nr_0)}^{-1} \sum_{l=1}^{r_0} (\phi_l^{r_0})^{-1} \lambda_l(\tilde{\bm{\theta}_{q}}^{r_0}) \bm{x}^{r_0}_{ql} (\bm{x}^{r_0}_{ql})^T$ and $i=1,\ldots,N$.\\
\textbf{Optimal sub-sampling and estimation:} Based on $\bm{\phi}^{mMSE}$ or $\bm{\phi}^{mV_c}$ draw a sub-sample of size $r$ from $F_{N}$, such that $S_r=\{h_q({\bm{x}_0}^r_{l}),y^r_l,\phi^r_l\}_{l=1}^r = (h_q(\bm{X}_0^r),\bm{y}^r_q, \bm{\phi}^r_q)$ for $q=1,\ldots,Q$.\\ \nonl
Combine $S_{r_0}$, $S_r$ and form $S_{(r_0+r)}$, and obtain:
\begin{align}
\tilde{\bm{\theta}}_q = & \argmaxA_{\bm{\theta}_q} ~ \log{L(\bm{\theta}_q| \bm{X}^{r_0}_q, \bm{y}^{r_0}_q, \bm{\phi}^{r_0}_q, \bm{X}^r_q,\bm{y}^r_q, \bm{\phi}^r_q)} \notag \\
\equiv & \argmaxA_{\bm{\theta}_q} ~ \frac{1}{r_0+r} \sum_{k \in \{r_0,r\}} \sum_{l=1}^{k} \Big[\frac{y^k_l \bm{\theta}^T_q \bm{x}^k_{ql} - \lambda_l(\bm{\theta}_q) - \log{(y^k_l!)}}{\phi^k_l} \Big]. \notag
\end{align} \\ \nonl
Estimate the variance-covariance matrix of $\tilde{\bm{\theta}}_q$, $\tilde{\bm{V}}_q$, by $\tilde{\bm{J}}^{-1}_{\bm{X}_q} \tilde{\bm{V}}_{q_c} \tilde{\bm{J}}^{-1}_{\bm{X}_q}$, here,
\begin{align}
\tilde{\bm{J}}_{\bm{X}_q} = & \frac{1}{N(r_0+r)} \sum_{k \in \{r_0,r\}} \sum_{l=1}^{k} \frac{\lambda_l(\tilde{\bm{\theta}_q}) \bm{x}^k_{ql} (\bm{x}^k_{ql})^T}{ \phi^k_l} \notag \\ \mbox{and} & \quad \tilde{\bm{V}}_{q_c} = \frac{1}{N^2(r_0+r)^2} \sum_{k \in \{r_0,r\}} \sum_{l=1}^{k} \frac{(y^k_l-{\lambda_l}(\tilde{\bm{\theta}_q}))^2 \bm{x}^k_{ql} (\bm{x}^k_{ql})^T}{ ({\phi^k_l})^2}. \notag
\end{align} \\
\textbf{Output :} $\tilde{\bm{\theta}}_q, S_{(r_0+r)}$ and $\tilde{\bm{V}}_q$ for $q=1,\ldots,Q$.
\caption{Two stage model robust optimal sub-sampling algorithm for Poisson regression.}\label{Algo:OSPACMR}
\end{algorithm}
\section{Code materials for the simulation setup and real world applications}\label{Appendix:Code}
\begin{enumerate}
\item Logistic Regression
\begin{enumerate}
\item \href{https://github.com/Amalan-ConStat/RS_OS_MROS_LR_TwoVar_Exponential}{Two covariate model with covariate data generated by the Exponential distribution.}
\item \href{https://github.com/Amalan-ConStat/RS_OS_MROS_LR_TwoVar_Normal}{Two covariate model with covariate data generated by the Normal distribution.}
\item \href{https://github.com/Amalan-ConStat/RS_OS_MROS_LR_RWS_SkinData}{Skin segmentation data.}
\end{enumerate}
\item Poisson Regression
\begin{enumerate}
\item \href{https://github.com/Amalan-ConStat/RS_OS_MROS_PR_TwoVar_Normal}{Two covariate model with covariate data generated by the Normal distribution.}
\item \href{https://github.com/Amalan-ConStat/RS_OS_MROS_PR_TwoVar_Uniform}{Two covariate model with covariate data generated by the Uniform distribution.}
\item \href{https://github.com/Amalan-ConStat/RS_OS_MROS_PR_RWS_NYC_Taxi_Fare_2013_Full}{New York City taxi fare data.}
\end{enumerate}
\end{enumerate}
\section{Figures}\label{Appendix:Figures}
\begin{figure}
\caption{(a) Model information (larger is better) and (b) SMSE (smaller is better) for the sub-sampling methods for the logistic regression model under $mMSE$. Covariate data were generated from the Exponential distribution.}
\label{Fig:RWA_LR_TV_Exp}
\end{figure}
\begin{figure}
\caption{(a) Model information (larger is better) and (b) SMSE (smaller is better) for the sub-sampling methods for the Poisson regression model under $mMSE$. Covariate data were generated from the Uniform distribution.}
\label{Fig:RWA_PR_TV_Uni}
\end{figure}
\end{appendices}
\end{document} |
\begin{document}
\title{The Distinguishing Number and Distinguishing Chromatic Number for Posets}
\begin{abstract}
In this paper we introduce the concepts of the distinguishing number and the distinguishing chromatic number of a poset. For a distributive lattice $L$ and its set $Q_L$ of join-irreducibles, we use classic lattice theory to show that any linear extension of $Q_L$ generates a distinguishing 2-coloring of $L$. We prove general upper bounds for the distinguishing chromatic number and particular upper bounds for the Boolean lattice and for divisibility lattices. In addition, we show that the distinguishing number of any twin-free Cohen-Macaulay planar lattice is at most 2.
\end{abstract}
\noindent \textbf{Keywords: distributive lattice, distinguishing number, distinguishing chromatic number, Birkhoff's theorem}
\section{Introduction}
\label{sect-intro}
The distinguishing number of a graph, introduced by Albertson and Collins \cite{AlCo96}, is the smallest integer $k$ for which the vertices can be colored using $k$ colors so that the only automorphism of the graph that preserves colors is the identity. The distinguishing chromatic number, introduced by Collins and Trenk \cite{CoTr06}, has the additional requirement that the coloring of the vertices is proper, that is, adjacent vertices get different colors. The distinguishing number of graph $G$ is denoted by $D(G)$ and the distinguishing chromatic number by $\chi_D(G)$. These and related topics have received considerable attention by many authors in recent years; see, for example, \cite{AlSo18,
BaPa17-b,
Cr18,
ImSmTuWa15,
Le17,
Pi17-b}.
In this paper, we introduce the distinguishing number and the distinguishing chromatic number of a poset.
There are several challenges in studying these parameters. A distinguishing coloring of a graph or poset does not always yield a distinguishing coloring of induced subgraphs or subposets. It is possible to have an graph $H$ induced in graph $G$ for which $D(H) > D(G)$, and the same holds for posets. We provide an example of this following Definition~\ref{disting-def}.
In addition, the structure inherent in posets makes these parameters qualitatively different from the graph versions.
We end this section with an overview of the rest of the paper. In Section~\ref{sect-prelim}, we provide background material about posets, lattices, and distributive lattices. We introduce the distinguishing number of a poset in Section~\ref{sect-dist-nos}, and prove results about sums of chains, distributive lattices, and divisibility lattices. In Section~\ref{sect-dist-chr}, we study the distinguishing chromatic number, giving upper bounds for the distinguishing chromatic number of distributive lattices, divisibility lattices, and Boolean lattices. We also show that there exist posets $P$ for which the gap between the distinguishing chromatic number of $P$ and that of its comparability graph is arbitrarily large. We return to the distinguishing number in Section~\ref{sect-planar} and focus on ranked planar lattices (equivalently, Cohen-Macaulay) that are rank-connected. We conclude with several open questions.
\section{Preliminaries} \label{sect-prelim}
In this section we provide definitions related to posets and lattices, and present Birkhoff's classic lattice theorem, which we use as a tool in Section~\ref{sect-dist-nos} and Section~\ref{sect-dist-chr}. For additional details and background, see \cite{St12}.
\subsection{General poset definitions}\label{sect-poset-defn}
The posets we consider are finite and reflexive. If $P$ is the poset $(X,\preceq)$, we call $X$ the \emph{ground set} of $P$ and refer to the elements of $X$ (which we also call the elements of $P$) as \emph{points}. We write $x \prec y$ if $x \preceq y$ and $x \neq y$. If $x \preceq y$ or $y \preceq x$, we say points $x$ and $y$ are \emph{comparable}, and otherwise they are \emph{incomparable}. We say that $y $ \emph{covers} $x$
if $x \prec y$ and there is no other point $v$ with $x \prec v \prec y$. An \emph{automorphism} of poset $P = (X,\preceq)$ is a bijection from $X$ to $X$ that preserves the relation $\preceq$.
A set of pairwise comparable points in a poset is called a \emph{chain}, and if the points are pairwise incomparable they form an \emph{antichain}. An $r$-chain is a chain with $r$ points, and such a chain has \emph{length} $r-1$. The \emph{height} of a poset is the size of a maximum chain and the \emph{width} is the size of a maximum antichain.
If a poset has a unique minimal element, we call this element $\hat{0}$ and if it has a unique maximal element we call it $\hat{1}$. We say that a poset with a $\hat{0}$ and $\hat{1}$ is \emph{ranked} if every maximal chain from $\hat{0}$ to $\hat{1}$ has the same length. The \emph{rank} of a point $x$ in a poset, denoted by $rank(x)$ or $r(x)$, is the length of a longest chain that has $x$ as its largest element. For example, in Figure~\ref{ex-fig}, each of posets $L_{pq^2}, L_{p^2q^2}, M$ and $L_{pqr}$ has a $\hat{0}$ and a $\hat{1}$, while poset $S_4$ has neither, and
$L_{pq^2}, L_{p^2q^2}$ and $L_{pqr}$ are ranked, while poset $M$ is not.
A poset is \emph{planar} if its Hasse diagram can be drawn in the plane with no edges crossing and so that the edge from $a$ to $b$ has strictly increasing $y$-coordinate when $a \prec b$. In Figure~\ref{ex-fig}, $L_{pq^2}, L_{p^2q^2}, M$ and $S_4$ are planar, even though the drawing shown of $S_4$ has edges crossing.
We demonstrate in Section \ref{sect-planar} that poset $L_{pqr}$ is not planar (see Remark~\ref{B-not-planar}).
\begin{figure}
\caption{Examples of posets with distinguishing labelings ($p$, $q$ and $r$ are distinct primes).}
\label{ex-fig}
\end{figure}
\subsection{Lattice definitions}
A point $z $ in poset $P$ is called the \emph{meet} of $x$ and $y$ in $P$, and denoted by $x\wedge y$, if it is the unique largest element in $ P$ such that $z\preceq x$ and $z\preceq y$. Thus, if $x\wedge y$ exists and $a\preceq x$ and $a\preceq y$, then $a\preceq x\wedge y$. Similarly, a point $w \in P$ is called the \emph{join} of $x$ and $y$ in $P$, and denoted by $x\vee y$, if it is the unique smallest element $w\in P$ such that $w\succeq x$ and $w\succeq y$. Thus, if $x\vee y$ exists and $a\succeq x$ and $a\succeq y$, then $a\succeq x\vee y$.
A poset $L$ is a \emph{lattice} if $x\wedge y$ and $x\vee y$ both exist for all points $x$ and $y$ in L. Furthermore, $L$ is a \emph{distributive lattice} if $\wedge$ and $\vee$ satisfy the distributive laws
\[(x\wedge y)\vee z = (x\vee z)\wedge (y\vee z)\]
\[(x\vee y)\wedge z = (x\wedge z)\vee (y\wedge z)\]
for all $x,y,z\in L$.
For example, all the posets in Figure~\ref{ex-fig} are lattices except for $S_4$, and all the lattices are distributive except for $M$. A lattice need not be ranked ($M$ is not ranked), but a distributive lattice is ranked, as a consequence of Birkhoff's Theorem (Theorem~\ref{birkhoff-thm} in Section~\ref{sect-birktheorem}).
A point $x$ in a lattice is called \emph{join-irreducible} if in the Hasse diagram of the lattice $x$ has exactly one downward edge.
For example, in Figure~\ref{ex-fig}, the join-irreducible points of $L_{pq^2}$ are $p,q,$ and $q^2$, while there are no join-irreducible points in poset $S_4$.
As we will see in Birkhoff's Theorem, the join-irreducible points of a distributive lattice generate all the elements in the lattice by the join operation. In this way, they act like the prime numbers in the prime factorization of an integer.
\subsection{Birkhoff's Theorem} \label{sect-birktheorem}
In this section, we present a fundamental theorem due to Birkhoff (Theorem~\ref{birkhoff-thm}) and a corollary, both of which will be used as tools in later sections of the paper.
Let poset $P= (X,\preceq)$.
The \emph{downset} of a point $a \in X$ is defined as $down(a) = \{x\in X: x \preceq a \}$ and the downset of a subset $A \subseteq X$ is defined as
$down(A) = \{x \in X: x \preceq a \hbox{ for some } a \in A\}$.
\begin{Def}{\rm Let $P= (X,\preceq)$ be a poset. The \emph{downset lattice\/} $J(P)$ has ground set $\{down(S): S\subseteq X\}$ and the relation is $\subseteq$.
}
\end{Def}
Observe that if $P$ is a poset, then $J(P)$ is a distributive lattice, in which the meet of elements $S$ and $T$ is $S \cap T$ and the join of these elements is $S \cup T$.
\begin{example} {\rm
In Figure~\ref{L-150-fig}, the join-irreducible elements of $ L_{150}$ are $a = 2$, $b=3$, $c=5$, and $d=25$. When these are ordered using the ordering induced by $L_{150}$ they produce the poset labeled $Q_L$ also shown in Figure~\ref{L-150-fig}. There are 16 subsets of elements of $Q_L$, producing 12 distinct downsets, which are given in the following table. When these 12 downsets are ordered by set inclusion, we obtain the downset lattice $J(Q_L)$ which is isomorphic to the original lattice $L_{150}$. }
\small
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline
$S$ & $\emptyset$ & $a$ &$ b$ & $c$ & $d$ & $ab$ & $ac$ & $ad$ & $bc$ &$ bd$ & $cd$ & $abc$ & $abd$ & $acd$ & $bcd$ & $abcd$ \\ \hline
$down(S)$& $\emptyset $ & $a$ & $b$ & $c$ & $cd$ & $ab$ & $ac$ & $acd$ & $bc$ & $bcd$ & $cd$ & $abc$ & $abcd$ & $acd$ & $bcd$ & $abcd$ \\ \hline
\end{tabular}
\end{center}
\normalsize
\label{JP2-example}
\end{example}
\begin{figure}
\caption{The lattice $L_{150}
\label{L-150-fig}
\end{figure}
This example illustrates the following classic theorem due to Birkhoff \cite{Bi33} and called the \emph{Fundamental Theorem of Distributive Lattices} in \cite{St12}.
\begin{Thm} If $L $ is a distributive lattice and $Q_L$ is the poset induced by the join-irreducible points of $L$, then $J(Q_L)$ is isomorphic to $L$.
Indeed, the function $f:L \to J(Q_L)$ defined by $f(w) = \{y \in Q_L: y \preceq w \}$ is an isomorphism.
\label{birkhoff-thm}
\end{Thm}
The following notation will be helpful as we use Theorem~\ref{birkhoff-thm} repeatedly.
\begin{Def} {\rm For a distributive lattice $L$, denote by $Q_L$ the induced poset of all join-irreducible points of $L$. }
\end{Def}
Birkhoff's theorem is fundamental in several ways. First it provides a method for checking whether a poset $L$ is a distributive lattice without having to verify that every pair of points has a meet and a join, namely, construct the induced poset $Q_L$ and check whether the mapping $f$ from Theorem~\ref{birkhoff-thm} is an isomorphism. Additionally, any distributive lattice can be generated by starting with a poset $P$ and constructing $J(P)$. We utilize Theorem~\ref{birkhoff-thm} in proving that all distributive lattices have distinguishing number at most two (Theorem~\ref{distrib-lattice-thm}) and in characterizing those that have distinguishing number one (Theorem~\ref{D-one-thm}). A proof of Theorem~\ref{birkhoff-thm} appears in \cite{St12}.
Observe that in lattice $L_{150}$ shown in Figure~\ref{L-150-fig}, the point 75 can be written as the join of all the join-irreducibles less or equal to it, namely $75=3\vee 5\vee 25$. Equivalently, in $J(Q_L)$, $\{b,c,d\} = \{b\}\cup \{c\}\cup \{c,d\}$. The next corollary shows this holds in general, that is, every point $w\in L$ can be written as the join of a unique subset of join-irreducibles of $L$.
It is a well-known consequence of Birkhoff's Theorem and we provide a proof for completeness.
\begin{Cor} \label{tough-cor}
Let $L$ be a distributive lattice and $f:L\rightarrow J(Q_L)$ be the isomorphism from Theorem~\ref{birkhoff-thm}.
If $w\in L$, then $w=\bigvee _{z\in f(w)} z$.
\end{Cor}
\begin{proof}
Fix $w \in L$ and let $f(w) = \{y_1, y_2, \ldots, y_t\}$. For each $i: 1 \le i \le t$, the reflexive property implies that $y_i \in f(y_i)$, and the fact that $f(w)$ is a downset implies that $f(y_i) \subseteq f(w)$. Therefore, $f(w) = f(y_1) \cup f(y_2) \cup \cdots \cup f(y_t)$. However, $f$ is an isomorphism, so applying $f^{-1}$ to both sides yields $w = y_1 \vee y_2 \vee \cdots \vee y_t$, as desired.
\end{proof}
For a distributive lattice $L$, the points of $J(Q_L)$ are downsets, and the rank of each point is the cardinality of its downset. We record this below.
\begin{remark}
\label{rank-rem}
Every distributive lattice $L$ is ranked and the rank of a point $w$ is $|\{z \in Q_L: z \preceq w\}|.$
\end{remark}
\section{Distinguishing numbers}\label{sect-dist-nos}
We begin with the definition of the distinguishing number of a poset and some examples.
\begin{Def} {\rm
A coloring of the points of poset $P$ is \emph{distinguishing} if the only automorphism of $P$ that preserves colors is the identity. The \emph{distinguishing number} of $P$, denoted $D(P)$ is the least integer $k$ so that $P$ has a distinguishing coloring using $k$ colors.
}
\label{disting-def}
\end{Def}
Distinguishing colorings are shown in Figure~\ref{ex-fig}, and the distinguishing numbers are the following: $D(L_{pq^2})=1$, $D(L_{p^2q^2})=2$, $D(M)=1$, $D(L_{pqr})=2$, $D(S_4)=2$. Note that while $D(M) = 1$, if we remove point $x$ from $M$, the resulting induced subposet $M-x$ has $D(M-x) = 2$; thus an induced subposet can have a larger distinguishing number than that of the original.
Observe that antichains are the only posets for which each point must receive a different color in a distinguishing coloring.
The \emph{comparability graph} of poset $P$ is the graph $G_P= (V,E)$ where $V$ is the ground set of $P$ and $xy \in E$ if and only if $x$ and $y$ are comparable in $P$. Any automorphism of a poset $P$ is also an automorphism of its comparability graph, $G_P$. This justifies the following remark.
\begin{remark}
$D(P) \le D(G_P)$.
\end{remark}
Some automorphisms of the graph $G_P$ are not automorphisms of the poset $P$ because they do not preserve the ordering of points in $P$. The following example shows that $D(P)$ can differ significantly from $D(G_P)$ and $D(\overline{G_P})$.
If $P$ is an $n$-chain then $D(P) = 1$. However, the comparability graph of $P$ is the complete graph $K_n$, and the incomparability graph of $P$ is its complement $\overline{K_n}$, and each of these has distinguishing number $n$.
\subsection{Sums of chains }
In the next two results, we find the distinguishing number for posets consisting of the sum of chains.
\begin{Prop}
Let $P$ be the poset consisting of the sum of $t$ chains, each consisting of $r$ points and let $k$ be the positive integer for which $(k-1)^r < t \le k^r$. Then $D(P) = k$.
\label{chains-lem}
\end{Prop}
\begin{proof}
First we find a distinguishing coloring of $P$ using $k$ colors. There are $k^r$ different ways to color the elements of an $r$-chain when $k$ colors are available. Coloring the elements of each $r$-chain differently is a distinguishing coloring since any automorphism of $P$ maps an $r$-chain to an $r$-chain. Thus $D(P) \le k$. We next show $D(P) > k-1$. For a contradiction, suppose there is a distinguishing coloring of $P$ using $k-1$ colors. There are $(k-1)^r$ ways to color each chain and since $t > (k-1)^r$, two chains have the same coloring. The automorphism that swaps those two chains is non-trivial, a contradiction.
\end{proof}
Combining Proposition~\ref{chains-lem} with the following proposition, allows us to compute the distinguishing number of any poset that consists of the sum of chains.
\begin{Prop} Let $P$ be the sum of chains and partition $P$ as $P_1 + P_2 + \cdots + P_m$ where $P_i$ consists of $t_i$ chains, each consisting of $r_i$ points, where $r_1, r_2, \cdots , r_m$ are distinct. Then $D(P) = \max \{D(P_i): 1 \le i \le m\}$.
\label{chains-prop}
\end{Prop}
\begin{proof}
The result follows immediately from the fact that any automorphism of $P$ will map $P_i$ to itself for each $i$.
\end{proof}
\subsection{Distributive lattices} \label{sect-dist-latt}
We find the distinguishing number of any distributive lattice in Theorems~\ref{D-one-thm} and \ref{distrib-lattice-thm}.
In showing that a coloring is distinguishing it can be helpful to analyze the points individually or in groups using the following concept of \emph{pinning}.
\begin{Def} {\rm Let $P$ be a poset with a color assigned to each point. We say that a point $x$ is \emph{pinned} if every automorphism of $P$ that preserves colors maps $x$ to itself. }
\end{Def}
Note that a coloring of the ground set of a poset $P$ is distinguishing precisely when every point is pinned.
\begin{Prop}\label{tough-pin-prop}
Let $\phi$ be any coloring of a distributive lattice $L$. If $\phi$ restricted to $Q_L$ pins every point of $Q_L$, then $\phi$ pins every point of $L$.
\end{Prop}
\begin{proof}
Let $\phi$ be a coloring of $L$ so that $\phi$ restricted to $Q_L$ pins every point of $Q_L$. By Corollary~\ref{tough-cor}, every element of $L$ is the join of a unique set of elements of $Q_L$. Since joins are preserved by isomorphism, it follows that every point of $L$ is pinned.
\end{proof}
We now have the tools to determine the distinguishing number of any distributive lattice. Theorem~\ref{distrib-lattice-thm} shows that any distributive lattice has distinguishing number at most two and Theorem~\ref{D-one-thm} characterizes those distributive lattices whose distinguishing number is one. The proof of Theorem~\ref{distrib-lattice-thm} is illustrated in Examples~\ref{example-birkhoff} and \ref{example-birkhoff-2}.
\begin{Thm} If $L$ is a distributive lattice, then $D(L) = 1$ if and only if $D(Q_L)=1$.
\label{D-one-thm}
\end{Thm}
\begin{proof}
By definition, any poset has distinguishing number equal to $1$ if and only if it has no non-trivial automorphisms. Let $\phi$ be a coloring of $L$ in which every vertex is colored the same. If $Q_L$ has no non-trivial automorphisms, then every point in $Q_L$ is pinned by $\phi$ and by Proposition~\ref{tough-pin-prop}, every point in $L$ is pinned.
Conversely, if $Q_L$ has a non-trivial automorphism $\sigma$, then by Corollary~\ref{tough-cor}, $\sigma$ can be extended to a non-trivial automorphism of $L$, contradicting $D(L)=1$.
\end{proof}
\begin{Thm} If $L= (X, \preceq)$ is a distributive lattice, then $D(L)\leq 2$ and $D(L) = 2$ if and only if $D(Q_L)>1$.
\label{distrib-lattice-thm}
\end{Thm}
\begin{proof}
Let $L= (X \preceq)$ be a distributive lattice and $Q_L = (Y, \prec)$ where $Y = \{y_1,y_2, \ldots, y_t\}$. By Theorem~\ref{birkhoff-thm}, $L$ is isomorphic to $J(Q_L)$. We will provide a distinguishing coloring of $J(Q_L)$ using two colors, showing $D(L)\leq 2$. The remainder of the theorem follows from Theorem~\ref{D-one-thm}.
Let $f:L \to J(Q_L)$ be the isomorphism defined in Theorem~\ref{birkhoff-thm}, and let $f(Y) = \{f(y_1),f(y_2), \ldots, f(y_t)\}$. The property of being join-irreducible is preserved under isomorphism, thus $f(Y)$ is the set of join-irreducible points of $J(Q_L)$.
Let $E:y_1 \prec y_2 \prec y_3 \prec \cdots \prec y_t$ be a linear extension of $Q_L$. Color the following chain of elements of $J(Q_L)$ using the color red: \\ $ \{f(y_1)\}, \ \{f(y_1), f(y_2)\}, \
\{f(y_1), f(y_2), f(y_3)\}, \ \cdots ,\ \{f(y_1), f(y_2), f(y_3), \cdots f(y_{t-1})\}.$ Color the remaining elements green. We show this is a distinguishing coloring of $J(Q_L)$ by showing that every nontrivial automorphism of $J(Q_L)$ preserves colors.
Since poset automorphisms preserve rank and there is at most one red vertex at each rank of $J(Q_L)$, we know the red vertices are pinned. Next we show all green points in $f(Y)$ are pinned. For $2\leq i\leq t-1$, each $f(y_i)\in f(Y)$ is less than a unique lowest red point in the chain of red vertices of $J(Q_L)$. In particular, $\{f(y_i)\} \preceq \{f(y_1), f(y_2), f(y_3), \cdots , f(y_i)\}$ but $\{f(y_i)\} $ is incomparable to all lower ranked red points. Hence each is pinned. Also, $f(y_t)$ is the only point in $f(Y)$ that is not less than any red vertex, hence it is pinned. Thus the green points in $f(Y)$ are pinned. By Corollary~\ref{tough-cor}, every point of $J(Q_L)$ that is not in $f(Y)$ is the join of a unique set of elements of $f(Y)$, and hence is pinned. Thus all points are pinned and the coloring is distinguishing.
\end{proof}
The next two examples illustrate the proof of Theorem~\ref{distrib-lattice-thm}.
\begin{example}
{\rm For the distributive lattice $L = L_{150}$ in Figure~\ref{L-150-fig}, the set of join-irreducible points is $Y = \{a,b,c,d\}$, where $a=2,\ b=3, \ c=5 $ and $d=25$. For the linear extension
$E: a \prec c \prec d \prec b$ of $Q_L$,
the chain of points in $J(Q_L)$ colored red in the proof of Theorem~\ref{distrib-lattice-thm}
is $ a \prec ac \prec acd $ and the remaining vertices are green. Observe that each join-irreducible point in $J(Q_L)$ except $b$ is indeed less than or equal to a unique lowest red point: $a \preceq a $ (rank 1), \
$c \preceq ac $ (rank 2), \
$cd \preceq acd $ (rank 3).
}
\label{example-birkhoff}
\end{example}
\begin{example}
{\rm For the distributive lattice in Figure~\ref{fig-distrib-lat}, the set of join-irreducible points is $Y = \{a,b,c,d\}$, where $a = w_1$, $b = w_4$, $c = w_5$, and $d = w_2$. For the linear extension
$E: d \prec a \prec b \prec c$ of $Q_L$,
the chain of points in $J(Q_L)$ colored red in the proof of Theorem~\ref{distrib-lattice-thm} is $ d \prec ad \prec abd $ and the remaining vertices are green. Observe that each join-irreducible point of $J(Q_L)$ except $c$ is indeed less than or equal to a unique lowest red point: $a \preceq ad $ (rank 2), \
$ab \preceq abd $ (rank 3), \
$d \preceq d $ (rank 1).
Each point of $J(Q_L)$ is the join of a unique set of join-irreducible points of $J(Q_L)$, for example, point $acd$ is the join of $a,d,ac$.
}
\label{example-birkhoff-2}
\end{example}
Our proof of Theorem~\ref{distrib-lattice-thm}
provides a distinguishing coloring of $L$ for each linear extension of $Q_L$. We record this in Corollary~\ref{count-coloring-cor}.
\begin{Cor}
For any distributive lattice $L$, each linear extension of $Q_L$ leads to a distinguishing coloring of $L$ using two colors, one of which appears on exactly $|Q_L| - 1$ points.
\label{count-coloring-cor}
\end{Cor}
\begin{figure}
\caption{A lattice $L$, its poset $Q_L$ of join-irreducibles, and the downset lattice $J(Q_L)$, together with a distinguishing coloring of $J(Q_L)$.}
\label{fig-distrib-lat}
\end{figure}
\subsection{Divisibility lattices}
Divisibility lattices form a subset of the set of distributive lattices. The meet of two integers is their greatest common divisor and their join is their least common multiple. For positive integer $n$, the \emph{divisibility lattice} is the poset $L_n$ consisting of the positive integer divisors of $n$ ordered by divisibility. Figure~\ref{ex-fig} shows the poset $L_n$ for $n=pq^2$ and $n=p^2q^2$ when $p$ and $q$ are distinct primes. As illustrated by this figure, the structure of $L_n$ is determined by the prime factorization of $n$.
Let $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k} $ where the $p_i$ are distinct primes and each $a_i$ is a positive integer.
It is straightforward to check that if $f$ is an automorphism of divisibility lattice $L_n$ and $f(p_i) = p_j$ then $a_i = a_j$. The join-irreducible elements of $L_n$ are the factors of $n$ of the form $p_i^{b_i}$.
When Theorem~\ref{distrib-lattice-thm} is translated to divisibility lattices, we can say precisely when $D(L)=1$ and when $D(L)=2$.
\begin{Thm}
Let $n$ be an integer greater than 1 and write $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k} $ where $p_i$ are distinct primes and $a_i \ge 1$ for each $i$. The divisibility lattice $L_n$ has $D(L_n) = 1$ if the $a_i$ are distinct and $D(L_n)= 2 $ otherwise.
\label{divis-lat-thm}
\end{Thm}
When $D(L_n) = 2$, the coloring in the proof of Theorem~\ref{distrib-lattice-thm} does not always use a minimum number of red vertices, as seen in the following example.
\begin{example}{\rm
Consider the divisibility lattice $L_n$ where $n = 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11$. The join-irreducibles of $L_{2310}$ are the primes $2,3,5,7$ and 11. Each of these points has rank 1.
Thus, in the proof of Theorem~\ref{divis-lat-thm}, the points colored red are the four in the chain
$2 \prec 2\cdot 3 \prec 2 \cdot 3 \cdot 5 \prec 2 \cdot 3 \cdot 5 \cdot 7 $, while the remaining vertices are colored green.
Instead, we could color the three points $5\cdot 7$, \ $ 7 \cdot 11$,\ $3 \cdot 5 \cdot 7$ red and the remaining points green. Each of the rank 1 points is pinned as follows: 2 is pinned since it is the only rank one point not below any red point, 3 is pinned since it below a rank 3 red point and no others, 5 is pinned since it is below one rank 2 red point and one rank 3 red point, 7 is pinned since it is below all three red points, and 11 is pinned since it is below one rank 2 red point and no others.}
\end{example}
\section{Distinguishing Chromatic Number}\label{sect-dist-chr}
The distinguishing chromatic number $\chi_D(G)$ of a graph $G$ was introduced in \cite{CoTr06} and studied further by other authors, see for example \cite{BaPa17-a, BaPa17-b, CoHoTr09,ImKaPiSh17}.
It is defined as the minimum number of colors needed to properly color the vertices of $G$ so that the only automorphism that preserves colors is the identity.
We next define an analogous parameter for posets.
\begin{Def}{\rm A coloring of the points of poset $P$ is \emph{proper} if comparable points are assigned different colors, that is, each color class induces an antichain. The \emph{distinguishing chromatic number} of poset $P$, denoted $\chi_D(P)$, is the least integer $k$ for which there is a coloring of $P$ that is both proper and distinguishing.
}
\end{Def}
For example, for the poset $M$ in Figure~\ref{ex-fig}, $\chi_D(M)=4$ and one proper distinguishing coloring is the following: color 1 for $z$, color 2 for $y$ and $w$, color 3 for $x$, and and color 4 for $v$.
The fact that any automorphism of a poset $P$ is also an automorphism of its comparability graph $G_P$ justifies the following remark.
\begin{remark}
$\chi_D(P) \le \chi_D(G_P)$.
\end{remark}
However, some automorphisms of the graph $G_P$ are not automorphisms of the poset $P$ because they do not preserve the ordering of points in $P$.
The following result shows that $\chi_D(P)$ can differ significantly from $\chi_D(G_P)$.
\begin{Prop}
There exist posets $P$ for which the gap between $\chi_D(P)$ and $\chi_D(G_P)$ is arbitrarily large.
\end{Prop}
\begin{proof}
Let $j\geq 3$ be a positive integer and $P$ be the poset consisting of $\binom{2j}{j}$ disjoint $j$-chains. The graph $G_P$ consists of $\binom{2j}{j}$ copies of the graph $K_j$, and thus $\chi_D(G_P) = 2j$. However, $\chi_D(P)=t $, where $t$ is the smallest integer such that $(t)_j=t(t-1)(t-2)\ldots (t-j+1)\geq \binom{2j}{j}$.
We will show that $\chi_D(P) \le j+1$, demonstrating that the gap between $\chi_D(G_P)$ and $\chi_D(P)$ can be made arbitrarily large.
It remains to show $t \le j+1$. For the initial value $j=3$, we have $(j+1)_j = 4\cdot 3\cdot 2 = 24> 20 = \binom{6}{3} = \binom{2j}{j}$, so $t \le j+1$. Observe that $$\binom{2j+2}{j+1} = \frac{2(j+1)(2j+1)}{(j+1)^2}\binom{2j}{j}<4 \binom{2j}{j}$$ whereas $(t)_{j+1} = (j+1)(t)_j$, so
$(t)_j$ grows at a faster rate than $\binom{2j}{j}$ for $j \ge 3$. Thus $t \le j+1$ for $j \ge 3$ as desired.
\end{proof}
\noindent
{\bf Application:} \ The definition of $\chi_D(P)$ is related to the following problem of designing a student's course schedule. Form a poset $P$ in which the points of $P$ are the courses a student plans to take to complete a major, and $x\prec y$ if course $x$ is a prerequisite for course $y$. In a proper coloring, if two courses receive the same color, neither is a prerequisite of the other and they can be taken in the same semester. The minimum number of colors needed for a proper coloring of $P$ is the minimum number of semesters needed to complete the major. If the coloring is also distinguishing, then $P$ together with its coloring will uniquely identify the courses as well as specify which ones are taken in which semester.
The next result can be used to determine the distinguishing chromatic number of posets consisting of the sum of chains.
We denote the falling factorial as $(k)_r = k(k-1)(k-2) \cdots (k-r+1)$.
\begin{Prop}
(i) If $P$ is the poset consisting of the sum of $t$ chains in which each chain contains $r$ elements, and $k$ is the positive integer for which $(k-1)_r < t \le (k)_r$, then $\chi_D(P) = k$.
(ii) Let $P$ be the sum of chains and partition $P$ as $P_1 + P_2 + \cdots + P_m$ where $P_i$ consists of $t_i$ chains, each consisting of $r_i$ points, where $r_1, r_2, \cdots , r_m$ are distinct. Then $\chi_D(P) = \max \{\chi_D(P_i): 1 \le i \le m\}$.
\label{chi-chains-prop}
\end{Prop}
\begin{proof}
Use the arguments given in the proofs of Propositions~\ref{chains-lem} and \ref{chains-prop},
except here the vertices of a chain must get different colors, so there are $(k)_r$ ways to color a chain of $r$ points if there are $k$ colors available.
\end{proof}
An alternate way to properly color the points of a poset is to color two points distinctly if they are \emph{incomparable}, or equivalently, so that each color class induces a chain. We call this a \emph{chain-proper} coloring. A coloring that is both chain-proper and distinguishing
is related to the following problem of assigning rooms to a set of scheduled events.
{\bf Application:} Represent a set of events as a poset $P$ in which the events are the points of $P$ and $x\prec y$ if event $x$ ends before event $y$ begins. In a chain-proper coloring, each color class is a set of events that can be assigned to the same room, and thus the minimum number of color classes is the number of rooms needed to schedule all of the events. If the coloring is distinguishing as well as proper, then the poset together with its coloring will uniquely identify the events as well as specifying which room each would occupy.
As an example, the poset $M$ in Figure~\ref{ex-fig}, requires two colors for a chain-proper coloring that is distinguishing: color $x$, $y$, and $z$ red, and color $v$ and $w$ blue.
The next proposition is a lovely consequence of Dilworth's theorem.
\begin{Prop} For any poset $P$, the minimum number of colors needed for a chain-proper coloring that is also distinguishing is the width of $P$.
\label{chi-D-thm}
\end{Prop}
\begin{proof}
Let $k$ be the width of $P$ and let $A$ be an antichain of $P$ with $|A| =k$. Coloring the points of $A$ properly requires $k$ colors, hence at least $k$ colors are required. To show that $k$ colors suffice, use Dilworth's theorem to partition the points of $P$ into $k$ sets, each of which induces a chain in $P$. Color all points on chain $i$ using color $i$ for $i = 1,2,3 \ldots, k$. By definition, this coloring is proper. Observe that all chain-proper colorings are distinguishing because each point on chain $i$ has a unique height on that chain, and height is preserved by automorphisms. \end{proof}
\subsection{Bounds for distributive lattices}
In our next result, we again use Birkhoff's Theorem, this time to relate the distinguishing chromatic number of a distributive lattice to that of its poset of join-irreducibles. Note that Lemma~\ref{chi-D-bound} is tight for $L_{pq}$ when $p$ and $q$ are distinct primes.
\begin{lemma} If $L$ is a distributive lattice, then $\chi_D(L) \le \chi_D(Q_L) + |Q_L|$.
\label{chi-D-bound}
\end{lemma}
\begin{proof}
First color each point of $L$ at rank $j$ using color $j$ for $0 \le j \le |Q_L|$. This provides a proper coloring of $L$ and by Remark~\ref{rank-rem}, it uses $|Q_L| + 1$ colors. Next, recolor the points of $Q_L$ using a proper and distinguishing coloring with $\chi_D(Q_L)$ new colors. The resulting coloring of $L$ is still proper. It is also distinguishing since by Corollary~\ref{tough-cor}
any point of $L$ can be written uniquely as the join of elements of $Q_L$, and automorphisms preserve joins. All rank 1 points of $L$ are in $Q_L$, so the color 1 is never used in the final coloring, thus we have a proper and distinguishing coloring of $L$ using $ \chi_D(Q_L) + |Q_L|$ colors.
\end{proof}
The lattice $L$ in Figure~\ref{fig-distrib-lat} has $\chi_D(Q_L) = 3$ and $|Q_L| = 4$ The proof of Lemma~\ref{chi-D-bound} provides a proper and distinguishing coloring of $L$ using 7 colors. We can show that $\chi_D(L) \ge 6$ as follows. At least 5 colors are needed for a proper coloring, and any proper coloring using 5 colors assigns the same color to $w_4$ and $w_5$, and thus is not distinguishing. The reverse inequality, $\chi_D(L) \le 6$, follows from our next theorem, and thus lattice $L$ is an example that shows the bound in Theorem~\ref{chi-D-bound-minus} is tight.
\begin{Thm} If $L$ is a distributive lattice and $\chi_D(Q_L)\ge 3$, then $\chi_D(L)\leq |Q_L|+ \chi_D(Q_L) - 1$.
\label{chi-D-bound-minus}
\end{Thm}
\begin{proof}
Let $d=\chi_D(Q_L)$ and let $\phi$ be a proper and distinguishing coloring of
$Q_L$, using the colors in the set $A = \{a_1, a_2, \ldots, a_d\}$. Let $A_i$ be the set of points in $Q_L$ with color $a_i$ for $1 \le i \le d$, that is $A_i = \{x \in Q_L: \phi(x) = a_i\}$. We use a different set of colors, the \emph{rank colors}, for the remaining points of $L$. For each point $x$ in $L$, let $r(x)$ be its rank in $L$. As before, each uncolored point $x$ in $L$ has $r(x) $ in the set $R = \{0,2,3,4, \ldots, |Q_L|\}$, and we color point $x$ using color $r(x)$.
As before, this is a proper and distinguishing coloring of $L$ using $|Q_L|+d$ colors. We will
construct a new coloring that uses one fewer color by eliminating the color 2.
We define three subsets of points of $Q_L$ as follows.
\[S_1 = \{z \in A_1: r(z) \ge 3\}\]
\[S_2 = \{z \in A_2 : r(z) \ge 3 \hbox { and } \exists \ y \in A_1 \hbox{ with } r(y) = 1 \hbox { and } \ y \prec z\} \]
\[S_3 = \{z \in A_3: r(z) \ge 3 \hbox { and } \exists \ y \in A_1 \hbox{ and } \exists \ w \in A_2 \hbox{ with }\\ r(y) = r(w) = 1, \hbox { and } \ y,w \prec z \} \]
Define a new coloring $\psi$ on points $x$ of $Q_L$ as follows.
\begin{equation*}
\psi(x) =
\begin{cases}
r(x) & \text{if } x\in S_1\cup S_2\cup S_3\\
\phi(x) & \text{otherwise.}\\
\end{cases}
\end{equation*}
The coloring $\psi$ is proper on the points of $Q_L$ because $\phi$ gives a proper coloring of the points not in $S_1 \cup S_2 \cup S_3$, the rank function gives a proper coloring of the points in $S_1 \cup S_2 \cup S_3$, and the colors in $R$ are different from the colors in $A$. We next show that $\psi$ is a distinguishing coloring of $Q_L$. Let $h$ be an automorphism of $Q_L$ that preserves the coloring $\psi$.
First we show that $h(S_1) = S_1$, $h(S_2) = S_2$ and $h(S_3) = S_3$, that is, $h$ preserves membership in each of the sets $S_1$, $S_2$, $S_3$. For $z \in S_1$ we have $\psi(z) = r(z)$, so $h(z)$ has a color in the set $ R$ and thus $h(z) \in S_1 \cup S_2 \cup S_3$.
By the definition of $S_1$, we know $\phi(z) = a_1$, and because $\phi$ is a proper coloring, the point $z$ is incomparable to all other points of color $a_1$. Each point in $S_2\cup S_3$ is comparable to a rank 1 point with color $a_1$. Since $h$ preserves coloring $\psi$, we know $h(z) \not\in S_2\cup S_3$. Hence $h(z)\in S_1$, and $h(S_1) = S_1$.
For $z\in S_2$, we have $\psi(z) = r(z) $, so $h(z)$ also has a color in the set $ R$, and thus $h(z) \in S_1 \cup S_2 \cup S_3$. However, $h$ is an automorphism and $h(S_1) = S_1$, hence $h(z) \in S_2\cup S_3$. Each point in $S_3$ is comparable to a rank 1 point with color $a_2$, and since $h$ preserves $\psi$, we know $h(z)\not \in S_3$. Hence $h(z)\in S_2$ and $h(S_2)=S_2$. Similarly, each point in $S_3$ has a color in $R$ and thus $h(S_3)=S_3$ as well.
Since $h$ preserves $\psi$ and $h(S_i)=S_i$, then $h(A_i-S_i)=A_i-S_i$ and $h(A_i)=A_i$ for $1\leq i\leq 3$. We know $h(A_i)=A_i$ for $i\geq 4$ because $h$ preserves $\psi$. Thus, $\psi$ preserves $\phi$.
Since $\phi$ is a distinguishing coloring of $Q_L$, we conclude that $h$ is the identity automorphism. Thus we have shown that $\psi$ is a distinguishing coloring of $Q_L$.
Now we extend $\psi$ to $L$. For each pair of distinct rank 1 points $x,y$ of $Q_L$, define
\begin{equation*}
g(x,y) =
\begin{cases}
a_1 & $\mbox{if $a_1\not \in \{ \phi(x),\phi(y)\}$ }$\\
a_2 &$\mbox{if $a_1\in \{ \phi(x),\phi(y)\}$ and $a_2\not\in \{ \phi(x),\phi(y)\}$ }$\\
a_3 &\{ \phi(x),\phi(y)\} = \{a_1, a_2\}\\
\end{cases}
\end{equation*}
We now extend $\psi$ to the elements of $L$ that are not in $Q_L$. By Remark~\ref{rank-rem},
each rank 2 point in $L$ that is not in $Q_L$ covers exactly two rank 1 points of $L$. This allows us to define $\psi(z)$ when $r(z) = 2$ and $z \not\in Q_L$ as follows.
\begin{equation*}
\psi(z) =
\begin{cases}
r(z) & z\not\in Q_L \mbox{ and } r(z)>2 \mbox{ or } r(z)=0\\
g(x,y) & z\not \in Q_L,\;r(z)=2 \mbox{ and } z \mbox { covers } x,y. \\
\end{cases}
\end{equation*}
Observe that $\psi$ uses colors from the set $\{0,3,4,\ldots, |Q_L|\} \cup A$, for a total of $d+|Q_L|-1$ colors.
Since $\psi$ is distinguishing on $Q_L$, it is distinguishing on $L$. We need additionally to show that $\psi$ is proper on $L$. The color $j$, for $j=0,3,4,\ldots, |Q_L|$, is used only on points of rank $j$, so the use of the rank colors is proper. We need to show that the set of points of $L$ colored $a_1$ by $\psi$ form an antichain, and similarly for the points colored $a_2$ and the points colored $a_3$. We partition the set of $z\in L$ with $r(z) =2$ as $T_1 \cup T_2 \cup T_3$ as follows, where the rank 1 points covered by $z$ are denoted $x$ and $y$:
(i) $z \in T_1$ if $a_1 \not\in \{ \phi(x), \phi(y)\}$
(ii) $z \in T_2$ if $a_1 \in \{ \phi(x), \phi(y)\}$ and $a_2 \not\in \{ \phi(x), \phi(y)\}$
(iii) $z \in T_3$ if $a_1,a_2 \in \{ \phi(x), \phi(y)\}$
Then the $a_i$-color class of $\psi$ is $T_i\cup (A_i-S_i)$, for $1\leq i\leq 3$.
Since $\phi$ is proper, $A_i$ is an antichain, hence $A_i-S_i$ is an antichain. Since every point in $T_i$ has rank 2, $T_i$ is an antichain.
Suppose that $z\in T_i$ and $w\in A_i-S_i$ are comparable. Since $w\in Q_L$, $r(w)\neq 0$. Since $z$ covers only $x,y$ and neither is in $A_i$, then $r(w)\neq 1$. Since $r(z)=2$, and points of the same rank are not comparable, then $r(w)\neq 2$. Hence $r(w)\geq 3$ and $w\succ z$.
Case 1: Let $i=1$. Then $\psi(z)=a_1$ and $w\in A_1-S_1$, so $\psi(w)\neq a_1$ by the definition of $S_1$. Therefore, $\psi(z)\neq \psi(w)$.
Case 2: Let $i=2$. Then $\psi(z)=a_2$ and $w\in A_2-S_2$. Thus $w$ is not above any rank 1 point colored $a_1$, but $z\in T_2$ and $z\succ x $ and $\psi(x)=a_1$. By transitivity of the order relation, this contradicts the assumption that $w $ and $z$ are comparable.
Case 3: Let $i=3$. Then $\psi(z)=a_3$ and $w\in A_3-S_3$. Thus, $w$ cannot be above both a rank 1 point colored $a_1$ and a rank 1 point colored $a_2$, but $z\in T_3$ and $z\succ x $, $\psi(x)=a_1$ and $z\succ y$, $\psi(y)=a_2$. By transitivity of the order relation, this contradicts the assumption that $w $ and $z$ are comparable.
Thus sets of the points colored $a_1, a_2, a_3$, respectively are antichains, and $\psi$ is proper. Since we have shown it is distinguishing, $\chi_D(L)\leq d+|Q_L|-1$, as desired.
\end{proof}
Theorem \ref{chi-D-bound-minus} is tight, as we will see in Section~\ref{Boolean}.
\subsection{Bounds for divisibility lattices}
In the next theorem, we provide an alternative bound in Theorem~\ref{chi-D-bound-minus} in the instance when the distributive lattice is a divisibility lattice. We begin by coloring each point by its rank, and then recolor the join-irredicuble points using the method in Proposition~\ref{chi-chains-prop}.
Not all join-irredicuble points need to receive new colors, so we can use fewer colors than the number needed in Proposition~\ref{chi-chains-prop}. Recall the falling factorial function is $(n)_k= n(n-1)(n-2)\cdots (n-k+1)$ where $(n)_0 = 1$.
\begin{Thm} \label{better-when-smaller}
Let $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k} $ where the $p_i$ are distinct primes and each $a_i$ is a positive integer and let $Q = Q_{L_n}$. Partition $Q$ as $Q_1 + Q_2 + \cdots + Q_j$ where $Q_i$ consists of $t_i$ chains, each consisting of $r_i$ points, where $r_1, r_2, \cdots , r_j$ are distinct.
For $1\leq i\leq k$, let $m_i$ be the smallest integer such that $t_i \leq \sum_{\ell=0}^{\min\{r_i, m_i\}}(^{r_i}_{\ell}) (m_i)_{\ell}$. Let $m=\max \{m_i: 1 \le i \le j\}$.
Then $\chi_D(L_n) \leq m + |Q|+1 $.
\end{Thm}
\begin{proof}
We begin by coloring each point in $L_n$ with its rank. There are $|Q|+1$ ranks in $L_n$.
We then re-color some points in $Q$ with $m$ new colors as follows.
We can choose $\ell$ points on each $r_i$-chain to recolor in $(^{r_i}_{\ell})$ ways, and the number of ways to recolor these points with $m_i$ colors is $(m_i)_{\ell}$. Thus the total number of ways to recolor $\ell$ points of an $r_i$-chain is $(^{r_i}_{\ell}) (m_i)_{\ell}$. Since two chains with a different number of points recolored will not have the same coloring, the total number of ways to recolor the points of $Q_i$ using $m_i$ colors is $\sum_{\ell=0}^{\min\{r_i, m_i\}}(^{r_i}_{\ell}) (m_i)_{\ell}$. By our choice of $m_i$, we can color the $t_i$ chains, each containing $r_i$ points, differently. Similarly, by our choice of $m$, there are enough colors for each value of $i$. Chains of different lengths in $Q$ can not map to one another under any automorphism, hence every point in $Q$ is pinned by this coloring. By Proposition~\ref{tough-pin-prop}, every point in $L_n$ is pinned and our coloring is distinguishing.
\end{proof}
As seen from the proof of the theorem, chains of different lengths may be considered independently. Let $Q_L$ have $t$ chains, each of length $r$, and
let $m$ be the smallest integer such that $t \leq \sum_{\ell=0}^{\min\{r, m\}}(^{r}_{\ell}) (m)_{\ell}$. Then the upper bound in Theorem~\ref{chi-D-bound-minus} for $\chi_D(L)$ is $\chi_D(Q_L)+|Q_L|-1$, and since an $r$-chain needs at least $r$ colors, the smallest value of $\chi_D(Q_L)$ is $r$.
The coloring in Theorem~\ref{better-when-smaller} allows us to use fewer than $r$ new colors, and thus can be a better bound when $t$ is not too large. For example, let $r=5$ and $t=31$, then Theorem~\ref{better-when-smaller} allows us to use only two new colors, whereas the proof in Theorem~\ref{chi-D-bound-minus} uses at least five. Thus, Theorem~\ref{chi-D-bound-minus} gives an upper bound of $5+5\cdot 31-1=159$, whereas Theorem~\ref{better-when-smaller} gives an upper bound of $2+5\cdot 31+1 = 158$.
The formula $\sum_{\ell=0}^{\min\{r, m\}}(^{r}_{\ell}) (m)_{\ell}$ is a well-known formula for the number of ways of placing $\ell $ rooks on an $m\times r$ chessboard \cite{KaRi46}.
\subsection{Bounds for Boolean lattices} \label{Boolean}
The Boolean lattice $B_n$ is the lattice of
subsets of $\{1,2,3, \ldots, n\}$, ordered by
inclusion. It is a distributive lattice and its
join-irreducibles are the singletons $\{ \{k\}\;|\;1\leq k\leq
n\}$. The number of points in any longest chain of $B_n$ is $n+1$.
By Theorem~\ref{chi-D-bound-minus}, $\chi_D(B_n)\leq n+n=2n$. The next theorem has a tighter bound.
\begin{Thm} \label{boolean-th} Let $B_n$ be the Boolean lattice of $\{1,2,3,\ldots, n\}$. Then
$\chi_D(B_n)\leq n+3$.
\end{Thm}
\begin{proof} Let $n$ be odd. We initially color each element $x$ with $r(x)$, for $0\leq r(x)\leq n$. This coloring is proper. Next we will recolor some of the vertices to obtain a distinguishing coloring, using two new colors, $a$ and $b$.
Let $S_i = \{i, i+1, i+2, \ldots, 2i-1\}$ for $1 \le i \le \frac{n+1}{2}$. For $1\le i < j \le \frac{n+1}{2}$, we have $i \in S_i - S_j$ and $2j-1 \in S_j - S_i$, thus the $S_i$ form an antichain. We change the color of each of the $S_i$ to $a$.
Similarly, let $T_i = \{n-(i-1), n-i, n-(i+1), \ldots, n-2(i-1)\}$ for $1 \le i \le \frac{n+1}{2}$. For
$1\le i < j \le \frac{n+1}{2}$, we have $n-(i-1) \in T_i - T_j$ and $n-2(j-1) \in T_j - T_i$, thus the $T_i$ form an antichain. We change the color of each of the $T_i$ to $b$.
For example, let $n=7$. Then we color $\{1\}, \{2,3\}, \{3,4,5\}, \{4,5,6,7\}$ with $a$, and
we color $\{7\}, \{6,5\}, \{5,4,3\},\{4, 3, 2, 1\}$ with $b$
An alternative description of the red elements
is, for each $1\leq i\leq \frac{n+1}{2}$, $S_i
$ contains the $i$ smallest elements greater
than or equal to $i$. Similarly, the blue
elements are described as, for each $1\leq j\leq
\frac{n+1}{2}$, $T_j$ contains the $j$ largest
elements less than or equal to $n-j$.
We show that the coloring is distinguishing. Each point of $B_n$ colored $a$ is pinned because it is the only point colored $a$ in its rank. Similarly, each point of $B_n$ colored $b$ is pinned. Since $Q_{B_n}$ is the set of rank 1 points, it is enough to show that the rank 1 points are pinned.
Given $i$, $1\leq i\leq \frac{n+1}{2}$, then the
highest ranked point colored $a$ that contains $i$ is
$S_i$. For $\frac{n+1}{2}\leq j$, the highest
ranked point colored $a$ that contains $j$ is
$S_{\frac{n+1}{2}}$. Thus $\{i\}$ is pinned for
$1\leq i\leq \frac{n-1}{2}$.
Similarly, given $j$, $\frac{n+1}{2}\leq j\leq
n$, the lowest ranked point colored $b$ that contains
$j$ is $T_j$. The lowest ranked point colored $b$
that contains $i$ is $T_{\frac{n+1}{2}}$, for
$1\leq i\leq \frac{n+1}{2}$. Thus $\{j\}$ is
pinned for $\frac{n+3}{2}\leq j\leq n$. Now
$\frac{n+1}{2} $ is the only element that is contained in both a point colored $a$ and a point colored $b$
at rank $\frac{n+1}{2}$. Thus, $\{ \frac{n+1}{2}\}$ is
pinned. Hence all the join-irreducibles of $B_n$
are pinned, and this coloring is
distinguishing.
Let $n$ be even. Then color all the elements by
their rank and then alter the coloring using the
same red and blue elements as in the case for
$B_{n-1}$. In this coloring, $n$ will not appear
in any point colored $a$ or $b$, but each of
$1,2,\ldots n-1$ will do so. Thus $\{n\}$ is
pinned. The other join-irreducibles of $B_n$ are
pinned for the same reasons as in the previous
argument.
\end{proof}
The examples in the next two propositions show that the bound in Theorem~\ref{chi-D-bound-minus} is tight when $\chi_D(Q_L) = 3$ or $4$.
\begin{Prop} The Boolean Lattice $B_3$ has $\chi_D(B_3)= 5$.
\end{Prop}
\begin{proof}
Observe that $|Q_{B_3}| = 3$ and $\chi_D(Q_{B_3}) = 3$, so by Theorem~\ref{boolean-th}, $\chi_D(B_3)\leq 5$. The following is a proper and distinguishing coloring of $B_3$ using five colors: $\emptyset $ is red, $\{1\}$ and $\{2,3\}$ are blue, $\{3\}$ and $\{1,2\}$ are green, $\{2\}$ and $\{1,3\}$ are yellow, and $\{1,2,3\}$ is purple.
Thus, $\chi_D(B_3)= 5$.
\end{proof}
\begin{Prop}
The Boolean lattice $B_4$ has
$\chi_D(B_4)= 7$.
\end{Prop}
\begin{proof}
Observe that $|Q_{B_4}| = 4$ and $\chi_D(Q_{B_4}) = 4$, so by Theorem~\ref{boolean-th}, $\chi_D(B_4)\leq 7$.
Thus we must show that $\chi_D(B_4) >6$. Suppose for a contradiction that we have a proper and distinguishing coloring $\phi$ of $B_4$ using 6 colors. One color is used for $\emptyset$ and another for $\{1,2,3,4\}$, so the remaining points must be colored using four colors: red, blue, yellow and green. We consider cases depending on the number of colors used on the rank 1 points.
\noindent {\bf Case 1: } Four colors are used on the rank 1 points.
Without loss of generality we may assume $\{1\}$ is red, $\{2\}$ is blue, $\{3\}$ is green, and $\{4\}$ is yellow. Since $\phi$ is proper, we know $\{1,2,3\}$ must be yellow and $\{1,2,4\}$ must be green. Now no color is available for $\{1,2\}$ since it is comparable to points that use all four colors.
\noindent {\bf Case 2:} Three colors are used on the rank 1 points.
Without loss of generality we may assume $\{1\}$ is red, $\{2\}$ is blue, and $\{3\}$ and $\{4\}$ are yellow. Since $\phi$ is proper, the colors are forced on all points except $\{3,4\}$, namely, $\{1,2,3\}$ and $\{1,2,4\}$ are green, $\{1,2\}$ is yellow, $\{2,3\}$ is red, $\{1,3\}$ is blue, $\{2,4\}$ is red, $\{1,3,4\}$ and $\{2,3,4\}$ are green, and $\{1,4\}$ is blue. Regardless of whether $\{3,4\}$ is red or blue, the automorphism that swaps all instances of $3$ and $4$ preserves colors, contradicting our assumption that $\phi$ is distinguishing.
\noindent {\bf Case 3:} Two colors are used on the rank 1 points.
First consider the instance that the two colors each appear on two rank 1 points.
Without loss of generality we may assume $\{1\}$ and $\{2\}$ are red, $\{3\}$ and $\{4\}$ are blue, and $\{1,3\}$ is green. Since $\phi$ is proper, the colors are forced on all points except $\{1,2\}$ and $\{3,4\}$ as follows: $\{1,2,3\}$ and $\{1,3,4\}$ are yellow, $\{2,3\}$ is green, $\{1,4\}$ is green, $\{1,2,4\}$ and $\{2,3,4\}$ are yellow, and $\{2,4\}$ is green. Regardless of the colors of $\{1,2\}$ and $\{3,4\}$, the automorphism that swaps 1 and 2 and also swaps 3 and 4 preserves colors, contradicting our assumption that $\phi$ is distinguishing.
Now consider the instance that one color appears on three of the rank 1 points and the other appears on one. Without loss of generality we may assume $\{1\}$ and $\{2\}$ and $\{3\}$ are red, $\{4\}$ is blue, and $\{3,4\}$ is green. Since $\phi$ is proper, the colors are forced on all points except $\{1,2\}$, $\{1,3\}$, $\{2,3\}$, and $\{1,2,3\}$ as follows: $\{1,3,4\}$ and $\{2,3,4\}$ are yellow, $\{2,4\}$ and $\{1,4\}$ are green, and $\{1,2,4\}$ is yellow. The remaining points $\{1,2\}$, $\{1,3\}$, $\{2,3\}$ must be green or blue, and two of them must be the same color. Without loss of generality, $\{1,2\}$ and $\{1,3\}$ are the same color, but then the automorphism that swaps 2 and 3 preserves colors, contradicting our assumption that $\phi$ is distinguishing.
\noindent {\bf Case 4:} The rank 1 points use one color.
If the rank 3 points use two or more colors, then we reach a contradiction using previous cases since $\phi$ is also a proper and distinguishing coloring of the dual of $B_4$. Thus without loss of generality we may assume $\{1\}$, $\{2\}$, $\{3\}$ and $\{4\}$ are red, and $\{1,2,3\}$, $\{1,2,4\}$, $\{1,3,4\}$, and$\{2,3,4\}$ are blue, and the rank 2 points are green and yellow. Any such coloring is proper but not distinguishing. Coloring the rank 2 points using green and yellow is equivalent to giving a distinguishing coloring to the edges of the $K_4$ graph using green and yellow. We show this is impossible.
If such a coloring were possible, without loss of generality, at least three edges are green. First suppose there is a triangle of green edges, say $\{1,2\}$, $\{1,3\}$, and $\{2,3\}$ are green. If the remaining edges are yellow, the automorphism $(123)(4)$ preserves colors. If one additional edge is green, say $\{1,4\}$, then the automorphism $(23)(1)(4)$ preserves colors. If two additional edges are green, say $\{1,4\}$ and $\{2,4\}$ then the automorphism $(12)(3)(4)$ preserves colors. If there is no triangle of green edges, then without loss of generality, $\{1,2\}$, $\{2,3\}$ and $\{3,4\}$ are green and the remaining three edges are yellow. In this instance, the automorphism $(14)(23)$ preserves colors.
\end{proof}
\section{Rank-Connected Planar Posets} \label{sect-planar}
In this final section of results, we consider rank-connected planar lattices.
In Section~\ref{sect-poset-defn} we defined planar posets and ranked posets. Note that a planar poset is a lattice if it has both a minimal and maximal element.
\begin{Def} {\rm A ranked poset is \emph{rank-connected} if every pair of consecutive ranks, considered as a vertex-induced subgraph is connected. }\end{Def}
\begin{Def}{\rm Incomparable points $x$ and $y$ are \emph{twins} if they have the same relationship to all other points of the poset. A poset is \emph{twin-free} if it has no twins. }
\end{Def}
Cohen-Macaulay posets, are a well-known class of posets in the study of flag $f$-vectors of simplicial complexes, for example, see \cite{St96}. Although it is difficult to obtain a complete characterization of the set of flag $f$-vectors of Cohen-Macaulay posets, many subclasses of this set are lexicographically shellable and thus have an explicit shelling. Collins \cite{Co92} has shown that for ranked, planar lattices, being Cohen-Macaulay is the same as being rank-connected. In this section, we show that the distinguishing number of a rank-connected, twin-free, planar lattice is less than or equal to 2. The proof is completely different from the proof of Theorem~\ref{distrib-lattice-thm}, which uses the regular structure of a distributive lattice. However, the 2-coloring of the points is again a chain in the poset.
We say that a Hasse diagram for ranked planar poset is a \emph{standard diagram} if it is planar, all points at a given rank have the same $y$-coordinate, and all edges are straight line segments.
The following result appears to be part of the folklore of the field \cite{Tr19}. We include a proof for completeness.
\begin{Prop}
If $P$ is a ranked planar poset with $\hat{0}$ and $ \hat{1}$ then $P$ has a standard~diagram.
\label{folklore-prop}
\end{Prop}
\begin{proof}
Partition the points of $P$ by rank so that $R_i$ is the set of points of rank $i$ for $0 \le i \le n$. Since $P$ is ranked, each covering edge in $P$ is between points at consecutive ranks. Suppose we have a planar Hasse diagram for $P$ in which the points of each rank have the same $y$-coordinate for ranks $k$ and lower and every edge between points of rank at most $k$ is a straight line segment. If $k = n$ we are done, so assume $k < n$.
Let $w_1, w_2, w_3, \ldots, w_r$ be the elements of $R_k$ listed from left to right in the Hasse diagram. If $n = k+1$ then $R_{k+1} = \{\hat{1}\}$ and we can draw a straight line segment from each element of $R_k$ to $\hat{1}$, and this completes the proof. Otherwise, $n \ge k+2$.
Let $L_k$ be the horizontal line containing the points of $R_k$ and $L_{k+1}$ the horizontal line containing the point(s) of $R_{k+1}$ with lowest $y$-coordinate. In Figure~\ref{fig-planar}, the only point of $R_{k+1}$ with lowest $y$-coordinate is $z_2$. Order the edges between $R_k$ and $R_{k+1}$ from left to right as $e_1, e_2, \ldots, e_t$ by their order in the strip between lines $L_k$ and $L_{k+1}$. This order is uniquely determined because the diagram is planar. This ordering of edges induces an ordering of the points in $R_{k+1}$ as follows: for any $u,v\in R_{k+1}$, we order $u$ before $v$ if the leftmost edge $e_i$ incident to $u$ is to the left of the leftmost edge $e_j$ incident to $v$. The resulting order is $z_1, z_2, z_3, \ldots , z_s$ and this is illustrated by the example in Figure~\ref{fig-planar}.
For each $z_i$ that is above line $L_{k+1}$, choose an edge $e_j$ incident to $z_i$ and relocate $z_i$ to the point $z_i'$ where edge $e_j$ meets line $L_{k+1}$. In Figure~\ref{fig-planar}, in each case the first such edge was selected. For each $z_i \in R_{k+1}$, let $N(z_i)$ be the set of points in $R_k$ covered by $z_i$. The points in $N(z_i)$ have consecutive indices, for otherwise, there would be a point in $R_k$ with no upward route to $\hat{1}$. Thus we can think of $z_i$ and its edges to $N(z_i)$ as forming a cone. If $N(z_i) = \{w_r, w_{r+1}, \ldots, w_t\}$, then $z_i$ is the only member of $R_{k+1}$ that covers any of the internal points $w_{r+1}, w_{j+2}, \ldots, w_{t-1}$, for any other such element in $R_{k+1}$ would have no upward path to $\hat{1}$. Thus cones intersect only at their outermost points. Indeed, if $w_r \neq w_t$, then no other $z_j$ of $R_{k+1}$ can cover \emph{both} $w_r$ and $w_t$, because this would imply that one of $z_i, z_j$ would have no upward path to $\hat{1}$. Thus for all $z_i, z_j\in R_{k+1}$ with $i \neq j$ we have, $|N(z_i)\cap N(z_j)|\leq 1$. Furthermore, if $i<j$ and $N(z_i)\cap N(z_j)\neq \emptyset$, then by the planarity assumption and the way we indexed the $z_i$'s, the unique point in $N(z_i)\cap N(z_j)$ is the rightmost element of $R_k$ incident to $z_i$ and also is the leftmost element of $R_k$ incident to $z_j$.
Starting with edges incident to $z_1 $ and continuing rightward, we can draw the edges between $R_{k+1}$ and $R_k$ as straight line segments from the points $z_1', z_2', \ldots, z_s'$ on $L_{k+1}$ to the points $w_1, w_2, \ldots w_r$ on $L_k$ and by the properties above we know that none of these segments cross. This is illustrated in the right portion of Figure~\ref{fig-planar}.
It remains to reroute the edges between $R_{k+1}$ and $R_{k+2}$. For each $z_i' \in R_{k+1}$, form a narrow band from $z_i'$ to $z_i$ along the original edge $e_j$ that is incident to $z_i'$ (see Figure~\ref{fig-planar}). Each edge between $z_i$ and $ R_{k+2}$ will now start at $z_i'$, travel through this band to $z_i$ and continue on its original path to its destination in $R_{k+2}$. The result is a planar Hasse diagram for $P$ in which points at each rank are each located on a horizontal line for ranks $k+1$ and lower, and edges between points of ranks at most $k+1$ are straight line segments. The result follows by induction.
\end{proof}
\begin{figure}
\caption{An illustration of the induction step in the proof of Proposition~\ref{folklore-prop}
\label{fig-planar}
\end{figure}
\begin{remark}
Using Proposition~\ref{folklore-prop},
it is not hard to show that the poset $L_{pqr}$ in Figure~\ref{ex-fig} is not planar.
\label{B-not-planar}
\end{remark}
The next lemma appears in \cite{Co92} and is helpful to us in the induction proof of Theorem~\ref{twin-free}.
\begin{lemma}
\label{karen-lem}
Given a standard diagram of a planar rank-connected poset that has a $\hat{0}$ and a $\hat{1}$, there exists a rank in which the leftmost element $x$ covers exactly one element, $a$, and is covered by exactly one element, $b$. Furthermore, if there is an element immediately to the right of $x$, it also covers $a$ and is covered by $b$.
\end{lemma}
\begin{Thm} If $P$ is a twin-free, rank-connected planar lattice, then $D(P) \le 2$. \label{twin-free}
\end{Thm}
\begin{proof}
Let $r$ be the maximum rank of points in $P$. In any automorphism of $P$, rank is preserved. If there exists a point $x$ in $P$ (other than $\hat{0}$ and $\hat{1}$) for which $x$ is the only element of its rank, then $x$ is pinned. Thus we need only pin the elements below $x$ and separately the elements above $x$. So without loss of generality, we may assume $P$ has at least two points at each rank other the lowest and highest ranks.
By Proposition~\ref{folklore-prop}, we may fix a standard diagram of $P$. At each rank, there is a leftmost point in the diagram, and because the diagram is planar and $P$ is rank-connected, the union of these points forms maximal chain $C_0$ from $\hat{0}$ to $\hat{1}$.
Color the points on chain $C_0$ red, except for its minimal and maximal elements. Since there is at most one red point at each rank, these points are pinned. Color the remaining points blue. We show this coloring is distinguishing.
We apply Lemma~\ref{karen-lem}, repeatedly to obtain a sequence of chains
$C_0, C_1, \ldots, C_n$, each from $\hat{0}$ to $\hat{1}$, so that $C_i$ and $C_{i+1}$ are identical except for two points $x_i \in C_i$ and $x_{i+1} \in C_{i+1}$ where $x_i$ and $x_{i+1}$ have the same rank in $P$ and $x_{i+1}$ is immediately to the right of $x_i$ in the diagram.
We know that $x_0$ is pinned since it is red and we proceed by induction.
Assume the points $x_0, x_1, \ldots, x_{j-1}$ are pinned, thus the points in $C_{j-1}$ are pinned. If there are no points at $x_j$'s rank that lie to the right of $x_j$ then $x_j$ is pinned since all remaining points at that rank are already pinned. Otherwise, there exists one or more points at $x_j$'s rank that lie to the right of $x_j$.
Let $a$ be the point immediately below $x_j$ on chain $C_j$ and $b$ the point immediately above $x_j$ on $C_j$. Suppose $x_j$ is not pinned, so thus there exists a nontrivial automorphism $\phi$ of $P$ with $\phi(x_j) = w_j$ for some $w_j \neq x_j$. We know $w_j$ has the same rank as $x_j$ and is located to the right of $x_j$ since the points to the left of $x_j$ are already pinned. Since $\phi$ is an automorphism, $\phi(a) \prec \phi(x_j) \prec \phi(b)$ and since $a$ and $b$ are pinned we have $a \prec w_j \prec b$.
Partition the set of points with $b$'s rank as $B_1 \cup B_2 \cup \{b\}$ where the points in $B_1$ lie to the left of $b$ and the points in $B_2$ lie to the right of $b$. By planarity, $w_j$ is not adjacent to any point in $B_1$. However, the points in $B_1 \cup \{b\}$ are pinned by our induction hypothesis, and $\phi(x_j) = w_j$, so $x_j$ cannot be adjacent to any point of $B_1$ either. Also by planarity, $x_j$ is not adjacent to any points in $B_2$, thus the only point at $b$'s rank that is adjacent to $x_j$ is the point $b$. Similarly, the only point at $a$'s rank adjacent to $x_j$ is $a$. So $x_j$ is adjacent only to $a$ and $b$. The same must be true of $w_j$ since $\phi(x_j) = w_j$ and $a$ and $b$ are pinned.
This means $x_j$ and $w_j$ are twins in $P$, a contradiction. Thus $x_j$ is pinned and this completes the induction.
\end{proof}
\section{Open Questions}
We conclude with some open questions.
\begin{Ques} \rm Is Theorem~\ref{boolean-th} tight for all $n\geq 5$?
\end{Ques}
\begin{Ques} \rm Many theorems about 2-distinguishability can be proven using the Motion Lemma, proved by Russell and Sundaram \cite{RuSu98}.
Is there a proof of Theorem~\ref{distrib-lattice-thm} using the Motion Lemma?
\end{Ques}
\begin{Ques} \rm Hadjicostas \cite{Ha19} has found the generating function for the number of distinguishing 2-colorings of an $n$-cycle. Given a distributive lattice $L$, what is the generating function of the number of distinguishing 2-colorings of $L$?
\end{Ques}
\begin{Ques} \rm The {\it cost} of 2-distinguishing a graph $G$ is the minimum size of a color class in any 2-distinguishing labeling of $G$, see \cite{Bo13}. What is the cost of 2-distinguishing a distributive lattice? The cost may be smaller than the minimum sizes of color classes in the 2-labelings we have used in our proofs. For example, the cost of 2-distinguishing $L_{p^2q^2}$ is 1, as can be seen by coloring point $p$ red and the remaining points blue, whereas the cost of 2-distinguishing $L_{pqr}$ is 2, because there will still be an automorphism that preserves the colors even if one point is fixed.
\end{Ques}
\end{document} |
\begin{document}
\title{\bf Motional Squashed States}
\author{Stefano Mancini$^{1\,\dag}$,
David Vitali$^{2\,\dag}$, and Paolo Tombesi$^{2\,\dag}$}
\address{
$^{1}$ Dipartimento di Fisica, Universit\`a di Milano,
Via Celoria 16, I-20133 Milano, Italy\\
$^{2}$ Dipartimento di Matematica e Fisica, Universit\`a di
Camerino, via Madonna delle Carceri, I-62032 Camerino, Italy \\
$^{\dag}$ Istituto Nazionale per la Fisica della Materia, Italy}
\date{\today}
\maketitle
\begin{abstract}
We show that by using a feedback loop it is possible to
reduce the fluctuations in one quadrature of the vibrational degree
of freedom of a trapped ion below the quantum limit.
The stationary state is not a proper squeezed state,
but rather a {\it squashed} state,
since the uncertainty in the orthogonal quadrature, which is larger
than the standard quantum limit,
is unaffected by the feedback action.
\end{abstract}
\pacs{PACS number(s): 03.65.-w, 32.80.Pj, 42.50.Dv}
\section{Introduction}
In recent years there has been an increasing interest on trapping
phenomena
and related cooling techniques \cite{pg}.
A number of recent theoretical and experimental papers have investigated
the ability to coherently control or ``engineer" atomic quantum states.
Experiments on trapped ions, where the zero point of motion was closely approached trough laser
cooling \cite{wineprl},
already showed the effects of nonclassical motion in the absorption spectrum
\cite{wineprl,monprl}.
More recent experiments report the generation of Fock, squeezed and Schr\"odinger cat states
\cite{winestates,wine}.
These states appear to be of fundamental physical interest and possibly of use for sensitive
detection of small forces \cite{smallforces}.
Moreover, the possibility to synthetize nonclassical motional states
gave rise to new models
in quantum computation \cite{ciraczoller}.
In this paper we present a
way to reach a stationary nonclassical motional state for a trapped particle \cite{my},
which is able to give a significant uncertainty contraction in one phase-space direction.
The scheme can be then applied to control the vibrational motion against
the heating processes
responsible for decoherence \cite{nist}.
This could be important to obtain high fidelity in quantum logic
operations \cite{nist}.
The basic idea of the scheme is to realize
an effective and continuous measurement of
a vibrational quadrature for the trapped particle and then apply a feedback
loop able to control, i.e. to reduce, its fluctuations even below the quantum limit.
The fact that a feedback loop may reduce the fluctuations in one quadrature of an
in-loop field without increasing the fluctuations in the other has been known for a
long time, and has recently been called ``squashing" \cite{squash} as opposed to ``squeezing"
of a free field, in which the conjugate fluctuations are increased.
In our scheme, the obtained stationary state results as a ``squashed" state;
however, the quadrature measurement
increases the noise in the orthogonal quadrature well above the
standard quantum limit and the squashing comes with
respect to the state one has in presence of the measurement
process and without the feedback action. In this way the uncertainty principle
is not violated.
The paper is organized as follows. In section II we show
how to realize the indirect continuous measurement of a vibrational quadrature
by coupling the trapped particle with a standing wave. In section III
we shall introduce the feedback loop, in section IV we shall
study the properties of the stationary state in the presence
of feedback and section V is for concluding remarks.
\section{Continuous monitoring of atomic motion}
We consider a generic particle trapped in an effective harmonic potential.
For simplicity we shall consider the one-dimensional
case, even if the method can be in principle generalized to the
three-dimensional case. This particle can be an ion trapped by a
rf-trap \cite{nist} or a neutral atom in an optical trap \cite{optlat,sara}.
Our scheme however does not depend on the specific trapping
method employed and therefore we shall always refer from now on
to a generic trapped ``atom''.
The trapped atom of mass $m$,
oscillating with frequency $\omega_a$ along the
$\hat{x}$ direction and with position
operator $x=x_{0}(a+a^{\dagger})$, $x_{0}=(\hbar/2m\omega_a)^{1/2}$,
is coupled to a standing wave in a cavity with frequency $\omega_{b}$,
wave-vector $k$ along $\hat{x}$ and
annihilation operator $b$. The standing wave is quasi-resonant with the
transition between two internal atomic levels $|\pm\rangle $
separated by $\hbar\omega_0$.
We consider also an external driving of the standing wave with a
laser at frequency $\omega_{B}$ and of the atomic center-of-mass motion
with a classical electric field along the $\hat{x}$ direction,
with frequency $\omega_{A}$. The resulting Hamiltonian of the system is
\begin{eqnarray}
&&H=\frac{\hbar \omega_{0}}{2}\sigma_{z} +\hbar\omega_a a^{\dagger}a+
\hbar\omega_{b} b^{\dagger}b +i\hbar \epsilon (\sigma_+ +\sigma_-)
(b-b^{\dagger}) \sin\left(kx+\phi\right) \nonumber \\
&&-qEx_0 (a+a^{\dagger})\sin(\omega_A t + \theta)+i\hbar
\left({\cal B}e^{-i\omega_B t} b^{\dagger} -{\cal B}^{*}
e^{i\omega_B t} b\right) \label{hiniz} \;,
\end{eqnarray}
where $\sigma_z= |+\rangle \langle +|-|-\rangle \langle -|$,
$\sigma_{\pm}=|\pm \rangle \langle \mp |$, and $\epsilon$
is the coupling constant.
In the interaction representation with respect to
$H_0=\hbar \omega_{B} \left(b^{\dagger} b +\frac{\sigma_{z}}{2}\right)$,
and making the rotating
wave approximation, that is, neglecting terms rapidly
oscillating at the driving laser frequency $\omega_{B}$,
this Hamiltonian becomes
\begin{eqnarray}
&& H=\frac{\hbar \Delta}{2}\sigma_{z} +\hbar\omega_a a^{\dag}a+
\hbar (\omega_b-\omega_B)b^{\dagger}b +i\hbar
\epsilon (\sigma_+ b - \sigma_-b^{\dagger})
\sin\left(kx+\phi\right) \nonumber \\
&& -qEx_0 (a+a^{\dagger})\sin(\omega_A t + \theta)+i\hbar
\left({\cal B} b^{\dagger} -{\cal B}^{*}
b\right) \;, \label{hrwa}
\end{eqnarray}
where $\Delta=\omega_0-\omega_B$ is the
atomic detuning.
This detuning can be set to be much larger than all the other parameters
$\Delta \gg \epsilon$, $\omega_{b}-\omega_{B}$,
and in this case, the excited level can be adiabatically eliminated,
so to get
the following effective Hamiltonian for the vibrational
motion of the atom and the standing wave mode alone \cite{qo}
\begin{eqnarray}\label{H2}
&& H=\hbar \left(\omega_b-\omega_{B}\right) b^{\dagger}b
+\hbar\omega_a a^{\dagger}a
-\hbar\frac{\epsilon^2}{\Delta} b^{\dag}b\sin^2
\left(kx+\phi\right) \nonumber \\
&& -qEx_0 (a+a^{\dagger})\sin(\omega_A t + \theta)+i\hbar
\left({\cal B} b^{\dagger} -{\cal B}^{*}
b\right) \;. \label{hdisp}
\end{eqnarray}
If we set the spatial phase $\phi=0$, and assume the
Lamb-Dicke regime, one can approximate $\sin^2
\left(kx+\phi\right) \simeq k^2 x^2$ in Eq.~(\ref{hdisp}). Then, in
the interaction representation with respect to $\hbar \omega_A a^{\dagger} a$
and making the rotating wave approximation, i.e., neglecting all the
terms oscillating at $\omega_A$ (which is of the order of 1 Mhz)
or faster because we are
interested in the dynamics at much larger times, we finally get
\begin{equation}\label{H2b}
H=\hbar \left(\omega_b-\omega_{\cal B}-G/2\right)
b^{\dagger}b
+\hbar(\omega_a-\omega_A) a^{\dagger}a
-\hbar G b^{\dag}b
a^{\dag} a +i\hbar
\left({\cal A} a^{\dagger} -{\cal A}^{*}
a\right) +i\hbar
\left({\cal B} b^{\dagger} -{\cal B}^{*}
b\right) \; ,
\end{equation}
where $G=2(\epsilon k x_0)^2/\Delta$ and $ {\cal A} = -q E e^{-i\theta}/2\hbar$.
This Hamiltonian gives rise
to a crossed Kerr-like effect which could be exploited to
generate nonclassical states analogously to
the all-optical case proposed in Ref. \cite{kerr}.
A similar approach was used for Schr\"odinger cat
motional states of atoms in cavity QED \cite{andrew}.
Here, instead, we are looking for stationary
nonclassical states.
The evolution of the density matrix $D$ of the whole system (vibrational
degree of freedom plus the cavity mode) is determined by the Hamiltonian
(\ref{H2b}) and by the terms describing the photon leakage out of the cavity with decay
rate $\kappa$ and the coupling of the vibrational motion
with the thermal environment ${\cal L}_{th}D$, that is
\begin{equation}\label{Dtot}
{\dot D}={\cal L}_{th}D-\frac{i}{\hbar}
\left[H,D\right]
+\frac{\kappa}{2}\left(2 b D b^{\dag}-b^{\dag} b
D-D b^{\dag} b\right) \;.
\end{equation}
For the determination of the damping term of the vibrational motion
${\cal L}_{th} D$, we note that it occurs at a
frequency $\omega_a $ of the order of MHz
and that the corresponding damping rate $\gamma$ is
usually much smaller \cite{wine}. It seems therefore
reasonable to
use the rotating wave approximation in the interaction
between the atom center-of-mass and its reservoir,
leading us to describe the damping of the vibrational degree of freedom
in terms of the quantum optical master equation (at nonzero
temperature) \cite{qnoise},
\begin{equation}
{\cal L}_{th}\rho=\frac{\gamma}{2}(n+1)
\left(2a\rho a^{\dag}-a^{\dag}a\rho-\rho a^{\dag}a\right)
+\frac{\gamma}{2}n
\left(2a^{\dag}\rho a-aa^{\dag}\rho-\rho aa^{\dag}\right)\,,
\label{liu}
\end{equation}
where
$n=\left[\exp\left(\hbar\nu/k_BT\right)-1\right]^{-1}\,,
$
is the number of thermal phonons ($k_B$ is the Boltzmann
constant and $T$ the equilibrium
temperature). An analogous treatment is considered in \cite{nist}.
We have to remark, however,
that the damping and heating mechanisms
of a trapped atom
are not yet well understood \cite{nist} and that different kinds of
ion-reservoir interaction have been proposed \cite{murao}.
The quantum Langevin equations \cite{qnoise}
corresponding to the master equation
(\ref{Dtot}) reads
\begin{eqnarray}\label{eqsba1}
{\dot b}&=& -i(\omega_b-\omega_{B}-G/2) b+iGa^{\dag} ab
-\frac{\kappa}{2} b+{\cal B}+ \sqrt{\kappa} b_{in}(t)\,,\\
{\dot a}&=& -i(\omega_a-\omega_{A}) a+iGb^{\dag} ba
-\frac{\gamma}{2} a+{\cal A}+\sqrt{\gamma} a_{in}(t)\,,
\label{eqsba2} \end{eqnarray}
where the input quantum noises $b_{in}(t)$ and $a_{in}(t)$ have
zero mean and the following correlation functions
\begin{eqnarray}
&& \langle b_{in}(t)b_{in}(t') \rangle = \langle
b_{in}^{\dagger}(t)b_{in}(t') \rangle
= 0 \\
&& \langle b_{in}(t)b_{in}^{\dagger}(t') \rangle = \delta(t-t') \\
&& \langle a_{in}(t)a_{in}(t') \rangle =0 \\
&& \langle a_{in}^{\dagger}(t)a_{in}(t') \rangle =
n\delta(t-t') \\
&& \langle a_{in}(t)a_{in}^{\dagger}(t') \rangle = (n+1)\delta(t-t') \;.
\end{eqnarray}
When the external driving terms described by ${\cal A}$ and ${\cal B}$
are sufficiently large, the stationary state of the system is quasi-classical,
that is, the standing wave is approximately in a coherent state with a
large amplitude
$\beta \gg 1$, and the atomic vibrational motion along $\hat{x}$
is approximately in a coherent state
with a large amplitude $\alpha \gg 1$. The values of $\alpha$ and
$\beta$ are given by the solutions
of the coupled nonlinear equations given by the semiclassical version
of the quantum Langevin equations Eqs.~(\ref{eqsba1}) and
(\ref{eqsba2}):
\begin{eqnarray}
0&=& -i\left(\omega_b-\omega_{B}- G/2-G|\alpha|^2\right)\beta-
\frac{\kappa}{2} \beta+{\cal B}\,,\label{sseqs1}\\
0&=& -i\left(\omega_a-\omega_{A}-G|\beta|^2\right)\alpha-
\frac{\gamma}{2} \alpha+{\cal A}
\,.\label{sseqs2}
\end{eqnarray}
Since $\alpha \simeq 2{\cal A}/\gamma $ and $\beta \simeq 2{\cal
B}/\kappa $, the semiclassical condition for the steady state is
satisfied when ${\cal A} \gg \gamma $ and ${\cal B} \gg \kappa$.
The fluctuations around this steady state are instead described by quantum
mechanics and their dynamics can be obtained by appropriately shifting both
modes, i.e., $b \rightarrow b+ \beta $ and $a \rightarrow a +\alpha $.
In the semiclassical limit $|\alpha |, |\beta | \gg 1$ it is reasonable to
linearize the equations, and since it is always possible to tune
the two driving frequencies $\omega_A$ and $\omega_B$ so to have zero
detunings, i.e., $|\alpha|^2=(\omega_b-\omega_{B})/G-1/2$,
$|\beta|^2=(\omega_a-\omega_{A})/G$, the linearized quantum Langevin equations
for the quantum fluctuations around the steady state can be written as
\begin{eqnarray}
{\dot b}&=&iG\beta(\alpha^*a+\alpha a^{\dag})-\frac{\kappa}{2} b
+\sqrt{\kappa} b_{in}(t) \,,\label{eqslin1}\\
{\dot a}&=&iG\alpha(\beta^* b+\beta b^{\dag})-\frac{\gamma}{2} a
+\sqrt{\gamma}a_{in}(t) \label{eqslin2}\,.
\end{eqnarray}
The effective linearized Hamiltonian leading to Eqs.(\ref{eqslin1}), (\ref{eqslin2}),
can be written as
\begin{equation}\label{Heff1}
H=\hbar\chi YX\,,
\end{equation}
where $\chi=-4G|\alpha||\beta |$,
$Y=(be^{-i\phi_{\beta}}+b^{\dag}e^{i\phi_{\beta}})/2$
is the standing wave field quadrature
with phase $\phi_{\beta}$ equal to the phase of the classical amplitude $\beta$,
and $X=(ae^{-i\phi_{\alpha}}+a^{\dag}e^{i\phi_{\alpha}})/2$ is
the vibrational quadrature
with phase $\phi_{\alpha}$ given by the phase of the classical amplitude $\alpha$.
For the sake of simplicity we shall consider
$\phi_{\alpha}=\phi_{\beta}=0$, i.e., the atomic position quadrature,
from now on, even if the following considerations can be easily
extended to the case of generic phases.
Note that in order to remain in the Lamb-Dicke regime
it is required that $kx_0|\alpha|\ll 1$;
however the linearisation is justified only when $|\alpha|\gg 1$,
and therefore we need $kx_0\ll\frac{1}{|\alpha|}\ll 1$.
Eq.~(\ref{Heff1}) implies that
an effective continuous, quantum non-demolition (QND)
measurement of the phonon quadrature $X$ is provided by
the homodyne measurement of the light outgoing from the cavity, which
plays the role of the ``meter''.
In fact, the homodyne photocurrent is \cite{quadra}
\begin{equation}\label{photoc}
I(t)=2\eta\kappa\langle Y_{\varphi}(t)\rangle_c+\sqrt{\eta\kappa} \xi(t)\,,
\end{equation}
where $Y_{\varphi}=(be^{-i\varphi}+b^{\dag}e^{i\varphi})/2$ is
the measured quadrature, the phase
$\varphi$ is related to the local oscillator, and $\eta$ is the
detection efficiency. The subscript
$c$ in Eq.~(\ref{photoc})
denotes the fact that the average is performed on the state
conditioned on
the results of the previous measurements and
$\xi(t)$ is a Gaussian white noise
\cite{quadra}.
In fact, the continuous monitoring of the field mode
performed through the homodyne measurement,
modifies the time evolution of the whole system. The
state conditioned on the result of
measurement, described by a stochastic conditioned density matrix
$D_c$, evolves
according to the following stochastic differential equation
(considered in the Ito sense)
\begin{eqnarray}\label{Dceq}
{\dot D}_c&=&{\cal L}_{th}D_c-\frac{i}{\hbar}
\left[H,D_c\right] +\frac{\kappa}{2}\left(2 b D_c b^{\dag}-b^{\dag} b
D_c-D_c b^{\dag} b\right)
\nonumber\\
&+&\sqrt{\eta\kappa}\,\xi(t)\left(
e^{-i\varphi} b D_c+e^{i\varphi}D_c b^{\dag}
-2\langle Y_{\varphi}\rangle_cD_c\right)\,.
\end{eqnarray}
We note that by performing the average over the white noise $\xi(t)$,
one gets the master equation of Eq.~(\ref{Dtot}).
It is now reasonable to assume that the standing wave mode is highly damped,
i.e. $\kappa \gg \chi$ (this does not conflict with the preceding
assumptions, since the coupling constant
$\chi = -8 \epsilon^{2}(kx_{0})^{2}|\alpha \beta
|/\Delta $ is usually smaller than the cavity decay rate).
This means that the radiation field
will almost always be
in its lower state $|0\rangle_b $ (displaced by an amount $\beta$).
This allows us to adiabatically eliminate the
field and to perform a perturbative calculation in the small
parameter $\chi /\kappa$, obtaining (see also Ref.~\cite{TV})
the following expansion for the total conditioned density matrix $D_c$
\begin{eqnarray}\label{Dofrho}
D_c&=&\left(\rho_c-\frac{\chi^2}{\kappa ^2}X\rho_c X\right)
\otimes|0\rangle_b{}_b\langle 0|
-i\frac{\chi}{\kappa}\left(
X\rho_c\otimes|1\rangle_b{}_b\langle 0|
-\rho_c X\otimes|0\rangle_b{}_b\langle 1|\right)\nonumber \\
&+& \frac{\chi^2}{\kappa ^2}X\rho_c X \otimes |1\rangle_b{}_b\langle 1|
- \frac{\chi^2}{\kappa^2 \sqrt{2}}\left(
X^2\rho_c\otimes|2\rangle_b{}_b\langle 0|
+\rho_c X^2\otimes|0\rangle_b{}_b\langle 2|\right) \,,
\end{eqnarray}
where $\rho ={\rm Tr}_{b}\,D$ is the reduced density matrix for
the vibrational motion. In the adiabatic regime, the internal dynamics
instantaneously follows the vibrational one and therefore
one gets
information
on $X$ by observing the quantity $Y_{\varphi}$.
The relationship between the conditioned mean values follows
from Eq.~(\ref{Dofrho})
\begin{equation}\label{homoflo}
\langle Y_{\varphi}(t)\rangle_c
=\frac{\chi}{\kappa}\langle X(t)\rangle_c \sin\varphi\,.
\end{equation}
Moreover, if we adopt the perturbative expansion (\ref{Dofrho}) for
$D_c$ in (\ref{Dceq}) and perform
the trace over the internal mode, we get an equation for the
reduced density matrix $\rho_c$
conditioned to the result of the measurement of the observable
$\langle Y_{\varphi}(t)\rangle _c$, and therefore
$\langle X(t)\rangle _c$
\begin{equation}\label{rhoceq}
{\dot\rho}_c = {\cal L}_{th}\rho_c
-\frac{\chi^2}{2\kappa}
\left[X,\left[X,\rho_c\right]\right]+
\sqrt{\eta\chi^2/\kappa}\,\xi(t)
\left(ie^{i\varphi}\rho_cX-ie^{-i\varphi}X\rho_c
+2\sin\varphi\langle X(t)\rangle_c\,\rho_c\right)\,.
\end{equation}
This equation describes the stochastic evolution of the vibrational state
of the trapped atom conditioned to the result of the continuous
homodyne measurement of the light field. The double commutator
with $X$ is typical of QND
measurements.
\section{The Feedback Loop}
We are now able to use the continuous record of the atom phonon quadrature
to control its motion through the application of a feedback loop.
We shall use the continous
feedback theory proposed by Wiseman and Milburn
\cite{wisemil}.
One has to take part
of the stochastic output homodyne
photocurrent $I(t)$, obtained from the continuous monitoring
of the meter mode, and
feed it back to the vibrational dynamics
(for example as a driving term)
in order to modify the evolution of the mode $a$.
To be more specific, the presence of feedback modifies the
evolution of the conditioned state
$\rho_c(t)$. It is reasonable to assume that the feedback
effect can be described by an additional
term in the master equation, linear in the photocurrent
$I(t)$, i.e. \cite{wisemil}
\begin{equation}\label{rhofb}
\left[{\dot\rho}_c(t)\right]_{fb}=\frac{I(t-\tau)}{\eta\chi}\,
{\cal K}\rho_c(t)\,,
\end{equation}
where $\tau$ is the time delay in the feedback loop
and ${\cal K}$
is a Liouville superoperator describing the way
in which the feedback signal acts
on the system of interest.
The feedback term (\ref{rhofb}) has to be considered in the
Stratonovich sense, since Eq.
(\ref{rhofb}) is introduced as limit of a real process, then
it should be transformed in the Ito
sense and added to the evolution equation (\ref{rhoceq}).
A successive average over the white noise
$\xi(t)$ yields the master equation for the reduced
density matrix $\rho={\rm Tr}_{b}D$
in the presence of feedback. In the general case of a nonzero
feedback delay time, one gets a non-Markovian master equation which is
very difficult to solve \cite{wisemil} (see however Ref.~\cite{delay}).
Most often however, the feedback
delay time is much shorter than the characteristic time of the
$a$ mode, which in the present case is given by the
energy relaxation time $\gamma ^{-1}$,
and in this case the dynamics in the presence of feedback can be
described by a Markovian master
equation \cite{wisemil}, which is given by
\begin{equation}
\dot{\rho }={\cal L}_{th} \rho
-\frac{\chi^2}{2\kappa}\left[X,\left[X
,\rho \right]\right]+{\cal K}\left(ie^{i\varphi }\rho X-
ie^{-i\varphi} X\rho\right)
+\frac{{\cal K}^{2}}{2\eta\chi^2/\kappa}\rho.
\label{qndfgen}
\end{equation}
The third term is the feedback term itself and
the fourth
term is a diffusion-like term, which is an unavoidable
consequence of the noise introduced
by the feedback itself.
Then,
since the Liouville superoperator ${\cal K}$ can only
be of Hamiltonian
form \cite{wisemil}, we choose it as
${\cal K}\rho =g
\left[a-a^{\dagger},
\rho \right]/2$ \cite{TV},
which means feeding back the measured homodyne photocurrent to
the vibrational oscillator with a driving
term in the Hamiltonian involving the
quadrature orthogonal to the
measured one;
$g$ is the feedback gain related to the
practical way of realizing the loop.
One could have chosen to feed
the system with a generic phase-dependent quadrature,
due to the homodyne current, however, it will turn
out that the above choice gives the best and simplest result.
Since the measured quadrature of the vibrational mode is
its position, the feedback will act as a driving for
the momentum.
Using the above expressions in
Eq.~(\ref{qndfgen}) and rearranging the terms in
an appropriate
way, we finally get the following master equation:
\begin{eqnarray}\label{totale}
\dot{\rho }&=& \frac{\Gamma}{2}(N+1)
\left(2a\rho a^{\dagger}-a^{\dagger}a\rho
-\rho a^{\dagger}a\right)
+\frac{\Gamma}{2}N
\left(2a^{\dagger}\rho a-aa^{\dagger}\rho
-\rho aa^{\dagger}\right)
\nonumber \\
&-&\frac{\Gamma}{2}M
\left(2a^{\dagger}\rho a^{\dagger}-a^{\dagger 2}\rho
-\rho a^{\dagger 2}
\right)
-\frac{\Gamma}{2}M^{*}
\left(2a\rho a-a^{2}\rho -\rho a^{2}\right)
\nonumber \\
&-&\frac{g}{4}\sin\varphi
\left[a^{2}-a^{\dag 2},\rho\right]\,,
\end{eqnarray}
where
\begin{eqnarray}\label{parameters}
\Gamma&=&\gamma-g\sin\varphi\,;\\
N&=&\frac{1}{\Gamma }\left[\gamma n
+\frac{\chi^2
}{4\kappa}+\frac{g^{2}}{4\eta\chi^2/\kappa}+\frac{g}{2}
\sin\varphi\right]\,;\\
M&=&-\frac{1}{\Gamma }\left[
\frac{\chi^2}{4\kappa}-\frac{g^{2}}{4\eta\chi^2/\kappa}
-i\frac{g}{2}\cos\varphi\right]\,.
\end{eqnarray}
Eq.~(\ref{totale}) is very instructive because
it clearly shows the
effects of the feedback loop on the
vibrational mode $a$.
The proposed
feedback mechanism, indeed,
not only introduces a parametric driving term proportional to
$g\sin \varphi$, but it also
simulates the presence of a squeezed bath,
characterized by an effective damping
constant $\Gamma $ and by the coefficients
$M$ and $N$, which are given
in terms of the feedback parameters \cite{TV}.
An interesting aspect of the effective bath described by
the first four
terms in the right hand side of (\ref{totale}) is that
it is characterized by
phase-sensitive fluctuations, depending upon the
experimentally adjustable
phase $\varphi$.
\section{The Stationary Solution}
Because of its linearity, the solution of
Eq. (\ref{totale}) can be easily obtained
by using the normally ordered
characteristic
function \cite{qnoise} ${\cal C}(\lambda,\lambda^*,t)$.
The partial differential equation corresponding to Eq.
(\ref{totale}) is
\begin{eqnarray}\label{chareq}
&&\left\{\partial_t+\frac{\Gamma}{2}\lambda\partial_{\lambda}
+\frac{\Gamma}{2}\lambda^*\partial_{\lambda^*}
+\frac{g}{2}\sin\varphi\left(\lambda\partial_{\lambda^*}
+\lambda^*\partial_{\lambda}\right)\right\}{\cal C}(\lambda,
\lambda^*,t)\nonumber\\
&&=\left\{-\Gamma N|\lambda|^2+
\left(\frac{\Gamma}{2}M+\frac{g}{4}\sin\varphi\right)(\lambda^*)^2
+\left(\frac{\Gamma}{2}M^*+\frac{g}{4}\sin\varphi\right)
\lambda^2\right\}
{\cal C}(\lambda,\lambda^*,t)\,,
\end{eqnarray}
The stationary state is reached only if the parameters
satisfy the stability condition,
i.e. $g\sin\varphi <\gamma $. In this case
the stationary solution has the following form
\begin{equation}\label{charsol}
{\cal C}(\lambda,\lambda^*,\infty)
=\exp\left[-\zeta|\lambda|^2+\frac{1}{2}\mu(\lambda^*)^2
+\frac{1}{2}\mu^*\lambda^2\right]\,,
\end{equation}
where
\begin{eqnarray}
\zeta&=&\frac{N\Gamma^2+g\sin\varphi(\Gamma {\rm Re}\{M\}+2\nu{\rm Im}\{M\})
+g^2\sin^2\varphi/2}
{\Gamma^2-g^2\sin^2\varphi}\,;\label{ze}\\
\mu&=&\Gamma\frac{(N+1/2)g\sin\varphi+\Gamma {\rm Re}\{M\}}
{\Gamma^2-g^2\sin^2\varphi}
+i\frac{(\Gamma^2-g^2\sin^2\varphi){\rm Im}\{M\}}
{\Gamma^2-g^2\sin^2\varphi}
\label{mu}\,.
\end{eqnarray}
Under the stability conditions and in the long time
limit $(t\to\infty)$
the variance of the generic quadrature operator
$X_{\theta}=(a e^{i\theta}+a^{\dag}e^{-i\theta})/2$ becomes
\begin{equation}\label{varXth}
\langle X^2_{\theta}\rangle=\frac{1}{2}\left[\frac{1}{2}+\zeta+
{\rm Re}\{\mu e^{2i\theta}\}\right]\,.
\end{equation}
\section{Discussions and Conclusions}
For the $X$ quadrature Eq.(\ref{varXth}) can be simply
written as
\begin{equation}\label{varXg}
\langle X^2\rangle=\frac{1}{2}\left[\frac{1}{2}+n_{eff}\right]\,,
\end{equation}
with
$
n_{eff}=\zeta+{\rm Re}\{\mu\}$.
In absence of feedback ($g=0$) we have $n_{eff}\equiv n$,
otherwise
$n_{eff}$ can be smaller than $n$, providing a {\it stochastic
localisation} in the $X$
quadrature. Depending on
the external parameters,
it can also be negative (but it is always $n_{eff} \ge -1/2$)
accounting for the possibility of going beyond the standard quantum limit. This is
a relevant result of the present feedback scheme since it is able to
reduce not only the thermal fluctuations but even the quantum ones.
The potentiality of this feedback mechanism is clearly shown
in Fig.1, where $n_{eff}$ goes well
below zero for increasing values of $\chi$.
Instead in Fig.2 we have sketched the phase space uncertainty
contours obtained by cutting the
Wigner function corresponding to Eq.(\ref{charsol}) at
$1/\sqrt{e}$ times its maximum height.
We see that the state resulting from the feedback action (solid line)
has a relevant contraction in the
$X_{\theta=0}$ direction, but the same uncertainty in the
$X_{\theta=\pi/2}$ direction with respect to the state of the
system undergoing measurement without feedback (dashed line).
We refer to this type of noise reduction produced
by feedback loop as squashing, whereas squeezing refers
to conventional quantum noise reduction \cite{squash}.
Summarizing, we have proposed a feedback scheme based
on an indirect (QND) measurement which is able not only to contain the
heating of the vibrational motion of a trapped ion, but also to produce nonclassical
motional states (squashed ones).
Up to this point we have not discussed the specific way
in which a particular feedback
Hamiltonian could be implemented.
In our case, it is important to be able to realize a term in
the feedback Hamiltonian
proportional to the quadrature orthogonal to $X$.
This is not straightforward, but could be realized by using the
feedback current to vary
an external potential applied to the atom without altering
the trapping potential \cite{kurt}.
In principle the model could be extended to the three
dimensional case,
one should only consider three orthogonal standing waves
far from resonant transitions.
In conclusion, although the experimental implementation of the
presented model may not be easy,
it is certainly a promising experimental challenge,
stimulated by the possibility of producing
nonclassical states for trapped atoms
and of controlling their heating to minimize decoherence
effects, especially in quantum information processing
\cite{pra}.
\begin{figure}
\caption{The quantity $n_{eff}
\label{fig1}
\end{figure}
\begin{figure}
\caption{The phase space uncertainty contours are represented for
$g=0$ (dashed line) and $g=0.025$ ${\rm s}
\label{fig2}
\end{figure}
\end{document} |
\begin{document}
\title{\LARGE \bf
Shaping up crowd of agents through controlling their statistical moments
}
\thispagestyle{empty}
\pagestyle{empty}
\begin{abstract}
In a crowd model based on leader-follower interactions, where positions of the leaders are viewed as the control input, up-to-date solutions rely on knowledge of the agents' coordinates. In practice, it is more realistic to exploit knowledge of statistical properties of the group of agents, rather than their exact positions. In order to shape the crowd, we study thus the problem of controlling the moments instead, since it is well known that shape can be determined by moments. An optimal control for the moments tracking problem is obtained by solving a modified Hamilton-Jacobi-Bellman (HJB) equation, which only uses the moments and leaders' states as feedback. The optimal solution can be solved fast enough for on-line implementations.
\end{abstract}
\section{Introduction}
With continuous urbanization of the global population, researchers are getting more and more aware that it is important to have a better understanding of crowd behavior under certain circumstances. Experts from different fields, including sociologists, psychologists, ecologists, physicists, mathematicians and computer scientists use their different perspectives to model and analyze the crowd. One typical and important application of the crowd behavior study is the evacuation problem in emergency situations. Although the human behavior can be very complicated in these situations, researchers have made lots of effort to model, analyze and simulate the crowd with different approaches for the purpose of minimizing the total societal loss.
From the pure social psychology point of view, there are models such as the theory of planned behavior form I. Ajzen \cite{Ajzen1991}. J.D. Sime linked the psychology part and the engineering part of the crowd behavior problem in \cite{Sime1995}. From the engineering aspect, one commonly used macroscopic approach is to consider the whole crowd as one entity that is described by a density function and use tools such as fluid mechanics and partial differential equations to conduct analysis. This continuum setting of the crowd has been used, e.g., in \cite{continuum} and \cite{flow}. The disadvantage of this type of approach relies on the fact that a density approximation of the human crowd is not adequate when the density is low. On the other hand, low computational cost is obviously the advantage of the approach, especially when the scale of the crowd is large enough.
Another method for crowd analysis is by using the tool of multi-agent systems theory that has rapidly developed in the past decades. Compared to the continuum setting, multi-agent system analysis can be seen as a microscopic model that focuses on individual behaviors. Part of the idea came from animal flocking observation and modeling such as Reynolds model in \cite{reynolds} in 1987. This kind of approach was used, e.g., in \cite{simulating}, \cite{Helbing2009}. One of the most widely used models, the social force model in \cite{Social force}, also uses this setting. With the theory development in multi-agent consensus problems, researchers do strict analysis for linear multi-agent systems, e.g., in \cite{Olfati-Saber2003}. However, many models for human crowd are highly nonlinear, which makes the analysis much harder due to the lack of theoretical support for nonlinear systems. The computation cost is also a big issue for numerical experiments. Nevertheless, the advantage is the accuracy of individual states if compared to macroscopic approaches.
Among the multi-agent crowd models, there is one type of model called leader-follower model that divides the crowd into two groups based on their roles. These models are very useful for example in the evacuation problem, where the rescue workers act as leaders and the general public can be considered as followers. Such models can be found for example in \cite{Morse2003}, \cite{Eren2005} and \cite{Olfati-Saber2006}. A central issue for leader-follower models is controllability. In \cite{magnus2006}, the controllability of linear leader-follower models is discussed in detail. Unfortunately, the system becomes uncontrollable in most cases of realistic linear models. In \cite{cdc2013}, an optimal control approach is used for control design even though the system is not controllable for the evacuation problem. In leader-follower based models, the formation of the followers is in general very hard to control by only the leaders, since the system is often uncontrollable. Unfortunately, the possibility of shaping the crowd is seldom discussed in the literature since the shape is less restrictive than the formation and maybe more elaborate to control. In this paper, we will study the moments of the crowd, which have a strong connection to the shape as showed in \cite{Milanfar1995} and \cite{Golub2004}. By using dynamic programming techniques, we will attack the question of shaping the crowd.
In the method introduced later in this article, we will design a controller so that the shape of the crowd will track approximately some desired shape during the process. Although the asymptotic behavior of the crowd is important, the transient phase is even more critical for crowd control problems. Lyapunov stability analysis does not cover the transient behavior of the system in a straightforward manner, thus is not used in the paper. We instead introduce a new control design approach by using optimal control theory. Moreover, the leaders will need only the moments information together with their own states to calculate their movement as time evolves even though the optimal control problem has all the followers' positions and velocities as its state variables. This make the method introduced in the paper more practical since individual states are in general very hard to measure. The numerical method introduced is shown to be efficient enough for on-line implementation as well.
The outline of this paper is as follows: in Section II, the basic leader-follower model is stated and the moments of the crowd are defined. There is a short discussion about the moments tracking problem in general and the motivation why the optimal control approach is used in this paper. The optimal control problem is introduced and approximately solved in Section III by using model predictive control tools. In Section IV, some improvements of the method are made to handle numerical issues while several detailed experiments are carried out to test the capability and robustness of the method. A short conclusion can be found in Section V together with a brief outline for future work.
\section{Problem formulation}
\subsection{Leader-follower model}
We will study the behavior of two types of agents in a two dimensional space. The first type of agents are called {\it followers}. We assume that there are $N$ followers in total, and their motion follows some simple rules that mostly depend on the relative positions and velocities among them and on those to the other type of the agents - the {\it leaders}. The $M$ leaders, on the other hand, have better knowledge of the whole environment and can plan their motion accordingly. Based on the situation of the whole crowd, the leaders should behave in an optimal way to guide the followers to reach certain goals. The position of follower $i$ is denoted by a vector $z_i\in \mathbb R^2$, and its velocity denoted by $v_i\in \mathbb R^2$. The position of leader $j$ is denoted by $z_{l_j}\in\mathbb R^2$, and its velocity by $v_{l_j}\in\mathbb R^2$.
Suppose that without the leaders, the influence from follower $k$ to follower $i$ is modeled as $f(z_i,v_i,z_k,v_k)$. By assuming unit mass for each follower, we can write
\begin{equation}
\dot v_i=\sum_{k=1\atop{k\ne i}}^{N} f(z_i,v_i,z_k,v_k)-p v_i,
\end{equation}
where $p$ is a coefficient to model the damping effect such as resistance or physical limit of the agent. When follower $i$ senses any of the leaders, the influence by the leader is added to the model:
\begin{equation}
\dot v_i=\sum_{j=1}^{M}g(z_i,v_i,z_{l_j},v_{l_j})+\sum_{k=1\atop k\ne i}^{N} f(z_i,v_i,z_k,v_k)-p v_i.
\end{equation}
The functions $f$ and $g$ will be defined or described later. Note that here we sum up all the terms with the $M$ leaders, meaning that the leader-follower interaction is state-based. There is no predesigned network. If $g$ has no zeros in its domain, then the followers can always sense all the leaders and be influenced more or less by all of them.
For the leader $j$, we have
\begin{equation}
\dot v_{l_j}=u_j-p v_j,
\end{equation}
so it has the same damping coefficient as the followers and $u_j$ is the input to be designed in order to fulfill certain tasks. We denote $$z=\begin{bmatrix}z_{1}\\z_{2}\\\vdots\\z_{N}\end{bmatrix},v=\begin{bmatrix}v_{1}\\v_{2}\\\vdots\\v_{N}\end{bmatrix},z_l=\begin{bmatrix}z_{l_1}\\z_{l_2}\\\vdots\\z_{l_M}\end{bmatrix},v_l=\begin{bmatrix}v_{l_1}\\v_{l_2}\\\vdots\\v_{l_M}\end{bmatrix}.$$
If there is a cost function defined over time and a terminal cost to be minimized, then we can formulate an optimal control problem:
\begin{equation}
\begin{array}{cl}
\displaystyle\min_{u}& \Phi(t_f,z(t_f),v(t_f),z_l(t_f),v_l(t_f))\\
&~~~~~~~~~+\displaystyle\int_{t_0}^{t_f}\mathcal L(t,z(t),v(t),z_l(t),v_l(t),u(t))dt\\
s.t.~ &\dot z_i=v_i,\\
&\dot v_i=\sum_{j=1}^{M}g(z_i,v_i,z_{l_j},v_{l_j})+\sum_{k=1}^{N} f(z_i,v_i,z_k,v_k)-p v_i,\\
&\dot z_{l_j}=v_{l_j},\\
&\dot z_{l_j}=u_j-pv_j,\\
&u_j\in U,~ z(t_0),v(t_0), z_l(t_0),v_l(t_0)\text{ given, }
\end{array}
\end{equation}
for $i=1,\cdots,N$, and $j=1,\cdots,M$, where the set $U\subset \mathbb R^2$ is the feasible that will be specified later. The solution to this problem $u=\begin{bmatrix}u_{1}&u_{2}&\cdots&u_{M}\end{bmatrix}^T$will give a proper control input for the leaders.
\subsection{Definitions of the moments}
In order to shape the crowd, we need at least some statistic properties of it. In this article, we make use of moments of the agents' position. There are two ways to define moments:
\subsubsection{Definition 1}
The \emph{(raw) moment} of {\it order} $k$ is defined by
\begin{equation}
\label{moments1.1}
M_{ab}=\frac{1}{N}\sum_{i=1}^{N}z_{i,x}^a z_{i,y}^b,
\end{equation}
where $z_{i,x}$ and $z_{i,y}$ are the $x$, $y$ coordinate for $z_i, and $$a, b\in\mathbb N$ with $a+b=k$.
\subsubsection{Definition 2}
The \emph{centralized moment} is defined by
\begin{equation}
\label{moments2.1}
\bar M_{10}=\frac{1}{N}\sum_{i=1}^{N}z_{i,x},\bar M_{01}=\frac{1}{N}\sum_{i=1}^{N}z_{i,y},
\end{equation}
and
\begin{equation}
\label{moments2.2}
\bar M_{ab}=\frac{1}{N}\sum_{i=1}^{N}(z_{i,x}-\bar M_{10})^a(z_{i,y}-\bar M_{01})^b, \text{ for }k>1, \text{ and }a+b=k.
\end{equation}
It is not hard to show that one can calculate all the centralized moments by knowing all the raw moments and vise versa. If not particularly stated, we will only use the raw moments in the rest of the paper.
The moments have a strong relationship with the shape of the crowd. If the shape and the distribution in the shape is known, one can approximate the moments by
$$M_{ab}=\int_{S}x^ay^b\rho(x,y)dxdy,$$
where $S$ is the shape, and $\rho(x,y)$ is the density function of the distribution. On the other hand, if all the moments $M_{ab}$ with $a+b\leq 2n-3$ are given, then one can approximate the shape, assuming it is convex and a uniform distribution, by a $n$-polygon by using a modified method similar to the one introduced in \cite{Milanfar1995}.
\subsection{Moments tracking}
It is noticeable that the scale of the leader-follower model:
\begin{align}
\begin{cases}
\dot z_i&=v_i,\notag\\
\dot v_i&=\sum_{j=1}^{M}g(z_i,v_i,z_{l_j},v_{l_j})+\sum_{k=1}^{N} f(z_i,v_i,z_k,v_k)-p v_i,\notag\\
\dot z_{l_j}&=v_{l_j},\notag\\
\dot z_{l_j}&=u_j-pv_j,
\end{cases}
\end{align}
for $i=1,\cdots,N$, and $j=1,\cdots,M$ becomes very large when the number of followers increases. Hence, individual control will be difficult even if the whole system is controllable, which is not always the case. This gives a motivation to look into the statistical properties of the crowd instead. Because of the strong connection between the moments and the shape, we can setup a moments tracking problem if we want to do shape tracking. Namely, from a desired shape ``signal" and a desired density distribution, we can calculate the corresponding desired moments. If moments of the crowd could track the desired moments, then the crowd should more or less follow the desired shape. The more moments we track, the better the performance should be.
Theoretically, if we can derive the evolution of the moments $M_{ab}$ with $a+b\leq m$ and their time derivatives by functions only using $M_{ab}$, $\frac{d}{dt}M_{ab}, a+b\leq m$ and $z_l,v_l,u$, then we will obtain a, probably nonlinear, system with $M_{ab}$ and $\frac{d}{dt}M_{ab}, a+b\leq m$ and $z_l,v_l$ as its states $u$ as its control. Furthermore, if this system is controllable, them for any continuous moments signal, we can get a minimum energy feedback control to track those moments. However, it is impossible in general to write down this type of system without using any information from $z$ and $v$ defined previously if no approximation is made. Even if for some special $f$ and $g$ functions, one can write down the system of moments, it is inaccessible for almost all cases that we have examined. This is still true even if one simplifies the leader-follower dynamics into a single integrator system. The analysis of some specific models are made in \cite{axiv} and is omitted here due to the space limitation.
Another way to deal with the tracking problem of an uncontrollable system is by setting a cost function for the system and using optimal control techniques to solve it. A very standard cost function is the quadratic errors between the real moments and the desired moments signals. Given a sequence of nonnegative scalars $\{c_{ab}\}$ with $a+b\leq m$, we can setup the following optimal control problem:
\begin{equation}
\label{OCP221}
\begin{array}{cl}
\displaystyle \min_{u} & \displaystyle\sum_{a+b\leq m}\displaystyle\int_{t_0}^{t_f}c_{ab}(M_{ab}(t)-M_{ab}^d(t))^2dt\\
s.t. &\dot z_i=v_i,\\
&\dot v_i=\sum_{j=1}^{M}g(z_i,v_i,z_{l_j},v_{l_j})+\sum_{k=1}^{N} f(z_i,v_i,z_k,v_k)-p v_i,\\
&\dot z_{l_j}=v_{l_j},\\
&\dot z_{l_j}=u_j-pv_j,\\
&\|u_j(t)\|\leq u_{max},\\
&z(t_0), v(t_0), z_l(t_0), v_l(t_0) \text{ given, }
\end{array}
\end{equation}
where $M_{ab}^d(t)$ is the given signal for $M_{ab}(t)$ to track.
The analytic solution is possible to find only if $f$ and $g$ are simple functions such as linear functions. One can use Pontryagin's minimum principle (PMP) to solve such type of problem similar to the method used in \cite{cdc2013}. However, these two functions are usually nonlinear in practical models, which implies that PMP is to hard to solve. For the dynamic programming approach, if we can find a cost to go function $J$ that satisfies the Hamilton-Jacobi-Bellman equation, then the optimal control can be calculated by partial derivatives of $J$. The HJB equation of (\ref{OCP221}) can be written as:
\begin{align}
-\frac{\partial J}{\partial t}=&\min_{\|u\|\leq u_{max}} \{ \sum_{a+b\leq m}c_{ab}(M_{ab}(t)-M_{ab}^d(t))^2+ \frac{\partial J}{\partial z}^T v\notag\\
+&\sum_{i=1}^N\frac{\partial J}{\partial v_i}(\sum_{j=1}^{M}g(z_i,v_i,z_{l_j},v_{l_j})+\sum_{k=1}^{N} f(z_i,v_i,z_k,v_k)-p v_i)\notag\\
+& \frac{\partial J}{\partial z_l}^T v_l+\frac{\partial J}{\partial v_l}^T (u-p v_l)\},
\label{HJB1}
\end{align}
with the boundary condition \begin{equation}J(t_f,z,v,z_l,v_l)=0.\end{equation}
The minimum of the right-hand side in equation (\ref{HJB1}) is reached when $u_j^*=-u_{max}\lambda_j$ where $\lambda_j$ is the unit vector of $\frac{\partial J}{\partial v_{l_j}}$. If we plug this back to (\ref{HJB1}), we get
\begin{align}
-\frac{\partial J}{\partial t}=& \sum_{a+b\leq m} c_{ab}(M_{ab}(t)-M_{ab}^d(t))^2+ \frac{\partial J}{\partial z}^T v\notag\\
+&\sum_{i=1}^N\frac{\partial J}{\partial v_i}(\sum_{j=1}^{M}g(z_i,v_i,z_{l_j},v_{l_j})+\sum_{k=1}^{N} f(z_i,v_i,z_k,v_k)-p v_i)\notag\\
+& \frac{\partial J}{\partial z_l}^T v_l-\sum_{j=1}^{M}u_{max}\|\frac{\partial J}{\partial v_{l_j}}\|-p \frac{\partial J}{\partial v_l}^T v_l,
\label{HJB2}
\end{align}
Unfortunately, this partial differential equation is also very difficult to solve since the number of variables is proportional to the number of followers, which means the complex of solving the PDE becomes very high. Meanwhile, even if the HJB equation is sovable, one needs all the followers' states to calculate the optimal control, which is very difficult to implement in practice. Hence we will introduce a suboptimal control by using moments information as feedback while the complexity of the algorithm is low enough.
\section{Feedback control using moments information only}
In this section, we will study a special problem, which gives a reasonable model for crowd behavior. Let us make the following assumption for $g$:
\begin{itemize}
{\item The function $g$ is composed of two parts, a position consensus part and a velocity alignment part. There are two functions that give different weight to these two parts which only depend on the positions of the leader and the follower, i.e.,
$$g(z_i,v_i,z_{l_j},v_{l_j})=g_1(z_i,z_{l_j})(z_{l_j}-z_i)+g_2(z_i,z_{l_j})(v_{l_j}-v_i),$$
where $g_1$ and $g_2$ are real valued functions. Ideally, both $g_1$ and $g_2$ should be functions of the distance $d=\|z_i-z_{l_j}\|$ to make the model reasonable in practice. Moreover, $g_1$ should be relatively large when $d$ is large while $g_2$ should dominate when $d$ is close to zero. This is due to the fact that catching up the leader is more important when the distance is large while moving in the same direction makes more sense when the distance is already short enough.}
\end{itemize}
With these assumptions we can rewrite the HJB equation (\ref{HJB2}) as
\begin{align}
-\frac{\partial J}{\partial t}=& \mathcal L(t) + \frac{\partial J}{\partial z}^T v+\sum_{i=1}^N\frac{\partial J}{\partial v_i}\Big(\sum_{j=1}^{M}g_1(z_i,z_{l_j})(z_{l_j}-z_i)\notag\\
&+g_2(z_i,z_{l_j})(v_{l_j}-v_i)+f_i-p v_i\Big)+ \frac{\partial J}{\partial z_l}^T v_l\notag\\
&-\sum_{j=1}^{M}u_{max}\|\frac{\partial J}{\partial v_{l_j}}\|-p \frac{\partial J}{\partial v_l}^T v_l,
\label{HJB3}
\end{align}
where the notations $\mathcal L(t)=\sum_{a+b\leq m} c_{ab}(M_{ab}(t)-M_{ab}^d(t))^2$ and $f_i=\sum_{k=1}^{N}f(z_i,v_i,z_k,v_k)$ are introduced for simplicity.
\subsection{HJB simplification}
In order to avoid using all the followers' states $z$ and $v$, we want to derive $J$ as a function of $M_{ab}$ with $a+b\leq m$, $\dot M_{10}$, $\dot M_{01}$, $z_l$ and $v_l$ only, which means that the followers' positions only appear in the moments and their velocities only appear in the time derivative of the first order moments. In this section, we will show that this can be achieved with an approximated HJB equation.
\begin{proposition}
The Hamilton-Jacobi-Bellman equation (\ref{HJB3}) can be approximated in such a way so that the cost-to-go function $J$ has $t, M_{ab}$ with $a+b\leq m$, $\dot M_{10}$, $\dot M_{01}$, $z_l$ and $v_l$ as its variables.
\end{proposition}
Assuming that there is a function $\bar J(t,M_{ab}, \dot M_{10}, \dot M_{01}, z_l, v_l)$ that satisfies (\ref{HJB3}), then we have the following equations because of the chain rule:
$$\frac{\partial \bar J}{\partial z_i}=\sum_{a+b\leq m}\frac{\partial \bar J}{\partial M_{ab}}\frac{\partial M_{ab}}{\partial z_i}=\frac{1}{N}\sum_{a+b\leq m}\frac{\partial \bar J}{\partial M_{ab}}\begin{bmatrix}az_{i,x}^{a-1}z_{i,y}^b
\\bz_{i,x}^{a}z_{i,y}^{b-1}\end{bmatrix}.$$
$$\frac{\partial \bar J}{\partial v_i}=\frac{\partial \bar J}{\partial \dot M_{10}}\frac{\partial \dot M_{10}}{\partial v_i}+\frac{\partial \bar J}{\partial \dot M_{01}}\frac{\partial \dot M_{01}}{\partial v_i}=\frac{1}{N}\begin{bmatrix}\frac{\partial\bar J}{\partial \dot M_{10}}
\\\frac{\partial J}{\partial \dot M_{01}}\end{bmatrix}.$$
Then we can write the HJB equation for $\bar J$ as
\begin{align}
-\frac{\partial \bar J}{\partial t}= &\mathcal L(t) +\frac{1}{N}\sum_{a+b\leq m}\frac{\partial \bar J}{\partial M_{ab}}\sum_{i=1}^N\begin{bmatrix}az_{i,x}^{a-1}z_{i,y}^b &bz_{i,x}^{a}z_{i,y}^{b-1}\end{bmatrix}v_i\notag\\
&+\frac{1}{N}\begin{bmatrix}\frac{\partial \bar J}{\partial \dot M_{10}}&\frac{\partial \bar J}{\partial \dot M_{01}}\end{bmatrix} \Big( \sum_{j=1}^{M} \big(\sum_{i=1}^N\big(g_1(z_i,z_{l_j})(z_{l_j}-z_i)\notag\\
&+g_2(z_i,z_{l_j})(v_{l_j}-v_i)\big)\big)+\sum_{i=1}^Nf_i(z,v)-p\sum_{i=1}^N v_i\Big)\notag\\
&+\frac{\partial \bar J}{\partial z_l}^T v_l-p \frac{\partial \bar J}{\partial v_l}^T v_l-\sum_{j=1}^{M}u_{max}\|\frac{\partial \bar J}{\partial v_{l_j}}\|.
\label{HJB4}
\end{align}
Unfortunately, there are still terms on the right-hand side of the equation that contain $z$ and $v$, which implies that the assumption does not hold in general. We need to approximate those terms in order to get rid of $z$ and $v$. In order to do the approximation, we need to make some further assumptions:
\begin{itemize}
\item{$ f(z_i,v_i,z_k,v_k)=- f(z_k,v_k,z_i,v_i)$, meaning that the follower-follower interactions are symmetric. As a result, we get $\sum_{i=1}^N f_i(z,v)=0$.}
\item{The terms of the form $\sum_{i=1}^N h(z_i)v_i$ can be approximated by $(\sum_{i=1}^N h(z_i))\dot M_1 $ for any scalar function $h(\cdot)$, where $\dot M_1=\begin{bmatrix}\dot M_{10} & \dot M_{01}\end{bmatrix}^T$. This leads to the following approximation $$\sum_{i=1}^N z_{i,x}^az_{i,y}^b v_i\approx (\sum_{i=1}^N z_{i,x}^az_{i,y}^b )\dot M_1=NM_{ab}\dot M_1,$$ and $$\sum_{i=1}^{N}(g_2(z_i,z_{l_j})(v_{l_j}-v_i))\approx\Big(\sum_{i=1}^{N}(g_2(z_i,z_{l_j})\Big)(v_{l_j}-\dot M_1)$$ If we regard $z$ and $v$ as random variables, then this assumption is equivalent to saying that $z$ and $v$ are independent.}
\item{The functions $g_1(z_1,z_2)$ and $g_2(z_1,z_2)$ can be approximated by polynomials of $z_1$ with degree less than $m$ for a given $z_2$, i.e.,
\begin{align*}
&g_1(z_1,z_2)=\sum_{a=0}^{m-1}\sum_{b=0}^{m-a-1}\alpha_{ab}(z_2)z_{1x}^az_{1y}^b,\\
&g_2(z_1,z_2)=\sum_{a=0}^{m-1}\sum_{b=0}^{m-a-1}\beta_{ab}(z_2)z_{1x}^az_{1y}^b.
\end{align*}
This approximation can be usually achieved by Taylor expansion around certain points if the function $g_1$ and $g_2$ are ``good" enough (smooth and not very steep). Once we can use polynomials to approximate $g_1$ and $g_2$, we will have
\begin{align*}
&\sum_{i=1}^Ng_1(z_i,z_{l_j})(z_{l_j}-z_i)\\
\approx&\sum_{i=1}^N \sum_{a=0}^{m-1}\sum_{b=0}^{m-a-1}\alpha_{ab}(z_{l_j})z_{i,x}^az_{i,y}^b(z_{l_j}-z_i)\\
=&\sum_{a=0}^{m-1}\sum_{b=0}^{m-a-1} \alpha_{ab}(z_{l_j}) \sum_{i=1}^N z_{i,x}^az_{i,y}^b(z_{l_j}-z_i)\\
=&N\sum_{a=0}^{m-1}\sum_{b=0}^{m-a-1} \alpha_{ab}(z_{l_j}) \left(M_{ab}z_{l_j}-\begin{bmatrix}M_{a+1,b}\\M_{a,b+1}\end{bmatrix}\right)
\end{align*}
And similarly we have
$$\sum_{i=1}^N g_2(z_i,z_{l_j})= N\sum_{a=0}^{m-1}\sum_{b=0}^{m-a-1} \beta_{ab}(z_{l_j})M_{ab}.$$}
\end{itemize}
With the above assumptions the HJB equation (\ref{HJB4}) can be finally written as:
\begin{align}
-\frac{\partial \tilde{J}}{\partial t}&= \mathcal L(t) +\frac{\partial \tilde{J}}{\partial z_l}^T v_l-p \frac{\partial \tilde{J}}{\partial v_l}^T v_l-u_{max}\|\frac{\partial \tilde{J}}{\partial v_l}\|\notag\\
&+\sum_{a=0}^{m}\sum_{b=0}^{m-a}\frac{\partial \tilde{J}}{\partial M_{ab}}\begin{bmatrix}aM_{a-1,b} &bM_{a,b-1}\end{bmatrix}\dot M_1\notag\\
&+\begin{bmatrix}\frac{\partial \tilde{J}}{\partial \dot M_{10}}&\frac{\partial \tilde{J}}{\partial \dot M_{01}}\end{bmatrix} \Big( \sum_{j=1}^{M} \big( \sum_{a=0}^{m-1}\sum_{b=0}^{m-a-1}\alpha_{ab}(z_{l_j})M_{ab}z_{l_j}\notag\\
&-\sum_{a=0}^{m-1}\sum_{b=0}^{m-a-1}\alpha_{ab}(z_{l_j})\begin{bmatrix}M_{a+1,b}
\\M_{a,b+1}\end{bmatrix}\notag\\
&+\sum_{a=0}^{m-1}\sum_{b=0}^{m-a-1}\beta_{ab}(z_{l_j})M_{ab}(v_{l_j}-\dot M_1)\big)-p\dot M_1\Big)
\label{HJB5}
\end{align}
with the boundary condition
\begin{equation}\tilde J(t_f,M_{ab}, \dot M_{10}, \dot M_{01}, z_l, v_l)=0.\end{equation}
\begin{remark}
If there are upper bounds for the errors we made during the approximations and if the original HJB equation (\ref{HJB3}) is stable backward in time, then the solution $\tilde J$ to (\ref{HJB5}) is close to the solution to (\ref{HJB3}).
\end{remark}
\subsection{Optimal control in an MPC setting}
Even though we can approximate the HJB equation so that it has all the moments and leaders' states as variables, solving this type of PDE numerically is still difficult. There is no efficient method for solving general PDEs with more than four variables. At the same time, the approximation of the HJB equation performs poorly if the time horizon is big since the errors grow
larger in this case. We use model predictive control as a remedy. In particular, we optimize over a relatively short time period and apply the derived optimal control for only a few steps and repeat the procedure. At time $t=\tau$ and for a certain integer $q \in \mathbb{Z}$, we optimize over $q\Delta t$ time length in the future and apply the obtained optimal control for $\Delta t$. Then the sub-problem becomes:
\begin{equation}
\begin{array}{cl}
\displaystyle\min_{u}~~ & \bar{\Phi}(z(\tau+q\Delta t))+\displaystyle\int_{\tau}^{\tau+q\Delta t}\bar{\mathcal L}(t)dt\\
s.t.~~ &\dot z_i=v_i\\
&\dot v_i=\sum_{j=1}^{M}\left(g_1(z_i,z_{l_j})(z_{l_j}-z_i)+g_2(z_i,z_{l_j})(v_{l_j}-v_i)\right)\\
&~~~~~+f_i-p v_i,\text{ for }i=1,\cdots,N,\\
&\dot z_{l_j}=v_{l_j}\\
&\dot v_{l_j}=u_j-pv_{l_j},\\
&\|u_j\|\leq u_{max},~~~~~~~~~~\text{ for }j=1,\cdots,M,\\
&z(\tau), v(\tau),z_l(\tau),v_l(\tau)\text{ given, }
\end{array}
\label{MPC1}
\end{equation}
Note that we have a terminal cost here and the cost function $\bar {\mathcal L}$ may also be different form the original problem. There is no universal way to model the new cost functions $\bar{\Phi}$ and $\bar {\mathcal L}$. Since we still want to avoid using individual state information for the followers, we should choose $\bar{\Phi}$ and $\bar {\mathcal L}$ to be functions of the moments only. Here, since the goal is still moments tracking, we keep using the old cost function $\bar{\mathcal L(t)}$ during the process, which equals to $\sum_{a+b\leq m}c_{ab}(M_{ab}(t)-M_{ab}^d(t))^2$. The terminal cost should be used to approximate the tail optimal cost of the states, which theoretically should be the solution of the HJB equation at time $t=t+q\Delta t$. However, this solution is unavailable beforehand. Here we add the final errors for the derivative of the first order moments, i.e., $(\dot M_{01}-\dot M_{01}^d)^2+(\dot M_{10}-\dot M_{10}^d)^2$, in addition to all the moments errors as an approximation of the tail cost.
The corresponding approximated HJB equation of problem (\ref{MPC1}) will be
\begin{align}
-\frac{\partial \tilde J}{\partial t}&= \bar{\mathcal L}(t) +\frac{\partial \tilde J}{\partial z_l}^T v_l-p \frac{\partial \tilde J}{\partial v_l}^T v_l-u_{max}\|\frac{\partial \tilde J}{\partial v_l}\|\notag\\
&+\sum_{a=0}^{m}\sum_{b=0}^{m-a}\frac{\partial \tilde{J}}{\partial M_{ab}}\begin{bmatrix}aM_{a-1,b} &bM_{a,b-1}\end{bmatrix}\dot M_1\notag\\
&+\begin{bmatrix}\frac{\partial \tilde{J}}{\partial \dot M_{10}}&\frac{\partial \tilde{J}}{\partial \dot M_{01}}\end{bmatrix} \Big( \sum_{j=1}^{M} \big( \sum_{a=0}^{m-1}\sum_{b=0}^{m-a-1}\alpha_{ab}(z_{l_j})M_{ab}z_{l_j}\notag\\
&-\sum_{a=0}^{m-1}\sum_{b=0}^{m-a-1}\alpha_{ab}(z_{l_j})\begin{bmatrix}M_{a+1,b}
\\M_{a,b+1}\end{bmatrix}\notag\\
&+\sum_{a=0}^{m-1}\sum_{b=0}^{m-a-1}\beta_{ab}(z_{l_j})M_{ab}(v_{l_j}-\dot M_1)\big)-p\dot M_1\Big)
\label{HJBMPC}
\end{align}
with the boundary condition:
\begin{equation}
\tilde J(t+q\Delta t)=\bar{\Phi}(z).
\label{boundaryHJBMPC}
\end{equation}
The complexity of numerically solving the partial differential equation (\ref{HJBMPC}) grows exponentially with $q$, which means we cannot handle a long period prediction. On the other hand, the whole MPC approach may lead to an unstable closed-loop system if $q$ is too small. There is a tradeoff between the numerical complexity and the stability of the approach, which will be discussed and examined with a numerical experiment in the next section.
\section{Numerical experiments}
We will implement the optimal control approach on a specific model to show the capability and robustness of it. Let us use the following settings:
$$N=100,M=4,m=5,$$
which means there are 100 followers and 4 leaders. The moments we want to track is up to the fifth orders, i.e., $M_{ab}$ with $a+b\leq 5$ because we will deal with 4-polygon now. The reference signal for the moments are calculated from integrating $x^ay^b$ over a moving and shrinking square when assuming a uniform distribution. Namely, we want the followers uniformly spread inside a moving square while following the leaders. The center of the square should initially be located at the origin and slowly move to a point called ``exit" at the position of $\begin{bmatrix}120 & 120\end{bmatrix}^T$. In this case, there is a potential numerical problem of the approach in that for high order moments, the scale of their value becomes quite big. For example, the 5-order moment $M_{40}$ may become $120^{5}\approx 2\times 10^10$, which is much bigger than $M_{10}\approx 120$ when the followers reach the exit. Therefore, we change to a new coordinate system in each MPC iteration as below.
\subsection{Re-coordination and centralized moments}
When we want to solve (\ref{MPC1}) in each MPC iteration, we change the coordinates in order to make the current center-of-mass of the follower crowd, $\begin{bmatrix} M_{10} & M_{01}\end{bmatrix}^T$ to be the new origin. By doing this, the measured moments then become the centralized moments we defined in (\ref{moments2.2}). The reference signal needs to be updated as well and there are two ways doing that:
\begin{itemize}
{\item[1. ] Integrate the functions $x^ay^b$ over the original desired shape in the new coordinate system. Some extra calculations are needed in each iteration step.}
{\item[2. ] Track the original first order moments $M_{10}$ and $M_{01}$ in the new coordinate system while the other (centralized) moments track signals generated by integrating over a shape centered on the origin. Since those reference signals can be calculated off-line, this approach will speed up the calculation. However, the objective is changed from ``the crowd should follow `this' shape trajectory" to ``the center of the crowd should follow `this' path while the whole crowd should form `this' shape".}
\end{itemize}
The second approach alters the objective a little while not changing the ultimate goal of shape tracking, and is used in the simulation to simplify the computation.
We use the following functions for the model:
$$g_1(d)=0.3+20e^{-\frac{d}{10}}-\frac{20}{d+0.1}, \text{ and }g_2(d)=\frac{20}{d+0.1}.$$
$$f(z_i,v_i,z_k,v_k)=\frac{8}{d}e^{-0.2d}(z_i-z_k)$$
\begin{align*}
\bar{\Phi}=&c_0((\dot M_{10}^d-\dot M_{01})^2+(\dot M_{01}^d-\dot M_{01})^2)\notag\\
&~~~~~~~~~~~~+\sum_{a=0}^{m-1}\sum_{b=0}^{m-a-1}c_{ab}(M_{ab}^d- M_{ab})^2,
\end{align*}
$$\bar{\mathcal L(t)}=\sum_{a=0}^{m-1}\sum_{b=0}^{m-a-1}\bar c_{ab}(M_{ab}^d(t)- M_{ab})^2,$$
where $c_{ab}$ and $\bar c_{ab}$ can be tuned for the test purpose. Note that the moments we write here should already be transformed accordingly as we mentioned above.
We still need to do polynomial approximation of the functions $g_1$ and $g_2$ by using Taylor expansions. The expansion should be taken around the center-of-mass of the follower crowd, which is now the origin in the new coordinate system, and the degree of the polynomial should be $m-1=4$. The expansion will look like:
\begin{align*}
g_s(z,z_l)&\approx g_s(l)+g_{sx}(l)z_{x}+g_{sy}(l)z_{y}+\cdots \notag\\
&+\frac{1}{24}\Big(g_{sxxxx}(l)z_{x}^4+4g_{sxxxy}(l)z_{x}^3z_{y}\notag\\
&+6g_{sxxyy}(l)z_{x}^2z_{y}^2+4g_{sxxyy}(l)z_{x}z_{y}^3+g_{syyyy}(l)z_{y}^4\Big),
\end{align*}
where $l=\|z_l\|$, $g_{sx}=\frac{\partial g_s}{\partial x}$, $g_{sy}=\frac{\partial g_s}{\partial y}$, and the higher order partial derivatives are denoted in the same pattern, for $s=1,2$. Note that both the variables $z$, $z_l$ and the functions $g_1$, $g_2$ should be already transformed into the new coordinate system here.
\subsection{Numerical solution to HJB equation}
Once we have all the information needed, we should solve (\ref{HJBMPC}) with the boundary condition (\ref{boundaryHJBMPC}). We use the backward explicit finite difference method to solve the PDE by approximating the partial derivatives as:
$$\frac{\partial \tilde J(t+\Delta t)}{\partial t}\approx \frac{\tilde J(t+\Delta t)-\tilde J(t)}{\Delta t},$$
and
$$\frac{\partial \tilde J(\cdot,\xi)}{\partial \xi}\approx \frac{\tilde J(\cdot,\xi+\Delta \xi)-\tilde J(\cdot,\xi-\Delta \xi)}{2\Delta \xi},$$
where $\xi$ can be $M_{ab}$, $\dot M_{10}$, $\dot M_{01}$, $z_j$, or $v_j$. Then we can solve for the function value of $\tilde J$ at time $\tau$.
With the setting we made, there are 38 non-time variables for $\tilde J$ and each partial derivative needs function values at two points in the later time to approximate, which results to a complexity of $O(76^q)$ that grows exponentially with $q$ as mentioned above. Fortunately, it is still fast to solve with $q=3$. The simulations we make below assume $q=3$ and $\Delta t=0.1$.
\subsection{Simulation 1}
\begin{figure}
\caption{Snapshots for simulation 1. The blue dots indicate the position of the 100 followers and the red crosses are the 4 leaders. The squares are the desired shape in each plot. The black arrows are the velocity of the leaders while the red arrows are the acceleration of the leaders which come from the solution of (\ref{MPC1}
\label{Fig.sub.11}
\label{Fig.sub.12}
\label{Fig.sub.13}
\label{Fig.sub.14}
\label{sim1}
\end{figure}
In this simulation, the 100 followers are initially randomly distributed in the area of $[-50,50]\times[-50,50]$ while the 4 leaders start at $\begin{bmatrix}\pm25 & \pm25\end{bmatrix}^T$. The desired shape is a shrinking square first moving from the origin to the point $\begin{bmatrix}0 & 120\end{bmatrix}^T$, and then to the point $\begin{bmatrix}120 & 120\end{bmatrix}^T$ where the exit is. When the square reaches the exit, it will stay there with the size $10\times10$ for a while. We want the crowd to be uniformly spread in the desired shape. The coefficients in the cost functions are set to be \begin{equation} c_{ab}=\bar c_{ab}=\begin{cases}1000, & k=1,\\10^{2(2-k)},&k>1,\end{cases}\end{equation}where $a+b=k$. The total length of the simulation is 120s.
Figure \ref{sim1} gives four snapshots of the simulation and Figure \ref{sim1e} gives the weighted tracking errors, where $e_{ab}=c_{ab}\|M_{ab}(t)-M^d_{ab}(t)\|^2$, up to the fourth order. The oscillation of the tracking error is expected since it is a double integrator model and the prediction horizon $q\Delta t$ is short.
\begin{figure}
\caption{Weighted tracking errors for different moments with the definition $e_{ab}
\label{sim1e}
\end{figure}
\subsection{Simulation2}
In this experiment, we have the same setting of the followers and the leaders but add some obstacles in the environment as a disturbance to test the robustness of the method. The leaders will not take the extra disturbance from the obstacles into account when they calculate the optimal control, while both followers and leaders will react to the obstacles in their dynamics with an additional obstacle avoidance term:
\begin{equation*}
\dot v_i=\cdots+\sum_{k}h_k(z_i), ~~\dot v_{l_j}=\cdots+\sum_{k}h_k(z_{l_j}),
\end{equation*}
where $h_k$ models the obstacle avoidance force for the obstacle $k$. In this simulation, we set two round obstacles and model the forces as $$h_k(z)=\begin{cases}\frac{\kappa}{\|z-z_{ob_k}\|-r_k}(z-z_{ob_k}),&\text{if } \|z-z_{ob_k}\|\leq r_k+\delta,\\0,&\text{otherwise}\end{cases}$$where $z_{ob_k}$ is the center of the obstacle and $r_k$ is its radius, for $k=1,2$. We choose
$$z_{ob_1}=\begin{bmatrix}5 \\ 60\end{bmatrix}, r_1=15, z_{ob_2}=\begin{bmatrix}80 \\ 80\end{bmatrix}, r_2=30,\delta=2.$$
\begin{figure}
\caption{Snapshots for simulation 2. The blue dots indicate the position of the 100 followers and the red crosses are the 8 leaders. The squares are the desired shape in each plot. The black arrows are the velocity of the leaders while the red arrows are the acceleration of the leaders which come from the solution of (\ref{MPC1}
\label{Fig.sub.21}
\label{Fig.sub.22}
\label{Fig.sub.23}
\label{Fig.sub.24}
\label{Fig.sub.25}
\label{Fig.sub.26}
\label{sim2}
\end{figure}
Figure \ref{sim2} shows that the method is still able to give a correct control signal when the followers and the leaders are passing by the obstacle.
\subsection{simulation 3}
In this experiment, we use almost the same setting as that in the first simulation except that we increase the number of leaders to 8.
The increase of the amount of leaders does decrease the errors slightly for all the orders. However, the performance improvement in terms of shape is hardly noticeable. In Figure \ref{sim3}, the followers form a circle in the end instead of the desired square, which could be considered to be worse than the shape generated by 4 leaders in Figure \ref{sim1} although the errors in Figure \ref{sim3e} turn to be smaller than those in Figure \ref{sim1e}.
\begin{figure}
\caption{Snapshots for simulation 3. The blue dots indicate the position of the 100 followers and the red crosses are the 8 leaders. The squares are the desired shape in each plot. The black arrows are the velocity of the leaders while the red arrows are the acceleration of the leaders which come from the solution of (\ref{MPC1}
\label{Fig.sub.21}
\label{Fig.sub.22}
\label{Fig.sub.23}
\label{Fig.sub.24}
\label{sim3}
\end{figure}
\section{Conclusions and future work}
Since the moments of the crowd have a strong connection to its shape, we use moments to control the shape of the crowd. An optimal moments tracking problem is introduced in the paper and solved numerically with only using moments information as feedback. In order to further reduce the computational complexity, a model predictive control algorithm is used. Three numerical experiments show that the method solves the moments tracking problem efficiently enough and can handle certain disturbance in the model. The performance of the optimal controller is acceptable even during the transient phase in the simulations.
Future work involves using local measurements of the moments to design a distributed leading strategy for the leaders. A better understanding of the relationship between the moments and shapes may give a more practical cost function to achieve a better performance in terms of shaping.
\begin{figure}
\caption{Weighted tracking errors for different moments with the definition $e_{ab}
\label{sim3e}
\end{figure}
\end{document} |
\begin{document}
\begin{abstract}
The aim of this paper is to survey and extend recent results concerning bounds
for the Euclidean minima of algebraic number fields. In particular, we give upper bounds for the Euclidean minima of abelian fields of prime power conductor.
\end{abstract}
\maketitle
\section{Introduction}
Let $K$ be an algebraic number field, and let $\mathcal{O}k$ be its ring of integers. We denote by ${\rm N}: K \to \mathbb{Q}$
the absolute value of the norm map. The number field $K$ is said to be {\it Euclidean} (with respect to the norm) if for every $a,b \in \mathcal{O}k$ with $b \not = 0$ there exist
$c, d \in \mathcal{O}k$ such that $a = bc + d$ and ${\rm N}(d) < {\rm N}(b)$.
It is easy to check that $K$ is Euclidean if and only if for every $x \in K$ there exists $c \in \mathcal{O}k$ such that ${\rm N}(x-c) < 1$. This suggests to look at
$$
M(K) = {\rm sup}_{x \in K} {\rm inf}_{c \in \mathcal{O}k} {\rm N}(x-c),
$$
called the {\it Euclidean minimum} of $K$.
The study of Euclidean number fields and Euclidean minima is a classical one. However, little is known about the precise value of $M(K)$ (see
for instance [Le 95] for a survey, and the tables of Cerri [C 07] for some numerical results). Hence, it is natural to look for {\it upper bounds} for $M(K)$.
This is also a classical topic, for which a survey can be found in [Le 95].
Let $n$ be the degree of $K$ and $D_K$ the absolute value of its discriminant.
It is shown in [BF 06] that for any number field $K$, we have $M(K) \le 2^{-n}D_K$.
The case of {\it totally real} fields is especially interesting, and has been the
subject matter of several papers. In particular, a conjecture attributed to Minkowski
states that if $K$ is totally real, then $M(K) \le 2^{-n} \sqrt {D_K}.$ This conjecture
is proved for $n \le 8$, cf. [Mc 05], [HGRS 09], [HGRS 11].
Several recent results concern the case of {\it abelian fields}. In [BFM 13], upper bounds are given for abelian fields of conductor $p^r$, where
$p$ is an odd prime. The present paper complements these results by handling the case of abelian fields of conductor a power of $2$.
In particular, we have
\noindent
{\bf Theorem.} {\it If $K$ is totally real of conductor $p^r$, where $p$ is a prime and $r \ge 2$, then $$M(K) \le 2^{-n} \sqrt {D_K}.$$ }
In other words, Minkowski's conjecture holds for such fields.
These results are based on the study of lattices associated to number fields (see [BF 99], [BF 02]). In \S 2, we recall some results on lattices and number fields,
and in \S 3 we survey the results of [BFM 13] concerning abelian fields of conductor an odd prime power. The case of abelian fields of power of $2$ conductor is the
subject matter of \S 4. Finally, we survey some results concerning cyclotomic fields and their maximal totally real subfields in \S 5.
\section{Lattices and number fields}
We start by recalling some standard notions concerning Euclidean lattices (see for instance
[CS 99] and [M 96]. A {\it lattice} is a pair $(L,q)$, where $L$ is a free $\mathbb{Z}$--module of finite rank, and $q : L_{\mathbb{R}} \times L_{\mathbb{R}} \to \mathbb{R}$
is a positive definite symmetric bilinear form, where $L_{\mathbb{R}} = L \otimes_\mathbb{Z} \mathbb{R}$.
If $(L,q)$ is a lattice and $a \in \mathbb{R}$, then we denote by $a(L,q)$ the lattice
$(L,aq)$. Two lattices $(L,q)$ and $(L',q')$ are said to be
{\it similar} if and only if there exists $a \in \mathbb{R}$ such that $(L',q')$ and $a(L,q)$ are isomorphic,
in other words if there exists an isomorphism of $\mathbb{Z}$-modules $f : L \to L'$ such that $q'(f(x),f(y)) = a q(x,y)$.
Let $(L,q)$ be a lattice, and set $q(x) = q(x,x)$. The {\it maximum }of $(L,q)$ is defined by
\[
{\rm max}(L,q) = \sup_{x \in L_{\mathbb{R}}} \inf_{c \in L} q(x-c).
\]
Note that ${\rm max}(L,q)$ is the square of the covering radius of the associated sphere covering.
The {\it determinant} of $(L,q)$ is denoted by ${\rm det}(L,q)$. It is by definition the determinant of the matrix of $q$ in a $\mathbb{Z}$--basis of $L$.
The {\it Hermite--like thickness} of $(L,q)$ is
$$
\tau(L,q) = {{\rm max}(L,q) \over {\rm det}(L,q)^{1/m}},
$$
where $m$ is the rank of $L$. Note that $\tau(L,q)$ only depends on the similarity class of the lattice $(L,q)$.
Next we introduce a family of lattices that naturally occur in connection with abelian fields (see \S 2),
and for which one has good upper bounds of the Hermite-like thickness. This family is defined as follows :
Let $m\in {\mathbb {N}}$, and $b\in {\mathbb{R}}$ with $b>m$.
Let $L=L_{b,m}$ be a lattice in ${\mathbb{R}} ^m$ with Gram matrix
$$b I_m - J_m = \left( \begin{array}{cccc}
b-1 & -1 & \ldots & -1 \\
-1 & \ddots & \ddots & \vdots \\
\vdots & \ddots & \ddots & -1 \\
-1 & \ldots & -1 & b-1
\end{array} \right),$$
where $I_m$ is the $m\times m$-identity matrix and $J_m$ is the all-ones matrix of size $m \times m$.
These lattices were defined in [BFN 05], (4.1).
Note that the lattice $L_{m+1,m} $ is similar to the dual lattice $A_m^{\#}$ of the
root lattice $A_m$ (see for instance [CS 99], Chapter 4, \S 6, or [M 96] for the definition of the root lattice $A_m$).
Let $K$ be an number field of degree $n$, and suppose that $K$ is either totally
real or totally complex. Let us denote by
$^{\overline {\ }} : K \to K$ the identity in the first case and the
complex conjugation in the second one, and let $P$ be the set of totally
positive elements of the fixed field of this involution. Let us denote by ${\rm Tr} : K \to \mathbb{Q}$ the trace
map. For any $\alpha \in P$, set $q_{\alpha}(x,y) = {\rm Tr}(\alpha x \overline y)$ for all $x, y \in K$.
Then $(\mathcal{O}k,q_{\alpha})$ is a lattice. Set
$$
\tau_{\rm min}(\mathcal{O}k) = {\rm inf} \{ \tau(\mathcal{O}k,q_{\alpha}) \ | \ \alpha \in P \}.
$$
If $D_K$ is the absolute value of the discriminant of $K$, then, by [BF 06], Corollary (5.2), we have
\begin{equation}\label{E:gen-est}
M(K) \leq \left(\mathcal{R}ac{\tau_{\min}(\mathcal{O}k)}{n}\right)^{\mathcal{R}ac{n}{2}} \sqrt{D_K},
\end{equation}
This is used in [BF 06], [BFN 05], [BFS 06] and [BFM 13] to give upper bounds of Euclidean minima (see also \S 3 and \S 5). The bounds
of \S 4 are also based on this result.
\section{Abelian fields of odd prime power conductor}
The set of all abelian extensions of $\mathbb{Q}$ of odd prime power conductor will be denoted by $\mathcal{A}$.
For $K \in \mathcal{A}$ we denote by
$n$ the degree of $K / \mathbb{Q}$, by
$D$ the absolute value of the discriminant of $K$, by $p$
the unique prime dividing the conductor of $K$, and by $r$ the $p$-adic additive valuation of the conductor of $K$.
If the dependence on the field $K$ needs to be emphazized, we shall add the index $K$ to the above symbols. For example, we shall write $n_K$ instead of $n$.
\begin{definition}
Let $\mathcal{D} \subset \mathcal{A}$, and let $\psi : \mathcal{D} \to \mathbb{R}$ be a function. We shall say that
$\psi_o \in \mathbb{R}$ is the limit of $\psi$ as $n_K$
goes to infinity and write
\[
\lim_{n_K \to \infty} \psi(K) = \psi_0
\]
if for every $\epsilon > 0$ there exists $N >0$ such that for every field $K \in \mathcal{D}$
\[
n_K > N \text{Im}plies |\psi(K) - \psi_0| < \epsilon.
\]
We shall also write
\[
\lim_{p_K \to \infty} \psi(K) = \psi_0
\]
if for every $\epsilon > 0$ there exists $N >0$ such that for every field $K \in \mathcal{D}$
\[
p_K > N \text{Im}plies |\psi(K) - \psi_0| < \epsilon.
\]
\end{definition}
The following is proved in [BFM 13], th. (3.1) and (3.2) :
\begin{theorem}\label{T:bound1}
Let $K \in \mathcal{A}$. Then there exist constants $\varepsilon=\varepsilon(K) \leq 2$ and $C=C(K) \leq \mathcal{R}ac{1}{3}$ such that
\[
M(K) \leq C^n \, (\sqrt{D_K})^{\varepsilon}.
\]
If $[K : \mathbb{Q}] > 2$, then one may choose $\varepsilon(K) < 2$. Moreover,
\[
\lim_{n_K \to \infty} \varepsilon(K) = 1.
\]
If $r_K \geq 2$, or $r_K=1$ and $[\cyclo:K]$ is constant, then we also have
\[
\lim_{p_K \to \infty} C(K) = \mathcal{R}ac{1}{2\sqrt{3}}.
\]
\end{theorem}
\begin{theorem}\label{T:bound2}
Let $K \in \mathcal{A}$. Then there is a constant $\omega=\omega(K)$ such that
\[
M(K) \leq \omega^n \sqrt{D_K}.
\]
If $r_K \geq 2$, or $r_K=1$ and $[\cyclo:K]$ is constant, then
\[
\lim_{p_K \to \infty} \omega(K) = \mathcal{R}ac{1}{2\sqrt{3}}.
\]
Moreover, if $r_K \geq 2$, then $\omega(K) \leq 3^{-2/3}$.
\end{theorem}
Note that this implies that Minkowski's conjecture holds for all totally real fields
$K \in \mathcal{A}$ with composite conductor :
\begin{corollary}\label{T:Minkowski}
Let $K \in \mathcal{A}$, and suppose that the conductor of $K$ is of the form $p^r$ with
$r > 1$. Then
\[
M(K) \leq 2^{-n} \sqrt{D_K}.
\]
\end{corollary}
This follows from Theorem~(\ref{T:bound2}), since $3^{-2/3} < 1/2$, and for $K$ totally real this is precisely
Minkowski's conjecture.
The proofs of these results are based on the method of [B 05], outlined in the previous section. For $K \in \mathcal{A}$, we denote by
$\mathcal{O}k$ is the ring of integers of $K$, and we consider the
lattice $(\mathcal{O}k,q)$, where $q$ is defined by $q(x,,y) = \textnormal{Tr}kq(x \overline y)$. As we have
seen in \S 4, the Hermite--like thickness of this lattice can be used to give an upper bound of
the Euclidean minimum of $K$.
Let $\zeta$ be a primitive root of unity
of order $p^r$, let us denote by $e$ the degree
$[\cyclo : K]$. Let $\Gamma_K$ be the orthogonal sum of
$\mathcal{R}ac{p^{r-1}-1}{e}$ copies of the lattice $p^{r-1}A_{p-1}^{\#}$.
Set $d=\mathcal{R}ac{p-1}{e}$, and let $\Lambda_K = e p^{r-1} L_{\mathcal{R}ac{p}{e},d}$ (note that
the scaling is taken in the sense of the previous section, that is it refers to multiplying the quadratic form by
the scaling factor). We have (see [BFM 13], Theorem (6.1)) :
\begin{theorem}\label{T:ortho-sum}
The lattice $(\mathcal{O}k,q)$ is isometric to the orthogonal sum of $\Gamma_K$ and of $\Lambda_K$.
\end{theorem}
This leads to the following
upper bound
of $\tau_{\rm min}(\mathcal{O}k)$ :
\begin{corollary}\label{c:tau-min}
We have
\[
\tau_{\rm min}(\mathcal{O}k) \leq \tau(\mathcal{O}k,q) \leq n \cdot p^{r-\mathcal{R}ac{\upsilon}{n}} \cdot \mathcal{R}ac{p^{r+1}+p^r+1-e^2}{12 p^{r+1}},
\]
where
\[
\upsilon = rn -\mathcal{R}ac{(p^{r-1}-1)}{e} -1.
\]
\end{corollary}
This is proved in [BFM 13], Corollary (6.7). The proof uses an upper bound for the Hermite--like thickness of the lattices $\Lambda_{b,m}$ proved in [BFN 05], (4.1).
Using this corollary and (2.1), one proves (3.2) and (3.3) as in [BFM 13], \S 7.
\section{Abelian fields of power of two conductor}
We keep the notation of \S 1 and 2. In particular, $K$ is a number field of degree $n = n_K$, and the absolute value of its disciriminant is
denoted by $D = D_K$.
Let $r \in \mathbb{N}$, and let $\zeta$ be a primitive $2^r$-th root of unity. A field $K$
is said to have {\it conductor $2^r$} if $K$ is contained in the cyclotomic field
$\cyclo$, but not in $\mathbb{Q}(\zeta^2)$. Note that there is no field of conductor $2$ and the only field of conductor $4$ is $\mathbb{Q}(i)$. For $r \geq 3$, we have
\begin{proposition}\label{P:bounds}
Suppose that $K$ is an abelian field of conductor $2^r$, where $r \geq 3$. Then we have
$$
K = \cyclo, \mr, \textnormal{ or } \mi.
$$
Moreover,
\begin{enumerate}
\item[(a)] If $K = \cyclo$ or $\mr$, then
$$
M(K) \le 2^{-n} \sqrt {D_K}.
$$
\item[(b)] If $K = \mi$, then
$$
M(K) \le 2^{-n}(2n-1)^{\mathcal{R}ac{n}{2}}.
$$
\end{enumerate}
\end{proposition}
\begin{proof}
Let us first prove that $K = \cyclo$, $\mathbb{Q}(\zeta + \zeta^{-1})$ or $\mi$.
Let $G = {\rm Gal}(\cyclo/\mathbb{Q})$. Then $G = \left<\sigma,\tau\right>$ where $\tau$ is the complex conjugation, and
$\sigma (\zeta) = \zeta^3$. The subgroups of order $2$ of $G$ are
$$
H_1 = \left<\tau\right>, H_2 = \left<\sigma^{2^{r-3}}\right>, \textnormal{ and } H_3 = \left<\sigma^{2^{r-3}} \tau\right>.
$$
It is easy to check that we have $\cyclo^{H_1} = \mr$, $\cyclo^{H_2} = \mathbb{Q}(\zeta^2)$ and $\cyclo^{H_3} = \mi$.
Note that any proper subfield of $\cyclo$ is a subfield of $\cyclo^{H_i}$ for $i = 1,2$ or $3$, and the statement easily follows from this observation.
Part (a) follows from Proposition (10.1) in [BF 06] for $K = \cyclo$, and from Corollary (4.3) in [BFN 05] for $K = \mathbb{Q}(\zeta + \zeta^{-1})$.
Let us prove (b). Suppose that $K = \mi$. Recall that $\cyclo^{H_3} = \mi$, and that the ring of integers of $\cyclo$ is $\mathbb{Z}[\zeta]$.
Therefore the ring of integers of $\mi$ is $\mathbb{Z}[\zeta -\zeta^{-1}]$. Set $e_i = \zeta + (-1)^i \zeta^{-1}$ for all $i \in \mathbb{Z}$ and $n = 2^{r-2}$. Then
an easy computation shows that the elements $1, e_1, \dots, e_{n-1}$ form an integral basis of $\mathcal{O}k = \mathbb{Z}[\zeta -\zeta^{-1}]$,
and that for $-2n \leq i \leq 2n$ we have
\[
\textnormal{Tr}kq(e_i) =
\begin{cases}
2n \quad &\textnormal{ if } i=0,\\
-2n \quad &\textnormal{ if } i=\pm 2n,\\
0 \quad &\textnormal{ otherwise. }
\end{cases}
\]
Recall that $\tau : K \to K$ is the complex conjugation, and
let
$$
q : \mathcal{O}k \times \mathcal{O}k \to \mathbb{Z}
$$
given by
$$
q(x,y) = \textnormal{Tr}kq (x \tau(y))
$$
be the trace form. Then the Gram matrix of $(\mathcal{O}k,q)$ with respect to the basis $1,e_1,\dots, e_n$ is
$$
{\rm diag}(n,2n,\dots,2n).
$$
Note that this implies that we have
$$D_K = n^n 2^{n-1}.$$ Since $1,e_1,\dots,e_{n-1}$ is an orthogonal basis for $(\mathcal{O}k,q)$, it
follows from the Pythagorean theorem that the point $x = \mathcal{R}ac{1}{2}(1 + e_1 + \dots + e_{n-1})$
is a deep hole of the lattice $(\mathcal{O}k,q)$. Thus we have
$$
\max(\mathcal{O}k,q) = \inf_{c \in \mathcal{O}k} q(x-c) = q(x) = \mathcal{R}ac{n(2n-1)}{4}.
$$
Set
$$
\tau(\mathcal{O}k,q) = \mathcal{R}ac{\max(\mathcal{O}k,q)}{\det(\mathcal{O}k,q)^{1/n}}.
$$
Then we have
$$
\tau(\mathcal{O}k,q) = \mathcal{R}ac{n (2n-1)}{4(n^n 2^{n-1})^{1/n}} = \sqrt[n]{2} \cdot \left(\mathcal{R}ac{2n - 1}{8}\right).
$$
By (2.1), we have
$$
M(K) \le \left( \mathcal{R}ac{\tau(\mathcal{O}k,q)}{n} \right)^{\mathcal{R}ac{n}{2}} \sqrt {D_K}.
$$
Thus we obtain
$$
M(K) \le 2^{-n}(2n-1)^{\mathcal{R}ac{n}{2}}.
$$
This completes the proof of the proposition.
\end{proof}
Let $r \geq 3$ and let $K$ be an abelian field of conductor $2^r$ of the form $\mi$. The following two corollaries show an asymptotic behavior of the bound obtained in Proposition~(\ref{P:bounds})(b).
\begin{corollary}\label{C:bound-with-varepsilon}
We have
\[
M(K) \leq 2^{-n} (\sqrt{D_K})^{1+\varepsilon(n)},
\]
where
\[
\varepsilon(n) \sim \mathcal{R}ac{\ln 2 - \mathcal{R}ac{1}{2}}{n \ln n}.
\]
\end{corollary}
\begin{proof}
We set
\[
\varepsilon(n) = \mathcal{R}ac{\ln a_n + \ln 2}{n \ln (2n) - \ln 2},
\]
where
\[
a_n = \left(1 - \mathcal{R}ac{1}{2n}\right)^n.
\]
Using the fact that $D_K = n^n 2^{n-1}$, the inequality of Proposition~(\ref{P:bounds})(b) can rewritten as
\[
M(K) \leq 2^{-n} (\sqrt{D_K})^{1+\varepsilon(n)}.
\]
A simple calculation shows that
\[
\lim_{n \to \infty} (n \ln n) \cdot \varepsilon(n) = \ln 2 - \mathcal{R}ac{1}{2}.
\]
The result follows.
\end{proof}
\begin{corollary} We have
\[
M(K) \leq (\sqrt{2} \, e^{-1/4}) \cdot 2^{-n} \sqrt{D_K}.
\]
\end{corollary}
\begin{proof}
Using the same notation as in the proof of Corollary~(\ref{C:bound-with-varepsilon}), we can rewrite the inequality of Proposition~(\ref{P:bounds})(b) as
\[
M(K) \leq (\sqrt{2 a_n}) \cdot 2^{-n} \sqrt{D_K}.
\]
The result follows from the fact that the sequence $(a_n)$ is increasing and its limit equals to $e^{-1/2}$.
\end{proof}
\section{Cyclotomic fields and their maximal totally real subfields}
Let $m \in \mathbb{N}$, and let $\zeta$ be a primitive $m$-th root of unity. Let $K = \cyclo$, and let $F = \mr$ be its maximal totally
real subfield.
Let us denote by $n_K$, respectively $n_F$, their degrees, and by $D_K$, respectively $D_F$, the absolute values of their discriminants.
The aim of this section is to survey some results concerning $M(K$) and $M(F)$.
\begin{theorem} We have
\[
M(K) \leq 2^{-n_K} \sqrt{D_K}
\]
\end{theorem}
\begin{proof} This is proved in [BF 06], Proposition (10.1).
\end{proof}
For certain values of $m$, one obtains better bounds :
\begin{theorem} We have
{\rm (i)} Suppose that $m$ is of the form $m = 2^r3^s5^t$, with $r \ge 0$,
$s \ge 1$ and $t \ge 1$; $m = 2^r 5^s$ with $r \ge 2$, $s \ge 1$;
$m = 2^r 3^s$ with $r \ge 3$, $s \ge 1$. Then
$$M(K) \le 8^{-n_K / 2} \sqrt{D_K}.$$
{\rm (ii)} Suppose that $m$ is of the form $m = 2^r 5^s 7^t$, with
$r \ge 0$, $s \ge 1, t \ge 1$; $m = 2^r 3^s 5^t$ with
$r \ge 0, s \ge 2, t \ge 1$; $m = 2^r 3^s 7^t$ with $r \ge 2, s \ge 1,
t \ge 1$. Then $$M(K) \le 12^{-n_K / 2} \sqrt{D_K}.$$
\end{theorem}
\begin{proof} This is proved in [BF 06], Proposition (10.2). The result follows from (2.1), and the fact that an orthogonal sum of lattices of type $E_8$ is defined over $K$
in the sense of \S 2 in case {\rm (i)}, and an orthogonal sum of lattices isomorphic to the Leech lattice in case {\rm (ii)}.
\end{proof}
The results are less complete for the maximal totally real subfields. We have
\begin{theorem} Suppose that $m = p^r$ where $p$ is a prime and $r \in \mathbb{N}$, or that $m = 4 k$ with $k \in \mathbb{N}$ odd. Then we have
\[
M(F) \leq 2^{-n_F} \sqrt{D_F}
\]
\end{theorem}
\begin{proof} This is proved in [BF 06], Proposition (8.4) for $m = p^r$ and $p$ and odd prime; in [BFN 05], Corollary (4.3) for $m = 2^r$; and in [BFS 06], Proposition (4.5) for $m = 4k$.
\end{proof}
{\bf Bibliography}
[BF 99] E. Bayer--Fluckiger, {\it Lattices and number Fields}, Contemp. Math., {\bf 241} (1999),
69--84.
[BF 02] E. Bayer-Fluckiger, {\it Ideal Lattices}, proceedings of the conference Number Theory
and Diophantine Geometry, (Zurich, 1999), Cambridge Univ. Press
(2002), 168--184.
[BF 06] E. Bayer-Fluckiger, {\it Upper bounds for Euclidean minima of algebraic number fields},
J. Number Theory {\bf 121} (2006), no. 2, 305-323.
[BFM 13] E. Bayer--Fluckiger, P. Maciak, {\it Upper bounds for the Euclidean minima of
abelian fields of odd prime power conductor}, Math. Ann. (to appear).
[BFN 05] E. Bayer-Fluckiger, G. Nebe, {\it On the euclidean minimum of some real number fields},
J. th. nombres Bordeaux, {\bf 17} (2005), 437-454.
[BFS 06], E. Bayer-Fluckiger, I. Suarez, {\it Ideal lattices over totally real number fields and Euclidean minima},
Archiv Math. {\bf86} (2006), 217-225.
[C 07] J.-P. Cerri, {\it Euclidean minima of totally real number fields. Algorithmic determination}, Math. Comp. {\bf 76} (2007), 1547-1575.
[CS 99] J. H. Conway, N. J. A. Sloane, {\it Sphere Packings, Lattices and Groups}, Third Edition., Springer-Verlag New York, Inc. (1999).
[HGRS 09] R. J. Hans-Gill, M. Raka and R. Sehmi, \textit{On Conjectures of Minkowski and Woods for $n = 7$},
J. Number Theory {\bf 129} (2009), 1011-1033.
[HGRS 11] R. J. Hans-Gill, M. Raka and R. Sehmi, \textit{On Conjectures of Minkowski and Woods for $n = 8$}, Acta Arith. {\bf 147} (2011), 337-385.
[Le 95] F. Lemmermeyer, \textit{The Euclidean algorithm in algebraic number fields}, Expo. Math. {\bf 13} (1995), 385-416.
[M 96] J. Martinet, {\it R\'eseaux parfaits des espaces euclidiens}, Masson (1996),
English translation: {\it Perfect lattices of Euclidean spaces}, Springer-Verlag, Grundlehren
der math. Wiss, {\bf 327} (2003).
[Mc 05] C.T. McMullen, \textit{Minkowski's conjecture, well-rounded lattices and topological dimension}, J. Amer. Math. Soc. {\bf 18} (2005) 711-734.
\end{document} |
\begin{document}
\title{Comprehensive proof of the Greenberger-Horne-Zeilinger
Theorem for the four-qubit system}
\author{Li Tang}
\email{tangli@wipm.ac.cn}
\affiliation{
State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics,
Chinese Academy of Sciences, Wuhan 430071, China}
\affiliation{
Center for Cold Atom Physics, Chinese Academy of Sciences,
Wuhan 430071, China}
\affiliation{
Graduate School, Chinese Academy of Sciences, Wuhan
430071, China}
\author{Jie Zhong}
\affiliation{
Laboratory of Mathematical Physics, Wuhan Institute of Physics and
Mathematics, Chinese Academy of Sciences, Wuhan 430071, China}
\affiliation{
Graduate School, Chinese Academy of Sciences, Wuhan
430071, China}
\author{Yaofeng Ren}
\affiliation{
Department of Mathematics, The Naval University of Engineering,
Wuhan 430033, China}
\author{Mingsheng Zhan}
\affiliation{
State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics,
Chinese Academy of Sciences, Wuhan 430071, China}
\affiliation{
Center for Cold Atom Physics, Chinese Academy of Sciences,
Wuhan 430071, China}
\author{Zeqian Chen}
\email{zqchen@wipm.ac.cn}
\affiliation{
State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics and Laboratory of Mathematical Physics, Wuhan
Institute of Physics and Mathematics, Chinese Academy of Sciences,
Wuhan 430071, China}
\date{\today}
\begin{abstract}
Greenberger-Horne-Zeilinger (GHZ) theorem asserts that there is a
set of mutually commuting nonlocal observables with a common
eigenstate on which those observables assume values that refute
the attempt to assign values only required to have them by the
local realism of Einstein, Podolsky, and Rosen (EPR). It is known
that for a three-qubit system there is only one form of the
GHZ-Mermin-like argument with equivalence up to a local unitary
transformation, which is exactly Mermin's version of the GHZ
theorem. In this paper, however, for a four-qubit system which was
originally studied by GHZ, we show that there are nine distinct
forms of the GHZ-Mermin-like argument. The proof is obtained by
using some geometric invariants to characterize the sets of
mutually commuting nonlocal spin observables on the four-qubit
system. It is proved that there are at most nine elements (except
for a different sign) in a set of mutually commuting nonlocal spin
observables in the four-qubit system, and each GHZ-Mermin-like
argument involves a set of at least five mutually commuting
four-qubit nonlocal spin observables with a GHZ state as a common
eigenstate in GHZ's theorem. Therefore, we present a complete
construction of the GHZ theorem for the four-qubit system.
\end{abstract}
\pacs{03.65.Ud, 03.67.-a}
\maketitle
\section{Introduction }
Bell's inequality \cite{Bell} indicates that certain statistical
correlations predicted by quantum mechanics for measurements on
two-qubit ensembles cannot be understood within a realistic
picture based on Einstein, Podolsky, and Rosen's (EPR's) notion of
local realism \cite{EPR}. There is an unsatisfactory feature in
the derivation of Bell's inequality that such a local realistic
and, consequently, classical picture can explain perfect
correlations and is only in conflict with statistical prediction
of quantum mechanics. Strikingly enough, the
Greenberger-Horne-Zeilinger (GHZ's) theorem exhibits that the
contradiction between quantum mechanics and local realistic
theories arises even for definite predictions on a four-qubit
system \cite{GHZ}. Mermin \cite{M90} subsequently refined the
original GHZ argument on a three-qubit system. Let us recall that
their approaches were characterized by the following premises:
(a) a set of mutually commuting nonlocal observables,
(b) a common eigenstate on which those observables assume values
that refute the attempt to assign values only required to have
them by EPR's local realism.
Based on this criterion of a GHZ-Mermin-like argument, we define a
GHZ-Mermin experiment by a set of mutually commuting nonlocal
observables with at least two different observables at each site.
(Note that, a common local observable does not provide a random
selection of measurements and so plays no role in the
GHZ-Mermin-type proof.) A GHZ-Mermin experiment presenting a
GHZ-Mermin-like argument on a certain common eigenstate is said to
be nontrivial. There is no nontrivial GHZ-Mermin experiment in the
two-qubit system, while in the three-qubit system there is only
one nontrivial GHZ-Mermin experiment (with equivalence up to a
local unitary transformation), which is exactly Mermin's version
of the GHZ theorem \cite{Chen}. On the other hand, the
GHZ-Mermin-like argument has been extended to $n$ qubits
\cite{PRC-C}, and to multiparty multilevel systems \cite{Cabello}.
So far, however, no complete construction of nontrivial GHZ-Mermin
experiments is presented beyond the three-qubit system as noted in
\cite{Chen}, there are only partial results \cite{R-Z}.
In this paper, we will construct all nontrivial GHZ-Mermin
experiments of the four-qubit system, for which the GHZ-like
argument was developed originally by GHZ \cite{GHZ}. It is proved
that there are nine distinct forms of the GHZ-Mermin-like argument
on the four-qubit system, and each GHZ-Mermin-like argument
involves a set of at least five mutually commuting four-qubit
nonlocal spin observables with a GHZ state as a common eigenstate
in GHZ's theorem. Precisely, we obtain the following results.
(i)~~All four-qubit GHZ-Mermin experiments of at most four
elements are trivial.
(ii)~~Four-qubit GHZ-Mermin experiments of five (6, 7, or 8)
elements possess 11 (9, 5, or 3) different forms, two of which are
nontrivial.
(iii)~~A four-qubit GHZ-Mermin experiment contains at most nine
elements and, the experiments of nine elements have two different
forms, one of which is trivial, while another one is nontrivial.
(iv)~~In every nontrivial GHZ-Mermin experiment for the four-qubit
system, the associated states exhibiting an ``all versus nothing"
contradiction between quantum mechanics and $\mathrm{EPR}$'s local
realism must be $\mathrm{GHZ}$ states.
Our proof is based on some subtle mathematical arguments. We first
classify the equivalence of GHZ-Mermin experiments by two basic
symmetries acting on them. Then, we define two geometric
invariants for a GHZ-Mermin experiment, which can be used to
distinguish two inequivalent experiments. These arguments can be
easily extended to $n$ qubits.
The structure of this paper is as follows. In Sec.II, we first
prove a lemma on the structure of two commuting nonlocal spin
observables of $n$ qubits. Then, we discuss two basic symmetries
($\mathrm{(S_1)}$ and $\mathrm{(S_2)}$) acting on GHZ-Mermin
experiments of $n$ qubits. By these two basic symmetries we define
the equivalence of GHZ-Mermin experiments. We illustrate that a
four-qubit GHZ-Mermin experiment of three elements must
equivalently be one of three different forms. Finally, we define
two geometric invariants ($\mathrm{C}$-invariants and
$\mathrm{R}$-invariants) for a GHZ-Mermin experiment. These two
geometric invariants are invariant under $\mathrm{(S_1)},
\mathrm{(S_2)},$ and local unitary transformations
($\mathrm{LU}$). They paly a crucial role in the equivalence of
GHZ-Mermin experiments. In Sec.III, we show that a four-qubit
GHZ-Mermin experiment of four elements must equivalently be one of
seven different forms. Then we prove that every four-qubit
GHZ-Mermin experiment of three or four elements is trivial. In
Sec.IV, we show that four-qubit GHZ-Mermin experiments of five (6,
7, or 8) elements possess 11 (9, 5, or 3) different forms. It is
proved that in each case there are two nontrivial GHZ-Mermin
experiments and, the associated states exhibiting an ``all versus
nothing" contradiction between quantum mechanics and
$\mathrm{EPR}$'s local realism are $\mathrm{GHZ}$ states. In
Sec.V, we prove that a four-qubit GHZ-Mermin experiment contains
at most nine elements and the experiments of nine elements have
two different forms, one of which is trivial while another one is
nontrivial. Finally, in Sec.VI we give some concluding remarks and
questions for further consideration.
\section{GHZ-Mermin Experiments and symmetries}
Let us consider a system of $n$ qubits labelled by
$1,2,\cdots,n.$ Let $A_j, A'_j$ denote spin observables on the
$j$th qubit, $j = 1, 2, \cdots, n.$ For $A^{(\prime)}_j=
\vec{a}^{(\prime)}_j \cdot \vec{\sigma}_j$ $(1\leq j \leq n),$ we
write$$(A_j,A'_j) = (\vec{a}_j, \vec{a}'_j ), A_j \times A'_j = (
\vec{a}_j \times \vec{a}'_j) \cdot \vec{\sigma}_j.$$Here
$\vec{\sigma}_j = (\sigma^j_x, \sigma^j_y, \sigma^j_z )$ are the
Pauli matrices for the $j$th qubit; the vectors
$\vec{a}^{(\prime)}_j$ are all unit vectors in $\mathbb{R}^3.$ It
is easy to check that\begin{equation}A_j A'_j = (A_j,A'_j) + i A_j
\times A'_j, \tag{2.1}\label{eq:2-1}\end{equation}\begin{equation}
A'_jA_j = (A_j, A'_j) - i A_j \times A'_j,
\tag{2.2}\label{eq:2-2}\end{equation}\begin{equation}\| A_j \times
A'_j \|^2=1- (A_j, A'_j)^2. \tag{2.3}\label{eq:2-3}
\end{equation}Also, $A_j \times A'_j = 0$ if and only if
$A_j = \pm A'_j,$ i.e., $A_j$ is parallel to $A'_j;$ $(A_j,A'_j) =
0$ if and only if $A_j$ is orthogonal to $A'_j,$ denoted by $A_j
\perp A'_j.$
We write $A^{(\prime)}_1 \cdots A^{(\prime)}_n,$ etc., as
shorthand for $A^{(\prime)}_1 \otimes \cdots \otimes
A^{(\prime)}_n.$ The following lemma clarifies the inner structure
of mutually commuting nonlocal spin observables of the $n$-qubit
system.
{\it Lemma: Two nonlocal $n$-qubit spin observables $A_1 \cdots
A_n$ and $A'_1 \cdots A'_n$ are commuting if and only if for every
$j = 1, 2, \cdots, n,$ $A_j$ is either parallel or orthogonal to
$A'_j,$ and the number of sites at which the corresponding local
spin observables are orthogonal to each other is even.}
{\it Proof.}~~The sufficiency is clear. Indeed, by Eqs.(2.1) and
(2.2), we have that $A_j A'_j = - A'_j A_j$ whenever $(A_j,A'_j) =
0.$ Since the number of elements of $\{j: A_j \perp A'_j \}$ is
even, it is immediately concluded that $A_1 \cdots A_n$ and $A'_1
\cdots A'_n$ are commuting.
To prove the necessity, suppose that $A_1 \cdots A_n$ and $A'_1
\cdots A'_n$ are commuting. For every unit vector $|u_1 \rangle
\otimes \cdots \otimes |u_n \rangle,$ one has\begin{equation*}
\prod_{j = 1}^n \| A_j A'_j |u_j \rangle \|^2 = \prod_{j = 1}^n
\langle A_j A'_j u_j |A'_j A_j u_j \rangle.\end{equation*}By
Eqs.(2.1) and (2.2), we have
\begin{eqnarray*}
\|A_j A'_j |u_j \rangle\|^{2} = (A_j, A'_j )^2 + \|A_j \times A'_j
|u_j \rangle\|^2,\end{eqnarray*}
\begin{eqnarray*} \langle A_j A'_j
u_j |A'_j A_j u_j \rangle = (A_j, A'_j )^2-\|A_j \times A'_j |u_j
\rangle \|^2
\\[0.4cm]
- 2i (A_j, A'_j ) \langle u_j |A_j \times A'_j | u_j \rangle.
\end{eqnarray*}Note that, if $A_j \times A'_j \neq 0,$ there correspond to two
eigenvalues $\pm \|A_j \times A'_j \|$ with the corresponding unit
eigenvectors $| u^{\pm}_j \rangle.$ In this case, we set $ | u_j
\rangle = ( | u^+_j \rangle + | u^-_j \rangle ) / \sqrt{2}$ and
obtain $\langle u_j |A_j \times A'_j |u_j \rangle = 0.$ Hence, we
have\begin{eqnarray*}\label{l} \prod_{j=1}^n \left [ (A_j, A'_j
)^{2} + \|A_j \times A'_j |u_j \rangle \|^2 \right ]\\[0.4cm]
= \prod_{j = 1}^n \left [ (A_j, A'_j )^2 -\|A_j \times A'_j |u_j
\rangle \|^2 \right ].\end{eqnarray*}
This immediately concludes
that either $A_j \times A'_j = 0$ or $(A_j,A'_j) = 0$ for each $j
= 1, 2, \cdots, n.$ On the other hand, by Eqs.(2.1) and (2.2) we
have that $A_j A'_j = - A'_j A_j$ whenever $(A_j,A'_j) = 0.$
Therefore, the number of elements of $A_j \perp A'_j$ is even.
The Lemma tells us that two commuting nonlocal spin observables of
the $n$-qubit system have a nice structure, which has been used to
clarify the geometric structure of GHZ-Mermin experiments of both
two-qubit and three-qubit systems in \cite{Chen}. For convenience,
we reformulate the Lemma in the case of four qubits that two
four-qubit nonlocal spin observables $A_1 A_2 A_3 A_4$ and $A'_1
A'_2 A'_3 A'_4$ are commuting if and only if one of the following
conditions is satisfied:
\begin{eqnarray*}
(1)&&A_{1}=\pm A'_1, A_2 = \pm A'_2, A_3 = \pm A'_3, A_4 = \pm
A'_4;
\\
(2)&&A_1 = \pm A'_1, A_2 = \pm A'_2, (A_3, A'_3 ) = (A_4, A'_4) =
0;
\\
(3)&&A_1 = \pm A'_1, A_3 = \pm A'_3, (A_2, A'_2 ) = (A_4, A'_4) =
0;
\\
(4)&&A_1 = \pm A'_1, A_4 = \pm A'_4, (A_2, A'_2) = (A_3, A'_3) =
0;
\\
(5)&&A_2 = \pm A'_2, A_3 = \pm A'_3, (A_1, A'_1) = (A_4, A'_4 ) =
0;
\\
(6)&&A_2 = \pm A'_2, A_4 = \pm A'_4, (A_1, A'_1) = (A_3, A'_3) =
0;
\\
(7)&&A_3 = \pm A'_3, A_4 = \pm A'_4, (A_1, A'_1) = (A_2, A'_2) =
0;
\\
(8)&&(A_1, A'_1) = (A_2, A'_2) = (A_3, A'_3) = (A_4, A'_4) = 0.
\end{eqnarray*}This concludes that\begin{equation}
\{A_1 A_2 A_3 A_4, A'_1 A'_2 A_3 A_4, A_1 A_2 A'_3 A'_4 \},
\tag{2.4}\label{eq:2-4}\end{equation}\begin{equation}
\{A_{1}A_{2}A_{3}A_{4},A'_{1}A'_{2}A_{3}A_{4},A''_{1}A''_{2}A'_{3}A'_{4}\},
\tag{2.5}\label{eq:2-5}\end{equation}and\begin{equation}
\{A_{1}A_{2}A_{3}A_{4},A'_{1}A'_{2}A'_{3}A'_{4},A''_{1}A''_{2}A''_{3}A''_{4}\},
\tag{2.6}\label{eq:2-6}\end{equation} are all GHZ-Mermin
experiments of the four-qubit system, where
$$(A_j,A'_j)=(A_j,A''_j)=(A'_j,A''_j)=0$$for $j=1,2,3.$ In this
article, we need to clarify the geometric structure of GHZ-Mermin
experiments of the four-qubit system.
Browsing through the sets of mutually commuting four-qubit
nonlocal spin observables we quickly get the feeling that there
are many rather similar ones, and also some sets which can be
obtained in a rather trivial way (e.g., add a common element) from
2-qubit and 3-qubit ones. Hence, there are many equivalent
GHZ-Mermin experiments. Here, we describe the grouping of
GHZ-Mermin experiments into ``essentially distinct ones." Some
symmetries acting on GHZ-Mermin experiments are obvious. There are
two basic symmetries leading to equivalent experiments as follows.
$\mathrm{(S_1)}$ Changing the labelling of the local observables
at each site.
$\mathrm{(S_2)}$ Permuting systems.
Here, we define as equivalent two GHZ-Mermin experiments ${\cal
A}$ and ${\cal B}$ if they can be transformed to each other by
symmetrical actions $\mathrm{(S_1)}$ and $\mathrm{(S_2)}$ or local
unitary operations $\mathrm{(LU)}.$ In this case, we denote by
${\cal A} \cong {\cal B}.$ For example,
$$\{A''_1 A''_2 A_3 A_4, A'_1 A'_2 A_3 A_4, A''_1 A''_2 A'_3 A'_4
\} \cong \mathrm{Eq.(2.4)}$$by changing the labelling of the local
observables with $A_1 \longleftrightarrow A''_1$ and $A_2
\longleftrightarrow A''_2.$
Also,$$\{A_{1}A_{2}A_{3}A_{4},A'_{1}A_{2}A_{3}A'_{4},
A''_{1}A'_{2}A'_{3}A''_{4}\} \cong \mathrm{Eq.(2.5)}$$by permuting
the system with qubit 2 $\Longleftrightarrow$ qubit 4.
Since $\textrm{SU}(2) \cong \textrm{SO}(3)$ through $U^{\dagger}
(\vec{a} \vec{\sigma}) U = (R \vec{a}) \vec{\sigma},$ there is a
local unitary transformation $U_j$ on the $j$th qubit such that
$A_j = U^*_j \sigma^j_x U_j, A'_j = U^*_j \sigma^j_y U_j,$ and
$A''_j = U^*_j \sigma^j_z U_j,$ provided
$(A_j,A'_j)=(A_j,A''_j)=(A'_j,A''_j)=0.$ Then, the GHZ-Mermin
experiments Eqs.(2.4)-(2.6) are respectively equivalent
to\begin{equation} \{ \sigma^1_x \sigma^2_x \sigma^3_x \sigma^4_x,
\sigma^1_y \sigma^2_y \sigma^3_x \sigma^4_x, \sigma^1_x \sigma^2_x
\sigma^3_y \sigma^4_y \},
\tag{2.7}\label{eq:2-7}\end{equation}\begin{equation} \{
\sigma^1_x \sigma^2_x \sigma^3_x \sigma^4_x, \sigma^1_y \sigma^2_y
\sigma^3_x \sigma^4_x, \sigma^1_z \sigma^2_z \sigma^3_y \sigma^4_y
\}, \tag{2.8}\label{eq:2-8}\end{equation}and\begin{equation} \{
\sigma^1_x \sigma^2_x \sigma^3_x \sigma^4_x, \sigma^1_y \sigma^2_y
\sigma^3_y \sigma^4_y, \sigma^1_z \sigma^2_z \sigma^3_z \sigma^4_z
\}. \tag{2.9}\label{eq:2-9}
\end{equation}Moreover, since local observables are either
parallel or orthogonal to each other in a GHZ-Mermin experiment of
$n$ qubits by the above Lemma, it then must be equivalent to a
GHZ-Mermin experiment with each local observable taking one of
$\sigma_x, \sigma_y$ and $\sigma_z.$ Therefore, for constructing a
GHZ-Mermin experiment of $n$ qubits we only need to choose
$\sigma_x, \sigma_y,$ or $\sigma_z$ as local observables.
Clearly, Eqs.(2.7)-(2.9) possess different geometric structure.
That is, there are two dichotomic observables per site in
Eq.(2.7), two triads $(\sigma^1_x, \sigma^1_y, \sigma^1_z)$ and
$(\sigma^2_x, \sigma^2_y, \sigma^2_z)$ in Eq.(2.8), and four ones
in Eq.(2.9). Since symmetrical actions $\mathrm{(S_1)}$ and
$\mathrm{(S_2)}$ and local unitary operations $\mathrm{(LU)}$ do
not change the geometric structure of GHZ-Mermin experiments,
Eqs.(2.7)-(2.9) are inequivalent to each other. Generally
speaking, every GHZ-Mermin experiment has two geometric
invariants. On one hand, the number of sites which has a triad is
invariant under $\mathrm{(S_1)}, \mathrm{(S_2)},$ and
$\mathrm{(LU)},$ denoted by $\mathrm{C}.$ Clearly, $\mathrm{C}
\leq n$ for the $n$-qubit system. On the another hand, for every
element of the experiment there corresponds to the number of
elements which are orthogonal to that element at two sites. The
set of those numbers is also invariant under $\mathrm{(S_1)},
\mathrm{(S_2)},$ and $\mathrm{(LU)},$ denoted by $\mathrm{R}.$ For
example, the $\mathrm{C}$ and $\mathrm{R}$ invariants of Eq.(2.7)
are respectively $0$ and $(2, 1, 1),$ the ones of Eq.(2.8) are $2$
and $(1, 1, 0),$ and the ones of Eq.(2.9) are $4$ and $(0, 0, 0).$
Eqs.(2.7)-(2.9) have different geometric invariants. In the
sequel, we show that each four-qubit GHZ-Mermin experiment of
three elements must equivalently be one of the forms
Eqs.(2.7)-(2.9) and hence, two four-qubit GHZ-Mermin experiments
of three elements are equivalent if and only if they have the same
geometric invariants.
To this end, by $\mathrm{(S_1)}$ we have that each GHZ-Mermin
experiment of three elements must be one of the forms
\begin{equation} \{ \sigma^1_x \sigma^2_x \sigma^3_x \sigma^4_x,
\sigma^1_y \sigma^2_y \sigma^3_x \sigma^4_x, \star \star \star
\star \}, \tag{2.10}\label{eq:2-10}\end{equation}\begin{equation}
\{ \sigma^1_x \sigma^2_x \sigma^3_x \sigma^4_x, \sigma^1_y
\sigma^2_y \sigma^3_y \sigma^4_y, \star \star \star \star \},
\tag{2.11}\label{eq:2-11}\end{equation}because\begin{eqnarray*} \{
\sigma^1_x \sigma^2_x \sigma^3_x \sigma^4_x, \sigma^1_y \sigma^2_x
\sigma^3_y \sigma^4_x, \star \star \star \star \},
\\
\{ \sigma^1_x \sigma^2_x \sigma^3_x \sigma^4_x, \sigma^1_y
\sigma^2_x \sigma^3_x \sigma^4_y, \star \star \star \star \},
\\
\{ \sigma^1_x \sigma^2_x \sigma^3_x \sigma^4_x, \sigma^1_x
\sigma^2_y \sigma^3_y \sigma^4_x, \star \star \star \star \},
\\
\{ \sigma^1_x \sigma^2_x \sigma^3_x \sigma^4_x, \sigma^1_x
\sigma^2_y \sigma^3_x \sigma^4_y, \star \star \star \star \},
\\
\{ \sigma^1_x \sigma^2_x \sigma^3_x \sigma^4_x, \sigma^1_x
\sigma^2_x \sigma^3_y \sigma^4_y, \star \star \star \star \},
\end{eqnarray*}
are all equivalent to Eq.(2.10) by $\mathrm{(S_2)}.$ Since there
are at least two distinct observables at each site, Eq.(2.10)
reduces to Eq.(2.7), Eq.(2.8), and\begin{equation} \{ \sigma^1_x
\sigma^2_x \sigma^3_x \sigma^4_x, \sigma^1_y \sigma^2_y \sigma^3_x
\sigma^4_x, \sigma^1_y \sigma^2_y \sigma^3_y \sigma^4_y \}.
\tag{2.12}\label{eq:2-12}\end{equation}However, Eq.(2.12) is
equivalent to Eq.(2.7) by $\mathrm{(S_1)}$ with $\sigma^1_x
\longleftrightarrow \sigma^1_y$ and $\sigma^2_x
\longleftrightarrow \sigma^2_y.$
On the other hand, Eq.(2.11) reduces to Eq.(2.9),
\begin{eqnarray*}\{ \sigma^1_x \sigma^2_x \sigma^3_x
\sigma^4_x, \sigma^1_y \sigma^2_y \sigma^3_y \sigma^4_y,
\sigma^1_z \sigma^2_z \sigma^3_y \sigma^4_y \} \cong
\mathrm{Eq.(2.8)},\end{eqnarray*}and\begin{eqnarray*}\{ \sigma^1_x
\sigma^2_x \sigma^3_x \sigma^4_x, \sigma^1_y \sigma^2_y \sigma^3_y
\sigma^4_y, \sigma^1_z \sigma^2_z \sigma^3_x \sigma^4_x \} \cong
\mathrm{Eq.(2.8)}. \end{eqnarray*}This concludes the required
result.
\section{Trivial GHZ-Mermin Experiments}
There are many trivial GHZ-Mermin experiments of four qubits. For
example, GHZ-Mermin experiments Eqs.(2.7)-(2.9) are all trivial.
Indeed, we will show in this section that each of Eqs.(2.7)-(2.9)
is included in a GHZ-Mermin experiment of four elements and all
four-qubit GHZ-Mermin experiments of four elements are trivial.
In the following, we write $\sigma^1_x \sigma^2_x \sigma^3_y
\sigma^4_y,$ etc., as shorthand for $x x y y$ or $x_1 x_2 y_3
y_4.$ We characterize all four-qubit GHZ-Mermin experiments of
four elements as follows.
{\it Proposition: A $\mathrm{GHZ-Mermin}$ experiment of four
elements for the four-qubit system must equivalently be one of the
following forms:\begin{equation} \{ xxxx, yyxx, zzxx, xxyy\},
\tag{3.1}\label{eq:3-1}\end{equation}
\begin{equation} \{ xxxx, yyxx, yxyx, xxyy \},
\tag{3.2}\label{eq:3-2}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, yxxy \},
\tag{3.3}\label{eq:3-3}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, zzzy \},
\tag{3.4}\label{eq:3-4}\end{equation}\begin{equation} \{ xxxx,
yyxx, xxyy, yyyy \},
\tag{3.5}\label{eq:3-5}\end{equation}\begin{equation} \{ xxxx,
yyxx, xxyy, zzyy \},
\tag{3.6}\label{eq:3-6}\end{equation}\begin{equation}\{ xxxx,
yyxx, xxyy, zzzz \},
\tag{3.7}\label{eq:3-7}\end{equation}\begin{equation}\{ xxxx,
yyxx, zzyy, zzzz \}.
\tag{3.8}\label{eq:3-8}\end{equation}Moreover, the geometric
invariants of $\mathrm{Eqs.(3.1)-(3.8)}$ are illustrated in
$\mathrm{Table~ I}.$ }
{\it Proof}:~~At first, a GHZ-Mermin experiment of four elements
for the four-qubit system must equivalently be one of the
forms\begin{equation} \{ xxxx, yyxx, \star \star \star \star,
\star \star \star \star \},
\tag{3.9}\label{eq:3-9}\end{equation}\begin{equation} \{ xxxx,
yyyy, \star \star \star \star, \star \star \star \star \},
\tag{3.10}\label{eq:3-10}\end{equation}Then, Eq.(3.9) reduces to
one of the following forms\begin{equation} \{ xxxx, yyxx, zzxx,
\star \star \star \star \},
\tag{3.9-1}\label{eq:3-9-1}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, \star \star \star \star \},
\tag{3.9-2}\label{eq:3-9-2}\end{equation}\begin{equation} \{ xxxx,
yyxx, xxyy, \star \star \star \star \},
\tag{3.9-3}\label{eq:3-9-3}\end{equation}\begin{equation} \{ xxxx,
yyxx, zzyy, \star \star \star \star \},
\tag{3.9-4}\label{eq:3-9-4}\end{equation}because $\{ x x x x, y y
x x, y x x y, \star \star \star \star \},$ $\{ xxxx, yyxx, xyyx,
\star \star \star \star \},$ and $\{ xxxx, yyxx, xyxy, \star \star
\star \star \}$ are all equivalent to Eq.(3.9-2), as well as $\{
xxxx, yyxx, yyyy, \star \star \star \star \} \cong
\mathrm{Eq.(3.9-3)}.$
(1)~~From Eq.(3.9-1) we obtain Eq.(3.1) and\begin{eqnarray*}\{
xxxx, yyxx, zzxx, yyyy\} \cong \mathrm{Eq.(3.1)},
\\
\{ xxxx, yyxx, zzxx, zzyy\} \cong \mathrm{Eq.(3.1)}.
\end{eqnarray*}
(2)~~From Eq.(3.9-2) we obtain Eqs.(3.2)-(3.4)
and\begin{eqnarray*}\{ xxxx, yyxx, yxyx, xyxy \} \cong
\mathrm{Eq.(3.2)},
\\
\{ xxxx, yyxx, yxyx, yyyy \} \cong \mathrm{Eq.(3.2)}.
\end{eqnarray*}
(3)~~From Eq.(3.9-3) we obtain that Eqs.(3.1), (3.2), (3.5)-(3.7),
and\begin{eqnarray*}\{ xxxx, yyxx, xxyy, xxzz \} \cong
\mathrm{Eq.(3.1)},
\\
\{ xxxx, yyxx, xxyy, yxxy \} \cong \mathrm{Eq.(3.2)},
\\
\{ xxxx, yyxx, xxyy, xyyx \} \cong \mathrm{Eq.(3.2)},
\\
\{ xxxx, yyxx, xxyy, xyxy \} \cong \mathrm{Eq.(3.2)},
\\
\{ xxxx, yyxx, xxyy, yyzz \} \cong \mathrm{Eq.(3.6)}.
\end{eqnarray*}
(4)~~From Eq.(3.9-4) we obtain that Eqs.(3.6), (3.8),
and\begin{eqnarray*} \{ xxxx, yyxx, zzyy, zzxx \} \cong
\mathrm{Eq.(3.1)},
\\
\{ xxxx, yyxx, zzyy, yyyy \} \cong \mathrm{Eq.(3.6)},
\\
\{ xxxx, yyxx, zzyy, yyzz \} \cong \mathrm{Eq.(3.7)},
\\
\{ xxxx, yyxx, zzyy, xxzz \} \cong \mathrm{Eq.(3.7)}.
\end{eqnarray*}
On the other hand, since\begin{eqnarray*}\{ xxxx, yyyy, yyxx,
\star \star \star \star \},
\\
\{ xxxx, yyyy, zzxx, \star \star \star \star \},
\\
\{ xxxx, yyyy, zzyy, \star \star \star \star \},
\end{eqnarray*}and their variants are all included in Eq.(3.9), it
is concluded that Eq.(3.10) reduces to \begin{equation} \{ xxxx,
yyyy, zzzz, \star \star \star \star \},
\tag{3.10-1}\label{eq:3-10-1}\end{equation}.From Eq.(3.10-1) we
obtain\begin{eqnarray*}\{ xxxx, yyyy, zzzz, yyxx \} \cong
\mathrm{Eq.(3.7)},
\\
\{ xxxx, yyyy, zzzz, zzxx \} \cong
\mathrm{Eq.(3.7)},
\\
\{ xxxx, yyyy, zzzz, zzyy \} \cong
\mathrm{Eq.(3.7)}.\end{eqnarray*}
The proof is complete.
\begin{table}
\caption{\label{tab:table1}Here, we denote by $j (= 1,2,3,4)$ the
$j$-th element of the experiments. The numbers in the $\mathrm{C}$
column are $\mathrm{C}$-invariants, while the numbers in $1-4$'s
columns are $\mathrm{R}$-invariants.}
\begin{ruledtabular}
\begin{tabular}{cccccc}
& $\mathrm{C}$ & 1 & 2 & 3 & 4 \\
\hline (3.1) & 2 & 3 & 2 & 2 & 1 \\
(3.2) & 0 & 3 & 2 & 3 & 2 \\
(3.3) & 0 & 3 & 3 & 3 & 3 \\
(3.4) & 3 & 2 & 2 & 2 & 0 \\
(3.5) & 0 & 2 & 2 & 2 & 2 \\
(3.6) & 2 & 2 & 1 & 2 & 1 \\
(3.7) & 4 & 2 & 1 & 1 & 0 \\
(3.8) & 4 & 1 & 1 & 1 & 1 \\
\end{tabular}
\end{ruledtabular}
\end{table}
From the above Proposition, it suffices to consider
Eqs.(3.1)-(3.8) for showing that every GHZ-Mermin experiments of
four elements for the four-qubit system is trivial. Let us recall
that the scenario for the GHZ-Mermin proof is the following:
Particles 1, 2, 3, and 4 move away from each other. At a given
time, an observer, Alice, has access to particle 1, a second
observer, Bob, has access to particle 2, a third observer,
Charlie, has access to particle 3, and a fourth observer, Davis,
has access to particle 4. For example, in the case of Eq.(3.5), by
introducing $(\cdot )$ to separate operators that can be viewed as
EPR's local elements of reality and for any common eigenstate
$|\varphi \rangle$ of Eq.(3.5), we have\begin{eqnarray*}
x_1 \cdot x_2 \cdot x_3 \cdot x_4 |\varphi \rangle & =
& \varepsilon_1 |\varphi \rangle,\\
y_1 \cdot y_2 \cdot x_3 \cdot x_4 |\varphi \rangle & =
& \varepsilon_2 |\varphi \rangle,\\
x_1 \cdot x_2 \cdot y_3 \cdot y_4 |\varphi \rangle & =
& \varepsilon_3 |\varphi \rangle,\\
y_1 \cdot y_2 \cdot y_3 \cdot y_4 |\varphi \rangle & = &
\varepsilon_4 |\varphi \rangle,
\end{eqnarray*}where $\varepsilon_j = \pm 1.$
According to EPR's criterion of local realism \cite{EPR}, Eq.(3.5)
allows Alice, Bob, Charlie, and Davis to predict the following
relations between the values of the elements of reality:
\begin{eqnarray*}
\nu( x_1 ) \nu( x_2 ) \nu( x_3 ) \nu( x_4 ) & = & \varepsilon_1,\\
\nu( y_1 ) \nu( y_2 ) \nu( x_3 ) \nu( x_4 ) & = & \varepsilon_2,\\
\nu( x_1 ) \nu( x_2 ) \nu( y_3 ) \nu( y_4 ) & = & \varepsilon_3,\\
\nu( y_1 ) \nu( y_2 ) \nu( y_3 ) \nu( y_4 ) & = & \varepsilon_4.
\end{eqnarray*}Since $ ( x_1 x_2 x_3 x_4 ) \times ( y_1 y_2 x_3 x_4) \times
( x_1 x_2 y_3 y_4 ) \times ( y_1 y_2 y_3 y_4 ) = 1,$ we have that
$\varepsilon_1 \varepsilon_2 \varepsilon_3 \varepsilon_4 = 1.$ In
this case, one can assign values $\nu( x_1 )= \varepsilon_1,$
$\nu( y_1 ) = \varepsilon_2, \nu( y_3 ) = \varepsilon_1
\varepsilon_3,$ and the remaining ones $\nu(\cdot) = 1.$ Thus, the
GHZ-Mermin proof is nullified in the case of Eq.(3.5).
The other cases are illustrated in Table II.
\begin{table}
\caption{\label{tab:table2} In every case, we can assign the
values as in the case of Eq.(3.5), the elements not indicated all
take the value $\nu(\cdot) = 1.$ }
\begin{ruledtabular}
\begin{tabular}{ccccc}
(3.1)& $\nu( x_1 ) = \varepsilon_1$ & $\nu( y_1 ) = \varepsilon_2$
& $ \nu( z_1 ) = \varepsilon_3$ &
$\nu( y_3 ) = \varepsilon_1 \varepsilon_4$\\
(3.2) & $\nu( x_1 ) = \varepsilon_1$ & $\nu( y_2 ) = \varepsilon_2
\varepsilon_3 $ & $\nu( y_1 ) = \varepsilon_3$ &
$\nu( y_4 ) = \varepsilon_1 \varepsilon_4 $\\
(3.3) & $\nu( x_1 ) = \varepsilon_1$ & $\nu( y_2 ) =
\varepsilon_2$ & $\nu( y_3 ) = \varepsilon_3$ &
$\nu( y_4 ) = \varepsilon_4$\\
(3.4) & $\nu( x_1 ) = \varepsilon_1$ & $\nu( y_2 ) =
\varepsilon_2$ & $\nu( y_3 ) = \varepsilon_3$ &
$\nu( z_1 ) = \varepsilon_4$\\
(3.6) & $\nu( x_1 ) = \varepsilon_1$ & $\nu( y_1 ) =
\varepsilon_2$ & $\nu( y_3 ) = \varepsilon_1 \varepsilon_3$ &
$\nu( z_1 ) = \varepsilon_1 \varepsilon_3 \varepsilon_4$\\
(3.7) & $\nu( x_1 ) = \varepsilon_1$ & $\nu( y_1 ) =
\varepsilon_2$ & $\nu( y_3 ) = \varepsilon_1 \varepsilon_3$ &
$\nu( z_1 ) = \varepsilon_4$\\
(3.8) & $\nu( x_1 ) = \varepsilon_1$ & $\nu( y_1 ) =
\varepsilon_2$ & $\nu( y_3 ) = \varepsilon_3$ &
$\nu( z_3 ) = \varepsilon_4$
\end{tabular}
\end{ruledtabular}
\end{table}
Finally, we note that Eq.(2.7) is included in Eq.(3.1), Eq.(2.8)
in Eq.(3.6), and Eq.(2.9) (equivalently) in Eq.(3.7). This
concludes that the four-qubit GHZ-Mermin experiments of three
elements are all trivial. Therefore, a nontrivial GHZ-Mermin
experiment of four qubits must have at least five elements.
\section{Nontrivial GHZ-Mermin Experiments}
In this section, we will present a complete construction of
nontrivial four-qubit GHZ-Mermin experiments of five (6, 7, 8)
elements. We show that the experiments of five (6, 7, 8) elements
possess 11 (9, 5, 3) different forms. It is proved that in each
case there are two nontrivial GHZ-Mermin experiments and, the
associated states exhibiting an ``all versus nothing"
contradiction between quantum mechanics and $\mathrm{EPR}$'s local
realism are $\mathrm{GHZ}$ states.
\subsection{The case of five elements}
We first characterize all four-qubit GHZ-Mermin experiments of
five elements as follows.
{\it Proposition: A $\mathrm{GHZ-Mermin}$ experiment of five
elements for the four-qubit system must equivalently be one of the
following forms:\begin{equation} \{ xxxx, yyxx, zzxx, xxyy, xxzz
\}, \tag{4A.1}\label{eq:4A-1}\end{equation}\begin{equation} \{
xxxx, yyxx, zzxx, xxyy, yyyy \},
\tag{4A.2}\label{eq:4A-2}\end{equation}\begin{equation} \{ xxxx,
yyxx, zzxx, xxyy, yyzz \}, \tag{4A.3}\label{eq:4A-3}\end{equation}
\begin{equation} \{ xxxx, yyxx, yxyx, xxyy, yxxy \},
\tag{4A.4}\label{eq:4A-4}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, xxyy, xyxy \},
\tag{4A.5}\label{eq:4A-5}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, xxyy, zzzz \},
\tag{4A.6}\label{eq:4A-6}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, yxxy, zzzz \},
\tag{4A.7}\label{eq:4A-7}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, zzzy, xyyx \},
\tag{4A.8}\label{eq:4A-8}\end{equation}\begin{equation} \{ xxxx,
yyxx, xxyy, yyyy, zzzz \},
\tag{4A.9}\label{eq:4A-9}\end{equation}\begin{equation} \{ xxxx,
yyxx, xxyy, zzyy, yyzz \},
\tag{4A.10}\label{eq:4A-10}\end{equation}\begin{equation} \{ xxxx,
yyxx, xxyy, zzzz, yyzz \}.
\tag{4A.11}\label{eq:4A-11}\end{equation}Moreover, the geometric
invariants of $\mathrm{Eqs.(4A.1)-(4A.11)}$ are illustrated in
$\mathrm{Table~ III}.$ }
\begin{table}
\caption{\label{tab:table3}The numbers in the $\mathrm{C}$ line
are $\mathrm{C}$-invariants, while the numbers in $1-5$'s lines
are $\mathrm{R}$-invariants.}
\begin{ruledtabular}
\begin{tabular}{cccccccccccc}
$\mathrm{4A}$& 1 & 2 & 3 & 4
& 5 & 6 & 7 & 8 & 9 & 10 & 11\\
\hline $\mathrm{C}$ & 4 & 2 & 4 & 0 & 0 & 4 & 4 & 3 & 4 & 4 & 4 \\
\hline 1 & 4 & 3 & 3 & 4 & 4 & 3 & 3 & 3 & 2 & 2 & 2 \\
2 & 2 & 3 & 3 & 3 & 3 & 2 & 3 & 3 & 2 & 2 & 2 \\
3 & 2 & 2 & 2 & 4 & 3 & 3 & 3 & 3 & 2 & 2 & 1 \\
4 & 2 & 2 & 1 & 3 & 3 & 2 & 3 & 0 & 2 & 1 & 1 \\
5 & 2 & 2 & 1 & 4 & 3 & 0 & 0 & 3 & 0 & 1 & 2 \\
\end{tabular}
\end{ruledtabular}
\end{table}
{\it Proof}:~~By repeating the proof of the Proposition in Section
III, we find that every subset of four elements in a GHZ-Mermin
experiment of five elements for the four-qubit system is a
GHZ-Mermin experiment, i.e., a set of four mutually commuting
nonlocal spin observables with at least two different observables
at each site. By the Proposition in Sec.III, this concludes that a
four-qubit GHZ-Mermin experiment of five elements must
equivalently be one of the forms\begin{equation} \{ xxxx, yyxx,
zzxx, xxyy, \star \star \star \star \},
\tag{4A.12}\label{eq:4A-12}\end{equation}
\begin{equation} \{ xxxx, yyxx, yxyx, xxyy, \star \star \star \star \},
\tag{4A.13}\label{eq:4A-13}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, yxxy, \star \star \star \star \},
\tag{4A.14}\label{eq:4A-14}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, zzzy, \star \star \star \star \},
\tag{4A.15}\label{eq:4A-15}\end{equation}\begin{equation} \{ xxxx,
yyxx, xxyy, yyyy, \star \star \star \star \},
\tag{4A.16}\label{eq:4A-16}\end{equation}\begin{equation} \{ xxxx,
yyxx, xxyy, zzyy, \star \star \star \star \},
\tag{4A.17}\label{eq:4A-17}\end{equation}\begin{equation} \{ xxxx,
yyxx, xxyy, zzzz, \star \star \star \star \},
\tag{4A.18}\label{eq:4A-18}\end{equation}\begin{equation} \{ xxxx,
yyxx, zzyy, zzzz, \star \star \star \star \}.
\tag{4A.19}\label{eq:4A-19}\end{equation}
(1)~~From Eq.(4A.12), we obtain Eqs.(4A.1)-(4A.3),
and\begin{eqnarray*}\{ xxxx, yyxx, zzxx, xxyy, zzyy \} \cong
\mathrm{Eq.(4A.2)},\\[0.3cm]
\{ xxxx, yyxx, zzxx, xxyy, zzzz \} \cong \mathrm{Eq.(4A.3)}.
\end{eqnarray*}
(2)~~From Eq.(4A.13), we obtain Eqs.(4A.4)-(4A.6),
and\begin{eqnarray*}\{ xxxx, yyxx, yxyx, xxyy, xyyx \} \cong
\mathrm{Eq.(4A.4)},\\[0.3cm]
\{ xxxx, yyxx, yxyx, xxyy, yyyy \} \cong \mathrm{Eq.(4A.5)}.
\end{eqnarray*}
(3)~~From Eq.(4A.14), we obtain Eqs.(4A.4), (4A.7),
and\begin{eqnarray*}\{ xxxx, yyxx, yxyx, yxxy, xyyx \} \cong
\mathrm{Eq.(4A.4)},\\[0.3cm]
\{ xxxx, yyxx, yxyx, yxxy, xyxy \} \cong
\mathrm{Eq.(4A.4)},\\[0.3cm]
\{ xxxx, yyxx, yxyx, yxxy, yyyy \} \cong \mathrm{Eq.(4A.4)}.
\end{eqnarray*}
(4)~~From Eq.(4A.15), we obtain Eq.(4A.8) and\begin{eqnarray*}\{
xxxx, yyxx,yxyx, zzzy, yyyz \} \cong
\mathrm{Eq.(4A.6)},\\[0.3cm]
\{ xxxx, yyxx, yxyx, zzzy, yxxz \} \cong
\mathrm{Eq.(4A.7)},\\[0.3cm]
\{ xxxx, yyxx, yxyx, zzzy, xyxz \} \cong
\mathrm{Eq.(4A.6)},\\[0.3cm]
\{ xxxx, yyxx, yxyx, zzzy, xxyz \} \cong \mathrm{Eq.(4A.6)}.
\end{eqnarray*}
(5)~~From Eq.(4A.16), we obtain Eqs.(4A.2), (4A.9),
and\begin{eqnarray*} \{ xxxx, yyxx, xxyy, yyyy, yxyx \} \cong
\mathrm{Eq.(4A.5)},\\[0.3cm]
\{ xxxx, yyxx, xxyy, yyyy, yxxy \} \cong
\mathrm{Eq.(4A.5)},\\[0.3cm]
\{ xxxx, yyxx, xxyy, yyyy, xyyx \} \cong
\mathrm{Eq.(4A.5)},\\[0.3cm]
\{ xxxx, yyxx, xxyy, yyyy, xyxy \} \cong
\mathrm{Eq.(4A.5)},\\[0.3cm]
\{ xxxx, yyxx, xxyy, yyyy, xxzz \} \cong
\mathrm{Eq.(4A.2)},\\[0.3cm]
\{ xxxx, yyxx, xxyy, yyyy, yyzz \} \cong
\mathrm{Eq.(4A.2)},\\[0.3cm]
\{ xxxx, yyxx, xxyy, yyyy, zzyy \} \cong \mathrm{Eq.(4A.2)}.
\end{eqnarray*}
(6)~~From Eq.(4A.17), we obtain Eq.(4A.10) and\begin{eqnarray*}\{
xxxx, yyxx, xxyy, zzyy, zzxx \} \cong
\mathrm{Eq.(4A.2)}, \\[0.3cm]
\{ xxxx, yyxx, xxyy, zzyy, xxzz \} \cong
\mathrm{Eq.(4A.3)}, \\[0.3cm]
\{ xxxx, yyxx, xxyy, zzyy, yyyy \} \cong
\mathrm{Eq.(4A.2)}, \\[0.3cm]
\{ xxxx, yyxx, xxyy, zzyy, zzzz \} \cong \mathrm{Eq.(4A.10)}.
\end{eqnarray*}
(7)~~From Eq.(4A.18), we obtain Eqs.(4A.6), (4A.9), (4A.11),
and\begin{eqnarray*}\{ xxxx, yyxx, xxyy, zzzz, yxxy \} \cong
\mathrm{Eq.(4A.6)}, \\[0.3cm]
\{ xxxx, yyxx, xxyy, zzzz, xyyx \} \cong
\mathrm{Eq.(4A.6)}, \\[0.3cm]
\{ xxxx, yyxx, xxyy, zzzz, xyxy \} \cong
\mathrm{Eq.(4A.6)}, \\[0.3cm]
\{ xxxx, yyxx, xxyy, zzzz, zzxx \} \cong
\mathrm{Eq.(4A.3)}, \\[0.3cm]
\{ xxxx, yyxx, xxyy, zzzz, zzyy \} \cong \mathrm{Eq.(4A.11)},
\\[0.3cm]
\{ xxxx, yyxx, xxyy, zzzz, xxzz \} \cong
\mathrm{Eq.(4A.3)}.
\end{eqnarray*}
(8)~~From Eq.(4A.19), we obtain
\begin{eqnarray*}\{ xxxx, yyxx, zzyy, zzzz, zzxx \} \cong
\mathrm{Eq.(4A.1)}, \\[0.3cm]
\{ xxxx, yyxx, zzyy, zzzz, xxyy \} \cong
\mathrm{Eq.(4A.11)}, \\[0.3cm]
\{ xxxx, yyxx, zzyy, zzzz, xxzz \} \cong
\mathrm{Eq.(4A.11)}, \\[0.3cm]
\{ xxxx, yyxx, zzyy, zzzz, yyzz \} \cong
\mathrm{Eq.(4A.11)}, \\[0.3cm]
\{ xxxx, yyxx, xxyy, zzzz, yyyy \} \cong
\mathrm{Eq.(4A.11)}.
\end{eqnarray*}
From Table III we find that except for Eqs.(4A.10) and (4A.11),
each of Eqs.(4A.1)-(4A.11) has different geometric invariants and
hence, they are inequivalent. In order to distinguish Eq.(4A.10)
from Eq.(4A.11), we need to use other geometric invariants. Note
that, for every subset of three elements in a GHZ-Mermin
experiment there corresponds the number of sites at which there is
a triad. Those numbers are invariant under $\mathrm{(S_1)}$ and
$\mathrm{(S_2)}.$ For example, $\{xxxx, yyzz, zzyy\}$ in
Eq.(4A.10) has four triads, while $\{xxxx, zzzz, yyzz\}$ in
Eq.(4A.11) has two triads. It is evident that there is no subset
of three elements in Eq.(4A.11) possessing four triads. This
concludes that Eqs.(4A.10) and (4A.11) are inequivalent. The proof
is complete.
\begin{table}
\caption{\label{tab:table4} The elements not indicated all take
the value $\nu(\cdot) = 1.$ }
\begin{ruledtabular}
\begin{tabular}{cccc}
(4A.1)& $\nu( x_1 ) = \varepsilon_1$ & $\nu( y_1 ) =
\varepsilon_2$ & $ \nu( z_1 ) = \varepsilon_3$\\
& $\nu( y_3 ) = \varepsilon_1 \varepsilon_4$ &
$\nu( z_3 ) = \varepsilon_1 \varepsilon_5 $ &\\
\hline (4A.3) & $\nu( x_1 ) = \varepsilon_1$ & $\nu( y_1 ) =
\varepsilon_2 $ & $\nu( z_1 ) = \varepsilon_3$\\
& $\nu( y_3 ) = \varepsilon_1 \varepsilon_4 $ &
$\nu( z_3 ) = \varepsilon_2 \varepsilon_5$ & \\
\hline (4A.6) & $\nu( x_1 ) = \varepsilon_1$ & $\nu( y_2 ) =
\varepsilon_2$ & $\nu( y_3 ) = \varepsilon_3$\\
& $\nu( y_4 ) = \varepsilon_1 \varepsilon_3 \varepsilon_4$ &
$\nu( z_1 ) = \varepsilon_5$ & \\
\hline (4A.7) & $\nu( x_1 ) = \varepsilon_1$ & $\nu( y_2 ) =
\varepsilon_2$ & $\nu( y_3 ) = \varepsilon_3$ \\
& $\nu( y_4 ) = \varepsilon_4$ & $\nu( z_1 ) = \varepsilon_5$ &\\
\hline (4A.10) & $\nu( x_1 ) = \varepsilon_1$ & $\nu( y_1 ) =
\varepsilon_2$ & $\nu( y_3 ) = \varepsilon_1 \varepsilon_3$ \\
& $\nu( z_1 ) = \varepsilon_1 \varepsilon_3 \varepsilon_4$ &
$\nu( z_3 ) = \varepsilon_2 \varepsilon_5$ &\\
\hline (4A.11) & $\nu( x_1 ) = \varepsilon_1$ & $\nu( y_1 ) =
\varepsilon_2$ & $\nu( y_3 ) = \varepsilon_1 \varepsilon_3$ \\
& $\nu( z_1 ) = \varepsilon_2 \varepsilon_4 \varepsilon_5$ & $\nu(
z_3 ) = \varepsilon_2 \varepsilon_5$ &
\end{tabular}
\end{ruledtabular}
\end{table}
It is easy to see that Eqs.(4A.1), (4A.3), (4A.6), (4A.7),
(4A.10), and (4A.11) are trivial, whose assigned values are
illustrated in Table IV. As follows, we show that Eqs.(4A.2),
(4A.5), and (4A.9) are also trivial. Indeed, we note that $(x_1
x_2 x_3 x_4) \times (y_1 y_2 x_3 x_4) \times (x_1 x_2 y_3 y_4)
\times (y_1 y_2 y_3 y_4) = 1.$ This concludes that for Eq.(4A.2),
$\varepsilon_1 \varepsilon_2 \varepsilon_4 \varepsilon_5 = 1$ and
thus, one can assign $\nu (x_1) = \varepsilon_1, \nu (y_1) =
\varepsilon_2, \nu (z_1) = \varepsilon_3, \nu (y_3) =
\varepsilon_1 \varepsilon_4,$ and the remaining ones $\nu ( \cdot
) =1.$ Similarly, since $(y_1 y_2 x_3 x_4) \times (y_1 x_2 y_3
x_4) \times (x_1 y_2 x_3 y_4) \times (x_1 x_2 y_3 y_4) = 1$ in
Eq.(4A.5) and $(x_1 x_2 x_3 x_4) \times (y_1 y_2 x_3 x_4) \times
(x_1 x_2 y_3 y_4) \times (y_1 y_2 y_3 y_4) = 1$ in Eq.(4A.9)
respectively, it is easily concluded that Eqs.(4A.5) and (4A.9)
are both trivial.
In the sequel, we prove that both Eqs.(4A.4) and (4A.8) are
nontrivial, and the associated states exhibiting 100\% violation
between quantum mechanics and EPR's local realism are GHZ states.
{\it Theorem: Nontrivial $\mathrm{GHZ-Mermin}$ experiments of five
elements for the four-qubit system must equivalently be either
$\mathrm{Eq.(4A.4)}$ or $\mathrm{Eq.(4A.8)}.$ Moreover, the
associated states exhibiting an ``all versus nothing"
contradiction between quantum mechanics and $\mathrm{EPR}$'s local
realism are $\mathrm{GHZ}$ states.}
{\it Proof}.~~At first, for $| \varphi \rangle =
\frac{1}{\sqrt{2}} \left ( |0000 \rangle - |1111 \rangle \right )$
we have\begin{equation} x_1 x_2 x_3 x_4 |\varphi \rangle = - |
\varphi \rangle,
\tag{4A.19}\label{eq:4A-19}\end{equation}\begin{equation} y_1 y_2
x_3 x_4 | \varphi \rangle = | \varphi \rangle,
\tag{4A.20}\label{eq:4A-20}\end{equation}\begin{equation} y_1 x_2
y_3 x_4 | \varphi \rangle = | \varphi \rangle,
\tag{4A.21}\label{eq:4A-21}\end{equation}\begin{equation} x_1 x_2
y_3 y_4 | \varphi \rangle = | \varphi \rangle,
\tag{4A.22}\label{eq:4A-22}\end{equation}\begin{equation} y_1 x_2
x_3 y_4 | \varphi \rangle = | \varphi \rangle.
\tag{4A.23}\label{eq:4A-23}\end{equation}By the GHZ-Mermin
argument based on EPR's local realism, one has\begin{equation} \nu
(x_1) \nu (x_2) \nu (x_3) \nu (x_4) = - 1,
\tag{4A.24}\label{eq:4A-24}\end{equation}\begin{equation} \nu
(y_1) \nu (y_2) \nu (x_3) \nu (x_4) = 1,
\tag{4A.25}\label{eq:4A-25}\end{equation}\begin{equation} \nu
(y_1) \nu (x_2) \nu (y_3) \nu (x_4) = 1,
\tag{4A.26}\label{eq:4A-26}\end{equation}\begin{equation} \nu
(x_1) \nu (x_2) \nu (y_3) \nu (y_4) = 1,
\tag{4A.27}\label{eq:4A-27}\end{equation}\begin{equation} \nu
(y_1) \nu (x_2) \nu (x_3) \nu (y_4) = 1.
\tag{4A.28}\label{eq:4A-28}\end{equation}However,
Eqs.(4A.24)-(4A.28) are inconsistent, because when we take the
product of Eqs.(4A.24) and (4A.26)-(4A.28), the value of the
left-hand side is one, while the right-hand side is $-1.$ This
concludes that Eq.(4A.4) is nontrivial.
Although the inconsistence of Eqs.(4A.24)-(4A.28) is concluded
from Eqs.(4A.24) and (4A.26)-(4A.28), the subset of $\{ xxxx,
yxyx, xxyy, yxxy\}$ in Eq.(4A.4) is not a GHZ-Mermin experiment of
four qubits at whose second site there is only one measurement. On
the other hand, there are some states other than GHZ's states
satisfying Eqs.(4A.19) and (4A.21)-(4A.23), such as $| \psi
\rangle = a \left ( | 0000 \rangle - | 1111 \rangle \right ) + b
\left ( | 0100 \rangle - | 1011 \rangle \right )$ with $|a |^2 +
|b |^2 = 1/2.$ However, we will show that the states exhibiting
the GHZ-Mermin proof in Eq.(4A.4) are the GHZ states. Thus,
Eq.(4A.20) and so $yyxx$ plays a crucial role in the GHZ-Mermin
experiment Eq.(4A.4).
Similarly, we can prove that Eq.(4A.8) is also nontrivial and omit
the details. In the sequel, we prove that the GHZ state is the
unique state with equivalence up to a local unitary transformation
which presents the GHZ-Mermin proof in both Eqs.(4A.4) and (4A.8).
To this end, we consider the generic form of Eq.(4A.8) and,
suppose $|\varphi \rangle$ is the common eigenstate of five
commuting nonlocal spin observables such that
\begin{equation}
A_1 A_2 A_3 A_4 |\varphi\rangle = \varepsilon_1 |\varphi\rangle,
\tag{4A.29}\label{eq:4A-29}\end{equation}\begin{equation} A'_1
A'_2 A_3 A_4 |\varphi\rangle = \varepsilon_2 |\varphi\rangle,
\tag{4A.30}\label{eq:4A-30}\end{equation}\begin{equation} A'_1 A_2
A'_3 A_4 |\varphi\rangle = \varepsilon_3 |\varphi\rangle,
\tag{4A.31}\label{eq:4A-31}\end{equation}\begin{equation} A_1 A'_2
A'_3 A_4 |\varphi\rangle = \varepsilon_4 |\varphi\rangle,
\tag{4A.32}\label{eq:4A-32}\end{equation}\begin{equation} A''_1
A''_2 A''_3 A'_4 |\varphi\rangle = \varepsilon_5 |\varphi\rangle,
\tag{4A.33}\label{eq:4A-33}\end{equation}where $(A, A') = (A, A'')
= (A', A'') =0.$ According to GHZ-Mermin's analysis based on EPR's
local realism, it is concluded that
\begin{equation}
\nu (A_1 ) \nu (A_2) \nu (A_3) \nu (A_4) =
\varepsilon_1,\tag{4A.34}\label{eq:4A-34}\end{equation}\begin{equation}
\nu(A'_1) \nu(A'_2) \nu(A_3) \nu(A_4) = \varepsilon_2,
\tag{4A.35}\label{eq:4A-35}\end{equation}\begin{equation}
\nu(A'_1) \nu(A_2) \nu(A'_3) \nu(A_4) = \varepsilon_3,
\tag{4A.36}\label{eq:4A-36}\end{equation}\begin{equation} \nu(A_1)
\nu(A'_2) \nu(A'_3) \nu(A_4) = \varepsilon_4,
\tag{4A.37}\label{eq:4A-37}\end{equation}\begin{equation}
\nu(A''_1) \nu(A''_2) \nu(A''_3) \nu(A'_4) = \varepsilon_5.
\tag{4A.38}\label{eq:4A-38}\end{equation}When $\varepsilon_1
\varepsilon_2 \varepsilon_3 \varepsilon_4 = 1,$ one can assign
$\nu(A_1) = \varepsilon_1, \nu(A'_2) = \varepsilon_2, \nu(A'_3) =
\varepsilon_3, \nu(A'_4) = \varepsilon_5,$ and the remaining ones
$\nu(\cdot)=1.$ Therefore, the necessary condition for $|\varphi
\rangle$ presenting a GHZ-Mermin-type proof is
\begin{equation}
\varepsilon_1 \varepsilon_2 \varepsilon_3 \varepsilon_4 = - 1.
\tag{4A.39}\label{eq:4A-39}\end{equation}
On the other hand, suppose Eq.(4A.39) holds, then it is impossible
to assign values, either 1 or -1, that satisfy Eqs.(4A.35)-(4A.38)
because when take the product of Eqs.(4A.35)-(4A.38), the value of
the left hand is equal to $\varepsilon_5$ while the right hand is
$-\varepsilon_5.$ Thus the condition Eq.(4A.39) is the necessary
and sufficient condition for $|\varphi\rangle$ presenting a
GHZ-Mermin-type proof. In this case, by changing the signs of
local observables $A_j$ and $A'_j$ ($A_1 \rightarrow -
\varepsilon_1 A_1, A'_2 \rightarrow \varepsilon_2 A'_2, A'_3
\rightarrow \varepsilon_3 A'_3,$ and $A'_4 \rightarrow
\varepsilon_5 A'_4$), we have that
\begin{equation}
A_1 A_2 A_3 A_4 |\varphi \rangle = -|\varphi \rangle,
\tag{4A.40}\label{eq:4A-40}\end{equation}\begin{equation} A'_1
A'_2 A_3 A_4 |\varphi \rangle = |\varphi \rangle,
\tag{4A.41}\label{eq:4A-41}\end{equation}\begin{equation} A'_1 A_2
A'_3 A_4 |\varphi \rangle = |\varphi \rangle,
\tag{4A.42}\label{eq:4A-42}\end{equation}\begin{equation} A_1 A'_2
A'_3 A_4 |\varphi \rangle = |\varphi \rangle,
\tag{4A.43}\label{eq:4A-43}\end{equation}\begin{equation} A''_1
A''_2 A''_3 A'_4 |\varphi \rangle = |\varphi\rangle.
\tag{4A.44}\label{eq:4A-44}\end{equation}
By Eqs.(2.1)-(2.3), one has that
\begin{eqnarray*}
A_j A'_j = -A'_j A_j = i A''_j,\\[0.3cm]
A'_j A''_j = - A''_j A'_j = i A_j,\\[0.3cm]
A''_j A_j = -A_j A''_j = i A'_j,\\[0.3cm]
A^2_j = (A'_j )^2 = (A''_j )^2 = 1.
\end{eqnarray*}
Hence, $A_j, A'_j,$ and $A''_j$ satisfy the algebraic identities
of Pauli's matrices \cite{Pauli}. Therefore, choosing $A''_j$
representation $\{|0 \rangle_j,|1 \rangle_j \},$ i.e., $ A''_j |0
\rangle_j = |0 \rangle_j, A''_j |1 \rangle_j = - |1 \rangle_j,$ we
have that
\begin{eqnarray*}A_j |0 \rangle_j = e^{-i\alpha_j} |1 \rangle_j,
~~A_j |1 \rangle_j = e^{i\alpha_j} |0 \rangle_j,\\[0.3cm]
A'_j |0 \rangle_j =i e^{-i\alpha_j} |1 \rangle_j,~~ A'_j |1
\rangle_j = -ie^{i\alpha_j} |0 \rangle_j,
\end{eqnarray*}where $0 \leq \alpha_j \leq 2 \pi.$ We write
$|0100 \rangle,$ etc., as shorthand for $|0 \rangle_1 \otimes |1
\rangle_2 \otimes |0 \rangle_3 \otimes |0 \rangle_4.$ Since
$\{|\epsilon_1 \epsilon_2 \epsilon_3 \epsilon_4 \rangle :
\epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4 =0,1 \}$ is an
orthogonal basis of the four-qubit system. We can uniquely write:
\begin{eqnarray*}
|\varphi \rangle = \sum_{\epsilon_1, \epsilon_2, \epsilon_3,
\epsilon_4 =0,1} \lambda_{\epsilon_1 \epsilon_2 \epsilon_3
\epsilon_4 }|\epsilon_1 \epsilon_2 \epsilon_3 \epsilon_4 \rangle
\end{eqnarray*}
with $\sum|\lambda_{\epsilon_1 \epsilon_2 \epsilon_3
\epsilon_4}|^2 = 1.$ We define the four-qubit operator
\begin{eqnarray*}
\mathcal{B} = -A_1 A_2 A_3 A_4 + A'_1 A'_2 A_3 A_4\\
+ A'_1 A_2 A'_3 A_4+ A_1 A'_2 A'_3 A_4.
\end{eqnarray*}
Then by Eqs.(4A.40)-(4A.43), one has that $\mathcal{B} |\varphi
\rangle = 4|\varphi \rangle.$ This conclude that
\begin{equation}
\mathcal{B}^2 |\varphi \rangle = 16 |\varphi \rangle.
\tag{4A.45}\label{eq:4A-45}\end{equation}However, a simple
computation yields that
\begin{eqnarray*}
\mathcal{B}^2 = 4 + 4 (A''_1 A''_2 + A''_1 A''_3 + A''_2 A''_3 ).
\end{eqnarray*}
Then, by using Eq.(4A.45) we conclude that
\begin{eqnarray*}
|\varphi \rangle = a |0000 \rangle + b |0001 \rangle + c |1110
\rangle + d |1111 \rangle
\end{eqnarray*}where $a = \lambda_{0000 }, b = \lambda_{0001},
c = \lambda_{1110},$ and $d = \lambda_{1111}.$ From Eqs.(4A.40)
and (4A.44) it is concluded that $a = -d e^{i ( \alpha_1 + \alpha_2
+ \alpha_3 + \alpha_4) }, b = -id e^{i ( \alpha_1 + \alpha_2 +
\alpha_3 )},$ and $ c = i d e^{i \alpha_4}.$ Therefore
\begin{eqnarray*}
|\varphi \rangle = -d e^{i ( \alpha_1 + \alpha_2 + \alpha_3 )}(
e^{i \alpha_4} |0000 \rangle + i |0001 \rangle ) \\
+ d ( i e^{i
\alpha_4} |1110 \rangle + |1111 \rangle )\\
= \frac{1}{\sqrt{2}} ( e^{i \theta} | 000 \rangle | u \rangle +
e^{i \phi}| 111 \rangle | v \rangle )
\end{eqnarray*}where $ | u \rangle = \frac{1}{\sqrt{2}} ( e^{i \alpha_4} | 0 \rangle + i
| 1 \rangle ), | v \rangle = \frac{1}{\sqrt{2}} ( i e^{i
\alpha_4} | 0 \rangle + | 1 \rangle),$ and $0 \leq \theta, \phi
\leq 2 \pi.$ Since $\langle u | v \rangle =0,$ $|\varphi \rangle$
is a GHZ state.
The proof for Eq.(4A.4) is similar and omitted.
\subsection{The case of six elements}
We first characterize all four-qubit GHZ-Mermin experiments of six
elements as follows.
{\it Proposition: A $\mathrm{GHZ-Mermin}$ experiment of six
elements for the four-qubit system must equivalently be one of the
following forms:\begin{equation} \{ xxxx, yyxx, zzxx, xxyy, xxzz,
yyyy \}, \tag{4B.1}\label{eq:4B-1}\end{equation}\begin{equation}
\{ xxxx, yyxx, zzxx, xxyy, yyyy, zzyy \},
\tag{4B.2}\label{eq:4B-2}\end{equation}\begin{equation} \{ xxxx,
yyxx, zzxx, xxyy, yyyy, zzzz \},
\tag{4B.3}\label{eq:4B-3}\end{equation}\begin{equation} \{ xxxx,
yyxx, zzxx, xxyy, yyzz, zzzz \},
\tag{4B.4}\label{eq:4B-4}\end{equation}
\begin{equation} \{ xxxx, yyxx, yxyx, xxyy, yxxy, xyyx \},
\tag{4B.5}\label{eq:4B-5}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, xxyy, yxxy, zzzz \},
\tag{4B.6}\label{eq:4B-6}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, xxyy, xyxy, yyyy \},
\tag{4B.7}\label{eq:4B-7}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, xxyy, xyxy, zzzz \},
\tag{4B.8}\label{eq:4B-8}\end{equation}\begin{equation} \{ xxxx,
yyxx, xxyy, zzyy, yyzz, zzzz \}.
\tag{4B.9}\label{eq:4B-9}\end{equation}Moreover, the geometric
invariants of $\mathrm{Eqs.(4B.1)-(4B.9)}$ are illustrated in
$\mathrm{Table~ V}.$ }
\begin{table}
\caption{\label{tab:table5}The numbers in the $\mathrm{C}$ line
are $\mathrm{C}$-invariants, while the numbers in $1-6$'s lines
are $\mathrm{R}$-invariants.}
\begin{ruledtabular}
\begin{tabular}{cccccccccc}
$\mathrm{4B}$& 1 & 2 & 3 & 4
& 5 & 6 & 7 & 8 & 9\\
\hline $\mathrm{C}$ & 4 & 2 & 4 & 4 & 0 & 4 & 0 & 4 & 4\\
\hline 1 & 4 & 3 & 3 & 3 & 5 & 4 & 4 & 4 & 2 \\
2 & 3 & 3 & 3 & 3 & 4 & 3 & 4 & 3 & 2 \\
3 & 2 & 3 & 3 & 3 & 5 & 4 & 4 & 3 & 2 \\
4 & 3 & 3 & 2 & 1 & 4 & 3 & 4 & 3 & 2 \\
5 & 2 & 3 & 2 & 2 & 4 & 4 & 4 & 3 & 2 \\
6 & 2 & 3 & 1 & 2 & 4 & 0 & 4 & 0 & 2 \\
\end{tabular}
\end{ruledtabular}
\end{table}
{\it Proof}:~~By the same argument in the case of five elements,
it is concluded that a subset of five elements in a GHZ-Mermin
experiment of six elements for the four-qubit system is a
four-qubit GHZ-Mermin experiment of five elements. Then, by the
Proposition in Section IV.A, a four-qubit GHZ-Mermin experiment of
six elements must equivalently be one of the forms\begin{equation}
\{ xxxx, yyxx, zzxx, xxyy, xxzz, \star \star \star \star \},
\tag{4B.10}\label{eq:4B-10}\end{equation}\begin{equation} \{ xxxx,
yyxx, zzxx, xxyy, yyyy, \star \star \star \star \},
\tag{4B.11}\label{eq:4B-11}\end{equation}\begin{equation} \{ xxxx,
yyxx, zzxx, xxyy, yyzz, \star \star \star \star \},
\tag{4B.12}\label{eq:4B-12}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, xxyy, yxxy, \star \star \star \star \},
\tag{4B.13}\label{eq:4B-13}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, xxyy, xyxy, \star \star \star \star \},
\tag{4B.14}\label{eq:4B-14}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, xxyy, zzzz, \star \star \star \star \},
\tag{4B.15}\label{eq:4B-15}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, yxxy, zzzz, \star \star \star \star \},
\tag{4B.16}\label{eq:4B-16}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, zzzy, xyyx, \star \star \star \star \},
\tag{4B.17}\label{eq:4B-17}\end{equation}\begin{equation} \{ xxxx,
yyxx, xxyy, yyyy, zzzz, \star \star \star \star \},
\tag{4B.18}\label{eq:4B-18}\end{equation}\begin{equation} \{ xxxx,
yyxx, xxyy, zzyy, yyzz, \star \star \star \star \},
\tag{4B.19}\label{eq:4B-19}\end{equation}\begin{equation} \{ xxxx,
yyxx, xxyy, zzzz, yyzz, \star \star \star \star \}.
\tag{4B.20}\label{eq:4B-20}\end{equation}
(1)~~From Eq.(4B.10) we obtain Eq.(4B.1), and\begin{eqnarray*} \{
xxxx, yyxx, zzxx, xxyy, xxzz, yyzz \} \cong
\mathrm{Eq.(4B.1)}, \\[0.3cm] \{ xxxx, yyxx, zzxx, xxyy,
xxzz, zzyy \} \cong
\mathrm{Eq.(4B.1)}, \\[0.3cm]
\{ xxxx, yyxx, zzxx, xxyy, xxzz, zzzz \} \cong
\mathrm{Eq.(4B.1)}.\end{eqnarray*}
(2)~~From Eq.(4B.11) we obtain Eqs.(4B.2), (4B.3),
and\begin{eqnarray*}\{ xxxx, yyxx, zzxx, xxyy, yyyy, xxzz \} \cong
\mathrm{Eq.(4B.1)}, \\[0.3cm]
\{ xxxx, yyxx, zzxx, xxyy, yyyy, yyzz \} \cong
\mathrm{Eq.(4B.1)}.\end{eqnarray*}
(3)~~From Eq.(4B.12) we obtain Eq.(4B.4), and\begin{eqnarray*}\{
xxxx, yyxx, zzxx, xxyy, yyzz, xxzz \} \cong
\mathrm{Eq.(4B.1)}, \\[0.3cm]
\{ xxxx, yyxx, zzxx, xxyy, yyzz, yyyy \} \cong
\mathrm{Eq.(4B.1)}, \\[0.3cm]
\{ xxxx, yyxx, zzxx, xxyy, yyzz, zzyy \} \cong
\mathrm{Eq.(4B.3)}.\end{eqnarray*}
(4)~~From Eq.(4B.13) we obtain Eqs.(4B.5), (4B.6),
and\begin{eqnarray*} \{ xxxx, yyxx, yxyx, xxyy, yxxy, xyxy \}
\cong \mathrm{Eq.(4B.5)}, \\[0.3cm]
\{ xxxx, yyxx, yxyx, xxyy, yxxy, yyyy \} \cong
\mathrm{Eq.(4B.5)}.\end{eqnarray*}
(5)~~From Eq.(4B.14) we obtain Eqs.(4B.7), (4B.8),
and\begin{eqnarray*} \{ xxxx, yyxx, yxyx, xxyy, xyxy, yxxy \}
\cong \mathrm{Eq.(4B.5)}, \\[0.3cm]
\{ xxxx, yyxx, yxyx, xxyy, xyxy, xyyx \} \cong
\mathrm{Eq.(4B.5)}.\end{eqnarray*}
(6)~~From Eq.(4B.15) we obtain Eqs.(4B.6), (4B.8),
and\begin{eqnarray*} \{ xxxx, yyxx, yxyx, xxyy, zzzz, xyyx \}
\cong \mathrm{Eq.(4B.6)}, \\[0.3cm]
\{ xxxx, yyxx, yxyx, xxyy, zzzz, yyyy \} \cong
\mathrm{Eq.(4B.8)}.\end{eqnarray*}
(7)~~From Eq.(4B.16) we obtain Eq.(4B.6) and\begin{eqnarray*} \{
xxxx, yyxx, yxyx, yxxy, zzzz, xyyx \}
\cong \mathrm{Eq.(4B.6)}, \\[0.3cm]
\{ xxxx, yyxx, yxyx, yxxy, zzzz, xyxy \}
\cong \mathrm{Eq.(4B.6)}, \\[0.3cm]
\{ xxxx, yyxx, yxyx, yxxy, zzzz, yyyy \} \cong
\mathrm{Eq.(4B.6)}.\end{eqnarray*}
(8)~~From Eq.(4B.17) we obtain\begin{eqnarray*} \{ xxxx, yyxx,
yxyx, zzzy, xyyx, yyyz \} \cong \mathrm{Eq.(4B.6)},\\[0.3cm]
\{ xxxx, yyxx, yxyx, zzzy, xyyx, yxxz \}
\cong \mathrm{Eq.(4B.6)},\\[0.3cm]
\{ xxxx, yyxx, yxyx, zzzy, xyyx, xyxz \}
\cong \mathrm{Eq.(4B.6)},
\\[0.3cm]
\{ xxxx, yyxx, yxyx, zzzy, xyyx, xxyz \}
\cong \mathrm{Eq.(4B.6)}.
\end{eqnarray*}
(9)~~From Eq.(4B.18) we obtain Eq.(4B.3) and\begin{eqnarray*} \{
xxxx, yyxx, xxyy, yyyy, zzzz, yxyx \}
\cong \mathrm{Eq.(4B.8)}, \\[0.3cm]
\{ xxxx, yyxx, xxyy, yyyy, zzzz, yxxy \}
\cong \mathrm{Eq.(4B.8)}, \\[0.3cm]
\{ xxxx, yyxx, xxyy, yyyy, zzzz, xyyx \}
\cong \mathrm{Eq.(4B.8)}, \\[0.3cm]
\{ xxxx, yyxx, xxyy, yyyy, zzzz, xyxy \}
\cong \mathrm{Eq.(4B.8)}, \\[0.3cm]
\{ xxxx, yyxx, xxyy, yyyy, zzzz, xxzz \}
\cong \mathrm{Eq.(4B.3)}, \\[0.3cm]
\{ xxxx, yyxx, xxyy, yyyy, zzzz, yyzz \}
\cong \mathrm{Eq.(4B.3)}, \\[0.3cm]
\{ xxxx, yyxx, xxyy, yyyy, zzzz, zzyy \} \cong
\mathrm{Eq.(4B.3)}.\end{eqnarray*}
(10)~~From Eq.(4B.19) we obtain Eq.(4B.9) and\begin{eqnarray*} \{
xxxx, yyxx, xxyy, zzyy, yyzz, zzxx \} \cong \mathrm{Eq.(4B.3)},
\\[0.3cm]
\{ xxxx, yyxx, xxyy, zzyy, yyzz, xxzz \} \cong \mathrm{Eq.(4B.3)},
\\[0.3cm]
\{ xxxx, yyxx, xxyy, zzyy, yyzz, yyyy \} \cong \mathrm{Eq.(4B.1)},
\\[0.3cm] \{ xxxx, yyxx, xxyy,
zzyy, yyzz, xxzz \} \cong \mathrm{Eq.(4B.3)}.\end{eqnarray*}
(11)~~From Eq.(4B.20) we obtain Eqs.(4B.4), (4B.9)
and\begin{eqnarray*} \{ xxxx, yyxx, xxyy, zzzz, yyzz, xxzz \}
\cong \mathrm{Eq.(4B.1)},\\[0.3cm]
\{ xxxx, yyxx, xxyy, zzzz, yyzz, yyyy \} \cong
\mathrm{Eq.(4B.3)}.\end{eqnarray*}
From Table V we find that except for Eqs.(4B.3) and (4B.4), each
of Eqs.(4A.1)-(4A.9) has different geometric invariants and hence,
they are inequivalent. However, $\{xxxx, yyzz, zzyy\}$ in
Eq.(4B.3) has four triads, while there is no subset of three
elements in Eq.(4B.4) possessing four triads, as noted in the case
of Eqs.(4A.10) and (4A.11) this concludes that Eqs.(4B.3) and
(4B.4) are inequivalent. The proof is complete.
{\it Theorem: Nontrivial $\mathrm{GHZ-Mermin}$ experiments of six
elements for the four-qubit system must equivalently be either
$\mathrm{Eq.(4B.5)}$ or $\mathrm{Eq.(4B.6)}.$ Moreover, the
associated states exhibiting an ``all versus nothing"
contradiction between quantum mechanics and $\mathrm{EPR}$'s local
realism are $\mathrm{GHZ}$ states.}
{\it Proof}.~~By the above Proposition, it suffices to show that
Eqs.(4B.1)-(4B.4) and (4B.7)-(4B.9) are all trivial, while
Eqs.(4B.5) and (4B.6) are both nontrivial.
(1)~~Since $(xxxx)\times (yyxx) \times (xxyy) \times (yyyy) = 1,$
for Eq.(4B.1) we have $\varepsilon_1 \varepsilon_2 \varepsilon_4
\varepsilon_6 =1.$ Then, one can assign $\nu (x_1) =
\varepsilon_1, \nu (y_1) = \varepsilon_2, \nu (z_1) =
\varepsilon_3, \nu (y_3) = \varepsilon_1 \varepsilon_4, \nu (z_3)
= \varepsilon_1 \varepsilon_5,$ and the remaining ones $v( \cdot )
=1.$
(2)~~Since $(xxxx)\times (yyxx) \times (xxyy) \times (yyyy) = 1$
and $(xxxx)\times (zzxx) \times (xxyy) \times (zzyy) = 1,$ for
Eq.(4B.2) we have $\varepsilon_1 \varepsilon_2 \varepsilon_4
\varepsilon_5 =1$ and $\varepsilon_1 \varepsilon_3 \varepsilon_4
\varepsilon_6 =1,$ respectively. Then, one can assign $\nu (x_1) =
\varepsilon_1, \nu (y_1) = \varepsilon_2, \nu (z_1) =
\varepsilon_3, \nu (y_3) = \varepsilon_1 \varepsilon_4,$ and the
remaining ones $v( \cdot ) =1.$
(3)~~Since $(xxxx)\times (yyxx) \times (xxyy) \times (yyyy) = 1,$
for Eq.(4B.3) we have $\varepsilon_1 \varepsilon_2 \varepsilon_4
\varepsilon_5 = 1.$ Then, one can assign $\nu (x_1) =
\varepsilon_1, \nu (y_1) = \varepsilon_2, \nu (z_1) =
\varepsilon_3, \nu (y_3) = \varepsilon_1 \varepsilon_4, \nu (z_3)
= \varepsilon_3 \varepsilon_6,$ and the remaining ones $v( \cdot )
=1.$
(4)~~Since $(yyxx)\times (zzxx) \times (yyzz) \times (zzzz) = 1,$
for Eq.(4B.4) we have $\varepsilon_2 \varepsilon_3 \varepsilon_5
\varepsilon_6 = 1.$ Then, one can assign $\nu (x_1) =
\varepsilon_1, \nu (y_1) = \varepsilon_2, \nu (z_1) =
\varepsilon_3, \nu (y_3) = \varepsilon_1 \varepsilon_4, \nu (z_3)
= \varepsilon_2 \varepsilon_5,$ and the remaining ones $v( \cdot )
=1.$
(5)~~Since $(yyxx)\times (yxyx) \times (xxyy) \times (xyxy) = 1$
and $(xxxx)\times (yxyx) \times (xyxy) \times (yyyy) = 1,$ for
Eq.(4B.7) we have $\varepsilon_2 \varepsilon_3 \varepsilon_4
\varepsilon_5 =1$ and $\varepsilon_1 \varepsilon_3 \varepsilon_5
\varepsilon_6 =1,$ respectively. Then, one can assign $\nu (x_1) =
\varepsilon_1, \nu (y_2) = \varepsilon_2, \nu (y_3) =
\varepsilon_3, \nu (y_4) = \varepsilon_1 \varepsilon_3
\varepsilon_4,$ and the remaining ones $v( \cdot ) =1.$
(6)~~Since $(yyxx)\times (yxyx) \times (xxyy) \times (xyxy) = 1,$
for Eq.(4B.8) we have $\varepsilon_2 \varepsilon_3 \varepsilon_4
\varepsilon_5 = 1.$ Then, one can assign $\nu (x_1) =
\varepsilon_1, \nu (y_2) = \varepsilon_2, \nu (y_3) =
\varepsilon_3, \nu (y_4) = \varepsilon_1 \varepsilon_3
\varepsilon_4, \nu (z_1) = \varepsilon_6,$ and the remaining ones
$v( \cdot ) =1.$
(7)~~Since $(xxxx) \times (yyxx) \times (xxyy) \times (zzxx)
\times (yyzz) \times (zzzz) = 1,$ for Eq.(4B.9) we have
$\varepsilon_1 \varepsilon_2 \varepsilon_3 \varepsilon_4
\varepsilon_5 \varepsilon_6 = 1.$ Then, one can assign $\nu (x_1)
= \varepsilon_1, \nu (y_1) = \varepsilon_2, \nu (y_3) =
\varepsilon_1 \varepsilon_3, \nu (z_1) = \varepsilon_1
\varepsilon_3 \varepsilon_4, \nu (z_3) = \varepsilon_2
\varepsilon_5,$ and the remaining ones $v( \cdot ) =1.$
Since Eq.(4A.4) is included in Eqs.(4B.5) and (4B.6) and $|
\varphi \rangle = \frac{1}{\sqrt{2}} \left ( |0000 \rangle - |1111
\rangle \right )$ is a common eigenstate of both Eqs.(4B.5) and
(4B.6), it is concluded that Eqs.(4B.5) and (4B.6) are both
nontrivial. Moreover, as shown in Section IV.A that the GHZ state
is the unique state with equivalence up to a local unitary
transformation which presents the GHZ-Mermin proof in Eq.(4A.4),
we conclude the same result for Eqs.(4B.5) and (4B.6). This
completes the proof.
\subsection{The case of seven elements}
We characterize all four-qubit GHZ-Mermin experiments of seven
elements as follows.
{\it Proposition: A $\mathrm{GHZ-Mermin}$ experiment of seven
elements for the four-qubit system must equivalently be one of the
following forms:\begin{equation} \{ xxxx, yyxx, zzxx, xxyy, xxzz,
yyyy, zzzz \}, \tag{4C.1}\label{eq:4C-1}
\end{equation}\begin{equation} \{ xxxx, yyxx, zzxx, xxyy, xxzz,
yyyy, zzyy \}, \tag{4C.2}\label{eq:4C-2}
\end{equation}\begin{equation} \{ xxxx, yyxx, yxyx, xxyy, yxxy,
xyyx, xyxy \}, \tag{4C.3}\label{eq:4C-3}
\end{equation}\begin{equation} \{ xxxx, yyxx, yxyx, xxyy, yxxy,
xyyx, zzzz \}, \tag{4C.4}\label{eq:4C-4}
\end{equation}\begin{equation} \{ xxxx, yyxx, yxyx, xxyy, xyxy,
yyyy, zzzz \}. \tag{4C.5}\label{eq:4C-5}
\end{equation}Moreover, the geometric
invariants of $\mathrm{Eqs.(4C.1)-(4C.5)}$ are illustrated in
$\mathrm{Table~ VI}.$ }
\begin{table}
\caption{\label{tab:table6}The numbers in the $\mathrm{C}$ column
are $\mathrm{C}$-invariants, while the numbers in $1-7$'s columns
are $\mathrm{R}$-invariants.}
\begin{ruledtabular}
\begin{tabular}{ccccccccc}
& $\mathrm{C}$ & 1 & 2 & 3 & 4 & 5 & 6 & 7\\
\hline (4C.1) & 4 & 4 & 3 & 3 & 3 & 3 & 2 & 2 \\
(4C.2) & 4 & 4 & 3 & 3 & 4 & 2 & 3 & 3 \\
(4C.3) & 0 & 6 & 5 & 5 & 5 & 5 & 5 & 5 \\
(4C.4) & 4 & 5 & 4 & 5 & 4 & 4 & 4 & 0 \\
(4C.5) & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 0 \\
\end{tabular}
\end{ruledtabular}
\end{table}
{\it Proof}:~~As similar as above, a subset of six elements in a
GHZ-Mermin experiment of seven elements for the four-qubit system
is a four-qubit GHZ-Mermin experiment of six elements. Then, by
the Proposition in Section IV.B, a four-qubit GHZ-Mermin
experiment of seven elements must equivalently be one of the
forms\begin{equation} \{ xxxx, yyxx, zzxx, xxyy, xxzz, yyyy, \star
\star \star \star \},
\tag{4C.6}\label{eq:4C-6}\end{equation}\begin{equation} \{ xxxx,
yyxx, zzxx, xxyy, yyyy, zzyy, \star \star \star \star \},
\tag{4C.7}\label{eq:4C-7}\end{equation}\begin{equation} \{ xxxx,
yyxx, zzxx, xxyy, yyyy, zzzz, \star \star \star \star \},
\tag{4C.8}\label{eq:4C-8}\end{equation}\begin{equation} \{ xxxx,
yyxx, zzxx, xxyy, yyzz, zzzz, \star \star \star \star \},
\tag{4C.9}\label{eq:4C-9}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, xxyy, yxxy, xyyx, \star \star \star \star \},
\tag{4C.10}\label{eq:4C-10}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, xxyy, yxxy, zzzz, \star \star \star \star \},
\tag{4C.11}\label{eq:4C-11}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, xxyy, xyxy, yyyy, \star \star \star \star \},
\tag{4C.12}\label{eq:4C-12}\end{equation}\begin{equation} \{ xxxx,
yyxx, yxyx, xxyy, xyxy, zzzz, \star \star \star \star \},
\tag{4C.13}\label{eq:4C-13}\end{equation}\begin{equation} \{ xxxx,
yyxx, xxyy, zzyy, yyzz, zzzz, \star \star \star \star \}.
\tag{4C.14}\label{eq:4C-14}\end{equation}
(1)~~From Eq.(4C.6) we obtain Eqs.(4C.1), (4C.2),
and\begin{eqnarray*} \{ xxxx, yyxx, zzxx, xxyy, xxzz, yyyy, yyzz
\} \cong \mathrm{Eq.(4C.2)}.\end{eqnarray*}
(2)~~From Eq.(4C.7) we obtain Eq.(4C.2) and\begin{eqnarray*} \{
xxxx, yyxx, zzxx, xxyy, yyyy, zzyy, zzzz \}\cong
\mathrm{Eq.(4C.2)}, \\[0.3cm]
\{ xxxx, yyxx, zzxx, xxyy, yyyy, zzyy, yyzz \}\cong
\mathrm{Eq.(4C.2)}.\end{eqnarray*}
(3)~~From Eq.(4C.8) we obtain Eq.(4C.1) and\begin{eqnarray*} \{
xxxx, yyxx, zzxx, xxyy, yyyy, zzzz, yyzz \} \cong
\mathrm{Eq.(4C.1)}, \\[0.3cm]
\{ xxxx, yyxx, zzxx, xxyy, yyyy, zzzz, zzyy \} \cong
\mathrm{Eq.(4C.2)}.\end{eqnarray*}
(4)~~From Eq.(4C.9) we obtain\begin{eqnarray*}\{ xxxx, yyxx, zzxx,
xxyy, yyzz, zzzz, xxzz \} \cong \mathrm{Eq.(4C.2)}, \\[0.3cm]
\{ xxxx, yyxx, zzxx, xxyy, yyzz, zzzz, yyyy \} \cong
\mathrm{Eq.(4C.1)}, \\[0.3cm]
\{ xxxx, yyxx, zzxx, xxyy, yyzz, zzzz, zzyy \} \cong
\mathrm{Eq.(4C.1)}.\end{eqnarray*}
(5)~~From Eq.(4C.10) we obtain Eqs.(4C.3), (4C.4),
and\begin{eqnarray*} \{ xxxx, yyxx, yxyx, xxyy, yxxy, xyyx, yyyy
\} \cong \mathrm{Eq.(4C.3)}.\end{eqnarray*}
(6)~~From Eq.(4C.11) we obtain Eq.(4C.4) and\begin{eqnarray*}\{
xxxx, yyxx, yxyx, xxyy, yxxy, zzzz, xyxy \}
\cong \mathrm{Eq.(4C.4)}, \\[0.3cm]
\{ xxxx, yyxx, yxyx, xxyy, yxxy, zzzz, yyyy \} \cong
\mathrm{Eq.(4C.4)}.\end{eqnarray*}
(7)~~From Eq.(4C.12) we obtain Eq.(4C.5) and\begin{eqnarray*}\{
xxxx, yyxx, yxyx, xxyy, xyxy, yyyy, yxxy \}
\cong \mathrm{Eq.(4C.3)}, \\[0.3cm]
\{ xxxx, yyxx, yxyx, xxyy, xyxy, yyyy, xyyx \} \cong
\mathrm{Eq.(4C.3)}.\end{eqnarray*}
(8)~~From Eq.(4C.13) we obtain Eq.(4C.5), and\begin{eqnarray*} \{
xxxx, yyxx, yxyx, xxyy, xyxy, zzzz, yxxy
\} \cong \mathrm{Eq.(4C.4)}, \\[0.3cm]
\{ xxxx, yyxx, yxyx, xxyy, xyxy, zzzz, xyyx \} \cong
\mathrm{Eq.(4C.4)}.
\end{eqnarray*}
(9)~~From Eq.(4C.14) we obtain\begin{eqnarray*}\{ xxxx, yyxx,
xxyy, zzyy, yyzz, zzzz, zzxx \} \cong
\mathrm{Eq.(4C.1)}, \\[0.3cm]
\{ xxxx, yyxx, xxyy, zzyy, yyzz, zzzz, xxzz \} \cong
\mathrm{Eq.(4C.1)}, \\[0.3cm]
\{ xxxx, yyxx, xxyy, zzyy, yyzz, zzzz, yyyy \} \cong
\mathrm{Eq.(4C.1)}.\end{eqnarray*}
From Table VI we find that each of Eqs.(4C.1)-(4C.5) has different
geometric invariants and hence, they are all inequivalent. The
proof is complete.
{\it Theorem: Nontrivial $\mathrm{GHZ-Mermin}$ experiments of
seven elements for the four-qubit system must equivalently be
either $\mathrm{Eq.(4C.3)}$ or $\mathrm{Eq.(4C.4)}.$ Moreover, the
associated states exhibiting an ``all versus nothing"
contradiction between quantum mechanics and $\mathrm{EPR}$'s local
realism are $\mathrm{GHZ}$ states.}
{\it Proof}.~~By the above Proposition, it suffices to show that
Eqs.(4C.1), (4C.2), and (4C.5) are all trivial, while Eqs.(4C.3)
and (4C.4) are both nontrivial.
(1)~~Since $(xxxx)\times (yyxx) \times (xxyy) \times (yyyy) = 1$
and $(xxxx)\times (zzxx) \times (xxzz) \times (zzzz) = 1,$ for
Eq.(4C.1) we have $\varepsilon_1 \varepsilon_2 \varepsilon_4
\varepsilon_6 =1$ and $\varepsilon_1 \varepsilon_3 \varepsilon_5
\varepsilon_7 =1,$ respectively. Then, one can assign $\nu (x_1) =
\varepsilon_1, \nu (y_1) = \varepsilon_2, \nu (z_1) =
\varepsilon_3, \nu (y_3) = \varepsilon_1 \varepsilon_4, \nu (z_3)
= \varepsilon_1 \varepsilon_5,$ and the remaining ones $v( \cdot )
=1.$
(2)~~Since $(xxxx)\times (yyxx) \times (xxyy) \times (yyyy) = 1$
and $(xxxx)\times (zzxx) \times (xxyy) \times (zzyy) = 1,$ for
Eq.(4C.2) we have $\varepsilon_1 \varepsilon_2 \varepsilon_4
\varepsilon_6 =1$ and $\varepsilon_1 \varepsilon_3 \varepsilon_4
\varepsilon_7 =1,$ respectively. Then, one can assign $\nu (x_1) =
\varepsilon_1, \nu (y_1) = \varepsilon_2, \nu (z_1) =
\varepsilon_3, \nu (y_3) = \varepsilon_1 \varepsilon_4, \nu (z_3)
= \varepsilon_1 \varepsilon_5,$ and the remaining ones $v( \cdot )
=1.$
(3)~~Since $(xxxx)\times (yxyx) \times (xyxy) \times (yyyy) = 1$
and $(yyxx)\times (yxyx) \times (xxyy) \times (xyxy) = 1,$ for
Eq.(4C.5) we have $\varepsilon_1 \varepsilon_3 \varepsilon_5
\varepsilon_6 =1$ and $\varepsilon_2 \varepsilon_3 \varepsilon_4
\varepsilon_5 =1,$ respectively. Then, one can assign $\nu (x_1) =
\varepsilon_1, \nu (y_2) = \varepsilon_2, \nu (y_3) =
\varepsilon_3, \nu (y_4) = \varepsilon_1 \varepsilon_3
\varepsilon_4, \nu (z_1) = \varepsilon_7,$ and the remaining ones
$v( \cdot ) =1.$
Since Eq.(4B.5) is included in Eqs.(4C.3) and (4C.4) and $|
\varphi \rangle = \frac{1}{\sqrt{2}} \left ( |0000 \rangle - |1111
\rangle \right )$ is a common eigenstate of both Eqs.(4C.3) and
(4B.4), it is concluded that Eqs.(4C.3) and (4C.4) are both
nontrivial. Moreover, as shown in Section IV.B that the GHZ state
is the unique state with equivalence up to a local unitary
transformation which presents the GHZ-Mermin proof in Eq.(4B.5),
we conclude the same result for Eqs.(4C.3) and (4C.4). This
completes the proof.
\subsection{The case of eight elements}
We characterize all four-qubit GHZ-Mermin experiments of eight
elements as follows.
{\it Proposition: A $\mathrm{GHZ-Mermin}$ experiment of eight
elements for the four-qubit system must equivalently be one of the
following forms:\begin{equation} \{ xxxx, yyxx, zzxx, xxyy, xxzz,
yyyy, zzzz, yyzz \}, \tag{4D.1}\label{eq:4D-1}
\end{equation}\begin{equation} \{ xxxx, yyxx, yxyx, xxyy, yxxy,
xyyx, xyxy, yyyy \}, \tag{4D.2}\label{eq:4D-2}
\end{equation}\begin{equation} \{ xxxx, yyxx, yxyx, xxyy, yxxy,
xyyx, xyxy, zzzz \}. \tag{4D.3}\label{eq:4D-3}
\end{equation}Moreover, the geometric
invariants of $\mathrm{Eqs.(4D.1)-(4D.3)}$ are illustrated in
$\mathrm{Table~ VII}.$ }
\begin{table}
\caption{\label{tab:table7}The numbers in the $\mathrm{C}$ column
are $\mathrm{C}$-invariants, while the numbers in $1-8$'s columns
are $\mathrm{R}$-invariants.}
\begin{ruledtabular}
\begin{tabular}{cccccccccc}
& $\mathrm{C}$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline (4D.1) & 4 & 4 & 4 & 3 & 2 & 4 & 3 & 3 & 4 \\
(4D.2) & 0 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 \\
(4D.3) & 4 & 6 & 5 & 5 & 5 & 5 & 5 & 5 & 0 \\
\end{tabular}
\end{ruledtabular}
\end{table}
{\it Proof}:~~As similar as above, a subset of seven elements in a
GHZ-Mermin experiment of eight elements for the four-qubit system
is a four-qubit GHZ-Mermin experiment of seven elements. Then, by
the Proposition in Section IV.C, a four-qubit GHZ-Mermin
experiment of eight elements must equivalently be one of the
forms\begin{equation} \{ xxxx, yyxx, zzxx, xxyy, xxzz, yyyy, zzzz,
\star \star \star \star \}, \tag{4D.4}\label{eq:4D-4}
\end{equation}\begin{equation} \{ xxxx, yyxx, zzxx, xxyy, xxzz,
yyyy, zzyy, \star \star \star \star \}, \tag{4D.5}\label{eq:4D-5}
\end{equation}\begin{equation} \{ xxxx, yyxx, yxyx, xxyy, yxxy,
xyyx, xyxy, \star \star \star \star \}, \tag{4D.6}\label{eq:4D-6}
\end{equation}\begin{equation} \{ xxxx, yyxx, yxyx, xxyy, yxxy,
xyyx, zzzz, \star \star \star \star \}, \tag{4D.7}\label{eq:4D-7}
\end{equation}\begin{equation} \{ xxxx, yyxx, yxyx, xxyy, xyxy,
yyyy, zzzz, \star \star \star \star \}. \tag{4D.8}\label{eq:4D-8}
\end{equation}
(1)~~From Eq.(4D.4) we obtain Eq.(4D.1) and\begin{eqnarray*}\{
xxxx, yyxx, zzxx, xxyy, xxzz, yyyy, zzzz, zzyy \}\\[0.3cm] \cong
\mathrm{Eq.(4D.1)}.\end{eqnarray*}
(2)~~From Eq.(4D.5) we obtain\begin{eqnarray*}\{ xxxx, yyxx, zzxx,
xxyy, xxzz, yyyy, zzyy, zzzz \} \\[0.3cm] \cong
\mathrm{Eq.(4D.1)},\\[0.3cm]
\{ xxxx, yyxx, zzxx, xxyy, xxzz, yyyy, zzyy, yyzz \} \\[0.3cm] \cong
\mathrm{Eq.(4D.1)} .\end{eqnarray*}
(3)~~From Eq.(4D.6) we obtain Eqs.(4D.2) and (4D.3).
(4)~~From Eq.(4D.7) we obtain Eq.(4D.3) and\begin{eqnarray*}\{
xxxx, yyxx, yxyx, xxyy, yxxy, xyyx, zzzz, yyyy \} \\[0.3cm] \cong
\mathrm{Eq.(4D.3)}.\end{eqnarray*}
(5)~~From Eq.(4D.8) we obtain\begin{eqnarray*}\{ xxxx, yyxx, yxyx,
xxyy, xyxy, yyyy, zzzz, yxxy \} \\[0.3cm] \cong
\mathrm{Eq.(4D.3)},\\[0.3cm]
\{ xxxx, yyxx, yxyx, xxyy, xyxy, yyyy, zzzz, xyyx \} \\[0.3cm] \cong
\mathrm{Eq.(4D.3)}.\end{eqnarray*}
From Table VII we find that each of Eqs.(4D.1)-(4D.3) has
different geometric invariants and hence, they are all
inequivalent. The proof is complete.
{\it Theorem: Nontrivial $\mathrm{GHZ-Mermin}$ experiments of
eight elements for the four-qubit system must equivalently be
either $\mathrm{Eq.(4D.2)}$ or $\mathrm{Eq.(4D.3)}.$ Moreover, the
associated states exhibiting an ``all versus nothing"
contradiction between quantum mechanics and $\mathrm{EPR}$'s local
realism are $\mathrm{GHZ}$ states.}
{\it Proof}.~~By the above Proposition, it suffices to show that
Eq.(4D.1) is trivial, while Eqs.(4D.2) and (4D.3) are both
nontrivial.
Indeed, since $(xxxx)\times (yyxx) \times (xxyy) \times (yyyy) =
1, (xxxx)\times (zzxx) \times (xxzz) \times (zzzz) = 1, $ and
$(xxxx)\times (yyxx) \times (xxzz) \times (yyzz) = 1,$ for
Eq.(4D.1) we have $\varepsilon_1 \varepsilon_2 \varepsilon_4
\varepsilon_6 =1, \varepsilon_1 \varepsilon_3 \varepsilon_5
\varepsilon_7 =1,$ and $\varepsilon_1 \varepsilon_2 \varepsilon_5
\varepsilon_8 =1.$ Then, one can assign $\nu (x_1) =
\varepsilon_1, \nu (y_1) = \varepsilon_2, \nu (z_1) =
\varepsilon_3, \nu (y_3) = \varepsilon_1 \varepsilon_4, \nu (z_3)
= \varepsilon_1 \varepsilon_5,$ and the remaining ones $v( \cdot )
=1.$
On the other hand, Eq.(4C.3) is included in Eqs.(4D.2) and (4D.3)
and $| \varphi \rangle = \frac{1}{\sqrt{2}} \left ( |0000 \rangle
- |1111 \rangle \right )$ is a common eigenstate of both
Eqs.(4D.2) and (4D.3), it is concluded that Eqs.(4D.2) and (4D.3)
are both nontrivial. Moreover, as shown in Section IV.C that the
GHZ state is the unique state with equivalence up to a local
unitary transformation which presents the GHZ-Mermin proof in
Eq.(4C.3), we conclude the same result for Eqs.(4D.2) and (4D.3).
This completes the proof.
\section{Maximal GHZ-Mermin Experiments}
In this section, we show that a GHZ-Mermin experiment of the
four-qubit system contains at most nine elements and those maximal
GHZ-Mermin experiments of nine elements have two different forms,
one of which is trivial, while another one is nontrivial.
{\it Proposition: A $\mathrm{GHZ-Mermin}$ experiment of the
four-qubit system contains at most nine elements, and a four-qubit
$\mathrm{GHZ-Mermin}$ experiment of nine elements must
equivalently be one of the following forms:\begin{equation} \{
xxxx, yyxx, zzxx, xxyy, xxzz, yyyy, zzzz, yyzz, zzyy \},
\tag{5.1}\label{eq:5-1}
\end{equation}\begin{equation} \{ xxxx, yyxx, yxyx, xxyy, yxxy,
xyyx, xyxy, yyyy, zzzz \}. \tag{5.2}\label{eq:5-2}
\end{equation}Moreover, the geometric
invariants of $\mathrm{Eqs.(5.1)}$ and $\mathrm{(5.2)}$ are
illustrated in $\mathrm{Table~ VIII}.$ }
\begin{table}
\caption{\label{tab:table8}The numbers in the $\mathrm{C}$ column
are $\mathrm{C}$-invariants, while the numbers in $1-9$'s columns
are $\mathrm{R}$-invariants.}
\begin{ruledtabular}
\begin{tabular}{ccccccccccc}
& $\mathrm{C}$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\
\hline (5.1) & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 \\
(5.2) & 4 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 0 \\
\end{tabular}
\end{ruledtabular}
\end{table}
{\it Proof}:~~Indeed, as similar as above, a subset of eight
elements in a GHZ-Mermin experiment of nine elements for the
four-qubit system is a four-qubit GHZ-Mermin experiment of eight
elements. Then, by the Proposition in Section IV.D, a four-qubit
GHZ-Mermin experiment of nine elements must equivalently be one of
the forms\begin{equation} \{ xxxx, yyxx, zzxx, xxyy, xxzz, yyyy,
zzzz, yyzz, \star \star \star \star \}, \tag{5.3}\label{eq:5-3}
\end{equation}\begin{equation} \{ xxxx, yyxx, yxyx, xxyy, yxxy,
xyyx, xyxy, yyyy, \star \star \star \star \},
\tag{5.4}\label{eq:5-4}
\end{equation}\begin{equation} \{ xxxx, yyxx, yxyx, xxyy, yxxy,
xyyx, xyxy, zzzz, \star \star \star \star \}.
\tag{5.5}\label{eq:5-5}
\end{equation}
From Eq.(5.3) we obtain Eq.(5.1), as well from Eqs.(5.4) and (5.5)
obtain Eq.(5.2).
On the other hand, it is evident that one cannot add a element
into Eq.(5.1) or (5.2) for obtaining a larger GHZ-Mermin
experiment. This completes the proof.
{\it Theorem: Nontrivial $\mathrm{GHZ-Mermin}$ experiments of nine
elements for the four-qubit system must equivalently be
$\mathrm{Eq.(5.2)}.$ Moreover, the associated states exhibiting an
``all versus nothing" contradiction between quantum mechanics and
$\mathrm{EPR}$'s local realism are $\mathrm{GHZ}$ states.}
{\it Proof}.~~By the above Proposition, it suffices to show that
Eq.(5.1) is trivial, while Eq.(5.2) is nontrivial.
Indeed, since $(xxxx)\times (yyxx) \times (xxyy) \times (yyyy) =
1, (xxxx)\times (zzxx) \times (xxzz) \times (zzzz) = 1,
(xxxx)\times (yyxx) \times (xxzz) \times (yyzz) = 1,$ and
$(xxxx)\times (zzxx) \times (xxyy) \times (zzyy) = 1,$ for
Eq.(5.1) we have $\varepsilon_1 \varepsilon_2 \varepsilon_4
\varepsilon_6 =1, \varepsilon_1 \varepsilon_3 \varepsilon_5
\varepsilon_7 =1, \varepsilon_1 \varepsilon_2 \varepsilon_5
\varepsilon_8 =1,$ and $\varepsilon_1 \varepsilon_3 \varepsilon_4
\varepsilon_9 =1.$ Then, one can assign $\nu (x_1) =
\varepsilon_1, \nu (y_1) = \varepsilon_2, \nu (z_1) =
\varepsilon_3, \nu (y_3) = \varepsilon_1 \varepsilon_4, \nu (z_3)
= \varepsilon_1 \varepsilon_5,$ and the remaining ones $v( \cdot )
=1.$
On the other hand, Eq.(4D.2) is included in Eq.(5.2) and $|
\varphi \rangle = \frac{1}{\sqrt{2}} \left ( |0000 \rangle - |1111
\rangle \right )$ is a common eigenstate of Eq.(5.2), it is
concluded that Eq.(5.2) is nontrivial. Moreover, as shown in
Section IV.D that the GHZ state is the unique state with
equivalence up to a local unitary transformation which presents
the GHZ-Mermin proof in Eq.(4D.2), we conclude the same result for
Eq.(5.2). This completes the proof.
\section{Conclusion}
By using some subtle mathematical arguments, we present a complete
construction of the GHZ theorem for the four-qubit system. Two
geometric invariants play a crucial role in our argument. We have
shown that a GHZ-Mermin experiment of the four-qubit system
contains at most nine elements and a four-qubit GHZ-Mermin
experiment presenting the GHZ-Mermin-like proof contains at least
five elements. We have exhibited all four-qubit GHZ-Mermin
experiments of 3-9 elements.
In particular, we have proved that the four-qubit states
exhibiting 100\% violation between quantum mechanics and EPR's
local realism are equivalent to $| \mathrm{GHZ} \rangle =
\frac{1}{\sqrt{2}} \left ( |0000 \rangle - |1111 \rangle \right )$
up to a local unitary transformation, which maximally violate the
following Bell inequality\begin{equation}\langle \mathcal{B}
\rangle \leq 9, \tag{6.1}\label{eq:6-1}
\end{equation}where $\mathcal{B} = - x_1 x_2 x_3 x_4 + y_1 y_2 x_3 x_4
+ y_1 x_2 y_3 x_4 + x_1 x_2 y_3 y_4 + y_1 x_2 x_3 y_4 + x_1 y_2
y_3 x_4 + x_1 y_2 x_3 y_4 - y_1 y_2 y_3 y_4 + z_1 z_2 z_3 z_4.$ On
the other hand, as shown in Theorem in Sec.V, the state maximally
violating Eq.(6.1) is unique and equal to $| \mathrm{GHZ}
\rangle.$ This yields that the maximal violation of statistical
predictions is equivalent to (and so implies) the violation of
definite predictions between quantum mechanics and EPR's local
realism for the four-qubit system, as similar to the three-qubit
system \cite{Chen}. Therefore, from the view of EPR's local
realism one concludes that the maximally entangled states of four
qubits should be just the GHZ state \cite{GB}. We would like to
expect the same result holds true for $n$ qubits, that is, all
states exhibiting 100\% violation between quantum mechanics and
EPR's local realism must be GHZ's states up to a local unitary
transformation. This will provides a natural definition of GHZ
states and hence clarifies the maximally entangled states of $n$
qubits.
Note that Eq.(6.1) is not a standard Bell inequality which have
two observables at each site \cite{WW-ZB}. From the viewpoint of
GHZ's theorem we need to study Bell inequalities of $n$ qubits in
which measurements on each particle can be chosen among three spin
observables. This should be helpful to reveal the close
relationship among entanglement, Bell inequalities, and EPR's
local realism \cite{ZBLW}. Since the Bell inequalities and GHZ's
theorem are two main theme on the violation of EPR's local
realism, it turns out that GHZ's theorem and Bell-type
inequalities can be used to reveal what the term maximally
entangled states should actually mean in the multipartite and/or
higher dimensional quantum systems.
\end{document} |
\begin{equation}} \def\ee{\end{equation}gin{document}
\title{A Review of First-Passage Theory for the Segerdahl Risk Process and Extensions}
\newcommand{0000-0002-2615-0950}{0000-0002-2615-0950}
\author{{Florin Avram}
and {Jose-Luis Perez-Garmendia}
}
\maketitle
absolute } \def\sub{subordinator } \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ups{ruin probabilities } \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uen{renewal equation tract{The Segerdahl process (Segerdahl (1955)), {characterized by}
exponential claims and
affine drift, has drawn a~considerable amount of interest---see, for example, (Tichy (1984); Avram and Usabel (2008); Albrecher et al. (2013); Marciniak and Palmowski (2016)), due to its
economic interest (it is the simplest risk process which takes into account the effect of interest
rates). See (Albrecher and Asmussen 2010, Chapter 8) for an excellent overview, including extensions to processes with state dependent drift. It is also the simplest
non-Lévy, non-diffusion example of a spectrally negative Markov risk model. Note that for both
spectrally negative Lévy and diffusion processes, first passage theories which are based on identifying
two “basic” monotone harmonic functions/martingales have been developed. This means that for
these processes many control problems involving dividends, capital injections, etc., may be solved
explicitly once the two basic functions have been obtained. Furthermore, extensions to general
spectrally negative Markov processes are possible (Landriault et al. (2017), Avram et al. (2018);
Avram and Goreac (2019); Avram et al. (2019b)). Unfortunately, methods for computing the basic
functions are still lacking outside the Lévy and diffusion classes, with the notable exception of the Segerdahl process, for which the
ruin probability has been computed (Paulsen and Gjessing (1997). As a consequence,
the $W$ scale function may be computed as well, via simple {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionroba \ arguments which apply a priori to all {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrocs\ with \mathbb{E}} \def\subsection} \def\Kol{Kolmogorov {\subsection} \def\Kol{Kolmogorov poj\ (and may be extended to phase-type jumps as well). Further work going beyond \mathbb{E}} \def\subsection} \def\Kol{Kolmogorov {\subsection} \def\Kol{Kolmogorov poj\ and linear drifts has been provided in provided in (Avram and Usabel (2008)) and (Czarna et al. (2017)), respectively.
However, there is a
striking lack of numerical results in both cases. This motivated us to review these approaches, with the
purpose of drawing attention to connections between them, and underlining open problems.}
{\bf Keywords:} {Segerdahl process; affine coefficients; first passage; spectrally negative Markov process; scale functions; hypergeometric functions}
\sec{Introduction and Brief Review of First Passage Theory for Spectrally Negative
\Mar \ Processes}
To set the stage for our topic and future research,
consider a~\sn
jump diffusion on a~filtered probability space $(\Omega, \{\mathcal{F}_t\}_{t \gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq 0}, P)$, which~ satisfies the~SDE:
\begin{equation}} \def\ee{\end{equation}gin{eqnarray} \label{jd}
d X_t&= & c(X_t) d t + \sigma(X_t) d B_t - d J_t, \; J_t=\sum_{i=1}^{N_\l(t)} C_i, \; \for X_t > 0
\epsilon} \newcommand{\ol}{\overlinend{eqnarray}
and is {absorbed}
when leaving a half line $(l,\infty} \def\Eq{\Leftrightarrow)$.\symbolfootnote[4]{The boundary point $l$ may be a natural barrier, like the largest root of $\s(x)$. Or, when $\s(x)=0$ and $c(x)$ is increasing, $l$ may be the largest root of $c(x)$, called absolute ruin point.
Or, it can be a point below which the process is artificially killed.}
Here, $C_i$ are nonnegative } \def\fr{\frac} \def\Y{Y} \def\t{T} \def\ta{T_{a,-}} \def\tb{T_{b,+}} \def\tl{T_{l,-} \; i.i.d. r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uvs with distribution measure $ F_C(dz)$ and finite mean, $B_t$ is an independent standard Brownian motion, $\s(x) >0, c(x)>0, \for x > l$, $N_\l(t)$ is an independent Poisson process of intensity $\l$. {The functions $c(x), \; v(x):=\fr{\s^2(x)}2$ and $\Pi} \def\x{\xi} \def\ith{it holds that (d z)=\l F_C(dz)$ are referred to as the
L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims-Khinchine
characteristics of $X_t$.}
Note that we assume that all jumps go in the same direction and have constant intensity so that we can take advantage of potential simplifications of the first passage theory in this case. See \cite{paulsen2010ruin} and \cite[Chapter 8]{AA} for further information on risk processes with state dependent drift, and~in particular the two pages of historical notes and references in the last reference.
{ {The Segerdahl process}} is the simplest example outside the spectrally negative L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims and diffusion classes. It is obtained by assuming $v(x)=0$ in \epsilon} \newcommand{\ol}{\overlineqr{jd}, and~$C_k$ to be exponential i.i.d random variables with density $f(x)=\mu e^{-\mu x}$--see \cite{Seg} for the case $c(x)=c + rx, r>0, c \gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq 0$, and see also~\cite{Tichy84} for nonlinear $c(x)$.
The Segerdahl\ \ {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionroc \ \sats\ thus the SDE
\begin{equation}} \def\ee{\end{equation}a d X_t= {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionr{\c + r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{U X_t} dt -d {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionr{\sum_{i=1}^{N_t}C_i}, \quad} \def\for{\forall r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{U>0, \c \gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq 0. \epsilon} \newcommand{\ol}{\overlineea
It is given explicitly by
\begin{equation}} \def\ee{\end{equation}gin{align*}
X_t=X_0+r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{U \int_0^{t}X_s ds+K_t, \quad} \def\for{\forallad {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!E}} \defOrnstein-Uhlenbeck } \def\difs{diffusions{Ornstein-Uhlenbeck } \def\difs{diffusionsq \quad} \def\for{\forallad
X_t=X_0e^{ r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{U t} +e^{ r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{U t} \int_0^t e^{-r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{U s} d K_s,
\epsilon} \newcommand{\ol}{\overlinend{align*}
with $K_t=\c t -\sum_{i=1}^{N_t}C_i$ being a Cram\'er-Lundberg process, whose Laplace exponent is
\[
\kappa} \def\l{\lambda} \def\a{\alpha(\theta):=\frac{1}{t}\log \mathbb{E}[e^{\th K_t}]=\c \th + \lambda(\frac{\mu }{\mu +\th}-1).
\]
\begin{equation}} \def\ee{\end{equation}R An essential point for the Segerdahl process is the fact that the point $-\fr{\c}{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{U}
$ is an absolute } \def\sub{subordinator } \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ups{ruin probabilities } \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uen{renewal equation ruin~level, in the sense that after a jump below this point, the process will never cross back. We may assume \wlo that the absolute } \def\sub{subordinator } \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ups{ruin probabilities } \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uen{renewal equation ruin~level is $0$, or, equivalently, that $\c=0$. \epsilon} \newcommand{\ol}{\overlineeR
{{First passage theory}} concerns the first passage times above and below fixed levels. For any process $\Xt$, these are defined by
\begin{equation}} \def\ee{\end{equation}gin{equation}\la{fpt}
\begin{equation}} \def\ee{\end{equation}gin{aligned}
\tb &= \tb^{X}=\inf\{t\gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq 0: X_t> b\},\\
T_{a,-}&=T_{a,-}^X=\inf\{t\gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq 0: X_t< a\},
\epsilon} \newcommand{\ol}{\overlinend{aligned}
\epsilon} \newcommand{\ol}{\overlinend{equation}
with $\inf\epsilon} \newcommand{\ol}{\overlinemptyset=+\infty$, and~the upper script $X$ typically omitted. Since $a$ is typically fixed below,
we will often write for simplicity $T $ instead of $\ta$.
First passage times are important in the control of reserves/risk
processes. The~rough idea is that when below low levels $a$, reserves processes should be replenished at some cost, and~when above high levels $b$, they should be partly invested to yield income---see, for example, the comprehensive textbook \cite{AA}.
The most important first passage functions are the two-sided upward and downward exit functions from a~bounded interval $[a,b]$, defined respectively by
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{Rui}
\bc \sRui^{b}_{q}(x,a) & :=E_{x}\left} \def\ri{\rightft[ e^{-q\tb}{\mathbf 1}_{\left} \def\ri{\rightft\{
\tb<\tar} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight\} }r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight]=P_{x}\left} \def\ri{\rightft[ \tb<\min(\ta, \mathbf{e}} \def{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrocs{processes} \def{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionroc{process_q)r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight] \\
\Psi} \def\sRui{\overline{\Rui}_{q}^{b}(x,a) & :=E_{x}\left} \def\ri{\rightft[ e^{-q\ta}{\mathbf 1}_{\left} \def\ri{\rightft\{ \ta<\tbr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight\} }r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight]=P_{x}\left} \def\ri{\rightft[ \ta<\min(\tb, \mathbf{e}} \def{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrocs{processes} \def{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionroc{process_q)r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight]\epsilon} \newcommand{\ol}{\overlinec \; \; q\gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq0, a\left} \def\ri{\rightq x\left} \def\ri{\rightq b,
\epsilon} \newcommand{\ol}{\overlineeq
where $\mathbf{e}} \def{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrocs{processes} \def{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionroc{process_q$ is an~independent \mathbb{E}} \def\subsection} \def\Kol{Kolmogorov {\subsection} \def\Kol{Kolmogorov po r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uv of rate $q$. We will call them (killed) {{survival}} and {{ruin}} probabilities, respectively\footnote{See \cite{Ivakil} for a~nice exposition of killing.}, but the qualifier killed will be usually dropped below. The~absence of killing will be indicated by omitting the subindex $q$.
Note that in the context of potential theory, \epsilon} \newcommand{\ol}{\overlineqr{Rui} are called equilibrium potentials \cite{blumenthal2007markov} (of the capacitors $\{b,a\}$ and~$\{a,b\}$).
{{\bf Beyond r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ups: scale functions, dividends, capital gains, etc.} Recall that for ``completely asymmetric L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims" {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrocs, with jumps going all in the same direction, a~large variety of first passage {{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrobs \ may} be reduced to the computation of the two monotone ``scale functions'' $W_\q,Z_\q$---see, \fe, \cite{Suprun,Ber97,Ber,AKP,APP,APP15,APY,IP,AIZ,LZ17,LP,AZ}, and see \cite{AGV} for a~recent compilation of more than 20 laws expressed in terms of $W_\q,Z_q$.}
For example, for {spectrally negative L\'evy processes},
the {killed survival} probability has a~well known simple factorization\footnote{The fact that the survival probability has the multiplicative structure~\epsilon} \newcommand{\ol}{\overlineqr{Wfac} is equivalent to the absence of positive jumps, by~the strong Markov property; this is the famous ``gambler's winning'' formula \cite{Kyp}.}:
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{Wfac} \sRui_{\q}^{b}(x,a) & =\fr{W_{\q}(x-a)}{W_{\q}(b-a)}.
\epsilon} \newcommand{\ol}{\overlineeq
For a~second example, the~\cite{deF} discounted dividends fixed barrier objective for {spectrally negative L\'evy processes} has a~simple expression in terms of either the $W_\q $ scale function or of its logarithmic derivative $\nu_{\q}=\fr{W'_\q}{W_\q}$ \cite{APP}.\footnote{$\nu_\q$ may be more useful than $W_\q$ in the \sn \Mar framework \cite{AG}.}
\begin{equation}} \def\ee{\end{equation}gin{equation} V^{b}(x)= \bc \frac{W_\q( x)}{W_{\q}^ {{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrime}(b)}=e^{-\int_{x}^{b} \nu_\q(m)dm}\fr 1{\nu_\q(b)}& 0 \left} \def\ri{\rightq x \left} \def\ri{\rightq b\\V^{b}(x)=x-b+ V^{b}(b) & x>b \epsilon} \newcommand{\ol}{\overlinec. \la{divL}
\epsilon} \newcommand{\ol}{\overlinend{equation}
Maximizing over the reflecting barrier $b$ is simply achieved by finding the roots of
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{smf} W_\q'' (b) =0 {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!E}} \defOrnstein-Uhlenbeck } \def\difs{diffusions{Ornstein-Uhlenbeck } \def\difs{diffusionsq {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusiond {}{b} \Big[\fr 1{\nu_\q(b)} \Big]={r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusiond {}{b} \Big[ V^{b}(b)\Big]=1.\epsilon} \newcommand{\ol}{\overlineeq
Since~results for \sn L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrocs \; (like the de Finetti } \def\app{approximation {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrob \; mentioned above)
require often not much more than the \str, it is natural to attempt to extend them
to the \sn strong Markov case.
As~expected, everything worked out almost smoothly for ``{L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims}-type cases'' like random walks \cite{AV}, Markov additive processes \cite{IP}, etc. Recently, it~was discovered that $W,Z$ formulas continue to hold a~priori for spectrally negative \Mar processes \cite{LLZ17b}, \cite{ALL}. The main difference is that in equations like \epsilon} \newcommand{\ol}{\overlineqr{Wfac}, $W_{\q}(x-a)$ must be replaced by a two-variable \fun \ $W_{\q}(x,a)$ (which reduces in the L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims case to $W_{\q} (x,y)=\T W_{\q} (x-y)$, with $\T W_q$ being the scale function of the L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims process). The same holds of course for the second scale function $Z_{\q,\th}(x-a)$ \cite{APP15,IP}. This unifying structure has lead to recent progress for the optimal {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um d}} \deffirst passage {first passage vs \ problem for \sn \Mar {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrocs \ -- see \cite{AG};
however, as of today, we~are not aware of other results on the control of the process \epsilon} \newcommand{\ol}{\overlineqr{jd} which have succeeded to exploit the $W,Z$ formalism.
Several approaches may allow handling particular cases of \sn \Mar {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrocs:
\begin{equation}} \def\ee{\end{equation}gin{enumerate}
\item {for the Segerdahl\ ~ {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionroc, the~direct IDE solving approach is successful
for computing the r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up ---see (\cite{Pau}}) and Theorem 1, Section r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uef{s:dir}.
\item{for L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims driven Langevin-type processes, renewal \epsilon} \newcommand{\ol}{\overlineqs \ have been provided in \cite{CPRY}} ---see Section r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uef{ex:fb}.
\item for processes with operators having affine drift and volatility, an~explicit integrating factor for the \LT \ may be found in \cite{AU}---see Section r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uef{s:AU}.
\item with phase-type jumps, there is Asmussen's embedding into a~regime switching diffusion {{\cite{asmussen1995stationary}}---see Section r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uef{s:CLSa}, and~the complex integral representations of {\cite{JJ}}.}\\
\epsilon} \newcommand{\ol}{\overlinend{enumerate}
We will review and complete here the first approach, and also review and discuss the second and third approaches.
Asmussen's approach is also recalled, because we believe it has considerable potential.
We end this introduction with an~example of {a} still open problem we would like to solve in the future:
\begin{equation}} \def\ee{\end{equation}Q Find the optimal dividend policy for the Segerdahl\ process in the presence of capital injections and bankruptcy (in particular, investigate the extensions of Equations \epsilon} \newcommand{\ol}{\overlineqr{divL} and \epsilon} \newcommand{\ol}{\overlineqr{smf}).
\epsilon} \newcommand{\ol}{\overlineeQ
{\bf {Contents}}. Section r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uef{s:ALL} explains the simplicity of \sn \Mar {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrocs \ with negative exponential and phase-type jumps, already sketched in \cite{ALL}.
{Section r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uef{s:dir} reviews the direct classic Kolmogorov approach for solving first passage problems. The~discounted r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up ($q>0$) \epsilon} \newcommand{\ol}{\overlineqr{Pau} for the Segerdahl~ process is obtained, following \cite{Pau}, by transforming the r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uen \epsilon} \newcommand{\ol}{\overlineqr{reneq} into the ODE \epsilon} \newcommand{\ol}{\overlineqr{Paueq}, which~is hypergeometric of order $2$.\footnote{This result due to Paulsen has stopped short further research
for more general
mixed exponential jumps, since it seems to require a~separate ``look-up'' of hypergeometric solutions for each particular
problem.} We also complete the study of this process by providing its $W$ scale function, using the results in section r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uef{s:ALL}.}
{Section r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uef{ex:fb} reviews the recent approach based on renewal
equations due to \cite{CPRY} (which needs still be justified for increasing premiums satisfying \epsilon} \newcommand{\ol}{\overlineqr{intc}). An important renewal (Equation \epsilon} \newcommand{\ol}{\overlineqr{reneqC}) for the ``scale derivative'' $\w$ is recalled here, and~a~new result relating the scale derivative to the { integrating factor} defined in \epsilon} \newcommand{\ol}{\overlineqr{defI0} is offered---see Theorem 4.}
{Section r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uef{s:AU} reviews older computations of \cite{AU} for more general {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrocs \ with affine operator, and~provides explicit formulas for the Laplace transforms } \def\q{q} \def\R{{\mathbb R} \ of the \sur and r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up \epsilon} \newcommand{\ol}{\overlineqr{seg0q}, in~terms of the same{ integrating factor} \epsilon} \newcommand{\ol}{\overlineqr{defI0} and its antiderivative.}
{Section r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uef{s:Segr} checks that our integrating factor approach recovers various results for Segerdahl's process, when $\q=0$ or $x=0$.}
{Section r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uef{s:CLSa} reviews Asmussen's approach for solving first passage problems with
phase-type jumps, and~illustrates the simple structure of the \sur and r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up of the Segerdahl-Tichy process, in~terms of the scale derivative $\w$. This approach yields quasi-explicit results when $q=0$.}
{Section r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uef{s:a} reviews necessary hypergeometric identities (used in section r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uef{s:dir}).}
{Finally, Section r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uef{s:con} outlines further promising directions of research.}
\sec{Two-sided first passage {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionros \ for \sn \Mar {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrocs \ with negative exponential and phase-type jumps reduce to computing two one sided first passage \funs~$H_\q,\Psi} \def\sRui{\overline{\Rui}_\q$ \la{s:ALL}}
\begin{equation}} \def\ee{\end{equation}R The results in this section apply a priori to a large class of \sn \Mar {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrocs \ \epsilon} \newcommand{\ol}{\overlineqr{jd} with negative exponential jumps, and are formulated as such. In particular, \BM\ may be present, and hence creeping downwards is possible. \epsilon} \newcommand{\ol}{\overlineeR
Our study of \sn \Mar {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrocs \ with negative exponential jumps is based on an increasing
$q$- harmonic function of our process $H_\q(x), l \left} \def\ri{\rightq x $ \satg\ \epsilon} \newcommand{\ol}{\overlineqr{H}, and a decreasing one $ \Psi} \def\sRui{\overline{\Rui}_{\q}(x,a), x \gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq a$, defined in \epsilon} \newcommand{\ol}{\overlineqr{Rui}.
For the Segerdahl\ {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionroc,
these functions, to be denoted by $K_1(x), K_2(x), x \gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq 0$, turn out to be related to the increasing and decreasing Kummer hypergeometric functions $M$ and $U$, r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uesp. Note that $K_1(0)=0$, which renders \epsilon} \newcommand{\ol}{\overlineqr{H} immediate. Subsequently, other
useful functions like $W_q(x,a),Z_q(x,a)$ will be identified by simple {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionroba \ arguments which apply a priori to all {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrocs\ with \mathbb{E}} \def\subsection} \def\Kol{Kolmogorov {\subsection} \def\Kol{Kolmogorov poj, and may be extended to phase-type jumps -- see Lammas 1,2. With $W$ and $Z$ computed, one may hope to solve complicated control {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrobs\ involving dividends and capital injections, by applying similar arguments as in the L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims case.
Our first step is to investigate the existence of a \fac\ formula for two-sided first passage {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionros \ {\bf upwards}, with lower limit at the boundary domain $l$:
\begin{equation}} \def\ee{\end{equation} \sRui_{\q}^b(x,l)=\mathbb{E}_x\left} \def\ri{\rightft[ e^{-q\tb}; \tb \left} \def\ri{\rightq\tlr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight]:=
\frac{W_{\q}(x,l_-)}{W_{\q}(b,l_-)}:=\frac{H_{\q}(x)}{H_{\q}(b)}, l \left} \def\ri{\rightq x \left} \def\ri{\rightq b,\label{H} \epsilon} \newcommand{\ol}{\overlinee
which defines the \fun\ $H_q$, \upm, and up to the existence of the limits when $x\to l_-$ (the latter is a delicate point which must be resolved separately in each particular case).
The existence of such a function $H$ is suggested by spectral negativity and the \str. To emphasize that this property holds a priori outside the L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims framework, we provide a justification, based on the trick of adding a point $c >b$, and starting with
\begin{equation}} \def\ee{\end{equation}a \sRui_{\q}^{c}(x,l)=\sRui_{\q}^{b}(x,l) \sRui_{\q}^{c}(b,l),\epsilon} \newcommand{\ol}{\overlineea
where only the absence of positive jumps and
the strong Markov property were used. Therefore,
\begin{equation}} \def\ee{\end{equation}a \sRui_{\q}^{b}(x,l)=\fr{\sRui_{\q}^{c}(x,l)}{ \sRui_{\q}^{c}(b,l)};\epsilon} \newcommand{\ol}{\overlineea
So, the quotient decomposition is trivial as long as we stay on a fixed interval $[0,c]$, with $c$ arbitrarily big.
In the L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims case, the dependence on $l$ cancels, and $H_\q(x)=e^{\Fq x}$, where $\Fq$ is the unique nonnegative } \def\fr{\frac} \def\Y{Y} \def\t{T} \def\ta{T_{a,-}} \def\tb{T_{b,+}} \def\tl{T_{l,-}\ root of the Cram\`er-Lundberg } \def\WH{Wiener-Hopf } \def\iof{l^{-1}(a) equation. For diffusions, $H_{\q}(x)$ is the increasing solution of the \SL equation $({\mathcal G}-\q) f(x)=0$ (see \fe \cite{BS}). In the general \std\ case, to provide a \fac\ independent of $c$, it suffices to obtain an increasing
$q$- harmonic function of our process $H_\q(x), l \left} \def\ri{\rightq x $ \satg\ $H_q(0)=0$, as is the case with the Segerdahl\ {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionroc; Doob's optional stopping theorem yields then the \fac.
When the claims are exponential, computing two-sided } \defwell-known} \def\ts{two sided exit {well-known } \defprobabilities } \def\fund{fundamental {probabilities first passage {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionros\ on $[a,b], a \gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq l$, may be reduced to computing $H_\q(x), x \gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq l,$ and the first passage ruin {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionros\ $\Psi} \def\sRui{\overline{\Rui}(x,a)$, cf. \cite{ALL}.
\begin{equation}} \def\ee{\end{equation}L Let $\Psi} \def\sRui{\overline{\Rui}_{\q,J}(x,a), \Psi} \def\sRui{\overline{\Rui}_{\q,C}(x,a)$ denote the killed r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ups\ by jump and by creeping, r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uesp\ (more precisely Laplace transforms } \def\q{q} \def\R{{\mathbb R}, but Laplace transforms } \def\q{q} \def\R{{\mathbb R} are just r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ups\ \wr a process where the inter-arrivals are killed-- see \cite{Ivakil} for generalizations and applications to risk theory). For the process \epsilon} \newcommand{\ol}{\overlineqr{jd} with negative \mathbb{E}} \def\subsection} \def\Kol{Kolmogorov {\subsection} \def\Kol{Kolmogorov poj, no upward jumps and $a>l$, \ith
\begin{equation}} \def\ee{\end{equation}\sRui_{\q}^b(x,a) =\frac{H_\q(x)-\Psi} \def\sRui{\overline{\Rui}_{\q,C}(x,a)H_\q(a)-\Psi} \def\sRui{\overline{\Rui}_{\q,J}(x,a)\int_0^{ {a- l}} \mu e^{-\mu y}H_\q(a-y)dy}{H_\q(b)-\Psi} \def\sRui{\overline{\Rui}_{\q,C}(b,a)H_\q(a)-\Psi} \def\sRui{\overline{\Rui}_{\q,J}(b,a)\int_0^{ {a- l}} \mu e^{-\mu y}H_\q(a-y)dy}:=\frac{W_\q(x,a)} {W_\q(b,a)}. \la{We}
\epsilon} \newcommand{\ol}{\overlinee
Also
\begin{equation}} \def\ee{\end{equation}gin{align}
\Psi} \def\sRui{\overline{\Rui}_{\q}^b(x,a,dy) &=\mathbb{E}_{x}\left} \def\ri{\rightft[ e^{-q\ta} 1_{\left} \def\ri{\rightft\{ \ta<\tb , a -X_{\ta} \in dy r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight\} }r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight] \nonumber \\&=\Psi} \def\sRui{\overline{\Rui}_{\q}(x,a,dy) -\sRui_{\q}^b(x,a) \Psi} \def\sRui{\overline{\Rui}_{\q}(b,a,dy) ={r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionp{\Psi} \def\sRui{\overline{\Rui}_{\q}(x,a) -\frac{ W_q(x,a)}{ W_q(b,a)} \Psi} \def\sRui{\overline{\Rui}_{\q}(b,a)}\; {\mu} e^{-\mu y} dy , \la{Ze}
\epsilon} \newcommand{\ol}{\overlinend{align}
and \begin{equation}} \def\ee{\end{equation}gin{align}
\Psi} \def\sRui{\overline{\Rui}_{\q}^b(x,a,\th) &={r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionp{\Psi} \def\sRui{\overline{\Rui}_{\q}(x,a) -\frac{ W_q(x,a)}{ W_q(b,a)} \Psi} \def\sRui{\overline{\Rui}_{\q}(b,a)}\; \fr{\mu} {\mu +\th} . \la{Zet}
\epsilon} \newcommand{\ol}{\overlinend{align}
\epsilon} \newcommand{\ol}{\overlineeL
\begin{equation}} \def\ee{\end{equation}R When $a=l$ this result holds as well, provided that $H_{\q}(l)=0$. \epsilon} \newcommand{\ol}{\overlineeR
{{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrf}
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{12e}
&&\sRui_{\q}^b(x,a) =\mathbb{E}_{x}\left} \def\ri{\rightft[ e^{-q\tb}1_{\left} \def\ri{\rightft\{\tb<\tar} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight\} }r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight] \nonumber\\
&&= \mathbb{E}_{x} \left} \def\ri{\rightft[ e^{-q \tb} r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight] -
\mathbb{E}_{x}\left} \def\ri{\rightft[e^ {-q\ta }1_{\{X_{\ta}=a, \ta<\tb\} }r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight]
\mathbb{E}_{a}\left} \def\ri{\rightft[ e^{-q \tb}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight]\nonumber \\&&
-\int_0^{ {a- l}}\mathbb{E}_{x}\left} \def\ri{\rightft[e^ {-q\ta }1_{\{a-X_{\ta}\in dy, \ta<\tb\} }r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight] \mathbb{E}_{a-y}\left} \def\ri{\rightft[ e^{-q \tb}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight]\nonumber \\
&&= \sRui_\q^b(x) -
\Psi} \def\sRui{\overline{\Rui}_{\q,C}^b(x,a)
\sRui_\q^b(a)
-\Psi} \def\sRui{\overline{\Rui}_{\q,J}^b(x,a)\int_0^{ {a- l}} \sRui_\q^b(a-y) \mu e^{-\mu y} \mathrm d} \def\mathcal I} \def\divs{dividends {\mathcal I} \def\divs{dividends y \nonumber\\
&& =\frac 1 {H_\q(b)}
\Big[H_\q(x) -\Psi} \def\sRui{\overline{\Rui}_{\q,C}^b(x,a) H_\q(a)
-\Psi} \def\sRui{\overline{\Rui}_{\q,J}^b(x,a)\int_0^{ {a- l}} H_\q(a-y) \mu e^{-\mu y} \mathrm d} \def\mathcal I} \def\divs{dividends {\mathcal I} \def\divs{dividends y\Big] \la{int}
\epsilon} \newcommand{\ol}{\overlineeq
where $\Psi} \def\sRui{\overline{\Rui}_{\q,C}, \Psi} \def\sRui{\overline{\Rui}_{\q,J}$ denote respectively ruin by creeping and by jumps.
Similarly,
\begin{equation}} \def\ee{\end{equation}a \Psi} \def\sRui{\overline{\Rui}_{\q,C}^b(x,a)=\Psi} \def\sRui{\overline{\Rui}_{\q,C}(x,a)-\sRui_{\q,J}^b(x,a)
\Psi} \def\sRui{\overline{\Rui}_{\q,C}(b,a), \Psi} \def\sRui{\overline{\Rui}_{\q,J}^b(x,a)=\Psi} \def\sRui{\overline{\Rui}_{\q,J}(x,a)-\sRui_{\q,J}^b(x,a)
\Psi} \def\sRui{\overline{\Rui}_{\q,J}(b,a).\epsilon} \newcommand{\ol}{\overlineea
Plugging the last equality into \epsilon} \newcommand{\ol}{\overlineqr{int} and putting
$$W_q(x,a)=H_\q(x)-\Psi} \def\sRui{\overline{\Rui}_{\q,C}(x,a)H_\q(a)-\Psi} \def\sRui{\overline{\Rui}_{\q,J}(x,a)\int_0^{ {a- l}} \mu e^{-\mu y}H_\q(a-y)dy$$
yields
\begin{equation}} \def\ee{\end{equation}a
&&{H_\q(b)} \sRui_{\q}^b(x,a) =
W_\q(x) +\sRui_{\q}^b(x,a) {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionp{H_\q(a) \Psi} \def\sRui{\overline{\Rui}_{\q,C}(b,a)
+ \Psi} \def\sRui{\overline{\Rui}_{\q,J}(b,a) \int_0^{ {a- l}} H_\q(a-y) \mu e^{-\mu y} \mathrm d} \def\mathcal I} \def\divs{dividends {\mathcal I} \def\divs{dividends y}
\epsilon} \newcommand{\ol}{\overlineea
and solving for $\sRui_{\q}^b(x,a)$ yields
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray}&&\sRui_{\q}^b(x,a) =\frac{H_\q(x)-\Psi} \def\sRui{\overline{\Rui}_{\q,C}(x,a)H_\q(a)-\Psi} \def\sRui{\overline{\Rui}_{\q,J}(x,a)\int_0^{ {a- l}} \mu e^{-\mu y}H_\q(a-y)dy}{H_\q(b)-\Psi} \def\sRui{\overline{\Rui}_{\q,C}(b,a)H_\q(a)-\Psi} \def\sRui{\overline{\Rui}_{\q,J}(b,a)\int_0^{ {a- l}} \mu e^{-\mu y}H_\q(a-y)dy}. \la{We}
\epsilon} \newcommand{\ol}{\overlineeq
\begin{equation}} \def\ee{\end{equation}R
\epsilon} \newcommand{\ol}{\overlineqr{12e}, \epsilon} \newcommand{\ol}{\overlineqr{Ze} show that, with negative \mathbb{E}} \def\subsection} \def\Kol{Kolmogorov {\subsection} \def\Kol{Kolmogorov po \ jumps, both two-sided } \defwell-known} \def\ts{two sided exit {well-known } \defprobabilities } \def\fund{fundamental {probabilities first passage {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionros \ may be constructed using three functions $H_\q, \Psi} \def\sRui{\overline{\Rui}_{\q,C}, \Psi} \def\sRui{\overline{\Rui}_{\q,J}$ from the one-sided theory. If down-crossing continuously is impossible, only two functions $H_\q, \Psi} \def\sRui{\overline{\Rui}_{\q,J}$ are necessary.
The extension to downwards jumps of phase-type $(\vec \b, B)$ (a dense family) is immediate. \epsilon} \newcommand{\ol}{\overlineeR
\begin{equation}} \def\ee{\end{equation}L For {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrocs \ with downwards jumps of phase-type $(\vec \b, B)$ \epsilon} \newcommand{\ol}{\overlineqr{12e} becomes:
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray}
&&\sRui_{\q}^b(x,a) =\mathbb{E}_{x}\left} \def\ri{\rightft[ e^{-q\tb}1_{\left} \def\ri{\rightft\{\tb<\tar} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight\} }r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight] \nonumber \\&&=\frac{H_\q(x)-\Psi} \def\sRui{\overline{\Rui}_{\q,C}(x,a)H_\q(a)-\vec \Psi} \def\sRui{\overline{\Rui}_{\q,J}(x,a)\int_0^{ {a- l}} e^{B y} \bff b H_\q(a-y)dy}
{H_\q(b)-\Psi} \def\sRui{\overline{\Rui}_{\q,C}(b,a)H_\q(a)-\vec \Psi} \def\sRui{\overline{\Rui}_{\q,J}(b,a)\int_0^{ {a- l}} e^{B y} \bff b H_\q(a-y)dy}:=\frac{ W_q(x,a)}{ W_q(b,a)}, \la{WeP}
\epsilon} \newcommand{\ol}{\overlineeq
where $\vec \Psi} \def\sRui{\overline{\Rui}_{\q,J}(x,a)$ is the lign vector of r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ups \ whose $k$-th component is the r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up \ when crossing of $x$ axis occurs during phase $k$, and $\bff b=(-B) \bff 1$ .
Similarly, \epsilon} \newcommand{\ol}{\overlineqr{Ze} becomes
\begin{equation}} \def\ee{\end{equation} \Psi} \def\sRui{\overline{\Rui}_{\q}^b(x,a,dy) = {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionp{\vec \Psi} \def\sRui{\overline{\Rui}_{\q}(x,a) -\sRui_{\q}^b(x,a) \vec \Psi} \def\sRui{\overline{\Rui}_{\q}(b,a)} e^{ B y} \bff b dy . \la{ZeP} \epsilon} \newcommand{\ol}{\overlinee
\epsilon} \newcommand{\ol}{\overlineeL
{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrf The same ideas as in the \mathbb{E}} \def\subsection} \def\Kol{Kolmogorov {\subsection} \def\Kol{Kolmogorov po case apply, except that
now we must take into account the ``conditional memory-less property of phase-type variables":
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{12eP}
&&\sRui_{\q}^b(x,a) = \sRui_\q^b(x) -
\Psi} \def\sRui{\overline{\Rui}_{\q,C}^b(x,a)
\sRui_\q^b(a)
-\int_0^{ {a- l}}\sum_{k=1}^K \mathbb{E}_{x}\left} \def\ri{\rightft[e^ {-q\tb }1_{\{\ta<\tb, J_c=k,X_{\ta}\in a-dy, \tb <\infty} \def\Eq{\Leftrightarrow\} }r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight],\nonumber \epsilon} \newcommand{\ol}{\overlineeq
where $J_c$ is the phase when down-crossing $a$. Now the last term may be written as
\begin{equation}} \def\ee{\end{equation}a&&\sum_{k=1}^K \int_0^{ {a- l}} \mathbb{E}_{x}\left} \def\ri{\rightft[e^ {-q\tb }1_{\{\ta<\tb, J_c=k,X_{\ta}\in a-dy, \tb <\infty} \def\Eq{\Leftrightarrow\} }r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight]=\\&&\sum_{k=1}^K \int_0^{ {a- l}} \mathbb{E}_{x}\left} \def\ri{\rightft[e^ {-q\ta }1_{\{\ta<\tb, J_c=k,X_{\ta}\in a-dy\}} r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight] \mathbb{E}_{a-y}\left} \def\ri{\rightft[ e^{-q \tb}; \tb <\infty} \def\Eq{\Leftrightarrow r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight] \\
&&\sum_{k=1}^K \Psi} \def\sRui{\overline{\Rui}_{\q,J}(x,a,k) \int_0^{ {a- l}} {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionp{e^{B y} \bff b}_k \sRui_\q^b(a-y) \mathrm d} \def\mathcal I} \def\divs{dividends {\mathcal I} \def\divs{dividends y=
\vec \Psi} \def\sRui{\overline{\Rui}_{\q,J}(x,a)\int_0^{ {a- l}} e^{B y} \bff b \fr{H_\q(a-y)dy}{H_\q(b)},\epsilon} \newcommand{\ol}{\overlineea
where $\Psi} \def\sRui{\overline{\Rui}_{\q,J}(x,a,k)$ denotes the r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up with crossing in phase $k$,
and where the conditional memory-less property was applied.
The rest of the proof must be modified similarly. \qed
\iffalse
\begin{equation}} \def\ee{\end{equation}Xa For the Cram\'{e}r-Lundberg model with exponential jumps stopped at $0$, $l=-\infty} \def\Eq{\Leftrightarrow$ the ``\upc" formula
$$W_q(x) {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionropto H_\q(x)-\Psi} \def\sRui{\overline{\Rui}_{\q}(x) \int_{0 }^\infty} \def\Eq{\Leftrightarrow \mu e^{-\mu z} H_\q(-z)dz $$ yields
\begin{equation}} \def\ee{\end{equation}a && e^{x r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Upl} -(1+ \frac{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Umi
}{\mu}) e^{ x r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Umi} \int_{0 }^\infty} \def\Eq{\Leftrightarrow \mu e^{-\mu z} e^{-r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Upl z}dz =e^{x r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Upl} - \frac{\mu +r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Umi
}{\mu} e^{ x r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Umi} \fr {\mu}{\mu + r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Upl } \\&&=e^{x r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Upl} - \frac{\mu +r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Umi
}{\mu + r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Upl } e^{ x r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Umi} = e^{x r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Upl} - \frac{A_-
}{A_+ } e^{ x r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Umi} {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionropto W_q(x)= {A_+ } e^{x r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Upl} - {A_-
} e^{ x r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Umi},\epsilon} \newcommand{\ol}{\overlineea
where $r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Upl,r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Umi$ are the roots of the Cram\`er-Lundberg } \def\WH{Wiener-Hopf } \def\iof{l^{-1}(a) equation, recovering thus the well-known} \def\ts{two sided exit \ result.
\epsilon} \newcommand{\ol}{\overlineeXa
\fi
\begin{equation}} \def\ee{\end{equation}R
Note that the formula
$$H_\q(x)= W_q(x,a)+\Psi} \def\sRui{\overline{\Rui}_{\q,C}(x,a)H_\q(a)+\vec \Psi} \def\sRui{\overline{\Rui}_{\q,J}(x,a) \int_{0 }^{ {a- l}} e^{B z} \bff b H_\q(a-z)dz$$ has a clear heuristic probabilistic interpretation: {the ``total weight", starting from $x$, of all paths converging to $\infty} \def\Eq{\Leftrightarrow$ equals the ``total weight" of all paths not reaching $a$ + the ``total weight" of all paths dropping to some $a-z >l, 0 \left} \def\ri{\rightq z$, and converging to $\infty} \def\Eq{\Leftrightarrow$ afterwards}. Note that in the presence of a lower limit $l$, converging to $\infty} \def\Eq{\Leftrightarrow$ in the heuristic may be replaced by never reaching $l$.
\epsilon} \newcommand{\ol}{\overlineeR
\section{Direct Conversion to an~ODE of Kolmogorov's Integro-Differential Equation for the Discounted Ruin Probability} \la{s:dir}
One may associate to the process \epsilon} \newcommand{\ol}{\overlineqr{jd}
a~Markovian semi-group with generator
\begin{equation}} \def\ee{\end{equation}gin{equation*} \label{genrisk}
{\mathcal G} h(x) = v(x) h''(x) + c(x) h'(x) + \int_{(0,\infty} \def\Eq{\Leftrightarrow)}
[h(x-y) - h(x) ] \Pi} \def\x{\xi} \def\ith{it holds that (d y)
\epsilon} \newcommand{\ol}{\overlinend{equation*}
acting
on $h\in C^2_{(0,\infty} \def\Eq{\Leftrightarrow)}$, up to the minimum between its explosion and exit time {$T_{0,-}$}.
The classic approach for computing the ruin, survival, optimal dividends, and other similar functions
starts with the well-known} \def\ts{two sided exit \ Kolmogorov integro-differential equations associated to this operator. With jumps having a rational \LT, one may remove the integral term in \Kol's equation } \def\BC{boundary condition } \def\fund{fundamental } \def\boun{boundary } \def\con{condition } \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uv{random variable } \def\ind{independent above by applying to it the differential operator $n(D)$ given by the denominator of the Laplace exponent } \def\LT{Laplace transform} \def\BM{Brownian motion $\kappa} \def\l{\lambda} \def\a{\alpha(D)$. For example, with \mathbb{E}} \def\subsection} \def\Kol{Kolmogorov {\subsection} \def\Kol{Kolmogorov po claims, we would apply the operator
$\mu +D$.
\subsection} \def\Kol{Kolmogorov {Ruin probabilities for Segerdahl's Process with Exponential Jumps {Paulsen and
Gjessing} (1997), ex. 2.1 \la{s:PG1}}
When $v(x)=0$ and $C_k$ in \epsilon} \newcommand{\ol}{\overlineqr{jd} are exponential i.i.d random variables with density $f(x)=\mu e^{-\mu x}$, the Kolmogorov integro-differential \epsilon} \newcommand{\ol}{\overlineq \ for the r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up \ \ is:
{\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} && \la{reneq} c(x) \Psi} \def\sRui{\overline{\Rui}_\q(x,a)' + \lambda \mu \int_a^x e^{-\mu(x-z)} \Psi} \def\sRui{\overline{\Rui}_\q(z,a) dz -(\lambda +q) \Psi} \def\sRui{\overline{\Rui}_\q(x,a)+ \l e^{-\mu x}=0, \nonumber \\&& \Psi} \def\sRui{\overline{\Rui}_\q(b,a)=1, \Psi} \def\sRui{\overline{\Rui}_\q(x,a)=0, x<a.\epsilon} \newcommand{\ol}{\overlineeq}
To remove the convolution term $\Psi} \def\sRui{\overline{\Rui}_\q* f_C$, apply the operator
$\mu +D$, which~replaces the convolution term by $\l \mu \Psi} \def\sRui{\overline{\Rui}_\q(x)$\footnote{More generally, for any \PH jumps $C_i$ with \LT $\H f_C(s)=\fr{a(s)}{b(s)}$, it may be checked that $\Psi} \def\sRui{\overline{\Rui}_\q* f_C=\H f_C(D) \Psi} \def\sRui{\overline{\Rui}_\q$ in the sense that $b(D) \Psi} \def\sRui{\overline{\Rui}_\q* f_C=a(D) \Psi} \def\sRui{\overline{\Rui}_\q$,~thus removing the convolution by applying the denominator
$b(D)$.} yielding finally
\begin{equation}} \def\ee{\end{equation}a \left} \def\ri{\right(c(x) D^2 + (c'(x) + \mu c(x)-(\lambda + \q)) D - \q \mu)r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui) \Psi} \def\sRui{\overline{\Rui}_\q(x) =0\epsilon} \newcommand{\ol}{\overlineea
When $c(x)=c+ r x, a=0, b=\infty} \def\Eq{\Leftrightarrow$, the r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up \sats:
\begin{equation}} \def\ee{\end{equation}gin{eqnarray} \la{Paueq}
&&\left} \def\ri{\right[(\Tc + x) \,D^2 + (1+ \mu (\Tc +x) -\Tq - \Tl) D - \mu
\Tqr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui]
\Psi} \def\sRui{\overline{\Rui}_{q}(x) =0, \nonumber \\&&(-c \,D + \l +\q)
\Psi} \def\sRui{\overline{\Rui}_{q}(0) =\l\footnote{this is implied by the Kolmogorov integro-differential equation $({\mathcal G} -\l - q) \Psi} \def\sRui{\overline{\Rui}_\q(x)+ \lambda \overline F(x)=0, x \gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq 0$}, \quad} \def\for{\forallad \Psi} \def\sRui{\overline{\Rui}_{q}(\infty} \def\Eq{\Leftrightarrow)=0 \la{bc} \epsilon} \newcommand{\ol}{\overlineeq
see \cite[{(2.14)},{(2.15)}]{Pau}, where $ \Tl =\fr{\lambda}r,
\Tq =\fr{\q}r$, and~$-\Tc:=-\fr{c}r$ is the absolute } \def\sub{subordinator } \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ups{ruin probabilities } \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uen{renewal equation ruin~level.
Changing the origin to $-\Tc$ by $z=\mu(x+ \Tc), \Psi} \def\sRui{\overline{\Rui}_{q}(x)=y(z)$ brings this to the
form
{ \begin{equation}} \def\ee{\end{equation}gin{equation} \la{conhypeq}
z y''(z) + (z+1-n) y'(z) - \Tq y(z)=0, \; n= \Tl + \Tq, \epsilon} \newcommand{\ol}{\overlinee}
\nonumberindent(we corrected here two wrong minuses in \cite{Pau}),
which corresponds to the process killed at the absolute ruin, with claims rate $\mu=1$.
Note that the (Sturm-Liouville) Equation~\epsilon} \newcommand{\ol}{\overlineqr{conhypeq} intervenes also in the study of the squared radial Ornstein-Uhlenbeck } \def\difs{diffusions diffusion (also called \CIR process) \cite[p. 140, Chapter II.8]{BS}.
{ Let $K_i(z)=K_i(\Tq,n,z), i=1,2, n=\Tq + \Tl$ denote the
(unique up to a~constant) increasing/decreasing solutions for $z\in (0,\infty} \def\Eq{\Leftrightarrow)$ of the confluent hypergeometric Equation \epsilon} \newcommand{\ol}{\overlineqr{conhypeq}. The~solution of \epsilon} \newcommand{\ol}{\overlineqr{conhypeq} is thus}
{ \begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{Ki} c_1 K_1(\Tq,n, z) + c_2 K_2(\Tq,n, z)=c_1 z^n e^{-z}M(\Tq+1,n+1,z) +c_2 z^n e^{-z}
U(\Tq+1,n+1,z),\epsilon} \newcommand{\ol}{\overlineeq}
\nonumberindent where \cite[{13.2.5}]{AS}
$U[a, a+c, z] = \fr 1{\Gammaamma[a]} \int_0^\infty} \def\Eq{\Leftrightarrow e^{-z t} t^{a - 1} (t + 1)^{c - 1} dt, Re[z] > 0, Re[a] > 0$ is Tricomi's decreasing hypergeometric U function
and
$M\left} \def\ri{\right(a,a+c,zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui)= \, _1 {F}_1 (a,a+c;z)
$ is Kummer's increasing nonnegative } \def\fr{\frac} \def\Y{Y} \def\t{T} \def\ta{T_{a,-}} \def\tb{T_{b,+}} \def\tl{T_{l,-} \ confluent hypergeometric function of the
first kind.\footnote{$M(a,b,z)$ and $U(a,b,z)$ are the increasing/decreasing solutions of the
to
Weiler's canonical form of Kummer equation $z f''(z) + (b-z) f'(z) - a~f(z)=0$, which~is obtained via the substitution $y(z) =e^{-z} z^{n} f(z),$ with $a=\Tq +1, b=n +1$. Some computer systems use instead of $M$ the Laguerre function defined by $M(a,b,z)=L_{-a}^{b-1}(z)\fr{\Gammaamma(1-a) \Gammaamma(b)}{\Gamma(b-a)}$, which~yields for natural $-a$ the Laguerre polynomial of degree $-a$.}
The killed ruin probability must be combination of $U$ and $M$, but the fact that it decreases to $0$
suggests the absence of the function $K_1$. The~next result shows that this is indeed the case: the r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up is proportional to $K_2(\mu(x+ c(a)/r))$ on an~arbitrary interval $[a,\infty} \def\Eq{\Leftrightarrow), a>-\Tc$.
$K_1$ yields the absolute \sur {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionro \ (and scale function) on $[-\Tc,\infty} \def\Eq{\Leftrightarrow)$, but over an~arbitrary interval we must use a~combination of $K_1$ and $K_2$.
\begin{equation}} \def\ee{\end{equation}gin{Thm} \la{t:W}
Put $z(x)=\mu (\Tc +x), \Tc=c(a)/r$. The~r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up on $[a,\infty} \def\Eq{\Leftrightarrow)$ is
{\begin{equation}} \def\ee{\end{equation}gin{equation}\label{Pau}
\Psi} \def\sRui{\overline{\Rui}_{q}(x,a) =E_x [e^{-q \ta}]=\fr{\Tl}{\Tc \mu} \frac{e^{-\mu
x }(1+ x/\Tc)^{(\Tq+\Tl) }\; U\big(1+ \Tq, 1+ \Tq+\Tl,\mu (\Tc +x)\big)}
{U\big(1+ \Tq, 2+ \Tq+\Tl,\mu \Tc \big)}
\epsilon} \newcommand{\ol}{\overlinend{equation}}
\nonumberindent (when $q=0,$ $K_2(0,n,z)=\Gamma(\Tl,z)$ and \epsilon} \newcommand{\ol}{\overlineqr{Pau} reduces to $\fr{\Gamma(\Tl,\mu (\Tc +x))}{\Gamma(\Tl+1,\mu \Tc ))})$.\footnote{Note that we have corrected Paulsen's original denominator by using the identity \cite[{13.4.18}]{AS}
$ U[a-1, b, z]+(b-a) U[a, b, z]=z U[a, b+1,z], a>1.$}
\iffalse
(b) The {\bf scale} function $W_\q(x,a)$ on $[a,\infty} \def\Eq{\Leftrightarrow)$ is
up to a~proportionality constant $${K_1(\Tq,\Tl,z(x))}- k K_2(\Tq,\Tl,z(x)),$$
with $k$ defined in \epsilon} \newcommand{\ol}{\overlineqr{k}.
\fi
\epsilon} \newcommand{\ol}{\overlinend{Thm}
\begin{equation}} \def\ee{\end{equation}gin{proof}
Following \cite[ex. 2.1]{Pau}, note that the limits $\lim_{z\to \infty} \def\Eq{\Leftrightarrow} U(z)=0, \lim_{z\to \infty} \def\Eq{\Leftrightarrow} M(z)=\infty} \def\Eq{\Leftrightarrow$ imply
\begin{equation}} \def\ee{\end{equation}a \Psi} \def\sRui{\overline{\Rui}_{q}(x)= k \; K_2(z)=k {e^{-
z} z^{\T \q + \Tl} U\left} \def\ri{\rightft( \T \q+1
,\T \q +\Tl+1,zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) }, \quad} \def\for{\forall z=\mu(x + \Tc).\epsilon} \newcommand{\ol}{\overlineea
The proportionality constant $k$ is obtained from the boundary condition \epsilon} \newcommand{\ol}{\overlineqr{bc}.
Putting $G_b [h](x):= [c(x) (h)'(x)- (\lambda + q) h(x)]_{x=0},$
\begin{equation}} \def\ee{\end{equation}a G_b [ \Psi} \def\sRui{\overline{\Rui}_q](x)+\lambda=0 \Lra k=\fr{\l}{-G_b [K_2](z(x))}, \epsilon} \newcommand{\ol}{\overlineea
\begin{equation}} \def\ee{\end{equation}a =-e^{-z} z^{\Tq + \Tl+1}U\left} \def\ri{\rightft(
\Tq+1,\Tq +\Tl+2,z r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)\epsilon} \newcommand{\ol}{\overlineea
Putting $z_0= \mu c$, we find
\begin{equation}} \def\ee{\end{equation}a &&-G_b [K_2](z(x))={z_0 e^{-z_0}
z_0^{\T \q + \Tl-1} U \left} \def\ri{\right(\T \q, \T \q + \Tl, z_0r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui) +(\T \q + \Tl)e^{-z_0}
z_0^{\T \q + \Tl}U \left} \def\ri{\right(\T \q+1, \T \q + \Tl+1, z_0r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui)}\\&&={ e^{-z_0}
z_0^{\T \q + \Tl} (U \left} \def\ri{\right(\T \q, \T \q + \Tl, z_0r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui) +(\T \q + \Tl)U \left} \def\ri{\right(\T \q+1, \T \q + \Tl+1, z_0r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui))}, \epsilon} \newcommand{\ol}{\overlineea
where we have used the
identity \cite[p. 640]{BS}
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray}
K_2'(z)&=&- e^{-z}
z^{\T \q + \Tl-1} \
U \left} \def\ri{\right(\T \q, \T \q + \Tl, zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui). \la{Kids}\epsilon} \newcommand{\ol}{\overlineeq
This may be further simplified to
\begin{equation}} \def\ee{\end{equation}a &&-G_b [K_2](z(x))={ e^{-z_0}
z_0^{\T \q + \Tl+1} U \left} \def\ri{\right(\T \q+1, \T \q + \Tl+2, z_0r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui))},\epsilon} \newcommand{\ol}{\overlineea
by using the identity
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{sPau} U[a, b, z]+b U[a+1, b+1, z]=z U[a+1, b+2,z], a>1,\epsilon} \newcommand{\ol}{\overlineeq
which is itself a~consequence of the identities \cite[{13.4.16,13.4.18}]{AS}
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{abr} &&(b - a) U[a, b, z] +
z U[a, 2 + b, z] = (z + b) U[a, 1 + b,
z]\\&&U[a, b, z]+(b-a-1) U[a+1, b, z]=z U[a+1, b+1,z]\epsilon} \newcommand{\ol}{\overlineeq
(replace $a$ by $a+1$ in the first identity, and~subtract the second).
Finally, we
obtain:
\begin{equation}} \def\ee{\end{equation}gin{eqnarray*} \label{Pauruin}
&&\Psi} \def\sRui{\overline{\Rui}_{q}(x)= \left} \def\ri{\rightft( \frac{\Tl}{\Tc \mu }r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) \frac{e^{-
\mu x} (1+ \fr{x}{\Tc} )^{\T \q + \Tl} U\left} \def\ri{\rightft( \T \q+1
,\T \q + 1+\Tl,\mu (x + \Tc )r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) }{U\left} \def\ri{\rightft( \T \q+1,\T \q +1+
\Tl+1,\mu \Tc r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) }\\&&=\left} \def\ri{\rightft( \frac{\Tl}{ \mu }r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)
\frac{\int_x^{\infty} ({s-x})^{\T \q} \, ({s}+{\Tc} )^{ \Tl-1} \,
e^{- \mu s}
{d s} }{\int_{0}^{\infty}~s^{\T \q}~(s+ \Tc)^{ \Tl}~e^{ - \mu s } ds }
,\nonumber\epsilon} \newcommand{\ol}{\overlinend{eqnarray*}
and
\begin{equation}} \def\ee{\end{equation}gin{align*}
&&\Psi} \def\sRui{\overline{\Rui}_{q}(0) =\left} \def\ri{\rightft( \frac{\lambda}{c\mu}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) \frac{U\left} \def\ri{\rightft(
\T \q+1 ,\Tl + \T \q+1,\Tc \mur} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) }{U\left} \def\ri{\rightft(\T \q+1,\Tl + \T \q + 2,\Tc
\mur} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) }=\left} \def\ri{\rightft( \frac{\lambda}{c\mu}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)
\frac{\int_{0}^{\infty}
\,t^{\T \q}~(1+t)^{
\Tl -1}\,e^{-\Tc \mu t }~dt}{\int_{0}^{\infty}\,\,\,t^{\T \q}
~(1+t)^{\Tl} \,e^{-\Tc \mu t } ~dt}\\&&=\left} \def\ri{\rightft( \frac{\Tl}{ \mu
}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) \frac{\int_0^{\infty} {s}^{\T \q} \, ({s}+{\Tc} )^{ \Tl-1} \,
e^{- \mu s}
{d s} }{\int_{0}^{\infty}~s^{\T \q}~(s+ \Tc)^{ \Tl}~
e^{ - \mu s } ds } =\left} \def\ri{\rightft( \frac{\Tl}{\Tc}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)
\frac{\int
_{0}^{\infty}\,t^{\T \q}
~(t+\mu)^{\Tl-1}\,e^{-\Tc t}~dt}{\int_{0}^{\infty}\,\,\,t^
{\T \q}~(t+\mu)^{\Tl} \,e^{-\Tc t}
dt}.
\epsilon} \newcommand{\ol}{\overlinend{align*}
For $(a,\infty} \def\Eq{\Leftrightarrow), a>-\Tc$, the~same proof works after replacing $c,z(0)$ by $c(a),z(a)$.
\iffalse
(b) On $(0,\infty} \def\Eq{\Leftrightarrow)$, we must determine,
up to proportionality,
a~linear combination $ W_\q(x)= K_1(\Tq,\Tl,z(x)) - k K_2(\Tq,\Tl,z(x))
$ satisfying
the boundary condition \begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} G_b W_q(x)=0 \Lra k=\fr{G_b[K_1](0)}{G_b[K_2](0)}, G_b h(x):= [c(x) (h)'(x)- (\lambda + q) h(x)]_{x=0}.\epsilon} \newcommand{\ol}{\overlineeq
{Recall we have already computed $G_b[K_2](z(0))=-e^{-z_0} z_0^{\Tq + \Tl+1}U\left} \def\ri{\rightft(
\Tq+1,\Tq +\Tl+2,z_0 r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)$ in the proof of~(a).
Similarly, using \cite[p. 640]{BS} (reproduced for convenience in \epsilon} \newcommand{\ol}{\overlineqr{Kids1} below)}
\begin{equation}} \def\ee{\end{equation}a &&-G_b[K_1](z(0))= (\Tl + \Tq) e^{-z_0} z_0^{\Tq + \Tl} \big[ M\left} \def\ri{\rightft(
\Tq+1,\Tq +\Tl+1,z_0 r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)- M\left} \def\ri{\rightft(
\Tq,\Tq +\Tl,z_0r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)\big].
\epsilon} \newcommand{\ol}{\overlineea
Hence\footnote{
Putting $M_{++}= M\left} \def\ri{\rightft( \Tq+2,\Tq + \Tl+2,\mu \Tc r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight), U_{++}=
U\left} \def\ri{\rightft( \Tq+2,\Tq + \Tl+2,\mu \Tc r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight),$ we must solve the
equation \begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} && M - l U= \fr{\Tq+1} {\Tq+\Tl +1} M_{++}+ l
(\Tq+1) U_{++}\nonumber \\&& {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!E}} \defOrnstein-Uhlenbeck } \def\difs{diffusions{Ornstein-Uhlenbeck } \def\difs{diffusionsq l=\fr{({\Tq+\Tl +1}) M - ({\Tq+1})
M_{++}}{({\Tq+\Tl +1}) \left} \def\ri{\right(U+ (\Tq+1) U_{++}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui)}=\fr{\Tl}{\Tq+\Tl
+1} \fr{M_+}{U_{+}},\epsilon} \newcommand{\ol}{\overlineeq where we put $M_+=M\left} \def\ri{\rightft( \Tq+1,\Tq +
\Tl+2,\mu \Tc r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight), U_+=U\left} \def\ri{\rightft( \Tq+1,\Tq + \Tl+2,\mu \Tc
r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)$, and~applied the identities \cite[{13.4.3, 13.4.4}]{AS}.}
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{k} && k=\fr{\Tl+ \Tq}{z_0} \fr { M\left} \def\ri{\rightft(
\Tq +1,\Tq +\Tl+1,z_0r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)-M\left} \def\ri{\rightft(
\Tq,\Tq +\Tl,z_0 r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)}{ U(\Tq +1,\Tq + \Tl +2,z_0).}
\epsilon} \newcommand{\ol}{\overlineeq
\fi
\epsilon} \newcommand{\ol}{\overlinend{proof}
\iffalse
\begin{equation}} \def\ee{\end{equation}R Note that on $(-\Tc,\infty} \def\Eq{\Leftrightarrow)$, choosing $-\Tc$ as the origin yields $z_0=0=k$ (since $M\left} \def\ri{\rightft(
\Tq,\Tq +\Tl,0 r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)=1$ \cite[{13.1.2}]{AS}) and the scale \fun is proportional to $K_1(z)$, which~follows also from the {uniqueness of the nondecreasing solution}.
\epsilon} \newcommand{\ol}{\overlineeR
\begin{equation}} \def\ee{\end{equation}C (a) Differentiating the {scale} function yields
\begin{equation}} \def\ee{\end{equation}a &&\w_{\q}(x)=e^{-z} z^{\Tq+\Tl-1}\Big((\Tq+\Tl) M(\Tq,\Tq+\Tl,z)+ k U(\Tq,{\Tq+\Tl},z)\Big).\epsilon} \newcommand{\ol}{\overlineea
(b) The scale function is increasing.
\epsilon} \newcommand{\ol}{\overlineeC
\fi
\iffalse
\subsection} \def\Kol{Kolmogorov {M,U in a nutshell}
Consider the Segerdahl\ \ process with $r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{U=1$, absolute ruin point $-\Tc=0$ and $ \mu=1.$
We recall now how $M,U$ intervene in our prob; for more detail, see Section r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uef{s:dir}.
After removing the integral term in Kolmogorov's equation for our {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionroc, we arrive to a confluent hypergeometric equation \epsilon} \newcommand{\ol}{\overlineqr{conhypeq}.
The~general solution of the ODE \epsilon} \newcommand{\ol}{\overlineqr{conhypeq} may be written as
{ \begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{Ki} && c_1 K_1(\q,n, z) + c_2 K_2(\q,n, z)=e^{-z} z^n {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionr{c_1 M(\q+1,n+1,z) +
c_2 U(\q+1,n+1,z)},
\epsilon} \newcommand{\ol}{\overlineeq}
where $K_i(z)=K_i(\q,n,z), i=1,2, n=\q + \l $ denote the
(unique up to a~constant) {\bf increasing/decreasing} solutions for $z\in (0,\infty} \def\Eq{\Leftrightarrow)$,
and where $M(a,b,z)$ and $U(a,b,z)$ are the increasing/decreasing solutions of
Weiler's canonical form \begin{equation}} \def\ee{\end{equation} \la{Wei} z f''(z) + (b-z) f'(z) - d~f(z)=0\epsilon} \newcommand{\ol}{\overlinee of Kummer's confluent hypergeometric equation (which~is obtained via the substitution $K(z) =e^{-z} z^{n} f(z),$ with $d=\q +1, b=n +1$).
Note the integral representations
$$M\left} \def\ri{\right(a,b,zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui)= \, _1 {F}_1 (a,b;z)=\fr{\Gamma(b)}{2 {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusioni i}\oint_C \fr{dt e^t} {t^{b-a} (t-z)^{a}},$$ where the contour $C$ surrounds the branch cut from $t=0$ to $t=z$ \cite[(14)]{hecht2000rayleigh}, and
$$U[a, a+c, z] = \frac 1{\Gammaamma[a]} \int_0^\infty} \def\Eq{\Leftrightarrow e^{-z t} t^{a - 1} (t + 1)^{c - 1} dt, Re[z] > 0, Re[a] > 0$$
\cite[{13.2.5}]{AS}.
\begin{equation}} \def\ee{\end{equation}R Note that when $q=0$ $U,M$ reduce to the incomplete and lower incomplete gamma function \epsilon} \newcommand{\ol}{\overlineqr{MUG}, r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uesp, which checks with Segerdahl's result. Note also that the $U$ function may be expressed in terms of $\Gammaa$ when $c=\fr{\l}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{U \in {\mathbb N},$
which checks with Sundt's results \cite{sundt1995ruin}.\epsilon} \newcommand{\ol}{\overlineeR
\begin{equation}} \def\ee{\end{equation}R For including the case $a=l$ in our computations, note that the $U$ function may be expressed in terms of $\Gammaa$ when $c\in {\mathbb N},$ that $\lim_{z->0}z^n U\left} \def\ri{\rightft(
q+1,n+1,z r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)=\fr{\Gamma(n)}{\Gamma(1+\q)}$, and that $M\left} \def\ri{\rightft(
a,a+c,0 r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)=1$ \cite[{13.1.2}]{AS}).\symbolfootnote[4]{In the derivation below, we will use mostly the $K_1,K_2$ functions; however, for simplifications and numerical computations with Mathematica/Maple, one needs to convert to the reference forms $M,U$.Some computer systems use instead of $M$ the Laguerre function defined by $M(a,b,z)=L_{-a}^{b-1}(z)\fr{\Gammaamma(1-a) \Gammaamma(b)}{\Gamma(b-a)}$, which~yields for natural $-a$ the Laguerre polynomial of degree $-a$.}\epsilon} \newcommand{\ol}{\overlineeR
\fi
\subsection} \def\Kol{Kolmogorov {Essentials of first passage theory for the Segerdahl\ \ {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionroc \la{s:fpS}}
We gather now together the most basic first passage results for the Segerdahl\ \ process with $r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{U=1$ and $\c=0$ (so that the absolute ruin point is $-\Tc=0$), and $ \mu=1.$
The general case with $\c \neq 0, r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{U \neq 0, \mu \neq 1, $ can be obtained by replacing $z,a$ in the theorem below by $z(x):=\mu (x+ \fr{\c}{c_1})$ and $z(a)$ -- see Section r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uef{s:PG1}.
\begin{equation}} \def\ee{\end{equation}T \la{tK}When $r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{U =1, \c = 0,$ the following formulas hold, with $n=\q+\l$:
\begin{equation}} \def\ee{\end{equation}gin{enumerate}} \def\BI{\begin{equation}} \def\ee{\end{equation}gin{itemize} \im
The function $ H_\q$ is up to a {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrc
\begin{equation}} \def\ee{\end{equation}
H_\q(z)=K_1(z)\sim z^n e^{-z} M(\q+1,n+1,z)=z^n M(\l ,n+1,-z), \epsilon} \newcommand{\ol}{\overlinee
where $K_1(z)$ is the unique
$q$-harmonic function which increases on $(0,\infty} \def\Eq{\Leftrightarrow)$. The last expression, obtained via a Kummer transformation, is sometimes more stable numerically.
\im For $a >0$, the ruin function is
\begin{equation}} \def\ee{\end{equation} \Psi} \def\sRui{\overline{\Rui}_{\q}(z,a )=E_z [e^{-q T_{a ,-}}]=\l \fr{K_2(\q,n,z)}{K_2(\q,n+1,a )} = \bc \fr{\l }{\mu } \fr{e^{-z} z^n U(\q+1,n+1,z)}{e^{-a } a ^{n+1} U(\q+1,n+2,a )}, & z \gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq a >0\\\frac{\l \Gammaamma \left} \def\ri{\rightft(1+\qr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) }{ \Gammaamma \left} \def\ri{\rightft(1+\q +\lr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)}e^{-z} z^n U(\q+1,n+1,z),&a=0\epsilon} \newcommand{\ol}{\overlinec
\la{Pau}\epsilon} \newcommand{\ol}{\overlinee
({where we used $\lim_{z \to 0} z^{\q + 1 + \l} U[1 + \q, \q + 2 + \l,z] =\frac{\Gammaamma \left} \def\ri{\rightft(1+\q +\lr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) }{ \Gammaamma \left} \def\ri{\rightft(1+\qr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)}$} for the case $a=0$).
\im $$\Psi} \def\sRui{\overline{\Rui}_{\q}(z,a ,\theta)=E_z [e^{-q T_{a ,-}+\th X_{\ta}}]=\Psi} \def\sRui{\overline{\Rui}_{\q}(z,a )\fr{\mu}{\mu + \th}$$
(by the memoryless property of \mathbb{E}} \def\subsection} \def\Kol{Kolmogorov {\subsection} \def\Kol{Kolmogorov po \ claims).
\im For $z \gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq a > 0$, cf. \epsilon} \newcommand{\ol}{\overlineqr{We}, the~scale function is given by
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} && \la{WSeg} W_q(z,a )=H_\q(z)-\Psi} \def\sRui{\overline{\Rui}_{\q}(z,a )\int_0^{a } \mu e^{-\mu y}H_\q(a -y)dy=K_1(q,n,z)-\Psi} \def\sRui{\overline{\Rui}_{\q}(z,a ) \fr{K_1(\q,n+1,a )}{n+1}\nonumber \\&&=K_1(q,n,z)-\fr{\l}{n+1} \fr{K_1(\q,n+1,a )}
{K_2(\q,n+1,a )} K_2(q,n,z)\\\nonumber
&&=z^n e^{-z} {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionp{ M(\q+1,n+1,z)-\fr{\l }{n + 1} U(\q+1,n+1,z) \, \fr{M(q+1,n+2,a ) }{ U(\q+1,n+2,a )} }.
\epsilon} \newcommand{\ol}{\overlineeq
Since this is only determined \upc, we may and will usually take \symbolfootnoteo
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} &&W_\q(z,a )= z^n e^{-z}\begin{equation}} \def\ee{\end{equation}gin{vmatrix} M(1+\q,\l+1+\q,z) & M(1+\q,\l+2+\q,a ) \\ \l U(1+\q,\l+1+\q,z) & (\l + 1+\q) U(1+\q,\l+2+\q,a ) \epsilon} \newcommand{\ol}{\overlinend{vmatrix}=\\&& \nonumber z^n e^{-z}{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionp{
{(\l +\q + 1) M(\q+1,n+1,z) U(\q+1,n+2,a )-\l U(\q+1,n+1,z) \, M(q+1,n+2,a ) }}. \la{Wee}\epsilon} \newcommand{\ol}{\overlineeq
The second derivative is
\begin{equation}} \def\ee{\end{equation}a &&{W_\q''(z,a)}=\mu^2 {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionr{K_1''(q,n,z)-\fr{\l}{n+1} \fr{K_1(\q,n+1,a )}
{K_2(\q,n+1,a )} K_2''(q,n,z)} \Lra {W_\q''(z,a)} e^{a} a^{-n+2} / \Gammaamma (n) \\&&=\frac{\,
_1\tilde{F}_1(q+1;n+2;a) \left} \def\ri{\rightft(\frac{\lambda n \left} \def\ri{\rightft(a
n-a q-n^2+nr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) U(q+1,n+1,a)}{U(q+1,n+2,a)}+a (a-n+1)
((n+1) (n-q)-\lambda n)r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)}{n+1}+\\&&\left} \def\ri{\rightft(-(a+1) n+a
q+n^2r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) \, _1\tilde{F}_1(q+1;n+1;a)\epsilon} \newcommand{\ol}{\overlineea
and \sats $$W_\q''(0,0) =\frac{(n-1) n \left} \def\ri{\rightft(1-\frac{\lambda n U(q+1,n+1,0)}{(n+1)^2
U(q+1,n+2,0)}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)}{\Gammaamma (n+1)}<0$$
iff
\begin{equation}} \def\ee{\end{equation} \q + \l <1.\epsilon} \newcommand{\ol}{\overlinee
\im The two-sided ruin function \epsilon} \newcommand{\ol}{\overlineqr{Rui} with stopping at an upper bound $b$ \sats
\begin{equation}} \def\ee{\end{equation}a &&\Psi} \def\sRui{\overline{\Rui}_{\q}^b(z,a )=\Psi} \def\sRui{\overline{\Rui}_{\q}(z,a ) -\fr{W_\q(z,a )}{W_\q(b,a )} \Psi} \def\sRui{\overline{\Rui}_{\q}(b,a )=
\fr{\l }{a } \fr{e^{-z} z^n U(\q+1,n+1,z)}{e^{-a} a ^n U(\q+1,n+2,a )} -\fr{\l }{a } \fr{e^{-b} b^n U(\q+1,n+1,b)}{e^{-a} a ^n U(\q+1,n+2,a )} \times \\
&&
(\frac z b)^n e^{b-z}\fr{(\l +\q + 1) M(\q+1,n+1,z) U(\q+1,n+2,a )-\l U(\q+1,n+1,z) \, M(q+1,n+2,a ) }
{(\l +\q + 1) M(\q+1,n+1,b) U(\q+1,n+2,a )-\l U(\q+1,n+1,b) \, M(q+1,n+2,a ) }. \la{Zee}
\epsilon} \newcommand{\ol}{\overlineea
{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!E}} \defOrnstein-Uhlenbeck } \def\difs{diffusions{Ornstein-Uhlenbeck } \def\difs{diffusionsEN
\epsilon} \newcommand{\ol}{\overlineeT
\begin{equation}} \def\ee{\end{equation}gin{proof} 1. holds since $K_1(z)$ is the unique
solution which increases on $(0,\infty} \def\Eq{\Leftrightarrow)$.
2. is a particular case of Theorem 1.
3. This follows from the memoryless property of the \mathbb{E}} \def\subsection} \def\Kol{Kolmogorov {\subsection} \def\Kol{Kolmogorov poj.
4. Apply \epsilon} \newcommand{\ol}{\overlineqr{We}. More precisely, } \def\sats{satisfies} \def\sat{satisfy} \def\fun{function } \def\expl{explicit } \def\funs{functions} \def\thr{therefore } \def\wf{we find that } \def\inc{increasing} \def\resp{respectively} \def\proc{process} \def\eq{equation} \def\eqs{equations} \def\cd{(\cdot)} \def\sat{satisfy}\def\fac{factorization}\def\fno{from now on} \def\upc{up to a constant}\def\std{state dependent }\def\satg{satisfying }\def\c{c}\def\SL{Sturm Liouville }\def\Fq{\Phi_\q}\def\prc{proportionality constant} \def\Itt{It turns out that
\begin{equation}} \def\ee{\end{equation}a && W_q(z,a )=H_\q(z)-\Psi} \def\sRui{\overline{\Rui}_{\q}(x,a )\int_0^{a } e^{y- a }H_\q(y)dy=K_1(z,n)-\Psi} \def\sRui{\overline{\Rui}_{\q}(x,a ) \fr{K_1(a,n+1)}{n+1}\\&&=z^n e^{-z} M(\q+1,n+1,z)-{\l } \fr{e^{-z} z^n U(\q+1,n+1,z)}{ a ^{n+1} U(\q+1,n+2,a )} \int_0^{a } y^n M (
\q+ 1,n + 1,y )dy \nonumber
\\&&=z^n e^{-z} M(\q+1,n+1,z)-{\l } \fr{e^{-z} z^n U(\q+1,n+1,z)}{ a ^{n+1} U(\q+1,n+2,a )} \fr{a ^{n+1}}
{n+1} \, M(q+1,n+2,a) \nonumber
\\&&=z^n e^{-z} {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionp{ M(\q+1,n+1,z)-\fr{\l }{n + 1} U(\q+1,n+1,z) \, \fr{M(q+1,n+2,a) }{ U(\q+1,n+2,a )} },
\epsilon} \newcommand{\ol}{\overlineea
where we used $\int_0^{a } y^n M (
\q+ 1,n + 1,y )dy=\fr{a ^{n+1}}
{n+1} \, M(q+1,n+2,a)$ -- see \cite{AS}.
5. This result is immediate.
\epsilon} \newcommand{\ol}{\overlinend{proof}
\iffalse
The computation of $W$ allows solving the de Finetti } \def\app{approximation dividend problem.
Its solution is however trivial.
\figu{Wp}{$W_\q'(z,a), a=1, \q+ \l >1$. Optimal policy is $b^*=0$ (TMS)}{.4}
\figu{Wi}{$W_\q'(z,a), W_\q''(z,a), a=0,q+ \l <1$. Optimal policy is $b^*=\infty} \def\Eq{\Leftrightarrow$}{.4}
\begin{equation}} \def\ee{\end{equation}C The solution of the de Finetti } \def\app{approximation dividends barrier problem is
$b^*=0$ if $\q+ \l >1$, and $b^*=\infty} \def\Eq{\Leftrightarrow$ else.
\epsilon} \newcommand{\ol}{\overlineeC
\fi
\begin{equation}} \def\ee{\end{equation}R
The apparent singularity in \epsilon} \newcommand{\ol}{\overlineqr{Pau} when $a,z\to 0$ may be removed, since \begin{equation}} \def\ee{\end{equation} \la{ru00} \Psi} \def\sRui{\overline{\Rui}_{q}(0,0)=\lim_{z->0} \Psi} \def\sRui{\overline{\Rui}_{q}(z,z) =\lim_{z->0} \left} \def\ri{\rightft( \frac{\l }{z}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) \frac{U\left} \def\ri{\rightft(
\q +1 ,\l + \q +1,zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) }{U\left} \def\ri{\rightft(\q +1,\l + \q + 2,zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) }=\frac{\l \Gammaamma (\q+\l )}{\Gammaamma (\q+\l +1)}=\frac{\l }{\l + \q}.\epsilon} \newcommand{\ol}{\overlinee
This result has a clear probabilistic interpretation and holds in fact
clearly for any L\'evy measure } \def\thf{therefore } \def\mgf{moment generating function of finite negative intensity $\l$.
\epsilon} \newcommand{\ol}{\overlineeR
\begin{equation}} \def\ee{\end{equation}R
When $\q=0$, \epsilon} \newcommand{\ol}{\overlineqr{Wee} and
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \bc M(1,1+\l,z)= \lambda e^x x^{-\lambda } \gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}ta(\l,z)\\ U(1,1+\l,z)= e^x x^{-\l} \Gamma(\l,z)=e^x E_{1-\lambda }(z) \epsilon} \newcommand{\ol}{\overlinec \la{MUG}\epsilon} \newcommand{\ol}{\overlineeq
where $\Gamma(\l,y)=\int_y^\infty} \def\Eq{\Leftrightarrow {x^{\l -1}} e^{-x} dx$ is the incomplete gamma function, $\gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}ta(\l,y)=\int_0^y {x^{\l -1}} e^{-x} dx=\Gamma(\l)-\Gamma(\l,y)$ is the lower incomplete gamma function,
and $E_{\lambda }(z)=\int_1^\infty} \def\Eq{\Leftrightarrow t^{-\l} e^{-x t} dt$ is the ExpIntegral function, yield
$$W(z,a )= z^{\l} e^{-z} \begin{equation}} \def\ee{\end{equation}gin{vmatrix} M(1,\l+1,z) & M(1,\l+2,a ) \\ \l U(1,\l+1,z) & (\l + 1) U(1,\l+2,a ) \epsilon} \newcommand{\ol}{\overlinend{vmatrix}=e^a \left} \def\ri{\rightft(-a^{-\lambda -1}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) \Gammaamma (\lambda +2)\lambda ( \Gammaamma (\lambda
,z)-\Gammaamma (\lambda,a )).$$
Up to a constant, we have $$W(z,a ) \sim \Gammaamma (\lambda,a )-\Gammaamma (\lambda
,z)=\int_a^z t^{\l -1} e^{-t} dt \Lra W'(z,a ) \sim e^{-z} z^{\l -1}, $$ a particular case of the formula
$W'(z,a ) \sim e^{-z} c(z)^{\l -1},$
which will be rederived below.
\epsilon} \newcommand{\ol}{\overlineeR
\sec{{The Renewal Equation for the Scale Derivative of L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims Driven Langevin Processes \la{ex:fb} {Czarna~et~al.}~(2017)}}
One tractable extension of the Segerdahl-Tichy process is provided by {is} the ``Langevin-type'' risk process
defined by
\begin{equation}} \def\ee{\end{equation}gin{equation} \la{Lang} X_t=x+\int_0^t c(X_s)\;ds + Y_t,
\epsilon} \newcommand{\ol}{\overlinend{equation}
where $Y_t$ is a~spectrally negative L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionroc,
and $ c(u)$ is a~nonnegative } \def\fr{\frac} \def\Y{Y} \def\t{T} \def\ta{T_{a,-}} \def\tb{T_{b,+}} \def\tl{T_{l,-}
premium function {satisfying}
\begin{equation}} \def\ee{\end{equation} \la{intc} u >0 \Lra c(u) >0, \; \int_{x_0}^\infty} \def\Eq{\Leftrightarrow\frac{1}{c(u)}\,du = \infty} \def\Eq{\Leftrightarrow, \; \for x_0 >0.\epsilon} \newcommand{\ol}{\overlinee
The integrability condition above is necessary to preclude explosions.
Indeed when $Y_t$ is a compound Poisson process, in~ between jumps (claims) the risk process \epsilon} \newcommand{\ol}{\overlineqr{Lang} moves
deterministically along the curves $x_t$ determined by the vector field
\begin{equation}} \def\ee{\end{equation}a \la{vf}\fr {dx}{dt}=c(x) {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!E}} \defOrnstein-Uhlenbeck } \def\difs{diffusions{Ornstein-Uhlenbeck } \def\difs{diffusionsq t=\int_{x_0}^{x} \fr {du}{c(u)}:=C(x;x_0), \for x_0 >0.\epsilon} \newcommand{\ol}{\overlineea
From the last equality, it may be noted
that if $C(x;x_0)$ satisfies $\lim_{x \to \infty} \def\Eq{\Leftrightarrow} C(x;x_0)< \infty} \def\Eq{\Leftrightarrow$, then $x_t$ must explode, and~the stochastic process $X_t$ may explode.
\iffalse
${r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionhi(u,t)$, which~satisfies the
partial differential equation
\[
\frac{{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionartial {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionhi}{{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionartial t} = c(u)\frac{{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionartial {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionhi}{{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionartial u};\quad} \def\for{\forallad {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionhi(x_0,0)=x_0,
\]
with characteristic lines
Fixing an~arbitrary $x_0>0$ and putting $C(x)=\int_{x_0}^{x} \fr {du}{c(u)}$, the~solution of \epsilon} \newcommand{\ol}{\overlineqr{vf} starting from $x_0$
is $x_t(x_0)=C^{-1}(t+ C(x_0))$.
\fi
The case of Langevin processes
has been tackled recently in \cite{CPRY}, who provide the construction of the process \epsilon} \newcommand{\ol}{\overlineqr{Lang} in the particular case of { non-increasing} functions $c\cd$. This setup can be used to model dividend payments, and~other mathematical finance applications.
\cite{CPRY} showed that the $W,Z$ scale \funs \; which provide a~basis for first passage problems
of L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims spectrally positive } \def\dif{diffusion negative processes have two variables extensions $\mW,\mZ$ for the process \epsilon} \newcommand{\ol}{\overlineqr{Lang}, which~satisfy integral equations. The~equation for $\mW$, obtained by putting ${r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionhi(x)=c(a) -c(x)$ in \cite[eqn.~(40)]{CPRY}, is:
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{reneqg} &&\mW_{\q}(x,a) =W_{\q}(x-a) + \int_a^x (c(a)-c(z)) W_{\q}(x-z) \mW_{\q}'(z;a) dz,
\epsilon} \newcommand{\ol}{\overlineeq
where $W_{\q}$ is the scale function of the L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims process
obtained by replacing $c(x)$ with $c(a)$.
It follows that the { scale derivative}
\begin{equation}} \def\ee{\end{equation}a
\w_{\q}(x,a)={r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusiond{}{x}\mW_\q(x,a)\epsilon} \newcommand{\ol}{\overlineea of the scale function of the process \epsilon} \newcommand{\ol}{\overlineqr{Lang}
\sats \ a~Volterra renewal equation \cite[eqn.~(41)]{CPRY}:
\begin{equation}} \def\ee{\end{equation} \la{reneqw} \w_{\q}(x,a)\left} \def\ri{\right({1+ (c(x)-c(a)) W_\q(0)} r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui) ={w_{\q}(x-a)}
+ \int_a^x (c(a)- c(z)) w_{\q}(x-z) \w_{\q}(z;a) dz,
\epsilon} \newcommand{\ol}{\overlinee
where $w_{\q}$ is the derivative of the scale function of the L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims process
$Y_t=Y_t^{(a)}$ obtained by replacing $c(x)$ with $c(a)$.
This may further be written as:
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{reneqC} && w_{\q}(x-a) + \int_a^x w_{\q}(x-z) \w_{\q}(z;a) (c(a)-c(z)) dz =\bc \w_{\q}(x,a),& Y_t \text{ of unbounded variation}\\
\w_{\q}(x,a)\fr {c(x)}{c(a)},& Y_t \text{ of bounded variation}\epsilon} \newcommand{\ol}{\overlinec.
\epsilon} \newcommand{\ol}{\overlineeq
\iffalse
\begin{equation}} \def\ee{\end{equation}R In the bounded variation case, recalling that $W_\q$ is only defined up to a~constant, we could further simplify to:
\begin{equation}} \def\ee{\end{equation}a \w_{\q}(x,a)c(x) = \T w_{\q}(x-a) + \int_a^x \T w_{\q}(x-z) \w_{\q}(z;a) (c(a)-c(z)) dz, \epsilon} \newcommand{\ol}{\overlineea by choosing $\T w_{\q}$ to be the derivative of the \textcolor[rgb]{0.00,0.00,1.00}{renormalized scale function $\T W_{\q}$ of the L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims process $X$, which~\sats \ $\T W_{\q}(0)=1$.} \epsilon} \newcommand{\ol}{\overlineeR
\fi
\begin{equation}} \def\ee{\end{equation}Q It is natural to conjecture that the formula \epsilon} \newcommand{\ol}{\overlineqr{reneqC} holds for all drifts satisfying \epsilon} \newcommand{\ol}{\overlineqr{intc}, but this is an~open problem for now.
\epsilon} \newcommand{\ol}{\overlineeQ
\begin{equation}} \def\ee{\end{equation}R\label{rem_lt} Note that renewal equations are a~more appropriate tool than Laplace transforms for the general Langevin problem. Indeed, taking ``shifted \LT" $\mL_a f(x)=\int_a^\infty} \def\Eq{\Leftrightarrow e^{-s (y-a)} f(y) dy$ of \epsilon} \newcommand{\ol}{\overlineqr{reneqC}, putting
\begin{equation}} \def\ee{\end{equation}a \bc \H \w_{\q}(s,a)=\int_a^\infty} \def\Eq{\Leftrightarrow e^{-s { (y-a)}} \w_{\q}(y,a) dy,\\ \H \w_{q,c}(s,a)=\int_a^\infty} \def\Eq{\Leftrightarrow e^{-s{ (y-a)}} \w_{\q}(y,a) c(y)dy \\ {\H w_{\q}(s)=\int_0^{\infty}e^{-s y} w_{\q}(y)dy}\epsilon} \newcommand{\ol}{\overlinec,\epsilon} \newcommand{\ol}{\overlineea
and using $$\mL_a [\int_a^x f(x-y) l(y) dy](s)=\mL_0 f(s) \mL_a l(s)$$ yields equations with two unknowns:
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{LTCPRY} \H w_{\q}(s)(1+c(a)\H \w_{\q}(s,a)-\H \w_{q,c}(s,a))=\bc \H \w_{\q}(s,a)& \text{ unbounded variation case}\\ \fr{\H \w_{\q,c}(s,a)}{c(a)} & \text{ bounded variation case} \epsilon} \newcommand{\ol}{\overlinec,\epsilon} \newcommand{\ol}{\overlineeq
whose solution is not obvious.
\epsilon} \newcommand{\ol}{\overlineeR
\subsection} \def\Kol{Kolmogorov *{{The Linear Case} $c(x)=rx+c$}
To get explicit Laplace transforms } \def\q{q} \def\R{{\mathbb R}, we will turn next to Ornstein-Uhlenbeck } \def\difs{diffusions type processes\footnote{For some background first passage results on these processes, see \fe \cite{borovkov2008exit,LoefPat}.} $X\cd$, with $c(x)= c(a) + r (x-a)$,
which implies \begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray}\label{aux_jl_2}
\H \w_{\q,c}(s,a)=\int_a^\infty} \def\Eq{\Leftrightarrow e^{-s {(y-a)}} \w_{\q}(y,a) (r {(y-a)}+c(a)) dy=-r\H \w_{\q}'(s,a)+c(a) \H \w_{\q}(s,a).
\epsilon} \newcommand{\ol}{\overlineeq
Equation \epsilon} \newcommand{\ol}{\overlineqr{LTCPRY} simplify then to:
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{AUCPRY} \H w_{\q}(s)(1+r\H \w_{\q}'(s,a))=\bc \H \w_{\q}(s,a)
& \text{ unbdd variation case}\\\H \w_{\q}(s,a)-\fr{r}{c(a) }\H \w_{\q}'(s,a)& \text{ bdd variation case} \epsilon} \newcommand{\ol}{\overlinec.\epsilon} \newcommand{\ol}{\overlineeq
\begin{equation}} \def\ee{\end{equation}R Note that the only dependence on $a$ in this equation is via $c(a)$, and~via the shifted \LT. Since $a$ is fixed, we may and will from now on simplify by assuming w.l.o.g. $a=0$, and~write $c=c(a)$. \epsilon} \newcommand{\ol}{\overlineeR
Let now
$$ \kappa} \def\l{\lambda} \def\a{\alpha(s)= \a_0 s^2 + c s- s \H {\overline \Pi} \def\x{\xi} \def\ith{it holds that }(s) -q, \a_0 >0, $$ denote the Laplace exponent } \def\LT{Laplace transform} \def\BM{Brownian motion \ or symbol of the L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims process $Y_t=\sqrt{2\alpha_0}B_t-J_t+ct$,
and recall that
\begin{equation}} \def\ee{\end{equation}a w_{\q}(s)=\bc \fr s{\kappa} \def\l{\lambda} \def\a{\alpha(s)}&\text{ unbdd variation case}\\
\fr s{\kappa} \def\l{\lambda} \def\a{\alpha(s)}-\fr 1 c&\text{ bdd variation case}\epsilon} \newcommand{\ol}{\overlinec \epsilon} \newcommand{\ol}{\overlineea
(where we have used that $W_\q(0)=0(\fr 1 c)$ in the two cases, respectively).
We obtain now from \epsilon} \newcommand{\ol}{\overlineqr{AUCPRY}
the following ODE
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray}\label{odeCPRY}
r\H \w_{\q}'(s,a)-\frac{\kappa} \def\l{\lambda} \def\a{\alpha(s)}{s}\H \w_{\q}(s,a)=-1+\fr{\kappa} \def\l{\lambda} \def\a{\alpha(s)}{ s}W_\q(0)=\bc-1&\text{ unbdd variation case}\\-1+\fr{\kappa} \def\l{\lambda} \def\a{\alpha(s)}{c s}:=-\fr{h(s)}{c}&\text{ bdd variation case}\epsilon} \newcommand{\ol}{\overlinec,
\epsilon} \newcommand{\ol}{\overlineeq
where $$h(s)=\H {\overline \Pi} \def\x{\xi} \def\ith{it holds that }(s) +\fr \q{ s}.$$
\begin{equation}} \def\ee{\end{equation}R \la{r:int}
The Equation \epsilon} \newcommand{\ol}{\overlineqr{odeCPRY} is easily solved multiplying by an~integrating factor
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{defI0} I_{q}(s,s_0)=e^{-\int_{s_0}^{s}\,\frac{ \kappa} \def\l{\lambda} \def\a{\alpha(z)/z}r dz }=e^{-\int_{s_0}^{s}\,\frac{\a_0 z+c -\woF
(z)-\q/z}r dz }, \epsilon} \newcommand{\ol}{\overlineeq where $s_0 > 0$ is an~arbitrary integration limit chosen so that
the integral converges (the formula \epsilon} \newcommand{\ol}{\overlineqr{defI0} appeared first in \cite{AU}).
To simplify, we may choose $s_0=0$ to integrate
the first part $\a_0 z+c -\woF(z)$, and~ a~different lower bound $s_0=1$ to integrate $q/z$.
Putting $\Tq=\fr q r, \Tc=\fr c r, \T \a_0 =\fr {\a_0} r$, we~get~that
\begin{equation}} \def\ee{\end{equation}gin{equation} \label{defI}
I_{\q}(s)=e^{-\int_{\cdot}^{s}\,\frac{ \kappa} \def\l{\lambda} \def\a{\alpha(z)/z}r dz }= s^{\Tq }
e^{-\left} \def\ri{\rightft[ \left} \def\ri{\rightft( \frac{\T \a_0}
{2}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) s^{2}+\Tc sr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight] + \frac{1}{r \, } \int_0^s \woF(z) dz} := s^{\Tq } I(s):=e^{-\Tc s} i_{\q}(s),
\epsilon} \newcommand{\ol}{\overlinend{equation}
where we replaced $s_0$ by $\cdot$ to indicate that two different lower bounds are in fact used, and~we put $I(s)= I_0(s)$ (the subscript $0$ will be omitted when $q=0$).
\epsilon} \newcommand{\ol}{\overlineeR
Solving \epsilon} \newcommand{\ol}{\overlineqr{odeCPRY} yields:
\begin{equation}} \def\ee{\end{equation}T \la{cor:wI} Fix $a$ and put $\overline I_{\q}(s)= \int_s^\infty} \def\Eq{\Leftrightarrow I_{\q}(y) dy$. Then, the \LT \ \ of the \sd of an~Ornstein-Uhlenbeck } \def\difs{diffusions type process \epsilon} \newcommand{\ol}{\overlineqr{Lang} satisfies:
\begin{equation}} \def\ee{\end{equation}gin{equation} \la{wI} \H \w_{\q}(s,a) = \fr{ \overline I_{\q}(s) }{r I_{\q}(s)}- W_\q(0)=\bc \fr{ \overline I_{\q}(s) }{r I_{\q}(s)},&
\text{ in the unbounded variation case}\\
\fr{ \overline I_{\q}(s) }{r I_{\q}(s)} -\fr 1 c,&\text{ in the bounded variation case}\epsilon} \newcommand{\ol}{\overlinec.\epsilon} \newcommand{\ol}{\overlinend{equation}
\epsilon} \newcommand{\ol}{\overlineeT
\begin{equation}} \def\ee{\end{equation}gin{proof}
{In the unbounded}
variation case, applying the integrating factor to \epsilon} \newcommand{\ol}{\overlineqr{odeCPRY} yields immediately:
\begin{equation}} \def\ee{\end{equation}a \nonumber &&\H \w_{\q}(s,a) I_{\q}(s)= r^{-1} \int_s^\infty} \def\Eq{\Leftrightarrow {I_{\q}(y)} dy=r^{-1} \overline I_{\q}(s).\epsilon} \newcommand{\ol}{\overlineea
In the bounded variation case, we observe that $$i_{\q}'(s)=\fr{h(s)}r i_{\q}(s),$$ where $i_q$ is defined in \epsilon} \newcommand{\ol}{\overlineqr{defI}. An integration by parts now yields
\begin{equation}} \def\ee{\end{equation}a \nonumber &&\H \w_{\q}(s,a) I_{\q}(s)= \int_s^\infty} \def\Eq{\Leftrightarrow \fr{h(y)}{c r} {I_{\q}(y)} dy=\int_s^\infty} \def\Eq{\Leftrightarrow \fr{h(y)}{c r} e^{-\Tc y} i_{\q}(y) dy\\&&=c^{-1} \int_s^\infty} \def\Eq{\Leftrightarrow e^{-\Tc y} i_{\q}'(y) dy=c^{-1}(-I_{\q}(s)+\Tc \int_s^\infty} \def\Eq{\Leftrightarrow e^{-\Tc y} i_{\q}(y) dy)=r^{-1} \overline I_{\q}(s) dy-c^{-1} I_{\q}(s).\epsilon} \newcommand{\ol}{\overlineea
\epsilon} \newcommand{\ol}{\overlinend{proof}
\begin{equation}} \def\ee{\end{equation}R The result \epsilon} \newcommand{\ol}{\overlineqr{wI} is quite similar to the \LT \ for the \sur and r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up\ (Gerber-Shiu functions) derived in \cite[p. 470]{AU}---see \epsilon} \newcommand{\ol}{\overlineqr{AU}, \epsilon} \newcommand{\ol}{\overlineqr{seg0q} below; the main difference is that in that case additional effort was needed for finding the values $\sRui(a,a), \Psi} \def\sRui{\overline{\Rui}(a,a)$. \epsilon} \newcommand{\ol}{\overlineeR
\sec{The \LT -Integrating Factor Approach for Jump-Diffusions with Affine Operator
{Avram and Usabel} (2008) \la{s:AU}}
We summarize now for comparison the results of \cite{AU} for the still tractable, more general extension of the Segerdahl-Tichy process provided by
jump-diffusions with
{ affine premium and volatility}
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{aff}
\bc c(x)= r x + c\\ \fr{\s^2(x)}2= \a_1 x + \a_0, \; \a_1, \a_0 \gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq 0.
\epsilon} \newcommand{\ol}{\overlinec \epsilon} \newcommand{\ol}{\overlineeq
\iffalse
first passage problems for Kolmogorov-Wong-Pearson (KWP) operators with spectrally one-sided jumps\footnote{the most often used name in the literature today is Pearson operators; however, these diffusions have been introduced by Kolmogorov \cite{kolmogoroff1931analytischen}, see also \cite{avram2009series}, \cite{avram2013spectral}
for some recent work.} with
{\bf quadratic
diffusion rate} and {\bf
linear drift} \begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{qco}
&&v(x)=\fr{\s^2(x)}{2}=\a_{2}x^{2}+
\a_1 x+ \a_0 \nonumber \\&& c(x) = r x + c\epsilon} \newcommand{\ol}{\overlineeq
\fi
Besides Ornstein-Uhlenbeck } \def\difs{diffusions type processes, \epsilon} \newcommand{\ol}{\overlineqr{aff} includes
another famous particular case, \CIR (CIR) type processes, obtained {when $\a_1 >0$}.
Introduce now a~{ combined ruin-survival} expected payoff at
time $t$
\begin{equation}} \def\ee{\end{equation}gin{equation}
{V}(t,u)=\mathbb{E}_{ X_{0}=u }\left} \def\ri{\rightft[{w}\left} \def\ri{\rightft( X_{T }r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)
\,1_{\left} \def\ri{\rightft\{ T \,<\,tr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight\} } + \,p(X_{t}
)\,\,1_{\left} \def\ri{\rightft\{ T \,\gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq\,tr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight\} }r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight] \label{payoff}
\epsilon} \newcommand{\ol}{\overlinend{equation}
where ${w}, p$
represent, respectively:
\begin{equation}} \def\ee{\end{equation}gin{itemize}
\item A penalty ${w}(X_{T })$ at a~stopping time ${T }$, \; $\,
{w}:\;\mathbb{R} \mathbb{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uightarrow R}$
\item A reward for survival after $t$ years: $p(X_{t}
),\,\,{\epsilon} \newcommand{\ol}{\overlinemph{p}}:
\;\mathbb{R} \mathbb{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uightarrow R}^{+}$.
\epsilon} \newcommand{\ol}{\overlinend{itemize}
Some particular cases of interest are the survival probability
for $t$ years, obtained with
\[
w(X_{T })=0,\;\;p(X_{t})=1_{\left} \def\ri{\rightft\{ X_{t}\,\,\gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq\,0\;r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight\}
}\;\;
\]
and the ruin probability with deficit larger in absolute value than
$y$, obtained with
\[
w(X_{T })=1_{\left} \def\ri{\rightft\{ X_{T }\,<\,-y\,\;r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight\} },\;\;p(X_{t})=0\;\;
\]
Let \begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} {V}_{q}(x)=\int_0^\infty} \def\Eq{\Leftrightarrow q e^{-q t} V(t,x) dt =E_x \left} \def\ri{\rightft[{w}\left} \def\ri{\rightft( X_{T }r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)
\,1_{\left} \def\ri{\rightft\{ T \,<\,\mathbf{e}} \def{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrocs{processes} \def{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionroc{process_qr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight\} } + \,p(X_{\mathbf{e}} \def{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrocs{processes} \def{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionroc{process_q}
)\,\,1_{\left} \def\ri{\rightft\{ T \,\gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq\,\mathbf{e}} \def{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrocs{processes} \def{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionroc{process_qr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight\} }r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight],\la{LCGS} \epsilon} \newcommand{\ol}{\overlineeq
denote a~ ``Laplace-Carson''/``Gerber Shiu'' discounted penalty/pay-off.
\iffalse
\subsection} \def\Kol{Kolmogorov {TThe Decomposition of the Gerber Shiu Discounted Penalty \la{s:GSdec} }
It is natural
to separate the Gerber-Shiu } \def\Fr{Furthermore, } \deffurthermore } \def\nonh{nonhomogeneous } \def\td{\tau} \def\probas{probabilities{furthermore }\def\vars{random variables }\newcommand{\bff}[1]{{\mbox{\boldmath$#1$}} discounted penalty \epsilon} \newcommand{\ol}{\overlineqr{LCGS} into three
parts:
$${V}_{q}(x)=\int_0^\infty} \def\Eq{\Leftrightarrow q e^{-q t} V(t,x) dt={V}_{q}^{(j)}(x)+ {V}_{q}^{(d)}(x)+P_q(x)$$
where ${V}_{q}^{(j)}(x), {V}_{q}^{(d)}(x)$ represent, respectively,
the payoffs in the case of having crossed the boundary via a~jump or
continuously \cite{CY05}, and~$P_q(x)$ represents the payoff in case of survival.
Let
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}&&
G=G_j + G_d, \nonumber \\
&& G_d \; V(x)=[c(x) \,\frac{{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionartial }{{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionartial x}+ v(x)
\frac{{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionartial^{2}}{{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionartial x^{2}}] \; V(x)
:=A(x,D) \; V(x) \nonumber \\&&
\la{jGe} G_j \; V(x) =\int_{0}^x (V(x-z) -V(x)) \Pi} \def\x{\xi} \def\ith{it holds that (dz) - \l V(x) \la{dGe} \epsilon} \newcommand{\ol}{\overlinend{eqnarray}
where $A(x,D)=c(x) \, D+ v(x) D^{2}$ with $D=\frac{{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionartial
}{{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionartial x}$ denotes the diffusion part of our operator, and
$G_j$ denotes the jump part.
\begin{equation}} \def\ee{\end{equation}gin{Ass} We assume that the function $p(x)$ is bounded
and has a~limit at infinity $p=p(\infty)$, assumed w.l.o.g. to be
either $1$ or $0$.
\epsilon} \newcommand{\ol}{\overlinend{Ass}
\begin{equation}} \def\ee{\end{equation}L
Under Assumption A, the~functions ${V}_{q}^{(j)}(x), {V}_{q}^{(d)}(x)$ satisfy,
respectively:
\begin{equation}} \def\ee{\end{equation}a \begin{equation}} \def\ee{\end{equation}gin{cases} (G -\q) \; \,{V}_{q}^{(j)}(x)
+w_\Pi} \def\x{\xi} \def\ith{it holds that (x)=0, \;
\text{ for $x\gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq0$}
\\{V}_{q}^{(j)}(0)=0
\\ {V}_{q}^{(j)}(\infty)=0
\\A(0,D) {V}_{q}^{(j)}(0) = - h(0),
\nonumber \epsilon} \newcommand{\ol}{\overlinend{cases} \epsilon} \newcommand{\ol}{\overlineea
where $w_\Pi} \def\x{\xi} \def\ith{it holds that (x)= \int_{x}^\infty} \def\Eq{\Leftrightarrow w(x -u) \Pi} \def\x{\xi} \def\ith{it holds that (du), w_\Pi} \def\x{\xi} \def\ith{it holds that (0)=\int_{0}^{\infty}w(-u) \Pi} \def\x{\xi} \def\ith{it holds that (du),$
\begin{equation}} \def\ee{\end{equation}a\begin{equation}} \def\ee{\end{equation}gin{cases} (G -\q)\;
\,{V}_{q}^{(d)}(x) + q p(x)=0, \; \nonumbernumber
\text{ for $x\gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq0$}
\\{V}_{q}^{(d)}(0)=w(0_{-})\qquad \text{ if }\a_0>0 \nonumber
\\ {V}_{q}^{(d)}(\infty)=p(\infty)
\\A(0,D) {V}_{q}^{(d)}(0) = h(0)
\nonumber \epsilon} \newcommand{\ol}{\overlinend{cases}\epsilon} \newcommand{\ol}{\overlineea
where $\l=\Pi} \def\x{\xi} \def\ith{it holds that (0,\infty} \def\Eq{\Leftrightarrow)$, and
their sum satisfies: \begin{equation}} \def\ee{\end{equation}gin{equation} \la{IDE} \begin{equation}} \def\ee{\end{equation}gin{cases} (G -\q) \; \,{V}_{q}(x) +h(x)=0, \;
\text{ for $x\gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq0$}
\\{V}_{q}(0)=w(0_{-})\qquad \text{ if }\a_0>0
\\ {V}_{q}(\infty)=p(\infty)
\\A(0,D) {V}_{q}(0) = 0
\epsilon} \newcommand{\ol}{\overlinend{cases}\epsilon} \newcommand{\ol}{\overlinend{equation}
where $h(x):=w_\Pi} \def\x{\xi} \def\ith{it holds that (x)+q p(x)$ denotes a~combination of the two payoffs.
\epsilon} \newcommand{\ol}{\overlineeL
{\bf Proof:} For the case $q=0,p(x)= 0$, see \cite[Thm 3.1-3.3]{CY05}.
For $q>0,$ the proof is similar to \cite[Lem 1]{AU}.
\fi
\begin{equation}} \def\ee{\end{equation}gin{Pro} \cite[Lem. 1, Thm. 2]{AU} {(a)} Consider the {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionroc\ \ \epsilon} \newcommand{\ol}{\overlineqr{aff}. Let $V_q(x)$ denote the corresponding Gerber-Shiu } \def\Fr{Furthermore, } \deffurthermore } \def\nonh{nonhomogeneous } \def\td{\tau} \def\probas{probabilities{furthermore }\def\vars{random variables }\newcommand{\bff}[1]{{\mbox{\boldmath$#1$}} function \epsilon} \newcommand{\ol}{\overlineqr{LCGS}, let $w_\Pi} \def\x{\xi} \def\ith{it holds that (x)= \int_{x}^\infty} \def\Eq{\Leftrightarrow w(x -u) \Pi} \def\x{\xi} \def\ith{it holds that (du)$ denote the expected payoff at ruin,
and let $g(x):=w_\Pi} \def\x{\xi} \def\ith{it holds that (x)+q p(x), \H g(s)$ denote the combination of the two payoffs and its \LT; note that the particular cases $$\widehat{g}(s)=\fr \q s, \; \widehat{g}(s)=\l \overline F(s)$$ correspond to the \sur and r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up, respectively \cite{AU}.
Then, the~Laplace transform of the derivative
$$V_*(x)=\int_0^{\infty}e^{-sx}dV_q(x)={s}{\H V_q(s)- V_q(0)}$$
satisfies the ODE
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{AU*} && \left} \def\ri{\rightft( \a_1 s + rr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) V_*(s)^{{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrime}- (\fr{\kappa} \def\l{\lambda} \def\a{\alpha(s)}{s}-\a_1)
V_*(s) = - h(s) {V}_{q}(0) -{\a_0}\,{V}_{q}^{{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrime}(0) +\widehat{g}(s) \Lra \nonumber\\&&
V_*(s) I_{\q}(s)=\int_s^\infty} \def\Eq{\Leftrightarrow
I_{\q}(y)\fr{h(y) V_{\q}(0) +{\a_0}\,{V}_{q}^{{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrime}(0) - \widehat{g} (y)}{r +\a_1 y}\; dy,\epsilon} \newcommand{\ol}{\overlineeq
where $h(s)=\H {\overline \Pi} \def\x{\xi} \def\ith{it holds that }(s) +\fr \q{ s}$ (this corrects a~typo in \cite[eqn. (9)]{AU}), and where the integrating factor is obtained from \epsilon} \newcommand{\ol}{\overlineqr{defI0} by replacing $c$ with $c-\a_1$ \cite[eqn. (11)]{AU}.
Equivalently,
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{AU} && r \left} \def\ri{\rightft( s \H V_{\q}(s)r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)^{{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrime}- \fr{\kappa} \def\l{\lambda} \def\a{\alpha(s)}{s}
\,s \H V_{\q}(s) = - (c +\a_0 s) {V}_{q}(0) -{\a_0}\,{V}_{q}^{{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrime}(0) +\widehat{g}(s)\Lra \nonumber\\&&s \H V_{\q}(s) I_{\q}(s)=\int_s^\infty} \def\Eq{\Leftrightarrow I_{\q}(y)\fr{(c +\a_0 s) V_{\q}(0) +{\a_0}\,{V}_{q}^{{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrime}(0) - \widehat{g} (y)}{r +\a_1 y} \; dy.\epsilon} \newcommand{\ol}{\overlineeq
(b) If $\a_0=0=\a_1$ and $q>0$, the \sur probability satisfies
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \label{seg0q} && \sRui_{\q}(0) =\fr{\Tq \overline I_{q-1}(0)}{\Tc \overline I_{q}(0)} \la{s0} \\&&s \H{\sRui}_{\q}(s) I_{\q}(s) = \int_s^\infty} \def\Eq{\Leftrightarrow I_{\q}(y)(\Tc \sRui_{\q}(0)- \fr {\Tq}y) dy=\Tc \sRui_{\q}(0) \overline I_{\q}(s)-\Tq \overline I_{q-1}(s)=\Tq\left} \def\ri{\right(\fr{ \overline I_{q-1}(0)}{ \overline I_{q}(0)} \overline I_{q}(s) - \overline I_{q-1}(s)r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui) \nonumber
\epsilon} \newcommand{\ol}{\overlineeq
\epsilon} \newcommand{\ol}{\overlinend{Pro}
\begin{equation}} \def\ee{\end{equation}gin{proof}
{(b) The survival {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionro \ follow from (a), by plugging $\widehat{g}(y)=\fr q y$. Indeed, the Equation~\epsilon} \newcommand{\ol}{\overlineqr{AU} becomes for the \sur {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionroba}
\[s \H \sRui_{\q}(s) I_{\q}(s)=\int_s^\infty} \def\Eq{\Leftrightarrow I_{\q}(y)(\Tc \sRui_{\q}(0) - \fr \Tq y) \; dy = \Tc \sRui_{\q}(0) \overline I_{\q}(s)-\Tq \overline I_{\q-1}(s).\]
Letting $s \to 0$ in this equation yields $\sRui_{\q}(0) =\fr{\Tq \overline I_{q-1}(0)}{\Tc \overline I_{q}(0)}$.
As a~check, let us verify also Equation \epsilon} \newcommand{\ol}{\overlineqr{AU} for the ruin {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionroba, by plugging $\widehat{g}(y)=\l \overline F(y)$:
\begin{equation}} \def\ee{\end{equation}a &&s \H \Psi} \def\sRui{\overline{\Rui}_{\q}(s) I_{\q}(s) = \int_s^\infty} \def\Eq{\Leftrightarrow I_{\q}(y)(\Tc \Psi} \def\sRui{\overline{\Rui}_{\q}(0)-\l \overline F(y)) dy=
\Tc \Psi} \def\sRui{\overline{\Rui}_{\q}(0) \bar I_{\q}(y)-J(y),\\&& J(y)=\int_s^\infty} \def\Eq{\Leftrightarrow y^{\Tq }
e^{- \Tc y} j'(y) dy, \; j(y):=e^{\Tl \int_0^y \bar F(z) dz}.\epsilon} \newcommand{\ol}{\overlineea
Integrating by parts, $J(y)= -I_{q}(s) +\Tc \overline I_{\q}(s)-\Tq \overline I_{q-1}(s)$. Finally,
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} &&s \H \Psi} \def\sRui{\overline{\Rui}_{\q}(s) I_{\q}(s) = \Tc (1-\sRui_{\q}(0)) \overline I_{q}(s)-\Big(-I_{q}(s) +\Tc \overline I_{\q}(s)-\Tq \overline I_{q-1}(s)\Big)=\nonumber\\&&I_{q}(s)+\Tq \overline I_{q-1}(s)-\Tc \sRui_{\q}(0) \overline I_{\q}(s)=I_{q}(s) -s \H{\sRui}_{\q}(s) I_{\q}(s).\epsilon} \newcommand{\ol}{\overlineeq
\epsilon} \newcommand{\ol}{\overlinend{proof}
\iffalse
\begin{equation}} \def\ee{\end{equation}a &&
\H \w_{\q}(s) I_{\q}(s)=
\int_s^\infty} \def\Eq{\Leftrightarrow (\fr{\Tl}{y+ \mu}+\fr{\Tq}{y}) ({1 +y/\mu})^{\Tl} \; {y^q} e^{-\Tc y} \; dy=\\&&\Tl \int_s^\infty} \def\Eq{\Leftrightarrow ({\mu +y})^{\Tl-1} \; {y^q} e^{-\Tc y} \; dy+\Tq \int_s^\infty} \def\Eq{\Leftrightarrow ({\mu +y})^{\Tl} \; {y^{q-1}} e^{-\Tc y} \; dy\epsilon} \newcommand{\ol}{\overlineea
Let us put $g_{q,\Tl}(s)=\int_s^\infty} \def\Eq{\Leftrightarrow y^{\Tq} e^{- \Tc y}(1+y/\mu)^{\Tl} dy.$ The equations
$ s \H V_{\q}(s) I_{\q}(s)=\int_s^\infty} \def\Eq{\Leftrightarrow I_{\q}(y)(c V_0 - \H h(y))/r dy, \Psi} \def\sRui{\overline{\Rui}_{\q}(0)=\fr{\lambda} {\mu c} \fr{g_{q,\Tl-1}(0)}{g_{q,\Tl}(0)} $ yield
\begin{equation}} \def\ee{\end{equation}a &&s \H \Psi} \def\sRui{\overline{\Rui}_{\q}(s) I_{\q}(s)={\int_s^\infty} \def\Eq{\Leftrightarrow (\Tc \Psi} \def\sRui{\overline{\Rui}_{\q}(0)-\fr{\Tl}{y+\mu})I_{\q}(y) dy}= \fr{\Tl}{\mu} \left} \def\ri{\right(g_{\Tq, \Tl-1}(0)\fr{g_{\Tq,\Tl}(s)}{g_{\Tq,\Tl}(0)}
- g_{\Tq,\Tl-1}(s)r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui).\epsilon} \newcommand{\ol}{\overlineea
\fi
\subsection} \def\Kol{Kolmogorov *{Segerdahl's Process via the Laplace Transform\ Integrating Factor}
We revisit now the particular case of Segerdahl's process with \mathbb{E}} \def\subsection} \def\Kol{Kolmogorov {\subsection} \def\Kol{Kolmogorov po \ claims of rate $\mu$ and $\a_0=\a_1=0$.
Using
$\overline \Pi} \def\x{\xi} \def\ith{it holds that (y)=\l F_C(y) dy=\fr \l{y + \mu}$ we find that
for Segerdahl's process the integrand in the exponent is
$$\fr{\kappa} \def\l{\lambda} \def\a{\alpha(s)}{rs}= \Tc -\Tl/(s+ \mu)-\Tq/s,$$ and the integrating factor \epsilon} \newcommand{\ol}{\overlineqr{defI} may be taken as
$$I_{\q}(x)=x^{\Tq} e^{- \Tc x}(1+x/\mu)^{\Tl}.$$
The antiderivative $\bar I_\q(x)$ is not explicit, except
for:
\begin{equation}} \def\ee{\end{equation}gin{enumerate}
\item $x=0$, when \ith
$$ \bar I_\q(0)= \mu^{\Tq +1 } U (\Tq +1,\Tq +\Tl +2,\Tc \mu ),$$
where \cite[{13.2.5}]{AS}
\footnote{Note that when $c=1$, this function
reduces
to a~power:
$
U\left} \def\ri{\rightft( a,a+1,zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) =\frac{\int_{0}^{\infty}~t^{a-1}~e^{-z
t}~dt}{\Gammaamma(a)}=z^{-a}.
$}
$$U[a, a+c, z] = \fr 1{\Gammaamma[a]} \int_0^\infty} \def\Eq{\Leftrightarrow e^{-z t} t^{a - 1} (t + 1)^{c - 1} dt, Re[z] > 0, Re[a] > 0.$$
\item for $q=0$, when \ith
$$I(x)=e^{- \Tc x}(1+x/\mu)^{\Tl}, \; \bar I(x)=\int_x^\infty} \def\Eq{\Leftrightarrow I(y) dy=\frac{e^{\Tc \mu } (\Tc \mu)^{-\Tl } \Gammaamma (\Tl +1,\Tc (x+\mu ))}{\Tc}.
$$
\epsilon} \newcommand{\ol}{\overlinend{enumerate}
However, the~Laplace transforms } \def\q{q} \def\R{{\mathbb R} of the integrating factor $I_{\q}(x)$ and its primitive are explicit:
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \nonumber &&\H I_{\q}(s)=\int_0^\infty} \def\Eq{\Leftrightarrow e^{-(s+\Tc) x} x^{\Tq} (1+x/\mu)^{\Tl}=\Gamma(\Tq +1)U(\Tq +1, \Tq +\Tl +2, \mu (\Tc+s),\\&& \H {\overline I}_{\q}(s)= \Gamma(\Tq +1)\fr{U(\Tq +1, \Tq +\Tl +2, \mu \Tc)-U(\Tq +1, \Tq +\Tl +2, \mu (\Tc+s)}s. \la{LTs}\epsilon} \newcommand{\ol}{\overlineeq
Finally, we may compute:
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} &&\sRui_{\q}(0) =\fr{\Tq \overline I_{q-1}(0)}{\Tc \overline I_{q}(0)}=\fr{\Tq U (\Tq,\Tq +\Tl +1,\Tc \mu )}{\Tc \mu U (\Tq +1,\Tq +\Tl +2,\Tc \mu )}\nonumber \\&&\Psi} \def\sRui{\overline{\Rui}_{\q}(0) =1-\sRui_{\q}(0)=1- \fr{\Tq U (\Tq,\Tq +\Tl +1,\Tc \mu )}{\Tc \mu U (\Tq +1,\Tq +\Tl +2,\Tc \mu )} \nonumber \\&&=\fr{\Tc \mu U (\Tq +1,\Tq +\Tl +2,\Tc \mu )- {\Tq} U (\Tq,\Tq +\Tl +1,\Tc \mu )}{\Tc \mu U (\Tq +1,\Tq +\Tl +2,\Tc \mu )} \nonumber \\&&=\left} \def\ri{\rightft( \frac{\l}{c \mu }r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) \frac{ U\left} \def\ri{\rightft( \T \q+1
,\T \q + 1+\Tl,\mu \Tc r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) }{U\left} \def\ri{\rightft( \T \q+1,\T \q +
\Tl+2,\mu \Tc r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) },\epsilon} \newcommand{\ol}{\overlineeq
where we used the identity \cite[{13.4.18}]{AS}
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{sPau} U[a-1, b, z]+(b-a) U[a, b, z]=z U[a, b+1,z], a>1,\epsilon} \newcommand{\ol}{\overlineeq
with $a =\Tq+1, b=\Tq + \Tl +1$.
This checks the (corrected) Paulsen result \epsilon} \newcommand{\ol}{\overlineqr{Pauruin} for $x=0$.
\begin{equation}} \def\ee{\end{equation}R We can now numerically answer Problem 1: (a) obtain the antiderivative $\bar I_\q(x)$ by numerical integration; (b) compute the \LT \ of the scale derivative by \epsilon} \newcommand{\ol}{\overlineqr{wI}; c) Invert the \LT. \epsilon} \newcommand{\ol}{\overlineeR
The example above raises the question:
\begin{equation}} \def\ee{\end{equation}Q Is it possible to compute explicitly the Laplace transforms } \def\q{q} \def\R{{\mathbb R} of the integrating factor $I_{\q}(x)$ and its primitive for affine processes
with phase-type jumps? \epsilon} \newcommand{\ol}{\overlineeQ
\sec{Revisiting Segerdahl's Process via the Scale Derivative/Integrating Factor Approach \la{s:Segr}}
Despite the new scale derivative/integrating factor approach, we were not able to produce further explicit results beyond \epsilon} \newcommand{\ol}{\overlineqr{Pau}, due to the fact that neither the scale derivative, nor the integral of the integrating factor are explicit when $\q >0$ (this is in line with \cite{avram2010lie}). \epsilon} \newcommand{\ol}{\overlineqr{Pau} remains thus for now an~outstanding, not well-understood exception.
\begin{equation}} \def\ee{\end{equation}Q Are there other explicit first passage results for Segerdahl's process when $\q>0$? \epsilon} \newcommand{\ol}{\overlineeQ
In the next subsections, we show that via the scale derivative/integrating factor approach, we may rederive well-known} \def\ts{two sided exit \ results for $q=0$.
\subsection} \def\Kol{Kolmogorov {Laplace Transforms of the Eventual Ruin and Survival Probabilities \la{s:Ak}}
For $\q=0$, both Laplace transforms } \def\q{q} \def\R{{\mathbb R} and their inverses are explicit, and~several classic results may be easily checked. The~scale derivative may be obtained using Proposition r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uef{cor:wI} and
$ \Gammaamma (\Tl +1,v)=e^{-v} v^{\Tl}+ \lambda \Gammaamma (\Tl,v) $ with $v=\Tc (s+\mu )$.
We find \begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{scSeg} &&\widehat {\mathbf w}(s,a) =\fr{{e^{\Tc \mu } (\Tc \mu )^{-\lambda } \Gammaamma (\lambda +1,\Tc
(s+\mu ))}}{e^{- \Tc s}(1+s/\mu)^{\Tl}}-1=1+ \lambda e^{v } (v )^{-\Tl } \Gammaamma (\Tl,v) -1=\Tl U(1,1+ \Tl,\Tc (s+\mu ))\nonumber \\&& \Lra {\w}(x,a)=
\frac{\Tl}{{\Tc}}\left} \def\ri{\rightft(1+ \frac{x}{\tilde{c}}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)^{\Tl-1}
e^{-\mu x},\epsilon} \newcommand{\ol}{\overlineeq
which checks \epsilon} \newcommand{\ol}{\overlineqr{scder}.
Using again $\H {\w}(s)=\Tc \fr{ \overline I(y)}{ I(y)} -1$ yields the ruin and \sur probabilities:
\begin{equation}} \def\ee{\end{equation}a &&s \H {\sRui}(s) =\fr{ \int_s^\infty} \def\Eq{\Leftrightarrow \Tc \sRui(0)I(y) dy}{I(s)}=\sRui(0) ({\H \w}(s)+1)\\&&s \H \Psi} \def\sRui{\overline{\Rui}(s) = \fr{\int_s^\infty} \def\Eq{\Leftrightarrow (\Tc \Psi} \def\sRui{\overline{\Rui}(0)-\fr{\Tl}{y+\mu})I(y) dy}{I(s)}=\Psi} \def\sRui{\overline{\Rui}(0) ({\H \w}(s)+1) - \H {\w}(s).
\epsilon} \newcommand{\ol}{\overlineea
Letting $s \to 0$ yields
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \label{seg0} \nonumber &&\Psi} \def\sRui{\overline{\Rui}(0) =\fr{\H {\mathbf w}(0)}{\H {\mathbf w}(0)+1}=
\frac{\Tl U\big(1, 1+\Tl,\mu \Tc\big)}
{\mu \Tc \; U\big(1, 2+\Tl,\mu \Tc\big)}=\fr{\Tl \Gammaamma (\Tl,\Tc \mu )} { \Gammaamma (\Tl +1,\Tc \mu )} {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!E}} \defOrnstein-Uhlenbeck } \def\difs{diffusions{Ornstein-Uhlenbeck } \def\difs{diffusionsq\\&& \sRui(0) =\fr{\lim_{s \to 0} s \H {\sRui}(s)}{\H
{\mathbf w}(0)+1} = \frac{\sRui(\infty} \def\Eq{\Leftrightarrow)}
{1+ \Tl \; U\big(1, 1+\Tl,\mu \Tc\big)}=\frac{1}
{\mu \Tc \; U\big(1, 2+\Tl,\mu \Tc\big)}\epsilon} \newcommand{\ol}{\overlineeq
\iffalse
{ and then} \begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} && s \H \Psi} \def\sRui{\overline{\Rui}(s) =\fr{ e^{\Tc \mu } (c \mu )^{-\Tl }}{I(s)}\Big(\Psi} \def\sRui{\overline{\Rui}(0) \Gammaamma (\Tl +1,\Tc (s+\mu ))- \Tl \Gammaamma (\Tl,\Tc (s+\mu ))\Big) \nonumber
\\&& =\Psi} \def\sRui{\overline{\Rui}(0) {\Tc(\mu +s)} U(1,2+\Tl,\Tc (s+\mu )-{\Tl}{U(1,1+\Tl,\Tc (s+\mu ))}\nonumber
\\&& =\Psi} \def\sRui{\overline{\Rui}(0) (1+\Tl U(1,1+\Tl,\Tc (s+\mu ))- \Tl {U(1,1+\Tl,\Tc (s+\mu ))}\nonumber\\&& =\Psi} \def\sRui{\overline{\Rui}(0) -\Tl (1-\Psi} \def\sRui{\overline{\Rui}(0))U(1,1+\Tl,\Tc (s+\mu )), \la{segTI}
\epsilon} \newcommand{\ol}{\overlineeq
\fi
For the survival probability, we finally find
\begin{equation}} \def\ee{\end{equation}a &&s \H {\sRui}(s) =
\sRui(0) (1+ \H {\mathbf w}(s))=
\fr{1 +\Tl U\big(1, 1+\Tl,\mu (\Tc+s)\big)}{1 +\Tl U\big(1, 1+\Tl,\mu \Tc\big)}=\fr{\Tc (\mu +s) U\big(1, 2+\Tl,\mu(\Tc+s)\big)}{\Tc \mu U\big(1, 2+\Tl,\mu \Tc\big)},\epsilon} \newcommand{\ol}{\overlineea
which checks with the \LT \ of the Segerdahl result \epsilon} \newcommand{\ol}{\overlineqr{segLT}.
\iffalse
For the ruin probability, we find
\begin{equation}} \def\ee{\end{equation}a &&s \H \Psi} \def\sRui{\overline{\Rui}(s) I(s)={\int_s^\infty} \def\Eq{\Leftrightarrow (\Tc \Psi} \def\sRui{\overline{\Rui}(0)-\fr{\Tl}{y+\mu})I(y) dy}=\\&&\Psi} \def\sRui{\overline{\Rui}(0){e^{\Tc \mu } (\Tc \mu )^{-\Tl } \Gammaamma (\Tl +1,\Tc (s+\mu ))}-\Tl{e^{\Tc \mu } (c \mu )^{-\Tl } \Gammaamma (\Tl,\Tc (s+\mu ))}.\epsilon} \newcommand{\ol}{\overlineea
{ and then} \begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} && s \H \Psi} \def\sRui{\overline{\Rui}(s) =\fr{ e^{\Tc \mu } (c \mu )^{-\Tl }}{I(s)}\Big(\Psi} \def\sRui{\overline{\Rui}(0) \Gammaamma (\Tl +1,\Tc (s+\mu ))- \Tl \Gammaamma (\Tl,\Tc (s+\mu ))\Big) \nonumber
\\&& =\Psi} \def\sRui{\overline{\Rui}(0) {\Tc(\mu +s)} U(1,2+\Tl,\Tc (s+\mu )-{\Tl}{U(1,1+\Tl,\Tc (s+\mu ))}\nonumber
\\&& =\Psi} \def\sRui{\overline{\Rui}(0) (1+\Tl U(1,1+\Tl,\Tc (s+\mu ))- \Tl {U(1,1+\Tl,\Tc (s+\mu ))}\nonumber\\&& =\Psi} \def\sRui{\overline{\Rui}(0) -\Tl (1-\Psi} \def\sRui{\overline{\Rui}(0))U(1,1+\Tl,\Tc (s+\mu )), \la{segTI}
\epsilon} \newcommand{\ol}{\overlineeq
which checks with the \LT \ of the Segerdahl result \epsilon} \newcommand{\ol}{\overlineqr{segLT}.
\fi
\subsection} \def\Kol{Kolmogorov {The Eventual Ruin and \sur {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionros \la{s:Ak}}
These may also be obtained directly by integrating the explicit scale derivative ${\w}(x,a)=
\frac{\Tl}{{\Tc}}\left} \def\ri{\rightft(1+ \frac{x}{\tilde{c}}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)^{\Tl-1}
e^{-\mu x}$ \epsilon} \newcommand{\ol}{\overlineqr{scSeg}
Indeed,
\begin{equation}} \def\ee{\end{equation}a &&\int_u^\infty} \def\Eq{\Leftrightarrow {\mathbf w}(x) dx=\int_u^\infty} \def\Eq{\Leftrightarrow \frac{\Tl}{\tilde{c}}\left} \def\ri{\rightft(1+ \frac{x}{\tilde{c}}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)^{\Tl-1}
e^{-\mu x}dx= \Tl e^{\mu \Tc}\int_{1+ \frac{u}{\tilde{c}}}^\infty} \def\Eq{\Leftrightarrow y^{\Tl-1} e^{\mu \Tc y}dy\\&&=\Tl e^{\mu \Tc}
\fr {1}{(\mu \T c)^{\Tl}}
\int_{\mu(\T c+ u)}^\infty} \def\Eq{\Leftrightarrow t^{\Tl-1}
e^{- t}dt=\Tl e^{\mu \Tc}
(\mu \T c)^{-\Tl} \Gammaamma(\Tl,\mu(\T c+ u)),
\epsilon} \newcommand{\ol}{\overlineea
{where $\Gammaamma(\epsilon} \newcommand{\ol}{\overlineta, x) = \int_x^{\infty} t^{\epsilon} \newcommand{\ol}{\overlineta -1}e^{-t}dt$ is the
incomplete gamma function.}
The
r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up\ is \cite{Seg}, \cite[ex. 2.1]{Pau}:
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}\label{segint}
&&\Psi} \def\sRui{\overline{\Rui}(x) = \Tl\frac{\mathbb{E}} \def\subsection} \def\Kol{Kolmogorov {\subsection} \def\Kol{Kolmogorov p(\mu
\Tc )(\mu \Tc)^{-\Tl }\Gammaamma\big(\Tl,\mu (\Tc + x)\big)}
{1+\Tl\mathbb{E}} \def\subsection} \def\Kol{Kolmogorov {\subsection} \def\Kol{Kolmogorov p(\mu
\Tc )(\mu \Tc)^{-\Tl }\Gammaamma(\Tl,\mu \Tc)}= \Tl\frac{e^{-\mu
x }(1+ x/\Tc)^{\Tl}\; U\big(1, 1+\Tl,\mu (\Tc +x)\big)}
{1+ \Tl U\big(1, 1+\Tl,\mu \Tc \big)} \nonumber\\&&=\fr{\Tl}{\mu \Tc} \frac{e^{-\mu
x }(1+ x/\Tc)^{\Tl }\; U\big(1, 1+\Tl,\mu (\Tc +x)\big)}
{ U\big(1, 2+\Tl,\mu \Tc \big)}=\fr{\Tl \Gammaamma (\Tl, \mu(\Tc+x) )} { \Gammaamma (\Tl +1,\Tc \mu )},
\epsilon} \newcommand{\ol}{\overlinend{eqnarray}
where we used
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{TriG} U\big(1, 1+\Tl,v)= e^v v^{-\Tl } \Gammaamma (\Tl,v)\epsilon} \newcommand{\ol}{\overlineeq
and \begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{sU} 1+ \Tl U\big(1, 1+\Tl,v \big)= v U\big(1, 2+\Tl,v \big),\epsilon} \newcommand{\ol}{\overlineeq which holds by integration by parts.
A simpler formula holds for the rate of ruin $r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uui(x)$ and its \LT
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{segLT}&& r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uui(x)=-\Psi} \def\sRui{\overline{\Rui}'(x)=\fr{{\mathbf w}(x)}{1+ \int_0^\infty} \def\Eq{\Leftrightarrow {\mathbf w}(x) dx}=
\fr{\Tl}{ \Gammaamma (\Tl +1,\Tc \mu )} \mu (\mu(\Tc+x) )^{\Tl-1}e^{-\mu(\Tc+x)} =e^{-\mu \Tc}\gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}ta_{\Tl,\mu}(x + \Tc) {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!E}} \defOrnstein-Uhlenbeck } \def\difs{diffusions{Ornstein-Uhlenbeck } \def\difs{diffusionsq \nonumber \\&&
\H r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uui(s)=\sRui(0) \H{\mathbf w}(s)=\bc
\fr{\Tl U(1,1+ \Tl,\Tc (s+\mu ))}{\Tc \mu U(1,2+ \Tl,\Tc \mu )}, &c> 0\\ (1+s/\mu )^{-\Tl }, &c= 0\epsilon} \newcommand{\ol}{\overlinec,\epsilon} \newcommand{\ol}{\overlineeq where $\gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}ta$ denotes a~(shifted) Gamma density.
Of course, the~case $c>0$ simplifies to a~Gamma density when moving the origin to the “absolute ruin'' point $-\Tc=-\fr c r,$, i.e., by putting $y=x+ \Tc, Y_t=X_t + \Tc$,
where the process $Y_t$ has drift rate $r Y_t$.
\begin{equation}} \def\ee{\end{equation}Q Find a~relation between the ruin derivative $r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uui_\q(x)=-\Psi} \def\sRui{\overline{\Rui}_q'(x)$ and the scale derivative ${\mathbf w_\q}(x)$ when $q>0$. \epsilon} \newcommand{\ol}{\overlineeQ
\section{Asmussen's Embedding Approach for Solving Kolmogorov's Integro-Differential Equation
with Phase-Type Jumps \la{s:CLSa}} One of the most convenient approaches to get rid of the integral term in \epsilon} \newcommand{\ol}{\overlineqr{reneq} is a~probabilistic transformation which gets rid of the jumps as in \cite{asmussen1995stationary}, when the downward
phase-type jumps have a
survival \fun $$\bar{F}_C(x)=\int_x^{\infty} f_C(u) du= {\vec \b}e^{B x}
\bff 1,$$ where ${ B}$ is a~$n\times n$ stochastic generating matrix
(nonnegative off-diagonal elements and nonpositive row sums),
$\vec \b=(\b_1,\ldots,\b_n)$ is a~row probability vector (with nonnegative elements and
$\sum_{j=1}^n\b_j=1$), and
$\bff 1=(1,1,...,1)$ is a~column probability vector.
The density is ${f_C(x)}=\vec \b e^{-B x} \bff b$, where ${\bff b}=(-B) \bff 1$
is a~column vector, and~the Laplace transform is
$$\hat{b}(s)={\vec \b}(s I -\bff B)^{-1} \bff b.$$
Asmussen's approach \cite{asmussen1995stationary,asmussen2002erlangian} replaces the negative jumps by segments of slope $-1$, embedding the original \sn L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims process into a~continuous Markov modulated L\'evy } \def\mL{{\mathcal L}} \defr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up{ruin probability } \def\epsilon} \newcommand{\ol}{\overlinexpc{exponential claims process. For the new process we have auxiliary unknowns {$A_i(x)$} representing ruin or survival probabilities (or, more generally, Gerber-Shiu functions) when starting at $x$ conditioned {on} a~phase $i$ with drift downwards (i.e., in one of the “auxiliary stages of artificial time'' introduced by changing the jumps to segments of slope $-1$).
Let ${\bf A}$ denote the column vector
with components $A_1,\ldots,A_n$. The~ Kolmogorov integro-differential equation turns then into a~system of ODE's, due to the continuity of the embedding process.
\begin{equation}} \def\ee{\end{equation}gin{equation}\label{Line}
\left} \def\ri{\rightft(
\begin{equation}} \def\ee{\end{equation}gin{aligned}
\Psi} \def\sRui{\overline{\Rui}_\q'(x)\\
{\bf A}'(x)\\
\epsilon} \newcommand{\ol}{\overlinend{aligned}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)=
\left} \def\ri{\rightft(
\begin{equation}} \def\ee{\end{equation}gin{matrix}
\frac{\lambda+q}{c(x)} & -\frac{\lambda}{c(x)}{\vec \b}\\
{\bf b} & {\bf B}\\
\epsilon} \newcommand{\ol}{\overlinend{matrix}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)
\left} \def\ri{\rightft(
\begin{equation}} \def\ee{\end{equation}gin{aligned}
\Psi} \def\sRui{\overline{\Rui}_\q(x)\\
{\bf A}(x)\\
\epsilon} \newcommand{\ol}{\overlinend{aligned}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight),\ {x\gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq 0.}
\epsilon} \newcommand{\ol}{\overlinend{equation}
For the r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up\ with exponential jumps {of rate $\mu$} for example, there is only one downward phase, and~the system is:
\begin{equation}} \def\ee{\end{equation}gin{equation} \label{expoLine} \left} \def\ri{\rightft(\begin{equation}} \def\ee{\end{equation}gin{array}{c}
\ {\Psi} \def\sRui{\overline{\Rui}}_\q'(x)\\ A'(x)
\epsilon} \newcommand{\ol}{\overlinend{array} r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) = \left} \def\ri{\rightft(\begin{equation}} \def\ee{\end{equation}gin{array}{cc}
\frac{\lambda +q}{c(x)} &- \frac{\lambda }{c(x)} \\ \mu & -\mu
\epsilon} \newcommand{\ol}{\overlinend{array} r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) \left} \def\ri{\rightft(\begin{equation}} \def\ee{\end{equation}gin{array}{c}
\Psi} \def\sRui{\overline{\Rui}_\q(x)\\ A(x)
\epsilon} \newcommand{\ol}{\overlinend{array} r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)\ {x\gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq 0}.\epsilon} \newcommand{\ol}{\overlinend{equation}
{For survival probabilities,
one only needs to modify the boundary conditions---see the following~section.}
\subsection} \def\Kol{Kolmogorov {Exit Problems for the
Segerdahl-Tichy process, with $\q=0$}
Asmussen's approach is particular convenient for solving exit problems for the
Segerdahl-Tichy process.
\begin{equation}} \def\ee{\end{equation}Xa {\bf The eventual ruin probability}.
{{\bf When $q=0,$} the system for the ruin probabilities {with $x\gamma} \def\d{\delta} \def\de{\delta} \def\b{\begin{equation}} \def\ee{\end{equation}taeq0$} is:}
\begin{equation}} \def\ee{\end{equation}gin{equation}
\left} \def\ri{\rightft\{\begin{equation}} \def\ee{\end{equation}gin{aligned} {\Psi} \def\sRui{\overline{\Rui}}'(x)&= \frac{\lambda }{c(x)} \
(\Psi} \def\sRui{\overline{\Rui}(x) - A(x)), \quad} \def\for{\forallad \; \; \Psi} \def\sRui{\overline{\Rui}(\infty)&=
A(\infty)= 0\\
A'(x)&= \mu \ (\Psi} \def\sRui{\overline{\Rui}(x)- A(x)), \
\quad} \def\for{\forallad \quad} \def\for{\forallad \; \; &A(0)= 1
\epsilon} \newcommand{\ol}{\overlinend{aligned}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight.
\epsilon} \newcommand{\ol}{\overlinend{equation}
This may be solved by subtracting the equations. Putting \begin{equation}} \def\ee{\end{equation}a K(x)=e^{-\mu
x + \int_0^x \frac{\lambda} {c(v)} d v}, \epsilon} \newcommand{\ol}{\overlineea we find:
\begin{equation}} \def\ee{\end{equation}gin{equation} \label{ZSeg}
\bc \Psi} \def\sRui{\overline{\Rui}(x)-A(x)&=(\Psi} \def\sRui{\overline{\Rui}(0)-A(0))
K(x), \\
A(x)&= \mu (A(0)-\Psi} \def\sRui{\overline{\Rui}(0)) \int_{x}^\infty} \def\Eq{\Leftrightarrow K(v) d v,
\epsilon} \newcommand{\ol}{\overlinec,
\epsilon} \newcommand{\ol}{\overlinend{equation}
whenever $K(v)$ is integrable at $\infty$.
The boundary
condition $
A(0)=1$ implies that $1-\Psi} \def\sRui{\overline{\Rui}(0)=\fr{1}{\mu \int_0^{\infty} K(v) d
v}$ and
\begin{equation}} \def\ee{\end{equation}a &&
\quad} \def\for{\forallad A(x)= \mu (1-\Psi} \def\sRui{\overline{\Rui}(0)) \int_x^{\infty} K(v)
d v=\fr{\int_x^{\infty} K(v) d v}{\int_0^{\infty} K(v)
d v},\\&&\Psi} \def\sRui{\overline{\Rui}(x)-A(x)={-\fr{K(x)}{\mu \int_0^{\infty} K(v) d v}}.\epsilon} \newcommand{\ol}{\overlineea
Finally,
\begin{equation}} \def\ee{\end{equation}a &&\Psi} \def\sRui{\overline{\Rui}(x)=A(x)+\left} \def\ri{\rightft(\Psi} \def\sRui{\overline{\Rui}(x)-A(x)r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)=\fr{ \mu
\int_x^\infty} \def\Eq{\Leftrightarrow K(v) d v - K(x)}{\mu \int_0^{\infty} K(v) d v},
\epsilon} \newcommand{\ol}{\overlineea
and {for the survival probability $\sRui$,}
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{seg1} && \sRui(x)=\fr{\mu
\int_0^{x} K(v) d v + K(x)}{\mu \int_0^{\infty} K(v) d v}:= \sRui(0) {\mathbf W}(x)=\frac{{\mathbf W}(x)}{{\mathbf
W}(\infty )}, \\&& {\mathbf W}(x)=\mu \int_0^x
K(v) d v + K(x), \nonumber\epsilon} \newcommand{\ol}{\overlineeq
where $\sRui(0) =\frac{1}{{\mathbf
W}(\infty )}$ by plugging ${\mathbf
W}(0 )=1$ in the first and last terms in \epsilon} \newcommand{\ol}{\overlineqr{seg1}.
We may also rewrite \epsilon} \newcommand{\ol}{\overlineqr{seg1} as:
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}\la{scder} &&\sRui(x)=\frac{1+\int_0^x \w(v) d v}{1+\int_0^\infty
\w(v) d v} \Leftrightarrow \Psi} \def\sRui{\overline{\Rui}(x)=\frac{\int_x^\infty \w(v) d v}{1+\int_0^\infty \w(v)
d v}, \w(x):={\mathbf W}'(x)=\fr{\lambda K(x)}{c(x)}\epsilon} \newcommand{\ol}{\overlinend{eqnarray}
Note that $\w(x) >0$ implies that the scale function ${\mathbf W}(x)$ is
nondecreasing.
\epsilon} \newcommand{\ol}{\overlineeXa
\begin{equation}} \def\ee{\end{equation}Xa For the two sided exit problem on $[a,b]$,
a similar derivation yields the
scale function
$${\mathbf W}(x,a)=\mu \int_a^x
\fr{K(v)}{K(a)} d v + \fr{K(x)}{K(a)}=1+\fr{1}{K(a)} \int_a^x \w(y) dy,$$
with scale derivative derivative
$ \w(x,a)=\fr{1}{K(a)}\w(x)$,
where
$\w(x)$ given by \epsilon} \newcommand{\ol}{\overlineqr{scder} does not depend on $a$.
Indeed, the~ analog of \epsilon} \newcommand{\ol}{\overlineqref{ZSeg} is:
\begin{equation}} \def\ee{\end{equation}gin{equation}
\left} \def\ri{\rightft\{\begin{equation}} \def\ee{\end{equation}gin{aligned} \label{ZSegc} \sRui^{b}(x,a)-A^{b}(x)&=\sRui^{b}(a,a) \fr{K(x)}{K(a)}, \nonumbernumber\\
A^{b}(x)&= \mu \sRui^{b}(a,a) \int_a^x \fr{K(v)}{K(a)} d v,
\epsilon} \newcommand{\ol}{\overlinend{aligned}r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight.\epsilon} \newcommand{\ol}{\overlinend{equation}
implying {by the fact that $\sRui^{b}(b,a)=1$ that}
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} &&\sRui^{b}(x,a)=\sRui^{b}(a,a) \left} \def\ri{\right(\fr{K(x)}{K(a)}+\mu \int_a^x \fr{K(v)}{K(a)} d vr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui)=\fr{{\mathbf W}(x,a)}
{{\mathbf W}(b,a)}=\fr{1+ \fr{1}{K(a)}\int_a^x\w(u) d u} {1+\fr{1}{K(a)}\int_a^b\w(u)
d u} {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!E}} \defOrnstein-Uhlenbeck } \def\difs{diffusions{Ornstein-Uhlenbeck } \def\difs{diffusionsq \nonumber \\ && \Psi} \def\sRui{\overline{\Rui}^{b}(x,a)= \fr{\int_x^b\w(u) d u} {K(a)+\int_a^b\w(u)
d u} {r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!E}} \defOrnstein-Uhlenbeck } \def\difs{diffusions{Ornstein-Uhlenbeck } \def\difs{diffusionsq \la{pdscale} \\&& r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uui^{b}(x,a):=- (\Psi} \def\sRui{\overline{\Rui}^{b})'(x,a)= \fr{ \w(x)} {K(a)+\int_a^b\w(u)
d u}=\w(x,a)\fr{ \sRui(a,a)} {\sRui(b,a)}. \nonumber\epsilon} \newcommand{\ol}{\overlineeq
\epsilon} \newcommand{\ol}{\overlineeXa
\begin{equation}} \def\ee{\end{equation}R The definition adopted in this section for the scale function ${\mathbf W}(x,a)$ uses the normalization ${\mathbf{W}(a,a)}=1$, which~is only appropriate in the absence of Brownian motion.
\epsilon} \newcommand{\ol}{\overlineeR
\begin{equation}} \def\ee{\end{equation}Q Extend the \epsilon} \newcommand{\ol}{\overlineqs \; for the \sur and r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Up\ of the Segerdahl-Tichy process in terms of the scale derivative
$\w_\q$, when $q>0$.
Essentially, this requires obtaining
$$T_q(x)=E_x\Big[ e^{-q [T_{a,-} \min T_{b,+}]} \Big ]$$
\epsilon} \newcommand{\ol}{\overlineeQ
\sec{Further Details on the Identities Used in the Proof of Theorem r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uef{t:W} \la{s:a}}
{We recall first some continuity and differentiation relations needed here \cite{AS}}
\begin{equation}} \def\ee{\end{equation}P Using the notation $M = M(a, b, z), M(a+) = M(a + 1, b, z), M(+,+) = M(a + 1, b+1, z)$, and~so on,
the Kummer and
Tricomi functions satisfy the following identities:
\begin{equation}} \def\ee{\end{equation}gin{equation*}
b M+ (a- b) M\left} \def\ri{\rightft(
b+ r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)=a M\left} \def\ri{\rightft(a+ r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) \tag{13.4.3}\epsilon} \newcommand{\ol}{\overlinend{equation*}
\begin{equation}} \def\ee{\end{equation}gin{equation*} b\big(M(a+)- M \big) =z M\left} \def\ri{\rightft(+,+ r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) \tag{13.4.4}
\epsilon} \newcommand{\ol}{\overlinend{equation*}
\begin{equation}} \def\ee{\end{equation}gin{equation*}(b - a) U +
z U( b+2 ) = (z + b) U(b+1) \tag{13.4.16} \epsilon} \newcommand{\ol}{\overlinend{equation*} \begin{equation}} \def\ee{\end{equation}gin{equation*}
U+ a~U\left} \def\ri{\rightft(+,+ r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) = U\left} \def\ri{\rightft(
b+ r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) \tag{13.4.17}\epsilon} \newcommand{\ol}{\overlinend{equation*}\begin{equation}} \def\ee{\end{equation}gin{equation*}U+(b-a-1) U(a+1)=z U(+,+) \tag{13.4.18}\epsilon} \newcommand{\ol}{\overlinend{equation*}
(see corresponding equations in \cite{AS}).
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} \la{der}
U'=-a U\left} \def\ri{\right(+,+r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui), \quad} \def\for{\forallad M'=\fr{a}{b}
M\left} \def\ri{\right(+,+r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui).\epsilon} \newcommand{\ol}{\overlineeq
\epsilon} \newcommand{\ol}{\overlineeP
\begin{equation}} \def\ee{\end{equation}P
The functions $K_i(\Tq,\Tl,z)$ defined by \epsilon} \newcommand{\ol}{\overlineqr{Ki} satisfy
the identities
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} K_1'(\Tq,n, z)&=&(\Tq+\Tl)e^{-z} z^{\Tq + \Tl-1} \
M \left} \def\ri{\right(\Tq, \Tq + \Tl, zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui)=(\Tq + \Tl) K_1(\Tq-1,\Tl,z) \la{Kids1}\\
K_2'(\Tq,n, z)&=&- e^{-z}
z^{\Tq + \Tl-1} \
U \left} \def\ri{\right(\Tq, \Tq + \Tl, zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui)=- K_2(\Tq-1,n, z) \la{Kids2}\epsilon} \newcommand{\ol}{\overlineeq
\begin{equation}} \def\ee{\end{equation}gin{eqnarray}} \def\eeq{\end{eqnarray} K_2(\Tq,n, z)= \int_z^{\infty}
({y-z})^{\T \q} \, (y )^{ n-\Tq-1} \,
e^{- y}
{d y}
\label{UIdent}
\epsilon} \newcommand{\ol}{\overlinend{eqnarray}
\epsilon} \newcommand{\ol}{\overlineeP
{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrf
For the first identity, note, using (\cite{AS} \cite{AS}, {13.4.3, 13.4.4}), that
\begin{equation}} \def\ee{\end{equation}a \fr{e^{z} }{z ^{\Tq + \Tl-1}} K_1'(z)&=& ({\Tq+\Tl}-{z}) M\left} \def\ri{\rightft( \Tq+1,\Tq
+1+ \Tl,zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) + z \fr{\Tq + 1}{\Tq + \Tl + 1} M\left} \def\ri{\rightft( \Tq+2,\Tq
+2+ \Tl,zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)
\\&=& ({\Tq+\Tl})M\left} \def\ri{\rightft( \Tq+1,\Tq +1+ \Tl,zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)\\&+& \fr{z}{\Tq + \Tl + 1} \left} \def\ri{\right((\Tq + 1) M\left} \def\ri{\rightft( \Tq+2,\Tq +2+ \Tl,zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)-({\Tq + \Tl +1})
M\left} \def\ri{\rightft( \Tq+1,\Tq +1+ \Tl,zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)\\&=& ({\Tq+\Tl})M\left} \def\ri{\rightft( \Tq+1,\Tq +1+ \Tl,zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)- \fr{z}{\Tq + \Tl + 1} \Tl
M\left} \def\ri{\rightft( \Tq+1,\Tq +2+ \Tl,zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) \\&=& ({\Tq+\Tl})M\left} \def\ri{\rightft( \Tq+1,\Tq +1+ \Tl,zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)- \Tl\left} \def\ri{\right(M\left} \def\ri{\rightft( \Tq+1,\Tq +1+ \Tl,zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)-M\left} \def\ri{\rightft( \Tq,\Tq +1+ \Tl,zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight)r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui)
\\&=& \Tq M\left} \def\ri{\rightft( \Tq+1,\Tq +1+ \Tl,zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight) + \Tl
M\left} \def\ri{\rightft( \Tq,\Tq +1+ \Tl,zr} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Uight).\epsilon} \newcommand{\ol}{\overlineea
The second formula may be derived similarly using 13.4.17, or by considering the
function
\begin{equation}} \def\ee{\end{equation}a _z {\T U}(\Tq+1, \Tq+1+\Tl,\mu):=
\Gammaamma(q+1) K_2(z) =\int_z^{\infty} ({s-z})^{\Tq} \,
({s} )^{ \Tl-1} \,
e^{- \mu s} {d s} \epsilon} \newcommand{\ol}{\overlineea
appearing in the numerator of the last form of \epsilon} \newcommand{\ol}{\overlineqref{UIdent}.
An integration by parts yields
\begin{equation}} \def\ee{\end{equation}a &&_z {\T U}'(\Tq+1, \Tq+1+\Tl,1)=\int_z^{\infty} ({s-z})^{\Tq} \,
\fr{d}{d z}[
({s} )^{ \Tl-1} \,
e^{- s}] {d s}\nonumber \\&&= (\Tl -1)
_z {\T U}(\Tq+1,\Tq+ \Tl,1) - _z {\T U}(\Tq+1, \Tq+\Tl+1,1), \; \Lra
\\&&
K_2'(\Tq+1, \Tl,z)= e^{-z}
z^{\Tq + \Tl-1}\left} \def\ri{\right((\Tl -1)
U(\Tq+1, \Tq +\Tl,z) -
U (\Tq+1, \Tq +\Tl+1,z)r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Ui) \epsilon} \newcommand{\ol}{\overlineea
and the result follows by (\cite{AS} \cite{AS},
{13.4.18.})\footnote{See also \cite[p. 640]{BS}, where however the first formula has a~typo.}
The third formula is obtained by the substitution $y=z(t+1)$.
\sec{Conclusions and Future Work} \la{s:con}
Two promising fundamental functions have been proposed for working with generalizations of Segerdahl's process: (a) the scale derivative $\w$ \cite{CPRY} and (b) the integrating factor $I$ \cite{AU}, and~they are shown to be related via Thm. 1.
Segerdahl's process per se is worthy of further investigation.
A priori, many risk problems (with absorbtion/reflection at a~barrier $b$ or with double reflection, etc.) might be solved by combinations of the hypergeometric functions $U$ and $M$.
However, this approach leads to an~impasse for more complicated jump structures, which~ will lead to more complicated hypergeometric functions.
In that case, we would prefer answers expressed in terms of the fundamental functions $\w$ or $I$.
We conclude by mentioning two promising numeric approaches, not discussed here. One due to \cite{JJ} bypasses the need to deal with high-order hypergeometric solutions by employing complex contour integral representations. The~second one uses Laguerre-Erlang expansions---see \cite {abate1996laguerre,ALR}.
Further effort of comparing their results
with those of the methods discussed above seems worthwhile.
\small
\epsilon} \newcommand{\ol}{\overlinend{document}
\begin{equation}} \def\ee{\end{equation}gin{thebibliography}{999}
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Abramowitz and Stegun}{Abramowitz and
Stegun}{1965}]{AS}
Abramowitz, Milton and Irene Ann Stegun. 1965.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Handbook of Mathematical Functions: With Formulas, Graphs and
Mathematical Tables}.
\newblock Mineola: Dover Publications, vol.~55.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Albrecher and Asmussen}{Albrecher and
Asmussen}{2010}]{AA}
Albrecher, Hansj{\"o}rg and S{\"o}ren Asmussen. 2010.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Ruin Probabilities}. Singapore:
\newblock World Scientific, vol.~14.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Albrecher, Ivanovs and ~Zhou}{Albrecher
et~al.}{2016}]{AIZ}
Albrecher, Hansj{\"o}rg, Jevgenijs~Ivanovs and ~Xiaowen~Zhou. 2016.
\newblock Exit identities for {L\'e}vy processes observed at {P}oisson arrival
times.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Bernoulli\/}~{22}: 1364--82.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Albrecher, Constantinescu, Palmowski, Regensburger and Rosenkranz}{Albrecher
et~al.}{2013}]{ACPR}
\hl{Albrecher, Hansj}{\"o}rg, Corina Constantinescu, Zbigniew Palmowski, Georg Regensburger and Markus Rosenkranz. 2013.
\newblock Exact and asymptotic results for insurance risk models with surplus-dependent premiums.
\newblock {\epsilon} \newcommand{\ol}{\overlinem SIAM Journal on Applied Mathematics\/}~{73}:
47--66.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Asmussen}{Asmussen}{1995}]{asmussen1995stationary}
Asmussen, S{\o}ren. 1995.
\newblock {Stationary distributions for fluid flow models with or without
brownian noise.}
\newblock {\epsilon} \newcommand{\ol}{\overlinem Communications in Statistics Stochastic Models\/}~{11}: 21--49.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Asmussen, Avram and ~Usabel}{Asmussen
et~al.}{2002}]{asmussen2002erlangian}
Asmussen, Soren, Florin Avram and ~Miguel Usabel. 2002.
\newblock Erlangian approximations for finite-horizon ruin probabilities.
\newblock {\epsilon} \newcommand{\ol}{\overlinem ASTIN Bulletin: The Journal of the IAA\/}~{32}: 267--81.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Abate, Choudhury and Whitt}{Abate, Choudhury and Whitt}{1996}]{abate1996laguerre}
Abate, Joseph, Choudhury, Gagan
and Whitt, Whitt. 1996.
\newblock On the Laguerre method for numerically inverting Laplace transforms.
\newblock {\epsilon} \newcommand{\ol}{\overlinem INFORMS Journal on Computing\/}~{8}: 413--27.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Avram, Cari{\~n}ena and ~de~Lucas}{Avram
et~al.}{2010}]{avram2010lie}
Florin~Avram, Jos{\'e}~F Cari{\~n}ena and ~Javier de~Lucas. 2010.
\newblock A lie systems approach for the first passage-time of piecewise
deterministic processes. \epsilon} \newcommand{\ol}{\overlinemph{Mathematics} {arXiv:1008.2625}.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Avram and Goreac}{Avram and Goreac}{2019}]{AG}
Florin~Avram and Dan Goreac. 2019.
\newblock A pontryaghin maximum principle approach for the optimization of
dividends/consumption of spectrally negative markov processes, until a
generalized draw-down time.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Scandinavian Actuarial Journal\/}~{2019}: 1--25.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Avram, Grahovac and ~Vardar-Acar}{Avram
et~al.}{2019a}]{AGV}
{Florin~Avram, Danijel~Grahovac and ~Ceren~Vardar-Acar. 2019a. {The $W,Z$ scale functions kit for first passage problems of
spectrally negative L\'evy processes and ~applications to the optimization of
dividends}. \epsilon} \newcommand{\ol}{\overlinemph{Mathematics} arXiv:1706.06841.}
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Avram, Grahovac and ~Vardar-Acar}{Avram
et~al.}{2019b}]{avram2019w}
\hl{Florin~Avram, Danijel Grahovac} and ~Ceren Vardar-Acar. 2019b.
\newblock The w, z/$\nu$, $\delta$ paradigm for the first passage of strong
markov processes without positive jumps.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Risks\/}~{7}: 18.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Avram, Kyprianou and ~Pistorius}{Avram
et~al.}{2004}]{AKP}
Florin Avram, Andreas~Kyprianou and ~Martijn~Pistorius. 2004.
\newblock Exit problems for spectrally negative {L\'e}vy processes and
applications to ({C}anadized) {R}ussian options.
\newblock {\epsilon} \newcommand{\ol}{\overlinem The Annals of Applied Probability\/}~{14}: 215--38.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Avram, Li and ~Li}{Avram et~al.}{2018}]{ALL}
Florin~Avram, Bin Li and ~Shu Li. 2018.
\newblock A unified analysis of taxed draw-down spectrally negative markov
processes. \epsilon} \newcommand{\ol}{\overlinemph{Risks} 7: 18.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Avram, Palmowski and ~Pistorius}{Avram
et~al.}{2007}]{APP}
Florin Avram, Zbigniew Palmowski and ~Martijn Pistorius. 2007.
\newblock On the optimal dividend problem for a~spectrally negative {L\'e}vy
process.
\newblock {\epsilon} \newcommand{\ol}{\overlinem The Annals of Applied Probability\/}~{17}: 156--80.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Avram, Leonenko, and Rabehasaina}{Avram
et~al.}{2009}]{avram09series}
Florin~Avram, Nikolai Leonenko and Landy Rabehasaina. 2009.~Series Expansions for the First Passage Distribution of Wong--Pearson Jump-Diffusions
\newblock {\epsilon} \newcommand{\ol}{\overlinem Stochastic Analysis and Applications}~{27}: 770--96.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Avram, Palmowski and ~Pistorius}{Avram
et~al.}{2015}]{APP15}
Avram Florin, Zbigniew Palmowski and ~Martijn Pistorius. 2015.
\newblock On {G}erber--{S}hiu functions and optimal dividend distribution for a
{L\'e}vy risk process in the presence of a~penalty function.
\newblock {\epsilon} \newcommand{\ol}{\overlinem The Annals of Applied Probability\/}~{25}: 1868--935.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Avram, P{\'e}rez and ~Yamazaki}{Avram
et~al.}{2016}]{APY}
Florin~Avram, Jos{\'e}~Luis P{\'e}rez and ~Kazutoshi Yamazaki. 2016.
\newblock Spectrally negative L\'evy processes with parisian
reflection below and classical reflection above. \epsilon} \newcommand{\ol}{\overlinemph{Mathematics} {arXiv:1604.01436}.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Avram and Usabel}{Avram and Usabel}{2008}]{AU}
Florin~Avram and Miguel Usabel. 2008.
\newblock The gerber-shiu expected discounted penalty-reward function under an~ affine jump-diffusion model.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Astin Bulletin\/}~{38}: 461--81.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Avram and Vidmar}{Avram and Vidmar}{2017}]{AV}
Florin~Avram and Matija Vidmar. 2017.
\newblock First passage problems for upwards skip-free random walks via the
$phi, w, z $ paradigm. \epsilon} \newcommand{\ol}{\overlinemph{Mathematics} arXiv:1708.06080.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Avram and Zhou}{Avram and Zhou}{2017}]{AZ}
Florin~Avram and Xiaowen Zhou. 2017.
\newblock On fluctuation theory for spectrally negative L\'evy processes with
parisian reflection below and ~applications.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Theory of Probability and Mathematical Statistics} 95:~{17--40}.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Bertoin}{Bertoin}{1997}]{Ber97}
Bertoin, Jean. 1997.
\newblock Exponential decay and ergodicity of completely asymmetric {L\'e}vy
processes in a~finite interval.
\newblock \epsilon} \newcommand{\ol}{\overlinemph{{The Annals of Applied Probability}}~{7}: 156--69.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Bertoin}{Bertoin}{1998}]{Ber}
Bertoin, Jean. 1998.
\newblock {\epsilon} \newcommand{\ol}{\overlinem L{\'e}vy Processes}.
\newblock Cambridge: Cambridge University Press, vol. 121.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Blumenthal and Getoor}{Blumenthal and
Getoor}{2007}]{blumenthal2007markov}
Blumenthal, Robert~McCallum and Ronald~Kay Getoor. 2007.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Markov Processes and Potential Theory}.
\newblock Chelmsford: Courier Corporation.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Borodin and Salminen}{Borodin and
Salminen}{2012}]{BS}
Borodin, Andrei~N. and Paavo Salminen. 2012.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Handbook of Brownian Motion-Facts and Formulae}.
\newblock Basel: Birkh{\"a}user.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Borovkov and Novikov}{Borovkov and
Novikov}{2008}]{borovkov2008exit}
Borovkov, Konstantin and Alexander Novikov. 2008.
\newblock On exit times of levy-driven ornstein--uhlenbeck processes.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Statistics \& Probability Letters\/}~{78}: 1517--25.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Czarna, P{\'e}rez, Rolski and
Yamazaki}{Czarna et~al.}{2017}]{CPRY}
Czarna, Irmina, Jos{\'e}-Luis P{\'e}rez, Tomasz Rolski and ~Kazutoshi Yamazaki.
2017.
\newblock Fluctuation theory for level-dependent L\'evy risk
processes. \epsilon} \newcommand{\ol}{\overlinemph{Mathematics} {arXiv:1712.00050}.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{de~Finetti}{de~Finetti}{1957}]{deF}
de~Finetti, B.. 1957.
\newblock Su un'impostazione alternativa della teoria collettiva del rischio.
\newblock In {\epsilon} \newcommand{\ol}{\overlinem Transactions of the XVth International Congress of
Actuaries}. New York: Mallon, vol.~2, pp. 433--43.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Ivanovs}{Ivanovs}{2013}]{Ivakil}
Ivanovs, Jevgenijs. 2013.
\newblock A note on killing with applications in risk theory.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Insurance: Mathematics and Economics\/}~52: 29--34.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Ivanovs and Palmowski}{Ivanovs and
Palmowski}{2012}]{IP}
Ivanovs, Jevgenijs and Zbigniew~Palmowski. 2012.
\newblock Occupation densities in solving exit problems for {M}arkov additive
processes and their reflections.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Stochastic Processes and Their Applications\/}~122:
3342--60.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Jacobsen and Jensen}{Jacobsen and
Jensen}{2007}]{JJ}
Jacobsen, Martin and Anders Tolver Jensen. 2007.
\newblock Exit times for a~class of piecewise exponential markov processes with
two-sided jumps.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Stochastic Processes and Their Applications}~{117}: 1330--56.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Jiang, Li and ~Wang}{Jiang
et~al.}{2019}]{jiang2019exit}
Jiang, Pingping, Bo~Li and ~Yongjin Wang. 2019.
\newblock Exit times, undershoots and overshoots for reflected cir process with
two-sided jumps.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Methodology and Computing in Applied Probability\/} {2019}: 1--18.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Kyprianou}{Kyprianou}{2014}]{Kyp}
Kyprianou, Andreas. 2014.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Fluctuations of L{\'e}vy Processes with Applications:
Introductory Lectures}.
\newblock Berlin/Heidelberg: Springer Science \& Business Media.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Landriault, Li and ~Zhang}{Landriault
et~al.}{2017}]{LLZ17b}
Landriault, David, Bin Li and ~Hongzhong Zhang. 2017.
\newblock A unified approach for drawdown (drawup) of time-homogeneous markov
processes.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Journal of Applied Probability\/}~{54}: 603--26.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Li and Palmowski}{Li and Palmowski}{2016}]{LP}
Li, Bo and Zbigniew Palmowski. 2016.
\newblock Fluctuations of omega-killed spectrally negative L\'evy
processes. \epsilon} \newcommand{\ol}{\overlinemph{Mathematics} {arXiv:1603.07967}.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Li and Zhou}{Li and Zhou}{2017}]{LZ17}
Li, Bo and Xiaowen Zhou. 2017.
\newblock On weighted occupation times for refracted spectrally negative
L\'evy processes. \epsilon} \newcommand{\ol}{\overlinemph{Mathematics} {arXiv:1703.05952}.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Loeffen and Patie}{Loeffen and
Patie}{2010}]{LoefPat}
Loeffen, Ronnie~{Lambertus} and Pierre Patie. 2010.
\newblock Absolute ruin in the ornstein-uhlenbeck type risk model. \epsilon} \newcommand{\ol}{\overlinemph{arXiv} arXiv:1006.2712.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Marciniak and Palmowski}{Marciniak and Palmowski}{2016}]{marciniak2016optimal} \hl{Marciniak, Ewa and Zbigniew Palmowski. 2016.}
\newblock On the optimal dividend problem for insurance risk models with surplus-dependent premiums.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Journal of Optimization Theory and Applications \/}~{168}: 723--42.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Paulsen and Gjessing}{Paulsen and
Gjessing}{1997}]{Pau}
Paulsen, Jostein and Hakon Gjessing. 1997.
\newblock Ruin theory with stochastic return on investments.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Advances in Applied Probability} 29: 965--85.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Paulsen}{Paulsen}{2010}]
{paulsen2010ruin} {Paulsen, Jostein}. 2010.
\newblock Ruin models with investment income.
\newblock In {\epsilon} \newcommand{\ol}{\overlinem Encyclopedia of Quantitative Finance}. Hoboken: Wiley Online Library.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Segerdahl}{Segerdahl}{1955}]{Seg}
Segerdahl, {C.-O.} 1955.
\newblock When does ruin occur in the collective theory of risk?
\newblock {\epsilon} \newcommand{\ol}{\overlinem Scandinavian Actuarial Journal}~38: 22--36.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Suprun}{Suprun}{1976}]{Suprun}
Suprun, {V.~N.} 1976.
\newblock Problem of destruction and resolvent of a~terminating process with
independent increments.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Ukrainian Mathematical Journal\/}~28: 39--51.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Tichy}{Tichy}{1984}]{Tichy84}
Tichy, Roland. 1984.
\newblock {U}ber eine zahlentheoretische {M}ethode zur numerischen
{I}ntegration und zur {B}ehandlung von {I}ntegralgleichungen.
\newblock {\epsilon} \newcommand{\ol}{\overlinem Osterreichische Akademie der Wissenschaften Mathematisch-Naturwissenschaftliche Klasse Sitzungsberichte II}~{193}: 329--58.
\bibitem[{r} \def\s{\sigma} \def\F{\Phi} \def\f{\varphi}\def\L{L} \def\U{Um I \!P}} \defspectrally positive } \def\dif{diffusion{spectrally positive } \def\dif{diffusionrotect\cite{Zhang and Cui}{Zhang and Cui}{2019}]{zhang2019Laguerre} Zhang, Zhimin and Cui, Zhenyu. 2019.
\newblock Laguerre series expansion for scale functions and
applications in risk theory. Preprint.
\epsilon} \newcommand{\ol}{\overlinend{thebibliography}
\epsilon} \newcommand{\ol}{\overlinend{document} |
\begin{document}
\title{A Selection Principle and Products in Topological Groups}
\begin{abstract}
We consider the preservation under products, finite powers, and forcing, of a selection principle based covering property of $T_0$ topological groups. Though the paper is in part a survey, it contributes some new information, including:
\begin{enumerate}
\item{The product of a strictly o-bounded group with an o-bounded group is an o-bounded group - Corollary \ref{cor:stroboundedproducts}}
\item{In the generic extension by a finite support iteration of $\aleph_1$ Hechler reals the product of any o-bounded group with a ground model $\aleph_0$ bounded group is an o-bounded group - Theorem \ref{thm:hechleriteration}}
\item{In the generic extension by a countable support iteration of Mathias reals the product of any o-bounded group with a ground model $\aleph_0$ bounded group is an o-bounded group - Theorem \ref{thm:mathiasiteration}}
\end{enumerate}
\end{abstract}
\section*{Introduction}
In this paper we consider selection principles for open covers on topological space under the imposition of three major constraints: The topological spaces are assumed to be $\textsf{T}_0$, are assumed to be topological groups, and these topological groups are $\aleph_0$-bounded (a notion due to Guran, and defined below).
Even under these three constraints there is a broad range of considerations regarding the relevant selection principles, and we shall also confine attention to a specific class of selection principles, and specific concerns regarding these. To give an initial indication of the scope of work considered here, recall: The following two selection principles, among several, are historically well-studied in several mathematical contexts: Let families $\mathcal{A}$ and $\mathcal{B}$ of sets be given. The symbol ${\sf S}_{fin}(\mathcal{A},\mathcal{B})$ denotes the statement that there is for each sequence $(A_n:n\in\mathbb{N})$ of members of the family $\mathcal{A}$, a corresponding sequence $(B_n:n\in\mathbb{N})$ such that for each $n$, $B_n$ is a finite subset of $A_n$, and $\bigcup\{B_n:n\in\mathbb{N}\}$ is a set in the family $\mathcal{B}$. The symbol ${\sf S}_1(\mathcal{A},\mathcal{B})$ denotes the statement that there is for each sequence $(A_n:n\in\mathbb{N})$ of members of the family $\mathcal{A}$, a corresponding sequence $(B_n:n\in\mathbb{N})$ such that for each $n$, $B_n$ is a member of $A_n$, and $\{B_n:n\in\mathbb{N}\}$ is a set in the family $\mathcal{B}$. It is well known that if $\mathcal{A} \subseteq \mathcal{B}$ and if $\mathcal{C}\subseteq \mathcal{D}$, then the following implications (more broadly illustrated in Figure \ref{fig:monotone}) hold: ${\sf S}_{fin}(\mathcal{B},\mathcal{C}) \Rightarrow {\sf S}_{fin}(\mathcal{A},\mathcal{D})$, ${\sf S}_1(\mathcal{B},\mathcal{C}) \Rightarrow {\sf S}_1(\mathcal{A},\mathcal{D})$, and ${\sf S}_1(\mathcal{A},\mathcal{B})\Rightarrow {\sf S}_{fin}(\mathcal{A},\mathcal{B})$.
\begin{figure}
\caption{Monotonicity Properties: $\mathcal{A}
\label{fig:monotone}
\end{figure}
If instead of giving an entire antecedent sequence $(A_n:n\in\mathbb{N})$ of items from family $\mathcal{A}$ all at once for a selection principle and then producing a consequent sequence $(B_n:n\in\mathbb{N})$ to confirm that for example ${\sf S}_{fin}(\mathcal{A},\mathcal{B})$ (or ${\sf S}_1(\mathcal{A},\mathcal{B})$) holds, one can define a competition between two players, named ONE and TWO, where in inning $n$ ONE chooses an element $A_n$ from $\mathcal{A}$, and TWO responds with a $B_n$ from TWO's eligible choices. The players play an inning per positive integer $n$, producing a play
\begin{equation}\label{eq:play}
A_1\; B_1\; A_2\; B_2\; \cdots\; A_m\; B_m\; \cdots
\end{equation}
In the game named ${\sf G}_{fin}(\mathcal{A},\mathcal{B})$ the play in (\ref{eq:play}) is won by TWO if for each $n$, $B_n$ is a finite subset of $A_n$ and $\bigcup\{B_n:n\in\mathbb{N}\}$ is an element of $\mathcal{B}$ - otherwise, ONE wins. In the game named ${\sf G}_1(\mathcal{A},\mathcal{B})$ the play in (\ref{eq:play}) is won by TWO if for each $n$ $B_n\in A_n$, and $\{B_n:n\in\mathbb{N}\}$ is an element of $\mathcal{B}$. Observe that when ONE does not have a winning strategy in the game ${\sf G}_{fin}(\mathcal{A},\mathcal{B})$, then ${\sf S}_{fin}(\mathcal{A},\mathcal{B})$ is true. Similarly, when ONE does not have a winning strategy in the game ${\sf G}_1(\mathcal{A},\mathcal{B})$, then ${\sf S}_1(\mathcal{A},\mathcal{B})$ is true. The relationship between existence of winning strategies of a player and the corresponding properties of the associated selection principle is a fundamental question, and answers often reveal significant mathematical information.
In this paper we consider the selection principle ${\sf S}_{fin}(\mathcal{A},\mathcal{B})$
in the context where the families $\mathcal{A}$ and $\mathcal{B}$ are types of open covers arising in the study of topological groups. In the context of topological groups and the classes of open covers of these considered, there are some equivalences between the ${\sf S}_{fin}(\cdot,\cdot)$ and ${\sf S}_1(\cdot,\cdot)$ selection principles, as will be pointed out.
As noted initially, we throughout assume that the topological groups being considered have the $\textsf{T}_0$ separation property and thus, by the following classical theorem, the $\textsf{T}_{3\frac{1}{2}}$ separation property:
\begin{theorem}[Kakutani, Pontryagin] \label{thm:T0} Any $T_0$ topological group is $T_{3\frac{1}{2}}$.
\end{theorem}
In Section \ref{sec:aleph0bded} we briefly describe the resilience of $\aleph_0$-bounded groups under certain mathematical constructions and contrast these with the more constrained classical Lindel\"of property. In Section \ref{sec:products} we consider, for groups satisfying the targeted instance of the selection principle ${\sf S}_{fin}(\mathcal{A},\mathcal{B})$,
the preservation of the
selection property under the product construction. Though that there is a significant extant body of work on this topic, only some of these works and motivating mathematical questions relevant to the topic of Section \ref{sec:products} will be mentioned. In Section \ref{sec:cardinality} we make a brief excursion into exploring the cardinality of a class of groups emerging from product considerations in Section \ref{sec:products}. In Section \ref{sec:finpowers} we focus attention on groups for which finite powers satisfy the instance of ${\sf S}_{fin}(\mathcal{A},\mathcal{B})$ being considered in this paper.
In Section \ref{sec:Pgroups} we briefly return to a specific class of $\aleph_0$ bounded topological groups featured earlier in the paper.
For background on topological groups we refer the reader to \cite{HR} and to \cite{Tkachenko}. For relevant background on forcing we refer the reader to \cite{Baumgartner}, \cite{Jech}, \cite{JechMF} and \cite{Kunen}.
Finally, as the reader will notice, this paper is part survey of known results, part investigation of refining or providing additional context for known results. The author would like to thank the editor of the volume for the flexibility in time to construct this paper.
\section{Open Covers and Fundamental Theorems}\label{sec:aleph0bded}
Besides the typical types of open covers considered for general topological spaces, there are also specific types of open covers considered in the context of topological groups. We introduce notation here for efficient reference to the various types of open covers relevant to this paper. Thus, let $(G,\otimes)$ denote a generic $\textsf{T}_0$ topological group, where $G$ is the set of elements of the group and $\otimes$ is the group operation. The symbol $id$ will denote the identity element of the group. It is also common practice to talk about the group $G$, without mentioning an explicit symbol for the operation.
For an element $x$ of the group $G$ and for a nonempty subset $S$ of $G$, define
\begin{equation}\label{eq:pttimesset}
x\otimes S = \{x\otimes g: g\in S\}.
\end{equation}
Observe that if $(G,\otimes)$ is a topological group, then when $S$ is an open subset of $G$, so is $x\otimes S$ for each element $x$ of $G$. Moreover, if $id$ is an element of $S$, then $x$ is an element of $x\otimes S$. Moreover, when $S$ and $T$ are nonempty subsets of $G$,
\begin{equation}\label{eq:settimesset}
S\otimes T = \{x\otimes y:x\in S \mbox{ and }y\in T\}.
\end{equation}
Now we introduce notation for types of open covers of $(G,\otimes)$ to be considered here.
\begin{itemize}
\item{$\mathcal{O}$: The set of all open covers of $G$.}
\item{$\mathcal{O}_{nbd}$: The set of all open covers of $G$ of the following form: For a neighborhood $U$ of $id$, $\mathcal{O}(U)$ denotes the open cover $\{x\otimes U:x\in G\}$ of $G$, and $\mathcal{O}_{nbd}$ denotes the collection $\{\mathcal{O}(U): U \mbox{ a neighborhood of } id\}$.}
\item{$\Omega$: An open cover $\mathcal{U}$ of $G$ is said to be an $\omega$-cover (originally defined in \cite{GN}) if $G$ itself is not a member of $\mathcal{U}$, and for each finite subset $F$ of $G$ there is a $U\in\mathcal{U}$ such that $F\subseteq U$. The symbol $\Omega$ denotes the set $\{\mathcal{U}:\; \mathcal{U} \mbox{ an }\omega \mbox{ cover of }G\}$}
\item{$\Omega_{nbd}$: The set of all open covers of $G$ of the following form: For a neighborhood $U$ of $id$, $\Omega(U)$ denotes the open cover $\{F\otimes U: F\subset G \mbox{ a finite set}\}$. The symbol $\Omega_{nbd}$ denotes the set of all open covers of the form $\Omega(U)$ of $G$.}
\item{$\Gamma$: An open cover $\mathcal{U}$ is a $\gamma$-cover (also introduced in \cite{GN}) if it is infinite and for each $x\in G$, $x$ is a member for all but finitely sets in $\mathcal{U}$. The symbol $\Gamma$ denotes the set $\{\mathcal{U}:\; \mathcal{U} \mbox{ a }\gamma \mbox{ cover of }G\}$}
\item{$\Lambda$: An open cover $\mathcal{U}$ is a large cover if for each $x\in G$, $x$ is a member of infinitely sets in $\mathcal{U}$. $\Lambda$ denotes the collection of large covers of $G$.}
\end{itemize}
Targeted properties related to topological objects, such as for example the preservation of a property of factor spaces in product spaces, have led to the identification of several additional types of open covers for topological spaces. Some of these used in this paper are as follows:
\begin{itemize}
\item{$\mathcal{O}^{gp}$: An open cover $\mathcal{U}$ is an element of $\mathcal{O}^{gp}$ if it is infinite, and there is a partition $\mathcal{U} = \bigcup\{\mathcal{U}_n:n\in\mathbb{N}\}$ where for each $n$ the set $\mathcal{U}_n$ is finite, for all $m\neq n$, we have $\mathcal{U}_m\cap\mathcal{U}_n = \emptyset$, and each element of the underlying space is in each but finitely many of the sets $\bigcup\mathcal{U}_n$. We say that $\mathcal{U}$ is a \emph{groupable} cover.}
\item{$\mathcal{O}^{wgp}$: An open cover $\mathcal{U}$ is an element of $\mathcal{O}^{wgp}$ if it is infinite, and there is a partition $\mathcal{U} = \bigcup\{\mathcal{U}_n:n\in\mathbb{N}\}$ where for each $n$ the set $\mathcal{U}_n$ is finite, for all $m\neq n$, we have $\mathcal{U}_m\cap\mathcal{U}_n = \emptyset$, and for each finite subset $F$ of the underlying space there is an $n$ such that $F\subseteq \bigcup\mathcal{U}_n$. We say that $\mathcal{U}$ is a \emph{weakly groupable} cover.}
\end{itemize}
From the definitions it is evident that the following inclusions hold among these types of open covers:
$\Gamma\subset \mathcal{O}^{gp}\subset \mathcal{O}^{wgp} \subset \Lambda\subset \mathcal{O}$,
$\Gamma \subset \Omega\subset \mathcal{O}^{wgp}\subset \mathcal{O}$,
$\Omega_{nbd}\subset \Omega$, $\Omega_{nbd}\subset \mathcal{O}_{nbd}$ and $\mathcal{O}_{nbd}\subset \mathcal{O}$.
For several traditional covering properties of topological spaces, natural counterparts are defined in the domain of topological groups by restricting the types of open covers considered in defining the covering properties. For example:
\begin{definition} A topological group is said to be
\begin{enumerate}
\item{$\aleph_0$-bounded if it has the Lindel\"of property with respect to the family $\mathcal{O}_{nbd}$ of open covers: That is, each member of $\mathcal{O}_{nbd}$ has a countable subset that covers the group.}
\item{totally bounded (or pre-compact) if it is compact with respect to the family $\mathcal{O}_{nbd}$ of open covers: That is, each element of $\mathcal{O}_{nbd}$ has a finite subset covering the group.}
\item{$\sigma$-bounded if it is a union of countably many totally bounded subsets.}
\end{enumerate}
\end{definition}
Many of the properties of $\aleph_0$-bounded groups can be obtained from the following fundamental result:
\begin{theorem}[Guran]\label{thm:Guran1} A topological group is $\aleph_0$-bounded if, and only if, it embeds as a topological group into the Tychonoff product of second countable groups.
\end{theorem}
The $\aleph_0$-boundedness property is resilient under several mathematical constructions. For example, any subgroup of an $\aleph_0$-bounded group is $\aleph_0$-bounded. Also, the Tychonoff product of any number of $\aleph_0$-bounded groups is an $\aleph_0$-bounded group. These two facts in particular imply:
\begin{lemma}\label{lemma:cardinality} There is for each infinite cardinal number $\kappa$ an $\aleph_0$-bounded group of cardinality $\kappa$.
\end{lemma}
The $\aleph_0$-boundedness property and the total boundedness property are also resilient under forcing extensions of the set theoretic universe:
\begin{theorem}\label{boundedpreserve} If $(G,\otimes)$ is an $\aleph_0$-bounded (totally bounded) topological group and $({\mathbb P},<)$ is a forcing notion, then
\[
{\mathbf 1}_{\mathbb P}\mathrel{\|}\joinrel\mathrel{-}``(\check{G},\otimes) \mbox{ is $\aleph_0$-bounded (respectively totally bounded)}".
\]
\end{theorem}
\begin{proof} Let $\dot{U}$ be a ${\mathbb P}$-name such that ${\mathbf 1}_{\mathbb P}\mathrel{\|}\joinrel\mathrel{-}``\dot{U} \mbox{ is a neighborhood of the identity}"$. Choose a maximal antchain $A$ for ${\mathbb P}$ and for each $q\in A$ a neighborhood $U_q$ of the identity such that $q\mathrel{\|}\joinrel\mathrel{-}``\dot{U} = \check{U}_q"$. We give an argument for $\aleph_0$-boundedness. The argument for totally bounded is similar.
Since $(G,\otimes)$ is $\aleph_0$-bounded, choose for each $q\in A$ a countable set $X_q:=\{x^q_n:n<\omega\}$ of elements of $G$ such that $X_q\otimes U_q=G$. Define $\dot{X} = \{(\check{x}^q_n,q):n<\omega \mbox{ and }q\in A\}$. Then $\dot{X}$ is a ${\mathbb P}$-name and
\[
{\mathbf 1}_{\mathbb P}\mathrel{\|}\joinrel\mathrel{-}``\dot{X}\subseteq\check{G} \mbox{ is countable and }\dot{X}\otimes\dot{U} = \check{G}"
\]
Thus, ${\mathbf 1}_{\mathbb P}\mathrel{\|}\joinrel\mathrel{-}``(\check{G},\otimes) \mbox{ is $\aleph_0$-bounded}"$.
\end{proof}
Proper forcing posets also preserves the property of not being $\aleph_0$-bounded:
\begin{theorem}\label{nonboundedpreserve}
Let $(G,\otimes)$ be a topological group which is not $\aleph_0$ bounded. Let $({\mathbb P},<)$ be a proper partially ordered set. Then
\[
{\mathbf 1}_{\mathbb P}\mathrel{\|}\joinrel\mathrel{-}``(\check{G},\otimes) \mbox{ is not $\aleph_0$-bounded}".
\]
\end{theorem}
{\flushleft{\bf Proof:}} For let $U$ be a neighborhood of the identity witnessing that $(G,\otimes)$ is not $\aleph_0$-bounded. Suppose that $p\in{\mathbb P}$ and ${\mathbb P}$-name $\dot{X}$ are such that $p\mathrel{\|}\joinrel\mathrel{-}``\dot{X}\otimes \check{U} = \check{G} \mbox{ and }\dot{X}\subseteq\check{G} \mbox{ is countable}"$. Since ${\mathbb P}$ is a proper poset there is a countable set $C\subseteq G$ such that $p\mathrel{\|}\joinrel\mathrel{-} \dot{X}\subseteq \check{C}"$ - \cite{JechMF}, Proposition 4.1. But then $p\mathrel{\|}\joinrel\mathrel{-}``\check{C}\otimes \check{U}=\check{G}"$. Since all the parameters in the sentence forced by $p$ are in the ground model, we find the contradiction that $G = C\otimes U$.
$\Box$
Thus, when forcing with a proper forcing notion, a ground model topological group is $\aleph_0$-bounded in the generic extension if, and only if, it is $\aleph_0$-bounded in the ground model. Note, incidentally, that the same argument shows
\begin{theorem}\label{nonlindelofpreserve}
Let $(X,\tau)$ be a topological space which is not Lindel\"of. Let $({\mathbb P},<)$ be a proper partially ordered set. Then
\[
{\mathbf 1}_{\mathbb P}\mathrel{\|}\joinrel\mathrel{-}``(\check{X},\tau) \mbox{ is not Lindel\"of}".
\]
\end{theorem}
When considering a strengthening of the $\aleph_0$-boundedness property, resilience of the stronger property under a corresponding mathematical constructions is more subtle. For example, the Lindel\"of property requires that for any open cover (not only ones from $\mathcal{O}_{nbd}$) there is a countable subset that still is a cover. Every Lindel\"of group is an $\aleph_0$-bounded group, but not conversely. The Lindel\"of property is not in general preserved by subspaces, products, or forcing extensions. Similarly, for subclasses (determined by selection principles) of the family of $\aleph_0$-bounded groups, preservation of membership to the subclass under Tychonoff products and behavior under forcing is more subtle. Also questions regarding cardinality of members of the more restricted family are more delicate.
\section{Products and Groups with the Property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$}\label{sec:products}
In the notation established here, a topological group is said to be \emph{o - bounded} if it has the property ${\sf S}_1(\Omega_{nbd},\mathcal{O})$. In the literature the notion of an o-bounded group is attributed to Okunev. In Theorem 3 of \cite{BKS} it is proven that for a topological group the three properties ${\sf S}_1(\Omega_{nbd},\mathcal{O})$, ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$ and ${\sf S}_{fin}(\Omega_{nbd},\mathcal{O})$ are equivalent. The property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$ is also known as Menger boundedness.
In Problem 5.2 of \cite{Hernandez} Hernandez asked:
\begin{problem}\label{problem:obdproducts}
Is the product of two topological groups, each satisfying the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$, a topological group satisfying the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$?
\end{problem}
Subsequently it was discovered - see Example 2.12 of \cite{HRT} - that there are groups $G$ and $H$, each satisfying the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$, for which the group $G\times H$ does not satisfy the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$. Since subgroups of a group satisfying ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$ inherit the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$, for $G\times H$ to have the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$, each of the groups $G$ and $H$ must have at least the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$. Thus Example 2.12 of \cite{HRT} demonstrates that $G$ or $H$ should satisfy additional hypotheses to guarantee that the product has the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$. Under what conditions on $G$ and $H$ would the product group $G\times H$ satisfy the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$?
A number of \emph{ad hoc} additional conditions on a topological group $G$ that guarantee that its product with a group $H$ also has the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$ have been discovered. Here are two examples of such conditions:
\begin{theorem}[\cite{Hernandez} Theorem 5.3]\label{thm:sigmacpt} If $G$ is a subgroup of a $\sigma$-compact
topological group and $H$ is an ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$ group, then $G\times H$ satisfies ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$.
\end{theorem}
For the next example recall that a topological group is a $P$ group if, and only if, the intersection of countably many open neighborhoods of the identity element still is an open neighborhood of the identity element. More generally a topological space is a $P$ space if each countable intersection of open sets is an open set.
\begin{theorem}[\cite{HRT}, Theorem 2.4]\label{thm:Pgp} If $G$ is an $\aleph_0$-bounded $P$ group and the group $H$ satisfies the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$, then $G\times H$ satisfies ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$.
\end{theorem}
Though the conditions in Theorems \ref{thm:sigmacpt} and \ref{thm:Pgp} at first glance seem very different, a single unifying property in the literature implies both results, namely:
\begin{theorem}[\cite{LB1}, Theorem 6]\label{thm:productgroups}
Let $(G,\otimes)$ be a $T_0$ topological group satisfying the selection principle ${\sf S}_1(\Omega_{nbd},\Gamma)$. Let $\mathcal{A}$ be any of $\mathcal{O}$, $\Omega$ or $\Gamma$. If $(H,\triangle)$ is a topological group satisfying ${\sf S}_1(\Omega_{nbd},\mathcal{A})$, then the product group $G\times H$ satisfies ${\sf S}_1(\Omega_{nbd},\mathcal{A})$.
\end{theorem}
To obtain Theorem \ref{thm:sigmacpt} from Theorem \ref{thm:productgroups}, observe
\begin{lemma}\label{lemma:Th5.3}
An infinite $\sigma$-compact group, as well as any infinite subgroup of it, has the property ${\sf S}_1(\Omega_{nbd},\Gamma)$.
\end{lemma}
\begin{proof} We give the argument for infinite $\sigma$-compact groups, leaving the proof for subgroups of such groups to the reader.
Assume that $G$ is $\sigma$-compact, and write $G$ as the union $\bigcup\{G_n:n\in\mathbb{N}\}$, where for each $n$ $G_n$ is compact and $G_n\subset G_{n+1}$. Let $(\mathcal{U}_n:n\in\mathbb{N})$ be a sequence of $\Omega_{nbd}$ covers of $G$. For each $n$ fix a neighborhood $U_n$ of the identity element such that $\mathcal{U}_n = \{F\otimes U_n: F\subset G \mbox{ finite}\}$.
For each $n$, as $G_n$ is compact, choose a finite set $F_n\subset G$ such that $G_n\subseteq S_n = F_n\otimes U_n \in \mathcal{U}_n$. Then the sequence $(S_n:n\in\mathbb{N})$ witnesses ${\sf S}_1(\Omega_{nbd},\Gamma)$ for the given sequence $(\mathcal{U}_n:n\in\mathbb{N})$ of $\Omega_{nbd}$ covers of $G$.
\end{proof}
Next we show how to derive Theorem \ref{thm:Pgp} from Theorem \ref{thm:productgroups}. First, using the argument in Theorem 2.4 of \cite{Hernandez},
\begin{lemma}\label{lemma:obPgp1}
If $(G,\otimes)$ is an $\aleph_0$-bounded $P$ group, then it has the property ${\sf S}_1(\mathcal{O}_{nbd},\mathcal{O})$
\end{lemma}
\begin{proof}
Let a sequence $(\mathcal{U}_n:n\in\mathbb{N})$ of $\mathcal{O}_{nbd}$-covers of $G$ be given. For each $n$ choose a neighborhood $M_n$ of the identity element such that $\mathcal{U}_n = \mathcal{O}(M_n)$. Since $G$ is a $P$ group, $M = \bigcap\{M_n:n\in\mathbb{N}\}$ is an open set, and neighborhood of the identity element. Then $\mathcal{U} = \{x\otimes M:x\in G\}$ is a member of $\mathcal{O}_{nbd}$. Since $G$ is $\aleph_0$-bounded, fix a countable set $\{x_n:n\in \mathbb{N}\}$ of elements of $G$ such that $\{x_n\otimes M:n\in\mathbb{N}\}$ is a cover of $G$. Then for each $n$ also $x_n\otimes M \subseteq x_n\otimes M_n$. Thus $\{x_n\otimes M_n:n\in\mathbb{N}\}$ witnesses for the sequence $(\mathcal{U}_n:n\in\mathbb{N}\}$ that $(G,\otimes)$ has the property ${\sf S}_1(\mathcal{O}_{nbd},\mathcal{O})$.
\end{proof}
Since ${\sf S}_1(\mathcal{O}_{nbd},\mathcal{O})$ implies ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$, the following lemma extends the conclusion of Lemma \ref{lemma:obPgp1}:
\begin{lemma}\label{lemma:obPgp2}
If $(G,\otimes)$ is an $\aleph_0$-bounded $P$ group, then it has the property ${\sf S}_1(\Omega_{nbd},\Omega)$
\end{lemma}
\begin{proof}
Finite products of $P$ spaces are P spaces. But any (Tychonoff) product of $\aleph_0$-bounded groups is $\aleph_0$-bounded (see for example Proposition 3.2 in the survey \cite{Tkachenko}). Thus, any finite product of $\aleph_0$-bounded $P$ groups is an $\aleph_0$ bounded P-group. By \cite{HRT} Theorem 2.4, finite products of $\aleph_0$-bounded P groups are ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$. By Theorems 2 and 4 of \cite{BKS}, $G$ satisfies ${\sf S}_1(\Omega_{nbd},\Omega)$.
\end{proof}
And finally, we strengthen the conclusion of Lemma \ref{lemma:obPgp2}.
\begin{theorem}\label{thm:PgpSel}\footnote{An alternative proof of Theorem \ref{thm:PgpSel} is given below by Lemmas \ref{lemma:obPgp3} and \ref{lemma:gptogamma}.}
Any $\aleph_0$-bounded $P$ group has the property ${\sf S}_1(\Omega_{nbd},\Gamma)$
\end{theorem}
\begin{proof} Let $(G,\otimes)$ be an $\aleph_0$-bounded $P$ group.
Let $(V_n:n\in\mathbb{N}$ be a sequence of neighborhoods of the identity element of $G$. For each $n$ choose a finite set $F_n$ such that $\{F_n\otimes V_n:n\in\mathbb{N}\}$ is an $\omega$-cover of $G$. Put $V = \bigcap\{U_n:n\in\mathbb{N}\}$. Since $G$ is a $P$-space, $V$ is an open neighborhood of the identity. Also for each $n$ $\mathcal{U} = \{F\otimes V:F\subset G \mbox{ finite}\}$ is an $\Omega_{nbd}$ cover refining $\mathcal{U}_n = \{F\otimes V_n: F\subset G \mbox{ finite}\}$. Applying ${\sf S}_1(\Omega_{nbd},\Omega)$ to the sequence $(\mathcal{U},\mathcal{U},\mathcal{U},\cdots)$ fix for each $n$ a finite set $S_n\subset G$ such that $\{S_n\otimes V:n\in\mathbb{N}\}$ is an $\omega$-cover. For each $n$, set $G_n = \bigcup\{S_j:j\le n\}$. Then for each $n$, $G_n\otimes V_n\in \mathcal{U}_n$, and $\{G_n\otimes V_n:n\in\mathbb{N}\}$ is a $\gamma$-cover of $G$.
\end{proof}
{\flushleft Finally, Theorem \ref{thm:productgroups} and Theorem \ref{thm:PgpSel} imply Theorem \ref{thm:Pgp}.}
Continuing with the theme of providing a single unifying property for questions and claims regarding preserving the property ${\sf S}_1(\Omega_{nbd},\mathcal{O})$ in products, we also give a result on a question from the literature. Tkachenko defined a topological group to be \emph{strictly o-bounded} if player TWO has a winning strategy in the game ${\sf G}_1(\Omega_{nbd},\mathcal{O})$\footnote{Equivalently, TWO has a winning strategy in the game ${\sf G}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$} - \cite{Hernandez}.
In Problem 2.4 of \cite{HRT} the authors ask
\begin{problem}\label{problem:HRT} Is it true that whenever $(G,\otimes)$ is a strictly o-bounded group and $(H,\triangle)$ satisfies the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$, then $G\times H$ also satisfies the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$?
\end{problem}
Problem \ref{problem:HRT} was partially answered in Corollary 8 of \cite{LB1} for the case when the strictly o-bounded group $(G,\otimes)$ is metrizable. Towards answering Problem \ref{problem:HRT} we generalize a part of Theorem 5 of \cite{LB1}.
\begin{theorem}\label{thm:sobishb} If player TWO has a winning strategy in the game ${\sf G}_1(\Omega_{nbd},\mathcal{O})$ played on a $\textsf{T}_0$ topological group, then that group has the property ${\sf S}_1(\Omega_{nbd},\Gamma)$.
\end{theorem}
\begin{proof} Let $(G,\otimes)$ be a strictly o-bounded group. By Lemma \ref{lemma:Th5.3} we may assume it is not totally bounded. Assume that TWO has a winning strategy in the game, say it is $\sigma$. Let $(U_n:n\in\mathbb{N})$ be a sequence of neighborhoods of the identity, each witnessing that the group is not totally bounded. For each $n$ let $\Omega(U_n)$ denote $\{F\otimes U_n:F\subset G\mbox{ finite}\}$, an element of $\Omega_{nbd}$ for $G$..
Then $(\Omega(U_n):n\in\mathbb{N})$ is a sequence of elements of $\Omega_{nbd}$. In the game ${\sf G}_1(\Omega_{nbd},\mathcal{O})$ ONE chooses elements of $\Omega_{nbd}$ and TWO selects members of ONE's moves. Following the construction in the proof of 1$\Rightarrow 2$ of Theorem 5 of \cite{LB1}, define the following subsets of $G$:
\begin{equation}\label{firstmove}
G_{\emptyset} = \bigcap_{n\in\mathbb{N}}\sigma(\Omega(U_n)).
\end{equation}
For $\tau = (n_1,\cdots,n_k)$ a finite sequence of positive integers, define
\begin{equation}\label{latermove}
G_{\tau} = \bigcap_{n\in\mathbb{N}}\sigma(\Omega(U_{n_1}),\, \cdots,\, \Omega(U_{n_k}),\, \Omega(U_n)).
\end{equation}
{\flushleft{\bf Claim 1: }} $G = \bigcup_{\tau\in\,^{<\omega}\mathbb{N}} G_{\tau}$\\
For suppose on the contrary that $x\in G$ is not an element of the union $\bigcup_{\tau\in\,^{<\omega}\mathbb{N}} G_{\tau}$. As $x$ is not in $G_{\emptyset}$, choose $n_1$ with $x\not\in \sigma(\Omega(U_{n_1}))$. Then as $x$ is not in $G_{n_1}$ choose an $n_2$ with $x\not\in \sigma(\Omega(U_{n_1}),\Omega(U_{n_2}))$, and so on. In this way we find a $\sigma$-play of the game during which TWO never covered $x$, contradicting the hypothesis that $\sigma$ is a winning strategy for TWO.
{\flushleft{\bf Claim 2:}} For each finite sequence $\tau$ of positive integers, and for each $n$ there is a finite set $F\subseteq G$ such that $G_{\tau}\subseteq F*U_n$. \\
For let $\tau=(n_1,\cdots,n_k)$ as well as $n$ be given. Then
\[
G_{\tau}\subseteq \sigma(\Omega(U_{n_1}),\cdots,\Omega(U_{n_k}),\Omega(U_n))\in\Omega(U_n).
\]
Finally, enumerate the set of finite sequences of positive integers as $\tau_1,\, \tau_2,\, \cdots,\, \tau_n,\,\cdots$. Choose finite subsets $F_1,\, F_2,\, \cdots,\, F_n,\,\cdots$ of $G$ so that for each $k$ we have
\[
G_{\tau_1}\cup\cdots\cup G_{\tau_k}\subseteq F_k\otimes U_k \in \Omega(U_k).
\]
The sequence $(F_k*U_k:k\in\mathbb{N})$ witnesses ${\sf S}_1(\Omega_{nbd},\Gamma)$ for the given sequence of neighborhoods of the identity.
\end{proof}
The following corollary answers Problem \ref{problem:HRT}:
\begin{corollary}\label{cor:stroboundedproducts}
If $(G,\otimes)$ is a strictly o-bounded $\textsf{T}_0$ group, and $(H,\triangle)$ is a $\textsf{T}_0$ group with the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$, then $G\times H$ has the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$.
\end{corollary}
\begin{proof}
Let $(G,\otimes)$ and $(H,\triangle)$ be as in the hypotheses. By Theorem \ref{thm:sobishb} the group $(G,\otimes)$ has the property ${\sf S}_1(\Omega_{nbd},\Gamma)$. Then by Theorem \ref{thm:productgroups}, $G\times H$ has the property ${\sf S}_1(\Omega_{nbd},\mathcal{O})$.
\end{proof}
\section{The cardinality of $T_0$ groups with the property ${\sf S}_1(\Omega_{nbd},\Gamma)$.} \label{sec:cardinality}
Next we briefly consider the cardinality of topological groups satisfying the property ${\sf S}_1(\Omega_{nbd},\Gamma)$. It is useful to first catalogue a few basic behaviors of the property ${\sf S}_1(\Omega_{nbd},\Gamma)$ under some standard forcing notions. Although one can prove that in general any forcing iteration of length of uncountable cofinality which cofinally often adds a dominating real converts any ground model $\aleph_0$-bounded group into a group satisfying ${\sf S}_1(\Omega_{nbd},\Gamma)$, we prove it here for a specific partially ordered set:
\begin{theorem}\label{thm:hechleriteration} Let $\kappa$ be a cardinal number of uncountable cofinality. Let $({\mathbb P}, <)$ be the finite support iteration by $\kappa$ Hechler reals. If $(G,\otimes)$ is $\aleph_0$-bounded in the ground model, then
\[
{\mathbf 1}_{{\mathbb P}}\mathrel{\|}\joinrel\mathrel{-} ``\check{G} \mbox{ has the property }{\sf S}_1(\Omega_{nbd},\Gamma)".
\]
\end{theorem}
{\flushleft{\bf Proof:}} By Theorem \ref{boundedpreserve} ${\mathbf 1}_{{\mathbb P}}\mathrel{\|}\joinrel\mathrel{-} ``\check{(G,\otimes)} \mbox{ is $\aleph_0$ bounded}"$. Thus, as ${\mathbb P}$ has the countable chain condition, if we take a ${\mathbb P}$-name $(\dot{\mathcal{U}}_n:n<\omega)$ for a sequence of $\mathcal{O}_{nbd}$ members we may assume that this sequence is present in the ground model, since it is a name in an initial segment of the iteration, and we can factor the iteration at this initial segment. Since in this initial segment $(G,\otimes)$ is $\aleph_0$-bounded we may choose for each $n$ a countable subset $X_n$ of $G$ such that $G = \bigcup_{n<\omega}X_n\otimes U_n$, where $\mathcal{U}_n = \mathcal{O}(U_n)$. Define for each $x\in G$ a function $f_x$ from $\omega$ to $\omega$ as follows: Enumerate $X_n$ as $(x^n_m:m<\omega)$. Then
\[
f_x(n) = \min\{m:x\in \{x^n_1,\cdots,x^n_m\}\otimes U_n\}.
\]
The family $\{f_x:x\in G\}$ is in an initial segment of the iteration,
and so the next Hechler real added eventually dominates each $f_x$. Let $g$ be the next Hechler real. Then $\{x^n_j:j\le g(n)\}\otimes U_n \subseteq \mathcal{U}_n$ is a finite subset of $\mathcal{U}_n$, and for each $x$, for all but finitely many $n$, $x\in \{x^n_j:j\le g(n)\} \otimes U_n$. It follows that the group $(G,\otimes)$ has the property ${\sf S}_1(\Omega_{nbd},\Gamma)$.
$\Box$
Incidentally, the Hechler reals partially ordered set does not preserve the Lindel\"of property. In Remark 5 of \cite{IG} Gorelic points out that the points ${\sf G}_{\delta}$ Lindel\"of subspace in this model fails to be Lindel\"of in the generic extension that forces MA plus not-CH. Indeed, this can be accomplished by a finite support iteration of $\omega_2$ or more Hechler reals over a model of CH. Readers could consult the original paper by Hechler \cite{Hechler}, or for example \cite{Palumbo} on Hechler real generic extensions
\begin{theorem}\label{thm:mathiasiteration} Let $({\mathbb P}, <)$ be the countable support iteration by $\aleph_2$ Mathias reals over a model of CH. If $(G,\otimes)$ is $\aleph_0$-bounded in the ground model, then
\[
{\mathbf 1}_{{\mathbb P}}\mathrel{\|}\joinrel\mathrel{-} ``\check{(G,\otimes)} \mbox{ has the property } {\sf S}_1(\Omega_{nbd},\Gamma)".
\]
\end{theorem}
\begin{proof} By Theorem \ref{boundedpreserve} ${\mathbf 1}_{{\mathbb P}}\mathrel{\|}\joinrel\mathrel{-} ``\check{G} \mbox{ is $\aleph_0$ bounded}"$. Thus, as CH holds and antichains of the poset $(\mathbb{P},<)$ have cardinality at most $\aleph_1$, for any ${\mathbb P}$-name $(\dot{U}_n:n<\omega)$ for a sequence of neighborhoods of the identity, we may assume that this sequence of neighborhoods of the identity is present in the ground model (the name of the sequence is a name in an initial segment of the iteration), and factor the iteration over this initial segment. Since by Theorem \ref{boundedpreserve} $(G,\otimes)$ is $\aleph_0$-bounded in this initial segment choose (in the generic extension by this initial segment) for each $n$ a countable subset $X_n$ of $G$ such that $G = \bigcup_{n<\omega}X_n\otimes U_n$, where $\mathcal{U}_n = \mathcal{O}(U_n)$. Define for each $x\in G$ a function $f_x$ from $\omega$ to $\omega$ as follows: Enumerate $X_n$ as $(x^n_m:m<\omega)$. Then
\[
f_x(n) = \min\{m:x\in \{x^n_1,\cdots,x^n_m\}\otimes U_n\}.
\]
The family $\{f_x:x\in G\}$ is in the generic extension by the initial segment (the ``ground model" fooro the remaining generic extension), and so the next Mathias real added by the generic extension eventually dominates each $f_x$. Let $g$ be such a dominating real. Then $\{x^n_j:j\le g(n)\}\otimes U_n \subseteq \mathcal{U}_n$ is a finite subset of $\mathcal{U}_n$, and for each $x$, for all but finitely many $n$, $x\in \{x^n_j:j\le g(n)\} \otimes U_n$. It follows that in the generic extension $(G,\otimes)$ has the property ${\sf S}_1(\Omega_{nbd},\Gamma)$.
\end{proof}
As a consequence we obtain
\begin{theorem}\label{thm:Hurewiczcards}
It is consistent, relative to the consistency of \textsf{ZFC}, that there is for each cardinal number $\kappa$ a group with property $ {\sf S}_1(\Omega_{nbd},\Gamma)$.
\end{theorem}
\begin{proof}
By Lemma \ref{lemma:cardinality} there exists for each infinite cardinal number $\kappa$ an $\aleph_0$-bounded group of cardinality $\kappa$. By either of Theorem \ref{thm:hechleriteration} or Theorem \ref{thm:mathiasiteration}, in the corresponding generic extension each ground model $\aleph_0$-bounded group has property ${\sf S}_1(\Omega_{nbd},\Gamma)$. Since the forcing partially ordered set in either case preserves cardinal numbers, the result follows.
\end{proof}
To round off the consideration of the property ${\sf S}_1(\Omega_{nbd},\Gamma)$ under forcing it is worth recording for the record that
\begin{theorem}\label{thm:Hurewiczcccpreserve} If the group $(G,\otimes)$ has the property ${\sf S}_1(\Omega_{nbd},\Gamma)$ and if $({\mathbb P},<)$ is a partially ordered set with the countable chain condition,, then
\[
{\mathbf 1}_{{\mathbb P}}\mathrel{\|}\joinrel\mathrel{-} ``(\check{G},\check{\otimes}) \mbox{ has the property }{\sf S}_1(\Omega_{nbd},\Gamma)".
\]
\end{theorem}
\begin{proof}
Let $(\dot{U}_n:n<\omega)$ be a ${\mathbb P}$-name for a sequence of neighborhoods of the identity element of the group $(G,\otimes)$.
For each $n$ choose (in the ground model) a sequence $(U^n_m:m<\omega)$ of neighborhoods of the identity element of $(G,\otimes)$, and a maximal antichain $(q^n_m:m<\omega)$ of ${\mathbb P}$ such that for each $n$ and $m$
$ q^n_m\mathrel{\|}\joinrel\mathrel{-} ``\check{U}^n_m\subseteq \dot{U}_n".
$
Then
$ \dot{V}_n = \{(\check{U}^n_m,q^n_m):m<\omega\}
$
is a ${\mathbb P}$-name and
\[
{\mathbf 1}_{{\mathbb P}}\mathrel{\|}\joinrel\mathrel{-}``\dot{V}_n\subseteq\dot{U}_n \mbox{ is a neighborhood of the identity element of } \check{G}"
\]
For each $n$ define
$ N_n = \bigcap_{k,\ell\le n}U^k_{\ell},
$
a (ground model) neighborhood of the identity element of $(G,\otimes)$.
Applying the property ${\sf S}_1(\Omega_{nbd},\Gamma)$, choose finite sets $F_1\subseteq F_2\subseteq\cdots$ such that for each $x\in G$, for all but finitely many $k$, $x$ is a member of $F_k\otimes N_k$.
Then for each $n$ define the ${\mathbb P}$-name $\dot{F}_n$ fooor a finite subset of $\check{G}$ by $\{(\check{F}_{n+m},q^n_m):m<\omega\}$.
{\flushleft{\bf Claim:}} ${\mathbf 1}_{{\mathbb P}}\mathrel{\|}\joinrel\mathrel{-}``(\forall x\in\check{G})(\forall^{\infty}_n)(x\in \dot{F}_n\otimes\dot{V}_n)"$.\\
For let $H$ be a ${\mathbb P}$-generic filter. For each $n$ choose $m_n$ with $q^n_{m_n}\in H$. Then we have that for each $n$,
$ (\dot{F}_n)_{H} = F_{n+m_n}.
$]
Consider any $x\in G$. Choose $k$ s0 large that for $n\ge k$ we have $n+m_n>k$ and $x\in F_{n+m_n}\otimes N_{n+m_n}$. Since
$ N_{n+m_n}\subseteq U^n_{m_n} = (\dot{V})_H
$
it follows that $x\in (\dot{F}_n\otimes\dot{V}_n)_H$.
\end{proof}
\section{Finite powers of Groups with the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$}\label{sec:finpowers}
Consider a topological group $(G,\otimes)$ has the property that whenever $(H,\triangle)$ is a topological group with the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$, then the product group $G\times H$ also has the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$. Then necessarily the group $G\times G$ has the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$: Indeed, every finite power of the group $(G,\otimes)$ has the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$.
Recall Example 2.12 of \cite{HRT} which illustrates that the product of two groups, each with the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$, does not necessarily have the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$. This example in fact gives a group $(G,\otimes)$ which has property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$ but $(G,\otimes)\times (G,\otimes)$ does not have property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$ (and the group is even metrizable).
One might ask whether the phenomenon exhibited by this example - $(G,\otimes)$ has property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$, but $(G,\otimes)\times(G,\otimes)$ does not - is the only obstruction to a topological group $(G,\otimes)$ having a property such as
\begin{itemize}
\item[(A)]{the product of $(G,\otimes)$ with any group with property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$ has the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$}
\item[(B)]{each finite power of $(G,\otimes)$ has property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$.}
\end{itemize}
The following two prior results shed significant light on version (B) of this question:
\begin{theorem}[Banakh and Zdomskyy, Mildenberger and Shelah]\label{thm:finpowersprod} The following statement is consistent, relative to the consistency of \textsf{ZFC}:
{\flushleft{For}} each $\textsf{T}_0$ group $(G,\otimes)$, if $(G,\otimes)\times(G,\otimes)$ has the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$, then the group $(G,\otimes)$ in fact has the property ${\sf S}_{fin}(\Omega_{nbd},\mathcal{O}^{wgp})$.
\end{theorem}
Regarding Theorem \ref{thm:finpowersprod}: Prior results (Theorems 3, 6 and 7 of \cite{BKS}) that show that every finite power of a topological group has property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$ if, and only if, the group has the property ${\sf S}_1(\Omega_{nbd},\mathcal{O}^{wgp})$. Moreover,
\begin{lemma}\label{lemma:sfinsone} For a topological group $(G,\otimes)$ the following are equivalent:
\begin{enumerate}
\item{$(G,\otimes)$ has the property ${\sf S}_{fin}(\Omega_{nbd},\mathcal{O}^{wgp})$}
\item{$(G,\otimes)$ has the property ${\sf S}_1(\Omega_{nbd},\mathcal{O}^{wgp})$}
\end{enumerate}
\end{lemma}
\begin{proof}
We must show that $(1)$ implies $(2)$. Thus, let $(\mathcal{U}_n:n\in\mathbb{N})$ be a sequence of elements of $\Omega_{nbd}$, say for each $n$ the set $V_n$ is a neighborhood of the identity element of $G$ and $\mathcal{V}_n = \{F\otimes V_n:F\subset G \mbox{ finite}\}$.
Then for each $n$ define $U_n = \bigcap\{V_j:j\le n\}$, a neighborhood of the identity of $G$, and define $\mathcal{U}_n = \{F\otimes U_n:n\in \mathbb{N}\}$. As each $\mathcal{U}_n$ is an element of $\Omega_{nbd}$, apply ${\sf S}_{fin}(\Omega_{nbd},\mathcal{O}^{wgp})$ to the sequence $(\mathcal{U}_n:n\in\mathbb{N})$. For each $n$ choose a finite subset $\mathcal{F}_n$ of $\mathcal{U}_n$ such that $\mathcal{G} = \bigcup\{\mathcal{F}_n:n\in\mathbb{N}\}$ is a weakly groupable cover of $G$.
Fix a partition $(\mathcal{G}_n:n\in\mathbb{N})$ of $\mathcal{G}$ into finite sets $\mathcal{G}_n$ such that there is for each finite subset $S$ of $G$ an $n$ with $S\subset \bigcup\mathcal{G}_n$.
\end{proof}
Thus, Theorem \ref{thm:finpowersprod} establishes the consistency of the statement that if a $\textsf{T}_0$ group $(G,\otimes)$ is such that $(G,\otimes) \times (G,\otimes)$ has the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$, then every finite power of $(G,\otimes)$ has the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$. It turns out that in fact this statement is independent of \textsf{ZFC}, since on the other hand:
\begin{theorem}[\cite{MST}, Theorem 11]\label{thm:kk+1}
It is consistent, relative to the consistency of \textsf{ZFC}, that there is for each positive integer $k$ a separable metrizable topological group $(G,\otimes)$ such that $G^k$ has the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$ while $G^{k+1}$ does not have the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$.
\end{theorem}
Less is known about version (A) of the question above. Interestingly, for the subclass of metrizable groups which have the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$ in all finite powers, it is consistent that a product of finitely many groups in this subclass is still in this subclass. In fact, an equiconsistency criterion has been identified.
\begin{theorem}[He, Tsaban and Zang \cite{HTZ}, Theorem 2.1]\label{thm:powertoprooducts}
The following statements are equivalent:
\begin{enumerate}
\item{\textsf{NCF}}
\item{The product of two metrizable groups, each with the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O}^{wgp})$, is a topological group with the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O}^{wgp})$.}
\end{enumerate}
\end{theorem}
This result raises the following potentially more modest analogue of the version (A) question:
\begin{problem}
Is it consistent that product of any two $\textsf{T}_0$ groups, each with the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O}^{wgp})$, is a topological group with the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O}^{wgp})$?
\end{problem}
\section{Further remarks on $\aleph_0$ bounded P groups}\label{sec:Pgroups}
Towards further strengthening results about $\aleph_0$-bounded $P$ groups we next consider products of topological groups with the property ${\sf S}_1(\mathcal{O}_{nbd},\mathcal{O})$, a stronger property than ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$.
For finite powers there is the following prior result
\begin{theorem}[\cite{BKS}, Theorem 15] \label{thm:sfinpowers} For a topological group $(G,\otimes)$ the following are equivalent:
\begin{enumerate}
\item{Each finite power of $(G,\otimes)$ has the property ${\sf S}_1(\mathcal{O}_{nbd},\mathcal{O})$}
\item{$(G,\otimes)$ has the property ${\sf S}_1(\mathcal{O}_{nbd},\mathcal{O}^{wgp})$}
\end{enumerate}
\end{theorem}
{\flushleft Lemma \ref{lemma:obPgp1} can be strengthened as follows:}
\begin{lemma}\label{lemma:obPgp3}
Any $\aleph_0$-bounded $P$ group has the property ${\sf S}_1(\mathcal{O}_{nbd},\mathcal{O}^{gp})$
\end{lemma}
\begin{proof}
Let $(G,\otimes)$ be an $\aleph_0$-bounded $P$ group. Let $(\mathcal{U}_n:n\in\mathbb{N})$ be a sequence of $\mathcal{O}_{nbd}$-covers of $G$. For each $n$ choose a neighborhood $U_n$ of the identity such that $\mathcal{U}_n = \mathcal{O}(U_n) = \{g\otimes U_n:g\in G\}$. Since $G$ is a $P$ space the $\textsf{G}_{\delta}$ set $U = \bigcap\{U_n:n\in\mathbb{N}\}$ is an open neighborhood of the identity, and the $\mathcal{O}_{nbd}$-cover $\{g\otimes U:g\in G\}$ of $G$ is a refinement of each of the $\mathcal{O}_{nbd}$ covers $\mathcal{U}_n$. For each $n$ set $\mathcal{V}_n$ be the $\Omega_{nbd}$ cover $\{F\otimes U:\; F\subset G \mbox{ finite}\}$.
As was shown in Theorem \ref{thm:PgpSel}, this group has the selection property ${\sf S}_1(\Omega_{nbd},\Gamma)$. Applying this selection property of $G$ to the sequence $(\mathcal{V}_n:n\in\mathbb{N})$ we find for each $n$ a set $V_n\in\mathcal{V}_n$ such that $\{V_n:n\in\mathbb{N}\}$ is a $\gamma$-cover of $G$ - that is, for each $g\in G$ we have for all but finitely many $n$ that $g\in V_n$. For each $n$ fix the finite set $F_n\subset G$ such that $V_n = F_n\otimes U$. Now let $n_1,\; n_2,\; \cdots,\; n_k,\; \cdots$ be natural numbers such that for each $i$ we have $n_i = \vert F_i\vert$. Next choose elements $g_1,\; g_2,\; \cdots,\; g_n,\; \cdots$ from $G$ as follows: $g_1,\cdots,g_{n_1}$ lists the distinct elements of $F_1$, $g_{n_1+1},\cdots,g_{n_1+n_2}$ lists the distinct elements of $F_2$, and in general
$\{g_{n_1+\cdots+n_{k-1}+1},\cdots,g_{n_1+n_2+\cdots+n_k}\}$ lists the distinct elements of $F_k$, and so on.
Thus for each $k$ we have $V_k = \cup\{g_i\otimes U: n_1+\cdots+n_{k-1} < i \le n_1+\cdots+n_k\}$.
{\flushleft{\bf Claim: }} $\{g_i\otimes U_i:i\in\mathbb{N}\}$ is a groupable open cover of $G$.
For let an $h\in G$ be given. Since $(V_n:n\in\mathbb{N})$ is a $\gamma$ cover of $G$, fix a $k$ such that for all $m\ge k$ it is true that $h\in V_k$. Then for all $m\ge k$, the element $g$ of $G$ is in $\bigcup\{g_i\otimes U_i:n_1+\cdots+n_{k-1}<i\le n_1+\cdots+n_k\}$, confirming that the selector $(g_i\otimes U_i:i\in\mathbb{N})$ of the original sequence of $\mathcal{O}_{nbd}$ covers is a groupable open cover of $G$.
\end{proof}
Lemma \ref{lemma:obPgp3} provides the following alternative derivation that $\aleph_0$-bounded $P$ groups have the property ${\sf S}_1(\Omega_{nbd},\Gamma)$:
\begin{lemma}\label{lemma:gptogamma}
If a topological group has the property ${\sf S}_1(\mathcal{O}_{nbd},\mathcal{O}^{gp})$ then it has the property ${\sf S}_1(\Omega_{nbd},\Gamma)$.
\end{lemma}
\begin{proof}
Let $(G,\otimes)$ be a topological group which has the property ${\sf S}_1(\mathcal{O}_{nbd},\mathcal{O}^{gp})$. Let $(\mathcal{U}_n:n\in\mathbb{N})$ be a sequence of $\Omega_{nbd}$ covers of $G$. For each $n$ fix $U_n$, the neighborhood of the identity element for which $\mathcal{U}_n = \{F\otimes U_n:F\subset G \mbox{ finite}\}$.
For each $n$ set $V_n = \bigcap\{U_j:j\le n\}$, a neighborhood if the identity element of the group $(G,\otimes)$. Set $\mathcal{V}_n = \{F\otimes V_n:F\subset G \mbox{ finite}\}$, a member of $\Omega_{nbd}$ that refines $\mathcal{U}_n$.
Now apply to selection principle ${\sf S}_1(\mathcal{O}_{nbd},\mathcal{O}^{gp})$ to each of the $\mathcal{O}_{nbd}$ covers $\mathcal{A}_n = \{g\otimes V_n: n\in\mathbb{N}\}$: For each $n$ choose
an $A_n\in\mathcal{A}_n$ such that $\{A_n:n\in\mathbb{N}\} \in\mathcal{O}^{gp}$. Fix a sequence $n_1<n_2<\cdots<n_k<\cdots$ of natural numbers such that for each $g\in G$, for all but finitely many $k$, $g$ is an element of $\bigcup\{A_m:n_{k-1}<m\le n_k\}$. For each $m$ fix $g_m\in G$ such that $A_m = g_m\otimes V_m$. Then define finite sets $F_1 = \{g_n:n \le n_1\}$ and for each $k$, $F_k = \{g_j: n_{k-1}<j \le n_k\}$.
For each $k$ set $U_k = F_k\otimes V_k$, an element of $\mathcal{U}_k$. Then $\{U_k:k\in\mathbb{N}\}$ is a $\gamma$-cover of $G$,
for let an $x\in G$ be given. Choose $k$ so large that for all $m\ge k$, $x$ is a member of $\bigcup_{n_{m-1}<j\le n_m}g_j\otimes V_j$. Since $\bigcup_{n_{m-1}<j\le n_m}g_j\otimes V_j \subseteq U_m$, it follows that for all $m\ge k$, $x$ is a member of $U_m$.
\end{proof}
\section{Conclusion}
In this paper we merely touched on four extensively explored topics in the arena of $\aleph_0$-bounded groups. Among the numerous exploration possibilities we pose here only the following one about cardinalities:
We have noted that there are no \emph{a priori} theoretical restrictions on the cardinality that a $T_0$ group with the property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$, or even ${\sf S}_1(\Omega_{nbd},\Gamma)$, can have. Each infinite cardinality is possible. As was pointed out in Theorems 8 and Corollary 17 of \cite{MS1}, the same holds for $T_0$ groups with the property ${\sf S}_1(\mathcal{O}_{nbd},\mathcal{O})$, or even the much stronger property that the group is an $\aleph_0$-bounded P group, or a group in which TWO has a winning strategy in the game ${\sf G}_1(\mathcal{O}_{nbd},\mathcal{O})$. However, the following issue regarding the achievable cardinality for a given type of $\aleph_0$-bounded group is much more subtle\footnote{Also, in the class of Lindel\"of spaces, for example, there are no constraints on the cardinalities achievable in the class oof $T_0$ Lindel\"of spaces, yet there are constraints on the cardinalities of subspaces that are Lindel\"of, as can for example be gleaned from \cite{KT}.
}: Let an $\aleph_0$-bounded $T_0$ group be given. It necessarily has subgroups with properties ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$, ${\sf S}_1(\Omega_{nbd},\Gamma)$, ${\sf S}_1(\mathcal{O}_{nbd},\mathcal{O})$, and any of the other nonempty selection based classes obtained by varying the types of open covers appearing. The question of what cardinality restrictions there may be on subgroups of an $\aleph_0$ bounded $T_0$ group has been extensively studied in the case of the property ${\sf S}_1(\mathcal{O}_{nbd},\mathcal{O})$. For example in \cite{GS} and in \cite{MS2} the following hypothesis\footnote{This hypothesis is a generalization of the classical Borel Conjecture} is investigated:
\begin{quote}
Each subgroup with property ${\sf S}_1(\mathcal{O}_{nbd},\mathcal{O})$ of an $\aleph_0$-bounded group of weight $\kappa$ has cardinality at most $\kappa$
\end{quote}
It would be interesting to know if there are similar feasible hypotheses of cardinality bounds for subgroups with property ${\sf S}_1(\Omega_{nbd},\Gamma)$ or with property ${\sf S}_{fin}(\mathcal{O}_{nbd},\mathcal{O})$ of $\aleph_0$ bounded groups that are not $\sigma$ totally bounded.
\end{document} |
\betaegin{document}
\makeatletter
{\delta}ef\inftymod#1{\alphallowbreak\mkern10mu({{\omega}perator@font mod}\,\,#1)}
\makeatother
\alphauthor{Alexander Berkovich}
\alphaddress{Department of Mathematics, University of Florida, 358 Little Hall, Gainesville FL 32611, USA}
\epsilonmail{alexb@ufl.edu}
\alphauthor{Frank Patane}
\alphaddress{Department of Mathematics, University of Florida, 358 Little Hall, Gainesville FL 32611, USA}
\epsilonmail{frankpatane@ufl.edu}
\varphiitle[{\sigma}calebox{.9}{Essentially Unique Representations by Certain Ternary Quadratic Forms}]{Essentially Unique Representations by Certain Ternary Quadratic Forms}
\betaegin{abstract}
In this paper we generalize the idea of ``essentially unique'' representations by ternary quadratic forms. We employ the Siegel formula, along with the complete classification of imaginary quadratic fields of class number less than or equal to 8, to deduce the set of integers which are represented in essentially one way by a given form which is alone in its genus. We consider a variety of forms which illustrate how this method applies to any of the 794 ternary quadratic forms which are alone in their genus. As a consequence, we resolve some conjectures of Kaplansky regarding unique representation by the forms $x^2 +y^2 +3z^2$, $x^2 +3y^2 +3z^2$, and $x^2 +2y^2 +3z^2$ \cite{kap}.
\epsilonnd{abstract}
\chieywords{representations of integers, sum of three squares, local densities, ternary quadratic forms, Siegel formula}
{\sigma}ubjclass[2010]{11B65, 11E16, 11E20, 11E25, 11E41, 11F37}
{\delta}ate{\varphioday}
\maketitle
{\sigma}ection{Introduction}
\lambdaabel{intro}
The concept of connecting the number of representations of a ternary quadratic form to the class number for binary quadratic forms dates back to Gauss \cite{gauss}. Gauss was the first to introduce many fundamental concepts such as discriminant, positive definite form, and equivalence of forms. After introducing these fundamental notions, he related the number of representations of an integer by $x^2 + y^2 +z^2$ to what is essentially the class number for binary quadratic forms.\\
Representation of integers by $x^2 +y^2 +z^2$, has been studied by many mathematicians since the time of Gauss. Building on the work of Hardy, Bateman \cite{bate} derived and
proved the formula for the number of representations of a positive integer as the sum of three squares. We point out that this representation formula is a special case of the more general Siegel formula which can be found in \cite{siegel}. We mention this since our treatment often relies on the Siegel formula, which we will describe in the next section.\\
Rather than discussing the total number of representations by a quadratic form, one can identify solutions according to a given relation. In the case of the form $x^2 +y^2 +z^2$, identifying solutions which are the same up to order and sign is equivalent to partitioning a number into three squares. In 1948, Lehmer considered partitions of an integer into $k$ squares \cite{lehm}. We refer the reader to \cite{sel} for a recent (2004) discussion of this topic.\\
In 1984, Bateman and Grosswald essentially classified all integers which have one representation up to order and sign by $x^2 +y^2 +z^2$ \cite{batgros}. Their proof assumed they had the complete list of discriminants of binary quadratic forms with class number less than or equal to 4. In 1992, Arno completely classified all discriminants of binary quadratic forms with class number less than or equal to 4 \cite{arno}. Bateman and Grosswald's assumption was proven correct.\\
In 1997, Kaplansky considered the forms $x^2 +y^2 +2z^2$, $x^2 +2y^2 +2z^2$, and $x^2 +2y^2 +4z^2$ \cite{kap}. He identified solutions which are the same up to ``order and sign'', and deduced which numbers are represented in essentially one way by the aforementioned forms. He utilized the completed list of discriminants of binary quadratic forms with class number less than or equal to 4 to deduce the integers with essentially unique representation by the forms he considered. Kaplansky then conjectured about the numbers which are represented in essentially one way by the forms $x^2 +2y^2 +3z^2$, $x^2 +3y^2 +3z^2$, and $x^2 +y^2 +3z^2$.\\
In this paper we employ the results of Watkins \cite{watkins} to resolve Kaplansky's conjectures. Furthermore, we extend the idea of essentially unique representation beyond diagonal forms, where we must consider more than ``order and sign''. Our treatment will apply to any of the 794 ternary quadratic forms which are alone in their genus. The determination of these forms was first explored by Watson \cite{wat1}, with the final touch delivered by Jagy, Kaplansky, and Schiemann \cite{jagykap}. (See also \cite{lorch}.) For the remainder of this paper, we call a form idoneal when it is alone in its genus. We point out that $x^2 +y^2 +z^2$, along with all the ternary forms discussed in \cite{kap}, are idoneal.\\
In Section {\rho}ef{notprim}, we will give the necessary definitions and notation as well as discuss automorphs and ``essentially unique'' representations. We then outline the general approach of how to use the Siegel formula along with class number bounds to derive integers which are represented in essentially one way by an idoneal ternary form. The largest class number bound we utilize in this paper is 8. We have compiled tables in the Appendix which list all discriminants of binary quadratic forms with class number less than or equal to 8 according to class group type. \\
In Section {\rho}ef{ex1}, we will consider the non-diagonal forms $x^2 +y^2 +z^2 +yz +xz +xy$, $3x^2 +3y^2 +3z^2 -2yz +2xz +2xy,$ and $x^2 +3y^2 +3z^2 +2yz$. These forms are selected and grouped together in Section {\rho}ef{ex1} because we treat these forms by relating them to $x^2+y^2+z^2$. This generalizes the approach of Kaplansky \cite{kap} to non-diagonal forms, and we comment that the three selected forms are among many which can be handled in a similar fashion. In particular, if $f$ is an idoneal form of discriminant $\Delta=2^k$, then one can find the integers which are uniquely represented by $f$ by reducing $f$ to $x^2+y^2+z^2$.\\
In Section {\rho}ef{ex2}, we will examine the non-diagonal forms $5x^2 +13y^2 +20z^2 -12yz +4xz +2xy$ and $7x^2 +15y^2 +23z^2 +10yz +2xz +6xy$. Both of these forms can be treated by the methods of Section {\rho}ef{ex1}, however we chose to use the Siegel formula along with local density considerations to derive the integers which they represent in essentially one way. \\
In Section {\rho}ef{ex3}, we resolve the aforementioned conjectures of Kaplansky. Explicitly, we find the integers which are represented in essentially one way by $x^2 +y^2 +3z^2$, $x^2 +3y^2 +3z^2$, and $x^2 +2y^2 +3z^2$, by applying the method outlined in Section {\rho}ef{notprim}.\\
Section {\rho}ef{concl}, also called the Outlook, contains our concluding remarks. In the Outlook we consider the form $x^2 +3y^2 +3z^2 +yz +xy$ which is not idoneal. We sketch the proof that we have found all integers which are represented in essentially one way by this form. We conclude this paper with prospects for future work.
{\sigma}ection{Notation and Preliminaries}
\lambdaabel{notprim}
We use the notation $(a,b,c,d,e,f)$ to represent the positive ternary quadratic form $ax^2 +by^2 +cz^2 +dyz +exz +fxy$. We remark that this paper only considers positive ternary quadratic forms. We use $(a,b,c,d,e,f;n)$ to denote the total number of representations of $n$ by $(a,b,c,d,e,f)$. We take $(a,b,c,d,e,f;n)=0$ when $n \not\inftyn \mathbb{N}$.
The associated theta series to the form $(a,b,c,d,e,f)$ is
\betaegin{equation}
\lambdaabel{thet}
\vartheta(a,b,c,d,e,f,q):={\sigma}um_{x,y,z}q^{ax^2 +by^2 +cz^2 +dyz +exz +fxy} = {\sigma}um_{n\gammaeq 0}(a,b,c,d,e,f;n)q^n.
\epsilonnd{equation}
The discriminant $\Delta$ of $(a,b,c,d,e,f)$ is defined as
\[
\Delta:= \frac{1}{2} \mbox{det}\lambdaeft( \betaegin{array}{lll}
2a& f &e \\
f & 2b &d \\
e&d &2c \\\epsilonnd{array} {\rho}ight) = 4abc +def -ad^2 -be^2 -cf^2.
\]
We note that the discriminant $\Delta>0$ for a positive ternary quadratic form.
Two ternary quadratic forms of discriminant $\Delta$ are in the same genus if they are equivalent over $\mathbb{Q}$ via a transformation matrix in $SL(3,\mathbb{Q})$ whose entries have denominators coprime to $2\Delta$.\\
Let $A$ be a 3 by 3 matrix of determinant $\pm 1$. $A$ is an automorph for the form $(a,b,c,d,e,f)$ if the action of $A$ on $(a,b,c,d,e,f)$ leaves $(a,b,c,d,e,f)$ unchanged. We denote the set of automorphs of $(a,b,c,d,e,f)$ by Aut$(a,b,c,d,e,f)$. A discussion of automorphs for ternary quadratic forms is given in \cite{dickson}. We use Sage 5.1 to explicitly compute the automorphs for the forms considered in this paper. We now give a brief example by considering the 8 automorphs of the form $(1,3,4,3,1,0)$. We have
\betaegin{align*}
\mbox{Aut}(1,3,4,3,1,0)=
\Bigg\{&\lambdaeft( \betaegin{array}{lll}
$1$& $0$ &$0$ \\
$0$ & $1$ &$0$ \\
$0$&$0$ &$1$ \\\epsilonnd{array} {\rho}ight),\lambdaeft( \betaegin{array}{lll}
$-$1$$& $0$ &$0$ \\
$0$ & $-$1$$ &$0$ \\
$0$&$0$ &$-$1$$ \\\epsilonnd{array} {\rho}ight),\lambdaeft( \betaegin{array}{lll}
$1$& $0$ &$1$ \\
$0$ & $-$1$$ &$0$ \\
$0$&$0$ &$-$1$$ \\\epsilonnd{array} {\rho}ight),\lambdaeft( \betaegin{array}{lll}
$-$1$$& $0$ &$-$1$$ \\
$0$ & $1$ &$0$ \\
$0$&$0$ &$1$ \\\epsilonnd{array} {\rho}ight),\\
& \lambdaeft( \betaegin{array}{lll}
$1$& $0$ &$0$ \\
$0$ & $-$1$$ &$-$1$$ \\
$0$&$0$ &$1$ \\\epsilonnd{array} {\rho}ight),\lambdaeft( \betaegin{array}{lll}
$-$1$$& $0$ &$0$ \\
$0$ & $1$ &$1$ \\
$0$&$0$ &$-$1$$ \\\epsilonnd{array} {\rho}ight),\lambdaeft( \betaegin{array}{lll}
$1$& $0$ &$1$ \\
$0$ & $1$ &$1$ \\
$0$&$0$ &$-$1$$ \\\epsilonnd{array} {\rho}ight),\lambdaeft( \betaegin{array}{lll}
$-$1$$& $0$ &$-$1$$ \\
$0$ & $-$1$$ &$-$1$$ \\
$0$&$0$ &$1$ \\\epsilonnd{array} {\rho}ight)\Bigg\}.
\epsilonnd{align*}
To give a further illustration, we note that $(1,3,4,3,1,0;19)=12$. Under the action of \\Aut$(1,3,4,3,1,0)$, the solutions form two orbits:
\betaegin{align*}
O_1:&=\{(-4, -1, 0), (-4, 1, 0), (4, -1, 0), (4, 1, 0)\},\\
O_2:&=\{(-3, -2, 2), (-3, 0, 2), (-1, 0, -2), (-1, 2, -2), (1, -2, 2), (1, 0,2), (3, 0, -2), (3, 2, -2)\}.
\epsilonnd{align*}
The solutions in $O_1$ are easily identified as the solution $(4,1,0)$ up to sign. However the solutions in $O_2$ are not so readily identified as being equivalent under the action of automorphs.\\
Identifying solutions which are equivalent under the action of automorphs is the way to generalize previous authors' (\cite{batgros}, \cite{kap}, \cite{lehm}) notion of solutions being equivalent up to ``order and sign''.\\
When the solutions form exactly $k$ orbits under the action of automorphs, we say the form represents the integer in essentially $k$ ways. We say an integer has an essentially unique representation when the solutions form 1 orbit under the action of automorphs. We also note that if $f$ is any ternary quadratic form then $f(x,y,z)=n$ implies $f(-x,-y,-z)=n$. Thus if $f$ represents a positive integer $n$, then $f$ represents $n$ in at least two ways. Hence there is little ambiguity if we say that $f$ uniquely represents an integer when the solutions form 1 orbit under the action of automorphs.\\
The focus of this paper is concerned with finding integers which have an essentially unique representation by a given idoneal form. In particular, we give a method which enables one to find all integers which are represented in essentially one way, by an idoneal ternary quadratic form. We now introduce an essential tool to our method, the celebrated Siegel theorem for positive ternary quadratic forms.
\betaegin{theorem}
\lambdaabel{sw}
Let $G$ be a genus of positive ternary quadratic forms of discriminant $\Delta$. Then
\betaegin{equation}
{\sigma}um_{t \inftyn G} \frac{R_t(n)}{|\varphiextnormal{Aut}(t)|} = 4\pi M(G) {\sigma}qrt{\frac{n}{\Delta}}\prod_{p} d_{G,p}(n),
\epsilonnd{equation}
\epsilonnd{theorem}
\noindent
where $R_t(n)$ denotes the total number of representations of $n$ by $t$, and it is understood that the sum on the left is over representatives of each equivalence class in the genus $G$. The product on the right is over all primes $p$, and the mass of $G$ is defined as
\[
M(G):={\sigma}um_{t \inftyn G} \frac{1}{|\mbox{Aut}(t)|}.
\]
Let $t:=ax^2 +by^2 +cz^2 +dyz +exz +fxy$ be any form in $G$. Then the $p$-adic local density $d_{G,p}$ (also called $d_{t,p}$) is
\betaegin{equation}
\lambdaabel{locald}
d_{t,p}(n):=\lambdaim_{k\varphio \inftynfty} p^{-2k}|\{(x,y,z)\inftyn \mathbb{Z}^3 : ax^2 +by^2 +cz^2 +dyz +exz +fxy\epsilonquiv n \inftymod{p^k}\}|.
\epsilonnd{equation}
We remark that the limit in \epsilonqref{locald} can be removed as long as we take $k$ large.\\
Theorem {\rho}ef{sw} is a special case of the general Siegel theorem given in \cite{siegel}. When the form $f$ is idoneal, Theorem {\rho}ef{sw} gives an explicit formula for $R_f(n)$.
\betaegin{corollary}
\lambdaabel{sws}
Let $f$ be an idoneal ternary quadratic form of discriminant $\Delta$. Then
\[
R_{f}(n) = 4\pi {\sigma}qrt{{\delta}frac{n}{\Delta}} \prod_{p} d_{f,p}(n),
\]
with all notation as previous.
\epsilonnd{corollary}
\noindent
In \cite{siegel} Siegel shows that when $(2\Delta,p)=1$ we have
\betaegin{equation}
\lambdaabel{exsi}
d_{t,p}(n) = \lambdaeft\{ \betaegin{array}{ll} 1+\frac{1}{p} + \frac{1}{p^{k+1}}\lambdaeft( \lambdaeft(\frac{-m\Delta}{p}{\rho}ight)-1{\rho}ight)& n=mp^{2k}, p \nmid m,\\
&\\
\lambdaeft(\frac{1}{p}+1{\rho}ight)\lambdaeft(1-\frac{1}{p^{k+1}}{\rho}ight)& n=mp^{2k+1}, p \nmid m,\\
\epsilonnd{array}
{\rho}ight.
\epsilonnd{equation}
where $\lambdaeft(\varphifrac{r}{p}{\rho}ight)$ is the Legendre symbol.
When we use Corollary {\rho}ef{sws} along with equation \epsilonqref{exsi}, we obtain a very explicit formula for the total number of representations of an integer by an idoneal form. Employing \epsilonqref{exsi} it can be shown that
\betaegin{equation}
\lambdaabel{sp}
\prod_{p\nmid 2\Delta} d_{f,p}(n) = {\delta}frac{8}{\pi^2}L(1,\chi(\Delta n)) P(n,\Delta) \prod_{2<p \mid \Delta}{\delta}frac{1}{1-\varphifrac{1}{p^2}},
\epsilonnd{equation}
where $L(1,\chi(n))$ is given by
\betaegin{equation}
\lambdaabel{el}
L(1,\chi(n)):={\sigma}um_{m=1}^{\inftynfty}{\delta}frac{\lambdaeft(\varphifrac{-4n}{m}{\rho}ight)}{m}=\prod_{p>2}{\delta}frac{1}{1-\varphifrac{\lambdaeft(\varphifrac{-n}{p}{\rho}ight)}{p}},
\epsilonnd{equation}
and $\chi(n):=\lambdaeft(\frac{-4n}{\betaullet}{\rho}ight)$. Lastly $P(n,\Delta)$ is given by the finite product
\betaegin{equation}
\lambdaabel{p}
P(n,\Delta) := \prod_{{\sigma}ubstack{(p^{2})^b\mid\mid n,\\p \nmid 2\Delta}}\lambdaeft( 1+ {\delta}frac{1}{p} + {\delta}frac{1}{p^2}+\cdots +{\delta}frac{1}{p^{b-1}} + {\delta}frac{1}{p^b (1 - \lambdaeft(\varphifrac{-\Delta np^{-2b}}{p}{\rho}ight)p^{-1})}{\rho}ight),
\epsilonnd{equation}
where the product is over all primes $p \nmid 2\Delta$ such that $p^2\mid n$, and $b$ is the largest integer such that $p^{2b}\mid n$. We note that the only property of $P(n,\Delta)$ that we use is $P(n,\Delta)$ is a finite product with $1\lambdaeq P(n,\Delta)$. Lastly, $P(n,4^k)=P(n,4)=P(n,1)$ and so we define the abbreviated $P(n):=P(n,1)$.\\
Combining Corollary {\rho}ef{sws} with \epsilonqref{sp} yields
\betaegin{equation}
\lambdaabel{mf}
R_{f}(n) = \frac{32{\sigma}qrt{n}}{\pi{\sigma}qrt{\Delta}} L(1,\chi(\Delta n))\cdot P(n,\Delta)\cdot \prod_{p\mid 2\Delta} d_{f,p}(n)\prod_{2<p \mid \Delta}{\delta}frac{1}{1-\varphifrac{1}{p^2}},
\epsilonnd{equation}
where $f$ is an idoneal ternary quadratic form of discriminant $\Delta$.\\
If we factor $n$ as $n=4^a \cdot m\cdot d^2$ with $2 \nmid d$ and $m$ squarefree, then
\betaegin{align}
\lambdaabel{lphi}
L(1,\chi(n)) &=L(1,\chi(m))\prod_{p\mid d}1-\frac{\lambdaeft(\varphifrac{-m}{p}{\rho}ight)}{p}.
\epsilonnd{align}
Dirichlet gives a wonderful connection between $L(1,\chi(m))$ and $h(D)$, the number of reduced primitive binary quadratic forms of discriminant $D=-m$ or $D=-4m$ \cite{dir}. We call $h(D)$ the class number of discriminant $D$. The relationship between $L(1,\chi(m))$ and $h(D)$ is given in the following theorem.
\betaegin{theorem}
\lambdaabel{tthg}
For $m$ squarefree and $\chi(m):=\lambdaeft(\frac{-4m}{\betaullet}{\rho}ight)$, we have
\betaegin{equation}
\lambdaabel{h}
L(1,\chi(m)) = \lambdaeft\{ \betaegin{array}{ll}
{\delta}fracrac{\pi}{4}& m=1,\\
&\\
{\delta}fracrac{\pi}{2{\sigma}qrt{3}} & m=3,\\
&\\
{\delta}fracrac{3\pi}{2{\sigma}qrt{m}}h(-m) & 3<m\epsilonquiv 3 \inftymod{8},\\
&\\
{\delta}fracrac{\pi}{2{\sigma}qrt{m}}h(-m) & m \epsilonquiv 7 \inftymod{8},\\
&\\
{\delta}fracrac{\pi}{2{\sigma}qrt{m}}h(-4m) & 1< m\epsilonquiv 1,2 \inftymod{4}.\\
\epsilonnd{array}
{\rho}ight.
\epsilonnd{equation}
\epsilonnd{theorem}
\noindent
See \cite{buell} for details.\\
Let $f$ be an idoneal ternary quadratic form. A necessary but not sufficient condition for $f$ to uniquely represent $n$ up to the action of automorphs, is
\betaegin{equation}
\lambdaabel{formula}
0<R_{f}(n) \lambdaeq |\varphiext{Aut}(f)|.
\epsilonnd{equation}
We employ \epsilonqref{mf} along with Theorem {\rho}ef{tthg} to give an explicit lower bound for $R_f(n)$ in terms of the class number. We then classify the $n$ which satisfy \epsilonqref{formula}, and call this set the prelist of $f$, denoted by $\varphiext{Prelist}(f)$. We need only check which elements of $\varphiext{Prelist}(f)$ have the property that their solutions form one orbit under Aut$(f)$. There are only a finite number of elements of $\varphiext{Prelist}(f)$ we need to check, and we employ Maple V.15 to compute the number of orbits of the solutions. Of course, solving for the elements of $\varphiext{Prelist}(f)$ requires one to have information on the bounds of the class number. Hence we now discuss class number considerations.\\
Attempting to solve $h(d)=k$ dates back to Gauss \cite[Section V, Article 303]{gauss}, where Gauss conjectured the list of imaginary quadratic fields of small class number.
The case $h(d)=1$ was essentially first completed by Heegner in 1951 \cite{heg}. In 1967, Baker and Stark gave independent proofs for the $h(d)=1$ case as well. Many mathematicians have done extensive work towards solving $h(d)=k$ for $k\lambdaeq 8$. (See \cite{arno}, \cite{baker}, \cite{gold}, and \cite{gross}.)\\
The latest developments are given by Watkins \cite{watkins}. According to Watkins, the largest (in magnitude) fundamental discriminant $d$ with class number less than or equal to $8$, is $d=-6307$. We can enumerate those with non-fundamental discriminant by employing the formula
\betaegin{equation}
\lambdaabel{veq}
h(D) = h(d) \cdot f\cdot \frac{w_D}{w_d} \prod_{p \mid f} 1 - \varphifrac{\lambdaeft(\varphifrac{d}{p}{\rho}ight)}{p},
\epsilonnd{equation}
where $D=d\cdot f^2$, $f$ is the conductor of $D$, and $w_\mathfrak{D}$ is given by
\[
w_\mathfrak{D} :=\lambdaeft\{ \betaegin{array}{ll}
6& \mathfrak{D} =-3,\\
4& \mathfrak{D} =-4,\\
2& \mathfrak{D} <-4.\\
\epsilonnd{array}
{\rho}ight.
\]
Equation \epsilonqref{veq} is Lemma 2.13 in \cite{voight}. An application of \epsilonqref{veq} is $h(d\cdot f^2)\lambdaeq 8$ and $d=-3$, $d=-4$, $|d| > 4$, implies $f \lambdaeq 90, 60, 30$, respectively.\\
Another important use of \epsilonqref{veq}, is we can combine \epsilonqref{lphi} with \epsilonqref{veq} to remove the restriction of $m$ being squarefree in Theorem {\rho}ef{tthg}.\\
In the Appendix, we include tables of the 527 discriminants (fundamental and non-fundamental) with class number $\lambdaeq 8$ organized by isomorphism class of the class group. We generated this complete set by utilizing the bounds found in \cite{watkins} along with \epsilonqref{veq}. We then used PARI/GP V.2.7.0 to identify the isomorphism class of each class group, and compile the tables.\\
We now move on to Section {\rho}ef{ex1}, where we consider the form $(1,1,1,0,0,0)$ and derive the corresponding prelist. We then use this prelist to find the integers which are uniquely represented by the forms $(1,1,1,1,1,1), (3,3,3,-2,2,2),$ and $(1,3,3,2,0,0)$.
{\sigma}ection{Some ternary forms of discriminant $2,4,32,64$ and their relation to $x^2+y^2+z^2$}
\lambdaabel{ex1}
In \cite{bate}, Bateman shows
\betaegin{equation}
\lambdaabel{thm}
(1,1,1,0,0,0;n)={\delta}fracrac{16{\sigma}qrt{n}}{\pi}d_{s,2}(n) L(1,\chi(n))P(n),
\epsilonnd{equation}
where $s:=x^2 +y^2 +z^2$, and all other notation is given in Section {\rho}ef{notprim}. We comment that equation \epsilonqref{thm} follows from \epsilonqref{mf} as well.
Since $|$Aut$(s)|=48$, the prelist of $s$ consists of all $n$ with $0<(1,1,1,0,0,0;n)\lambdaeq 48$. By congruence considerations it is easy to see that $(1,1,1,0,0,0;n)=(1,1,1,0,0,0;4n)$, and so we restrict to $4 \nmid n$.\\
We refer to \cite[Equation (1.5)]{berk}, for the function $d_{s,2}(n)$. We have
\betaegin{equation}
\lambdaabel{psi}
d_{s,2}(n) = \lambdaeft\{ \betaegin{array}{ll}
3/2 & n\epsilonquiv 1,2\inftymod{4}, \\
1 & n\epsilonquiv 3\inftymod{8}, \\
0 & n\epsilonquiv 7\inftymod{8}, \\
\epsilonnd{array}
{\rho}ight.
\epsilonnd{equation}
for $n \not\epsilonquiv 0 \inftymod{4}$. We need not consider $n \epsilonquiv 7 \inftymod{8}$ since \epsilonqref{psi} and \epsilonqref{thm} imply\\ $(1,1,1,0,0,0;n)=0$ for such $n$.\\
Both $n=1,3$ satisfy $0<(1,1,1,0,0,0;n)\lambdaeq 48$. Employing \epsilonqref{h}, \epsilonqref{thm}, and \epsilonqref{psi}, we find
\betaegin{equation}
\lambdaabel{sss1}
(1,1,1,0,0,0;n)\gammaeq \lambdaeft\{ \betaegin{array}{ll}
24h(-n) & 3<n\epsilonquiv 3 \inftymod{8},\\
12h(-4n) & 1< n\epsilonquiv 1,2 \inftymod{4}.\\
\epsilonnd{array}
{\rho}ight.
\epsilonnd{equation}
Our goal is to solve for $n$ which satisfy
\betaegin{equation}
\lambdaabel{sss2}
48\gammaeq \lambdaeft\{ \betaegin{array}{ll}
24h(-n) & 3<n\epsilonquiv 3 \inftymod{8},\\
12h(-4n) & 1< n\epsilonquiv 1,2 \inftymod{4}.\\
\epsilonnd{array}
{\rho}ight.
\epsilonnd{equation}
We point out that to solve \epsilonqref{sss2} we need information regarding discriminants with class number 4. Employing the tables in the Appendix, we solve \epsilonqref{sss2} and find there are a total of 53 integers $n$ with $4 \nmid n$, and $0<(1,1,1,0,0,0;n) \lambdaeq 48$. We remark that three spurious solutions, $n=49, 75, 99$, satisfy \epsilonqref{sss2} yet $(1,1,1,0,0,0;n) > 48$. $\varphiext{Prelist}(1,1,1,0,0,0)$ consists of the integers $4^k \cdot v$, $k \gammaeq 0$, and\\ $v\inftyn$
\{1, 2, 3, 5, 6, 9, 10, 11, 13, 14, 17, 18, 19, 21, 22, 25, 27, 30, 33,
34, 35, 37, 42, 43, 46, 51, 57, 58, 67, 70, 73, 78, 82, 85, 91, 93,
97, 102, 115, 123, 130, 133, 142, 163, 177, 187, 190, 193, 235, 253,
267, 403, 427\}.
In Table {\rho}ef{prel} we tabulate the integers $4 \nmid n$ with $0<(1,1,1,0,0,0;n) \lambdaeq k$ for select values of $k$.
\betaegin{table}[htb]
\caption{} \lambdaabel{prel}
\betaegin{center}
\betaegin{tabular}{|l|l|} \hline
$(1,1,1,0,0,0;n)$ & $4\nmid n$\\ \hline
$0<(1,1,1,0,0,0;n)<6$ & $n \inftyn \{\}$\\ \hline
$(1,1,1,0,0,0;n)=6$ & $n \inftyn \{1\}$\\ \hline
$6<(1,1,1,0,0,0;n)\lambdaeq 8$ & $n \inftyn \{3\}$\\ \hline
$8<(1,1,1,0,0,0;n)\lambdaeq 12$ & $n \inftyn \{2\}$\\ \hline
$12<(1,1,1,0,0,0;n)< 24$ & $n \inftyn \{\}$\\ \hline
$(1,1,1,0,0,0;n)= 24$ & $n \inftyn \{5, 6, 10, 11, 13, 19, 22, 37, 43, 58, 67, 163\}$\\ \hline
$24<(1,1,1,0,0,0;n)< 48$ & $n \inftyn \{9, 18, 25, 27\}$\\ \hline
$(1,1,1,0,0,0;n)=48$ & $n \inftyn \{$14, 17, 21, 30, 33, 34, 35, 42, 46,\\
& 51, 57, 70, 73, 78, 82, 85, 91, 93, 97, 102, 115, 123, \\
& 130, 133, 142, 177, 187, 190, 193, 235, 253, 267, 403, 427$\}$\\ \hline
\epsilonnd{tabular}
\betaegin{flushright}
.
\epsilonnd{flushright}
\epsilonnd{center}
\epsilonnd{table}
We now check which integers in $\varphiext{Prelist}(1,1,1,0,0,0)$ are represented in an essentially unique way, and arrive at the following theorem.
\betaegin{theorem}
\lambdaabel{ttz}
The form $(1,1,1,0,0,0)$ uniquely represents $n$ (up to action of automorphs) if and only if $n=4^k\cdot v$, $k\gammaeq 0$, and \\ $v \inftyn$
\{$1$, $2$, $3$, $5$, $6$, $10$, $11$, $13$, $14$, $19$, $21$, $22$, $30$, $35$, $37$, $42$, $43$, $46$, $58$, $67$, $70$, $78$, $91$, $93$, $115$, $133$, $142$, $163$, $190$, $235$, $253$, $403$, $427$\}.
\epsilonnd{theorem}
Theorem {\rho}ef{ttz} was previously established in \cite{batgros}. However we additionally derived Table {\rho}ef{prel} in the process of proving Theorem {\rho}ef{ttz}, and we employ Table {\rho}ef{prel} to find the integers which are uniquely represented by certain forms that are considered in subsequent sections.\\
The idea of relating a form to $x^2+y^2+z^2$ to count the number of representations is not new. Indeed, this idea was developed by Dickson \cite{mdickson}, and is the main technique of Kaplansky \cite{kap} where he determined the integers which are represented in an essentially unique way by the forms $x^2+y^2+2z^2$, $x^2+2y^2+2z^2$, and $x^2+2y^2+4z^2$. We mention that although all the Theorems listed in \cite{kap} are correct, some of the main Lemmas (Lemma 3.1--3.3) can easily be misinterpreted. We chose to restate and augment these important Lemmas of Kaplansky.
\betaegin{lemma}[Kaplansky]
\lambdaabel{kap1l}
Let $n$ be a nonnegative integer. We have
\betaegin{equation}
\lambdaabel{kap1}
(1,1,2,0,0,0;2n)= (1,1,1,0,0,0;n),
\epsilonnd{equation}
\betaegin{equation}
\lambdaabel{kap1b}
3(1,1,2,0,0,0;2n+1)= (1,1,1,0,0,0;4n+2).
\epsilonnd{equation}
\epsilonnd{lemma}
\betaegin{lemma}[Kaplansky]
\lambdaabel{kap2l}
Let $n$ be a nonnegative integer. We have
\betaegin{equation}
\lambdaabel{kap2}
(1,2,2,0,0,0;4n)= (1,1,1,0,0,0;n),
\epsilonnd{equation}
\betaegin{equation}
\lambdaabel{kap2b}
3(1,2,2,0,0,0;4n+1)= (1,1,1,0,0,0;4n+1),
\epsilonnd{equation}
\betaegin{equation}
\lambdaabel{kap2c}
3(1,2,2,0,0,0;4n+2)= (1,1,1,0,0,0;4n+2),
\epsilonnd{equation}
\betaegin{equation}
\lambdaabel{kap2d}
(1,2,2,0,0,0;4n+3)= (1,1,1,0,0,0;4n+3).
\epsilonnd{equation}
\epsilonnd{lemma}
\betaegin{lemma}[Kaplansky]
\lambdaabel{kap3l}
Let $n$ be a nonnegative integer. We have
\betaegin{equation}
\lambdaabel{kap3}
(1,2,4,0,0,0;8n)= (1,1,1,0,0,0;n),
\epsilonnd{equation}
\betaegin{equation}
\lambdaabel{kap3c}
3(1,2,4,0,0,0;8n+2)= (1,1,1,0,0,0;4n+1),
\epsilonnd{equation}
\betaegin{equation}
\lambdaabel{kap3d}
3(1,2,4,0,0,0;8n+4)= (1,1,1,0,0,0;4n+2),
\epsilonnd{equation}
\betaegin{equation}
\lambdaabel{kap3e}
(1,2,4,0,0,0;8n+6)= (1,1,1,0,0,0;4n+3),
\epsilonnd{equation}
\betaegin{equation}
\lambdaabel{kap3b}
6(1,2,4,0,0,0;2n+1)= (1,1,1,0,0,0;4n+2).
\epsilonnd{equation}
\epsilonnd{lemma}
We now go beyond Kaplansky's forms and relate the non-diagonal forms $(1,1,1,1,1,1),$\\ $(3,3,3,-2,2,2)$, and $(1,3,3,2,0,0)$ to $x^2+y^2+z^2$ to find the integers which they uniquely represent. \\
The form $(1,1,1,1,1,1)$ has discriminant 2 and 48 automorphs. We write $f(x,y,z):=x^2 +y^2+z^2 +yz +xz +xy$. It is easy to check that
\[
f(x,y,z) \epsilonquiv 0 \inftymod{2},
\]
if and only if $x \epsilonquiv y \epsilonquiv z \inftymod{2}$.\\
For any $x,y,z$ we must have $x \epsilonquiv y \epsilonquiv z \inftymod{2}$ or we have one of the following: $x \epsilonquiv y \not\epsilonquiv z \inftymod{2}$, $x \epsilonquiv z \not\epsilonquiv y \inftymod{2}$, or $y \epsilonquiv z \not\epsilonquiv x \inftymod{2}$. We note that
\betaegin{equation}
\lambdaabel{lee}
{\sigma}um_{x\epsilonquiv y \not\epsilonquiv z\inftymod{2}}q^{f(x,y,z)}={\sigma}um_{x\epsilonquiv z \not\epsilonquiv y\inftymod{2}}q^{f(x,y,z)}={\sigma}um_{y\epsilonquiv z \not\epsilonquiv x\inftymod{2}}q^{f(x,y,z)},
\epsilonnd{equation}
and thus we have
\betaegin{equation}
\lambdaabel{dis1}
{\sigma}um_{x,y,z}q^{f(x,y,z)}={\sigma}um_{x\epsilonquiv y \epsilonquiv z\inftymod{2}}q^{f(x,y,z)} + 3{\sigma}um_{x\epsilonquiv y \not\epsilonquiv z\inftymod{2}}q^{f(x,y,z)}.
\epsilonnd{equation}
The substitution $x \mapsto (-x +y+z)$, $y \mapsto (x-y+z)$, and $z \mapsto (x+y-z)$, guarantees the condition $x \epsilonquiv y \epsilonquiv z \inftymod{2}$, hence
\betaegin{equation}
\lambdaabel{dis2}
{\sigma}um_{x\epsilonquiv y \epsilonquiv z\inftymod{2}}q^{f(x,y,z)}={\sigma}um_{x,y,z}q^{f(-x +y+z,x-y+z,x+y-z)}={\sigma}um_{x,y,z}q^{2(x^2+y^2+z^2)}.
\epsilonnd{equation}
The substitution $x \mapsto (-x +y+z)$, $y \mapsto (x-y+z)$, and $z \mapsto (x+y-z+1)$, gives $x\epsilonquiv y \not\epsilonquiv z\inftymod{2}$, and we have
\betaegin{equation}
\lambdaabel{dis22}
{\sigma}um_{x\epsilonquiv y \not\epsilonquiv z\inftymod{2}}q^{f(x,y,z)}={\sigma}um_{x,y,z}q^{f(-x +y+z,x-y+z,x+y-z+1)}=q{\sigma}um_{x,y,z}q^{2(x^2+y^2+z^2)+2(y+z)}.
\epsilonnd{equation}
We have now proven the following lemma.
\betaegin{lemma}
\lambdaabel{tot}
\betaegin{equation}
{\sigma}um_{x,y,z}q^{x^2 +y^2+z^2 +yz +xz +xy} = {\sigma}um_{x,y,z}q^{2(x^2+y^2+z^2)} + 3q{\sigma}um_{x,y,z}q^{2(x^2+y^2+z^2)+2(y+z)}.
\epsilonnd{equation}
\epsilonnd{lemma}
Lemma {\rho}ef{tot} implies
\betaegin{equation}
\lambdaabel{odd1}
(1,1,1,1,1,1;2n+1) = (1,1,1,0,0,0;4n+2),
\epsilonnd{equation}
and
\betaegin{equation}
\lambdaabel{even1}
(1,1,1,1,1,1;2n) = (1,1,1,0,0,0;n),
\epsilonnd{equation}
for any nonnegative integer $n$.\\
Equations \epsilonqref{odd1} and \epsilonqref{even1} relate $(1,1,1,1,1,1;n)$ to $(1,1,1,0,0,0;n)$ for any nonnegative integer. We can use Table {\rho}ef{prel} to determine the solutions to
\betaegin{equation}
\lambdaabel{oddl}
0<(1,1,1,1,1,1;2n+1)=(1,1,1,0,0,0;4n+2) \lambdaeq 48,
\epsilonnd{equation}
and
\betaegin{equation}
\lambdaabel{evenl}
0<(1,1,1,1,1,1;2n)=(1,1,1,0,0,0;n) \lambdaeq 48.
\epsilonnd{equation}
We find the solutions to $0<(1,1,1,1,1,1;2n) \lambdaeq 48$ to be $2n =4^k \cdot v$, $k\gammaeq 0$, and $v$ is in the set
\{2, 4, 6, 10, 12, 18, 20, 22, 26, 28, 34, 36, 38, 42, 44, 50, 54, 60, 66, 68, 70, 74, 84, 86, 92, 102, 114, 116, 134, 140, 146, 156, 164, 170, 182, 186, 194, 204, 230, 246, 260, 266, 284, 326, 354, 374, 380, 386, 470, 506, 534, 806, 854\}.
We find the solutions to $0<(1,1,1,1,1,1;2n+1) \lambdaeq 48$ to be
\betaegin{equation}
\lambdaabel{oddsol}
2n+1 =1, 3, 5, 7, 9, 11, 15, 17, 21, 23, 29, 35, 39, 41, 51, 65, 71, 95.
\epsilonnd{equation}
We use Maple V.15 to check the above candidates for unique representation, and arrive at the following theorem.
\betaegin{theorem}
\lambdaabel{333oe}
The form $(1,1,1,1,1,1)$ uniquely represents the integer $n$ (up to action of automorphs) if and only if $n=4^k\cdot v$, $k\gammaeq 0$, and \\ $v \inftyn$
\{$1$, $ 2$, $ 3$, $ 5$, $ 6$, $ 7$, $ 10$, $ 11$, $ 15$, $ 21$, $ 22$, $ 23$, $ 26$, $ 29$, $ 35$, $ 38$, $ 39$, $ 42$, $ 70$, $ 71$, $ 74$, $ 86$, $ 95$, $ 134$, $ 182$, $ 186$, $ 230$, $ 266$, $ 326$, $ 470$, $ 506$, $ 806$, $ 854$\}.
\epsilonnd{theorem}
We now consider the form $g:=(3,3,3,-2,2,2)$ of discriminant $\Delta =64$ and $|$Aut($g$)$|$=48. We see that $g(x,y,z):=3x^2+3y^2+3z^2 - 2yz+2xz+2xy$ is even if and only if $x+y+z \epsilonquiv 0 \inftymod{2}$. The substitution $x \mapsto (y+z),$ $y \mapsto (x-z),$ and $z \mapsto (x-y)$, ensures $x+y+z \epsilonquiv 0 \inftymod{2}$, and we have
\betaegin{equation}
\lambdaabel{dis21}
{\sigma}um_{x+y+z \epsilonquiv 0 \inftymod{2}}q^{g(x,y,z)}={\sigma}um_{x,y,z}q^{g(y+z,x-z,x-y)}={\sigma}um_{x,y,z}q^{4(x^2+ y^2+z^2)}.
\epsilonnd{equation}
The substitution $x \mapsto (y+z+1),$ $y \mapsto (x-z)$, and $z \mapsto (x-y)$, guarantees the condition $x+y+z \epsilonquiv 1 \inftymod{2}$, and we have
\betaegin{equation}
\lambdaabel{dis211}
{\sigma}um_{x+y+z \epsilonquiv 1 \inftymod{2}}q^{g(x,y,z)}={\sigma}um_{x,y,z}q^{g(y+z+1,x-z,x-y)}={\sigma}um_{x,y,z}q^{(2x+1)^2 +(2y+1)^2 + (2z+1)^2}.
\epsilonnd{equation}
Combining \epsilonqref{dis21} and \epsilonqref{dis211} yields the following lemma.
\betaegin{lemma}
\lambdaabel{sif}
\betaegin{equation}
{\sigma}um_{x,y,z}q^{3x^2+3y^2+3z^2 - 2yz+2xz+2xy} = {\sigma}um_{x,y,z}q^{4(x^2+ y^2+z^2)} + {\sigma}um_{x,y,z}q^{(2x+1)^2 +(2y+1)^2 + (2z+1)^2}.
\epsilonnd{equation}
\epsilonnd{lemma}
Lemma {\rho}ef{sif} implies
\betaegin{equation}
\lambdaabel{low3}
(3,3,3,-2,2,2;4n)=(1,1,1,0,0,0;n),
\epsilonnd{equation}
\betaegin{equation}
\lambdaabel{low3b}
(3,3,3,-2,2,2;8n+3)=(1,1,1,0,0,0;8n+3),
\epsilonnd{equation}
and $(3,3,3,-2,2,2;n)=0$ for any $n \not\epsilonquiv 0,3,4 \inftymod{8}$.\\
A necessary condition that the integer $n$ is uniquely represented by $(3,3,3,-2,2,2)$ is
\[
0<(3,3,3,-2,2,2;n) \lambdaeq |\mbox{Aut}((3,3,3,-2,2,2))|=48.
\]
We use $\varphiext{Prelist}(1,1,1,0,0,0)$ to determine the solutions to
\betaegin{equation}
\lambdaabel{evenl2}
0<(3,3,3,-2,2,2;4n)=(1,1,1,0,0,0;n)\lambdaeq 48,
\epsilonnd{equation}
and
\betaegin{equation}
\lambdaabel{oddl2}
0<(3,3,3,-2,2,2;8n+3)=(1,1,1,0,0,0;8n+3)\lambdaeq 48.
\epsilonnd{equation}
The integers $n$ which satisfy \epsilonqref{evenl2} is exactly $\varphiext{Prelist}(1,1,1,0,0,0)$, which we derived earlier in the section. The integers which satisfy \epsilonqref{oddl2} is the finite subset of integers which are congruent to $3$ modulo $8$ and are in $\varphiext{Prelist}(1,1,1,0,0,0)$.\\
Using Maple V.15 we check these solutions for unique representation, and find the following theorem.
\betaegin{theorem}
\lambdaabel{333oe}
The form $(3,3,3,-2,2,2)$ uniquely represents the integer $n$ (up to action of automorphs) if and only if $n=4^k\cdot v$, $k\gammaeq 0, 4 \nmid v$, and \\ $v \inftyn$
\{$3$, $ 4$, $ 8$, $ 11$, $ 19$, $ 20$, $ 24$, $ 35$, $ 40$, $ 43$, $ 52$, $ 56$, $ 67$, $ 84$, $ 88$, $ 91$, $ 115$, $ 120$, $ 148$, $ 163$, $ 168$, $ 184$, $ 232$, $ 235$, $ 280$, $ 312$, $ 372$, $ 403$, $ 427$, $ 532$, $ 568$, $ 760$, $ 1012$\}.
\epsilonnd{theorem}
The three forms considered so far, $(1,1,1,0,0,0), (1,1,1,1,1,1),$ and $(3,3,3,-2,2,2)$, all have the maximum number of automorphs: 48. Let us briefly consider the form $(1,3,3,2,0,0)$ which has 8 automorphs.\\
The form $h=(1,3,3,2,0,0)$ is of discriminant $\Delta=32$ and $|$Aut($h$)$|$=8. To connect\\ $(1,3,3,2,0,0;n)$ to $(1,1,1,0,0,0;n)$ we note that $x^2 +3y^2 +3z^2 +2yz = x^2 +(y-z)^2 +2(y+z)^2$, and we can use Lemma {\rho}ef{kap1l} to reduce $(1,1,2,0,0,0)$ to $(1,1,1,0,0,0)$. Let $h(x,y,z)=x^2 +3y^2 +3z^2 +2yz$. We have
\betaegin{equation}
\lambdaabel{133200}
\betaegin{aligned}
{\sigma}um_{x,y,z} q^{h(x,y,z)} &= {\sigma}um_{{\sigma}ubstack{x,\\ y\epsilonquiv z \inftymod{2}}} q^{x^2 +y^2 +2z^2}\\
&={\sigma}um_{x,y,z} q^{x^2 +2y^2 +8z^2}+{\sigma}um_{{\sigma}ubstack{x,\\ y\epsilonquiv z \epsilonquiv 1 \inftymod{2}}} q^{x^2 +y^2 +2z^2}\\
&={\sigma}um_{{\sigma}ubstack{x \epsilonquiv 1\inftymod{2},\\y,z}} q^{x^2 +2y^2 +4z^2}+{\sigma}um_{x,y,z} q^{4(x^2 +y^2 +2z^2)}+{\sigma}um_{x \epsilonquiv y \epsilonquiv z \epsilonquiv 1 \inftymod{2}} q^{x^2 +y^2 +2z^2}\\
&={\sigma}um_{{\sigma}ubstack{x \epsilonquiv 1\inftymod{2},\\y,z}} q^{x^2 +2y^2 +4z^2}+{\sigma}um_{x,y,z} q^{8(x^2 +y^2 +z^2)}+{\delta}frac{3}{2}{\sigma}um_{x \epsilonquiv y \epsilonquiv z \epsilonquiv 1 \inftymod{2}} q^{x^2 +y^2 +2z^2},\\
\epsilonnd{aligned}
\epsilonnd{equation}
where we employed the identity
\[
4{\sigma}um_{{\sigma}ubstack{x \epsilonquiv 1\inftymod{2},\\y,z}} q^{4(x^2 +2y^2 +4z^2)} = {\sigma}um_{x \epsilonquiv y \epsilonquiv z \epsilonquiv 1 \inftymod{2}} q^{x^2 +y^2 +2z^2}.
\]
Making use of \epsilonqref{kap3b} and observing that
\[
{\sigma}um_{x \epsilonquiv y \epsilonquiv z \epsilonquiv 1 \inftymod{2}}q^{x^2 +y^2 +2z^2} = 2{\sigma}um_{{\sigma}ubstack{x\epsilonquiv y\epsilonquiv 1 \inftymod{2}\\zeta \epsilonquiv 0 \inftymod{2}}}q^{2(x^2 +y^2 +z^2)},
\]
we find
\betaegin{equation}
\lambdaabel{l333}
(1,3,3,2,0,0;8n)=(1,1,1,0,0,0;n),
\epsilonnd{equation}
\betaegin{equation}
\lambdaabel{l333b}
(1,3,3,2,0,0;8n+4)=(1,1,1,0,0,0;4n+2),
\epsilonnd{equation}
\betaegin{equation}
\lambdaabel{l333c}
(1,3,3,2,0,0;4n+2)=0,
\epsilonnd{equation}
\betaegin{equation}
\lambdaabel{l333d}
6(1,3,3,2,0,0;2n+1)=(1,1,1,0,0,0;4n+2).
\epsilonnd{equation}
Employing \epsilonqref{l333} -- \epsilonqref{l333d} along with Table {\rho}ef{prel} allows us to solve $0<(1,3,3,2,0,0;n) \lambdaeq 8$.\\
According to Table {\rho}ef{prel} we have $0<(1,1,1,0,0,0;n)\lambdaeq 8$ and $4\nmid n$ if and only if $n=1,3$. Hence \epsilonqref{l333} implies $0<(1,3,3,2,0,0;8\cdot 4^k) \lambdaeq 8$ and $0<(1,3,3,2,0,0;24\cdot 4^k) \lambdaeq 8$. We see $(1,1,1,0,0,0;4n+2) \lambdaeq 48$ has the solutions $2n+1=1, 3, 5, 7, 9, 11, 15, 17, 21, 23, 29, 35, 39, 41, 51, 65, 71, 95$.\\
We now have all solutions to $0<(1,3,3,2,0,0;n) \lambdaeq 8$. Using Maple V.15 we directly check these integers for unique representation to arrive at the following theorem.
\betaegin{theorem}
\lambdaabel{1333thm}
The form $(1,3,3,2,0,0)$ uniquely represents the integer $n$ (up to action of automorphs) if and only if $n= 1, 3, 5, 7, 11, 15, 21, 23, 29, 35, 39, 71,$ or $95$.
\epsilonnd{theorem}
{\sigma}ection{Some ternary forms of discriminant $4096$ and $8192$}
\lambdaabel{ex2}
The discriminant $4096$ is the largest discriminant which is a power of 4 and contains a ternary idoneal form \cite{jagykap}. Indeed the two forms $(5,13,20,-12,4,2)$ and $(5,12,20,8,4,4)$ are both idoneal and of discriminant $4096$. These forms are alike in the sense that they can be connected to $(1,1,1,0,0,0)$ to find the integers which they uniquely represent. The form $(5,12,20,8,4,4)$ has only 2 automorphs, and can be handled similarly to $(5,13,20,-12,4,2)$. We now deduce the integers uniquely represented by $(5,13,20,-12,4,2)$.\\
As mentioned, we can relate $(5,13,20,-12,4,2;n)$ to $(1,1,1,0,0,0;n)$ as given in the following theorem.
\betaegin{theorem}
\lambdaabel{4096link}
Let $n$ be a nonnegative integer. We have
\betaegin{equation}
\lambdaabel{zel}
(5,13,20,-12,4,2;64n)= (1,1,1,0,0,0;n),
\epsilonnd{equation}
\betaegin{equation}
3(5,13,20,-12,4,2;32(2n+1))= (1,1,1,0,0,0;32(2n+1)),
\epsilonnd{equation}
\betaegin{equation}
3(5,13,20,-12,4,2;16(4n+1))= (1,1,1,0,0,0;16(4n+1)),
\epsilonnd{equation}
\betaegin{equation}
(5,13,20,-12,4,2;16(4n+3))= (1,1,1,0,0,0;16(4n+3)),
\epsilonnd{equation}
\betaegin{equation}
3(5,13,20,-12,4,2;4(8n+5))= (1,1,1,0,0,0;8n+5),
\epsilonnd{equation}
\betaegin{equation}
\lambdaabel{endd}
12(5,13,20,-12,4,2;8n+5)= (1,1,1,0,0,0;8n+5),
\epsilonnd{equation}
and $(5,13,20,-12,4,2;k)=0$ for any $k$ not covered by \epsilonqref{4096link} -- \epsilonqref{endd}.
\epsilonnd{theorem}
Proof of the above theorem is elementary, but contains many details. Employing Theorem {\rho}ef{4096link} in conjunction with Table {\rho}ef{prel} gives all integers $n$ such that
\[
(5,13,20,-12,4,2;n) \lambdaeq | \varphiext{Aut}(5,13,20,-12,4,2)|=4,
\]
and so we can check which integers are uniquely represented by $(5,13,20,-12,4,2)$. In particular we point out that Table {\rho}ef{prel} implies there is no integer $n \epsilonquiv 0 \inftymod{64}$ with $(1,1,1,0,0,0,\varphifrac{n}{64}) \lambdaeq 4$, and employing \epsilonqref{zel} implies $(5,13,20,-12,4,2)$ does not uniquely represent infinitely many integers.\\
Instead of using Theorem {\rho}ef{4096link} to deduce the integers which are uniquely represented by \\$(5,13,20,-12,4,2)$, we employ the Siegel formula along with the method described in Section {\rho}ef{notprim}.\\
Using \epsilonqref{mf} with $f:=(5,13,20,-12,4,2)$ we find
\betaegin{equation}
\lambdaabel{thm4096}
(5,13,20,-12,4,2;n)={\delta}fracrac{{\sigma}qrt{n}}{2\pi}\cdot d_{f,2}(n)\cdot L(1,\chi(n))\cdot P(n).
\epsilonnd{equation}
Letting $n=4^av$, with $4\nmid v$, we see \epsilonqref{thm4096} implies the bound
\betaegin{equation}
\lambdaabel{thm22}
(5,13,20,-12,4,2;n)\gammaeq {\delta}fracrac{2^{a-1}{\sigma}qrt{v}}{\pi}\cdot d_{f,2}(n)\cdot L(1,\chi(v)).
\epsilonnd{equation}
The local 2-adic density of $f$ is given by
\betaegin{equation}
\lambdaabel{4096ld1}
d_{f,2}(n) = \lambdaeft\{ \betaegin{array}{ll}
\frac{3}{2^{a-4}} & a \gammaeq 3, ~v\epsilonquiv 1,2 \inftymod{4},\\
\frac{1}{2^{a-5}} & a \gammaeq 3, ~ v\epsilonquiv 3 \inftymod{8},\\
0 & v\epsilonquiv 7 \inftymod{8}.
\epsilonnd{array}
{\rho}ight.
\epsilonnd{equation}
The values of $d_{f,2}(n)$ not covered in \epsilonqref{4096ld1}, are listed in Table {\rho}ef{prelll}.
\betaegin{table}[htb]
\caption{} \lambdaabel{prelll}
\betaegin{center}
\betaegin{tabular}{ c | c | c | c }
&$a=0$& $a=1$& $a=2$ \\
\hline
$v \epsilonquiv 1\inftymod{8}$ & 0& 0 & 4 \\ \hline
$v \epsilonquiv 3\inftymod{8}$ &0& 0 & 8 \\ \hline
$v \epsilonquiv 5\inftymod{8}$ & 4& 8 & 4 \\ \hline
$v \epsilonquiv 2\inftymod{4}$ & 0& 0 & 4 . \\
\hline
\epsilonnd{tabular}
\betaegin{flushright}
.
\epsilonnd{flushright}
\epsilonnd{center}
\epsilonnd{table}
We used Sage 5.1 in computing the above densities. We refer the reader to \cite{berk} and \cite{hanke} for more details on computing local densities.\\
We can use \epsilonqref{thm22} along with \epsilonqref{4096ld1}, \epsilonqref{h}, and Table {\rho}ef{prelll}, to find all $n$ that satisfy\\$0<(5,13,20,-12,4,2;n) \lambdaeq 4$. We write $n=4^av$ with $4\nmid v$ and split our analysis according to the cases $a=0,1;$ $a=2;$ or $a\gammaeq 3$.\\
\varphiextbf{\lambdaarge Case 1:} $a=0,1.$\\
Since $a=0,1$ we see $(5,13,20,-12,4,2;n)=0$ unless $v \epsilonquiv 5 \inftymod{8}$. Hence we only consider $n=v$ or $n=4v$ with $v \epsilonquiv 5 \inftymod{8}$. In the case $n \epsilonquiv 5 \inftymod{8}$, we employ \epsilonqref{h}, \epsilonqref{thm22}, \epsilonqref{4096ld1}, and Table {\rho}ef{prelll}, to find
\betaegin{equation}
\lambdaabel{thm223}
(5,13,20,-12,4,2;n)\gammaeq h(-4n).
\epsilonnd{equation}
We find $n=5, 13, 21, 37, 45, 85, 93, 133, 253$ are the solutions to $n \epsilonquiv 5 \inftymod{8}$ and $h(-4n) \lambdaeq 4$. Noting that $(5,13,20,-12,4,2;45)>4$, we see $n=45$ is the only spurious solution.
In the case $n=4v$ with $v \epsilonquiv 5 \inftymod{8}$, we employ \epsilonqref{h}, \epsilonqref{thm22}, \epsilonqref{4096ld1}, and Table {\rho}ef{prelll}, to find
\betaegin{equation}
\lambdaabel{thm224}
(5,13,20,-12,4,2;4v)\gammaeq 4h(-4v).
\epsilonnd{equation}
There are no solutions to $h(-4v)=1$, and thus no solutions to $0<(5,13,20,-12,4,2;4v)\lambdaeq 4$ with $v \epsilonquiv 5 \inftymod{8}$.\\
\varphiextbf{\lambdaarge Case 2:} $a=2.$\\
Since $a=2$, we must consider $n=16v$ with $4 \nmid v$. Particularly, when $v=1$, we have $n=16$, and $(5,13,20,-12,4,2;16)=2$, so $n=16$ is a candidate for unique representation by $f$. However, $v=3$ implies $n=48$, and $(5,13,20,-12,4,2;48)>4$, so we need not consider $v=3$. Using \epsilonqref{h}, \epsilonqref{thm22}, \epsilonqref{4096ld1}, and Table {\rho}ef{prelll}, we have
\betaegin{equation}
\lambdaabel{p}
(5,13,20,-12,4,2;16v)\gammaeq \lambdaeft\{ \betaegin{array}{ll}
24\cdot h(-v) & 3<v\epsilonquiv 3 \inftymod{8},\\
4\cdot h(-4v) & 1< v\epsilonquiv 1,2 \inftymod{4}.\\
\epsilonnd{array}
{\rho}ight.
\epsilonnd{equation}
Inequality \epsilonqref{p} shows that we need only consider $v\epsilonquiv 1,2 \inftymod{4}$ when solving\\$0<(5,13,20,-12,4,2;16v) \lambdaeq 4$. We are left to solve $h(-4v)=1$ with $1<v\epsilonquiv 1,2 \inftymod{4}$, and we find the only solution is $v=2$. Therefore, we see that $n=32$ is a candidate for unique representation. \\
\varphiextbf{\lambdaarge Case 3:} $a\gammaeq 3.$\\
We directly consider $n=4^a v$ with $4 \nmid v$ and $a \gammaeq 3$. Using \epsilonqref{h}, \epsilonqref{thm22}, \epsilonqref{4096ld1}, and Table {\rho}ef{prelll}, we have
\betaegin{equation}
\lambdaabel{s}
(5,13,20,-12,4,2;4^a v)\gammaeq \lambdaeft\{ \betaegin{array}{ll}
6 & v=1,\\
8 & v=3,\\
24h(-v) & 3<v\epsilonquiv 3 \inftymod{8},\\
12h(-4v) & 1< v\epsilonquiv 1,2 \inftymod{4}.\\
\epsilonnd{array}
{\rho}ight.
\epsilonnd{equation}
From \epsilonqref{s}, we see there are no solutions to $(5,13,20,-12,4,2;4^a v) \lambdaeq 4$ and $a \gammaeq 3$.\\
Combining the above case shows $\varphiext{Prelist}(5,13,20,-12,4,2) =\{5, 13, 16, 21, 32, 37, 85, 93, 133, 253\}$. Employing Maple V.15 we easily check the elements of $\varphiext{Prelist}(5,13,20,-12,4,2)$ for unique representation by $(5,13,20,-12,4,2)$ and we find the following theorem.
\betaegin{theorem}
\lambdaabel{tt}
The form $(5,13,20,-12,4,2)$ uniquely represents $n$ (up to action of automorphs) if and only if $n=5, 13, 16, 21, 32, 37, 93, 133,$ or $253$.
\epsilonnd{theorem}
We now treat the form $g:=(7,15,23,10,2,6)$. $g$ has discriminant $\Delta =2^{13} =8192$ and $|$Aut($g$)$|$=2. We remark that 8192 is the largest discriminant which is a power of two and contains a ternary idoneal form \cite{jagykap}. We can relate $(7,15,23,10,2,6;n)$ to $(1,1,1,0,0,0;n)$ as given in the following theorem.
\betaegin{theorem}
\lambdaabel{8192link}
Let $n$ be a nonnegative integer. We have
\betaegin{equation}
\lambdaabel{81}
(7,15,23,10,2,6;128n)=(1,1,1,0,0,0;n),
\epsilonnd{equation}
\betaegin{equation}
3(7,15,23,10,2,6;64(2n+1))=(1,1,1,0,0,0;4n+2),
\epsilonnd{equation}
\betaegin{equation}
3(7,15,23,10,2,6;32(4n+1))=(1,1,1,0,0,0;4n+1),
\epsilonnd{equation}
\betaegin{equation}
(7,15,23,10,2,6;32(4n+3))=(1,1,1,0,0,0;4n+3),
\epsilonnd{equation}
\betaegin{equation}
6(7,15,23,10,2,6;16(2n+1))=(1,1,1,0,0,0;4n+2),
\epsilonnd{equation}
\betaegin{equation}
6(7,15,23,10,2,6;4(8n+7))=(1,1,1,0,0,0;2(8n+7)),
\epsilonnd{equation}
\betaegin{equation}
\lambdaabel{81b}
24(7,15,23,10,2,6;8n+7)=(1,1,1,0,0,0;2(8n+7)),
\epsilonnd{equation}
and $(7,15,23,10,2,6;k)=0$ for any $k$ not covered by \epsilonqref{81} -- \epsilonqref{81b}.
\epsilonnd{theorem}
Employing Table {\rho}ef{prel} along with Theorem {\rho}ef{8192link} implies $(7,15,23,10,2,6)$ does not uniquely represent infinitely many integers. We chose to use the method of Section {\rho}ef{notprim} to treat the form $(7,15,23,10,2,6)$.
Using \epsilonqref{mf} we have
\betaegin{equation}
\lambdaabel{thm8192}
(7,15,23,10,2,6;n)={\delta}fracrac{{\sigma}qrt{n}}{ 2\pi{\sigma}qrt{2}}d_{g,2}(n)\cdot L(1,\chi(2n))\cdot P(n,2).
\epsilonnd{equation}
Let us write $n=4^av$, with $4\nmid v$. We find the 2-adic local densitiy of $g$ to be
\betaegin{equation}
\lambdaabel{8192ld1}
d_{g,2}(n) = \lambdaeft\{ \betaegin{array}{ll}
\frac{3}{2^{a-5}} & a \gammaeq 4, ~v\epsilonquiv 1 \inftymod{2},\\
\frac{3}{2^{a-4}} & a \gammaeq 4, ~ v\epsilonquiv 2 \inftymod{8},\\
\frac{1}{2^{a-5}} & a \gammaeq 4, ~ v\epsilonquiv 6 \inftymod{16},\\
0 & v\epsilonquiv 14 \inftymod{16}.
\epsilonnd{array}
{\rho}ight.
\epsilonnd{equation}
The values of $d_{g,2}(n)$ not covered in \epsilonqref{8192ld1} are listed in Table {\rho}ef{prellll}.
\betaegin{table}[htb]
\caption{} \lambdaabel{prellll}
\betaegin{center}
\betaegin{tabular}{ c | c | c | c|c }
&$a=0$& $a=1$& $a=2$& $a=3$\\
\hline
$v \epsilonquiv 1,3,5\inftymod{8}$ & 0& 0 & 4 &4 \\ \hline
$v \epsilonquiv 7\inftymod{8}$ & 4& 8 & 4 & 4 \\ \hline
$v \epsilonquiv 2\inftymod{8}$ & 0& 0 & 4 &6 \\ \hline
$v \epsilonquiv 6\inftymod{16}$& 0& 0 & 8 &4. \\
\hline
\epsilonnd{tabular}
\betaegin{flushright}
.
\epsilonnd{flushright}
\epsilonnd{center}
\epsilonnd{table}
We now break our analysis into two cases depending on the parity of the order of 2 in $n$.\\
\varphiextbf{\lambdaarge Case 1:} $n=4^a \cdot v$ with $v$ odd.\\
Employing \epsilonqref{h}, we have
\betaegin{equation}
\lambdaabel{1l}
L(1,\chi(2n))=L(1,\chi(2v))=\frac{\pi}{2{\sigma}qrt{2v}}\cdot h(-8v),
\epsilonnd{equation}
and combining \epsilonqref{thm8192} with \epsilonqref{1l} yields
\betaegin{equation}
\lambdaabel{1thmm8192}
(7,15,23,10,2,6;n)\gammaeq 2^{a-3}d_{g,2}(n)\cdot h(-8v).
\epsilonnd{equation}
The form $(7,15,23,10,2,6)$ has 2 automorphs, so $0<(7,15,23,10,2,6;n)\lambdaeq 2$ is a necessary condition that $n$ be uniquely represented up to the action of automorphs.\\
Solving
\betaegin{equation}
\lambdaabel{m}
2^{a-3}d_{g,2}(n)\cdot h(-8v) \lambdaeq 2
\epsilonnd{equation}
requires class number information up to class number 4. We remark that finding the solutions to \epsilonqref{m} is similar to the process we used when we treated $(5,12,20,8,4,4)$ earlier in this section. We find $n=7, 15, 16, 23, 39, 71, 95$ are solutions to $0<(7,15,23,10,2,6;n)\lambdaeq 2$.\\
\varphiextbf{\lambdaarge Case 2:} $n=4^a \cdot 2\cdot v$ with $v$ odd.\\
In this case we see $L(1,\chi(2n))=L(1,\chi(v))$, and \epsilonqref{thm8192} becomes
\betaegin{equation}
\lambdaabel{thm88192}
(7,15,23,10,2,6;n)\gammaeq{\delta}fracrac{{\sigma}qrt{n}}{ 2{\sigma}qrt{2}\pi}d_{g,2}(n)\cdot L(1,\chi(v)).
\epsilonnd{equation}
Employing \epsilonqref{h} and \epsilonqref{thm88192} yields
\betaegin{equation}
\lambdaabel{thmm81922}
(7,15,23,10,2,6;n)\gammaeq \lambdaeft\{ \betaegin{array}{ll}
2^{a-3}\cdot d_{g,2}(n)& v=1,\\
&\\
2^{a-2}\cdot d_{g,2}(n) & v=3,\\
&\\
2^{a-2}\cdot d_{g,2}(n)\cdot h(-4v) & 1<v\epsilonquiv 1 \inftymod{4},\\
&\\
3\cdot 2^{a-2}\cdot d_{g,2}(n)\cdot h(-v) & 3<v\epsilonquiv 3 \inftymod{8},\\
&\\
2^{a-2}\cdot d_{g,2}(n)\cdot h(-v) & v \epsilonquiv 7 \inftymod{8}.\\
\epsilonnd{array}
{\rho}ight.
\epsilonnd{equation}
We employ \epsilonqref{8192ld1}, \epsilonqref{thmm81922}, Table {\rho}ef{prellll}, and the tables in the Appendix to find the only number $n=4^a \cdot 2\cdot v$ with $v$ odd, and $0<(7,15,23,10,2,6;n)\lambdaeq 2$, is $n=32$.
Combining Case 1 and Case 2 yields $\varphiext{Prelist}(7,15,23,10,2,6) =\{7, 15, 16, 23, 32, 39, 71, 95\}$. Employing Maple V.15 we check the integers $7, 15, 16, 23, 32, 39, 71, 95,$ for unique representation and arrive at the following theorem.
\betaegin{theorem}
\lambdaabel{t3t}
The form $(7,15,23,10,2,6) $ uniquely represents $n$ (up to action of automorphs) if and only if $n=7, 15, 16, 23, 32, 39, 71,$ or $95$.
\epsilonnd{theorem}
{\sigma}ection{Resolving Some Conjectures of Kaplansky}
\lambdaabel{ex3}
In the concluding remarks of \cite{kap}, Kaplansky regards $x^2 +y^2 +3z^2$ as ``the next challenge''. He computationally found the integers which are uniquely represented by $x^2 +y^2 +3z^2$. We now supply the proof of this conjecture.\\
In this section we deduce the integers which are uniquely represented by the forms $(1,3,3,0,0,0)$ and $(1,1,3,0,0,0)$. These two forms are intertwined with each other since for any nonnegative integer $n$, we have $(1,3,3,0,0,0;3n)=(1,1,3,0,0,0;n)$ and $(1,3,3,0,0,0;n)=(1,1,3,0,0,0;3n)$. Hence we also have $(1,3,3,0,0,0;n)=(1,3,3,0,0,0;9n)$ and $(1,1,3,0,0,0;n)=(1,1,3,0,0,0;9n)$.\\
Let $f:=(1,3,3,0,0,0)$ which is of discriminant 36 and has $|$Aut($f$)$|$=16. Equation \epsilonqref{mf} gives
\betaegin{equation}
\lambdaabel{thm36}
(1,3,3,0,0,0;n)={\delta}fracrac{6}{\pi}{\sigma}qrt{n}\cdot d_{f,2}(n)\cdot d_{f,3}(n)\cdot L(1,\chi(9n))\cdot P(n,9),
\epsilonnd{equation}
with all notation as defined in Section {\rho}ef{notprim}. Since $(1,3,3,0,0,0;n)=(1,3,3,0,0,0;9n)$, we only consider $9 \nmid n$. We write $n=4^a v$ with $4\nmid v$ and $9\nmid v$. We refer to \cite{berk} and \cite{hanke} for the following local density results:
\betaegin{equation}
\lambdaabel{ld133}
d_{f,3}(n) = \lambdaeft\{ \betaegin{array}{ll}
2 & v\epsilonquiv 1\pmod{3}, \\
0 & v\epsilonquiv 2\pmod{3}, \\
\frac{4}{3} & v\epsilonquiv 3,6\pmod{9}, \\
\epsilonnd{array}
{\rho}ight.
\epsilonnd{equation}
and
\betaegin{equation}
\lambdaabel{lld2}
d_{f,2}(n) = \lambdaeft\{ \betaegin{array}{ll}
\frac{2^{a+2}-3}{2^{a+1}} & v\epsilonquiv 1,2\inftymod{4}, \\
&\\
\frac{2^{a+1}-1}{2^{a}} & v\epsilonquiv 3\inftymod{8}, \\
&\\
2 & v\epsilonquiv 7\inftymod{8}. \\
\epsilonnd{array}
{\rho}ight.
\epsilonnd{equation}
Either by congruence considerations or by employing \epsilonqref{ld133}, it is clear that $(1,3,3,0,0,0;n)=0$ when $n\epsilonquiv 2 \inftymod{3}$, so we do not consider such $n$. Using \epsilonqref{el} we see
\betaegin{equation}
\lambdaabel{l9}
L(1,\chi(9n))=\lambdaeft\{ \betaegin{array}{ll}
\varphifrac{4}{3} L(1,\chi(n)) & n\epsilonquiv 1 \inftymod{3},\\
&\\
L(1,\chi(n)) & n\epsilonquiv 3,6 \inftymod{9}.\\
\epsilonnd{array}
{\rho}ight.
\epsilonnd{equation}
Employing \epsilonqref{thm36}, \epsilonqref{ld133}, and \epsilonqref{l9}, we find
\betaegin{equation}
\lambdaabel{n2}
(1,3,3,0,0,0;n)\gammaeq \lambdaeft\{ \betaegin{array}{ll}
\varphifrac{16}{\pi}{\sigma}qrt{n}\cdot d_{f,2}(n)\cdot L(1,\chi(n)) & n\epsilonquiv 1 \inftymod{3},\\
&\\
\varphifrac{8}{\pi}{\sigma}qrt{n}\cdot d_{f,2}(n)\cdot L(1,\chi(n)) & n\epsilonquiv 3,6 \inftymod{9}.\\
\epsilonnd{array}
{\rho}ight.
\epsilonnd{equation}
Let $n=4^a v$ with $4\nmid v$, $9\nmid v$. Using \epsilonqref{h} and \epsilonqref{lld2} we have
\betaegin{equation}
\lambdaabel{n3}
\varphifrac{1}{\pi}{\sigma}qrt{n}\cdot d_{f,2}(n)\cdot L(1,\chi(n))= \lambdaeft\{ \betaegin{array}{ll}
\varphifrac{2^{a+2}-3}{8} & v=1,\\
&\\
\varphifrac{2^{a+1}-1}{2} & v=3,\\
&\\
\varphifrac{3(2^{a+1}-1)}{2}\cdot h(-v) & 3<v\epsilonquiv 3\inftymod{8},\\
&\\
2^a \cdot h(-v) & v\epsilonquiv 7\inftymod{8},\\
&\\
\varphifrac{2^{a+2}-3}{4}\cdot h(-4v) & 1<v\epsilonquiv 1,2\inftymod{4}.\\
\epsilonnd{array}
{\rho}ight.
\epsilonnd{equation}
Combining \epsilonqref{n2} and \epsilonqref{n3}, we find our lower bound for $(1,3,3,0,0,0;n)$ in terms of the class number, where $n=4^a \cdot v$ with $4\nmid v$, $9\nmid v$.
\betaegin{equation}
\lambdaabel{n4}
(1,3,3,0,0,0;n)\gammaeq \lambdaeft\{ \betaegin{array}{ll}
2(2^{a+2}-3) & v=1,\\
&\\
4(2^{a+1}-1) & v=3,\\
&\\
24(2^{a+1}-1)\cdot h(-v) & v\epsilonquiv 19\inftymod{24},\\
&\\
2^{a+4}\cdot h(-v) & v\epsilonquiv 7\inftymod{24},\\
&\\
4(2^{a+2}-3)\cdot h(-4v) & 1<v\epsilonquiv 1,10\inftymod{12},\\
&\\
12(2^{a+1}-1)\cdot h(-v) & 3<v\epsilonquiv 3,51\inftymod{72},\\
&\\
2^{a+3}\cdot h(-v) & v\epsilonquiv 15,39\inftymod{72},\\
&\\
2(2^{a+2}-3)\cdot h(-4v) & v\epsilonquiv 6,21,30,33 \inftymod{36}.\\
\epsilonnd{array}
{\rho}ight.
\epsilonnd{equation}
The form $(1,3,3,0,0,0)$ has 16 automorphs. Employing \epsilonqref{n4} along with the tables given in the Appendix, gives that there are exactly 53 numbers $n$ with $9 \nmid n$ and $0<(1,3,3,0,0,0;n)\lambdaeq 16$. These 53 numbers are the numbers in the set\\
$S=$\{1, 3, 4, 6, 7, 10, 12, 13, 15, 21, 22, 25, 30, 33, 34, 37, 42, 46, 57, 58, 66, 69, 70, 73, 78, 82, 85, 93, 97, 102, 105, 114, 130, 133, 138, 141, 142, 165, 177, 190, 193, 210, 213, 253, 258, 273, 282, 330, 345, 357, 438, 462, 498\}. We comment that we solving $0<(1,3,3,0,0,0;n)\lambdaeq 16$ requires class number information up to class number 8. We can use Maple V.15 to check the elements of $S$ for unique representation.
\betaegin{theorem}
\lambdaabel{tt133}
The form $(1,3,3,0,0,0)$ uniquely represents $n$ (up to action of automorphs) if and only if $n=9^k\cdot v$, $k\gammaeq 0$, with\\ $v\inftyn$\{$1$, $ 3$, $ 6$, $ 10$, $ 13$, $ 21$, $ 22$, $ 30$, $ 33$, $ 34$, $ 37$, $ 42$, $ 46$, $ 57$, $ 58$, $ 66$, $ 69$, $ 78$, $ 82$, $ 85$, $ 93$, $ 102$, $ 114$, $ 130$, $ 138$, $ 141$, $ 142$, $ 165$, $ 177$, $ 190$, $ 210$, $ 213$, $ 253$, $ 258$, $ 282$, $ 345$, $ 357$, $ 462$, $ 498$\}.
\epsilonnd{theorem}
From the comments at the beginning of this section, Theorem {\rho}ef{tt133} directly implies the following theorem.
\betaegin{theorem}
The form $(1,1,3,0,0,0)$ uniquely represents $n$ (up to action of automorphs) if and only if $n=9^k\cdot v$, $k\gammaeq 0$, with\\ $v\inftyn$\{$1$, $ 2$, $ 3$, $ 7$, $ 10$, $ 11$, $ 14$, $ 19$, $ 22$, $ 23$, $ 26$, $ 30$, $ 31$, $ 34$, $ 38$, $ 39$, $ 46$, $ 47$, $ 55$, $ 59$, $ 66$, $ 70$, $ 71$, $ 86$, $ 94$, $ 102$, $ 111$, $ 115$, $ 119$, $ 138$, $ 154$, $ 166$, $ 174$, $ 246$, $ 255$, $ 390$, $ 426$, $ 570$, $ 759$\}.
\epsilonnd{theorem}
\noindent
We have now resolved the conjecture of Kaplansky, concerning the forms $(1,3,3,0,0,0)$ and\\ $(1,1,3,0,0,0)$. We comment that an almost identical analysis holds for the other idoneal forms of discriminant 36. We move on to treat a second conjecture of Kaplansky given in the concluding remarks of \cite{kap}.
In the concluding remarks of \cite{kap}, Kaplansky considers the forms $x^2+2y^2+3z^2$. His Theorem 7.2 states that the even integers which are uniquely represented by $x^2 +2y^2 +3z^2$ are the odd powers of 2. He does not show the proof of this theorem, but instead offers it as an exercise to the reader. Kaplansky computationally found the odd integers which are uniquely represented by $x^2 +2y^2 +3z^2$, but admits that the proof was not yet accessible.\\
We use the method of Section {\rho}ef{notprim} to find the odd integers which are uniquely represented by $x^2 +2y^2 +3z^2$. For the rest of our consideration of $(1,2,3,0,0,0;n)$ we take $n$ to be odd.\\
Let $g=(1,2,3,0,0,0)$ which is of discriminant 24 and has 8 automorphs. Employing \epsilonqref{mf}, we find
\betaegin{equation}
\lambdaabel{thm123}
(1,2,3,0,0,0;n)={\delta}fracrac{18{\sigma}qrt{n}}{\pi{\sigma}qrt{6}}\cdot d_{g,2}(n)\cdot d_{g,3}(n)\cdot L(1,\chi(6n))\cdot P(n,6),
\epsilonnd{equation}
with all notation as defined in Section {\rho}ef{notprim}.
We find that for $n$ odd, we have $d_{g,2}(n)=1$. The 3-adic local density for $g$ is given by the following Lemma.
\betaegin{lemma}
\lambdaabel{3lden}
Let $n=9^{b}v$ with $9\nmid v$. We have
\[
d_{g,3}(n) = \lambdaeft\{ \betaegin{array}{ll}
\frac{2(3^{b+1}-2)}{3^{b+1}} & v\epsilonquiv 1,2\inftymod{3}, \\
&\\
2 & v\epsilonquiv 3\inftymod{9}, \\
&\\
\frac{2(3^{b+1}-1)}{3^{b+1}} & v\epsilonquiv 6\inftymod{9}. \\
\epsilonnd{array}
{\rho}ight.
\]
\epsilonnd{lemma}
Let $n=9^b\cdot v$ with $v$ odd and $9\nmid v$. Employing \epsilonqref{h} we have
\betaegin{equation}
\lambdaabel{l3}
L(1,\chi(6n))=\frac{\pi}{2{\sigma}qrt{6n}}\cdot h(-24n).
\epsilonnd{equation}
Combining \epsilonqref{3lden}, \epsilonqref{thm123}, and \epsilonqref{l3} we arrive at
\betaegin{equation}
\lambdaabel{thmm4}
(1,2,3,0,0,0;n)\gammaeq \lambdaeft\{ \betaegin{array}{ll}
\lambdaeft(3-\varphifrac{2}{3^b}{\rho}ight)h(-24n) & v\epsilonquiv 1,2\inftymod{3}, \\
&\\
3h(-24n) & v\epsilonquiv 3\inftymod{9}, \\
&\\
\lambdaeft(3-\varphifrac{1}{3^b}{\rho}ight)h(-24n) & v\epsilonquiv 6\inftymod{9}. \\
\epsilonnd{array}
{\rho}ight.
\epsilonnd{equation}
The form $(1,2,3,0,0,0)$ has 8 automorphs, and so we solve for all odd $n$ with $0<(1,2,3,0,0,0;n)\lambdaeq 8$ which requires class number information up to 8. Utilizing \epsilonqref{thmm4} along with the tables of the Appendix, we find the only odd $n$ with $0<(1,2,3,0,0,0;n)\lambdaeq 8$ to be
\[
n= 1, 3, 5, 7, 11, 13, 17, 19, 23, 35, 43, 47, 55, 73, 77, 83.
\]
Maple V.15 can be used to check these 16 numbers for unique representation.
\betaegin{theorem}
\lambdaabel{tt123}
The form $(1,2,3,0,0,0)$ uniquely represents odd $n$ (up to action of automorphs) if and only if $n=1,5,7,13,17,23,47,$ or $55$.
\epsilonnd{theorem}
We have now confirmed and proven the observation of Kaplansky regarding $(1,2,3,0,0,0)$ \cite{kap}.
{\sigma}ection{Outlook}
\lambdaabel{concl}
The method of employing the Siegel formula along with class number bounds easily extends to classifying the integers which are represented in essentially $k$ ways by an idoneal ternary quadratic form. One can use this paper as a guide to deduce the integers which are represented in an essentially unique way by any of the 794 idoneal ternary quadratic forms. It would be interesting to find the class number bounds that are necessary to classify the integers which are represented in an essentially unique way by any of the 794 idoneal ternary quadratic forms.\\
The restriction of the form being idoneal is not always necessary to find the integers which are uniquely represented by that form. To demonstrate this, we consider the form $(1,3,3,1,0,1)$ which is not idoneal, since $(1,1,11,1,1,1)$ shares the same genus. It can be shown that
\betaegin{equation}
\lambdaabel{etap}
\vartheta(1,1,11,1,1,1,q) - \vartheta(1,3,3,1,0,1,q) = 4qE(q^4)^2E(q^{16}),
\epsilonnd{equation}
where
\[
E(q):= \prod_{n=1}^{\inftynfty}(1-q^n).
\]
Equation \epsilonqref{etap} shows that for $n\not\epsilonquiv 1 \inftymod{4}$ we have $(1,3,3,1,0,1;n)=(1,1,11,1,1,1;n)$. So when $n\not\epsilonquiv 1 \inftymod{4}$ we have
\betaegin{equation}
\lambdaabel{cr1}
{\delta}frac{(1,3,3,1,0,1;n)}{4} + {\delta}frac{(1,1,11,1,1,1;n)}{12}= {\delta}frac{(1,3,3,1,0,1;n)}{3},
\epsilonnd{equation}
where we have used $|$Aut$(1,3,3,1,0,1)|=4$, and $|$Aut$(1,1,11,1,1,1)|=12$. Employing the Siegel formula gives\\
\betaegin{equation}
\lambdaabel{a1}
(1,3,3,1,0,1; 32n) = (1,1,1,0,0,0;n),
\epsilonnd{equation}
\betaegin{equation}
(1,3,3,1,0,1; 16(2n+1) ) = (1,1,1,0,0,0;2(2n+1)),
\epsilonnd{equation}
\betaegin{equation}
(1,3,3,1,0,1; 8(2n+1) ) = 0,
\epsilonnd{equation}
\betaegin{equation}
2(1,3,3,1,0,1; 4(2n+1) ) = (1,1,1,0,0,0;2(2n+1)),
\epsilonnd{equation}
\betaegin{equation}
(1,3,3,1,0,1; 2(2n+1) ) = 0,
\epsilonnd{equation}
\betaegin{equation}
\lambdaabel{ao}
4(1,3,3,1,0,1; 4n+3 ) = (1,1,1,0,0,0;2(4n+3)).
\epsilonnd{equation}
Using $|$Aut$(1,3,3,1,0,1)|=4$, Table {\rho}ef{prel}, and \epsilonqref{a1} -- \epsilonqref{ao}, we see $(1,3,3,1,0,1)$ does not uniquely represent any $n \not\epsilonquiv 1\inftymod{4}$. For $n \epsilonquiv 1 \inftymod{4}$, the Siegel formula gives
\[
3(1,3,3,1,0,1; n ) + (1,1,11,1,1,1;n) = (1,1,1,0,0,0; 2n).
\]
Thus we obtain $0<(1,3,3,1,0,1; n) \lambdaeq \varphifrac{1}{3}(1,1,1,0,0,0; 2n)$ for $n \epsilonquiv 1 \inftymod{4}$. We are left to solve $(1,1,1,0,0,0; 2n) \lambdaeq 12$ for $n \epsilonquiv 1 \inftymod{4}$, and we find only $n=1$ as a solution. Indeed, $n=1$ is the only integer which is represented uniquely (up to the action of Aut$(1,3,3,1,0,1)$) by the form $(1,3,3,1,0,1)$.\\
It is of interest to see which ternary quadratic forms uniquely represent (up to the action of automorphs) only a finite number of integers. The diagonal form $(1,2,4,0,0,0)$ considered by Kaplansky has this property, as well as the forms $(1,3,3,2,0,0), (5,13,20,-12,4,2), (7,15,23,10,2,6),$ and $(1,3,3,1,0,1)$, which were considered in this paper. The authors intend to address this topic in a subsequent paper.
{\sigma}ection{Acknowledgements}
\lambdaabel{ack}
We are grateful to William Jagy, Li--Chien Shen, John Voight, and Kenneth Williams for helpful discussions. We would like to thank George Andrews and James Sellers for their kind interest and encouragement. We are indebted to Keith Grizzell and Sue--Yen Patane for a careful reading of the manuscript and for many valuable suggestions.
{\sigma}ection{Appendix}
Below we list the negative of the 527 discriminants (fundamental and nonfundamental) of binary quadratic forms with class number $\lambdaeq 8$ according to the isomorphism class of the class group. We denote the class group of discriminant $D$ by $H(D)$, and we denote the cyclic group of order $n$ by $\mathbb{Z}_n$. We remark that we generated the tables below by utilizing the bounds found in \cite{watkins} along with equation \epsilonqref{veq}. We then used PARI/GP V.2.7.0 to identify the isomorphism class of each class group.\\
\betaegin{table}[htb]
\caption*{$H(D)\cong \mathbb{Z}_1 $}
\betaegin{tabular}{ccccccccccccc}
3,& 4,& 7,& 8,& 11,& 12,& 16,& 19,& 27,& 28,& 43,& 67,& 163\\
\epsilonnd{tabular}
\epsilonnd{table}
\betaegin{table}[htb]
\caption*{$H(D)\cong \mathbb{Z}_2 $}
\betaegin{tabular}{cccccccccc}
15,& 20,& 24,& 32,& 35,& 36,& 40,& 48,& 51,& 52,\\
60,& 64,& 72,& 75,& 88,& 91,& 99,& 100,& 112,& 115,\\
123,& 147,& 148,& 187,& 232,& 235,& 267,& 403,& 427&
\epsilonnd{tabular}
\epsilonnd{table}
\betaegin{table}[htb]
\caption*{$H(D)\cong \mathbb{Z}_3 $}
\betaegin{tabular}{cccccccccc}
23,& 31,& 44,& 59,& 76,& 83,& 92,& 107,& 108,& 124,\\
139,& 172,& 211,& 243,& 268,& 283,& 307,& 331,& 379,& 499,\\
547,& 643,& 652,& 883,& 907&&&&&
\epsilonnd{tabular}
\epsilonnd{table}
\betaegin{table}[htb]
\caption*{$H(D)\cong \mathbb{Z}_2 \varphiimes \mathbb{Z}_2 $}
\betaegin{tabular}{cccccccccc}
84,& 96,& 120,& 132,& 160,& 168,& 180,& 192,& 195,& 228,\\
240,& 280,& 288,& 312,& 315,& 340,& 352,& 372,& 408,& 435,\\
448,& 483,& 520,& 532,& 555,& 595,& 627,& 708,& 715,& 760,\\
795,& 928,& 1012,& 1435&&&&&&
\epsilonnd{tabular}
\epsilonnd{table}
\betaegin{table}[htb]
\caption*{$H(D)\cong \mathbb{Z}_4 $}
\betaegin{tabular}{cccccccccccccc}
39,& 55,& 56,& 63,& 68,& 80,& 128,& 136,& 144,& 155,& 156,& 171,& 184,& 196,\\
203,& 208,& 219,& 220,& 252,& 256,& 259,& 275,& 291,& 292,& 323,& 328,& 355,& 363,\\
387,& 388,& 400,& 475,& 507,& 568,& 592,& 603,& 667,& 723,& 763,& 772,& 955,& 1003,\\
1027,& 1227,& 1243,& 1387,& 1411,& 1467,& 1507,& 1555&&&&&&
\epsilonnd{tabular}
\epsilonnd{table}
\betaegin{table}[htb]
\caption*{$H(D)\cong \mathbb{Z}_5 $}
\betaegin{tabular}{cccccccccc}
47,& 79,& 103,& 127,& 131,& 179,& 188,& 227,& 316,& 347,\\
412,& 443,& 508,& 523,& 571,& 619,& 683,& 691,& 739,& 787,\\
947,& 1051,& 1123,& 1723,& 1747,& 1867,& 2203,& 2347,& 2683&
\epsilonnd{tabular}
\epsilonnd{table}
\betaegin{table}[htb]
\caption*{$H(D)\cong \mathbb{Z}_6 $}
\betaegin{tabular}{cccccccccccccc}
87,& 104,& 116,& 135,& 140,& 152,& 175,& 176,& 200,& 204,& 207,& 212,& 216,\\
244,& 247,& 300,& 304,& 324,& 339,& 348,& 364,& 368,& 396,& 411,& 424,& 432,\\
436,& 451,& 459,& 460,& 472,& 484,& 492,& 496,& 515,& 531,& 540,& 588,& 628,\\
648,& 675,& 676,& 688,& 700,& 707,& 747,& 748,& 771,& 808,& 828,& 835,& 843,\\
856,& 867,& 891,& 931,& 940,& 963,& 988,& 1048,& 1059,& 1068,& 1072,& 1075,\\
1083,& 1099,& 1107,& 1108,& 1147,& 1192,& 1203,& 1219,& 1267,& 1315,& 1323,\\
1347,& 1363,& 1432,& 1563,& 1588,& 1603,& 1612,& 1675,& 1708,& 1843,& 1915,\\
1963,& 2227,& 2283,& 2403,& 2443,& 2515,& 2563,& 2608,& 2787,& 2923,& 3235,\\
3427,& 3523,& 3763,& 4075&&&&&&&
\epsilonnd{tabular}
\epsilonnd{table}
\betaegin{table}[htb]
\caption*{$H(D)\cong \mathbb{Z}_7 $}
\betaegin{tabular}{cccccccccc}
71,& 151,& 223,& 251,& 284,& 343,& 463,& 467,& 487,& 587,\\
604,& 811,& 827,& 859,& 892,& 1163,& 1171,& 1372,& 1483,& 1523,\\
1627,& 1787,& 1852,& 1948,& 1987,& 2011,& 2083,& 2179,& 2251,& 2467,\\
2707,& 3019,& 3067,& 3187,& 3907,& 4603,& 5107,& 5923&&
\epsilonnd{tabular}
\epsilonnd{table}
\betaegin{table}[htb]
\caption*{$H(D)\cong \mathbb{Z}_8 $}
\betaegin{tabular}{ccccccccccccc}
95,& 111,& 164,& 183,& 248,& 272,& 295,& 299,& 371,& 376,& 380,& 392,& 395,\\
444,& 452,& 512,& 539,& 548,& 579,& 583,& 632,& 712,& 732,& 784,& 904,& 939,\\
979,& 995,&1024,& 1043,& 1156,& 1168,& 1180,& 1195,& 1252,& 1299,& 1339,& 1348,& 1528,\\
1552,& 1587,& 1651,& 1731,& 1795,& 1803,& 1828,& 1864,& 1912,& 1939,& 2059,& 2107,& 2248,\\
2307,& 2308,& 2323,& 2332,& 2395,& 2419,& 2587,& 2611,& 2827,& 2947,& 2995,& 3088,& 3283,\\
3403,& 3448,& 3595,& 3787,& 3883,& 3963,& 4195,& 4267,& 4387,& 4747,& 4843,& 4867,& 5587,\\
5707,& 5947,& 7987&&&&&&&&&&
\epsilonnd{tabular}
\epsilonnd{table}
\betaegin{table}[htb]
\caption*{$H(D)\cong \mathbb{Z}_4 \varphiimes \mathbb{Z}_2 $}
\betaegin{tabular}{ccccccccccccc}
224,& 260,& 264,& 276,& 308,& 320,& 336,& 360,& 384,& 456,& 468,& 504,& 528,\\
544,& 552,& 564,& 576,& 580,& 600,& 612,& 616,& 624,& 640,& 651,& 720,& 736,\\
768,& 792,& 819,& 820,& 832,& 852,& 868,& 880,& 900,& 912,& 915,& 952,& 987,\\
1008,& 1032,& 1035,& 1060,& 1128,& 1131,& 1152,& 1204,& 1240,& 1275,& 1288,& 1312,& 1332,\\
1360,& 1395,& 1408,& 1443,& 1488,& 1600,& 1635,& 1659,& 1672,& 1683,& 1752,& 1768,& 1771,\\
1780,& 1792,& 1827,& 1947,& 1992,& 2020,& 2035,& 2067,& 2088,& 2115,& 2128,& 2139,& 2163,\\
2212,& 2272,& 2275,& 2368,& 2392,& 2451,& 2475,& 2632,& 2667,& 2715,& 2755,& 2788,& 2832,\\
2907,& 2968,& 3172,& 3243,& 3355,& 3507,& 3627,& 3712,& 3843,& 4048,& 4123,& 4323,& 5083,\\
5467,& 6307&&&&&&&&&&&
\epsilonnd{tabular}
\epsilonnd{table}
\betaegin{table}[htb]
\caption*{$H(D)\cong \mathbb{Z}_2\varphiimes \mathbb{Z}_2\varphiimes \mathbb{Z}_2 $}
\betaegin{tabular}{cccccccccc}
420,& 480,& 660,& 672,& 840,& 960,& 1092,& 1120,& 1155,& 1248,\\
1320,& 1380,& 1428,& 1540,& 1632,& 1848,& 1995,& 2080,& 3003,& 3040,\\
3315&&&&&&&&&
\epsilonnd{tabular}
\epsilonnd{table}
\betaegin{thebibliography}{30}
\betaibitem{arno}
S.~Arno, The imaginary quadratic fields of class number 4. Acta Arithmetica \varphiextbf{60} (1992), 321--334
\betaibitem{baker}
A. Baker, Imaginary quadratic fields with class number 2. Ann. Math.\varphiextbf{94}, (1971), 139--152.
\betaibitem{bate}
P.~T. Bateman, On the representations of a number as the sum of three squares. Trans. Amer. Math. Soc. \varphiextbf{71} (1951), 70–-101.
\betaibitem{batgros}
P.~T. Bateman, E. Grosswald, Positive integers expressible as a sum of three squares in essentially only one way. J. Number Theory, \varphiextbf{19}, (1984), 301--308.
\betaibitem{berk}
A.~Berkovich, W.~.C.~Jagy, On representation of an integer as the sum of three squares and the ternary quadratic forms with the discriminants $p^2$, $16p^2$. (2011)
\betaibitem{buell}
D. Buell, \epsilonmph{Binary Quadratic Forms: Classical Theory and Modern Computations}, Springer-Verlag, New York, 1990.
\betaibitem{cox}
D.~A. Cox, \epsilonmph{Primes of the Form $x^2 +ny^2$: Fermat Class Field Theory and Complex Multiplication}, John Wiley \& Sons, New York, 1989.
\betaibitem{mdickson}
L.~E. Dickson, \epsilonmph{Modern Elementary Theory of Numbers}, Univ. of Chicago Press, 1939.
\betaibitem{dickson}
L.~E. Dickson, \epsilonmph{Studies in the Theory of Numbers}, Chelsea Publishing Company, New York, 1957.
\betaibitem{dir}
P.~G.~ Dirichlet, \epsilonmph{Lectures on Number Theory}, American Mathematical Society, 1999.
\betaibitem{gauss}
C.~F. Gauss, \epsilonmph{Disquisitiones Arithmeticae}, Fleischer, Leipzig, 1801.
\betaibitem{gold}
D.~ M. Goldfeld, Gauss' class number problem for imaginary quadratic fields. Bull. Amer. Math. Soc. \varphiextbf{13}, (1985), 23--37.
\betaibitem{gross}
B.~ Gross, D.~ Zagier, Points de Heegner et d{\'e}riv{\'e}es de fonctions L, C.R. Acad. Sci, Paris, \varphiextbf{297} (1983), 85--87.
\betaibitem{hanke}
J.~ Hanke. Local densities and explicit bounds for representability by a quadratic form. Duke Math. J., \varphiextbf{124(2)} (2004), 351–-388.
\betaibitem{sel}
M.~ Hirschhorn, J.~ Sellers. On a problem of Lehmer on partitions into squares, Ramanujan Journal \varphiextbf{8}, no. 3 (2004), 279--288
\betaibitem{heg}
K. Heegner, Diophantische analysis und modulfunktionen. Mathematische Zeitschrift, \varphiextbf{56}(3) (1952), 227–-253
\betaibitem{jagykap}
W.~C Jagy, I.~Kaplansky, and A.~Schiemann. There are 913 regular ternary quadratic forms. Mathematika, \varphiextbf{44} (1997), 332--341.
\betaibitem{kap}
I. Kaplansky, Integers uniquely represented by certain ternary quadratic forms. \epsilonmph{The mathematics of Paul Erd\H os}, vol. II. Edited by S.~ Butler, R.~L. Graham, J. Ne\v set\v ril. Springer, New York, (2013), 71--79.
\betaibitem{lehm}
D.~H. Lehmer, On the partition of numbers into squares. Amer. Math. Monthly, \varphiextbf{55,} (1948), 476–-481.
\betaibitem{lorch}
D.~Lorch, M.~Kirschmer, Single-class genera of positive integral lattices. LMS Journal of Computation and Mathematics, \varphiextbf{16}, (2013), 172--186.
\betaibitem{siegel}
C.~L. Siegel, Lectures on the Analytical Theory of Quadratic Forms, Notes by Morgan Ward. Third revised edition. Buchhandlung Robert Peppm\"uller, G\"ottingen, 1963.
\betaibitem{voight}
J. Voight, \epsilonmph{Quadratic forms that represent almost the same primes}, Math. Comp., \varphiextbf{76 }(2007), 1589--1617.
\betaibitem{watkins}
M. Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation \varphiextbf{73}, (2004), 907–-938
\betaibitem{wat1}
G.~L. Watson, Transformations of a quadratic form which do not increase the class-number, Proc. London Math. Soc. \varphiextbf{12}, (1962), 577–-587.
\epsilonnd{thebibliography}
\epsilonnd{document} |
\begin{document}
\title[Improvements of the Katznelson-Tzafriri theorem]{Some improvements of the Katznelson-\\Tzafriri theorem on Hilbert space}
\subjclass[2010]{Primary: 47D03; secondary: 43A45, 43A46, 47A35.}
\author{David Seifert}
\address{Mathematical Institute, 24--29 St Giles', Oxford\;\;OX1 3LB, United Kingdom}
\curraddr{Balliol College, Oxford\;\;OX1 3BJ, United Kingdom}
\mathrm{e}mail{david.seifert@balliol.ox.ac.uk}
\date{27 March 2013}
\begin{abstract}
This paper extends two recent improvements in the Hilbert space setting of the well-known Katznelson-Tzafriri theorem by establishing both a version of the result valid for bounded representations of a large class of abelian semigroups and a quantified version for contractive representations. The paper concludes with an outline of an improved version of the Katznelson-Tzafriri theorem for individual orbits, whose validity extends even to certain unbounded representations.
\mathrm{e}nd{abstract}
\maketitle
\section{Introduction}
In \cite{KT}, Katznelson and Tzafriri proved that, given a power-bounded operator $T$ on a complex Banach space $X$, $\|T^n(I-T)\|\rightarrow 0$ as $n\rightarrow\infty$ if (and only if) $\sigma(T)\cap\mathbb{T}\subset\{1\}$, where $\sigma(T)$ denotes the spectrum of $T$ and $\mathbb{T}$ is the unit circle. Given a sequence $a\in\mathrm{e}ll^1(\mathbb{Z}_+)$, define $\widehat{a}(\lambda):=\sum_{n=0}^\infty a(n)\lambda^n$ ($|\lambda|\leq1$) and $\widehat{a}(T):=\sum_{n=0}^\infty a(n)T^n$, which is a bounded linear operator on $X$. Katznelson and Tzafriri also showed, in the same paper, that \begin{equation}\label{discrete_KT}\lim_{n\rightarrow\infty}\|T^n\widehat{a}(T)\|=0\mathrm{e}nd{equation} provided there exists a sequence $(a_n)$ in $\mathrm{e}ll^1(\mathbb{Z}_+)$ such that each $\widehat{a_n}$ vanishes on an open neighbourhood of $\sigma(T)\cap\mathbb{T}$ and $\|a-a_n\|_1\rightarrow0$ as $n\rightarrow\infty$. This result has itself subsequently been extended, first to the case of $C_0$-semigroups (see \cite[Th\'eor\`eme~III.4]{ESZ} and \cite[Theorem~3.2]{Vu}) and later to more general semigroup representations (see \cite[Theorem~4.3]{BV} and \cite{Vu}).
These results are optimal in various senses (see \cite[Section~5]{CT}), but improvements are possible when $X$ is assumed to be a Hilbert space. It is shown in \cite[Corollary~2.12]{ESZ2}, for instance, that the weaker (and necessary) condition that $\widehat{a}$ vanish on $\sigma(T)\cap\mathbb{T}$ is sufficient for \mathrm{e}qref{discrete_KT} to hold, at least when $T$ is a contraction; see \cite[Proposition~1.6]{KvN} for a slightly more general result. This result has in turn been improved upon in two recent papers. In \cite{Leka}, L\'eka has extended this result to power-bounded operators on Hilbert space, and Zarrabi in \cite{Zarr} has shown that for contractions, and likewise for pairs of commuting contractions and for contractive $C_0$-semigroups, the limit appearing in \mathrm{e}qref{discrete_KT} is given, more generally, by $\sup\{ |\widehat{a}(\lambda)|:\lambda\in\sigma(T)\cap\mathbb{T}\}$ or the appropriate analogue; for related results see \cite{AR}, \cite{BBG}, \cite{Berc2}, \cite{Berc} and \cite{Mus}.
The purpose of this paper is to improve both L\'eka's and Zarrabi's versions of the Katznelson-Tzafriri theorem by extending them to representations of a significantly larger class of semigroups. The main result, Theorem~\ref{Thm}, is a Katznelson-Tzafriri type theorem which holds for bounded (as opposed to contractive) representations and thus includes both \cite[Theorem~2.1]{Leka} and various results contained as special cases in \cite{Zarr}. The main implication of Theorem~\ref{Thm} is proved first via a certain ergodic condition as in \cite{Leka} and then by a more direct argument. The second method does not involve the ergodic condition and, as is shown in Theorem~\ref{ub}, leads naturally to an extension of Zarrabi's quantified results for contractive representations. Section~5, finally, contains a brief exposition of an improved version of the Katznelson-Tzafriri theorem for individual orbits. First of all, though, Section~2 sets out the necessary preliminary material.
\section{Preliminaries}
The setting throughout, even when not stated explicitly, will be that of an abelian semigroup $S$ contained in a locally compact abelian group $G$ satisfying $G=S-S$. The Haar measure on $G$ is denoted by $\mu$, and it is assumed that $S$ is Haar-measurable and hence itself becomes a measure space with respect to the restriction of $\mu$. Assume furthermore that the interior $S^\circ$ of $S$ (in the topology induced by $G$) is non-empty. The semigroup $S$ becomes a directed set under the relation $\succeq$, where $s\succeq t$ for $s,t\in S$ whenever $s-t\in S\cup\{0\}$; this makes it possible to speak of limits as $s\rightarrow\infty$ through $S$. The dual group of $G$, consisting of all continuous bounded characters $\chi:G\rightarrow\mathbb{C}$, is denoted by $\Gamma$, the set of continuous bounded characters on $S$ by $S^*$. It follows from the assumption that $S$ spans $G$ that the subset of $S^*$ of characters taking values in the unit circle $\mathbb{T}$ can (and will throughout) be identified with $\Gamma$. Two important examples of the above are the (semi)groups $\mathbb{Z}_{(+)}$ with counting measure and $\mathbb{R}_{(+)}$ with Lebesgue measure. Here $S^*$ can be identified in a natural way with $\{\lambda\in\mathbb{C}:|\lambda|\leq1\}$ and $\{\lambda\in\mathbb{C}:\R \lambda\leq0\}$, respectively, and the dual group $\Gamma$ is $\mathbb{T}$ and $\mathrm{i}\mathbb{R}$ in each case.
For $\Omega=G$ or $S,$ let $L^1(\Omega)$ denote the algebra (under convolution) of functions $a:\Omega\rightarrow\mathbb{C}$ that are integrable with respect to (the restriction of) Haar measure and, given $a\in L^1(\Omega)$, define its Fourier transform by $$\widehat{a}(\chi):=\int_\Omega a(s)\chi(s)\,\mathrm{d}\mu(s),$$ where $\chi$ is an element of $\Gamma$ or $S^*$, as appropriate. Given a closed subset $\Lambda$ of $\Gamma$, define $J_\Lambda:=\{a\in L^1(G): \widehat{a}\mathrm{e}quiv0\;\mbox{in a neighbourhood of $\chi\in\Lambda$}\}$ and $K_\Lambda:=\{a\in L^1(G): \widehat{a}(\chi)=0\;\mbox{for all $\chi\in\Lambda$}\}$. An element of $L^1(G)$ is said to be of \textsl{spectral synthesis} with respect to $\Lambda$ if it lies in the closure of $J_\Lambda$. Since $K_\Lambda$ is closed, any such function must be an element of $K_\Lambda$. If $K_\Lambda$ coincides with the closure of $J_\Lambda$, the set $\Lambda$ is said to be of \textsl{spectral synthesis}.
For any closed subset $\Lambda$ of $\Gamma$, the map $W_\Lambda: a\mapsto \widehat{a}|_\Lambda$ is a well-defined contractive algebra homomorphism from $L^1(G)$ into $ C_0(\Lambda)$ whose kernel is $K_\Lambda$ and whose range, by the Stone-Weierstrass theorem, is dense in $C_0(\Lambda)$; see \cite[Theorem~1.2.4]{Rudin}. Since $K_\Lambda(G)$ is a closed ideal of $L^1(G)$, $W_\Lambda$ induces a well-defined injective algebra homomorphism $ U_\Lambda:L^1(G)/K_\Lambda(G)\rightarrow C_0(\Lambda)$ which, by the Inverse Mapping Theorem, is an isomorphism precisely when it is surjective. Since its range is dense in $C_0(\Lambda)$, this is the case if and only if the map is an isomorphic embedding (which in turn is equivalent to the dual operator $ U_\Lambda':M(\Lambda)\rightarrow K_\Lambda^\perp$ being either a surjection or an isomorphic embedding, where $M(\Lambda)$ denotes the set of complex-valued regular measures on $\Lambda$ which have finite total variation; see for instance \cite[Appendix~C11]{Rudin}). When these conditions are satisfied, $\Lambda$ is said to be a \textsl{Helson set}, and the quantity $\alpha(\Lambda):=\| U_\Lambda^{-1}\|$ is known as its \textsl{Helson constant}. Since $U_\Lambda$ is contractive, $\alpha(\Lambda)\geq1$ for any Helson set $\Lambda$. For some examples of Helson sets, see \cite[Remark~5.6]{Zarr}.
In what follows, it will be assumed that the set $\{\widehat{a}:a\in L^1(S)\}$ separates points both from each other and from zero and, furthermore, that the interior $S^\circ$ is dense in $S$. For further details and discussion of these conditions, see for instance \cite{BV}. These assumptions ensure, in particular, that there exists a net $(\Omega_\alpha)$, known as a \textsl{F\o lner net}, of compact, measurable, non-null subsets of $S$ satisfying $$\lim_{\alpha\rightarrow\infty}\frac{\mu\big(\Omega_\alpha\bigtriangleup(\Omega_\alpha+s)\big)}{\mu(\Omega_\alpha)}=0,$$uniformly for $s$ in compact subsets of $S$.
Given a Banach space $X$ and $\Omega=G$ or $S$ for $S$ and $G$ as above, a \textsl{representation} of $\Omega$ on $X$ is a strongly continuous homomorphism $T:\Omega\rightarrow\mathcal{B}(X)$ which, if $0\in \Omega$, satisfies $T(0)=I.$ The representation is said to be \textsl{bounded} if $\sup\{\|T(s)\|:s\in\Omega\}<\infty$ and in this case, given $a\in L^1(\Omega)$, the operator $\widehat{a}(T)\in\mathcal{B}(X)$ is defined, for each $x\in X$, by $$\widehat{a}(T)x:=\int_\Omega a(s)T(s)x\,\mathrm{d}\mu(s).$$
Given a bounded representation $T$ of $G$ on a Banach space $X$ and a closed subspace $\Lambda$ of the dual group $\Gamma$, the corresponding \textsl{spectral subspace} $M_T(\Lambda)$ is defined as $$M_T(\Lambda):=\bigcap\nolimits_{a\in J_\Lambda}\Ker\widehat{a}(T),$$ the \textsl{(Arveson) spectrum} $\Sp(T)$ of $T$ as $$\Sp(T):=\bigcap\big\{\Lambda\subset\Gamma:\mbox{$\Lambda$ is closed and $M_T(\Lambda)=X$}\big\}.$$ Thus the spectrum is a closed subset of $\Gamma$, and it is shown in \cite[Theorems~8.1.4 and 8.1.12]{Ped}, respectively, that $\Sp(T)$ is non-empty whenever $X$ is non-trivial and that it is compact if and only if $T$ is continuous with respect to the norm topology on $\mathcal{B}(X)$. If $T$ is a representation by isometries, this notion of spectrum coincides with the \textsl{finite L-spectrum} of \cite[Section~5.2]{Lyu}. By \cite[Proposition~8.1.9]{Ped}, furthermore, $\Sp(T)$ has the alternative description $$\Sp(T)=\big\{\chi\in\Gamma:|\widehat{a}(\chi)|\leq\|\widehat{a}(T)\|\;\mbox{for all $a\in L^1(G)$}\big\}.$$ Accordingly, given a bounded representation $T$ of a semigroup $S$ on a Banach space $X$, the \textsl{spectrum} $\Sp(T)$ of $T$ is defined as $$\Sp(T):=\big\{\chi\in S^*:|\widehat{a}(\chi)|\leq\|\widehat{a}(T)\|\;\mbox{for all $a\in L^1(S)$}\big\},$$ and the \textsl{unitary spectrum} of $T$ is given by $\Spu(T):=\Sp(T)\cap\Gamma$; see \cite{BV} for details. In the examples mentioned above, bounded semigroup representations correspond to a single power-bounded operator $T\in\mathcal{B}(X)$ if $S=\mathbb{Z}_+$ and to a bounded $C_0$-semigroup if $S=\mathbb{R}_+$. The spectrum is given by $\sigma(T)$ and $\sigma(A)$, respectively, where $A$ denotes the generator of the semigroup.
\section{A general Katznelson-Tzafriri type result}
The aim of this section is to prove the following generalisation of \cite[Theorem~2.1]{Leka}.
\begin{mythm}\label{Thm}
Let $T$ be a bounded representation of a semigroup $S$ on a Hilbert space $X$, and suppose that $a\in L^1(S)$. Then the following are equivalent:
\begin{enumerate}
\item[(i)]$\widehat{a}(\chi)=0$ for every $\chi\in\Spu(T);$
\item[(ii)] Given any F\o lner net $(\Omega_\alpha)$ for $S$ and any $\chi\in \Spu(T)$, \begin{equation}\label{erg}\lim_{\alpha\rightarrow\infty}\frac{1}{\mu(\Omega_\alpha)}\left\|\int_{\Omega_\alpha}\overline{\chi(s)}T(s)\widehat{a}(T)\,\mathrm{d}\mu(s)\right\|=0;\mathrm{e}nd{equation}
\item[(iii)] $\|T(s)\widehat{a}(T)\|\rightarrow0$ as $s\rightarrow\infty$.
\mathrm{e}nd{enumerate}
\mathrm{e}nd{mythm}
\begin{rem}\label{Q_rem}
In \cite[Theorem~2.1]{Leka}, conditions (ii) and (iii) above are presented in a slightly more general form, with the operator $\widehat{a}(T)$ replaced by an arbitrary $Q\in\mathcal{B}(X)$ that commutes with the representation. The presentation here is restricted to the case $Q=\widehat{a}(T)$ purely for simplicity.
\mathrm{e}nd{rem}
The proof of this result will be broken up into a number of separate steps, all of which correspond to some part of the proof of \cite[Theorem~2.1]{Leka} given by L\'eka but typically with some modifications to accommodate the more general setting in which the representation need not be norm continuous. The following lemma constitutes the main step towards proving that (i)$\implies$(ii); it corresponds to \cite[Lemma~2.2]{Leka}. Note that the Hilbert space assumption is not required for this part of the argument.
\begin{mylem}\label{int_lem} Let $T$ be a bounded representation of a semigroup $S$ on a Banach space $X$, and let $a\in L^1(S)$. Then, for all $\chi\in\Gamma$, $$\lim_{\alpha\rightarrow\infty}\frac{1}{\mu(\Omega_\alpha)}\left\|\int_{\Omega_\alpha}\overline{\chi(s)}T(s)\big(\widehat{a}(T)-\widehat{a}(\chi)\big)\,\mathrm{d}\mu(s)\right\|=0,$$ where $(\Omega_\alpha)$ is any F\o lner net for $S$ and the integral is taken in the strong sense.
\mathrm{e}nd{mylem}
\begin{proof}[\textsc{Proof}]
With $a\in L^1(S)$ and $\chi\in\Gamma$ fixed, let $x\in X$ have unit norm and set $$I_x(\alpha):=\frac{1}{\mu(\Omega_\alpha)}\left\|\int_{\Omega_\alpha}\overline{\chi(s)}T(s)\big(\widehat{a}(T)x-\widehat{a}(\chi)x\big)\,\mathrm{d}\mu(s)\right\|.$$ Then, by a simple application of Fubini's theorem, $$\begin{aligned}I_x(\alpha)&\leq M\int_S\frac{\mu\big(\Omega_\alpha\bigtriangleup(\Omega_\alpha+s)\big)}{\mu(\Omega_\alpha)}|a(s)|\,\mathrm{d}\mu(s),\mathrm{e}nd{aligned}$$ where $M:=\sup\{\|T(s)\|:s\in S\}$. Let $\varepsilon>0$. Since $a\in L^1(S)$, there exists a compact subset $K$ of $S$ such that $\int_{S\backslash K}|a(s)|\,\mathrm{d}\mu(s)<\varepsilon/4M.$ Defining $$\xi_K(\alpha):=\sup\left\{\frac{\mu(\Omega_\alpha\bigtriangleup(\Omega_\alpha+s))}{\mu(\Omega_\alpha)}:s\in K\right\},$$ it follows from the definition of a F\o lner net that $\xi_K(\alpha)\rightarrow0$ as $\alpha\rightarrow\infty$. Since $$I_x(\alpha)\leq M\|a\|_1\xi_K(\alpha)+2M\int_{S\backslash K}|a(s)|\,\mathrm{d}\mu(s),$$ $I_x(\alpha)<\varepsilon$ for all sufficiently large $\alpha$, and the result follows.
\mathrm{e}nd{proof}
\begin{mycor}\label{cor:erg_cond}
Let $T$ be a bounded representation of a semigroup $S$ on a Banach space $X$, and let $\chi\in\Gamma$. Suppose that $a\in L^1(S)$ is such that $\widehat{a}(\chi)=0$. Then \mathrm{e}qref{erg} holds for any F\o lner net $(\Omega_\alpha)$ for $S$.
\mathrm{e}nd{mycor}
The next result is an important step towards establishing the implication (ii)$\implies$(iii) in Theorem~\ref{Thm} and should be compared with \cite[Lemma~2.4]{Leka}.
\begin{myprp}\label{zero}
Let $S$ be a semigroup and $T$ a representation of a group $G=S-S$ by unitary operators on a Hilbert space $X$. Suppose that $a\in L^1(G)$ and that, for each $\chi\in \Sp(T)$, $$\lim_{\alpha\rightarrow\infty}\frac{1}{\mu(\Omega_\alpha)}\left\|\int_{\Omega_\alpha}\overline{\chi(s)}T(s)\widehat{a}(T)\,\mathrm{d}\mu(s)\right\|=0,$$ where $(\Omega_\alpha)$ is any F\o lner net for $S$. Then $\widehat{a}(T)=0$.
\mathrm{e}nd{myprp}
\begin{proof}[\textsc{Proof}]
Writing $B(\Gamma)$ for the set of Borel subsets of the dual group $\Gamma$, let $E:B(\Gamma)\rightarrow \mathcal{B}(X)$ denote the spectral measure associated with $T$ (see \cite[Theorem~8.3.2]{Ped}) and, for $s\in G$ and $\Lambda\in B(\Gamma)$, let $T_\Lambda(s):=T(s)E(\Lambda)$. Then \begin{equation*}\label{spectral1}T_\Lambda(s):=\int_\Lambda \chi(s)\,\mathrm{d}E(\chi),\mathrm{e}nd{equation*} the integral being taken in the weak sense, and, by Fubini's theorem, \begin{equation}\label{spectral}\widehat{b}(T_\Lambda)=\int_\Lambda \widehat{b}(\chi)\,\mathrm{d}E(\chi)\mathrm{e}nd{equation} for all $b\in L^1(G)$. Thus if $\Lambda\in B(\Gamma)$ is closed and $b\in J_\Lambda(G)$, then $\widehat{b}(T_\Lambda)=0$, and it follows that $M_{T_\Lambda}(\Lambda)=X$, so that $\Sp(T_\Lambda)\subset\Lambda$. Choosing $\Lambda\in B(\Gamma)$ to be compact ensures that the representation $T_\Lambda$ of $G$ on $X$ is norm continuous.
Set $Q:=\widehat{a}(T)$ and, for a given compact subset $\Lambda$ of $\Sp(T)$, define $Q_\Lambda:=QE(\Lambda)$, noting that $Q_\Lambda$ is normal and that $Q_\Lambda\rightarrow Q$ in the weak (and indeed the strong) operator topology as $\Lambda$ approaches $\Sp(T)$ through compact subsets. Furthermore, let $\mathcal{A}_\Lambda$ denote the commutative unital $C^*$-algebra generated by $\{Q_\Lambda,Q_\Lambda^*\}\cup\{T_\Lambda(s):s\in G\}$, and let $\Delta(\mathcal{A}_\Lambda)$ denote its character space. Write $\Phi_\Lambda:\mathcal{A}_\Lambda\rightarrow C_0(\Delta(\mathcal{A}_\Lambda))$ for the Gelfand transform of $\mathcal{A}_\Lambda$, which is an isometric $*$-isomorphism, and consider the map $\chi_\xi:G\rightarrow\mathbb{C}\backslash\{0\}$ given, for $\xi\in\Delta(\mathcal{A}_\Lambda)$ and $s\in G$, by $\chi_\xi(s):= \Phi_\Lambda(T_\Lambda(s))(\xi).$ Since the representation $T_\Lambda$ is norm continuous, $\chi_\xi$ is a continuous group homomorphism, and the fact that each $\xi\in\Delta(\mathcal{A}_\Lambda)$ is a bounded linear functional on $\mathcal{A}_\Lambda$ with $\|\xi\|=|\xi(E(\Lambda))|=1$ implies that $|\chi_\xi(s)|\leq1$ for all $s\in G$. Hence $\chi_\xi\in\Gamma$. Moreover, if $b\in L^1(G)$, then $$\big|\widehat{b}(\chi_\xi)\big|=\left|\xi\left(\int_Gb(s)T_\Lambda(s)\,\mathrm{d}\mu(s)\right)\right|\leq\big\|\widehat{b}(T_\Lambda)\big\|,$$ which is to say that $\chi_\xi\in \Sp(T_\Lambda)$, and hence $\chi_\xi\in\Sp(T)$. Let $g_\Lambda:=\Phi_\Lambda(Q_\Lambda)$. Then $$\begin{aligned}|g_\Lambda(\xi)|&=\frac{1}{\mu(\Omega_\alpha)}\left|\int_{\Omega_\alpha}|\chi_\xi(s)|^2g_\Lambda(\xi)\,\mathrm{d}\mu(s)\right|\\&\leq \frac{1}{\mu(\Omega_\alpha)}\left\|\Phi_\Lambda\left(\int_{\Omega_\alpha}\overline{\chi_\xi(s)}T_\Lambda(s)Q_\Lambda\,\mathrm{d}\mu(s)\right)\right\|_\infty\\&=\frac{1}{\mu(\Omega_\alpha)}\left\|\int_{\Omega_\alpha}\overline{\chi_\xi(s)}T_\Lambda(s)Q_\Lambda\,\mathrm{d}\mu(s)\right\|\\&\leq \frac{1}{\mu(\Omega_\alpha)}\left\|\int_{\Omega_\alpha}\overline{\chi_\xi(s)}T(s)Q\,\mathrm{d}\mu(s)\right\|,\mathrm{e}nd{aligned}$$for any $\xi\in \Delta(\mathcal{A}_\Lambda)$ and letting $\alpha\rightarrow\infty$ shows that $g_\Lambda=0.$ Since $\Phi_\Lambda$ is an isometry, it follows that $Q_\Lambda=0$, and allowing $\Lambda$ to approach $\Sp(T)$ through compact subsets gives $Q=0$, as required.
\mathrm{e}nd{proof}
\begin{rem}
The result remains true when $\widehat{a}(T)$ is replaced by any $Q\in\mathcal{B}(X)$ which commutes with $T$. If $Q$ is normal, this follows from the same argument as above; the general case can then be obtained by considering the operator $Q^* Q$; see also \cite[Lemma~2.4]{Leka}.
\mathrm{e}nd{rem}
Propositions~\ref{pwlim} and \ref{lim} below correspond in essence to the two main stages in the proof of \cite[Theorem~2.1]{Leka} and show, via an intermediate result for the strong operator topology, that (ii)$\implies$(iii) in Theorem~\ref{Thm}.
\begin{myprp}\label{pwlim}
Let $T$ be a bounded representation of a semigroup $S$ on a Hilbert space $X$, and let $a\in L^1(S)$. Suppose that, for some F\o lner net $(\Omega_\alpha)$ for $S$, \mathrm{e}qref{erg} holds for all $\chi\in\Spu(T)$. Then, for all $x\in X$, $\|T(s)\widehat{a}(T)x\|\rightarrow0$ as $s\rightarrow\infty$.
\mathrm{e}nd{myprp}
\begin{proof}[\textsc{Proof}]
Fix a Banach limit $\phi$ on $L^\infty(S)$. A construction analogous to \cite[Proposition~3.1]{BV} and \cite[Section~1]{KvN} shows that there exist a Hilbert space $X_\phi$, a representation $T_\phi$ of $S$ on $X_\phi$ by isometries with $\Sp(T_\phi)\subset\Sp(T)$ and an operator $\pi_\phi:X\rightarrow X_\phi$ with the following properties: $\pi_\phi$ is bounded with norm $\|\pi_\phi\|\leq \sup\{\|T(s)\|:s\in S\}$, $\|\pi_\phi(x)\|^2=\phi(\|T(\cdot)x\|^2)$ for all $x\in X$, $\Ran\pi_\phi$ is dense in $X_\phi$, $\Ker\pi_\phi=\{x\in X:\|T(s)x\|\rightarrow0\;\mbox{as}\;s\rightarrow\infty\}$
and $\pi_\phi T(s)= T_\phi(s)\pi_\phi$ for all $s\in S$ (so $\pi_\phi$ is an \textsl{intertwining operator}). In particular, for any operator $Q\in\mathcal{B}(X)$ that commutes with $T$, the operator $Q_\phi\in\mathcal{B}(X_\phi)$ defined by $\pi_\phi Q=Q_\phi\pi_\phi$ satisfes $\|Q_\phi\|\leq\|Q\|$. By a construction analogous to \cite[Proposition~2.1]{BG} (see also \cite[Proposition~3.2]{BV}, \cite{BY}, \cite{Dou} and \cite{Ito}), there exist a further Hilbert space $Y_\phi$, a representation $T_G$ of the group $G=S-S$ by unitary operators on $Y_\phi$ with $\Sp(T_G)=\Spu(T_\phi)$ and an isometric intertwining operator $\pi_G:X_\phi\rightarrow Y_\phi$ such that $\{T_G(-s)\pi_G(x):s\in S, x\in X_\phi\}$ is dense in $Y_\phi$. The latter implies, in particular, that $\|Q_G\|=\|Q_\phi\|$ for all $Q_\phi\in\mathcal{B}(X_\phi)$ and all $Q_G\in\mathcal{B}(X_G)$ which commute with $T_G$ and satisfy $\pi_G Q_\phi=Q_G\pi_G$. Thus it is possible to assume, sacrificing only the density condition on the range of the intertwining operator, that $T_\phi$ itself is in fact a representation of $G$ by unitary operators on $X_\phi$.
Now, given $\chi\in\Sp(T_\phi)$, define operators $Q_\alpha\in\mathcal{B}(X)$ and $Q_{\phi,\alpha}\in\mathcal{B}(X_\phi)$ as \begin{equation}\label{Ba}Q_\alpha:=\frac{1}{\mu(\Omega_\alpha)}\int_{\Omega_\alpha}\overline{\chi(s)}T(s)\widehat{a}(T)\,\mathrm{d}\mu(s)\mathrm{e}nd{equation} and $$Q_{\phi,\alpha}:=\frac{1}{\mu(\Omega_\alpha)} \int_{\Omega_\alpha}\overline{\chi(s)}T_\phi(s)\widehat{a}(T_\phi)\,\mathrm{d}\mu(s).$$ Then $\pi_\phi Q_\alpha=Q_{\phi,\alpha}\pi_\phi$, from which it follows that $\|Q_{\phi,\alpha}\|\leq\|Q_\alpha\|$. In particular, $\|Q_{\phi,\alpha}\|\rightarrow0$ as $\alpha\rightarrow\infty$. Identifying $L^1(S)$ in the natural way with a subset of $L^1(G)$, it follows from Proposition~\ref{zero} applied to $T_\phi$, $X_\phi$ and $a$ that $\widehat{a}(T_\phi)=0$. In particular, $\pi_\phi(\widehat{a}(T)x)=\widehat{a}(T_\phi)\pi_\phi(x)=0$ for any $x\in X$, so the result follows from the description of $\Ker\pi_\phi$.\mathrm{e}nd{proof}
\begin{myprp}\label{lim}
Let $T$ be a bounded representation of a semigroup $S$ on a Hilbert space $X$, and let $a\in L^1(S)$. Suppose that, for some F\o lner net $(\Omega_\alpha)$ for $S$, equation~\mathrm{e}qref{erg} is satisfied for all $\chi\in\Spu(T)$. Then $\|T(s)\widehat{a}(T)\|\rightarrow0$ as $s\rightarrow\infty.$
\mathrm{e}nd{myprp}
\begin{proof}[\textsc{Proof}]
Suppose, for the sake of contradiction, that \mathrm{e}qref{erg} holds for all $\chi\in\Spu(T)$ and some F\o lner net $(\Omega_\alpha)$ but that there exist $\varepsilon>0$ and a net $(s_\beta)$ in $S$, with indexing set $A$ say, such that $s_\beta\rightarrow\infty$ as $\beta\rightarrow\infty$ and, for some suitable sequence $(y_\beta)$ of unit vectors in $X$, $\|T(s_\beta)\widehat{a}(T)y_\beta\|\geq \varepsilon$ for all $\beta\in A$. Letting $M:=\sup\{\|T(s)\|:s\in S\}$, it follows that $\|T(s)\widehat{a}(T)y_\beta\|\geq \varepsilon M^{-1}$ whenever $s_\beta-s\in S$. Fix $t\in S^\circ$ and, essentially as in \cite[Section~1.1.8]{Rudin}, let $b\in L^1(S)$ satisfy $\|b\|_1=1$ and $\|a*b-a_{t}\|_1<\varepsilon/2M^3$, where $a_{t}\in L^1(S)$ is given by $a_{t}(s)=a(s-t)$ if $s-t\in S$ and $a_{t}(s)=0$ otherwise. Now, by a construction similar those contained in \cite{HR}, \cite{NR} and \cite{RW}, there exist
\begin{enumerate}
\item[(a)] a Banach space $X_A^\infty$, which is contained in the set $\mathrm{e}ll^\infty(A;X)$ of $X$-valued nets indexed by $A$ and contains all nets of the form $(\widehat{c}(T)x_\alpha)$ with $c\in L^1(S)$ and $(x_\alpha)\in \mathrm{e}ll^\infty(A;X)$, and a bounded representation $T_A^\infty$ of $S$ on $X_A^\infty$ with $\Sp(T_A^\infty)=\Sp(T)$;
\item[(b)]a Hilbert space $X_A$, a bounded representation $T_A$ of $S$ on $X_A$ with $\Sp(T_A)\subset \Sp(T_A^\infty)$ and a surjective intertwining operator $\pi_A:X_A^\infty\rightarrow X_A$ which is contractive and such that $\|\pi_A(x_\alpha)\|$ is given, for each $(x_\alpha)\in X_A^\infty$, by the limit of the net $(\|x_\alpha\|)$ along some ultrafilter on $A$ which contains the filter generated by the sets $\{\alpha\in A:\alpha\succeq\beta\}$ with $\beta\in A$.
\mathrm{e}nd{enumerate}
Note, in particular, that $\Sp(T_A)\subset \Sp(T)$. Consider the element $(x_\beta)$ of $X_A^\infty$, where $x_\beta:=\widehat{b}(T)y_\beta$. Then, writing $c:=a*b-a_{t}$, \begin{equation*}\label{long}\begin{aligned}\|T_A(s)\widehat{a}(T_A)\pi_A(x_\beta)\|&=\|\pi_AT_A^\infty(s)\widehat{a}(T_A^\infty)(x_\beta)\|\\&=\big\|\pi_A\big(T(s)\widehat{a*b}(T)y_\beta\big)\big\|\\&\geq\big\|\pi_A\big(T(s+t)\widehat{a}(T)y_\beta\big)\big\|-\big\|\pi_A\big(T(s)\widehat{c}\,(T)y_\beta\big)\big\|\\&\geq\liminf_{\beta\rightarrow\infty}\|T(s+t)\widehat{a}(T)y_\beta\|-M^2\|c\|_1
\mathrm{e}nd{aligned}\mathrm{e}nd{equation*} for all $s\in S$, where the last line follows from the definition of the norm on $X_A$. Thus $\|T_A(s)\widehat{a}(T_A)\pi_A(x_\beta)\|\geq \varepsilon/2M$ for all $s\in S$.
Fix $\chi\in\Spu(T_A)$ and define the operators $Q_{A,\alpha}^\infty\in\mathcal{B}(X_A^\infty)$ and $Q_{A,\alpha}\in\mathcal{B}(X_A)$ as $$Q_{A,\alpha}^\infty:=\frac{1}{\mu(\Omega_\alpha)}\int_{\Omega_\alpha}\overline{\chi(s)}T_A^\infty(s)\widehat{a}(T_A^\infty)\,\mathrm{d}\mu(s)$$ and $$Q_{A,\alpha}:=\frac{1}{\mu(\Omega_\alpha)}\int_{\Omega_\alpha}\overline{\chi(s)}T_A(s)\widehat{a}(T_A)\,\mathrm{d}\mu(s).$$ Then $\pi_A Q_{A,\alpha}^\infty=Q_{A,\alpha}\pi_A$, so the properties of $\pi_A$ and the fact that $Q_{A,\alpha}^\infty$ acts on $X_A^\infty$ by entrywise application of the operator $Q_\alpha$, as defined in equation~\mathrm{e}qref{Ba}, imply that $\|Q_{A,\alpha}\|\leq\|Q_\alpha\|$. Hence $\|Q_{A,\alpha}\|\rightarrow0$ as $\alpha\rightarrow\infty$, and Proposition~\ref{pwlim} applied to $T_A$ and $X_A$ leads to the required contradiction.
\mathrm{e}nd{proof}
Corollary~\ref{cor:erg_cond} and Proposition~\ref{lim} together prove the implications (i)$\implies$(ii)$\implies$(iii) of Theorem~\ref{Thm}. The following simple lemma, which follows immediately from the definition of the spectrum of a semigroup representation $T$ along with the observation that $\widehat{a_s}(T)=T(s)\widehat{a}(T)$ for all $a\in L^1(S)$ and $s\in S$, shows that (iii)$\implies$(i), thus completing the proof of the main result.
\begin{mylem}\label{converse2}
Let $T$ be a bounded representation of a semigroup $S$ on a Banach space $X$, and let $a\in L^1(S)$. Then $|\widehat{a}(\chi)|\leq\|T(s)\widehat{a}(T)\|$ for all $\chi\in\Spu(T)$ and all $s\in S$.
\mathrm{e}nd{mylem}
\begin{rem}
There is a direct proof of the implication (iii)$\implies$(ii) in Theorem~\ref{Thm}. Indeed, if $T$ is a bounded representation of a semigroup $S$ on any Banach space $X$, if $a\in L^1(S)$ and if $(\Omega_\alpha)$ is any F\o lner net for $S$, then $$\|Q_\alpha\|\leq \sup\{\|T(s)\widehat{a}(T)\|:s\succeq t\}+ M^2\|a\|_1\frac{\mu\big(\Omega_\alpha\bigtriangleup(\Omega_\alpha+t)\big)}{\mu(\Omega_\alpha)}$$ for any $t\in S$, where $Q_\alpha$ is as in \mathrm{e}qref{Ba} and $M=\sup\{\|T(s)\|:s\in S\}$. Hence (iii)$\implies$(ii) by definition of a F\o lner net. Moreover, it is possible, at least in special cases, to show directly that (ii)$\implies$(i). When $S=\mathbb{Z}_+$, this follows from Corollary~\ref{cor:erg_cond} and the uniform ergodic theorem (see \cite[Corollary~2.3]{Leka}), and a similar argument works when $S=\mathbb{R}_+$.
\mathrm{e}nd{rem}
If one is interested in establishing only the equivalence of statements (i) and (iii) of Theorem~\ref{Thm}, there is a shorter argument which may be of independent interest. Recall the classical fact that, given a representation $T$ of a group $G$ by isometries on a Banach space $X$, one has $\widehat{a}(T)=0$ for all $a\in L^1(G)$ which are of spectral synthesis with respect to $\Sp(T)$. This is a simple consequence of the definition of the spectrum (see also \cite[Chapter~8]{Dav}, \cite[Chapter~5]{Lyu} and \cite[Lemma~2.4.3]{vNBirk}) and is used (together with constructions analogous to those used in the proof of Proposition~\ref{pwlim}) in \cite[Theorem~4.3]{BV} to derive a general form of the Katznelson-Tzafriri theorem on Banach space. Corollary~\ref{synth0} below, which is an improved version of this result when $X$ is a Hilbert space, makes it possible to obtain the implication (i)$\implies$(iii) of Theorem~\ref{Thm} by an analogous argument which bypasses Proposition~\ref{zero}. It is a special case of the following more general statement.
\begin{myprp}\label{synth}
Let $T$ be a representation of a group $G$ by unitary operators on a Hilbert space $X$, and let $a\in L^1(G)$. Then $\|\widehat{a}(T)\|=\sup\{|\widehat{a}(\chi)|:\chi\in\Sp(T)\}.$
\mathrm{e}nd{myprp}
\begin{proof}[\textsc{Proof}]Let $\mathcal{A}_T$ denote the norm closure in $\mathcal{B}(X)$ of $\{\widehat{b}(T):g\in L^1(G)\}$. Then $\mathcal{A}_T$ is a commutative $C^*$-algebra and hence, writing $\Delta(\mathcal{A}_T)$ for the character space of $\mathcal{A}_T$, the Gelfand transform $\Phi:\mathcal{A}_T\rightarrow C_0(\Delta(\mathcal{A}_T))$ is an isometric $*$-isomorphism. By \cite[Proposition~2.4]{BV}, the map sending a character $\chi\in\Sp(T)$ to the character $\xi_\chi$ on $\mathcal{A}_T$ defined, on the dense subspace $\{\widehat{b}(T):g\in L^1(G)\}$, by $\xi_\chi(\widehat{b}(T)):=\widehat{b}(\chi)$ is a bijection, and hence $\|\widehat{a}(T)\|=\|\Phi(\widehat{a}(T))\|_\infty=\sup\{|\widehat{a}(\chi)\big|:\chi\in\Sp(T)\}$.
\mathrm{e}nd{proof}
\begin{rem}
This result can also be proved using \mathrm{e}qref{spectral} with $\Lambda=\Sp(T)$.
\mathrm{e}nd{rem}
\begin{mycor}\label{synth0}
Let $T$ be a representation of a group $G$ by unitary operators on a Hilbert space $X$ with spectrum $\Lambda:=\Sp(T)$, and let $a\in L^1(G)$. Then $\widehat{a}(T)=0$ if and only if $a\in K_\Lambda$.
\mathrm{e}nd{mycor}
\begin{rem}
This follows also from Corollary~\ref{cor:erg_cond} and Proposition~\ref{zero} together with Lemma~\ref{converse2}.
\mathrm{e}nd{rem}
\section{Quantified results for contractive representations}
The purpose of this section is to study the limit of $\|T(s)\widehat{a}(T)\|$ as $s\rightarrow\infty$ in the case where $T$ is a contractive representation of a semigroup $S$ on a Hilbert space $X$ and $a$ is an element of $L^1(S)$ whose Fourier transform $\widehat{a}$ does not necessarily vanish on the unitary spectrum $\Spu(T)$ of $T$. Theorem~\ref{ub} below constitutes an important step towards this aim and can be viewed as a sharper version of \cite[Proposition~5.5]{BBG} which holds on general Banach space. It follows from the following result for individual orbits.
\begin{myprp}\label{pwub}
Let $T$ be a representation of a semigroup $S$ by contractions on a Hilbert space $X$ with unitary spectrum $\Lambda:=\Spu(T)$, and let $a\in L^1(S)$. Then$$\lim_{s\rightarrow\infty}\|T(s)\widehat{a}(T)x\|\leq \|a+K_\Lambda\|\|x\|$$for all $x\in X$.
\mathrm{e}nd{myprp}
\begin{proof}[\textsc{Proof}]
Fix any Banach limit $\phi$ on $L^\infty(S)$ and let $X_\phi$, $T_\phi$ and $\pi_\phi$ be as in the proof of Proposition~\ref{pwlim}. Then, since $T$ is contractive, $\|\pi_\phi(x)\|=\lim_{s\rightarrow\infty}\|T(s)x\|$ for all $x\in X$ and it is possible, as before, to assume that $T_\phi$ is in fact a representation of the group $G=S-S$ on $X_\phi$ by unitary operators. It follows from Corollary~\ref{synth0} that $\widehat{b}(T_\phi)=0$ for all $b\in K_\Lambda$. Hence $
\|\widehat{a}(T_\phi)\|\leq \|a-b\|_1$
for any such $b$, which implies that $\|\widehat{a}(T_\phi)\|\leq \|a+K_\Lambda(G)\|$. Thus, given any $x\in X$,
$$\begin{aligned}
\lim_{s\rightarrow\infty}\|T(s)\widehat{a}(T)x\|&=\|\pi_\phi(\widehat{a}(T)x)\|\\&=\|\widehat{a}(T_\phi)\pi_\phi(x)\|\\&\leq \|a+K_\Lambda(G)\| \|\pi_\phi(x)\|,
\mathrm{e}nd{aligned}$$and the result follows since $\pi_\phi$ is a contraction.
\mathrm{e}nd{proof}
\begin{mythm}\label{ub}
Let $T$ be a representation of $S$ by contractions on a Hilbert space $X$ with unitary spectrum $\Lambda:=\Spu(T)$, and let $a\in L^1(S)$. Then $$\lim_{s\rightarrow\infty}\|T(s)\widehat{a}(T)\|\leq \|a+K_\Lambda\|.$$
\mathrm{e}nd{mythm}
\begin{proof}[\textsc{Proof}]
Suppose not. Then there exist $\varepsilon>0$ and a net $(s_\beta)$ in $S$, with indexing set $A$, say, and satisfying $s_\beta\rightarrow\infty$ as $\beta\rightarrow\infty$, as well as a net of unit vectors $(y_\beta)$ in $X$ such that $\|T(s_\beta)\widehat{a}(T)y_\beta\|\geq \|a+K_\Lambda\|+\varepsilon$ for all $\beta\in A$. In particular, $\|T(s)\widehat{a}(T)y_\beta\|\geq \|a+K_\Lambda\|+\varepsilon$ whenever $s_\beta-s\in S$.
Let $X_A$, $X_A^\infty$ , $T_A$ and $\pi_A$ be as in the proof of Proposition~\ref{lim}, choose, for a fixed $t\in S^\circ$, $b\in L^1(S)$ such that $\|b\|_1=1$ and $\|a*b-a_{t}\|_1<\varepsilon/2$, and again define $(x_\beta)\in X_A^\infty$ by setting $x_\beta:=\widehat{b}(T)y_\beta$. It then follows from a calculation analogous to the one in the proof of Proposition~\ref{lim} that $\|T_A(s)\widehat{a}(T_A)\pi_A(x_\beta)\| > \|a+K_\Lambda\|+\varepsilon/2$ for all $s\in S$. However, applying Proposition~\ref{pwub} to the contractive representation $T_A$ of $S$ on $X_A$, and using the fact that $\|\pi_A(x_\beta)\|\leq \|\widehat{b}(T)\|\leq1,$ leads to $$\begin{aligned}\lim_{s\rightarrow\infty}\|T_A(s)\widehat{a}(T_A)\pi_A(x_\beta)\|&\leq \|a+K_{\Lambda_A}\| \|\pi_A(x_\beta)\|\leq \|a+K_{\Lambda}\|,\mathrm{e}nd{aligned}$$ where $\Lambda_A:=\Spu(T_A)\subset \Lambda$. This gives the required contradiction.
\mathrm{e}nd{proof}
The final result is a special instance Theorem~\ref{ub} which applies to representations whose unitary spectrum is a Helson set. It is a simple consequence of the definition of a Helson set and should be compared with the results in \cite[Section~5]{Zarr}, which hold on Banach space but in addition assume the unitary spectrum to be of spectral synthesis.
\begin{mycor}\label{Helub}
Let $T$ be a representation of a semigroup $S$ by contractions on a Hilbert space $X$, and let $a\in L^1(S)$. Suppose that the unitary spectrum $\Lambda:=\Spu(T)$ of $T$ is a Helson set. Then \begin{equation*}\label{UB} \sup\{|\widehat{a}(\chi)|:\chi\in \Lambda\}\leq \lim_{s\rightarrow\infty}\|T(s)\widehat{a}(T)\|\leq \alpha(\Lambda) \sup\{|\widehat{a}(\chi)|:\chi\in \Lambda\}.\mathrm{e}nd{equation*}
\mathrm{e}nd{mycor}
\begin{rem}
The first inequality holds irrespective of whether the unitary spectrum is a Helson set, and indeed of whether $X$ is a Hilbert space. It is an immediate consequence of Lemma~\ref{converse2}.
\mathrm{e}nd{rem}
\section{Local results}
This final section gives a brief account of how some of the aforementioned improvements of the Katznelson-Tzafriri theorem on Hilbert space carry over to orbitwise, or `local', versions of the result. More specifically, the aim is to obtain spectral conditions which ensure, given a bounded representation $T$ of a semigroup on a Hilbert space $X$ and an element $a$ of $L^1(S)$, that $\|T(s)\widehat{a}(T)x\|\rightarrow0$ as $s\rightarrow\infty$ for a \textsl{particular} point $x\in X$. Such results are of particular interest in the context of $C_0$-semigroups, where orbits correspond to solutions of the associated Cauchy problem, and they have been studied for instance in \cite{BvNR}, \cite[Section~4]{BY2} and \cite{KvN}.
The notion of \textsl{spectrum} that is most appropriate to this context goes back to \cite{Alb}. Consider a bounded representation $T$ of a semigroup $S$ on some Banach space $X$, and let $x\in X$ be given. A character $\chi\in \Gamma$ will be said to be \textsl{locally regular at $x$} if there exist $n\in\mathbb{N}$, $a_1,\dotsc,a_n\in L^1(S)$, a neighbourhood $\Omega$ of the point $\lambda_0:=(\widehat{a_1}(\chi),\dotsc,\widehat{a_n}(\chi))$ in $\mathbb{C}^n$ and holomorphic functions $g_1,\dotsc,g_n:\Omega\rightarrow X$ such that $\sum_{k=0}^n(\lambda_k-\widehat{a_k}(T))g_k(\lambda)=x$ for all $\lambda=(\lambda_1,\dotsc,\lambda_n)\in\Omega$. The \textsl{unitary local (Albrecht) spectrum} $\Spu(T;x)$ of $T$ at $x$ is then defined as the set of all characters $\chi\in \Gamma$ which fail to be locally regular at $x$. It is easy to see that $\Spu(T;x)\subset\Spu(T)$ for each $x\in X$. For further details on the unitary local spectrum and its relation to other natural notions of local spectrum, see \cite[Section~4]{BY2}.
The main result of this section is the following theorem, which improves \cite[Theorem~5.1]{BvNR} in the Hilbert space setting.
\begin{mythm}\label{local_KT}
Let $T$ be a bounded representation of a semigroup $S$ on a Hilbert space $X$. Furthermore, let $x\in X$ and $a\in L^1(S)$. Then $\|T(s)\widehat{a}(T)x\|\rightarrow0$ as $s\rightarrow\infty$ provided $\widehat{a}(\chi)=0$ for all $\chi\in\Spu(T;x)$.
\mathrm{e}nd{mythm}
\begin{proof}[\textsc{Proof}]
Fix a Banach limit $\phi$ on $L^\infty(S)$ and let $X_\phi$, $T_\phi$ and $\pi_\phi$ be as in the proof of Proposition~\ref{pwlim}, so that $T_\phi$ may again be assumed to be a representation of the group $G=S-S$ on $X_\phi$ by unitary operators. By \cite[Proposition~5.1]{BY2}, $\Spu(T_\phi;\pi_\phi(x))\subset\Spu(T;x)$. Moreover, writing $X_x$ for the closed linear span of the set $\{T_\phi(s)\pi_\phi(x):s\in G\}$ in $X_\phi$ and $T_x$ for the representation of $G$ on $X_x$ obtained by restricting $T_\phi$, \cite[Theorem~4.5]{BY2} gives $\Sp(T_x)=\Spu(T_\phi;\pi_\phi(x))$ and hence $\Sp(T_x)\subset\Spu(T;x)$. Thus the assumption on $a$ implies that $\widehat{a}(\chi)=0$ for all $\chi\in \Sp(T_x)$, from which it follows by Corollary~\ref{synth0} that $\widehat{a}(T_x)=0$. Hence $\pi_\phi(\widehat{a}(T)x)=\widehat{a}(T_\phi)\pi_\phi(x)=0$, which is to say $\widehat{a}(T)x\in\Ker\pi_\phi$, as required.
\mathrm{e}nd{proof}
Theorem~\ref{local_KT} in fact holds even for certain unbounded representations provided the growth of the norm is sufficiently slow and regular. Given a semigroup $S$, a measurable function $w:S\rightarrow[1,\infty)$ is said to be a \textsl{weight} if it is bounded on compact subsets of $S$ and satisfies $w(s+t)\leq w(s)w(t)$ for all $s,t\in S$. Given a weight $w$ and a representation $T$ of $S$ on a Banach space $X$ which satisfies $\|T(s)\|\leq w(s)$ for all $s\in S$, it is possible, essentially by replacing any occurrence of $L^1(S)$ with the Beurling algebra $L^1_w(S)$, to define the modified unitary local (Albrecht) spectrum $\Spuw(T;x)$ for any point $x\in X$; see \cite{BY} and \cite{Lyu} for details.
An argument entirely analogous to the proof of Theorem~\ref{local_KT}, but this time using the full strength of the results in \cite{BY2}, then leads to the following result. Here the additional regularity assumption on the weight $w$ ensures that the representation corresponding to $T_\phi$ in the above proof is again isometric; see \cite[Proposition~3.1]{BY2}. Similar non-local results may be found in \cite[Theorem~3.4]{BY2} and \cite[Theorem~8]{Vu2}.
\begin{mythm}\label{local_nqa_KT}
Let $T$ be a representation of a semigroup $S$ on a Hilbert space $X$ and suppose that $T$ is dominated by a weight $w$ such that, for every $t\in S$, $w(s)^{-1}w(s+t)\rightarrow1$ as $s\rightarrow\infty$. Furthermore, let $x\in X$ and suppose that $a\in L_w^1(S)$ is such that $\widehat{a}(\chi)=0$ for all $\chi\in \Spuw(T;x)$. Then $\|T(s)\widehat{a}(T)x\|=o(w(s))$ as $s\rightarrow\infty$.
\mathrm{e}nd{mythm}
\begin{thebibliography}{10}
\bibitem{Alb}
E.~Albrecht.
\newblock Spectral decompositions for systems of commuting operators.
\newblock {\mathrm{e}m Math. Proc. R. Ir. Acad. (Section A)}, 81:81--98, 1981.
\bibitem{AR}
G.R. Allan and T.J. Ransford.
\newblock {Power-dominated elements in a Banach algebra}.
\newblock {\mathrm{e}m Studia Math.}, 94:63--79, 1989.
\bibitem{BBG}
C.J.K. Batty, Z.~Brze\'zniak, and D.A. Greenfield.
\newblock A quantative asymptotic theorem for contraction semigroups with
countable unitary spectrum.
\newblock {\mathrm{e}m Studia Math.}, 121(2):167--183, 1996.
\bibitem{BG}
C.J.K. Batty and D.A. Greenfield.
\newblock On the invertibility of isometric semigroup representations.
\newblock {\mathrm{e}m Studia Math.}, 110(3):235--250, 1994.
\bibitem{BvNR}
C.J.K. Batty, J.M.A.M. van Neerven, and F.~R\"abiger.
\newblock {Local spectra and individual stability of uniformly bounded
$C_0$-semigroups}.
\newblock {\mathrm{e}m Trans. Amer. Math. Soc.}, 350(5):2071--2085, 1998.
\bibitem{BV}
C.J.K. Batty and Q.P. V$\tilde{\mbox{u}}$.
\newblock {Stability of strongly continuous representations of abelian
semigroups}.
\newblock {\mathrm{e}m Math. Z.}, 209:75--88, 1992.
\bibitem{BY2}
C.J.K. Batty and S.B. Yeates.
\newblock Weighted and local stability of semigroups of operators.
\newblock {\mathrm{e}m Math. Proc. Cambridge Philos. Soc.}, 129:85--98, 2000.
\bibitem{BY}
C.J.K. Batty and S.B. Yeates.
\newblock Extensions of semigroups of operators.
\newblock {\mathrm{e}m J. Operator Theory}, 46:139--157, 2001.
\bibitem{Berc2}
H.~Bercovici.
\newblock {The asymptotic behaviour of bounded semigroups in a Banach algebra}.
\newblock {\mathrm{e}m Math. Proc. Cambridge Philos. Soc.}, 108(2):365--370, 1990.
\bibitem{Berc}
H.~Bercovici.
\newblock Commuting power-bounded operators.
\newblock {\mathrm{e}m Acta Sci. Math. (Szeged)}, 57(1-4):55--64, 1993.
\bibitem{CT}
R.~Chill and Y.~Tomilov.
\newblock Stability of operator semigroups: ideas and results.
\newblock In {\mathrm{e}m Perspectives in Operator Theory}. Banach Center Publications,
Volume 75, Polish Academy of Sciences, Warsaw, 2007.
\bibitem{Dav}
E.B. Davies.
\newblock {\mathrm{e}m One-Parameter Semigroups}.
\newblock Academic Press, London, 1980.
\bibitem{Dou}
R.G. Douglas.
\newblock On extending commutative semigroups of isometries.
\newblock {\mathrm{e}m Bull. Lond. Math. Soc.}, 1:157--159, 1969.
\bibitem{ESZ2}
J.~Esterle, E.~Strouse, and F.~Zouakia.
\newblock {Theorems of Katznelson-Tzafriri type for contractions}.
\newblock {\mathrm{e}m J. Funct. Anal.}, 94:273--287, 1990.
\bibitem{ESZ}
J.~Esterle, E.~Strouse, and F.~Zouakia.
\newblock Stabilit\'e asymptotique de certaines semi-groupes d'op\'erateurs et
ideaux primaires de {$L^1(\mathbb{R}^+)$}.
\newblock {\mathrm{e}m J. Operator Theory}, 28:203--227, 1992.
\bibitem{HR}
S.-Z. Huang and F.~R\"abiger.
\newblock {Superstable $C_0$-semigroups on Banach spaces}.
\newblock In P.~Cl\'ement and G.~Lumer, editors, {\mathrm{e}m Evolution Equations,
Control Theory and Biomathematics}, volume 155 of {\mathrm{e}m Lecture Notes in Pure
and Applied Mathematics}, pages 291--300. Dekker, New York, 1994.
\bibitem{Ito}
T.~Ito.
\newblock On the commutative family of subnormal operators.
\newblock {\mathrm{e}m Journal of the Faculty of Science, Hokkaido University (Series
1)}, 14:1--15, 1958.
\bibitem{KT}
Y.~Katznelson and L.~Tzafriri.
\newblock On power bounded operators.
\newblock {\mathrm{e}m J. Funct. Anal.}, 68:313--328, 1986.
\bibitem{KvN}
L.~K{\'e}rchy and J.M.A.M. van Neerven.
\newblock Polynomially bounded operators whose spectrum on the unit circle has
measure zero.
\newblock {\mathrm{e}m Acta Sci. Math. (Szeged)}, 63:551--562, 1997.
\bibitem{Leka}
Z.~L\'eka.
\newblock {A Katznelson-Tzafriri type theorem in Hilbert spaces}.
\newblock {\mathrm{e}m Proc. Amer. Math. Soc.}, 137(11):3763--3768, 2009.
\bibitem{Lyu}
Y.I. Lyubich.
\newblock {\mathrm{e}m Introduction to the Theory of Banach Representations of Groups}.
\newblock Birkh\"auser, Basel, 1988.
\bibitem{Mus}
H.S. Mustafayev.
\newblock Asymptotic behaviour of polynomially bounded operators.
\newblock {\mathrm{e}m C. R. Math. Acad. Sci. Paris}, 348:517--520, 2010.
\bibitem{NR}
R.~Nagel and F.~R\"abiger.
\newblock Superstable operators on {Banach spaces}.
\newblock {\mathrm{e}m Israel J. Math.}, 81:213--226, 1993.
\bibitem{Ped}
G.K. Pedersen.
\newblock {\mathrm{e}m C*-Algebras and their Automorphism Groups}.
\newblock Academic Press, San Diego, 1979.
\bibitem{RW}
F.~R\"abiger and M.P.H. Wolff.
\newblock Superstable semigroups of operators.
\newblock {\mathrm{e}m Indag. Math. (N.S.)}, 6(4):481--494, 1995.
\bibitem{Rudin}
W.~Rudin.
\newblock {\mathrm{e}m Fourier Analysis on Groups}.
\newblock Wiley, New York, 1962.
\bibitem{vNBirk}
J.M.A.M. van Neerven.
\newblock {\mathrm{e}m The Asymptotic Behaviour of Semigroups of Linear Operators}.
\newblock Birkh\"auser, Basel, 1996.
\bibitem{Vu}
Q.P. V$\tilde{\mathrm{u}}$.
\newblock {Theorems of Katznelson-Tzafriri type for semigroups of operators}.
\newblock {\mathrm{e}m J. Funct. Anal.}, 103:74--84, 1992.
\bibitem{Vu2}
Q.P. V$\tilde{\mathrm{u}}$.
\newblock Semigroups with nonquasianalytic growth.
\newblock {\mathrm{e}m Studia Math.}, 103:229--241, 1993.
\bibitem{Zarr}
M.~Zarrabi.
\newblock {Some results of Katznelson-Tzafriri type}.
\newblock {\mathrm{e}m J. Math. Anal. Appl.}, 397:109--118, 2013.
\mathrm{e}nd{thebibliography}
\mathrm{e}nd{document} |
\begin{document}
\title{Stability and triviality of the transverse invariant from Khovanov homology}
\author[D. Hubbard]{Diana Hubbard}
\author[C. Lee]{Christine Ruey Shan Lee}
\address[]{Department of Mathematics, Brooklyn College, Brooklyn, NY 11210}
\email[]{diana.hubbard@brooklyn.cuny.edu}
\address[]{Department of Mathematics and Statistics, University of South Alabama, Mobile AL 36608}
\email[]{crslee@southalabama.edu}
\graphicspath{{Pics/}}
\thanks{Hubbard was partially supported by an NSF RTG grant 1045119 and an AMS-Simons travel grant during the completion of this work. Lee was partially supported by an NSF postdoctoral fellowship DMS 1502860 and an NSF grant DMS 1907010.}
\begin{abstract}
We explore properties of braids such as their fractional Dehn twist coefficients, right-veeringness, and quasipositivity, in relation to the transverse invariant from Khovanov homology defined by Plamenevskaya for their closures, which are naturally transverse links in the standard contact $3$-sphere. For any $3$-braid $\beta$, we show that the transverse invariant of its closure does not vanish whenever the fractional Dehn twist coefficient of $\beta$ is strictly greater than one. We show that Plamenevskaya's transverse invariant is stable under adding full twists on $n$ or fewer strands to any $n$-braid, and use this to detect families of braids that are not quasipositive. Motivated by the question of understanding the relationship between the smooth isotopy class of a knot and its transverse isotopy class, we also exhibit an infinite family of pretzel knots for which the transverse invariant vanishes for every transverse representative, and conclude that these knots are not quasipositive.
\end{abstract}
\maketitle
\section{Introduction}
Khovanov homology is an invariant for knots and links smoothly embedded in $\mathbb{S}^{3}$ considered up to smooth isotopy. It was defined by Khovanov in \cite{khovanov1999categorification} to be a categorification of the Jones polynomial. Khovanov homology and related theories have had numerous topological applications, including a purely combinatorial proof due to Rasmussen {\cite{rasmussen2010khovanov} of the Milnor conjecture (for other applications see for instance \cite{ng2005legendrian} and \cite{kronheimer2011khovanov})}. In this paper we will consider Khovanov homology calculated with coefficients in $\mathbb{Z}$ and reduced Khovanov homology with coefficients in $\mathbb{Z}/2\mathbb{Z}$.
Transverse links in the contact $3$-sphere are links that are everywhere transverse to the standard contact structure induced by the contact form $\xi_{st} = dz + r^{2} d \theta$. Bennequin proved in \cite{bennequin1983entrelacements} that every transverse link is transversely isotopic to the closure of some braid. Furthermore, Orevkov and Shevchisin \cite{orevkov2003markov}, and independently Wrinkle \cite{wrinkle2002markov}, showed that there is a one-to-one correspondence between transverse links (up to transverse isotopy) and braids (up to braid relations, conjugation, and positive stabilization). Hence we can study transverse links by studying braids. Plamenevskaya used this to observe in \cite{Pla06} that Khovanov homology can be used to define an invariant of transverse links. Given a braid $\beta$ whose closure $\widehat{\beta}$ is transversely isotopic to a transverse link $K$, she showed that there is a distinguished element $\widetilde{\psi}(\beta)$ in the Khovanov chain complex $CKh(\widehat{\beta})$ whose homology class $\psi(\beta)$ in the Khovanov homology of $K$ is a transverse invariant that encodes the classical self-linking number. Plamenevskaya also defined a version of this transverse invariant in reduced Khovanov homology, which we will denote as $\psi'(\beta)$, see Section \ref{subsec:redpsi}.
A transverse invariant is called \emph{effective} if it can distinguish between a pair of smoothly isotopic but not tranversely isotopic links with the same self-linking number. It is an open question whether $\psi$ is an effective transverse invariant. Several efforts \cite{lipshitz2015transverse, wu2008braids, hubbard2016annular, collari2017transverse} have been made to both understand the effectiveness of $\psi$ and to define new invariants related to $\psi$ in the hope that one of these would be effective. Thus far these efforts have not yielded any transverse invariants arising from Khovanov-type constructions that are known to be effective or not. However, $\psi$ has other applications. For instance, $\psi$ and one of its refinements provide new solutions to the word problem in the braid group \cite{baldwin2015categorified}, \cite{hubbard2016annular}.
One of the goals in this paper is to explore the following question:
\begin{que}\label{psiproperties} Given a transverse knot $K$, what properties of $K$ does $\psi(K)$ detect?
\end{que}
In \cite{plamenevskaya2015transverse}, Plamenevskaya explored Question \ref{psiproperties} for another transverse invariant, $\hat\theta$, arising from knot Floer homology \cite{ozsvath2008legendrian}, which she computed using $\mathbb{Z}/2\mathbb{Z}$ coefficients. In contrast to $\psi$, $\hat\theta$ is known to be effective \cite{ng2008transverse}. Plamenevskaya showed that given a transverse link $K$ with a braid representative $\beta$, the behavior of $\hat\theta(K)$ is related to dynamical properties of $\beta$ when $\beta$ is viewed as acting on the $n$-punctured disk $D_{n}$.
\begin{thm}\label{RVPlamenevskaya} [\cite{plamenevskaya2015transverse}, Theorem 1.2] Suppose $K$ is a transverse knot that has a $3$-braid representative $\beta$. Every braid representative of $K$ is right-veering if and only if $\hat\theta(K) \neq 0$.
\end{thm}
\begin{thm}\label{FDTCPlamenevskaya} [\cite{plamenevskaya2015transverse}, Theorem 1.3] Suppose $K$ is a transverse knot that has an n-braid representative $\beta$ with fractional Dehn twist coefficient $\tau(\beta) > 1$. Then $\hat\theta(K) \neq 0$.
\end{thm}
Informally, the fractional Dehn twist coefficient of a braid $\beta$ measures the amount of rotation $\beta$ effects on the boundary of the punctured disk $D_{n}$, see Section \ref{subsec:defnsofRVQPFDTC}. The fractional Dehn twist coefficient can be defined in general for elements in the mapping class group of any surface $\Sigma$ with a single boundary component. As all right-veering braids have fractional Dehn twist coefficient greater than or equal to $0$, see \cite{etnyre2015monoids}, Theorem \ref{FDTCPlamenevskaya} allows us to conclude that, roughly, ``most" right-veering braids have non-vanishing $\hat\theta$. Theorem \ref{FDTCPlamenevskaya} is similar in flavor to a previous result about contact structures: work of Honda, Kazez, and Mati{\'c} in \cite{honda2008right}, together with that of Ozsv{\'a}th and Szab{\'o} in \cite{ozsvath2004holomorphic} proves that a contact structure supported by an open book decomposition with connected binding where the pseudo-Anosov monodromy has fractional Dehn twist coefficient greater than or equal to one has non-vanishing Heegaard Floer twisted contact invariant.
In this paper we first consider the behavior of $\psi$ and $\psi'$ with respect to the property of being right-veering and the fractional Dehn twist coefficient, and study the extent to which the analogous statements of Theorem \ref{RVPlamenevskaya} and \ref{FDTCPlamenevskaya} hold. We note that several facts were already known relating the behavior of $\psi$ to properties of braids: that $\psi$ does not vanish for transverse links that have a quasipositive braid representative \cite{Pla06} and that it does vanish for transverse links with non-right-veering braid representatives \cite{baldwin2015categorified} and links with $n$-braid representatives that are negative stabilizations of an $(n-1)$-braid \cite{Pla06}. See Section \ref{subsec:defnsofRVQPFDTC} for more background. These properties also hold for $\psi'$.
A calculation (see Section \ref{sec:appsofgenstability}) shows that the statement corresponding to Theorem \ref{RVPlamenevskaya} is not true for $\psi$ (nor $\psi'$): there exist right-veering $3$-braids, namely the family $\Delta^{2}\sigma_{2}^{-k}$ for sufficiently large $k \in \mathbb{N}$, for which $\psi$ and $\psi'$ vanish on their closures.
However, we show that the analogue of Theorem
\ref{FDTCPlamenevskaya} holds for 3-braids:
\begin{thm}\label{3braidFDTC} Suppose $K$ is a transverse knot that has a $3$-braid representative $\beta$ with fractional Dehn twist coefficient $\tau(\beta) > 1$. Then $\psi(K) \neq 0$ (when computed over $\mathbb{Q}$, $\mathbb{Z}$, and $\mathbb{Z}/2\mathbb{Z}$ coefficients) and $\psi'(K) \neq 0$.
\end{thm}
Our second result shows a ``stability" property of $\psi$ under adding a sufficient number of negative or positive twists on any number of strands to an arbitrary braid word.
\begin{thm} \label{thm:stab} Let $L$ be any closed braid $\widehat{\beta \alpha^{\pm}}$ with $\beta$ of strand number $b$ and $\alpha^{\pm}$ of strand number $2\leq a<b$ consisting of positive/negative sub-full twists
$$\alpha^{\pm} = (\sigma_{i}^{\pm} \sigma_{i+1}^{\pm}\cdots \sigma^{\pm}_{i+a-2})^a. $$ where $1 \leq i \leq b-a+1$. Denote by $L^m_{\pm}$ the closed braid $\widehat{\beta (\alpha^{\pm})^m}$. There is some $N$ for which we have that for all $m > N$, $\psi(L_{\pm}^m) =0$ if and only if $\psi(L_{\pm}^{m+1})=0$. Similarly, $\psi'(L_{\pm}^m) = 0$ if and only if $\psi'(L_{\pm}^{m+1})=0$.
\end{thm}
The behavior of $\psi$ has several topological applications \cite{Pla06, baldwin2010khovanov}. However, in many cases $\psi$ is very difficult to compute. Theorem \ref{thm:stab} implies that, given a vanishing/non-vanishing result for $\psi$ of a braid closure, we can immediately extend it to an infinite family obtained by adding sub-full twists.
Furthermore, we have concrete bounds for $N$ based on the number of negative or positive crossings in $\beta$. Throughout the rest of the paper, we will refer to these sorts of ``full" twists on fewer than the full number of strands as ``sub-full" twists. This theorem means that after adding a large enough number of sub-full twists, the transverse invariant stabilizes. This echoes the results by \cite{CK05} which demonstrates a stability behavior of the Jones polynomial of a braid under adding full twists on any number of strands, and \cite{Sto07} which considers stability in the Khovanov homology of infinite torus braids.
Note that Theorem \ref{thm:stab} is immediate if the twists we add are full twists on $b$ strands instead of sub-full twists. Indeed, adding sufficiently many positive full twists to any braid will result in a positive braid, which has non-vanishing $\psi$. Similarly, adding sufficiently many negative full twists to any braid will result in a non-right-veering braid, which has vanishing $\psi$, and the non-right-veeringness will be preserved under adding even more negative twists.
As an application of Theorem \ref{thm:stab}, we find many examples of braids which are right-veering but not quasipositive, see Section \ref{sec:appsofgenstability}, which shows that $\psi$ may be used to detect quasipositivity. In general, it is of interest - particularly to contact geometers - to detect braids that are right-veering but not quasipositive. Indeed, this was a main theme of the work of Honda, Kazez, and Mati\'c in \cite{honda2008right}: the idea is that the difference between right-veering and quasipositive braids reflects the difference between tight and Stein fillable contact structures. These braids are often constructed by adding sub-full negative twists to right-veering braids.
For $4$-braids we have:
\begin{prop}\label{4braidexample} There exist families of $4$-braids, $\alpha_{k} = \Delta^{2}\sigma_{2}^{-k}$, $\beta_{k} = \Delta^{2}\sigma_{3}^{-k}$, and $\eta_{k} = \Delta^{2}(\sigma_{2}\sigma_{3})^{-k}$ such that for $k \in \mathbb{N}$:
\begin{enumerate}
\item $\tau(\alpha_{k}) = \tau(\beta_{k}) = \tau(\eta_{k}) = 1$.
\item $\hat\theta(\widehat{\alpha_{k}}) \neq 0$, $\hat\theta(\widehat{\beta_{k}}) \neq 0$, $\hat\theta(\widehat{\eta_{k}}) \neq 0$. (Plamenevskaya, proof of Theorem \ref{FDTCPlamenevskaya} in \cite{plamenevskaya2015transverse}).
\item For $k \geq 12$, $\alpha_{k}$, $\beta_{k}$, and $\eta_{k}$ are not quasipositive \footnote{This is due to a simple calculation of the writhe. These braids may be not quasipositive for some values of $k < 12$.}, and $\psi/\psi'(\alpha_{k}) \neq 0$, $\psi/\psi'(\beta_{k}) \neq 0$, but $\psi/\psi'(\eta_{k})=0$.
\end{enumerate}
\end{prop}
Proposition \ref{4braidexample} allows us to conclude that an infinite collection of non-quasipositive $4$-braids with fractional Dehn twist coefficient greater than one have non-vanishing $\psi$. Using functoriality allows us to conclude that any braid that has a word of the form $\Delta^{2} \sigma$ where $\sigma$ contains only positive powers of $\sigma_{1}$ and $\sigma_{2}$ but arbitrarily many negative powers of $\sigma_{3}$ has non-vanishing $\psi$. Many such braids are not quasipositive, and thus $\psi$ does not primarily detect quasipositivity. We remark that in general, it is not known whether sufficiently large fractional Dehn twist coefficient guarantees non-vanishing $\psi$. For instance, it may be that $n$ being large enough for the braid $\triangle^{2n}(\sigma_2\sigma_3)^{-k}$ guarantees $\psi(\triangle^{2n}(\sigma_2\sigma_3)^{-k}) \not=0$ regardless of $k$. The behavior of $\psi$ and $\psi'$ for the family $\eta_{k}$ allows us to conclude that $\hat\theta$ and $\psi$ can differ for braids with more than three strands.
A different perspective on Question \ref{psiproperties} is whether one can characterize \emph{smooth} link types for which \emph{every} transverse representative has vanishing $\psi$. In some sense, this question is asking about properties of smooth link types in which $\psi$ has no chance of distinguishing between distinct transverse representatives. Notice that every link type has infinitely many distinct transverse representatives, and \emph{some} transverse representative for which $\psi$ vanishes. For instance, one can always negatively stabilize a braid $\beta$ to yield $\beta'$ with $\psi(\beta')=0$, another braid representative of the link represented by $\hat{\beta}$. The transverse link $\hat{\beta'}$ is not transversely isotopic to $\hat{\beta}$ as their self-linking numbers differ by two \cite{orevkov2003markov}, \cite{wrinkle2002markov}.
One way to explore this question is by examining the relationship between the Khovanov homology of a smooth link type $L$ and its maximal self-linking number, see Definition \ref{d.msl}. This quantity is of natural interest since it provides bounds on several topological link invariants, including the slice genus, see \cite{rudolph1993quasipositivity} and \cite{Ng08}. The distinguished element $\widetilde{\psi}$ in the Khovanov chain complex of a transverse representative $\hat\beta$ of $L$ lives in homological grading $0$ and quantum grading the self-linking number of $\hat\beta$. We have the following immediate observation.
\begin{rem} \label{rem:msl0} Suppose the maximal self-linking number of $L$ is $n$. If every nontrivial homology class in homological grading $0$ of the Khovanov homology of $L$ has quantum grading strictly greater than $n$, then $\psi$ vanishes for every transverse representative of $L$.
\end{rem}
\begin{eg}\label{Ngexample} According to Proposition 4 of \cite{ng2012arc}, the mirror of the knot $11n33$, which we denote $\overline{11n33}$, has maximal self-linking number $-7$. Using the Khovanov polynomial for $\overline{11n33}$ in KnotInfo \cite{knotinfo}, we see that in homological grading $0$, the Khovanov homology of $\overline{11n33}$ is empty for $q$-grading less than $-3$. We can conclude that every transverse representative of $\overline{11n33}$ has vanishing $\psi$.
\end{eg}
We describe an infinite family of 3-tangle pretzel knots for which $\psi=0$ for every transverse representative by Remark \ref{rem:msl0}. In particular, we give conditions on the parameters of pretzel knots that guarantee that $\psi$ vanishes for every transverse representative of $L$ using the bound on the maximal self-linking number by Franks-William \cite{FW87}, Morton \cite{Mor86}, and Ng \cite{ng2005legendrian} from the HOMFLY-PT polynomial.
\begin{thm} \label{thm:homflypretzel}
Let $K=P(r, -q, -q)$ be a pretzel knot with $q>0$ odd and $r\geq 2$ even, then $\psi = 0$ for every transverse link representative of $K$.
\end{thm}
We conclude that every such pretzel knot has no quasipositive braid representatives and hence is not quasipositive, see Corollary \ref{cor:pretzelQP} and the following discussion. Preliminary computational evidence based on the braid representatives of $K$ from the program by Hilary Hunt available at \url{https://tqft.net/web/research/students/HilaryHunt/} \cite{Hilaryhunt} implementing the Yamada-Vogel algorithm suggests that the fractional Dehn twist coefficient of this family may always be less than or equal to one. This is also true for another braid representative $\sigma_1\sigma_1\sigma_1\sigma_2^{-1}\sigma_1^{-1}\sigma_1^{-1}\sigma_3\sigma_2\sigma_2\sigma^{-1}_4\sigma_3\sigma^{-1}_4$\cite{knotinfo} for $\overline{11n33}$ from Example \ref{Ngexample} which in fact has fractional Dehn twist coefficient equal to $0$. It seems possible that this is a characterization of links for which every single braid representative has $\psi=0$, and we will address this question in the future.
\subsection*{Organization} This paper is organized as follows: we give the preliminary background on Khovanov homology, reduced Khovanov homology, and the transverse invariant in Section \ref{sec:bg}, and we prove Theorem \ref{3braidFDTC} in Section \ref{sec:3-braids}. Theorem \ref{thm:stab} is proven in Section \ref{sec:genstability}, and we collect a few examples and prove Proposition \ref{4braidexample} in Section \ref{sec:appsofgenstability}. Finally, we prove Theorem \ref{thm:homflypretzel} in Section \ref{maxselflinking}, where the necessary results on the maximal self-linking number and the HOMFLY-PT polynomial are summarized.
\subsection{Acknowledgments} Both authors would like to thank John Baldwin for his computer program computing $\psi'$ available at \url{https://www2.bc.edu/john-baldwin/Programs.html} as it was very helpful. We would also like to thank Hilary Hunt for her computer program implementing the Yamada-Vogel algorithm. In addition both authors would like to acknowledge the database KnotInfo, and KnotAtlas, which makes available the KnotTheory package. The first author would also like to acknowledge helpful private correspondence with Olga Plamenevskaya and Peter Feller and several interesting discussions with Adam Saltz and John Baldwin. The second author would like to thank Matt Hogancamp for interesting conversations on the stable Khovanov homology of torus knots.
\section{Background} \label{sec:bg}
In this section we will set our conventions and briefly review Khovanov homology, the transverse invariant defined by Plamenevskaya \cite{Pla06}, and standard tools used for computing the invariant. We will also review the definitions of quasipositivity, right-veering, and the fractional Dehn twist coefficient.
\subsection{Khovanov homology} The readers may refer to \cite{Bar02}, \cite{Tur17} for excellent introductions to the subject.
Given a crossing in an oriented link diagram $D$, a Kauffman state chooses the $0$-resolution or the $1$-resolution as depicted in the following figure, which replaces the crossing by a set of two arcs.
Number the crossings of $D$ from $1,\ldots, n$. Each Kauffman state $\sigma$ on $D$ can be represented by a string of 0 and 1's in $\{0, 1\}^n$ where $0$ at the $i$th position means that the $0$-resolution is chosen at the $i$th crossing of $D$ and similarly $1$ at the $i$th position means that the $1$-resolution is chosen at the $i$th crossing.
The bi-graded chain complex $CKh(D)$ is generated by a direct sum of $\mathbb{Z}$-vector spaces associated to a Kauffman state $\sigma$.
\[CKh(D) := \bigoplus_{\sigma} CKh(D_{\sigma}), \]
where $CKh(D_{\sigma})$ is defined as follows. Let $s_{\sigma}(D)$ be the set of disjoint circles resulting from applying the Kauffman state $\sigma$ to $D$, and let $|s_{\sigma}(D)|$ be the number of circles. Then
\[CKh(D_{\sigma}) := V^{\otimes|s_{\sigma}(D)|}, \]
where $V$ is the free graded $\mathbb{Z}$-module generated by two elements $v_-$ and $v_+$ with grading $p$ such that $p(v_{\pm}) = \pm 1$. The grading is extended to the tensor product by the rule $p(v\otimes v') = p(v) + p(v')$.
Let $r(\sigma)$ be the number of 1's in the string in $\{0, 1\}^n$ representing a Kauffman state $\sigma$. Two gradings $i, j$ are defined on $CKh(D)$ as follows. The \emph{homological grading} $i$ is defined by $i(v) = r(\sigma)- n_-(D)$, where $\sigma$ is the state giving rise to the vector space $V^{\otimes |s_{\sigma}(D)| }$ containing $v$, while the \emph{quantum grading} $j$ is defined by $j(v) = p(v) + i(v) + n_+(D)-n_-(D)$, where $n_{+}(D)$ and $n_{-}(D)$ are the number of positive and negative crossings in $D$, respectively.
We shall indicate the $\mathbb{Z}$-vector space with bi-grading $(i,j)$ in $CKh(D)$ as $CKh^i_j(D)$.
For the differential of the chain complex, we first define a map $d_c$ on $CKh(D)$ from $\sigma$ to $\sigma_c$:
\[d_c: V^{\otimes|s_{\sigma}(D)|} \rightarrow V^{\otimes|s_{\sigma_c}(D)|}, \]
where $\sigma$ and $\sigma_c$ differ in their resolution at exactly one crossing $c$ where $\sigma$ chooses the $0$-resolution and $\sigma_c$ chooses the $1$-resolution. From $\sigma$ to $\sigma_c$, either two circles merge into one or a circle splits into two. In the first case, the map $d_c$ contracts $V\otimes V$, representing the pair of circles in $s_{\sigma}(D)$, to $V$, representing the resulting circle in $s_{\sigma_c}(D)$ by the merging map $m$ as defined below.
\begin{align*}
&m(v_+\otimes v_+) = v_+ \\
&m(v_+\otimes v_-)= m(v_-\otimes v_+) = v_- \\
&m(v_-\otimes v_-) = 0.
\end{align*}
In the other case where a circle splits into two, $d_c$ is given by the splitting map $\triangle$ taking $V\rightarrow V\otimes V$ as follows.
\begin{align*}
&\triangle(v_+) = v_+\otimes v_- + v_-\otimes v_+ \\
&\triangle(v_-) = v_-\otimes v_-.
\end{align*}
Now on $CKh^i_{*}(D_{\sigma})$ the differential $d$ is defined by
\[d = \sum_{c \text{ \ crossing in $D$ on which $\sigma$ chooses the 0-resolution}} (-1)^{\text{sgn}(\sigma, \sigma_c)}d_c.\]
We extend $d$ by linearity. The sign $\text{sgn}(\sigma, \sigma_c)$ is chosen so that $d\circ d = 0$; for instance one can choose that $\text{sgn}(\sigma, \sigma_c)$ is the number of 1's in the string representing $\sigma$ before $c$. The resulting homology groups $Kh(D)$ are independent of this choice.
Khovanov \cite{khovanov1999categorification} defined and showed that $Kh(D)$ is independent of the diagram chosen for the link $L$, so $Kh(L) = Kh(D)$ is a link invariant.
\subsubsection{Reduced Khovanov homology with $\mathbb{Z}/2\mathbb{Z}$-coefficients} \label{ss.reduced}
Given an oriented link diagram $D$ with a marked point on a link strand, consider the Khovanov complex $CKh(D)$. A circle in a state $s_{\sigma}(D)$ is marked if it contains the marked point. We denote the marked circle of a state $\sigma$ by $\sigma(m)$. Consider the sub-complex
\[CKh(D, -):= \bigoplus_{\sigma} V_1 \otimes \cdots \otimes V_{\sigma(m)-1} \otimes v_- \otimes V_{\sigma(m)+1}\cdots \otimes V_{|s_\sigma(D)|}, \] where $v_-$ is the element in the vector space assigned to $\sigma(m)$, and let $\overline{CKh}(D)$ be the quotient complex $CKh(D)/CKh(D, -)$. Reduced Khovanov homology is generated by $\overline{CKh}(D)$ with the differential $d$ of $CKh(D)$ descending to the differential $d'$ on the quotient complex. When the vector space $CKh(D)$ is generated as a direct sum of $\mathbb{Z}/2\mathbb{Z}$-vector spaces, $\overline{Kh}(D)$ is an invariant of the link represented by $L$, independent of the placement of the marked point \cite{Kho03}. Thus, we will denote by $\overline{Kh}(L)$ the reduced Khovanov homology of $L$ with $\mathbb{Z}/2\mathbb{Z}$-coefficients. We will also adopt the convention of shifting the quantum grading by -1, see for example \cite{baldwin2010khovanov} so that the homology of the unknot is at grading $i=0, j=0$ in this theory.
\subsection{The transverse element} Here we follow the conventions of Plamenevskaya \cite{Pla06} except for a minor change in notation. In her paper the bi-grading is indicated as $Kh_{i, j}$, whereas in this paper the homological grading is placed on top as $Kh_j^i$.
Let $\beta$ be a braid representative of a link $L$ giving a closed braid diagram $\hat{\beta}$ of $L$. Consider the \emph{oriented resolution}, the Kauffman state $\sigma_{\beta}$ of $\hat{\beta}$ where we take the 0-resolution for each positive crossing and the 1-resolution for each negative crossing.
\begin{defn} The \emph{transverse invariant} of a closed braid representative $\widehat{\beta}$ of $L$, denoted by $\psi(\hat{\beta})$, is the homology class in $Kh(L)$ of the following element in the vector space associated to $\sigma_{\beta}$:
\[ \widetilde{\psi}(\beta) := v_- \otimes v_- \otimes \cdots \otimes v_- \in V^{\otimes|s_{\sigma_{\beta}}(\hat{\beta})|}=CKh(\hat{\beta}_{\sigma_{\beta}}).\]
\end{defn}
Plamenevskaya has shown that this is, up to sign, a well-defined homology class in $Kh(L)$ \cite[Proposition 1]{Pla06} under transverse link isotopy. Thus $\psi(\hat{\beta})$ is a transverse link invariant which lies in $ Kh^0_{\text{sl}(\hat{\beta})}(L)$, where $\text{sl}(\hat{\beta})$ is the \emph{self-linking number} of a transverse link represented by a closed braid $\hat{\beta}$ defined as follows.
\begin{defn}\label{defn:selflinking} The \emph{self-linking number} of the transverse link $\hat{\beta}$ is given by
\[\text{sl}(\hat{\beta})= -b + n_+(\hat{\beta})-n_{-}(\hat{\beta}).\]
Note $b$ is the strand number of $\beta$.
\end{defn}
To simplify the notation, we will omit the hat $`` \ \hat{} \ "$ in $\psi(\hat{\beta})$ and simply write $\psi(\beta)$.
\subsubsection{The reduced version}\label{subsec:redpsi}
In the reduced setting, the transverse invariant of a closed braid representative $\hat{\beta}$ of $L$, denoted by $\psi'(\beta)$, is the homology class (up to sign) in $\overline{Kh}(L)$ of the following element in the vector space associated to $\sigma_{\beta}$
\[ \widetilde{\psi}'(\beta) := v_- \otimes \cdots \otimes v_+ \otimes \cdots \otimes v_- \in V^{\otimes|s_{\sigma_{\beta}}(\hat{\beta})|}=\overline{CKh}(\hat{\beta}_{\sigma_{\beta}}), \] where $v_+$ corresponds to the element in the vector space $V$ associated to the marked circle of $\sigma_{\beta}$. Note that $\widetilde{\psi}'(\beta)$ lives in quantum grading $\text{sl}(\hat{\beta}) + 1$.
\subsection{Functoriality and properties of $\psi$.}
Using the map on Khovanov homology induced by a cobordism between a pair of links, Plamenevskaya proved the following useful result for computing $\psi$.
\begin{thm}{\cite[Theorem 4]{Pla06}} Suppose that the transverse link $\hat{\beta}^-$ represented by the closure of the braid $\beta^-$ is obtained from another transverse link $\hat{\beta}$, also represented by a closed braid, by resolving a positive crossing (note that it has to be the 0-resolution). Let $S$ be the resolution cobordism, and $f_S: Kh(\hat{\beta}) \rightarrow Kh(\hat{\beta}^-)$ be the associated map on homology, then
\[f_S(\psi(\beta)) = \pm \psi(\beta^-). \]
\end{thm}
A consequence of this is that if $\psi(\beta) =0$ then $\psi(\beta^-)=0$. Similarly, suppose that $\hat{\beta}^+$ is obtained from $\hat{\beta}$ by resolving a negative crossing, then $\psi(\beta) \not=0$ implies that $\psi(\beta^+) \not=0$. When we use this property in our computations, we will often cite it as ``functoriality". Furthermore, this property holds for both the unreduced and reduced versions of the transverse element in the corresponding versions of Khovanov homology.
\subsection{Skein exact sequence}
Let $D$ be a link diagram and let $D_0$ and $D_1$ be link diagrams differing locally in the 0-resolution and the 1-resolution, respectively, at a negative crossing $c$ of $D$.
Let $D_1$ inherit the orientation from $D$ and let $u = n_-(D_0) - n_-(D)$ be the difference in the number of negative crossings of the two diagrams, where we pick an orientation on $D_0$. Consider the short exact sequence given by the following maps.
\[\alpha: CKh^{i}_{j+1}(D_1) \rightarrow CKh_j^i(D) \text{ and } \gamma: CKh^i_j(D) \rightarrow CKh^{i-u}_{j-3u-1}(D_0), \] where $\alpha$ is induced by inclusion, and $\gamma$ is induced by the quotient map.
We have the induced long exact sequence below \cite{watson2007knots}, also called the ``skein exact sequence." See \cite{Ras05} for an alternate formulation using the oriented skein relation for the Jones polynomial.
\begin{equation} \label{eq:ssesn} \cdots \rightarrow Kh^{i}_{j+1}(D_1) \stackrel{\alpha}{\rightarrow} Kh^i_{j}(D) \stackrel{\gamma}{\rightarrow} Kh^{i-u}_{j-3u-1}(D_0) \rightarrow Kh^{i+1}_{j+1}(D_1) \rightarrow \cdots. \end{equation}
For a chosen positive crossing $c$ and with $u = n_-(D_1) - n_-(D)$, we have instead
\begin{equation} \label{eq:ssesp} \cdots \rightarrow Kh^{i-u-1}_{j-3u-2}(D_1) \stackrel{\alpha}{\rightarrow} Kh^i_{j}(D) \stackrel{\gamma}{\rightarrow} Kh^{i}_{j-1}(D_0) \rightarrow Kh^{i-u}_{j-3u-2}(D_1) \rightarrow \cdots. \end{equation}
These grading shifts can be understood by first considering the shifts of the maps in the exact sequences \emph{before} incorporating the final shifts in $i$ of $-n_{-}$ and in $j$ of $n_{+}-2n_{-}$, and then incorporating those final shifts carefully, keeping in mind that the number of positive and negative crossings in the different diagrams is not the same. Note that the same long exact sequence will hold for reduced Khovanov homology, and over different coefficients.
\subsection{Quasipositivity, right-veeringness, and the fractional Dehn twist coefficient}\label{subsec:defnsofRVQPFDTC}
\begin{defn} A \emph{quasipositive} $n$-braid is a braid that can be expressed as a product of conjugates of the standard positive Artin generators $\sigma_{1}, \ldots, \sigma_{n-1}$.
\end{defn}
A link is called quasipositive if it is the closure of a quasipositive braid. One reason that quasipositive knots are of interest is that their slice genus can be computed from one of their quasipositive braid representatives \cite{rudolph1993quasipositivity}. While there are obstructions to quasipositivity, there are no known algorithms for determining whether a given link is quasipositive or not.
We now define the concept of a ``right-veering" braid. To do this we consider the action of the $n$-braid monodromy on the disk $D_{n}$ with $n$ punctures. (Recall that the braid group $B_{n}$ is naturally isomorphic to the mapping class group of $D_{n}$, see for instance \cite{birman_brendle_braids}.) We call an arc in $D_{n}$ starting at a point on $\partial D_{n}$ and ending at a puncture while avoiding all other punctures simply an ``arc on $D_{n}$". We say that an arc $\eta$ is ``to the right" of an arc $\gamma$ in $D_{n}$ if, after pulling tight to eliminate non-essential intersections, $\eta$ and $\gamma$ originate from the same point on $\partial D_{n}$, and the pair of tangent vectors $(\dot{\eta}, \dot{\gamma})$ at their initial point induces the original orientation on the disk. See \cite{baldwin2015categorified} and \cite{plamenevskaya2015transverse} for more details. With this terminology in place, we have:
\begin{defn} An $n$-braid is \emph{right-veering} if, under the action of the braid, every arc on $D_{n}$ is sent either to an arc isotopic to itself or to an arc to the right of itself.
\end{defn}
All quasipositive braids are right-veering, but not all right-veering braids are quasipositive. Recall from the introduction that detecting braids that are right-veering but not quasipositive is generally of interest - see \cite{honda2008right}.
Finally, we discuss the fractional Dehn twist coefficient, which we will often abbreviate from here on out as the FDTC. If $h$ is any element of the mapping class group of a surface with one boundary component, we denote its FDTC by $\tau(h)$. It roughly measures the amount of twisting effected by the mapping class about the boundary component of the surface. The concept first appeared (though in quite different language) in the work of Gabai and Oertel in \cite{gabai_oertel}. We give here a non-classical definition for braids involving a left order on the braid group as it requires little background. There are several other more geometrically flavored definitions that generalize easily beyond braids to more general mapping class groups. For instance, one way to define the FDTC involves using lifts of the braid to the universal cover of the punctured disk to define a map $\Theta: B_{n} \to \widetilde{Homeo^{+}(S^{1})}$; the FDTC is then defined to be the translation number of $\Theta$. For a more thorough discussion of this and alternate definitions, see \cite{malyutin2004writhe}, \cite{ito_kawamuro_FDTC}, and \cite{plamenevskaya2015transverse}.
First, a $\sigma_{i}$-positive $n$-braid is one that, for some $i$ such that $1 \leq i < n$, can be written with no $\sigma_{j}^{\pm 1}$'s for $j < i$ and only positive powers of $\sigma_{i}$. We say that a braid $\beta \in B_{n}$ is Dehornoy positive, that is, $\beta > 1$, if it can be written as a $\sigma_{i}$-positive word. Dehornoy proved in \cite{dehornoy} that this can be used to define a total left-order on the braid group (an order on all of the elements of the braid group that is invariant by multiplication on the left) via the following: we say $\alpha < \beta$ if $\alpha^{-1}\beta > 1$. This order is often called the Dehornoy order on the braid group.
The element $(\sigma_{1} \cdots \sigma_{n-1})^{n}$ is referred to as the full twist in the braid group $B_{n}$, and is denoted by $\Delta^{2}$. The existence of the Dehornoy order on the braid group implies that for every braid $\beta \in B_{n}$, there is a unique integer $m$ such that $\Delta^{2m} \leq \beta < \Delta^{2m+2}$. We denote $m$ as $\lfloor \beta \rfloor$. Malyutin observed in \cite{malyutin2004writhe} that:
\begin{defn}
The fractional Dehn twist coefficient is, for each $\beta \in B_{n}$: $$\tau(\beta) = \displaystyle\lim_{k \to \infty} \frac{\lfloor \beta^{k} \rfloor}{k}.$$
\end{defn}
The following proposition summarizes some basic properties of the FDTC:
\begin{prop}\label{FDTCprops} [See for instance \cite{hedden_mark}.] For any two braids $\alpha$, $\beta$ in $B_{n}$, we have:
\begin{itemize}
\item $|\tau(\alpha\beta) - \tau(\alpha) - \tau(\beta)| \leq 1$.
\item $\tau(\alpha^{-1}\beta\alpha) = \tau(\beta)$.
\item $\tau(\beta^{n}) = n\tau(\beta)$.
\item $\tau(\Delta^{2}\beta) = 1 + \tau(\beta)$.
\end{itemize}
\end{prop}
We also have the following result due to Malyutin.
\begin{prop}\label{FDTCestimate} [Malyutin, \cite{malyutin2004writhe}, Lemma 5.4 and Proposition 13.1] If a braid $\beta \in B_{n}$ is represented by a word that contains $r$ occurrences of $\sigma_{i}$ and $s$ occurrences of $\sigma_{i}^{-1}$ for some $i \in \{1, \ldots, n-1\}$, then $-s \leq \tau(\beta) \leq r$. In particular, if a braid word $\beta \in B_{n}$ is $\sigma_{i}$-free (meaning: it contains no $\sigma_{i}$ or $\sigma_{i}^{-1}$) for some $i \in \{1,\ldots,n-1\}$, then $\tau(\beta)$ is $0$.
\end{prop}
This follows immediately:
\begin{prop}\label{examples} Suppose an $n$-braid $\beta_{n,k}$ has a word of the form $\Delta^{2n}(\alpha)^{k}$ for $\alpha$ any $\sigma_{i}$-free word, $n$ and $k$ integers. Then $\tau(\beta_{n,k})$ is $n$.
\end{prop}
\section{3-braids} \label{sec:3-braids}
We first determine $\psi, \psi'$ for closed 3-braids of the form $\widehat{\triangle^2\sigma_1\sigma_2^{-k}}$ for $k>0$. The reader may skip to Section \ref{subsec:pRVP} to see how these braids come up in the proof of Theorem \ref{3braidFDTC}.
\subsection{The transverse element for the braid $\widehat{\triangle^2\sigma_1\sigma_2^{-k}}$}
\begin{thm}\label{FDTCfamily} Working in the $3$-braid setting, for $k \in \mathbb{Z}$, $k \geq 0$,
$$\psi(\Delta^{2}\sigma_{1}\sigma_{2}^{-k}) \neq 0$$
when computed over $\mathbb{Q}$, $\mathbb{Z}$, and $\mathbb{Z}/2\mathbb{Z}$ coefficients, and
$$\psi'(\Delta^{2}\sigma_{1}\sigma_{2}^{-k}) \neq 0.$$
\end{thm}
Recall that $\psi'$ is the reduced version of the transverse invariant briefly defined in subsection \ref{ss.reduced}. For the definitions, details, and background of many of the notions used in this proof, see the references cited.
\begin{proof}
Notice first that for $k$ odd, the $3$-braid $\Delta^{2}\sigma_{1}\sigma_{2}^{-k}$ closes to a knot rather than to a link. In \cite{baldwin2008heegaard}, Baldwin showed that the family of $3$-braids
$$\Delta^{2d}\sigma_{1}\sigma_{2}^{-a_{1}}\sigma_{1}\sigma_{2}^{-a_{2}}\cdots\sigma_{1}\sigma_{2}^{-a_{n}}$$
where the $a_{i} \geq 0$ and some $a_{j} \neq 0$ is quasi-alternating if and only if $d \in \{-1,0,1\}$. By work of Manolescu and Ozsv\'ath in \cite{manolescu2007khovanov}, quasi-alternating links are Khovanov homologically $\sigma$-thin. This means that the reduced Khovanov homology over $\mathbb{Z}$ takes a particularly simple form: supported on only one diagonal $j-2i$ grading where $j-2i = \sigma$ the signature of the link \footnote{There are two convention discrepancies between this definition and the cited paper; we are using here the conventions that seem to now be in most common use. In \cite{manolescu2007khovanov}, they consider the grading $j'-i$ instead of $j-2i$ where $j'=\frac{j}{2}$. Their theorem as stated in the paper is that the reduced Khovanov homology over $\mathbb{Z}$ is supported only in grading $j'-i = -\frac{\sigma}{2}$ for quasi-alternating links, or in our grading notation, $j-2i = -\sigma$. The sign discrepancy is explained by the fact that they take the opposite sign convention for the signature as we do: we take as our convention that positive knots have positive signature.}\footnote{We note that in this proof, the symbol $\sigma$ is used to denote only the signature of a link and should not be confused with Kauffman states or braid generators.}.
As a consequence of the long exact sequence established by Asaeda and Przytycki in \cite{asaeda_przytycki}, Lowrance observed that \cite[Corollary 2.3]{lowrance_khovanov_width} reduced Khovanov homology over $\mathbb{Z}$ has support in grading $j-2i = \sigma$ if and only if Khovanov homology over $\mathbb{Z}$ has support in gradings $j-2i = \sigma+1$ and $j-2i = \sigma-1$. This implies that the Khovanov homology over $\mathbb{Q}$ also only has support in the same gradings. Recall also that Rasmussen's s-invariant \cite{rasmussen2010khovanov} is defined to be the maximum $j$-grading minus one (inherited from the Khovanov complex over $\mathbb{Q}$) of the element in the Lee complex that contributes to Lee homology \cite{lee_khovanov}. Lee homology for knots is particularly simple, and is only supported in $i$-grading $0$ \cite{lee_khovanov}. Hence for Khovanov $\sigma$-thin links, Rasmussen's s-invariant is defined to be the signature $\sigma$.
Next, notice that for these knots, $\text{sl} = \sigma - 1$ \cite[Remark 7.6]{baldwin2010khovanov}. Thus $\text{sl} = s-1$. Then work of Baldwin and Plamenevskaya \cite[Theorem 1.2]{baldwin2010khovanov} \footnote{This theorem as stated is for reduced Khovanov homology over $\mathbb{Z}/2\mathbb{Z}$ coefficients. However, notice that the proof also explicitly covers the case for Khovanov homology over $\mathbb{Q}$ coefficients and $\mathbb{Z}/2\mathbb{Z}$ coefficients.} implies both that $\psi' \neq 0$ and that $\psi \neq 0$ in Khovanov homology over $\mathbb{Z}/2\mathbb{Z}$ coefficients and $\mathbb{Q}$. This last fact implies that $\psi \neq 0$ in Khovanov homology over $\mathbb{Z}$ coefficients as well.
Finally, suppose that $k$ is even. Then $\psi(\Delta^{2}\sigma_{1}\sigma_{2}^{-k-1}) \neq 0$ and $\psi'(\Delta^{2}\sigma_{1}\sigma_{2}^{-k-1}) \neq 0$ by what we just proved. By functoriality, $\psi(\Delta^{2}\sigma_{1}\sigma_{2}^{-k}) \neq 0$ and $\psi'(\Delta^{2}\sigma_{1}\sigma_{2}^{-k}) \neq 0$.
\end{proof}
\subsection{Proof of Theorem \ref{3braidFDTC}} \label{subsec:pRVP} We restate the theorem for convenience:
\begin{restate2} Suppose $K$ is a transverse knot that has a $3$-braid representative $\beta$ with fractional Dehn twist coefficient $\tau(\beta) > 1$. Then $\psi(K) \neq 0$ when computed over $\mathbb{Q}$, $\mathbb{Z}$, and $\mathbb{Z}/2\mathbb{Z}$ coefficients, and $\psi'(K) \neq 0$.
\end{restate2}
\begin{proof}
We will write this proof only for $\psi$; it is identical for $\psi'$. According to Murasugi's classification of 3-braids \cite{murasugi1974closed}, every $\sigma \in B_{3}$ comes in the following types up to conjugation:
\begin{enumerate}[a)]
\item $\Delta^{2d}\sigma_{1}\sigma_{2}^{-a_{1}}\sigma_{1}\sigma_{2}^{-a_{2}}\cdots\sigma_{1}\sigma_{2}^{-a_{n}}$ where $a_{i} \geq 0$ for all $i$ and some $a_{i} > 0$,
\item $\Delta^{2d}\sigma_{2}^{m}$ where $m \in \mathbb{Z}$, and
\item $\Delta^{2d}\sigma_{1}^{m}\sigma_{2}^{-1}$ where $m=-1, -2, -3$,
\end{enumerate}
where $d$ can take on any integer value.
Recall that $\psi$ is invariant under conjugation, so whichever of these conjugacy classes $\sigma$ belongs to determines $\psi(\sigma)$. The FDTC is also invariant under conjugation, and hence whichever of these conjugacy classes $\sigma$ belongs to determines its FDTC.
All braids in classes (a) and (b) have FDTC $d$, since $$\tau(\sigma_{2}^{-a_{1}}\sigma_{1}\sigma_{2}^{-a_{2}}\cdots\sigma_{1}\sigma_{2}^{-a_{n}}) = 0$$ and $\tau(\sigma_{2}^{m}) = 0$ by Proposition \ref{FDTCestimate}, and for any braid $\beta$, $\tau(\Delta^{2}\beta) = 1 + \tau(\beta)$. All braids in class (c) have FDTC less than or equal to $d$, since by Proposition \ref{FDTCestimate}, $\tau(\sigma_{1}^{m}\sigma_{2}^{-1}) \leq 0$ for negative values of $m$.
Hence for each of the classes, we need only consider $d > 1$. Since by Theorem \ref{FDTCfamily} we know that the model braid $\psi(\Delta^{2}\sigma_{1}\sigma_{2}^{-k}) \neq 0$ for all positive $k$ then every other braid in (a) and (b) with $d > 1$ has $\psi \neq 0$ by functoriality. Indeed: by making $k$ possibly quite large, we can achieve every other braid in (a) and (b) with $d > 1$ by inserting positive crossings.
Finally, a straightforward manipulation of the braid words yields that the braids in (c) with $d > 1$ are all quasipositive. Hence for the braids in (c) with $d > 1$, $\psi \neq 0$ (\cite{Pla06}).
\end{proof}
\section{General stability}\label{sec:genstability}
We prove Theorem \ref{thm:stab} in this section. Note in general that we have the following bounds on $Kh^i_j$. If $D$ is a diagram of a link $K$, then $Kh^i_j(D) = 0$ if $i$ or $j$ are outside of the following bounds:
\begin{align*}
-n_-(D) &\leq i \leq n_+(D) \\
n_+(D)-2n_-(D)-|s_0(D)| &\leq j \leq |s_1(D)| + 2n_+(D) - n_-(D).
\end{align*}
\subsection{Negative sub-full twists }
Let $\beta$ be a braid of strand number $b$ and let $2\leq a < b$. Let $k$ be a positive integer, and write $k = (a-1) \ell + r$, so $r = k \mod (a-1)$. We consider the closed braid $D^k$ obtained by adding to $\beta$ the following braid
\[ \alpha'_-= (\sigma_{i}^{-1}\sigma_{i+1}^{-1}\cdots\sigma_{i+a-2}^{-1})^{\ell}(\sigma_{i}^{-1}\sigma_{i+1}^{-1} \cdots \sigma_{i+r-1}^{-1})\] of strand number $a$, with $1\leq i \leq b-a+1$, and then taking the closure.
\begin{figure}
\caption{\label{fig:schema}
\label{fig:schema}
\end{figure}
We denote by $D_0^k$ and $D_1^k$ the link diagrams obtained by taking the 0-resolution and the 1-resolution, respectively, at the crossing $\sigma_{i+r-1}^{-1}$.
\begin{figure}
\caption{\label{fig:count}
\label{fig:count}
\end{figure}
\begin{lem} \label{l.exceed}
Let $n_+'(D_0^k)$ be the number of positive crossings of $D_0^k$ in the subset $\alpha'_-$.
Then
\begin{equation} \label{eq:est} n_+'(D_0^k) - n_+'(\widetilde{D_0^k}) \geq \ell, \end{equation}
where $\widetilde{D_0^k}$ is the diagram isotopic to $D_0^k$, obtained by isotoping the cap through the rest of the braid $\alpha_-'$ resulting from choosing the $0$-resolution at the crossing $\sigma^{-1}_{i+r-1}$ in $D^k$ to get $D_0^k$. See Figure \ref{fig:count} for an example.
\end{lem}
\begin{proof}
Denote the strands of the braid $D^k$ by $S_1, \ldots, S_b$. We follow the isotopy of the cap resulting from choosing the $0$-resolution at the crossing $\sigma_{i+r-1}^{-1}$ through the $\ell$ copies of $(\sigma_{i}^{-1}\sigma_{i+1}^{-1}\cdots \sigma_{i+a-2}^{-1})$ as shown above in Figure \ref{fig:count} in an example where $r=1$. Initially, the cap joins the strands $S_{i}$ and $S_{i+r}$. Regardless of the choice of orientation on $D_0^k$, we end up decreasing the number of positive crossings by one for each set of $(\sigma_{i}^{-1}\sigma_{i+1}^{-1} \cdots \sigma_{i+a-2}^{-1})$ in $\alpha'$ through the isotopy. See also Figure \ref{f.orientation}.
\begin{figure}
\caption{\label{f.orientation}
\label{f.orientation}
\end{figure}
\end{proof}
We use the long exact sequence in Khovanov homology at a distinguished negative crossing \eqref{eq:ssesn}. Let $u = n_-(D_0^k) - n_-(D^k)$, the long exact sequence takes the form:
\[\cdots \rightarrow Kh^{i-1-u}_{j-3u-1}(D_0^k) \rightarrow Kh^{i}_{j+1}(D^k_1)\rightarrow Kh^i_j(D^k) \rightarrow Kh^{i-u}_{j-3u-1}(D_0^k) \rightarrow Kh^{i+1}_{j+1}(D_1^k) \rightarrow \cdots \]
Then we show that with $k$ large enough, $Kh^{-1-u}_*(D_0^k)= 0$ and $Kh^{-u}_*(D_0^k)=0$ by showing
$-u -1> n_+(\widetilde{D_0^k})$. This implies
\[Kh^0_{j+1}(D_1^k) \stackrel{\alpha}{\cong} Kh^0_{j}(D^k), \] with
\[ \alpha(\widetilde{\psi}(D_1^k)) = \widetilde{\psi}(D^k) .\]
\begin{lem} \label{thm:ltrivial} Given $D^k = \widehat{\beta \alpha'_-}$, there exists a fixed number $N> 0 $ such that for all $k\geq N$, the homology groups
\[Kh^{-1-u}_*(D_0^k) \text{ and } Kh^{-u}_*(D_0^k) \] are both trivial.
\end{lem}
\begin{proof}
We need to show
\begin{align*}
n_-(D^k) - n_-(D_0^k)-1 &> n_+(\widetilde{D_0^k}).
\intertext{Let $n'_-(D^k)$ be the number of negative crossings of $D^k$ in the subset $\alpha'_-$ and $n^{\beta}_{\pm}(D^k) = n_{\pm}(D_k) - n'_{\pm}(D^k)$ be the number of positive/negative crossings of $D^k$ in the subset $\beta$. We rewrite the inequality as}
n'_-(D^k) - n'_-(D_0^k) + n^{\beta}_-(D^k) - n^{\beta}_-(D_0^k)-1 &> n'_+(\widetilde{D_0^k}) + n_+^{\beta}(\widetilde{D_0^k}). \\
\intertext{The following inequality obtained from rewriting $n_-'(D^k) - n_-'(D_0^k) = n'_+(D_0^k)+1$ and using Lemma \ref{l.exceed} implies the desired inequality above.}
n'_+(D_0^k) + n^{\beta}_-(D^k) - n^{\beta}_-(D_0^k) &> n'_+(D_0^k)- \ell + n_+^{\beta}(D^k) + n^{\beta}_-(D^k) - n^{\beta}_-(D_0^k).\\
\intertext{This simplifies to}
0 &> - \ell + n_+^{\beta}(D^k).
\end{align*}
We can certainly make the last inequality true by making $k$ large enough so that $\ell > n_+(D^k)$, since $n_+^{\beta}(D^k)$ is constant.
\end{proof}
\subsection{Proof of Theorem \ref{thm:stab} for adding negative sub-full twists}
\begin{proof}
Let $L^m_{\pm} = \widehat{\beta (\alpha^{\pm})^m}$ as in the statement of the theorem.
Choose large enough $m$ so that
\[ \psi(L^{m}_-) = \psi(\beta(\alpha^-)^m\sigma_i^{-1}) =\psi(\beta(\alpha^-)^m\sigma_i^{-1} \sigma_{i+1}^{-1}) = \cdots = \psi(\beta(\alpha^-)^m\sigma_i^{-1}\sigma_{i+1}^{-1}\cdots \sigma^{-1}_{i+a-2}) = \psi(L^{m+1}_-)\]
by Lemma \ref{thm:ltrivial}. The conclusion of the theorem follows.
\end{proof}
\subsection{Positive sub-full twists}
\noindent This proof is analogous to the one for negative sub-full twists; the primary difference is that we use the long exact sequence at a distinguished positive crossing \eqref{eq:ssesp} with $u = n_-(D_1^k) - n_-(D^k)$, and we show that $-u < -n_{-}(\widetilde{D_{1}^{k}})$ instead of the bound on the homological degree on the other side as the isotopy that simplifies $D_{1}^{k}$ to $\widetilde{D_{1}^{k}}$ will reduce the number of negative crossings. We use the same notation as before for indicating the positive/negative crossings in different regions of the braid.
We consider the closed braid $D^k$ obtained from adding to $\beta$ the following braid
\[ \alpha'_+ = (\sigma_{i}\sigma_{i+1}\cdots\sigma_{i+a-2})^{\ell}(\sigma_{i}\cdots \sigma_{i+r-1}), \] of strand number $2\leq a< b$, with $k = (a-1) \ell+r$, and then taking the closure. Let $D_1^k$ be the link diagram obtained by choosing the 1-resolution at the crossing $\sigma_{i+r-1}$. We obtain the analogous statement $n_-'(D_1^k) - n_-'(\widetilde{D^k_1}) \geq \ell$
to Lemma \ref{l.exceed} by replacing $n'_+(D^k_0)$ with $n'_-(D^k_1)$ and $n_+(\widetilde{D^k_0})$ with $n_-(\widetilde{D^k_1})$. The argument is similar except that the cap from choosing the $1$-resolution at $\sigma_{i+r-1}$ is now over the other braid strands. The inequality follows that
$$ n_-(D^k) - n_-(D_1^k)< -n_-(\widetilde{D_1^k}), $$ whenever $\ell > n_-(D^k)$.
\subsection{Stability for the reduced version}
Note that we have the same bounds for reduced Khovanov homology with $\mathbb{Z}/2\mathbb{Z}$ coefficients on the homological grading: $\overline{Kh}^{i}_{j}(D) = 0$ if $i$ does not satisfy $-n_-(D)\leq i \leq n_+(D)$. Thus the same proof as above goes through to show stability for $\psi'$ under adding positive/negative sub-full twists using the long exact sequence for the reduced version.
\section{Applications and examples}\label{sec:appsofgenstability}
In this section we apply the collection of tools we now have to determine the behavior of $\psi$ and $\psi'$ for a few families of closed braids, and draw conclusions about their quasipositivity and right-veeringness. For $\psi'$, we use Baldwin's program together with the stability behavior of $\psi'$ proved in Section \ref{sec:genstability}; for $\psi$, we will use by-hand computation with stability. For instance, one can determine the behavior of $\psi'$ and $\psi$ for the $3$-braid family from Theorem \ref{FDTCfamily} in this way, as it is possible to check that $\psi'$ does not vanish for $\Delta^{2}\sigma_{1}\sigma_{2}^{-8}$ using Baldwin's program, and $\psi\not=0$ by hand. In cases where the bound in Section \ref{sec:genstability} would require checking an example with too many crossings for Baldwin's program to handle, it is sometimes still possible to use the same general approach to get more precise information, as we do in subsection \ref{subsec:4braidex} for $\psi'$ for a family of 4-braids.
\subsection{A collection of examples}\label{subsec:appsofgenstability}
The first four columns of Table \ref{table:qprv} denote the number of strands of the braid, the word template for the braid family that we consider, the behavior of $\psi$ and $\psi'$ for these braids that we are able to determine\footnote{While we have no examples where the behaviors of $\psi$ and $\psi'$ differ, we know of no mathematical reason why their behaviors should \emph{always} match.}, and the methods used to obtain these results: ``Prog." stands for Baldwin's program for $\psi'$ and ``Comp." stands for a by-hand computation for $\psi$. Wherever we claim that $\psi$ dies due to a by-hand computation, we provide the element that kills it in subsection \ref{subsec:killpsi}. The fifth column gives the writhe of the braid, and the sixth and seventh columns determine whether the braid is quasipositive and/or right-veering, if possible, along with the method used. We have:
\begin{itemize}
\item Braid families that are right-veering but not quasipositive (the first six).
\item Braid families that are not quasipositive and have positive writhes (the last three).
\end{itemize}
\begin{table}[H]
\begin{center}
\resizebox{1\textwidth}{!}{
\begin{tabular}{||c | c | c | p{22mm} | c | p{22mm} | p{19mm}||}
\hline
$n$ & Braid in $B_{n}$ & $\psi, \psi'$ & Method & Writhe & Quasipositive & Right-veering \\
\hline\hline
3 & $\Delta^{2}\sigma_{2}^{-k}, k > 4$ & $\psi, \psi'=0$ & Prog,./Comp., functoriality & $6-k$ & No, $k > 6$, writhe & Yes, FDTC \\
\hline
3 & $\Delta^{2}\sigma_{1}\sigma_{2}^{-k}, k \in \mathbb{N}$ & $\psi, \psi' \neq 0$ & See Thm \ref{FDTCfamily} & $7-k$ & No, $k > 7$, writhe & Yes, $\psi/\psi'$ or FDTC \\
\hline
4 & $\Delta^{2}\sigma_{2}^{-k}, k \in \mathbb{N}$ & $\psi, \psi' \neq 0$ & Example above, functoriality & $12-k$ & No, $k > 12$, writhe & Yes, $\psi/\psi'$ or FDTC \\
\hline
4 & $\Delta^{2}\sigma_{3}^{-k}, k \in \mathbb{N}$ & $\psi' \neq 0$ & See subsection \ref{subsec:4braidex} & $12-k$ & No, $k > 12$, writhe & Yes, $\psi'$ or FDTC \\
\hline
4 & $\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{3}\sigma_{2}\sigma_{1}\sigma_{3}^{-k}, k > 2$ & $\psi, \psi' =0$ & Prog./Comp., functoriality & $6-k$ & No, $k > 6$, writhe & Yes, FDTC \\
\hline
4 & $\Delta^{2}(\sigma_{2}\sigma_{3})^{-k}, k > 5$ & $\psi, \psi' =0$ & Example above\footnotemark, functoriality & $12-2k$ & No, $k > 6$, writhe & Yes, FDTC \\
\hline
4 & $(\sigma_{1})^2\sigma_{2}^{-1}\sigma_{3}\sigma_{2}^{-1}\sigma_{1}^{-1}\sigma_{2}(\sigma_{3})^2(\sigma_{2}\sigma_{3})^{k}, k \in \mathbb{N}$ & $\psi, \psi' =0$ & Prog./Comp., stability & $3+2k$ & No, all $k$, $\psi/\psi'$ & ? \\
\hline
5 & $\sigma_{1}\sigma_{2}^{-1}\sigma_{3}\sigma_{4}^{-1}\sigma_{2}^{-1}\sigma_{1}^{-1}(\sigma_{2})^{2}\sigma_{3}(\sigma_{4})^2(\sigma_{2}\sigma_{3})^{k}, k \in \mathbb{N}$ & $\psi, \psi' =0$ & Prog./Comp., stability & $3+2k$ & No, all $k$, $\psi/\psi'$ & ? \\
\hline
6 & $\sigma_{4}\sigma_{1}\sigma_{2}\sigma_{4}\sigma_{5}^{-1}\sigma_{4}^{-1}\sigma_{3}\sigma_{5}\sigma_{1}^{-1}\sigma_{2}(\sigma_{2}\sigma_{3})^{k}, k \in \mathbb{N}$ & $\psi, \psi' =0$ & Prog./Comp., stability & $4+2k$ & No, all $k$, $\psi/\psi'$ & ? \\
\hline
\end{tabular}
}
\end{center}
\caption{Calculations of $\psi$ and $\psi'$ for various braids on $n$ strands, together with their calculation method as well as other properties of these braids.}\label{table:qprv}
\end{table}
\vspace*{3mm}
\footnotetext{{Using the fact that $\Delta^{2}(\sigma_{2}\sigma_{3})^{-3} = \sigma_{1}\sigma_{2}\sigma_{3}\sigma_{3}\sigma_{2}\sigma_{1}$}.}
Recall that it is of interest to detect braids that are right-veering but not quasipositive - see subsection \ref{subsec:defnsofRVQPFDTC}. Baldwin and Grigsby proved in \cite{baldwin2015categorified} that if a braid is not right-veering, then it has vanishing $\psi$. Their proof would apply just as well to $\psi'$. Hence if a braid has non-vanishing $\psi$ or $\psi'$, it is guaranteed to be right-veering. Notice also that if one has a braid with negative writhe, then it cannot be quasipositive. Thus $\psi$ or $\psi'$ together with the writhe can be used to detect braids that are right-veering but not quasipositive, as is done in the second through fourth examples in the table. However, it is also possible for $\psi$ and $\psi'$ to vanish for braids that are right-veering but not quasipositive, as can be seen in the first, fifth, and sixth examples in the table. In these cases we were able to determine that the FDTCs for these braid families were greater than or equal to one, which implies that these braids are indeed right-veering \cite{honda2008right}.
In the case where a braid has positive writhe, $\psi$ or $\psi'$ can be of use to detect non-quasipositivity. Indeed, Plamenevskaya proved in \cite{Pla06} that if a braid is quasipositive, then it has non-vanishing $\psi$, and her proof applies equally well to $\psi'$. The last three examples in the table have arbitrarily large writhes but also have vanishing $\psi$ and $\psi'$, and hence are not quasipositive. We chose these examples as it is not obvious by simply manipulating the braid words that they are not quasipositive; there should be many more such examples.
\subsection{Justification for $\psi=0$ in Table \ref{table:qprv}}\label{subsec:killpsi}
For each braid $\beta$ in Table \ref{table:qprv} where we state that ``comp." is the method for showing $\psi(\beta) = 0$, we justify the claim by giving an explicit element that can be directly verified to kill $\psi$, that is, we give an element $\Phi \in CKh^{-1}_{\text{sl}(\hat{\beta})} = CKh^{-1}_{\text{sl}(\hat{\beta})}(\hat{\beta})$ such that $d(\Phi) = \psi$. These braids are (in order of their appearance from top to bottom in Table \ref{table:qprv}):
\begin{enumerate}
\item $\triangle^2 \sigma_2^{-k}$, $k>4$
\item $\sigma_1\sigma_2\sigma_3\sigma_3\sigma_2\sigma_1\sigma_3^{-k}$, $k>2$
\item $(\sigma_1)^2\sigma_2^{-1}\sigma_3\sigma_2^{-1}\sigma_1^{-1}\sigma_2(\sigma_3)^2(\sigma_2\sigma_3)^k$, $k\in \mathbb{N}$
\item $\sigma_1\sigma_2^{-1}\sigma_3\sigma_4^{-1}\sigma_2^{-1}\sigma_1^{-1} (\sigma_2)^2\sigma_3(\sigma_4)^2(\sigma_2\sigma_3)^k, k\in \mathbb{N}$
\item $\sigma_4\sigma_1\sigma_2\sigma_4\sigma_5^{-1}\sigma_4^{-1}\sigma_3\sigma_5\sigma_1^{-1}\sigma_2(\sigma_2\sigma_3)^k, k \in \mathbb{N}$
\end{enumerate}
To represent an element in $CKh^{-1}_{\text{sl}(\hat{\beta})}$ that kills $\psi$, we represent a generator in $CKh$ by first giving its Kauffman state by decorating the braid word $\beta$ with dots. A dot on top of an Artin generator indicates that the $1$-resolution is chosen at the corresponding crossing. Without the dot, the $0$-resolution is chosen. Then, we number the state circles of the Kauffman state and indicate between parentheses which circle is marked with a +, corresponding to $v_+\in V$, in the grading $i=-1$, $j = \text{sl}(\hat{\beta})$. Each circle in the state not indicated between parentheses is marked with a $-$, corresponding to $v_-\in V$. If there are two numbers in the parentheses then the corresponding circles are both marked with a $+$. In the figures, a segment between state circles indicates where the crossing is before the Kauffman state is applied. Thickened (or blue) segments indicate that the $1$-resolution is chosen, and thin (or red) segments indicate that the $0$-resolution is chosen. See the following figure for an example of how to interpret a dotted braid word and how to read off the generator from a Kauffman state on the closed braid where circles are labeled.
\begin{figure}
\caption{On the left: the closure of a 4-braid. On the right: a generator in a Kauffman state on the closed 4-braid. The tensor product $V^{\otimes 5}
\end{figure}
\begin{enumerate}
\item $\triangle^2 \sigma_2^{-k}$, $k>4$.
By the braid relations, we get $\triangle^2\sigma_2^{-5} = \sigma_1\sigma_2\sigma_2\sigma_1\sigma_2^{-3}$. If we show $\psi = 0$ for the closure of $\sigma_1\sigma_2\sigma_2\sigma_1\sigma_2^{-3}$, then functoriality would show that $\psi = 0$ for $\widehat{\triangle^2\sigma_2^{-k}}$ for all $k>5$. We claim
\begin{align*} \label{3psi1}
&\psi(\sigma_1\sigma_2\sigma_2\sigma_1\sigma_2^{-3})\\
&= d(-\dot{\sigma_1}\sigma_2\sigma_2\dot{\sigma_1}\sigma_2^{-3}(4) + \sigma_1\dot{\sigma_2}\sigma_2\dot{\sigma_1}\sigma_2^{-3}(4) - \sigma_1\dot{\sigma_2}\sigma_2\sigma_1\sigma_2^{-1}\dot{\sigma_2}^{-1}\sigma_2^{-1}(1) + \sigma_1\sigma_2\dot{\sigma_2}\dot{\sigma_1}\sigma_2^{-3}(4) \\ \notag
&+ \sigma_1\dot{\sigma_2}\sigma_2\sigma_1\sigma_2^{-1}\dot{\sigma_2}^{-1}\sigma_2^{-1}(4)-\sigma_1\dot{\sigma_2}\sigma_2\dot{\sigma_1}\sigma_2^{-3}(3)-\sigma_1\sigma_2\dot{\sigma}_2\sigma_1\sigma_2^{-1}\dot{\sigma}_2^{-1}\sigma_2^{-1}(1) + \sigma_1\sigma_2\dot{\sigma}_2\sigma_1\sigma_2^{-1}\dot{\sigma}_2^{-1}\sigma_2^{-1}(4)\\ \notag
&-\sigma_1\sigma_2\dot{\sigma_2}\dot{\sigma_1}\sigma_2^{-3}(3)-\sigma_1\dot{\sigma_2}\dot{\sigma_2}\sigma_1\sigma_2^{-3}(45)+ \sigma_1\dot{\sigma_2}\dot{\sigma_2}\sigma_1\sigma_2^{-3}(35)+ \sigma_1\sigma_2\sigma_2\sigma_1\sigma_2^{-1}\dot{\sigma_2}^{-1}\dot{\sigma_2}^{-1}),
\end{align*}
where $\psi$ is the image of $d$ on the element indicated on the right hand side of the equation. See Figure \ref{f.3-braid} for the labeling of circles of the states in $CKh^{-1}_{\text{sl}(\widehat{\sigma_1\sigma_2\sigma_2\sigma_1\sigma_2^{-3})})}$.
\begin{figure}
\caption{\label{f.3-braid}
\label{f.3-braid}
\end{figure}
\item $\sigma_1\sigma_2\sigma_3\sigma_3\sigma_2\sigma_1\sigma_3^{-k}$, $k>2$. Also by functoriality, it suffices to show $\psi$ vanishes for the closure of $\sigma_1\sigma_2\sigma_3\sigma_3\sigma_2\sigma_1\sigma_3^{-3}$. We claim
\begin{align*}
&\psi(\sigma_1\sigma_2\sigma_3\sigma_3\sigma_2\sigma_1\sigma_3^{-3})\\
&= d(\sigma_1\sigma_2\sigma_3\sigma_3\sigma_2\sigma_1 \sigma_3^{-1}\dot{\sigma}_3^{-1}\dot{\sigma}_3^{-1}-\sigma_1\dot{\sigma}_2\sigma_3\sigma_3\dot{\sigma}_2\sigma_1 \sigma_3^{-3}(4)+\sigma_1\sigma_2\dot{\sigma}_3\sigma_3\dot{\sigma}_2\sigma_1 \sigma_3^{-3}(4) \\
&-\sigma_1\sigma_2\dot{\sigma}_3\sigma_3\dot{\sigma}_2\sigma_1 \sigma_3^{-3}(5)+\sigma_1\sigma_2\sigma_3\dot{\sigma}_3\dot{\sigma}_2\sigma_1 \sigma_3^{-3}(4)-\sigma_1\sigma_2\sigma_3\dot{\sigma}_3\dot{\sigma}_2\sigma_1 \sigma_3^{-3}(5)-\sigma_1\sigma_2\dot{\sigma}_3\sigma_3\sigma_2\sigma_1 \dot{\sigma}_3^{-1}\sigma_3^{-2}(3)\\
&-\sigma_1\sigma_2\dot{\sigma}_3\sigma_3\sigma_2\sigma_1 \sigma_3^{-1}\dot{\sigma}_3^{-1}\sigma_3^{-1}(2)-\sigma_1\sigma_2\sigma_3\dot{\sigma}_3\sigma_2\sigma_1 \sigma_3^{-1}\dot{\sigma}_3^{-1}\sigma_3^{-1}(2)-\sigma_1\sigma_2\sigma_3\dot{\sigma}_3\sigma_2\sigma_1\dot{\sigma}_3^{-1}\sigma_3^{-2}(3)\\
&-\sigma_1\sigma_2\dot{\sigma}_3\dot{\sigma}_3\sigma_2\sigma_1 \sigma_3^{-3}(45)).
\end{align*}
See Figure \ref{f.4braid1} for the labeling of circles of the states in $CKh^{-1}_{\text{sl}(\widehat{\sigma_1\sigma_2\sigma_3\sigma_3\sigma_2\sigma_1\sigma_3^{-3}})}$.
\begin{figure}
\caption{\label{f.4braid1}
\label{f.4braid1}
\end{figure}
\item $(\sigma_1)^2\sigma_2^{-1}\sigma_3\sigma_2^{-1}\sigma_1^{-1}\sigma_2(\sigma_3)^2(\sigma_2\sigma_3)^k$, $k\in \mathbb{N}$. We claim for all $k\in \mathbb{N}$,
\begin{align*}
&\psi((\sigma_1)^2\sigma_2^{-1}\sigma_3\sigma_2^{-1}\sigma_1^{-1}\sigma_2(\sigma_3)^2(\sigma_2\sigma_3)^k) \\
&= d((\sigma_1)^2\dot{\sigma}_2^{-1}\sigma_3\dot{\sigma}_2^{-1}\sigma_1^{-1}\sigma_2(\sigma_3)^2(\sigma_2\sigma_3)^k-\sigma_1\dot{\sigma}_1\dot{\sigma}_2^{-1}\sigma_3\sigma_2^{-1}\sigma_1^{-1}\sigma_2(\sigma_3)^2(\sigma_2\sigma_3)^k \\
&-\dot{\sigma}_1\sigma_1\dot{\sigma}_2^{-1}\sigma_3\sigma_2^{-1}\sigma_1^{-1}\sigma_2(\sigma_3)^2(\sigma_2\sigma_3)^k).
\end{align*}
All the circles of all these states are marked with a $-$. Note that the same elements would map to $\psi$ regardless of $k$.
\item $\sigma_1\sigma_2^{-1}\sigma_3\sigma_4^{-1}\sigma_2^{-1}\sigma_1^{-1} (\sigma_2)^2\sigma_3(\sigma_4)^2(\sigma_2\sigma_3)^k, k\in \mathbb{N}$. We claim for all $k\in \mathbb{N}$,
\begin{align*}
&\psi(\sigma_1\sigma_2^{-1}\sigma_3\sigma_4^{-1}\sigma_2^{-1}\sigma_1^{-1} (\sigma_2)^2\sigma_3(\sigma_4)^2(\sigma_2\sigma_3)^k)\\
&= d(\sigma_1\dot{\sigma}_2^{-1}\sigma_3\dot{\sigma_4}^{-1}\dot{\sigma}_2^{-1}\sigma_1^{-1}\sigma_2^2\sigma_3\sigma_4^2(\sigma_2\sigma_3)^k-\dot{\sigma}_1\dot{\sigma}_2^{-1}\sigma_3\dot{\sigma}_4^{-1}\sigma_2^{-1}\sigma_1^{-1}\sigma_2^2\sigma_3\sigma_4^2(\sigma_2\sigma_3)^k).
\end{align*}
All the circles of all these states are marked with a $-$.
\item $\sigma_4\sigma_1\sigma_2\sigma_4\sigma_5^{-1}\sigma_4^{-1}\sigma_3\sigma_5\sigma_1^{-1}\sigma_2(\sigma_2\sigma_3)^k, k \in \mathbb{N}$. We claim for all $k\in \mathbb{N}$,
\begin{align*}
&\psi(\sigma_4\sigma_1\sigma_2\sigma_4\sigma_5^{-1}\sigma_4^{-1}\sigma_3\sigma_5\sigma_1^{-1}\sigma_2(\sigma_2\sigma_3)^k) \\
&= d(\sigma_4\sigma_1\sigma_2\sigma_4\sigma_5^{-1}\dot{\sigma_4}^{-1}\sigma_3\sigma_5\dot{\sigma}_1^{-1}\sigma_2(\sigma_2\sigma_3)^k+\sigma_4\sigma_1\sigma_2\sigma_4\sigma_5^{-1}\sigma_4^{-1}\sigma_3\dot{\sigma}_5\dot{\sigma}_1^{-1}\sigma_2(\sigma_2\sigma_3)^k).
\end{align*}
All the circles of all these states are marked with a $-$.
\end{enumerate}
\subsection{A four-braid example} \label{subsec:4braidex}
Proposition \ref{4braidexample} follows from applying functoriality and stability to $\alpha_k = \triangle^2 \sigma_2^{-k}$ and $\eta_k = \triangle^2(\sigma_2\sigma_3)^{-k}$ in Table \ref{table:qprv} and the following theorem.
\begin{thm}\label{4braid} The $4$-braid family
$$\beta_{k} = \Delta^{2}\sigma_{3}^{-k}$$
where $k \in \mathbb{N}$ satisfies
$$\psi'(\beta_{k}) \neq 0.$$
\end{thm}
Notice that for $k > 12$, $\beta_{k}$ is not quasipositive since the writhe is $12 - k < 0$,
so Theorem \ref{4braid} gives an infinite family of non-quasipositive braids with non-vanishing $\psi'$.
\begin{proof}
First, using Baldwin's computer program, we determine that
$$\psi'(\beta_{9})\neq 0.$$
By functoriality this guarantees that $\psi'(\beta_{k})\neq 0$ for all $1 \leq k < 9$. We will induct on $k$ for all $k>9$.
Similar to the notation introduced in Section \ref{sec:genstability}, let $D_{0}^{k}$ and $D_{1}^{k}$ be the knots or links that are obtained by replacing the last crossing of $\beta_{k}$ with its 0- and 1-resolutions, respectively, and taking the closure, where $D_1^{k} = \widehat{\beta_{k-1}}$. We orient $D_{1}^{k}$ with the same orientation as $\widehat{\beta_{k-1}}$ (all strands oriented downwards). We orient $D_{0}^{k}$ so that all three outer strands are oriented downwards above the braid word. We first observe that for all $k > 1$, $D_{0}^{k}$ is isotopic to the disjoint union of the unknot, oriented counter-clockwise, and the Hopf link $\sigma_{1}^{2}$. So a quick computation shows that the reduced Khovanov homology over $\mathbb{Z}/2\mathbb{Z}$ of $D_{0}^{k}$ is as shown in Table \ref{table:Kh}.
\begin{table}
\begin{center}
\begin{tabular}{||c | c | c | c||}
\hline
$j$ & $i=0$ & $i=2$ \\ [0.5ex]
\hline\hline
0 & $\mathbb{Z}/2\mathbb{Z}$ & \\
\hline
2 & $\mathbb{Z}/2\mathbb{Z}$ & \\
\hline
4 & & $\mathbb{Z}/2\mathbb{Z}$ \\
\hline
6 & & $\mathbb{Z}/2\mathbb{Z}$ \\[1ex]
\hline
\end{tabular}
\caption{The reduced Khovanov homology over $\mathbb{Z}/2\mathbb{Z}$ of $D_{0}^{k}$.}\label{table:Kh}
\end{center}
\end{table}
In addition, the number of negative crossings in $D_{0}^{k}$ is $6$ for all $k > 1$. We are interested
in $\overline{Kh}(\widehat{\beta_{k}})$ in homological grading $0$ and $q$-grading the self-linking number of $\beta_{k}$ plus one, so $i = 0$ and $j = -4+12-k+1 = 9-k$. The long exact sequence \eqref{eq:ssesn} for reduced Khovanov homology corresponding to taking the resolution of the last negative crossing in the word then takes the following form:
$$ \cdots \longrightarrow \overline{Kh}^{k-7}_{2k-10}(D_{0}^{k}) \longrightarrow \overline{Kh}^{0}_{10-k}(D_{1}^{k}) \longrightarrow \overline{Kh}^{0}_{9-k}(\widehat{\beta_{k}}) \longrightarrow \overline{Kh}^{k-6}_{2k-10}(D_{0}^{k}) \longrightarrow \cdots,$$
where the $u$ from \eqref{eq:ssesn} is $u = n_{-}(D_{0}^{k}) - n_{-}(\widehat{\beta_{k}}) = 6 - k$.
For $k\geq 10$, both $k-7, k-6>2$. Using the information on the reduced Khovanov homology over $\mathbb{Z}/2\mathbb{Z}$ of $D_0^k$, the long exact sequence becomes
$$ \cdots \longrightarrow 0 \longrightarrow \overline{Kh}^{0}_{10-k}(D_{1}^{k}) \longrightarrow \overline{Kh}^{0}_{9-k}(\widehat{\beta_{10}}) \longrightarrow 0 \longrightarrow \cdots$$
The map on chain complexes yields an isomorphism
$$\overline{Kh}^0_{10-k}(\widehat{\beta_{k-1}}) \cong \overline{Kh}^{0}_{10-k}(D_{1}^{k}) \\\stackrel{\cong}{\longrightarrow} \overline{Kh}^{0}_{9-k}(\widehat{\beta_{k}}).$$ This isomorphism is induced by the map that naturally sends $\widetilde{\psi}'(\beta_{k-1})$ to $\widetilde{\psi}'(\beta_{k})$. Hence since $\psi'(\beta_9) \in \overline{Kh}^{0}_{0}(\widehat{\beta_{9}})$ is non-zero as computed earlier, the isomorphism implies that $\psi'(\beta_k) \in \overline{Kh}^{0}_{{9-k}}(\widehat{\beta_{k}})$ is non-zero for all $k\geq 10$.
\end{proof}
\section{Bennequin-type inequalities and the maximum self-linking number}\label{maxselflinking}
In this section we prove Theorem \ref{thm:homflypretzel}. We will first give the necessary background on the maximal self-linking number and recall some results which bound the maximal self-linking number using the HOMFLY-PT polynomial of the link. We follow the conventions of the Knot Atlas \cite{kat} for the HOMFLY-PT polynomial.
Recall the self-linking number $\text{sl}(\overline{L})$ of a transverse link $\overline{L}$ as defined in Definition \ref{defn:selflinking}.
\begin{defn} \label{d.msl} The \emph{maximal self-linking number}, $\overline{\text{sl}}(L)$ of a smooth link $L$ is the maximum of $\text{sl} (\overline{L})$ taken over all transverse link representatives $\overline{L}$ of $L$.
\end{defn}
Let $P_L(a, z)$ be the HOMFLY-PT polynomial of a smooth link $L$, normalized so that $P=1$ for the unknot and defined by the following skein relation.
\begin{equation} \label{eq:homfly}
\vcenter{\hbox{
\def.5\columnwidth{.5\columnwidth}
\input{homfly.pdf_tex} }}
\end{equation}
The pictures $\vcenter{\hbox{\def .5\columnwidth{.03\columnwidth} \input{Pics/s1.pdf_tex}}}$, $\vcenter{\hbox{\def .5\columnwidth{.03\columnwidth} \input{Pics/s2.pdf_tex}}}$, and $\vcenter{\hbox{\def .5\columnwidth{.03\columnwidth} \input{Pics/s0.pdf_tex}}}$ indicate smooth links $L_+, L_-$, and $L_0$ where $L_+$ and $L_-$ differ by switching a crossing, and $L_0$ is the link resulting from choosing the oriented resolution at the crossing.
By \cite{FW87} and \cite{Mor86}, we have the following inequality.
\begin{thm}{(\cite{FW87}, \cite{Mor86})} \label{thm:mslbhomfly}
\[\overline{\text{sl}}(L) \leq -\deg_a(P_L(a, z))-1, \]
where $\deg_a(P_L(a, z))$ is the maximum degree in $a$ of $P_L(a, z)$.
\end{thm} Ng also provides a skein-theoretic proof that unifies several similiar inequalities in \cite{Ng08}. The transverse element $\widetilde{\psi}(\beta)$ for a braid representative $\beta$ of $L$ is always supported in the grading $i=0$ and $j=\text{sl}(\hat{\beta})$ in $Kh(L)$ \cite[Proposition 2]{Pla06}. Recall that by Remark \ref{rem:msl0}, this means that if $Kh^0_j(L) = 0$ for all $j \leq \overline{\text{sl}}(L)$, then $\psi(\beta) = 0$ for every braid representative $\beta$ of $L$.
Consider the 3-tangle pretzel knots $K=P(r, -s, -t)$ where $r>0$ is even and $s, t> 0$ are odd. Our convention is illustrated in Figure \ref{fig:pconvention} below.
\begin{figure}
\caption{\label{fig:pconvention}
\label{fig:pconvention}
\end{figure}
Since $K$ is a negative knot by the standard pretzel diagram $D$, which means that all the crossings are negative in $D$ with an orientation, there is a single state, the all-1 state, which chooses the $1$-resolution on all the crossings of $D$ and gives the generators for the chain complex at homological grading $i=0$. Recall $|s_1(D)|$ is the number of state circles in the all-$1$ state. Since $K$ is a negative knot, the state graph $s_1(D)$ has no one-edged loops, so $K$ is adequate on one side by definition. It is known, see for example the proof of \cite[Proposition 5.1]{Kho03}, that this implies that in $i=0$, there are only two possible nontrivial homology groups at $j=|s_1(D)|-n_-(D)$ and $j = |s_1(D)|-n_-(D)-2$. It is also possible to see this for these pretzel knots by direct computation. Note that Manion gives an explicit characterization of the Khovanov homology of 3-tangle pretzel knots in \cite{Man14}.
Now
\[|s_1(D)| = r+1. \]
Thus
\[ |s_1(D)|-n_-(D)= 1-s-t, \]
and $Kh(K)$ can only have nontrivial homology groups for $i=0$ at $j = 1-s-t, 1-s-t-2$.
Before proving Theorem \ref{thm:homflypretzel}, it is helpful to see an example in the $P(2, -5, -5)$ pretzel knot.
\begin{eg} \label{eg:n255}
The pretzel knot $K=P(2, -5, -5)$.
$Kh(K)$ has nontrivial homology groups supported in $i=0, j=-11$ and $i=0, j=-9$, and trivial homology groups for all other $j$ when $i=0$.
It has HOMFLY-PT polynomial \cite{kat}
\begin{align*} P_K(a, z) &= 10 a^{10} - 13 a^{12} + 4 a^{14} + 39 a^{10} z^{2} - 32 a^{12} z^2 + 4 a^{14} z^2 +
57 a^{10} z^4 - 27 a^{12} z^4 + a^{14} z^4 \\
&+36 a^{10} z^6 - 9 a^{12} z^6 + 10 a^{10} z^8 - a^{12} z^8 + a^{10} z^{10}. \end{align*}
Using Theorem \ref{thm:mslbhomfly} gives that $\overline{\text{sl}}(P(2, -5, -5)) \leq -(14)-1 < -11.$ Therefore, $\psi=0$ for every braid representative of $P(2, -5, -5)$.
\end{eg}
We generalize the above examples using the computation for the HOMFLY-PT polynomial for torus knots by Jones \cite{Jon87}. For our purpose it is enough to have the following lemma which we prove here by inducting on the defining skein relation.
\begin{lem} \label{lem:toruspqhomfly} Let $T_{2, -q}$ denote the negative $2q$ torus link with all negative crossings. If $q>1$ is odd, then
\[\deg_a(P_{T_{2, -q}}(a, z)) = q+1, \] with all negative coefficients.
If $q>1$ is even, then with the orientation given that makes all the crossings negative,
\[\deg_a(P_{T_{2, -q}}(a, z)) = q+1, \] with all positive coefficients.
\end{lem}
\begin{proof} We give a proof here by induction. The base cases are $q=2$ and $q=3$. We see respectively \cite{kat} that $P_{T_{2, -2}}(a, z)$ has all positive coefficients with the term of the highest $a$-degree given by $+\frac{a^3}{z}$.
Similarly, $P_{T_{2, -3}}(a, z)$ has all negative coefficients with the term of the highest $a$-degree: $-a^4$.
For $P_{T_{2, -q}}(a, z)$, where $q>3$, we expand a single crossing by \eqref{eq:homfly}. This gives that
\begin{equation} \label{eq:thomfly}
P_{T_{2, -q}} = a^2 P_{T_{2, -(q-2)}} - azP_{T_{2, -(q-1)}}.
\end{equation}
Assuming the induction hypothesis, we have $\deg_a P_{T_{2, -(q-2)}} (a, z) = q-1$ with all positive/negative coefficients for even/odd $q-2$. Similarly, we have
$\deg_a P_{T_{2, -(q-1)}}(a, z) = q$ with all negative/positive leading coefficients for odd/even $q-1$. Plugging this into \eqref{eq:homfly} gives that there is no cancellation between the terms with the maximal $a$-degree $q+1$, and the coefficients are either all positive when $q$ is even, or all negative when $q$ is odd.
\end{proof}
Now we show Theorem \ref{thm:homflypretzel}, which we reprint here for reference.
\begin{restate}
Let $K=P(r, -q, -q)$ be a pretzel knot with $q>0$ odd and $r\geq 2$ even, then $\psi = 0$ for every transverse link representative of $K$.
\end{restate}
\begin{proof}
We apply relation \eqref{eq:homfly} to the top left negative crossing of $P(r, -q, -q)$. Denote the diagram obtained by switching the crossing by $D_+$ and the diagram obtained by resolving the crossing following the orientation by $D_0$. See Figure \ref{fig:npretzel}.
\begin{figure}
\caption{\label{fig:npretzel}
\label{fig:npretzel}
\end{figure}
Then we have
\[ a^{-1}(P_D(a, z)) = a(P_{D_+}(a, z)) -z(P_{D_0}(a, z)),\]
and $D_0$ is the diagram of the torus link $T_{2, -2q}$ with the orientation as in Figure \ref{fig:npretzel}. By Lemma \ref{lem:toruspqhomfly} we know that $\deg_a P_{T_{2, -2q}}(a, z) = 2q+1$.
We first consider $P(2, -q, -q)$. Switching the top left negative crossing results in $D_+$ being a connected sum of 2 $T_{2, -q}$'s, and $D_0$ is $T_{2, -2q}$. Therefore $\deg_a P_{D_+}(a, z)=2q+2$ as the HOMFLY-PT polynomial of a connected sum is the product of the individual HOMFLY-PT polynomials, and $\deg_a P_{D_0}(a, z) = 2q+1$. This clearly shows
\[\deg_a P_D(a, z) = 2 + \deg_aP_{D_+}(a, z)=2q+4. \]
For even $r>2$ we may now induct on $r$ with the hypothesis that
\[\deg_a(P_{P(r, -q, -q)}(a, z)) = 2+r+2q. \]
Indeed, notice that switching the top left negative crossing of $P(r,-q,-q)$ yields that $D_{+}$ is simply $P(r-2,-q,-q)$ and that $D_{0}$ is still $T_{2,-2q}$. Thus we obtain that
\[\overline{\text{sl}}(P(r, -q, -q)) \leq -\deg_a(P_K(a, z))-1 \leq -2-r-2q-1 \]
by Theorem \ref{thm:mslbhomfly}.
On the other hand, there are only two possible nontrivial homology groups for $i=0$ with $j$-grading equal to $|s_1(D)|-n_-(D) = 1-2q$ and $-1-2q$ in $Kh(P(r, -q, -q))$. We apply Remark \ref{rem:msl0} to finish the proof of the theorem.
\end{proof}
Recall that any transverse link $L$ with a quasipositive braid representative $\beta$ satisfies that $\psi(L) \neq 0$ \cite{Pla06}. Thus Theorem \ref{thm:homflypretzel} directly implies that no transverse representative of such $P(r,-q,-q)$ has a quasipositive braid representative. We can conclude that as a smooth link, $P(r,-q,-q)$ is not the closure of a quasipositive braid, and so:
\begin{cor}\label{cor:pretzelQP} Every $3$-tangle pretzel knot of the form $P(r,-q,-q)$ with $q>0$ odd and $r\geq 2$ even is not quasipositive.
\end{cor}
Recall that $P(r,-q,-q)$ is a negative knot. We thank Peter Feller for the observation that another argument can be made to show that, in general, \emph{any} negative knot that is quasipositive must be the unknot (using the fact that any negative knot is strongly quasinegative \cite{nakamura_positive}, \cite{rudolph_stronglyQP} and facts about the behavior of the Ozsv\'ath-Szab\'o concordance invariant $\tau$ \cite{ozsvath_szabo_tau} for quasipositive and strongly quasinegative knots). Our proof method for Corollary \ref{cor:pretzelQP} is clearly of a very different flavor, as we do not depend on tools from Heegaard Floer homology or four-dimensional topology. See also \cite{boileau_quasipositive} for a discussion of the strong quasipositivity of 3-tangle pretzel knots, up to mirror images.
To understand these examples better, we consider the FDTCs of some braid representatives of these pretzel knots which do not admit a transverse representative with non-vanishing $\psi$.
\begin{eg} Table \ref{table:pretzel} gives some examples of pretzel knots satisfying the conditions of Theorem \ref{thm:homflypretzel} and braid representatives, using Hunt's program. \\
\begin{table}
\begin{tabular}{|p{2cm}|p{13cm}|}
\hline
Knot & Braid representative \\
\hline
$P(2, -5, -5)$ & $\sigma_1^{-1} \sigma_2^{-5} \sigma_1^{-1} \sigma_2^{-5}$ \\
\hline
$P(4, -5, -5)$ & $\sigma_1^{-1} \sigma_2^{-5} \sigma_3^{-1} \sigma_2^{-5} \sigma_1 \sigma_2 \sigma_3^{-1} \sigma_4 \sigma_3^{-3} \sigma_2^{-1} \sigma_3^{-1} \sigma_4^{-1}$ \\
\hline
$P(6, -5, -5)$ & \text{$\sigma_1^{-1} \sigma_2 \sigma_3^{-1} \sigma_4^{-1} \sigma_3^{-1} \sigma_2^{-1} \sigma_3^{-1} \sigma_4 \sigma_5 \sigma_6 \sigma_1 \sigma_2^{-1} \sigma_3^{-1}\sigma_4\sigma_5\sigma_4^{-5}\sigma_3^{-1}\sigma_4^{5}\sigma_5^{-1}\sigma_6^{-1}\sigma_2\sigma_3^{-1}\sigma_4^{-1}\sigma_5^{-1}$} \\
\hline
$P(8, -5, -5)$ & $\sigma_1^{-1}\sigma_2^{-1}\sigma_3\sigma_4^{-1}\sigma_5^{-1}\sigma_6^{-1}\sigma_7^{-1} \sigma_4^{-1}\sigma_5^{-5}\sigma_6^{-1}\sigma_3^{-1} \sigma_4^{-1}\sigma_5^{-1}\sigma_4^{-1}\sigma_2\sigma_3^{-1}\sigma_4^{-1}\sigma_5\sigma_6\sigma_7\sigma_8
\sigma_5^{-5}$ \\
& $\sigma_6 \sigma_7\sigma_5\sigma_6\sigma_1\sigma_2\sigma_3^{-1}\sigma_4^{-1}\sigma_5^{-1}\sigma_4^{-1}\sigma_3\sigma_4^{-1}\sigma_5\sigma_6^{-1}\sigma_7^{-1}\sigma_8^{-1}\sigma_2^{-1}\sigma_3$ \\
\hline
\end{tabular}
\caption{Examples of pretzel knots satisfying the conditions of Theorem \ref{thm:homflypretzel}, together with braid word representatives.}\label{table:pretzel}
\end{table}
\end{eg}
\begin{rem}
Each of these braid representatives has at most a single $\sigma_{1}$ and a single $\sigma_{1}^{-1}$. By Proposition \ref{FDTCestimate}, this implies that each of their FDTCs lies in the interval $[-1,1]$. Notice that every transverse link has \emph{some} braid representative with FDTC in $[-1,1]$, since any $n$-braid that is a positive stabilization of some $(n-1)$-braid has FDTC lying in $[0,1]$.
\end{rem}
\begin{que}\label{psizerobraidrep1} Suppose $K$ is a smooth link such that $\psi(\beta) = 0$ for all braid representatives $\beta$ of $K$. Then is the FDTC of each braid representative of $K$ less than or equal to one?
\end{que}
An affirmative answer to Question \ref{psizerobraidrep1} would prove a statement similar in flavor to Theorem \ref{FDTCPlamenevskaya} in the setting of Khovanov homology. In particular, its contrapositive would state that if a link $K$ has some braid representative whose FDTC is strictly greater than one, then it has some transverse representative for which $\psi$ does not vanish.
\end{document} |
\begin{document}
\baselineskip 16pt
\phantom{.} \vskip 5cm
\begin{center}
{\LARGE \bf Eccentric Connectivity Index \\[12pt]
of Chemical Trees} \\
{\large \sc Aleksandar Ili\'c\ \footnotemark[3] }
{\em Faculty of Sciences and Mathematics, Vi\v{s}egradska 33, 18 000 Ni\v{s}, Serbia} \\
e-mail: {\tt aleksandari@gmail.com}
{\large \sc Ivan Gutman }
{\em Faculty of Science, University of Kragujevac, P.O. Box 60, 34000 Kragujevac, Serbia} \\
e-mail: {\tt gutman@kg.ac.rs}
{\small (Received May 25, 2009)}
\end{center}
\begin{abstract}
The eccentric connectivity index $\xi^c$ is a distance--based molecular
structure descriptor that was recently used for mathematical modeling of
biological activities of diverse nature. We prove that the broom has maximum
$\xi^c$ among trees with a fixed maximum vertex degree, and characterize such
trees with minimum $\xi^c$\,. In addition, we propose a simple linear
algorithm for calculating $\xi^c$ of trees.
\end{abstract}
\footnotetext[3] {Corresponding author.}
\baselineskip=0.30in
\section{Introduction}
Let $G$ be a simple connected graph with $n = |V|$ vertices.
For a vertex $v \in V (G)$\,, $deg (v)$ denotes the degree of $v$\,.
For vertices $v, u \in V$\,, the distance $d (v, u)$ is defined as
the length of a shortest path between $v$ and $u$ in $G$\,.
The eccentricity $\varepsilon (v)$ of a vertex $v$ is the maximum
distance from $v$ to any other vertex.
Sharma, Goswami and Madan \cite{ShGoMa97} introduced a distance--based molecular
structure descriptor, which they named ``{\it eccentric connectivity index\/}''
and which they defined as
$$
\xi^c = \xi^c (G) = \sum_{v \in V (G)} deg (v) \cdot \varepsilon (v) \ .
$$
The index $\xi^c$ was successfully used for mathematical modeling of
biological activities of diverse nature \cite{DuGuMa08,GuSiMa02,KuSaMa04,SaMa00,SaMa03}.
Some mathematical properties of $\xi^c$ were recently reported in \cite{ZhDu09}.
Chemical trees (trees with maximum vertex degree at most four) provide the graph
representation of alkanes \cite{GuPo86}. It is therefore a natural problem to
study trees with bounded maximum degree.
Denote by $\Delta = \Delta(T)$ the maximum vertex degree of a tree $T$\,. The
path $P_n$ is the unique $n$-vertex tree with $\Delta = 2$\,, while the star $S_n$ is
the unique tree with $\Delta = n-1$\,. Therefore, we can assume that
$3 \leq \Delta \leq n - 2$\,.
For an arbitrary tree $T$ on $n$ vertices \cite{ZhDu09},
$$
\left \lfloor \frac{3 (n - 1)^2 + 1}{2} \right \rfloor = \xi^c (P_n)
\geq \xi^c (T) \geq \xi^c (S_n) = 3 (n - 1) \ .
$$
\section{Chemical trees with maximum eccentric connectivity index}
The broom $B_{n, \Delta}$ is a tree consisting of a star $S_{\Delta
+ 1}$ and a path of length $n - \Delta - 1$ attached to a
pendent vertex of the star. It is proven
in \cite{LiGu07} that among trees with maximum vertex degree equal to
$\Delta$\,, the broom $B_{n, \Delta}$ uniquely minimizes the largest eigenvalue of the
adjacency matrix. Further, within the same class of trees, the broom has minimum
Wiener index and Laplacian-energy like invariant \cite{St09}. In \cite{YaYe05} and
\cite{YuLv06} it was demonstrated that the broom has minimum energy among trees with,
respectively, fixed diameter and fixed number of pendent vertices.
The $\Delta$-starlike tree $T(n_1,n_2,\ldots,n_\Delta)$ is a
tree composed of the root $v$\,, and the paths $P_{n_1}$\,,
$P_{n_2}$\,, \ldots, $P_{n_\Delta}$\,, attached to $v$\,.
The number of vertices of $T(n_1,n_2,\ldots,n_{\Delta})$ is
thus equal to $n_1 + n_2 + \cdots + n_{\Delta} +
1$\,. Notice that the broom $B_{n, \Delta}$ is
a $\Delta$-starlike tree, $B_{n, \Delta} \cong T(n-\Delta,1,1,\ldots,1)$\,.
\begin{thm}
\label{thm-pi} Let $w$ be a vertex of a nontrivial connected graph
$G$\,. For nonnegative integers $p$ and $q$\,, let $G (p, q)$ denote
the graph obtained from $G$ by attaching to the vertex $w$ pendent paths
$P = w v_1 v_2 \ldots v_p$ and $Q = w u_1 u_2 \dots u_q$ of lengths $p$
and~$q$\,, respectively. If $p \geq q \geq 1$\,, then
$$
\label{eq-pi} \xi^c (G (p, q)) < \xi^c (G (p + 1, q - 1)) \ .
$$
\end{thm}
\begin{proof}
The degrees of vertices $u_{q - 1}$ and $v_p$ are changed, while all other vertices have the same
degree in $G (p + 1, q - 1)$ as in $G (p, q)$\,. Since after this transformation
the longer path has increased, the eccentricity of vertices from $G$ are either
the same or increased by one. We will consider three cases based on the longest path
from the vertex $w$ in the graph $G$\,. Denote by $deg' (v)$ and $\varepsilon'(v)$
the degree and eccentricity of vertex $v$ in $G(p+1,q-1)$\,.
\noindent
{\bf Case 1. } The length of the longest path from the vertex $w$ in $G$ is greater than $p$\,.
This means that the pendent vertex of $G$\,, most distant from $w$ is the most
distant vertex for all vertices of $P$ and $Q$\,. It follows that
$\varepsilon_{G (p + 1, q - 1)} (v) = \varepsilon_{G (p, q)} (v)$ for all vertices
$w, v_1, v_2, \ldots, v_p, u_1, u_2, \ldots, u_{q - 1}$\,, while the eccentricity
of $u_q$ increased by $p + 1 - q$\,.
\begin{eqnarray*}
\xi^c (G (p + 1, q - 1)) - \xi^c (G (p, q)) &\geq&
\left[ deg' (u_{q - 1})\,\varepsilon' (u_{q - 1})
+ deg' (u_{q})\,\varepsilon'\,(u_{q}) + deg' (v_{p})\,\varepsilon' (v_{p}) \right] \\
&-& \left[ deg (u_{q - 1})\,\varepsilon (u_{q - 1}) + deg (u_{q})\,\varepsilon (u_{q}) +
deg (v_{p})\,\varepsilon (v_{p}) \right] \\
&=& - \varepsilon (u_{q - 1}) + (p - q + 1) + \varepsilon (v_p) > 0 \ .
\end{eqnarray*}
\noindent {\bf Case 2. } The length of the longest path from the vertex $w$ in $G$ is less than or
equal to $p$ and greater than $q$\,. This means that either the vertex of $G$ that is most distant
from $w$ or the vertex $v_p$ is the most distant vertex for all vertices of $P$\,, while for
vertices $w, u_1, u_2, \ldots, u_q$ the most distant vertex is $v_p$\,. It follows that
$\varepsilon_{G (p + 1, q - 1)} (v) = \varepsilon_{G (p, q)} (v)$ for vertices $v_1, v_2, \ldots,
v_p$\,, while $\varepsilon_{G (p + 1, q - 1)} (v) = \varepsilon_{G (p, q)} (v) + 1$ for vertices
$w, u_1, u_2, \ldots, u_{q - 1}$\,. The eccentricity of $u_q$ increased by at least $1$\,.
\begin{eqnarray*}
\xi^c (G (p + 1, q - 1)) - \xi^c (G (p, q)) &\geq& deg' (w)\,\varepsilon' (w) +
deg' (v_{p})\,\varepsilon' (v_{p}) + \sum_{j = 1}^q deg' (u_{j})\,\varepsilon' (u_{j})\\
&-& deg (w)\,\varepsilon (w) - deg (v_{p}) \,\varepsilon (v_{p}) -
\sum_{j = 1}^q deg (u_{j})\,\varepsilon (u_{j})\\
&\geq& q + \left[ \varepsilon (u_{q - 1}) + 1 \right]
\left[ deg (u_{q - 1}) - 1 \right] -
\varepsilon (u_{q - 1})\,deg (u_{q - 1}) + \varepsilon (v_p)\\
&>& \varepsilon (v_p) - \varepsilon (u_{q - 1}) > 0 \ .
\end{eqnarray*}
\noindent
{\bf Case 3. } The length of the longest path from the vertex $w$ in $G$ is
less than or equal to $q$\,. This means that the pendent vertex most distant from
the vertices of $P$ and $Q$ is either $v_p$ or $u_q$\,, depending on the position.
Using the formula for eccentric connectivity index of a path, we have
\begin{eqnarray*}
\xi^c (G (p + 1, q - 1)) - \xi^c (G (p, q)) &>& \xi^c (P_{p + q + 1}) +
[deg (w) - 2]\,\varepsilon' (w) \\
&-& \xi^c (P_{p + q + 1}) - [deg (w) - 2]\,\varepsilon (w) \\
&=& deg (w) - 2 \geq 0 \ .
\end{eqnarray*}
Since $G$ is a nontrivial graph with at least one vertex, we have strict inequality.
This completes the proof.
\end{proof}
\begin{thm}
\label{thm-broom} Let $T \not \cong B_{n, \Delta}$ be an arbitrary
tree on $n$ vertices with maximum vertex degree $\Delta$\,. Then
$$
\xi^c (B_{n, \Delta}) > \xi^c (T) \ .
$$
\end{thm}
\begin{proof}
Fix a vertex $v$ of degree $\Delta$ as a root and let
$T_1, T_2, \ldots, T_{\Delta}$ be the trees
attached at~$v$\,. We can repeatedly apply the transformation described
in Theorem \ref{thm-pi} to any vertex of degree at least three with
greatest eccentricity from the root in every tree~$T_i$\,, as long as
$T_i$ does not become a path. When all trees $T_{1} ,T_{2},\dots, T_{\Delta}$
turn into paths, we can again apply transformation from Theorem~\ref{thm-pi}
at the vertex~$v$ as long as there exists at least two paths of length greater
than one, further decreasing the eccentric connectivity index. Finally, we arrive
at the broom $B_{n, \Delta}$ as the unique tree with maximum eccentric
connectivity index.
\end{proof}
By direct verification, it holds
$$
\xi^c (BT_{n, \Delta}) = \left \lfloor \frac{3n^2 - 2\Delta n - 2n -
\Delta^2 +4\Delta}{2} \right \rfloor .
$$
From the above proof, we also get that
$B'_{n,\Delta} = T(n-\Delta-1,2,1,\ldots,1)$ has the second minimal
$\xi^c$ among trees with maximum vertex degree $\Delta$\,.
It was proven in \cite{ZhDu09} that the path $P_n$ has maximum and the
star $S_n$ minimum $\xi^c$-value among connected graphs on $n$ vertices.
From Theorem~\ref{thm-broom} we know that the maximum eccentric connectivity
index among trees on $n$~vertices is achieved for one of the brooms
$B_{n,\Delta}$\,. If $\Delta>2$\,, we
can apply the transformation from Theorem~\ref{thm-pi} at the
vertex of degree~$\Delta$ in $B_{n, \Delta}$ and obtain
$B_{n, \Delta-1}$\,. Thus, it follows
$$
EE(S_{n}) = EE(B_{n,n-1}) < EE(B_{n,n-2}) < \cdots < EE(B_{n,3})<
EE(B_{n,2})=EE(P_{n}) \ .
$$
Also, it follows that $B_{n, 3}$ has the second maximum eccentric connectivity
index among trees on $n$ vertices.
\section{The minimum eccentric connectivity index of trees with fixed \\radius}
Vertices of minimum eccentricity form the center. A tree has exactly one
or two adjacent center vertices; in this latter case one speaks of a
bicenter. In what follows, if a tree has a bicenter, then our considerations
apply to any of its center vertices.
For a tree $T$ with radius $r(T)$\,,
$$
\label{tree_center} d (T) = \left\{
\begin{array}{l l}
2\,r(T) - 1 & \quad \mbox{if $T$ has a bicenter }\\[3mm]
2\,r (T) & \quad \mbox{if $T$ has has a center. }\\
\end{array} \right.
$$
Let $T_{(n, d)}$ be the set of $n$-vertex trees obtained from the
path $P_{d + 1} = v_0 v_1 \ldots v_d$ by attaching
$n - d - 1$ pendent vertices to $v_{\lfloor d/2 \rfloor}$ and/or
$v_{\lceil d/2 \rceil}$\,, where $2 \leq d \leq n - 2$\,.
Zhou and Du in \cite{ZhDu09} proved that for arbitrary tree $T$ on $n$
vertices and diameter $d$\,,
$$
\xi^c (T) \geq \xi (T^*)\ , \quad T^* \in T_{(n, d)}
$$
with equality if and only if $T \in T_{(n, d)}$\,.
Using the transformation from Theorem \ref{thm-pi} and applying it to a
center vertex, it follows that $\xi^c (T') < \xi^c (T'')$ for $T' \in T_{(n, 2r-1)}$
and $T'' \in T_{(n, 2r)}$\,.
\begin{cor}
Let $T$ be an arbitrary tree on $n$ vertices with radius $r$\,. Then
$$
\xi^c (T) \geq 3r (2r - 1) + 2 + (n - 2r)(2r + 1)
$$
with equality if and only if $T \in T_{(n, 2r-1)}$\,.
\end{cor}
\section{The maximum eccentric connectivity index of trees with \\ perfect matchings}
A graph possessing perfect matchings must have an even number of vertices.
Therefore throughout this section we assume that $n$ is even.
It is well known that if a tree $T$ has a perfect matching,
then this perfect matching $M$ is unique:
namely, a pendent vertex $v$ has to be matched with its unique neighbor $w$\,,
and then $M-\{vw\}$ forms the perfect matching of $T-v-w$\,.
Let $A_{n, \Delta}$ be a $\Delta$-starlike tree $T(n-2\,\Delta,2,2,\ldots,2,1)$
consisting of a central vertex $v$\,,
a pendent vertex, a pendent path of length $n-2\,\Delta$\,, and
$\Delta - 2$ pendant paths of length $2$\,, all attached to $v$\,.
\begin{thm}
The tree $A_{n,\Delta}$ has maximum eccentric connectivity index among trees
with perfect matching and maximum vertex degree $\Delta$\,.
\end{thm}
\begin{proof}
Let $T$ be an arbitrary tree with perfect matching and let
$v$ be a vertex of degree $\Delta$\,,
with neighbors $v_1, v_2, \ldots, v_{\Delta}$\,.
Let $T_1, T_2, \ldots, T_{\Delta}$ be the maximal subtrees
rooted at $v_1, v_2, \ldots, v_{\Delta}$\,, respectively.
Then at most one of the numbers $|T_1|, |T_2|, \ldots, |T_{\Delta}|$
can be odd (if $T_i$ and $T_j$ have odd number of vertices, then their
roots $v_i$ and $v_j$ will be unmatched). Since the number of vertices
of $T$ is even, there exists exactly one among
$T_1, T_2, \ldots,T_{\Delta}$ with odd number of vertices.
Using Theorem \ref{thm-pi}, we may transform each $T_i$ into a path
attached to $v$ -- while simultaneously decreasing $\xi^c$ and keeping the
existence of a perfect matching.
Assume that $T_{\Delta}$ has odd number of vertices,
while the remaining trees have even number of vertices.
We apply a transformation similar to the one in Theorem \ref{thm-pi},
but instead of moving one vertex, we move two vertices
in order to keep the existence of a perfect matching.
Thus, if $p \geq q \geq 2$ then
$$
\xi^c (G (p, q)) < \xi^c (G (p + 2, q - 2)) \ .
$$
Using this transformation we may reduce $T_{\Delta}$ to one vertex,
the trees $T_2, \ldots, T_{\Delta - 1}$ to two vertices,
leaving $T_1$ with $n - 2\Delta$ vertices, and thus obtaining $A_{n,\Delta}$\,.
Since all times we strictly decreased $\xi^c$\,,
we conclude that $A_{n, \Delta}$ has minimum eccentric connectivity index
among the trees with perfect matching and maximum vertex degree $\Delta$\,.
\end{proof}
The path $P_n \cong A_{n,2}$ has maximum, while $A_{n,n/2}$ has minimum
eccentric connectivity index among trees with perfect matchings.
\section{The minimum eccentric connectivity index of trees \\
with fixed number of pendent vertices}
In \cite{ZhDu09} the authors determinate the $n$-vertex trees with $p$ pendent vertices,
$2 \leq p \leq n - 1$\,, with the maximum eccentric connectivity index, and, consecutively, the extremal trees
with the maximum, second-maximum and third-maximum eccentric connectivity index for $n \geq 6$\,.
For the completeness, here we determine the $n$-vertex trees with $2 \leq p \leq n - 1$ pendent vertices
that have minimum eccentric connectivity index.
\begin{de}
Let $v$ be a vertex of a tree $T$ of degree $m + 1$\,. Suppose that $P_1, P_2, \ldots, P_m$ are
pendent paths incident with $v$\,, with lengths $1 \leq n_1 \leq n_2 \leq \ldots \leq n_m$\,. Let
$w$ be the neighbor of $v$ distinct from the starting vertices of paths $v_1, v_2, \ldots, v_m$\,,
respectively. We form a tree $T' = \delta (T, v)$ by removing the edges $v v_1, v v_2, \ldots, v
v_{m - 1}$ from $T$ and adding $m - 1$ new edges $w v_1, w v_2, \ldots, w v_{m - 1}$ incident with
$w$\,. We say that $T'$ is a $\delta$-transform of $T$ and write $T' = \delta (T, v)$\,.
\end{de}
\begin{thm}
\label{thm-delta} Let $T' = \delta (T, v)$ be a $\delta$-transform of a tree $T$ of order $n$\,.
Let $v$ be a non-central vertex, furthest from the root among all branching vertices (with degree
greater than~$2$). Then
$$
\xi^c (T) > \xi^c (T')\,.
$$
\end{thm}
\begin{proof}
The degrees of vertices $v$ and $w$ have changed -- namely, $deg (v) - deg' (v) = deg' (w) - deg
(w) = m - 1$\,. Since the furthest vertex from $v$ does not belong to $P_1, P_2, \ldots, P_m$ and
$n_m \geq n_i$ for $i = 1, 2, \ldots, m - 1$\,, it follows that the eccentricities of all vertices
different from $P_1, P_2, \ldots, P_{m - 1}, P_m$ do not change after $\delta$ transformation. The
eccentricities of vertices from $P_m$ also remain the same, while the eccentricities of vertices
from $P_1, P_2, \ldots, P_{m - 1}$ decrease by one. Using the equality $\varepsilon (v) =
\varepsilon (w) + 1$\,, it follows that
\begin{eqnarray*}
\xi^c (T) - \xi^c (T') &=& \sum_{i = 1}^{m - 1} (1 + 2(n_i - 1)) + (m - 1) \cdot \varepsilon (v) - (m - 1) \cdot \varepsilon (w) \\
&=& 2 \left ( n_1 + n_2 + \ldots + n_{m - 1} \right) - (m - 1) + (m - 1)(\varepsilon (v) - \varepsilon (w)) \\
&=& 2 \left ( n_1 + n_2 + \ldots + n_{m - 1} \right) > 0\,.
\end{eqnarray*}
This completes the proof.
\end{proof}
The $p$-starlike tree $SB_{n, p} = T(n_1, n_2, \ldots, n_p)$ is {\it balanced\/} if all paths have
almost equal lengths, i.e., $|n_i - n_j| \leq 1$ for every $1 \leq i \leq j \leq p$\,.
\begin{thm}
The balanced $p$-starlike tree $SB_{n, p}$ has minimum eccentric connectivity index among trees
with $p$ pendent vertices, $2 < p < n - 1$\,.
\end{thm}
\begin{proof}
Let $T$ be a rooted $n$-vertex tree with $p$ pendent vertices. If
$T$ contains only one vertex of degree greater than two, we can
apply Theorem \ref{thm-pi} in order to arrive at the balanced starlike
tree $SB_{n, p}$\,, without changing the
number of pendent vertices. If $T$ has several vertices of degree
greater than $2$\,, such that there are only pendent paths attached
below them, then we take the one most distant from the center vertex of $T$\,.
By repetitive application of the $\delta$ transformation and
balancing pendant paths, the eccentric connectivity index decreases.
Assume that we arrived at a tree with two centers $C = \{v, w\}$ with
only pendent paths attached at both centers.
If all pendent paths have equal lengths, then $n = k p + 2$\,. Since we
can reattach $p - 2$ pendent paths at any central vertex
without changing $\xi^c (T)$\,, it follows that there are exactly
$\lfloor p/2 \rfloor$ extremal trees with minimum
eccentric connectivity index in this special case.
Now, let $R$ be the path with length $r = r (T) - 1$ attached to
$v$ and let $Q$ be the shortest path of length $q$ attached to
$w$\,. After applying the $\delta$ transformation at vertex $v$\,, the
eccentric connectivity index remains the same. If we apply the transformation
from Theorem \ref{thm-pi} to two pendant paths of lengths $r + 1$ and $q$ attached at $w$\,,
we will strictly decrease the eccentric connectivity index. Finally, we conclude
that $SB_{n, p}$ is the unique extremal tree that minimizes $\xi^c$ among
$n$-vertex trees with $p$ pendent vertices for $n \not \equiv 2 \pmod p$\,.
\end{proof}
\section{Chemical trees with minimal eccentric connectivity index}
\begin{thm}
\label{thm-rot}
Let $T$ be a rooted tree, with a center vertex $c$ as root. Let $u$ be the vertex
closest to the root vertex, such that $deg (u) < \Delta$\,. Let $w$ be the pendent
vertex most distant from the root, adjacent to vertex $v$\,, such that
$\varepsilon (v) > \varepsilon (u)$\,. Construct a tree $T'$ by deleting the edge $vw$ and
inserting the new edge $uw$\,. Then
$$
\xi^c (T) > \xi^c (T') \ .
$$
\end{thm}
\begin{proof}
In the transformation $T \to T'$ the degrees of vertices other than $u$
and $v$ remain the same, while $deg' (u) = deg (u) + 1$ and $deg' (v) = deg (v) - 1$\,.
Since the tree is rooted at the center vertex, the radius of $T$ is equal
to $r (T) = d (c, w)$\,. Furthermore, there exists a vertex $w'$ in a different
subtree attached to the center vertex, such that
$d (c, w') = r (T)$ or $d (c, w') = r (T) - 1$\,.
From the condition $\varepsilon (v) > \varepsilon (u)$\,, it follows that
$d (c, w') > d (c, u)$ and $w' \neq u$\,.
By rotating the edge $vw$ to $uw$\,, the eccentricity of vertices other than $w$ decrease if and
only if $w$ is the only vertex at distance $r (T)$ from the center vertex. Otherwise, the
eccentricities remain the same. In both cases, we have
\begin{eqnarray*}
\xi^c (T) - \xi^c (T') &\geq& deg (v)\,\varepsilon (v) + deg (w)\,\varepsilon (w)
+ deg (u)\,\varepsilon (u)\\
&-& \left[ deg' (v)\,\varepsilon' (v) + deg' (w)\,\varepsilon' (w) +
deg' (u)\,\varepsilon' (u) \right] \\
&\geq& \varepsilon (v) + (\varepsilon (v) - \varepsilon (u)) - \varepsilon (u)
= 2 (\varepsilon (v) - \varepsilon (u)) > 0 \ .
\end{eqnarray*}
This completes the proof.
\end{proof}
The Volkmann tree $VT (n, \Delta)$ is a tree on $n$ vertices and
maximum vertex degree $\Delta$\,, defined as follows \cite{770,FiHo02}.
Start with the root having $\Delta$ children. Every vertex different
from the root, which is not in one of the last two levels, has exactly
$\Delta -1$ children. In the last level, while not all vertices need
to exist, the vertices that do exist fill the level consecutively.
Thus, at most one vertex on the level second to last has its degree
different from $\Delta$ and $1$\,. For more details on Volkmann trees
see \cite{770,FiHo02,GuFMG07}. In \cite{770,FiHo02} it was shown that
among trees with fixed $n$ and $\Delta$\,, the Volkmann tree has
minimum Wiener index. Volkmann trees have also other extremal
properties among trees with fixed $n$ and $\Delta$
\cite{GuFMG07,790,SiTo05,YuLu08}.
\begin{thm}
\label{thm-volkman} Let $T$ be an arbitrary
tree on $n$ vertices with maximum vertex degree~$\Delta$\,. Then
$$
\xi^c (T) \geq \xi^c (VT_{n, \Delta}).
$$
\end{thm}
\begin{proof}
Among $n$-vertex trees with maximum degree $\Delta$\,, let $T^*$ be the extremal tree with minimum
eccentric connectivity index. Assume that $u$ is a vertex closest to the root vertex $c$\,, with
$deg (u) < \Delta$ and let $w$ be the pendent vertex most distant from the root, adjacent to vertex
$v$\,. Also, let $k$ be the greatest integer, such that
$$
n \geq 1 + \Delta + \Delta (\Delta - 1) + \Delta (\Delta - 1)^2 +
\cdots + \Delta (\Delta - 1)^{k - 1} \ .
$$
First, we will show that the radius of $T^*$ has to be less than or equal to $k + 1$\,.
Assume that $r (T^*) = d (c, w) > k + 1$\,. Since the distance from the
center vertex to $u$ is less than or equal to $k$\,, it follows that
$$
\varepsilon (v) \geq 2 r (T^*) - 2 \geq k + r (T^*) \geq \varepsilon (u) \ .
$$
If strict inequality holds, then we can apply Theorem \ref{thm-rot} and decrease the eccentric
connectivity index -- which contradicts to the assumption that $T^*$ is the tree with minimum
$\xi^c$\,. Therefore, $\varepsilon (v) = \varepsilon (u)$ and after performing the transformation
from Theorem \ref{thm-rot}, the eccentric connectivity index does not change. According to the
definition of the number $k$\,, after finitely many transformations, the vertex~$w$ will be the
only vertex at distance $r(T)$ from the center vertex and we will strictly decrease $\xi^c
(T^*)$\,. Also, this means that for the case $n = 1 + \Delta + \Delta (\Delta - 1) + \Delta (\Delta
- 1)^2 + \cdots + \Delta (\Delta - 1)^{k - 1}$\,, the Volkmann tree is the unique tree with minimum
eccentric connectivity index.
Now, we can assume that the radius of $T^*$ is equal $k + 1$\,.
If the distance $d (c, u)$ is less than $k - 1$\,, it follows again
that $\varepsilon (v) > \varepsilon (u)$\,, which is
impossible. Therefore, the levels $1, 2, \ldots, k - 1$ are full
(level $i$ contains exactly $\Delta (\Delta - 1)^{i - 1}$ vertices),
while the $k$-th and $(k + 1)$-th levels contain
$$
L = n - \left[ 1 + \Delta + \Delta (\Delta - 1) + \Delta (\Delta - 1)^2 +
\cdots + \Delta (\Delta - 1)^{k - 1} \right]
$$
vertices.
Assume that $T^*$ has only one center vertex -- then $d (c, w) = k + 1$ and
$\varepsilon (v) = 2 r (T^*) - 1$\,.
If $d (c, u) = k - 1$\,, we can apply the transformation from Theorem \ref{thm-rot}
and strictly decrease $\xi^c$\,. Thus,
for $L > (\Delta - 1)^k$\,, the $k$-th level is also full and the pendent vertices in
the $(k + 1)$-th level can be
arbitrarily assigned. Using the same argument, for $L \leq (\Delta - 1)^k$\,, the extremal
trees are bicentral. By completing the $k$-th level,
we do not change the eccentric connectivity index -- since $\varepsilon (v) = \varepsilon (u)$\,.
Finally, $\xi (T^*) = \xi (VT (n, \Delta))$ and the result follows.
\end{proof}
In Table 1 we give the minimum value of eccentric connectivity index
among $n$ vertex trees with maximum vertex degree $\Delta$\,, together
with the number of such extremal trees (of which one is the Volkman tree).
Note that for $n \leq 2 \Delta$ the number of extremal trees is $1$\,, and for
$\Delta > 2$ holds $\xi^c (VT (n, \Delta - 1)) \geq \xi^c (VT (n, \Delta))$\,.
\section{A linear algorithm for calculating the eccentric connectivity index of a tree}
Let $T$ be a rooted tree, with a center vertex as root.
Let $c_1, c_2, \ldots, c_k$ be the
neighbors of the center vertex $c$\,, and $T_1, T_2, \ldots, T_k$ be the
corresponding rooted subtrees. Let $r_i$ be the length of the longest path
from $c_i$ in the subtree $T_i$\,, $i = 1, 2, \ldots, k$\,.
\begin{lemma}
\label{le-ecc}
The eccentricity of the vertex $v \in V (T_i)$ equals
$$
\varepsilon (v) = d (v, c) + 1 + \max_{i \neq k} r_k \ .
$$
\end{lemma}
\begin{tabular} {||c|ccccccccc||} \hline
$n$ & $\Delta=2$ & $\Delta=3$ & $\Delta=4$ & $\Delta=5$ & $\Delta=6$ & $\Delta=7$ & $\Delta=8$
& $\Delta=9$ & $\Delta=10$ \\[2mm] \hline
11 & 150\,;\,1 & 79\,;\,3 & 62\,;\,5 & 60\,;\,6 & 49\,;\,1 & 49\,;\,1 & 49\,;\,1 & 49\,;\,1 & 30\,;\,1 \\[1mm]
12 & 182\,;\,1 & 88\,;\,3 & 69\,;\,4 & 67\,;\,8 & 54\,;\,1 & 54\,;\,1 & 54\,;\,1 & 54\,;\,1 & 54\,;\,1 \\[1mm]
13 & 216\,;\,1 & 97\,;\,1 & 76\,;\,4 & 74\,;\,9 & 72\,;\,10 &59\,;\,1 & 59\,;\,1 & 59\,;\,1 & 59\,;\,1 \\[1mm]
14 & 254\,;\,1 & 106\,;\,1 & 83\,;\,3 & 81\,;\,11 & 79\,;\,12 & 64\,;\,1 & 64\,;\,1 & 64\,;\,1 & 64\,;\,1 \\[1mm]
15 & 294\,;\,1 & 130\,;\,7 & 90\,;\,2 & 88\,;\,11 & 86\,;\,16 & 84\,;\,14 & 69\,;\,1 & 69\,;\,1 & 69\,;\,1 \\[1mm]
16 & 338\,;\,1 & 141\,;\,10 & 97\,;\,1 & 95\,;\,12 & 93\,;\,19 & 91\,;\,19 & 74\,;\,1 & 74\,;\,1 & 74\,;\,1 \\[1mm]
17 & 384\,;\,1 & 152\,;\,7 & 104\,;\,1 & 102\,;\,11 & 100\,;\,23 & 98\,;\,24 & 96\,;\,21 & 79\,;\,1 & 79\,;\,1 \\[1mm]
18 & 434\,;\,1 & 163\,;\,7 & 138\,;\,24 & 109\,;\,11 & 107\,;\,25 & 105\,;\,31 & 103\,;\,27 & 84\,;\,1 & 84\,;\,1 \\[1mm]
19 & 486\,;\,1 & 174\,;\,4 & 147\,;\,20 & 116\,;\,9 & 114\,;\,29 & 112\,;\,37 & 110\,;\,36 & 108\,;\,29 & 89\,;\,1 \\[1mm]
20 & 542\,;\,1 & 185\,;\,3 & 156\,;\,18 & 123\,;\,8 & 121\,;\,30 & 119\,;\,46 & 117\,;\,45 & 115\,;\,39 & 94\,;\,1 \\[1mm]
&&&&&&&&& \\ \hline \hline
$n$ & $\Delta=11$ & $\Delta=12$ & $\Delta=13$ & $\Delta=14$ & $\Delta=15$ & $\Delta=16$ &
$\Delta=17$ & $\Delta=18$ & $\Delta=19$ \\[2mm] \hline
11 & & & & & & & & & \\[1mm]
12 & 33\,;\,1 & & & & & & & & \\[1mm]
13 & 59\,;\,1 & 36\,;\,1 & & & & & & & \\[1mm]
14 & 64\,;\,1 & 64\,;\,1 & 39\,;\,1 & & & & & & \\[1mm]
15 & 69\,;\,1 & 69\,;\,1 & 69\,;\,1 & 42\,;\,1 & & & & & \\[1mm]
16 & 74\,;\,1 & 74\,;\,1 & 74\,;\,1 & 74\,;\,1 & 45\,;\,1 & & & & \\[1mm]
17 & 79\,;\,1 & 79\,;\,1 & 79\,;\,1 & 79\,;\,1 & 79\,;\,1 & 48\,;\,1 & & & \\[1mm]
18 & 84\,;\,1 & 84\,;\,1 & 84\,;\,1 & 84\,;\,1 & 84\,;\,1 & 84\,;\,1 & 51\,;\,1 & & \\[1mm]
19 & 89\,;\,1 & 89\,;\,1 & 89\,;\,1 & 89\,;\,1 & 89\,;\,1 & 89\,;\,1 & 89\,;\,1 & 54\,;\,1 & \\[1mm]
20 & 94\,;\,1 & 94\,;\,1 & 94\,;\,1 & 94\,;\,1 & 94\,;\,1 & 94\,;\,1 & 94\,;\,1 & 94\,;\,1 & 57\,;\,1 \\[1mm]
\hline \hline
\end{tabular}
\baselineskip=0.20in
\noindent
{\bf Table 1.} The minimal value of the eccentricity connectivity index of trees
with $n$ vertices and maximum vertex degree $\Delta$\,, and the number of such extremal trees.
\baselineskip=0.30in
\begin{proof}
We show that the longest path starting at vertex $v$ has to traverse the center vertex $c$\,.
This means that the eccentricity of $v$ is equal to the sum of $d (v, c)$ and the
longest path starting at $c$ and not contained in $T_i$\,. Assume that the longest path
$P$ from $v$ stays in the subtree $T_i$\,, and let $w$ be the
vertex from $P$ at the smallest distance from the root $c$\,. Then
$d (v, c) \geq d (v, w) + 1$\,. Since the root vertex is a center of $T$\,, we have
$\max\limits_{k \neq i} r_k + 1 \geq r_i$ and consequently
$$
d (v, c) + \max_{k \neq i} r_k \geq d (v, w) + r_i \geq |P| \ .
$$
This means that $d (v, c) + 1 + \max\limits_{k \neq i} r_k$ is strictly greater
than $|P|$\,, which is a contradiction.
\end{proof}
We now present a simple linear algorithm for calculating the eccentric connectivity index of a
tree~$T$\,. First, find a center vertex of a tree -- this can be done in time $O (n)$ (see
\cite{CoLRS01} for details). For every vertex $v$\,, we have to find the length of the longest path
from $v$ in the subtree rooted at $v$\,. This can be done inductively using depth--first search,
also in time $O (n)$\,. If $r [v]$ represents the length of the longest path in the subtree rooted
at $v$\,, then
$$
r [v] = 1 + \max_{(v, w) \in E (T), \ w \neq p [v]} r [w]
$$
where $p [v]$ denotes the parent of vertex $v$ in $T$\,.
For all neighbors $c_i$ of the center vertex $c$\,, we can calculate the maximum
$\max\limits_{i \neq j} r [c_j]$\,. Finally, for every vertex $v$ we calculate the
eccentricity $\varepsilon (v)$ in $O (1)$ using Lemma \ref{le-ecc},
and sum $deg (v) \cdot \varepsilon (v)$\,.
The time complexity of the algorithm is linear $O (n)$\,, and the memory used
is $O (n)$\,, since we need three additional arrays of length $n$\,.
\noindent
{\it Acknowledgement.\/} This work was supported by the
research grants 144015G and 144007 of the Serbian Ministry of Science
and Technological Development.
\end{document} |
\begin{enumerate}gin{document}
\renewcommand{\fnsymbol{footnote}}{\fnsymbol{footnote}}
\setcounter{MaxMatrixCols}{20}
\include{BEcommandsExpInlineVar}
\begin{enumerate}gin{center} {\bf \Large Functions realising as abelian group automorphisms} \end{center}
\begin{enumerate}gin{center} {\bf B-E de Klerk, JH Meyer, J Szigeti, L van Wyk} \end{center}
{\bf Abstract:} Let $A$ be a set and $f:A\rightarrow A$ a bijective function. Necessary and sufficient conditions on $f$ are determined which makes it possible to endow $A$ with a binary operation $*$ such that $(A,*)$ is a cyclic group and $f\in \mbox{Aut}(A)$. This result is extended to all abelian groups in case $|A|=p^2, \ p$ a prime. Finally, in case $A$ is countably infinite, those $f$ for which it is possible to turn $A$ into a group $(A,*)$ isomorphic to $\Bbb Z^n$ for some $n\ge 1$, and with $f\in \mbox{Aut} (A)$, are completely characterised. \\
{\sl Keywords:} Automorphism, abelian group
{\sl 2010 Mathematics Subject Classification:} 20K30, 20K01, 20E34\\
{\bf 1. Introduction}
The question on which functions from a set to itself (selfmaps) appear as functions with a certain structural property, has been addressed by various authors. In particular, in \cite{FolSzi} and \cite{Szigeti} those selfmaps which appear as lattice endomorphisms or lattice anti-endomorphisms have been characterised. In \cite{BouFonKeYeh} a similar study was done for infra-endomorphisms of the groups $\Bbb Z_n$ and $D_n$. In this paper we characterise those selfmaps that appear as automorphisms of certain abelian groups, namely the cyclic groups, the group $\Bbb Z_p\times \Bbb Z_p$, $p$ prime, and the group $\Bbb Z^n$ for some $n\ge1$.
For a given set $A$, let us agree to say that a bijection $f:A\rightarrow A$ has the {\it auto-property} if it is possible to find a binary operation $*$ on $A$ such that $(A,*)$ is an abelian group and $f\in \mbox{Aut}(A)$.
If $A$ is finite, such an $f$ necessarily gives rise to cycles, i.e., (disjoint) finite sequences $a_1,a_2,\ldots,a_m$ from $A$ such that $f(a_i)=a_{i+1}$ for $1\le i \le m-1$ and $f(a_m)=a_1$. Every element of $A$ belongs to some cycle. The number of elements in a cycle is its {\it length}. So a fixed point of $f$ is a cycle of length $1$. The {\it cycle structure} of $f$ is a description of how many cycles of each length $f$ has. A convenient notation for this structure will be developed and used in Section 2.
On the other hand, if $A$ is infinite, then, apart from possible cycles, there is also the possibility of $f$ having {\it chains}, i.e., infinite sequences $\ldots, a_i, a_{i+1}, a_{i+2},\ldots$ from $A$ such that $f(a_i) = a_{i+1}$ for all $i$. The number of cycles of various lengths, as well as the number of chains, will be referred to simply as the {\it structure} of $f$. This infinite case will be discussed in Section 3. \\
{\bf 2. Cyclic groups and groups of order $p^2$, $p$ prime}
For this section, we always assume that $A$ is a finite set and that $f:A\rightarrow A$ is a bijection.
If $f$ has $c_i$ cycles of length $t_i$ ($1\le i \le k$), then we say $f$ has the {\it cycle structure} $\left[\begin{array}{cccc}c_1&c_2&\cdots&c_k\\t_1&t_2&\cdots&t_k\end{array}\right]$, where, for consistency, we always take $t_1>t_2>\cdots>t_k$. We stress that it is possible that some $c_i$ could be $0$, and these columns can just as well be omitted from the array.
Also note that $\sum_{i=1}^k c_it_i=|A|$, and that the identity map has cycle structure $\left[\begin{array}{c}|A|\\1\end{array}\right]$.
It is evident that if we want to investigate the conditions $f$ has to satisfy to have the auto-property, then it suffices to find the cycle structures of all possible automorphisms on all abelian groups of order $|A|$.
These cycle structures completely determine all those $f$ having the auto-property.
We begin by doing this for cyclic groups.
We determine all possible cycle structures of automorphisms $f:\Bbb Z_n \rightarrow \Bbb Z_n$, for the additive (cyclic) group $\Bbb Z_n = \{0,1,\ldots,n-1\}$.
For the ring $\Bbb Z_n = \{0,1,\ldots,n-1\}$, consider the group of units $U_n = \{k_1,k_2,\ldots,k_{\phi(n)}\} = \{k \in \Bbb Z_n : (k,n)=1\}$ (where we take $k_1 = 1$). Let $T_n = (\Bbb Z_n\setminus U_n)\setminus \{0\}$ and
for $z\in T_n$, put $z' = \frac{n}{(z,n)}$.
Let $f:\Bbb Z_n \rightarrow \Bbb Z_n$ be an automorphism. Then $f(1)\in U_n$, otherwise, if $f(1) = z\in T_n$, then $f(z') = 0 = f(0)$, a contradiction. If $f(1) = 1 = k_1$, then $f$ is the identity map.
Let $2\le i \le \phi(n)$, and assume that $f(1) = k_i$. Then $1, k_i, k_i^2,\ldots,k_i^{\ell_i-1}$ is a cycle of length $\ell_i = \mbox{ord}_n(k_i)$
(the least $x\in\Bbb N$ such that $k_i^x\equiv 1(\mbox{mod } n)$), and consisting exactly of the elements of the subgroup $\langle k_i \rightarrowngle$ of $U_n$. If $\langle k_i \rightarrowngle \ne U_n$, choose any $k_j \in U_n\setminus \langle k_i \rightarrowngle$. Then $k_j, k_jk_i, k_jk_i^2,\ldots,k_jk_i^{\ell_i-1}$ is another cycle of length $\ell_i$, consisting exactly of the coset $k_j\langle k_i \rightarrowngle$ of $\langle k_i \rightarrowngle$ in $U_n$. Continuing in this manner, we obtain $[U_n : \langle k_i \rightarrowngle]$ cycles of this type, exhausting all the elements of $U_n$.
Now consider any $z \in T_n$. Then the cycle $z, zk_i, zk_i^2,\ldots,zk_i^{\lambda-1}$ is obtained, where the length of the cycle is the least $\lambda\in\Bbb N$ such that $n\,|\,z(k_i^{\lambda}-1)$. This means that $\lambda=\mbox{ord}_{z'}(k_i)$. Note that $\lambda|\ell_i$. Also note that each member of this cycle is in $T_n$. Other elements of $T_n$, not in this cycle, might give rise to cycles of the same length $\lambda$. Hence, the total number of cycles of length $\lambda$ is given by $\frac1{\lambda}\left|\{z\in T_n : \mbox{ord}_{z'}(k_i)=\lambda\}\right|=:L_{i,\lambda}$. Finally, cycles of length $1$ obtained in this way exclude the fixed point $0$, so that there are $\left|\{z\in T_n : \mbox{ord}_{z'}(k_i)=1\}\right|+1$ cycles of length $1$.
Hence we have one direction of the following theorem:
{\bf 2.1. Theorem.\ } Let $|A|=n$ and let $f:A\rightarrow A$ be a bijection. Then there exists a binary operation $*$ on $A$ such that $(A,*)$ is a cyclic group and $f\in \mbox{Aut}(A)$ if and only if either $f$ is the identity map, or there is an $i,\ 2\le i \le \phi(n)$, such that $f$ has the cycle structure
$$\left[\begin{array}{ccccc}[U_n:\langle k_i \rightarrowngle]+L_{i,\ell_i}&L_{i,\lambda_1}&\cdots&L_{i,\lambda_{t}}&L_{i,1} + 1\\\ell_i&\lambda_1&\cdots&\lambda_t&1\end{array}\right],$$ where $\ell_i > \lambda_1 > \cdots > \lambda_t>1$ denotes the complete list of (positive) divisors of $\ell_i=\mbox{ord}_n(k_i)$.
{\bf Proof: } It remains to show how to turn $A$ into an abelian group with $f\in \mbox{Aut}(A)$, given that $f$ satisfies the stated conditions . This can be done {\it via} the so-called {\it structural graph} of $f$. Let ${\cal G} = (V,E)$ be a directed graph with $|V| = n$ and $\rho : A\rightarrow V$ a bijection. Then ${\cal G}$ is called a {\it structural graph} of $f$ if $(u,v) \in E \Leftrightarrow(\exists a\in A : u = \rho(a) \wedge v = \rho(f(a)))$. $\rho$ is called a {\it graph projection} of $f$.
Now, if there exists a group automorphism $h : G \rightarrow G$ for some abelian group $G$ such that the structural graphs of $f$ and $h$ are isomorphic (as graphs), then one easily sees that $A$ can be endowed with and abelian group structure such that $f$ is a group automorphism.
In particular, let $\rho_f$ and $\rho_h$ be graph projections of $f$ and $h$ respectively, and let $\psi$ be a graph isomorphism from the codomain of $\rho_f$ to the codomain of $\rho_h$. Define $\eta : A\rightarrow G$ by $\eta = \rho_h^{-1}\psi\rho_f$. Then it is routine to check that $(A,*)$ is an abelian group, where $\alpha * \begin{enumerate}ta = \eta^{-1}(\eta(\alpha)\cdot_G \eta(\begin{enumerate}ta))$ for all $\alpha, \begin{enumerate}ta \in A$. The identity is $1_A = \eta^{-1}(1_G)$. It is also routine to check that $f\in \mbox{Aut}(A)$.
\ \bb
{\bf 2.2. Corollary.\ } If $|A|=n$, then there are at most $\phi(n)$ cycle structures for a bijection $f:A\rightarrow A$ that will turn $A$ into a cyclic group, with $f\in\mbox{Aut}(A)$.
{\bf Proof: \ } Apart from the identity map, the possible cycle structures of automorphisms are determined by $2\le i \le \phi(n)$. But note that distinct $i$'s could give rise to the same cycle structure of an automorphism. \ \bb
{\bf 2.3. Example.\ }
\begin{enumerate}
\item[(a)] If $|A|=p$, where $p$ is a prime, then $f:A\rightarrow A$ has the auto-property if and only if it has the cycle structure $\left[\begin{array}{cc}d&1\\\frac{p-1}{d}&1\end{array}\right]$ for some divisor $d$ of $p-1$. (Note that in case $d=p-1$, we get that $\left[\begin{array}{cc}d&1\\\frac{p-1}{d}&1\end{array}\right] = \left[\begin{array}{c} p\\1\end{array}\right]$, representing the identity map.)
\item[(b)] Let $|A|=12$. Then $U_{12}=\{1,5,7,11\}$, so that $(k_1,k_2,k_3,k_4)=(1,5,7,11)$. Then we have
$\ell_2 = \mbox{ord}_{12}(5)=2$. $L_{2,1}=|\{3,6,9\}|=3, \ L_{2,2}=\frac12\cdot|\{2,4,8,10\}|=2$. This gives the cycle structure $\left[\begin{array}{cc} [U_{12}:\langle 5\rightarrowngle] +L_{2,2} & L_{2,1}+1\\ 2&1\end{array}\right] = \left[\begin{array}{cc} 4 & 4\\ 2&1\end{array}\right]$.
Similarly, for $\ell_3 = 2$ we obtain the cycle structure $\left[\begin{array}{cc} 3 & 6\\ 2&1\end{array}\right]$ and for $\ell_4 = \ell_{\phi(12)}= 2$ we obtain the cycle structure $\left[\begin{array}{cc} 5 & 2\\ 2&1\end{array}\right]$.
Hence, $A$ can be turned into a cyclic group with $f\in \mbox{Aut}(A)$ if and only if $f$ is the identity map, or $f$ has one of the three cycle structures above.
\item[(c)] Let $|A|=p^2$, with $p$ prime. Then $z' = p$ for all $z\in T_{p^2}= \{p,2p,\ldots,(p-1)p\}$. This implies that
\[ L_{i,\lambda} = \textstyle\frac1{\lambda}\left|\{z\in T_{p^2} : \mbox{ord}_{p}(k_i)=\lambda\}\right| = \left\{\begin{array}{cl } \frac{p-1}{\lambda} & \mbox{if } \mbox{ord}_p(k_i)=\lambda\\ 0 & \mbox{otherwise}\end{array}\right.\]
for every divisor $\lambda$ of $\ell_i = \mbox{ord}_{p^2}(k_i)$, where $2\le i \le p^2-p$.
For instance, if $p=3$, then $(k_1,k_2,\ldots,k_6) = (1,2,4,5,7,8)$. For $i=2$ we have $\ell_2 = \mbox{ord}_9(2) = 6$, and since $\mbox{ord}_3(2) = 2$, it follows that $L_{2,2} = \frac22 = 1$ and $L_{2,1} = L_{2,3} = L_{2,6} = 0$. Also, since $k_2 = 2$ is a generator of the group $U_9$, $[U_9 : \langle 2 \rightarrowngle] = [U_9:U_9] = 1$. So (for the case $i=2$) we obtain, by Theorem 2.1, the cycle structure $\left[\begin{array}{cccc} 1 &0& 1&1\\ 6&3&2&1\end{array}\right] =\left[\begin{array}{ccc} 1 & 1&1\\ 6&2&1\end{array}\right]$.
Similarly, for $i=3$ we get the cycle structure $\left[\begin{array}{cc} 2 & 3\\ 3&1\end{array}\right]$, for $i=4$ we get $\left[\begin{array}{ccc} 1&1&1\\ 6&2&1\end{array}\right]$, for $i=5$ we get $\left[\begin{array}{cc} 2&3\\ 3&1\end{array}\right]$, and finally, for $i=6=\phi(9)$ we get $\left[\begin{array}{cc} 4&1\\ 2&1\end{array}\right]$.
Consequently, if $|A|=9$, it can be turned into a cyclic group with $f\in \mbox{Aut}(A)$ if and only if $f$ has one of the cycle structures $\left[\begin{array}{ccc} 1&1&1\\ 6&2&1\end{array}\right], \ \left[\begin{array}{cc} 2 & 3\\ 3&1\end{array}\right], \ \left[\begin{array}{cc} 4&1\\ 2&1\end{array}\right]$ or $\left[\begin{array}{c} 9\\ 1\end{array}\right]$ (the identity). Note that other cycle structures are indeed possible in the non-cyclic case (see Theorem 2.5).
\end{enumerate}
We now turn our attention to the case $|A|=p^2$, $p$ prime, and completely determine when $f$ has the auto-property in this case. Theorem 2.1 takes care of the case when $A$ is cyclic. We will therefore focus here only on the group $\Bbb Z_p^2= \Bbb Z_p \times \Bbb Z_p$, with Aut$(\Bbb Z_p \times \Bbb Z_p) \cong GL_2(\Bbb Z_p)$. Our aim is to determine the cycle structures of all the elements of $GL_2(\Bbb Z_p)$, when acting on the elements of $\Bbb Z_p^2$.
We recall that conjugate permutations have the same cycle structures, and we formalize this in
{\bf 2.4. Lemma.\ } If $F$ is a finite field, and $A,B\in GL_2(F)$ are similar, then they determine the same cycle structure on the group $F^2$. \ \bb
Henceforth, for $\alpha$ in the finite field $F$, we use the notation $o^+(\alpha)$ for the (additive) order of $\alpha\in F$ and we use $o^\times(\alpha)$ for the (multiplicative) order of $\alpha\in F^*$.
In \cite{glg} it is given that there exist elements of order $d$ in $GL_2(\Bbb Z_p)$, for any $d$ that divides $p^2-1$, as well as of order $pd$ for any $d\,|\, p-1$. Furthermore, by virtue of Lemma 2.4, we only have to study the Jordan normal forms of the matrices in $GL_2(\Bbb Z_p)$. We do it by considering three cases:
\begin{enumerate}
\item[I.] Here, we only consider those matrices $A$ in $GL_2(\Bbb Z_p)$ having Jordan normal form $\left(\begin{array}{cc}\alpha_1 & 0\\0 & \alpha_2\end{array}\right)$, where $\alpha_1,\alpha_2\in U_p$. The order of such an $A$ is $d$, where $d\,|\,p-1$.
First, if $\alpha_1=\alpha_2=\alpha$ (say), with $o^\times(\alpha)=d$, then $A=\left(\begin{array}{cc}\alpha & 0\\0 & \alpha\end{array}\right)$ has $\frac{p^2-1}{d}$ cycles of the form $$\left(\begin{array}{c} x\\y \end{array}\right), \left(\begin{array}{c}\alpha x\\\alpha y \end{array}\right),\ldots, \left(\begin{array}{c} \alpha^{d-1}x\\\alpha^{d-1}y \end{array}\right),$$ each of length $d$ and where $x,y\in\Bbb Z_p$, not both $0$.
Second, let $\alpha_1\ne \alpha_2$, with $o^\times(\alpha_1)=d_1$ and $o^\times(\alpha_2)=d_2$, where $d_1$ and $d_2$ are divisors of $p-1$. Here, $A=\left(\begin{array}{cc}\alpha_1 & 0\\0 & \alpha_2\end{array}\right)$ has
$\frac{p-1}{d_1}$ cycles of the form
$$\left(\begin{array}{c} x\\0 \end{array}\right), \left(\begin{array}{c}\alpha_1 x\\0 \end{array}\right),\ldots, \left(\begin{array}{c} \alpha_1^{d_1-1}x\\0 \end{array}\right),$$
each of length $d_1$, where $x\in U_p$;
$\frac{p-1}{d_2}$ cycles of the form
$$\left(\begin{array}{c} 0\\y \end{array}\right), \left(\begin{array}{c}0 \\\alpha_2 y \end{array}\right),\ldots, \left(\begin{array}{c} 0\\\alpha_2^{d_2-1}y \end{array}\right),$$
each of length $d_2$, where $y\in U_p$;
$\frac{(p-1)^2}{\mbox{\footnotesize lcm}(d_1,d_2)}$ cycles of the form
$$\left(\begin{array}{c} x\\y \end{array}\right), \left(\begin{array}{c}\alpha_1 x \\\alpha_2 y \end{array}\right),\ldots,\left(\begin{array}{c} \alpha_1^{K-1}x\\\alpha_2^{K-1}y \end{array}\right),$$
each of length $K=\mbox{lcm}(d_1,d_2)$, where $x,y\in U_p$.
\item[II.] Now we consider those $A\in GL_2(\Bbb Z_p)$ with Jordan normal form $\left(\begin{array}{cc}\alpha & 1\\0 & \alpha\end{array}\right)$, where $\alpha\in U_p$. The order of $A$ is $pd$, where $d=o^\times(\alpha)$ is a divisor of $p-1$, and for any $d\,|\,p-1$, there exists such an $A$.
Then $A$ has $\frac{p-1}{d}$ cycles of the form
$$\left(\begin{array}{c} x\\0 \end{array}\right), \left(\begin{array}{c}\alpha x\\0 \end{array}\right),\ldots, \left(\begin{array}{c} \alpha^{d-1}x\\0 \end{array}\right),$$
each of length $d$, where $x\in U_p$;
$A$ has $\frac{p-1}{d}$ cycles of the form
$$\left(\begin{array}{c} x\\y \end{array}\right), \left(\begin{array}{c}\alpha x+y \\ \alpha y \end{array}\right),\ldots,\left(\begin{array}{c}\alpha^k x+k\alpha^{k-1} y\\ \alpha^k y \end{array}\right),\ldots, \left(\begin{array}{c}\alpha^{pd-1}x+ (pd-1)\alpha^{pd-2}y\\\alpha^{pd-1}y \end{array}\right),$$
each of length $pd$, where $x,y\in \Bbb Z_p$, with $y\ne0$. (Note that $o^+(d\alpha^{pd-1}y)=p$.)
\item[III.] The only remaining case is where $A\in GL_2(\Bbb Z_p)$ has Jordan normal form $\tilde{A}$ $ \ =\left(\begin{array}{cc}\begin{enumerate}ta & 0\\0&\overline{\begin{enumerate}ta} \end{array}\right)$, where $\begin{enumerate}ta,\overline{\begin{enumerate}ta}\in \Bbb Z_p(\begin{enumerate}ta)$, a quadratic field extension of $\Bbb Z_p$ (and $\begin{enumerate}ta$ and $\overline{\begin{enumerate}ta}$ are conjugate roots of an irreducible quadratic polynomial over $\Bbb Z_p$). For any $d$ such that $d\,|\,p^2-1$ but $d\nmid p-1$, there exists such an $A$ (and hence $\tilde{A}$) having order $d$.
It follows that all cycles in $\Bbb Z_p(\begin{enumerate}ta)^2$, except the trivial one $\left(\begin{array}{c} 0\\0 \end{array}\right)$, have length $d$. Note that the orbit of $\tilde{A}$ on $\left(\begin{array}{c} x\\y \end{array}\right)$, for $x,y\in \Bbb Z_p(\begin{enumerate}ta)$, not both $0$, is given by
$$\left(\begin{array}{c} x\\y \end{array}\right), \left(\begin{array}{c}\begin{enumerate}ta x \\ \overline{\begin{enumerate}ta} y \end{array}\right),\left(\begin{array}{c}\begin{enumerate}ta^2 x \\ \overline{\begin{enumerate}ta}^2 y \end{array}\right),\ldots, \left(\begin{array}{c}\begin{enumerate}ta^{d-1} x\\ \overline{\begin{enumerate}ta}^{d-1} y \end{array}\right),$$
of length $d$, since $o^\times(\overline{\begin{enumerate}ta}) = o^\times(\begin{enumerate}ta) = d$. Hence, by Lemma 2.4, all nontrivial cycles of $A$ in $\Bbb Z_p^2$ also have length $d$.
\end{enumerate}
We are now ready to characterise all automorphisms of $\Bbb Z_p^2$.
{\bf 2.5. Theorem.\ } Let $|A|=p^2$ where $p$ is prime, and let $f:A\rightarrow A$ be a bijection. Then $A$ can be turned into a group isomorphic to $\Bbb Z_p^2$, with $f\in \mbox{Aut}(\Bbb Z_p^2)$, if and only if $f$ is the identity map, or $f$ has any one of the following cycle structures:
\begin{enumerate}
\item[(a)] $\left[\begin{array}{cc} \frac{p^2-1}{d} & 1\\ d&1\end{array}\right]$ for some divisor $d$ of $p^2-1$;
\item[(b)] $\left[\begin{array}{ccc} \frac{p-1}{d} & \frac{p-1}{d}& 1\\ pd&d&1\end{array}\right]$ for some divisor $d$ of $p-1$;
\item[(c)] $\left[\begin{array}{cccc} \frac{(p-1)^2}{\mbox{\footnotesize lcm}(d_1, d_2)} & \frac{p-1}{d_1}&\frac{p-1}{d_2}& 1\\ \mbox{\footnotesize lcm}(d_1, d_2)&d_1&d_2&1\end{array}\right]$ for divisors $d_1$ and $d_2$ of $p-1$;
\end{enumerate}
{\bf Proof:\ } From case III in the discussion preceding the theorem, an automorphism $f$ with cycle structure $\left[\begin{array}{cc} 1 & 1\\ p^2-1&1\end{array}\right]$ exists. By letting $d$ vary over all divisors of $p^2-1$, and by considering the corresponding automorphisms $f^d$ ($f$ composed with itself $d$ times), we obtain all the possible cycle structures given in (a). The cycle structures in (b) and (c) follow from cases II and I respectively. Also note that, if it happens that $d_1=d_2 = d$ in (c), where $d\,|\,p-1$, then the cycle structure in (a) is obtained. In particular, $d_1=d_2=1$ gives the identity map.\ \bb
{\bf 2.6. Example.\ } Let $p=7$. The divisors of $p^2-1=48$ that are not divisors of $p-1=6$, are given by $d\in \{4, 8,12,16,24,48\}$. For these divisors we obtain, from Theorem 2.5(a), the following corresponding cycle structures:
$$\left[\begin{array}{cc} 12 & 1\\ 4&1\end{array}\right], \ \left[\begin{array}{cc} 6 & 1\\ 8&1\end{array}\right], \ \left[\begin{array}{cc} 4 & 1\\ 12&1\end{array}\right], \ \left[\begin{array}{cc} 3 & 1\\ 16&1\end{array}\right], \ \left[\begin{array}{cc} 2 & 1\\ 24&1\end{array}\right], \ \left[\begin{array}{cc} 1 & 1\\ 48&1\end{array}\right].$$
The divisors of $p-1=6$ are $d\in \{1,2,3,6\}$, so Theorem 2.5(b) gives the corresponding cycle structures
$$\left[\begin{array}{ccc} 6 & 6 & 1\\ 7&1&1\end{array}\right]=\left[\begin{array}{cc} 6 & 7\\ 7&1\end{array}\right], \ \left[\begin{array}{ccc} 3 & 3 & 1\\ 14&2&1\end{array}\right], \ \left[\begin{array}{ccc} 2 & 2 & 1\\ 21&3&1\end{array}\right], \ \left[\begin{array}{ccc} 1 & 1 & 1\\ 42&6&1\end{array}\right].$$
Finally, for the remaining cases, we consider Theorem 2.5(c), where we take $d_1,d_2\in\{1,2,3,6\}$ and we may assume that $1\le d_1 \le d_2\le 6$. We obtain the cycles
$$\left[\begin{array}{cccc} 36 & 6 & 6& 1\\ 1&1&1&1\end{array}\right] = \left[\begin{array}{c} 49\\ 1\end{array}\right], \ \left[\begin{array}{cccc} 18 & 6 & 3& 1\\ 2&1&2&1\end{array}\right]=\left[\begin{array}{cc} 21 & 7\\ 2&1\end{array}\right], \
\left[\begin{array}{cccc} 12 & 6 & 2& 1\\ 3&1&3&1\end{array}\right] = \left[\begin{array}{cc} 14 & 7\\ 3&1\end{array}\right], $$
$$\left[\begin{array}{cccc} 6 & 6 & 1& 1\\ 6&1&6&1\end{array}\right] = \left[\begin{array}{cc} 7 & 7\\ 6&1\end{array}\right], \
\left[\begin{array}{cccc} 18 & 3 & 3& 1\\ 2&2&2&1\end{array}\right] = \left[\begin{array}{cc} 24 & 1\\ 2&1\end{array}\right], \
\left[\begin{array}{cccc} 6 & 3 & 2& 1\\ 6&2&3&1\end{array}\right] = \left[\begin{array}{cccc} 6 & 2&3&1\\ 6&3&2&1\end{array}\right], $$
$$ \left[\begin{array}{cccc} 6 & 3 & 1& 1\\ 6&2&6&1\end{array}\right] = \left[\begin{array}{ccc} 7 & 3&1\\ 6&2&1\end{array}\right], \
\left[\begin{array}{cccc} 12 & 2 & 2& 1\\ 3&3&3&1\end{array}\right] = \left[\begin{array}{cc} 16&1\\ 3&1\end{array}\right], \
\left[\begin{array}{cccc} 6 & 2 & 1& 1\\ 6&3&6&1\end{array}\right] = \left[\begin{array}{ccc} 7 & 2&1\\ 6&3&1\end{array}\right], $$
$$ \left[\begin{array}{cccc} 6 & 1 & 1& 1\\ 6&6&6&1\end{array}\right] = \left[\begin{array}{cc} 8 & 1\\ 6&1\end{array}\right].$$
Consequently, if $|A|=49$, then $A$ can be made into a group isomorphic to $\Bbb Z_7^2$, with $f\in \mbox{Aut}(\Bbb Z_7^2)$, if and only if $f$ has one of the $20$ cycle structures shown here.
One immediately raises the question of how the cycle structures of automorphisms on $\Bbb Z_{p^2}$ relate to the cycle structures of automorphisms on $\Bbb Z_p \times \Bbb Z_p$. It turns out that the former forms a subset of the latter.
{\bf 2.7. Theorem.\ } Let $|A|=p^2$, with $p$ prime, and let $f:A\rightarrow A$ be a bijection. Then $f$ has the auto-property if and only if $f$ has one of the cycle structures of an automorphism of $\Bbb Z_p^2$, given by Theorem 2.5.
{\bf Proof: \ } It suffices to show that every cycle structure that appears in Example 2.3(c), also appears in Theorem 2.5. For the parameters $\ell_i=\mbox{ord}_{p^2}(k_i),\ 2\le i\le p^2-p$, and $\lambda = \mbox{ord}_p(k_i)$, we have $\ell_i\mid \phi(p^2)$ and $\lambda\mid \phi(p)$ (see \cite[Theorem 2.14]{Nat}), and $\lambda\mid \ell_i$ (see Example 2.3(c)). So we have the two possibilities:
\begin{enumerate}
\item $\ell_i=\lambda$. This gives the cycle structure $\left[\begin{array}{cc} \frac{p^2-p}{\lambda}+\frac{p-1}{\lambda} & 1\\ \lambda &1\end{array}\right] = \left[\begin{array}{cc} \frac{p^2-1}{\lambda} & 1\\ \lambda &1\end{array}\right]$, which agrees with the cycle structure of Theorem 2.5(c) with $d_1=d_2=\lambda$.
\item $\ell_i = p\lambda$. This gives the cycle structure $\left[\begin{array}{ccc} \frac{p^2-p}{p\lambda}&\frac{p-1}{\lambda} & 1\\ p\lambda & \lambda&1\end{array}\right] = \left[\begin{array}{ccc} \frac{p-1}{\lambda}&\frac{p-1}{\lambda} & 1\\ p\lambda & \lambda&1\end{array}\right]$, which agrees with the cycle structure of Theorem 2.5(b) with $d=\lambda$. \ \bb
\end{enumerate}
A natural question is whether this result holds in a more general setting, i.e., whether, for a given prime $p$ and an integer $n\ge2$, the cycle structures of the automorphisms of $\Bbb Z_p^n$ already contain all possible cycle structures of all abelian groups of order $p^n$. This is unfortunately not the case, as the next example shows:
{\bf 2.8 Example.\ } The group $\mathbb{Z}_8$ has an automorphism of which the cycle structure is different from that of all automorphisms of $\mathbb{Z}_2^3$.
{\bf Proof: \ } Consider $f:\mathbb{Z}_8\rightarrow \mathbb{Z}_8$ defined by $f(x)=-x$. The cycle structure of $f$ is $\left[\begin{array}{cc} 2 & 3\\ 1 &2\end{array}\right]$. Assume that there is an automorphism $g$ of $\mathbb{Z}_2^3$ which has the same cycle structure.
Consider the Jordan canonical form of $g$. Then $g$ has minimal polynomial $(x+1)^2$ since all elements of $\Bbb Z_2^3$ lie within cycles of length at most $2$. The characteristic polynomial of $g$ is therefore $(x+1)^3$, so that there are two blocks in Jordan form, one of size $2\times 2$ and one of size $1\times 1$. Consequently, the eigenspace related to the eigenvalue $-1 = 1$ must be of dimension at least $2$, implying that there will be at least four elements of $\Bbb Z_2^3$ in cycles of length $1$, a contradiction. \ \bb \\
{\bf 3. Groups isomorphic to $\Bbb Z^n$}
We will now investigate the cycle structures of the automorphisms of all groups of the form $\mathbb{Z}^n$. We must clearly still have the trivial cycle of length $1$, representing $0\mapsto 0$. From now on, we will refer to this cycle as the {\it zero cycle} of the map. Since $\mathbb{Z}$ is infinite there is the possibility of not only having (finite) cycles such as with the cases in Section 2, but also {\it chains}, i.e., distinct elements $\ldots, a_i, a_{i+1},a_{i+2},\ldots$ from $A$, such that $f(a_i) = a_{i+1}$ for all $i$.
One of the major tools that we used to investigate the automorphisms of the finite groups was the fact that the elements of the general linear group were much more than just matrices over rings, but they were actually matrices over fields, which allowed us to use the Jordan normal form to form conjugacy classes which partitioned the general linear group. In the case of matrices over $\mathbb{Z}$ this cannot be done, as $\mathbb{Z}$ is not a field. But even though we have lost the Jordan normal forms, we still have that the automorphism group is isomorphic to $GL(\mathbb{Z},n)$. These are clearly all the $n\times n$ integer matrices with determinant equal to $\pm1 $ (\cite{Newman_IM}).
For $1\le i \le n$, $e_i$ is used to denote the element $(0,\ldots,1,\ldots,0) \in \Bbb Z^n$, with $1$ in the $i$-th coordinate and zeros elsewhere.
{\bf 3.1 Proposition.\ } \label{NesZn} Suppose $f\in \mbox{Aut}(\Bbb Z^n)$ for some positive integer $n$. Then the following conditions must hold:
\begin{enumerate}gin{enumerate}
\item If $f$ has a cycle, apart from the zero cycle, of any length $k$, then $f$ has infinitely many cycles of length $k$.
\item If $f$ has a chain, it has infinitely many chains.
\item If all $e_i,i\in\{1,2,\ldots,n\}$ are in cycles of $f$, then all elements of $\Bbb Z^n$ are in cycles, i.e., $f$ has no chains.
\end{enumerate}
{\bf Proof:}\
Let the matrix representation of $f$ be $M$, and represent the elements of the group $\Bbb Z^n$ as columns.
\begin{enumerate}gin{enumerate}
\item Consider any non-zero cycle $T=(x,Mx,M^2x,\ldots,M^{k-1}x)$ of length $k$. Let $S_T$ be the (finite) set of all the absolute values of the non-zero components of the members of $T$. There exists a (non-zero) smallest element in $S_T$. Now, for any positive integer $m$, we see that $mT=(mx,M(mx),M^2(mx),\ldots,M^{k-1}(mx))=(mx,mMx,mM^2x,\ldots,mM^{k-1}x)$ is a cycle of length $k$, with $S_{mT}=mS_T$, from which it follows that the cycles $mT$ are disjoint for different $m\in\Bbb N$ as the minimum components are all distinct from one another. Consequently there are infinitely many cycles of length $k$.
\item The proof is roughly the same as above. The cycle $T=(x,Mx,\ldots,M^{k-1}x)$ is just replaced by the chain $T = (\ldots, M^{-2}x,M^{-1}x,x,Mx,M^2x,\ldots).$ Here, the existence of the smallest (non-zero) element is guaranteed by the well-ordering principle on $\Bbb N$.
\item Suppose all the $e_i$ are in cycles with the cycle containing $e_i$ of length $k_i$. Any $x\in\Bbb Z^n$ can be represented as $x=\sum_{i=1}^n\alpha_ie_i,\alpha_i\in\Bbb Z$. Denote the least common multiple of the set $\{k_j,j\in\{1,2,\ldots,n\}\}$ by $\ell$, and define $q_i=\frac{\ell}{k_i}$. Then
\[
M^\ell x = \sum_{i=1}^n \alpha_iM^\ell e_i
= \sum_{i=1}^n \alpha_i e_i = x,
\]
as $M^{\ell}e_i=e_i$ for all $i = 1,2,\ldots,n$. Hence $x$ lies in a cycle of length dividing $\ell$. \ \bb
\end{enumerate}
Proposition 3.1 tells us that if the structure of an automorphism consists of cycles only, then there is only a finite number of possible cycle lengths, as all cycles must be of length dividing the least common multiple of the lengths of the cycles of the $e_i$'s. However, it is still possible, in principle, for an infinite number of distinct cycle lengths to exist for an automorphism, but then some $e_i$ must lie in a chain. The following result shows that not even this is possible.\\
{\bf 3.2 Proposition.}\
The structure of any automorphism of $\Bbb Z^n$ possesses at most finitely many distinct cycle lengths.
{\bf Proof:} \
Suppose that $f$ has infinitely many distinct cycle lengths. Let the matrix representation of $f$ be given by the $n\times n$ matrix $M$.
If $n=1$, then $f(1)=1$ or $f(1)=-1$, as $\det(M)=\pm 1$. The former case is simply the identity mapping, and the latter has a cycle structure consisting of only cycles of length two, together with the zero-cycle.
So assume that $n\geq 2$ for the remainder of the proof. Let, for $1\le i \le n$,
\[
x_i=\begin{enumerate}gin{bmatrix}
x_{1i}\\
x_{2i} \\
\vdots \\
x_{ni}
\end{bmatrix}
\]
be any $n$ distinct non-zero elements of $\Bbb Z^n$ occurring in cycles. For each $i$, let the cycle length of $x_i$ be $s_i$. Let $U = [x_1|x_2|\ldots|x_n]$, the $n\times n$ matrix with columns $x_i, i=1,\ldots,n$. From Proposition~3.1 it follows that at least one of the $e_i$'s belongs to a chain, say it is $e_1$. Also, let $U(r,c)$ denote the $(r,c)$-minor of $U$. We now consider
\[y = \sum_{i=1}^n(-1)^{i-1}U(1,i)x_i.\]
For each $i\in\{2,\ldots,n\}$, the entry in the $i$-th row of $y$ is clearly the determinant of the matrix
\begin{enumerate}gin{center}
$
\begin{enumerate}gin{bmatrix}
x_{i1} & x_{i2} & \ldots & x_{in} \\
x_{21} & x_{22} & \ldots & x_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
x_{i1} & x_{i2} & \ldots & x_{in} \\
\vdots & \vdots & \ddots & \vdots \\
x_{n1} & x_{n2} & \ldots & x_{nn}
\end{bmatrix}, $
\end{center}
which is zero, as the $i$-th row is identical to the first row. This means all the components of $y$, except perhaps the first, are equal to zero.
In the same way we see that the first component of $y$ is simply the determinant of $U$. By denoting the least common multiple of $\{s_i, i \in\{1,2,\ldots,n\}\}$ by $\ell$, it is clear that
\[
M^\ell y = \sum_{i=1}^n (-1)^{i-1}U(1,i)M^\ell x_i \\
= \sum_{i=1}^n (-1)^{i-1}U(1,i)x_i \\
= y,
\]
which means that $y$ belongs to a cycle. However, the element $e_1$ lies in a chain, implying that all non-zero elements with only their first components non-zero, lie in a chain. Consequently, $y$ must be $0$, from which it follows that $\det(U)=0$. This means that the columns of $U$ are linearly dependent, and for fixed $x_1,x_2,\ldots,x_{n-1}$, any other element $z$ that lies in some cycle, can be expressed as $z = \sum_{i=1}^{n-1} \gamma_ix_i$, with $\gamma_i\in\Bbb Q$. Hence $z$ must belong to a cycle having a length dividing the least common multiple of the set $\{s_1,s_2,\ldots,s_{n-1}\}$. Since this holds for all $z$ occurring in cycles, we see that the structure of any automorphism has only finitely many distinct cycle lengths. \ \bb
{\bf 3.3 Example.} \ The structure of the automorphism on $\Bbb Z^2$ represented by $M=\left[\begin{array}{cc}1&1\\1&0\end{array}\right]$ does not have any non-zero cycles, hence it consists only of the zero cycle, and infinitely many chains.
{\bf Proof:}\ First we notice that for any positive integer $n$, $M^n= \left[\begin{array}{cc}F_{n+1}&F_n\\F_n&F_{n-1}\end{array}\right]$ with $F_n$ the $n$-th number in the Fibonacci sequence. Suppose that the structure of $M$ contains a cycle of length $n\in\Bbb N$. Then there exist $a,b\in\Bbb Z$ such that
\begin{enumerate}gin{align*}
aF_{n+1} + bF_n &= a \\
aF_n+bF_{n-1} &= b,
\end{align*}
which can be written as
\begin{enumerate}gin{align*}
(F_{n+1}-1)a + F_nb &= 0 \\
F_na+(F_{n-1}-1)b &= 0.
\end{align*}
The determinant of this system is $(F_{n+1}-1)(F_{n-1}-1)-F_n^2$, which reduces to $(F_{n+1}^2-F_{n+1}F_n-F_n^2) + 1 - (F_{n+1}+F_{n-1})$. Using the identity $F_{n+1}^2-F_{n+1}F_n-F_n^2= (-1)^n$, we see that the system has a non-zero determinant, and conclude that $\begin{enumerate}gin{bmatrix} a \\ b \end{bmatrix}=\begin{enumerate}gin{bmatrix} 0 \\ 0 \end{bmatrix}$ is the only solution.\ \bb
{\bf 3.4 Definition.} \
Suppose the structure of an automorphism of $\Bbb Z^n$ contains a cycle of length $k$. Then this cycle is called a {\it primitive cycle} of the structure if for any proper divisor $d$ of $k$, there are no non-zero cycles of length $d$ in this structure. In this case, we call $k$ a {\it primitive cycle length} of the structure of the automorphism.
We shall now investigate whether for any natural number $k$, there exists an automorphism for which all the non-zero cycles are of length $k$. In order to do so, we shall first take some inspiration on the construction of cycles from larger ones. Suppose, for example, the automorphism $M$ of $\Bbb Z^n$ has a cycle $(x,Mx,M^2x,\ldots,M^5x)$ of length $6$.
The cycle generated by
$x+M^2x+M^4x$ is $(x+M^2x+M^4x,Mx+M^3x+M^5x)$, hence has length $1$ or $2$. Similarly, the cycle $(x+M^3x,Mx+M^4x,M^2x+M^5x)$, generated by $x+M^3x$, must have a length that divides $3$.
Note that these cycle lengths are not necessarily of lengths $2$ and $3$ respectively. They could also be of length $1$. At first it seems that this could severely restrict the possibilities on the numbers which could be primitive cycle lengths. However, surprisingly, this result does not eventually restrict the numbers which are primitive cycle lengths, but rather tells us how to construct automorphisms with exactly those primitive cycle lengths. For our $6$-cycle case, for example, if we can somehow find an invertible integer matrix $M$ such that $I+M^2+M^4=I+M^3=0$, then the constructed elements which could have cycle lengths of $2$ and $3$ will actually turn out to be the zero element, and the cycle reduces to the zero-cycle.
{\bf 3.5 Example.}\ The automorphism on $\Bbb Z^2$ with matrix representation $M=\left[\begin{array}{cc}0&1\\-1&1\end{array}\right]$ has all of its non-zero cycle lengths equal to $6$.
{\bf Proof: \ }
We have that $I+M^2+M^4=0$ and $I+M^3=0$, and also that $M^6=I$, which means that all cycles are of length dividing $6$. Consider an arbitrary $\begin{enumerate}gin{bmatrix}x \\ y\end{bmatrix}\in\Bbb Z^2$. This gives the cycle
\begin{enumerate}gin{center}
$\left( \begin{enumerate}gin{bmatrix}x \\ y\end{bmatrix}, \begin{enumerate}gin{bmatrix}y \\ y-x\end{bmatrix}, \begin{enumerate}gin{bmatrix}y-x \\ -x\end{bmatrix}, \begin{enumerate}gin{bmatrix}-x \\ -y\end{bmatrix}, \begin{enumerate}gin{bmatrix}-y \\ x-y\end{bmatrix}, \begin{enumerate}gin{bmatrix}x-y \\ x\end{bmatrix} \right)$.
\end{center}
This is an explicit example of an automorphism of which the structure consists of the zero cycle, no chains and all non-zero cycles of length $6$, implying that $6$ is a primitive length with respect to this structure. Note that if any of these cycles were to collapse into a cycle of length less than $6$, then we must have that
\begin{enumerate}gin{center}
$\begin{enumerate}gin{bmatrix}x \\ y\end{bmatrix}\in
\left\{ \begin{enumerate}gin{bmatrix}y \\ y-x\end{bmatrix}, \begin{enumerate}gin{bmatrix}-x \\ -y\end{bmatrix}, \begin{enumerate}gin{bmatrix}y-x \\ -x\end{bmatrix}\right\}.$
\end{center}
All these possibilities lead to the zero cycle. \ \bb
This example also paves the way towards establishing a technique that will allow us, for any positive integer $k$, the construction of an automorphism on some $\Bbb Z^n$ of which the structure has all of its non-zero cycles of length $k$. The next theorem is the first step towards this goal:
{\bf 3.6 Theorem.\ } For any $n>1$, let $n=\prod_{i=1}^k p_i^{\alpha_i}$ be the prime factorization of $n$, where we assume that $p_1>p_2>\cdots >p_k$. Define, for each $i\in\{1,\ldots, k\}$ the polynomial $Q_i$ by
\[Q_i(\lambda)=\sum_{j=0}^{p_i-1}\lambda^{\frac{n\cdot j}{p_i}}.\]
Then the $n$-th cyclotomic polynomial $\Phi_n$ divides $Q_i$ for all $i\in\{1,2,\ldots,k\}$. Moreover, $\Phi_n$ is the only non-constant polynomial that divides all the $Q_i$.
{\bf Proof: \ }
First we notice that $\lambda^n-1=(\lambda^{\frac{n}{p_i}}-1)Q_i$ for any $i\in\{1,2,\ldots,k\}$. Let $\zeta$ be a primitive $n$-th root of unity. From $(\zeta^{\frac{n}{p_i}}-1)Q_i(\zeta)=0$ and $\zeta^{\frac{n}{p_i}}-1\neq 0$ it follows that $Q_i(\zeta)=0$. An immediate consequence is that $\lambda-\zeta$ is a factor of $Q_i$ for all primitive roots $\zeta$ of unity, so the $n$-th cyclotomic polynomial $\Phi_n$ divides all of the $Q_i$.
Now suppose that there is another non-constant polynomial $R$ which is a factor of all the $Q_i$'s but with a root $\eta$ which is not a primitive $n$-th root of unity. As the roots of $R$ must all be $n$-th roots of unity, it follows that $\eta=\zeta^m$ for some $m\in\{1,2,\ldots,n\}$ and such that $\mboxox{gcd}(m,n)\neq 1$. However, then there exists an $i$ such that $\eta^\frac{n}{p_i}-1=0$, and as $Q_i(\eta)=0$, it follows that $\eta$ is a root of $\lambda^n-1$ of multiplicity at least two. This is a contradiction, as all roots of $\lambda^n-1$ have multiplicity $1$. \ \bb
Combining Theorem 3.6 and The Cayley-Hamilton Theorem, it is clear that
if we can find an $n\times n$ matrix $M$ with characteristic polynomial $\Phi_n$, then $M$ is a root of $\Phi_n$, and thus of all the $Q_i$'s.
We now have:
{\bf 3.7 Proposition.\ } Let $n\in \Bbb N$. Then there exists an automorphism $f_n:\mathbb{Z}^m\rightarrow\mathbb{Z}^m$, for some positive integer $m$, such that the structure of $f_n$ consists of only the zero cycle and infinitely many cycles of length $n$.
{\bf Proof:\ }
Theorem 3.6 shows that the $n$-th cyclotomic polynomial is the (non-constant) greatest common divisor of the $Q_i$'s. Let $C_{\Phi_n}$ be its companion matrix (so that $C_{\Phi_n}$ has characteristic polynomial $\Phi_n$). Since the constant term of $\Phi_n$ is either $1$ or $-1$, we have that $\det C_{\Phi_n}=\pm1$. Hence $C_{\Phi_n}$ is invertible, making it the matrix representation of an automorphism. $C_{\Phi_n}$ is a root of $\Phi_n$, and since $\Phi_n$ divides all the $Q_i$'s, we have that $C_{\Phi_n}$ is a root of all the $Q_i$'s.
Since all the $Q_i$'s divide $\lambda^n-1$, all cycles associated with $C_{\Phi_n}$ have lengths dividing $n$. Any cycle length $d$ properly dividing $n$, would have to divide $\frac{n}{p_i}$ for some $p_i$. By letting $\begin{enumerate}gin{bmatrix} x \\ y \end{bmatrix}$ be a non-zero element in any cycle of length $d$, we note that
\[
Q_i(C_{\Phi_n})\begin{enumerate}gin{bmatrix}x \\y \end{bmatrix} =\sum_{j=0}^{p_i-1}(C_{\Phi_n})^{\frac{nj}{p_i}}\begin{enumerate}gin{bmatrix}x \\y \end{bmatrix}
= \sum_{j=0}^{p_i-1} \begin{enumerate}gin{bmatrix}x \\y \end{bmatrix}
= p_i\begin{enumerate}gin{bmatrix}x \\y \end{bmatrix}
= \begin{enumerate}gin{bmatrix}p_i x \\p_iy \end{bmatrix},
\]
which clearly cannot hold, since $Q_i(C_{\Phi_n})=0$. The automorphism on $\Bbb Z^m$, where $m=\phi(n)$, of which $C_{\Phi_n}$ is the matrix representation consequently has a structure consisting of the zero-cycle, no chains, and only cycles of length $n$. Note, we cannot use Theorem 3.6 if $n=1$, but, of course, an automorphism with all its (non-zero) cycles of length $1$ does exist -- simply take the identity map on the group $\Bbb Z^m$, for any $m\ge1$.\ \bb
For each $n\in \Bbb N$, we shall call the automorphism described in Proposition 3.7 a {\it pure $n$-cyclic automorphism} and denote its matrix representation by $P_n$.
We now proceed to investigate automorphisms on $\Bbb Z^n$ with cycles of different lengths.
{\bf 3.8 Theorem.\ }
Suppose the structure of an automorphism on $\Bbb Z^n$ has non-zero cycles of lengths $\alpha$ and $\begin{enumerate}ta$. Then the structure also has a cycle of length $[\alpha,\begin{enumerate}ta]$ (the least common multiple of $\alpha$ and $\begin{enumerate}ta$).
{\bf Proof:\ }
Let $M$ be the matrix representation of the automorphism. Suppose $x$ lies in a cycle of length $\alpha$ and $y$ in a cycle of length $\begin{enumerate}ta$. It is clear that for each positive integer $k$,
\[
M^{[\alpha,\begin{enumerate}ta]}(x+ky) = M^{[\alpha,\begin{enumerate}ta]}x + kM^{[\alpha,\begin{enumerate}ta]}y
= x+ky,
\]
as $\alpha|[\alpha,\begin{enumerate}ta]$ and $\begin{enumerate}ta|[\alpha,\begin{enumerate}ta]$. Denote the cycle length of $x+ky$ by $\gamma_k$ for all $k\in\Bbb N$. Clearly, $\gamma_k|[\alpha,\begin{enumerate}ta]$, so there exist distinct $k,j\in\Bbb N$ with $\gamma_k=\gamma_j$. Denote this common value by $\gamma$. Consider the two cycles $(x+ky,M(x+ky),\ldots, M^{\gamma-1}(x+ky))$ and $(x+jy,M(x+jy),\ldots, M^{\gamma-1}(x+jy))$. Since matrix multiplication is distributive over matrix summation we can subtract these two cycles term by term to obtain a new cycle $((j-k)y,M(j-k)y,\ldots,M^{\gamma-1}(j-k)y)$. Note though, that the cycle length of $(j-k)y$ need not be $\gamma$. It is possible that the newly formed cycle actually fully traverses the cycle containing $(j-k)y$ several times. However, the cycle length of $(j-k)y$ must divide $\gamma$. Since $k\neq j$ and $M(j-k)y=(j-k)My$, it is clear that $(j-k)y$ must be in a cycle of the same length as $y$, and hence $\begin{enumerate}ta|\gamma$.
Let $\gamma=q\alpha+r,\ 0\leq r<\alpha$. Then
\[
x+ky = M^\gamma(x+ky) =M^rx+kM^\gamma y
= M^rx + ky. \\
\]
So $M^rx=x$, but since the cycle containing $x$ is of length $\alpha$, it follows that $r=0$, and so $\alpha\mid\gamma$. It follows that $[\alpha,\begin{enumerate}ta]|\gamma$, and we conclude that $\gamma=[\alpha,\begin{enumerate}ta]$. \ \bb
We can now give a complete structural characterization of all functions having the auto-property with underlying group $\Bbb Z^n$.
{\bf 3.9 Theorem.\ } Let $A$ be a countably infinite set.
A bijective function $f:A\rightarrow A$ possesses the auto-property with underlying group structure $(\Bbb Z^n,+)$ (for some $n\ge1$) if and only if the structure of $f$ satisfies all of the following:
\begin{enumerate}gin{enumerate}
\item[(1)] It contains at least one cycle of length $1$.
\item[(2)] The number of distinct cycle lengths is finite.
\item[(3)] If it contains a non-zero cycle then it contains infinitely many cycles of this length.
\item[(4)] If it contains a chain, it contains infinitely many chains.
\item[(5)] If it contains non-zero cycles of length $\alpha$ and $\begin{enumerate}ta$, then it contains a cycle of length $[\alpha,\begin{enumerate}ta]$.
\end{enumerate}
{\bf Proof:\ }
Propositions 3.1, 3.2 and Theorem 3.8 show that the conditions listed above are necessary.
Let $f$ be a function on a countably infinite set satisfying all the conditions listed in the theorem. We show that $f$ has the auto-property by constructing an invertible integer matrix representing $f$. Condition (2) allows the construction of a finite set $\mathcal{L}=\{n_1,n_2,\ldots,n_s\}$ consisting of the distinct cycle lengths occurring in the structure of $f$. For each $n_i\in\mathcal{L}$, Proposition 3.7 shows the existence of a pure $n_i$-cyclic automorphism. If $f$ has no chains, construct the integer matrix
\[M=\begin{enumerate}gin{bmatrix}
P_{n_1} & 0 & 0 & \cdots & 0 \\
0 & P_{n_2} & 0 & \cdots & 0 \\
0 & 0 & P_{n_3}& \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & P_{n_s}
\end{bmatrix}\]
which is a block diagonal matrix obtained by placing the matrices $P_{n_i}$, as defined in Proposition 3.7 (as blocks) along the diagonal of $M$ and all other entries equal to $0$. If $f$ has chains, simply append the matrix $\left[\begin{array}{cc}1&1\\1&0\end{array}\right]$ along the diagonal of $M$, say at the bottom on the right.
Since all of the $P_{n_i}$'s are along the diagonal, it follows that $\det(M) = \det(P_{n_1})\det(P_{n_2})\cdots \det(P_{n_s})$ is either $1$ or $-1$, as all the $P_{n_i}$'s are invertible. This shows that $M$ is invertible, and represents an automorphism $f_M:\Bbb Z^m\rightarrow\Bbb Z^m$ for some positive integer $m$. Let the number of rows of $P_{n_i}$ be denoted by $x_i$. For $n_i$, it is clear that the cycle containing the element $e_{x_1+\cdots+x_{i-1}+1}$ is of length $n_i$ in the structure of $f_M$, as the cycle of $e_1$ is of length $n_i$ in the structure of the pure ${n_i}$-cycle represented by the matrix $P_{n_i}$. The structure of $f_M$ thus contains cycles of length $n_i$ for each $n_i\in\mathcal{L}$. If $f$ contains a chain, the last matrix embedded in the diagonal of $M$ is $\left[\begin{array}{cc}1&1\\1&0\end{array}\right]$. Example 3.3 then shows that $e_{x_1+\cdots+x_s+1}$ lies in a chain. It is now clear that a non-zero cycle of length $n_i$ (or a chain) occurs in the structure of $f$ only if one also occurs in the structure of $f_M$.
Given any element $z\in\Bbb Z^m$, written as a column, we can decompose $z$ as the sum $z=z_1+z_2+\cdots+z_s+\hat{z}$ with each $z_n$ being a column of length $m$, with its $j$-th entry equal to that of $z$, for all $j\in\left\{\left( \sum_{i=1}^{n-1}x_i\right) +1,\ldots,\left( \sum_{i=1}^{n-1}x_i\right)+x_n \right\}$, and zeros elsewhere. If $f$ has chains, $\hat{z}$ is a column of length $m$, with the first $m-2$ entries equal to $0$, and the last two entries equal to the corresponding entries of $z$; otherwise put $\hat{z}$ equal to the zero column of length $m$, i.e., all its entries are equal to $0$. We will refer to $z_i$ as the $n_i$-cycle component of $z$, and to $\hat{z}$ as the chain component of $z$. Since $M^{\ell}z=\sum_{j=1}^s M^{\ell}z_j+ M^{\ell}\hat{z}$ for all $\ell\ge1$, and each $z_i$ is in a cycle of length dividing $n_i$ \ ($1\le i \le s$), it is clear that $z$ is in a chain if and only if $\hat{z}$ is in a chain, which is the case for exactly all non-zero $\hat{z}$. Consequently, if $f_M$ has a chain, then $f$ must also have had one (since otherwise $\hat{z}=0$ for all $z\in\Bbb Z^m$). Now take any $z$ in a non-zero cycle of $f_M$. As discussed above, $\hat{z}$ must be the zero column. However, since $M$ acts on $z_k$ in the same way as the pure $n_k$-cycle would on a column consisting of the $\left( \left( \sum_{i=1}^{k-1}x_i\right) +1\right)$-th up to $\left( \left( \sum_{i=1}^{k-1}x_i\right) +x_k\right)$-th entries of $z_k$, it follows that the cycle of $z_k$ is either the zero-cycle, or of length $n_k$. Since the $z_i$'s are linearly independent, the cycle length of $z$ is equal to the least common multiple of the $n_i$'s for which the corresponding $z_i$'s are not zero columns. It now follows that any non-zero cycle of $f_M$ has length the least common multiple of $n_{\sigma(1)},n_{\sigma(2)},\ldots,n_{\sigma(k)}$ for some permutation $\sigma$ of $(1,2,\ldots,s)$, with $k\leq s $, and (by condition $5$) of the same length as some cycle of $f$. Consequently, a cycle of length $n$ (or a chain) occurs in the structure of $f_M$ only if one also occurs in that of $f$. We now have that the structures of $f$ and $f_M$ have cycles of the same distinct lengths (as well as chains) if and only if the other one has, and by conditions (1), (3) and (4), infinitely many of them, apart from the zero-cycle. It follows that $f$ has the auto-property. \ \bb
{\bf 3.11 Example.} \ Suppose we want to construct a matrix which represents an automorphism with chains, and cycles of lengths $6$ and $15$.
As there are cycles of length $6$ and $15$, there must be a cycle of length $30$. We proceed to find $P_6,P_{15}$ and $P_{30}$.
{\bf Pure $\mathbf{6}$-cycle}: $Q_1(\lambda)=1+\lambda^2+\lambda^4$ and $Q_2(\lambda)=1+\lambda^3$. The $\gcd$ of the $Q_i$'s is $\Phi_6(\lambda)=1-\lambda+\lambda^2$. The companion matrix of this polynomial is:
\[P_6=\begin{enumerate}gin{bmatrix}
\phantom{-}0 & 1 \\
-1 & 1
\end{bmatrix}.\]
{\bf Pure $\mathbf{15}$-cycle}: $Q_1(\lambda)=1+\lambda^3+\lambda^6+\lambda^9+\lambda^{12}$ and $Q_2(\lambda)=1+\lambda^5+\lambda^{10}$. The $\gcd$ of the $Q_i$'s is $\Phi_{15}(\lambda)=1-\lambda+\lambda^3-\lambda^4+\lambda^5-\lambda^7+\lambda^8$. The companion matrix of this polynomial is:
\[P_{15}=\begin{enumerate}gin{bmatrix}
\phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1\\
-1 & \phantom{-}1 & \phantom{-}0 & -1 & \phantom{-}1 & -1 & \phantom{-}0 & \phantom{-}1
\end{bmatrix}.\]
{\bf Pure $\mathbf{30}$-cycle}: $Q_1(\lambda)=1+\lambda^6+\lambda^{12}+\lambda^{18}+\lambda^{24}$, $Q_2(\lambda)=1+\lambda^{10}+\lambda^{20}$ and $Q_3(\lambda)=1+\lambda^{15}$. The $\gcd$ of the $Q_i$'s is $\Phi_{30}(\lambda)=1+\lambda-\lambda^3-\lambda^4-\lambda^5+\lambda^7+\lambda^8$. The companion matrix of this polynomial is:
\[P_{30}=\begin{enumerate}gin{bmatrix}
\phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1\\
-1 & -1 & \phantom{-}0 & \phantom{-}1 & \phantom{-}1 & \phantom{-}1 & \phantom{-}0 & -1
\end{bmatrix}.\]
The matrix which represents the desired automorphism (on $\Bbb Z^{20}$) is
\begin{enumerate}gin{center}
\resizebox{.9\hsize}{!}{
$\begin{enumerate}gin{bmatrix}
\phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 \\
-1 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 \\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & -1 & \phantom{-}1 & \phantom{-}0 & -1 & \phantom{-}1 & -1 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 \\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0& \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0& \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0& \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0& \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0& \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0& \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0& \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0& \phantom{-}0 & -1 & -1 & \phantom{-}0 & \phantom{-}1 & \phantom{-}1 & \phantom{-}1 & \phantom{-}0 & -1 & \phantom{-}0 & \phantom{-}0 \\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}1 \\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 \\
\end{bmatrix}.$}
\end{center}
{\bf Acknowledgement.}\ The first, second and fourth authors would like to thank the National Research Foundation of South Africa for financial assistance. All the authors would also like to thank the referees for valuable comments and suggestions.
\begin{enumerate}gin{thebibliography}{99}
\bibitem{glg} http://groupprops.subwiki.org/wiki/Element\_structure\_of\_general\_linear\_group\_of\_degree\_two\_\linebreak over\_a\_finite\_field
\bibitem{BouFonKeYeh} P. Bouchard, Y. Fong, W.-F. Ke and Y.-N. Yeh, Counting $f$ such that $f\circ g = g\circ f$, Result. Math. {\bf 31} (1997), 14-27.
\bibitem{FolSzi} S. Foldes and J. Szigeti, Which self-maps appear as lattice anti-endomorphisms?, Algebra Univers. {\bf 75} (2016), 439-449.
\bibitem{Nat} M.B. Nathanson, \textit{Elementary Methods in Number Theory}, Springer, 2000.
\bibitem{Newman_IM} I. Newman, \textit{Integral matrices}, Academic Press, 1972.
\bibitem{Szigeti} J. Szigeti, Which self-maps appear as lattice endomorphisms?, Discrete Math., \textbf{321} (2014), 53-56.
\end{thebibliography}
\end{document} |
{\beta}gin{document}
\title{Markov Chain Approximations to Singular Stable-like Processes}
\date{\empty }
\author{Fangjun Xu{\theta}anks{F. Xu is supported in part by the Robert Adams Fund.}\\
Department of Mathematics \\
University of Kansas \\
Lawrence, Kansas, 66045 USA}
\maketitle
{\beta}gin{abstract}
\noindent We consider the Markov chain approximations for a class of singular stable-like processes. First we obtain properties of some Markov chains. Then we construct the approximating Markov chains and give a necessary condition for the weak convergence of these chains to the singular stable-like processes.\vskip.2cm
\noindent {\it Keywords}: Markov chain approximation, Weighted Poincar\'{e} inequality, Lower bound, Exit time. \vskip.2cm
\noindent {\it Subject Classification}: Primary 60B10, 60J27;
Secondary 60J75.
\end{abstract}
\section{Introduction}
A class of singular stable-like processes $X$ is considered in \cite{xu}. These processes $X$
correspond to the Dirichlet forms
{\beta}gin{equation} {\lambda}bel{form}
\left\{
{\beta}gin{array}{ll}
\mathcal{E}(f,f)=\int_{{\mathbb R}^d}\int_{{\mathbb R}^d}\big(f(y)-f(x)\big)^2J(x,y)\,m(dy)\,dx,\\
\\
\mathcal{F}=\big\{f\in L^2({\mathbb R}^d):\mathcal{E}(f,f)<\infty\big\},
\end{array}\right.
\end{equation}
where $m(dy)$ is the measure on the union of coordinate axes $\cup^d_{i=1}{\mathbb R}_i$ with ${\mathbb R}_i$ being the i-th coordinate axis of ${\mathbb R}^d$ and $m$ restricted to each ${\mathbb R}_i$ being one-dimensional Lebesgue measure on ${\mathbb R}$. The jump kernel $J(x,y)$
satisfies
\[
J(x,y)=\left\{
{\beta}gin{array}{rl}
\frac{c(x,y)}{|x-y|^{1+{\alpha}pha}}, &~\text{if}~ y-x\in
\cup^d_{i=1}{\mathbb R}_i\backslash\{0\};\\
0, &~\text{otherwise},
\end{array}\right.
\]
where $c(x,y)=c(y,x)$ and $0<\kappa_1\leq c(x,y)\leq
\kappa_2<\infty$ for all $x$ and $y$ in ${\mathbb R}^d$.
In this paper, we consider the Markov chain approximations for the processes $X$ in \cite{xu}. In the last few years, Markov chain approximations for symmetric Markov
processes have received a lot of attention. Stroock and Zheng proved the Markov chain approximations to symmetric diffusions in \cite{stroock_zheng}. We refer to \cite{bass_kumagai2}, \cite{husseini_kassmann} and \cite{chen_kim_kumagai} for Markov chain approximations to general symmetric Markov processes and \cite{deuschel_kumagai} for Markov chain approximations to non-symmetric diffusions. The Markov chain approximations to $X$ are not considered in the above references since the processes $X$ have singular jump kernels. It is natural to ask whether $X$ could be approximated by Markov chains. If so, under what conditions, such approximation holds. The main difficulty is to get the near diagonal lower bounds in Proposition \ref{lower}. We use weighted Poincar\'{e} inequalities to obtain these lower bounds. This technique first appeared in \cite{stroock_saloff}, see also \cite{saloff_coste} and \cite{xu1}.
The paper is organized as follows. In Section 2 we introduce
notation and define Markov chains related to $X$. In Section 3, we first construct a sequence of Markov chains. Then we obtain heat kernel estimates, exit time estimates and the regularity for these chains. In Section 4 we show the Markov chain approximations for processes $X$. Throughout this paper, if not mentioned otherwise, the letter $c$ with or without a subscript denotes a positive finite constant whose exact value is unimportant and may change from line to line.
\section{Preliminaries}
Let $C(\cdot,\cdot):{\mathbb Z}^d\times{\mathbb Z}^d\to [0,\infty)$ be the function satisfying
{\beta}gin{itemize}
\item[(a)] $C(x,y)=C(y,x)$ for all $x,y\in{\mathbb Z}^d$;
\item[(b)] There exist positive constants
$\kappa_1$ and $\kappa_2$ such that
\[
\left\{
{\beta}gin{array}{cl}
\frac{\kappa_1}{|y-x|^{1+{\alpha}pha}}\leq C(x,y)\leq
\frac{\kappa_2}{|y-x|^{1+{\alpha}pha}}, &\qquad\text{if}~y-x\in\cup^d_{i=1}{\mathbb Z}_i\backslash\{0\},\\
C(x,y)=0, &\qquad\text{otherwise},
\end{array}\right.
\]
where ${\mathbb Z}_i={\mathbb Z} e_i$ with $e_i$ being the i-th vector in ${\mathbb R}^d$.
\end{itemize}
For any $x$ and $y$ in ${\mathbb Z}$, $C(x,y)$ is called the conductance between $x$ and $y$. Set
\[
G_x:=\sum_{y\in{\mathbb Z}^d}C(x,y)=\sum_{z\in \cup^d_{i=1}{\mathbb Z}_i}C(x,x+z).
\]
We define a symmetric Markov chain $\widetilde{Y}$ on
${\mathbb Z}^d$ by
\[
{\mathbb P}(\widetilde{Y}_1=y\,|\,\widetilde{Y}_0=x)=\frac{C(x,y)}{G_x},\quad
\text{for}\; x,y\in{\mathbb Z}^d.
\]
The Markov chain $\widetilde{Y}$ is discrete in time and in space.
We next introduce the continuous time version of $\widetilde{Y}$. Let
$Y$ be a process that waits at a point in ${\mathbb Z}^d$ for a length of
time that is exponential with parameter 1, then jumps according to
the jump probabilities of $\widetilde{Y}$. After that, the process
$Y$ waits at the new point for a length of time that is exponential
with parameter 1 and independent of what has gone before, and so on.
The process $Y$ defined above is the continuous time version of
$\widetilde{Y}$. The continuous time and continuous state process
closely related to both $\widetilde{Y}_n$ and $Y_t$ is the process
$X$ corresponding to the Dirichlet form $(\mathcal{E},\mathcal{F})$ in \eref{form}.
Let $q_Y(t,x,y)$ be the transition density of $Y$. Since the conductance function of $Y$ satisfies conditions (A1)-(A4) in \cite{xu1}, Proposition 2.2 in \cite{xu1} implies the following result.
{\beta}gin{proposition} {\lambda}bel{bound1} For all $x$ and $y$ in ${\mathbb Z}^d$, there exists a positive constant $c_1$ such that
\[
q_Y(t,x,y)\leq c_1(t^{-d/{\alpha}pha}\wedge 1),\quad \text{for all}\; t>0.
\]
\end{proposition}
\section{Heat Kernel Estimates and Regularity}
In this section, we first define a sequence of Markov chains from $Y$. Then we obtain heat kernel estimates, exit time estimates and the regularity result for these chains.
For each $\rho\geq 1$, set $\mathcal{S}=\rho^{-1}\mathbb{Z}^d$. For each $x\in\mathcal{S}$ and $A\subset\mathcal{S}$, let $\mu^{\rho}_x=\rho^{-d}$
and $\mu^{\rho}(A)=\sum\limits_{y\in A}\mu^{\rho}_y$. Define the rescaled process $V$ as
\[
V_t=\rho^{-1}Y_{\rho^{{\alpha}pha}t},\quad\text{for}\; t\geq 0.
\]
We see that the Dirichlet form corresponding to $V$ is
\[
\left\{
{\beta}gin{array}{rl}
\mathcal{E}^{\rho}(f,f)= \sum\limits_{\mathcal{S}}\sum\limits_{\mathcal{S}}\big(f(y)-f(x)\big)^2C^{\rho}(x,y),\\
\mathcal{F}_{\rho}= \big\{f\in L^2(\mathcal{S},\mu^{\rho}):\;\mathcal{E}^{\rho}(f,f)<\infty\big\},
\end{array}\right.
\]
where $C^{\rho}(x,y)=\rho^{{\alpha}pha-d}C(\rho x,\rho y)$ for all $x,y\in\mathcal{S}$.
Write $p(t,x,y)$ for the transition density of $V$. Then
{\beta}gin{equation} {\lambda}bel{scale}
p(t,x,y)=\rho^{d}q_Y(\rho^{{\alpha}pha}t,\rho x,\rho y)
\end{equation}
for all $x,y\in{\mathbb Z}^d$ and $t>0$.
{\beta}gin{proposition} For all $\rho\geq 1$, there exists $c_1$ such that
\[
p(t,x,y)\leq c_1t^{-d/{\alpha}pha}.
\]
\end{proposition}
{\beta}gin{proof} This follows from \eref{scale} and Proposition \ref{bound1}.
\end{proof}
For each ${{\lambda}mbda}bda\geq 1$, let $V^{{\lambda}mbda}bda$ be the process $V$ with
jumps greater than ${{\lambda}mbda}bda$ removed. Write $p^{{{\lambda}mbda}bda}(t,x,y)$ for
the transition density of the truncated process $V^{{{\lambda}mbda}bda}$. The argument in the proof of Lemma 2.5 in \cite{xu1} gives the following off-diagonal upper bound for $p^{{{\lambda}mbda}bda}(t,x,y)$.
{\beta}gin{lemma} {\lambda}bel{bound2} For all $t>0$ and $x,y\in\mathcal{S}$, there exist $c_1$ and $c_2$
such that
\[
p^{{{\lambda}mbda}bda}(t,x,y)\leq c_1t^{-d/{\alpha}pha}e^{c_2t-|x-y|/{{\lambda}mbda}bda}.
\]
\end{lemma}
For any set $A\subset\mathcal{S}$, define
{\beta}gin{equation*}
T_A(V)=\inf\big\{t\geq 0: V_t\notin A\big\}\quad\text{and}\quad
\tau_A(V)=\inf\big\{t\geq 0: V_t\in A\big\}.
\end{equation*}
The upper bound in Lemma \ref{bound2} implies the following exiting time
estimates for $V$, whose proof can be found in Proposition 3.4 of
\cite{bass_kumagai1} and Proposition 4.1 of \cite{chen_kumagai}.
{\beta}gin{theorem} {\lambda}bel{exit1}
For $a>0$ and $0<b<1$, there exists ${\gamma}mma={\gamma}mma(a,b)\in(0,1)$
such that for any $R\geq 1$ and $x\in\mathcal{S}$,
\[
{\mathbb P}^x\big(\tau_{(x,aR)}(V)<{\gamma}mma R^{\alpha}pha\big)\leq b.
\]
\end{theorem}
Recall the definition of the rescaled process $V$. Using Proposition 2.7, Remark 2.8 and Theorem 2.11 in \cite{xu1}, we obtain the following near diagonal lower bound for $p(t,x,y)$.
{\beta}gin{proposition} {\lambda}bel{lower}
There exists $c>0$ such that
\[
p(t,x,y)\geq c t^{-d/{\alpha}pha},
\]
for all $t\geq \rho^{-{\alpha}pha}$ and
$|x-y|<2t^{1/{\alpha}pha}$.
\end{proposition}
Theorem \ref{exit1} and the proof of Lemma 4.5 in
\cite{bass_kumagai1} imply the following lemma.
{\beta}gin{lemma} {\lambda}bel{hit}
Given ${\partial}ta>0$ there exists $\kappa>0$ such that if
$x,y\in\mathcal{S}$, and $A\subset\mathcal{S}$ with $dist(x,A)$ and
$dist(y,A)$ both larger than $\kappa t^{1/2}$, then
\[
{\mathbb P}^x(V_t=y, T_A\leq t)\leq {\partial}ta t^{-d/{\alpha}pha}\rho^{-d}.
\]
\end{lemma}
{\beta}gin{proposition} {\lambda}bel{exit2} For all $t\geq\rho^{-{\alpha}pha}$, there exist $c_1>0$ and ${\theta}eta\in(0,1)$ such that if $|x-z|$,
$|y-z|\leq t^{1/{\alpha}pha}$, $x,y,z\in\mathcal{S}$,
and $r\geq t^{1/{\alpha}pha}/{\theta}eta$, then
{\beta}gin{equation} {\lambda}bel{eq1}
{\mathbb P}^x(V_t=y,\tau_{B_{(z,r)}}>t)\geq c_1 t^{-d/{\alpha}pha}\rho^{-d}.
\end{equation}
\end{proposition}
{\beta}gin{proof} This follows easily from Proposition \ref{lower} and Lemma \ref{hit}.
\end{proof}
{\beta}gin{remark}
The above proposition still holds if we replace `` $|x-z|$, $|y-z|\leq t^{1/{\alpha}pha}$, $x,y,z\in\mathcal{S}$ '' with `` $|x-y|\leq 2t^{1/{\alpha}pha}$, $x,y\in\mathcal{S}$ '' and `` z '' in \eref{eq1} with `` x '', respectively.
\end{remark}
As an application of Proposition \ref{exit2}, we have
{\beta}gin{corollary} {\lambda}bel{exit3}
For each $0<{\varepsilon}psilonilon<1$, there exists
${\theta}eta={\theta}eta({\varepsilon}psilonilon)\in(0,1)$ with the following property: if
$x,y\in\mathcal{S}$ with $|x-y|\leq t^{1/{\alpha}pha}$, $t\in[0,{\theta}eta^{{\alpha}pha}r^{\alpha}pha)$, and ${\cal G}amma\subset B(y,
t^{1/{\alpha}pha})\cap\mathcal{S}$ satisfies
$\mu^{\rho}({\cal G}amma)t^{-d/{\alpha}pha}\geq{\varepsilon}psilonilon$, then
\[
{\mathbb P}^x(V_t\in{\cal G}amma~and~\tau_{B(y,r)}>t)>c_1{\varepsilon}psilonilon.
\]
\end{corollary}
In the remaining of this section we show the regularity result for $V$. Since $V$ is a Hunt process, there is a L\'{e}vy system formula for it. We refer to \cite{chen_kumagai} for its proof.
{\beta}gin{lemma} {\lambda}bel{levy}
Let $f:{\mathbb R}^+\times \mathcal{S}\times\mathcal{S}\to{\mathbb R}^+$ be a bounded
measurable function vanishing on the diagonal. Then, for all
$x\in\mathcal{S}$ and predictable stopping time $T$, we have
\[
{{\mathbb E}\,}^x\big[\sum_{s\leq
T}f(s,V_{s-},V_s)\big]={{\mathbb E}\,}^x\Big[\int^T_0 \sum_{y\in\mathcal{S}}f(s,V_s,y)\, C^{\rho}(V_s,y)\rho^{d}\, ds\Big].
\]
\end{lemma}
Let $W_t=W_0+t$ be a deterministic process. Then $Z=(W_t, V_t)$ is the
space-time process on ${\mathbb R}^+\times\mathcal{S}$ associated with $V$.
We say that a nonnegative Borel measurable function $h(t,x)$ on
${\mathbb R}^+\times\mathcal{S}$ is parabolic in an open set
$B\subset{\mathbb R}^+\times\mathcal{S}$ if for all open relative compact
sets $B'\subset B$ and $(t,x)\in B'$,
\[
q(t,x)={{\mathbb E}\,}^{(t,x)}\big[h(Z_{\tau(B'; Z_s)})\big].
\]
For any $t_0>0$, by Lemma 4.5 in \cite{chen_kumagai}, the function
$q^{\rho}(t,x)=p(t_0-t,x,y)$ is parabolic in $[0,t_0)\times\mathcal{S}$.
{\beta}gin{lemma} {\lambda}bel{lem1}
For each ${\partial}ta\in(0,1)$, there exists
${\gamma}mma={\gamma}mma_{{\partial}ta}\in(0,1)$ such that for $t>0$, and
$x\in\mathcal{S}$, if $A\subset
Q^{\rho}_{{\gamma}mma}(t,x,r):=[t,t+{\gamma}mma
r^{{\alpha}pha}]\times (B(x,r)\cap \mathcal{S})$ satisfies
$\frac{m\otimes\mu^{\rho}(A)}{m\otimes\mu^{\rho}(Q^{\rho}_{{\gamma}mma}(t,x,r))}\geq{\partial}ta$,
then
\[
{\mathbb P}^{(t,x)}\big( T_A(Z)<\tau_{Q^{\rho}_{{\gamma}mma}(t,x,r)}(Z)\big)\geq c_1{\partial}ta.
\]
\end{lemma}
{\beta}gin{proof} Thanks to Corollary \ref{exit3}, this follows from using similar arguments in the proof of Lemma 4.7 in \cite{bass_kumagai1}.
\end{proof}
{\beta}gin{lemma} {\lambda}bel{lem2}
There exists a positive constant $c_1$ such that for $s>2r$ and
$(t,x)\in[0,\infty)\times\mathcal{S}$
{\beta}gin{equation*}
{\mathbb P}^{(t,x)}\big(Z_{\tau_{Q^{\rho}(t,x,r)}}\notin
Q^{\rho}(t,x,s)\big) \leq c_1\frac{r^{{\alpha}pha}}{s^{{\alpha}pha}}.
\end{equation*}
\end{lemma}
{\beta}gin{proof} For simplicity of notation, we write
$\tau$ for $\tau_{Q^{\rho}(t,x,r)}$. Note that
{\beta}gin{align*}
{\mathbb P}^{(t,x)}\big(Z_{\tau}\notin
Q^{\rho}(t,x,s)\big)={\mathbb P}^{x}\big(V_{\tau}\notin B(x,s)\cap\mathcal{S}
;\tau\leq{\gamma}mma r^{{\alpha}pha}\big).
\end{align*}
By Lemma \ref{levy},
{\beta}gin{equation*}
{\mathbb P}^{x}\big(V_{\tau}\notin B(x,s)\cap\mathcal{S}\big)
={{\mathbb E}\,}^x\big[\int^{\tau}_0 \sum_{|y-x|\geq
s}C^{\rho}(V_{t},y)\, \rho^{d}\, dt\big]\leq
c_2s^{-{\alpha}pha}{{\mathbb E}\,}^x(\tau).
\end{equation*}
On the other hand,
{\beta}gin{equation*}
1\geq{\mathbb P}^{x}\big(V_{\tau}\notin B(x,r)\cap\mathcal{S})
={{\mathbb E}\,}^x\big[\int^{\tau}_0 \sum_{|y-x|\geq
r}C^{\rho}(V_{t},y)\,\rho^{d}\, dt\big]\geq
c_3r^{-{\alpha}pha}{{\mathbb E}\,}^x(\tau).
\end{equation*}
Combining these estimates gives the required inequality.
\end{proof}
We next derive the regularity result for $V$, which is also needed in the Markov chain approximations.
{\beta}gin{theorem} {\lambda}bel{regular}
There exist $c>0$ and ${\beta}ta>0$ (independent of $R$ and $\rho$) such
that for every bounded parabolic function $q$ in
$Q^{\rho}(0,x_0,4R)$,
{\beta}gin{equation} {\lambda}bel{eq2}
|q(s,x)-q(t,y)|\leq c\, \|q\|_{\infty, R}R^{-{\beta}ta}(|t-s|^{1/{\alpha}pha}+|y-x|)^{{\beta}ta}
\end{equation}
holds for $(s,x), (t,y)\in Q^{\rho}(0,x_0,R)$, where $\|q\|_{\infty, R}:=\sup\limits_{(t,y)\in [0,{\gamma}mma(4R)^2]\times\mathcal{S}}|q(t,y)|$. In
particular,
\[
|p(s,x_1,y_1)-p(t,x_2,y_2)|\ \leq c(t\wedge
s)^{-(d+{\beta}ta)/{\alpha}pha}(|t-s|^{1/{\alpha}pha}+|x_1-x_2|+|y_1-y_2|)^{{\beta}ta}.
\]
\end{theorem}
{\beta}gin{proof} With the help of Lemmas \ref{lem1} and \ref{lem2}, we can prove \eref{eq2} in the same
way as Theorem 4.9 of \cite{bass_kumagai1}.
\end{proof}
\section{Approximations}
In this section, we first construct the approximating Markov chains and then give a necessary condition for the weak convergence of these chains to singular stable-like processes $X$ corresponding to the Dirichlet forms $(\mathcal{E},\mathcal{F})$ in \eref{form}.
For $x\in{\mathbb R}^d$ and $n\in\bN$, define
\[
[x]_n=\big([nx_1]/n,\dots,[nx_d]/n\big),\; \mathcal{S}_n=\big\{[x]_n:
x\in{\mathbb R}^d\big\}\; \text{and}\; \mathcal{S}'_n=\big\{[x]_n:
x\in\cup^d_{i=1}{\mathbb R}_i\big\}.
\]
For any $x$ and $y$ in $\mathcal{S}_n$, let $C_n(x,y)$ be conductance on
$\mathcal{S}_n\times\mathcal{S}_n$ satisfying
\[
\left\{
{\beta}gin{array}{cl}
\frac{\kappa_1}{|y-x|^{1+{\alpha}pha}}\leq C_n(x,y)\leq
\frac{\kappa_2}{|y-x|^{1+{\alpha}pha}}, &\quad\text{if}~y-x\in\mathcal{S}'_n-\{0\};\\
C_n(x,y)=0, &\quad\text{otherwise},
\end{array}\right.
\]
$Y^n$ the Markov chain associated with $C_n(x,y)$ and $(\mathcal{E}^n,\mathcal{F}_n)$ the Dirichlet form corresponding to $Y^n$. Let
$p^n(t,x,y)$ be the transition density of $Y^n$. We can extend
$C_n(x,y)$ to ${\mathbb R}^d\times{\mathbb R}^d$ as follows:
\[
C_n(x,y)=C_n([x]_n,[y]_n),\quad\text{for}\; x,y\in{\mathbb R}^d.
\]
If $f$ is a function on ${\mathbb R}^d$, we define its restriction to
$\mathcal{S}_n$ by $R_n f(x)=f(x)$ for $x\in\mathcal{S}_n$. For ${{\lambda}mbda}bda>0$, let
$U^{{{\lambda}mbda}bda}_n$ be the ${{\lambda}mbda}bda$-resolvent for $Y^n$ and $U^{{{\lambda}mbda}bda}$ the ${{\lambda}mbda}bda$-resolvent for $X$.
For any $f$ and $g$ in $L^2(\mathcal{S}_n)$, set $(f,g)_n=\sum_{x\in \mathcal{S}_n} f(x)g(x)\, n^{-d}$ and $\|f\|_{2,n}=\sqrt{(f,f)_n}$.
We next prove the Markov chain approximations to singular stable-like processes $X$. The proof of the following result is similar to those in \cite{husseini_kassmann}, \cite{bass_kumagai2} and references therein.
{\beta}gin{theorem} Suppose that for each $N\geq 1$,
{\beta}gin{equation*}
C_n([x]_n,[y]_n)1_{[N^{-1},N]}(|x-y|)\,dy\,dx\rightarrow
J(x,y)1_{[N^{-1},N]}(|x-y|)\,m(dy)\, dx
\end{equation*}
weakly in the sense of measures as $n\to\infty$. Then for each $x\in{\mathbb R}^d$ and each $t_0>0$
the ${\mathbb P}^{[x]_n}$-laws of $\{Y^n_t; 0\leq t\leq t_0 \}$ converge
weakly to the ${\mathbb P}^x$-law of $\{X_t; 0\leq t\leq t_0 \}$ which corresponds to the Dirichlet form \eref{form}.
\end{theorem}
{\beta}gin{proof} The proof will be done in several steps.
\noindent
\textbf{Step 1} \quad We show that any subsequence $\{n_j\}$ has a further
subsequence $\{n_{j_k}\}$ such that
$\{U^{{{\lambda}mbda}bda}_{n_{j_k}}R_{n_{j_k}}f\}$ converges uniformly on
compact sets whenever $f\in C_0({\mathbb R}^d)$.
For each $x\in\mathcal{S}_n$, let $Q_n(x)=\prod^d_{i=1}[x_i,
x_i+1/n]$. If $f$ is a function on $\mathcal{S}_n$, we define its
extension to ${\mathbb R}^d$ by $E_nf$ which is a Lipschitz-continuous
function ${\mathbb R}^d\to {\mathbb R}$ and satisfies conditions (a) $E_n f(x)=f(x)$ for $x\in\mathcal{S}_n$ and (b) $E_nf$ is linear in each $Q_n(x)$. A construction of such function $E_nf$ is available in \cite{bass_kumagai1}.
For fixed $f\in C_0({\mathbb R}^d)$,
\[
||U^{{{\lambda}mbda}bda}_n(R_n(f))||_{\infty}\leq ||
R_n(f)||_{\infty}/{{\lambda}mbda}bda\leq || f||_{\infty}/{{\lambda}mbda}bda.
\]
Therefore $\{U^{{{\lambda}mbda}bda}_n(R_nf)\}$ is uniformly bounded, so is
$\{E_nU^{{{\lambda}mbda}bda}_n R_nf\}$. For any $x$ and $y$ in ${\mathbb R}^d$,
{\beta}gin{align*}
\big|U^{{{\lambda}mbda}bda}_nR_nf([y]_n)-U^{{{\lambda}mbda}bda}_nR_nf([x]_n)\big|
\leq& \int^{t_0}_0e^{-{{\lambda}mbda}bda t}\sum_{z\in\mathcal{S}_n}|f(z)|\big|
p^n(t,[x]_n,z)-p^n(t,[y]_n,z)\big|\,n^{-d}\,dt\\
&\quad+\int^{\infty}_{t_0}e^{-{{\lambda}mbda}bda t}\sum_{z\in\mathcal{S}_n}\mid
f(z)|\big|p^n(t,[x]_n,z)-p^n(t,[y]_n,z)\big|\,n^{-d}\,dt\\
\leq &\, 2\,\| f\|_{\infty}\,t_0+c_1\,{t_0}^{-(d+{\beta}ta)/{\alpha}pha}|[x]_n-[y]_n|^{{\beta}ta},
\end{align*}
where we used Theorem
\ref{regular} in the last inequality and $c_1$ is a constant independent of $n$. For any ${\varepsilon}psilonilon>0$, we choose $t_0$ small enough such that the first term
is less than ${\varepsilon}psilonilon/3$. Fix such $t_0$, we next estimate the second term. Note that
$|[x]_n-[y]_n|\leq |x-y|+2\sqrt{d}/n$. We obtain
\[
|[x]_n-[y]_n|^{{\beta}ta}\leq c_2\big(|x-y|^{{\beta}ta}+n^{-{\beta}ta}\big).
\]
For the fixed $t_0$, there exists $n_0\in\bN$ such that
$c_1\,{t_0}^{-(d+{\beta}ta)/{\alpha}pha}\,c_2\,n^{-{\beta}ta}<{\varepsilon}psilonilon/3$ for all
$n\geq n_0$. Hence
\[
|U^{{{\lambda}mbda}bda}_nR_nf([y]_n)-U^{{{\lambda}mbda}bda}_nR_nf([x]_n)|\leq
{\varepsilon}psilonilon
\]
for all $n\geq n_0$ and $|y-x|\leq 1/n_0$. Since
$|[x]_n-x|\leq\sqrt{d}/n$, by the definition of $E_n$ and Theorem
\ref{regular},
\[
\big|E_nU^{{{\lambda}mbda}bda}_nR_nf(x)-U^{{{\lambda}mbda}bda}_n R_nf([x]_n)\big|\leq
c_3\,n^{-{\beta}ta},\quad\text{for all}\; x\in{\mathbb R}^d.
\]
Therefore, for any ${\varepsilon}psilonilon>0$, there exists $n_1\in\bN$ such that
{\beta}gin{equation} {\lambda}bel{eq3}
|E_nU^{{{\lambda}mbda}bda}_nR_nf(y)-E_nU^{{{\lambda}mbda}bda}_nR_nf(x)|\leq {\varepsilon}psilonilon
\end{equation}
for all $n\geq n_1$ and $|y-x|\leq 1/n_1$. This implies that $\{E_nU^{{{\lambda}mbda}bda}_nR_nf\}$
is equicontinuous on ${\mathbb R}^d$. By the Arzel\`{a}-Ascoli Theorem,
any subsequence of $\{E_nU^{{{\lambda}mbda}bda}_nR_nf\}$ has a convergent further subsequence. Therefore, any
subsequence $\{n_j\}$ has a further subsequence $\{n_{j_k}\}$ such
that $\{U^{{{\lambda}mbda}bda}_{n_{j_k}}R_{n_{j_k}}f\}$ converges uniformly on
compact sets whenever $f\in C_0({\mathbb R}^d)$.
\noindent
\textbf{Step 2} \quad Suppose that $\{n'\}$ is a subsequence that
$\{U^{{{\lambda}mbda}bda}_{n'}R_{n'}f\}$ converges uniformly to some $H$. We show
$H\in\mathcal{F}$.
For ${{\lambda}mbda}bda>0$, let $u_n=U^{{{\lambda}mbda}bda}_nR_nf$. Then
{\beta}gin{equation} {\lambda}bel{eq4}
\mathcal{E}^n(u_n,u_n)=(R_n f,u_n)_n-{{\lambda}mbda}bda\|u_n\|^2_{2,n}.
\end{equation}
Moreover,
\[
\|{{\lambda}mbda}bda u_n\|_{2,n}=\|{{\lambda}mbda}bda U^{{{\lambda}mbda}bda}_n R_n f\|_{2,n}\leq
\|R_n f\|_{2,n}\leq \sup_n \|R_n f\|_{2,n}<\infty,
\]
where we used $\lim\limits_{n\to\infty}\|R_n f\|_{2,n}=\|f\|_2<\infty$
in the last inequality.
Therefore the right hand side of \eref{eq4} is
bounded by
\[
|(R_nf,u_n)_n|+{{\lambda}mbda}bda\|u\|^2_{2,n}\leq \frac{1}{{{\lambda}mbda}bda}\|R_n
f\|_{2,n}\|{{\lambda}mbda}bda u_n\|_{2,n}+\frac{1}{{{\lambda}mbda}bda}\|{{\lambda}mbda}bda
u_n\|^2_{2,n}\leq\frac{2}{{{\lambda}mbda}bda}\sup_n\|R_nf\|^2_{2,n}.
\]
This implies that $\{\mathcal{E}^n(u_n,u_n)\}$ is uniformly bounded.
Since $u_{n'}$ converges uniformly to $H$ on $\overline{B(0,N)}$ for
$N>0$, by assumption,
{\beta}gin{align*}
& \int\int_{N^{-1}\leq |y-x|\leq
N}\big(H(y)-H(x)\big)^2J(x,y)\,m(dy)\,dx\\
&\leq \lim\sup_{n'\to\infty}
\sum_{x,y\in\mathcal{S}_{n'},|y-x|\leq N}\big(u_{n'}(y)-u_{n'}(x)\big)^2C_{n'}(x,y)(n')^{-1-d}\\
&\leq \limsup_{n'\to\infty}\mathcal{S}^{n'}(u_{n'},u_{n'})\\
&<\infty.
\end{align*}
On the other hand,
\[
\int_{\overline{B(0,N)}}H^2(x)\,dx\leq \frac{1}{{{\lambda}mbda}bda^2}\sup_n\|R_nf\|^2_{2,n}<\infty.
\]
Combining these estimates and letting $N\to\infty$, we have
\[
\mathcal{E}(H,H)+\|H\|^2_2<\infty
\]
and thus $H\in\mathcal{F}$.
\noindent
\textbf{Step 3} \quad We show
$\lim\limits_{n'\to\infty}\mathcal{E}^{n'}(u_{n'},g)=\mathcal{E}(H,g)$ for all $g\in C^1_0({\mathbb R}^d)$.
Since $g\in C^1_0({\mathbb R}^d)$, we can choose $K$ large enough so that the
support of $g$ is contained in $B(0,K)$. By the Cauchy-Schwartz inequality,
{\beta}gin{align*}
&\Big|\sum_{x,y\in\mathcal{S}_n,|y-x|>N}(u_n(y)-u_n(x))(
g(y)-g(x)) C_n(x,y)\,n^{-1-d}\Big|\\
&\leq \big(\mathcal{E}^{n}(u_n,u_n)\big)^{1/2}\Big(\sum_{x,y\in\mathcal{S}_n,|y-x|>N}(g(y)-g(x))^2C_n(x,y)\,n^{-1-d}\Big)^{1/2}\\
&\leq 2\,\|g\|_{\infty}\big(\mathcal{E}^n(u_n,u_n)\big)^{1/2}\Big(\sum_{x\in
B(0,K)\cap\mathcal{S}_n}\sum_{|y-x|>N}
C_n(x,y)\, n^{-1-d}\Big)^{1/2}\\
&\leq c_4\|g\|_{\infty}K^dN^{-{\alpha}pha}\big(\mathcal{E}^n(u_n,u_n)\big)^{1/2}\\
&\leq c_5\,N^{-{\alpha}pha}.
\end{align*}
Similarly,
{\beta}gin{align*}
&\Big|\sum_{x,y\in\mathcal{S}_n,|y-x|<N^{-1}}(u_n(y)-u_n(x))(
g(y)-g(x))C_n(x,y)\,n^{-1-d}\Big|\\
&\leq \big(\mathcal{E}^{n}(u_n,u_n)\big)^{1/2}\Big(\sum_{x,y\in\mathcal{S}_n,|y-x|<N^{-1}}(g(y)-g(x))^2C_n(x,y)\,n^{-1-d}\Big)^{1/2}\\
&\leq \|\nabla
g\|_{\infty}\big(\mathcal{E}^n(u_n,u_n)\big)^{1/2}\Big(\sum_{x\in
B(0,K)\cap\mathcal{S}}\sum_{|y-x|<N^{-1}}|y-x|^2C_n(x,y)\,n^{-1-d}\Big)^{1/2}\\
&\leq c_6\, \|\nabla
g\|_{\infty}K^dN^{{\alpha}pha-2}(\mathcal{E}^n(u_n,u_n))^{1/2}\\
&\leq c_7\,N^{{\alpha}pha-2}.
\end{align*}
Since $H\in\mathcal{F}$, we can choose $N$ large enough such that
\[
\big|\int\int_{|y-x|\notin[N^{-1},N]}(H(y)-H(x))(g(y)-g(x))J(x,y)\, m(dy)\, dx\big|
\]
is small. Recall that $\{n'\}$ is a
subsequence of $\{n\}$ and $U^{{{\lambda}mbda}bda}_{n'}R_{n'}f$ converges
uniformly to $H$ on compact sets. Therefore,
{\beta}gin{align*}
& \sum_{x,y\in\mathcal{S}_{n'},N^{-1}\leq |y-x|\leq
N}(u_{n'}(y)-u_{n'}(x))(g(y)-g(x))C_{n'}(x,y)\,(n')^{-1-d}\\
&\rightarrow\int\int_{N^{-1}\leq |y-x|\leq N}
(H(y)-H(x))(g(y)-g(x))J(x,y)\, m(dy)\, dx.
\end{align*}
Combining these estimates gives $\lim\limits_{n'\to\infty}\mathcal{E}^{n'}(u_{n'},g)=\mathcal{E}(H,g)$.
\noindent
\textbf{Step 4} \quad We show that $\mathcal{E}(H,g)=(f,g)-{{\lambda}mbda}bda(H,g)$ for all
$g\in\mathcal{F}$ and $H=U^{{{\lambda}mbda}bda}f$.
From the above three steps,
\[
\mathcal{E}(H,g)=\lim_{n'\to\infty}\mathcal{E}^{n'}(u_{n'},g)=\lim_{n'\to
\infty}(f,g)_{n'}-{{\lambda}mbda}bda(u_{n'},g)_{n'}=(f,g)-{{\lambda}mbda}bda(H,g)
\]
for all $g\in C^1_0({\mathbb R}^d)$. Note that $C^1_0({\mathbb R}^d)$ is dense in
$\mathcal{F}$ with respect to the norm
$(\mathcal{E}(\cdot,\cdot)+\|\cdot\|^2_2)^{1/2}$, see Theorem 3.9 in \cite{xu}.
\[
\mathcal{E}(H,g)=(f,g)-{{\lambda}mbda}bda(H,g),\quad\text{for
all}\;g\in\mathcal{F}.
\]
This implies that $H$ is the ${{\lambda}mbda}bda$-resolvent of $f$ for the
process corresponding to the Dirichlet form
$(\mathcal{E},\mathcal{F})$, that is, $H=U^{{{\lambda}mbda}bda}f$. According to
what we have obtained so far, we know that every subsequence of
$\{U^{{{\lambda}mbda}bda}_nR_nf\}$ has a convergent further subsequence with limit
$U^{{{\lambda}mbda}bda}f$. Therefore, the whole sequence $U^{{{\lambda}mbda}bda}_nR_nf$
converges to $U^{{{\lambda}mbda}bda}f$ whenever $f\in C_0({\mathbb R}^d)$, that is,
$\lim\limits_{n\to\infty}U^{{{\lambda}mbda}bda}_nR_nf=U^{{{\lambda}mbda}bda}f$ for $f\in
C_0({\mathbb R}^d)$.
\noindent
\textbf{Step 5} \quad For each $f\in C_0({\mathbb R}^d)$, we show $\lim\limits_{n\to\infty}{\mathbb P}^n_t
R_n f={\mathbb P}_t f$.
Using the same argument as in \textbf{Step 1}, we see that any sequence of
$\{{\mathbb P}^{n}_tR_nf\}$ has a uniformly convergent subsequence whenever
$f\in C_0({\mathbb R}^d)$. Suppose we have a subsequence $\{n'\}$ such that
$\lim\limits_{n'\to\infty}{\mathbb P}^{n'}_t R_n f$ exists. Note that
\[
U^{{{\lambda}mbda}bda}_n R_nf=\int^{\infty}_0e^{-{{\lambda}mbda}bda
t}\,{\mathbb P}^n_tR_nf\, dt\quad\text{and}\quad
U^{{{\lambda}mbda}bda}f=\int^{\infty}_0e^{-{{\lambda}mbda}bda t}\,{\mathbb P}_tf\,dt.
\]
Using the uniqueness of Laplace transform and the fact
$\lim\limits_{n\to\infty}U^{{{\lambda}mbda}bda}_nR_nf=U^{{{\lambda}mbda}bda}f$ for $f\in
C_0({\mathbb R}^d)$, we obtain that the whole sequence $\{{\mathbb P}^{n}_t R_n
f\}$ converges to ${\mathbb P}_t f$ whenever $f\in C_0({\mathbb R}^d)$.
\noindent
\textbf{Step 6} \quad We show the weak convergence of the ${\mathbb P}^{[x]_n}$-laws of
$\{Y^n_t; 0\leq t\leq t_0\}$ for each $t_0>0$.
It suffices to show the tightness of $\{Y^{n}_t; 0\leq t\leq t_0\}$
in the space $D[0,t_0]$ and that finite-dimensional distributions of
$\{Y^n_t; 0\leq t\leq t_0\}$ converge to those of $\{X_t; 0\leq
t\leq t_0\}$. Let $\tau_n$ be stopping times bounded by $t_0$ and
$\{{\partial}ta_n\}$ a sequence of positive real numbers converging to
0. Then, by Theorem \ref{exit1} and the strong Markov property,
\[
{\mathbb P}^{[x]_n}\big(|Y^n_{\tau_n+{\partial}ta_n}-Y^n_{\tau_n}|>a\big)
= {\mathbb P}^{[x]_n}\big(|Y^n_{{\partial}ta_n}-Y^n_0|>a\big)\leq {\mathbb P}^{[x]_n}\big(\tau_{([x]_n,A)}(Y^n)<{\gamma}mma(a,b)\big)\leq b
\]
for all $n$ large enough such that ${\partial}ta_n\leq {\gamma}mma(a,b)$.
Moreover, $[x]_n\to x$ implies the tightness of the starting
distributions and Theorem \ref{exit1} implies the tightness of
$\max_{t\in[0,t_0]}|Y^n_t-Y^n_{t-}|$ both under ${\mathbb P}^{[x]_n}$. By
Theorem 1 in \cite{aldous}, we have the tightness of the
${\mathbb P}^{[x]_n}$-laws of $\{Y^n_t; 0\leq t\leq t_0\}$. Suppose $f\in
C_0({\mathbb R}^d)$. Then, for each $t\in[0,t_0]$,
\[
{{\mathbb E}\,}^xf(X_s)={\mathbb P}_tf=\lim_{n\to\infty}{\mathbb P}^{n}_tR_{n}f=\lim_{n\to\infty}{{\mathbb E}\,}^{[x]_{n}}R_{n}f(Y^n_t).
\]
Thus the one-dimensional distributions of $\{Y^n_t; 0\leq t\leq
t_0\}$ converge to those of $\{X_t; 0\leq t\leq t_0\}$. Similarly,
we can prove the finite-dimensional case using the Markov property and the time-homogeneity of $Y^n$ and the result in \textbf{Step 5}.
\end{proof}
{\beta}gin{thebibliography}{99}
\bibitem{aldous} D. Aldous, Stopping times and tightness,
\emph{Ann. Probab.}, \textbf{6} (1978), 335--340.
\bibitem{bass_kumagai1} R.F. Bass and T. Kumagai,
Symmstric Markov chains on ${\mathbb Z}^d$ with unbouded range, \emph{Trans.
Amer. Math. Soc.}, \textbf{360} (2008), 2041--2075.
\bibitem{bass_kumagai2} R.F. Bass, T. Kumagai and T. Uemura,
Convergence of symmstric Markov chains on ${\mathbb Z}^d$, \emph{Probab. Theory Relat. Fields}, \text{148} (2010), 107--140.
\bibitem{chen_kumagai} Z.-Q. Chen and T. Kumagai, Heat kernel
estimates for stable-like processes on $d$-sets, \emph{Stoch. Proc.
Applic.}, \textbf{108} (2003), 27--62.
\bibitem{chen_kim_kumagai} Z.-Q. Chen, P. Kim and T. Kumagai, Discrete approximation of symmetric jump processes on metric measure spaces, \emph{Probab. Theory Relat. Fields}, to appear.
\bibitem{deuschel_kumagai} J.-D. Deuschel and T. Kumagai, Markov chain approximations to non-symmetric diffusions with bounded coefficients, \emph{Comm. Pure Appl. Math.}, to appear.
\bibitem{husseini_kassmann} R. Husseini and M. Kassmann, Markov chain
approximations for symmetric jump processes, \emph{Potential Anal},
\textbf{27} (2007), 353--380.
\bibitem{saloff_coste} L. Saloff-Coste, {\it Aspects of Sobolev-Type Inequalities}, Cambridge University Press, 2002.
\bibitem{stroock_saloff} L. Saloff-Coste and D. W. Stroock, Op\'{e}rateurs uniform\'{e}ment sous-elliptiques sur
les groupes de Lie, {\it J. Funct. Anal.}, \textbf{98} (1991), 97--121.
\bibitem{stroock_zheng} D.W. Stroock and W. Zheng, Markov chain
approximations to symmetric diffusions, \emph{Ann. Inst. Henri.
Poincar\'{e}-Probab. Statist.}, \textbf{33} (1997), 619--649.
\bibitem{xu} F. Xu, A class of singular symmetric Markov processes, \emph{Potential Analysis}, accepted.
\bibitem{xu1} F. Xu, Regularity of harmonic functions for some Markov chains with unbounded range, \emph{Journal of Theoretical Probability}, accepted.
\end{thebibliography}
\end{document} |
\begin{document}
\begin{abstract}
This paper studies a class of binomial ideals associated to graphs with finite vertex sets. They generalize the
binomial edge ideals, and they arise in the study of conditional independence ideals. A Gröbner basis can be
computed by studying paths in the graph. Since these Gröbner bases are square-free, generalized binomial edge ideals
are radical. To find the primary decomposition a combinatorial problem involving the connected components of
subgraphs has to be solved. The irreducible components of the solution variety are all rational.
\mathrm{e}nd{abstract}
\title{Generalized Binomial Edge Ideals}
\section{Introduction}
\label{sec:introduction}
Let $\mathcal{X}_{0}$ and $\Xcal_\text{\textup{in}}$ be finite sets, $d_{0}=|\mathcal{X}_{0}|>1$, and denote $\Xcal=\mathcal{X}_{0}\times\Xcal_\text{\textup{in}}$. Let $\mathrm{e}nsuremath{\mathbb{K}}$
be a field, and consider the polynomial ring $\mathrm{e}nsuremath{\mathfrak{R}} = \mathrm{e}nsuremath{\mathbb{K}}[p_{x}:x\mathrm{i}n\Xcal]$ with $|\Xcal|$ unknowns $p_{x}$ indexed by
$\Xcal$. For all $i,j\mathrm{i}n\mathcal{X}_{0}$ and all $x,y\mathrm{i}n\Xcal_\text{\textup{in}}$ let
\begin{equation*}
f^{ij}_{xy} = p_{ix} p_{jy} - p_{iy} p_{jx}.
\mathrm{e}nd{equation*}
For any graph $G$ on $\Xcal_\text{\textup{in}}$ the ideal $I_{G}$ in $\mathrm{e}nsuremath{\mathfrak{R}}$ generated by the binomials $f^{ij}_{xy}$ for all
$i,j\mathrm{i}n\mathcal{X}_{0}$ and all edges $(x,y)$ in $G$ is called the $d_{0}$th \mathrm{e}mph{binomial edge ideal} of $G$ over $\mathrm{e}nsuremath{\mathbb{K}}$.
This is a direct generalization of~\perp\!\!\!\perpte{HHHKR10:Binomial_Edge_Ideals} and~\perp\!\!\!\perpte{Ohtani11:Ideals_of_some_2-minors},
where the same ideals have been considered in the special case $d_{0}=2$. For a comparison of the results of the
present paper to previous results see Remark~\ref{rem:comparison-to-HHH}.
One motivation to look at generalized binomial edge ideals comes from the study of conditional independence ideals.
Given $n+1$ random variables $X_{0},X_{1},\mathrm{d}\:\!ots,X_{n}$, generalized binomial edge ideals correspond to a collection of
statements of the form (see~\perp\!\!\!\perpte{HHHKR10:Binomial_Edge_Ideals} for an explanation of the notation and further details)
\begin{equation*}
\CI{X_{0}}{X_{R}}[X_{S}=x_{S}],
\mathrm{e}nd{equation*}
where $R\cup S=\{1,\mathrm{d}\:\!ots,n\}$. Such statements naturally occur in the study of robustness. Implications of the
algebraic study of generalized binomial edge ideals will be studied in another paper~\perp\!\!\!\perpte{RauhAy12:Robustness_and_CI},
see also~\perp\!\!\!\perpte[Section~4]{HHHKR10:Binomial_Edge_Ideals}. Generalized binomial edge ideals also cover the conditional
independence ideals associated with the intersection axiom in~\perp\!\!\!\perpte{Fink11:Binomial_ideal_of_intersection_axiom}. A
different generalization of the results in~\perp\!\!\!\perpte{Fink11:Binomial_ideal_of_intersection_axiom} was recently studied
in~\perp\!\!\!\perpte{SwansonTaylor11:Minimial_Primes_of_CI_Ideals}. The ideals $I^{\langle 1\rangle}$ defined
in~\perp\!\!\!\perpte{SwansonTaylor11:Minimial_Primes_of_CI_Ideals} are special cases of binomial edge ideal.
\section{The Gröbner basis}
\label{sec:grobner}
Choose a total order $>$ on $\Xcal_\text{\textup{in}}$ (e.g.~choose a bijection $\Xcal_\text{\textup{in}}\cong[N]$). This induces a lexicographic monomial
order on~$\mathrm{e}nsuremath{\mathfrak{R}}$, also denoted by $>$, via
\begin{equation*}
p_{ix}> p_{jy} \qquad\Longleftrightarrow\qquad
\begin{cases}
\text{ either } & i > j,\\
\text{ or } & i = j \text{ and } x > y.
\mathrm{e}nd{cases}
\mathrm{e}nd{equation*}
To construct a Gröbner basis for $I_{G}$ with respect to this order the following definitions are needed:
\begin{defi}
\label{def:admissible-path}
A path $\pi:x=x_0,x_1,\ldots,x_r=y$ from $x$ to $y$ in $G$ is called \mathrm{e}mph{admissible} if
\begin{enumerate}
\mathrm{i}tem[(i)] $x_s\neq x_t$ for $s\neq t$, and $x < y$;
\mathrm{i}tem[(ii)] for each $k=1,\ldots,r-1$ either $x_k<x$ or $x_k>y$;
\mathrm{i}tem[(iii)] for any proper subset $\{y_1,\ldots,y_s\}$ of $\{x_1,\ldots,x_{r-1}\}$, the sequence $x,y_1,\ldots,y_s,y$
is not a path.
\mathrm{e}nd{enumerate}
A function $\kappa: \{0,\mathrm{d}\:\!ots,r\}\to [d]$ is called \mathrm{e}mph{$\pi$-antitone} if it satisfies
\begin{equation*}
x_{s} < x_{t} \Longrightarrow \kappa(s) \ge \kappa(t),\text{ for all }0 \le s,t \le r.
\mathrm{e}nd{equation*}
$\kappa$ is \mathrm{e}mph{strictly $\pi$-antitone} if it is $\pi$-antitone and satisfies $\kappa(0) > \kappa(r)$.
\mathrm{e}nd{defi}
The notion of $\pi$-antitonicity also applies to paths which are not necessarily admissible. However, since admissible
paths are \mathrm{e}mph{injective} (i.e.~they only pass at most once at each vertex), in the admissible case it is possible to
write $\kappa(\mathrm{e}ll)$ instead of $\kappa(s)$, if $\mathrm{e}ll=x_{s}$.
To any $x<y$, any path $\pi: x=x_0,x_1,\ldots,x_r=y$ from $x$ to $y$ and any function $\kappa:\{0,\mathrm{d}\:\!ots,r\}\to\mathcal{X}_{0}$
associate the monomial
\begin{equation*}
u_{\pi}^{\kappa}= \prod_{k=1}^{r-1}p_{\kappa(k)x_k}.
\mathrm{e}nd{equation*}
\begin{thm}
\label{thm:Gbasis}
The set of binomials
\begin{multline*}
{\mathcal G}
= \smash{\bigcup_{i<j} \,\bigg\{}
\,u_{\pi}^{\kappa}f_{xy}^{\kappa(y)\kappa(x)}\,:\; x<y,\; \pi \text{ is an admissible path in }G\text{ from $x$ to $y$},
\\
\kappa\text{ is strictly $\pi$-antitone}\,\smash[t]{\bigg\}}
\mathrm{e}nd{multline*}
is a reduced Gröbner basis of $I_G$ with respect to the monomial order introduced above.
\mathrm{e}nd{thm}
The role of $\pi$-antitonicity is the following: In smaller monomials $\prod_{k=1}^{r} p_{i_{k}x_{k}}$, smaller indices
$i_{k}$ are associated to larger points~$x_{k}$. Hence the initial term of
$u_{\pi}^{\kappa}f_{xy}^{\kappa(y)\kappa(x)}$ is $u_{\pi}^{\kappa}p_{\kappa(y)x}p_{\kappa(x)y}$. This explains why in
the definition of $\mathcal{G}$ the point $x$ is associated to the index $\kappa(y)$, and vice versa. The main idea of
the proof of Theorem~\ref{thm:Gbasis} is that reduction modulo $\mathcal{G}$ changes the association of the indices
$\{i_{k}\}$ and the points $\{x_{k}\}$ until the resulting monomial is minimal. The following lemma is a first step:
\begin{lemma}
\label{lem:pi-antiton}
Let $\pi:x_{0},\mathrm{d}\:\!ots,x_{r}$ be a path in $G$, and let $\kappa:\{0,\mathrm{d}\:\!ots,r\}\to [d]$ be an arbitrary function. If
$\kappa$ is not $\pi$-antitone, then there exists $g\mathrm{i}n\mathcal{G}$ such that $\mathrm{e}nsuremath{\operatorname{ini}_{<}}(g)$ divides the initial term of
$u_{\pi}^{\kappa} f_{xy}^{\kappa(y)\kappa(x)}$.
\mathrm{e}nd{lemma}
\begin{proof}
Let $\tau:y_{0},\mathrm{d}\:\!ots,y_{s}$ be a minimal subpath of $\pi$ with respect to the property that the restriction of
$\kappa$ to $\tau$ is not $\tau$-antitone. This means that $\kappa$ is $\tau_{0}$-antitone and $\tau_{s}$-antitone,
where $\tau_{0}=y_{1},\mathrm{d}\:\!ots,y_{s}$ and $\tau_{s}=y_{0},\mathrm{d}\:\!ots,y_{s-1}$. Assume without loss of generality that
$y_{0}<y_{s}$, otherwise reverse $\tau$. The minimality implies that $\kappa(y_{0})<\kappa(y_{s})$. It follows that
$\tau$ is admissible: By minimality, if $y_{0}<y_{k}<y_{s}$, then $\kappa(y_{k}) \ge \kappa(y_{s}) > \kappa(y_{0}) \ge
\kappa(y_{k})$, a contradiction. Define
\begin{equation*}
\tilde\kappa(k) =
\begin{cases}
\kappa(s), & \text{ if }k=0, \\
\kappa(0), & \text{ if }k=s, \\
\kappa(k), & \text{ if }0<k<s.
\mathrm{e}nd{cases}
\mathrm{e}nd{equation*}
Then $\tilde\kappa$ is $\tau$-antitone, and
$\mathrm{e}nsuremath{\operatorname{ini}_{<}}(u_{\tau}^{\tilde\kappa}f_{y_{0}y_{s}}^{\tilde\kappa(y_{s})\tilde\kappa(y_{0})})$ divides
$\mathrm{e}nsuremath{\operatorname{ini}_{<}}(u_{\pi}^{\tilde\kappa}f_{xy}^{\tilde\kappa(y)\tilde\kappa(x)})$.
\mathrm{e}nd{proof}
\begin{proof}[Proof of Theorem~\ref{thm:Gbasis}]
The proof is organized in three steps.
\noindent
{\mathrm{e}m Step 1: ${\mathcal G}$ is a subset of $I_G$.}
Let $\pi: x=x_0,x_1,\ldots,x_{r-1},x_r=y$ be an admissible path in~$G$. The proof that $u_\pi^{\kappa}
f_{xy}^{\kappa(j)\kappa(i)}$ belongs to $I_G$ is by induction on~$r$. Clearly the assertion is true if $r = 1$, so
assume $r > 1$. Let $A = \{ x_k : x_k < x \}$ and $B = \{ x_\mathrm{e}ll : x_\mathrm{e}ll > y \}$. Then either $A \neq \mathrm{e}mptyset$ or
$B \neq \mathrm{e}mptyset$.
Suppose $A \neq \mathrm{e}mptyset$ and set $x_{k} = \max A$. The two paths $\pi_1 : x_{k}, x_{k-1}, \ldots, x_1, x_0=x$
and $\pi_2 : x_{k}, x_{k+1}, \ldots, x_{r-1}, x_r = y$ in $G$ are admissible.
Let $\kappa_{1}$ and $\kappa_{2}$ be the restrictions of $\kappa$ to $\pi_{1}$ and $\pi_{2}$. Let $a=\kappa(r)$,
$b=\kappa(0)$ and $c=\kappa(k)$. The calculation
\begin{multline*}
(p_{by}p_{ax}-p_{bx}p_{ay})p_{cx_{k}}
\\
=
(p_{bx_{k}}p_{cx} - p_{bx}p_{cx_{k}})p_{ay}
+ (p_{ax_{k}}p_{by} - p_{ay}p_{bx_{k}})p_{cx}
- (p_{ax_{k}}p_{cx} - p_{ax}p_{cx_{k}})p_{by}
\mathrm{e}nd{multline*}
implies that $u_{\pi}^{\kappa}f_{xy}^{ab}$ lies in the ideal generated by $u_{\pi_{1}}^{\kappa_{1}}f_{x_{k}x}^{bc}$,
$u_{\pi_{2}}^{\kappa_{2}}f_{x_{k}y}^{ab}$ and $u_{\pi_{1}}^{\kappa_{1}}f_{x_{k}x}^{ac}$. By induction it lies in
$I_{G}$.
The case $B \neq \mathrm{e}mptyset$ can be treated similarly.
\noindent
{\mathrm{e}m Step 2: ${\mathcal G}$ is a Gröbner basis of $I_G$.}
Let $\pi:x_0,\mathrm{d}\:\!ots,x_{r}$ and $\sigma:y_{0},\mathrm{d}\:\!ots,y_{s}$ be admissible paths in $G$ with $x_{0}<x_{r}$ and
$y_{0}<y_{s}$, and let $\kappa$ and $\mu$ be $\pi$- and $\sigma$-antitone. By Buchberger's criterion it suffices to
show that the $S$-pairs $S := S(u_\pi^{\kappa} f_{x_{0}x_{r}}^{\kappa(r)\kappa(0)}, u_\sigma^{\mu}
f_{y_{0}y_{s}}^{\mu(s)\mu(0})$ reduces to zero.
If $S\neq 0$, then $S$ is a binomial. Write $S=S_{1}-S_{2}$, where $S_{1}=\mathrm{e}nsuremath{\operatorname{ini}_{<}}(S)$.
$S$ is homogeneous with respect to the multidegrees given by
\begin{equation*}
\mathrm{d}\:\!eg(p_{ix})_{j} = \mathrm{d}\:\!elta_{ij} =
\begin{cases}
1, & \text{ if } i=j, \\
0, & \text{ else,}
\mathrm{e}nd{cases}
\mathrm{e}nd{equation*}
and
\begin{equation*}
\mathrm{d}\:\!eg(p_{ix})_{y} = \mathrm{d}\:\!elta_{xy} =
\begin{cases}
1, & \text{ if } x=y, \\
0, & \text{ else}
\mathrm{e}nd{cases}
\mathrm{e}nd{equation*}
(this is a multidegree with $|\mathcal{X}_{0}| + |\Xcal_\text{\textup{in}}|$ components).
If $\pi$ and $\sigma$ are disjoint paths, then $S$ trivially reduces to zero, since
$u_\pi^{\kappa}f_{x_{0}x_{r}}^{\kappa(r)\kappa(0)}$ and $u_\sigma^{\mu} f_{y_{0}y_{s}}^{\mu(s)\mu(0)}$ contain
different variables.
So assume that $\pi$ and $\sigma$ meet and that $S\neq 0$. Then $S_{1}$ and $S_{2}$ are monomials, and the unknowns
$p_{ix}$ occurring in $S_{1}$ and $S_{2}$ satisfy $x\mathrm{i}n\pi\cup\sigma$.
Assume that there are $x<y$ such that $D_{x}:=\min\{i\mathrm{i}n\mathcal{X}_{0}: p_{ix}\,|\, S_{1}\} < \max\{i\mathrm{i}n\mathcal{X}_{0}:
p_{iy}\,|\,S_{1}\}=:D_{y}$. Since $\pi\cup\sigma$ is connected there is an injective path $\tau:z_{0},\mathrm{d}\:\!ots,z_{s}$
from $x=z_{0}$ to $y=z_{s}$ in $\pi\cup\sigma$. Choose a map $\lambda:\{0,\mathrm{d}\:\!ots,s\}\to\mathcal{X}_{0}$ such that
$\lambda(0) = D_{x}$, $\lambda(s) = D_{y}$ and $p_{\lambda(a)a}\,|\,S_{1}$ for all $0\le a\le s$. Then
$u_{\tau}^{\lambda}$ divides $S_{1}$, and $\lambda$ is not $\tau$-antitone. So Lemma~\ref{lem:pi-antiton} applies,
and $S$ can be reduced to a smaller binomial.
Let $S'$ be the reduction of $S$ modulo $\mathcal{G}$. If $S'\neq 0$, then let $S'_{1}=\mathrm{e}nsuremath{\operatorname{ini}_{<}}(S')$. The above argument shows
that $\min\{i\mathrm{i}n\mathcal{X}_{0}: p_{ix}\,|\, S'_{1}\} \ge \max\{i\mathrm{i}n\mathcal{X}_{0}: p_{iy}\,|\,S'_{1}\}$ for all $x<y$. This
property characterizes $S'_{1}$ as the unique minimal monomial in $\mathrm{e}nsuremath{\mathfrak{R}}$ with multidegree $\mathrm{d}\:\!eg(S'_{1}) = \mathrm{d}\:\!eg(S)$. But
since the reduction algorithm turns binomials into binomials, $S'-S'_{1}$ is also a monomial of multidegree $\mathrm{d}\:\!eg(S)$,
and smaller than $\mathrm{d}\:\!eg(S'_{1})$. This contradiction shows $S'=0$.
\noindent
{\mathrm{e}m Step 3: $\mathcal{G}$ is reduced.}
Let $\pi:x_0,\mathrm{d}\:\!ots,x_{r}$ and $\sigma:y_{0},\mathrm{d}\:\!ots,y_{s}$ be admissible paths in $G$ with $x_{0}<x_{r}$ and
$y_{0}<y_{s}$, and let $\kappa$ and $\mu$ be $\pi$- and $\sigma$-antitone.
Suppose that $u_\pi^{\kappa} p_{\kappa(r)x_{0}}p_{\kappa(0)x_{r}}$ divides either $u_\sigma^{\mu} p_{\mu(s)y_{0}}p_{\mu(0)y_{s}}$ or
$u_\sigma^{\mu} p_{\mu(s)y_{s}} p_{\mu(0)y_{0}}$. Then $\{ x_0, \ldots, x_r \}$ is a
subset of $\{ y_0, \ldots, y_s \}$, and $\kappa(b) = \mu(\sigma^{-1}(x_{b}))$ for $0<b<r$.
From admissibility follows $x_{0}\le y_{0}<y_{s}\le x_{r}$ and $\kappa(0)\ge\mu(0) > \mu(s)\ge\kappa(r)$.
If $x_0<y_{0}$, then $p_{\kappa(r)x_{0}}$ divides $u_{\sigma}^{\mu}$, and so $x_{0}=y_{t}$ for some $t<s$ with
$\mu(t)=u=\kappa(r)$. On the other hand, since $y_{t}\le y_{0}$, it follows that $\mu(t)\ge\mu(0) > \kappa(r)$, a
contradiction. Hence $x_{0}=y_{0}$. Similarly, by a symmetric argument, $x_{r}=y_{s}$.
This means that $\pi$ is a sub-path of $\sigma$. By Definition~\ref{def:admissible-path}, $\pi$ equals $\sigma$.
Therefore, $u_\pi^{\kappa} f_{x_{0}x_{r}}^{\kappa(r)\kappa(0)}$ and $u_\sigma^{\mu} f_{y_{0}y_{s}}^{\mu(s)\mu(0)}$ have the
same (total) degree, and hence they agree.
\mathrm{e}nd{proof}
\begin{cor}
\label{cor:radical}
$I_G$ is a radical ideal.
\mathrm{e}nd{cor}
\begin{proof}
The assertion follows from Theorem~\ref{thm:Gbasis} and the following general fact: A homogeneous ideal that has a
Gröbner basis with square-free initial terms is radical. See the proof
of~\perp\!\!\!\perpte[Corollary~2.2]{HHHKR10:Binomial_Edge_Ideals} for details.
\mathrm{e}nd{proof}
\section{The primary decomposition}
\label{sec:primdec}
Since $I_{G}$ is radical, in order to compute the primary decomposition of the ideal it is enough to compute the minimal
primes. From this it will be easy to deduce the irreducible decomposition of the variety $V_{G}$ of $I_{G}$ in the case
of characteristic zero.
The following definition is needed: Two vectors $v,w$ (living in the same $\mathrm{e}nsuremath{\mathbb{K}}$-vector space) are \mathrm{e}mph{proportional}
whenever $v=\lambda w$ or $w=\lambda v$ for some $\lambda\mathrm{i}n\mathrm{e}nsuremath{\mathbb{K}}$. A set of vectors is \mathrm{e}mph{proportional} if each pair
is proportional. Since $\lambda=0$ is allowed, proportionality is not transitive: If $v$ and $w$ are proportional and
if $u$ and $v$ are proportional, then
$u$ and $w$ need not proportional, because $v$ may vanish.
Let $V_{G}$ be the variety of $I_{G}$, which is a subset of $\mathrm{e}nsuremath{\mathbb{K}}^{\mathcal{X}_{0}\times\Xcal_\text{\textup{in}}}$. As usual, elements of
$\mathrm{e}nsuremath{\mathbb{K}}^{\mathcal{X}_{0}\times\Xcal_\text{\textup{in}}}$ will be denoted with the same symbol $p=(p_{ix})_{i\mathrm{i}n\mathcal{X}_{0},x\mathrm{i}n\Xcal_\text{\textup{in}}}$ as the
unknowns in the polynomial ring $\mathrm{e}nsuremath{\mathfrak{R}} = \mathrm{e}nsuremath{\mathbb{K}}[p_{ix}:(i,x)\mathrm{i}n\mathcal{X}_{0}\times\Xcal_\text{\textup{in}}]$. Any
$p\mathrm{i}n\mathrm{e}nsuremath{\mathbb{K}}^{\mathcal{X}_{0}\times\Xcal_\text{\textup{in}}}$ can be written as a $d_{0}\times|\Xcal_\text{\textup{in}}|$-matrix. Each binomial equation in $I_{G}$
imposes conditions on this matrix saying that certain submatrices have rank 1. For a fixed edge $(x,y)$ in $G$ the
equations $f^{ij}_{xy}=0$ for all $i,j\mathrm{i}n\mathcal{X}_{0}$ require that the submatrix $(p_{kz})_{k\mathrm{i}n\mathcal{X}_{0},z\mathrm{i}n\{x,y\}}$
has rank one. More generally, if $K\subseteq G$ is a clique (i.e.~a complete subgraph), then the submatrix
$(p_{kz})_{k\mathrm{i}n\mathcal{X}_{0},z\mathrm{i}n K}$ has rank one. This means that all columns of this submatrix are proportional. The
columns of $p$ will be denoted by $\tilde p_{x}$, $x\mathrm{i}n\Xcal_\text{\textup{in}}$. A point $p$ lies in $V_{G}$ if and only if $\tilde p_{x}$
and $\tilde p_{y}$ are proportional for all edges $(x,y)$ of $G$.
Even if the graph $G$ is connected, not all columns $\tilde p_{x}$ must be proportional to each other, since
proportionality is not a transitive relation. Instead, there are ``blocks'' of columns such that all columns within one
block are proportional.
For any subset $\mathcal{Y}\subseteqeq\Xcal_\text{\textup{in}}$ denote by $G_{\mathcal{Y}}$ the subgraph of $G$ induced by $\mathcal{Y}$.
Then:
\begin{itemize}
\mathrm{i}tem A point $p$ lies in $V_{G}$ if and only if $\tilde p_{x}$ and $\tilde p_{y}$ are proportional whenever
$x,y\mathrm{i}n\mathcal{S}$ lie in the same connected component of $G_{\mathcal{S}}$, where $\mathcal{S}=\{x\mathrm{i}n\Xcal_\text{\textup{in}}:\tilde p_{x}\neq 0\}$.
\mathrm{e}nd{itemize}
Let $V_{G,\mathcal{Y}}$ be the set of all $p\mathrm{i}n\mathrm{e}nsuremath{\mathbb{K}}^{\mathcal{X}_{0}\times\Xcal_\text{\textup{in}}}$ for which
$\tilde p_{x}=0$ for all $x\mathrm{i}n\Xcal_\text{\textup{in}}\setminus\mathcal{Y}$ and for which $\tilde p_{x}$ and $\tilde p_{y}$ are
proportional whenever $x,y\mathrm{i}n\Xcal_\text{\textup{in}}$ lie in the same connected component of $G_{\mathcal{Y}}$. Then
\begin{equation}
\label{eq:decomposition-VG}
V_{G} = \bigcup_{\mathcal{Y}\subseteqeq\Xcal_\text{\textup{in}}} V_{G,\mathcal{Y}}.
\mathrm{e}nd{equation}
The sets $V_{G,\mathcal{Y}}$ are rational irreducible algebraic varieties:
\begin{lemma}
\label{lem:VGY-is-sirreducible}
For any $\mathcal{Y}\subseteqeq\Xcal_\text{\textup{in}}$ the set $V_{G,\mathcal{Y}}$ is the variety of the ideal $I_{G,\mathcal{Y}}$
generated by the monomials
\begin{equation}
\label{eq:monomials}
p_{ix} \qquad\text{for all }x\mathrm{i}n\Xcal_\text{\textup{in}}\setminus\mathcal{Y}\text{ and }i\mathrm{i}n\mathcal{X}_{0},
\mathrm{e}nd{equation}
and the binomials $f_{xy}^{ij}$ for all $i,j\mathrm{i}n\mathcal{X}_{0}$ and all $x,y\mathrm{i}n\mathcal{Y}$ that lie in the same connected
component of $G_{\mathcal{Y}}$. The ideal $I_{G,\mathcal{Y}}$ is prime, and the variety $V_{G,\mathcal{Y}}$ is rational.
\mathrm{e}nd{lemma}
\begin{proof}
The first statement follows from the definition of $V_{G,\mathcal{Y}}$. Write $I^{1}_{G,\mathcal{Y}}$ for the ideal generated by
all monomials \mathrm{e}qref{eq:monomials}, and for any $\mathcal{Z}\subseteqeq\mathcal{Y}$ write $I^{2}_{\mathcal{Z}}$ for the ideal generated by
the binomials $f_{xy}^{ij}$, with $i,j\mathrm{i}n\mathcal{X}_{0}$ and $x,y\mathrm{i}n\mathcal{Z}$. Then $I^{1}_{G,\mathcal{Y}}$ is obviously prime.
Each of the $I^{2}_{\mathcal{Z}}$ is a $2\times2$ determinantal ideal. It is a classical (but difficult) result that this
ideal is the defining ideal of a Segre embedding, and that it is prime (see~\perp\!\!\!\perpte{Sturmfels91:GB_of_toric_varieties}
for a rather modern proof). In fact, both $I^{1}_{G,\mathcal{Y}}$ and $I^{2}_{\mathcal{Z}}$ are geometrically prime, i.e.~they
remain prime over any field extension. Hence the ideal $I_{G,\mathcal{Y}}$ is the sum of the geometrically prime ideals
$I^{1}_{G,\mathcal{Y}}$ and $I^{2}_{\mathcal{Z}}$ for all connected components $\mathcal{Z}$ of $G_{\mathcal{Y}}$, and since the defining
equations of all these ideals involve disjoint sets of unknowns, $I_{G,\mathcal{Y}}$ itself is prime. $V_{G,\mathcal{Y}}$ is
rational, since the varieties of $I^{1}_{G,\mathcal{Y}}$ and $I^{2}_{\mathcal{Z}}$ are rational.
\mathrm{e}nd{proof}
The decomposition~\mathrm{e}qref{eq:decomposition-VG} is not the irreducible decomposition of $V_{G}$, because the union is
redundant. The redundant components can be removed using Lemma~\ref{lem:VGY-is-sirreducible}:
\begin{lemma}
\label{lem:VGYcontainsVGZ}
Let $\mathcal{Y},\mathcal{Z}\subseteqeq\Xcal_\text{\textup{in}}$. Then $V_{G,\mathcal{Y}}$ is contained in $V_{G,\mathcal{Z}}$ if and only
if the following two conditions are satisfied:
\begin{itemize}
\mathrm{i}tem $\mathcal{Y}\subseteqeq\mathcal{Z}$.
\mathrm{i}tem If $x,y\mathrm{i}n\mathcal{Y}$ are connected in $G_{\mathcal{Z}}$, then they are connected in $G_{\mathcal{Y}}$.
\mathrm{e}nd{itemize}
\mathrm{e}nd{lemma}
\begin{proof}
Assume that $V_{G,\mathcal{Y}}\subseteqeq V_{G,\mathcal{Z}}$. Then $I_{G,\mathcal{Y}}\supseteqeq I_{G,\mathcal{Z}}$. For
any $x\mathrm{i}n\Xcal_\text{\textup{in}}\setminus\mathcal{Z}$ and any $i\mathrm{i}n\mathcal{X}_{0}$ this implies $p_{ix}\mathrm{i}n I_{G,\mathcal{Y}}$. On the
other hand, Lemma~\ref{lem:VGY-is-sirreducible} shows that the point with coordinates
\begin{equation*}
p_{iy}=
\begin{cases}
1, & \qquad\text{if }y\mathrm{i}n\mathcal{Y},\\
0, & \qquad\text{else},
\mathrm{e}nd{cases}
\mathrm{e}nd{equation*}
lies in $V_{G,\mathcal{Y}}$ and hence in $V_{G,\mathcal{Z}}$. This implies $x\mathrm{i}n\Xcal_\text{\textup{in}}\setminus\mathcal{Y}$; and so
$\mathcal{Y}\subseteqeq\mathcal{Z}$.
Let $x\mathrm{i}n\mathcal{Y}$.
Choose two linearly independent non-zero vectors $v,w\mathrm{i}n\mathrm{e}nsuremath{\mathbb{K}}^{d_{0}}$. By Lemma~\ref{lem:VGY-is-sirreducible} the
matrix with columns
\begin{equation*}
\tilde p_{y}=
\begin{cases}
v, & \qquad\text{if }y\mathrm{i}n\mathcal{Y}\text{ is connected to }x\text{ in }G_{\mathcal{Y}},\\
w, & \qquad\text{if }y\mathrm{i}n\mathcal{Y}\text{ is not connected to }x\text{ in }G_{\mathcal{Y}},\\
0, & \qquad\text{else},
\mathrm{e}nd{cases}
\mathrm{e}nd{equation*}
is contained in $V_{G,\mathcal{Y}}$ and hence in $V_{G,\mathcal{Z}}$. Therefore, if $z$ is connected to $x$ in
$G_{\mathcal{Z}}$, then it is connected to $x$ in $G_{\mathcal{Y}}$.
Conversely, if both conditions are satisfied, then all defining equations of $I_{G,\mathcal{Z}}$ lie in~$I_{G,\mathcal{Y}}$.
\mathrm{e}nd{proof}
\begin{thm}
\label{thm:primary-decomposition}
The primary decomposition of $V_{G}$ is
$I_{G} = \bigcap_{\mathcal{Y}} I_{G,\mathcal{Y}}$,
where the intersection is over all $\mathcal{Y}\subseteqeq\Xcal_\text{\textup{in}}$ such that the following holds: For any
$x\mathrm{i}n\Xcal_\text{\textup{in}}\setminus\mathcal{Y}$ there are edges $(x,y)$, $(x,z)$ in $G$ such that $y,z\mathrm{i}n\mathcal{Y}$ are not
connected in $G_{\mathcal{Y}}$. Equivalently, for any $x\mathrm{i}n\Xcal_\text{\textup{in}}\setminus\mathcal{Y}$ the induced subgraph
$G_{\mathcal{Y}\cup\{x\}}$ has fewer connected components than $G_{\mathcal{Y}}$.
\mathrm{e}nd{thm}
\begin{proof}
First, assume that $\mathrm{e}nsuremath{\mathbb{K}}$ is algebraically closed. By~\mathrm{e}qref{eq:decomposition-VG} and
Lemma~\ref{lem:VGY-is-sirreducible} it suffices to show that the condition on $\mathcal{Y}$ stated in the theorem
characterizes the maximal sets $V_{G,\mathcal{Y}}$ in the union~\mathrm{e}qref{eq:decomposition-VG} (with respect to inclusion).
This follows from Lemma~\ref{lem:VGYcontainsVGZ}.
If $\mathrm{e}nsuremath{\mathbb{K}}$ is not algebraically closed, then one can argue as follows: By~\perp\!\!\!\perpte{EisenbudSturmfels96:Binomial_Ideals} a
binomial ideal has a binomial primary decomposition over some algebraic extension field
$\hat\mathrm{e}nsuremath{\mathbb{K}}=\mathrm{e}nsuremath{\mathbb{K}}[\alpha_{1},\mathrm{d}\:\!ots,\alpha_{k}]$. The algebraic numbers $\alpha_{1},\mathrm{d}\:\!ots,\alpha_{k}$ are coefficients of
the defining equations of the primary components. Let $\mathrm{e}nsuremath{\overline{\mathbb{K}}}$ be the algebraic closure of $\mathrm{e}nsuremath{\mathbb{K}}$. Since the ideals
$I_{G,\mathcal{Y}}$ are defined by pure differences and since the ideals $\mathrm{e}nsuremath{\overline{\mathbb{K}}}\otimes I_{G,\mathcal{Y}}$ are the primary components
of $\mathrm{e}nsuremath{\overline{\mathbb{K}}}\otimes I_{G,\mathcal{Y}}$ in $\mathrm{e}nsuremath{\overline{\mathbb{K}}}\otimes\mathrm{e}nsuremath{\mathfrak{R}}$ it follows that the ideals $I_{G,\mathcal{Y}}$ are already the primary
components of $I_{G}$ (in other words, the primary decomposition is independent of the base field).
\mathrm{e}nd{proof}
\begin{rem}[Comparison to~\perp\!\!\!\perpte{Ohtani11:Ideals_of_some_2-minors,HHHKR10:Binomial_Edge_Ideals}]
\label{rem:comparison-to-HHH}
Both~\perp\!\!\!\perpte{Ohtani11:Ideals_of_some_2-minors} and~\perp\!\!\!\perpte{HHHKR10:Binomial_Edge_Ideals} discuss Gröbner bases and primary
decompositions of binomial edge ideals with $d_{0}=2$.
Theorem~\ref{thm:Gbasis} generalizes Theorems~2.1 from~\perp\!\!\!\perpte{HHHKR10:Binomial_Edge_Ideals} and Theorem~3.2
in~\perp\!\!\!\perpte{Ohtani11:Ideals_of_some_2-minors}. While the proofs in~\perp\!\!\!\perpte{HHHKR10:Binomial_Edge_Ideals}
and~\perp\!\!\!\perpte{Ohtani11:Ideals_of_some_2-minors} use a case by case analysis, the proof of Theorem~\ref{thm:Gbasis} is more
conceptual.
The primary decomposition in Theorem~\ref{thm:primary-decomposition} generalizes Theorem~3.2
from~\perp\!\!\!\perpte{HHHKR10:Binomial_Edge_Ideals}. The proof of Theorem~\ref{thm:primary-decomposition} relied on the
irreducible decomposition of the corresponding variety, while the proof
in~\perp\!\!\!\perpte{HHHKR10:Binomial_Edge_Ideals} directly shows the equality of the two ideals.
Instead of describing the primary decomposition explicitly, \perp\!\!\!\perpte{Ohtani11:Ideals_of_some_2-minors} presents an
algorithm to compute the primary decomposition. Since the primary decomposition of a binomial edge ideal is
independent of~$d_{0}$, the same algorithm applies for all~$d_{0}$. A nice feature of the algorithm is that it works
graph-theoretically.
\mathrm{e}nd{rem}
\mathrm{e}nd{document} |
\begin{document}
\title{Improving fidelity of continuous-variable teleportation via local
operations}
\author{Jarom\'{\i}r Fiur\'{a}\v{s}ek}
\affiliation{Department of Optics, Palack\'{y} University, 17. listopadu
50, 77200 Olomouc, Czech Republic}
\begin{abstract}
We study the Braunstein-Kimble setup for teleportation of quantum state
of a single mode of optical field. We assume that the sender and receiver
share a two-mode Gaussian state and we identify optimum local Gaussian
operations that maximize the teleportation fidelity.
We consider fidelity of teleportation of pure Gaussian states and
we also introduce fidelity of the teleportation transformation.
We show on an explicit example that in some cases the optimum local operation
is not a simple unitary symplectic transformation but some more
general completely positive map.
\end{abstract}
\pacs{03.67.-a, 42.50.Dv}
\maketitle
\section{Introduction}
Quantum state teleportation is undoubtedly one of the most exciting
developments in the rapidly growing field of Quantum Information
Processing. In quantum teleportation, the information about the teleported
quantum state is transferred from the sender, Alice,
to the receiver, Bob, via dual classical and quantum EPR
channels \cite{Bennett93}. The latter is
established via an entangled state shared by Alice and Bob.
The teleportation protocol goes as follows:
Alice carries out a Bell-type measurement on the state she wants
to teleport and her part of the shared entangled state. She sends the result
of her measurement via classical channel to Bob, who applies to his part of
entangled state a transformation which depends on the classical information
received from Alice.
The teleportation is perfect and Bob recovers an exact copy of the state
teleported to him by Alice only if the quantum channel is ideal
maximally entangled state. If we deal with qubits represented by polarization
states of photons, then we can employ pair of polarization-entangled photons
generated by means of spontaneous parametric down-conversion,
where the entanglement is almost perfect \cite{Zeilinger97,Boschi98}.
However, in case of continuous quantum variables
\cite{Vaidman94,Braunstein98},
an ideal EPR channel is an unphysical infinitely squeezed state.
In quantum optics, the available resource is a two-mode squeezed
vacuum state with some finite degree of squeezing
\cite{Braunstein98,Furusawa98}. Moreover,
the parts of the entangled state must be distributed among Alice and
Bob, e.g., through optical fibers. This transmission inevitably
introduces losses and noise and the entangled state shared by Alice and
Bob will be some mixed state in general.
An important question is whether one can somehow improve the quality of
the teleportation by means of local operations on the parts of the
shared entangled state. Recently, this problem has been studied
for teleportation of qubits and it was shown that local
transformations may indeed be helpful
\cite{Banaszek00,Badziag00,Rehacek01}. Moreover, it was demonstrated that the
optimum local transformation that maximizes the average teleportation
fidelity need not be simple unitary transformation, but some completely
positive (CP) map \cite{Badziag00,Rehacek01}. In other words,
it may be advantageous to let the parts of the shared quantum state
interact with local ancillas.
In this paper, we investigate how to improve the fidelity
of teleportation of continuous quantum variables by means of local operations
on sender's and receiver's side. The first steps in this direction were
already taken. Bowen {\em et al.} showed that in certain cases
the fidelity of teleportation of coherent or squeezed states may be improved
when Alice and Bob locally apply squeezing transformations to their parts of
the shared quantum state \cite{Bowen01}. Kim and Lee considered
an asymmetric mixed quantum channel and showed that in that case
the fidelity of teleportation of coherent states may be enhanced
when a local transformation accompanied by decoherence is applied to
one part of the quantum channel \cite{Kim01}.
To make the problem tractable,
we restrict ourselves to the class of trace-preserving Gaussian CP
maps \cite{Demoen77,Lindblad00}. These maps preserve the Gaussian shape
of the Wigner function of the transformed state. The restriction
to Gaussian CP maps is very reasonable from the experimental point of view,
because these maps can be implemented in the laboratory
as a unitary symplectic transformation (linear canonical
transformation of quadrature operators) on the signal mode and auxiliary
modes initially prepared in some Gaussian states.
In quantum optical setups this can be done with the help of phase shifters,
beam splitters and squeezers. Gaussian CP maps were recently applied to
description of cloning of continuous quantum variables
\cite{Lindblad00}. Another recent paper discussed the conditions under
which a given two-mode shared Gaussian state can be transformed into another
Gaussian state by means of local Gaussian CP maps \cite{Eisert01}.
The paper is organized as follows. In Sec. II we will briefly describe the
Braunstein-Kimble teleportation setup and we will derive compact
formulas for fidelities of teleportation of any pure Gaussian state.
We shall also introduce a fidelity for the teleportation operation itself.
It will turn out that this latter fidelity can be interpreted as a fidelity of
entanglement swapping. In Sec. III we will briefly review the properties of
Gaussian CP maps and we will derive optimum local Gaussian CP map
which maximizes a chosen teleportation fidelity. We shall consider two
scenarios: in the first case the transformation is applied only on one
side, in the second case both Alice and Bob may locally apply some
CP maps. In Sec. IV we present an example of our optimization procedure.
Finally, Sec. V contains conclusions.
\section{Fidelities}
We shall consider the Braunstein-Kimble setup for
teleportation of a single mode of optical field \cite{Braunstein98}.
The quantum channel between Alice and Bob is established via two-mode
entangled state $\rho_{AB}$ fully described by its Wigner function
$W_{AB}(x_A,p_A,x_B,p_B)$. Alice mixes the mode whose state she wants to
teleport with her part of entangled state on balanced beam splitter and
she carries out a homodyne detection on each output mode thereby
measuring two commuting quadratures $X_{+}=(x_{\rm in}+x_A)\sqrt{2}$ and
$P_{-}=(p_{\rm in}-p_A)/\sqrt{2}$. After receiving the measured values of
$X_{+}$ and $P_{-}$ from Alice,
Bob displaces his part of entangled state as follows:
$x_B\rightarrow x_B+\sqrt{2}X_{+}$, $p_B\rightarrow p_B+\sqrt{2}P_{-}$.
We assume ideal homodyne detectors on Alice's side and a zero coherent
component of the entangled state $\rho_{AB}$ (mean values of all quadratures
$x_{A,B}$, $p_{A,B}$ vanish). Under these conditions the resulting state
on Bob's side possesses the same coherent component as the original state
teleported to him by Alice and the teleportation is invariant under
displacement transformation.
Ide {\em et al.} \cite{Ide01} showed that the fidelity of continuous
variable (CV) teleportation
can be improved by optimizing the gain $g$ in the modulation of the output
field whose quadratures are displaced by the amount $gX_{+}$ and $gP_{-}$.
However, in this case the teleportation is not
in general invariant under displacement transformation and, for
instance, the fidelity of teleportation
of coherent state $|\alpha\rangle$ depends on the intensity
$|\alpha|^2$. Here we keep the gain $g$ fixed and improve the
teleportation fidelity by suitable local transformations of the shared
entangled state. Assuming fixed gain $g=\sqrt{2}$, the relation between input
and output Wigner functions of the teleported state is given by convolution
\cite{Chizhov02}
\begin{equation}
W_{\rm out}(x_2,p_2)=\int_{-\infty}^{\infty}K(x_2-x_1,p_2-p_1) W_{\rm in}(x_1,p_1)
dx_1 d p_1.
\label{WOUT}
\end{equation}
In order to express the kernel $K$ it is convenient to rewrite $W_{AB}$
as a function of the variables $x_{\pm}=x_A \pm x_B$ and
$p_{\pm}=p_A \pm p_B$,
\begin{equation}
W_{AB}(x_A,p_A,x_B,p_B)= {\cal{W}}_{AB}(x_+, p_+,x_-,p_-).
\label{WABCAL}
\end{equation}
With the help of ${\cal{W}}_{AB}$ we can write
\begin{equation}
K(x_+,p_-)= \frac{1}{4}\int_{-\infty}^{\infty}{\cal{W}}_{AB}(x_+, p_+,x_-,-p_-) dx_- d p_+ .
\label{K}
\end{equation}
In what follows we shall assume that the shared quantum state $\rho_{AB}$
is two-mode Gaussian state. This is reasonable assumption since this
class of states can be prepared in the lab.
It is computationally convenient to deal with characteristic function
of this state, defined as Fourier transform of the Wigner function,
\begin{equation}
W_{AB}({\bm r})=\frac{1}{(2\pi)^4}\int_{-\infty}^{\infty}w_{AB}({\bm q})\exp(i {\bm q}
\cdot {\bm r}) \,d^4 {\bm q} ,
\label{WABFOURIER}
\end{equation}
where ${\bm r}=(x_A, p_A, x_B, p_B)$
and ${\bm q}=(\xi_A, \eta_A,\xi_B,\eta_B)$ are real vectors.
For Gaussian state with vanishing coherent component we have
\cite{Perina91}
\begin{equation}
w_{AB}({\bm q})= \exp\left[-\frac{1}{4} {\bm q} {\bm \Gamma}_{AB}
{\bm q}^T\right].
\label{WABGAUSS}
\end{equation}
The elements of the covariance matrix $\bm \Gamma_{AB}$ are given by
\begin{equation}
\Gamma_{AB,ij}= \langle \Delta r_i \Delta r_j\rangle +\langle
\Delta r_j \Delta r_i \rangle,
\end{equation}
where $\Delta r_j =r_j-\langle r_j\rangle $ (note that if the coherent
component of the state vanishes then $\langle r_j \rangle=0$).
We can express the real covariance matrix ${\bm\Gamma}_{AB}$ in terms
of three $2\times 2$ matrices $\bm A$, $\bm B$, and $\bm C$,
\begin{equation}
{\bm\Gamma}_{AB}= \left(
\begin{array}{cc}
{\bm A} & {\bm C} \\
{\bm C}^{T} & {\bm B}
\end{array}
\right).
\label{GAMMAAB}
\end{equation}
Here $\bm A$ and $\bm B$ are covariance matrices of the single modes on
Alice's and Bob's side, respectively, and $\bm C$ contains the
inter-modal correlations.
Now consider teleportation of a pure single-mode Gaussian state with
covariance matrix $\bm D$.
Since the teleportation is invariant under displacement
transformation, all states with the same covariance matrix but
different coherent components are teleported with the same fidelity. It thus
suffices to consider state with vanishing coherent amplitude,
whose characteristic function reads
\begin{equation}
w_{\rm in}({\bm q}_{\rm in})= \exp\left[-\frac{1}{4} {\bm q}_{\rm in} \bm D
{\bm q}_{\rm in}^T\right],
\label{WINGAUSS}
\end{equation}
where ${\bm q}_{\rm in}=(\xi_{\rm in}, \eta_{\rm in})$.
Fidelity of teleportation of a pure state can be calculated as an
overlap integral of input and output Wigner functions
over the whole phase space
\begin{equation}
F= 2\pi \int_{-\infty}^{\infty}W_{\rm in}(x,p) W_{\rm out}(x,p) dx dp.
\label{FDEF}
\end{equation}
After making use of the formulas (\ref{WOUT}) and (\ref{K}),
expressing all Wigner functions as Fourier transforms of the characteristic
functions and carrying out all integrals, we arrive at a compact formula
for the teleportation fidelity,
\begin{equation}
F= \frac{2}{\sqrt{\det \bm{E}}},
\label{F}
\end{equation}
where the matrix $\bm E$ reads
\begin{equation}
{\bm E}=2 {\bm D}+ {\bm R \bm A \bm R}^T + {\bm R \bm C}+
{\bm C}^T{\bm R}^T +{\bm B}
\label{E}
\end{equation}
and
\begin{equation}
{\bm R}=\left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right).
\label{R}
\end{equation}
Besides the fidelity of teleportation of certain class of Gaussian
states, one can introduce the fidelity of the teleportation process
itself. How this can be accomplished becomes clear when one notices that
the teleportation transformation (\ref{WOUT}) is a trace-preserving CP
map \cite{Takeoka02}.
Any CP map can be represented by positive
semidefinite operator $\chi$ on a Hilbert space which is a tensor product
of the Hilbert space of input states $\cal{H}$ and Hilbert space
of output states $\cal{K}$ \cite{Jamiolkowski72}.
This representation is not only mathematical, the state $\chi$ can be
actually prepared in the lab if we first prepare a maximally entangled
state on Hilbert space ${\cal{H}}^{\otimes 2}$
and then apply the CP map to one part of the entangled state.
In case of CV teleportation, the maximally entangled state is the EPR
state
\begin{equation}
W_{\rm EPR}=\frac{1}{2\pi}\delta(x_1-x_2)\delta(p_1+p_2)
\label{WEPR}
\end{equation}
and the teleportation of one part of that state can
be interpreted as an entanglement swapping \cite{Loock99,Tan99}.
Hence the fidelity we obtain in this way is the fidelity of
entanglement swapping of the EPR state.
Formally, the CP map that transforms input density matrix $\rho_{\rm in}$
onto output density matrix $\rho_{\rm out}$ can be written as a partial
trace over the input Hilbert space,
\begin{equation}
\rho_{\rm out} = {\rm Tr}_{\cal H}[\chi\, \rho_{\rm in}^T \otimes
\openone_{\cal{K}}].
\label{RHOOUT}
\end{equation}
In our case, the Wigner function $W_{\rm tel}$ of the teleportation
CP map $\chi_{\rm tel}$
is closely related to the kernel $K$ because the convolution (\ref{WOUT}) is
essentially the partial trace (\ref{RHOOUT}) rewritten in terms of
Wigner functions,
\begin{equation}
W_{\rm tel}= \frac{1}{2\pi}K(x_2-x_1,p_2+p_1).
\label{WTEL}
\end{equation}
Notice the change of sign in front of $p_1$ which reflects the transposition
in Eq. (\ref{RHOOUT}). The ideal teleportation is an identity map represented
by the EPR state (\ref{WEPR}). Now since the CP maps
are represented by positive semidefinite operators and since the ideal
transformation is represented by a pure state (\ref{WEPR}), we can calculate
the fidelity between the ideal and actual teleportation as fidelity of these
two states \cite{Raginsky01}. Thus we can write
\begin{eqnarray}
{\cal{F}}_\chi&=& 4\pi^2\int_{-\infty}^{\infty}W_{\rm EPR}(x_1,p_1,x_2,p_2) \nonumber \\
&&\qquad\quad\times W_{\rm tel}(x_1,p_1,x_2,p_2) d x_1 d p_1 dx_2 d p_2.
\nonumber \\
\label{FSWAPDEF}
\end{eqnarray}
On inserting the explicit formulas (\ref{WEPR}) and (\ref{WTEL}) into
Eq. (\ref{FSWAPDEF}) we obtain
\begin{equation}
{\cal{F}}_\chi= K(0,0) \int_{-\infty}^{\infty}d x d p.
\label{FSWAP}
\end{equation}
We can see that $\cal{F}_\chi$ is infinite, as could have been expected since
we work in infinite dimensional Hilbert space. Nevertheless,
the fidelity (\ref{FSWAP}) can be renormalized. If we drop an infinite
constant proportional to Dirac delta function and multiply by $2\pi$,
then we obtain
\begin{equation}
{\cal{F}}=2\pi K(0,0).
\label{FSWAPREN}
\end{equation}
If we insert the explicit formula (\ref{K}) for kernel $K$
into Eq. (\ref{FSWAPREN}), then we find that
\begin{equation}
{\cal{F}}=2\pi\int_{-\infty}^{\infty}W_{AB}(x,p,-x,p) dx dp.
\label{FSWAPBLA}
\end{equation}
For Gaussian quantum channels (\ref{WABGAUSS}), this formula simplifies to
\begin{equation}
{\cal{F}}= \frac{2}{\sqrt{\det {\bm E}\bm '}}.
\label{FSWAPLAST}
\end{equation}
where the matrix $\bm E\bm '$ reads
\begin{equation}
{\bm E \bm '}=\bm R \bm A {\bm R}^T + \bm R \bm C
+ {\bm C}^T {\bm R}^T +\bm B.
\label{ETILDE}
\end{equation}
The expression for the fidelity $\cal{F}$ is
a rather special case of the formula for the fidelity of teleportation
of pure Gaussian states (\ref{F}) where we set the covariance matrix
$\bm D$ equal to zero.
Of course, this means that $\cal{F}$ is unbounded. Nevertheless, $\cal{F}$
is a good measure of the quality of teleportation. For instance it can be
shown that ${\cal{F}}>1$ only if the state $\rho_{AB}$ is entangled
(see Appendix). In particular, if $\rho_{AB}$ is two-mode squeezed vacuum
state parametrized by squeezing constant $r$ then one gets
\begin{equation}
{\cal{F}}= \exp(2r),
\label{FSWAPSQUEEZED}
\end{equation}
hence the fidelity monotonically exponentially grows with the squeezing.
\section{Optimum local Gaussian CP map}
Our task is to maximize the fidelity of teleportation (either $F$ or
$\cal{F}$) by means of local Gaussian trace-preserving CP maps
on Alice's and Bob's side. We shall consider two scenarios:
in the first, simpler scenario the CP map is applied only on Bob's side
while in the second case both Alice and Bob may locally apply some CP maps.
Gaussian CP maps are those maps for which the Wigner function of the
corresponding operator $\chi$ has a Gaussian form.
The teleportation with Gaussian quantum channel is an example
of Gaussian CP map \cite{Takeoka02}.
The Wigner function representing single-mode trace-preserving
Gaussian CP map reads
\begin{equation}
W_\chi=\frac{1}{2 \pi^{2}\sqrt{\det \bm G}} \exp\left(
-\Delta \bm r^T \bm G^{-1} \Delta \bm r \right),
\label{WCHI}
\end{equation}
where $\bm S$ and $\bm G$ are real $2\times 2$ matrices,
moreover, $\bm G$ is symmetric positive semidefinite matrix,
\begin{equation}
\Delta \bm r = \bm r_{\rm out}- \bm S \bm r_{\rm in}^\ast ,
\end{equation}
and
\[
\bm r_{\rm out}=\left(
\begin{array}{c}
x_{\rm out} \\
p_{\rm out}
\end{array}
\right), \qquad
\bm r_{\rm in}^\ast = \left(
\begin{array}{c}
x_{\rm in} \\
-p_{\rm in}
\end{array}
\right)
\]
are column vectors of output
and input quadratures, respectively.
Since we deal with Gaussian
states whose form is invariant under Gaussian CP maps, it suffices to
provide rule for transformation of the covariance matrix $\bm \Gamma$.
The relation beteween input and output single-mode
covariance matrices $\bm\Gamma_{\rm in}$ and $\bm \Gamma_{\rm out}$ is
given by a simple linear map \cite{Lindblad00}
\begin{equation}
\bm \Gamma_{\rm out} =\bm S\bm \Gamma_{\rm in} \bm S^T + \bm G.
\label{CPMAP}
\end{equation}
The map (\ref{WCHI}) is completely positive
if and only if $\bm S$ and $\bm G$ satisfy an inequality
\cite{Lindblad00}
\begin{equation}
\bm G +i\bm \Sigma-i \bm S \bm \Sigma {\bm S}^T \geq 0,
\label{CPCOND}
\end{equation}
where
\begin{equation}
\bm \Sigma= \left(
\begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array}
\right).
\label{SIGMA}
\end{equation}
The condition (\ref{CPCOND}) can be derived as follows:
the Wigner function (\ref{WCHI}) must represent
a positive semidefinite operator, which imposes constraint on the
covariance matrix $\bm G$ \cite{Holevo82}. Namely, the matrix
\begin{equation}
M_{ij}=G_{ij}+[\Delta r_i,\Delta r_j],
\label{MIJ}
\end{equation}
must be positive semidefinite, where $[,]$ stands for commutator.
Making use of canonical commutation relations for the quadratures
$x_{\rm in},$ $p_{\rm in}$ and $x_{\rm out},$ $p_{\rm out}$
one arrives after some algebra at the inequality
(\ref{CPCOND}).
Assume now that a Gaussian CP map (\ref{WCHI}) is applied to Bob's part
of shared two-mode state $\rho_{AB}$. This modifies the covariance matrix
$\bm \Gamma_{AB}$,
\begin{equation}
\bm \Gamma_{AB}=\left(
\begin{array}{cc}
\bm A & \bm C\bm S^T \\
\bm S\bm C^T & \bm S\bm B\bm S^T+\bm G
\end{array}
\right),
\label{GAMMACP}
\end{equation}
The maximization of the fidelity then amounts to the
minimization of the determinant
\begin{equation}
{\cal{D}}=\det[2 \bm D+\bm R \bm A \bm R^T + \bm R\bm C\bm S^T
+ \bm S\bm C^T \bm R^T +\bm S\bm B\bm S^T+\bm G]
\label{DET}
\end{equation}
under the constraints (\ref{CPCOND}). Recall that on setting $\bm D=0$
we obtain as a special case the fidelity of entanglement
swapping (\ref{FSWAPLAST}).
We divide the optimization of the CP map into two steps. In the first
step we find optimum $\bm G$ for a given matrix $\bm S$ and then we shall
optimize over all possible matrices $\bm S$. Since the matrices $\bm S$
and $\bm G$ have altogether seven independent elements,
\begin{equation}
\bm G=\left(
\begin{array}{cc}
g_{11} & g_{12} \\
g_{12} & g_{22}
\end{array}
\right),
\qquad
\bm S=\left(
\begin{array}{cc}
s_{11} & s_{12} \\
s_{21} & s_{22}
\end{array}
\right),
\label{GSMATRIX}
\end{equation}
we have to find a global minimum of a function of seven
real variables under the constraint (\ref{CPCOND}),
which can be equivalently expressed as
\begin{equation}
g_{11}\geq 0 , \qquad g_{22}\geq 0
\label{GPOS}
\end{equation}
and
\begin{equation}
g_{11}g_{22}-g_{12}^2 -(1-s)^2 \geq 0,
\label{DETPOS}
\end{equation}
where $s=s_{11}s_{22}-s_{12}s_{21}$.
We introduce a short-hand notation for the elements of matrix
\begin{equation}
2 \bm D+\bm R \bm A \bm R^T + \bm R\bm C\bm S^T+ \bm S\bm C^T \bm R^T
+\bm S\bm B\bm S^T=
\left( \begin{array}{cc}
\alpha & \gamma \\
\gamma & \beta
\end{array}
\right).
\label{MATRIX}
\end{equation}
Notice that this matrix is, by definition, positive semidefinite,
and its elements $\alpha$, $\beta$, $\gamma$ are functions of $s_{ij}$.
Thus we can write the determinant (\ref{DET}) in a compact form,
\begin{equation}
{\cal{D}}=(\alpha+g_{11})(\beta+g_{22}) -(g_{12}+\gamma)^2.
\label{DETA}
\end{equation}
It is always optimal to choose ``extremal'' matrix $\bm G$
that satisfies the inequality (\ref{DETPOS}) as an equality. Indeed,
if a sharp inequality holds in (\ref{DETPOS}), then we can reduce the value of
diagonal elements $g_{11}$ and $g_{22}$ until the equality is reached in
(\ref{DETPOS}) and this would obviously reduce also the value of $\cal{D}$.
Hence we can write
\begin{equation}
g_{12}=\pm\sqrt{g_{11}g_{22}-(1-s)^2}
\label{GOFFDIAG}
\end{equation}
and insert into (\ref{DETA}). Furthermore, we can see that it is optimal to
choose the sign of $g_{12}$ the same as the sign of $\gamma$ and we have,
\begin{equation}
{\cal{D}}=(\alpha+g_{11})(\beta+g_{22})
-(\sqrt{g_{11}g_{22}-(1-s)^2}+|\gamma|)^2.
\label{DETB}
\end{equation}
Upon solving the set of two nonlinear extremal equations
\begin{equation}
\frac{\partial {\cal{D}}}{\partial g_{11}}=0, \qquad
\frac{\partial {\cal{D}}}{\partial g_{22}}=0,
\label{PARTIAL}
\end{equation}
we find that the optimum matrix $\bm G$ is proportional to the matrix
(\ref{MATRIX}),
\begin{eqnarray}
\bm G=\frac{|1-s|}{\sqrt{\alpha\beta-\gamma^2}}
\left(
\begin{array}{cc}
\alpha & \gamma \\
\gamma & \beta
\end{array}
\right).
\label{GOPT}
\end{eqnarray}
On inserting the elements of the optimum $\bm G$ back into Eq. (\ref{DETB})
we finally obtain
\begin{equation}
{\cal{D}}= \left(|1-s|+\sqrt{\alpha\beta-\gamma^2}\right)^2.
\label{DETC}
\end{equation}
Now $\cal{D}$ is a function of four variables $s_{11}$, $s_{22}$,
$s_{12}$ and $s_{21}$ and we have to find its {\em global} minimum.
In general, such optimization is a hard task and can be solved only
numerically. However, we shall see that when making some assumptions
we will be able to solve this problem analytically.
It is well known that by means of local symplectic transformations it is
possible to bring any two-mode covariance matrix $\bm \Gamma_{AB}$ into
tridiagonal form \cite{Simon00},
\begin{equation}
\bm \Gamma_{AB}=
\left(
\begin{array}{cccc}
a & 0 & c_1 & 0 \\
0 & a & 0 & c_2 \\
c_1 & 0 & b & 0 \\
0 & c_2 & 0 & b
\end{array}
\right).
\label{GAMMAABBIG}
\end{equation}
It suffices to consider Gaussian quantum channels for which
all the matrices $\bm A$, $\bm B$ and $\bm C$ in (\ref{GAMMAAB}) are
diagonal. Further assume that also the covariance matrix $\bm D$ of the
teleported state is diagonal, ${\bm D}={\rm diag}(d_{11},d_{22})$.
Note that this assumption is not a serious restriction,
because any $\bm D$ can be diagonalized by means
of reversible symplectic transformation.
In this case it can be shown that the necessary conditions on extremum
\begin{equation}
\frac{\partial {\cal{D}}}{\partial s_{12}}=0, \qquad
\frac{\partial {\cal{D}}}{\partial s_{21}}=0,
\label{PARTIALS}
\end{equation}
are satisfied when $s_{12}=s_{21}=0$. One has to be a bit
careful here because there is an absolute value in Eq. (\ref{DETC})
and three cases must be distinguished: (i) $1-s>0$,
(ii) $1-s<0$, and (iii) $s=1$.
In all cases, the conditons on extremum (\ref{PARTIALS})
are satisfied when $s_{12}=s_{21}=0$.
We are thus lead to make the hypothesis that if the matrices
$\bm A,\bm B, \bm C, \bm D$
are diagonal, then the optimum $\bm G$ and $\bm S$ are also diagonal.
When $\bm S$ is diagonal, then $\gamma=0$, and the matrix elements $\alpha$
and $\beta$ become quadratic functions of $s_{11}$ and $s_{22}$,
respectively,
\begin{equation}
\begin{array}{c}
\alpha(s_{11})= 2d_{11}+a+2c_1 s_{11} +bs_{11}^2, \\[2mm]
\beta(s_{22})= 2d_{22}+a-2c_2 s_{22} +bs_{22}^2.
\end{array}
\label{ALPHABETA}
\end{equation}
For the sake of notational simplicity we define $x=s_{11}$ and $y=s_{22}$
and we must minimize the function
\begin{equation}
f(x,y)= |1-xy|+\sqrt{\alpha(x)\beta(y)}.
\label{FXY}
\end{equation}
The extremal equations are obtained by setting the partial derivatives
of $f(x,y)$ equal to zero,
\begin{equation}
x=\pm \sqrt{\frac{\alpha(x)}{\beta(y)}} (by-c_2),
\qquad
y=\pm \sqrt{\frac{\beta(y)}{\alpha(x)}} (bx+c_1),
\label{XY}
\end{equation}
where the signs $+$ and $-$ correspond to the cases when $1-xy>0$ and
$1-xy<0$, respectively.
From the product of the formulas for $x$ and $y$, we can express $y$
in terms of $x$,
\begin{equation}
y=\frac{c_2(bx+c_1)}{x(b^2-1)+bc_1}.
\label{Y}
\end{equation}
Substituting this formula back into the second Eq. (\ref{XY}) and
squaring that equation, we arrive at
\begin{equation}
c_2^2 \, \alpha(x)=[x(b^2-1)+bc_1]^2\beta\left(
\frac{c_2(bx+c_1)}{x(b^2-1)+bc_1} \right).
\label{XEQUATION}
\end{equation}
This is a quadratic equation for $x$ and can be solved analytically.
In this way we identify all potential minima outside the boundary
$xy=1$. It remains to localize minima on the boundary where
$y=1/x$ and we must minimize the function
$\alpha(x)\beta(1/x)$. The condition on extremum
\begin{equation}
\frac{d}{dx} \left[\alpha(x)\beta\left(\frac{1}{x}\right)\right]=0
\label{QUARTIC}
\end{equation}
reduces to quartic equation for $x$. Upon solving this equation we get
positions of all possible minima on the boundary, i.e., we determine all
potentially optimum symplectic transformations.
Let us now consider a more general protocol, where both Alice and Bob
are allowed to apply some local Gaussian CP maps.
To make the problem tractable, we do not assume any communication
between Alice and Bob at this stage, hence they both apply
their local operations independently. Furthermore, we shall assume
that all relevant matrices are diagonal,
hence we shall seek the optimum two-mode CP map in the form
\begin{equation}
\bm S=\left(
\begin{array}{cccc}
u & 0 & 0 & 0 \\
0 & v & 0 & 0 \\
0 & 0 & x & 0 \\
0 & 0 & 0 & y
\end{array}
\right),
\end{equation}
\begin{equation}
\bm G=\left(
\begin{array}{cccc}
g_{A,11} & 0 & 0 & 0 \\
0 & g_{A,22} & 0 & 0 \\
0 & 0 & g_{B,11} & 0 \\
0 & 0 & 0 & g_{B,22}
\end{array}
\right).
\label{SGTWOMODE}
\end{equation}
The covariance matrix $\bm \Gamma_{AB}$ transforms according to
\begin{equation}
\bm \Gamma_{AB} \rightarrow \bm S\bm \Gamma_{AB} \bm S^T +\bm G.
\label{CPMAPTWOMODE}
\end{equation}
From Eq. (\ref{DETPOS}) where the equality should hold and
where $g_{12}=0$, we obtain the following relations between the
elements of the optimum matrix $\bm G$,
\begin{equation}
g_{A,11}g_{A,22}=(1-uv)^2, \qquad
g_{B,11}g_{B,22}=(1-xy)^2.
\label{GABMIN}
\end{equation}
The determinant $\cal{D}$ can be expressed as
\begin{equation}
{\cal{D}}=(\alpha+g_{A,11}+g_{B,11})\left(\beta+\frac{(1-uv)^2}{g_{A,11}}
+\frac{(1-xy)^2}{g_{B,11}}\right).
\label{DETD}
\end{equation}
This function attains its global minimum when
\begin{equation}
g_{A,11}= |1-uv|\sqrt{\frac{\alpha}{\beta}}, \qquad
g_{B,11}= |1-xy|\sqrt{\frac{\alpha}{\beta}}.
\label{GABOPT}
\end{equation}
On inserting these expressions back into Eq. (\ref{DETD}), we get
\begin{equation}
{\cal{D}}=(|1-xy|+|1-uv|+\sqrt{\alpha\beta})^2,
\label{DETE}
\end{equation}
where $\alpha$ and $\beta$ are functions of four real variables
$u,v,x,y$, the elements of matrix $\bm S$. In general, the minimum of the
function (\ref{DETE}) must be found numerically. In what follows we shall
focus on the fidelity of entanglement swapping and we shall see that in
this case one can find the global minimum analytically.
The square root of the determinant (\ref{DETE}) that we must minimize
reads in this case ($d_{11}=d_{22}=0$)
\begin{eqnarray}
&&f(u,v,x,y)=|1-uv|+|1-xy| \nonumber \\
&&\qquad +[(u^2a+2uxc_1+x^2b)(v^2a-2vyc_2+y^2b)]^{1/2}.
\nonumber \\
\label{FUVXY}
\end{eqnarray}
This is actually a function of only three variables. This becomes
apparent when we make the following substitutions
\begin{eqnarray}
uv \rightarrow w, \qquad x\rightarrow x/v, \qquad y\rightarrow yv.
\label{SUBST}
\end{eqnarray}
The function (\ref{FUVXY}) then reads
\begin{eqnarray}
&&f(w,x,y)=|1-w|+|1-xy| \nonumber \\
& &\quad \qquad +[(w^2a+2wxc_1+x^2b)(a-2yc_2+y^2b)]^{1/2}.
\nonumber \\
\label{FWXY}
\end{eqnarray}
After another substitution $x=qw$ the function (\ref{FWXY})
becomes a linear function of $w$:
\begin{eqnarray}
&&f(w,q,y)=|1-w|+|1-wqy|\nonumber \\
& &\quad \qquad +|w|[(a+2qc_1+q^2b)(a-2yc_2+y^2b)]^{1/2}.
\nonumber \\
\label{FWQY}
\end{eqnarray}
From the linearity of (\ref{FWQY}) it is clear that the extrema are
localized at points, where one absolute value is equal to zero.
Hence we have to consider three different possibilities:
(i) $w=1$, no operation is applied on Alice's side and a CP map
is applied on Bob's side.
(ii) $wqy=1$, a symplectic transformation is applied on Bob's side.
However, this symplectic transformation can be in our case ``absorbed''
into CP map on Alice's side, hence another possibly optimum strategy is
to do nothing on Bob's side and to apply a CP map on Alice's side.
(iii) $w=0$, this means that both Alice and Bob throw away their parts of
shared quantum state and replace them with vacuum states.
Clearly, this strategy is optimum if the quantum channel is not in
entangled state, because with vacuum state at both sides
one gets maximum fidelity $\cal{F}$ obtainable without the aid of
entanglement, ${\cal{F}}_{\rm max, class}=1$.
One may object that the substitution $x=wq$ is problematic
when $w=0$ and $x\neq 0$. However, a detailed analysis reveals
that if one of the four parameters $u,v,x,y$ is set equal to zero
and the three remaining parameters are optimized,
then we once again arrive at the above listed alternatives (i)--(iii).
The strategies (i) and (ii) represent a CP map on
only one side, while nothing is performed on the other side. It was
shown above that these optimum one-sided CP maps can be found
analytically. The search for optimum CP map would thus consist of
three parts: find one-sided optimum Gaussian CP maps on Alice's side,
on Bob's side and also consider replacement of the shared quantum state
with vacuum state and choose the optimum alternative that yields
maximum fidelity.
\section{Example of optimization}
To illustrate how the optimization works in practice,
let us assume that the covariance matrix $\bm\Gamma_{AB}$ has the
tridiagonal structure given by Eq. (\ref{GAMMAABBIG}) and the nonzero
elements read
\begin{equation}
\begin{array}{lcl}
a=1+2\sinh^2 r & \qquad & c_1=-\sinh(2r), \\
b=1+2\sinh^2 r +b_0 & \qquad & c_2=\sinh(2r).
\end{array}
\label{abc}
\end{equation}
For $b_0=0$ we recover the covariance matrix of pure two-mode squeezed
vacuum state and the quantum channel is in a mixed state for any $b_0>0$.
Let us analyse how the fidelity of teleportation of coherent state
can be improved by means of local transformations on Bob's side.
Since $d_{11}=d_{22}=1$ and also $c_1=-c_2$ [c.f. Eq. (\ref{abc})],
the solution of Eq. (\ref{XEQUATION})
simplifies considerably because the functions $\alpha$ and $\beta$ are
identical. The optimum $x$ and $y$ are equal, $x=y$, and the two roots
of Eq. (\ref{XEQUATION}) read
\begin{equation}
x_1= \frac{c_2}{b-1}, \qquad x_2 = \frac{c_2}{b+1}.
\label{XONETWO}
\end{equation}
Furthermore, the quartic equation (\ref{QUARTIC}) for optimum symplectic
transformation splits into two quadratic quations that have only two real
roots $x=\pm 1$.
\begin{figure}
\caption{Fidelity of teleportation of coherent state for $b_0=0.5$ and
variable squeezing $r$. The solid line shows the maximum fidelity achievable
via local CP map on Bob's side and the dashed line shows the maximum
fidelity achievable via local symplectic transformations on Bob's side.
Both curves coincide when the squeezing is higher than the
threshold $r_{\rm th}
\end{figure}
We must evaluate the fidelity of teleportation of coherent state
for all these potentially optimum transformations on Bob's side,
and choose the maximum value.
The resulting fidelity is plotted in Fig. 1 for $b_0=1/2$ and a variable
degree of squeezing $r$. It turns out that if the squeezing is lower
than certain threshold $r_{\rm th}=-[\ln(1-b_0)]/2$,
then the optimum transformation on Bob's
side is a CP map with $x$ given by $x_1$ in Eq. (\ref{XONETWO}).
The parameter $x_1$ grows from zero for $r=0$ to the value $x_1=1$
that is attained when $r=r_{\rm th}$.
For higher squeezing, the best strategy is to do nothing,
i.e., the optimum operation is a symplectic transformation with $x=y=1$.
The optimum CP map for $r<r_{\rm th}$ is a simple damping process which
can be implemented with the help of a beam splitter with amplitude
transmittance $t=c_2/(b-1)$ whose two input ports are fed with Bob's part of
entangled state and a vacuum state, respectively.
This transformation reduces the noise represented by $b_0$
in Eq. (\ref{abc}), which in turn improves the teleportation
fidelity.
With the help of CP map on Bob's side, the fidelity of teleportation of
coherent state is always larger than the maximum fidelity $1/2$
achievable without the aid of entanglement. On the other hand,
if we allow only for unitary symplectic transformations on Bob's part of
the state, then there is a region of squeezing where the maximum
achievable fidelity is lower than $1/2$. This example clearly
illustrates that in certain cases it is advantageous to couple the
shared state to the local environment \cite{Badziag00,Rehacek01}.
\begin{figure}
\caption{The same as Fig. 1 but the fidelity of entanglement
swapping $\cal{F}
\end{figure}
Similar results are obtained for the fidelity of entanglement swapping
$\cal{F}$. In this case we can optimize over all
local Gaussian CP maps on both Alice's and Bob's side,
because the problem reduces to the
optimization of a Gaussian CP map on only one side, as discussed in the
previous Section. For our specific example, the optimum CP map
is actually the same as that for the fidelity of teleportation of
coherent state.
Also the dependence of the fidelity $\cal{F}$ on $r$ is qualitatively
similar to that shown in Fig. 1, see Fig. 2. In particular, there is a
region where ${\cal{F}}>1$ if Bob applies the optimum CP map,
but a restriction to local symplectic transformation results in ${\cal{F}}<1$.
\section{Conclusions}
In this paper we have shown that one can improve the fidelity of
teleportation of continuous quantum variables by means of local
operations on the sender's and receiver's parts of the shared entangled
state $\rho_{AB}$ (quantum channel). We have considered
the fidelity of teleportation of pure Gaussian states and we have also
introduced a fidelity measure for the teleportation transformation.
The latter fidelity was interpreted as a fidelity of
entanglement swapping of infinitely squeezed EPR state.
We have restricted ourselves to the class of local trace-preserving
Gaussian completely positive maps and we have shown that in this case
the optimization problem can be solved analytically.
We have demonstrated on a simple example that the optimum
local operation need not be a unitary symplectic transformation but
some more general CP map.
\vspace*{6mm}
\acknowledgments
I would like to thank R. Filip, L. Mi\v{s}ta, Jr., and J. Pe\v{r}ina
for valuable comments.
This work was supported by Grant No LN00A015 and Research Project
CEZ:J14/98 of the Czech Ministry of Education and by the EU grant under
QIPC, project IST-1999-13071 (QUICOV).
\appendix
\section{}
Here we prove that the inequality ${\cal{F}}\leq 1$ holds if the shared
quantum state $W_{AB}(x_A,p_A,x_B,p_B)$ is separable.
Our starting point is the the formula (\ref{FSWAPBLA}):
\begin{equation}
{\cal{F}}= 2\pi \int_{-\infty}^\infty W_{AB}(x,p,-x,p)d x dp.
\label{FSWAPWAB}
\end{equation}
Density matrix $\rho_{AB}$ of any separable state can be written as
a convex mixture of product states,
\begin{equation}
\rho_{AB}=\sum_j p_j \, \rho_{A,j}\otimes \rho_{B,j},
\label{SEPARABILITY}
\end{equation}
where $p_j> 0$ and
$
\sum_{j} p_j =1.
$
Formula (\ref{SEPARABILITY}) implies that
\begin{equation}
W_{AB}(x_A,p_A,x_B,p_B)=\sum_j p_j W_{A,j}(x_A,p_A)W_{B,j}(x_B,p_B).
\end{equation}
If $W_{B,j}(x_B,p_B)$ is Wigner function of the quantum state $\rho_{B,j}$,
then $W_{B,j}(-x_B,p_B)$ is a Wigner function of the quantum state
$\rho_{B,j}^T$, because the transformation $x \rightarrow -x$ and
$p\rightarrow p$ is the transposition.
For separable state (\ref{SEPARABILITY}), the formula (\ref{FSWAPWAB})
thus reduces to
\begin{equation}
{\cal{F}} = \sum_j p_j {\rm Tr}[ \rho_{A,j} \rho_{B,j}^T],
\end{equation}
where we used that
\begin{equation}
{\rm Tr}[\rho_A \rho_B]=2\pi\int_{-\infty}^\infty W_{A}(x,p)W_{B}(x,p)dx dp.
\end{equation}
The Schwarz inequality implies that
${\rm Tr}[ \rho_{A,j} \rho_{B,j}^T] \leq 1$ and with the help of
normalization of $p_j$ we finally obtain
\begin{equation}
{\cal{F}} \leq 1.
\end{equation}
The fidelity ${\cal{F}}=1$ forms a boundary between
classical information transfer and quantum teleportation. The state
$\rho_{AB}$ must be entangled in order to achieve ${\cal{F}}>1$.
This illustrates the essential and central role of the entanglement
in the teleportation.
\end{document} |
\begin{document}
\title{Probabilistic aspects of critical \\ growth-fragmentation equations}
\authorone[University of Zurich]{Jean Bertoin}
\addressone{Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Z\"urich, Switzerland}
\authortwo[University of Manchester]{Alexander R. Watson}
\addresstwo{School of Mathematics, University of Manchester, Manchester, M13 9PL, UK}
\begin{abstract}
The self-similar growth-fragmentation equation describes the evolution
of a medium in which particles grow and divide as time proceeds, with the
growth and splitting of each particle depending only upon its size.
The critical case of the equation, in which the growth and division rates
balance one another, was
considered by Doumic and Escobedo \cite{DouEsc} in the
homogeneous case where the rates do not depend on the particle size.
Here, we study the general self-similar case, using a probabilistic approach
based on L\'evy processes and positive self-similar Markov processes
which also permits us to analyse quite general splitting rates.
Whereas existence and uniqueness of the solution are rather easy to establish
in the homogeneous case, the equation in the non-homogeneous case
has some surprising features.
In particular, using the fact that certain
self-similar Markov processes can enter $(0,\infty)$ continuously from either $0$
or $\infty$, we exhibit unexpected spontaneous generation of mass in the solutions.
\end{abstract}
\keywords{Growth-fragmentation equation;
self-similarity;
self-similar Markov process;
branching process.}
\ams{35Q92}{45K05; 60G18; 60G51}
\section{Introduction}
The growth-fragmentation equation is a linear differential equation
intended to describe the evolution of a medium in which particles grow and split as time passes.
It is frequently expressed in terms of the concentration of particles with size $x>0$ at time $t$,
say $u(t,x)$, as follows:
\begin{equation}\label{EqGFG}
\partial_tu(t,x)+ \partial_x(\tau(x)u(t,x))+B(x)u(t,x)=\int_x^{\infty}k(y,x)B(y) u(t,y) \d y,
\end{equation}
where $\tau(x)$ is the speed of growth of a particle with size $x$,
$B(x)$ the rate at which a particle of size $x$ splits, and $k(y,x)=k(x-y,x)$
twice the probability density that a particle with size $x$ splits into two
particles with size $yx$ and $(1-y)x$ (the factor $2$ is due to the symmetry of the splitting events).
This type of equation
has a variety of applications in mathematical modeling,
notably in biology where particles should be thought of as cells,
and has motivated several works in the recent years;
see, for example, \cite{DouEsc}, which also contains a summary of some
recent literature.
We are interested here in the situation $\tau(x)=c x^{\alpha+1}$, $B(x)=x^{\alpha}$
for some $\alpha\in\mathbb{R}$ and $k$ has the form $k(y,x)=x^{-1}k_0(y/x)$;
for these parameters, \eqref{EqGFG} possesses a useful self-similarity property. This is referred to
as the \emph{critical} case by Doumic and Escobedo \cite{DouEsc}, who studied in depth the
situation when, additionally, $\alpha=0$.
For our purposes, it will be more convenient to write the equation in
weak form, as follows.
For $x>0$ and $y\in[1/2,1)$, we write $(y\mid x)$ for the pair $\{yx,(1-y)x\}$, which we view
as the dislocation of a mass $x$ into two smaller masses, and then for every function $f\from (0,\infty)\to \mathbb{R}$, we set
$$f(y\mid x) \coloneqq f(yx)+f((1-y)x).$$
Consider test functions $f\in{\mathcal C}^{\infty}_c(0,\infty)$,
that is, $f$ is infinitely differentiable with compact support. For a measure
$\mu$ on $(0,\infty)$, we write $\ip{\mu}{f} \coloneqq \int_{(0,\infty)} f(x) \, \mu(\dd x)$.
By integrating \eqref{EqGFG}, we obtain the equation
\begin{equation}\label{EqFG}
\partial_t \langle \mu_t, f{\mathbf{r}}angle
= \left\langle \mu_t, {\mathcal L}f{\mathbf{r}}ight{\mathbf{r}}angle,
\end{equation}
where $\mu_t(\d x) = u(t,x)\d x$ and the operator ${\mathcal L}$ has the form
\begin{equation}\label{Eqop0}
{\mathcal L}f(x) = x^{\alpha}\left( cxf'(x) + \int_{[1/2,1)}\left(f(y\mid x)-f(x){\mathbf{r}}ight) K(\d y){\mathbf{r}}ight) ,
\end{equation}
where
$$K(\d y) \coloneqq k_0(y)\d y = k_0(1-y)\d y\,,\qquad y\in[1/2,1),$$
is referred to as the \emph{dislocation measure}.
The advantage of this formulation is that we do not require
absolute continuity of the solution $\mu_t$ or the dislocation measure $K$.
More generally, one might also consider non-binary dislocation measures,
but we refrain from doing so in this work in order to simplify the presentation.
\skippar
In this article, we study the equation \eqref{EqFG} for operators of the form
\begin{equation} \label{Eqop1}
{\mathcal L}_{\alpha}f(x)
\coloneqq
x^{\alpha}\biggl( a x^2 f''(x)+ b xf'(x)
+ \int_{[1/2,1)}\bigl(f(y\mid x)-f(x)+xf'(x)(1-y)\bigr) K(\d y)\biggr) ,
\end{equation}
where $a\geq 0$, $b\in\mathbb{R}$, and we now only assume that the measure $K$ satisfies the weaker requirement
\begin{equation}\label{Eqcondnu}
\int_{[1/2,1)}(1-y)^2 K(\d y)<\infty.
\end{equation}
Our notion of a \emph{solution} of \eqref{EqFG} is a collection of locally finite
measures $(\mu_t)_{t\ge 0}$ on $(0,\infty)$ such that,
for every $f \in \mathbb{C}test(0,\infty)$ and $t\geq 0$, there is the identity
$$ \langle \mu_t, f{\mathbf{r}}angle = \langle \mu_0, f{\mathbf{r}}angle + \int_0^t \left\langle \mu_s, {\mathcal L}f{\mathbf{r}}ight{\mathbf{r}}angle \d s .$$
(This requires implicitly that
$s\mapsto \left\langle \mu_s, {\mathcal L}f{\mathbf{r}}ight {\mathbf{r}}angle$ be a well-defined, locally integrable function,
and in particular the family $(\mu_t)_{t\ge 0}$ is then vaguely continuous).
We offer a comparison between the original operator \eqref{Eqop0} and our operator \eqref{Eqop1}.
Besides the appearance of a second order derivative,
there is a new term $xf'(x)(1-y)$ in the integral in \eqref{Eqop1}.
The latter should be interpreted as an additional growth term which, in some sense,
balances the accumulation of small dislocations.
We stress that \eqref{Eqcondnu} is the necessary and sufficient condition for \eqref{Eqop1} to be
well-defined, and that when the measure $K$ is finite (or at least fulfills $\int_{[1/2,1)}(1-y) K(\d y)<\infty$),
every operator of the form \eqref{Eqop0} can also be expressed in the form
\eqref{Eqop1}. Our motivation for considering this more general setting stems from the recent
work \cite{BeCF}, in which a new class of growth-fragmentation stochastic processes is constructed
such that, loosely speaking, the strong rates of dislocation that would instantaneously shatter the
entire mass can be somehow compensated by an intense growth; the dislocation measure associated
with such a fragmentation process need only satisfy \eqref{Eqcondnu}.
In short, the purpose of this work is to demonstrate the usefulness of
some probabilistic methods for the study of these critical growth-fragmentation equations.
More precisely, we shall see that solutions to
\eqref{EqFG}
for ${\mathcal L}={\mathcal L}_{\alpha}$ can be related to
the one-dimensional distributions of certain self-similar Markov processes,
and this will enable us to reveal some rather unexpected features of the former.
Although in the homogeneous case $\alpha=0$,
we establish existence and uniqueness of the solution in full generality,
this feature is lost for $\alpha\neq 0$.
In particular, we shall see that
under a fairly general assumption on the parameters of the model,
the critical growth-fragmentation equation permits spontaneous generation,
i.e. there exist non-degenerate solutions starting from the null initial condition.
We need some notation before describing more precisely our main results.
We first introduce the function $\kappa \from [0,\infty)\to (-\infty,\infty]$ which plays a major role in our approach and is given by:
\begin{equation}\label{eqkappa}
\kappa(q)
\coloneqq
a q^2 + (b-a) q + \int_{[1/2,1)}\left(y^q +(1-y)^q- 1 +q(1-y){\mathbf{r}}ight) K(\d y) , \qquad q\geq 0.
\end{equation}
There are two principal `Malthusian hypotheses' which we will require when $\alpha\neq 0$:
\begin{enumerate}
\itemlab{$(M_+)$}{i:M1}
$\inf_{q\geq 0} \kappa(q) <0.$
\itemlab{$(M_-)$}{i:M2}
There exist $0\le \omega_{_-}< \omega_+$ and $\epsilon>0$ such that
$\kappa(\omega_{_-})=\kappa(\omega_+)=0$ and
\linebreak
$\kappa(\omega_--\epsilon)<\infty$.
\end{enumerate}
We now summarise our main results,
deferring their proofs to the body of the article.
\begin{itemize}
\item
For $\alpha = 0$, the equation \eqref{EqFG} with operator \eqref{Eqop1}
and initial condition $\mu_0 = \delta_1$ has a unique solution.
\item
For $\alpha < 0$: suppose that {\mathbf{r}}ef{i:M1} holds. Then,
the equation \eqref{EqFG} with operator \eqref{Eqop1}
has a solution with initial condition $\mu_0 = \delta_1$.
There exists further a non-degenerate solution started from $\mu_0=0$;
in particular, uniqueness fails.
\item
For $\alpha > 0$: if {\mathbf{r}}ef{i:M1} holds, then
the equation \eqref{EqFG} with operator \eqref{Eqop1}
has a solution with initial condition $\mu_0 = \delta_1$.
If {\mathbf{r}}ef{i:M2} holds,
then there also exists a non-degenerate solution started from $\mu_0=0$;
again, uniqueness fails.
\end{itemize}
We shall also observe that under essentially the converse assumption to {\mathbf{r}}ef{i:M1},
namely that $\inf_{q\ge0} \kappa(q) >0$,
the particle system that corresponds to the stochastic version of the model
may explode in finite time almost surely.
This is a strong indication that \eqref{EqFG} should have no global solution in the latter case.
\skippar
The rest of this article is organized as follows.
The next section provides brief preliminaries on the function $\kappa$ and the use of the Mellin transform
in the study of growth-fragmentation equations.
Section {\mathbf{r}}ef{s:hom} is devoted to the homogeneous case $\alpha=0$, and then the general self-similar case
$\alpha \neq 0$ is presented in section {\mathbf{r}}ef{s:ss}.
In section {\mathbf{r}}ef{s:explosion},
we investigate a stochastic model related to the growth-fragmentation equation,
to demonstrate that explosion may occur when the Malthusian hypothesis fails.
Finally, in section {\mathbf{r}}ef{s:bps},
we briefly discuss another interpretation of the growth-fragmentation equation
in terms of branching particle systems and many-to-one formulas,
placing the results of sections {\mathbf{r}}ef{s:hom} and {\mathbf{r}}ef{s:ss} in context.
\section{The Mellin transform and the growth-fragmentation equation}
\label{s:prelim}
We observe first that, for any $\alpha \in \mathbb{R}$, the operator
${\mathcal L}_{\alpha}$
fulfills a self-similarity property.
Specifically,
for every $c>0$,
if we denote by $\varphi_c(x)=cx$ the dilation function with factor $c$,
then for a generic $f\in{\mathcal C}^{\infty}_c(0,\infty)$,
there is the identity
\begin{equation}\label{EqSSL}
{\mathcal L}_{\alpha}(f\circ \varphi_c)=c^{-\alpha}\left({\mathcal L}_{\alpha}f{\mathbf{r}}ight)\circ \varphi_c.
\end{equation}
As a consequence, if $(\mu_t)_{t\geq 0}$ is a solution to \eqref{EqFG} for all
$f\in{\mathcal C}^{\infty}_c(0,\infty)$
with initial condition $\mu_0=\delta_1$,
and if $\tilde \mu_t$ denotes the image of $\mu_{t}$ by the dilation $\varphi_c$,
then $(\tilde \mu_{c^{\alpha}t})_{t\geq 0}$ is a solution to \eqref{EqFG}
for all $f\in{\mathcal C}^{\infty}_c(0,\infty)$ with initial condition $\tilde\mu_0=\delta_c$.
For the sake of simplicity,
we shall therefore focus on the growth-fragmentation equation
with initial condition $\mu_0=\delta_1$,
since this does not induce any loss of generality.
Recall that the function $\kappa$ has been introduced in \eqref{eqkappa};
its domain is clarified by the following result.
\begin{lem} \mbox{} \label{Lbounds}
\begin{enumerate}[(i)]
\item{\mathbf{r}}elax\label{i:Lbounds:def}
For every $q\geq 0$, $\kappa(q)$ is well-defined with values in $(-\infty,\infty]$.
The function $\kappa$ is convex, and
we define $\dom \kappa = \{ q \geq 0 : \kappa(q)<\infty\}$.
\item{\mathbf{r}}elax\label{i:Lbounds:dom}
For $q\geq 0$, $\kappa(q)<\infty$ if and only if $\int_{[1/2,1)}(1-y)^qK(\d y)<\infty$,
and in particular $[2,\infty)\subseteq \dom \kappa$.
\item{\mathbf{r}}elax\label{i:Lbounds:asym}
For every function $f$ in ${\mathcal C}^{\infty}_c(0,\infty)$, ${\mathcal L}_{\alpha}f$
is a continuous function on $(0,\infty)$ and is identically $0$ in some neighborhood of $0$.
Furthermore, ${\mathcal L}_{\alpha}f(x)=o(x^{q+\alpha})$ as $x\to \infty$ for every
$q\in \dom \kappa$, and thus in particular for $q=2$.
\end{enumerate}
\end{lem}
\begin{proof}
\begin{itemize}
\item[(i--ii)]
First, the integral $\int_{[1/2,1)}\left(y^q - 1 +q(1-y){\mathbf{r}}ight) K(\d y)$ converges
absolutely thanks to \eqref{Eqcondnu}, since $y^q - 1 +q(1-y)=O((1-y)^2)$.
It follows that $\kappa(q)$ is then well-defined with values in $(-\infty,\infty)$
if and only if $\int_{[1/2,1)}(1-y)^qK(\d y)<\infty$, and otherwise $\kappa(q)=\infty$.
\item[(iii)]
The first assertions are straightforward, and so we check only the last one.
Take $q\in \dom \kappa$ and recall from above that $\int_{[1/2,1)}(1-y)^qK(\d y)<\infty$.
This entails $K([1/2,1-\varepsilon))=o(\varepsilon^{-q})$ as $\varepsilon\to 0+$.
Since $f$ has compact support in $(0,\infty)$, we have for $x$ sufficiently large that
$\mathcal L_0f(x)=\int_{[1/2,1)}f(x(1-y))K(\d y)$, and we easily conclude that ${\mathcal L}_0f(x)=o(x^q)$.
\qedhere
\end{itemize}
\end{proof}
Doumic and Escobedo \cite{DouEsc} studied certain growth-fragmentation equations
with homogeneous operators given by \eqref{Eqop0} for $\alpha=0$, and
observed that the Mellin transform plays an important role.
In this direction, it is useful to introduce the notation
$h_q\from (0,\infty)\to (0,\infty)$, $h_q(x)=x^q$, for the power function with exponent $q$, and
recall that the Mellin transform of a measurable function $f\colon(0,\infty)\to \mathbb{R}$ is defined for $z\in\mathbb{C}$ by
$${\mathcal M}f(z)\coloneqq \int_0^{\infty} f(x) x^{z-1}\d x $$
whenever the integral in the right-hand side converges.
It follows from \autoref{Lbounds} that the Mellin transform of ${\mathcal L}_{\alpha}f$
is well defined for all $z < -2-\alpha$, or more generally for all $z$ such that $-z-\alpha \in (\dom \kappa)^\circ$.
The role of $\kappa$ in this study stems from the following lemma,
which is easily checked from elementary properties of the Mellin transform;
see \cite[\S 1.1]{DouEsc} and \cite[\S 12.3]{Davies}.
\begin{lem}\mbox{} \label{L0}
\begin{enumerate}[(i)]
\item{\mathbf{r}}elax\label{i:L0:eigenfunction}
Let $q \in \dom \kappa$. Then,
$$ {\mathcal L}_{\alpha}h_q(x)= \kappa(q) h_{q+\alpha}(x), \qquad x > 0.$$
In particular, for $\alpha=0$, $h_q$ is an eigenfunction for ${\mathcal L}_{0}$ with eigenvalue $\kappa(q)$.
\item For every $q$ such that $q-\alpha \in (\dom \kappa)^\circ$
and every $f\in{\mathcal C}^{\infty}_c(0,\infty)$, there is the identity
$${\mathcal M}({\mathcal L}_{\alpha}f)(-q)= \kappa(q-\alpha) {\mathcal M}f(-q+\alpha).$$
\end{enumerate}
\end{lem}
\section{The homogeneous case}
\label{s:hom}
Throughout this section, we assume that $\alpha=0$, and refer to this case as \textit{homogeneous}.
Recall that when $a=0$ and the dislocation measure $K$ fulfills the stronger condition
\begin{equation}\label{Eqcondnu1}
\int_{[1/2,1)}(1-y)K(\d y)<\infty,
\end{equation}
then we can express the operator ${\mathcal L}_{0}$ in the simpler form
\begin{equation}\label{e:L-c-nu}
{\mathcal L}_{c,K}f(x) \coloneqq cxf'(x) + \int_{[1/2,1)}\left(f(y\mid x)-f(x){\mathbf{r}}ight) K(\d y)
\end{equation}
with $c=b+\int_{[1/2,1)}(1-y) K(\d y)$.
This situation was considered in depth by \citet{DouEsc}, and most of the results of this section
should be viewed as extensions of those in \cite{DouEsc} to the case when either $a>0$, or $K$
fulfills \eqref{Eqcondnu} but not \eqref{Eqcondnu1}. Furthermore, the case
$c \le 0$ was considered by \citet{Haas} using the same method that we employ below.
\subsection{Main results}
\label{s:zero-main}
The key observation in \autorefpref{L0}{i:L0:eigenfunction} that
power functions $h_q$ are eigenfunctions of the operator
${\mathcal L}_{0}$, underlies the analysis of the homogeneous case.
Specifically, if we knew that $(\mu_t)_{t\geq 0}$ solves \eqref{EqFG} with $f=h_q$ for $q\geq 2$,
then the Mellin transform of $\mu_t$,
${M}_t(z)=\langle \mu_t, h_{z-1}{\mathbf{r}}angle$, would solve the linear equation
\begin{equation}\label{EqMelh}
\partial {M}_t(q+1) = \kappa(q){M}_t(q+1).
\end{equation}
Focussing for simplicity on the initial condition $\mu_0=\delta_1$, so that ${M}_0(q)=1$, we would find
\begin{equation}\label{SolMelh}
{M}_t(q+1)=\exp(t\kappa(q)).
\end{equation}
In order to check that \eqref{SolMelh} is indeed the Mellin transform of a positive measure, we define,
for every $\omega \in \dom \kappa$,
a new function by shifting $\kappa$:
\[
\mathbb{P}hi_{\omega}(q)\coloneqq \kappa(\omega+q)-\kappa(\omega),\qquad q\geq 0.
\]
This is a smooth, convex function, and it has a simple probabilistic interpretation,
which will play a major role throughout. In this direction, recall first that
a \define{L\'evy process} is a stochastic process
issued from the origin
with stationary and independent increments and
c\`adl\`ag paths. It is further called \define{spectrally negative} if all its jumps are negative.
If ${\mathbf{x}}i: = ({\mathbf{x}}i(t))_{t\geq 0}$ is a spectrally negative L\'evy process
with law $\lP$, then for all $t\geq 0$ and $\theta\in \mathbb{R}$,
the \emph{Laplace exponent} $\mathbb{P}hi$, given by
\[
\mathbb{E}\bigl[ \exp(q {\mathbf{x}}i(1)) \bigr] = \exp(\mathbb{P}hi(q)) ,
\]
is well defined (and finite) for all $q\geq 0$, and
satisfies the classical L\'evy-Khintchin formula
\[
\mathbb{P}hi(q) = \mathtt{a}q + \frac{1}{2}\sigma^2q^2 +
\int_{(-\infty, 0)} ( \e^{qx} -1 + q x\Indic{|x|\leq 1})\mathbb{U}psilon(\dd x) , \qquad q \ge 0,
\]
where $\mathtt{a}\in\mathbb{R}$, $\sigma\geq 0$, and $\mathbb{U}psilon$ is a measure
(the \define{L\'evy measure}) on
$(-\infty, 0)$ such that
$$\int_{(-\infty, 0)}(1\wedge x^2)\mathbb{U}psilon(\dd x)<\infty.$$
is in fact the Laplace exponent of a spectrally negative Lévy process;
see \cite[Theorem 8.1]{Sato}.
\begin{lem}{\mathbf{r}}elax \label{L1}
Let $\omega \in \dom\kappa$. Then:
\begin{enumerate}[(i)]
\item
the function $\mathbb{P}hi_\omega$ is the Laplace
exponent of a spectrally negative Lévy process, which we will call\
${\mathbf{x}}i_\omega = ({\mathbf{x}}i_\omega(t))_{t\ge 0}$.
\item
for every $t\geq 0$, there exists a unique probability measure
${\mathbf{r}}ho^{[\omega]}_t$ on $(0,\infty)$ with Mellin transform given by
\begin{equation}\label{Eqrho}
\mathcal{M} {\mathbf{r}}ho^{[\omega]}_t (q+1)
\coloneqq
\int_{(0,\infty)} x^q {\mathbf{r}}ho^{[\omega]}_t(\d x) = \exp(t\mathbb{P}hi_\omega(q))\,, \qquad q\geq 0.
\end{equation}
The family of measures has the representation ${\mathbf{r}}ho^{[\omega]}_t = \mathbb{P}(\exp({\mathbf{x}}i_\omega(t)) \in \cdot)$, for $t\ge 0$.
\end{enumerate}
\end{lem}
\begin{proof}
We prove both parts simultaneously, and focus first on the case $\omega=2$, where we write $\mathbb{P}hi \coloneqq \mathbb{P}hi_2$.
We can express $\mathbb{P}hi$ in the form
\begin{equation}\label{EqLKPhi}
\mathbb{P}hi(q)= a q^2 +b'q + \int_{[1/2,1)}(y^q -1+q(1-y))y^2K(\d y) + \int_{[1/2,1)}((1-y)^q -1)(1-y)^2K(\d y)
\end{equation}
with
$$
b'=3a+b+\int_{[1/2,1)}(1-y)(1-y^2)K(\d y).
$$
Let us denote by $\Lambda(\d x)$ the image of $y^2K(\d y)$ by the map $y\mapsto x=\ln y$,
and $\mathbb{P}i(\d x)$ the image of $(1-y)^2K(\d y)$ by the map $y\mapsto x=\ln (1-y)$.
Then, thanks to \eqref{Eqcondnu}, $\Lambda$ is a measure on $[-\ln 2,0)$ with $\int x^2\Lambda(\d x)<\infty$,
and $\mathbb{P}i$ is a finite measure on $(-\infty,-\ln 2]$, and there are the identities
$$ \int_{[1/2,1)}(y^q -1+q(1-y))y^2K(\d y)= \int_{[-\ln 2,0)}(\e^{qx}-1+q(1-\e^x))\Lambda(\d x)$$
and
$$\int_{[1/2,1)}((1-y)^q -1)(1-y)^2K(\d y)=\int_{(-\infty, -\ln 2]}(\e^{qx}-1)\mathbb{P}i(\d x).$$
This shows that $\mathbb{P}hi$ is given by a L\'evy-Khintchin formula, and therefore, $\mathbb{P}hi$
can be viewed as the Laplace exponent of a spectrally negative L\'evy process ${\mathbf{x}}i=({\mathbf{x}}i(t))_{t\geq 0}$
(see Chapter VI in \cite{BeLP} for background), i.e.,
$$\mathbb{E}\left( \exp(q{\mathbf{x}}i(t)){\mathbf{r}}ight) = \exp(t\mathbb{P}hi(q))\,,\qquad t, q\geq 0.$$
We conclude that \eqref{Eqrho} does indeed determine a probability measure
${\mathbf{r}}ho_t$ which arises as the distribution of $\exp({\mathbf{x}}i(t))$.
Finally, if $\omega \ne 2$, we observe that the function $\mathbb{P}hi_\omega$ can
be written
\linebreak
$\mathbb{P}hi_\omega(q) = \mathbb{P}hi(q+\omega-2)-\mathbb{P}hi(\omega-2)$, which implies that
$\mathbb{P}hi_\omega$ is given by an Esscher transform of $\mathbb{P}hi$, and hence is also
the Laplace exponent of a spectrally negative Lévy process;
see \cite[Theorem 3.9]{Kypr2} or \cite[Theorem 33.1]{Sato}.
\end{proof}
\begin{rem}
More generally, if ${\mathbf{z}}eta$ denotes a random time having an exponential distribution,
say with parameter $\mathtt{k}\geq 0$, which is further independent of ${\mathbf{x}}i$,
then the process
$$
{\mathbf{x}}i_{\dagger}(t)=\left\{ \begin{matrix} {\mathbf{x}}i(t) &\hbox{ if }& t<{\mathbf{z}}eta\\
-\infty &\hbox{ if }& t\geq {\mathbf{z}}eta \end{matrix}{\mathbf{r}}ight.
$$
is referred to as a killed L\'evy process. Note that if we set $\mathbb{P}hi_{\dagger}(q))= \mathtt{k} + \mathbb{P}hi(q)$, then
\[ \mathbb{E}\bigl[ \exp(q {\mathbf{x}}i_{\dagger} (t)) \bigr] = \exp(t\mathbb{P}hi_{\dagger}(q)) ,\]
with the convention that $\exp(q {\mathbf{x}}i_{\dagger} (t)) =0$ for $t\geq {\mathbf{z}}eta$.
So \autoref{L1} shows that whenever $\kappa(\omega)\leq 0$, the function $q \mapsto \kappa(\omega+q)$
can be viewed as the Laplace exponent of a spectrally negative L\'evy process killed at an
independent exponential time with parameter $-\kappa(\omega)$.
\end{rem}
Recall from \autorefpref{Lbounds}{i:Lbounds:dom} that $2 \in \dom\kappa$ always;
we will write ${\mathbf{r}}ho_t$ for ${\mathbf{r}}ho^{[2]}_t$. Since ${\mathbf{r}}ho_t$
is guaranteed to exist, this collection of measures will play a particular
role in the case $\alpha=0$. We stress that in the cases $\alpha < 0$ and $\alpha >0$,
we will need to choose different values of $\omega$,
and the notation ${\mathbf{r}}ho_t$ will then refer to a different distribution.
We point out the following property of the probability measures ${\mathbf{r}}ho^{[\omega]}_t$,
which essentially rephrases Kolmogorov's forward equation.
\begin{cor}{\mathbf{r}}elax\label{C0}
The family of probability measures $({\mathbf{r}}ho^{[\omega]}_t)_{t\geq 0}$ defined in
\autoref{L1} depends continuously on the parameter $t$ for the topology of weak convergence.
Further, for every $g\in {\mathcal C}^{\infty}_c(0,\infty)$, the function
$t\mapsto \langle {\mathbf{r}}ho^{[\omega]}_t, g{\mathbf{r}}angle$ is differentiable with derivative
$\partial_t\langle {\mathbf{r}}ho^{[\omega]}_t, g{\mathbf{r}}angle= \langle {\mathbf{r}}ho^{[\omega]}_t, {\mathcal A}g{\mathbf{r}}angle$,
where
$${\mathcal A}g(x) \coloneqq x^{-\omega} {\mathcal L}_0(h_\omega g)(x)-\kappa(\omega)g(x)\,, \qquad x>0.$$
\end{cor}
\begin{proof}
Recall that every L\'evy processes fulfills the Feller property and in particular, for every function
$\varphi\in{\mathcal C}_0(\mathbb{R})$, the map $t\mapsto \mathbb{E}(\varphi({\mathbf{x}}i_\omega(t)))$ is continuous.
Taking
\linebreak
$\varphi(x)=g(\e^x)$ with $g\in{\mathcal C}_0(0,\infty)$ yields
the weak continuity of the map $t\mapsto {\mathbf{r}}ho^{[\omega]}_t$.
Further, it is well-known that the domain of the infinitesimal generator of a L\'evy process contains
${\mathcal C}^{\infty}_c(\mathbb{R})$ (see, e.g., Theorem 31.5 in \citet{Sato}), and it follows similarly
that for $g\in {\mathcal C}^{\infty}_c(0,\infty)$, the map
$t\mapsto \langle {\mathbf{r}}ho^{[\omega]}_t, g{\mathbf{r}}angle$ is differentiable.
To compute the derivative, that is to find the infinitesimal generator, take $q\geq 0$ and recall that
$h_q(x)=x^q$ for $x>0$. Then simply observe from \eqref{Eqrho} that
$$\partial_t\langle {\mathbf{r}}ho_t, h_q{\mathbf{r}}angle
= \mathbb{P}hi_\omega(q) \exp(t\mathbb{P}hi_\omega(q))
= \langle {\mathbf{r}}ho^{[\omega]}_t, \mathbb{P}hi_\omega(q)h_q{\mathbf{r}}angle
=\langle {\mathbf{r}}ho^{[\omega]}_t, \kappa(q+\omega)h_q-\kappa(\omega)h_q{\mathbf{r}}angle.$$
Using \autorefpref{L0}{i:L0:eigenfunction}, we can express
$$\kappa(q+\omega)h_q=h_{-\omega}\kappa(q+\omega)h_{q+\omega}=h_{-\omega}{\mathcal L}_0(h_\omega h_q),$$
which shows that
$\partial_t\langle {\mathbf{r}}ho^{[\omega]}_t, h_q{\mathbf{r}}angle= \langle {\mathbf{r}}ho^{[\omega]}_t, {\mathcal A}h_q{\mathbf{r}}angle$
for all $q\geq 0$.
That the same holds when $h_q$ is replaced by a function $g\in{\mathcal C}^{\infty}_c$ now follows from standard arguments, using linear combinations of $h_q$.
\end{proof}
We would now like to invoke \autoref{L1} to invert the Mellin transform \eqref{SolMelh}, observing that
(using \eqref{Eqrho} with ${\mathbf{r}}ho = {\mathbf{r}}ho^{[2]}$)
$$\exp(t\kappa(q))= \exp(t\kappa(2))\langle {\mathbf{r}}ho_t, x^{q-2}{\mathbf{r}}angle,$$
and conclude that
$$\mu_t(\d x) = \e^{t\kappa(2)} x^{-2}{\mathbf{r}}ho_t(\d x).$$
However $h_q\not \in {\mathcal C}^{\infty}_c(0,\infty)$ and we cannot directly apply this simple argument.
Nonetheless we claim the following.
\begin{thm}{\mathbf{r}}elax\label{T1}
The equation \eqref{EqFG}, for $f\in{\mathcal C}^{\infty}_c(0,\infty)$
and with ${\mathcal L}={\mathcal L}_{0}$, has a unique solution started from
$\mu_0 = \delta_1$, given by
\[ \mu_t(\d x) = \e^{t\kappa(2)} x^{-2}{\mathbf{r}}ho_t(\d x), \qquad t \ge 0, \]
where ${\mathbf{r}}ho_t$ is the probability measure on $(0,\infty)$ defined by \eqref{Eqrho} for $\omega=2$.
\end{thm}
\begin{rem}{\mathbf{r}}elax \label{r:LE}
In particular, the unique solution in \autoref{T1} fulfills
\linebreak
$\langle \mu_t,h_q{\mathbf{r}}angle = \exp(t\kappa(q))$, as we expected from \eqref{SolMelh}.
From a probabilistic perspective, this does not come as a surprise. In \cite{BeCF},
a homogeneous growth-fragmentation stochastic process
${\mathbf{Z}}b(t)=(Z_1(t), Z_2(t), \dotsc)$ was constructed whose evolution
is, informally speaking, governed by the stochastic growth-fragmentation dynamics
described in the introduction. Using a spine technique, it may be shown
(we omit the proof)
that the solution $(\mu_t)_{t\ge 0}$ has the representation $\ip{\mu_t}{f} = \mathbb{E} \bigl[\sum_{i=1}^\infty f(Z_i(t)) \bigr]$,
for any $f$ for which the right-hand side is finite; and
in \cite[Theorem 1]{BeCF}, it is proved that
$\mathbb{E}\bigl[\sum_{i=1}^{\infty} Z^q_i(t)\bigr]= \exp(t\kappa(q))$ for all $q\geq 2$.
We offer a more detailed discussion of the spine technique in \autoref{s:bps}.
\end{rem}
\begin{proof}[Proof of \autoref*{T1}]
It is straightforward to check that
$\mu_t(\d x) = \e^{t\kappa(2)} x^{-2}{\mathbf{r}}ho_t(\d x)$ is indeed a solution.
Specifically, we deduce from \eqref{Eqrho}, that
$$\langle \mu_t, h_q{\mathbf{r}}angle = \exp(t\kappa(2)) \exp(t\mathbb{P}hi(q-2)) = \exp(t\kappa(q)).$$
We thus see that $(\mu_t)_{t\geq 0}$ solves \eqref{EqFG} with ${\mathcal L}= {\mathcal L}_{0}$
and $f=h_q$ for every $q\geq 0$,
and it follows from classical properties of the Mellin transform that this entails that
$(\mu_t)_{t\geq 0}$ solves \eqref{EqFG} more generally for all $f\in{\mathcal C}^{\infty}_c(0,\infty)$.
Conversely, given a solution $(\mu_t)_{t\geq 0}$ to \eqref{EqFG} with $\mu_0=\delta_1$,
set
\[ \tilde {\mathbf{r}}ho_t(\d x) = \e^{-t\kappa(2)} x^{2}\mu_t(\d x) . \]
Take $g\in{\mathcal C}^{\infty}_c(0,\infty)$ and define $f(x)=x^2g(x)$ for $x>0$,
so $f\in{\mathcal C}^{\infty}_c(0,\infty)$. Then we have
$\langle \tilde {\mathbf{r}}ho_t, g{\mathbf{r}}angle = \e^{-t\kappa(2)}\langle \mu_t, f{\mathbf{r}}angle$ and
$$
\partial_t \langle \tilde {\mathbf{r}}ho_t, g{\mathbf{r}}angle
= -\kappa(2)\langle \tilde {\mathbf{r}}ho_t, g{\mathbf{r}}angle + \e^{-t\kappa(2)}\langle \mu_t, {\mathcal L}_{0}f{\mathbf{r}}angle,
$$
that is,
\begin{equation} \label{Eqgenexp}
\partial_t \langle \tilde {\mathbf{r}}ho_t, g{\mathbf{r}}angle = \langle \tilde {\mathbf{r}}ho_t, {\mathcal A}g{\mathbf{r}}angle,
\end{equation}
with
$${\mathcal A}g(x)= x^{-2} {\mathcal L}_{0}f (x)- \kappa(2)g(x),$$
as in the notation of \autoref{C0}. We can thus
interpret \eqref{Eqgenexp} as Kolmogorov's forward equation for the infinitesimal generator
of the Feller process $(\exp({\mathbf{x}}i(t)))_{t\geq 0}$.
This will in turn enable us to identify
$\tilde {\mathbf{r}}ho_t= {\mathbf{r}}ho_t$.
To be precise, examining the proof of \cite[Proposition 4.9.18]{EK-mp},
we see that \eqref{Eqgenexp} for
all $g\in {\mathcal C}^{\infty}_c(0,\infty)$ has at most one solution (in the sense of a
vaguely right-continuous collection
of measures $(\tilde {\mathbf{r}}ho_t)_{t\ge 0}$)
so long as the image of ${\mathcal C}^{\infty}_c(0,\infty)$ by
$\lambda-{\mathcal A}$ is separating (see \cite[p.~112]{EK-mp}) for each $\lambda >0$.
Since ${\mathcal A}$ is the generator of a Feller process and
${\mathcal C}^{\infty}_c(0,\infty)$ is a core (cf. Theorem 31.5 in Sato \cite{Sato}),
we know that the image of ${\mathcal C}^{\infty}_c(0,\infty)$ by
$\lambda-{\mathcal A}$ is a dense subset of $\mathcal C_0$,
and this implies that it is separating.
If $(\tilde {\mathbf{r}}ho_t)_{t \ge 0}$ is a collection of measures solving \eqref{Eqgenexp},
then for any $g \in \mathcal C_c^\infty$, the function $t \mapsto \langle \tilde {\mathbf{r}}ho_t,g{\mathbf{r}}angle$ is right-continuous.
Hence, the solution of
\eqref{Eqgenexp} restricted to $\mathcal C_c^\infty$ is unique, and this transfers to
\eqref{EqFG}.
\end{proof}
\subsection{Some properties of solutions}
\label{s:zero-properties}
We next present some properties of the solution identified in \autoref{T1},
by means of translating known results on Lévy processes.
We first point out that, depending on whether \eqref{Eqcondnu1} holds and $a=0$,
the support of the solution $\mu_t$ is bounded or not.
Specifically, if $a=0$ and \eqref{Eqcondnu1} holds,
we set
$$d \coloneqq b + \int_{[1/2,1)}(1-y) K(\d y),$$
and otherwise $d=\infty$. It is easy to verify that $d=\lim_{q\to \infty} q^{-1}\kappa(q)$.
\begin{cor}\label{C1}
If $a=0$ and \eqref{Eqcondnu1} holds,
then for every $t>0$, $\e^{dt}$ is the supremum of the support of $\mu_t$,
i.e., we have for every $\varepsilon >0$,
$$\mu_t((\e^{t d}, \infty))=0 \ \hbox{and}\ \mu_t((\e^{t d}-\varepsilon,\e^{t d}])>0.$$
\end{cor}
\begin{proof}
The spectrally
negative L\'evy process ${\mathbf{x}}i = {\mathbf{x}}i_2$, arising in \autoref{L1},
has bounded variation with drift coefficient $d$ exactly when the conditions of the result hold,
and it is then well-known that
$td$ is the maximum of the support of the distribution of ${\mathbf{x}}i(t)$.
Therefore we have ${\mathbf{r}}ho_t((\e^{t d}, \infty))=0$ and ${\mathbf{r}}ho_t((\e^{t d}-\varepsilon,\e^{t d}])>0$,
and our claim follows from \autoref{T1}.
\end{proof}
In the case when the assumptions of \autoref{C1} are not fulfilled, we have the following
large deviations estimates for the tail
$ \bar \mu_t(x) \coloneqq \mu_t((x,\infty))$ of $\mu_t$.
Recall that $\kappa$ is a convex function, and observe that
$\lim_{q \to +\infty} \kappa'(q)=+\infty$ when either $a>0$ or \eqref{Eqcondnu1} fails.
Thus for every $r$ sufficiently large, the equation $\kappa'(q)=r$ has a unique solution
which we denote by $\theta(r)$, and the Legendre-Fenchel transform of $\kappa$ is given by
$$\kappa^*(r)\coloneqq\sup_{q>0}\{rq-\kappa(q)\}=r\theta(r)-\kappa(\theta(r)).$$
\begin{cor}\label{C2}
Suppose that $a>0$ or \eqref{Eqcondnu1} fails.
Then for every $r>0$ sufficiently large, we have
$$\lim_{t\to \infty} t^{-1}\ln \bar \mu_t(\e^{tr}) = -\kappa^*(r).$$
\end{cor}
\begin{proof}
This follows easily from the identity $\langle \mu_t, h_q{\mathbf{r}}angle = \exp(t\kappa(q))$
by adapting the classical arguments of Cramér and Chernoff; see, for instance, Theorem 1 in \citet{Bigg}.
\end{proof}
The estimate of \autoref{C2} can easily be reinforced by using the local central limit theorem.
Here is a typical example (compare with Theorem 1.3 in \cite{DouEsc}).
\begin{cor}\label{C3}
Suppose that $a>0$ or \eqref{Eqcondnu1} fails,
and further that $\kappa'(q)<0$ for some $q$.
Then $\theta(0)$ is well-defined, $0<\kappa''(-\theta(0))<\infty$,
and for every $f\in{\mathcal C}_c$, we have
$$\langle \mu_t, f{\mathbf{r}}angle
\sim
\frac{\e^{t\kappa(\theta(0))}}{\sqrt{2\pi t \kappa''(\theta(0))}}\int_0^{\infty} f(x) x^{\theta(0)-1} \d x.$$
\end{cor}
\begin{proof}
The first assertion about the existence of $\theta(0)$ and $\kappa''(\theta(0))$ are immediate from the convexity
of $\kappa$ and the fact that $\lim_{+\infty} \kappa=+\infty$.
The function $\tilde \mathbb{P}hi(q)\coloneqq\kappa(q+\theta(0))-\kappa(\theta(0)) = \mathbb{P}hi(q+\theta(0))-\mathbb{P}hi(\theta(0))$
is the Laplace exponent of a spectrally negative L\'evy process $(\tilde {\mathbf{x}}i(t))_{t\geq 0}$ which is centered and
has finite variance $\kappa''(\theta(0))$.
Further, we see from Esscher transform and \autoref{T1} that
$$\mu_t(\d x) = \e^{t\kappa(\theta(0))} x^{-\theta(0)} \mathbb{P}(\exp(\tilde {\mathbf{x}}i(t))\in \d x).$$
Our claim then follows readily from the local central limit theorem for the L\'evy process.
\end{proof}
\begin{cor}\label{C4}
If $a > 0$ or the absolutely continuous component of $K(\d y)$ has an infinite total mass,
then $\mu_t$ is absolutely continuous for every $t>0$.
\end{cor}
\begin{proof}
Using \citet[Theorem 27.7 and Lemma 27.1]{Sato}, it follows
from the assumptions of the statement that
the one-dimensional distributions of the L\'evy process ${\mathbf{x}}i(t)$ are
absolutely continuous for every $t>0$.
Our claim follows from the representation in \autoref{T1}.
\end{proof}
\section{The self-similar case}
\label{s:ss}
We now turn our attention to the growth-fragmentation equation \eqref{EqFG}
for ${\mathcal L}={\mathcal L}_{\alpha}$ given by \eqref{Eqop1} and $\alpha\neq 0$.
We first point out that the function $\kappa$ is non-increasing if and only if $a=0$,
the dislocation measure $K$ fulfills \eqref{Eqcondnu1}, and
$$
b+ \int_{[1/2,1)}(1-y) K(\d y) \leq 0.
$$
In this case, the operator ${\mathcal L}_{\alpha}$ can be expressed in the form \eqref{Eqop0} with
$c\leq 0$, and \eqref{EqFG} is then a pure fragmentation equation as studied by Haas \cite{Haas}.
To avoid duplication of existing literature, this case will be implicitly excluded hereafter.
Recall the notation $h_q(x)=x^q$ for $x>0$. In the self-similar case, power functions are no
longer eigenfunctions of the operator ${\mathcal L}_{\alpha}$; however, there is the simple relation
\begin{equation}\label{Eqvpss}
{\mathcal L}_{\alpha}h_q=\kappa(q) h_{q+\alpha}, \qquad q \in \dom \kappa;
\end{equation}
see \autorefpref{L0}{i:L0:eigenfunction}.
Hence, if \eqref{EqFG} applies to power functions, the linear equation \eqref{EqMelh} for the
Mellin transform $ {M}_t(z) = \langle \mu_t, h_{z-1}{\mathbf{r}}angle$ in the homogeneous case has
to be replaced by the system
\begin{equation}\label{EqMelss}
\partial_t {M}_t(1+q) = \kappa(q){M}_t(1+q+\alpha).
\end{equation}
We make the fundamental assumption, that
\begin{equation}\label{Eqnegspeed}
\inf_{q\geq 0} \kappa(q) < 0,
\end{equation}
which is implicitly enforced throughout this section.
The role and the importance of \eqref{Eqnegspeed} shall become clear in the sequel.
Recall that $\kappa$ is a convex function on $\mathbb{R}$, and is ultimately increasing,
since we are excluding the case when $\kappa$ is non-increasing throughout \autoref{s:ss}.
Hence, condition \eqref{Eqnegspeed}
ensures the existence of a unique $\omega_{_+}\in\mathbb{R}$ with
\begin{equation}\label{EqMalthus}
\kappa(\omega_{_+})=0 \hbox{ and } \kappa'(\omega_{_+})>0.
\end{equation}
We refer to $\omega_{_+}$ as the \emph{Malthusian parameter}.
The sign of the scaling parameter $\alpha$ plays a crucial role, and we shall study the two cases
separately, even though some ideas are similar.
\subsection{The case \texorpdfstring{$\alpha < 0$}{alpha < 0}}
\label{s:alpha-neg}
We now focus on the case $\alpha <0$.
We start by observing that the existence of a Malthusian parameter enables us to view \eqref{EqMelss} as a closed
system for an arithmetic sequence, and thus
to solve it.
\begin{lem}\label{P3}
Consider a sequence of functions
${M}_{\bullet}(1+q)\from [0,\infty)\to (0,\infty)$ for
\linebreak
$q=\omega_{_+}-k\alpha$, $k=-1,0,1, \ldots$,
with ${M}_{0}(1+q)=1$.
Suppose that \eqref{EqMelss} and \eqref{Eqnegspeed} hold and recall that $\omega_+$ is the Malthusian parameter
defined by \eqref{EqMalthus}. Then ${M}_t(1+\omega_{_+})= 1$ for all $t\geq 0$ and
for $k=1,2, \ldots$, we have
\[ {M}_t(1+\omega_{_+}-k\alpha)
= 1 + \sum_{\ell=1}^k \frac{\kappa(\omega_{_+}-\alpha k)\cdots \kappa(\omega_{_+}-\alpha(k-\ell+1)) }{\ell!}\, t^{\ell} . \]
\end{lem}
\begin{proof}
The equation \eqref{EqMelss} applied to the Malthusian exponent $\omega_{_+}$
implies that the function $t\mapsto {M}_t(1+\omega_{_+})$ is constant.
We can then solve
\eqref{EqMelss} for $q=\omega_{_+}-\alpha k$ and $k=1, 2, \ldots$ by induction in order to obtain the given formula.
\end{proof}
In comparison with the homogeneous case, \autoref{P3} is a much weaker result than \eqref{SolMelh},
as we are not able to compute the whole Mellin transform of a solution, but merely its moments for orders
forming an arithmetic sequence. There is hence an additional crucial issue: it does not suffice
to find a family of measures having the desired moments, but also to ensure that the moment problem is determining.
It turns out that moment calculations which were performed in \cite{BY-moments}
for self-similar Markov processes enable us to solve the moment problem in \autoref{P3},
and check that this indeed yields a solution to \eqref{Eqop1}.
Similar calculations also point at a rather surprising result,
namely that the self-similar growth-fragmentation permits
spontaneous generation!
\begin{thm}\label{T2}
Assume \eqref{Eqnegspeed} and $\alpha <0$.
\begin{enumerate}[(i)]
\item{\mathbf{r}}elax\label{i:T2:pos}
For every $t\geq 0$, there exists a unique measure $\mu^{ }_t$ on $(0,\infty)$ such that
\linebreak
$\langle \mu^{ }_t, h_{\omega_{_+}}{\mathbf{r}}angle=1$ and for every integer $k\geq 1$,
$$\langle \mu^{ }_t, h_{\omega_{_+}-k\alpha}{\mathbf{r}}angle
= 1 + \sum_{\ell=1}^k
\frac{\kappa(\omega_{_+}-\alpha k)\cdots \kappa(\omega_{_+}-\alpha(k-\ell+1)) }{\ell!}\, t^{\ell} .$$
In particular, $\mu^{ }_0=\delta_1$ and the family $(\mu^{ }_t)_{t\geq0}$
solves \eqref{EqFG} for all $f\in{\mathcal C}^{\infty}_c(0,\infty)$
when ${\mathcal L}={\mathcal L}_{\alpha}$ is given by \eqref{Eqop1}.
\item{\mathbf{r}}elax\label{i:T2:zero}
For every $t>0$, there exists a unique measure $\gamma^{ }_t$ on
$(0,\infty)$ such that $\langle \gamma^{ }_t, h_{\omega_{_+}}{\mathbf{r}}angle=1$ and
$$\langle \gamma^{ }_t, h_{\omega_{_+}-k\alpha}{\mathbf{r}}angle
= t^k\frac{\kappa(\omega_{_+}-\alpha)\cdots \kappa(\omega_{_+}-\alpha k)}{k!}
\qquad \hbox{for every integer $k\geq 1$}.$$
If we further set $\gamma^{ }_0\equiv 0$, then the family $(\gamma^{ }_t)_{t\geq0}$ solves \eqref{EqFG} for all $f\in{\mathcal C}^{\infty}_c(0,\infty)$
when ${\mathcal L}={\mathcal L}_{\alpha}$ is given by \eqref{Eqop1}.
\end{enumerate}
\end{thm}
\begin{proof}
\begin{enumerate}[(i)]
\item
Let us define $\mathbb{P}hi_{_+} \coloneqq \mathbb{P}hi_{\omega_+} = \kappa(\cdot +\omega_+)$,
which, as we saw in \autoref{L1}, is the Laplace exponent of the Lévy
process ${\mathbf{x}}i_{_+} \coloneqq {\mathbf{x}}i_{\omega_+}$.
Observe that
\linebreak
$\mathbb{P}hi_{_+}'(0)=\kappa'(\omega_{_+})>0$,
so this L\'evy process has a strictly positive and finite first moment.
Proposition 1 in \cite{BY-moments} then ensures, for every $t>0$,
the existence of a unique probability measure ${\mathbf{r}}ho^{ }_t$ on $(0,\infty)$,
such that for every integer $k\geq 1$,
$$\langle {\mathbf{r}}ho^{ }_t, h_{-\alpha k}{\mathbf{r}}angle
= 1 + \sum_{\ell=1}^k
\frac{\kappa(\omega_{_+}-\alpha k)\cdots \kappa(\omega_{_+}-\alpha(k-\ell+1)) }{\ell!}\, t^{\ell} \,,$$
so that in particular ${\mathbf{r}}ho^{ }_0=\delta_1$.
Thus we may set $\mu^{ }_t(\d x) = x^{-\omega_{_+}} {\mathbf{r}}ho^{ }_t(\d x)$,
and then $\langle \mu^{ }_t, h_{\omega_{_+}-k\alpha}{\mathbf{r}}angle$ is given
as in the statement for every integer $k\geq 0$. That this determines $\mu^{ }_t$
derives from the uniqueness of ${\mathbf{r}}ho^{ }_t$.
Now, using \eqref{Eqvpss}, we immediately check that
$\langle \mu_t, h_{\omega_{_+}-k\alpha}{\mathbf{r}}angle$ satisfies \eqref{EqMelss}.
It then follows that $(\mu^{ }_t)_{t\geq 0}$ solves \eqref{EqFG} for every $f\in{\mathcal C}^{\infty}_c(0,\infty)$
(recall that the probability measure ${\mathbf{r}}ho^{ }_t$ is determined by its entire moments).
Finally, the map $t\mapsto \langle {\mathbf{r}}ho^{ }_t, h_{-\alpha k}{\mathbf{r}}angle$ is continuous, and we deduce that
$({\mathbf{r}}ho^{ }_t)_{t\geq 0}$ is vaguely continuous (using again the fact that ${\mathbf{r}}ho^{ }_t$ is determined
by its moments $\langle {\mathbf{r}}ho^{ }_t, h_{-\alpha k}{\mathbf{r}}angle$ for $k\in\mathbb{N}$).
Hence the same holds for $(\mu^{ }_t)_{t\geq 0}$.
\item
Recall from above that $\mathbb{P}hi_{_+}=\kappa(\omega_{_+}+\cdot)$ is the Laplace exponent of a
spectrally negative L\'evy process which has strictly positive and finite first moments.
Proposition 1 in \cite{BY-moments} ensures for every $t>0$ the existence of a unique probability measure
$\pi^{ }_t$ on $(0,\infty)$ such that its
moments are given by
\[ \langle \pi^{ }_t, h_{-\alpha k}{\mathbf{r}}angle
= t^k\frac{\mathbb{P}hi_{_+}(-\alpha)\cdots \mathbb{P}hi_{_+}(-\alpha k)}{k!}
= t^k\frac{\kappa(\omega_{_+}-\alpha)\cdots \kappa(\omega_{_+}-\alpha k)}{k!}, \]
for $k=1,2, \dotsc$,
and this determines $\pi^{ }_t$.
It follows immediately that $(\pi^{ }_t)_{t\geq 0}$ is vaguely continuous (recall that $\pi^{ }_0=0$).
We then define for $t>0$
$$\gamma^{ }_t(\d x) = x^{-\omega_{_+}} \pi^{ }_t(\d x)\,, \qquad x>0,$$
so
$$\langle \gamma^{ }_t, h_{\omega_{_+}-k\alpha}{\mathbf{r}}angle
= t^k\frac{\kappa(\omega_{_+}-\alpha)\cdots \kappa(\omega_{_+}-\alpha k)}{k!}.$$
Then \eqref{Eqvpss} entails that for every integer $k\geq 1$, there is the identity
\begin{eqnarr*}
\partial_t\langle \gamma^{ }_t, h_{\omega_{_+}-k\alpha}{\mathbf{r}}angle
&=&
t^{k-1}\frac{\kappa(\omega_{_+}-\alpha)\cdots \kappa(\omega_{_+}-\alpha k)}{(k-1)!}\\
&=&
\kappa(\omega_{_+}-\alpha k)\langle \gamma^{ }_t, h_{\omega_{_+}-(k-1)\alpha}{\mathbf{r}}angle
= \langle \gamma^{ }_t, \mathcal{L}h_{\omega_{_+}-k\alpha}{\mathbf{r}}angle,
\end{eqnarr*}
and the conclusion follows just as in (i). \qedhere
\end{enumerate}
\end{proof}
\autorefpref{T2}{i:T2:zero} entails that uniqueness of the solution fails
when one only requires \eqref{Eqop1} to be fulfilled for all
$f\in{\mathcal C}^{\infty}_c(0,\infty)$,
which contrasts sharply with the results of Haas \cite{Haas} for the pure-fragmentation equation.
We conjecture that the solution $\mu^{ }_t$ given in \autorefpref{T2}{i:T2:pos} is minimal,
in the sense that if $(\tilde \mu_t)_{t\geq 0}$ is another solution
with the same initial condition $\tilde \mu_0=\delta_1$,
then $\mu^{ }_t \leq \tilde \mu_t$ for every $t>0$.
We also stress that uniqueness of the solution can be restored by
requiring \eqref{EqFG} to hold for the functions $h_q$ with $q\geq \omega_{_+}+\alpha$;
see \autoref{P3} and \autorefpref{T2}{i:T2:pos}.
\skippar
We now present a different approach to \autoref{T2}.
In the homogeneous case $\alpha=0$, we saw in the preceding section
that the equation \eqref{Eqop1} bears a close relationship with certain exponential L\'evy processes.
It turns out that in the self-similar case with $\alpha<0$, the vital connection
is with positive
self-similar Markov processes, and is made via the Lamperti transformation which associates these
with the class of L\'evy processes. We first provide some background in this area.
A \define{positive self-similar Markov process} (\define{pssMp}) with
\define{self-similarity index} $\gamma \in \mathbb{R}$ is a standard Markov process
$R = (R_t)_{t\geq 0}$ with associated filtration ${\mathcal F}Ft$ and probability laws
$(\ssP_x)_{x \in (0,\infty)}$, on $[0,\infty]$, which has $0$ and $\infty$ as
absorbing states and
which satisfies the \define{scaling property}, that for every $x, c > 0$,
\[
\label{scaling prop}
\text{ the law of } (cR_{t c^{-\alpha}})_{t \ge 0}
\text{ under } \ssP_x \text{ is } \ssP_{cx} \text{.}
\]
Here, we mean ``standard'' in the sense of \cite{BG-mppt},
which is to say, ${\mathcal F}Ft$ is a complete, right-continuous filtration,
and $R$ has c\`adl\`ag paths and is strong Markov
and quasi-left-continuous.
In the seminal paper \cite{Lamp}, Lamperti describes a one-to-one correspondence
between pssMps and (possibly killed) L\'evy processes, which we now outline.
It may be worth noting that we have presented a slightly
different definition of pssMp from Lamperti; for the connection, see
\cite[\S 0]{VA-Ito}.
Let
$S(t) = \int_0^t (R_u)^{-\gamma}\, \dd u .$
This process is continuous and strictly increasing until $R$ reaches zero.
Let $(T(s))_{s \ge 0}$ be its inverse, and define
\[ \eta_s = \log R_{T(s)} \qquad s\geq 0.
\]
Then $\eta : = (\eta_s)_{s\geq 0}$ is a (possibly killed) L\'evy process started
at position $\log x$, possibly killed at an independent
exponential time; the law of the L\'evy process and the
rate of killing do not depend
on the value of $x$. The real-valued process
$\eta$ with probability laws
$(\LevP_y)_{y \in \mathbb{R}}$ is called the
\define{L\'evy process associated to $R$},
or the \define{Lamperti transform of $R$}.
An equivalent definition of $S$ and $T$, in terms of $\eta$ instead
of $R$, is given by taking
$T(s) = \int_0^s \exp(\gamma \eta_u)\, \dd u$
and $S$ as its inverse. Then,
\begin{equation*}
\label{Lamp repr}
R_t = \exp(\eta_{S(t)})
\end{equation*}
for all $t\geq 0$, and this shows that the Lamperti transform is a bijection.
A useful fact is that, as a consequence of the definitions we have just given,
it holds that $\dd t = \exp(-\gamma \eta_{S(t)}) \, \dd S(t)$.
Most of the literature on pssMps (including Lamperti's
original paper) assumes that $\gamma>0$, and much of it is also given for $\gamma=1$.
Indeed it is easy to change the index of self-similarity.
If $R$ is a pssMp of index $\gamma$ and corresponding to the
Lévy process $\eta$, then, for any
$\gamma' \in \mathbb{R}$, the process $R^{\gamma'} = (R_t^{\gamma'})_{t\ge 0}$
is a pssMp with index $\gamma/\gamma'$, corresponding to
the Lévy process $\gamma' \eta$. It is also useful to note that the
time-changes appearing in the Lamperti transformation are a.s.\ equal
for $R$ and $R^{\gamma'}$.
We should point out that the case $\gamma=0$ is special, since
in this case the time-change does not have any effect, and the
pssMps of index $0$ are just exponential Lévy processes.
Note that, if the Lévy process process $\eta$ is killed at time ${\mathbf{z}}eta$, then we define
$R_t = 0$ for $t \ge T({\mathbf{z}}eta)$ if $\gamma \ge 0$, and
$R_t = +\infty$ for $t \ge T({\mathbf{z}}eta)$ if $\gamma < 0$.
Recall \autoref{L1}, and define $\mathbb{P}hi_{_+} \coloneqq \mathbb{P}hi_\omega =\kappa(\cdot+\omega_{_+})$,
which is the Laplace exponent of the spectrally negative L\'evy process ${\mathbf{x}}i_{_+} \coloneqq {\mathbf{x}}i_{\omega_+}$.
Let us denote by $X_+$ the pssMp with index $-\alpha$ associated to ${\mathbf{x}}i_+$ by the Lamperti
transformation. Note that, because ${\mathbf{x}}i_+$ has positive mean, the process $X_+$ never
reaches the absorbing boundaries $0$ or $+\infty$.
We define the measure ${\mathbf{r}}ho_t^{ }$ to be the distribution of $X_{_+}(t)$ under $\ssP_1$, that is,
the probability measure on $(0,\infty)$ defined by
$$\langle {\mathbf{r}}ho_t^{ }, f {\mathbf{r}}angle = \ssE_1(f(X_{_+}(t)))\,,\qquad f\in{\mathcal C}_0(0,\infty)$$
and give first the following analogue of \autoref{C0}:
\begin{lem}\label{LanC0}
The family of probability measures $({\mathbf{r}}ho^{ }_t)_{t\geq 0}$
depends continuously on the parameter $t$ for the topology of weak convergence.
Further, for every $g\in {\mathcal C}^{\infty}_c(0,\infty)$,
the function $t\mapsto \langle {\mathbf{r}}ho^{ }_t, g{\mathbf{r}}angle$ is differentiable with derivative
$\partial_t\langle {\mathbf{r}}ho^{ }_t, g{\mathbf{r}}angle= \langle {\mathbf{r}}ho^{ }_t, {\mathcal A}^{(\alpha)}_{_+}g{\mathbf{r}}angle$,
where
$${\mathcal A}^{(\alpha)}_{_+}g(x) \coloneqq x^{-\omega_{_+}} {\mathcal L}_{\alpha}(h_{\omega_{_+}}g)(x)\,, \qquad x>0.$$
\end{lem}
\begin{proof}
The first assertion follows easily from the Feller property of self-similar Markov processes;
see Theorem 2.1 in Lamperti \cite{Lamp} and the remark on page 212.
In order to establish the second, we work with the L\'evy process ${\mathbf{x}}i_{_+}$
The exponential L\'evy process $\exp({\mathbf{x}}i_{_+}(\cdot))$ is a Feller process in $(0,\infty)$,
and the same calculation as in the proof of \autoref{C0} shows that its infinitesimal generator
${\mathcal A}_{_+}$ is given by
$${\mathcal A}_{_+}g(x)
= x^{-\omega_{_+}} {\mathcal L}_{0}(h_{\omega_{_+}}g)(x)\,,\qquad g\in{\mathcal C}^{\infty}_c(0,\infty).$$
According to Dynkin's formula (see, e.g., Proposition 4.1.7 of \cite{EK-mp}),
for every
\linebreak
$g\in{\mathcal C}^{\infty}_c(0,\infty)$, the process
$$g(\exp({\mathbf{x}}i_{_+}(t)))- \int_0^t {\mathcal A}_{_+}g(\exp({\mathbf{x}}i_{_+}(s))) \d s$$
is a martingale. Recall that by definition, $X_{_+}$ arises as the transform of $\exp({\mathbf{x}}i_{_+}(\cdot))$
by the time substitution ${S}$, which is given as the inverse of the additive functional
$\int_0^t h^{-1}_{\alpha}\left( \exp({\mathbf{x}}i_{_+}(s){\mathbf{r}}ight)\d s $, and we have the identity
$$
g(X_{_+}(t))- \int_0^{{S}(t)} {\mathcal A}_{_+}g(\exp({\mathbf{x}}i_{_+}(s))) \d s
= g(X_{_+}(t))- \int_0^{t} h_{\alpha}(X_{_+}(s)) {\mathcal A}_{_+}g(X_{_+}(s)) \d s.
$$
A priori, the time-substitution above changes a martingale into a local martingale.
However, using \autorefpref{Lbounds}{i:Lbounds:asym} and the fact that $\alpha <0$, we see that ${\mathcal A}^{(\alpha)}_{_+}g\coloneqq h_{\alpha} {\mathcal A}_{_+}g$ is bounded, and it follows that the process
$$g(X_{_+}(t))- \int_0^{t} {\mathcal A}^{(\alpha)}_{_+}g(X_{_+}(s)) \d s$$
is a true martingale. Taking expectations, we arrive at
$$\langle {\mathbf{r}}ho^{ }_t, g{\mathbf{r}}angle - g(1)=\int_0^t \langle {\mathbf{r}}ho^{ }_s, {\mathcal A}^{(\alpha)}_{_+}g{\mathbf{r}}angle\d s,$$
and our claim follows.
\end{proof}
The connection with solutions of the growth-fragmentation equation is the following:
\begin{cor}\label{c:mu-alpha-neg}
Let
\[ \tilde \mu^{ }_t = h_{-\omega_{_+}}{\mathbf{r}}ho_t^{ }, \qquad t\geq 0. \]
Then, $(\tilde \mu^{ }_t)_{t\geq 0}$ is
equal to the solution $(\mu^{ }_t)_{t \ge 0}$ of the growth-fragmentation equation appearing in \autorefpref{T2}{i:T2:pos}.
\end{cor}
\begin{proof}
We deduce immediately from \autoref{LanC0} that for every $f\in{\mathcal C}^{\infty}_c$
$$\partial_t\langle \tilde\mu^{ }_t, f{\mathbf{r}}angle
= \partial_t\langle {\mathbf{r}}ho^{ }_t, h^{-1}_{\omega_{_+}}f{\mathbf{r}}angle
= \langle {\mathbf{r}}ho^{ }_t, h^{-1}_{\omega_{_+}}{\mathcal L}_{\alpha}f{\mathbf{r}}angle
= \langle \tilde\mu^{}_t, {\mathcal L}_{\alpha}f{\mathbf{r}}angle\,,$$
that is, the family $\bigl(\tilde\mu^{ }_t\bigr)_{t\geq 0}$ solves
\eqref{EqFG} with ${\mathcal L}={\mathcal L}_{\alpha}$.
That $\mu^{ }_t$ coincides with the measure appearing in \autorefpref{T2}{i:T2:pos},
and that the notation
${\mathbf{r}}ho_t^{ }$ for the distribution of $X_{_+}(t)$ is consistent with that used in the proof of \autorefpref{T2}{i:T2:pos},
follows from Proposition 1 of \cite{BY-moments}.
\end{proof}
This approach could also be adapted to prove the existence of $(\gamma_t^{ })_{t\geq 0}$
using the process $X_+(t)$ started from zero, and indeed, this will be our method for
the case $\alpha>0$ in \autoref{s:alpha-pos}.
\skippar
We conclude the section by offering some results on the asymptotic behaviour of the solution
$(\mu_t)_{t\ge 0}$ given by the previous theorem.
Our first result in this direction indicates that, thanks to the self-similarity property \eqref{EqSSL} of
the equation \eqref{EqFG}, the solution starting
from zero given above can
be used to describe the asymptotic behaviour of $\mu^{ }_t$ as $t\to\infty$.
\begin{prop}\label{p:asymptotic-negative}
For any $f \in \mathbb{C}b$,
\[ \int f(t^{-1/\abs{\alpha}} z) z^{\omega_+} \mu_t^{ }(\dd z) \to \int f(z) z^{\omega_+} \gamma_1^{ }(\dd z) . \]
\end{prop}
\begin{proof}
Since $\kappa'(\omega_+) >0$, it is possible to extend
the definition of ${\mathbf{X}}p$ in order to allow it to start
from ${\mathbf{X}}p(0)=0$, such that it is a Feller process on the state space $[0,\infty)$;
this is a consequence of \cite[Theorem 1]{BY-entrance}.
For $x \ge 0$, we will denote by $\ssP_x$ the law of the process with ${\mathbf{X}}p(0) = x$.
It then follows from the scaling property that $\ssE_x[ f(t^{1/\alpha} {\mathbf{X}}p(t))] = \ssE_{x t^{1/\alpha}}[f({\mathbf{X}}p(1))]$,
and then the convergence
\[ \ssE_x[f(t^{1/\alpha} {\mathbf{X}}p(t)) ] \to \ssE_0[f({\mathbf{X}}p(1))], \qquad t \to \infty, \]
follows from the scaling property of ${\mathbf{X}}p$.
Finally, we know from the reference \cite{BY-moments}, which we used in
the proof of \autoref{T2}, that
the measures $x^{\omega_+}\mu_t^{ }(\dd x)$ and $x^{\omega_+} \gamma_t^{ }(\dd x)$ are,
respectively, equal to $\ssP_1(X_+ (t) \in \dd x)$ and $\ssP_0(X_+(t) \in \dd x)$.
Our claim follows immediately.
\end{proof}
We remark that the statement of the proposition can easily be extended to solutions
based on $(\mu_t^{ })$ whose initial value is a measure with compact support
in $(0,\infty)$.
\skippar
Suppose now that the equation $\kappa(q) = 0$ has two solutions,
$\omega_{_-}$ and $\omega_+$, with $\omega_{_-}<\omega_+$. Then we can say a little more.
Let $X_-$ be the $(-\alpha)$-pssMp associated with the Lévy process
${\mathbf{x}}i_- \coloneqq ({\mathbf{x}}i_-(t))_{t\ge 0}$
having Laplace exponent $\mathbb{P}hi_- \coloneqq \mathbb{P}hi_{\omega_{_-}}$.
Recall that we say the Lévy process ${\mathbf{x}}i_-$ is \emph{lattice} if,
for some $r \in \mathbb{R}$, the support
of ${\mathbf{x}}i_-(1)$ a.s.\ lies in $r{\mathbf{Z}}Z$;
otherwise, we say that ${\mathbf{x}}i_-$ is \emph{non-lattice}.
If we define
the random variable
\[ I = \int_0^\infty \e^{\abs{\alpha}{\mathbf{x}}i_-(t)}\, \dd t, \]
then it is known from \cite[Lemma 4]{Riv-re1} that, so long as
${\mathbf{x}}i_-$ is non-lattice,
\[ \lim_{t\to \infty} t^{(\omega_+-\omega_{_-})/\abs{\alpha}} \mathbb{P}_0(I>t) = C , \]
for some $0 < C < \infty$.
We obtain from \citet{HR-Yaglom} the following result.
\begin{prop}
Let $f \in \mathcal{C}_0(0,\infty)$ and assume that ${\mathbf{x}}i_-$ is non-lattice. Then,
\[ \dfrac{\dint x^{\omega_{_-}} f(t^{-1}x^{\abs{\alpha}}) \, \mu_t^{ }(\dd x)}{\mathbb{P}_0(I>t)}
\to \int f(x) \, \nu(\dd x), \qquad t \to \infty,
\]
where $\nu$ is the distribution of the random variable
$J_{(\omega_+-\omega_{_-})/\abs{\alpha}}$ in equation (13) of \cite{HR-Yaglom}.
\end{prop}
\begin{proof}
As remarked in the proof of \autoref{p:asymptotic-negative}, the equation
\[ x^{\omega_+}\mu_t^{ }(\dd x) = \ssP_1(X_+(t) \in \dd x) \] holds as an identity of
probability measures, where $X_+$ is the $(-\alpha)$-pssMp corresponding to
the Lévy process ${\mathbf{x}}i_+$ with Laplace exponent $\mathbb{P}hi_+ = \mathbb{P}hi_{\omega_+}$.
We now wish to use the `Esscher transform' for pssMps, which is essentially
obtained by standard arguments from the Esscher transform of Lévy processes
given in \cite[Theorem 3.9]{Kypr2} (see, for instance, the discussion around
\cite[Theorem 14]{CR-cond} for an application in the context of pssMps.)
This allows
us to perform a change of measure to switch from the
process $X_+$, related to the Laplace exponent $\mathbb{P}hi_+$,
to the process $X_-$, related to the Laplace exponent $\mathbb{P}hi_- = \mathbb{P}hi_+(\cdot + \omega_{_-} - \omega_+)$.
Specifically, we have:
\[ x^{\omega_{_-}}\mu_t^{ }(\dd x)
= x^{\omega_{_-}-\omega_+}x^{\omega_+}\mu_t^{ }(\dd x)
= x^{\omega_{_-}-\omega_+} \ssP_1(X_+(t) \in \dd x)
= \ssP_1(X_-(t) \in \dd x),
\]
for $x > 0$.
The process
${\mathbf{x}}i_-$ (which corresponds to $X_-$) is a Lévy process with non-monotone paths
and which satisfies the conditions of \citet[Theorem~1.6]{HR-Yaglom}. Applying
this theorem gives the result.
\end{proof}
\subsection{The case \texorpdfstring{$\alpha >0$}{alpha > 0}}
\label{s:alpha-pos}
In the case $\alpha>0$, the equation \eqref{EqMelss} for the Mellin transform is unfortunately
much less useful, for the following reasons.
Firstly, the analogue of \autoref{P3} would require to assume that
$\langle \mu_t,h_q{\mathbf{r}}angle <\infty$ for all $q$ sufficiently negative.
Roughly speaking, this would force the scarcity of small particles,
and this phenomenon only occurs for a very restricted class of dislocation measures $K$
(informally, dislocations should not generate too many small particles, and in particular
the total intensity of dislocations must be finite).
Secondly, even if one were able to get an expression for the (negative) moments
$\langle \mu^{ }_t, h_{\omega_{_+}-k\alpha}{\mathbf{r}}angle$ with $k\in\mathbb{N}$,
this moment problem would be in general indeterminate, and the arguments used
in the preceding section would thus collapse.
Nonetheless, we have just seen from \autoref{LanC0} that, for $\alpha<0$,
self-similar growth-fragmentation equations have a close connection
with certain self-similar Markov processes, and using the intuition that we gained,
we are able to offer a very similar set of results when $\alpha > 0$.
Recall that $\kappa\from [0,\infty)\to (-\infty,\infty]$ is a convex function,
and that, since we are assuming \eqref{Eqnegspeed},
we may pick $\omega>0$ such that $\mathtt{k}\coloneqq -\kappa(\omega)>0$.
Then, the function
$$\mathbb{P}hi_{\dagger}(q)\coloneqq \mathbb{P}hi_{\omega}(q) + \mathtt{k}= \kappa(\omega+q) \,,\qquad q\geq 0,$$
is the Laplace exponent of a spectrally negative L\'evy process killed at an independent exponential
time with parameter $\mathtt{k}$, say
${\mathbf{x}}i_{\dagger}$, and we denote by $X_{\dagger}$ the $(-\alpha)$-pssMp
associated with ${\mathbf{x}}i_\dagger$ via the Lamperti transformation.
Note that $X_{\dagger}$ hits the absorbing state $+\infty$ by a jump.
We write ${\mathbf{r}}ho_t^{ }$
for the sub-probability measure on $(0,\infty)$ induced by the distribution of $X_{\dagger}(t)$
and study its properties in the following result, which
mirrors \autoref{LanC0}.
\begin{lem}{\mathbf{r}}elax\label{LanC0-}
Suppose \eqref{Eqnegspeed} holds and $\alpha >0$. Then we have, in the notation above:
\begin{enumerate}[(i)]
\item{\mathbf{r}}elax\label{i:LanC0-:sup}
$\mathbb{E}( \sup_{t \ge 0} X_{\dagger}(t)^q) <\infty$
for all $0\leq q <\omega_{_+}-\omega$.
\item{\mathbf{r}}elax\label{i:LanC0-:ef}
$\mathbb{E}\biggl[ \displaystyle\int_0^\infty X_{\dagger}(u)^p \, \dd u \biggr] < \infty$
for all
$0 < p -\alpha < \omega_{_+} - \omega$.
\item{\mathbf{r}}elax\label{i:LanC0-:sol}
The family $({\mathbf{r}}ho^{ }_t)_{t\geq 0}$ depends continuously on the parameter $t$ for the topology
of weak convergence.
For every
$g\in {\mathcal C}^{\infty}_c(0,\infty)$, the function $t\mapsto \langle {\mathbf{r}}ho^{ }_t, g{\mathbf{r}}angle$
is differentiable with derivative
$\partial_t\langle {\mathbf{r}}ho^{ }_t, g{\mathbf{r}}angle= \langle {\mathbf{r}}ho^{ }_t, {\mathcal A}^{(\alpha)}_{\dagger}g{\mathbf{r}}angle$,
where
\[ {\mathcal A}^{(\alpha)}_{\dagger}g(x)
\coloneqq x^{-\omega} {\mathcal L}_{\alpha}(h_{\omega}g)(x)\,, \qquad x>0.
\]
\item
$({\mathbf{r}}ho^{ }_t)_{t\ge 0}$ solves the above equation also for $g(x) = x^q$ with $0 < q < \omega_{_+} - \omega$.
\end{enumerate}
\end{lem}
\begin{proof}
\begin{enumerate}[(i)]
\item
From the very construction of $X_{\dagger}$, the overall supremum
\linebreak
$\bar X_{\dagger}\coloneqq \sup_{t\geq 0} X_{\dagger}(t)$ is given by $\bar X_{\dagger}=\exp(\bar{\mathbf{x}}i_{\dagger})$,
with $\bar{\mathbf{x}}i_{\dagger}\coloneqq \sup_{t\geq 0} {\mathbf{x}}i_{\dagger}(t)$. We infer from Corollary VII.2
in \cite{BeLP} that $\bar{\mathbf{x}}i_{\dagger}$ has the exponential distribution with parameter $\omega_{_+}-\omega$
(which is the positive root to the equation $\mathbb{P}hi_{\dagger}(q)=0$), and our claim follows.
\item
We begin with the following calculation, using the discussion of pssMps in the preceding section.
Recall that $S$ is the time-change appearing in the Lamperti transform relating $X_{\dagger}$ and ${\mathbf{x}}i_{\dagger}$, and that there is the identity $\dd S(t)=\exp(\alpha {\mathbf{x}}i_{\dagger}(t))\dd t$. We have therefore
\begin{align*}
\int_0^\infty X_{\dagger}(u)^p \, \dd u
&= \int_0^\infty \e^{p{\mathbf{x}}i_{\dagger}(S(u))} \, \dd u \\
&= \int_0^\infty \e^{(p-\alpha){\mathbf{x}}i_{\dagger}(S(u))} \, \dd S(u)
= \int_0^{\infty} \e^{(p-\alpha){\mathbf{x}}i_{\dagger}(s)} \, \dd s.
\end{align*}
But now we can consider the expectation:
\begin{align*} \mathbb{E}_x\biggl[ \int_0^\infty X_{\dagger}(u)^p \, \dd u \biggr]
&= x^{p-\alpha} \mathbb{E}\biggl[ \int_0^\infty \e^{(p-\alpha){\mathbf{x}}i_{\dagger}(s)}\, \dd s \biggr] \\
&= \begin{cases}
\dfrac{x^{p-\alpha}}{\kappa(p-\alpha+\omega)}, & \text{if } \kappa(p-\alpha+\omega) < 0, \\
\infty, & \text{otherwise}.
\end{cases}
\end{align*}
We complete the proof by recalling the definition of $\omega_{_+}$.
\item
Just as in the proof of \autoref{LanC0}, the first assertion follows from the
Feller property of self-similar Markov processes, and the process
\begin{equation}\label{e:time-change-local-martingale}
N_t \coloneqq g(X_{\dagger}(t))-g(1)-\int_0^t {\mathcal A}^{(\alpha)}_{\dagger} g(X_{\dagger}(s)) \d s
\end{equation}
is a local martingale for every $g\in{\mathcal C}^{\infty}_c(0,\infty)$.
We will show that \[ \mathbb{E}\bigl(\textstyle\sup_{s \le t} \abs{N_s}\bigr) < \infty, \qquad t \ge 0, \]
which implies that $N$ is a true martingale;
see \cite[Theorem I.51]{Pro-stoch-calc}.
Since $g$ is bounded, certainly $\sup_{s\le t}\abs{g(X_{\dagger}(s))-g(x)}$ is in
$L^1(\mathbb{P})$. In constrast to \autoref{LanC0}, the function
$\mathcal{A}^{(\alpha)}_{\dagger}g$ may be unbounded for $\alpha > 0$;
however, we do know from
\autorefpref{Lbounds}{i:Lbounds:asym} that ${\mathcal A}^{(\alpha)}_{\dagger}g$
is zero on some neighborhood of $0$ and,
for any $q \in \dom \kappa$, fulfills ${\mathcal A}^{(\alpha)}_{\dagger}g= o(x^{q+\alpha-\omega})$
as $x\to \infty$.
We let $\omega < q < \omega_{_+}$ and keep
it fixed for the rest of the proof.
For some $K>0$ we then have
\begin{align}
\mathbb{E} \biggl[\sup_{u\le t} \abs[\bigg]{\int_0^u {\mathcal A}^{(\alpha)}_{\dagger}g(X_{\dagger}(s))\, \dd s} \biggr]
&\le
\mathbb{E} \biggl[\int_0^t \abs[\big]{ {\mathcal A}^{(\alpha)}_{\dagger}g(X_{\dagger}(s))}\,\dd s \biggr] \label{e:mg-crit-ef} \\
&\le t \sup_{[0,K]} \abs{{\mathcal A}^{(\alpha)}_{\dagger}g}
+ \mathbb{E} \biggl[ \int_0^t X_{\dagger}(s)^{q+\alpha-\omega} \, \Indic{X_{\dagger}(s) > K} \, \dd s \biggr]. \nonumber
\end{align}
Setting $p = q+\alpha-\omega$ in part \eqref{i:LanC0-:ef}, we see that the right-hand
side is finite.
We have thus shown that $N$ is a true martingale, and
\begin{equation*}\label{e:Ag-Fubini}
\mathbb{E}\left( \int_0^t |{\mathcal A}^{(\alpha)}_{\dagger}g(X_{\dagger}(s)) |\d s {\mathbf{r}}ight)<\infty.
\end{equation*}
Taking expectations in \eqref{e:time-change-local-martingale}
and applying Fubini's theorem, we obtain
\[
\langle {\mathbf{r}}ho^{ }_t, g{\mathbf{r}}angle - g(1)=\int_0^t \langle {\mathbf{r}}ho^{ }_s,
\mathcal{A}^{(\alpha)}_{\dagger}g{\mathbf{r}}angle\d s,
\]
which completes the proof.
\item
This part is proved by setting $g(x) = x^q$ in the previous part, as follows.
Using the Markov property one sees immediately that the process
\[ M_t = \e^{q{\mathbf{x}}i_{\dagger}(t)} - 1 - \kappa(\omega_{_-}+q)\int_0^t \e^{q{\mathbf{x}}i_{\dagger}(s)}\,\dd s, \qquad t \ge 0, \]
is a martingale in the filtration of ${\mathbf{x}}i_{\dagger}$ for every $q \ge 0$.
Applying the same reasoning with the time-change as in \autoref{LanC0}, it follows
that
\[
N_t = X_{\dagger}(t)^q - 1 - \kappa(\omega+q)\int_0^t X_{\dagger}(s)^{q+\alpha}\, \dd s, \qquad t \ge 0,
\]
is a local martingale.
(For our choice of $g$, we have
${\mathcal A}^{(\alpha)}_{\dagger}g(x) = \kappa(\omega+q) x^{q+\alpha}$,
so this is consistent with the proof of part (iii).)
We observe that
\[ \sup_{t \ge 0} \abs{N_t}
\le
1 + \sup_{t \ge 0} X_{\dagger}(t)^q
- \kappa(\omega+q) \int_0^\infty X_{\dagger}(s)^{q+\alpha} \, \dd s .
\]
We now apply directly parts (i) and (ii) of this lemma in order to show
that $\mathbb{E} [ \sup_{t \ge 0} \abs{N_t}] < \infty$. This is a sufficient criterion
for $N$ to be a uniformly integrable martingale (see \cite[Theorem I.51]{Pro-stoch-calc}),
and the remainder of the proof follows in the same way as in part (iii). \qedhere
\end{enumerate}
\end{proof}
We can then repeat the calculations which were made after the proof of \autoref{LanC0},
and arrive at:
\begin{cor} \label{c:mu-alpha-pos}
Suppose that \eqref{Eqnegspeed} holds.
Define $\mu^{ }_t = h_{-\omega}{\mathbf{r}}ho_t^{ }$, for $t\geq 0$.
Then the vaguely continuous family of measures $(\mu^{ }_t)_{t\geq 0}$ solves
\eqref{EqFG} with ${\mathcal L}={\mathcal L}_{\alpha}$ both for
\linebreak
$f \in \mathbb{C}test(0,\infty)$ and
for $f=h_q$, for any $\omega<q<\omega_+$.
\end{cor}
An interesting contrast with the case $\alpha < 0$
is that we do \emph{not} show that $\mu_t^{ }$
solves \eqref{EqFG} for all power functions.
\noindent
We now give the basis of a solution to the growth-fragmentation equation starting from
the zero measure; in this case, the solution should be interpreted, not as spontaneous generation
from infinitely small masses, but as starting from infinite
mass and breaking apart instantaneously. Recall that the equation $\kappa(q)=0$ has at most two solutions.
More precisely, we have already seen that there is always a unique solution
$\omega_{_+}$ with $\kappa'(\omega_{_+}) > 0$
(this is the Malthusian exponent defined by \eqref{EqMalthus}).
When a second solution, say $\omega_{_-}$, exists,
then $\omega_{_-}<\omega_{_+}$ and $\kappa'(\omega_{_-}) \in[-\infty, 0)$.
We give the results under the assumptions:
\begin{equation}\label{EqMalthus-}
\hbox{
the equation $\kappa(q)=0$ with $q\geq 0$
has two solutions $\omega_{_-}<\omega_{_+}$, and $\kappa'(\omega_{_-})>-\infty$
}
\end{equation}
which is thus stronger than \eqref{EqMalthus}.
We write ${\mathbf{x}}i_-$ for the spectrally negative L\'evy process with Laplace exponent
$\mathbb{P}hi_-(q)\coloneqq \mathbb{P}hi_{\omega_-} = \kappa(q+\omega_{_-})$, and then $X_-$ for the pssMp
with index $-\alpha$ associated to ${\mathbf{x}}i_-$ by Lamperti's transform.
\begin{lem}\label{Lentrance}
Assume that \eqref{EqMalthus-} holds.
Then there exists a c\`adl\`ag process $({\mathcal X}(t))_{t> 0}$ with values in
$(0,\infty)$ and $\lim_{t\to 0+}{\mathcal X}(t)=\infty$ a.s.\ such that:
\begin{itemize}
\item
For every $s>0$, conditionally on ${\mathcal X}(s)=x$, the shifted process $({\mathcal X}(s+t))_{t\geq 0}$
has the law $\ssP_x$ of the pssMp $X_-$ started from $x$.
\item
For all $0< \varepsilon < (\omega_+-\omega_{_-})/\alpha$, there is $c(\varepsilon)<\infty$ such that
$$\mathbb{E}\left({\mathcal X}(t)^{\alpha(1-\varepsilon)}{\mathbf{r}}ight) = c(\varepsilon) t^{\varepsilon-1},\qquad t>0.$$
\end{itemize}
\end{lem}
\begin{proof}
Let $Y$ denote the pssMp with self-similarity index $\alpha$ associated to
the L\'evy process $-{\mathbf{x}}i_-$, so $Y$ has the same law as $1/X_-$.
Because \[ \mathbb{E}(-{\mathbf{x}}i_-(1))=-\mathbb{P}hi'_-(0+)\coloneqq m\in(0,\infty), \]
\cite{BY-entrance} shows that $0+$ is an entrance boundary for $Y$,
i.e., there exists a c\`adl\`ag process $({\mathcal Y}(t))_{t> 0}$
with values in $(0,\infty)$ and $\lim_{t\to 0+}{\mathcal Y}(t)=0$ a.s., such that,
for every $s>0$, conditionally on ${\mathcal Y}(s)=y$, the shifted process
$({\mathcal Y}(s+t))_{t\geq 0}$ has the law $Y$ started from $y$.
Our first claim follows by setting $\mathcal{X}(t) = 1/\mathcal{Y}(t)$.
Further, according to Theorem 1 in \cite{BY-entrance}, there is the identity
$$\mathbb{E}\left( {\mathcal Y}^{\alpha(\varepsilon-1)}(t){\mathbf{r}}ight) = \frac{1}{\alpha m}\mathbb{E}\left(I^{-1}(t/I)^{\varepsilon-1}{\mathbf{r}}ight)$$
where $I\coloneqq \int_0^{\infty}\exp(\alpha{\mathbf{x}}i_-(s))\dd s$. It thus follows
$$\mathbb{E}\left( {\mathcal X}^{\alpha(1-\varepsilon)}(t){\mathbf{r}}ight) = c(\varepsilon) t^{\varepsilon-1}$$
where $c(\varepsilon)= \mathbb{E}\left(I^{\varepsilon}{\mathbf{r}}ight)/ \alpha m \in (0,\infty]$.
For every $0< \varepsilon < (\omega_+-\omega_{_-})/\alpha$, the Laplace exponent
$q\mapsto \mathbb{P}hi_-(\alpha q)$ of $\alpha {\mathbf{x}}i_-$ fulfills $\mathbb{P}hi_-(\alpha \varepsilon)<0$, and according to
Lemma 3 in Rivero \cite{Rivero}, this ensures that $\mathbb{E}\left(I^{\varepsilon}{\mathbf{r}}ight)<\infty$.
\end{proof}
We next further require that
$\omega_{_-} \in (\dom \kappa)^\circ$. This is only a little stronger than the
condition $\kappa'(\omega_{_-})>-\infty$, which is necessary and sufficient for $\mathcal X$ to exist.
We write $\Aa_-$ for the operator $\Aa_{\dagger}$ given in \autorefpref{LanC0-}{i:LanC0-:sol} for
$\omega=\omega_{_-}$, and deduce the following.
\begin{cor}\label{Centrance}
Assume that \eqref{EqMalthus-} holds and that $\omega_{_-} \in (\dom \kappa)^\circ$.
For $t>0$, write $\pi_t$ for the distribution of ${\mathcal X}(t)$. Then, for every $f\in{\mathcal C}^{\infty}_c(0,\infty)$, we have
$$\int_0^t \abs[\big]{\langle \pi_s, \Aa_-f{\mathbf{r}}angle } \, \dd s <\infty,$$
and there is the identity
$$\langle \pi_t, f{\mathbf{r}}angle= \int_0^t \langle \pi_s, \Aa_-f{\mathbf{r}}angle \, \dd s.$$
\end{cor}
\begin{proof}
Recall from \autorefpref{LanC0-}{i:LanC0-:sol} that
$h_{\omega_{_-}}\Aa_-f={\mathcal L}_{\alpha}(h_{\omega_{_-}}f)$,
so \autorefpref{Lbounds}{i:Lbounds:asym} and the assumption that
$\omega_{_-}-\alpha \varepsilon \in \dom \kappa$ for some $\varepsilon>0$ entail
$$\Aa_-f(x) = o\Bigl(x^{-\omega_{_-}+\alpha+\omega_{_-}-\alpha \varepsilon} \Bigr) = o\Bigl(x^{\alpha(1-\varepsilon)} \Bigr) .$$
It now follows from the preceding lemma that
$|\langle \pi_s, \Aa_-f{\mathbf{r}}angle | \leq C(s^{\varepsilon-1}+1)$, where $C$ is a finite constant
depend ending only on $f$ and $\varepsilon$. Our first claim follows.
Recall then from the proof of \autorefpref{LanC0-}{i:LanC0-:sol} that
$$f(X_-(t))-\int_0^t \Aa_-f(X_-(s))\dd s\,,\qquad t\geq 0$$
is a local martingale, and thus, thanks to \autoref{Lentrance}, so is
$$f({\mathcal X}(t+s))-f({\mathcal X}(s))-\int_0^t \Aa_-f({\mathcal X}(r+s))\dd r\,,\qquad t\geq 0$$
for all $s>0$. Since
$$\mathbb{E}\left( \int_0^t |\Aa_-f({\mathcal X}(r+s))|\dd r {\mathbf{r}}ight) = \int_s^{t+s}|\langle \pi_r, \Aa_- f{\mathbf{r}}angle | \dd r < \infty,$$
the above process is actually a true martingale, and taking expectations, we arrive at the identity
$$\langle \pi_t, f{\mathbf{r}}angle- \langle \pi_s, f{\mathbf{r}}angle= \int_s^t \langle \pi_r, \Aa_-f{\mathbf{r}}angle \dd r.$$
With the observation above, we can let $s\to 0+$, and since $\langle \pi_s, f{\mathbf{r}}angle\to 0$
thanks to \autoref{Lentrance}, we conclude that
$$\langle \pi_t, f{\mathbf{r}}angle= \int_0^t \langle \pi_r, \Aa_-f{\mathbf{r}}angle \dd r.$$
\end{proof}
We conclude this section with the following corollary, which
demonstrates the existence of a solution to the growth-fragmentation equation
started from zero mass.
\begin{cor}
Suppose that the hypotheses of \autoref{Centrance} are fulfilled.
Let
\linebreak
$\gamma_t(\dd x)=x^{-\omega_{_-}}\pi_t(\dd x)$
for $t>0$, and set $\gamma^{ }_0\equiv 0$. Then, the family $(\gamma^{ }_t)_{t\geq0}$
solves \eqref{EqFG} for all $f\in{\mathcal C}^{\infty}_c(0,\infty)$
when ${\mathcal L}={\mathcal L}_{\alpha}$ is given by \eqref{Eqop1}.
\end{cor}
\section{Explosion of the stochastic model}
\label{s:explosion}
In this section, we discuss the behaviour of the stochastic growth-fragmentation
process in a simplified setting.
In particular, we point out that,
when the Malthusian hypotheses from \autoref{s:ss} do not hold,
this stochastic model may experience explosion, in the sense that some arbitrarily small
compact sets contain infinitely many particles after a finite time.
We will focus on the case where $\alpha < 0$, though similar arguments
can be made for $\alpha > 0$.
\skippar
Assume that the measure $K$ is a probability measure on $[1/2,1)$
which is not equal to $\delta_{\frac{1}{2}}$, and denote by $Y$
a random variable with law $K$. Choose $c \in \mathbb{R}$ such that
$\mathbb{E}[\log(1-Y)] + c < 0 < \mathbb{E}[\log Y] + c$.
We now set up the stochastic model. Let $\tree = \bigcup_{n \ge 0} \{L,R\}^n$,
where $\{L,R\}^0 = \{\varnothing\}$.
We view this as a binary tree, as follows. The root node $\eve$ gives rise
to child nodes $L$ and $R$; the former then has children $LL$ and $LR$, while
the latter has children $RL$ and $RR$, and so on. We introduce the
ancestry relationship `$\parenteq$' by saying that,
for individuals $u,u' \in \tree$, $u \parenteq u'$
if and only if there exists $u'' \in \tree$ such that $uu'' = u'$;
we also define the strict relation $u \parent u'$ to mean that
$u \parenteq u'$ but $u \ne u'$.
Let ${\mathbf{r}}ays = \{L,R\}^\mathbb{N}$.
This is the set of infinite lines of descent,
or rays,
in $\tree$. For instance, $LLRRLRLRRRRL\dotsb \in {\mathbf{r}}ays$
traces a line of descent starting at individual $\eve$,
and proceeding to $L$, then $LL$, then $LLR$, and so on.
If $u \in \tree$ and $v \in {\mathbf{r}}ays$, we say (by slight
abuse of notation) that $u \parent v$ if and only if
there exists $v' \in {\mathbf{r}}ays$ such that $uv' = v$.
To each $u \in \tree$, we assign, independently
of everything else,
a \emph{lifetime} $T_u$ which is
has an exponential distribution of rate 1,
and an \emph{offspring distribution} $Y_u$
which is distributed with law $K$. We then recursively
assign positions ${\mathbf{z}}eta_u$ to the individuals in $\tree$.
The root is positioned at a given point $x \in \mathbb{R}$, that is,
${\mathbf{z}}eta_{\eve} = x$.
Its descendents are positioned as follows:
\[
{\mathbf{z}}eta_{uL} = {\mathbf{z}}eta_u + \log(1-Y_u) + cT_u \quad \text{and} \quad
{\mathbf{z}}eta_{uR} = {\mathbf{z}}eta_u + \log(Y_u) + cT_u, \qquad u \in \tree.
\]
This gives a model in which each individual lives an exponential time,
dies, and (on average) scatters one child to the left and one to the right.
It will be convenient to introduce a model in which individuals also
move continuously, as follows.
For $u \in \tree$, define its birth time $a_u = \sum_{u'\parent u} T_{u'}$
and its death time $b_u = \sum_{u'\parenteq u} T_{u'} = a_u + T_u$. Its position
between those times is then given by ${\mathbf{x}}i_u(t) = {\mathbf{z}}eta_u + c(t-a_u)$,
for $a_u \le t < b_u$. By another abuse of notation, let us define also
the positions of a ray: for $v \in {\mathbf{r}}ays$, let ${\mathbf{x}}i_v(t) = {\mathbf{x}}i_u(t)$, where
$u \in \tree$ is the unique individual with $a_u \le t < b_u$ and $u \parent v$.
We may now see the model as containing individuals which move to the right
at constant rate $c$, until an exponential clock rings and the individual
dies, scattering offspring to the left.
The model may also be viewed as a stochastic process
${\mathbf{Y}}y = ({\mathbf{Y}}y(t))_{t\ge 0}$, with
$${\mathbf{Y}}y(t) = \sum_{u \in \tree} \delta_{\exp({\mathbf{x}}i_u(t))} \Indic{a_u\le t< b_u},$$
taking values in the space $\mathbb{N}n$ of locally finite point measures
and such that ${\mathbf{Y}}y(0) = \delta_{\e^x}$.
With this perspective, it is
an important and useful fact that the process has the branching property.
Loosely speaking, this means that the behaviour of $({\mathbf{Y}}y(t+s))_{s\ge 0}$
is given by collecting the atoms of ${\mathbf{Y}}y(t)$ and running from
each one an independent copy of ${\mathbf{Y}}y$;
for a precise statement and proof see, for instance, \cite[Proposition 2]{BeCF}.
The process ${\mathbf{Y}}y$ we have just described is a stochastic model
corresponding to the homogeneous fragmentation equation.
In particular, if we define a collection of measures $(\mu_t)_{t\ge 0}$ via
\[ \ip{\mu_t}{f} = \mathbb{E}\bigl[\ip{{\mathbf{Y}}y(t)}{f}\bigr] = \mathbb{E}\biggl[ \sum_{u \in \tree} f(\exp({\mathbf{x}}i_u(t)))\Indic{a_u \le t < b_u} \biggr],
\qquad f \in \mathbb{C}test,
\]
then we obtain a solution to \eqref{EqFG} with $\alpha=0$ and $\mathcal{L}$ given as in \eqref{Eqop0};
we give some more details on this in \autoref{s:bps}.
This corresponds to
the function $\kappa$ satisfying, for $q \ge 0$,
\[ \kappa(q) = cq + \int_{[\frac{1}{2},1)} \bigl[ y^q+(1-y)^q-1 \bigr] \, K(\dd y)
> q\Big(c + \int_{[\frac{1}{2},1)} \log y \, K(\dd y) \Bigr) > 0.
\]
Here, the first inequality holds because it holds for the integrands, and
the second inequality is by our assumption about $c$ at the beginning
of the section. Also, $\kappa(0) = 1$.
Thus, $\inf_{q \ge 0} \kappa(q) > 0$, and so the Malthusian hypothesis \eqref{EqMalthus},
which was an important assumption in \autoref{s:ss}, is
not satisfied for our model.
We now give the model corresponding to the self-similar equation. To do this,
we should first introduce the notion of a \emph{stopping line}. We say
that $S = (S_v)_{v\in{\mathbf{r}}ays}$ is a stopping line if:
\begin{enumerate}[(i)]
\item For every $v \in {\mathbf{r}}ays$, $S_v$ is a stopping time for the natural filtration of ${\mathbf{x}}i_v$;
\item If $u \in \tree$ and $v, v' \in {\mathbf{r}}ays$ such that $u \parent v,v'$, then $\mathbb{P}(S_v = S_{v'} \given a_u \le S_v < b_u) = 1$.
\end{enumerate}
Now, for $v \in {\mathbf{r}}ays$ and $\alpha \in \mathbb{R}$, let
\[ T_v(s) = \int_0^s \e^{-\alpha {\mathbf{x}}i_v(r)} \, \dd r , \]
and denote its inverse by $S_v$. Then, $S(t) = (S_v(t))_{v \in {\mathbf{r}}ays}$ is
a stopping line for every $t \ge 0$.
If $u \in \tree$ is an individual
such that, for some $v \in {\mathbf{r}}ays$ with $u\parent v$, $a_u\le S_v(t)< b_u$ holds,
then we define $S_u(t)$ to be equal to $S_v(t)$;
by property (ii) of the definition of a stopping line,
this does not depend on the choice of $v$.
We define
\[ X_v(t) = \exp({\mathbf{x}}i_v(S_v(t))), \qquad v \in {\mathbf{r}}ays, \; t \ge 0 , \]
and
\begin{equation}\label{e:def-Xf}
{\mathbf{X}}f(t) = \sum_{u \in \tree} \delta_{\exp {\mathbf{x}}i_u(S_u(t))} \Indic{a_u \le S_u(t) < b_u}, \qquad t \ge 0,
\end{equation}
where the sum is over only those $u$ for which $S_u(t)$ is defined.
The process ${\mathbf{X}}f$ is called the
$\alpha$-self-similar fragmentation process.
The stopping-line nature of $S$ means that the process ${\mathbf{X}}f$ retains
the branching property; however, it is not clear that it should be locally finite,
and indeed,
our main result in this section is that ${\mathbf{X}}f$ a.s.\ does \emph{not}
remain locally finite for all time:
\begin{prop}
Let $\alpha < 0$. For any $a > 0$, there exists some random time $\sigma$ such that
there are infinitely many individuals of ${\mathbf{X}}f(\sigma)$ in the compact set $[1,1+a]$.
\end{prop}
\begin{proof}
We will study rays $p_k = L^kR^\infty$ which follow the left-hand offspring
for $k$ steps, and the right-hand offspring thereafter. Our first remark
is that, if we define
\[ \tau_0(v) = \inf\{ t\ge 0 : X_v(t) = 0\} , \qquad v \in {\mathbf{r}}ays, \]
then we have
\[
\tau_0 \coloneqq \tau_0(L^\infty)
= T_{L^\infty}(\infty) = \int_0^\infty \e^{-\alpha {\mathbf{x}}i_{L^{\infty}}(t)}\, \dd t.
\]
Since ${\mathbf{x}}i_{L^\infty}$ is a Lévy process with negative mean (by our assumption on $c$)
we obtain that $\tau_0 < \infty$ almost surely.
Consequently, for any $\eta > 0$ there exists some infinite set $C$ such that,
for $k \in C$,
$T_{p_k}(b_{L^k}) \in (\tau_0-\eta, \tau_0)$ and $X_{p_k}(T_{p_k}(b_{L^k})) \to 0$ as $k \to \infty$.
We define
\[ L_1^+(v) = \sup\{ t\ge 0 : X_v(t) \le 1 \} , \qquad v \in {\mathbf{r}}ays, \]
which is the \emph{last passage time} of the level 1 by the process $X_v$;
then, for $k \in C$, we have
\[ L_1^+(p_k) = T_{p_k}(b_{L^k}) + \tilde L_1^+(R^\infty) , \]
where $\tilde L_1^+(R^\infty)$ is the last passage time of the level 1
by $\tilde X_{R^\infty}$, computed for an
independent self-similar fragmentation process started at $X_{p_k}(b_{L^k})$.
We are therefore reduced to studying first passage times of the
$(-\alpha)$-pssMp $X_{R^\infty}$ corresponding to a spectrally negative Lévy process ${\mathbf{x}}i_{R^\infty}$
started from a level $x < 0$ with Laplace
exponent
\[ \mathbb{P}hi_{R^\infty}(q) = cq + \int_{[\frac{1}{2},1)} (y^q-1) K(\dd y). \]
We seek $t$ such that, with positive probability (not depending
on $x$), $\tilde L_1^+({R^\infty}) \le t$.
The first observation is that $\mathbb{P}hi'_{R^{\infty}}(0+)>0$, which implies that the process $X_{R^\infty}$
can be extended to start at zero; it is then Feller on $[0,\infty)$. Furthermore,
the pssMp drifts to $+\infty$ as $t\to \infty$.
Hence, we pick $\epsilon>0$ and $t \ge 0$ such that $\mathbb{P}_0(L_1^+ \le t) \ge 2\epsilon$.
By the Portmanteau theorem, $\liminf_{x \to 0}\mathbb{P}_{x}(L_1^+ \le t) \ge 2\epsilon$ also;
and therefore for $x$ sufficiently close to zero, $\mathbb{P}_x(L_1^+ \le t) \ge \epsilon$.
Applying the Borel-Cantelli lemma to the paths referred to above,
there are then infinitely many $k \in C$ such that
\[ L_1^+(p_k) \le T_{p_k}(b_{L^k}) + t \le \tau_0 + t \]
with probability 1.
We therefore have infinitely many paths whose last passage time of $1$
occurs in a (random) compact interval. In particular, there must exist
some finite random time $\sigma$ such that for every $\delta> 0$,
there are infinitely many paths which cross $1$ for the last
time in $(\sigma-\delta,\sigma)$.
Since the particles are large, the time-change is bounded,
in that $S_{p_k}(L^+_1(p_k)+u)-S_{p_k}(L_1^+(p_k)) \le u$ for all $u\ge 0$.
Furthermore, the processes ${\mathbf{x}}i_{p_k}$
can grow at most at rate $c$;
thus, picking $\delta < c^{-1}\log(1+a)$
ensures that,
at time $\sigma$, all the selected particles are in $[1,1+a]$,
which completes the proof.
\end{proof}
\skippar
This result illustrates one example where the Malthusian hypothesis, under we
examined the growth-fragmentation equation, fails, and where the stochastic model ${\mathbf{X}}f$
(whose mean measure
could otherwise be expected to give a solution)
does not remain locally finite. However, since
uniqueness generally fails for the self-similar growth-fragmentation equation, it
does not immediately imply that there is no global solution of \eqref{EqFG}.
The procedure
of creating the growth-fragmentation process ${\mathbf{Y}}y$ can be carried out under
the general conditions on $a$, $b$ and $K$ given in the main body of the paper
(this is done in \citet{BeCF}) and the self-similar
time-change \eqref{e:def-Xf} can also be applied in this general context for any $\alpha \ne 0$
(see \cite[Corollary 2]{BeMGF}.) However,
it remains an open problem to determine necessary and sufficient conditions for
the process ${\mathbf{X}}f$ to be locally finite at all times, and to decide when global solutions
of \eqref{EqFG} exist.
\section{Branching particle system and many-to-one formulas}
\label{s:bps}
In this concluding section, we aim to clarify the connection between
our work and the
`spine' or `tagged fragment' approach to branching particle systems
and fragmentation processes, with the hope of explaining the source
of the solutions obtained in
\autoref{T1} and Corollaries {\mathbf{r}}ef{c:mu-alpha-neg} and {\mathbf{r}}ef{c:mu-alpha-pos}.
We refer the reader to the survey of \citet{HH-spine} for results in the context
of branching processes, \citet[\S 3.2.2]{Ber-frag} for the background on
fragmentation processes, and \citet{Haas} for the use of the tagged fragment
in solving the classical fragmentation equation.
For the sake of simplicity, we focus on the case when
the dislocation measure $K$ is finite
and the operator ${\mathcal L}$ has the form \eqref{Eqop0}.
\noindent
Let us assume that we can construct a system of branching particles in $(0,\infty)$,
with the following dynamics: particles evolve independently of one other,
each particle located at $x>0$ grows at rate $cx^{\alpha+1}$,
and a particle located at $x>0$ is replaced by
two particles located respectively at $xy$ and $x(1-y)$ at rate $x^{\alpha} K(\d y)$.
Let $\mathbf{Z}(t)=(Z_i(t))_{i\ge 1}$ denote the collection of particles in the system at time $t\geq 0$,
starting at time $0$ from a single particle located at $1$.
Informally, the verbal description of the dynamics of the particle system suggests
that for every test function $f\in{\mathcal C}^{\infty}_c(0,\infty)$,
the functional $F(\mathbf{z})=\sum_i f(z_i)$ for $\mathbf{z}=(z_i)_{i\ge 1}$
belongs to the domain of the infinitesimal generator ${\mathcal G}$
of the process $(\mathbf{Z}(t))_{t\geq 0}$, and that
\[
\mathcal{G}F(\mathbf{z})=\sum_{i\ge 1} z_i^{\alpha} \biggl( c z_i f'(z_i)+\int_{[1/2,1)}(f(y\mid z_i)-f(z_i))K(\d y)\biggr).
\]
Therefore, if we write $\mu_t$ for the intensity measure of $\mathbf{Z}(t)$, i.e.
\[ \langle \mu_t,f{\mathbf{r}}angle = \mathbb{E}(F(\mathbf{Z}(t)) = \mathbb{E}\biggl(\sum_i f(Z_i(t))\biggr), \]
then Kolmogorov's forward equation entails that $(\mu_t)_{t\geq 0}$ solves the fragmentation equation \eqref{EqFG}.
The analysis of the system $(\mathbf{Z}(t))_{t\ge 0}$ can be significantly
simplified by identifying a \emph{spine} among the particles, and formulating
questions about the entire system in terms of just the spine particle via a \emph{many-to-one}
formula. This proceeds roughly as follows. Suppose that we can identify
a function $(t,z) \mapsto \varphi(t,z) > 0$ such that
the process $M(t) = \sum_{i \ge 1} \varphi(t,Z_i(t))$ is a martingale. We introduce
a new probability measure $\tilde \mathbb{P}$ by means of a martingale change of measure using $M$,
simultaneously identifying one of the particles to be the spine; specifically, we
identify particle $i$ of ${\mathbf{Z}}b(t)$ as the spine with $\tilde \mathbb{P}$-probability
proportional to $\varphi(t,Z_i(t))$.
At each time $0 \le s\le t$, the spine particle has a unique ancestor,
and we define the random variable $W(s)$
to be the position of this ancestor.
We now aim to identify the law of certain functionals of ${\mathbf{Z}}b$
in terms of the law of $W$. In particular, if we define
${\mathbf{r}}ho_t$ to be the law of $W(t)$, we obtain
\[
\ip{\mu_t}{f} = \mathbb{E}\biggl( \sum_{i\ge 1} f(Z_i(t)) \biggr)
= \tilde \mathbb{E}\biggl(\frac{f(W(t))}{\varphi(t,W(t))} \biggr)
= \ip{{\mathbf{r}}ho_t}{f/\varphi(t,\cdot)},
\]
which is known as a many-to-one formula.
The spine method for solving \eqref{EqFG}
can be summarised as follows. We first use the dynamics of the branching particle system
and the effect of the martingale change of measure to identify the process $W$.
The one-dimensional distributions of $W$ give the collection of measures
$({\mathbf{r}}ho_t)_{t \ge 0}$, and then the many-to-one formula
gives us an explicit description of $(\mu_t)_{t \ge 0}$.
\noindent
The method we have sketched can be made rigorous in the homogeneous case $\alpha=0$,
even in the more general situation when the dislocation measure $K$ is infinite and fulfills \eqref{Eqcondnu}.
More precisely,
one can take $\varphi(t,z)=\exp(-\kappa(\omega)t)z^{\omega}$
for any $\omega\in \dom \kappa$,
and then the process $W(t)$ is the exponential of a L\'evy process
with no positive jumps and Laplace exponent $\mathbb{P}hi_{\omega} = \kappa(\omega+\cdot)-\kappa(\omega)$;
this is the justification for \autoref{r:LE}.
We stress, however, that the general self-similar case $\alpha\neq 0$ is far less simple.
In particular, it is not clear whether the branching particle system can indeed
be constructed since, as noted in the previous section, explosion may occur.
A fairly general class of growth-fragmentation processes was introduced recently
in \cite{BeMGF} by means of a Crump-Mode-Jagers process;
however, although it is expected to be related to growth-fragmentation equations as described above,
so far no many-to-one formula is known to make the connection rigorous.
\acks
This work was submitted while the second author was at the University of Zurich, Switzerland.
We thank Robin Stephenson for drawing to our attention some mistakes in an earlier draft,
and the anonymous referee for their helpful comments.
\end{document} |
\begin{document}
\title{Primes of height one and a class of Noetherian finitely presented
algebras\thanks{Research partially supported by the Onderzoeksraad
of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk
Onderzoek (Flanders), Flemish-Polish bilateral agreement
BIL2005/VUB/06 and a MNiSW research grant N201 004 32/0088
(Poland).}}
\author{ Isabel Goffa\footnote{Research funded by a Ph.D grant of
the Institute for the Promotion of Innovation
through Science and Technology in Flanders
(IWT-Vlaanderen). \newline 2000 Mathematics
Subject Classification. Primary 16P40, 16H05,
16S36; Secondary 20M25, 16S15. } \and Eric
Jespers \and Jan Okni\'{n}ski} \maketitle
\date{}
\begin{abstract}
Constructions are given of Noetherian maximal orders that are
finitely presented algebras over a field $K$, defined by monomial
relations. In order to do this, it is shown that the underlying
homogeneous information determines the algebraic structure of the
algebra. So, it is natural to consider such algebras as semigroup
algebras $K[S]$ and to investigate the structure of the monoid
$S$. The relationship between the prime ideals of the algebra and
those of the monoid $S$ is one of the main tools. Results
analogous to fundamental facts known for the prime spectrum of
algebras graded by a finite group are obtained. This is then
applied to characterize a large class of prime Noetherian maximal
orders that satisfy a polynomial identity, based on a special
class of submonoids of polycyclic-by-finite groups. The main
results are illustrated with new constructions of concrete classes
of finitely presented algebras of this type.
\end{abstract}
Because of the role of Noetherian algebras in the
algebraic approach in noncommutative geometry, new
concrete classes of finitely presented algebras
recently gained a lot of interest. Via the
applications to the solutions of the Yang-Baxter
equation, one such class also widens the interest
into other fields, such as mathematical physics.
These algebras are finitely generated, say by
elements $x_{1},\ldots ,x_{n}$, and they have a
presentation defined by ${n\choose 2}$ monomial
relations of the form $x_{i}x_{j}=x_{k}x_{l}$ so
that every word $x_{i}x_{j}$ appears at most once in
a relation. Such algebras have been extensively
studied, for example in
\cite{eting-gur,eting,gat-van,jes-okn-bin,rump}. It
was shown that they have a rich algebraic structure
that resembles that of a polynomial algebra in
finitely many commuting generators; in particular,
they are prime Noetherian maximal orders. Clearly,
these algebras can be considered as semigroup
algebras $K[S]$, where $S$ is a monoid defined via a
presentation as that of the algebra. It turns out
that these algebras are closely related to group
algebras since $S$ is a submonoid of a torsion-free
abelian-by-finite group.
In this paper we further explore new constructions
of Noetherian maximal orders that are finitely
presented algebras which are defined via monomial
relations. In other words, we look for new
constructions of semigroup algebras $K[S]$ of this
type. In general, it remain unsolved problems to
characterize when an arbitrary semigroup algebra is
Noetherian and when it is a prime Noetherian maximal
order. Recall that the former question even for
group algebras has been unresolved. The only class
of groups for which a positive answer has been given
is that of the polycyclic-by-finite groups. Hence it
is natural to consider the problems first for
semigroup algebras $K[S]$ of submonoids $S$ of
polycyclic-by-finite groups. A structural
characterization of such algebras $K[S]$ that are
right Noetherian was obtained by the authors in
\cite{jes-okn-poly,jes-okn-acc}. Namely, $S$ has a
group of quotients $G=SS^{-1}$ in which there is a
normal subgroup $H$ of finite index so that
$[H,H]\subseteq S$ and $S\cap H$ is finitely
generated, or equivalently, $S$ has a group of
quotients $G$ that contains normal subgroups $F$ and
$N$ so that $F\subseteq N$, $F$ is a subgroup of
finite index in the unit group ${\rm U}(S)$ of $S$, the
group $N/F$ is abelian, $G/N$ is finite and $S\cap
N$ is finitely generated. It follows that $K[S]$ is
right Noetherian if and only if $K[S]$ is left
Noetherian. We simply say that $K[S]$ is Noetherian.
Also, if $S$ is a submonoid of a finitely generated
group $G$ that has an abelian subgroup $A$ of finite
index, then $K[S]$ is Noetherian if and only if
$S\cap A$ is finitely generated. Recall that if
$K[S]$ is Noetherian then $S$ is finitely generated
and $K[S\cap G_{1}]$ is Noetherian for every
subgroup $G_{1}$ of $G$. If, furthermore, $G_{1}$ is
of finite index in $G$ then $K[S]$ is a finitely
generated right (and left) $K[S\cap G_{1}]$-module.
Due to a result of Chin and Quinn \cite{chin-quinn}
on rings graded by polycyclic-by-finite groups,
$K[S]$ is Noetherian if and only if $S$ satisfies
the ascending chain condition on right
(equivalently, left) ideals.
Concerning the second problem. If $S$ is an
abelian monoid, Anderson \cite{and1,and} proved
that $K[S]$ is a prime Noetherian maximal order
if and only if $S$ is a submonoid of a finitely
generated torsion free abelian group, $S$
satisfies the ascending chain condition on ideals
and $S$ is a maximal order in its group of
quotients. More generally, Chouinard \cite{cho}
proved that a commutative monoid algebra $K[S]$
is a Krull domain if and only if $S$ is a
submonoid of a torsion free abelian group which
satisfies the ascending chain condition on cyclic
subgroups and $S$ is a Krull order in its group
of quotients. In particular, it turns out that
the height one prime ideals of $K[S]$ determined
by the minimal primes of $S$ are crucial. Brown
characterized group algebras $K[G]$ of a
polycyclic-by-finite group $G$ that are
Noetherian prime maximal orders
\cite{brown1,brown2}. This turns out to be always
the case if $G$ is a finitely generated torsion
free abelian-by-finite group (equivalently,
$K[G]$ is a Noetherian domain that satisfies a
polynomial identity, PI for short).
In this situation all height one primes are
principally generated by a normal element. So, in the terminology
of Chatters and Jordan \cite{cha-jor}, $K[G]$ is a unique
factorization ring. The authors described in \cite{jes-okn-max}
when a semigroup algebra of a submonoid $S$ of a finitely
generated abelian-by-finite group is a Noetherian maximal order
that is a domain. The description is fully in terms of the monoid
$S$. Examples of such monoids are the binomial monoids and more
general monoids of $I$-type. As mentioned earlier in the
introduction, the latter were introduced by Gateva-Ivanova and Van
den Bergh in \cite{gat-van} and studied in several papers (see for
example \cite{eting-gur,eting,gat-sol,jes-okn-bin,rump}) and
generalized to monoids of $IG$-type in \cite{gof-jes}.
In this paper we continue the investigations of
prime Noetherian maximal orders $K[S]$ for
submonoids $S$ of a polycyclic-by-finite group
$G=SS^{-1}$. Of course, knowledge of prime ideals
is fundamental. The primes not intersecting $S$
come from primes in the group ring $K[SS^{-1}]$.
These are rather well understood through the work
of Roseblade (see for example \cite{passman}),
and in particular, the height one primes can be
handled via Brown's result. The crucial point in
our investigations are thus the primes of $K[S]$
intersecting $S$ non-trivially. In case $K[S]$ is
Noetherian and $G=SS^{-1}$ is torsion free, the
following information was proved in
\cite{jes-okn-poly}.
\begin{enumerate}
\item If $Q$ is a prime ideal of $S$ then $K[Q]$ is a prime ideal of $K[S]$.
\item The height one prime ideals of $K[S]$ intersecting $S$
non-trivially are precisely the ideals $K[Q]$
with $Q$ a minimal prime ideal of $S$.
\end{enumerate}
Note that $G$ being torsion free is equivalent
with $K[G]$ being a domain (see
\cite[Theorem~37.5]{pas-cro}).
In the first part of this paper we continue the
study of prime ideals in case $K[S]$ is prime (and
thus $G$ is not necessarily torsion free). We show
that the first property does not remain valid,
however the second part on primes of height one
still remains true. As a consequence, we establish
going up and going down properties between prime
ideals of $S$ and prime ideals of $S\cap H$, where
$H$ is a subgroup of finite index in $G$. These are
the analogs of the important results (see
\cite[Theorem~17.9]{pas-cro}) known on the prime
ideal behaviour between a ring graded by a finite
group and its homogeneous component of degree $e$
(the identity of the grading group). As an
application it is shown that the classical Krull
dimension ${\rm clKdim} (K[S])$ of $K[S]$ is the sum of
the prime dimension of $S$ and the plinth length of
the unit group ${\rm U}(S)$. Also, a result of Schelter
is extended to the monoid $S$: the prime dimension
of $S$ is the sum of the height and depth of any
prime ideal of $S$.
The information obtained on primes of height one
then allows us in the second section to determine
when a semigroup algebra $K[S]$ is a prime
Noetherian maximal order provided that $G=SS^{-1}$
is a finitely generated abelian-by-finite group
(that is, $K[S]$ satisfies a polynomial identity).
The result reduces the problem to the structure of
the monoid $S$ (in particular $S$ has to be a
maximal order within its group of quotients $G$) and
to that of $G$. The characterization obtained
generalizes the one given in \cite{jes-okn-max} in
case $G$ is torsion free and it shows that the
action of $G$ on minimal primes of some abelian
submonoid of $S$ is very important (as was also
discovered for the special case where $S$ is a
finitely generated monoid of $IG$-type in
\cite{gof-jes}). Finally, in the last section, we
prove a useful criterion for verifying the maximal
order property of such $S$. We then show how this,
together with our main result, can be used to build
new examples of finitely presented algebras $R$
(defined by monomial relations) that are maximal
orders. Since the main result of the paper deals
with semigroup algebras $K[S]$ of submonoids $S$ of
groups, we first have to check that $R$ is defined
by such submonoids. So, in particular, in the third
section we show how to go from the language of
presentations of algebras to the language of monoids
and their semigroup algebras.
\section{Prime spectrum}
Let $K$ be a field. Recall that if $S$ is a
monoid with a group of quotients $G=SS^{-1}$ then
the semigroup algebra $K[S]$ is prime if and only
if $K[G]$ is prime
(\cite[Theorem~7.19]{okn-book1}), or equivalently
$G$ does not contain nontrivial finite normal
subgroups. The latter can also be stated as
$\Delta^{+}(G) =\{ 1 \}$, where $\Delta^{+}(G)$
is the characteristic subgroup of $G$ consisting
of the periodic elements with finitely many
conjugates (see \cite[Theorem 5.5]{pas-cro}). By
$\Delta (G)$ one denotes the subgroup of $G$
consisting of the elements with finitely many
conjugates.
Also recall that an equivalence relation $\rho$
on a semigroup $S$ is said to be a congruence if
$s\rho t$ implies $su\rho tu$ and $us\rho ut$ for
every $u\in S$. The set of $\rho$-classes,
denoted $S/\rho$, has a natural semigroup
structure inherited from $S$. In case $S$ has a
group of quotients $G$ and $H$ is a normal
subgroup of $G$ we denote by $\sim_{H}$ the
congruence on $S$ defined by: $s\sim_{H}t$ if and
only if $s=ht$ for some $h\in H$. If $G$ is a
polycyclic-by-finite group then by ${\rm h} (G)$ we
mean the Hirsch rank of $G$.
We often will make use of the following fact (\cite[Lemma~3.1]{search}). If $S$ is a submonoid of
a finitely generated abelian-by-finite group $G$ then $S$ has a group of quotients $S{\rm Z}(S)^{-1}=\{
sz^{-1} \mid s\in S,\; z\in {\rm Z}(S)\}$. It follows that if $T$ is a submonoid of a group $G$ that
contains a normal subgroup $F$ so that $F\subseteq T$ and $G/F$ is finitely generated and
abelian-by-finite (for example $T\subseteq G$, $G$ is polycyclic-by-finite and $T$ satisfies the
ascending chain condition on right ideals, see \cite{jes-okn-acc}) then $T$ has a group of
quotients that is obtained by inverting the elements $u$ of $T$ that are central modulo $F$. In
particular, every (nonempty) ideal of $T$ contains such an element $u$.
\begin{theorem} \label{thm-primes}
Let $S$ be a submonoid of a polycyclic-by-finite
group, say with a group of quotients $G$, and let
$K$ be a field. Assume that $S$ satisfies the
ascending chain condition on right ideals and $G$
does not contain nontrivial finite normal
subgroups. Then the height one prime ideals $P$
of $K[S]$ such that $P\cap S \neq \emptyset$ are
exactly the ideals of the form $P=K[Q]$, where
$Q$ is a minimal prime ideal of $S$.
\end{theorem}
\begin{proof}
Let $P$ be a height one prime ideal of $K[S]$ such that $Q=S\cap P \neq \emptyset$. By
\cite{okn-prime} (or see \cite[Theorem~4.5.2]{jes-okn-book}, and also
\cite[Corollary~4.5.7]{jes-okn-book}), $S/Q$ embeds into ${\mathcal M}_{n}(H)$, the semigroup of
$n\times n$ monomial matrices over a group $H=TT^{-1}$ for a subsemigroup $T$ of $S$ such that
$T\cap Q =\emptyset$. Furthermore, $K[S]/K[Q]$ embeds in the matrix algebra $M_{n}(K[H])$ and the
latter is a localization of $K[S]/K[Q]$ with respect to an Ore set that does not intersect
$P/K[Q]$. Subsemigroups of $S$ not intersecting $Q$ can be identified with subsemigroups of
${\mathcal M}_{n}(H)$. It is also known that there exists an ideal $I$ of $S$ containing $Q$ such
that the nonzero elements of $I/Q$ are the matrices in $S/Q\subseteq {\mathcal M}_{n}(H)$ with
exactly one nonzero entry. Furthermore, $T$ may be chosen so that $T\subseteq I\setminus Q$ and
actually $T$ may be identified with $(e_{11}{\mathcal M}_{n}(H)e_{11}\cap (S/Q))\setminus \{0\}$,
where $e_{11}$ is a diagonal idempotent of rank one in ${\mathcal M}_{n}(H)$. Clearly $I/Q$ is an
essential ideal in $S/Q$. We know that $S$ contains a normal subgroup $F$ of $G$ such that $G/F$
is abelian-by-finite. Then $H\subseteq FH$ and clearly $FH\cap Q=\emptyset$ and $FH=FT(FT)^{-1}$.
Clearly $FTTF\subseteq FHF=FH$, whence $(FT)(TF)\cap Q=\emptyset$. Since $T$ is a diagonal
component of $I$ and $FT,TF\subseteq I$, by the matrix pattern on the matrices of rank one in
${\mathcal M}_{n}(H)$ it follows that $TF=FT=T$. Hence $H={\rm gr} (T)={\rm gr} (TF)=HF$.
Notice that we have a natural homomorphism $\phi :K[S]/K[Q]\longrightarrow K[S]/P$. Since $K[S]$
is Noetherian and as $M_{n}(K[H])$ is a localization of $K[S]/K[Q]$ with respect to an Ore set of
regular elements that does not intersect $P/K[Q]$, there exists a prime ideal $R$ in $K[H]$ such
that $M_{n}(K[H]/R)$ is a localization of $K[S]/P$, see \cite[Theorem~9.22]{goo-war}. Moreover,
since $P/K[Q]$ is a prime ideal of $K[S]/K[Q]$, we also have that $M_{n}(K[H]/R)\subseteq
M_{n}(M_{t}(D))$ for a division ring $D$ and a positive integer $t$ such that $M_{t}(D)$ is the
ring of quotients of $K[H]/R$ and $\phi$ extends to a homomorphism $\phi
':M_{n}(K[H])\longrightarrow M_{n}(K[H]/R)$. Let $\rho_{P}$ be the congruence on $S$ defined by
the condition: $(x,y)\in \rho _{P}$ if $x-y\in P$. Since the image of a nonzero entry of a matrix
$s\in {\mathcal M}_{n}(H)$ under $\phi '$ is invertible in $K[H]/R$, the rank of the matrix $\phi
' (s)$ is equal to $t$ multiplied by the number of nonzero entries of $s$. Thus, the ideal $J$ of
matrices of rank at most $t$ in $S/\rho_{P}\subseteq M_{nt}(D)$ satisfies $\phi^{-1}(J)=I$.
Moreover, $S/\rho_{P}=\phi '(S/Q)\subseteq {\mathcal M}_{n}(GL_{t}(D))$ inherits the monomial
pattern of $S/Q\subseteq {\mathcal M}_{n}(H)$. From \cite[Corollary~4.4]{jes-okn-poly} we know
that there is a right Ore subsemigroup $U$ of $I$ such that ${\rm h} (UU^{-1})={\rm h} (G)-1$ and $U\cap
P=\emptyset $, and also the image $U'$ of $U$ in $S/\rho_{P}$ is contained in a diagonal component
of $J$ (viewed as a monomial semigroup over $GL_{t}(D)$). Then $U$ can be identified with a
subsemigroup of $S/Q$ and it is contained in a diagonal component of $S/Q$. It thus follows that
${\rm h} (H) \geq {\rm h}({\rm gr}(U))={\rm h} (G)-1$.
By the remark before the theorem we know that every ideal of $S$ contains an element of $S$ that
is central modulo $F$. So, choose $z\in Q$ such that $zx\in xzF$ for every $x\in S$. Suppose that
$z^{m}\in H$ for some positive integer $m$. Then $z^{m}=ab^{-1}$ for some $a,b\in T$. Hence
$z^{m}b\in T\cap P$. As $T\cap P=\emptyset$, we obtain a contradiction. It thus follows that ${\rm h}
({\rm gr}(z,H)) > {\rm h} (H)\geq {\rm h} (G)-1$. Therefore ${\rm h} ({\rm gr}(z,H))={\rm h} (G)$ and, because $G$ is
polycyclic-by-finite, we get $[G:{\rm gr} (z, H)]<\infty $. Since $zs\in szF$ for every $s\in S\cap H$,
it follows that $z(S\cap H)\subseteq (S\cap H)zF=(S\cap HF)z=(S\cap H)z$. As $H=(S\cap H)(S\cap
H)^{-1}$, it follows that $zH=Hz$. Therefore $\Delta^{+}(H)$ has finitely many conjugates in $G$.
Since $\Delta^{+}(H)$ is finite, we get that $\Delta^{+}(H)\subseteq \Delta^{+} (G)=\{ 1\}$ and
thus $K[H]$ is a prime algebra. Hence, the localization $M_{n}(K[H])$ of $K[S]/K[Q]$ is prime. So
$K[Q]$ is a prime ideal in $K[S]$. As $K[S]$ is prime and $P$ is of height one, we get that
$P=K[Q]$, as desired.
To prove the converse, let $Q$ be a minimal prime
ideal of $S$. Let $P$ be an ideal of $K[S]$
maximal with respect to $S\cap P = Q$. Clearly
$P$ is a prime ideal of $K[S]$. Again by the
remark before the theorem, we know that there
exists an element $z$ in $S$ that belongs to $Q$
and that is central modulo $F$. Then $zS=Sz$, so
$z$ is a normal element of $K[S]$. Since, by
assumption $K[S]$ is a prime Noetherian algebra,
the principal ideal theorem therefore yields a
prime ideal $P'$ of $K[S]$ that is of height one
so that $z\in P'$ and $P'\subseteq P$. By the
first part of the result, $P'=K[S\cap P']$. Since
$S\cap P'$ is a prime ideal of $S$ contained in
the minimal prime ideal $S\cap P = Q$, we get
that $S\cap P' =Q$. So $K[Q]=P'$ is a height one
prime ideal of $K[S]$.
\end{proof}
Recall that the rank ${\rm rk} (S)$ of a monoid $S$
(not necessarily cancellative) is the supremum of
the ranks of the free abelian subsemigroups of
$S$. The dimension $\dim (S)$ of $S$ is defined
in the following way. By definition $\dim S = 0$
if $S=\{ e\}$. If $S$ has no zero element, then
$\dim (S)$ is the maximal length $n$ of a chain
$Q_{0}\subset Q_{1}\subset Q_{2}\subset \cdots
\subset Q_{n}$, where $Q_{0}=\emptyset$ and
$Q_{i}$ are prime ideals of $S$ for $i>0$ (note
that primes are by definition different from
$S$), or $\infty$ if such $n$ does not exist. If
$\{ e \} \neq S$ has a zero element, then $\dim
(S)$ is the maximal length $n$ of such a chain
with all $Q_{i}$ ($i\geq 0$) prime ideals of $S$,
or $\infty$ if such $n$ does not exist. The
spectrum ${\rm Spec} (S)$ is the set of all prime
ideals of $S$.
We give an easy example that shows that
Theorem~\ref{thm-primes} can not be extended to
semigroup algebras that are not Noetherian. Let
$S=\{ x^{i}y^{j} \mid i> 0,\; j\in {\mathbb Z}\}
\cup \{ 1 \} $, a submonoid of the free abelian
group ${\rm gr} (x,y)$ of rank two. It is easy to see
that for a given $i>0$ and $j\in {\mathbb Z}$ the
ideal $ x^{i}y^{j}S$ contains the set $\{
x^{k}y^{j} \mid k> i,\; j\in {\mathbb Z}\} $.
Therefore $P=S\setminus \{ 1 \}$ is nil modulo
the ideal $x^{i}y^{j}S$. It follows that
$S\setminus \{ 1\}$ is the only prime ideal of
$S$. Moreover ${\rm rk} (S/P)=0< {\rm rk} (S) -1$, while
$P$ is a minimal prime ideal of $S$. Clearly
$K[P]$ is a prime ideal of $K[S]$. Let $P'$ be
the $K$-linear span of the set consisting of all
elements of the form $x^{i}(y^{j}-y^{k})$ with
$i>0$ and $j,k\in {\mathbb Z}$. Then $P'$ is an
ideal of $K[S]$ and $K[S]/P'$ is isomorphic to
the polynomial algebra $K[x]$. Therefore $K[P]$
is a prime ideal of height two (note that
${\rm clKdim} (K[S])={\rm rk} (S)=2$ ).
The latter equalities also follow from the
following result, which will be needed later in
the paper. Let $T$ be a cancellative semigroup
and $K$ a field. If $K[T]$ satisfies a polynomial
identity then ${\rm clKdim} (K[T])={\rm rk} (T)={\rm GK}(
K[T])$, \cite[Theorem~23.4]{okn-book1}. (By ${\rm GK}
(R)$ we denote the Gelfand-Kirillov dimension of
a $K$-algebra $R$.)
On the other hand, for Noetherian semigroup
algebras one can also give an example (see
Example~\ref{mainexample}) showing that
Theorem~\ref{thm-primes} cannot be extended to
prime ideals of height exceeding one.
In order to give some applications to the
behaviour of prime ideals of $S$ and those of
$S\cap H$ (with $H$ a normal subgroup of finite
index in $SS^{-1}$) we prove the following
technical lemma.
\begin{lemma} \label{invariant}
Let $S$ be a submonoid of a polycyclic-by-finite
group, say with a group of quotients $G$. Assume
that $S$ satisfies the ascending chain condition
on right ideals. Then $G$ has a poly-(infinite
cyclic) normal subgroup $H$ of finite index so
that $S\cap H$ is $G$-invariant and $H/{\rm U}(S\cap
H)$ is abelian. If $G$ is abelian-by-finite then
$H$ can be chosen to be an abelian subgroup.
\end{lemma}
\begin{proof}
It is well known that $G$ contains a
characteristic subgroup $C$ that is
poly-(infinite cyclic). Since $S$ satisfies the
ascending chain condition on right ideals, $G$
contains a normal subgroup $F$ so that $G/F$ is
abelian-by-finite and $F\subseteq S$. Hence
$C\cap F$ is a normal subgroup of $G$ that is
poly-(infinite cyclic), $C\cap F \subseteq {\rm U}(S)$
and $G/(C\cap F)$ is abelian-by-finite. It is
thus sufficient to prove the result for the
monoid $S/\sim_{C\cap F}$ and its group of
quotients $G/(C\cap F)$. In other words we may
assume that $G$ is abelian-by-finite.
So, let $A$ be a torsion free abelian and normal
subgroup of finite index in $G$. Then $S\cap A$
is a finitely generated abelian monoid and its
group of quotients is of finite index in $G$.
Since $S$ satisfies the ascending chain condition
on right ideals, we also know \cite{jes-okn-poly}
that for every $g\in S$ and $s\in S$ there exists
a positive integer $k$ so that $gs^{k}g^{-1}\in
S$. Because $G=S{\rm Z}(S)^{-1}$ (again by the remark
before Theorem~\ref{thm-primes}), the latter
property holds for all $g\in G$. As $S\cap A$ is
finitely generated abelian and $G/A$ is finite,
it follows that there exists a positive integer
$n$ so that
$$g(S\cap A)^{(n)}g^{-1}\subseteq S\cap A,$$
for all $g\in G$, where by definition $(S\cap
A)^{(n)}=\{ s^{n} \mid s\in S\cap A\}$. Hence
$$ T=\bigcap_{g\in G} g(S\cap A)^{(n)}g^{-1}$$
is a $G$-invariant submonoid of $S$. Because $A$
is the group of quotients of $S\cap A$, we get
that each $g(S\cap A)^{(n)}g^{-1}$ has a group of
quotients that is of finite index in $G$. Since
there are only finitely many such conjugates, it
is clear that $TT^{-1}$ is of finite index in
$G$. Hence, the result follows.
\end{proof}
The notion of height ${\rm h}ht (P)$ of a prime ideal
of a ring $R$ is well known. We now define the
height ${\rm h}ht (Q)$ of a prime ideal $Q$ of a
monoid $S$. If $S$ does not have a zero element
then ${\rm h}ht (Q)$ is the maximal length $n$ of a
chain $Q_{0}\subset Q_{1}\subset Q_{2}\subset
\cdots \subset Q_{n}=Q$, where $Q_{0}=\emptyset$
and $Q_{1},\ldots, Q_{n-1}$ are prime ideals of
$S$. On the other hand, if $S$ has a zero
element, then ${\rm h}ht (Q)$ is the maximal length of
such a chain with all $Q_{i}$ prime ideals of
$S$, $i\geq 0$. If such $n$ does not exist then
we say that the height of $P$ is infinite.
\begin{corollary} \label{primes-prop}
Let $S$ be a submonoid of a polycyclic-by-finite
group, say with a group of quotients $G$, and let
$K$ be a field. Assume that $S$ satisfies the
ascending chain condition on right ideals. If $H$
is a torsion free normal subgroup of finite index
in $G$, then
\begin{enumerate}
\item If $P$ is a prime ideal of $S$ then $P\cap H =
Q_{1}\cap \cdots \cap Q_{n}$, where $Q_{1},\ldots
, Q_{n}$ are all the primes of $S\cap H$ that are
minimal over $P\cap H$. Furthermore, $P=M\cap S$
for any prime ideal $M$ of $K[S]$ that is minimal
over $K[P]$, ${\rm h}ht (M)={\rm h}ht (K[Q_{1}]) =\cdots =
{\rm h}ht (K[Q_{n}])$ and $P={\mathcal B}(S(Q_{1}\cap
\cdots \cap Q_{n})S)$ (the prime radical of
$S(Q_{1}\cap \cdots \cap Q_{n})S$). If,
furthermore, $H\cap S$ is $G$-invariant then
$Q_{i}=Q_{1}^{g}$ for some $g\in G$, $1\leq i
\leq n$.
\item If $S\cap H$ is $G$-invariant and
$Q=Q_{1}$ is a prime ideal of $S\cap H$ then
there exists a prime ideal $P$ of $S$ so that
$P\cap H=Q_{1}\cap Q_{2}\cap \cdots \cap Q_{m}$,
where $Q_{1},\ldots , Q_{m}$ are all the prime
ideals of $S\cap H$ that are minimal over $P\cap
H$. One says that $P$ lies over $Q$. Moreover,
each $Q_{i}=Q^{g_{i}}$ for some $g_{i}\in G$.
\item {\bf Incomparability}
Suppose $S\cap H$ is $G$-invariant, $Q_{1}$ and
$Q_{2}$ are prime ideals of $S\cap H$, and
$P_{1}$ and $P_{2}$ are prime ideals of $S$. If
$P_{1}$ lies over $Q_{1}$ and $P_{2}$ lies over
$Q_{2}$ so that $Q_{1} \subseteq Q_{2}$ and
$P_{1}\subseteq P_{2}$, then $P_{1}=P_{2}$ if and
only if $Q_{1}=Q_{2}$.
\item {\bf Going up} Assume $S\cap H$ is $G$-invariant.
Suppose $Q_{2}$ is a prime ideal of $S\cap H$ and
$P_{2}$ is a prime ideal of $S$ lying over
$Q_{2}$.
\begin{enumerate}
\item
If $Q_{1}$ is a prime ideal of $S\cap H$
containing $Q_{2}$ then there exists a prime
ideal $P_{1}$ lying over $Q_{1}$ so that
$P_{2}\subseteq P_{1}$.
\item
If $P_{1}$ is a prime ideal of $S$ containing
$P_{2}$ then there exists a prime ideal $Q_{1}$
of $S\cap H$ containing $Q_{2}$ so that $P_{1}$
lies over $Q_{1}$.
\end{enumerate}
\item {\bf Going down} Assume $S\cap H$ is $G$-invariant. Suppose
$Q_{1}$ is a prime ideal of $S\cap H$ and $P_{1}$
is a prime ideal of $S$ lying over $Q_{1}$.
\begin{enumerate}
\item
If $Q_{2}$ is a prime ideal of $S\cap H$
contained in $Q_{1}$ then there exists a prime
ideal $P_{2}$ lying over $Q_{2}$ so that
$P_{2}\subseteq P_{1}$.
\item If $P_{2}$ is a prime ideal of $S$
contained in $P_{1}$ then there exists a prime
ideal $Q_{2}$ of $S\cap H$ contained in $Q_{1}$
so that $P_{2}$ lies over $Q_{2}$.
\end{enumerate}
\end{enumerate}
\end{corollary}
\begin{proof}
The algebra $K[S]$ has a natural gradation by the
finite group $G/H$. Its homogeneous component of
degree $e$ (the identity of the group $G$) is the
semigroup algebra $K[S\cap H]$. Let $P$ be a
prime ideal of $S$. Let $M$ be a prime ideal of
$K[S]$ minimal over $K[P]$. Note that then
$K[S]/K[P]$ inherits a natural $G/H$-gradation,
with component of degree $e$ the algebra $K[S\cap
H]/K[P\cap H]$. Because of Theorem~17.9 in
\cite{pas-cro} on going-up and down on prime
ideals of rings graded by finite groups, one gets
that $K[S\cap H] \cap M = P_{1}\cap \cdots \cap
P_{n}$, where $P_{1},\ldots , P_{n}$ are all the
prime ideals of $K[S\cap H]$ that are minimal
over $K[P]\cap K[S\cap H]=K[P\cap H]$.
Furthermore, ${\rm h}ht (P_{i})={\rm h}ht (M)$ for $1\leq
i\leq n$. Since $H$ is torsion free, we know that
$K[P_{i}\cap H]$ is a prime ideal of $K[S\cap H]$
(see the introduction). Since it clearly contains
$K[P\cap H]$, it follows that $P_{i}=K[Q_{i}]$,
with $Q_{i}=P_{i}\cap H$. Furthermore, since
$K[Q]$ is a prime ideal in $K[S\cap H]$ for every
prime ideal $Q$ of $S\cap H$, it follows that
$Q_{1}, \ldots , Q_{n}$ are all the prime ideals
of $S\cap H$ minimal over $P\cap H$. So $K[S\cap
H]\cap M =K[Q_{1}\cap \cdots \cap Q_{n}]$.
Because $K[S\cap H]/K[P\cap H]$ is Noetherian, we
know that its prime radical is nilpotent. Hence
$(Q_{1}\cap \cdots \cap Q_{n})^{k} \subseteq
P\cap H$ for some positive integer $k$. Since $H$
is of finite index in $G$, it then also follows
that $M\cap S$ is an ideal of $S$ that is nil
modulo $S(P\cap H)S$. Since $K[S]$ is Noetherian,
this yields that $(M\cap S)^{l}\subseteq S(P\cap
H)S\subseteq P$, for some positive integer $l$.
As $P$ is a prime ideal, we therefore obtain that
$M\cap S=P, P\cap H=Q_{1}\cap \cdots \cap Q_{n}$
and $P={\mathcal B}(S(Q_{1}\cap \cdots \cap
Q_{n})S)$.
Assume now that, furthermore, $S\cap H$ is
$G$-invariant. For an ideal $I$ of $S\cap H$ put
$I^{inv}=\bigcap_{g\in G} I^{g}$, the largest
invariant ideal of $S\cap H$ contained in $I$.
Clearly $SI^{inv}=I^{inv}S$ is an ideal of $S$
and $SI^{inv}\cap (S\cap H) =I^{inv}$. It follows
that
$$SQ_{1}^{inv} \cdots S Q_{n}^{inv}\subseteq
P.$$
Hence $Q_{i}^{inv} \subseteq P\cap (S\cap H)=Q_{1}\cap \cdots \cap Q_{n}$ for some $i$. Because
$S\cap H$ is invariant, it follows that every $Q_{i}^{g}$ is a prime ideal of $S\cap H$ (of the
same height as $Q_{i}$). Hence, for every $1\leq j \leq n$ there exists $g\in G$ with
$Q_{i}^{g}=Q_{j}$. This proves the first part of the result.
To prove the second part, let $Q$ be a prime ideal of $S\cap H$ and suppose that $S\cap H$ is
$G$-invariant. Then, each $Q^{g}$ is a prime ideal of $S\cap H$ and ${\rm h}ht (Q)={\rm h}ht (Q^{g})$. Now,
if $Q'$ is a prime ideal of $S\cap H$ containing $Q^{inv}$, then $Q^{g}\subseteq Q'$ for some
$g\in G$. Hence, if $Q'$ is a prime minimal over $Q^{inv}$ then $Q^{g}= Q'$. Clearly, for every
$h\in G$, the prime $Q^{h}$ contains a prime ideal $Q''$ of $S\cap H$ minimal over $Q^{inv}$.
Hence, by the previous, $Q''=Q^{g}\subseteq Q^{h}$, for some $g$. Since ${\rm h}ht(Q^{g})={\rm h}ht(Q^{h})$,
it thus follows that $Q^{h}=Q''$. So, we have shown that the ideals $Q^{g}$ are precisely the
prime ideals of $S\cap H$ that are minimal over $Q^{inv}$. Since $H$ is torsion free we thus get
(again see the introduction) that the ideals $K[Q^{g}]$ are precisely the prime ideals of $K[S\cap
H]$ that are minimal over $K[Q^{inv}]$. Since $SQ^{inv}=Q^{inv}S$ is an ideal of $S$, the algebra
$K[S]/K[SQ^{inv}]$ has a natural $G/H$-gradation, with $K[S\cap H]/K[Q^{inv}]$ as component of
degree $e$. Hence, by \cite[Theorem~17.9]{pas-cro}, there exists a prime ideal $M$ of $K[S]$ that
is minimal over $K[SQ^{inv}]$ and $M\cap K[S\cap H]=K[Q]\cap P_{2} \cap \cdots \cap P_{m}$, where
$K[Q], P_{2},\ldots , P_{m}$ are all the prime ideals that are minimal over $M\cap K[S\cap H]$
(and these ideals are of the same height as $M$). Hence each $P_{i}$ is minimal over $K[Q^{inv}]$
and thus is of the form $K[Q^{g}]$ for some $g$. It follows that $P=M\cap S$ is a prime ideal of
$S$ so that $P\cap H$ is an intersection of $Q$ and some
of the prime ideals $Q^{g}$, with $g\in
G$. This proves part two.
Parts (3), (4) and (5) are now immediate
consequences of parts (1) and (2) and of the
corresponding results on going up and down for
rings graded by finite groups (see for example
\cite[Theorem~17.9]{pas-cro}).
\end{proof}
A fundamental result (see
\cite[Theorem~19.6]{pas-cro}) on the group
algebra $K[G]$ of a polycyclic-by-finite group
says that ${\rm clKdim} (K[G]) ={\rm pl} (G)$. For the
definition on the plinth length ${\rm pl} (G)$ of $G$
we refer the reader also to \cite{pas-cro}. In
the following result we determine relations
between the considered invariants of $S$ and
$K[S]$ for Noetherian semigroup algebras $K[S]$
of submonoids $S$ of polycyclic-by-finite groups.
\begin{corollary} \label{primes-form}
Let $S$ be a submonoid of a polycyclic-by-finite
group and let $K$ be a field. Assume that $S$
satisfies the ascending chain condition on right
ideals and let $G$ be its group of quotients. Let
$F$ be a normal subgroup of $G$ so that
$F\subseteq S$, ${\rm U}(S)/F$ is finite and $G/F$ is
abelian-by-finite. The following properties hold.
\begin{enumerate}
\item $\dim (S) = {\rm rk} (G/F)$.
\item
$\dim (S)
=\dim (S\cap W)$ for any normal subgroup $W$ of
$G$ of finite index.
\item ${\rm Spec} (S)$ is finite.
\item $ {\rm clKdim}(K[S])=\dim (S)+{\rm pl}
({\rm U}(S))$.
\end{enumerate}
\end{corollary}
\begin{proof}
Because of Lemma~\ref{invariant}, the group $G$
has a poly-(infinite cyclic) normal subgroup $H$
of finite index so that $S\cap H$ is
$G$-invariant and $H/F$ is abelian for some
normal subgroup $F$ of $H$ that is contained in
$S$. Parts (1-5) of Corollary~\ref{primes-prop}
easily yield that $\dim (S)=\dim (S\cap H)$. It
also easily is verified that $\dim (S\cap H)
=\dim (T)$ where $T=(S\cap H) /\sim_{F}$. Clearly
$TT^{-1}$ is abelian and $TT^{-1}=H/F$ because
$[G:H]<\infty$. Since ${\rm U}(S)/F$ is finite, we
obtain that ${\rm U}(T)$ is finite. Hence
$\dim (T)={\rm rk} (T)$ (see \cite{jes-okn-poly}). So part (1) follows.
To prove part (2) let $W$ be a normal subgroup of
finite index in $G$. Then, $S\cap W$ inherits the
assumptions on $S$ and $W$ is the group of
quotients of $S\cap W$. So, from part (1) we
obtain that $\dim (S)={\rm rk} (G/F)={\rm rk} (W/(F\cap
W))= \dim (S\cap W)$ and thus $\dim (S)=\dim
(S\cap W)$.
It easily is seen that there is a natural
bijective map between ${\rm Spec} (S)$ and
${\rm Spec}(S/\sim_{F})$. Hence to prove part (3) we
may assume that $SS^{-1}$ is abelian-by-finite.
Because of Lemma~\ref{invariant} and part (1) of
Corollary~\ref{primes-prop}, we obtain that
${\rm Spec}(S)$ is finite if the corresponding
property holds for finitely generated abelian
monoids $A=\langle a_{1},\ldots, a_{n}\rangle$.
This is obviously satisfied. Indeed, if $P\in
{\rm Spec}(A)$ then $A\setminus P$ is a submonoid of
$A$ and $a_{i}\in P$ for some $i$. It follows
that $A\setminus P$ is generated by a proper
subset of $\{ a_{1},\ldots, a_{n} \}$, whence
$|{\rm Spec}(A)|<\infty$.
Finally, we prove part (4). From
\cite[Corollary~4.4]{jes-okn-poly} we know that
$${\rm clKdim}(K[S])= {\rm rk}(G/F)+{\rm pl} ({\rm U}(S)).$$ So the
statement follows at once from part (1).
\end{proof}
In the following proposition it is shown that the
prime spectra of $S$ and
$S/\sim_{\Delta^{+}(SS^{-1})}$ can be identified.
\begin{proposition} \label{reduction}
Let $S$ be a submonoid of a polycyclic-by-finite
group $G=SS^{-1}$ and assume that $S$ satisfies
the ascending chain condition on right ideals.
Let $H=\Delta^{+}(G)$, the maximal finite normal
subgroup of $G$. Let $\sim$ be the congruence
relation on $S$ determined by $H$, that is,
$s\sim t$ if and only if $sH=tH$. Then the map
${\rm Spec} (S) \longrightarrow {\rm Spec} (S/ \!\sim )$,
defined by $P\mapsto P/\!\sim $, is a bijection.
\end{proposition}
\begin{proof}
Let $\varphi : S \longrightarrow S/\! \sim \
\subseteq G/H$ be the natural epimorphism. Let
$P$ be a prime ideal of $S$.
We claim that $P=\varphi^{-1}(\varphi (P))$. One
inclusion is obvious. To prove the converse, let
$x\in \varphi^{-1}(\varphi (P))$. Then there
exists $p\in P$ such that $x\sim p$. So $x=ph$
for some $h\in H$. Let $s\in S$. Then
$$(hsp)^{n}= h h^{sp} h^{(sp)^{2}} \cdots h^{(sp)^{n-1}} (sp)^{n},$$
for every $n\geq 1$. Because $h\in H$, there
exists a positive integer $k$ so that
$h^{(sp)^{k}}=h$. Define $n=(k-1)|H| +1$. Then
$$(hsp)^{n} = (h h^{sp} h^{(sp)^{2}} \cdots
h^{(sp)^{k-1}})^{|H|} (sp)^{n} = (sp)^{n}\in P.$$ It follows that $xS$ is nil modulo $P$. Because
$K[S]$ is Noetherian, from \cite[Theorem~5.18]{goo-war} we get that $x\in P$. This proves the
claim.
The claim easily implies that $\varphi (P)$ is a
prime ideal of $S$ and the statement follows.
\end{proof}
We can now prove for the semigroups under
consideration an analogue of Schelter's theorem
on prime affine algebras that satisfy a
polynomial identity; this on its turn yields the
catenary property (see for example
\cite[Theorem~13.10.12 and
Corollary~13.10.13]{rob}).
\begin{proposition} Let $S$ be a submonoid of a
polycyclic-by-finite group, say with a group of
quotients $G$. Assume that $S$ satisfies the
ascending chain condition on right ideals. Let
$P$ be a prime ideal of $S$. Then $\dim (S/P)+
{\rm h}ht (P)=\dim (S)$. Furthermore, if ${\rm U}(S)$ is
finite, then $\dim (S/P)={\rm rk} (S/P)$.
\end{proposition}
\begin{proof}
Again let $F$ be a normal subgroup of $G$ so that
$F\subseteq {\rm U}(S)$, $[{\rm U}(S):F]<\infty$, $S\cap F$
is finitely generated and $G/F$ is
abelian-by-finite. Because of the natural
bijection between ${\rm Spec} (S)$ and ${\rm Spec}
(S/\sim_{F})$, we may replace $S$ by $S/\sim_{F}$
and thus we may assume that $G$ is
abelian-by-finite and ${\rm U} (S)$ is finite. Hence,
by Lemma~\ref{invariant}, $G$ contains a normal
torsion free abelian subgroup $A$ so that
$[G:A]<\infty$, $T=S\cap A$ is finitely generated
and $G$-invariant.
We now first prove that $\dim (T/Q)+ {\rm h}ht
(Q)=\dim (T)$ for a prime ideal $Q$ in the
abelian monoid $T$. Since $A$ is torsion free, we
know (see the introduction) that $K[Q]$ is a
prime ideal of $K[T]$ (which is of height one if
$Q$ is a minimal prime of $T$). Hence (by
Schelter's result for finitely generated
commutative algebras) ${\rm clKdim} (K[T])={\rm clKdim}
(K[T]/K[Q]) + {\rm h}ht (K[Q])$. Note that $T\setminus
Q$ is a submonoid of $A$ and ${\rm clKdim} (K[T]/K[Q])
={\rm clKdim} (K[T\setminus Q])$ (we will several
times use this fact without specific reference).
Consequently, by Corollary~\ref{primes-form},
$\dim (T)=\dim (T\setminus Q) + {\rm h}ht (K[Q])$.
Since $\dim (T\setminus Q) =\dim (T/Q)$, we need
to prove that ${\rm h}ht (K[Q]) ={\rm h}ht (Q)$. We prove
this by induction on ${\rm h}ht (Q)$ (note that by
Corollary~\ref{primes-form} the prime spectrum of
$T$ is finite and hence every prime contains a
minimal prime ideal of $T$). If ${\rm h}ht (Q)=1$ then
the statement holds. So, assume ${\rm h}ht (Q)>1$. Let
$Q_{1}$ be a prime of height one contained in
$Q$. Then, by the induction hypothesis, ${\rm h}ht
(Q/Q_{1}) ={\rm h}ht (K[Q/Q_{1}])$. Since
$K[(T/Q_{1})/(Q/Q_{1})]\cong K[T/Q]$, we thus get
from Schelter's result that ${\rm clKdim} (K[T]) -1=
{\rm clKdim} (K[T/Q_{1}]) ={\rm clKdim} (K[T/Q]) + {\rm h}ht
(Q/Q_{1}) ={\rm clKdim} (K[T]) -{\rm h}ht (K[Q]) +{\rm h}ht
(Q/Q_{1})$. Hence ${\rm h}ht (K[Q]) = {\rm h}ht (Q/Q_{1})
+1 \leq {\rm h}ht (Q)$. Since $K[M]$ is prime in
$K[T]$ if $M$ is prime in $T$, it is clear that
${\rm h}ht (Q)\leq {\rm h}ht (K[Q])$. Hence we obtain that
${\rm h}ht (K[Q]) ={\rm h}ht (Q)$, as desired.
Now let $P$ be a prime ideal of $S$. Because of
Corollary~\ref{primes-prop}, $\dim (S)=\dim
(S\cap A)$, $\dim (S/P)=\dim (T/Q)$ and ${\rm h}ht
(P)={\rm h}ht (Q)$, where $Q$ is a prime ideal of $T$
and $P$ lies over $Q$. From the previous it thus
follows that
$$\dim (S/P)+{\rm h}ht (P)=\dim (S).$$
So, only the last part of the statement of the
result remains to be proven. Let $Q_{1}=Q$ and
write $P\cap T=Q_{1}\cap \cdots \cap Q_{n}$,
where $Q_{1},\ldots , Q_{n}$ are all primes
minimal over $P\cap T$. We know that ${\rm clKdim}
(K[T/Q_{i}]) ={\rm rk} (T/Q_{i})$. Furthermore,
because $TT^{-1}$ is torsion free and $K[T]$ is
Noetherian, we also know that $K[Q_{i}]$ is a
prime ideal of $K[T]$ that is minimal over
$K[P\cap T]$ (see the introduction). Hence,
$K[Q_{i}/(P\cap T)]$ is a minimal prime ideal in
the finitely generated commutative algebra
$K[T/(P\cap T)]$. It follows that
\begin{eqnarray}
{\rm clKdim} (K[T/(P\cap T)]) &=&
{\rm clKdim} (K[T/Q_{i}]) + {\rm h}ht (K[Q_{i}/(P\cap
T)]) \nonumber\\
&=&{\rm clKdim} (K[T/Q_{i}]) ={\rm rk} (T/Q_{i}).
\label{equalb}
\end{eqnarray}
Since $K[T/(P\cap T)]$ is a finitely generated
commutative algebra, it is well known that
${\rm clKdim} (K[ T/(P\cap T)]) ={\rm GK} (K[T/(P\cap T)])$
(see for example \cite[Theorem~4.5]{krause}).
From \cite[Theorem~23.14]{okn-book1}) it follows
that
$${\rm rk} (T/(P\cap T))= {\rm clKdim} (K[T/(P\cap T)]).$$
Using (\ref{equalb}) we thus get
\begin{eqnarray*}
{\rm rk} (T/(P\cap T))&=&{\rm clKdim} (K[T]/(K[Q_{1}]\cap \cdots \cap K[Q_{n}]))\\
&=& \sup \{ {\rm clKdim} (K[T/Q_{i}])\mid 1\leq i \leq
n\}\\
&=&{\rm clKdim} (K[T/Q]) ={\rm rk} (T/Q).
\end{eqnarray*}
Since $TT^{-1}$ is of finite index in $SS^{-1}$,
it follows that
$${\rm rk} (S/P) ={\rm rk} (T/(T\cap P)) ={\rm rk}
(T/Q).$$ Because ${\rm U}(T/Q)$ is finite,
Corollary~\ref{primes-form} yields that ${\rm rk}
(T/Q)= \dim (T/Q)$ and thus we get that
\begin{eqnarray*}
{\rm rk}(S/P) ={\rm rk}
(T/Q) &=& \dim (T/Q)
=\dim (T)-{\rm h}ht (Q)\\
&=&\dim (S)-{\rm h}ht (P) = \dim (S/P).
\end{eqnarray*} This finishes the proof.
\end{proof}
\section{Maximal Orders}
In this section we describe when a semigroup
algebra $K[S]$ of a cancellative submonoid $S$ of
a polycyclic-by-finite group $G$ is a prime
Noetherian maximal order that satisfies a
polynomial identity. In case $G$ is torsion free
such a result was obtained in \cite{jes-okn-max}
and in case $S=G$ this was done by Brown in
\cite{brown1,brown2} (even without the
restriction that $K[S]$ has to be PI).
For completeness' sake we recall some notation
and terminology on (maximal) orders. We state
these in the semigroup context (see for example
\cite{gil} and \cite{wau-kru}) as these are
basically the same as in the more familiar ring
case. A cancellative monoid $S$ which has a left
and right group of quotients $G$ is called an
{\sl order}. Such a monoid $S$ is called a {\sl
maximal order} if there does not exist a
submonoid $S'$ of $G$ properly containing $S$ and
such that $aS'b\subseteq S$ for some $a,b\in G$.
For subsets $A,B$ of $G$ we define $(A:_{l}B)=\{
g\in G \; |\; gB\subseteq A\} $ and by
$(A:_{r}B)=\{ g\in G \; |\; Bg\subseteq A\}$.
Note that $S$ is a maximal order if and only if
$(I:_{l}I)=(I:_{r}I)=S$ for every {\sl fractional
ideal} $I$ of $S$. The latter means that
$SIS\subseteq I$ and $cI,Id\subseteq S$ for some
$c,d\in S$. If $S$ is a maximal order, then
$(S:_{l}I)=(S:_{r}I)$ for any fractional ideal
$I$; we simply denote this fractional ideal by
$(S:I)$ or by $I^{-1}$. Recall that then $I$ is
said to be {\sl divisorial} if $I=I^{*}$, where
$I^{*}=(S:(S:I))$. The divisorial product $I*J$
of two divisorial ideals $I$ and $J$ is defined
as $(IJ)^{*}$. Also recall that a fractional
ideal is said to be invertible if $IJ=JI=S$ for
some fractional ideal $J$ of $S$. In this case
$J=I^{-1}$ and $I$ is a divisorial ideal.
Recall then that (see for example \cite{wau-kru})
a cancellative monoid $S$ is said to be a Krull
order if and only if $S$ is a maximal order
satisfying the ascending chain condition on
divisorial integral ideals (the latter are the
fractional ideals contained in $S$). In this case
the set $D(S)$ of divisorial fractional ideals is
a free abelian group for the $*$ operation. If
$SS^{-1}$ is abelian-by-finite then (as said
before, see \cite[Lemma~1.1]{jes-wang}) every
ideal of $S$ contains a central element and it
follows that the minimal primes of $S$ form a
free basis for $D(S)$, \cite{wau-kru}.
Similarly a prime Goldie ring $R$ is said to be a
Krull order if $R$ is a maximal order that
satisfies the ascending chain condition on
divisorial integral ideals. Although there are
several notions of noncommutative Krull orders,
for rings satisfying a polynomial identity all
these notions are the same.
In the next theorem we collect some of the
essential properties of these orders. For details
we refer the reader to
\cite{chamarie-t,chamarie-a}. For a ring $R$ and
an Ore set $C$ of regular elements in $R$ we
denote by $R_{C}$ the classical localization of
$R$ with respect to $C$. The classical ring of
quotients of a prime Goldie ring $R$ is denoted
by $Q_{cl}(R)$. The prime spectrum of $R$ is
denoted by ${\rm Spec} (R)$, the set of height one
prime ideals of $R$ by $X^{1}(R)$.
\begin{theorem} \label{cha-kru}
Let $R$ be a prime Krull order satisfying a
polynomial identity. Then the following
properties hold.
\begin{enumerate}
\item
The divisorial ideals form a free abelian group with basis
$X^{1}(R)$, the height one primes of $R$.
\item If $P\in X^{1}(R)$ then $P\cap {\rm Z}(R)\in X^{1}({\rm Z}(R))$, and
furthermore, for any ideal $I$ of $R$, $I\subseteq P$ if and only
if $I\cap {\rm Z}(R) \subseteq P\cap {\rm Z}(R)$.
\item $R=\bigcap R_{Z(R)\setminus P}$, where the intersection is taken
over all height one primes of $R$, and every
regular element $r\in R$ is invertible in almost
all (that is, except possibly finitely many)
localizations $R_{{\rm Z}(R)\setminus P}$.
Furthermore, each $R_{{\rm Z}(R)\setminus P}$ is a
left and right principal ideal ring with a unique
nonzero prime ideal.
\item For a multiplicatively closed set of ideals ${\cal M}$ of $R$,
the (localized) ring $R_{{\cal M}}=\{ q\in
Q_{cl}(R) \mid Iq \subseteq R,\; \mbox{for some }
I\in {\cal M}\}$ is a Krull order, and
$$R_{{\cal M}} =\bigcap R_{{\rm Z}(R)\setminus P},$$
where the intersection is taken over those height
one primes $P$ for which $R_{{\cal M}} \subseteq
R_{{\rm Z}(R)\setminus P}$.
\end{enumerate}
\end{theorem}
Next we prove some necessary condition for $K[S]$
to be a prime Noetherian maximal order that
satisfies a polynomial identity.
\begin{lemma}\label{lem31}
Let $S$ be a submonoid of an abelian-by-finite
group $G=SS^{-1}$ and let $K$ be a field. Let
$A$ be an abelian subgroup that is normal and of
finite index in $G$ and let $P$ be a minimal
prime ideal of $S$. The following properties
hold.
\begin{enumerate}
\item
If $S$ is a maximal order then $S\cap A$ is
$G$-invariant and $S\cap A$ is a maximal order in
its group of quotients.
\item If
$K[S]$ is a prime Noetherian maximal order, then
$S$ is a maximal order, $S\cap A$ and $P\cap A$
are $G$-invariant.
\end{enumerate}
\end{lemma}
\begin{proof}
By \cite[Lemma~2.1]{jes-okn-max}, if $S$ is a maximal order then $S\cap A$ is $G$-invariant and
it is a maximal order in its group of quotients. Assume now that $K[S]$ is a prime Noetherian
maximal order. It is straightforward (as in the proof of Lemma~3.3 in \cite{jes-okn-max}) to
verify that then $S$ is a maximal order, and thus $S\cap A$ is $G$-invariant. Let $P$ be a minimal
prime ideal of $S$. Of course, $A\subseteq \Delta (G)$ and thus $P\cap A= (P\cap \Delta (G)) \cap
(A\cap S)$. Hence to prove that $P\cap A$ is $G$-invariant, we may assume in the remainder that
$A=\Delta (G)$. Indeed, since $K[S]$ is prime, by \cite[Theorem~7.19]{okn-book1}, $K[G]$ is
prime. Then, from \cite[Theorem~4.2.10]{passman} it follows that $\Delta(G)$ is abelian. Now, by
Corollary~\ref{primes-prop}, $P\cap A=Q_{1}\cap \cdots \cap Q_{n}$, an intersection of minimal
primes of $A$ that are $G$-conjugate. We need to show that $\{ Q_{1},\ldots , Q_{n}\} =\{
Q_{1}^{g}\mid g\in G\}$. Suppose the contrary, so assume $Q'$ is a minimal prime of $S\cap A$
that is conjugate to $Q_{1}$ but is different from all $Q_{i}$, for $1\leq i\leq n$. Then by
Corollary~\ref{primes-prop}, there exists a prime ideal $P'$ of $S$ so that $P'\cap A=Q'\cap
Q_{2}'\cap \cdots \cap Q_{m}'$ for some minimal primes $Q_{2}',\ldots , Q_{m}'$ of $S\cap A$.
Because of Theorem~\ref{thm-primes}, both $K[P]$ and $K[P']$ are distinct height one prime ideals
of $K[S]$. As, by assumption, $K[S]$ is a prime PI Noetherian maximal order, it follows from
Theorem~\ref{cha-kru} that there exists a central element of $K[S]$ that belongs to $K[P']$ but
not to $K[P]$. Since central elements of $K[S]$ are linear combinations of finite conjugacy class
sums of $K[G]$ we get that $P'$ contains a $G$-conjugacy class $C$ that does not belong to $P$.
Since $A=\Delta (G)$, we thus get that $C\subseteq A$. Hence, $C\subseteq Q'$. As $Q'$ is a
$G$-conjugate of $Q_{1}$ it follows that $C\subseteq Q_{1}\cap \cdots \cap Q_{n}=P\cap A$, a
contradiction.
\end{proof}
We need one more lemma in order to prove the main
theorem of this section. If $\alpha =\sum_{s\in
S} k_{s} s$ (with each $k_{s}\in K$) then we
denote by ${\rm supp} (\alpha ) =\{ s\in S \mid
k_{s}\neq 0\}$ the support of $\alpha$.
\begin{lemma}\label{lem32}
Let $S$ be a submonoid of an abelian-by-finite
group $G=SS^{-1}$. Let $K$ be a field and
suppose $K[S]$ is Noetherian. Let $A$ be a normal
abelian subgroup of finite index in $G$. Assume
that $P$ is a prime ideal of $S$ so that $K[P]$
is a prime ideal of $K[S]$ and $P\cap A$ is
$G$-invariant. Also assume that $S\cap A$ is
$G$-invariant. If $J$ is an ideal of $S$ not
contained in $P$ then $J\cap A$ contains a
$G$-conjugacy class $D$ such that $D\not
\subseteq P$ (and thus clearly $D\subseteq (J\cap
A)\setminus P$).
\end{lemma}
\begin{proof}
We may assume that $J$ is a proper ideal of $S$. The prime algebra $R=K[S]/K[P]$ has a natural
$G/A$-gradation, with identity component $R_{e}=K[S\cap A]/K[P\cap A]$, a semiprime commutative
algebra. Let $L$ be a nonzero ideal of $R$. It then follows from \cite[Theorem~1.7]{coh-row} that
$r_{e}\in L\cap R_{e}$, for some regular element $r_{e}$ of $R_{e}$. Because of the assumptions,
$G$ acts by conjugation in $R_{e}$. Clearly, $r$ has only finitely many such conjugates, say
$r_{1},\ldots , r_{m}$ and $r_{1}\cdots r_{m}\in L \cap R_{e}$ is central and nonzero. So, $L$
contains a non-trivial element in $R_{e}\cap {\rm Z}(R)$.
We apply the above to the ideal
$L=(K[J]+K[P])/K[P]$. So, let $\alpha \in K[J\cap
A]$ be such that the image $\overline{\alpha} \in
K[S]/K[P]$ is nonzero and lies in the center. So
$\alpha$ is regular modulo $K[P]$. We may assume
that ${\rm supp} (\alpha)\cap P=\emptyset$. Note also
that $1\not\in {\rm supp} (\alpha )$. Write $\alpha =
\alpha_{1}+\cdots + \alpha_{q}$, where
$\alpha_{i}$ have supports contained in different
$G$-conjugacy classes. Then $g\alpha_{1}+\cdots +
g\alpha_{q}-(\alpha_{1}g+\cdots + \alpha_{q}g)
\in K[P]$ for every $g\in S$. Clearly,
$g\alpha_{i}$ and $\alpha_{j}g$ have disjoint
supports if $i\neq j$. So we must have
$g\alpha_{i}-\alpha_{i}g\in K[P]$ for every $i$.
Then every $\alpha_{i}$ also lies in the center
modulo $K[P]$ and ${\rm supp}(\alpha _{i})$ is
contained in a $G$-conjugacy class $D_{i}$. By
the hypothesis, $D_{i}\subseteq (S\cap
A)\setminus P$. Hence, replacing $\alpha $ by
$\alpha_{1}$, we may assume that $\alpha =
\alpha_{1}$. Write $D=D_{1}$.
Because of the assumptions and
Corollary~\ref{primes-prop}, $P\cap A=Q_{1}\cap
\cdots \cap Q_{n}$ where $Q_{1},\ldots ,Q_{n}$ is
a full orbit of conjugate primes in $S\cap A$.
For $1\leq i \leq n$ let $A_{i}=\left(
\bigcap_{j,\; j\neq i} Q_{j}\right) \setminus
Q_{i}$. Each $A_{i}$ is a subsemigroup of
$A\setminus P$. Each $g\in G$ permutes the sets
$A_{1},\ldots ,A_{n}$ (by conjugation). Let
$I=\bigcup_{i=1}^{n} A_{i} \cup (P\cap A)$. Then
$I$ is $G$-invariant, $SI=IS$ and $SI\cap A=I$.
Replacing $J$ by $J\cap SI$ we may assume that
$J\subseteq SI$ and thus also $\alpha \in K[J\cap
A]\subseteq K[SI\cap A]=K[I]$. Hence, we can
write $\alpha = \gamma_{1}+\cdots +\gamma_{n}$,
with ${\rm supp} (\gamma_{i})\subseteq A_{i}$ for
$i=1,\ldots, n$. Notice that
${\rm supp}(\gamma_{i})\neq \emptyset$ for every $i$.
Indeed, suppose the contrary, that is, assume $
\gamma_{i}=0$ for some $i$. Then, $\alpha
A_{i}\subseteq P$, in contradiction with the
regularity of $\alpha$ modulo $K[P]$. So, indeed,
${\rm supp} (\gamma_{i})\neq \emptyset$ for every $i$.
Let $E_{i}=D\cap A_{i}$. Since also ${\rm supp}
(\gamma_{i})\subseteq D$, we get that
$\emptyset\neq {\rm supp} (\gamma_{i})\subseteq
E_{i}$. Put $a_{i}=\prod_{x\in E_{i}}x$. Clearly
$a_{i}\in A_{i}$. It follows that $a_{1}+\cdots
+a_{n}\in K[J]$. Let $1\leq i\leq n$ and $g\in
G$. Then $g^{-1}A_{i}g=A_{j}$ for some $j$. Hence
$g^{-1}E_{i}g=E_{j}$ and thus
$g^{-1}a_{i}g=a_{j}$. So, conjugation by elements
of $G$ permutes $a_{1},\ldots, a_{n}$. Therefore
the result follows.
\end{proof}
Recall from \cite{brown1,pas-cro} that a group
$G$ is dihedral free if, for every subgroup $D$
of $G$ isomorphic to the infinite dihedral group,
the normalizer $N_{G}(D)$ of $D$ in $G$ has
infinite index.
\begin{theorem}\label{thm33}
Let $K$ be a field and let $S$ be a submonoid of a finitely generated abelian-by-finite group.
Let $A$ be an abelian normal subgroup of finite index in $G=SS^{-1}$. The following conditions are
equivalent.
\begin{enumerate}
\item $K[S]$ is a prime Noetherian maximal order.
\item
$S$ is a maximal order that satisfies the
ascending chain condition on right ideals,
$\Delta^{+}(G)=\{ 1 \}$, $G$ is dihedral free and
for every minimal prime ideal $P$ of $S$ the set
$A\cap P$ is $G$-invariant.
\end{enumerate}
\end{theorem}
\begin{proof}
Assume that $K[S]$ is a prime Noetherian maximal
order. Then, by Theorem~\ref{cha-kru}, the
localization $K[G]=K[S{\rm Z}(S)^{-1}]$ of $K[S]$ also
is a maximal order. Because of Brown's result on
the description of group algebras of
polycyclic-by-finite groups that are maximal
orders, the latter holds if and only if
$\Delta^{+}(G)=\{ 1\}$ and $G$ is dihedral free.
Lemma~\ref{lem31} then yields that the other
conditions listed in (2) hold as well.
Conversely, assume that condition (2) holds. The assumption on
$\Delta^{+}(G)$ yields that $K[G]$ and thus $K[S]$ is prime.
Because $S$ is a maximal order, Lemma~\ref{lem31} gives that
$S\cap A$ is $G$-invariant for any abelian normal subgroup $A$ of
$G$ of finite index. To prove that $K[S]$ is a maximal order one
can follow the lines of the proof of Theorem~3.5 in
\cite{jes-okn-max}. We now give a simplified proof.
We begin by showing that if $P$ is a minimal prime ideal of $S$ then the localized ring
$R=K[S](({\rm Z}(K[S])\cap K[A])\setminus K[P])^{-1}$ is a maximal order. To do so, we show that $R$ is
a local ring with unique maximal ideal $RP$ and so that $RP$ is invertible and every proper
nonzero ideal of $R$ is of the form $(RP)^{n}$ for some positive integer $n$. First we show that
$RP$ is the only height one prime ideal of $R$. Of course if $Q'$ is a height one prime ideal of
$R$ then $Q=K[S]\cap Q'$ is a height one prime ideal of $K[S]$ that does not intersect
$({\rm Z}(K[S])\cap K[A])\setminus K[P]$. Because of Theorem~\ref{thm-primes}, either $Q=K[S\cap Q]$ or
$S\cap Q =\emptyset$. Because of Lemma~\ref{lem32}, the former implies that $S\cap Q \subseteq P$
and thus $Q=K[P]$, as desired. So assume $S\cap Q=\emptyset$, or equivalently, $A\cap Q
=\emptyset$. Because $S\cap A$ is $G$-invariant and $Q$ does not contain homogeneous elements, it
follows (see for example Lemma~7.1.4 in \cite{jes-okn-book}) that $\overline{Q}=\bigcap_{g\in G}
g^{-1}(Q\cap K[A])g$ is not contained in $K[P\cap A]$ . As $\overline{Q}$ is $G$-invariant, we get
that $K[S]\overline{Q}$ is an ideal of $K[S]$. So $(K[S]\overline{Q}+K[P])/K[P]$ is a nonzero
ideal of the Noetherian algebra $K[S]/K[P]$. This algebra has a natural $G/A$-gradation, with
component of degree $e$ the semiprime algebra $K[S\cap A]/K[P\cap A]$. By Theorem~1.7 in
\cite{coh-row}, the ring $K[S]/K[P]$ has a classical ring of quotients that is obtained by
inverting the regular elements of $K[S\cap A]/K[P\cap A]$. Hence the ideal
$(K[S]\overline{Q}+K[P])/K[P]$ contains a regular element $\overline{\alpha}$ that is contained in
$K[S\cap A]/K[P\cap A]$. Since $K[S\cap A]$ and $K[P\cap A]$ are $G$-invariant, conjugation
induces an action of $G$ on $K[S\cap A]/K[P\cap A]$. Hence the product of the finitely many
conjugates of $\overline{\alpha}$ also belongs to $K[S\cap A]/K[P\cap A]$. Since this element is
central, we thus may assume that $\overline{\alpha}$ also is central in $K[S]/K[P]$ and clearly
$\overline{\alpha}\in (\overline{Q}+K[P\cap A])/K[P\cap A]$. Write $\overline{\alpha} = \gamma +
K[P\cap A]$ for some $\gamma \in \overline{Q}$. Let $\beta$ be the product of the distinct
conjugates of $\gamma$. Then $\beta +K[P\cap A] =\gamma^{m}+K[P\cap A]$ for some positive integer
$m$. Since $K[P\cap A]$ is a semiprime ideal in $K[S\cap A]$, it follows that $\beta \not\in
K[P\cap A]$. Hence $\beta \in (Q\cap {\rm Z}(K[S]))\setminus K[P]$, a contradiction. This implies that
indeed $RP$ is the only height one prime ideal of $R$.
Because of Lemma~\ref{lem31}, $S\cap A$ is a finitely generated maximal order. Hence we know
that $K[S\cap A]$ is a Noetherian maximal order (\cite{and1,and}) and thus it is well known (or
use Theorem~\ref{cha-kru}) that $K[S\cap A](({\rm Z}(K[S]\cap K[A])\setminus P)^{-1}$ is a Noetherian
maximal order with only finitely many height one prime ideals. Hence it is a principal ideal
domain (see for example \cite{fossum}) and thus it has prime dimension one. As this is the
component of degree $e$ of the $G/A$-graded ring $R$, from \cite[Theorem~17.9]{pas-cro} it follows
that $R$ also has dimension one. Hence $RP$ is the only maximal ideal of $R$. As $R$ also is a PI
algebra, we then obtain that $RP$ is the Jacobson radical of $R$ and thus, by
\cite[Theorem~8.12]{goo-war}, $\bigcap_{n}(RP)^{n}=\{ 0\}$. Also note that $P(S:P)$ is an ideal of
$S$ that is not contained in $P$. Hence $RP(S:P)=R$ and thus the unique maximal ideal $RP$ is an
invertible ideal of $R$. It then easily follows that every proper nonzero ideal of $R$ is of the
form $(RP)^{n}$ for some unique positive integer $n$. This proves the desired properties of $R$.
Let $\alpha =\sum_{i=1}^{n}k_{i}g_{i} \in K[G]$,
where $0\neq k_{i} \in K$ and $g_{i}\in G$ for each
$1\leq i \leq n$ and $g_{i}\neq g_{j}$ for $i\neq j$
We now show that if $\alpha \in K[G] \cap
\bigcap_{P} K[S](({\rm Z}(K[S])\cap K[A])\setminus
K[P])^{-1}$ (where the intersection runs over all
minimal primes $P$ of $S$) then $\alpha \in K[S]$.
We prove this by induction on $n$. If $n=1$ then
$\alpha =k_{1}g_{1}$. For each minimal prime $P$ of
$S$ there then exists a central element $\delta$ of
$K[S]$ that belongs to $K[A]\setminus P$ so that
$\delta k_{1}g_{1}\in K[S]$. Hence there is a
$G$-conjugacy class $C(P)$ so that $C\subseteq
(S\cap A)\setminus P$ and $C(P)g_{1}\subseteq S$.
Let $C=\bigcup_{P} C(P)$. Then $SCg_{1}\subseteq S$
and $SC$ is an ideal of $S$ that is not contained in
any of the minimal prime ideals of $S$. Since $S$ is
a maximal order, it follows that $g_{1} \in S$, as
desired. Now assume $n>1$. Since $\bigcap_{P}
K[S](({\rm Z}(K[S])\cap K[A])\setminus K[P])^{-1}$ is a
$G/A$-graded ring, the induction hypothesis yields
that we may assume that $\alpha$ is
$G/A$-homogeneous, that is, each $g_{i}g_{j}^{-1}\in
A$. Since this statement holds for any normal
abelian subgroup of $G$ of finite index and because
$A$ is residually finite, we get that $g_{i}=g_{j}$
for all $i=j$. Hence we may assume $n=1$ and thus by
the above $\alpha \in K[S]$.
So we have shown that $K[S]= K[G] \cap \bigcap_{P} K[S](({\rm Z}(K[S])\cap K[A])\setminus K[P])^{-1}$.
Recall that by Theorem~F in \cite{brown1}, since $G$ is dihedral-free, $K[G]$ is a maximal order.
Since also each $K[S](({\rm Z}(K[S])\cap K[A])\setminus K[P])^{-1}$ is a maximal order and a central
localization of $K[S]$, it follows that $K[S]$ is a maximal order. This finishes the proof.
\end{proof}
\section{Constructing examples}
Theorem~\ref{thm33} reduces the problem of determining when $K[S]$
is a prime Noetherian maximal order to the algebraic structure of
$S$. It hence provides a strong tool for constructing new classes
of such algebras. For some examples the required conditions on $S$
can easily be verified, but on the other hand, for some examples
this still requires substantial work. In this section this is
illustrated with some concrete constructions.
A first class of examples consists of algebras
defined by monoids of $I$-type (see
\cite{gat-van,jes-okn-book}). Recall that, in
particular, these are quadratic algebras $R$ with
a presentation defined by $n$ generators
$x_{1},\ldots , x_{n}$ and with ${n \choose 2}$
relations of the form $x_{i}x_{j}=x_{k} x_{l}$ so
that every word $x_{i}x_{j}$ appears at most once
in one of the defining relations. Clearly
$R=K[S]$, where $S$ is the monoid defined by the
same presentation. It turns out that $S$ has a
group of quotients $G$ that is torsion free and
has a free abelian subgroup $A$ of finite index.
Furthermore, for any minimal prime ideal $P$ of
$S$ one has that $P=Ss=sS$, $P\cap A=(S\cap A)a$,
for some $s\in S,\; a\in A$, and $P\cap A$ is
$G$-invariant. Using Theorem~\ref{thm33} we then
immediately recover the known result that $R$ is
a maximal order. The only noncommutative algebra
of such type which is generated by two elements
is $K\langle x,y \mid x^{2}=y^{2}\rangle$
(\cite{gat-van}). A related example on three
generators that is not of this type is $K\langle
x,y,z\mid x^{2}=y^{2}=z^{2}, zx=yz,zy=xz\rangle$.
As an application of Theorem~\ref{thm33} one can
show by elementary calculations that this algebra
also is a prime Noetherian PI maximal order.
In the remainder of this section we discuss in full detail one
more construction that illustrates Theorem~\ref{thm33} but also
shows that certain assumptions in Theorem~\ref{thm-primes} are
essential. Before this, we establish a useful general method for
constructing nonabelian submonoids of abelian-by-finite groups
that are maximal orders, starting from abelian maximal orders.
\begin{proposition}\label{notesjan}
Let $A$ be an abelian normal subgroup of finite index in a group
$G$. Suppose that $B$ is a submonoid of $A$ so that $A=BB^{-1}$
and $B$ is a finitely generated maximal order. Let $S$ be a
submonoid of $G$ such that $G=SS^{-1}$ and $S\cap A=B$. Then $S$
is a maximal order that satisfies the ascending chain condition on
right ideals if and only if $S$ is maximal among all submonoids $T$
of $G$ with $T\cap A=B$.
\end{proposition}
\begin{proof}
First suppose $S$ is a maximal order that satisfies the ascending
chain condition on right ideals. Suppose that $S\subseteq
T\subseteq G$ for a submonoid $T$ of $G$ such that $S\cap
A=B=T\cap A$. By assumption, $B$ is finitely generated. Since also
$A$ is normal and of finite index in $G$, it thus follows (see the
introduction) that $K[T]$ is Noetherian and it is a finitely
generated right $K[B]$-module. Thus $T$ satisfies the ascending
chain condition on one-sided ideals and $T=\bigcup _{i=1}^{n}
t_{i}B$ for some $n\geq 1$ and $t_{i}\in T$. Since $G$ is finitely
generated and abelian-by-finite, we know that for every $i$ there
exists $z_{i}\in {\rm Z}(S)$ such that $z_{i}t_{i}\in S$. Let
$z=z_{1}\cdots z_{k}$. Then $zt_{i}\in S$ and therefore
$zT=\bigcup _{i} zt_{i}(S\cap A)\subseteq S$. Since $S$ is a
maximal order, this implies that $T=S$. So, we have shown that $S$
is maximal among all submonoids $T\subseteq SS^{-1}$ such that
$T\cap A=S\cap A$.
Conversely, assume that $S\subseteq G$ is maximal among all
submonoids $T$ of $G$ with $T\cap A=B$. As above, because $S\cap
A$ is finitely generated, $S$ satisfies the ascending chain
condition on right ideals. Suppose that $T\subseteq G$ is a
submonoid such that $S\subseteq T$ and $gTh\subseteq S$ for some
$g,h\in G$. There exist $s,t\in S$ such that $sg,ht\in {\rm Z}(G)\cap
B$. So $sgTht\subseteq S$ and therefore $Tz\subseteq S$ for some
$z\in {\rm Z}(G)\cap B$. In particular $(T\cap A)z\subseteq S\cap A=B$.
Since $S\cap A\subseteq T\cap A$ and $S\cap A$ is a maximal order,
it follows that $T\cap A=S\cap A$. Because $S\subseteq T$, the
assumption on $S$ then implies that $T=S$. Therefore $S$ is a
maximal order.
\end{proof}
In order to illustrate the above proposition with a
concrete example, we start with the following
construction of a monoid that contains the abelian
monoid generated by $a_{1},a_{2},a_{3},a_{4}$ and
defined by the extra relation
$a_{1}a_{2}=a_{3}a_{4}$. It was shown in \cite{and}
that the latter is a cancellative monoid that is a
maximal order.
\begin{example} \label{abel-max}
The abelian monoid $B=\langle
a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}\rangle$
defined by the relations
$a_{1}a_{2}=a_{3}a_{4}=a_{5}a_{6}$ is a
cancellative monoid that is a maximal order (in
its torsion free group of quotients).
\end{example}
\begin{proof}
Let $F= \langle x_{1},\ldots ,x_{8} \rangle $ be
a free abelian monoid of rank $8$. Define
$$b_{1}=x_{1}x_{2}x_{3}x_{4}, \,
b_{2}=x_{5}x_{6}x_{7}x_{8}, \,
b_{3}=x_{1}x_{2}x_{5}x_{6}, $$
$$b_{4}=x_{3}x_{4}x_{7}x_{8},\,
b_{5}=x_{1}x_{3}x_{5}x_{7}, \,
b_{6}=x_{2}x_{4}x_{6}x_{8}.$$ Clearly, these
$b_{i}$ satisfy the defining relations for $B$.
We claim that actually $B\cong \langle
b_{1},\ldots, b_{6}\rangle$ under the map
determined by $a_{i}\mapsto b_{i}$, $i=1,\ldots
,6$. In order to prove this, suppose that there
is a relation $w=v$, where $w,v$ are nontrivial
words in $b_{i}$. We need to show that this
relation follows from
$b_{1}b_{2}=b_{3}b_{4}=b_{5}b_{6}$. Cancelling in
$F$, if needed, we may assume that each $b_{i}$
appears at most on one side of the relation. We
also may assume that not both sides are
divisible in $\langle b_{1},\ldots, b_{6}\rangle$
by one of the equal words $b_{1}b_{2}$,
$b_{3}b_{4}$ and $b_{5}b_{6}$. Further, on both
sides of $v=w$ we need some $b_{i}$ with an even
$i$. Indeed, suppose the contrary, then $x_{8}$
is not involved in $v$ and $w$. Hence also
$b_{5}$ cannot occur because of $x_{7}$. But, as
$b_{1}$ and $b_{3}$ generate a free abelian
monoid of rank $2$, it then follows that $v$ and
$w$ are identical words in the $b_{i}$'s, as
desired. Hence, by symmetry we may assume that
$w$ contains $b_{2}^{i_{2}}$ with $i_{2}>0$ and
$w$ does not contain $b_{4}$ nor $b_{6}$ as a
factor, and $v$ does not contain $b_{2}$ as a
factor and contains $b_{4}^{i_{4}}b_{6}^{i_{6}}$
for some nonnegative $i_{4},i_{6}$ with
$i_{2}=i_{4}+i_{6}$ (the latter follows by taking
into the account the degree of $x_{8}$ in the
respective words). Looking at $x_{4}$ we then get
that $b_{1}$ appears in $w$. If $i_{6}=0$ then
(looking at $x_{6}$) we get that $b_{3}$ is in
$v$, a contradiction because
$b_{1}b_{2}=b_{3}b_{4}$ divides then both $v$ and
$w$. So $i_{6}>0$. Also $v$ must contain
$x_{1},x_{5}$ and so $b_{3}$ or $b_{5}$ is in
$v$. The latter is not possible because then $v$
and $w$ are divisible by $b_{1}b_{2}=b_{5}b_{6}$,
a contradiction. Thus, $b_{3}$ occurs in $v$.
Then we must have $i_{4}=0$ because otherwise
$b_{1}b_{2}=b_{3}b_{4}$ divides $v$ and $w$. So
$w=b_{1}^{i_{1}}b_{2}^{i_{2}}b_{5}^{i_{5}}=b_{3}^{i_{3}}b_{6}^{i_{6}}=v$
for some $i_{1},i_{2},i_{6}>0$ and
$i_{3},i_{5}\geq 0$. Then the exponents of
$x_{7}$ show that $i_{2}=i_{5}=0$, a
contradiction. The claim follows. Hence we may
indeed identify $a_{i}$ with $b_{i}$ and $B$ with
$\langle b_{1},\ldots, b_{6}\rangle$.
Next, suppose that
$w=a_{2}^{i_{2}}a_{3}^{i_{3}}a_{4}^{i_{4}}a_{5}^{i_{5}}\in
F\cap {\rm gr} (B)$. This is equivalent to the
conditions: $i_{3}+i_{5}\geq 0, i_{3}\geq 0,
i_{4}+i_{5}\geq 0,i_{4}\geq
0,i_{2}+i_{3}+i_{5}\geq 0, i_{2}+i_{3}\geq 0,
i_{2}+i_{4}+i_{5}\geq 0, i_{2}+i_{4}\geq 0$. Let
$j=\min \{i_{3},i_{4}\} \geq 0$. Then
$j+i_{2}+i_{5}\geq 0$. We choose $s,t\geq 0$ so
that $i_{2}+s,i_{5}+t \geq 0$ and $s+t=j$. Then
$w=a_{1}^{s}a_{2}^{i_{2}+s}a_{3}^{i_{3}-j}a_{4}^{i_{4}-j}a_{5}^{i_{5}+t}a_{6}^{t}$.
It is thus clear that $w\in B$. So $B=F\cap {\rm gr}
(B)$. Since $F$ is a maximal order it then easily
follows that $B$ is a maximal order.
\end{proof}
We conclude with the promised illustration of Theorem~\ref{thm33}.
This example also shows that Theorem~\ref{thm-primes} cannot be
extended to prime ideals of height exceeding $1$.
\begin{example} \label{mainexample}
Let $K$ be any field and let $R=K\langle x_{1},x_{2},x_{3},x_{4} \rangle $
be the algebra defined by the following relations:
\begin{eqnarray*} x_{1}x_{4}=x_{2}x_{3},
x_{1}x_{3}=x_{2}x_{4}, x_{3}x_{1}=x_{4}x_{2} \\
x_{3}x_{2}=x_{4}x_{1}, x_{1}x_{2}=x_{3}x_{4},
x_{2}x_{1}=x_{4}x_{3} . \end{eqnarray*} Clearly, $R=K[S]$ for the
monoid $S$ defined by the same presentation. Then $S$ is
cancellative (but the group $SS^{-1}$ is not torsion free) and $R$
is a prime Noetherian PI-algebra that is a maximal order.
Furthermore, there exists a prime ideal $P$ of $S$ so that $K[P]$
is not a prime ideal of $K[S]$.
\end{example}
\begin{proof}
Notice that each of the permutations $(12)(34),
(13)(24)$ and $(14)(23)$ determines an
automorphism of $S$. First we list some
equalities in $S$, namely all relations between
the elements of length $3$. For brevity, we use
the index $i$ in place of the generator $x_{i}$.
\begin{eqnarray*}
112=134=244, &
113=124=344, &
114=123=343=321=411 \\
221=243=133, &
224=213=433, &
223=214=434=412=322 \\
331=342=122, &
334=312=422, &
332=341=121=143=233 \\
442=431=211, &
443=421=311, &
441=432=212=234=144
\end{eqnarray*}
\begin{eqnarray*}
131=142=232=241, &
141=132=242=231 \\
313=423=414=324, &
323=314=424=413 . \end{eqnarray*}
It follows easily that $A=\langle
x_{1}^{2},x_{2}^{2},x_{3}^{2},x_{4}^{2}\rangle $
is an abelian submonoid and it is normal, that is
$xA=Ax$ for every $x\in S$. Moreover
$x_{1}^{2}x_{4}^{2}=x_{2}^{2}x_{3}^{2}$ because
$x_{2}x_{2}x_{3}x_{3}=x_{2}x_{3}x_{3}x_{2}=x_{1}x_{4}x_{4}x_{1}=
x_{4}x_{4}x_{1}x_{1}$.
Let $a_{1}=x_{1}x_{4}, a_{2}=x_{4}x_{1}, a_{3}= x_{1}^{2},
a_{4}=x_{4}^{2}, a_{5}=x_{2}^{2},a_{6}= x_{3}^{2}$ and $B=\langle
a_{1}, a_{2}, a_{3},a_{4},a_{5}, a_{6}\rangle $. From the above
equalities it follows that $B$ is abelian and $sB=Bs$ for every
$s\in S$.
Every element of $S$ that is a word of length $3$
is either of the form $xyy$ or of the form
$xx_{1}x_{4}, xx_{4}x_{1}$ for some $x$. It is
also easy to see that every element of $S$ is of
the form $zu_{1}w_{1}\cdots u_{r}w_{r}$ or
$zu_{1}w_{1}\cdots u_{r}w_{r}u_{r+1}$ or $zc$,
where $z\in A$ and $c\in S$ is an element of
length at most $2$ in the $x_{j}$ and either
$u_{i}\in \{ x_{1},x_{2}\}, w_{i} \in \{
x_{3},x_{4}\} $ or $u_{i}\in \{ x_{3},x_{4}\},
w_{i} \in \{ x_{1},x_{2} \} $ (for all $i$). So
we may assume that $c\in
\{x_{1}x_{2},x_{2}x_{1}\}$. Here $r$ is a
non-negative integer. Using the relations listed
above (especially
$x_{1}x_{4}x_{1}=x_{2}x_{4}x_{2},
x_{4}x_{1}x_{4}=x_{3}x_{1}x_{3}$) it may be
checked that the even powers of
$x_{1}x_{4}=x_{2}x_{3}$ and
$x_{2}x_{4}=x_{1}x_{3}$ are equal and the even
powers of $x_{4}x_{1}=x_{3}x_{2}$ and
$x_{3}x_{1}=x_{4}x_{2}$ are equal. Also
$\{x_{4}x_{1},x_{1}x_{4}\}\{x_{1}x_{2},x_{2}x_{1}\}\subseteq
A\{x_{1}x_{3},x_{3}x_{1}\}$. It follows that
possible forms of elements of $S$ are
\begin{eqnarray} z(x_{1}x_{4})^{i}, \ z(x_{1}x_{4})^{i}x_{1}, \
z(x_{1}x_{4})^{i}x_{2}, \
z(x_{1}x_{4})^{i}x_{1}x_{3}, \ zx_{1}x_{2}
\label{two}\\ z(x_{4}x_{1})^{i}, \
z(x_{4}x_{1})^{i}x_{4}, \ z(x_{4}x_{1})^{i}x_{3},
\ z(x_{4}x_{1})^{i}x_{3}x_{1}, \ zx_{2}x_{1},
\label{three}
\end{eqnarray}
with $z\in A, i\geq 0$. This leads to
\begin{eqnarray}
S&=&B\cup Bx_{1} \cup Bx_{2}\cup Bx_{3}\cup
Bx_{4} \cup Bx_{1}x_{3}\cup Bx_{3}x_{1} \cup
Bx_{1}x_{2} \cup Bx_{2}x_{1} . \label{semgrpel}
\end{eqnarray}
It follows that $K[S]$ is finite module over the
finitely generated commutative algebra $K[B]$.
Hence $K[S]$ is a Noetherian PI-algebra.
Let $C$ be the free abelian group of rank $4$
generated by elements $a,b,c,d$. Let $X$ be the
free monoid on $x_{1},x_{2},x_{3},x_{4}$.
Consider the monoid homomorphism $\phi
:X\longrightarrow M_{4}(K[C])$ defined by
\begin{eqnarray*}
x_{1}\mapsto \left(
\begin{array}{cccc}
0 & a & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & a^{-1}bc \\
0 & 0 & 1 & 0
\end{array} \right) , &&x_{2}\mapsto \left( \begin{array}{cccc}
0 & 0 & b & 0 \\
0 & 0 & 0 & a^{-1}bc \\
1 & 0 & 0 & 0 \\
0 & ab^{-1} & 0 & 0
\end{array} \right) ,\\
&&\\
x_{3}\mapsto \left( \begin{array}{cccc}
0 & 0 & bcd^{-1} & 0 \\
0 & 0 & 0 & a^{-1}bd \\
b^{-1}d & 0 & 0 & 0 \\
0 & ad^{-1} & 0 & 0
\end{array} \right) ,&& x_{4}\mapsto \left(
\begin{array}{cccc}
0 & bcd^{-1} & 0 & 0 \\
a^{-1}d & 0 & 0 & 0 \\
0 & 0 & 0 & d \\
0 & 0 & ad^{-1} & 0
\end{array} \right) .
\end{eqnarray*}
It is easy to check that these matrices satisfy
the defining relations of $S$, so $\phi $ can be
viewed as a homomorphism from $S$ to the group of
monomial matrices over $C$. Moreover
\begin{eqnarray*}a_{3}\mapsto \left(
\begin{array}{cccc}
a & 0 & 0 & 0 \\
0 & a & 0 & 0 \\
0 & 0 & a^{-1}bc & 0 \\
0 & 0 & 0 & a^{-1}bc
\end{array} \right) ,&&
a_{5}\mapsto \left( \begin{array}{cccc}
b & 0 & 0 & 0 \\
0 & c & 0 & 0 \\
0 & 0 & b & 0 \\
0 & 0 & 0 & c
\end{array} \right) \\
&&\\
a_{4}\mapsto \left( \begin{array}{cccc}
a^{-1}bc & 0 & 0 & 0 \\
0 & a^{-1}bc & 0 & 0 \\
0 & 0 & a & 0 \\
0 & 0 & 0 & a
\end{array} \right) ,&&
a_{6}\mapsto \left( \begin{array}{cccc}
c & 0 & 0 & 0 \\
0 & b & 0 & 0 \\
0 & 0 & c & 0 \\
0 & 0 & 0 & b
\end{array} \right) ,\\
&&\\
a_{1}\mapsto \left( \begin{array}{cccc}
d & 0 & 0 & 0 \\
0 & bcd^{-1} & 0 & 0 \\
0 & 0 & bcd^{-1} & 0 \\
0 & 0 & 0 & d
\end{array} \right) ,&&
a_{2}\mapsto \left( \begin{array}{cccc}
bcd^{-1} & 0 & 0 & 0 \\
0 & d & 0 & 0 \\
0 & 0 & d & 0 \\
0 & 0 & 0 & bcd^{-1}
\end{array} \right) .
\end{eqnarray*}
The projection of the group $C'$ generated by $\phi (a_{1}), \phi
(a_{3}), \phi (a_{5}), \phi (a_{6})$ onto the $(1,1)$-entry
contains $a,b,c,d$, whence it is free abelian of rank $4$. So
$C'\cong C$. In particular, $\phi$ is injective on $B_{0}=\langle
a_{1},a_{3},a_{5},a_{6}\rangle$, and thus $B_{0}$ is a free
abelian monoid of rank $4$. Let $B'$ be the abelian monoid with
presentation $\langle y_{1},y_{2},y_{3},y_{4},y_{5},y_{6}\mid
y_{1}y_{2}=y_{3}y_{4}=y_{5}y_{6}\rangle$. Clearly, we have natural
homomorphisms $B'\longrightarrow B\longrightarrow \phi (B)$. We
also know that $\phi (B)$ generates a free abelian group of rank
$\geq 4$ and $B'$ has a group of quotients that is free abelian of
rank $4$. So $B'$ and $\phi(B)$ must be isomorphic, since
otherwise under the map $B'\longrightarrow \phi(B)$ we have to
factor out an additional relation and the rank would decrease. It
follows that $\phi $ is injective on $B$ and thus, because of
Example~\ref{abel-max}, $B$ is a cancellative maximal order with
group of quotients $N= BB^{-1}\cong {\mathbb Z}^{4}$. Note that
$AA^{-1}={\rm gr} (a_{3},a_{5},a_{6})$. Using the defining relations
$a_{1}a_{2}=a_{3}a_{4}=a_{5}a_{6}$ for $B$, it is readily verified
that if $i,j,k\in {\mathbb Z}$ then
$a_{3}^{i}a_{5}^{j}a_{6}^{k}\in A=\langle
a_{3},a_{4},a_{5},a_{6}\rangle$ if and only if $j,k\geq 0$ and
$\min (j,k)+i\geq 0$.
The images under $\phi $ of the first $4$ types
listed in (\ref{two}) have different patterns of
nonzero entries in $M_{4}(K[C])$. The same
applies to the first $4$ types in (\ref{three}).
Notice that $\phi $ is injective on each of the
$10$ types. Suppose that
$s=z(x_{1}x_{4})^{i}x_{1},
t=z'(x_{4}x_{1})^{j}x_{4}$ have the same image.
Then $sx_{4}=z(x_{1}x_{4})^{i+1}$ and $tx_{4}=z
'(x_{4}x_{1})^{j}x_{4}^{2}$ have equal images.
This is not possible because
$(x_{1}x_{4})^{i+1}A\cap
(x_{4}x_{1})^{j}A=\emptyset$ by the above
description of the group $BB^{-1}\cong
\phi(B)\phi(B)^{-1}$. Similarly one deals with
elements of any two different types listed in
(\ref{two}) and (\ref{three}), showing that only
elements the form $s=zx_{1}x_{2},
t=z'x_{2}x_{1}$, where $z,z'\in A$, can have
equal images. Then
$\phi(zx_{1}x_{1}x_{4}x_{1})=\phi
(tx_{4}x_{2})=\phi(sx_{4}x_{2})=\phi(z'x_{4}x_{1}x_{2}x_{2})$
and cancellativity of $\phi (B)\cong B$ yields
$zx_{1}x_{1}=z'x_{2}x_{2}$. Write
$z=a_{3}^{i}a_{5}^{j}a_{6}^{k}$, with $i,j,k\in
{\mathbb Z}$. Since $z,z'\in A$, the above yields
that $z'=a_{3}^{i+1}a_{5}^{j-1}a_{6}^{k}$ and
$j,k\geq 0,-i$ and $j-1,k\geq 0,-(i+1)$. Notice
that
$a_{3}x_{2}x_{1}=x_{1}x_{1}x_{2}x_{1}=x_{2}x_{2}x_{1}x_{2}=a_{5}x_{1}x_{2}$.
Therefore, if $j-1\geq -i$ then
$a_{3}^{i}a_{5}^{j-1}a_{6}^{k}\in A$ and hence
$zx_{1}x_{2}=a_{3}^{i}a_{5}^{j-1}a_{6}^{k}a_{5}x_{1}x_{2}=
a_{3}^{i}a_{5}^{j-1}a_{6}^{k}a_{3}x_{2}x_{1}=z'x_{2}x_{1}$.
On the other hand, if $j=-i$ then
$a_{4}^{j-1}a_{6}^{k-j}\in A$ and
$a_{4}x_{1}x_{2}=a_{6}x_{2}x_{1}$. Hence we also
get
$zx_{1}x_{2}=a_{4}^{j}a_{6}^{k-j}x_{1}x_{2}=a_{4}^{j-1}a_{6}^{k-j}a_{4}x_{1}x_{2}=
a_{4}^{j-1}a_{6}^{k-j}a_{6}x_{2}x_{1}=z'x_{2}x_{1}$.
It follows that $\phi $ is injective on all
elements of types (\ref{two}),(\ref{three}).
Therefore $\phi $ is an embedding and thus $S$ is
cancellative.
We identify $S$ with $\phi (S)$. Put $G=SS^{-1}$. Then $G= N\cup
x_{1} N\cup x_{2}N\cup x_{1}x_{2}N$. Moreover $N\cong {\mathbb
Z}^{4}$ and $N$ is a normal subgroup with $G/N$ the four group.
From (\ref{semgrpel}) it follows that $S\cap N=B$.
We now show that $G$ is dihedral free. To prove
this, notice that $S$ acts by conjugation on $B$
and the generators $x_{i}$ of $S$ correspond to
the following permutations $\sigma _{i}$ of the
generating set of $B$ (the numbers $1,2,3,4,5,6$
correspond to the generators
$a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}$).
\begin{eqnarray} \label{permut}
\sigma_{1} =(12)(56), \, \sigma_{2}= (12)(34),\,
\sigma_{3}= (12)(34),\, \sigma_{4} =(12)(56) .
\end{eqnarray}
Suppose $D\subseteq G$ is an infinite dihedral group such that the normalizer $N_{G}(D)$ of $D$ in
$G$ is of finite index. Let $t\in D$ be an element of order $2$. Then there exists $k\geq 1$ such
that $a_{i}^{k}ta_{i}^{-k}t\in D$ for $i=1,2,\ldots ,6$. Clearly
$$a_{i}^{k}ta_{i}^{-k}t=a_{i}^{k}a_{\sigma(i)}^{k}t^{2}=a_{i}^{k}a_{\sigma(i)}^{k},$$
where $\sigma$ is the automorphism of $N$
determined by $t$. Since $t\in Nx_{1}\cup
Nx_{2}\cup Nx_{1}x_{2}$, it follows that $\sigma
$ is determined by the conjugation by
$x_{1},x_{2}$ or $x_{1}x_{2}$. So $\sigma $
permutes exactly two of the pairs $a_{1},a_{2};
a_{3},a_{4}$ and $a_{5},a_{6}$. It follows that
$a_{i}^{k}a_{j}^{k}, a_{p}^{k}a_{q}^{k}\in D$ for
two different pairs $i,j$ and $p,q$. Hence ${\rm rk}
(N\cap D)\geq 2$, a contradiction. Therefore $G$
indeed is a dihedral free group.
Next we show that $\Delta^{+}(G)$ is trivial. For this, let $F$ be
a finite normal subgroup of $G$. Since $N$ is torsion free, it is
clear that $F$ is isomorphic with a subgroup of ${\mathbb Z}
_{2}\times {\mathbb Z} _{2}$. A nontrivial element $t\in F$ must
be of order $2$, whence as above we get that
$a_{i}^{n}a_{j}^{n}t\in F$ for some $i\neq j$ and every $n\geq 1$.
Therefore $F$ is infinite, a contradiction. It follows that
$\Delta ^{+}(G)$ is trivial. Therefore $K[G]$ is prime and hence
$K[S]$ also is prime.
Note that in $G$ we have
$x_{2}^{-1}x_{1}=x_{3}x_{4}^{-1}=x_{4}x_{3}^{-1}$
and
$x_{1}x_{2}^{-1}=x_{3}^{-1}x_{4}=x_{4}^{-1}x_{3}$.
So $x_{2}^{-1}x_{1}$ is an element of order $2$.
We describe the minimal prime ideals of $S$. For
this we first notice that it is easy to see that
the minimal prime ideals of $B$ are:
\begin{eqnarray*}
Q=Q_{1}=(a_{1},a_{3},a_{5}), &&
Q_{2}=Q^{x_{1}}=(a_{2},a_{3},a_{6}),\\
Q_{3}=Q^{x_{2}}=(a_{2},a_{4},a_{5}), &&
Q_{4}=Q^{x_{1}x_{2}}=(a_{1},a_{4},a_{6}),\\
Q'=Q_{5}=(a_{2},a_{3},a_{5}), &&
Q_{6}=(Q')^{x_{1}}=(a_{1},a_{3},a_{6}),\\
Q_{7}=(Q')^{x_{2}}=(a_{1},a_{4},a_{5}),&&
Q_{8}=(Q')^{x_{1}x_{2}}=(a_{2},a_{4},a_{6}).
\end{eqnarray*}
Because of (\ref{permut}), it is easily verified
that $a_{1}a_{2}$ is a central element of $S$ and
that every ideal of $S$ contains a positive power
of $a_{1}a_{2}$. In particular, $a_{1}a_{2}$
belongs to every prime ideal of $S$. Consider in
$B$ the following $G$-invariant ideals:
$$M=(a_{1}a_{3}a_{5},
a_{2}a_{3}a_{6},a_{2}a_{4}a_{5},a_{1}a_{4}a_{6},a_{1}a_{2})$$
and $$M'=(a_{2}a_{3}a_{5},
a_{1}a_{4}a_{5},a_{1}a_{3}a_{6},a_{2}a_{4}a_{6},a_{1}a_{2}).$$
Again because of (\ref{permut}), it is easy to
see that $bSb' \subseteq a_{1}a_{2}S$ for every
defining generator $b$ of $M$ and $b'$ of $M'$.
It follows that a prime ideal of $S$ contains $M$
or $M'$.
Notice that $xa_{1}a_{3}a_{5}y\in xy\{
a_{1}a_{3}a_{5},a_{2}a_{4}a_{5},a_{2}a_{3}a_{6},a_{1}a_{4}a_{6}\}$ for every $x,y\in S$. Therefore
$Sa_{1}a_{3}a_{5}S\cap \langle a_{2},a_{3},a_{5}\rangle=\emptyset$ (for example,
$xya_{1}a_{3}a_{5}\not \in \langle a_{2},a_{3},a_{5}\rangle$ because otherwise
$xy=a_{1}^{-1}a_{3}^{-1}a_{5}^{-1}\langle a_{2},a_{3},a_{5}\rangle \cap (S\cap N)$, which is not
possible because $S\cap N=B$ and $N={\rm gr}(a_{1},a_{2},a_{3},a_{5})$ is free abelian of rank $4$). So
there exists a (unique) ideal $P$ of $S$ that is maximal with respect to the property $P\cap
\langle a_{2},a_{3},a_{5}\rangle =\emptyset$. It is easy to see that $P$ is a prime ideal of $S$.
Since $a_{2}a_{3}a_{5}\not\in P$, we get that $M'\not\subseteq P$, whence $M\subseteq P$.
Let $E=a_{2}a_{3}a_{5}\langle a_{2},a_{3},a_{5}\rangle $. Then
\begin{eqnarray} x_{4}\langle
a_{1},a_{3},a_{6}\rangle x_{1}\subseteq E, \,
x_{2}\langle a_{1},a_{4},a_{5}\rangle x_{2}
\subseteq E, \, x_{1}x_{3} \langle
a_{2},a_{4},a_{6} \rangle x_{3}x_{1}\subseteq E .
\end{eqnarray}
Therefore $P\cap \langle a_{1},a_{3},a_{6}\rangle=\emptyset, P\cap \langle
a_{1},a_{4},a_{5}\rangle=\emptyset, P\cap \langle a_{2},a_{4},a_{6}\rangle=\emptyset$. For every
element $b\in B\setminus M$ the ideal $bB$ intersects one of the sets
$$a_{2}a_{3}a_{5}\langle a_{2},a_{3},a_{5}\rangle ,a_{1}a_{3}a_{6}\langle
a_{1},a_{3},a_{6}\rangle, a_{1}a_{4}a_{5}\langle a_{1},a_{4},a_{5}\rangle, a_{2}a_{4}a_{6} \langle
a_{2},a_{4},a_{6} \rangle.$$ Since these sets do not intersect $P$, we get that $P\cap B=M$. Hence
it is $G$-invariant and, because $M\subseteq Q_{1}\cap Q_{2}\cap Q_{3}\cap Q_{4}$ and all $Q_{i}$
are minimal primes of $B$, Corollary~\ref{primes-prop} yields that $P\cap B=M=Q_{1}\cap Q_{2}\cap
Q_{3}\cap Q_{4}={\mathcal B}(SMS)$ and $P$ is a minimal prime ideal of $S$. A similar argument
shows that there exists an ideal $P'$ of S that is maximal with respect to the property $P' \cap
\langle a_{1},a_{3},a_{5}\rangle = \emptyset$, and it follows that $P'\cap B=M'=Q_{5}\cap
Q_{6}\cap Q_{7}\cap Q_{8}={\mathcal B}(SM'S)$ is the only other minimal prime of $S$.
In order to continue the proof we first show the following claim
on the representation of elements of $B$.
{\it Claim: Presentation}\\
An element $b
=a_{1}^{\alpha_{1}}a_{3}^{\alpha_{3}}a_{4}^{\alpha_{4}}a_{6}^{\alpha_{6}}$
of $N=BB^{-1}$ is in $B$ (with each $\alpha_{i} \in {\mathbb Z}$)
if and only if $\alpha_{3},\alpha_{4}\geq 0$ and either (i)
$\alpha_{1}\geq 0$ and $\min (\alpha_{3} ,\alpha_{4})+\alpha_{6}
\geq 0$ or (ii) $\alpha_{1}<0$, $\alpha_{3}+\alpha_{1},
\alpha_{4}+\alpha_{1}\geq 0$ and $\min (\alpha_{3} ,\alpha_{4})
+\alpha_{1} +\alpha_{6}\geq 0$, or equivalently, $\min (\alpha_{3}
+\alpha_{1},\alpha_{4} +\alpha_{1})\geq \max(0, - \alpha_{6})$.
That the first condition is sufficient is easily verified. For the
second condition, we rewrite $a_{1}^{\alpha_{1}}$ as
$a_{3}^{\alpha_{1}}a_{4}^{\alpha_{1}}a_{2}^{-\alpha_{1}}$ and the
result follows because of the first part (by interchanging $a_{1}$
with $a_{2}$).
To prove that they are necessary, suppose
$b=a_{1}^{\alpha_{1}}a_{3}^{\alpha_{3}}
a_{4}^{\alpha_{4}}a_{6}^{\alpha_{6}}\in B$, with each $\alpha_{i}
\in {\mathbb Z}$. Because $Ba_{4}\cap \langle
a_{1},a_{3},a_{6}\rangle =\emptyset$ and $Ba_{3}\cap \langle
a_{1},a_{4},a_{6}\rangle =\emptyset$, it follows that
$\alpha_{3},\alpha_{4}\geq 0$. Suppose now that $\alpha_{1}\geq
0$. Then
$$b=a_{1}^{\alpha_{1}} a_{3}^{\alpha_{3}-\min
(\alpha_{3},\alpha_{4}) } a_{4}^{\alpha_{4}-\min
(\alpha_{3},\alpha_{4}) } a_{5}^{\min (\alpha_{3},\alpha_{4})}
a_{6}^{\alpha_{6}+\min (\alpha_{3},\alpha_{4})}$$ and thus
$a_{1}^{\alpha_{1}} a_{3}^{\alpha_{3}-\min
(\alpha_{3},\alpha_{4})} a_{4}^{\alpha_{4}-\min
(\alpha_{3},\alpha_{4})}a_{5}^{\min (\alpha_{3},\alpha_{4})} \in
Ba_{6}^{-(\alpha_{6}+\min (\alpha_{3},\alpha_{4}))}$. Since the
exponent of $a_{3}$ or $a_{4}$ is $0$, this implies that
$\alpha_{6}+\min (\alpha_{3},\alpha_{4}) \geq 0$, as desired.
On the other hand, suppose that $\alpha_{1}<0$. Then $b=a_{2}^{-\alpha_{1}}
a_{3}^{\alpha_{3}+\alpha_{1}} a_{4}^{\alpha_{4}+\alpha_{1}}
a_{6}^{\alpha_{6}}$. The previous case yields that
$\alpha_{3}+\alpha_{1}\geq 0$, $\alpha_{4}+\alpha_{1} \geq 0$ and
$\min (\alpha_{3}+\alpha_{1},\alpha_{4}+\alpha_{1})
+\alpha_{6}\geq 0$, again as desired. This proves the claim {\it
Presentation}.
Next we show that $S$ is a maximal order. First note that $B$ is a
maximal order by Example~\ref{abel-max}. So, because of
Proposition~\ref{notesjan}, it is sufficient to prove that if
$s\in SS^{-1}\setminus S$ then $\langle S,s \rangle \cap N$
strictly contains $B$.
We know that $G=SS^{-1}= N\cup x_{1} N\cup x_{2}N\cup
x_{1}x_{2}N$. If $s\in N \setminus B$ then clearly $\langle S,s
\rangle \cap N$ strictly contains $B$. So, there are three cases
to be dealt with: (1) $s = bx_{1}\in G \setminus S$, (2) $s =
bx_{2}\in G\setminus S$, and (3) $s = bx_{1}x_{2}\in G \setminus
S$, where $b\in N\setminus B$.
Case (1): $s = bx_{1}\in G \setminus S$. Obviously, $bx_{1}x_{4} =
b a_{1}\in \langle S,s \rangle$. If $ba_{1} \in N\setminus B$,
then we are done. So suppose $ba_{1} \in B$ and $b \in N\setminus
B$. We write $b$ in the following form:
$$a_{1}^{\alpha_{1}}a_{3}^{\alpha_{3}}a_{4}^{\alpha_{4}}a_{6}^{\alpha_{6}}.$$
First, suppose that $\alpha_{1} \geq 0$. By the claim {\it Presentation}, from $ba_{1} \in B$ we
get that $\min(\alpha_{3},\alpha_{4})\geq \max(- \alpha_{6},0)$. But, also from $b\in N\setminus
B$ we get that $\min(\alpha_{3},\alpha_{4}) < \max(- \alpha_{6},0)$, a contradiction. So,
$\alpha_{1}$ has to be strictly negative.
Therefore, assume that $\alpha_{1} < 0$. Then $$b
=a_{1}^{\alpha_{1}}a_{3}^{\alpha_{3}}a_{4}^{\alpha_{4}}a_{6}^{\alpha_{6}}
=
a_{2}^{-\alpha_{1}}a_{3}^{\alpha_{3}+\alpha_{1}}a_{4}^{\alpha_{3}+\alpha_{1}}a_{6}^{\alpha_{6}}$$
and $$ba_{1} = a_{2}^{-\alpha_{1} -
1}a_{3}^{\alpha_{3}+\alpha_{1} +
1}a_{4}^{\alpha_{3}+\alpha_{1}+
1}a_{6}^{\alpha_{6}}.$$ By the claim {\it
Presentation} (by interchanging $a_{1}$ and
$a_{2}$), we get that $\min(\alpha_{3} +
\alpha_{1} + 1,\alpha_{4} + \alpha_{1} + 1)\geq
\max(-\alpha_{6},0)$, but also $\min(\alpha_{3}
+ \alpha_{1},\alpha_{4} + \alpha_{1}) <
\max(-\alpha_{6},0)$. Hence, $\min(\alpha_{3} +
\alpha_{1} + 1,\alpha_{4} + \alpha_{1} + 1) =
\max(-\alpha_{6},0)$. Suppose $\alpha_{6} \geq
0$. Then $\min(\alpha_{3} + \alpha_{1} +
1,\alpha_{4} + \alpha_{1} + 1) = 0$. Therefore,
$b
=a_{1}^{\alpha_{1}}a_{3}^{\alpha_{3}}a_{4}^{\alpha_{4}}a_{6}^{\alpha_{6}}
= a_{1}^{-1}a_{2}^{\alpha_{3}}a_{4}^{\alpha_{4} -
\alpha_{3}}a_{6}^{\alpha_{6}}$ and $\alpha_{4}>
\alpha_{3}$ or $b
=
a_{1}^{-1}a_{2}^{\alpha_{4}}a_{3}^{\alpha_{3} -
\alpha_{4}}a_{6}^{\alpha_{6}}$ and $\alpha_{3}\geq \alpha_{4}$. In
the first case we get that $bx_{1}\in S$, since
$x_{4}^{-1}x_{1}^{-1}x_{4}x_{4}x_{1} = x_{4}$. So this case gives
a contradiction and is hence impossible. In the second case,
$bx_{1}x_{1} = a_{1}^{-1}a_{2}^{\alpha_{4}}a_{3}^{\alpha_{3} -
\alpha_{4}+1}a_{6}^{\alpha_{6}} \in (\langle S,s \rangle \cap N
)\setminus B$, as desired. If $\alpha_{6} < 0$, $\min(\alpha_{3} +
\alpha_{1} + 1,\alpha_{4} + \alpha_{1} + 1) = -\alpha_{6}$ and
therefore $b =
a_{1}^{\alpha_{1}}a_{3}^{\alpha_{3}}a_{4}^{\alpha_{4}}a_{6}^{\alpha_{6}}
= a_{1} ^{- 1} a_{2}^{-\alpha_{1} - 1}a_{3}^{\alpha_{3} -
\alpha_{4}}a_{5}^{-\alpha_{6}}$ if $\alpha_{4}\leq \alpha_{3}$ or
$b = a_{1} ^{- 1} a_{2}^{-\alpha_{1} - 1}a_{4}^{\alpha_{4} -
\alpha_{3}}a_{5}^{-\alpha_{6}}$ if $\alpha_{3}< \alpha_{4}$. Since
$-\alpha_{1} - 1 \geq 0$ and $-\alpha_{6} > 0$, by interchanging
$a_{5}$ and $a_{6}$, this case is completely similar to the case
where $\alpha_{6}\geq 0$. This finishes the proof of Case (1).
Case (2): $s=bx_{2}\in G \setminus S$. The permutation $\sigma
=(12)(34)$ determines an automorphism $\sigma$ on $S$ with
$\sigma (B)=B$. Clearly, $\sigma (s) =b'x_{1}$ with $b'=\sigma
(b)\not\in B$. From Case (1) we get that $\langle S,\sigma
(s)\rangle \cap N $ properly contains $B$. Again applying $\sigma$
to the latter we get that $\langle S,s\rangle \cap N$ properly
contains $B$, as required.
Case (3): $s = bx_{1}x_{2}\in G\setminus S$.
Clearly, $bx_{1}x_{2}x_{2}x_{4} = b a_{1}a_{6}\in \langle S,s
\rangle$. If $ba_{1}a_{6} \in N\setminus B$, then we are done. So
suppose $ba_{1}a_{6} \in B$ and $b \in N\setminus B$. Write
$b=a_{1}^{\alpha_{1}}a_{3}^{\alpha_{3}}a_{4}^{\alpha_{4}}a_{6}^{\alpha_{6}}$.
We know that $\alpha_{3},\alpha_{4}\geq0$ and we consider again the two cases:
$\alpha_{1}\geq 0$ and $\alpha_{1} < 0$, separately.
First assume that $\alpha_{1}\geq 0$. Since $ba_{1}a_{6}\in B$ and $b\not \in B$, we get that
$\min(\alpha_{3},\alpha_{4}) = -\alpha_{6} - 1$. Then $b =
a_{1}^{\alpha_{1}}a_{3}^{\alpha_{3}}a_{4}^{\alpha_{4}}a_{6}^{\alpha_{6}} =
a_{1}^{\alpha_{1}}a_{3}^{\alpha_{3} - \alpha_{4}}a_{5}^{-\alpha_{6} - 1}a_{6}^{-1}$, if
$\alpha_{4}\leq \alpha_{3}$, or $b = a_{1}^{\alpha_{1}}a_{4}^{\alpha_{4} -
\alpha_{3}}a_{5}^{-\alpha_{6} - 1}a_{6}^{-1}$, if $\alpha_{3}< \alpha_{4}$. Now, let $\alpha_{1} >
0$. Then $bx_{1}x_{2}\in S$, since $x_{1}x_{4}x_{3}^{-1}x_{3}^{-1}x_{1}x_{2} = x_{2}x_{4}$, a
contradiction. So this case is impossible. Therefore, $\alpha_{1} = 0$. If $\alpha_{3} <
\alpha_{4}$, then $bx_{1}x_{2} \in S$, since $x_{4}x_{4}x_{3}^{-1}x_{3}^{-1}x_{3}x_{4} =
x_{2}x_{1}$, a contradiction, so this case is again impossible. If $\alpha_{4} \leq \alpha_{3}$,
then $bx_{1}x_{2}x_{3}x_{1} = ba_{2}a_{3} \in (\langle S,s \rangle \cap N)\setminus B$, as
desired.
Finally, suppose $\alpha_{1} < 0$. Then $b
=a_{1}^{\alpha_{1}}a_{3}^{\alpha_{3}}a_{4}^{\alpha_{4}}a_{6}^{\alpha_{6}}
=
a_{2}^{-\alpha_{1}}a_{3}^{\alpha_{3}+\alpha_{1}}a_{4}^{\alpha_{4}+\alpha_{1}}a_{6}^{\alpha_{6}}$
and $ba_{1}a_{6} = a_{2}^{-\alpha_{1} -
1}a_{3}^{\alpha_{3}+\alpha_{1} + 1}a_{4}^{\alpha_{4}+\alpha_{1}+
1}a_{6}^{\alpha_{6}+1}$. By the claim {\it Presentation} (by
interchanging $a_{1}$ and $a_{2}$), we get that $\min(\alpha_{3} +
\alpha_{1} + 1,\alpha_{4} + \alpha_{1} + 1)\geq \max(-\alpha_{6} -
1,0)$, but also $\min(\alpha_{3} + \alpha_{1},\alpha_{4} +
\alpha_{1}) < \max(-\alpha_{6},0)$. If $\alpha_{6} \geq 0$,
$\min(\alpha_{3},\alpha_{4}) + \alpha_{1} = -1$ and $b =
a_{1}^{-1}a_{2}^{\alpha_{4}}a_{3}^{\alpha_{3} -
\alpha_{4}}a_{6}^{\alpha_{6}}$, if $\alpha_{4}\leq \alpha_{3}$, or
$b = a_{1}^{-1}a_{2}^{\alpha_{3}}a_{4}^{\alpha_{4} -
\alpha_{3}}a_{6}^{\alpha_{6}}$, if $\alpha_{3}<\alpha_{4}$. So, we
have the same conditions here as in Case 1 and the result follows.
If $\alpha_{6} < 0$, there are two possibilities:
$\min(\alpha_{3},\alpha_{4}) + \alpha_{1} = -\alpha_{6} - 1$ or
$\min(\alpha_{3},\alpha_{4}) + \alpha_{1}= -\alpha_{6} - 2$. In
the first case, we get that $b = a_{1}^{-1}a_{2}^{-\alpha_{1} -
1}a_{3}^{\alpha_{3} - \alpha_{4}}a_{5}^{-\alpha_{6}}$ if
$\alpha_{4}\leq \alpha_{3}$, or $b = a_{1}^{-1}a_{2}^{-\alpha_{1}
- 1}a_{4}^{\alpha_{4} - \alpha_{3}}a_{5}^{-\alpha_{6}}$ if
$\alpha_{3}<\alpha_{4}$. Since $-\alpha_{1} -1 \geq 0$ and $-
\alpha_{6}> 0$, by interchanging $a_{5}$ and $a_{6}$, this case is
completely similar to the case where $\alpha_{6}\geq 0$ and hence
can also be treated as in Case 1. Finally, if the second
possibility holds, that is $\min(\alpha_{3},\alpha_{4}) +
\alpha_{1}= -\alpha_{6} - 2$, then $b = a_{1}^{- 2}a_{2}^{- 2 -
\alpha_{1}}a_{3}^{\alpha_{3} -\alpha_{4}}a_{5}^{-\alpha_{6}}$, if
$\alpha_{4}\leq \alpha_{3}$, or $b = a_{1}^{- 2}a_{2}^{- 2 -
\alpha_{1}}a_{4}^{\alpha_{4} -\alpha_{3}}a_{5}^{-\alpha_{6}}$, if
$\alpha_{4} > \alpha_{3}$, so we always get that $ba_{2}a_{3} \in
(\langle S,s \rangle \cap N)\setminus B$, as desired.
This finishes the proof of the fact that $S$ is a maximal order.
Since $P\cap B$ is invariant for every minimal prime $P$ of $S$,
it then follows from Theorem~\ref{thm33} that $K[S]$ is a maximal
order.
Let $V$ be the ideal of $S$ generated by the elements $a_{3},a_{4},a_{5},a_{6}, x_{1}x_{2},
x_{2}x_{1}$. It easily follows that the elements of $S\setminus V$ are of the form
\begin{eqnarray*} (x_{1}x_{4})^{i}, \ (x_{1}x_{4})^{i}x_{1}, \
(x_{1}x_{4})^{i}x_{2}, \
(x_{1}x_{4})^{i}x_{1}x_{3}, \\
(x_{4}x_{1})^{i}, \ (x_{4}x_{1})^{i}x_{4}, \ (x_{4}x_{1})^{i}x_{3}, \ (x_{4}x_{1})^{i}x_{3}x_{1},
\end{eqnarray*}
where $i$ is a non-negative integer. Then $S\setminus V= \{ 1\}\cup I$ where the set $I$ can be
written in matrix format as a union of disjoint sets:
$$\left( \begin{array}{cc}
I_{11} & I_{12}\\
I_{21} & I_{22}
\end{array} \right) ,$$
with $I_{11} =\langle a_{2}\rangle a_{2}\cup \langle a_{2}\rangle
x_{3}x_{1}$, $I_{12}=\langle a_{2}\rangle x_{3} \cup \langle
a_{2}\rangle x_{4}$, $I_{21} = x_{1} \langle a_{2}\rangle \cup
x_{2}\langle a_{2}\rangle$ and $I_{22} =\langle a_{1}\rangle
a_{1}\cup \langle a_{1}\rangle x_{1}x_{3}$. Moreover,
$I_{ij}I_{kl}\subseteq I_{il}$ if $j=k$, and is contained in $V$
otherwise. In $S/V$ the set $I\cup \{ 0\}$ is an ideal and the
semigroup $I_{11}$ (treated as a subsemigroup of $S$) has a group
of quotients $H={\rm gr} (a_{2},x_{3}x_{1})$. Since
$a_{2}^{2}=(x_{3}x_{1})^{2}$ and $a_{2} (x_{3}x_{1})
=(x_{3}x_{1})a_{2}$, we get that $H$ is isomorphic with ${\mathbb
Z} \times {\mathbb Z}_{2}$. From the matrix pattern of $I$ it
follows that $S/V$ is a prime semigroup. So $V$ is a prime ideal
of $S$. However, because $K[H]$ and thus $K[I_{11}]$ is not prime,
standard generalized matrix ring arguments yield that $K[S]/K[V]$
is not prime.
\end{proof}
\noindent
\begin{tabular}{ll}
I. Goffa and E. Jespers & J. Okni\'{n}ski\\
Department of Mathematics& Institute of Mathematics\\
Vrije Universiteit Brussel & Warsaw University\\
Pleinlaan 2& Banacha 2\\
1050 Brussel, Belgium& 02-097 Warsaw, Poland\\
efjesper@vub.ac.be and igoffa@vub.ac.be & okninski@mimuw.edu.pl
\end{tabular}
\end{document} |
\begin{document}
\defspacingset#1{\renewcommand{\baselinestretch}
{#1}small\normalsize} spacingset{1}
\if11
{
\title{\bf Extremal Dependence-Based Specification Testing of Time Series}
\author{Yannick Hoga\thanks{
The author is grateful to seminar participants at CREST and Erasmus University Rotterdam for valuable comments and suggestions, in particular Christian Francq, Jeroen Rombouts, Jean-Michel Zako\"{i}an and Chen Zhou. The author would also like to thank Christoph Hanck and Till Massing for their careful reading of an earlier version of this manuscript. This work was supported by the German Research Foundation (DFG) under Grant HO 6305/1-1.}\hspace{.2cm}\\
Faculty of Economics and Business Administration\\University of Duisburg-Essen
}
maketitle
} \fi
\if01
{
\begin{center}
{\LARGE\bf Extremal Dependence-Based Specification Testing of Time Series}
\end{center}
medskip
} \fi
\onehalfspacing
\begin{abstract}
\noindent We propose a specification test for conditional location--scale models based on extremal dependence properties of the standardized residuals. We do so comparing the left-over serial extremal dependence---as measured by the pre-asymptotic tail copula---with that arising under serial independence at different lags. Our main theoretical results show that the proposed Portmanteau-type test statistics have nuisance parameter-free asymptotic limits. The test statistics are easy to compute, as they only depend on the standardized residuals, and critical values are likewise easily obtained from the limiting distributions. This contrasts with extant tests (based, e.g., on autocorrelations of squared residuals), where test statistics depend on the parameter estimator of the model and critical values may need to be bootstrapped. We show that our test performs well in simulations. An empirical application to S\&P~500 constituents illustrates that our test can uncover violations of residual serial independence that are not picked up by standard autocorrelation-based specification tests, yet are relevant when the model is used for, e.g., risk forecasting.
\end{abstract}
\noindent
{\it Keywords:} Location--scale models, Serial extremal dependence, Specification test, Tail copula
section{Motivation}
\onehalfspacing
The dynamics of many economic and financial time series can be successfully captured by location--scale models, that allow for mean and variance changes in the conditional distribution. The benchmark models for incorporating mean changes are ARMA models, and GARCH-type processes have become the standard volatility models, since the seminal work of \citet{Eng82} and \citet{Bol86}. The main aim of this paper is to derive tests that check whether the fitted location--scale model adequately captures the time variation in the mean and variance of the conditional distribution. As is common for specification tests, a rejection points to a problem in the mean \textit{or} variance dynamics, but does not establish which of the two.\footnote{A rare exception is the work of \citet{FRZ06}.} While our tests may also pick up misspecification in the conditional mean, they are first and foremost designed to detect misspecified volatility dynamics. Thus, our main focus in the following will be on volatility models.
Testing for correctly specified volatility models is of high practical relevance. This is because volatility models are widely used to predict risk measures, such as the Value-at-Risk (VaR) and the Expected Shortfall (ES); see, e.g., \citet{MF00,Cea07,Hog18+}. Risk measure forecasts are key inputs in risk management procedures of financial institutions, due to regulatory requirements in the Basel framework of the \citet{BCBSBF19}. The Basel framework penalizes sustained underpredictions of risk by imposing higher capital requirements. On the other hand, if risk forecasts are too high, too much capital is put aside as a buffer against large losses. In both cases of over- and underprediction, some portion of the capital can no longer earn premiums, leading to foregone profits. Thus, accurate risk forecasts from correctly specified volatility models are important. Consequently, diagnostic tests should be routinely applied to the chosen volatility specification.
To this end, several specification tests are available. These tests typically verify whether the model residuals are independent, identically distributed (i.i.d.). However, standard tests of the i.i.d.~property---such as \citet{BP70} or \citet{LB78} tests---cannot be applied `as usual', since the residuals are only estimated. \citet{LM94} were the first to propose a Portmanteau-type test for the autocorrelations of squared residuals that corrects for this fact in (conditionally Gaussian) location--scale models. \citet{BHK03b} and \citet{LL97} extend the applicability of the \citet{LM94} test to more general GARCH and $N(0,1)$--FARIMA--GARCH models, respectively. Similarly, \citet{FG12} consider weighted Portmanteau statistics applied to ARCH residuals. \citet{HZ07} develop spectral-based tests for ARCH($\infty$) series. The limiting distributions for all these proposed test statistics have only been derived for the quasi-maximum likelihood (QML) estimator. Moreover, practical application of these tests is complicated by the fact that the limiting distributions depend on nuisance parameters induced by parameter estimation. Valid tests thus require consistent estimators of the nuisance parameters (that have to be computed on a case-by-case basis) or involved bootstrap-based procedures to compute critical values. By construction, all these tests diagnose the complete serial dependence structure of the standardized residuals.
Here, instead of considering the complete dependence structure, we take a different tack by focusing on the left-over serial \textit{extremal} dependence in the standardized residuals. When diagnosing the complete dependence structure, the effect of any remaining residual extremal dependence may be `washed-out'.
However, overlooked serial dependence in the extremes may be very harmful as it invalidates, e.g., the model's risk forecasts, such as VaR and ES forecasts \citep{Sea21+}. As pointed out above, misspecified risk forecasts are penalized under the Basel framework. Hence, a separate diagnostic for the extremes is desirable. In empirical work, \citet{DMC12} and \citet{DMZ13} use the pre-asymptotic (PA) extremogram as a diagnostic. For instance, to assess the GARCH fit for FTSE returns, \citet{DMC12} show that the PA-extremogram of the standardized residuals exhibits no signs of extremal dependence. However, such a model check lacks theoretical justification so far, because the central limit theory for extremal dependence measures has not been developed for model residuals. It is the main theoretical contribution of this paper to do so.
Specifically, we base our tests on the PA-tail copula \citep{SS06} of the (absolute values of the) standardized residuals. The tail copula---i.e., the limit of the PA-tail copula---has been used by, e.g., \citet{BJW15} to detect structural changes in tail dependence. The PA-tail copula is closely related to the PA-extremogram of \citet{DMC12}. However, for our purposes, working with the PA-tail copula gives rise to nuisance parameter-free limiting distributions, which would not be the case for the PA-extremogram.
It turns out that---unlike the above mentioned traditional specification tests---our tests are easy to apply: We only require the standardized residuals to compute the test statistics. In particular, it does not matter for our tests which parameter estimator (QML estimator, Self-Weighted QML estimator, Least Squares, etc.) was used in fitting the volatility model. This gives our tests wide practical appeal.
Testing for any left-over serial extremal dependence in the residuals directs the power of our specification tests to the tails. Thus, they are more sensitive to any remaining dependence in the tails. This may be important when, say, a GARCH model captures well the volatility dynamics in the body of the distribution, but not so much in the extremes (see also the simulations for such an example). It is exactly for these cases that our tests are designed. Thus, the diagnostic tests developed here are not meant to replace standard tests based (e.g.) on the autocorrelations of the squared residuals, but rather to supplement them.
We illustrate the good size and power of our tests in simulations. Specifically, we show that---as predicted by theory---in sufficiently large samples the size is indeed unaffected by parameter estimation effects. We also demonstrate that empirically relevant misspecifications can be detected more easily using our extremal dependence-based tests instead of autocorrela-tion-based tests.
The empirical application shows that for the S\&P~500 components an autocorrelation-based test can often not detect any significant departures from a fitted GARCH-type model. However, in many of these cases, applying our tests suggests some statistically significant serial extremal dependence left over in the residuals. This indicates that the fitted GARCH-type processes capture well the volatility changes in the center of the distribution, but not so much in the tails. This result is consistent with \citet{EM04}, who conclude from fitting their CAViaR models at different risk levels that `our findings suggest that the process governing the behavior of the tails might be different from that of the rest of the distribution.' Such a dynamic is particularly worrisome when using GARCH-type models to forecast risk measures, such as VaR and ES, far out in the tail.
The remainder of the paper proceeds as follows. In Section~\ref{Main results}, we present our model setup and propose our Portmanteau-type tests for serial extremal dependence. Section~\ref{Simulations} explores size and power of our tests in finite samples, while Section~\ref{Application} illustrates the benefits of our test for real data. Finally, Section~\ref{Conclusion} concludes. The supplementary Appendix contains all proofs and additional simulations.
section{Main Results}\label{Main results}
subsection{Location--Scale Model}\label{Location--Scale Model}
Denote by $Y_1,\ldots, Y_n$ the observations of interest to which some parametric model will be fitted. For instance, the $Y_1,\ldots, Y_n$ may denote log-returns on some speculative asset. We define the $sigma$-field generated by $Y_t,Y_{t-1},\ldots$ and some exogenous (possibly multivariate) variables $\bm x_{t},\bm x_{t-1},\ldots$ by $mathcal{F}_t=sigma(Y_t,Y_{t-1},\ldots;\bm x_{t},\bm x_{t-1},\ldots)$. We assume the $Y_t$ to follow the parametric conditional location--scale model
\begin{equation}\label{eq:ls model}
Y_t=mu_{t}(\bm \theta^\circ)+sigma_t(\bm \theta^\circ)\varepsilon_t,
\end{equation}
where $\bm \theta^\circ\in\bm \varTheta$ is the true parameter vector from the parameter space $\bm \varThetasubsetmathbb{R}^{m}$ ($m\inmathbb{N}$), $mu_{t}(\bm \theta^\circ)$ and $sigma_t(\bm \theta^\circ)$ are the $mathcal{F}_{t-1}$-measurable conditional mean and conditional standard deviation, and $\varepsilon_t$ is i.i.d.~with mean zero and unit variance (written: $\varepsilon_t\overset{\text{i.i.d.}}{sim}(0,1)$), independent of $mathcal{F}_{t-1}$.
In the following, we require a smoothness condition on the tail of the $|\varepsilon_t|$, whose distribution function (d.f.) we denote by $F(\cdot)$.
\begin{assumption}\label{ass:U distr 2}
The d.f.~$F(\cdot)$ of the $|\varepsilon_t|$ satisfies:
\begin{enumerate}
\item[(i)] The upper endpoint of $F(\cdot)$ is infinite.
\item[(ii)] There exists some constant $C_F>0$ such that $F(\cdot)$ is differentiable with density $f(\cdot)$ on $[C_F,\infty)$.
\item[(iii)] $\lim_{x\to\infty}xf(x)/[1-F(x)]=\alpha\in(0,\infty)$.
\end{enumerate}
\end{assumption}
Assumption~\ref{ass:U distr 2} is known as one of the \textit{von Mises'} conditions \citep[see, e.g.,][Rem.~1.1.15]{HF06} and ensures a sufficiently well-behaved tail of the $|\varepsilon_t|$. This is needed to justify the replacement of $|\varepsilon_t|$ in our test statistic with the standardized residuals. In their autocorrelation-based diagnostic test, \citet{BHK03b} impose a similar regularity condition; see in particular their equation~(2.4). Assumption~\ref{ass:U distr 2} is only needed to ensure the validity of Lemma~\ref{lem:Lem F} in Appendix~\ref{Proofs of Propositions}. Thus, it may be replaced by any other assumption on the tail decay, as long as the conclusions of Lemma~\ref{lem:Lem F} remain valid.
Next, we require a $sqrt{n}$-consistent estimator of the unknown $\bm \theta^{\circ}$:
\begin{assumption}\label{ass:estimator}
There exists an estimator $\widehat{\bm \theta}$ satisfying $n^{1/2}|\widehat{\bm \theta}-\bm \theta^{\circ}|=O_{\operatorname{P}}(1)$, as $n\to\infty$.
\end{assumption}
Root-$n$ consistent estimators are available under various regularity conditions for ARMA--GARCH and GARCH-type models; see, e.g., \citet{BH04}, \citet{FZ04}, \citet{PWT08}, \citet{FT19}.
Once the parameters have been estimated, volatility $\widehat{sigma}_t=\widehat{sigma}_t(\widehat{\bm \theta})$, the conditional mean $\widehat{mu}_t=\widehat{mu}_t(\widehat{\bm \theta})$ and the standardized residuals $\widehat{\varepsilon}_t=\widehat{\varepsilon}_t(\widehat{\bm \theta})=[Y_t-\widehat{mu}_t(\widehat{\bm \theta})]/\widehat{sigma}_t(\widehat{\bm \theta})$ can be computed. Note that $mu_t(\bm \theta)$ and $sigma_t(\bm \theta)$ may depend on the infinite past of $Y_t$ (and, possibly, $\bm x_t$) and, hence, can only be approximated via $\widehat{mu}_t(\bm \theta)$ and $\widehat{sigma}_t(\bm \theta)$ using some initial values; see, e.g., the truncated recursion \eqref{eq:vola} in Appendix~\ref{APARCH and ARMA--GARCH Models}.
For our final assumption, we define the neighborhood of the true $\bm \theta^\circ$ as
\[
N_n(\eta)=\left\{\bm \theta\ :\ n^{1/2}|\bm \theta-\bm \theta^{\circ}|\leq\eta\right\},\qquad\eta>0,
\]
where $|\cdot|$ is some norm on $\bm \varTheta$.
\begin{assumption}\label{ass:UA}
For any $\eta>0$, there exist r.v.s $m_t:=m_{n,t}(\eta)\geq0$ and $s_t:=s_{n,t}(\eta)\geq0$ with $max_{t=\ell_{n},\ldots,n}m_{t}=o_{\operatorname{P}}(1)$ and $max_{t=\ell_{n},\ldots,n}s_{t}=o_{\operatorname{P}}(1)$ for any $\ell_{n}\leq n$ with $\ell_n\to\infty$, such that with probability approaching 1, as $n\to\infty$,
\[
|\varepsilon_t|(1-s_t)-m_t\leq |\widehat{\varepsilon}_t(\bm \theta)| \leq |\varepsilon_t|(1+s_t)+m_t
\]
for all $t=\ell_{n},\ldots,n$ and all $\bm \theta\in N_n(\eta)$.
\end{assumption}
Assumption~\ref{ass:UA} imposes a uniform approximability of the innovations $|\varepsilon_t|$ by the $|\widehat{\varepsilon}_t(\bm \theta)|$ in a $n^{-1/2}$-neighborhood of the true parameter. However, Assumption~\ref{ass:UA} does not require the first $(\ell_{n}-1)$ innovations to be approximable. This is convenient as the first few residuals are often imprecise, due to initialization effects caused by using artificial initial values in the mean and variance recursions. The variables $m_t$ and $s_t$ bound the error in the $\widehat{\varepsilon}_t(\bm \theta)$ associated with approximating the conditional mean and conditional variance, respectively. Of course, Assumption~\ref{ass:UA} is a high-level condition that must be verified for each specific model on a case-by-case basis. In Appendix~\ref{APARCH and ARMA--GARCH Models}, we verify Assumption~\ref{ass:UA} for two popular model classes, namely APARCH and ARMA--GARCH models.
subsection{The Tail Copula and its Estimators}\label{The Tail Copula and its Estimators}
The \textit{survival copula} at lag $d$ of the $\{|\varepsilon_t|\}$ from Section~\ref{Location--Scale Model} is
\[
\overline{C}^{(d)}(u,v)=\operatorname{P}\big\{|\varepsilon_t|>F^{\leftarrow}(1-u),\ |\varepsilon_{t-d}|>F^{\leftarrow}(1-v)\big\},\qquad 0\leq u,v\leq 1,
\]
where $F^{\leftarrow}(\cdot)$ denotes the left-continuous inverse of $F(\cdot)$. Following \citet{SS06}, the (upper) \textit{tail copula} is the directional derivative of the survival copula at the origin with direction $(x,y)$, i.e.,
\[
\Lambda^{(d)}(x,y)=\lim_{s\to\infty}s\overline{C}^{(d)}(x/s, y/s),
\]
where the limit is assumed to exist. Thus, the tail copula describes the serial extremal dependence structure of the $|\varepsilon_t|$ at different lags $d$. For $x=y=1$, the tail copula simplifies to the \textit{tail dependence coefficient} of \citet{Sib60}. In this case, it also coincides with the most popular version of the \textit{extremogram} due to \citet{DM09}.
Let $k=k(n)$ be an intermediate sequence satisfying $k\to\infty$ and $k/n\to0$ as $n\to\infty$. Then, replacing $s$ with $n/k\to\infty$ and also replacing population quantities with empirical counterparts, one may estimate the tail copula non-parametrically via
\[
\widetilde{\Lambda}_{n}^{(d)}(x,y)=\frac{1}{k}sum_{t=d+1}^{n}I_{\{|\varepsilon_t|>|\varepsilon|_{(\lfloor kx\rfloor+1)},\ |\varepsilon_{t-d|}>|\varepsilon|_{(\lfloor ky\rfloor+1)}\}},
\]
where $|\varepsilon|_{(1)}\geq\ldots\geq |\varepsilon|_{(n)}$ denote the order statistics, and $\lfloor\cdot\rfloor$ rounds to the nearest smaller integer.
In the case of interest here, where the $\varepsilon_t$ from \eqref{eq:ls model} are i.i.d., the tail copula can easily shown to be identically zero, i.e., $\Lambda^{(d)}(x,y)\equiv0$. However, even when there is strong (non-extremal) dependence between $|\varepsilon_t|$ and $|\varepsilon_{t-d}|$, we have $\Lambda^{(d)}(x,y)\equiv0$. For instance, this is the case when the dependence between the two variables is governed by a Gaussian copula \textit{for any} correlation parameter $\rho\in(-1,1)$ \citep{Hef00}. Hence, while the relationship may be strong in non-extremal regions, it is non-existent in the limit as measured by the tail copula. Likewise, \citet{Hil11a} shows that the tail copula is identically zero if $\varepsilon_t$ follows a \textit{serially dependent} stochastic volatility model. These are well-known to display weaker extremal dependence than GARCH-type models \citep{DM09,DMZ13}. Thus, the tail copula cannot adequately discriminate between independent and dependent variables, which would result in compromised power of a specification test based on the tail copula.
For this reason, we prefer to base our test on what we term the \textit{PA-tail copula}:
\begin{equation}\label{eq:PA-tail copula}
\Lambda_{n}^{(d)}(x,y)=\frac{n}{k}\operatorname{P}\left\{|\varepsilon_t|>b\Big(\frac{n}{kx}\Big),\ |\varepsilon_{t-d}|>b\Big(\frac{n}{ky}\Big)\right\},
\end{equation}
where $b(x)=F^{\leftarrow}(1-1/x)$. Now, the PA-tail copula is $\Lambda_{n}^{(d)}(x,y)=(k/n)xy$ for serially independent $\varepsilon_t$, but $\Lambda_{n}^{(d)}(x,y)\neq (k/n)xy$ when $\varepsilon_t$ follows a stochastic volatility model \citep{Hil11a} or when $(|\varepsilon_t|, |\varepsilon_{t-d}|)^\operatorname{P}rime$ possesses a Gaussian copula. Thus, the PA-tail copula captures finer differences in pre-asymptotic levels of extremal dependence than the tail copula, ensuring that our test has power against more subtle misspecifications. For $x=y=1$, the PA-tail copula is the perhaps most popular version of the \textit{PA-extremogram}, due to \citet{DM09}. Even though the PA-tail copula differs from the tail copula, it may also be estimated via $\widetilde{\Lambda}_{n}^{(d)}(x,y)$.
subsection{A Portmanteau-Type Test for Residual Extremal Dependence}\label{A Portmanteau-Type Test for Residual Extremal Dependence}
In specification testing, one is interested in verifying whether observations $Y_1,\ldots,Y_n$ are generated by some parametric model \eqref{eq:ls model}. Formally, one would like to test
\[
H_0\ :\ \{Y_t\}_{t=1,2,\ldots}\ \text{are generated by the parametric model \eqref{eq:ls model}.}
\]
Typically, this is done by fitting the specific model \eqref{eq:ls model} to the $Y_1,\ldots,Y_n$. Under $H_0$, the residuals should then be approximately i.i.d. This is verified by testing some implication of the i.i.d.~property. For instance, Ljung--Box-type tests focus on the implication that the autocorrelation function is identically zero. As pointed out in the Motivation, we test the implication of $H_0$ that the residuals display no serial \textit{extremal} dependence. To this end, note that the PA-tail copula in \eqref{eq:PA-tail copula} equals $(k/n)xy$ under serial independence. Hence, we test the implication that $\Lambda_n^{(d)}(x,y)=(k/n)xy$ for $d=1,2,\ldots$.
However, for purposes of model checking, the estimator $\widetilde{\Lambda}_{n}^{(d)}(x,y)$ from the previous subsection is infeasible, as the $\varepsilon_t$ are unobserved. Hence, we rely on the feasible counterpart
\[
\widehat{\Lambda}_{n}^{(d)}(x,y)=\frac{1}{k}sum_{t=d+1}^{n}I_{\{|\widehat{\varepsilon}_t|>|\widehat{\varepsilon}|_{(\lfloor kx\rfloor+1)},\ |\widehat{\varepsilon}_{t-d}|>|\widehat{\varepsilon}|_{(\lfloor ky\rfloor+1)}\}}.
\]
With this estimator, we can detect deviations from $\Lambda_n^{(d)}(x,y)=(k/n)xy$ using the Portmanteau-type test statistic
\begin{equation*}
mathcal{P}_n^{(D)}(x,y) = \frac{n}{xy}sum_{d=1}^{D}\Big[\widehat{\Lambda}_n^{(d)}(x,y)-(k/n)xy\Big]^2,
\end{equation*}
where $D\inmathbb{N}$ is some fixed user-specified integer. Once again, the fact that we compare $\widehat{\Lambda}_n^{(d)}(x,y)$ with the null hypothetical value of the PA tail copula $(k/n)xy$ (and not with the null hypothetical value of the tail copula, i.e., $0$) gives our test power in situations, where there is extremal independence (as measured by the tail copula), but not independence (as measured by the PA tail copula).
In choosing $D$, there is the usual trade-off. Using a small $D$ possibly leads to undetected misspecifications at higher lags, resulting in a loss of power. However, choosing $D$ too large, the estimates $\widehat{\Lambda}_{n}^{(D)}(x,y)$ may be based on too few observations, thus distorting size. We explore the choice of $D$ in detail in the simulations in Appendix~\ref{Simulation Results for Varying $D$}. There, we show that $D=5$ typically leads to a good balance between size and power.
Regarding the sequence $k$, we impose
\begin{assumption}\label{ass:k}
For $n\to\infty$, the sequence $k$ satisfies $k\to\infty$, $k/n\to0$, and $k/sqrt{n}\to\infty$.
\end{assumption}
Consistent with the notion of extremal dependence, the requirement that $k/n\to0$ ensures that only a vanishing fraction of the residuals is used in estimation. On the other hand, sufficiently many extremes need to be used, as specified by $k/sqrt{n}\to\infty$. In the related context of estimating the \textit{tail event correlation}
\[
r_n^{(d)}=\frac{n}{k}\left[\operatorname{P}\big\{|\varepsilon_{t}|>b(n/k),\ |\varepsilon_{t-d}|>b(n/k)\big\}-\operatorname{P}\big\{|\varepsilon_{t}|>b(n/k)\big\}\operatorname{P}\big\{|\varepsilon_{t-d}|>b(n/k)\big\}\right],
\]
\citet[Sec.~5.2]{Hil11} even requires $k/n^{2/3}\to\infty$. Hence, assuming $k/sqrt{n}\to\infty$ may be regarded as a rather mild requirement.
The following is our first main theoretical result:
\begin{thm}\label{thm:mainresult}
Suppose Assumptions~\ref{ass:U distr 2}--\ref{ass:k} are met under $H_0$. Then, for any $x>0$ and $y>0$, as $n\to\infty$,
\[
mathcal{P}_n^{(D)}(x,y)\overset{d}{\longrightarrow}\chi^2_{D},
\]
where $\chi^2_{D}$ denotes the $\chi^2$-distribution with $D$ degrees of freedom.
\end{thm}
We defer the proof of Theorem~\ref{thm:mainresult} to Appendix~\ref{Proof of Theorem 1}. The proof reveals that the same $\chi^2$-limit as in Theorem~\ref{thm:mainresult} is obtained, if the $\widehat{\varepsilon}_t$'s are replaced with the true $\varepsilon_t$'s in $mathcal{P}_n^{(D)}(x,y)$. Thus, the effects of parameter estimation in $\widehat{\varepsilon}_t=\widehat{\varepsilon}_t(\widehat{\bm \theta})$ vanish asymptotically. In the context of residual-based estimation of univariate extremal quantities (such as the tail index or large quantiles), such vanishing estimation effects have been observed before by, e.g., \citet{Cea07}, \citet{Hil15}, and \citet{Hog18+}. The explanation is that the convergence rate of the parameter estimator is faster than that of the tail quantity, leading to vanishing parameter estimation effects.
For $x=y=1$ the PA-tail copula is the most prominent version of \citeauthor{DM09}'s \citeyearpar{DM09} PA-extremogram. Thus, the choice $x=y=1$ is the leading case for applications of $mathcal{P}_n^{(D)}(x,y)$. However, other choices of $x$ and $y$ may lead to different test results. Thus, the next section derives a test that combines the evidence across different values of $x$ and $y$. This may also lead to more powerful specification tests, as we will see in the simulations.
subsection{A Functional Portmanteau-Type Test}
To obtain powerful functional versions of $mathcal{P}_n^{(D)}(x,y)$, we need to integrate it over a suitable area in the $(x,y)^\operatorname{P}rime$-space. Observe that $\widehat{\Lambda}_n^{(d)}(x,y)$ estimates $\Lambda_n^{(d)}(x,y)$, which converges to the tail copula $\Lambda^{(d)}(x,y)$ (if it exists) for $n\to\infty$. From Theorem~1~(ii) of \citet{SS06}, we know that the tail copula is homogeneous, i.e., it satisfies $\Lambda^{(d)}(tx,ty)=t\Lambda^{(d)}(x,y)$ for all $t>0$. This suggests that $\widehat{\Lambda}_n^{(d)}(tx,ty)\approx t\widehat{\Lambda}_n^{(d)}(x,y)$, which implies that integrating over a larger space than some sphere $S_{c,\ \norm{\cdot}}:=\{(x,y)^\operatorname{P}rime\in[0,\infty)^2\ :\ \norm{(x,y)^\operatorname{P}rime}=c\}$ does not provide us with more information against the null. Here, $c>0$ is a constant and $\norm{\cdot}$ some norm on $mathbb{R}^2$. A typical choice in extreme value theory is the $\norm{\cdot}_1$-norm. Since---as pointed out above---putting $x=y=1$ is a natural choice and $\norm{(1,1)^\operatorname{P}rime}_1=2$, we fix $c=2$ and, thus, integrate over the sphere $S_{2,\ \norm{\cdot}_1}$. Using the $\norm{\cdot}_1$-norm in defining the sphere also has the added advantage of giving a limiting distribution in Theorem~\ref{thm:mainresult2} in terms of simple Brownian bridges.
These considerations lead to the functional Portmanteau-type test statistic
\[
mathcal{F}_n^{(D)}=nsum_{d=1}^{D}\int_{[\iota, 1-\iota]}\Big[\widehat{\Lambda}_n^{(d)}(2-2z,2z)-\frac{k}{n}(2-2z)2z\Big]^2\,mathrm{d} z,
\]
where $\iota\in(0,1/2)$ is some typically small constant chosen by the practitioner. The parameter $\iota$ serves to bound the $z$-values in the integral away 0 and 1, where our non-parametric estimator $\widehat{\Lambda}_n^{(d)}(2-2z,2z)$ may be unreliable as very extreme quantile thresholds are considered, above which hardly any observations lie. We mention that there is a trade-off between size and power involved in choosing $\iota$. A large value of $\iota$ leads to a smaller area being integrated over, thus leading to compromised power, yet estimates tend to be more stable, thus improving size. However, unreported simulations show that the differences in size and power are not large. In our numerical experiments, we choose $\iota=0.1$ and compute the integral in $mathcal{F}_n^{(D)}$ using standard numerical integration techniques, which allow to approximate integrals with any required accuracy.
\begin{thm}\label{thm:mainresult2}
Suppose Assumptions~\ref{ass:U distr 2}--\ref{ass:k} are met under $H_0$. Then, as $n\to\infty$,
\[
mathcal{F}_n^{(D)}\overset{d}{\longrightarrow}4sum_{d=1}^{D}\int_{[\iota,1-\iota]}B_d^2(z)\,mathrm{d} z,
\]
where $\{B_d(\cdot)\}_{d=1,\ldots,D}$ are mutually independent Brownian bridges.
\end{thm}
Theorem~\ref{thm:mainresult2} is proven in Appendix~\ref{Proof of Theorem 2}. While the limiting distribution in Theorem~\ref{thm:mainresult2} is non-standard, it is free of nuisance parameters. Thus, critical values can be computed to any desired degree of precision by simulating sufficiently often from the limit. Table~\ref{tab:cv} shows some selected critical values for $\iota=0.1$ and different numbers of lags $D$.\footnote{The critical values in Table~\ref{tab:cv} are based on 4,000,000 replications of a Brownian bridge simulated on 100,000 grid points.}
\begin{rem}
In the spirit of \citet{AD52}, it is also possible to consider a weighted version of the functional test statistic \`{a} la
\begin{equation}\label{eq:lim WF}
mathcal{WF}_n^{(D)}=nsum_{d=1}^{D}\int_{[\iota, 1-\iota]}\operatorname{P}si(z)\Big[\widehat{\Lambda}_n^{(d)}(2-2z,2z)-\frac{k}{n}(2-2z)2z\Big]^2\,mathrm{d} z\overset{d}{\longrightarrow}4sum_{d=1}^{D}\int_{[\iota,1-\iota]}\operatorname{P}si(z)B_d^2(z)\,mathrm{d} z,
\end{equation}
where $\operatorname{P}si(z)\geq0$ is a weight function satisfying some regularity conditions (e.g., continuity). The weight function allows to put more weight on certain $z$-values; e.g., $\operatorname{P}hi(z)=z(1-z)$ would accentuate $z$ that are further away from the boundaries. We refrain from exploring such possibilities. However, we mention that for $\operatorname{P}si(z)\equiv1/4$, $\iota=0$ and $D=1$, the limit in \eqref{eq:lim WF} equals the limit of the well-known Cram\'{e}r--von Mises goodness-of-fit test. For $\operatorname{P}si(z)=1/[z(1-z)]$, we obtain the limit of the Anderson--Darling test statistic.
\end{rem}
\begin{table}[!t]
\centering
\begin{tabular}{lrrrrrrrrrr}
\toprule
$\alpha$ & multicolumn{10}{c}{$D$}\\[0.25ex]
\cline{2-11}\\[-2.25ex]
& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
midrule
10\% & 1.340 & 2.336 & 3.231 & 4.077 & 4.896 & 5.694 & 6.477 & 7.249 & 8.011 & 8.766 \\
5\% & 1.791 & 2.890 & 3.859 & 4.765 & 5.636 & 6.480 & 7.306 & 8.117 & 8.916 & 9.705 \\
1\% & 2.905 & 4.178 & 5.273 & 6.286 & 7.248 & 8.178 & 9.082 & 9.964 & 10.832 & 11.683 \\
\bottomrule
\end{tabular}
\caption{Critical values $c_{\alpha}^{(D)}$ for significance level $\alpha\in(0,1)$ and $D$ for $\iota=0.1$ in the limiting distribution in Theorem~\ref{thm:mainresult2}.}
\label{tab:cv}
\end{table}
section{Simulations}\label{Simulations}
Here, we investigate the size and power of the specification tests based on $mathcal{P}_n^{(D)}$ (with the standard choice $x=y=1$) and $mathcal{F}_n^{(D)}$, and compare the results with those of classical Ljung--Box tests based on the autocorrelations of the squared residuals. We do so for different samples sizes $n\in\{500,\ 1000,\ 2000\}$ and different significance levels $\alpha\in\{1\%,\ 5\%,\ 10\%\}$. Furthermore, we pick $\iota=0.1$. Other choices of $\iota$ (e.g., $\iota=0.05$ or even $\iota=0.01$) do not materially change the results. All simulations use \textit{R version 4.0.3} (\citealp{R4.0.3}) and results are based on 10,000 replications.
The choice of $D$ is common to all tests and we pick $D=5$ in the following. This choice is supported by simulations in Appendix~\ref{Simulation Results for Varying $D$}, where we investigate the influence of $D$ on the tests. Here, we instead explore the effect of different $k$'s on our test statistics $mathcal{P}_n^{(D)}$ and $mathcal{F}_n^{(D)}$. In applications of extreme value methods the choice of $k$ is often a tricky issue, because the results may be very sensitive to the specific value of $k$ \citep{HF06}. Therefore, it is of interest to investigate the robustness of our test with respect to $k$. A very popular choice in applications of extreme value theory is to follow \citeauthor{DuM83}'s \citeyearpar{DuM83} rule by setting $k=\lfloor 0.1\cdot n\rfloor$ \citep{QFP01,MF00,MT11,CES14}. However, at least asymptotically, this is not a valid choice (here and elsewhere) since it fails the Assumption~\ref{ass:k} requirement that $k$ be a vanishing fraction of $n$ ($k/n\to0$). Nonetheless, its widespread use suggests some merit in finite samples. Thus, we consider $k=\lfloor\varrho n^{0.99}\rfloor$ for $\varrho\in[0.05,\, 0.15]$, where the choice $\varrho=0.11$ roughly corresponds to \citeauthor{DuM83}'s \citeyearpar{DuM83} rule. Note that $k=\lfloor\varrho n^{0.99}\rfloor$ satisfies both Assumption~\ref{ass:k} requirements.
For simplicity, we only consider model \eqref{eq:ls model} with zero conditional mean, $mu_t(\bm \theta^{\circ})\equiv0$. We do so, because unmodeled dynamics in the tails of the conditional distribution are most likely caused by misspecified volatility dynamics in practice. Throughout, we simulate from APARCH(1,1) models under the null of correct specifications, so that in particular our high-level Assumption~\ref{ass:UA} is satisfied; see Appendix~\ref{APARCH and ARMA--GARCH Models}.
We compare our tests with classical Ljung--Box tests when volatility dynamics are misspecified (Section~\ref{Misspecified Volatility}) and for non-i.i.d.~$\varepsilon_t$ (Section~\ref{Misspecified Innovations}). In both cases, we estimate our models using the \texttt{rugarch} package (\citealp{rugarch}).
subsection{Misspecified Volatility}\label{Misspecified Volatility}
We generate time series from an APARCH model with an exogenous covariate $x_t$ in the volatility equation. Specifically, we simulate from the APARCH--X(1,1) model with $\delta^{\circ}=1$ given by
\begin{align}
Y_t &= sigma_t(\bm \theta^{\circ}) \varepsilon_t,\qquad\varepsilon_t\overset{\text{i.i.d.}}{sim}(0,1)\notag\\
sigma_t(\bm \theta^{\circ}) &=\omega^{\circ} + \alpha_{+,1}^{\circ}(Y_{t-1})_{+}
+ \alpha_{-,1}^{\circ}(Y_{t-1})_{-}
+ \beta_{1}^{\circ}sigma_{t-1}(\bm \theta^{\circ})
+ \operatorname{P}i_{1}^{\circ}x_{t-1}.\label{eq:APARCH-X(1,1)}
\end{align}
Here, the $\varepsilon_t$ follow a (standardized) Student's $t$-distribution with $4.1$ degrees of freedom to ensure $\operatorname{E}|\varepsilon_t^4|<\infty$, which is required for $sqrt{n}$-consistent QML estimation in Assumption~\ref{ass:estimator} \citep{FT19}. The exogenous $x_t$ are stationary with
\begin{align*}
x_{t}&=\exp(z_t), \\
z_t &=0.9\cdot z_{t-1}+e_t,\qquad e_t\overset{\text{i.i.d.}}{sim}N(0,1).
\end{align*}
For the parameters, we take $\bm \theta^{\circ}:=\bm \theta^{\circ}_{s}:=(\omega^{\circ}, \alpha_{+,1}^{\circ}, \alpha_{-,1}^{\circ}, \beta_{1}^{\circ},\operatorname{P}i_{1}^{\circ})^\operatorname{P}rime=(0.046, 0.027, 0.092, 0.843, 0)^\operatorname{P}rime$ (to compute size) and $\bm \theta^{\circ}:=\bm \theta^{\circ}_{p}:=(0.046, 0.027, 0.092, 0.843, 0.089)^\operatorname{P}rime$ (to compute power).\footnote{The data-generating process is taken from \citet[Sec.~3.1]{FT19}, who obtained the parameters in \eqref{eq:APARCH-X(1,1)} from fitting an APARCH--X model to Boeing returns.} Irrespective of the choice of $\bm \theta^{\circ}$, we estimate an APARCH(1,1) model (i.e., we estimate \eqref{eq:APARCH-X(1,1)} imposing $\operatorname{P}i_{1}^{\circ}=0$) via QML. Thus, when $\bm \theta^{\circ}=\bm \theta^{\circ}_{s}$ we fit a correctly specified model (yielding size), yet when $\bm \theta^{\circ}=\bm \theta^{\circ}_{p}$ the fitted model is misspecified (yielding power). In the latter case, the exogenous variable serves to introduce some unmodeled dynamics in the variance equation, with infrequent outbursts of $z_t$ inducing some dependence in the tails of the residuals.\footnote{The second-to-top panel of Figure~\ref{fig:SimTS} in Appendix~\ref{Simulation Results for Varying $D$} shows a representative trajectory of $z_t$.} Hence, our simulation setup is well-suited to $mathcal{P}_n^{(D)}$ and $mathcal{F}_n^{(D)}$ and, at the same time, relevant in practice as exogenous volatility shocks are often empirically plausible; see \citet{HK14} for references.
For $v=10$, we generate $\{Y_t\}_{t=-v+1,\ldots,n}$ from \eqref{eq:APARCH-X(1,1)}. Since volatility estimates $\widehat{sigma}_{t}(\widehat{\bm \theta})$ may be imprecise for the first few $t$ due to initialization effects in the variance equation \citep[see also][]{HY03}, we discard the first $v$ standardized residuals and only consider $\{\widehat{\varepsilon}_t=Y_t/\widehat{sigma}_{t}(\widehat{\bm \theta})\}_{t=1,\ldots,n}$ in our test statistics $mathcal{P}_n^{(D)}$ and $mathcal{F}_n^{(D)}$.
Since we consider $k=\lfloor\varrho n^{0.99}\rfloor$ with $\varrho\in[0.05,\, 0.15]$, the $k$'s are between $[23, 70]$ / $[46, 139]$ / $[92, 278]$ for $n=500$ / $n=1000$ / $n=2000$. These values may be regarded as being sufficiently small relative to the sample size for extreme value methods to be applicable.
For purposes of comparison, we also include the results of a classical Ljung--Box test (with test statistic denoted by $\operatorname{LB}_{n}^{(D)}$) based on the first $D$ autocorrelations of the squared residuals. We employ the corrected Ljung--Box test statistic of \citet{CF11}, which ensures that $\operatorname{LB}_n^{(D)}$ is asymptotically $\chi_D^2$-distributed. Note that this correction is specific to APARCH models estimated via a Gaussian QML estimator.
\begin{figure}
\caption{Rejection frequencies at the 5\%-level under the null and alternative for $mathcal{P}
\label{fig:sev k}
\end{figure}
Figure~\ref{fig:sev k} displays the rejection frequencies of all tests at the 5\%-level. Both the $mathcal{P}_n^{(D)}$-test (black) and the $mathcal{F}_n^{(D)}$-test (red) have almost identical size, which---as the sample size increases---converges rapidly to the nominal level for all $k$. Larger differences in power between the two tests only emerge for large $n$. We also see that size and power are reasonably stable across different choices of $k$, suggesting that test results are quite robust to the particular value of $k$. This is encouraging given that extreme value methods can be very sensitive to the choice of the cutoff. Our approximation $k=\lfloor0.11\cdot n^{0.99}\rfloor$ to \citeauthor{DuM83}'s \citeyearpar{DuM83} rule---indicated by the dashed vertical lines in Figure~\ref{fig:sev k}---yields good results in terms of both size and power. Note that for $n=500$, where at first sight our heuristic does not lead to optimal power, size and power decrease with increasing $k$, such that the size-corrected power of our proposal for the choice of $k$ appears very close to optimal.
Finally, we compare our two tests with the $\operatorname{LB}_n^{(D)}$-test (green), which is of course independent of $k$. We find that---despite rejecting more often under the null---the $\operatorname{LB}_n^{(D)}$-test rejects less often under the alternative. Hence, the size-corrected power of our tests is even larger than the differences in the right-hand panels of Figure~\ref{fig:sev k} suggest.
subsection{Misspecified Innovations}\label{Misspecified Innovations}
Standard location--scale models as in \eqref{eq:ls model} can only dynamically model the conditional mean and the conditional variance. However, higher-order features of the conditional distribution---such as skewness and kurtosis---may sometimes change as well for real data \citep{Han94,JR03a,BMT08}. Here, we assess how our tests perform relative to a standard Ljung--Box test in such a setting. To account for estimation effects, we use the version of the Ljung--Box test based on Theorem~8.2 of \citet{FZ10}. Since no confusion can arise, we denote the corresponding test statistic by $\operatorname{LB}_n^{(D)}$, which again has a standard $\chi_{D}^2$-limit.
To carry out the comparison, we consider the following GARCH(1,1) model
\begin{align}
Y_t &= sigma_t(\bm \theta^{\circ}) \varepsilon_t,\qquad\varepsilon_tsim(0,1),\notag\\
sigma_t^2(\bm \theta^{\circ}) &=\omega^{\circ} + \alpha^{\circ}Y_{t-1}^2 + \beta^{\circ}sigma_{t-1}^2(\bm \theta^{\circ})\label{eq:GARCH(1,1)}
\end{align}
with $\bm \theta^{\circ}:=(\omega^{\circ}, \alpha^{\circ}, \beta^{\circ})^\operatorname{P}rime=(0.046, 0.127, 0.843)^\operatorname{P}rime$. We assume that $\varepsilon_t$ is distributed according to \citeauthor{Han94}'s \citeyearpar{Han94} skewed $t$-distribution (written: $\varepsilon_tsim st_{\lambda,\eta}$), i.e., it has density
\[
f_{\lambda,\eta}(x)=bc\Bigg[1+\frac{1}{\eta-2}\Big(\frac{bx+a}{1+\operatorname{sgn}(x+a/b)\lambda}\Big)^2\Bigg]^{-\frac{\eta+1}{2}},\qquad x\in(-\infty,\infty),\ \eta\in(2,\infty),\ \lambda\in(-1,1),
\]
where
\[
a=4\lambda c\frac{\eta-2}{\eta-1},\qquad b^2=1+3\lambda^2 - a^2,\qquad c=\frac{\Gamma((\eta+1)/2)}{sqrt{\operatorname{P}i(\eta-2)} \Gamma(\eta/2)}.
\]
For $\lambda=0$, $f_{\lambda,\eta}(\cdot)$ reduces to the standard $t$-distribution with degrees of freedom equal to $\eta$. \citet[Appendix~A]{JR03a} show that $f_{\lambda,\eta}(\cdot)$ has zero mean and unit variance. In their equations (2) and (3), they also derive formulas for the skewness and kurtosis as functions of $\lambda$ and $\eta$.
To introduce time variation in the skewness and kurtosis, we let $\lambda=\lambda_t$ and $\eta=\eta_t$ vary over time. To do so, we use Model M4 of \citet[Table~1]{JR03a}. That is, we let
\begin{align*}
\widetilde{\eta}_t &= a_1+b_1Y_{t-1}+c_1\widetilde{\eta}_{t-1},\\
\widetilde{\lambda}_t &= a_2+b_2Y_{t-1}+c_2\widetilde{\lambda}_{t-1},
\end{align*}
and restrain $\eta_t$ and $\lambda_t$ to the intervals $(2,30)$ and $(-1, 1)$ by the logistic transformations $\eta_t=g_{(2, 30)}(\widetilde{\eta}_t)$ and $\lambda_t=g_{(-1, 1)}(\widetilde{\lambda}_t)$, where $g_{(L,U)}(x)=L+(U-L) / [1+\exp(-x)]$. Under the null, we set $(a_1,b_1,c_1)^\operatorname{P}rime=(-3, 0, 0)^\operatorname{P}rime$ and $(a_2,b_2,c_2)^\operatorname{P}rime=(-1, 0, 0)^\operatorname{P}rime$, leading to constant $\eta_t\equiv28.67...$ and $\lambda_t\equiv0.45...$. Thus, the $\varepsilon_t$ are serially independent and $H_0$ is satisfied. Under the alternative, we introduce time-variation in the skewness and kurtosis by using non-zero $b_i$ and $c_i$ ($i=1,2$). Specifically, we let $(a_1,b_1,c_1)^\operatorname{P}rime=(-3, -6, 0.6)^\operatorname{P}rime$ and $(a_2,b_2,c_2)^\operatorname{P}rime=(-1, -2, 0.6)^\operatorname{P}rime$. Thus, while $\varepsilon_tsim(0,1)$, the $\varepsilon_t$ have time-varying skewness and kurtosis and, thus, are not i.i.d., as required under the null.
To estimate the model parameters, we again use Gaussian QML. \citet[Theorem~2]{Esc09} shows that the QML estimator is asymptotically normal under both the null and the alternative.
\begin{figure}
\caption{Rejection frequencies at the 5\%-level under the null and alternative for $mathcal{P}
\label{fig:sev k1}
\end{figure}
Figure~\ref{fig:sev k1} displays the results. For all tests, size is very close to the nominal level, such that power is directly comparable. While for the $\operatorname{LB}_n^{(D)}$-test power is low and even decreases below the nominal level for $n=2000$, the power of our tests is high and approaches one. The huge difference in power may be explained as follows. The $\operatorname{LB}_n^{(D)}$-test only looks at the autocorrelations of the squared residuals $\widehat{\varepsilon}_t^2$. However, the $\varepsilon_t^2$ satisfy $\operatorname{E}[\varepsilon_t^2midmathcal{F}_{t-1}]=1$, such that---despite being serially dependent---they have zero autocorrelations. Hence, the power of the $\operatorname{LB}_n^{(D)}$-test is seriously impaired. Comparing the $mathcal{P}_n^{(D)}$- and the $mathcal{F}_n^{(D)}$-test, we find that the latter has higher power for small $k$, yet the difference becomes negligible for larger $k$. For these two tests, the results are now more sensitive to the choice of $k$ than in Figure~\ref{fig:sev k}. Yet, our suggestion $k=\lfloor0.11\cdot n^{0.99}\rfloor$ once again leads to good size and power.
To summarize the simulation results, we find that size and power of our tests may depend somewhat on the choice of $k$. However, setting $k=\lfloor0.11\cdot n^{0.99}\rfloor$, which approximates \citeauthor{DuM83}'s \citeyearpar{DuM83} rule, gives good results---irrespective of the type of misspecification. Although our tests work well for $k=\lfloor0.11\cdot n^{0.99}\rfloor$, in practice we recommend to apply the tests for several values of $k$ as a `robustness check' (see Figure~\ref{fig:Fig6} in the empirical application for an example), much like results for standard Ljung--Box tests are also routinely reported for several lags. Our tests also depend on the number of included lags and, although we find $D=5$ to lead to good results in Appendix~\ref{Simulation Results for Varying $D$}, we again recommend---if possible---to report results for different $D$. We generally find that the $mathcal{F}_n^{(D)}$-test is more powerful than the $mathcal{P}_n^{(D)}$-test, yet both have markedly higher power than the $\operatorname{LB}_n^{(D)}$-test for the types of misspecifications considered here. As we have argued, these misspecifications may be empirically plausible, such that our tests should have substantial merit in practice. We investigate this next.
section{Diagnosing S\&P~500 Constituents}\label{Application}
Consider the 500 components of the Standard \& Poor's 500 (S\&P~500) as of 31/5/2021. We apply the $mathcal{F}_n^{(D)}$- and $\operatorname{LB}_n^{(D)}$-test to the log-returns of each of the constituents. We do so for the recommended values $D=5$, $k=\lfloor 0.11\cdot n^{0.99}\rfloor$ and $\iota=0.1$. For brevity, we do not report results for $mathcal{P}_n^{(D)}$, which showed slightly inferior performance to $mathcal{F}_n^{(D)}$ in the simulations. We consider the 495 components of the S\&P~500 for which we have complete data for our sample period from 1/1/2010 to 31/12/2019. We split the 10-year sample into an `in-sample' period of 8 years (used for specification testing) and an `out-of-sample' period of 2 years (used for backtesting VaR forecasts). The goal of our analysis is twofold. First, we want to illustrate that (similarly as in the simulations) cases may arise where $mathcal{F}_n^{(D)}$ warrants a rejection, but $\operatorname{LB}_{n}^{(D)}$ does not. Our second goal is to show that in these cases, a rejection by $mathcal{F}_n^{(D)}$ is not spurious, but---on the contrary---indicative of `out-of-sample' forecast failure of the model.
Specifically, we fit an APARCH(1,1) model without covariates as in \eqref{eq:APARCH-X(1,1)} (i.e., with $\operatorname{P}i_1^{\circ}=0$) based on the `in-sample' period, and use the fitted model for one-step-ahead VaR forecasting in the `out-of-sample' part. The VaR at level $\theta$ is the loss in $(t+1)$ that is only exceeded with probability $\theta$ given the current state of the market (embodied by some information set $mathcal{F}_t$). Formally, $\operatorname{VaR}_t$ is the $mathcal{F}_{t}$-measurable random variable satisfying
\[
\operatorname{P}\{Y_{t+1}\leq\operatorname{VaR}_tmidmathcal{F}_t\} = \theta.
\]
It immediately follows from the multiplicative structure in \eqref{eq:APARCH-X(1,1)} that $\operatorname{VaR}_t=sigma_{t+1}\operatorname{VaR}_{\varepsilon}$, where $\operatorname{VaR}_{\varepsilon}$ is the (unconditional) $\theta$-quantile of the $\varepsilon_t$. Thus, we can easily compute the `out-of-sample' VaR forecasts from the volatility forecasts of the fitted model (say, $\widehat{sigma}_{t+1}$) and an estimate of $\operatorname{VaR}_{\varepsilon}$ from the standardized residuals of the `in-sample' period. Then, we backtest the VaR forecasts using the DQ test of \citet{EM04}.\footnote{We use the DQ test as implemented in the \texttt{GAS} package (\citealp{GAS}) with 4 lags.}
\begin{table}[t!]
\begin{center}
\begin{tabular}{cccc}
\toprule
$\operatorname{LB}_{n}^{(D)}$ & $mathcal{F}_{n}^{(D)}$ & Total & DQ-test \\
midrule
0 & 0 & 181 & 60 \\
0 & 1 & 122 & 69 \\
1 & 0 & 113 & 31 \\
1 & 1 & 79 & 38 \\
\bottomrule
\end{tabular}
\end{center}
\caption{\label{tab:app}Results of specification and DQ tests carried out at 5\%-level.}
\end{table}
As we run two specification tests, there are four potential outcomes because the $\operatorname{LB}_{n}^{(D)}$- and the $mathcal{F}_{n}^{(D)}$-test can either accept (indicated by a 0 in Table~\ref{tab:app}) or reject (indicated by a 1 in Table~\ref{tab:app}). The column `Total' in Table~\ref{tab:app} indicates the total number of times each of the four cases occurs for the 495 S\&P~500 stocks. In 181 out of 495 cases, none of the tests reject. For these 181 stocks, the `out-of-sample' VaR forecasts are rejected 60 times by the DQ-test. One reason for this rather large number of rejections may be the long out-of-sample period of 2 years. Typically, models are re-estimated at least quarterly \citep{AH14}, yet we only fitted the model once based on the `in-sample' data. Nonetheless, the rather long out-of-sample period of 2 years (yielding roughly 500 daily observations) is needed for the DQ-test to have reasonable power. Table~\ref{tab:app} further shows that for 122 stocks, $mathcal{F}_{n}^{(D)}$ but not $\operatorname{LB}_{n}^{(D)}$ leads to a rejection. The fact that for these 122 stocks the DQ test rejects in 69 cases suggests some unmodeled dynamics in these stocks, that were picked up by our $mathcal{F}_{n}^{(D)}$-test, yet not the Ljung--Box test. When both tests agree in rejecting a model (which happens in 79 cases), the model produces inadequate VaR forecasts 38 times. For the remaining 113 stocks, only $\operatorname{LB}_{n}^{(D)}$ rejects the null, yet the DQ-test rejects only 31 times.
\begin{figure}
\caption{Top: Cisco returns ($Y_t$) in black and estimated volatility ($\widehat{sigma}
\label{fig:MRKy}
\end{figure}
Overall, we find that in most of the 198 cases when the DQ-test rejects, at least one of the specification test signals problems in advance. Of these 138 cases, 107 cases are identified by our $mathcal{F}_n^{(D)}$-test, yet the $\operatorname{LB}_n^{(D)}$-test rejects in only 69 of these cases. This suggests that our extreme value-based tests provide complementary information to more classical specification tests.
\begin{figure}
\caption{Top: Estimates $\widehat{\rho}
\label{fig:MRKacf}
\end{figure}
Next, we illustrate one of the cases where $\operatorname{LB}_{n}^{(D)}$ does not reject, yet $mathcal{F}_{n}^{(D)}$ does. We do so for Cisco log-returns, shown in the top panel of Figure~\ref{fig:MRKy}. Volatility estimates are superimposed in red. During our in-sample period from 2010-2017 there are only mild signs of volatility clustering. A cursory inspection of the residuals in the lower panel suggests that the model successfully captures this. This is also confirmed by the plots of the squared residual autocorrelations in the top part of Figure~\ref{fig:MRKacf}, which shows that autocorrelations up to lag 10 are insignificant. However, focusing on the extremes, we find that the $\widehat{\Lambda}_n^{(d)}(1,1)$-estimates in Figure~\ref{fig:MRKacf} show strong (positive) serial extremal dependence at all lags. This may be due to occasional exogenous shocks to the share price. For instance, the largest drop on 11/11/2010 followed a profit warning of Cisco. This and other outliers apparent from the plot of the $Y_t$ in Figure~\ref{fig:MRKy} seem to have been unpredictable from past information in $mathcal{F}_{t-1}$, thus violating the modeling paradigm in \eqref{eq:ls model}. Similarly as for model \eqref{eq:APARCH-X(1,1)} in the simulations, such exogenous shocks may have caused the serial extremal dependence in the standardized residuals.
\begin{figure}
\caption{$mathcal{F}
\label{fig:Fig6}
\end{figure}
The results for the S\&P~500 components of this section are for $k=\lfloor0.11\cdot n^{0.99}\rfloor$. For a single time series one may want to assess the robustness of the result with respect to $k$. To do so, it is common in extreme value theory to plot the results as a function of $k$. We illustrate this for the Cisco returns in Figure~\ref{fig:Fig6}. There, the test statistics $mathcal{F}_{n}^{(D)}$ and $mathcal{P}_{n}^{(D)}$ are plotted as functions of $k$. As in the simulations, the test result is quite robust to $k$ with any reasonable choice leading to a rejection.
section{Conclusion}\label{Conclusion}
We propose two specification tests based on the tail copula of time series residuals. By relying on the tail copula, our tests direct the power to any remaining serial \textit{extremal} dependence and, thus, complement the evidence by more classical Ljung--Box-type tests. The test statistics are easy to compute and their limiting distributions are free of nuisance parameters. Thus, our tests are simple to implement and enjoy broad applicability. This contrasts with more classical diagnostic tests, which require involved bootstrap procedures for valid inference and/or depend on the specific estimator used to fit the model, hence, limiting practical applications. Simulations demonstrate the good size of our proposals. They also show that when a misspecified model captures well the serial dependence in the body of the distribution but not in the tail, our tests have higher power than Ljung--Box tests. The misspecified models considered in the simulations are realistic in that they display exogenous shocks in the volatility equation or unmodeled higher-order dynamics in the innovations, both of which being empirically relevant sources of misspecification \citep{Han94,HK14}. We also exemplify the better detection properties of our tests on S\&P~500 constituents, where a rejection of our test is a more reliable indicator of poor out-of-sample risk predictions. Thus, our tests help to identify unsuitable risk forecasting models in advance, which is economically desirable under the Basel framework \citep{BCBSBF19}. Consequently, by focusing on the extremes, our tests are useful complements to standard specification tests, where valuable information on the serial extremal dependence may be `washed-out'.
\begin{center}
{\large\bf SUPPLEMENTARY MATERIAL}
\end{center}
\begin{description}
\item[Title:] Appendix containing verification of Assumption~\ref{ass:UA} for APARCH and ARMA--GARCH models (Appendices~\ref{APARCH and ARMA--GARCH Models} and \ref{Proof of Theorem}), the proofs of Theorems~\ref{thm:mainresult}--\ref{thm:mainresult2} (Appendices~\ref{Proof of Theorem 1}
and \ref{Proof of Theorem 2}), and additional simulations (Appendix~\ref{Simulation Results for Varying $D$}). (pdf-file)
\item[R code:] File containing the \texttt{R} code to reproduce the simulation study and the empirical application. (zipped file)
\end{description}
singlespacing
\onehalfspacing
\begin{appendices}
section{APARCH and ARMA--GARCH Models}\label{APARCH and ARMA--GARCH Models}
\renewcommand{D.\arabic{equation}}{A.\arabic{equation}}
setcounter{equation}{0}
setcounter{page}{1}
We first verify Assumption~\ref{ass:UA} for the asymmetric power ARCH (APARCH) model of \citet{DGE93}; see also \citet{PWT08}. To that end, consider
\begin{align}
Y_t &= sigma_t(\bm \theta^\circ) \varepsilon_t,\qquad \varepsilon_t\overset{\text{i.i.d.}}{sim}(0,1),\notag\\
sigma_t^{\delta^{\circ}}(\bm \theta^\circ) &= \omega^{\circ}+sum_{j=1}^{p}\left\{\alpha_{+,j}^{\circ}(Y_{t-j})_{+}^{\delta^{\circ}}+\alpha_{-,j}^{\circ}(Y_{t-j})_{-}^{\delta^{\circ}}\right\}+sum_{j=1}^{q}\beta_j^{\circ}sigma_{t-j}^{\delta^{\circ}}(\bm \theta^\circ),\label{eq:APARCH}
\end{align}
where $y_{+}=max\{y,0\}$, $y_{-}=max\{-y,0\}$, $\omega^{\circ}>0$, $\delta^{\circ}>0$, and $\alpha_{+,j}^{\circ}$, $\alpha_{-,j}^{\circ}$ and $\beta_{j}^{\circ}$ are non-negative. Note that $mu_t(\bm \theta^{\circ})\equiv0$ here. As the likelihood is typically quite flat in the $\delta^{\circ}$-direction, we follow \citet{FT19} and assume the power $\delta^{\circ}$ to be a fixed known constant.\footnote{\citet[Sec.~2.5]{FT19} provide some theoretically-backed guidance on the choice of $\delta^{\circ}$ in practice.} Thus, the vector of the true unknown parameters is $\bm \theta^{\circ}=(\omega^{\circ},\alpha_{+,1}^{\circ},\ldots,\alpha_{+,p}^{\circ},\alpha_{-,1}^{\circ},\ldots,\alpha_{-,p}^{\circ},\beta_1^{\circ},\ldots,\beta_{q}^{\circ})^\operatorname{P}rime$. A generic parameter vector from the parameter space $\bm \varThetasubset(0,\infty)\times[0,\infty)^{d-1}$ ($d=2p+q+1$) is denoted by
\[
\bm \theta=(\omega,\alpha_{+,1},\ldots,\alpha_{+,p},\alpha_{-,1},\ldots,\alpha_{-,p},\beta_1,\ldots,\beta_{q})^\operatorname{P}rime.
\]
The APARCH($p,q$) model in \eqref{eq:APARCH} nests several popular GARCH variants. We obtain \citeauthor{Zak94}'s \citeyearpar{Zak94} TARCH model for $\delta^{\circ}=1$, and the GJR--GARCH of \citet{GJR93} for $\delta^{\circ}=2$. When $\delta^{\circ}=2$ and $\alpha_{+,j}^{\circ}=\alpha_{-,j}^{\circ}$, \eqref{eq:APARCH} is the classic GARCH specification of \citet{Bol86}.
As volatility depends on the infinite past in the APARCH model, we rely on the truncated recursion
\begin{equation}\label{eq:vola}
\widehat{sigma}_t^{\delta^{\circ}}(\bm \theta)=\omega+sum_{j=1}^{p}\left\{\alpha_{+,j}(Y_{t-j})_{+}^{\delta^{\circ}}+\alpha_{-,j}(Y_{t-j})_{-}^{\delta^{\circ}}\right\}+sum_{j=1}^{q}\beta_j\widehat{sigma}_{t-j}^{\delta^{\circ}}(\bm \theta),\qquad t\geq1,
\end{equation}
with initial values $Y_{1-p}=\ldots=Y_0=0$ and $\widehat{sigma}_{1-q}^{\delta^{\circ}}(\bm \theta)=\ldots=\widehat{sigma}_{0}^{\delta^{\circ}}(\bm \theta)=0$.
\begin{thm}\label{thm:UA}
Suppose for the APARCH model in \eqref{eq:APARCH} that
\begin{enumerate}
\item[(a)] $min\{\omega^{\circ},\alpha_{+,1}^{\circ},\ldots,\alpha_{+,p}^{\circ},\alpha_{-,1}^{\circ},\ldots,\alpha_{-,p}^{\circ},\beta_1^{\circ},\ldots,\beta_{q}^{\circ}\}>0$, $sum_{i=1}^{q}\beta_{i}^{\circ}<1$;
\item[(b)] the top Lyapunov exponent $\gamma$ \citep[p.~41]{FT19} satisfies $\gamma<0$.
\end{enumerate}
Suppose further that Assumption~\ref{ass:estimator} is satisfied. Then, Assumption~\ref{ass:UA} holds for $\widehat{\varepsilon}_t(\bm \theta)=Y_t/\widehat{sigma}_t(\bm \theta)$ with $\widehat{sigma}_t(\bm \theta)$ defined in \eqref{eq:vola}.
\end{thm}
Appendix~\ref{Proof of Theorem} contains the proof of Theorem~\ref{thm:UA}. Condition~\textit{(b)} ensures strict stationarity of the APARCH model, while \textit{(a)} bounds the true parameters away from the boundary of the parameter space to guarantee the approximability of the innovations.
Next, we consider the ARMA--GARCH model
\begin{equation}\label{eq:ARMA-GARCH}
Y_t=sum_{j=1}^{\overline{p}}\operatorname{P}hi_j^{\circ}Y_{t-j}+ X_t-sum_{j=1}^{\overline{q}}\vartheta_j^{\circ}X_{t-j},
\end{equation}
where the $X_t$ follow the GARCH model
\begin{align}
X_t&=sigma_t(\bm \theta^{\circ})\varepsilon_t,\qquad\varepsilon_t\overset{\text{i.i.d.}}{sim}(0,1),\notag\\
sigma_t^2(\bm \theta^{\circ})&=\omega^{\circ}+sum_{j=1}^{p}\alpha_j^{\circ}X_{t-j}^2+sum_{j=1}^{q}\beta_j^{\circ}sigma_{t-j}^2(\bm \theta^{\circ}).\label{eq:GARCH part}
\end{align}
Here, $\bm \theta^{\circ}=(\operatorname{P}hi_1^{\circ}, \ldots, \operatorname{P}hi_{\overline{p}}^{\circ}, \vartheta_1^{\circ}, \ldots, \vartheta_{\overline{q}}^{\circ},\omega^{\circ}, \alpha_1^{\circ},\ldots, \alpha_p^{\circ}, \beta_1^{\circ},\ldots, \beta_q^{\circ})^\operatorname{P}rime$ is the true parameter vector contained is some parameter space $\bm \varTheta=\bm \varTheta_{\operatorname{ARMA}}\times\bm \varTheta_{\operatorname{GARCH}}$, where $\bm \varTheta_{\operatorname{ARMA}}subsetmathbb{R}^{\overline{p}+\overline{q}}$ and $\bm \varTheta_{\operatorname{GARCH}}subset(0,\infty)\times[0,\infty)^{p+q}$. One can show that the model can be rewritten in the form \eqref{eq:ls model} with $mu_t(\bm \theta^{\circ}) = sum_{j=1}^{\overline{p}}\operatorname{P}hi_j^{\circ}Y_{t-j} - sum_{j=1}^{\overline{q}}\vartheta_j^{\circ}[Y_{t-j}-mu_{t-j}(\bm \theta^{\circ})]$. We estimate the GARCH errors for a generic parameter vector $\bm \theta\in\bm \varTheta$ via
\[
\widehat{X}_t(\bm \theta)=Y_t-sum_{j=1}^{\overline{p}}\operatorname{P}hi_j Y_{t-j} + sum_{j=1}^{\overline{q}}\vartheta_j \widehat{X}_{t-j}(\bm \theta),
\]
where we artificially set $\widehat{X}_t(\bm \theta)=Y_t=0$ for $t\leq0$. Based on the $\widehat{X}_t(\bm \theta)$, we approximate volatility by the truncated recursion
\begin{equation}\label{eq:vola2}
\widehat{sigma}_t^2(\bm \theta)=\omega+sum_{j=1}^{p}\alpha_j\widehat{X}_{t-j}^2(\bm \theta)+sum_{j=1}^{q}\beta_j\widehat{sigma}_{t-j}^2(\bm \theta),
\end{equation}
where $\widehat{sigma}_t^2(\bm \theta)=0$ for $t\leq0$.
\begin{thm}\label{thm:UA2}
Suppose for the ARMA--GARCH model in \eqref{eq:ARMA-GARCH} that
\begin{itemize}
\item[(a)] the polynomials $\operatorname{P}hi(z)=1-sum_{j=1}^{\overline{p}}\operatorname{P}hi_j^{\circ} z^{j}$ and $\vartheta(z)=1-sum_{j=1}^{\overline{q}}\vartheta_j^{\circ} z^{j}$ have no common roots and no roots on the unit circle;
\item[(b)] the GARCH parameters satisfy $sum_{j=1}^{p}\alpha_j^{\circ}+sum_{j=1}^{q}\beta_j^{\circ}<1$;
\item[(c)] $\operatorname{E}|X_t|^{2+\delta}<\infty$ for some $\delta>0$.
\end{itemize}
Suppose further that Assumption~\ref{ass:estimator} is satisfied. Then, Assumption~\ref{ass:UA} holds for $\widehat{\varepsilon}_t(\bm \theta)=\widehat{X}_t(\bm \theta)/\widehat{sigma}_t(\bm \theta)$ with $\widehat{sigma}_t(\bm \theta)$ defined in \eqref{eq:vola2}.
\end{thm}
The proof of Theorem~\ref{thm:UA2} is also in Appendix~\ref{Proof of Theorem}. Assumption~\textit{(a)} is a standard stationarity, invertibility and identifiability condition for model~\eqref{eq:ARMA-GARCH}. Assumption~\textit{(b)} ensures a strictly stationary solution to the GARCH recurrence in \eqref{eq:GARCH part} with $\operatorname{E}|X_t|^{2}<\infty$. Thus, \textit{(c)} is only a mild additional requirement, for which Theorem~2.1 in \citet{LM02} provides necessary and sufficient conditions.
section{Proofs of Theorems~\ref{thm:UA} and \ref{thm:UA2}}\label{Proof of Theorem}
\renewcommand{D.\arabic{equation}}{B.\arabic{equation}}
setcounter{equation}{0}
All $o_{(\operatorname{P})}$- and $O_{(\operatorname{P})}$-symbols in Appendices~\ref{Proof of Theorem}--\ref{Proof of Theorem 2} are to be understood with respect to $n\to\infty$.
\begin{proof}[{\textbf{Proof of Theorem~\ref{thm:UA}:}}]
To avoid superscripts, we put $h_t(\bm \theta)=sigma_t^{\delta^{\circ}}(\bm \theta)$, $h_{t}=h_{t}(\bm \theta^{\circ})=sigma_{t}^{\delta^{\circ}}(\bm \theta^{\circ})$ and $\widehat{h}_t(\bm \theta)=\widehat{sigma}_{t}^{\delta^{\circ}}(\bm \theta)$, where $\widehat{sigma}_{t}^{\delta^{\circ}}(\bm \theta)$ is defined in \eqref{eq:vola} in the main paper. Hence, $\widehat{h}_t(\bm \theta)$ satisfies
\[
\widehat{h}_t(\bm \theta)=\omega+sum_{j=1}^{p}\left\{\alpha_{+,j}(Y_{t-j})_{+}^{\delta^{\circ}}+\alpha_{-,j}(Y_{t-j})_{-}^{\delta^{\circ}}\right\}+sum_{j=1}^{q}\beta_j\widehat{h}_{t-j}(\bm \theta),\qquad t\geq1,
\]
for initial values $Y_{1-p}=\ldots=Y_0=0$ and $\widehat{h}_{1-q}(\bm \theta)=\ldots=\widehat{h}_{0}(\bm \theta)=0$. Thus, while $h_t(\bm \theta)$ denotes the true volatility (raised to the power of $\delta^{\circ}$) if $\bm \theta$ were the true parameter vector, $\widehat{h}_t(\bm \theta)$ approximates $h_t(\bm \theta)$ using artificial initial values. Recalling that $h_t=h_t(\bm \theta^{\circ})$, write
\begin{align}
\widehat{\varepsilon}_t(\bm \theta) &= \frac{Y_t}{\widehat{sigma}_t(\bm \theta)}=\varepsilon_t\frac{h_t^{1/\delta^\circ}}{[\widehat{h}_t(\bm \theta)]^{1/\delta^\circ}}\notag\\
&=\varepsilon_t\Big[1+\frac{h_t(\bm \theta) - \widehat{h}_t(\bm \theta)}{\widehat{h}_t(\bm \theta)}\Big]^{1/\delta^\circ}\Big[1+\frac{h_t - h_t(\bm \theta)}{h_t(\bm \theta)}\Big]^{1/\delta^\circ}.\label{eq:(A.1)}
\end{align}
Note that the quantities in square brackets are positive almost surely and, hence, the power is well defined for $\delta^{\circ}>0$. Finally, define $N_n^{-}(\eta)=N_n(\eta)setminus\{\bm \theta^{\circ}\}$.
We require two results that follow directly from Lemma~13 and its proof in \citet{KL16}. First, for
\[
\,mathrm{d}elta_{n,t}=sup_{\bm \theta\in N_n^{-}(\eta)}\left|\frac{h_t-h_t(\bm \theta)}{|\bm \theta-\bm \theta^{\circ}|h_t(\bm \theta)}\right|,
\]
it holds that
\begin{equation}\label{eq:(3.11)}
max_{t=1,\ldots,n}\frac{\,mathrm{d}elta_{n,t}}{n^{1/2}}=o_{\operatorname{P}}(1).
\end{equation}
Second, there exists $r_0\in[0,1)$, such that for sufficiently large $n$
\begin{equation}\label{eq:(4.11)}
sup_{\bm \theta\in N_n(\eta)}\left|\frac{h_t(\bm \theta)-\widehat{h}_t(\bm \theta)}{\widehat{h}_t(\bm \theta)}\right|\leq r_0^{t}V_0,\qquad t=1,\ldots,n,
\end{equation}
where $V_0=V_0(\eta)\geq0$ with $\operatorname{E}|V_0|^{\nu_0}<\infty$ for some $\nu_0>0$. We have $max_{t=\ell_{n},\ldots,n}r_0^{t}V_0=r_0^{\ell_n}V_0=o_{\operatorname{P}}(1)$.
Consider the two right-hand side factors in \eqref{eq:(A.1)}. Observe that
\[
sup_{\bm \theta\in N_n(\eta)}\left|\frac{h_t - h_t(\bm \theta)}{h_t(\bm \theta)}\right|\leq sup_{\bm \theta\in N_n^{-}(\eta)}\big\{n^{1/2}|\bm \theta-\bm \theta^{\circ}|\big\}\,mathrm{d}elta_{n,t}/n^{1/2}\leq\eta \,mathrm{d}elta_{n,t} /n^{1/2}.
\]
Set $s_{1t}=r_0^t V_0$ and $s_{2t}=\eta\,mathrm{d}elta_{n,t}/n^{1/2}$, whence $max_{t=\ell_n,\ldots,n}s_{1t}=o_{\operatorname{P}}(1)$ and $max_{t=\ell_n,\ldots,n}s_{2t}=o_{\operatorname{P}}(1)$. Then, for sufficiently large $n$,
\begin{align*}
1-\underline{s}_{t}&:=[1-min\{s_{1t},1\}]^{1/\delta^{\circ}}[1-min\{s_{2t},1\}]^{1/\delta^{\circ}}\\
&\leq \Big[1+\frac{h_t(\bm \theta) - \widehat{h}_t(\bm \theta)}{\widehat{h}_t(\bm \theta)}\Big]^{1/\delta^\circ}\Big[1+\frac{h_t - h_t(\bm \theta)}{h_t(\bm \theta)}\Big]^{1/\delta^\circ}\\
&\leq [1+s_{1t}]^{1/\delta^{\circ}}[1+s_{2t}]^{1/\delta^{\circ}}=:1+\overline{s}_t.
\end{align*}
Put $s_t=max\{\underline{s}_{t},\ \overline{s}_t\}$. Then, $max_{t=\ell_n,\ldots,n}s_{t}=o_{\operatorname{P}}(1)$ and
\[
1-s_{t}\leq \Big[1+\frac{h_t(\bm \theta) - \widehat{h}_t(\bm \theta)}{\widehat{h}_t(\bm \theta)}\Big]^{1/\delta^\circ}\Big[1+\frac{h_t - h_t(\bm \theta)}{h_t(\bm \theta)}\Big]^{1/\delta^\circ} \leq 1+s_{t}
\]
holds with probability approaching 1 (w.p.a.~1), as $n\to\infty$. Hence, the conclusion follows with our choice of $s_t$ and $m_t\equiv0$.
\end{proof}
\begin{proof}[{\textbf{Proof of Theorem~\ref{thm:UA2}:}}]
Denoting true volatility by $sigma_t^2=sigma_t^{2}(\bm \theta^{\circ})$, we write
\begin{align}
\widehat{\varepsilon}_t(\bm \theta) &= \frac{\widehat{X}_t(\bm \theta)}{\widehat{sigma}_t(\bm \theta)}=\frac{X_t}{\widehat{sigma}_t(\bm \theta)} + \frac{\widehat{X}_t(\bm \theta)-X_t}{\widehat{sigma}_t(\bm \theta)}\notag\\
&= \varepsilon_t\Big[1+\frac{sigma_t^2(\bm \theta)-\widehat{sigma}_t^2(\bm \theta)}{\widehat{sigma}_t^2(\bm \theta)}\Big]^{1/2}\Big[1+\frac{sigma_t^2-sigma_t^2(\bm \theta)}{sigma_t^2(\bm \theta)}\Big]^{1/2}+\frac{\widehat{X}_t(\bm \theta)-X_t}{\widehat{sigma}_t(\bm \theta)}.\label{eq:decomp1}
\end{align}
We again set $N_n^{-}(\eta)=N_n(\eta)setminus\{\bm \theta^{\circ}\}$. \citet[p.~264]{KL16} show that for $\bm \theta\in N_n(\eta)$
\begin{equation}\label{eq:2ndterm}
\frac{|sigma_t^2-sigma_t^2(\bm \theta)|}{sigma_t^2(\bm \theta)}\leq|\bm \theta-\bm \theta^{\circ}|\Pi_{1,n,t}^{*} + |\bm \theta-\bm \theta^{\circ}|^2\Pi_{2,n,t}^{*}
\end{equation}
for some $\Pi_{i,n,t}^{*}=\Pi_{i,n,t}^{*}(\eta)\geq0$ ($i=1,2$), both independent of $\bm \theta$. Furthermore, define $X_t(\bm \theta)=sigma_t(\bm \theta)\varepsilon_t$ and set
\[
\Pi_{3,n,t}=\Pi_{3,n,t}(\eta)=sup_{\bm \theta\in N_n^{-}(\eta)}\frac{1}{|\bm \theta-\bm \theta^{\circ}|}\frac{|X_t(\bm \theta)-X_t|}{\widehat{sigma}_t(\bm \theta)}.
\]
By Equation~(A.26) in \citet{Hog18+}, there exists $v_0>2$ such that
\begin{equation}\label{eq:(A.9m)}
\limsup_{n\to\infty}\operatorname{E}|\Pi_{1,n,t}^{*}|^{v_0}<\infty,\quad\limsup_{n\to\infty}\operatorname{E}|\Pi_{2,n,t}^{*}|^{v_0/2}<\infty,\quad \limsup_{n\to\infty}\operatorname{E}|\Pi_{3,n,t}|^{v_0}<\infty
\end{equation}
uniformly in $t\inmathbb{N}$. From this it follows that
\begin{equation}\label{eq:op1 all}
max_{t=\ell_{n},\ldots,n}\Pi_{1,n,t}^{*}/sqrt{n}=o_{\operatorname{P}}(1),\quad max_{t=\ell_{n},\ldots,n}\Pi_{2,n,t}^{*}/n=o_{\operatorname{P}}(1),\quad max_{t=\ell_{n},\ldots,n}\Pi_{3,n,t}/sqrt{n}=o_{\operatorname{P}}(1).
\end{equation}
We only show the first claim in \eqref{eq:op1 all}, as the others follow along similar lines. Using subadditivity, Markov's inequality and \eqref{eq:(A.9m)}, we get
\begin{align*}
\operatorname{P}\Big\{ max_{t=\ell_{n},\ldots,n}\Pi_{1,n,t}^{*}/sqrt{n} \geq\epsilon \Big\} &= \operatorname{P}\Big\{ \bigcup_{t=\ell_{n},\ldots,n}\big\{\Pi_{1,n,t}^{*}/sqrt{n} \geq\epsilon\big\} \Big\}\\
&\leqsum_{t=\ell_{n}}^{n}\operatorname{P}\big\{ \Pi_{1,n,t}^{*}/sqrt{n} \geq\epsilon \big\}\\
&\leqsum_{t=\ell_{n}}^{n}\epsilon^{-v_0}n^{-v_0/2}\operatorname{E}|\Pi_{1,n,t}^{*}|^{v_0}=o(1).
\end{align*}
By Lemma~A~1 in \citet{Hog18+}, there exist $r\in(0,1)$ and r.v.s $V_t\geq0$ with $sup_{t\inmathbb{N}}\operatorname{E}|V_t|^{v}<\infty$ for some $v>0$, such that for sufficiently large $n$,
\begin{equation}\label{eq:boundXs}
sup_{\bm \theta\in N_n(\eta)}max\big\{|\widehat{X}_t(\bm \theta)-X_t(\bm \theta)|,\ |\widehat{sigma}_t^2(\bm \theta)-sigma_t^2(\bm \theta)|\big\}\leq r^{t}V_t\qquad\text{for all }t\inmathbb{N}.
\end{equation}
We easily obtain that $max_{t=\ell_{n},\ldots,n}r^{t}V_t=o_{\operatorname{P}}(1)$, since
\begin{align*}
\operatorname{P}\Big\{max_{t=\ell_{n},\ldots,n}r^{t}V_t\geq\epsilon\Big\} &=\operatorname{P}\Big\{\bigcup_{t=\ell_{n},\ldots,n}\big\{r^{t}V_t\geq\epsilon\big\}\Big\} \\
&\leqsum_{t=\ell_{n}}^{n}\operatorname{P}\big\{r^{t}V_t\geq\epsilon\big\} \\
&\leqsum_{t=\ell_{n}}^{n}\epsilon^{-v}(r^{t})^v\operatorname{E}[V_t^{v}] \\
&\leq C\Big\{sup_{t=\ell_{n},\ldots,n}\operatorname{E}[V_t^{v}]\Big\}sum_{t=\ell_{n}}^{n}(r^{t})^v=o(1),
\end{align*}
where the last equality follows from properties of the geometric series.
Equipped with these results, we now consider the terms on the right-hand side of \eqref{eq:decomp1} separately. First, by \eqref{eq:vola2} and the fact that $\omega^{\circ}>0$, we have for sufficiently large $n$ that $\widehat{sigma}_t^2(\bm \theta)\geq\underline{\omega}>0$ for all $\bm \theta\in N_n(\eta)$ and all $t$. Combine this with \eqref{eq:boundXs} to obtain
\begin{align}
max_{t=\ell_{n},\ldots,n}\frac{|sigma_t^2(\bm \theta)-\widehat{sigma}_t^2(\bm \theta)|}{\widehat{sigma}_t^2(\bm \theta)}&\leq\underline{\omega}^{-1}max_{t=\ell_{n},\ldots,n}r^{t}V_t=:max_{t=\ell_{n},\ldots,n}s_{1t}=o_{\operatorname{P}}(1),\label{eq:beg1}\\
max_{t=\ell_{n},\ldots,n}\frac{|\widehat{X}_t(\bm \theta)-X_t|}{\widehat{sigma}_t(\bm \theta)}&\leqmax_{t=\ell_{n},\ldots,n}\Big\{\underline{\omega}^{-1/2}|\widehat{X}_t(\bm \theta)-X_t(\bm \theta)|+ \frac{|X_t(\bm \theta)-X_t|}{\widehat{sigma}_t(\bm \theta)}\Big\}\notag\\
&\leq max_{t=\ell_{n},\ldots,n}\Big\{\underline{\omega}^{-1/2}r^{t}V_t + sup_{\bm \theta\in N_n^{-}(\eta)}\big\{sqrt{n}|\bm \theta-\bm \theta^{\circ}|\big\}\Pi_{3,n,t}/sqrt{n}\Big\}\notag\\
&\leq max_{t=\ell_{n},\ldots,n}\Big\{\underline{\omega}^{-1/2}r^{t}V_t + \eta\Pi_{3,n,t}/sqrt{n}\Big\}\notag\\
&=:max_{t=\ell_{n},\ldots,n}m_t=o_{\operatorname{P}}(1).\notag
\end{align}
Finally, we have from \eqref{eq:2ndterm} and \eqref{eq:op1 all} that
\begin{align}
max_{t=\ell_{n},\ldots,n}\frac{|sigma_t^2-sigma_t^2(\bm \theta)|}{sigma_t^2(\bm \theta)}&\leqmax_{t=\ell_{n},\ldots,n}\Big\{ sup_{\bm \theta\in N_n^{-}(\eta)}\big\{sqrt{n}|\bm \theta-\bm \theta^{\circ}|\big\}\Pi_{1,n,t}^{*}/sqrt{n} + sup_{\bm \theta\in N_n^{-}(\eta)}\big\{n|\bm \theta-\bm \theta^{\circ}|^2\big\}\Pi_{2,n,t}^{*}/n\Big\}\notag\\
&\leq max_{t=\ell_{n},\ldots,n}\Big\{\eta\Pi_{1,n,t}^{*}/sqrt{n} + \eta^2\Pi_{2,n,t}^{*}/n\Big\}\notag\\
&=:max_{t=\ell_{n},\ldots,n}s_{2t}=o_{\operatorname{P}}(1).\label{eq:beg3}
\end{align}
Combining \eqref{eq:beg1} and \eqref{eq:beg3}, we get
\begin{align*}
1-\underline{s}_{t}&:=[1-min\{s_{1t},1\}]^{1/2}[1-min\{s_{2t},1\}]^{1/2}\\
&\leq \Big[1+\frac{sigma_t^2(\bm \theta)-\widehat{sigma}_t^2(\bm \theta)}{\widehat{sigma}_t^2(\bm \theta)}\Big]^{1/2}\Big[1+\frac{sigma_t^2-sigma_t^2(\bm \theta)}{sigma_t^2(\bm \theta)}\Big]^{1/2}\\
&\leq [1+s_{1t}]^{1/2}[1+s_{2t}]^{1/2}=:1+\overline{s}_t.
\end{align*}
Hence, for $s_t=max\{\underline{s}_{t},\ \overline{s}_t\}$ with $max_{t=\ell_n,\ldots,n}s_{t}=o_{\operatorname{P}}(1)$, it holds w.p.a.~1, as $n\to\infty$, that
\[
1-s_{t}\leq \Big[1+\frac{sigma_t^2(\bm \theta)-\widehat{sigma}_t^2(\bm \theta)}{\widehat{sigma}_t^2(\bm \theta)}\Big]^{1/2}\Big[1+\frac{sigma_t^2-sigma_t^2(\bm \theta)}{sigma_t^2(\bm \theta)}\Big]^{1/2} \leq 1+s_{t}.
\]
Thus, the conclusion follows with the above choices of $m_t$ and $s_t$.
\end{proof}
section{Proof of Theorem~\ref{thm:mainresult}}\label{Proof of Theorem 1}
\renewcommand{D.\arabic{equation}}{C.\arabic{equation}}
setcounter{equation}{0}
Let $C>0$ denote a large constant that may change from line to line. For notational brevity, we put
\[
U_t=|\varepsilon_t|,\quad \widehat{U}_t=|\widehat{\varepsilon}_t|,\quad \widehat{U}_{(i)}=|\widehat{\varepsilon}|_{(i)}.
\]
We denote the survivor function of $U_t$ by $\overline{F}(\cdot)=1-F(\cdot)$, where $F(\cdot)$ is from Assumption~\ref{ass:U distr 2}. For $\bm x=(x,y)^\operatorname{P}rime\in(0,\infty)^2$ and $\bm \xi=(\xi_1, \xi_2)^\operatorname{P}rime\inmathbb{R}^2$, define
\begin{align*}
M_n(\bm \xi) &=M_n(\bm x,\bm \xi)= \frac{1}{k}sum_{t=d+1}^{n}I_{\big\{U_t>e^{\xi_1/sqrt{k}}b(\frac{n}{kx}),\ U_{t-d}>e^{\xi_2/sqrt{k}}b(\frac{n}{ky})\big\}},\\
\widehat{M}_n(\bm \xi) &=\widehat{M}_n(\bm x,\bm \xi)= \frac{1}{k}sum_{t=d+1}^{n}I_{\big\{\widehat{U}_t>e^{\xi_1/sqrt{k}}b(\frac{n}{kx}),\ \widehat{U}_{t-d}>e^{\xi_2/sqrt{k}}b(\frac{n}{ky})\big\}}.
\end{align*}
If not specified otherwise, we assume the conditions of Theorem~\ref{thm:mainresult} to hold for the following lemmas and propositions, even though some of them may hold under a subset of the assumptions of Theorem~\ref{thm:mainresult}.
The proof of Theorem~\ref{thm:mainresult} requires the following four propositions, which are proved in Subsection~\ref{Proofs of Propositions} of this Appendix. The first proposition derives an (infeasible) estimate of the PA-tail copula for the (unobserved) innovations. The limit theory developed by \citet{SS06} cannot be applied to prove the next proposition, since it only applies to tail dependent sequences of random variables. Recall for the following that $b(x)=F^{\leftarrow}(1-1/x)$.
\begin{prop}\label{prop:base convergence}
For i.i.d.~$U_t$ with d.f.~$F(\cdot)$ satisfying Assumption~\ref{ass:U distr 2}~(ii), it holds that, as $n\to\infty$,
\begin{equation*}
sqrt{\frac{n}{xy}}\begin{pmatrix}
\frac{1}{k}sum_{t=2}^{n}\Big[I_{\left\{U_t>b(\frac{n}{kx}),\ U_{t-1}>b(\frac{n}{ky})\right\}}
-\Big(\frac{k}{n}\Big)^2xy\Big]\\
\vdots \\
\frac{1}{k}sum_{t=D+1}^{n}\Big[I_{\left\{U_t>b(\frac{n}{kx}),\ U_{t-D}>b(\frac{n}{ky})\right\}}
-\Big(\frac{k}{n}\Big)^2xy\Big]\\
\end{pmatrix}\overset{d}{\longrightarrow}N(\boldsymbol{0},\bm I_{D\times D}),
\end{equation*}
where $\bm I_{D\times D}$ denotes the $(D\times D)$-identity matrix, and $\boldsymbol{0}$ is a $(D\times 1)$-vector of zeros.
\end{prop}
In the following, we show that in the indicators in Proposition~\ref{prop:base convergence} we can replace the $U_t$ with the $\widehat{U}_t$, and the $b(n/[kz])$ by the empirical analog $\widehat{U}_{(\lfloor kz\rfloor +1)}$ ($z\in\{x,y\}$) without changing the limit. Then, Theorem~\ref{thm:mainresult} follows immediately from an application of the continuous mapping theorem. The following three propositions serve to justify these replacements.
\begin{prop}\label{prop:xi convergence}
For any $K>0$, it holds that
\[
sup_{\bm \xi\in[-K, K]^2}|M_n(\bm \xi)-M_n(\boldsymbol{0})|=o_{\operatorname{P}}(n^{-1/2}).
\]
\end{prop}
\begin{prop}\label{prop:hat convergence}
For any $K>0$, it holds that
\[
sup_{\bm \xi\in[-K, K]^2}|\widehat{M}_n(\bm \xi)-M_n(\bm \xi)|=o_{\operatorname{P}}(n^{-1/2}).
\]
\end{prop}
\begin{prop}\label{prop:emp an}
For any $x>0$, it holds that
\[
|\xi_n(x)|=\left|sqrt{k}\log\Bigg(\frac{\widehat{U}_{(\lfloor kx\rfloor+1)}}{b(n/[kx])}\Bigg)\right|=O_{\operatorname{P}}(1).
\]
\end{prop}
Equipped with these four propositions, we proceed to prove Theorem~\ref{thm:mainresult}.
\begin{proof}[{\textbf{Proof of Theorem~\ref{thm:mainresult}:}}]
We show that for each $d\in\{1,\ldots,D\}$,
\begin{equation}\label{eq:to show1}
\Bigg|\frac{1}{k}sum_{t=d+1}^{n}I_{\{U_t>b(\frac{n}{kx}),\ U_{t-d}>b(\frac{n}{ky})\}}-\frac{1}{k}sum_{t=d+1}^{n}I_{\{\widehat{U}_t>\widehat{U}_{(\lfloor kx\rfloor+1)},\ \widehat{U}_{t-d}>\widehat{U}_{(\lfloor ky\rfloor+1)}\}}\Bigg|=o_{\operatorname{P}}(n^{-1/2}).
\end{equation}
Combining this with Proposition~\ref{prop:base convergence}, the desired convergence follows from Lemma~4.7 of \citet{Whi01} and the continuous mapping theorem \citep[e.g.,][Theorem~7.20]{Whi01}. Decompose
\begin{align}
M_n(\boldsymbol{0}) &= \frac{1}{k}sum_{t=d+1}^{n}I_{\{U_t>b(\frac{n}{kx}),\ U_{t-d}>b(\frac{n}{ky})\}} \notag\\
&= \frac{1}{k}sum_{t=d+1}^{n}I_{\{U_t>b(\frac{n}{kx}),\ U_{t-d}>b(\frac{n}{ky})\}}-\frac{1}{k}sum_{t=d+1}^{n}I_{\{U_t>e^{\xi_1/sqrt{k}}b(\frac{n}{kx}),\ U_{t-d}>e^{\xi_2/sqrt{k}}b(\frac{n}{ky})\}}\notag\\
&\hspace{0.4cm} +\frac{1}{k}sum_{t=d+1}^{n}I_{\{U_t>e^{\xi_1/sqrt{k}}b(\frac{n}{kx}),\ U_{t-d}>e^{\xi_2/sqrt{k}}b(\frac{n}{ky})\}} - \frac{1}{k}sum_{t=d+1}^{n}I_{\{\widehat{U}_t>e^{\xi_1/sqrt{k}}b(\frac{n}{kx}),\ \widehat{U}_{t-d}>e^{\xi_2/sqrt{k}}b(\frac{n}{ky})\}}\notag\\
&\hspace{0.4cm} +\frac{1}{k}sum_{t=d+1}^{n}I_{\{\widehat{U}_t>e^{\xi_1/sqrt{k}}b(\frac{n}{kx}),\ \widehat{U}_{t-d}>e^{\xi_2/sqrt{k}}b(\frac{n}{ky})\}}\notag\\
&=o_{\operatorname{P}}(n^{-1/2}) + \widehat{M}_n(\bm \xi),\label{eq:(2.1.1)}
\end{align}
where the $o_{\operatorname{P}}(n^{-1/2})$-term is uniform on $\bm \xi\in[-K,K]^2$ by Propositions~\ref{prop:xi convergence} and~\ref{prop:hat convergence}.
Define $\bm \xi_n=\big(\xi_n(x),\ \xi_n(y)\big)^\operatorname{P}rime$, where $\xi_n(z)=sqrt{k}\log\big(\widehat{U}_{(\lfloor kz\rfloor+1)}/b(n/[kz])\big)$ ($z\in\{x,y\}$) as in Proposition~\ref{prop:emp an}. Then, $\widehat{M}_n(\bm \xi_n)=\frac{1}{k}sum_{t=d+1}^{n}I_{\{\widehat{U}_t>\widehat{U}_{(\lfloor kx\rfloor+1)},\ \widehat{U}_{t-d}>\widehat{U}_{(\lfloor ky\rfloor+1)}\}}$. To show \eqref{eq:to show1} or, equivalently, that $\widehat{M}_n(\bm \xi_n)-M_n(\boldsymbol{0})=o_{\operatorname{P}}(n^{-1/2})$ we have to prove that for any $\varepsilon>0$ and $\delta>0$
\[
\operatorname{P}\left\{sqrt{n}|\widehat{M}_n(\bm \xi_n)-M_n(\boldsymbol{0})|>\varepsilon\right\}\leq\delta
\]
for sufficiently large $n$. By Proposition~\ref{prop:emp an}, we can choose $K>0$ such that $\operatorname{P}\{|\bm \xi_n|>K\}\leq\delta/2$ for all sufficiently large $n$. Furthermore, by \eqref{eq:(2.1.1)}, $\operatorname{P}\{sqrt{n}sup_{\bm \xi\in[-K, K]^2}|\widehat{M}_n(\bm \xi)-M_n(\boldsymbol{0})|>\varepsilon\}\leq\delta/2$ for large $n$. Using these two results, we obtain
\begin{align*}
\operatorname{P}\left\{sqrt{n}|\widehat{M}_n(\bm \xi_n)-M_n(\boldsymbol{0})|>\varepsilon\right\} &\leq\operatorname{P}\Big\{|\bm \xi_n|>K\Big\} + \operatorname{P}\Big\{sqrt{n}sup_{\bm \xi\in[-K, K]^2}|\widehat{M}_n(\bm \xi)-M_n(\boldsymbol{0})|>\varepsilon,\ |\bm \xi_n|\leq K\Big\}\\
&\leq\frac{\delta}{2}+\frac{\delta}{2}=\delta,
\end{align*}
i.e., \eqref{eq:to show1}. The conclusion follows.
\end{proof}
subsection{Proofs of Propositions~\ref{prop:base convergence}--\ref{prop:emp an}}\label{Proofs of Propositions}
\begin{proof}[{\textbf{Proof of Proposition~\ref{prop:base convergence}:}}]
For $d\in\{1,\ldots,D\}$, we define
\begin{align*}
Z_{n,t}^{(d)}&=Z_{n,t}^{(d)}(x,y)=\begin{cases}0, & t=1,\ldots,d,\\
\frac{n}{k}\left[I_{\left\{U_t>b(\frac{n}{kx}),\ U_{t-d}>b(\frac{n}{ky})\right\}}
-\operatorname{P}\left\{U_t>b\Big(\frac{n}{kx}\Big),\ U_{t-d}>b\Big(\frac{n}{ky}\Big)\right\}\right],& t>d,
\end{cases}\\
\bm Z_{n,t}&=\big(Z_{n,t}^{(1)},\ldots, Z_{n,t}^{(D)}\big)^\operatorname{P}rime.
\end{align*}
Note that by Assumption~\ref{ass:U distr 2}~(ii)
\[
\operatorname{P}\left\{U_t>b\Big(\frac{n}{kx}\Big),\ U_{t-d}>b\Big(\frac{n}{ky}\Big)\right\}=\operatorname{P}\left\{U_t>b\Big(\frac{n}{kx}\Big)\right\}\operatorname{P}\left\{U_{t-d}>b\Big(\frac{n}{ky}\Big)\right\}=\left(\frac{k}{n}\right)^2xy
\]
for sufficiently large $n$.
Let $\bm \lambda=(\lambda_1,\ldots,\lambda_D)^\operatorname{P}rime\inmathbb{R}^{D}$ be any vector with $\bm \lambda^\operatorname{P}rime\bm \lambda=1$. Then, by the Cram\'{e}r--Wold device \citep[e.g.,][Proposition~5.1]{Whi01} it suffices to show that $n^{-1/2}sum_{t=1}^{n}\bm \lambda^\operatorname{P}rime\bm Z_{n,t}\overset{d}{\longrightarrow}sqrt{xy}\bm \lambda^\operatorname{P}rime \bm Z$ for $\bm Zsim N(\boldsymbol{0},\bm I_{D\times D})$. To do so, we verify the conditions of Theorem~5.20 of \citet{Whi01}.
Since the $\bm \lambda^\operatorname{P}rime \bm Z_{n,t}$ are (row-wise) $D$-dependent, the array $\{\bm \lambda^\operatorname{P}rime \bm Z_{n,t}\}$ is obviously $\operatorname{P}hi$-mixing of any rate \citep[Sec.~3]{Whi01}. Furthermore, $\operatorname{E}[\bm \lambda^\operatorname{P}rime \bm Z_{n,t}]=0$ and
\begin{align*}
\operatorname{E}[(\bm \lambda^\operatorname{P}rime \bm Z_{n,t})^2] &= \operatorname{Var}(\bm \lambda^\operatorname{P}rime \bm Z_{n,t})=\operatorname{Var}\Big(sum_{d=1}^{D}\lambda_d Z_{n,t}^{(d)}\Big)\\
&=sum_{d=1}^{D}\lambda_d^2\operatorname{Var}( Z_{n,t}^{(d)}) + sum_{c\neq d}\lambda_c\lambda_d \operatorname{Cov}( Z_{n,t}^{(c)},\ Z_{n,t}^{(d)}).
\end{align*}
Consider the variances and covariances separately. For the variance we get for $t>d$
\begin{align}
\operatorname{Var}\big( Z_{n,t}^{(d)}\big) &= \frac{n^2}{k^2}\operatorname{Var}\Big(I_{\left\{U_t>b(\frac{n}{kx}),\ U_{t-d}>b(\frac{n}{ky})\right\}}\Big)\notag\\
&= \frac{n^2}{k^2}\Big(\frac{k}{n}\Big)^2xy\left[1-\Big(\frac{k}{n}\Big)^2xy\right]\notag\\
&=xy+o(1),\label{eq:Var1}
\end{align}
where the second line exploits that $I_{\left\{U_t>b(\frac{n}{kx}),\ U_{t-d}>b(\frac{n}{ky})\right\}}$ is Bernoulli-distributed with success probability $p=(k/n)^2xy$ and, hence,
\begin{equation*}
\operatorname{Var}\Big(I_{\left\{U_t>b(\frac{n}{kx}),\ U_{t-d}>b(\frac{n}{ky})\right\}}\Big)=\Big(\frac{k}{n}\Big)^2xy\left[1-\Big(\frac{k}{n}\Big)^2xy\right].
\end{equation*}
Using similar arguments, we obtain
\begin{align}
\operatorname{Cov}\big( Z_{n,t}^{(c)}, Z_{n,t}^{(d)}\big) &= O\left(\frac{n^2}{k^2}\operatorname{P}\left\{U_r>b\Big(\frac{n}{kx}\Big),\ U_{s}>b\Big(\frac{n}{ky}\Big),\ U_{t}>b\Big(\frac{n}{kz}\Big)\right\}\right)\notag\\
&=O\Big(\frac{n^2}{k^2}\frac{k^3}{n^3}\Big)=O\Big(\frac{k}{n}\Big)=o(1)\label{eq:cov bound in sum},
\end{align}
where $r,s,t$ are pairwise distinct. Overall, we have $\operatorname{E}[(\bm \lambda^\operatorname{P}rime \bm Z_{n,t})^2]\leq 2<\infty$ for sufficiently large $n$.
Finally, exploiting $D$-dependence of $\bm \lambda^\operatorname{P}rime\bm Z_{n,t}$ again,
\begin{align}
\overline{sigma}_n^2&:= \operatorname{Var}\Big(n^{-1/2}sum_{t=1}^{n}\bm \lambda^\operatorname{P}rime \bm Z_{n,t}\Big)\notag\\
&=\frac{1}{n}sum_{t=1}^{n}\operatorname{Var}(\bm \lambda^\operatorname{P}rime \bm Z_{n,t})+\frac{1}{n}sum_{s\neq t}\operatorname{Cov}(\bm \lambda^\operatorname{P}rime \bm Z_{n,s}, \bm \lambda^\operatorname{P}rime \bm Z_{n,t})\notag\\
&=\frac{1}{n}sum_{t=1}^{n}\operatorname{Var}(\bm \lambda^\operatorname{P}rime \bm Z_{n,t})+\frac{2}{n}sum_{s=1}^{n}sum_{t=s+1}^{min\{s+D, n\}}\operatorname{Cov}(\bm \lambda^\operatorname{P}rime \bm Z_{n,s}, \bm \lambda^\operatorname{P}rime \bm Z_{n,t}).\label{eq:(A.5-)}
\end{align}
It is easy to check that the sum on the right-hand side of \eqref{eq:(A.5-)} vanishes, because the covariance terms are of order $o(1)$ by \eqref{eq:cov bound in sum}. For the sum involving the variances, we get using \eqref{eq:Var1}, \eqref{eq:cov bound in sum} and $\bm \lambda^\operatorname{P}rime\bm \lambda=1$ that for $t>D$
\begin{align*}
\operatorname{Var}(\bm \lambda^\operatorname{P}rime \bm Z_{n,t}) &= \operatorname{Var}\Big(sum_{d=1}^{D}\lambda_{d} Z_{n,t}^{(d)}\Big)\\
&=sum_{d=1}^{D}\lambda_{d}^2\operatorname{Var}\big( Z_{n,t}^{(d)}\big) + sum_{c\neq d}\lambda_{c}\lambda_{d}\operatorname{Cov}\big(Z_{n,t}^{(c)}, Z_{n,t}^{(d)}\big)\\
&= sum_{d=1}^{D}\lambda_{d}^2[xy+o(1)]+o(1)=xy+o(1).
\end{align*}
Overall, we obtain that $\overline{sigma}_n^2=xy+o(1)\longrightarrow xy$, as $n\to\infty$.
Thus, we may apply Theorem~5.20 of \citet{Whi01}, yielding that $n^{-1/2}sum_{t=1}^{n}\bm \lambda^\operatorname{P}rime\bm Z_{n,t}\overset{d}{\longrightarrow}sqrt{xy}\bm \lambda^\operatorname{P}rime \bm Z$. This concludes the proof.
\end{proof}
The following lemma bounds the variation in the tail of the $U_t$, and is used throughout the proofs.
\begin{lem}\label{lem:Lem F}
Suppose Assumption~\ref{ass:U distr 2} holds. Then, for any $\nu\in(-1,\infty)$ there exists $\widetilde{\nu}$ between $1$ and $\nu$, such that
\begin{align*}
\overline{F}\Big((1+\nu)b\Big(\frac{n}{kx}\Big)\Big) &= \frac{kx}{n}\Big[1-\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu o(1)\Big],
\intertext{where the $o(1)$-term vanishes uniformly on compact $\nu$-sets in $(-1,\infty)$ and on compact $x$-sets in $(0,\infty)$. In particular,}
\overline{F}\Big(e^{\xi/sqrt{k}}b\Big(\frac{n}{kx}\Big)\Big) &= \frac{kx}{n}\Bigg[1-\frac{\alpha\xi}{sqrt{k}}+o\Big(\frac{1}{sqrt{k}}\Big)\Bigg]
\end{align*}
uniformly on compact $\xi$-sets in $mathbb{R}$ and on compact $x$-sets in $(0,\infty)$.
\end{lem}
Lemma~\ref{lem:Lem F} is proved in Appendix~\ref{Proofs of Auxiliary Lemmas}.
\begin{proof}[{\textbf{Proof of Proposition~\ref{prop:xi convergence}:}}]
We only consider the supremum over $\bm \xi\in[-K,0]^2$; that over the other three quadrants can be dealt with similarly. Observe that
\begin{align*}
sup_{\bm \xi\in[-K,0]^2}|M_n(\bm \xi)-M_n(\boldsymbol{0})|&\leq M_n(-(K,K)^\operatorname{P}rime)-M_n(\boldsymbol{0})\\
&=\frac{1}{k}sum_{t=d+1}^{n}\Big[I_{\{U_t>e^{-K/sqrt{k}}b(\frac{n}{kx}),\ U_{t-d}>e^{-K/sqrt{k}}b(\frac{n}{ky})\}}-I_{\{U_t>b(\frac{n}{kx}),\ U_{t-d}>b(\frac{n}{ky})\}}\Big]\\
&=:\frac{1}{k}sum_{t=d+1}^{n}I_{1t},
\end{align*}
where $I_{1t}\in\{0,1\}$. Markov's inequality implies
\begin{equation}\label{eq:Cheby}
\operatorname{P}\left\{sqrt{n}sup_{\bm \xi\in[-K,0]^2}|M_n(\bm \xi)-M_n(\boldsymbol{0})|>\varepsilon\right\}
\leq\frac{\varepsilon^{-4}n^2}{k^4}\operatorname{E}\Big[sum_{t=d+1}^{n}I_{1t}\Big]^4.
\end{equation}
Due to $d$-dependence, the $I_{1t}$ are trivially $\rho$-mixing, and thus we may apply Lemma~2.3 of \citet{Sha93} (for, in his notation, $q=4$) to deduce that
\begin{equation}\label{eq:Shao1}
\operatorname{E}\Big[sum_{t=d+1}^{n}I_{1t}\Big]^4\leq n^2 C \big\{\operatorname{E}[I_{1t}^2]\big\}^2+n C \operatorname{E}[I_{1t}^4].
\end{equation}
Since $I_{1t}\in\{0,1\}$ and, hence, $\operatorname{E}[I_{1t}^4]=\operatorname{E}[I_{1t}^2]=\operatorname{E}[I_{1t}]$, it suffices to compute $\operatorname{E}[I_{1t}]$. Using Lemma~\ref{lem:Lem F}, we get that
\begin{align*}
\operatorname{E}[I_{1t}] &=\operatorname{P}\Big\{U_t>e^{-K/sqrt{k}}b\Big(\frac{n}{kx}\Big)\Big\} \operatorname{P}\Big\{U_{t-d}>e^{-K/sqrt{k}}b\Big(\frac{n}{ky}\Big)\Big\}-\operatorname{P}\Big\{U_t>b\Big(\frac{n}{kx}\Big)\Big\}\operatorname{P}\Big\{U_{t-d}>b\Big(\frac{n}{ky}\Big)\Big\}\\
&=\frac{k^2}{n^2}xy\Big[1+\frac{\alpha K}{sqrt{k}}+o\Big(\frac{1}{sqrt{k}}\Big)\Big]^2-\frac{k^2}{n^2}xy\\
&=\frac{k^2}{n^2}xy\Big[1+\frac{2\alpha K}{sqrt{k}}+o\Big(\frac{1}{sqrt{k}}\Big)-1\Big]\\
&\leq C\frac{k^{3/2}}{n^2}xy,
\end{align*}
whence, from \eqref{eq:Shao1},
\[
\operatorname{E}\Big[sum_{t=d+1}^{n}I_{1t}\Big]^4\leq C\frac{k^3}{n^2}+C\frac{k^{3/2}}{n}.
\]
Together with \eqref{eq:Cheby}, this implies that
\[
\operatorname{P}\Big\{sqrt{n}sup_{\bm \xi\in[-K,0]^2}|M_n(\bm \xi)-M_n(\boldsymbol{0})|>\varepsilon\Big\}=O\Big(\frac{1}{k}+\frac{n}{k^{5/2}}\Big)=o(1)
\]
by Assumption~\ref{ass:k}. Hence, the conclusion follows.
\end{proof}
Define
\begin{align}
A_t(\bm \xi,\bm \theta) &= I_{\left\{\widehat{U}_t(\bm \theta)>e^{\xi_1/sqrt{k}}b(n/[kx]),\ \widehat{U}_{t-d}(\bm \theta)>e^{\xi_2/sqrt{k}}b(n/[ky])\right\}},\notag\\
A_t(\bm \xi) &= I_{\left\{U_t>e^{\xi_1/sqrt{k}}b(n/[kx]),\ U_{t-d}>e^{\xi_2/sqrt{k}}b(n/[ky])\right\}},\notag\\
A_t(\bm \xi,\eta,\eta_0) &= I_{\left\{U_t[1+\eta_0s_{t}]+\eta_0 m_t>e^{\xi_1/sqrt{k}}b(n/[kx]),\ U_{t-d}[1+\eta_0s_{t-d}]+\eta_0 m_{t-d}>e^{\xi_2/sqrt{k}}b(n/[ky])\right\}},\qquad \eta_0\in\{-1, 1\},\notag\\
B_t(\bm \xi,\eta,\eta_0) &= B_t(\bm x,\bm \xi,\eta,\eta_0)= A_t(\bm \xi,\eta,\eta_0) - A_t(\bm \xi),\label{eq:(A.10+)}
\end{align}
where $m_{t}=m_{n,t}(\eta)\geq0$ and $s_{t}=s_{n,t}(\eta)\geq0$ are from Assumption~\ref{ass:UA}.
The proof of Proposition~\ref{prop:hat convergence} requires two further preliminary lemmas, which are both proven in Appendix~\ref{Proofs of Auxiliary Lemmas}.
\begin{lem}\label{lem:wpa1}
Let $\eta>0$. Then, w.p.a.~1, as $n\to\infty$,
\begin{equation*}
A_t(\bm \xi,\eta,-1) \leq A_t(\bm \xi,\bm \theta) \leq A_t(\bm \xi,\eta, 1)
\end{equation*}
for all $\bm \theta\in N_n(\eta)$ and $t=\ell_{n},\ldots,n$.
\end{lem}
\begin{lem}\label{lem:Bop1}
Let $\eta>0$, $\eta_0\in\{-1, 1\}$. Then, for any $K>0$,
\begin{equation*}
sup_{\bm \xi\in[-K,K]^2}\left|\frac{1}{k}sum_{t=d+1}^{n}B_t(\bm \xi,\eta,\eta_0)\right|=o_{\operatorname{P}}(n^{-1/2}).
\end{equation*}
\end{lem}
\begin{proof}[{\textbf{Proof of Proposition~\ref{prop:hat convergence}:}}]
Let $\eta>0$ and $\ell_{n}\to\infty$ with $\ell_{n}=o(k/sqrt{n})$ as $n\to\infty$. Then, due to Lemma~\ref{lem:wpa1}, we get that w.p.a.~1, as $n\to\infty$,
\[
B_t(\bm \xi,\eta,-1)=A_t(\bm \xi,\eta,-1)-A_t(\bm \xi)\leq A_t(\bm \xi,\bm \theta)-A_t(\bm \xi) \leq A_t(\bm \xi,\eta,1)-A_t(\bm \xi)=B_t(\bm \xi,\eta,1)
\]
for all $\bm \theta\in N_n(\eta)$ and $t=\ell_{n},\ldots,n$. Hence, from Lemma~\ref{lem:Bop1},
\[
sup_{\bm \theta\in N_n(\eta)}sup_{\bm \xi\in[-K, K]^2}\frac{1}{k}\left|sum_{t=\ell_{n}}^{n}\left\{A_t(\bm \xi,\bm \theta)-A_t(\bm \xi)\right\}\right|=o_{\operatorname{P}}(n^{-1/2}).
\]
Using additionally that $\ell_{n}/k=o(n^{-1/2})=o(1)$, we also have that (since $|A_t(\bm \xi,\bm \theta)-A_t(\bm \xi)|\leq 1$)
\[
sup_{\bm \theta\in N_n(\eta)}sup_{\bm \xi\in[-K, K]^2}\frac{1}{k}\left|sum_{t=d+1}^{n}\left\{A_t(\bm \xi,\bm \theta)-A_t(\bm \xi)\right\}\right|=o_{\operatorname{P}}(n^{-1/2}).
\]
Because by Assumption~\ref{ass:estimator}, $\widehat{\bm \theta}\in N_n(\eta)$ w.p.a.~1 as $n\to\infty$ followed by $\eta\to\infty$, we get
\[
sup_{\bm \xi\in[-K, K]^2}\frac{1}{k}\left|sum_{t=d+1}^{n}\left\{A_t(\bm \xi,\widehat{\bm \theta})-A_t(\bm \xi)\right\}\right|=o_{\operatorname{P}}(n^{-1/2}).
\]
This, however, is just the conclusion.
\end{proof}
\begin{proof}[{\textbf{Proof of Proposition~\ref{prop:emp an}:}}]
The result follows immediately from Proposition~\ref{prop:unif emp an}, which is proven in Appendix~\ref{Proofs of Propositions 2}.
\end{proof}
subsection{Proofs of Lemmas~\ref{lem:Lem F}--\ref{lem:Bop1}}\label{Proofs of Auxiliary Lemmas}
\begin{proof}[{\textbf{Proof of Lemma~\ref{lem:Lem F}:}}]
By \citet[Theorem~1.1.11 and Theorem~B.1.10]{HF06}, we have that for any $\widetilde{\widetilde{\nu}}\in(-1,\infty)$, any $\varepsilon>0$ and any $\delta>0$
\begin{equation}\label{eq:unif in F}
\left|\frac{\overline{F}\Big((1+\widetilde{\widetilde{\nu}})b\Big(\frac{n}{kx}\Big)\Big)}{\overline{F}\Big(b\Big(\frac{n}{kx}\Big)\Big)}-(1+\widetilde{\widetilde{\nu}})^{-\alpha}\right|\leq\varepsilonmax\{(1+\widetilde{\widetilde{\nu}})^{-\alpha-\delta}, (1+\widetilde{\widetilde{\nu}})^{-\alpha+\delta}\}
\end{equation}
for sufficiently large $n$. Use the mean value theorem to deduce that there is some $\widetilde{\nu}$ between $1$ and $\nu$, such that
\begin{align}
\overline{F}\Big(b\Big(\frac{n}{kx}\Big)\Big) &- \overline{F}\Big((1+\nu)b\Big(\frac{n}{kx}\Big)\Big)\notag\\
&=F\Big((1+\nu)b\Big(\frac{n}{kx}\Big)\Big) - F\Big(b\Big(\frac{n}{kx}\Big)\Big)\notag\\
&=f\Big((1+\widetilde{\nu})b\Big(\frac{n}{kx}\Big)\Big)\left[(1+\nu)b\Big(\frac{n}{kx}\Big)-b\Big(\frac{n}{kx}\Big)\right]\notag\\
&= \frac{(1+\widetilde{\nu})b\Big(\frac{n}{kx}\Big)f\Big((1+\widetilde{\nu})b\Big(\frac{n}{kx}\Big)\Big)}{1-F\Big((1+\widetilde{\nu})b\Big(\frac{n}{kx}\Big)\Big)}\cdot\frac{(1+\nu)b\Big(\frac{n}{kx}\Big)-b\Big(\frac{n}{kx}\Big)}{(1+\widetilde{\nu})b\Big(\frac{n}{kx}\Big)}\cdot\frac{\overline{F}\Big((1+\widetilde{\nu})b\Big(\frac{n}{kx}\Big)\Big)}{\overline{F}\Big(b\Big(\frac{n}{kx}\Big)\Big)}\cdot\overline{F}\Big(b\Big(\frac{n}{kx}\Big)\Big)\notag\\
&=\big[\alpha+o(1)\big]\frac{1+\nu-1}{1+\widetilde{\nu}}\big[(1+\widetilde{\nu})^{-\alpha}+o(1)\big]\frac{kx}{n}\notag\\
&=\big[1+o(1)\big]\Bigg[\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu o(1)\Bigg]\frac{kx}{n}\notag\\
&=\Bigg[\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu o(1)\Bigg]\frac{kx}{n},\label{eq:(B.X)}
\end{align}
where the fourth step uses that the convergence
\[
\frac{\overline{F}\Big((1+\widetilde{\nu})b\Big(\frac{n}{kx}\Big)\Big)}{\overline{F}\Big(b\Big(\frac{n}{kx}\Big)\Big)}-(1+\widetilde{\nu})^{-\alpha}\longrightarrow0
\]
is uniform in $\widetilde{\nu}$ and uniform in $x$ by \eqref{eq:unif in F}. By continuity of $F(\cdot)$ in the tail (see Assumption~\ref{ass:U distr 2}~(ii)), \eqref{eq:(B.X)} implies
\begin{align*}
\overline{F}\Big((1+\nu)b\Big(\frac{n}{kx}\Big)\Big) &= \overline{F}\Big(b\Big(\frac{n}{kx}\Big)\Big)-\Big[\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu o(1)\Big]\frac{kx}{n}\\
&= \frac{kx}{n}\Big[1-\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu o(1)\Big].
\end{align*}
To prove the second statement, it suffices to recall the Taylor expansion $e^{\xi/sqrt{k}}=1+\xi/sqrt{k}+o(1/sqrt{k})$ and apply a Taylor expansion to $(1+\widetilde{\nu})^{1+\alpha}$.
\end{proof}
\begin{proof}[{\textbf{Proof of Lemma~\ref{lem:wpa1}:}}]
We have to show that w.p.a.~1, as $n\to\infty$,
\begin{align*}
&I_{\left\{U_t[1-s_{t}]-m_t>e^{\xi_1/sqrt{k}}b(n/[kx]),\ U_{t-d}[1-s_{t-d}]-m_{t-d}>e^{\xi_2/sqrt{k}}b(n/[ky])\right\}}\\
&\hspace{1cm}\leq I_{\left\{\widehat{U}_t(\bm \theta)>e^{\xi_1/sqrt{k}}b(n/[kx]),\ \widehat{U}_{t-d}(\bm \theta)>e^{\xi_2/sqrt{k}}b(n/[ky])\right\}}\\
&\hspace{1cm}\leq I_{\left\{U_t[1+s_{t}]+m_t>e^{\xi_1/sqrt{k}}b(n/[kx]),\ U_{t-d}[1+s_{t-d}]+m_{t-d}>e^{\xi_2/sqrt{k}}b(n/[ky])\right\}}
\end{align*}
for all $\bm \theta\in N_n(\eta)$ and $t=\ell_{n},\ldots,n$. However, this follows immediately from Assumption~\ref{ass:UA}.
\end{proof}
\begin{proof}[{\textbf{Proof of Lemma~\ref{lem:Bop1}:}}]
Let $\eta_0=1$; the case $\eta_0=-1$ can be dealt with similarly. Define
\[
w_t=I_{\left\{m_{t}\leq\varepsilon_0,\ m_{t-d}\leq\varepsilon_0,\ s_{t}\leq\varepsilon_0,\ s_{t-d}\leq\varepsilon_0\right\}},\qquad\varepsilon_0>0.
\]
If $w_t=1$, there exists $\nu=\nu(\varepsilon_0)>0$, such that
\begin{equation*}
1-\nu/2<(1+s_{t})^{-1}\leq1\qquad\text{and}\qquad 1-\nu/2<(1+s_{t-d})^{-1}\leq1.
\end{equation*}
Furthermore, $b(x)\to\infty$ for $x\to\infty$ by Assumption~\ref{ass:U distr 2}, such that
\[
1-\nu/2<1-e^{K/sqrt{k}}\varepsilon_0/b\big(n/[kmax\{x,\, y\}]\big)\leq1
\]
for sufficiently large $n$. Then, for $\bm \xi\in[-K,0]^2$ (the other quadrants can be dealt with similarly) and sufficiently large $n$,
\begin{align*}
&0\leq\frac{sqrt{n}}{k}sum_{t=\ell_{n}}^{n}w_tB_t(\bm \xi,\eta,\eta_0)\\
&=\frac{sqrt{n}}{k}sum_{t=\ell_{n}}^{n}w_t\Big[I_{\Big\{U_t>e^{\xi_1/sqrt{k}}(1+s_{t})^{-1}\big(1-e^{-\xi_1/sqrt{k}}\frac{m_t}{b(n/[kx])}\big) b(n/[kx]),}\\
&\hspace{3cm} _{U_{t-d}>e^{\xi_2/sqrt{k}}(1+s_{t-d})^{-1}\big(1-e^{-\xi_2/sqrt{k}}\frac{m_{t-d}}{b(n/[ky])}\big)b(n/[ky])\Big\}}\\
&\hspace{8.22cm}-I_{\left\{U_t>e^{\xi_1/sqrt{k}}b(n/[kx]),\ U_{t-d}>e^{\xi_2/sqrt{k}}b(n/[ky])\right\}}\Big]\\
&\leq \frac{sqrt{n}}{k}sum_{t=\ell_{n}}^{n}\Big[I_{\left\{U_t>(1-\nu) b(n/[kx]),\ U_{t-d}>(1-\nu)b(n/[ky])\right\}}-I_{\left\{U_t>b(n/[kx]),\ U_{t-d}>b(n/[ky])\right\}}\Big]\\
&=: \frac{sqrt{n}}{k}sum_{t=\ell_{n}}^{n}I_{2t}.
\end{align*}
Use Markov's inequality and Lemma~2.3 of \citet{Sha93}, to obtain that
\begin{align}
\operatorname{P}\Big\{\frac{sqrt{n}}{k}sum_{t=\ell_{n}}^{n}I_{2t}>\varepsilon\Big\}&\leq\varepsilon^{-4}\frac{n^2}{k^4}\operatorname{E}\Big[sum_{t=\ell_{n}}^{n}I_{2t}\Big]^4\notag\\
&=C\frac{n^2}{k^4}\Big\{n^2\big\{\operatorname{E}[I_{2t}^2]\big\}^2 + n\operatorname{E}[I_{2t}^4] \Big\}.\label{eq:Cheby2}
\end{align}
Similarly as for $I_{1t}$ in the proof of Proposition~\ref{prop:xi convergence}, we obtain using Lemma~\ref{lem:Lem F} that for sufficiently large $n$,
\begin{align*}
\operatorname{E}\big[I_{2t}\big] &= \operatorname{P}\left\{U_t>(1-\nu)b(n/[kx])\right\}\operatorname{P}\left\{U_{t-d}>(1-\nu)b(n/[ky])\right\}\\
&\hspace{6cm}-\operatorname{P}\left\{U_t>b(n/[kx])\right\}\operatorname{P}\left\{U_{t-d}>b(n/[ky])\right\}\\
&=\frac{k^2}{n^2}xy\Big[1+\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu o(1)\Big]^2-\frac{k^2}{n^2}xy\\
&\leq C\frac{k^2}{n^2}\nu.
\end{align*}
Thus, the right-hand side of \eqref{eq:Cheby2} can be bounded by
\begin{equation}\label{eq:HHH}
C\frac{n^2}{k^4}\Big[n^2\frac{k^4}{n^4}+n\frac{k^2}{n^2}\Big]\nu=O(1)\nu.
\end{equation}
For a suitable choice of $\varepsilon_0$ in $w_t$, $\nu$ in \eqref{eq:HHH} can be chosen arbitrarily close to zero. Thus, the expression on the right-hand side can be made arbitrarily small, yielding that
\begin{equation*}
sup_{\bm \xi\in[-K,0]^2}\frac{sqrt{n}}{k}\Bigg|sum_{t=\ell_{n}}^{n}w_tB_t(\bm \xi,\eta,\eta_0=1)\Bigg|=o_{\operatorname{P}}(1).
\end{equation*}
By Assumption~\ref{ass:UA}, $w_{\ell_{n}}=\ldots=w_{n}=1$ w.p.a.~1, as $n\to\infty$. Thus, we obtain
\begin{equation}\label{eq:(13.1)}
sup_{\bm \xi\in[-K,0]^2}\frac{sqrt{n}}{k}\Bigg|sum_{t=\ell_{n}}^{n}B_t(\bm \xi,\eta,\eta_0=1)\Bigg|=o_{\operatorname{P}}(1).
\end{equation}
By choosing $\ell_{n}\to\infty$ with $\ell_{n}=o(k/sqrt{n})$ (recall Assumption~\ref{ass:k}), we also have by boundedness of the $B_t(\bm \xi,\eta,\eta_0=1)$ that
\begin{equation}\label{eq:(13.2)}
sup_{\bm \xi\in[-K,0]^2}\frac{sqrt{n}}{k}\Bigg|sum_{t=d+1}^{\ell_{n}-1}B_t(\bm \xi,\eta,\eta_0=1)\Bigg|\leq \frac{sqrt{n}}{k}\ell_{n}=o(1).
\end{equation}
Combining \eqref{eq:(13.1)} and \eqref{eq:(13.2)} gives the desired conclusion.
\end{proof}
section{Proof of Theorem~\ref{thm:mainresult2}}\label{Proof of Theorem 2}
\renewcommand{D.\arabic{equation}}{D.\arabic{equation}}
setcounter{equation}{0}
The proof of Theorem~\ref{thm:mainresult2} is structured similarly as that of Theorem~\ref{thm:mainresult}. It requires four preliminary propositions, whose proofs are relegated to Appendix~\ref{Proofs of Propositions 2}. We follow the same notational conventions as in Appendix~\ref{Proof of Theorem 1}. Furthermore, we define $D[0,1]^2$ to be the space of $mathbb{R}^{D}$-valued functions on $[0,1]\times[0,1]$ that are continuous from above, with limits from below \citep[see][Sec.~3]{BW71}.
\begin{prop}\label{prop:base functional convergence}
For i.i.d.~$U_t$ with d.f.~$F(\cdot)$ satisfying Assumption~\ref{ass:U distr 2}~(ii), it holds that, as $n\to\infty$,
\begin{equation*}
\bm M_n(x,y)=sqrt{n}\begin{pmatrix}
\frac{1}{k}sum_{t=2}^{n}\Big[I_{\left\{U_t>b(\frac{n}{kx}),\ U_{t-1}>b(\frac{n}{ky})\right\}}
-\Big(\frac{k}{n}\Big)^2xy\Big]\\
\vdots \\
\frac{1}{k}sum_{t=D+1}^{n}\Big[I_{\left\{U_t>b(\frac{n}{kx}),\ U_{t-D}>b(\frac{n}{ky})\right\}}
-\Big(\frac{k}{n}\Big)^2xy\Big]\\
\end{pmatrix}\overset{d}{\longrightarrow}\bm B(x,y)\quad\text{in}\ D[0,1]^2,
\end{equation*}
where $\bm B=(B_1,\ldots,B_D)^\operatorname{P}rime$ is a $D$-dimensional Brownian sheet with independent components, i.e., a zero-mean Gaussian process with $\operatorname{Cov}\big(\bm B(x_1,y_1), \bm B(x_2,y_2)\big)=min(x_1,x_2)min(y_1,y_2)\bm I_{D\times D}$.
\end{prop}
For the proof of Theorem~\ref{thm:mainresult2}, we have to justify the replacement of $U_t$ by $\widehat{U}_t$, and $b(\frac{n}{kz})$ by $\widehat{U}_{(\lfloor kz\rfloor +1)}$ ($z\in\{x,y\}$) in the indicators in Proposition~\ref{prop:base functional convergence}. The following three propositions serve that purpose.
\begin{prop}\label{prop:Mn unif x xi}
For any $K>0$, it holds that
\[
sup_{\bm \xi\in[-K, K]^2}sup_{\bm x\in[0,K]^2}|M_n(\bm x,\bm \xi)-M_n(\bm x,\boldsymbol{0})|=o_{\operatorname{P}}(n^{-1/2}).
\]
\end{prop}
\begin{prop}\label{prop:hat unif convergence}
For any $K>0$, it holds that
\[
sup_{\bm \xi\in[-K, K]^2}sup_{\bm x\in[0,K]^2}|\widehat{M}_n(\bm x,\bm \xi)-M_n(\bm x,\bm \xi)|=o_{\operatorname{P}}(n^{-1/2}).
\]
\end{prop}
\begin{prop}\label{prop:unif emp an}
For any $0<\iota<K<\infty$, it holds that
\[
sup_{x\in[\iota,K]}|\xi_n(x)|=sup_{x\in[\iota,K]}\left|sqrt{k}\log\Bigg(\frac{\widehat{U}_{(\lfloor kx\rfloor+1)}}{b(n/[kx])}\Bigg)\right|=O_{\operatorname{P}}(1).
\]
\end{prop}
These four propositions allow us to prove Theorem~\ref{thm:mainresult2}.
\begin{proof}[{\textbf{Proof of Theorem~\ref{thm:mainresult2}:}}]
The outline of the proof is similar to that of Theorem~\ref{thm:mainresult}. We show that for each $d\in\{1,\ldots,D\}$,
\begin{equation}\label{eq:to show}
sup_{\bm x=(x,y)^\operatorname{P}rime\in[\iota,1]^2}\Bigg|\frac{1}{k}sum_{t=d+1}^{n}I_{\{U_t>b(\frac{n}{kx}),\ U_{t-d}>b(\frac{n}{ky})\}}-\frac{1}{k}sum_{t=d+1}^{n}I_{\{\widehat{U}_t>\widehat{U}_{(\lfloor kx\rfloor+1)},\ \widehat{U}_{t-d}>\widehat{U}_{(\lfloor ky\rfloor+1)}\}}\Bigg|=o_{\operatorname{P}}(n^{-1/2}).
\end{equation}
Combining this with Proposition~\ref{prop:base functional convergence}, Lemma~4.7 of \citet{Whi01} and the continuous mapping theorem \citep[e.g.,][Theorem~7.20]{Whi01} gives us
\[
mathcal{F}_n^{(D)}\overset{d}{\longrightarrow}sum_{d=1}^{D}\int_{[\iota,1-\iota]}W_d^2(2-2z, 2z)\,mathrm{d} z,
\]
where $\{W_d(\cdot,\cdot)\}_{d=1,\ldots,D}$ are mutually independent Brownian sheets, i.e., zero-mean Gaussian processes with covariance function $\operatorname{Cov}\big(W_d(x_1, y_1), W_d(x_2, y_2)\big)=min(x_1, x_2)min(y_1, y_2)$. Computing the covariance function of $\{W_d(2-2z, 2z)\}_{z\in[0,1]}$, we find that $\{W_d(2-2z, 2z)\}_{z\in[0,1]}\overset{d}{=}\{2B_d(z)\}_{z\in[0,1]}$, where $B_d(\cdot)$ is a standard Brownian bridge. Due to this,
\[
sum_{d=1}^{D}\int_{[\iota,1-\iota]}W_d^2(2-2z, 2z)\,mathrm{d} z\overset{d}{=}4sum_{d=1}^{D}\int_{[\iota,1-\iota]}B_d^2(z)\,mathrm{d} z
\]
and, hence, the limit is as claimed.
So it remains to show \eqref{eq:to show}. Decompose
\begin{align}
M_n(\bm x,\boldsymbol{0}) &= \frac{1}{k}sum_{t=d+1}^{n}I_{\{U_t>b(\frac{n}{kx}),\ U_{t-d}>b(\frac{n}{ky})\}} \notag\\
&= \frac{1}{k}sum_{t=d+1}^{n}I_{\{U_t>b(\frac{n}{kx}),\ U_{t-d}>b(\frac{n}{ky})\}}-\frac{1}{k}sum_{t=d+1}^{n}I_{\{U_t>e^{\xi_1/sqrt{k}}b(\frac{n}{kx}),\ U_{t-d}>e^{\xi_2/sqrt{k}}b(\frac{n}{ky})\}}\notag\\
&\hspace{0.4cm} +\frac{1}{k}sum_{t=d+1}^{n}I_{\{U_t>e^{\xi_1/sqrt{k}}b(\frac{n}{kx}),\ U_{t-d}>e^{\xi_2/sqrt{k}}b(\frac{n}{ky})\}} - \frac{1}{k}sum_{t=d+1}^{n}I_{\{\widehat{U}_t>e^{\xi_1/sqrt{k}}b(\frac{n}{kx}),\ \widehat{U}_{t-d}>e^{\xi_2/sqrt{k}}b(\frac{n}{ky})\}}\notag\\
&\hspace{0.4cm} +\frac{1}{k}sum_{t=d+1}^{n}I_{\{\widehat{U}_t>e^{\xi_1/sqrt{k}}b(\frac{n}{kx}),\ \widehat{U}_{t-d}>e^{\xi_2/sqrt{k}}b(\frac{n}{ky})\}}\notag\\
&=o_{\operatorname{P}}(n^{-1/2}) + \widehat{M}_n(\bm x,\bm \xi),\label{eq:(2.1)}
\end{align}
where the $o_{\operatorname{P}}(n^{-1/2})$-term is uniform on $\bm \xi\in[-K,K]^2$ and $\bm x\in[\iota,1]^2$ by Propositions~\ref{prop:Mn unif x xi} and~\ref{prop:hat unif convergence}.
Define $\bm \xi_n:=\bm \xi_n(\bm x):=\big(\xi_n(x),\ \xi_n(y)\big)^\operatorname{P}rime$, such that $\widehat{M}_n(\bm x,\bm \xi_n)=\frac{1}{k}sum_{t=d+1}^{n}I_{\{\widehat{U}_t>\widehat{U}_{(\lfloor kx\rfloor+1)},\ \widehat{U}_{t-d}>\widehat{U}_{(\lfloor ky\rfloor+1)}\}}$. Then, to show \eqref{eq:to show} or, equivalently, that $sup_{\bm x\in[\iota,1]^2}|\widehat{M}_n(\bm x,\bm \xi_n)-M_n(\bm x,\boldsymbol{0})|=o_{\operatorname{P}}(n^{-1/2})$, we have to prove that for any $\varepsilon>0$ and $\delta>0$
\[
\operatorname{P}\left\{sqrt{n}sup_{\bm x\in[\iota,1]^2}|\widehat{M}_n(\bm x,\bm \xi_n)-M_n(\bm x,\boldsymbol{0})|>\varepsilon\right\}\leq\delta
\]
for sufficiently large $n$. By Proposition~\ref{prop:unif emp an}, we can choose $K>0$ such that $\operatorname{P}\{sup_{\bm x\in[\iota,1]^2}|\bm \xi_n|>K\}\leq\delta/2$ for sufficiently large $n$. Furthermore, by \eqref{eq:(2.1)}, $\operatorname{P}\{sqrt{n}sup_{\bm \xi\in[-K, K]^2}sup_{\bm x\in[\iota,1]^2}|\widehat{M}_n(\bm x,\bm \xi)-M_n(\bm x,\boldsymbol{0})|>\varepsilon\}\leq\delta/2$ for large $n$. Using these two results, we obtain
\begin{align*}
\operatorname{P}&\Bigg\{sqrt{n}sup_{\bm x\in[\iota,1]^2}|\widehat{M}_n(\bm x,\bm \xi_n)-M_n(\bm x,\boldsymbol{0})|>\varepsilon\Bigg\}\\
&\leq\operatorname{P}\Bigg\{sup_{\bm x\in[\iota,1]^2}|\bm \xi_n|>K\Bigg\} + \operatorname{P}\Bigg\{sqrt{n}sup_{\bm \xi\in[-K, K]^2}sup_{\bm x\in[\iota,1]^2}|\widehat{M}_n(\bm x,\bm \xi)-M_n(\bm x,\boldsymbol{0})|>\varepsilon,\ sup_{\bm x\in[\iota,1]^2}|\bm \xi_n|\leq K\Bigg\}\\
&\leq\frac{\delta}{2}+\frac{\delta}{2}=\delta,
\end{align*}
i.e., \eqref{eq:to show}. The conclusion follows.
\end{proof}
subsection{Proofs of Propositions~\ref{prop:base functional convergence}--\ref{prop:unif emp an}}\label{Proofs of Propositions 2}
\begin{proof}[{\textbf{Proof of Proposition~\ref{prop:base functional convergence}:}}]
The corollary on p.~1664 of \citet{BW71} shows that to prove the claim, it suffices to show convergence of the finite-dimensional distributions and tightness. First, to prove convergence of the finite-dimensional distributions, we have to show that for all pairwise distinct $(x_i, y_i)\in[0,1]^2$, $i=1,\ldots,m$,
\begin{equation}\label{eq:(1.1)}
\big(\bm M_n(x_1,y_1),\ldots,\bm M_n(x_m, y_m)\big)^\operatorname{P}rime\overset{d}{\longrightarrow}\big(\bm B(x_1,y_1),\ldots,\bm B(x_m,y_m)\big)^\operatorname{P}rime.
\end{equation}
Second, to prove tightness it suffices by Theorem~26.23 of \citet{Dav94} to do so for each of the marginals separately. To that end, denote the components of $\bm M_n(x,y)$ by $M_n^{(d)}(x,y)$ $(d=1,\ldots,D)$ and define
\begin{align*}
B&= (x_1,x_2]\times (y_1, y_2],\\
M_n^{(d)}(B) &= M_n^{(d)}(x_2,y_2) - M_n^{(d)}(x_1,y_2) - M_n^{(d)}(x_2,y_1) + M_n^{(d)}(x_1,y_1).
\end{align*}
By their Theorem~3 and the subsequent comment together with their Equation~(3), \citet{BW71} show that for tightness it suffices to prove that for $d\in\{1,\ldots,D\}$
\begin{align}
\operatorname{E}\Big[\big\{M_n^{(d)}(B_1)\big\}^2\big\{M_n^{(d)}(B_2)\big\}^2\Big] &\leq K(x_2-x_1)^2(y_2-y)(y-y_1),\label{eq:(B.4m)}\\
\operatorname{E}\Big[\big\{M_n^{(d)}(B_3)\big\}^2\big\{M_n^{(d)}(B_4)\big\}^2\Big] &\leq K(x-x_1)(x_2-x)(y_2-y_1)^2,\label{eq:tight2}
\end{align}
where, for $0\leq x_1<x\leq x_2\leq 1$ and $0\leq y_1<y\leq y_2\leq 1$,
\begin{align*}
B_1&= (x_1,x_2]\times (y_1, y],\quad B_2= (x_1,x_2]\times (y, y_2],\\
B_3&= (x_1,x]\times (y_1, y_2],\quad B_4= (x,x_2]\times (y_1, y_2].
\end{align*}
We first show convergence of the finite-dimensional distributions, i.e., \eqref{eq:(1.1)}. We do so for $m=2$; the case $m>2$ being only notationally more complicated. By the Cram\'{e}r--Wold device \citep[Thm.~25.5]{Dav94} it suffices to show that, as $n\to\infty$,
\begin{equation*}
\bm \lambda_1^\operatorname{P}rime\bm M_n(x_1,y_1)+\bm \lambda_2^\operatorname{P}rime\bm M_n(x_2,y_2)\overset{d}{\longrightarrow}\bm \lambda_1^\operatorname{P}rime\bm B(x_1,y_1)+\bm \lambda_2^\operatorname{P}rime\bm B(x_2,y_2)
\end{equation*}
for any $\bm \lambda=(\bm \lambda_1^\operatorname{P}rime, \bm \lambda_2^\operatorname{P}rime)^\operatorname{P}rime=(\lambda_{1,1},\ldots,\lambda_{1,D},\lambda_{2,1},\ldots,\lambda_{2,D})^\operatorname{P}rime$ with $\bm \lambda^\operatorname{P}rime\bm \lambda=1$. The proof of this convergence is similar to that of Proposition~\ref{prop:base convergence}, and uses Theorem~5.20 of \citet{Whi01}. We re-define
\begin{align}
\overline{sigma}_n^2&=\operatorname{Var}\Big(\bm \lambda_1^\operatorname{P}rime\bm M_n(x_1,y_1)+\bm \lambda_2^\operatorname{P}rime\bm M_n(x_2,y_2)\Big)\notag\\
&= \operatorname{Var}\Big(\bm \lambda_1^\operatorname{P}rime\bm M_n(x_1,y_1)\Big)+\operatorname{Var}\Big(\bm \lambda_2^\operatorname{P}rime\bm M_n(x_2,y_2)\Big)+2\operatorname{Cov}\Big(\bm \lambda_1^\operatorname{P}rime\bm M_n(x_1,y_1),\ \bm \lambda_2^\operatorname{P}rime\bm M_n(x_2,y_2)\Big).\label{eq:(3.1)}
\end{align}
From the proof of Proposition~\ref{prop:base convergence}, we get that
\begin{equation}\label{eq:(C.14)}
\operatorname{Var}\Big(\bm \lambda_i^\operatorname{P}rime\bm M_n(x_i,y_i)\Big)=sum_{d=1}^{D}\lambda_{i,d}^2x_iy_i+o(1) \qquad (i=1,2).
\end{equation}
Thus, it remains to compute the covariance term. Recall from the proof of Proposition~\ref{prop:base convergence} that
\[
Z_{n,t}^{(d)}(x,y)=\begin{cases}0, & t=1,\ldots,d,\\
\frac{n}{k}\Big[I_{\left\{U_t>b(\frac{n}{kx}),\ U_{t-d}>b(\frac{n}{ky})\right\}}
-\Big(\frac{k}{n}\Big)^2xy\Big],& t>d.
\end{cases}
\]
With this
\begin{align}
&\operatorname{Cov}\Big(\bm \lambda_1^\operatorname{P}rime\bm M_n(x_1,y_1),\ \bm \lambda_2^\operatorname{P}rime\bm M_n(x_2,y_2)\Big) \notag\\
&\hspace{1cm}= \operatorname{Cov}\Big(sqrt{n}sum_{d_1=1}^{D}\frac{1}{n}sum_{t=1}^{n}\lambda_{1,d_1}Z_{n,t}^{(d_1)}(x_1,y_1),\ sqrt{n}sum_{d_2=1}^{D}\frac{1}{n}sum_{t=1}^{n}\lambda_{2,d_2}Z_{n,t}^{(d_2)}(x_2,y_2)\Big)\notag\\
&\hspace{1cm}= sum_{d_1=1}^{D}sum_{d_2=1}^{D}\frac{1}{n}\operatorname{Cov}\Big(sum_{t=1}^{n}\lambda_{1,d_1}Z_{n,t}^{(d_1)}(x_1,y_1),\ sum_{t=1}^{n}\lambda_{2,d_2}Z_{n,t}^{(d_2)}(x_2,y_2)\Big).\label{eq:(4.1)}
\end{align}
For $d_1\neq d_2$, we obtain that
\begin{align*}
\frac{1}{n}\operatorname{Cov}&\Big(sum_{t=1}^{n}\lambda_{1,d_1}Z_{n,t}^{(d_1)}(x_1,y_1),\ sum_{t=1}^{n}\lambda_{2,d_2}Z_{n,t}^{(d_2)}(x_2,y_2)\Big)\\
&= \frac{1}{n}sum_{s=1}^{n}sum_{t=1}^{n}\lambda_{1,d_1}\lambda_{2,d_2}\operatorname{Cov}\Big(Z_{n,s}^{(d_1)}(x_1,y_1),\ Z_{n,t}^{(d_2)}(x_2,y_2)\Big)\\
&=\frac{1}{n}\Bigg\{sum_{t=1}^{n}\lambda_{1,d_1}\lambda_{2,d_2}\operatorname{Cov}\Big(Z_{n,t}^{(d_1)}(x_1,y_1),\ Z_{n,t}^{(d_2)}(x_2,y_2)\Big)\\
&\hspace{1cm}+\Bigg(sum_{s-t=d_1} + sum_{t-s=d_2} + sum_{t-s=d_2-d_1} \Bigg)\lambda_{1,d_1}\lambda_{2,d_2}\operatorname{Cov}\Big(Z_{n,s}^{(d_1)}(x_1,y_1),\ Z_{n,t}^{(d_2)}(x_2,y_2)\Big)\Bigg\}\\
&=O(k/n)=o(1)
\end{align*}
by a relation similar to \eqref{eq:cov bound in sum}. This shows that the right-hand side of \eqref{eq:(4.1)} equals
\begin{align*}
sum_{d=1}^{D}\frac{1}{n}&\operatorname{Cov}\Big(sum_{t=1}^{n}\lambda_{1,d}Z_{n,t}^{(d)}(x_1,y_1),\ sum_{t=1}^{n}\lambda_{2,d}Z_{n,t}^{(d)}(x_2,y_2)\Big)+o(1)\\
&=sum_{d=1}^{D}\frac{1}{n}sum_{t=1}^{n}\lambda_{1,d}\lambda_{2,d}\operatorname{Cov}\Big(Z_{n,t}^{(d)}(x_1,y_1),\ Z_{n,t}^{(d)}(x_2,y_2)\Big)\\
&\hspace{1cm}+sum_{d=1}^{D}\frac{1}{n}sum_{|t-s|=d}\lambda_{1,d}\lambda_{2,d}\operatorname{Cov}\Big(Z_{n,s}^{(d)}(x_1,y_1),\ Z_{n,t}^{(d)}(x_2,y_2)\Big)+o(1)\\
&=sum_{d=1}^{D}\lambda_{1,d}\lambda_{2,d}min(x_1,x_2)min(y_1,y_2)+o(1),
\end{align*}
where the last line follows from \eqref{eq:cov bound in sum} and the fact that
\begin{align*}
\operatorname{Cov}&\Big(Z_{n,t}^{(d)}(x_1,y_1),\ Z_{n,t}^{(d)}(x_2,y_2)\Big)=\operatorname{E}\big[Z_{n,t}^{(d)}(x_1,y_1) Z_{n,t}^{(d)}(x_2,y_2)\big]-\operatorname{E}\big[Z_{n,t}^{(d)}(x_1,y_1)\big]\operatorname{E}\big[Z_{n,t}^{(d)}(x_2,y_2)\big]\\
&=\Big(\frac{n}{k}\Big)^2\operatorname{P}\Big\{U_t>b\big(n/[kmin(x_1,x_2)]\big),\ U_{t-d}>b\big(n/[kmin(y_1,y_2)]\big)\Big\}-\Big(\frac{n}{k}\Big)^2\Big(\frac{k}{n}\Big)^2x_1y_1 \Big(\frac{k}{n}\Big)^2x_2y_2\\
&= min(x_1,x_2)min(y_1,y_2)+o(1).
\end{align*}
Overall, we get that
\[
\operatorname{Cov}\Big(\bm \lambda_1^\operatorname{P}rime\bm M_n(x_1,y_1),\ \bm \lambda_2^\operatorname{P}rime\bm M_n(x_2,y_2)\Big)=sum_{d=1}^{D}\lambda_{1,d}\lambda_{2,d}min(x_1,x_2)min(y_1,y_2)+o(1).
\]
Plugging this and \eqref{eq:(C.14)} into \eqref{eq:(3.1)} gives
\begin{equation*}
\overline{sigma}_n^2 = sum_{d=1}^{D}\big[\lambda_{1,d}^2 x_1y_1 + \lambda_{2,d}^2x_2y_2+2\lambda_{1,d}\lambda_{2,d}min(x_1,x_2)min(y_1,y_2)\big]+o(1).
\end{equation*}
The limit of $\overline{sigma}_n^2$ then is easily seen to equal
\begin{align*}
\operatorname{Var}&\big(\bm \lambda_1^\operatorname{P}rime\bm B(x_1,y_1)+\bm \lambda_2^\operatorname{P}rime\bm B(x_2,y_2)\big)\\
&=\operatorname{Var}\big(\bm \lambda_1^\operatorname{P}rime\bm B(x_1,y_1)\big)+\operatorname{Var}\big(\bm \lambda_2^\operatorname{P}rime\bm B(x_2,y_2)\big)+2\operatorname{Cov}\big(\bm \lambda_1^\operatorname{P}rime\bm B(x_1,y_1),\ \bm \lambda_2^\operatorname{P}rime\bm B(x_2,y_2)\big)\\
&=sum_{d=1}^{D}\lambda_{1,d}^2\operatorname{Var}\big(B_d(x_1,y_1)\big) + sum_{d=1}^{D}\lambda_{2,d}^2\operatorname{Var}\big(B_d(x_1,y_1)\big)\\
&\hspace{5cm}+2\operatorname{Cov}\Big(sum_{d=1}^{D}\lambda_{1,d}B_d(x_1, y_1),\ sum_{d=1}^{D}\lambda_{2,d}B_d(x_2, y_2)\Big)\\
&=sum_{d=1}^{D}\big[\lambda_{1,d}^2 x_1y_1 + \lambda_{2,d}^2x_2y_2+2\lambda_{1,d}\lambda_{2,d}min(x_1,x_2)min(y_1,y_2)\big],
\end{align*}
as desired. The remaining conditions of Theorem~5.20 of \citet{Whi01} can be checked easily and, hence, \eqref{eq:(1.1)} follows.
It remains to show tightness of the marginals. We only show \eqref{eq:(B.4m)}, because \eqref{eq:tight2} can be proved in exactly the same manner. Set
\begin{align*}
G_t &= I_{\Big\{U_t\in\big(b(\frac{n}{kx_2}),\ b(\frac{n}{kx_1})\big],\ U_{t-d}\in\big(b(\frac{n}{ky}),\ b(\frac{n}{ky_1})\big]\Big\}}-\Big(\frac{k}{n}\Big)^2(x_2-x_1)(y-y_1)\\
&= I_{\Big\{U_t\in\big(b(\frac{n}{kx_2}),\ b(\frac{n}{kx_1})\big],\ U_{t-d}\in\big(b(\frac{n}{ky}),\ b(\frac{n}{ky_1})\big]\Big\}}-p_{G},\\
H_t &= I_{\Big\{U_t\in\big(b(\frac{n}{kx_2}),\ b(\frac{n}{kx_1})\big],\ U_{t-d}\in\big(b(\frac{n}{ky_2}),\ b(\frac{n}{ky})\big]\Big\}}-\Big(\frac{k}{n}\Big)^2(x_2-x_1)(y_2-y)\\
&= I_{\Big\{U_t\in\big(b(\frac{n}{kx_2}),\ b(\frac{n}{kx_1})\big],\ U_{t-d}\in\big(b(\frac{n}{ky_2}),\ b(\frac{n}{ky})\big]\Big\}}-p_{H}.
\end{align*}
It is easy to check that
\begin{align*}
M_n^{(d)}(B_1) &= M_n^{(d)}(x_2,y) - M_n^{(d)}(x_1,y) - M_n^{(d)}(x_2,y_1) + M_n^{(d)}(x_1,y_1)\\
&=\frac{sqrt{n}}{k}sum_{t=d+1}^{n}G_t,\\
M_n^{(d)}(B_2)&=\frac{sqrt{n}}{k}sum_{t=d+1}^{n}H_t.
\end{align*}
Thus, we can write
\begin{align}
\operatorname{E}&\Big[\big\{M_n^{(d)}(B_1)\big\}^2\big\{M_n^{(d)}(B_2)\big\}^2\Big]\notag\\
&=\frac{n^2}{k^4}sum_{t_1=d+1}^{n}sum_{t_2=d+1}^{n}sum_{t_3=d+1}^{n}sum_{t_4=d+1}^{n}\operatorname{E}[G_{t_1}G_{t_2}H_{t_3}H_{t_4}]\notag\\
&=\frac{n^2}{k^4}\left\{sum_{t=d+1}^{n}\operatorname{E}[G_{t}G_{t}H_{t}H_{t}]+ sum_{t_1\neq t_2}\operatorname{E}[G_{t_1}^2H_{t_2}^2]+2sum_{t_1\neq t_2}\operatorname{E}[G_{t_1}H_{t_1}G_{t_2}H_{t_2}]\right\}+R_n.\label{eq:(7.2)}
\end{align}
Here, the first term in curly brackets on the right-hand side of \eqref{eq:(7.2)} collects those terms that would arise for serially independent $\{(G_t,H_t)^\operatorname{P}rime\}$, whereas the remainder $R_n$ collects those terms that arise due to the $d$-dependence of $\{(G_t,H_t)^\operatorname{P}rime\}$. Exploiting the fact that $G_t$ and $H_t$ are centered Bernoulli-distributed random variables with respective success probabilities $p_G$ and $p_H$, the first term on the right-hand side of \eqref{eq:(7.2)} reduces to (with $\widetilde{n}=n-d$)
\begin{align*}
\frac{n^2}{k^4}&\left\{\widetilde{n}\operatorname{E}[G_1^2H_1^2]+\widetilde{n}(\widetilde{n}-1)\operatorname{E}[G_1^2]\operatorname{E}[H_1^2]+2\widetilde{n}(\widetilde{n}-1)\operatorname{E}^2[G_1H_1]\right\}\\
&=\frac{n^2}{k^4}\left\{\widetilde{n}[p_G^2p_{H}+p_G p_H^2+p_G^2 p_H^2]+\widetilde{n}(\widetilde{n}-1)[p_G p_H(1-p_G)(1-p_H)]+2\widetilde{n}(\widetilde{n}-1)p_G^2p_H^2\right\}\\
&\leq K(x_2-x_1)^2(y_2-y)(y-y_1).
\end{align*}
Lengthy but straightforward calculations establish a similar bound for $R_n$. In sum, we obtain
\[
\operatorname{E}\Big[\big\{M_n^{(d)}(B_1)\big\}^2\big\{M_n^{(d)}(B_2)\big\}^2\Big]\leq K(x_2-x_1)^2(y_2-y)(y-y_1).
\]
This completes the proof.
\end{proof}
\begin{proof}[{\textbf{Proof of Proposition~\ref{prop:Mn unif x xi}:}}]
Consider the supremum over $\bm \xi\in[-K,0]^2$; the cases where $\bm \xi$ lies in the other quadrants can be dealt with similarly. By definition of $M_n(\cdot,\cdot)$, it holds that
\begin{equation}\label{eq:Mn bound}
sup_{\bm \xi\in[-K, 0]^2}sup_{\bm x\in[0,K]^2}|M_n(\bm x,\bm \xi)-M_n(\bm x,\boldsymbol{0})|\leqsup_{\bm x\in[0,K]^2}|M_n(\bm x,-(K,K)^\operatorname{P}rime)-M_n(\bm x,\boldsymbol{0})|.
\end{equation}
Let $\rho>0$. Assume without loss of generality that $K/\rho\inmathbb{N}$. Put
\[
mathcal{L}=\left\{(l_1,l_2)^\operatorname{P}rime\inmathbb{N}_0^2\ :\ (l_1,l_2)^\operatorname{P}rime\in[0, K/\rho]^2\right\}.
\]
Then, bound the right-hand side of \eqref{eq:Mn bound} by
\begin{align*}
&max_{(l_1,l_2)^\operatorname{P}rime\inmathcal{L}}|M_n((l_1,l_2)^\operatorname{P}rime\rho,-(K,K)^\operatorname{P}rime)-M_n((l_1,l_2)^\operatorname{P}rime\rho,\boldsymbol{0})|\\
&\hspace{2cm}+sup_{|\bm x_1-\bm x_2|\leq\rho}\big|\big\{M_n(\bm x_1,-(K,K)^\operatorname{P}rime)-M_n(\bm x_1,\boldsymbol{0})\big\}-\big\{M_n(\bm x_2,-(K,K)^\operatorname{P}rime)-M_n(\bm x_2,\boldsymbol{0})\}\big|\\
&\leqmax_{(l_1,l_2)^\operatorname{P}rime\inmathcal{L}}|M_n((l_1,l_2)^\operatorname{P}rime\rho,-(K,K)^\operatorname{P}rime)-M_n((l_1,l_2)^\operatorname{P}rime\rho,\boldsymbol{0})|+sup_{|\bm x_1-\bm x_2|\leq\rho}|M_n(\bm x_1,\boldsymbol{0})-M_n(\bm x_2,\boldsymbol{0})|\\
&\hspace{2cm}+sup_{|\bm x_1-\bm x_2|\leq\rho}|M_n(\bm x_1,-(K,K)^\operatorname{P}rime)-M_n(\bm x_2,-(K,K)^\operatorname{P}rime)|\\
&=:A_{1n}+B_{1n}+C_{1n}.
\end{align*}
Here, $A_{1n}$ is the maximum of the function $\bm xmapsto|M_n(\bm x,-(K,K)^\operatorname{P}rime)-M_n(\bm x,\boldsymbol{0})|$ on the grid $mathcal{L}$, and $B_{1n}+C_{1n}$ bound the function's maximum absolute increase over a grid quadrangle.
By subadditivity and Proposition~\ref{prop:xi convergence}, we obtain that
\begin{align*}
\operatorname{P}\{sqrt{n}A_{1n}\geq\varepsilon\} &= \operatorname{P}\Big\{\cup_{(l_1,l_2)^\operatorname{P}rime\inmathcal{L}}\big\{sqrt{n}\big|M_n((l_1,l_2)^\operatorname{P}rime\rho,-(K,K)^\operatorname{P}rime)-M_n((l_1,l_2)^\operatorname{P}rime\rho,\boldsymbol{0})\big|\geq\varepsilon\big\}\Big\}\\
&\leqsum_{(l_1,l_2)^\operatorname{P}rime\inmathcal{L}}\operatorname{P}\Big\{sqrt{n}\big|M_n((l_1,l_2)^\operatorname{P}rime\rho,-(K,K)^\operatorname{P}rime)-M_n((l_1,l_2)^\operatorname{P}rime\rho,\boldsymbol{0})\big|\geq\varepsilon\Big\}=o(1).
\end{align*}
Thus, $A_{1n}=o_{\operatorname{P}}(n^{-1/2})$.
Since $B_{1n}=o_{\operatorname{P}}(n^{-1/2})$ can be shown similarly as $C_{1n}=o_{\operatorname{P}}(n^{-1/2})$, we only prove the former. Define
\[
\widetilde{mathcal{L}}=\left\{(l_1,l_2)^\operatorname{P}rime\inmathbb{N}_0^2\ :\ [l_1,l_1+2]\times[l_2,l_2+2]subset[0, K/\rho]^2\right\}.
\]
Use a monotonicity argument to deduce that
\begin{equation*}
B_{1n} \leq max_{(l_1,l_2)^\operatorname{P}rime\in\widetilde{mathcal{L}}}\big|M_n((l_1+2,l_2+2)^\operatorname{P}rime\rho,\boldsymbol{0})-M_n((l_1,l_2)^\operatorname{P}rime\rho,\boldsymbol{0})\big|.
\end{equation*}
Put
\begin{equation*}
I_{3t}:=I_{\Big\{U_t>b\big(\frac{n}{k[l_1+2]\rho}\big),\ U_{t-d}>b\big(\frac{n}{k[l_2+2]\rho}\big)\Big\}}-I_{\Big\{U_t>b\big(\frac{n}{kl_1\rho}\big),\ U_{t-d}>b\big(\frac{n}{kl_2\rho}\big)\Big\}}.
\end{equation*}
For fixed $(l_1,l_2)^\operatorname{P}rime\in\widetilde{mathcal{L}}$, use Markov's inequality to obtain that
\begin{align}
\operatorname{P}&\left\{sqrt{n}\big|M_n((l_1+2,l_2+2)^\operatorname{P}rime\rho,\boldsymbol{0})-M_n((l_1,l_2)^\operatorname{P}rime\rho,\boldsymbol{0})\big|\geq\varepsilon\right\}\notag\\
&=\operatorname{P}\left\{\frac{sqrt{n}}{k}sum_{t=d+1}^{n}I_{3t}\geq\varepsilon\right\}\leq \varepsilon^{-4}\frac{n^2}{k^4}\operatorname{E}\Big[sum_{t=d+1}^{n}I_{3t}\Big]^4.\label{eq:bound grid deriv}
\end{align}
Due to their $d$-dependence, the $I_{3t}$ are trivially $\rho$-mixing of any rate, and thus we may again apply Lemma~2.3 of \citet{Sha93} (for, in his notation, $q=4$) to deduce that
\begin{equation}\label{eq:Shao}
\operatorname{E}\Big[sum_{t=d+1}^{n}I_{3t}\Big]^4\leq n^2 C \big\{\operatorname{E}[I_{3t}^2]\big\}^2+n C \operatorname{E}[I_{3t}^4].
\end{equation}
Since $I_{3t}\in\{0,1\}$ and, hence, $\operatorname{E}[I_{3t}^4]=\operatorname{E}[I_{3t}^2]=\operatorname{E}[I_{3t}]$, it suffices to compute $\operatorname{E}[I_{3t}]$. Use Lemma~\ref{lem:Lem F} to obtain
\begin{align*}
\operatorname{E}[I_{3t}] &=\operatorname{P}\Big\{U_t>b\Big(\frac{n}{k[l_1+2]\rho}\Big)\Big\} \operatorname{P}\Big\{U_{t-d}>b\Big(\frac{n}{k[l_2+2]\rho}\Big)\Big\}\\
&\hspace{3cm}-\operatorname{P}\Big\{U_t>b\Big(\frac{n}{kl_1\rho}\Big)\Big\} \operatorname{P}\Big\{U_{t-d}>b\Big(\frac{n}{kl_2\rho}\Big)\Big\}\\
&=\frac{k^2}{n^2}\big[(l_1+2)(l_2+2)-l_1l_2\big]\rho^2\\
&\leq C\frac{k^2}{n^2}\rho^2.
\end{align*}
Together with \eqref{eq:Shao}, this implies
\[
\operatorname{E}\Big[sum_{t=d+1}^{n}I_{3t}\Big]^4\leq C \frac{k^4}{n^2}\rho^4+C\frac{k^2}{n}\rho^2.
\]
With \eqref{eq:bound grid deriv}, this gives
\[
\operatorname{P}\left\{sqrt{n}\big|M_n((l_1+2,l_2+2)^\operatorname{P}rime\rho,\boldsymbol{0})-M_n((l_1,l_2)^\operatorname{P}rime\rho,\boldsymbol{0})\big|\geq\varepsilon\right\}\leq C\rho^4+C\frac{n}{k^2}\rho^2.
\]
Thus, by subadditivity and using that the cardinality of $\widetilde{mathcal{L}}$ is of the order $\rho^{-2}$, we obtain with Assumption~\ref{ass:k} that
\begin{align*}
\operatorname{P}\{sqrt{n}B_{1n}\geq\varepsilon\} &= \operatorname{P}\Big\{\cup_{(l_1,l_2)^\operatorname{P}rime\inmathcal{L}}\big\{sqrt{n}\big|M_n((l_1+2,l_2+2)^\operatorname{P}rime\rho,\boldsymbol{0})-M_n((l_1,l_2)^\operatorname{P}rime\rho,\boldsymbol{0})\big|\geq\varepsilon\big\}\Big\}\\
&\leqsum_{(l_1,l_2)^\operatorname{P}rime\inmathcal{L}}\operatorname{P}\Big\{sqrt{n}\big|M_n((l_1+2,l_2+2)^\operatorname{P}rime\rho,\boldsymbol{0})-M_n((l_1,l_2)^\operatorname{P}rime\rho,\boldsymbol{0})\big|\geq\varepsilon\Big\}\\
&\leq O(\rho^{-2})\Big[C\rho^4+C\frac{n}{k^2}\rho^2\Big]\leq C\rho^2+o(1).
\end{align*}
Since $\rho$ can be chosen arbitrarily small, it follows that $B_{1n}=o_{\operatorname{P}}(n^{-1/2})$. This ends the proof.
\end{proof}
For the following lemma, recall the definition of $B_t(\bm x,\bm \xi,\eta,\eta_0)$ in \eqref{eq:(A.10+)}. The proof of the lemma is in Appendix~\ref{Auxiliary Lemmas for Proof of Theorem 2}.
\begin{lem}\label{lem:unif Bop1}
Let $\eta>0$, $\eta_0\in\{-1, 1\}$. Then, for any $K>0$,
\begin{equation*}
sup_{\bm \xi\in[-K,K]^2}sup_{\bm x\in[0,K]^2}\left|\frac{1}{k}sum_{t=d+1}^{n}B_t(\bm x,\bm \xi,\eta,\eta_0)\right|=o_{\operatorname{P}}(n^{-1/2}).
\end{equation*}
\end{lem}
\begin{proof}[{\textbf{Proof of Proposition~\ref{prop:hat unif convergence}:}}]
The proof is identical to that of Proposition~\ref{prop:hat convergence} using Lemma~\ref{lem:unif Bop1} instead of Lemma~\ref{lem:Bop1}.
\end{proof}
The next lemma, whose proof is in Appendix~\ref{Auxiliary Lemmas for Proof of Theorem 2}, is required for the proof of Proposition~\ref{prop:unif emp an}.
\begin{lem}\label{lem:univ unif}
For any $0<\iota<K<\infty$, it holds that, as $n\to\infty$,
\[
sup_{x\in[\iota,K]}\Bigg|\frac{1}{sqrt{k}}sum_{t=1}^{n}\Big[I_{\big\{\widehat{U}_t>b(n/[kx])\big\}}-I_{\big\{U_t>b(n/[kx])\big\}}\Big]\Bigg|=o_{\operatorname{P}}(1).
\]
\end{lem}
\begin{proof}[{\textbf{Proof of Proposition~\ref{prop:unif emp an}:}}]
Observe that for all $\xi\inmathbb{R}$
\begin{align}
&sqrt{k}\log\Bigg(\frac{\widehat{U}_{(\lfloor kx\rfloor+1)}}{b(n/[kx])}\Bigg)\leq\xi\quad\Longleftrightarrow \quad \widehat{U}_{(\lfloor kx\rfloor+1)}\leq e^{\xi/sqrt{k}}b(n/[kx])\notag\\
&\Longleftrightarrow\quad sum_{t=1}^{n}I_{\big\{\widehat{U}_t>e^{\xi/sqrt{k}}b(n/[kx])\big\}}\leq kx \notag\\
&\Longleftrightarrow\quad \frac{1}{sqrt{k}}sum_{t=1}^{n}\Big[I_{\big\{\widehat{U}_t>e^{\xi/sqrt{k}}b(n/[kx])\big\}}-\operatorname{P}\big\{U_t>e^{\xi/sqrt{k}}b(n/[kx])\big\}\Big]\leq\alpha\xi x+o(1),\label{eq:fin equiv}
\end{align}
where the $o(1)$-term is uniform in $x\in[\iota,K]$ by Lemma~\ref{lem:Lem F}.
Denote by $D[0,K]$ the space of real-valued functions on $[0,K]$ that are right-continuous with limits from below \citep{Dav94}. Then, it can be proven along similar lines as in Proposition~\ref{prop:base functional convergence} that
\[
\frac{1}{sqrt{k}}sum_{t=1}^{n}\Big[I_{\big\{U_t>b(n/[kx])\big\}}-\operatorname{P}\Big\{U_t>b(n/[kx])\Big\}\Big]\overset{d}{\longrightarrow}B(x)\qquad \text{in }D[0,K],
\]
where $B(\cdot)$ denotes a standard Brownian motion; see also the proof of Proposition~2.1 in \citet{RS97a}. This implies by the continuous mapping theorem that
\[
sup_{x\in[\iota,K]}\Bigg|\frac{1}{sqrt{k}}sum_{t=1}^{n}\Big[I_{\big\{U_t>b(n/[kx])\big\}}-\operatorname{P}\Big\{U_t>b(n/[kx])\Big\}\Big]\Bigg|=O_{\operatorname{P}}(1).
\]
From this, Lemma~\ref{lem:univ unif} and Assumption~\ref{ass:U distr 2} we deduce that
\[
sup_{x\in[\iota,K]}\Bigg
|\frac{1}{sqrt{k}}sum_{t=1}^{n}\Big[I_{\big\{\widehat{U}_t>e^{\xi/sqrt{k}}b(n/[kx])\big\}}-\operatorname{P}\Big\{U_t>e^{\xi/sqrt{k}}b(n/[kx])\Big\}\Big]\Bigg|=O_{\operatorname{P}}(1).
\]
Combining this with \eqref{eq:fin equiv}, the conclusion follows.
\end{proof}
subsection{Proofs of Lemmas~\ref{lem:unif Bop1}--\ref{lem:univ unif}}\label{Auxiliary Lemmas for Proof of Theorem 2}
\begin{proof}[{\textbf{Proof of Lemma~\ref{lem:unif Bop1}:}}]
It suffices to consider the supremum over $\bm \xi\in[-K,0]^2$; the cases where $\bm \xi$ lies in the other quadrants can be dealt with similarly. Using the same grid $mathcal{L}$ as in the proof of Proposition~\ref{prop:Mn unif x xi}, we get that
\begin{align*}
sup_{\bm \xi\in[-K,0]^2}&sup_{\bm x\in[0,K]^2}\left|\frac{1}{k}sum_{t=d+1}^{n}B_t(\bm x,\bm \xi,\eta,\eta_0)\right|\leqsup_{\bm x\in[0,K]^2}\left|\frac{1}{k}sum_{t=d+1}^{n}B_t(\bm x,-(K,K)^\operatorname{P}rime,\eta,\eta_0)\right|\\
&\leqmax_{(l_1,l_2)^\operatorname{P}rime\inmathcal{L}}\left|\frac{1}{k}sum_{t=d+1}^{n}B_t((l_1, l_2)^\operatorname{P}rime\rho,-(K,K)^\operatorname{P}rime,\eta,\eta_0)\right|\\
&\hspace{2cm}+sup_{|\bm x_1-\bm x_2|\leq\rho}\left|\frac{1}{k}sum_{t=d+1}^{n}B_t(\bm x_1,-(K,K)^\operatorname{P}rime,\eta,\eta_0)-B_t(\bm x_2,-(K,K)^\operatorname{P}rime,\eta,\eta_0)\right|\\
&=:A_{2n}+B_{2n}.
\end{align*}
By subadditivity and Lemma~\ref{lem:Bop1}, we obtain that
\begin{align*}
\operatorname{P}\{sqrt{n}A_{2n}\geq\varepsilon\} &= \operatorname{P}\Bigg\{\bigcup_{(l_1,l_2)^\operatorname{P}rime\inmathcal{L}}\Big\{sqrt{n}\Big|\frac{1}{k}sum_{t=d+1}^{n}B_t\big((l_1,l_2)^\operatorname{P}rime\rho,-(K,K)^\operatorname{P}rime,\eta,\eta_0\big)\Big|\geq\varepsilon\Big\}\Bigg\}\\
&\leqsum_{(l_1,l_2)^\operatorname{P}rime\inmathcal{L}}\operatorname{P}\Bigg\{sqrt{n}\Big|\frac{1}{k}sum_{t=d+1}^{n}B_t\big((l_1,l_2)^\operatorname{P}rime\rho,-(K,K)^\operatorname{P}rime,\eta,\eta_0\big)\Big|\geq\varepsilon\Bigg\}=o(1).
\end{align*}
Thus, $A_{2n}=o_{\operatorname{P}}(n^{-1/2})$.
To show that $B_{2n}=o_{\operatorname{P}}(n^{-1/2})$, use a monotonicity argument to deduce that
\begin{equation*}
B_{2n} \leq max_{(l_1,l_2)^\operatorname{P}rime\in\widetilde{mathcal{L}}}\left|\frac{1}{k}sum_{t=d+1}^{n}B_t\big((l_1+2,l_2+2)^\operatorname{P}rime\rho,-(K,K)^\operatorname{P}rime,\eta,\eta_0\big)-B_t\big((l_1,l_2)^\operatorname{P}rime\rho,-(K,K)^\operatorname{P}rime,\eta,\eta_0\big)\right|,
\end{equation*}
where $\widetilde{mathcal{L}}$ is again defined as in the proof of Proposition~\ref{prop:Mn unif x xi}. Let $\eta_0=1$; the case $\eta_0=-1$ can be treated analogously. Define $w_t$ and $\nu=\nu(\varepsilon_0)>0$ as in the proof of Lemma~\ref{lem:Bop1}. In particular, if $w_t=1$ then
\begin{equation*}
1-\nu/2<(1+s_{t})^{-1}\leq1\qquad\text{and}\qquad 1-\nu/2<(1+s_{t-d})^{-1}\leq1.
\end{equation*}
For sufficiently large $n$, such that $(1-\nu/2)<\big[1-e^{K/sqrt{k}}\varepsilon_0/b(n/[k K])\big]\leq1$,
\begin{align*}
&w_t\big|B_t\big((l_1+2,l_2+2)^\operatorname{P}rime\rho,-(K,K)^\operatorname{P}rime,\eta,\eta_0=1\big)-B_t\big((l_1,l_2)^\operatorname{P}rime\rho,-(K,K)^\operatorname{P}rime,\eta,\eta_0=1\big)\big| \\
&=w_t\Big|I_{\Big\{U_t> e^{-K/sqrt{k}}(1+s_{t})^{-1}\big(1-e^{K/sqrt{k}}\frac{m_{t}}{b(n/[k(l_1+2)\rho])}\big)b\big(\frac{n}{k[l_1+2]\rho}\big),}\\
& \hspace{1.5cm} _{U_{t-d}> e^{-K/sqrt{k}}(1+s_{t-d})^{-1}\big(1-e^{K/sqrt{k}}\frac{m_{t-d}}{b(n/[k(l_2+2)\rho])}\big)b\big(\frac{n}{k[l_2+2]\rho}\big)\Big\}}\\
&\hspace{8.4cm}-I_{\Big\{U_t> e^{-K/sqrt{k}}b\big(\frac{n}{k[l_1+2]\rho}\big),\ U_{t-d}> e^{-K/sqrt{k}}b\big(\frac{n}{k[l_2+2]\rho}\big)\Big\}}\\
&\hspace{0.5cm}-I_{\Big\{U_t> e^{-K/sqrt{k}}(1+s_{t})^{-1}\big(1-e^{K/sqrt{k}}\frac{m_{t}}{b(n/[kl_1\rho])}\big)b\big(\frac{n}{kl_1\rho}\big),}\\
&\hspace{1.5cm} _{U_{t-d}> e^{-K/sqrt{k}}(1+s_{t-d})^{-1}\big(1-e^{K/sqrt{k}}\frac{m_{t-d}}{b(n/[kl_2\rho])}\big)b\big(\frac{n}{kl_2\rho}\big)\Big\}}+I_{\Big\{U_t> e^{-K/sqrt{k}}b\big(\frac{n}{kl_1\rho}\big),\ U_{t-d}> e^{-K/sqrt{k}}b\big(\frac{n}{kl_2\rho}\big)\Big\}}\Big|\\
&\leq \Big|I_{\Big\{U_t> (1-\nu)b\big(\frac{n}{k[l_1+2]\rho}\big),\ U_{t-d}> (1-\nu)b\big(\frac{n}{k[l_2+2]\rho}\big)\Big\}}-I_{\Big\{U_t> b\big(\frac{n}{k[l_1+2]\rho}\big),\ U_{t-d}> b\big(\frac{n}{k[l_2+2]\rho}\big)\Big\}}\Big|\\
&\hspace{1cm}+\Big|I_{\Big\{U_t> (1-\nu)b\big(\frac{n}{kl_1\rho}\big),\ U_{t-d}> (1-\nu)b\big(\frac{n}{kl_2\rho}\big)\Big\}}-I_{\Big\{U_t> b\big(\frac{n}{kl_1\rho}\big),\ U_{t-d}> b\big(\frac{n}{kl_2\rho}\big)\Big\}}\Big|\\
&=:I_{4t}(l_1,l_2)+I_{5t}(l_1,l_2).
\end{align*}
Since $w_{\ell_{n}}=\ldots=w_n=1$ w.p.a.~1 as $n\to\infty$, to prove $B_{2n}=o_{\operatorname{P}}(n^{-1/2})$, it suffices to show that
\begin{equation}\label{eq:(C.21)}
max_{(l_1,l_2)^\operatorname{P}rime\in\widetilde{mathcal{L}}}\frac{1}{k}sum_{t=d+1}^{n}I_{jt}(l_1,l_2)=o_{\operatorname{P}}(n^{-1/2})\qquad(j=4,5).
\end{equation}
We only show this claim for $j=5$, as that for $j=4$ can be proven along similar lines. By subadditivity and Markov's inequality,
\begin{align*}
\operatorname{P}\Bigg\{sqrt{n}max_{(l_1,l_2)^\operatorname{P}rime\in\widetilde{mathcal{L}}}\frac{1}{k}sum_{t=d+1}^{n}I_{5t}(l_1,l_2)\geq\varepsilon\Bigg\}&=\operatorname{P}\Bigg\{\bigcup_{(l_1,l_2)^\operatorname{P}rime\in\widetilde{mathcal{L}}}\Big\{\frac{sqrt{n}}{k}sum_{t=d+1}^{n}I_{5t}(l_1,l_2)\geq\varepsilon\Big\}\Bigg\}\\
&=sum_{(l_1,l_2)^\operatorname{P}rime\in\widetilde{mathcal{L}}}\operatorname{P}\Big\{\frac{sqrt{n}}{k}sum_{t=d+1}^{n}I_{5t}(l_1,l_2)\geq\varepsilon\Big\}\\
&\leqsum_{(l_1,l_2)^\operatorname{P}rime\in\widetilde{mathcal{L}}}\varepsilon^{-4}\frac{n^2}{k^4}\operatorname{E}\Big[sum_{t=d+1}^{n}I_{5t}(l_1,l_2)\Big]^4.
\end{align*}
Use Lemma~\ref{lem:Lem F} to conclude that for sufficiently small $\nu>0$,
\begin{align*}
\operatorname{E}[I_{5t}(l_1,l_2)]&=\operatorname{P}\Big\{U_t> (1-\nu)b\big(\frac{n}{kl_1\rho}\big)\Big\}\operatorname{P}\Big\{U_{t-d}> (1-\nu)b\big(\frac{n}{kl_2\rho}\big)\Big\}\\
&\hspace{5cm}-\operatorname{P}\Big\{U_t> b\big(\frac{n}{kl_1\rho}\big)\Big\}\operatorname{P}\Big\{U_{t-d}> b\big(\frac{n}{kl_2\rho}\big)\Big\}\\
&=\frac{k^2}{n^2}\rho^2 l_1 l_2\Big[1+\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu o(1)\Big]^2-\frac{k^2}{n^2}\rho^2 l_1 l_2\\
&\leq C\frac{k^2}{n^2}\rho^2 l_1 l_2\Big[\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu o(1)\Big]\\
&\leq C\frac{k^2}{n^2}\rho^2\nu.
\end{align*}
Arguing similarly as after \eqref{eq:Shao}, we obtain
\[
\operatorname{E}\Big[sum_{t=d+1}^{n}I_{5t}(l_1,l_2)\Big]^4\leq C \frac{k^4}{n^2}\rho^4\nu+C\frac{k^2}{n}\rho^2\nu.
\]
Using Assumption~\ref{ass:k} and that the cardinality of $\widetilde{mathcal{L}}$ is of the order $\rho^{-2}$, we get from this that
\[
\operatorname{P}\Bigg\{sqrt{n}max_{(l_1,l_2)^\operatorname{P}rime\in\widetilde{mathcal{L}}}\frac{1}{k}sum_{t=d+1}^{n}I_{5t}(l_1,l_2)\geq\varepsilon\Bigg\} \leq C\rho^2\nu + C\frac{n}{k^2}\nu\leq C\rho^2\nu,
\]
which can be made arbitrarily small by a suitable choice of $\nu$. This establishes \eqref{eq:(C.21)}, concluding the proof.
\end{proof}
The following lemmas are required for the proof of Lemma~\ref{lem:univ unif}. Set
\begin{align*}
C_t(x,\bm \theta) &= I_{\big\{\widehat{U}_t(\bm \theta)>b(n/[kx])\big\}},\\
C_t(x) &= I_{\big\{U_t>b(n/[kx])\big\}},\\
C_t(x,\eta,\eta_0) &= I_{\left\{U_t[1+\eta_0s_{t}]+\eta_0m_t > b(n/[kx])\right\}},\qquad\eta_0\in\{-1, 1\},\\
D_t(x,\eta,\eta_0) &= C_t(x,\eta,\eta_0) - C_t(x),
\end{align*}
where $m_t=m_{n,t}(\eta)\geq0$ and $s_t=s_{n,t}(\eta)\geq0$ are from Assumption~\ref{ass:UA}.
\begin{lem}\label{lem:Ct bound}
Let $\eta>0$. Then, w.p.a.~1, as $n\to\infty$,
\begin{equation*}
C_t(x,\eta,-1) \leq C_t(x,\bm \theta) \leq C_t(x, \eta,1)
\end{equation*}
for all $\bm \theta\in N_n(\eta)$ and $t=\ell_{n},\ldots,n$.
\end{lem}
\begin{proof}
The proof closely resembles that of Lemma~\ref{lem:wpa1} and, hence, is omitted.
\end{proof}
\begin{lem}\label{lem:Dt unif}
Let $\eta>0$ and $\eta_0\in\{-1, 1\}$. Then, for any $0<\iota<K<\infty$,
\[
sup_{x\in[\iota,K]}\Bigg|\frac{1}{sqrt{k}}sum_{t=1}^{n}D_t(x,\eta,\eta_0)\Bigg|=o_{\operatorname{P}}(1).
\]
\end{lem}
\begin{proof}
The outline of the proof is similar to that of Lemma~\ref{lem:Bop1}. Let $\ell_{n}\to\infty$ with $\ell_{n}=o(sqrt{k})$. Consider the case $\eta_0=1$; the case $\eta_0=-1$ can be dealt with similarly. Define
\[
v_t=I_{\left\{m_{t}\leq\varepsilon_0,\ s_{t}\leq\varepsilon_0\right\}},\qquad\varepsilon_0>0.
\]
If $v_t=1$, there exists $\nu=\nu(\varepsilon_0)>0$, such that for $s_{t}$ from Assumption~\ref{ass:UA} it holds that
\begin{equation*}
1-\nu/2<(1+s_{t})^{-1}\leq1.
\end{equation*}
Then, for sufficiently large $n$, such that $1-\nu/2<1-\varepsilon_0/b(n/[kx])\leq1$ for all $x\in[\iota,K]$, we can bound
\begin{align*}
\Big|\frac{1}{sqrt{k}}sum_{t=\ell_{n}}^{n}v_tD_t(x,\eta,\eta_0=1)\Big|&=\Big|\frac{1}{sqrt{k}}sum_{t=\ell_{n}}^{n}v_t\big[I_{\left\{U_t>(1+s_{t})^{-1}(1-m_t/b(n/[kx])) b(n/[kx])\right\}}-I_{\left\{U_t>b(n/[kx])\right\}}\big]\\
&\leq\frac{1}{sqrt{k}}sum_{t=\ell_{n}}^{n}\big[I_{\left\{U_t>(1-\nu) b(n/[kx])\right\}}-I_{\left\{U_t>b(n/[kx])\right\}}\big]\\
&=\frac{1}{sqrt{k}}sum_{t=\ell_{n}}^{n}I_{\left\{U_t\in\big( (1-\nu)b(n/[kx]),\ b(n/[kx])\big]\right\}}\\
&=: \frac{1}{sqrt{k}}sum_{t=\ell_{n}}^{n}I_{6t}(x).
\end{align*}
For $\rho>0$, define $mathcal{M}=\{l\inmathbb{N}_0\ :\ l\in[0, K/\rho]\}$. Then,
\begin{align*}
\frac{1}{sqrt{k}}sum_{t=\ell_{n}}^{n}I_{6t}(x) &= max_{l\inmathcal{M}}\frac{1}{sqrt{k}}sum_{t=\ell_{n}}^{n}I_{6t}(l\rho)+sup_{|x_1-x_2|\leq\rho}\Big|\frac{1}{sqrt{k}}sum_{t=\ell_{n}}^{n}[I_{6t}(x_1)-I_{6t}(x_2)]\Big|\\
&=:A_{3n} + B_{3n}.
\end{align*}
Using in turn subadditivity, Markov's inequality, and Lemma~2.3 of \citet{Sha93}, we get that
\begin{align}
\operatorname{P}\{A_{3n}>\varepsilon\} &\leq sum_{l\inmathcal{M}}\operatorname{P}\Big\{\frac{1}{sqrt{k}}sum_{t=\ell_{n}}^{n}I_{6t}(l\rho)>\varepsilon\Big\}\notag\\
&\leq sum_{l\inmathcal{M}}\frac{1}{k^2}\operatorname{E}\Bigg[sum_{t=\ell_{n}}^{n}I_{6t}(l\rho)\Bigg]^4\notag\\
&\leq sum_{l\inmathcal{M}}\frac{C}{k^2}\Big\{n^2\big\{\operatorname{E}[I_{6t}^{2}(l\rho)]\big\}^2 + n\operatorname{E}[I_{6t}^4(l\rho)]\Big\}.\label{eq:(10.1)}
\end{align}
Since $I_{6t}(l\rho)\in\{0,1\}$, Lemma~\ref{lem:Lem F} implies
\begin{align}
\operatorname{E}[I_{6t}^{4}(l\rho)] &= \operatorname{E}[I_{6t}^{2}(l\rho)] = \operatorname{E}[I_{6t}(l\rho)]\notag\\
&= \operatorname{P}\Big\{U_t\in\big((1-\nu)b(n/[kl\rho]),\ b(n/[kl\rho])\big]\Big\}\notag\\
&= \operatorname{P}\Big\{U_t>(1-\nu)b(n/[kl\rho])\Big\} - \operatorname{P}\Big\{U_t>b(n/[kl\rho])\Big\}\notag\\
&= \frac{kl\rho}{n}\Big\{1+\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu o(1)-1\Big\}\notag\\
&= \frac{kl\rho}{n}\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu l\rho o\Big(\frac{k}{n}\Big)\label{eq:(10.2)}\\
&\leq C\nu\rho\frac{k}{n}.\notag
\end{align}
Plugging this into \eqref{eq:(10.1)} gives
\begin{align*}
\operatorname{P}\{A_{3n}>\varepsilon\} &\leq sum_{l\inmathcal{M}}C\left\{\frac{n^2}{k^2}[\nu\rho]^2\frac{k^2}{n^2}+\frac{n}{k^2}\nu\rho\frac{k}{n}\right\}\\
&\leq C \left\{\nu^2\rho^2 + \frac{\nu}{k}\rho\right\},
\end{align*}
which can be made arbitrarily small by a suitable choice of $\nu$. Thus, $A_{3n}=o_{\operatorname{P}}(1)$.
To show $B_{3n}=o_{\operatorname{P}}(1)$, define
\[
\widetilde{mathcal{M}} = \{l\inmathbb{N}_0\ :\ [l, l+2]subset[0,K/\rho]\}.
\]
Use a monotonicity argument to deduce that
\[
B_{3n}\leqmax_{l\in\widetilde{mathcal{M}}}\Big|\frac{1}{sqrt{k}}sum_{t=\ell_{n}}^{n}\big[I_{6t}([l+2]\rho)-I_{6t}(l\rho)\big]\Big|=:max_{l\in\widetilde{mathcal{M}}}\Big|\frac{1}{sqrt{k}}sum_{t=1}^{n}I_{7t}(l,\rho)\Big|.
\]
We get by Markov's inequality that
\begin{equation}\label{eq:(11.1)}
\operatorname{P}\Bigg\{\Big|\frac{1}{sqrt{k}}sum_{t=\ell_{n}}^{n}I_{7t}(l,\rho)\Big|>\varepsilon\Bigg\}\leq\frac{\varepsilon^{-4}}{k^2}\operatorname{E}\Big[sum_{t=\ell_{n}}^{n}I_{7t}(l,\rho)\Big]^4.
\end{equation}
Since, by \eqref{eq:(10.2)},
\begin{align*}
\operatorname{E}\Big[I_{7t}(l,\rho)\Big] &= 2\rho\frac{k}{n}\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+2\rho\nu o(k/n)\leq C\rho\nu\frac{k}{n},
\end{align*}
we obtain from Lemma~2.3 of \citet{Sha93} that
\begin{align*}
\operatorname{E}\Big[sum_{t=\ell_{n}}^{n}I_{7t}(l,\rho)\Big]^4 &\leq C\left\{n^2 \big\{\operatorname{E}[I_{7t}(l,\rho)]^2\big\}^2 + n \operatorname{E}[I_{7t}(l,\rho)]^4 \right\} \\
&\leq C \Big\{ n^2\Big[\rho\nu\frac{k}{n}\Big]^2 + n\Big[C\rho\nu\frac{k}{n}\Big]\Big\}.
\end{align*}
Thus, the right-hand side of \eqref{eq:(11.1)} can be bounded by $C\rho^2\nu^2+\frac{C}{k}\rho\nu$. Using this, we obtain
\begin{align*}
\operatorname{P}\{B_{3n}>\varepsilon\} &\leq sum_{l\in\widetilde{mathcal{M}}}\operatorname{P}\Big\{\Big|\frac{1}{sqrt{k}}sum_{t=\ell_n}^{n}I_{7t}(l,\rho)\Big|>\varepsilon\Big\}\\
&= sum_{l\in\widetilde{mathcal{M}}} \Big[C\rho^2\nu^2+\frac{C}{k}\rho\nu\Big]\\
&\leq C\{\rho\nu^2+\nu/k\},
\end{align*}
where we have used in the final step that the cardinality of $\widetilde{mathcal{M}}$ is of the order $1/\rho$. This proves
\[
\Big|\frac{1}{sqrt{k}}sum_{t=\ell_{n}}^{n}v_tD_t(x,\eta,\eta_0)\Big| = o_{\operatorname{P}}(1),
\]
since $\nu>0$ can be chosen arbitrarily small. By Assumption~\ref{ass:UA}, we have $v_{\ell_{n}}=\ldots=v_n=1$ w.p.a.~1, as $n\to\infty$. Hence, the above display also implies that
\[
\Big|\frac{1}{sqrt{k}}sum_{t=\ell_{n}}^{n}D_t(x,\eta,\eta_0)\Big| = o_{\operatorname{P}}(1).
\]
Moreover, by boundedness of the $D_t(x,\eta,\eta_0)$, we easily get $\Big|\frac{1}{sqrt{k}}sum_{t=1}^{\ell_{n}-1}D_t(x,\eta,\eta_0)\Big|=O(\ell_{n}/sqrt{k})=o(1)$. Putting these two results together, the conclusion follows.
\end{proof}
Now, we are in a position to prove Lemma~\ref{lem:univ unif}.
\begin{proof}[{\textbf{Proof of Lemma~\ref{lem:univ unif}:}}]
The proof is again similar to that of Proposition~\ref{prop:hat convergence}, where now we use Lemma~\ref{lem:Ct bound} (instead of Lemma~\ref{lem:wpa1}) and Lemma~\ref{lem:Dt unif} (instead of Lemma~\ref{lem:Bop1}).
\end{proof}
section{Simulation Results for Varying $D$}\label{Simulation Results for Varying $D$}
In light of the simulation results in Section~\ref{Simulations}, we pick $k=\lfloor 0.11\cdot n^{0.99}\rfloor$. We do so to focus here on the sensitivity of our tests with respect to the number of included lags $D\in\{1,\ldots,10\}$. We reconsider the models of Section~\ref{Simulations} with misspecified volatility dynamics (in Section~\ref{App: Misspecified Volatility}) and with misspecified innovations (in Section~\ref{App: Misspecified Innovations}).
subsection{Misspecified Volatility}\label{App: Misspecified Volatility}
\begin{table}[t!]
\begin{center}
\begin{tabular}{lllrrrrrrrrrr}
\toprule
$n$ & Test & $\alpha$&multicolumn{10}{c}{$D$} \\[0.25ex]
\cline{4-13}\\[-2.25ex]
& & & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
midrule
500 & $mathcal{P}_n^{(D)}$ & 1\% & 0.3 & 0.5 & 0.6 & 0.5 & 0.5 & 0.6 & 0.6 & 0.5 & 0.4 & 0.4 \\
& & 5\% & 2.6 & 2.5 & 2.5 & 2.5 & 2.2 & 2.3 & 2.1 & 1.9 & 1.8 & 1.7 \\
& & 10\% & 8.1 & 5.4 & 5.3 & 4.8 & 5.0 & 4.9 & 4.5 & 4.2 & 3.8 & 3.7 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $mathcal{F}_n^{(D)}$ & 1\% & 0.5 & 0.4 & 0.4 & 0.3 & 0.3 & 0.4 & 0.3 & 0.2 & 0.2 & 0.2 \\
& & 5\% & 2.0 & 1.7 & 1.8 & 1.8 & 1.6 & 1.6 & 1.4 & 1.2 & 1.0 & 0.8 \\
& & 10\% & 4.6 & 3.8 & 3.4 & 3.4 & 3.1 & 3.2 & 2.9 & 2.5 & 2.1 & 1.9 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $\operatorname{LB}_n^{(D)}$ & 1\% & 1.2 & 1.7 & 2.3 & 2.8 & 3.0 & 3.2 & 3.4 & 3.5 & 3.4 & 3.5 \\
& & 5\% & 3.2 & 3.9 & 4.8 & 5.3 & 5.8 & 6.1 & 6.5 & 6.6 & 6.9 & 6.9 \\
& & 10\% & 5.7 & 6.1 & 7.3 & 7.4 & 8.2 & 8.5 & 9.0 & 9.4 & 9.3 & 9.8 \\
midrule
1000& $mathcal{P}_n^{(D)}$ & 1\% & 0.8 & 1.0 & 0.9 & 0.9 & 0.7 & 0.5 & 0.5 & 0.6 & 0.5 & 0.5 \\
& & 5\% & 4.8 & 3.4 & 3.5 & 3.3 & 3.1 & 2.8 & 2.7 & 2.5 & 2.2 & 2.0 \\
& & 10\% & 9.7 & 7.0 & 6.5 & 6.1 & 5.9 & 5.7 & 5.4 & 4.9 & 4.4 & 4.0 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $mathcal{F}_n^{(D)}$ & 1\% & 0.7 & 0.7 & 0.7 & 0.6 & 0.5 & 0.5 & 0.6 & 0.5 & 0.4 & 0.4 \\
& & 5\% & 3.5 & 2.9 & 2.9 & 2.2 & 2.0 & 2.0 & 1.8 & 1.4 & 1.4 & 1.3 \\
& & 10\% & 6.9 & 5.4 & 5.2 & 4.4 & 4.2 & 3.9 & 3.7 & 3.4 & 2.7 & 2.4 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $\operatorname{LB}_n^{(D)}$ & 1\% & 1.3 & 2.0 & 2.5 & 2.5 & 2.9 & 3.0 & 3.3 & 3.7 & 3.7 & 3.7 \\
& & 5\% & 2.7 & 4.0 & 5.0 & 5.3 & 5.7 & 5.7 & 6.2 & 6.5 & 6.8 & 6.5 \\
& & 10\% & 4.8 & 6.4 & 7.2 & 7.6 & 8.3 & 8.5 & 8.6 & 8.7 & 9.1 & 9.3 \\
midrule
2000& $mathcal{P}_n^{(D)}$ & 1\% & 0.9 & 1.0 & 1.1 & 1.1 & 1.0 & 0.9 & 1.0 & 0.9 & 0.8 & 0.7 \\
& & 5\% & 4.2 & 4.0 & 3.8 & 3.6 & 3.4 & 3.0 & 2.9 & 2.8 & 2.5 & 2.6 \\
& & 10\% & 7.9 & 7.7 & 7.3 & 6.9 & 6.5 & 6.2 & 5.8 & 5.4 & 5.1 & 5.2 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $mathcal{F}_n^{(D)}$ & 1\% & 1.0 & 1.3 & 1.3 & 1.3 & 1.1 & 1.2 & 1.0 & 0.9 & 0.8 & 0.8 \\
& & 5\% & 3.9 & 3.5 & 3.6 & 3.5 & 3.2 & 3.0 & 2.7 & 2.3 & 2.2 & 2.1 \\
& & 10\% & 7.4 & 6.8 & 6.2 & 6.3 & 5.8 & 5.1 & 4.6 & 4.5 & 4.1 & 3.7 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $\operatorname{LB}_n^{(D)}$ & 1\% & 1.6 & 2.2 & 3.0 & 3.3 & 3.5 & 3.8 & 3.9 & 4.1 & 4.3 & 4.2 \\
& & 5\% & 3.2 & 4.3 & 5.2 & 5.8 & 6.0 & 6.4 & 6.7 & 7.0 & 7.4 & 7.5 \\
& & 10\% & 4.9 & 6.0 & 7.3 & 7.8 & 8.4 & 8.6 & 9.0 & 9.3 & 9.7 & 9.9 \\
\bottomrule
\end{tabular}
\end{center}
\caption{\label{tab:size}Size in \% of tests based on $mathcal{P}_n^{(D)}$, $mathcal{F}_n^{(D)}$ and $\operatorname{LB}_n^{(D)}$ for significance levels $\alpha\in\{1\%, 5\%, 10\%\}$ and $D=1,\ldots,10$. Results are for model \eqref{eq:APARCH-X(1,1)} with $\bm \theta^{\circ}=\bm \theta^{\circ}_{s}=(0.046, 0.027, 0.092, 0.843, 0)^\operatorname{P}rime$.}
\end{table}
Reconsider the setup of Section~\ref{Misspecified Volatility} in the main paper. The only difference is that we now consider a fixed $k=\lfloor 0.11\cdot n^{0.99}\rfloor$ and varying $D\in\{1,\ldots,10\}$. When $\bm \theta^{\circ}=\bm \theta^{\circ}_{s}$ in \eqref{eq:APARCH-X(1,1)}, the results correspond to size, which is displayed in Table~\ref{tab:size}. As a benchmark, we have again included the $\operatorname{LB}_{n}^{(D)}$-test with the estimation correction of \citet{CF11}. We see that, particularly for small sample sizes, our tests tend to be undersized. Yet, the size distortions substantially decrease with increasing $n$. Except for $D=1$, size is reasonably stable across different choices of the number $D$ of included lags.
\begin{table}[t!]
\begin{center}
\begin{tabular}{lllrrrrrrrrrr}
\toprule
$n$ & Test & $\alpha$&multicolumn{10}{c}{$D$} \\[0.25ex]
\cline{4-13}\\[-2.25ex]
& & & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
midrule
500 & $mathcal{P}_n^{(D)}$ & 1\% & 3.8 & 8.3 & 11.0 & 13.0 & 14.3 & 14.7 & 14.8 & 14.9 & 14.9 & 14.7 \\
& & 5\% & 12.1 & 15.6 & 20.1 & 22.2 & 23.9 & 25.0 & 24.9 & 25.0 & 24.6 & 24.0 \\
& & 10\% & 20.7 & 21.7 & 26.8 & 30.6 & 31.9 & 32.7 & 32.9 & 32.4 & 31.6 & 31.1 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $mathcal{F}_n^{(D)}$ & 1\% & 5.3 & 8.5 & 11.5 & 13.8 & 15.3 & 15.9 & 15.9 & 15.6 & 15.5 & 15.1 \\
& & 5\% & 11.2 & 16.4 & 20.6 & 23.7 & 25.2 & 26.1 & 26.1 & 25.8 & 25.0 & 24.3 \\
& & 10\% & 17.0 & 23.4 & 28.0 & 31.0 & 32.8 & 33.5 & 33.4 & 33.5 & 32.7 & 31.6 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $\operatorname{LB}_n^{(D)}$ & 1\% & 4.3 & 8.0 & 10.6 & 12.5 & 14.1 & 14.8 & 15.1 & 15.4 & 15.9 & 15.5 \\
& & 5\% & 7.3 & 12.3 & 15.2 & 17.6 & 19.5 & 20.8 & 20.9 & 20.8 & 21.1 & 20.9 \\
& & 10\% & 9.6 & 15.1 & 18.9 & 21.2 & 22.9 & 24.1 & 24.5 & 24.7 & 24.7 & 24.2 \\
midrule
1000& $mathcal{P}_n^{(D)}$ & 1\% & 9.5 & 13.8 & 19.9 & 24.0 & 27.4 & 29.8 & 31.2 & 31.4 & 31.6 & 31.0 \\
& & 5\% & 19.1 & 27.0 & 34.1 & 39.0 & 43.5 & 46.2 & 47.0 & 47.3 & 47.6 & 46.9 \\
& & 10\% & 27.2 & 34.1 & 43.0 & 49.1 & 53.3 & 55.6 & 56.5 & 56.8 & 56.8 & 56.6 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $mathcal{F}_n^{(D)}$ & 1\% & 10.0 & 16.9 & 23.9 & 29.9 & 33.7 & 35.9 & 37.6 & 37.7 & 37.4 & 37.1 \\
& & 5\% & 20.0 & 30.4 & 38.7 & 44.9 & 49.4 & 52.3 & 53.2 & 53.9 & 53.9 & 53.4 \\
& & 10\% & 27.5 & 39.0 & 48.5 & 55.5 & 59.2 & 61.1 & 62.1 & 62.2 & 62.3 & 61.8 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $\operatorname{LB}_n^{(D)}$ & 1\% & 5.9 & 10.3 & 13.7 & 15.6 & 17.3 & 18.3 & 18.8 & 19.2 & 19.1 & 19.0 \\
& & 5\% & 9.7 & 15.2 & 19.4 & 22.4 & 23.6 & 25.0 & 25.7 & 25.9 & 26.0 & 25.3 \\
& & 10\% & 12.4 & 18.7 & 23.3 & 26.1 & 27.7 & 28.8 & 29.2 & 29.7 & 29.4 & 29.1 \\
midrule
2000& $mathcal{P}_n^{(D)}$ & 1\% & 14.2 & 27.3 & 39.8 & 49.0 & 55.8 & 60.5 & 62.9 & 64.0 & 64.6 & 65.5 \\
& & 5\% & 26.7 & 44.4 & 58.1 & 67.1 & 73.1 & 76.3 & 78.2 & 79.2 & 79.5 & 79.5 \\
& & 10\% & 33.0 & 54.4 & 67.4 & 75.3 & 80.9 & 83.3 & 84.7 & 85.6 & 85.9 & 85.2 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $mathcal{F}_n^{(D)}$ & 1\% & 18.8 & 35.1 & 50.2 & 61.4 & 68.6 & 72.5 & 74.7 & 75.5 & 76.2 & 76.3 \\
& & 5\% & 31.5 & 52.3 & 67.7 & 77.2 & 82.5 & 85.4 & 87.2 & 87.2 & 87.4 & 87.1 \\
& & 10\% & 41.6 & 61.8 & 76.3 & 84.2 & 88.6 & 90.4 & 91.2 & 92.0 & 92.1 & 91.6 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $\operatorname{LB}_n^{(D)}$ & 1\% & 06.7 & 12.9 & 16.2 & 18.2 & 19.7 & 20.4 & 21.0 & 21.1 & 21.2 & 20.6 \\
& & 5\% & 10.4 & 18.2 & 22.4 & 24.2 & 25.3 & 26.0 & 26.7 & 26.9 & 26.9 & 26.5 \\
& & 10\% & 13.3 & 21.9 & 26.6 & 28.5 & 29.8 & 29.9 & 30.2 & 30.5 & 30.1 & 29.7 \\
\bottomrule
\end{tabular}
\end{center}
\caption{\label{tab:power}Power in \% of tests based on $mathcal{P}_n^{(D)}$, $mathcal{F}_n^{(D)}$ and $\operatorname{LB}_n^{(D)}$ for significance levels $\alpha\in\{1\%, 5\%, 10\%\}$ and $D=1,\ldots,10$. Results are for model \eqref{eq:APARCH-X(1,1)} with $\bm \theta^{\circ}=\bm \theta^{\circ}_{p}=(0.046, 0.027, 0.092, 0.843, 0.089)^\operatorname{P}rime$.}
\end{table}
Comparing our $mathcal{P}_n^{(D)}$- and $mathcal{F}_n^{(D)}$-test with the Ljung--Box test, we find that all tests are comparable in their sensitivity to the choice of $D$. Furthermore, while our tests may have inferior size at the 5\%- and 10\%-level for small samples, size tends to be better overall for $n=2000$. E.g., for the $\operatorname{LB}_n^{(D)}$-test, the empirical rejection frequencies at the 1\%-level vary between 1.6\% an 4.3\% for $n=2000$, while the fluctuations for the $mathcal{F}_n^{(D)}$-test are much smaller (size between 0.8\% and 1.3\%). The fact that size of our tests improves more for increasing $n$ than for the Ljung--Box test is as expected, because our tests are based on the \textit{tail} copula, thus, requiring reasonably large samples.
\begin{figure}
\caption{From top to bottom: $Y_t$; $z_t$, $sigma_t$ (black) and $\widehat{sigma}
\label{fig:SimTS}
\end{figure}
Now, we turn to a comparison of power by simulating from \eqref{eq:APARCH-X(1,1)} with $\bm \theta^{\circ}=\bm \theta^{\circ}_{p}$. Table~\ref{tab:power} displays the results. As expected, power increases for all tests the larger the sample. For almost all sample sizes and choices of $D$, our tests are more likely than the Ljung--Box test to signal a misspecified model. This difference in power increases markedly in $n$, because the power increase in $n$ is rather modest for $\operatorname{LB}_n^{(D)}$. Table~\ref{tab:power} reveals larger differences in empirical rejection frequencies across different $D$ than Table~\ref{tab:size}. Irrespective of the sample size, the power of our tests reaches the maximum for $D\approx7$. Comparing $mathcal{P}_n^{(D)}$ and $mathcal{F}_n^{(D)}$, we find that clear differences in power favoring $mathcal{F}_n^{(D)}$ only emerge in large samples.
\begin{figure}
\caption{Top: Estimates $\widehat{\rho}
\label{fig:ACF}
\end{figure}
Finally, we illustrate the difference between $\operatorname{LB}_n^{(D)}$ and our tests now for a representative sample from \eqref{eq:APARCH-X(1,1)} with $\bm \theta^{\circ}=\bm \theta^{\circ}_{p}$ and $n=500$. Figure~\ref{fig:SimTS} shows the time series $Y_t$. True volatility $sigma_t=sigma_t(\bm \theta^{\circ})$ and estimated volatility $\widehat{sigma}_t$ (estimated based on the misspecified APARCH(1,1) model) are shown in the middle panel, with the differences $sigma_t-\widehat{sigma}_t$ plotted below. While volatility seems to be captured well by the misspecified model during calm phases, this is not the case in turbulent periods. This, in turn, induces some serial dependence only in the extreme regions of the distribution of $\widehat{\varepsilon}_t$.
To see this more clearly, consider Figure~\ref{fig:ACF}. It shows lag $d$-estimates of the squared residual autocorrelations, $\widehat{\rho}_n^{(d)}$, and the tail copula, $\widehat{\Lambda}_n^{(d)}(1,1)$.\footnote{We have excluded the estimates $\widehat{\rho}_n^{(d)}=\widehat{\Lambda}_n^{(d)}(1,1)=1$ for $d=0$ to zoom in on the relevant autocorrelations in Figure~\ref{fig:ACF}.} The pointwise 95\%-confidence intervals are indicated by the red dashed lines in both panels. None of the autocorrelations are significant at the 5\%-level, yet the $\widehat{\Lambda}_n^{(d)}$ suggest two significant lags of the tail copula. Thus, while the linear dependence in the body of the distribution appears to be negligible, the misspecified volatility estimates nonetheless induce some serial \textit{extremal} dependence in the filtered residuals.
subsection{Misspecified Innovations}\label{App: Misspecified Innovations}
\begin{table}[t!]
\begin{center}
\begin{tabular}{lllrrrrrrrrrr}
\toprule
$n$ & Test & $\alpha$&multicolumn{10}{c}{$D$} \\[0.25ex]
\cline{4-13}\\[-2.25ex]
& & & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
midrule
500 & $mathcal{P}_n^{(D)}$ & 1\% & 1.4 & 2.9 & 3.2 & 3.4 & 3.6 & 3.6 & 3.5 & 3.4 & 3.5 & 3.3 \\
& & 5\% & 5.6 & 5.7 & 6.2 & 6.4 & 6.6 & 6.7 & 6.7 & 6.5 & 6.5 & 6.4 \\
& & 10\% & 12.0 & 9.3 & 9.7 & 9.8 & 9.8 & 10.0 & 10.0 & 9.5 & 9.3 & 9.1 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $mathcal{F}_n^{(D)}$ & 1\% & 2.4 & 3.2 & 3.9 & 4.1 & 4.2 & 4.2 & 4.1 & 4.1 & 4.0 & 3.8 \\
& & 5\% & 5.8 & 6.2 & 6.3 & 6.4 & 6.6 & 6.6 & 6.7 & 6.5 & 6.4 & 6.0 \\
& & 10\% & 9.4 & 9.7 & 9.3 & 9.3 & 9.2 & 8.7 & 8.8 & 8.4 & 8.4 & 8.0 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $\operatorname{LB}_n^{(D)}$ & 1\% & 3.2 & 5.2 & 5.5 & 5.8 & 5.9 & 5.6 & 5.5 & 5.7 & 5.7 & 5.3 \\
& & 5\% & 6.0 & 8.3 & 8.6 & 9.1 & 8.8 & 8.7 & 8.4 & 8.3 & 8.5 & 7.9 \\
& & 10\% & 8.3 & 10.7 & 11.4 & 11.3 & 11.3 & 10.7 & 10.4 & 10.4 & 10.5 & 9.9 \\
midrule
1000& $mathcal{P}_n^{(D)}$ & 1\% & 1.8 & 1.9 & 2.3 & 2.5 & 2.7 & 2.5 & 2.6 & 2.6 & 2.5 & 2.4 \\
& & 5\% & 6.1 & 4.8 & 5.5 & 5.4 & 5.5 & 5.5 & 5.6 & 5.4 & 5.5 & 5.4 \\
& & 10\% & 10.9 & 9.1 & 8.5 & 8.9 & 8.5 & 8.6 & 8.7 & 8.8 & 8.6 & 8.3 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $mathcal{F}_n^{(D)}$ & 1\% & 1.9 & 2.4 & 2.6 & 2.7 & 2.8 & 2.8 & 2.7 & 2.7 & 2.7 & 2.6 \\
& & 5\% & 5.0 & 5.5 & 5.6 & 5.3 & 5.1 & 4.7 & 4.8 & 4.8 & 4.9 & 4.8 \\
& & 10\% & 9.0 & 9.0 & 8.9 & 8.3 & 7.9 & 7.2 & 7.1 & 7.1 & 7.2 & 6.8 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $\operatorname{LB}_n^{(D)}$ & 1\% & 2.3 & 4.4 & 5.1 & 5.2 & 4.7 & 4.5 & 4.6 & 4.7 & 5.2 & 5.1 \\
& & 5\% & 4.2 & 7.3 & 7.7 & 7.3 & 6.3 & 6.3 & 6.7 & 6.8 & 7.3 & 7.3 \\
& & 10\% & 5.7 & 9.3 & 9.7 & 8.8 & 8.2 & 7.8 & 8.1 & 8.4 & 9.0 & 9.1 \\
midrule
2000& $mathcal{P}_n^{(D)}$ & 1\% & 1.1 & 1.6 & 1.7 & 1.7 & 1.7 & 1.5 & 1.6 & 1.7 & 1.7 & 1.5 \\
& & 5\% & 4.6 & 5.2 & 5.0 & 4.7 & 4.5 & 4.4 & 4.2 & 4.1 & 4.0 & 4.0 \\
& & 10\% & 9.7 & 9.2 & 9.1 & 8.7 & 8.3 & 7.5 & 7.6 & 7.4 & 7.2 & 7.0 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $mathcal{F}_n^{(D)}$ & 1\% & 1.8 & 2.0 & 2.0 & 2.0 & 2.1 & 2.0 & 1.9 & 1.9 & 1.8 & 1.8 \\
& & 5\% & 5.9 & 5.6 & 5.3 & 4.8 & 5.0 & 4.4 & 4.2 & 4.1 & 4.0 & 3.7 \\
& & 10\% & 10.2 & 9.4 & 9.2 & 8.2 & 7.8 & 7.3 & 6.8 & 6.3 & 6.3 & 5.9 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $\operatorname{LB}_n^{(D)}$ & 1\% & 1.3 & 3.6 & 4.9 & 4.6 & 4.4 & 3.8 & 3.9 & 3.8 & 4.2 & 4.1 \\
& & 5\% & 2.8 & 5.6 & 7.3 & 6.7 & 6.3 & 5.4 & 5.8 & 5.6 & 6.2 & 6.1 \\
& & 10\% & 4.1 & 7.2 & 8.9 & 8.5 & 7.7 & 6.7 & 7.0 & 7.0 & 7.6 & 7.5 \\
\bottomrule
\end{tabular}
\end{center}
\caption{\label{tab:size2} Size in \% of tests based on $mathcal{P}_n^{(D)}$, $mathcal{F}_n^{(D)}$ and $\operatorname{LB}_n^{(D)}$ for significance levels $\alpha\in\{1\%, 5\%, 10\%\}$ and $D=1,\ldots,10$. Results are for model \eqref{eq:GARCH(1,1)} with $(a_1,b_1,c_1)^\operatorname{P}rime=(-3, 0, 0)^\operatorname{P}rime$ and $(a_2,b_2,c_2)^\operatorname{P}rime=(-1, 0, 0)^\operatorname{P}rime$.}
\end{table}
Now, we reconsider the setup of Section~\ref{Misspecified Innovations} in the main paper with fixed $k=\lfloor 0.11\cdot n^{0.99}\rfloor$ and varying $D\in\{1,\ldots,10\}$. We first report size, where $(a_1,b_1,c_1)^\operatorname{P}rime=(-3, 0, 0)^\operatorname{P}rime$ and $(a_2,b_2,c_2)^\operatorname{P}rime=(-1, 0, 0)^\operatorname{P}rime$ for model \eqref{eq:GARCH(1,1)}. Table~\ref{tab:size2} displays the tests' size, together with that of the estimation effects-corrected Ljung--Box test \citep[Theorem 8.2]{FZ10}. Except perhaps for small samples and $\alpha=1\%$, the size of our tests is very good, and generally better than that of the Ljung--Box test. Moreover, the size of our tests is also very stable across different $D$.
\begin{table}[t!]
\begin{center}
\begin{tabular}{lllrrrrrrrrrr}
\toprule
$n$ & Test & $\alpha$&multicolumn{10}{c}{$D$} \\[0.25ex]
\cline{4-13}\\[-2.25ex]
& & & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
midrule
500 & $mathcal{P}_n^{(D)}$ & 1\% & 47.2 & 51.8 & 50.8 & 48.6 & 47.0 & 45.0 & 44.0 & 42.6 & 40.8 & 39.7 \\
& & 5\% & 63.2 & 61.8 & 61.5 & 59.0 & 56.8 & 55.5 & 53.8 & 52.8 & 51.2 & 50.1 \\
& & 10\% & 71.0 & 67.6 & 66.6 & 65.3 & 63.6 & 61.7 & 59.9 & 58.4 & 57.1 & 55.8 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $mathcal{F}_n^{(D)}$ & 1\% & 55.9 & 56.9 & 55.1 & 53.1 & 50.9 & 49.1 & 47.6 & 46.2 & 44.7 & 43.0 \\
& & 5\% & 66.6 & 67.0 & 65.2 & 62.9 & 60.7 & 59.0 & 57.3 & 55.5 & 54.1 & 52.6 \\
& & 10\% & 72.0 & 72.3 & 70.7 & 68.7 & 66.2 & 64.4 & 62.8 & 61.1 & 59.7 & 58.2 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $\operatorname{LB}_n^{(D)}$ & 1\% & 13.5 & 15.5 & 15.0 & 14.1 & 13.1 & 12.4 & 11.3 & 10.3 & 9.9 & 9.8 \\
& & 5\% & 16.9 & 18.0 & 17.0 & 16.3 & 15.4 & 14.7 & 13.7 & 12.9 & 12.3 & 12.2 \\
& & 10\% & 19.7 & 19.7 & 18.4 & 17.2 & 16.6 & 15.9 & 14.9 & 14.3 & 14.0 & 13.8 \\
midrule
1000& $mathcal{P}_n^{(D)}$ & 1\% & 67.5 & 68.0 & 67.9 & 66.2 & 64.6 & 62.5 & 60.7 & 58.8 & 57.4 & 56.2 \\
& & 5\% & 77.1 & 78.4 & 77.8 & 76.2 & 74.8 & 73.1 & 71.7 & 70.5 & 69.4 & 67.9 \\
& & 10\% & 81.6 & 82.5 & 82.5 & 81.4 & 79.7 & 78.4 & 76.9 & 75.8 & 74.7 & 73.7 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $mathcal{F}_n^{(D)}$ & 1\% & 72.5 & 74.7 & 73.4 & 71.5 & 70.0 & 68.2 & 66.3 & 64.9 & 63.4 & 62.2 \\
& & 5\% & 80.5 & 82.3 & 81.8 & 80.7 & 78.9 & 77.5 & 75.7 & 74.5 & 73.5 & 72.6 \\
& & 10\% & 83.6 & 85.6 & 85.3 & 84.6 & 83.3 & 81.6 & 80.5 & 79.1 & 78.0 & 76.8 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $\operatorname{LB}_n^{(D)}$ & 1\% & 6.2 & 7.5 & 7.7 & 7.7 & 7.5 & 7.2 & 7.1 & 6.9 & 6.7 & 6.6 \\
& & 5\% & 8.1 & 9.0 & 8.9 & 8.9 & 8.7 & 8.5 & 8.3 & 8.1 & 7.9 & 7.7 \\
& & 10\% & 9.3 & 9.8 & 9.7 & 9.4 & 9.2 & 9.2 & 9.0 & 8.8 & 8.7 & 8.5 \\
midrule
2000& $mathcal{P}_n^{(D)}$ & 1\% & 84.9 & 87.8 & 88.0 & 87.1 & 86.2 & 85.5 & 84.8 & 83.8 & 82.7 & 81.9 \\
& & 5\% & 90.6 & 92.6 & 92.4 & 92.4 & 91.7 & 90.8 & 90.1 & 89.4 & 88.9 & 88.4 \\
& & 10\% & 91.9 & 94.4 & 94.6 & 94.3 & 93.8 & 93.5 & 92.5 & 92.2 & 91.6 & 90.8 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $mathcal{F}_n^{(D)}$ & 1\% & 88.9 & 90.6 & 90.6 & 90.1 & 89.4 & 88.4 & 87.9 & 87.4 & 86.9 & 86.7 \\
& & 5\% & 92.5 & 93.9 & 94.1 & 93.9 & 93.4 & 92.6 & 92.2 & 92.0 & 91.4 & 90.9 \\
& & 10\% & 93.9 & 95.6 & 95.6 & 95.4 & 95.3 & 94.5 & 93.9 & 93.8 & 93.5 & 93.0 \\[0.25ex]
\cline{2-13}\\[-2.25ex]
& $\operatorname{LB}_n^{(D)}$ & 1\% & 1.8 & 2.4 & 2.5 & 2.3 & 2.4 & 2.5 & 2.7 & 2.7 & 2.8 & 3.0 \\
& & 5\% & 2.5 & 2.9 & 3.2 & 3.1 & 3.1 & 3.0 & 3.1 & 3.1 & 3.2 & 3.6 \\
& & 10\% & 3.0 & 3.7 & 3.6 & 3.5 & 3.5 & 3.4 & 3.6 & 3.5 & 3.5 & 3.9 \\
\bottomrule
\end{tabular}
\end{center}
\caption{\label{tab:power2}Power in \% of tests based on $mathcal{P}_n^{(D)}$, $mathcal{F}_n^{(D)}$ and $\operatorname{LB}_n^{(D)}$ for significance levels $\alpha\in\{1\%, 5\%, 10\%\}$ and $D=1,\ldots,10$. Results are for model \eqref{eq:GARCH(1,1)} with $(a_1,b_1,c_1)^\operatorname{P}rime=(-3, -6, 0.6)^\operatorname{P}rime$ and $(a_2,b_2,c_2)^\operatorname{P}rime=(-1, -2, 0.6)^\operatorname{P}rime$.}
\end{table}
Now, we turn to a comparison of power by simulating from \eqref{eq:GARCH(1,1)} with $(a_1,b_1,c_1)^\operatorname{P}rime=(-3, -6, 0.6)^\operatorname{P}rime$ and $(a_2,b_2,c_2)^\operatorname{P}rime=(-1, -2, 0.6)^\operatorname{P}rime$. Table~\ref{tab:power2} displays the results. Once more our tests are more likely than the Ljung--Box test to signal a misspecified model for all sample sizes and choices of $D$. This difference in power increases markedly in $n$, because the $\operatorname{LB}_n^{(D)}$-test even \textit{loses} power for increasing $n$. Table~\ref{tab:power2} reveals larger differences in empirical rejection frequencies across different $D$ than Table~\ref{tab:size}. There is a tendency for power to decrease for larger $D$. Yet, that power decrease is only small up until $D=5$. Comparing $mathcal{P}_n^{(D)}$ and $mathcal{F}_n^{(D)}$, we find that there are only minor differences favoring $mathcal{F}_n^{(D)}$.
Summing up the results of Appendices~\ref{App: Misspecified Volatility} and~\ref{App: Misspecified Innovations}, we recommend the $mathcal{F}_n^{(D)}$-test with $D=5$, due to its good size and power. By virtue of its simplicity, we can, however, also recommend the $mathcal{P}_n^{(D)}$-based test as a viable alternative. Although the choice $D=5$ leads to good results in both simulation setups, we recommend---as is common practice for other tests---to also report results for other choices of $D$.
\end{appendices}
\end{document} |
\begin{document}
\title{f Bounded linear operators in PN-spaces}
\begin{abstract}
The manner in which the strong completeness of probabilistic Banach spaces
is frequently exploited depends on the Baire theorem about strong complete
probabilistic metric spaces. By using the Baire theorem, in this paper,
we present the open mapping, closed Graph, Principle of Uniform Boundedness
and Banach-Steinhaus theorems in PN-spaces.
\noindent
{\bf Key words:}
$\check{S}$erstnev probabilistic normed spaces, bounded linear operators, the
open mapping theorem.\\
{\bf Mathematics Subject Classification (2010):} 46S50, 54E70.
\footnote{\noindent $^1$
Email: delavar.Varasteh@gmail.com}
\footnote{ $^2$ Corresponding author:
Mahdi Azhini. Email:m.azhini@srbiau.ac.ir}
\end{abstract}
\section*{1. Introduction}
Probabilistic Normed (briefly PN) spaces were first introduced by
$\check{S}$erstnev in a series of papers [1-4]. Then a new definition was
proposed by Alsina, Schweizer and Sklar [2]. Linear operators in probabilistic
normed spaces were first studied by Lafuerza-Guill$\acute{e}$n,
Rodrignez-Lallena, and Sempi in [5]. In section 2, we recall some notations
and definitions of probabilistic normed spaces according to those of [2] and
[8]. In section 3, we present another way of formulating a PN-space. In
section 4, by using this formulating, we prove all the known classical
theorems of functional analysis, such as Open Mapping and Closed Graph and
Principle of Uniform Boundedness and Banach-Steinhaus theorems in PN-spaces.
\section*{2. Preliminaries}
\paragraph{Definition 2.1.}
A distribution function (briefly a d.f.) is a function
$F:\bar{\mbox{$\mathbb{R}$}}\longrightarrow [0,1]$
that is nondecreasing and left-continuous on $\mbox{$\mathbb{R}$}$, Moreover, $F(-\infty)=0$
and $F(+\infty)=1$. Here $\bar{\mbox{$\mathbb{R}$}}=\mbox{$\mathbb{R}$}\cup\{-\infty,+\infty\}$. The set of
all the d.f.'s will be denoted by $\Delta$ and the subset of those d.f.'s
called distance d.f.'s, such that $F(0)=0$, by $\Delta^+$. We shall also
consider $\mathcal{D}$ and $\mathcal{D}^+$,
the subsets of $\Delta$ and $\Delta^+$,
respectively, formed by the proper d.f.'s, i.e., by those d.f.'s $F\in\Delta$
that satisfy the conditions
$$\lim_{x\longrightarrow -\infty}F(x)=0\quad \text{and} \quad
\lim_{x\longrightarrow
+\infty}F(x)=1.$$
The first of these is obviously satisfied in all of $\Delta^+$,
since, in it,
$F(0)=0$. By setting $F\leq G$ whenever $F(x)\leq G(x)$ for every $x\in\mbox{$\mathbb{R}$}$,
one introduces in natural ordering in $\Delta$ and in $\Delta^+$. The maximal
element for $\Delta^+$ in this order in the d.f., given by
$$H_a(x)=\begin{cases}
0 & \text{if }\quad x\leq a,\\
1 & \text{if} \quad x>a
\end{cases}$$
where $a\in\mbox{$\mathbb{R}$}$.
The space $\Delta$
can be metrized in several way [10], [12], [14], but we shall here
adopt the modified Levy metric $d_L$. If $F$ and $G$ are d.f.'s and $h$ is in
$(0,1]$, let $(F, G;h)$ denote the condition
$$F(x-h)-h\leq G(x)\leq F(x+h)+h\quad \text{for all}~ x\in
(-\frac{1}{h},\frac{1}{h}).$$
Then the modified Levy metric $d_L$ is defined by
$$d_L(F,G)=\inf\{h\in (0,1]:\text{both}(F,G;h)\text{and}(G,F;h)\text{hold}\},$$
under which a sequence of distribution functions $\{F_n\}_{n\in \mbox{$\mathbb{N}$}}$
converges to $F\in\Delta^+$ if and only if at each point $t\in\mbox{$\mathbb{R}$}$, where $F$
is
continuous, $F_n(t)\longrightarrow F(t)$ [7].
\paragraph{Definition 2.2.}
A triangle function is a binary operation on $\Delta^+$, namely a function
$\tau:\Delta^+\times \Delta^+\longrightarrow \Delta^+$ that is associative,
commutative,
nondecreasing and which has $H_0$ as unit, that is, for all $F,G,H\in\Delta^+$,
we have
\\
$\tau(\tau(F,G),H)=\tau(F,\tau(G,H)),$\\
$\tau(F,G)=\tau(G,F),$\\
$\tau(F,H)\leq \tau(G,H), \quad \text{if}\; F\leq G,$\\
$\tau(F,H_0)=F.$
\\
Typical continuous triangle function are convolution and the operators
$\tau_T$ and $\tau_{T^*}$, which are, respectively, given by
\\
$ \tau_T(F,G)(x)= \sup\{T(F(s),G(t))|s+t=x\}\quad \text{and}$\\
$ \tau_{T^*}(F,G)(x)= \inf\{T^*(F(s),G(t))|s+t=x\}$
\\
for all $F,G$ in $\Delta^+$ and all $x\in \mbox{$\mathbb{R}$}$ [7], here $T$ is a continuous
$t$-norm, i.e, a continuous binary operator on $[0,1]$ that is associative,
commutative, nondecreasing and has 1 as identity; $T^*$ is a continuous
$t$-conorm, namely a continuous binary operation on $[0,1]$ that is related to
continuous $t$-norm through
$$T^*(x,y)=1-T(1-x, 1-y).$$
The most important $t$-norms are the functions $W,\Pi$ and $M$ which are
defined, respectively by
$$\begin{array}{l}
W(a,b)=\max\{a+b-1,0\},\\
\Pi(a,b)=a.b,\\
M(a,b)=\min\{a,b\}.
\end{array}$$
Their corresponding $t$-conorms are given, respectively by
$$\begin{array}{l}
W^*(a,b)=\min\{a+b,1\},\\
\Pi^*(a,b)=a+b-ab,\\
M^*(a,b)=\max\{a,b\}.
\end{array}$$
It follows without difficulty that
$$\tau_T(H_a,H_b)=H_{a+b}=\tau_{T^*}(H_a,H_b),$$
for any continuous $t$-norm $T$, any continuous $t$-conorm $T^*$ and any
$a,b\geq 0$.
\paragraph{Definition 2.3.} [5]
A $\check{S}$erstnev PN-space or a $\check{S}$erstnev space is a triple
$(V,\nu,\tau)$, where $V$ is a (real or complex) vector space, $\nu$ is a
mapping from $V$ into $\Delta^+$ and $\tau$ is a continuous triangle function
and
the following conditions are satisfied for all $p$ and $q$ in $V$:
\\
$(N1)\
\nu_p=H_0, \; \text{if and only if,}~ p=\theta (\theta~ \text{is the
null vector in} V);$\\
$(N3) \ \nu_{p+q}\geq \tau(\nu_p,\nu_q);$ \\
$(\check{S})$
$\ \forall \alpha\in\mbox{$\mathbb{R}$}\backslash\{0\}, \forall x\in \bar{\mbox{$\mathbb{R}$}}_+,
\nu_{\alpha p}(x)=\nu_p(\frac{x}{|\alpha|}).$\\
$\ \text{Notice that condition} (\check{S}) \text{implies}$\\
$(N2) \ \forall p\in V, \nu_{-p}=\nu_p.$
\paragraph{Definition 2.4.} [5]
A PN-space is a quadruple $(V,\nu,\tau,\tau^*)$, where $V$ is a real vector
space, $\tau$ and $\tau^*$ are continuous triangle functions such that
$\tau\leq \tau^*$, and the mapping $\nu:V\longrightarrow \Delta^+$ satisfies,
for all $p$
and $q$ in $V$, the conditions:
\\
$(N1)\
\nu_p=H_0, \; \text{if and only if,} p=\theta (\theta\text{ is the null
vector in V)};$\\
$(N2)\ \forall p\in V, \; \nu_p=\nu_p;$\\
$(N3)\ \nu_{p+q}\geq \tau(\nu_p,\nu_q);$\\
$(N4)\ \forall \alpha\in [0,1], \nu_p\leq \tau^*(\nu_{\alpha p},
\nu_{(1-\alpha)p}).$
\\
The function $\nu$ is called the probabilistic norm.
If $\nu$ satisfies the condition, weaker than $(N1), \nu_\theta=H_0$, then
$(V,\nu,\tau,\tau^*)$ is called a Probabilistic Pseudo-Normed space (briefly,
a PPN space).
If $\nu$ satisfies the condition $(N1)$ and $(N2)$, then $(V,\nu,\tau,\tau^*)$
is said to be a Probabilistic seminormed space (briefly, PSN space).
If $\tau=\tau_T$ and $\tau^*=\tau_{T^*}$ for some continuous $t$-norm $T$ and
its $t$-conorm $T^*$, then $(V,\nu,\tau_T,\tau_{T^*})$ is denoted by
$(V,\nu,T)$ and is called a Menger space.
\paragraph{Definition 2.5.}[5]
There is a natural topology on a PN-space $(V,\nu,\tau,\tau^*)$, called the
strong topology, which is defined by the system of neighborhoods
$$N_p(t)=\{q\in V|~ \nu_{p-q}(t)>1-t\}=\{q\in
V|d_L(\nu_{p-q},H_0)<t\},\text{where}~ p\in V\text{and}~ t>0.$$
Let $(V,\nu,\tau,\tau^*)$ be a PN-space, then
\begin{itemize}
\item[i)]
A sequence $\{p_n\}$ in $V$ is said to be strongly convergent to a point $p$
in $V$, and we write $p_n\longrightarrow p$, if for each $t>0$, there exist a
positive
integer $m$ such that $p_n\in N_p(t)$, for $n\geq m$.
\item[ii)]
A sequence $\{p_n\}$ in $V$ is called a strong cauchy sequence if for every
$t>0$, there is a positive integer $N$ such that $\nu_{p_n-p_m}(t)>1-t$,
whenever $m,n>N$.
\item[iii)]
The PN-space $(V,\nu,\tau,\tau^*)$ is said to be distributionally compact
($D$-compact) if every sequence $\{p_n\}$ in $V$ has a convergent subsequence
$\{p_{n_k}\}$. A subset $A$ of a PN-space $(V,\nu,\tau,\tau^*)$ is said to be
$D$-compact if every sequence $\{p_m\}$ in $A$ has a subsequence $\{p_{m_k}\}$
that converges to a point $p\in A$.
\item[iv)]
In the strong topology, the closure $\overline{N_p(t)}$ of $N_p(t)$ is
defined by $\overline{N_p(t)}=N_p(t)\cup N'_p(t)$, where $N'_p(t)$ is the set
of limit points of all convergent sequence in $N_p(t)$.
\end{itemize}
In this paper, we will consider those PN-space for which
$\tau=\tau_M=\tau^*$, which are clearly $\check{S}$erstnev as well as Menger
PN-space also, and will denote such spaces simply by the pair $(V,\nu)$.
\section*{3. Bounded linear operators in SPN-spaces}
\paragraph{Definition 3.1.}
Let $\nabla$ be the set of all nondecreasing and right continuous functions
$f:[0,1]\longrightarrow \bar{\mbox{$\mathbb{R}$}}$, and $\nabla^+$ the subset consisting of
all
non-negative $f\in \nabla$. Suppose $\mathcal{R}^+$in the set of all $f\in
\nabla^+$ with $f(w)<+\infty$, for all $w\in (0,1)$. For $F\in \Delta$, define
$\hat{F}:[0,1]\longrightarrow \mbox{$\mathbb{R}$}$ by
$$\hat{F}(w)=\sup\{t\in \mbox{$\mathbb{R}$}| F(t)<w\}.$$
\paragraph{Theorem 3.2.}[7]
If $F\in \mathcal{D}^+$, then $\hat{F}\in \mathcal{R}^+$. Moreover, for $F$ and
$G$
in
$\Delta^+$,
\begin{itemize}
\item[i)]
if $F\leq G$, then $\hat{F}\geq \hat{G}$;
\item[ii)]
if $\hat{F}\geq \hat{G}$, then $F(t)\leq G(t+h)$ for all $t\in \mbox{$\mathbb{R}$}$ and
$h>0$;
\item[iii)]
$\widehat{(\tau_M(F,G))}=\hat{F}+\hat{G}$;
\item[iv)]
$\widehat{F(\frac{t}{h}))}=h\hat{F}(t), \text{for all } t\in \mbox{$\mathbb{R}$}~ \text{and}
~h>0.$
\end{itemize}
\paragraph{Theorem 3.3.}[7]
The map $\wedge:\Delta^+\longrightarrow \nabla^+$ is one-to-one.
\paragraph{Definition 3.4.}
Let $(V,\nu)$ be a PN-space. The map $V\longrightarrow \mathcal{R}^+$ given
through
the
composition of maps $V\overset{\nu}{\longrightarrow }
\Delta^+\overset{\widehat{ }}{\longrightarrow}\mathcal{R}^+$
will be denoted by $\|\cdot\|$, and for $x\in V$, the value of $\|x\|$ at
$w\in (0,1)$, simply by $\|x\|_w$, i.e, $\|x\|_w=(\widehat{\nu_x})(w)$.
The proof of the following Theorem is obtained, using the definition of
PN-space, Theorem 3.2.
\paragraph{Theorem 3.5.}
Suppose $(V,\nu)$ is a PN-space. The map $\|\cdot\|:V\longrightarrow \nabla^+$
has the
following properties:
\begin{itemize}
\item[i)]
For $x\in V$, $\|x\|\geq 0$, and $\|x\|=0$ if and only if $x=0$.
\item[ii)]
For $x\in V$ and $\alpha\in\mbox{$\mathbb{R}$}$, $\|\alpha x\|=|\alpha|\|x\|.$
\item[iii)]
For $x$ and $y$ in $V$, $\|x+y\|\leq \|x\|+\|y\|.$
\end{itemize}
\paragraph{Remark 3.6.}
Note that all assertion of theorem 3.5 are to be understood point-wise. Thus
for every $w\in (0,1)$, the map $\|\cdot\|_w:V\longrightarrow \mbox{$\mathbb{R}$}$ is a norm
on $V$.
\paragraph{Theorem 3.7.}[12]
Let $(V,\nu)$ be a PN-space. For $p\in V, r>0$, and $w\in (0,1)$ let
$B_w(p;r)$ be defined by
$B_w(p;r)=\{x\in V| \|x-p\|_w<r\}$. Then the family
$\{B_w(p;r)|p\in V, r>0, w\in (0,1)\}$ forms a basis for the strong topology
on $V$.
\paragraph{Theorem 3.8.}[6]
A PN-space $(V,\nu)$ is a topological vector space if and only if $\|x\|\in
\mathcal{R}^+$ for every $x\in V$. Moreover it is seen this topology is locally
convex.\\[4mm]
Now, we denote the strong topology of PN-space $(V,\nu)$ by $\tau$, and the
corresponding dual space by $V^*$. Hence $f\in V^*$ is a continuous
(equivalently bounded because of PN-space $(V,\nu)$ (see [15])) linear
functional on $V$.
\paragraph{Theorem 3.9.}[12]
Let $f:V\longrightarrow \mbox{$\mathbb{R}$}$ be linear. Then $f\in V^*$ if and only if there
is some $w\in
(0,1)$, with
$$\sup_{x\in B_{V,1-w}}|f(x)|<+\infty, \text{where } B_{V,1-w}=\{x\in
V|\|x\|_w\leq 1\}.$$
Moreover if $\|f\|:[0,1)\longrightarrow [0,+\infty)$ is defined by
$\|f\|_w=\inf_{w<w'}(\sup_{x\in B}|f(x)|)$, then $\|f\|$ is a nondecreasing
and right-continuous function, and
$$|f(x)|\leq \|f\|_w\|x\|_{w'}, \text{for all}~
x\in V \text{and}~ w\in (0,1).$$
By theorem 3.9, a linear functional $f:V\longrightarrow \mbox{$\mathbb{R}$}$ belongs to $V^*$
if and only
if $\|f\|_w<+\infty$ for some $w\in (0,1)$.
\\[4mm]
The following theorem introduces a simple way of constructing PN-space. We
recall that for a nondecreasing function $f:(0,1)\longrightarrow \mbox{$\mathbb{R}$}$, the
function
$l^-f:(0,1)\longrightarrow \mbox{$\mathbb{R}$}$ is defined by
$$l^-f(w_0)=\lim_{w\longrightarrow w_0^-}f(w)=\sup_{w<w_0}f(w).$$
Clearly $l^-f$ is nondecreasing and left-continuous. We also denote the
Lebesgue measure on $(0,1)$ by $m$.
\paragraph{Theorem 3.10 [13]}
Let $V$ be a real vector space. Suppose $p:V\times (0,1)\longrightarrow
[0,+\infty)$,
satisfies the following conditions:
\begin{itemize}
\item[i)]
for every $x\in V$, the function $P(x,.):(0,1)\longrightarrow [0,+\infty)$ is
nondecreasing,
\item[ii)]
for every $w\in (0,1)$, the map $p(.,w):V\longrightarrow [0,+\infty)$ is a
semi-norm on
$V$,
\item[iii)]
the family of semi-norms $\{p(.,w)|w\in (0,1)\}$ is separating on $V$.
\end{itemize}
then, there exists a unique map $\nu:V\longrightarrow \Delta^+$, defined by
$$\nu_x(t)=m(\{w\in (0,1)|p(x,w)<t\}),$$
such that $(V,\nu)$ is a PN-space. Moreover, $\|x\|_w=l^-p(x,w)$ for all $x\in
V$ and $w\in (0,1)$.
\paragraph{Theorem 3.11. [13] }
Let $V$ be a real vector space. Suppose $p:V\times(0,1)\longrightarrow
[0,+\infty)$
is a map
which satisfies conditions of theorem 3.10, with $\nu:V\longrightarrow
\Delta^+$
the
corresponding probabilistic norm on $V$. Then the strong topology on $V$ is
induced by the separating family of semi-norm
$\{p (.,w)|w\in (0,1)\cap Q\}$. hence $V$ under the strong topology is a
locally topological vector space.
\paragraph{Remark 3.12.}
Let $(V,\nu)$ be a PN-space, and $p:V\times (0,1)\longrightarrow [0,+\infty)$
be given by
$p(x,w)=\|x\|_w=\hat{\nu}_x(t)$. Then by Remak 3.6 and Theorem 3.11, the
family
$$\{B_{V,w}(0;r)|w\in (0,1), r>0\},$$
where $B_{V,w}(0;r)$ is given by $\{x\in V, \|x-x_0\|_w<r\}$, forms a local
basis for the strong topology. Moreover, a subset $E$ of $V$ is bounded if and
only if the map $p (.,w):V\longrightarrow [0,+\infty)$ is bounded on $E$, for
every $w\in
(0,1)$ (see [16] ).
Using theorem 3.10, we may construct many examples of PN-spaces.
\paragraph{Example 3.13.}
Let $(V,\|\cdot\|)$ be a real normed vector space. If we define
$p:V\times(0,1)\longrightarrow [0,+\infty)$ by
$$p(x,w)=\|x\|\quad \forall x\in V, \quad \forall w\in (0,1),$$
then $p$ satisfies all condition of theorem 3.10. Using $\nu_x(t)=m(\{w\in
(0,1)|p(x,w)<t\})$, we obtain $\nu_x=H_{\|x\|}$.
\paragraph{Example 3.14.}
Let $(V,\nu)$ and $(W,\mu)$ be two PN-spaces. Then, the map $\theta:V\times
W\longrightarrow
\Delta^+$ defined by
$$\theta(x,y)=\tau_M(\nu_x,\nu_y),$$
is a probabilistic norm on $V\times W$, i.e., $(V\times W,\theta)$ is a
PN-space. Because define $p:V\times W\times(0,1)\longrightarrow [0,+\infty)$
by
$$p(x,y,w)=\|x\|_{V,w}+\|y\|_{W,w}$$
where $\|\cdot\|_V:V\longrightarrow \mathcal{R}^+$
and $\|\cdot\|_W:W\longrightarrow
\mathcal{R}^+$ are
the composite
maps given by Definition 3.4. Then the desired is obtained by Theorem 3.10 and
Theorem 3.2 (iii).
\paragraph{Definition 3.15.}
Let $V$ and $W$ be two PN-spaces. We remind that, for topological vector space
$V$ and $W$, a linear operator $T:V\longrightarrow W$ is called bounded if
$T(E)$ is
bounded (in $W$) whenever $E$ is bounded subset of $V$. Metrisability of the
PN-spaces implies that $T$ is bounded if and only if $T$ is continuous (see
[16]
). As usual, the set of all bounded linear operators from $V$ to $W$ is
denoted by $B(V,W)$.
\paragraph{Definition 3.16.}
Let $V$ and $W$ be two PN-spaces.
For linear operator $T:V\longrightarrow W$ and a pair
$(w,w')\in J=\{(w,w')|w,w'\in(0,1)\}\subseteq \mbox{$\mathbb{R}$}^2$,
the norm $\|T\|_{(w,w')}$ is defined by
$$\|T\|_{(w,w')}=\sup\{\|Tx\|_{w'}|~x\in B_{V,w}\}, \text{where}
~B_{V,w}=\{x\in V| \|x\|_w\leq 1\}.$$
\paragraph{Theorem 3.17.} [13]
For a pair $(w,w')\in J$, if $\|T\|_{(w,w')}<+\infty$, then
$$\|Tx\|_{w'}\leq \|T\|_{(w,w')}\|x\|_{w}, \text{for all}~ x\in V.$$
\paragraph{Theorem 3.18.} [13]
Let $V$ and $W$ be two PN-spaces, and $T:V\longrightarrow W$ be a linear
operator. Then
$T\in B(V,W)$ if and only if, for every $w'\in (0,1)$ there exists $w\in
(0,1)$ such that $\|T\|_{(w,w')}<+\infty$.
\section*{4. Main Results}
Baire theorem is one of the two keystones of Linear Functional Analysis, the
other one being the Hahn-Banach theorem. Bair's theorem for reaching
consequences include such basic theorems as the Banach open mapping theorem,
the Banach closed graph theorem and the Banach-Steinhaus theorem. This chapter
concludes consequences of Bair's theorem in PN-spaces.
It is impossible to speak about PN-spaces without making reference to concept
of a Probabilistic Metric space (briefly a PM space).
PM spaces are studied in depth in the fundamental book [7] by Schweizer and
Sklar.
\paragraph{Definition 4.1.}
A triple $(V,\mbox{$\mathcal{F}$},\tau)$, where $V$ is a nonempty set and $\mbox{$\mathcal{F}$}$ is a function from
$V\times V$ into $\Delta^+$ and $\tau$ is a triangle function, is a
Probabilistic Metric space if the following conditions are satisfied for all
points $p,q$ and $r$ in $V$:\\
$(PM1) \ {\mbox{$\mathcal{F}$}}(p,p)=H_0;$\\
$(PM2)\ {\mbox{$\mathcal{F}$}}(p,q)\neq H_0~ \text{if}~ p\neq q;$\\
$(PM3)\ {\mbox{$\mathcal{F}$}}(p,q)={\mbox{$\mathcal{F}$}}(q,p);$\\
$(PM4)\ {\mbox{$\mathcal{F}$}}(p,r)\geq \tau({\mbox{$\mathcal{F}$}}(p,q),{\mbox{$\mathcal{F}$}}(q,r)).$
\\
Notice that, given PN space $(V,\nu)$, a probabilistic metric ${\mbox{$\mathcal{F}$}}:V\times
V\longrightarrow \Delta^+$ is defined via ${\mbox{$\mathcal{F}$}}(p,q)=\nu_{p-q}$, so that the
triple
$(V,{\mbox{$\mathcal{F}$}},\tau)$ becomes a PM space.
\paragraph{Theorem 4.2 (Baire Theorem in PM spaces).[11]}
Let $(V,{\mbox{$\mathcal{F}$}},\tau)$ be a strong complete PM space under a continuous triangle
function $\tau$. If $\{G_n\}$ is a sequence of dense and strongly open subset
of $V$, then $\cap_{n=1}^\infty G_n$ is not empty. (In fact it is dense in
$V$).
\paragraph{Corollary 4.3.}
Let $(V, {\mbox{$\mathcal{F}$}},\tau)$ be a strong complete PM space under a continuous
triangle function $\tau$. Then the following two equivalent properties hold:
\begin{itemize}
\item[a)]
Let $\{f_n\}_{n=0}^\infty$ be a sequence of strongly closed subsets of $V$
such that $\text{int}
F_n=\phi$ for all $n\geq 0$. Then $\text{int}(\cup_{n=0}^\infty
F_n)=\phi$.
\item[b)]
Let $\{G_n\}_{n=0}^\infty$ be a sequence of strongly open subsets of $V$
such that $G_n=V$ for all $n\geq 0$. Then $\overline{\cap_{n=0}^\infty
G_n}=V$.
\end{itemize}
\paragraph{Corollary 4.4.}
Let $(V,{\mbox{$\mathcal{F}$}},\tau)$ be a PM space under a continuous triangle function
$\tau$, and let $F_n$, $n\geq 0$, be strongly closed subsets of $V$ such that
$V=\cup_{n=0}^\infty F_n$.
\begin{itemize}
\item[a)]
If $\text{int} F_n=\phi$ for all $n\geq 0$, then $V$ is not strong complete.
\item[b)]
If $V$ is strong complete, there exists $n_0\geq 0$ such that $\text{int}
F_{n_0}\neq \phi$.
\end{itemize}
\paragraph{Theorem 4.5 (Open Mapping theorem in PN-spaces).}
If $T$ is a probabilistic bounded linear operator from a strong complete
PN-space $(V,\nu)$ onto a strong complete $(W,\mu)$, then $T$ is an open
mapping.
\paragraph{Proof.}
The Theorem will be proved by the following steps: \\
{\bf Step 1:} The set $T(B_{V,w}(0;1))$, where $B_{V,w}(0;r)=\{x\in
V|\|x\|_w<r\}$ for all $r>0$ and $w\in (0,1)$, contains an strong open ball.
Now fix $w\in (0,1)$. Given any $y\in W$,
there exists $x\in V$ such that $y=Tx$, since $T$ is sujective. Since $x\in
B_{V,w}(0;n)$ for some integer $n\geq 1$, this shows that
$W=\bigcup_{n=1}^\infty T(B_{V,w}(0;n))$, hence
$W=\bigcup_{n=1}^\infty \overline{T(B_{V,w}(0;n))}$.
The space $W$ being strong complete, hence Corollary 4.4 (b) shows that there
exists an integer $n_0\geq 1$ such that
$$\text{int}\overline{T(B_{V,w}(0;n_0))}\neq \phi.$$
Therefore $\text{int}\overline{T(B_{V,w}(0;1))}\neq \phi$, since
$\overline{T(B_{V,w}(0;1))}=\frac{1}{n_0}\overline{T(B_{V,w}(0;n_0))}$
by the linearity of $T$. Hence the set $\overline{T(B_{V,w}(0;1))}$ contains
on strong open ball. \\
{\bf Step 2:}
The set $\overline{T(B_{V,w}(0;1))}$ contains an strong open ball centered at
the origin of $W$. By Step 1, there exist $y\in W$ and $s>0$ such that
$B_{W,w}(y;2s)\subset \overline{T(B_{V,w}(0;1))},$ and hence
$$B_{W,w}(0;2s)=\{-y\}+B_{W,w}(y;2s)\subset
\{-y\}+\overline{T(B_{V,w}(0;1)))}.$$
Since $-y\in \overline{T(B_{V,w}(0;1))}$(because $y\in
\overline{T(B_{V,w}(0;1))}$ and $T$ is linear), it follows that
$$\{-y\}+\overline{T(B_{V,w}(0;1))}\subset 2T(B_{V,w}(0;1).$$
{\bf Step 3:}
The set $T(B_{V,w}(0;1))$ contains an strong open ball centered at the origin
of $W$. To prove this assertion, we will show that
$B_{W,w}(0;\frac{s}{2})\subset T(B_{V,w}(0;1))$, where $s>0$ is the radius of
the ball $B_{W,w}(0;s)$ found in Step 2. This means that, given any $y\in
B_{V,w}(0;\frac{s}{2})$, we need to find $x\in B_{V,w}(0;1)$ such that $y=Tx$.
So, let $y\in B_{W,w}(0;\frac{s}{2})$ be given. Since $y\in
B_{W,w}(0;\frac{s}{2})\subset \overline{T(B_{V,w}(0;\frac{1}{2}))}$ by Step 2,
there exists $x_1\in B_{V,w}(0;\frac{1}{2})$ such that
$\|y-Tx_1\|_w<\frac{s}{2^2}$, since
$y-Tx_1\in B_{W,w}(0;\frac{s}{2^2})\subset
\overline{T(B_{V,w}(0;\frac{1}{2^2}))},$ again by Step 2, thee exists
$x_2\in B_{V,w}(0;\frac{1}{2^2})$ such that $\|y-Tx_1-Tx_2\|_w<\frac{s}{2^3}$,
and so on. In this manner we construct a sequence $\{x_n\}_{n=1}^\infty$ of
points $x_n\in V$ with the following properties:
$x_n\in B_{V,w}(0;\frac{1}{2^n})$ and
$\|y-T(\sum_{k=1}^nx_k)\|_w<\frac{s}{2^{n+1}}$, for all $n\geq 1$
.
Since the series $\sum_{n=1}^\infty x_n$ is absolutely convergent (because
$\|x_n\|_w<\frac{1}{2^n}$, for all $n\geq 1$) and the space $V$ is strong
complete, the series $\sum_{n=1}^\infty x_n$ converges to a point $x\in V$ and
$$\|x\|_w\leq \sum_{n=1}^\infty \|x_n\|_w<\frac{1}{2}+\frac{1}{2^2}+\cdots +
\frac{1}{2^{n+1}}+\cdots=1,$$
by Theorem 3.5, so that $x\in B_{V,w}(0;1)$. Furthermore,
$y=\lim_{n\longrightarrow \infty}T(\sum_{k=1}^n x_k)=Tx,$ since $T$ is
continuous. Hence
the assertion is proved. \\
{\bf Step 4:}
The mapping $T$ is open strongly.
Given any strong open subset $U$ of $V$ and give any
$y\in T(U)$, we must find $\delta>0$ such that $B_{W,w}(y;\delta)\subseteq
T(U)$.
So let $x\in U$ be such that $y=Tx$. Since $U$ is strong open, thee exists
$r>0$ such that $B_{V,w}(x;r)=\{x\}+B_{V,w}(0;r)\subseteq U$, and by Step 3,
there exists $\delta>0$ such that $B_{W,w}(0;\delta)\subseteq T(B_{V,w}(0;r))$.
Hence
\begin{align*}
B_{W,w}(y;\delta)= \{y\}+B_{W,w}(0;\delta) & \subseteq \{y\}+T(B_{V,w}(0;r))\\
& = T(\{x\}+B_{V,w}(0;r))\\
& \subseteq T(U).
\qquad \qquad \qquad
\qquad \qquad \qquad
\qquad \qquad \qquad
\Box
\end{align*}
\paragraph{Corollary 4.6.}
Let $(V,u)$ and $(W,\mu)$ be two strong complete PN-spaces and let $T\in
B(V,W)$ be bijective. Then $T^{-1}\in B(W,V)$.
\paragraph{Proof.}
First fix $w\in (0;1)$. It is clear that $T^{-1}:W\longrightarrow V$ is
linear, since
$T\in B(V,W)$, hence by Definition 3.15, suffices to show that
$T^{-1}:W\longrightarrow V$
is continuous at the origin, i.e, that given any strong open ball
$B_{V,w}(0;\epsilon)\subseteq V$, there exists an strong open ball
$B_{V,w}(0;\delta)\subseteq W$, such that $T^{-1}(B_{W,w}(0;\delta))\subseteq
B_{V
,w}(0;\delta)$, by definition of continuouty at a point. But the inclusion
$T^{-1}(B_{W,w}(0;\delta))\subseteq B_{V,w}(0;\epsilon)$, is equivalent to the
inclusion $B_{W,w}(0;\delta)\subseteq T(B_{V,w}(0;\epsilon))$, since the
mapping
$T:V\longrightarrow W$ is bijective, and open Mapping Theorem precisely show
that this
last inclusion hold for some $\delta>0$.
$\Box$
\paragraph{Theorem 4.7.}
Let $\|\cdot \|_w$ and $\|\cdot\|'_w$, for all $w\in (0,1)$, be two norms on
the same vector space $V$, with the following properties: both
$(V,\|\cdot\|_w)$ and $(V,\|\cdot\|'_w)$, are strong complete
PN-spaces, and there exists
a constant $C$ such that $\|x\|'_w\leq C\|x\|_w$ for all $x\in V$. Then there
exists $w'\in (0,1)$ and $c_{w'}\in\mbox{$\mathbb{R}$}$ such that
$\|x\|_{w'}\leq c_{w'}\|x\|'_w$, for all $x\in V$.
\paragraph{Proof.}
Fix $w\in (0,1)$. The bijective and linear identity mapping
$i:(V,\|\cdot\|_w)\longrightarrow (V,\|\cdot\|'_w)$ belongs to $B(V,V)$ by
assumption.
Corollary 4.6, therefore shows that the inverse mapping
$i^{-1}:(V,\|\cdot\|'_w)\longrightarrow (V,\|\cdot\|_w)$, belongs to $B(V,V)$,
hence by Theorems 3.18 and 3.17,
there exists $w'\in (0,1)$ and $c_{w'}$ such that $\|x\|_{w'}\leq
c_{w'}\|x\|'_w$, for all $x\in V$.
$\Box$
\paragraph{Definition 4.8.}
Let $(V,\nu)$ and $(W,\mu)$ be two PN-spaces. The graph $GrT$ of a mapping
$T:V\longrightarrow W$ is the subset of the product $V\times W$ defined by
$$GrT=\{(x,Tx)\in V\times W; x\in V\}.$$
Let $(V,\nu)$ and $(W,\mu)$ be two PN-spaces. By Example 3.14 $(V\times
W,\theta)$ is a PN-space, where $\theta(x,y)=\tau_M(\nu_x,\mu_y)$, for all
$x\in V$ and $y\in W$, is a probabilistic norm on $V\times W$.
\paragraph{Definition 4.9.}
If $(V,\nu)$ and $(W,\mu)$ are PN-spaces, a mapping $T:V\longrightarrow W$ is
said to be
closed graph, if its graph $GrT$ is strong closed in the PN-space $V\times W$
(equipped with strong product topology, by Example 3.14 and Theorem 3.11).
Therefore a mapping $T:V\longrightarrow W$ is strongly closed if and only if
$x_n\longrightarrow x$
in $V$ and
$Tx_n\longrightarrow y$ in $W$ implies $y=Tx$.
\paragraph{Theorem 4.10 (Closed Graph Theorem in PN-spaces).}
Let $(V,\nu)$ and $(W,\mu)$ be strongly complete PN-spaces, and
$T:V\longrightarrow W$ be a
closed linear operator. Then $T\in B(V,W)$.
\paragraph{Proof.}
First fix $w\in (0,1)$ and
$w'\in(0,1)$. By using Theorem 3.5, we define another norm on $V$ by
$$\|x\|'_{V,w}=\|x\|_{V,w}+\|Tx\|_{W,w'},$$
and $x\in V$. On the other hand by the definition of norm $\|\cdot\|'_{V,w}$,
if $\{x_n\}_{n=1}^\infty$ is a Cauchy sequence with respect to
$\|\cdot\|'_{V,w}$, then $\{x_n\}_{n=1}^\infty$ and $\{Tx_n\}_{n=1}^\infty$
are Cauchy sequences in the PN-spaces $V$ and $W$, respectively.
Since both spaces are strongly
complete, there exist $x\in V$ and $y\in W$ such
that $x_n\longrightarrow x$ in $V$ and $Tx_n\longrightarrow y$ in $W$, and
thus $y=Tx$,
since $T$ is strongly
closed by assumption. Therefore,
$$\|x_n-x\|'_{V,w}=\|x_n-x\|_{V,w}+\|Tx_n-Tx\|_{W,w'}=\|x_n-x\|_{V,w}+
\|Tx_n-y\|_{W,w'}\longrightarrow 0$$
as $n\longrightarrow \infty$, which shows that $(V,\|\cdot\|'_w)$ is also
strongly complete.
Since $\|x\|_{V,w}\leq \|x\|'_{V,w}$, for all $x\in V$, Theorem 4.7 shows that
there exists $w''\in (0,1)$ and $C_{w''}$ such that
$\|Tx\|_{W,w'}\leq
\|x\|_{V,w}+\|Tx\|_{W,w'}=\|x\|'_{V,w''}\leq C_{w''}\|x\|_{V,w}$
for all $x\in V$. Therefore by Definition 3.16 we have
$\|T\|_{(w'',w')}<+\infty$, where
$\|T\|_{(w'',w')}=\sup\{\|Tx\|_{w'}|\|x\|_{w''}\leq 1\}$.
Hence by Theorem 3.18, $T\in B(V,W)$.
$\Box$
\paragraph{Theorem 4.11 (Uniform boundedness principle theorem in PN-spaces).}
Let $(V,\nu)$ be a strongly complete PN-space and let $(W,\mu)$ be a PN-space,
and let $\{T_n\}_{n\in\mbox{$\mathbb{N}$}}$ be a family of operators $T_n\in B(V,W)$, that
satisfy
$$\sup\{\|T_nx\|_{w'}:T_n\in B(V,W)\}<\infty,$$
for all $x\in V$ and $w'\in (0,1)$, then there exists $w\in (0,1)$ such that
$$\sup_{n\in\mbox{$\mathbb{N}$}}\|T_n\|_{(w,w')}<\infty$$
\paragraph{Proof.}
First fix $n\in\mbox{$\mathbb{N}$}$ and $w'\in(0,1)$, and define the set
$$F_n=\{x\in V|\sup\|T_nx\|_{w'}\leq n, \; \forall T_n\in B(V,W)\}.$$
Given any $x\in V$, $\sup_{n\in\mbox{$\mathbb{N}$}}\|T_nx\|_{w'}<\infty$, by assumption, hence
there exists an integer $n_x\geq 0$ such that
$\sup_{n\in\mbox{$\mathbb{N}$}\cup\{0\}}\|T_nx\|_{w'}\leq n_x$, which means that $x\in
F_{n_{x}}$,
hence $V=\cup_{n=0}^\infty F_n$. On the other hand $F_n$ is
strongly closed subset of $V$.
Since $V$ is strongly complete by Corollary 4.4 (b), there exists an integer
$n_0\geq 0$ such that $\text{int}
F_{n_0}\neq \phi$. Hence there exists $x_0\in
F_{n_0}$ and $r>0$ such that $\overline{B_{V,w}(x_0;r)}\subset F_{n_0}$, where
$$B_{V,w}(x_0;r)=\{x\in V|\|x-x_0\|_w<r\}, \text{for some}~ w\in (0,1).$$
By definition of $F_{n_0}$, and Theorem 3.18 for all $z\in
\overline{B_{V,w}(x_0;r)}$ and for all $n\in \mbox{$\mathbb{N}$}\cup\{0\}$
$\|T_nz\|_{w'}\leq n_0$
Since any nonzero $x\in V$ can be written as
$x=\frac{\|x\|_w}{r}(z-x_0)$ with $z=(x_0+\frac{r}{\|x\|_w}x)\in
\overline{B_{V,w}(x_0;r)}.$
By Theorem 3.5 we hae
\begin{align*}
\|T_nx\|_{w'} & \leq \frac{\|x\|_w}{r}(\|T_nz\|_{w'}+\|T_nx_0\|_{w'})\\
& \leq \frac{1}{r}(n_0+\|T_nx_0\|_{w'})\|x\|_w\\
& \leq \frac{1}{r}(n_0+\sup_{n\in\mbox{$\mathbb{N}$}\cup\{0\}}\|T_nx_0\|_{w'})\|x\|_w,
\end{align*}
for all $x\in V$.\\
Therefore, by definition of $\|T\|_{(w,w')}$;
$\sup_{n\in\mbox{$\mathbb{N}$}\cup\{0\}}\|T_n\|_{(w,w')}\leq \frac{1}{r}(n_0+
\sup_{n\in\mbox{$\mathbb{N}$}\cup\{0\}}\|T_nx_0\|_{w'})<\infty$, since
$\sup_{n\in\mbox{$\mathbb{N}$}\cup\{0\}}\|T_nx_0\|_{w'}<\infty$, by assumption.
$\Box$
\paragraph{Theorem 4.12 (Banach-Steinhaus Theorem in PN-spaces).}
Let $(V,\nu)$ be a strongly complete PN-space, let $(W,\mu)$ be a PN-space,
let $\{T_n\}_{n=1}^\infty$ be a family of operators $T_n\in B(V,W)$ such that,
for all $x\in V$, the sequence $\{T_nx\}_{n=1}^\infty$ strongly converges in
$W$, that is $T_nx\longrightarrow Tx$, as $n\longrightarrow \infty$, then
$T\in B(V,W).$
\paragraph{Proof.}
The linearity of each operators $T_n$, combined with the continuity of the
addition and scalar multiplication shows that the operator $T:V\longrightarrow
W$ defined
by $Tx=\lim_{n\longrightarrow \infty}T_nx$ for each $x\in V$ in linear. The
convergence of
each sequence $\{T_nx\}_{n=1}^\infty$ and Theorem 3.17 implies that
: $\sup_{n\in\mbox{$\mathbb{N}$}}\|T_nx\|_{w'}<\infty$, for all $x\in V$ and
$w'\in(0,1).$
By Theorem 4.11 there exist $M>0$ and $w\in(0,1)$ such that
$\|T_n\|_{(w,w')}\leq M$, for all $n\in\mbox{$\mathbb{N}$}$. Now by Theorem 3.17 and Theorem
3.5, for all $x\in \overline{B_{V,w}(0;1)}$ and $n\in\mbox{$\mathbb{N}$}$ we have
\begin{align*}
\|Tx\|_{w'} & \leq \|Tx-T_nx\|_{w'}+\|T_nx\|_{w'}\\
& \leq \|Tx-T_nx\|_{w'}+M.
\end{align*}
Hence $\|Tx\|_w'\leq M$, when $n\longrightarrow \infty$, for all $x\in
\overline{B_{V,w}(0;1)}$ therefore by definition of $\|T\|_{(w,w')}$, we have
$T\in B(V,W)$.
$\Box$
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\title[Trilinear Forms with Double Kloosterman Sums]{Trilinear Forms with Double Kloosterman Sums}
\author[I. E. Shparlinski] {Igor E. Shparlinski}
\thanks{This work was supported by ARC Grant~DP170100786.}
\address{Department of Pure Mathematics, University of New South Wales,
Sydney, NSW 2052, Australia}
{\mathbf{\,e}}mail{igor.shparlinski@unsw.edu.au}
\begin{abstract} We obtain several estimates for trilinear form with double Kloosterman sums.
In particular, these bounds show the existence of nontrivial cancellations between such sums.
{\mathbf{\,e}}nd{abstract}
\keywords{Double Kloosterman sum, cancellation, trilinear form}
\subjclass[2010]{11D79, 11L07}
\maketitle
\section{Introduction}
\label{sec:intro}
\subsection{Background and motivation}
Let $q$ be a positive
integer. We denote the residue ring modulo $q$ by $\Z_q$ and denote the group
of units of $\Z_q$ by $\Z_q^*$.
For integers ${\mathbf{\,e}}ll$, $m$ and $n$ we define
the {\it double Kloosterman sum\/}
$$
\cK_q(\ell, m,n) = \sum_{x,y \in \Z_q^*} {\mathbf{\,e}}q\left({\mathbf{\,e}}ll xy +m \xbar + n\ybar \right).
$$
where $\xbar$ is the multiplicative inverse of $x$ modulo $q$ and
$$
{\mathbf{\,e}}q(z) = {\mathbf{\,e}}xp(2 \pi i z/q).
$$
Given three sets
\begin{align*}
{\mathcal L} & = \{u+1, \ldots, u+L\}, \\
{\mathcal M} & = \{v+1, \ldots, v+M\}, \\
{\mathcal N} & = \{w+1, \ldots, w+N\},
{\mathbf{\,e}}nd{align*}
of $L$, $M$, $N$ consecutive integers
and a sequence of weights $\balpha = \{\alpha_{\mathbf{\,e}}ll\}_{{\mathbf{\,e}}ll\in {\mathcal L}}$,
we define the weighted triple sums of double Kloosterman sums
$$
\SALMNq = \sum_{{\mathbf{\,e}}ll \in {\mathcal L}} \alpha_{\mathbf{\,e}}ll \sum_{m\in {\mathcal M}} \sum_{n \in {\mathcal N}}\cK_q(\ell, m,n).
$$
Assuming that $\alpha_{\mathbf{\,e}}ll = 0$ if $\gcd({\mathbf{\,e}}ll, q)>1$ and using the Weil bound~\cite[Equation~(11.58)]{IwKow}, one can easily obtain
$$
\left| \SALMNq \right| \le MN q^{1+o(1)} \sum_{{\mathbf{\,e}}ll \in {\mathcal L}} |\alpha_{\mathbf{\,e}}ll|,
$$
which in the case $|\alpha_{{\mathbf{\,e}}ll}| \le 1$ takes form
\begin{equation}
\label{eq:trivial ALMN}
\left| \SALMNq \right| \le LMN q^{1+o(1)}.
{\mathbf{\,e}}nd{equation}
We are interested in studying cancellations amongst Kloosterman sums and
thus in improvements of the trivial bound~{\mathbf{\,e}}qref{eq:trivial ALMN}.
This question is partially motivated by a series of recent results concerning
various bilinear forms with single Kloosterman sums
$$
\cK_q(m,n) = \sum_{x,y \in \Z_q^*} {\mathbf{\,e}}q\left(m x +n \xbar \right),
$$
see~\cite{BFKMM1,BFKMM2,FKM,KMS,Shp,ShpZha} and references therein for various approaches,
and also for generalisation to bilinear forms with more general quantities.
The triple sums $\SALMNq $ seems to be a new object of study.
\subsection{Results}
Here we use some ideas from~\cite{Shp,ShpZha} to improve the trivial bound~{\mathbf{\,e}}qref{eq:trivial ALMN}.
Although the approach works in larger generality, to exhibit it in a simplest form we assume that
weights $\balpha$ supported only on ${\mathbf{\,e}}ll \in \Z_q^*$, that is, that $\alpha_{\mathbf{\,e}}ll = 0$ if $\gcd({\mathbf{\,e}}ll, q)=1$.
\begin{theorem}
\label{thm:SALMNq} For any integer $q \ge 1$, and weights
$\balpha = \{\alpha_{\mathbf{\,e}}ll\}_{{\mathbf{\,e}}ll\in {\mathcal L}}$ with $|\alpha_{{\mathbf{\,e}}ll}| \le 1$ and
supported only on ${\mathbf{\,e}}ll \in \Z_q^*$, we have,
\begin{align*}
&|\SALMNq| \\
& \qquad \le \min\bigl\{
\left(L + L^{1/2} M^{1/2}\right) N^{1/2} q^{3/2} , \\
& \qquad \qquad \qquad\qquad\qquad
\left(L + L^{3/4} M^{1/4}\right) \left( N^{1/8} q^{7/4} + N^{1/2} q^{3/2}\right) \bigr\}q^{o(1)}.
{\mathbf{\,e}}nd{align*}
{\mathbf{\,e}}nd{theorem}
\begin{theorem}
\label{thm:SALMNq Aver}
For any fixed real $\varepsilon > 0$ and integer $r\ge 2$, for any sufficiently large $Q \ge 1$, for all but at most $Q^{1-2r \varepsilon + o(1)}$ integers
$q\in [Q,2Q]$ and weights $\balpha_q=\{\alpha_{q,{\mathbf{\,e}}ll}\}_{{\mathbf{\,e}}ll\in {\mathcal L}}$, that may depend on $q$,
with $|\alpha_{q,{\mathbf{\,e}}ll}| \le 1$ and supported only on ${\mathbf{\,e}}ll \in \Z_q^*$, we have,
$$
|\SAqLMNq| \le \left(L + L^{1-1/2r} M^{1/2r}\right) \left(q^{2-1/2r} + N^{1/2} q^{3/2}\right) q^{o(1)}.
$$
{\mathbf{\,e}}nd{theorem}
Clearly, the roles of $M$ and $N$ can be interchanged in
Theorems~\ref{thm:SALMNq} and~\ref{thm:SALMNq Aver}.
Now, assuming that $N \le q^{2/3+o(1)}$ we have $N^{1/8} q^{7/4} \ge N^{1/2} q^{3/2+o(1)}$.
Hence by Theorems~\ref{thm:SALMNq} we have
$$
|\SALMNq| \le \left(L + L^{3/4} M^{1/4}\right) N^{1/8} q^{7/4+o(1)},
$$
which improves~{\mathbf{\,e}}qref{eq:trivial ALMN} for
$$
N \le q^{2/3+o(1)} \qquad \mbox{and} \qquad \min\{L,M\} M^3 N^{7/2} \ge q^{3+\varepsilon}
$$
for some fixed $\varepsilon>0$.
Thus in the symmetric case when $L \sim M \sim N$ this condition becomes
$q^{2/3+o(1)} \ge L \ge q^{2/5+\varepsilon}$.
\subsection{Possible generalisations and open problems}
Analysing the proofs of Theorems~\ref{thm:SALMNq} and~\ref{thm:SALMNq Aver} one can easily see that
they can be extended to more general sums of the form
$$
\sum_{x,y \in \Z_q^*} {\mathbf{\,e}}ta_{x} \kappa_y {\mathbf{\,e}}q\left({\mathbf{\,e}}ll xy +m \xbar + n\ybar \right),
$$
with complex weights satisfying $|{\mathbf{\,e}}ta_{x}|, |\kappa_y|\le 1$ (which may depend
on $q$ in the settings of Theorem~\ref{thm:SALMNq Aver}).
On the other hand, our approach does not work for the sums
$$
\sum_{{\mathbf{\,e}}ll \in {\mathcal L}} \alpha_{\mathbf{\,e}}ll \sum_{m\in {\mathcal M}} \beta_m \sum_{n \in {\mathcal N}}\cK_q(\ell, m,n)
\quad \text{and}\quad
\sum_{{\mathbf{\,e}}ll \in {\mathcal L}} \alpha_{\mathbf{\,e}}ll \sum_{m\in {\mathcal M}} \beta_m \sum_{n \in {\mathcal N}} \gamma_n\cK_q(\ell, m,n)
$$
with nontrivial weights attached to the variables $m$ and $n$. It is however possible that
one can apply to these sums the method of~\cite{BFKMM1,FKM,KMS}.
\section{Preliminaries}
\subsection{General notation}
We always assume that the sequence of weights $\balpha = \{\alpha_{\mathbf{\,e}}ll\}_{{\mathbf{\,e}}ll \in {\mathcal L}}$ is
supported only on ${\mathbf{\,e}}ll$ with $\gcd({\mathbf{\,e}}ll,q)=1$, that is, we have $\alpha_{\mathbf{\,e}}ll = 0$ if $\gcd({\mathbf{\,e}}ll,q)>1$ (and the same for the weights $\balpha_q=\{\alpha_{q,{\mathbf{\,e}}ll}\}_{{\mathbf{\,e}}ll\in {\mathcal L}}$
depending on $q$).
Throughout the paper, as usual $A\ll B$ and $B \gg A$ are equivalent to the inequality $|A|\le cB$
with some constant $c>0$, which
occasionally, where obvious, may depend on the real
parameter $\varepsilon>0$ and on the integer parameter $r \ge 1$, and
is absolute otherwise.
\subsection{Number of solutions to some multiplicative congruences}
We start with some estimates on power moments of character sums.
Let ${\mathcal X}_q$ be the set of all multiplicative characters $\chi$ modulo $q$ and
let ${\mathcal X}_q^* = {\mathcal X}_q \setminus \{\chi_0\}$ be the set of non-principal characters.
The first result is a special case of a bound of
Cochrane and Shi~\cite[Theorem~1] {CochShi}.
\begin{lemma}
\label{lem:4th-Moment}
For any integers $k$ and $H$ we have
$$
\sum_{\chi\in{\mathcal X}_q}
\left|\sum_{z=k+1}^{k+H}\chi(x)\right|^4 \le H^2 q^{o(1)}.
$$
{\mathbf{\,e}}nd{lemma}
We now derive our main technical tool.
\begin{lemma}
\label{lem:Energy}
For any sets
$$
{\mathcal A} = \{s+1, \ldots, s+A\} \qquad \mbox{and} \qquad {\mathcal B} = \{t+1,\ldots, t+B\}
$$
consisting of $A$ and $B$ consecutive integers, respectively, for
\begin{align*}
E({\mathcal A}, {\mathcal B}) = \{a_1 b_1 {\mathbf{\,e}}quiv a_2 b_2 \bmod q~:~
a_1, a_2 \in {\mathcal A} , &\ b_1, b_2 \in {\mathcal B}, \\
& \gcd(a_1a_2b_1b_2,q)=1 \}
{\mathbf{\,e}}nd{align*}
we have
$$
E({\mathcal A}, {\mathcal B}) \le \left(\frac{A^2B^2}{q} + AB\right) q^{o(1)}.
$$
{\mathbf{\,e}}nd{lemma}
\begin{proof}
Using the orthogonality of characters, we write
$$
E({\mathcal A}, {\mathcal B}) =
\sum_{\substack{a_1, a_2 \in {\mathcal A}\\
\gcd(a_1a_2,q)=1}} \ \sum_{\substack{b_1, b_2 \in {\mathcal A}\\
\gcd(b_1b_2,q)=1}} \frac{1}{\varphi(q)}\sum_{\chi\in {\mathcal X}_q} \chi\left(a_1a_2b_1^{-1}b_2^{-1}\right),
$$
where, as usual, $\varphi(q)$ denotes the
Euler function.
Changing the order summation, and separating the contribution
$$
\frac{1}{\varphi(q)} \sum_{\substack{a_1, a_2 \in {\mathcal A}\\
\gcd(a_1a_2,q)=1}} \sum_{\substack{b_1, b_2 \in {\mathcal B}\\
\gcd(b_1b_2,q)=1}} 1 \le \frac{A^2B^2}{\varphi(q)}
$$
from the principal character, we obtain
\begin{equation}
\label{eq:E R}
E({\mathcal A}, {\mathcal B}) \le \frac{A^2B^2}{\varphi(q)} + R,
{\mathbf{\,e}}nd{equation}
where
$$
R = = \frac{1}{\varphi(q)}\sum_{\chi\in {\mathcal X}_q*}
\sum_{\substack{a_1, a_2 \in {\mathcal A}\\
\gcd(a_1a_2,q)=1}} \ \sum_{\substack{b_1, b_2 \in {\mathcal B}\\
\gcd(b_1b_2,q)=1}} \chi\left(a_1a_2b_1^{-1}b_2^{-1}\right).
$$
Rearranging, we obtain
$$
R = \frac{1}{\varphi(q)}\sum_{\chi\in {\mathcal X}_q*}
\left(\sum_{a\in {\mathcal A}} \chi\left(a\right)\right)^2 \left(\sum_{b \in {\mathcal B}} \overline\chi\left(b\right)\right)^2,
$$
where $\overline\chi$ is the complex conjugate character (note the co-primality
conditions $\gcd(a,q)= \gcd(b,q)=1$ are abandoned from the last sum as redundant).
Now, using the Cauchy inequality and recalling Lemma~\ref{lem:4th-Moment}, we conclude that
$$
|R| \le AB q^{o(1)}.
$$
Substituting this in~{\mathbf{\,e}}qref{eq:E R}, and recalling the well-known lower bound
$$
\varphi(q) \gg \frac{q}{\log \log(q+2)}
$$\
see~\cite[Theorem~328]{HardyWright}, we conclude the proof.
{\mathbf{\,e}}nd{proof}
\subsection{Number of solutions to some congruences with reciprocals}
\label{sec:CongEqReipr}
An important tool in our argument is an upper bound on the number
of solutions $J_r(q;K)$ to the congruence
$$\frac{1}{x_1}+ \ldots+ \frac{1}{x_r}{\mathbf{\,e}}quiv \frac{1}{x_{r+1}}+ \ldots+\frac{1}{x_{2r}}\bmod q, \quad
1 \le x_1, \ldots, x_{2r} \le K,
$$
where $r =1, 2, \ldots$.
We start with the trivial bound $1 \le K \le q$ we have
\begin{equation}
\label{eq:J triv}
J_1(q;K) \ll K.
{\mathbf{\,e}}nd{equation}
For $r \ge 2$ and arbitrary $q$ and $K$, good upper bounds on $J_r(q;K)$
are known only for $r =2$ and are due to
Heath-Brown~\cite[Page~368]{H-B1}
(see the bound on the sums of quantities $m(s)^2$ in the
notation of~\cite{H-B1}). More precisely, we have:
\begin{lemma}
\label{lem:Cayley} For $1 \le K \le q$ we have
$$
J_2(q;K) \le \left(K^{7/2} q^{-1/2} + K^2\right)q^{o(1)}.
$$
{\mathbf{\,e}}nd{lemma}
It is also shown by Fouvry and Shparlinski~\cite[Lemma~2.3]{FouShp} that the bound of
Lemma~\ref{lem:Cayley}
can be improved on average over $q$ in a dyadic interval $[Q,2Q]$. The same argument
also works for $J_r(q;K)$ without any changes.
Indeed, let $J_r(K)$ be the number
of solutions to the equation
$$
\frac{1}{x_1}+ \ldots+ \frac{1}{x_r} = \frac{1}{x_{r+1}}+ \ldots+\frac{1}{x_{2r}}, \qquad
1 \le x_1, \ldots, x_{2r} \le K,
$$
where $r =1, 2, \ldots$. We recall that by the result of Karatsuba~\cite{Kar} (presented in
the proof of~\cite[Theorem~1]{Kar}), see also~\cite[Lemma~4]{BouGar1}, we have:
\begin{lemma}
\label{lem:Recipr Eq} For any fixed positive integer $r$, we have
$$
J_r(K) \le K^{r + o(1)}.
$$
{\mathbf{\,e}}nd{lemma}
Now repeating the argument of the proof of~\cite[Lemma~2.3]{FouShp} and using
Lemma~\ref{lem:Recipr Eq} in the appropriate place, we obtain:
\begin{lemma}
\label{lem:Cayley Aver} For any fixed positive integer $r$ and sufficiently large integers $1 \le K \le Q$,
we have
$$
\frac{1}{Q}\sum_{Q \le q \le 2Q} J_{r}(q;K) \le \left(K^{2r} Q^{-1} + K^r\right)Q^{o(1)}.
$$
{\mathbf{\,e}}nd{lemma}
\section{Proofs of Main Results}
\subsection{Proof of Theorem~\ref{thm:SALMNq}}
For an integer $u$ we define
$$
\langle u\rangle_q = \min_{k \in \Z} |u - kq|
$$
as the distance to the closest integer,
which is a multiple of $q$.
Changing the order of summation and then changing the variables
$$
x \mapsto \xbar \qquad \mbox{and} \qquad y \mapsto \ybar,
$$
we obtain
\begin{align*}
\SALMNq & = \sum_{{\mathbf{\,e}}ll \in {\mathcal L}} \alpha_{\mathbf{\,e}}ll \sum_{x\in \Z_q^*} \sum_{m\in {\mathcal M}} \sum_{n \in {\mathcal N}}
{\mathbf{\,e}}q\left({\mathbf{\,e}}ll \xbar \ybar +m x + ny \right)\\
& = \sum_{{\mathbf{\,e}}ll \in {\mathcal L}} \alpha_{\mathbf{\,e}}ll \sum_{x,y\in \Z_q^*} {\mathbf{\,e}}q\left({\mathbf{\,e}}ll \xbar \ybar\right) \sum_{m\in {\mathcal M}} {\mathbf{\,e}}q(mx) \sum_{n \in {\mathcal N}} {\mathbf{\,e}}q(n y).
{\mathbf{\,e}}nd{align*}
Hence
$$
\SALMNq = \sum_{{\mathbf{\,e}}ll \in {\mathcal L}} \alpha_{\mathbf{\,e}}ll
\sum_{x,y\in \Z_q^*} \mu_x \nu_y {\mathbf{\,e}}q\left({\mathbf{\,e}}ll \xbar \ybar\right),
$$
with some complex coefficients $\mu_x$ and $\nu_y$, satisfying
\begin{equation}
\label{eq:bound mu nu}
|\mu_x| \le \min\left\{M, \frac{q}{\langle x \rangle_q}\right \} \qquad \mbox{and} \qquad
|\nu_y| \le \min\left\{N, \frac{q}{\langle y \rangle_q}\right \},
{\mathbf{\,e}}nd{equation}
see~\cite[Bound~(8.6)]{IwKow}.
We now set $I = \rf{\log (M/2)}$ and define $2(I+1)$ the sets
\begin{equation}
\label{eq:Set Q}
\begin{split}
{\mathcal Q}_0^{\pm} & =\{x \in \Z~: ~ 0 < \pm x \le q/M,\ \gcd(x,q)=1\}, \\
{\mathcal Q}_i^{\pm} & = \{x \in \Z~: ~\min\{q/2, e^{i} q/M\}\ge \pm x > e^{i-1}q/M,\\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \gcd(x,q)=1 \},
{\mathbf{\,e}}nd{split}
{\mathbf{\,e}}nd{equation}
where $i = 1, \ldots, I$.
Similarly, we set $J = \rf{\log (N/2)}$ and define $2(J+1)$ the sets
\begin{equation}
\label{eq:Set R}
\begin{split}
{\mathcal R}_0^{\pm} & =\{x \in \Z~: ~ 0 < \pm y \le q/N,\ \gcd(x,q)=1\}, \\
{\mathcal R}_j^{\pm} & = \{y \in \Z~: ~\min\{q/2, e^{j} q/N\}\ge \pm y > e^{j-1}q/N,\\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \gcd(x,q)=1 \},
{\mathbf{\,e}}nd{split}
{\mathbf{\,e}}nd{equation}
where $j = 1, \ldots, J$.
Therefore,
\begin{equation}
\label{eq:SAIJ S0i}
\SALMNq \ \ll \sum_{i=0}^I \sum_{j=0}^J \left(\left|S_{i,j}^+\right | + \left|S_{i,j}^-\right|\right),
{\mathbf{\,e}}nd{equation}
where
$$
S_{i,j} ^{\pm} = \sum_{{\mathbf{\,e}}ll \in {\mathcal L}}\sum_{x \in {\mathcal Q}_i^\pm} \sum_{y \in {\mathcal R}_j^\pm}
\alpha_{\mathbf{\,e}}ll \mu_x \nu_y {\mathbf{\,e}}q\left({\mathbf{\,e}}ll \xbar \ybar\right), \qquad i = 0, \ldots, I, \ j = 0, \ldots J.
$$
For $\lambda \in \Z_q$ we denote
$$
T_i^{\pm}(\lambda) = \mathop{\sum\sum}_{\substack{{\mathbf{\,e}}ll \in {\mathcal L}, \ x \in {\mathcal Q}_i^\pm \\ {\mathbf{\,e}}ll x {\mathbf{\,e}}quiv \lambda \bmod q}} \alpha_{\mathbf{\,e}}ll \mu_x
$$
and note that $T_i^{\pm}(\lambda) =0 $ unless $\lambda \in \Z_q^*$.
Therefore,
$$
S_{i,j} ^{\pm} = \sum_{\lambda \in \Z_q^*} \sum_{y \in {\mathcal R}_j^\pm} T_i^{\pm}(\lambda)
\nu_y {\mathbf{\,e}}q\left(\lambda \ybar\right), \qquad i = 0, \ldots, I, \ j = 0, \ldots J.
$$
Let us fix some integers $i\in [0,I]$ and $j \in [0,J]$.
We now fix some integer $r \ge 1$.
Below we present the argument in a general form with an arbitrary integer $r \ge 1$.
We then apply it with $r =1$ and $r=2$ since we use Lemma~\ref{lem:Cayley}. However
in the proof of Theorem~\ref{thm:SALMNq Aver} we use it in full generality.
Writing
$$
\left |S_{i,j} ^{\pm} \right| \le \sum_{\lambda \in \Z_q^*} \left| T_i^{\pm}(\lambda)\right|^{1 -1/r}
\left| T_i^{\pm}(\lambda)^2\right|^{1/2r}
\left |\sum_{y \in {\mathcal R}_j^\pm} \nu_y {\mathbf{\,e}}q(m \ybar)\right|,
$$
by the H{\"o}lder inequality, for every choice of the
sign `$+$' or `$-$', we obtain
\begin{equation}
\label{eq:Holder S}
\begin{split}
\left |S_{i,j} ^{\pm} \right| & \le
\left( \sum_{\lambda \in \Z_q^*} \left| T_i^{\pm}(\lambda)\right|\right)^{1 -1/r}
\left( \sum_{\lambda \in \Z_q^*} \left| T_i^{\pm}(\lambda)\right|^2\right)^{1/2r} \\
& \qquad \qquad \qquad \qquad \qquad \left( \sum_{\lambda \in \Z_q}
\left |\sum_{y \in {\mathcal R}_j^\pm} \nu_y {\mathbf{\,e}}q(m \ybar)\right|^{2r} \right)^{1/2r}.
{\mathbf{\,e}}nd{split}
{\mathbf{\,e}}nd{equation}
We observe that by~{\mathbf{\,e}}qref{eq:bound mu nu} and~{\mathbf{\,e}}qref{eq:Set Q}, for $x \in {\mathcal Q}_i^\pm$ we have
$$
\mu_x \ll e^{-i} M.
$$
Hence
\begin{equation}
\label{eq:T1}
\begin{split}
\sum_{\lambda \in \Z_q^*} \left| T_i^{\pm}(\lambda)\right| & \ll
\sum_{\lambda \in \Z_q^*}
\mathop{\sum\sum}_{\substack{{\mathbf{\,e}}ll \in {\mathcal L}, \ x \in {\mathcal Q}_i^\pm \\ {\mathbf{\,e}}ll x {\mathbf{\,e}}quiv \lambda \bmod q}} \left| \alpha_{\mathbf{\,e}}ll \mu_x \right| \ll
e^{-i} M \sum_{\lambda \in \Z_q^*}
\mathop{\sum\sum}_{\substack{{\mathbf{\,e}}ll \in {\mathcal L}, \ x \in {\mathcal Q}_i^\pm \\ {\mathbf{\,e}}ll x {\mathbf{\,e}}quiv \lambda \bmod q}} 1 \\
& \ll
e^{-i} M \# {\mathcal L} \# {\mathcal Q}_i^\pm \ll e^{-i} L M \left(e^{i} q/M\right) = qL.
{\mathbf{\,e}}nd{split}
{\mathbf{\,e}}nd{equation}
Similarly,
\begin{align*}
\sum_{\lambda \in \Z_q^*} \left| T_i^{\pm}(\lambda)\right|^2
& \ll
e^{-2i} M^2 \sum_{\lambda \in \Z_q^*}
\left( \mathop{\sum\sum}_{\substack{{\mathbf{\,e}}ll \in {\mathcal L}, \ x \in {\mathcal Q}_i^\pm \\ {\mathbf{\,e}}ll x {\mathbf{\,e}}quiv \lambda \bmod q}} 1\right)^2
= e^{-2i} M^2 E( {\mathcal L}, {\mathcal Q}_i^\pm),
{\mathbf{\,e}}nd{align*}
where $E({\mathcal A}, {\mathcal B})$ is as defined in Lemma~\ref{lem:Energy}, which implies
\begin{equation}
\begin{split}
\label{eq:T2}
\sum_{\lambda \in \Z_q^*} \left| T_i^{\pm}(\lambda)\right|^2 & \le e^{-2i} M^2
\left(\frac{L^2 \left(e^{i} q/M\right) ^2}{q} + L \left(e^{i} q/M\right) \right) q^{o(1)}\\
& \le q^{1+o(1)} L^2 + e^{-i} q^{1+o(1)} LM.
{\mathbf{\,e}}nd{split}
{\mathbf{\,e}}nd{equation}
Next, opening up the inner exponential sum in~{\mathbf{\,e}}qref{eq:Holder S}, changing the order of summation and
using the orthogonality of exponential functions, we obtain
\begin{align*}
\sum_{\lambda \in \Z_q}&\left |\sum_{y \in {\mathcal R}_j^\pm} \nu_y {\mathbf{\,e}}q(\lambda \ybar)\right|^{4} \\
& \le\sum_{\lambda \in \Z_q} \mathop{\sum\ldots \sum}_{y_1, \ldots y_{2r} \in {\mathcal R}_j^\pm }
\prod_{j=1}^r \nu_{y_j} \overline{\nu_{y_{r + j}}}
{\mathbf{\,e}}q\left(\lambda \sum_{j=1}^r \left(\ybar_j - \ybar_{r + j}\right)\right)\\
& \le \mathop{\sum\ldots \sum}_{y_1, \ldots, y_{2r} \in{\mathcal R}_j^\pm}
\prod_{j=1}^r \nu_{y_j} \overline{\nu_{y_{r + j}}}
\sum_{\lambda \in \Z_q}{\mathbf{\,e}}q\left(m \sum_{j=1}^r \left(\ybar_j - \ybar_{r + j}\right)\right)\\
& = q \mathop{\sum\ldots \sum}_{\substack{y_1, \ldots, y_{2r} \in{\mathcal R}_j^\pm\\
\ybar_1 + \ldots + \ybar_r {\mathbf{\,e}}quiv \ybar_{r+1}+ \ldots + \ybar_{2r} \bmod q}}
\prod_{j=1}^r \nu_{y_j} \overline{\nu_{y_{r + j}}}.
{\mathbf{\,e}}nd{align*}
We observe that by~{\mathbf{\,e}}qref{eq:bound mu nu} and~{\mathbf{\,e}}qref{eq:Set R} for $y \in {\mathcal R}_j^\pm$ we have
$$
\nu_y \ll e^{-j} N.
$$
Hence,
\begin{equation}
\label{eq:S 2r}
\begin{split}
\sum_{\lambda \in \Z_q}\left |\sum_{y \in {\mathcal R}_j^\pm} \nu_y {\mathbf{\,e}}q(\lambda \ybar)\right|^{2r} &
\ll e^{-2r j} q N^{2r} \mathop{\sum\ldots \sum}_{\substack{y_1, \ldots, y_{2r} \in {\mathcal R}_j^\pm\\
\ybar_1 + \ldots + \ybar_r {\mathbf{\,e}}quiv \ybar_{r+1}+ \ldots + \ybar_{2r} \bmod q}}1\\
& \le e^{-2r j} q N^{2r} J_r(q;\fl{e^{j}q/N}).
{\mathbf{\,e}}nd{split}
{\mathbf{\,e}}nd{equation}
Substituting~{\mathbf{\,e}}qref{eq:T1}, {\mathbf{\,e}}qref{eq:T2} and~{\mathbf{\,e}}qref{eq:S 2r} in~{\mathbf{\,e}}qref{eq:Holder S}, we see that
\begin{equation}
\label{eq:Holder STT}
\begin{split}
\left |S_{i,j} ^{\pm} \right| & \le
\left( qL\right)^{1 -1/r}
\left(q^{1+o(1)} L^2 + e^{-i} q^{1+o(1)} LM\right)^{1/2r} \\
& \qquad \qquad \qquad \qquad \qquad \left( e^{-2r j} q N^{2r} J_r\left(q;\fl{e^{j}q/N}\right)\right)^{1/2r}\\
& \le e^{- j} LN
q^{1+o(1)} \left(1 + e^{-i} M/L\right)^{1/2r} J_r\left(q;\fl{e^{j}q/N}\right)^{1/2r}.
{\mathbf{\,e}}nd{split}
{\mathbf{\,e}}nd{equation}
Now using~{\mathbf{\,e}}qref{eq:Holder STT} with $r = 1$ and recalling~{\mathbf{\,e}}qref{eq:J triv}, we derive
\begin{equation}
\label{eq:Sij}
\begin{split}
\left |S_{i,j} ^{\pm} \right| & \le e^{- j} LN
q^{1+o(1)} \left(1 + e^{-i/2} (M/L)^{1/2}\right) e^{j/2} N^{-1/2} q^{1/2} \\
& \le e^{-j/2}
\left(L + e^{-i/2} L^{1/2} M^{1/2}\right) N^{1/2} q^{3/2+o(1)}.
{\mathbf{\,e}}nd{split}
{\mathbf{\,e}}nd{equation}
Summing over all admissible $i$ and $j$ yields
\begin{equation}
\label{eq:Bound1}
\SALMNq \le \left(L + L^{1/2} M^{1/2}\right) N^{1/2} q^{3/2+o(1)}.
{\mathbf{\,e}}nd{equation}
Next, using~{\mathbf{\,e}}qref{eq:Holder STT} with $r = 2$ and invoking Lemma~\ref{lem:Cayley}, we obtain
\begin{align*}
\left |S_{i,j} ^{\pm} \right| & \le e^{- j} LN
q^{1+o(1)} \left(1 + e^{-i/4} (M/L)^{1/4}\right) \\
& \qquad \qquad \qquad \qquad \quad \left(e^{7j/8} N^{-7/8} q^{3/4} + e^{j/2} N^{-1/2}q^{1/2} \right) \\
& \le
\left(L + e^{-i/4} L^{3/4} M^{1/4}\right) \left(e^{-j/8}N^{1/8} q^{7/4} + e^{-j/2} N^{1/2} q^{3/2}\right) q^{o(1)},
{\mathbf{\,e}}nd{align*}
and we now derive
\begin{equation}
\label{eq:Bound2}
\SALMNq \le
\left(L + L^{3/4} M^{1/4}\right) \left( N^{1/8} q^{7/4} + N^{1/2} q^{3/2}\right) q^{o(1)}.
{\mathbf{\,e}}nd{equation}
Combining the bounds~{\mathbf{\,e}}qref{eq:Bound1} and~{\mathbf{\,e}}qref{eq:Bound2}, we obtain the result.
\subsection{Proof of Theorem~\ref{thm:SALMNq Aver}}
We proceed as in the proof of Theorem~\ref{thm:SALMNq}, in particular,
we set $J = \rf{\log (N/2)}$. We also define $K_j = \fl{2 e^{i}Q/N}$
and replace $J_r(q;\fl{e^{j}q/N})$
with $J_r(q;K_j)$ in~{\mathbf{\,e}}qref{eq:Holder STT}, $j= 0, \ldots, J$.
We now see that by Lemma~\ref{lem:Cayley Aver} for every $j=0, \ldots, J$ for all
but at most $Q^{1-2r \varepsilon + o(1)}$ integers $q \in [Q,2Q]$ we have
\begin{equation}
\label{eq: Bound JqKi}
J_{r}(q;K_j) \le \left(K_j^{2r} q^{-1} + K_j^r\right)Q^{2 r \varepsilon}.
{\mathbf{\,e}}nd{equation}
Since $J = Q^{o(1)}$, for all
but at most $Q^{1-2r \varepsilon + o(1)}$ integers $q \in [Q,2Q]$,
the bound~{\mathbf{\,e}}qref{eq: Bound JqKi} holds for all $j=0, \ldots, J$
simultaneously.
Now, for every such intreger $q$, using~{\mathbf{\,e}}qref{eq: Bound JqKi} instead of the
bound of Lemma~\ref{lem:Cayley}, we obtain
\begin{align*}
\left |S_{i,j} ^{\pm} \right| & \le e^{- j} LN
q^{1+o(1)} \left(1 + e^{-i} M/L\right)^{1/2r} \\
& \qquad \qquad \qquad \qquad \quad \left(e^{j} N^{-1} q^{1-1/2r} + e^{j/2} N^{-1/2}q^{1/2} \right) \\
& \le
\left(L + e^{-i/2r} L^{1-1/2r} M^{1/2r}\right) \left(q^{2-1/2r} + e^{-j/2} N^{1/2} q^{3/2}\right) q^{o(1)},
{\mathbf{\,e}}nd{align*}
instead of~{\mathbf{\,e}}qref{eq:Sij} for every $i=0, \ldots, I$ and $j = 0, \ldots, J$.
Since $I, J= Q^{o(1)}$, the result now follows.
\begin{thebibliography}{9999}
\bibitem{BFKMM1} V. Blomer, {\'E}. Fouvry, E. Kowalski, P. Michel and D. Mili{\'c}evi{\'c},
`On moments of twisted $L$-functions',
{\it Amer. J. Math.\/}, {\bf 139} (2017), 707--768.
\bibitem{BFKMM2} V. Blomer, {\'E}. Fouvry, E. Kowalski, P. Michel and D. Mili{\'c}evi{\'c},
`Some applications of smooth bilinear forms with Kloosterman sums',
{\it Proc. Steklov Math. Inst.\/}, {\bf 296} (2017), 18--29.
\bibitem{BouGar1} J. Bourgain and M. Z. Garaev,
`Sumsets of reciprocals in prime fields and multilinear Kloosterman sums',
{\it Izv. Ross. Akad. Nauk Ser. Mat.\/}, {\bf 78} (2014), 9--72 (in Russian); translation in
{\it Izv. Math\/}, {\bf 78} (2014), 656--707.
\bibitem{CochShi} T. Cochrane and S. Shi,
`The congruence $x_1x_2 {\mathbf{\,e}}quiv x_3x_4 \bmod m$
and mean values of character sums',
{\it J. Number Theory\/}, {\bf 130} (2010), 767--785.
\bibitem{FKM} {\'E}. Fouvry, E. Kowalski and
P. Michel, `Algebraic trace functions over the primes',
{\it Duke Math. J.\/}, {\bf 163} (2014), 1683--1736.
\bibitem{FouShp} \'E. Fouvry and I.E. Shparlinski,
`On a ternary quadratic form over
primes', {\it Acta Arith.\/}, {\bf 150} (2011), 285--314.
\bibitem{HardyWright} G. H. Hardy and E. M. Wright, {\it An introduction to
the theory of numbers\/}, Oxford Univ. Press, Oxford, 2008.
\bibitem{H-B1}
D. R.~Heath-Brown,
`Almost primes in arithmetic progressions and short intervals',
{\it Math. Proc. Cambridge Philos. Soc.\/}, {\bf 83} (1978), 357--375.
\bibitem{IwKow} H. Iwaniec and E. Kowalski,
{\it Analytic number theory\/}, American Mathematical Society,
Providence, RI, 2004.
\bibitem{Kar} A. A. Karatsuba, `Analogues of Kloosterman sums',
{\it Izv. Ross. Akad. Nauk Ser. Mat. (Transl. as Russian Acad. Sci. Izv.
Math.)\/}, {\bf 55}(5) (1995),
93--102 (in Russian)
\bibitem{KMS} E. Kowalski, P. Michel and W. Sawin,
`Bilinear forms with Kloosterman sums and applications',
{\it Annals Math.\/}, (to appear).
\bibitem{Shp} I. E. Shparlinski, `On sums of Kloosterman and Gauss sums',
{\it Preprint\/}, 2016 (available
from \url{http://arxiv.org/abs/1608.06160}).
\bibitem{ShpZha} I. E. Shparlinski and T. P. Zhang,
`Cancellations amongst Kloosterman sums',
{\it Acta Arith.\/}, {\bf 176} (2016), 201--210.
{\mathbf{\,e}}nd{thebibliography}
{\mathbf{\,e}}nd{document} |
\begin{document}
\title{Existence of weak solutions to stochastic evolution inclusions}
\begin{abstract}
We prove the existence of a weak mild solution to the Cauchy problem
for the semilinear stochastic differential inclusion in a Hilbert
space
$$dX_t\in AX_t\,dt+F(t,X_t)\,dt+G(t,X_t)\,dW_t$$
where
$W$ is a Wiener process,
$A$ is a linear operator which generates a $C_0$-semigroup,
$F$ and $G$ are multifunctions with convex compact values satisfying
some growth condition and, with respect to the second variable,
a condition
weaker than the Lipschitz condition.
The weak solution is constructed in the sense of Young
measures.
\end{abstract}
\section{Introduction}
Ordinary and stochastic differential and inclusions in infinite dimensional
spaces have many important and interesting applications on which we
do not stop here, addressing the reader to the books and to the reviews (see for example
\cite{ahmed91book,dapratozabczyk92book,bogachev95deterministic}).
For ordinary differential equations, after the famous
example of Dieudonn\'e
\cite{dieudonne50exemples} and the fundamental results of A.~Godunov
\cite{godunov72counter,godunov74peano},
it became evident that,
in the case of infinite dimensional Banach spaces,
it is necessary to suppose an auxilary
condition on the right hand side of the equation
\begin{equation}\label{eq:1}
x'=f(t,x)
\end{equation}
in order to have an existence theorem for the initial value problem with the initial
condition
\begin{equation}\label{eq:2}
x(0)=x_0.
\end{equation}
It is well known that the estimation
\begin{equation}\label{eq:3}
\| f(t,x) - f(t,y)\|\le L(t, \|x-y\|),
\end{equation}
where $L$ is a real function such that the integral inequality
\begin{equation*}
u(t)\le \int_0^t L(s, u(s))\,ds,
\end{equation*}
has a unique solution $u(t)=0,$ gives the existence condition mentioned below.
In \cite{kibenko-krasno-mamedov61one-sided} it was shown that it is
possible to add to $f$ satisfying \eqref{eq:3} a continuous compact
operator. As generalisations of this fact many papers in the
seventies
were devoted to a condition of the form
\begin{equation}\label{eq:5}
\varphi (f(t, \Lambda))\le L(s,\varphi (\Lambda)),
\end{equation}
where $\varphi $ is a measure of noncompactness (see e.g.~\cite{akprs92book}). The abstract
fixed point theorem for condensing operators (see \cite{akprs92book}) was successfully
applied in this way.
In the end of the XXth century, in many papers (see
e.g.~\cite{ugowski85evolution,szufla86existence,banas87various}),
it was shown
that Condition \eqref{eq:5} implies the existence of solutions to
the semilinear equation
\begin{equation*}
x'=Ax+f(t,x)
\end{equation*}
and the semilinear inclusion (e.g.~\cite{obukhovski91controlled}, see
also the references in \cite{hu-papageorgiou00book})
\begin{equation}\label{eq:7}
x'\in Ax+f(t,x),
\end{equation}
with the initial condition \eqref{eq:2} and a linear operator $A$ generating a $C_0$-semigroup.
In the inclusion \eqref{eq:7}, $f$ is a multivalued map with convex compact values.
In the same period from the
seventies,
it was remarked that the existence of a
weak solution
and the condition of unicity of trajectories implies
the existence of a strong solution
to the stochastic differential equation
\begin{equation}\label{eq:8}
dX_t=a(t,X_t)\,dt+b(t,X_t)\,dW_t,
\end{equation}
where $W$ is a standard Wiener process (here,``weak'' and ``strong'' are taken
in the probabilistic sense; the existence of a weak solution
with only continuous $a$ and $b$ was known since Skorokhod
\cite{skorokhod65book}).
The direct
proof of the existence of a strong solution for \eqref{eq:8}
using a convenient measure of noncompactness
when $a$ and $b$ satisfy a condition like \eqref{eq:3} was presented
in \cite{rodkina84heredity}, see also
\cite{akprs92book}.
The passage from the finite dimensional case to the infinite
dimentional case with $a$ and $b$ satisfying
a Lipschitz condition is presented in
\cite{dapratozabczyk92book,bogachev95deterministic}
see there the
bibliography.
The generalisation to the case of a semilinear stochastic differential equation
\begin{equation*}
dX_t=AX_t\,dt+f(t,X_t)dt+\sigma (t,X_t)\,dW_t,
\end{equation*}
deals with the estimations for the stochastic convolution operator
$$ \int_0^te^{(t-s)A}v(s)\,dW_s $$
(see
\cite{tubaro84burk,kotelenez84stopped,
dapratozabczyk92book,dapratozabczyk92convolution,hausenblasseidler01stochconv})
and for Lipschitz $f$ and $\sigma$ is presented in
\cite{dapratozabczyk92book}.
In the work of
Da Prato and Frankowska \cite{dapratofrankowska94filippov},
the existence result is proved for the semilinear stochastic inclusion
\begin{equation}\label{eq:10}
dX_t\in AX_t\,dt+f(t,X_t)\,dt+\sigma (t,X_t)dW_t,
\end{equation}
with multivalued $f$ and $\sigma$ which are Lipschitz with respect to
the Hausdorff metric.
In the present paper we aim to prove the existence of a ``weak''
solution (or a ``solution measure'') for \eqref{eq:10} in the case where
$f$ and $\sigma$ are compact valued multifunctions satisfying a
condition like \eqref{eq:3} where the norm in the left hand side is
replaced by the Hausdorff metric.
More precisely,
we prove the existence of a ``weak'' mild
solution $\sol$ to the Cauchy problem
\begin{equation}\label{eq:generale}
\begin{syst}
d\sol_t\in A\sol_t+F(t,\sol_t)\,dt
+G(t,\sol_t)\,d\wien(t)\\
\sol(0)=\initial
\end{syst}
\end{equation}
where $\sol$ takes its values in a Hilbert space $\esp$,
$\wien$ is a Brownian motion on a Hilbert space $\hilbw$,
$A$ is a
linear operator on $\esp$ and $F$ and $G$ are
multivalued mappings with compact
convex values, continuous in the second variable,
and satisfy
an assumption which is much more general than the usual Lispchitz one.
Instead of constructing the weak solution with the help of Skorokhod's
representation theorem, we define our weak solution as a Young
measure (ie as a {\em solution measure} in the sense of Jacod and
M\'emin \cite{jacod-memin81weakstrong}).
As in the case of the ordinary differential equation \eqref{eq:1} with the
right hand side satisfying
\eqref{eq:5} (see \cite{kamenski72peano}) we apply the Tonelli scheme,
and prove that its solutions are uniformly tight.
We then
pass to the limit and we obtain the existence of a weak mild solution
to the initial problem.
To get rid of the Lipschitz assumption on $F$ and $G$ has a cost:
not only do we obtain ``weak'' solutions, but
our techniques based on compactness lead us to consider mappings $F$
and $G$ with convex compact values,
whereas the more geometric methods of
\cite{dapratofrankowska94filippov} deal
with unbounded closed valued mappings.
Furthermore, in our work, the multifunctions $F$ and $G$
are deterministic, whereas they are random in
\cite{dapratofrankowska94filippov}.
\section{Formulation of the problem, statement of the result}
\paragraph{Notations}
Throughout,
$0<\Time<+\infty$ is a fixed time
and
$\stbas=(\esprob,\tribu, \filtration_{t\in\Times},\pr)$ is a
stochastic basis satisfying the usual conditions.
The $\sigma$--algebra of predictable subsets of $\prodspace$ is denoted by
$\predict$.
If $\espmetricgeneric$ is a separable metric space, we denote
$\bor{\espmetricgeneric}$ its Borel $\sigma$--algebra, and by
$\laws{\espmetricgeneric}$ the space of probability laws on
$(\espmetricgeneric,\bor{\espmetricgeneric})$, endowed with the usual narrow (or weak)
topology.
The law of a random element $X$ of $\espmetricgeneric$ is denoted by
$\law{X}$.
The space of random elements of $\espmetricgeneric$ defined on
$(\esprob,\tribu)$ is denoted by $\randvar(\esprob;\espmetricgeneric)$
or simply $\randvar(\espmetricgeneric)$, and
it is endowed with the topology of convergence in
$\pr$--probability. We identify random elements which are equal
$\pr$--a.e.
Recall that a subset $\Lambda$ of $\randvar(\espmetricgeneric)$ is
said to be tight
if,
for any $\delta>0$, there exists a compact subset
$\mathcal K$ of $\espmetricgeneric$ such that, for every
$X\in\Lambda$, $\pr(X\in {\mathcal K})\geq 1-\delta$.
\paragraph{Spaces of processes}
In all this paper, $p> 2$ is a fixed number.
For any $t\in\Times$,
we denote by $\BTC{t}{\esp}$ the space of continuous
$\filtration$--adapted $\esp$--valued
processes $\sol$ such that
\begin{equation*}
\norm{\sol}^p_{ \BTC{t}{\esp} }
:=\expect\CCO{\sup_{0\leq s\leq t}\norm{\sol(s)}_\esp^p}<+\infty.
\end{equation*}
For any $t\in\Times$
we denote by
$\skoroC([0,t];\esp)$ the space of continuous mappings from $[0,t]$
to $\esp$. The space $\skoroC([0,t];\esp)$ is endowed with the
topology of uniform convergence defined by the distance
$\distC(u,v)=\sup_{s\in[0,t]}\norm{u(s)-v(s)}$.
The space $\BTC{t}{\esp}$ is thus a closed separable subspace of the space
$\randvar\CCO{\skoroC([0,t];\esp)}$ of $\skoroC([0,t];\esp)$--valued
random variables.
\paragraph{Measures of noncompactness}
If $(\espmetricgene,d)$ is a metric space and
$\EProc\subset\espmetricgene$, we say that a subset $\EProc'$ of
$\espmetricgene$ is an {\em$\epsilon$--net} of $\EProc$ if
\begin{equation*}
\inf_{\proc\in\EProc}d(\proc,\EProc')\leq \epsilon
\end{equation*}
(note that $\EProc'$ is not necessarily a subset of $\EProc$).
Let $\EProc$ be a subset of $\BTC{\Time}{\esp}$.
For any $s\in[0,\Time]$, we denote
by $\EProc\restr{s}$ the subset of $\BTC{s}{\esp}$ of
restrictions to $[0,s]$ of elements of $\EProc$.
We denote
\begin{equation*}
\mncp(\EProc)(s):=\inf\accol{\epsilon>0\tq
\text{$\EProc\restr{s}$ has a tight $\epsilon$--net in $\BTC{s}{\esp}$}}.
\end{equation*}
We shall see in Lemma \ref{lem:G}
that $\EProc$ is tight \iff $\mncp(\EProc)(\Time)=0$.
Note also that the mapping $s\mt\mncp(\EProc)(s)$ is nondecreasing.
We denote
\begin{equation*}
\mncp(\EProc):=\CCO{\mncp(\proc)(s)}_{0\leq s\leq \Time}.
\end{equation*}
The family $\CCO{\mncp(\proc)(s)}_{0\leq s\leq t}$ is called the
{\em measure of noncompactness} of $\EProc$ (see
\cite{akprs92book,koz01book} about measures of noncompactness).
\paragraph{Spaces of closed compact subsets}
For any metric space $\espgene$, the set of nonempty compact
(resp.~nonempty compact convex)
subsets of $\espgene$ is denoted by $\compactspaconv(\espgene)$ (resp.~$\compacts(\espgene)$).
We endow $\compactspaconv(\espgene)$ and its subspace
$\compacts(\espgene)$ with the Hausdorff distance
$$\Hausd[\espgene](B,C):=\max\CCO{\inf_{b\in B}d(b,C),\inf_{c\in C}d(c,B)}.$$
Recall that, if $\espgene$ is Polish, then $\compacts(\espgene)$ is
Polish too (see \eg~\cite[Corollary II-9]{castaingvaladier77book}).
\paragraph{Stable convergence and Young measures}
Let $\espmetricgeneric$ be a complete metric space.
We denote by $\youngs(\esprob,\tribu,\pr;\espmetricgeneric)$ (or simply
$\youngs(\espmetricgeneric)$) the set of measurable mappings
$$\mu :\,\left\{
\begin{matrix}\esprob&\mt&\laws{\espmetricgeneric}\\
\omega&\mt&\mu_\omega
\end{matrix}
\right.$$
Each element $\mu$ of $\youngs(\esprob,\tribu,\pr;\espmetricgeneric)$ can be
identified with the measure $\widetilde{\mu}$ on
$(\esprob\times\espmetricgeneric,\tribu\otimes
\bor{\espmetricgeneric})$
defined by $\widetilde{\mu}(A\times
B)=\int_A\mu_\omega(B)\,d\pr(\omega)$ (and the mapping
$\mu\mt\widetilde{\mu}$ is onto,
see \eg~\cite{valadier73desi}). In the sequel, we shall use freely this
identification. For instance, if $f:\,\esprob\times\espmetricgeneric\ra\R$
is a bounded measurable mapping, the notation $\mu(f)$ denotes
$\int_\esprob \mu_\omega(f(\omega,.))\,d\pr(\omega)$.
The elements of $\youngs(\esprob,\tribu,\pr;\espmetricgeneric)$ are called
{\sl Young measures} on $\esprob\times\espmetricgeneric$.
Let $\Cb{\espmetricgeneric}$ denote the set of continuous bounded real
valued functions defined on $\espmetricgeneric$.
The set $\youngs(\esprob,\tribu,\pr;\espmetricgeneric)$ is endowed
with a metrizable topology, such that
a sequence $(\mu^n)$ of Young measures converges to a Young measure $\mu$ if,
for each $A\in\tribu$ and each $f\in\Cb{\espmetricgeneric}$,
the sequence $(\mu^n(\un{A}\otimes f))$ converges to
$\mu(\un{A}\otimes f)$ (where $\un{A}$ is the indicator function of $A$).
We then say that $(\mu^n)$ converges {\em stably} to $\mu$.
Each element $X$ of $\randvar(\esprob;\espmetricgeneric)$ can (and
will sometimes) be
identified with the Young
measure
$$\ydirac{X} :\,\omega\mt\dirac{X(\omega)},$$
where, for any $x\in\espmetricgeneric$, $\dirac{x}$
denotes the probability concentrated on $x$.
Note that the restriction to $\randvar(\esprob;\espmetricgeneric)$ of
the topology of stable convergence is the topology of convergence in
probability.
If $(X_n)$ is a tight sequence in
$\randvar(\esprob;\espmetricgeneric)$,
then, by Prohorov's compactness criterion
for Young measures \cite{balder89Prohorov,cc-prf-valadier04book},
each subsequence of $(X_n)$ has a further subsequence, say $(X'_n)$,
which converges stably to some $\mu\in\youngs(\espmetricgeneric)$,
that is, for every $A\in\tribu$ and every $f\in\Cb{\espmetricgeneric}$,
$$\lim_n\int_A f(X'_n)\,d\pr=\mu(\un{A}\otimes f).$$
This entails in particular that $(X'_n)$ converges in ditribution to
the measure $\mu(\esprob\times.)\in\laws{\espmetricgeneric}$.
See \cite{valadier94course} or \cite{balder00lectures} for
an introduction to Young measures and their applications.
\paragraph{Hypothesis}
In the sequel,
we are given two separable Hilbert spaces $\esp$ and $\hilbw$ and
an $\filtration_{t\in\Times}$--Brownian
motion $\wien$ (possibly cylindrical) on $\hilbw$.
We denote by $\HSz$ the space of Hilbert--Schmidt
operators from $\hilbw$ to $\esp$.
We shall consider the following hypothesis:
\begin{list}{}{\leftmargin 4em\labelwidth 2em}
\item[\HySg] $A$ is the generator of a $\mathcal{C}_0$ semigroup
$\semigroupe$.
In particular
(see \eg~\cite[Theorem 1.3.1]{ahmed91book}), there exist
$\sgcontract>0$ and $\sgcoef\in]-\infty,+\infty[$ such that, for
every $t\geq 0$,
\begin{equation*}
\norm{\semigr{t}}\leq\sgcontract e^{\sgcoef t}.
\end{equation*}
For $t\in[0,\Time]$, we denote $\CteCauchy{t}=
\sup_{0\leq s\leq t}\sgcontract e^{\sgcoef s}$.
\item[\HyF]
$F :\,\esprob\times\Times\times\esp\ra \compacts(\esp)$
and $G :\,\esprob\times\Times\times\esp\ra \compacts(\HSz)$ are
measurable mappings which satisfy
the following conditions:
\begin{qromain}
\item \label{hyp:growth}
There exists a constant $\growthconstant>0$ such that, for all
$(t,x)\in\Times\times\esp$,
\begin{align*}
\Hausd[\esp](0,F(t,x))&\leq \growthconstant(1+\norm{x})\\
\Hausd[\HSz](0,G(t,x))&\leq \growthconstant(1+\norm{x}).
\end{align*}
\item\label{hyp:FL}
For all $(t,x,y)\in\Times\times\esp\times\esp$,
\begin{align*}
\CCO{\Hausd[\esp](F(t,x),F(t,y))}^p
&\leq \LL(t,\norm{x-y}^p)\\
\CCO{\Hausd[\HSz](G(t,x),G(t,y))}^p
&\leq \LL(t,\norm{x-y}^p),
\end{align*}
where $\LL :\,[0,\Time]\times [0,+\infty]\ra
[0,+\infty]$ is a given continuous mapping such that
\begin{portemanteau}alph
\item for every $t\in[0,\Time]$, the mapping $\LL(t,.)$ is
nondecreasing and convex,
\item\label{hyp:L}
for every measurable mapping $\mapR :\,
[0,\Time]\ra[0,+\infty] $ and for every constant $\ctegeneral>0$,
the following implication holds true:
\begin{equation}\label{eq:hyp:L}
\croche{\foreach{t\in[0,\Time]}
\mapR(t)\leq \ctegeneral \int_0^t \LL(s,\mapR(s))\,ds
}\Rightarrow \mapR =0.
\end{equation}
In particular, we have $\LL(0)=0$, thus Hypothesis \HyFG-\qref{hyp:FL}
entails that, for each $t\in\Times$, the mappings $F(t,.)$ and
$G(t,.)$ are continuous for the Hausdorff distances $\Hausd[\esp]$ and
$\Hausd[\HSz]$ respectively.
Such a function $\LL$ is considered in
e.g.~\cite{yamada81successive,rodkina84heredity,manthey90convergence,akprs92book,
taniguchi92successive,barbu98local-global,barbu-bocsan02approx}.
Concrete examples can be found in \cite[Section 6 of Chapter 3]{hartman73book}.
\end{qualpha}
\end{qromain}
\item[\HyI]
$\initial\in\Ellp{p}(\esprob,\sstribu{0},\pr\restr{\sstribu{0}};\esp)$.
\end{list}
Recall that, under Hypothesis \HySg,
there exists a constant $\CteConv$ such that, for any predictable process
$\procHS\in\Ellp{p}(\esprob\times\Times;\HSz)$, we have
\begin{equation}\label{stochconvcondense}
\expect\croche{\sup_{s\leq t}\norm{
\int_0^s \semigr{s-r}\procHS(r)\,d\wien(r) }^p}
\leq \CteConv\,t^{(p/2)-1}\,\expect\int_0^t\norm{\procHS(s)}_{\HSz}^p\,ds
\end{equation}
(see
\cite{dapratozabczyk92convolution,dapratozabczyk92book}, and also
\cite{hausenblasseidler01stochconv} for a strikingly short proof in the case
when $\smgr$ is contractive).
\paragraph{Weak and strong mild solutions}
We say that $\sol\in\BTC{\Time}{\esp}$ is a {\em (strong) mild solution}
to Equation \eqref{eq:generale} if
there exist two predictable processes $f$ and $g$ defined on $\stbas$ satisfying
\begin{equation}\label{eq:ids}
\left\{
\begin{aligned}
&\sol(t)= \semigr{t}\initial+\int_0^t
\semigr{t-s}f(s)\,ds+\int_0^t
\semigr{t-s} g(s)\,d\wien(s)\\
&f(s)\in F(s,\sol(s))\ \pr\text{-a.e.}\\
&g(s)\in G(s,\sol(s))\ \pr\text{-a.e.}.
\end{aligned}\right.
\end{equation}
So, ``mild solution'' refers to the variation of constant formula,
whereas
``strong'' refers to the fact that the solution is defined on the
given stochastic basis.
We say that a process $\sol$ is a {\em weak mild solution}
or a {\em mild solution-measure}
to
\eqref{eq:generale} if there exists a stochastic basis
$\underline{\stbas}$\allowbreak
$=(\esprobb,$\allowbreak$\tribuu,$\allowbreak$(\tribuu_t)_t,$\allowbreak$\mu)$
satisfying the following conditions:
\begin{enumerate}
\item
$\esprobb$ has the form $\esprobb=\esprob\times\esprob'$,
$\tribuu=\tribu\otimes\tribu'$ for some $\sigma$--algebra $\tribu'$ on
$\esprob'$, $\tribuu_t=\tribu_t\otimes\tribu'_t$ for some right
continuous filtration $(\tribu'_t)$ on $(\esprob',\tribu')$, and the
probability $\mu$ satisfies $\mu(A\times\esprob')=\pr(A)$ for every
$A\in\tribu$.
\item The process $\wien$ is a Brownian motion on $\underline{\stbas}$
(we identify here every random variable $X$ on $\Omega$ with the
random variable $(\omega,\omega')\mt X(\omega)$ defined on
$\esprobb$).
\item
$X\in\BTCC$
and there exist two predictable processes $f$ and $g$ defined on $\underline{\stbas}$
satisfying \eqref{eq:ids}.
\end{enumerate}
The terminology {\em solution-measure} is that of
\cite{jacod-memin81weakstrong}.
If $\tribu'$ is the Borel $\sigma$--algebra of some
topology on $\esprob'$, a solution-measure can also be seen as a
Young measure.
This is the point of view adopted by Pellaumail
\cite{pellaumail81solfaibles,pellaumail81weak}, who calls Young
measures {\em rules}.
\paragraph{Main result and corollaries}
We can now state the main result of this paper.
The proofs will be given in Section \ref{sect:proofs}.
\begin{theo}\label{theo:main}
\titre{Main result}
Under Hypothesis \HySg, \HyFG\ and \HyI, Equation \eqref{eq:generale} has a
weak mild solution.
\end{theo}
An easy adaptation of our reasoning also yields, as a by-product, a
well-known strong existence
result:
\begin{prop}\label{prop:single-valued}
\titre{Strong existence in the single valued case \cite{barbu98local-global,barbu-bocsan02approx}}
Under Hypothesis \HySg, \HyFG\ and \HyI,
if furthermore $F$ and $G$ are single-valued, then \eqref{eq:generale} has a
strong mild solution.
\end{prop}
Using the Steiner point for the choice of selections of $F$ and $G$,
we deduce the following
\begin{prop}\label{cor:finitedim}
\titre{Strong existence in the finite dimensional case}
Under Hypothesis \HySg, \HyFG\ and \HyI,
if furthermore
$\esp$ and $\hilbw$ are finite dimensional, then \eqref{eq:generale} has a
strong mild solution.
\end{prop}
\section{Preliminary results}
\paragraph{Tightness results and boundedness results}
We start with a very simple and useful lemma.
\begin{lem}\label{lem:G}
\titre{A tightness criterion}
Let $(\espmet,\dist)$ be a separable complete metric space.
Let $\EProc$ be a set of random
elements of $\espmet$ defined on $\oap$.
Let $r\geq 1$.
Assume that, for every $\epsilon>0$, there exists a tight subset
$\EProc_\epsilon$ such that
\begin{equation*}
\EProc\subset\EProc_\epsilon^{[\epsilon]}:=\{X\in \randvar\CCO{\espmet}\tq \exists
Y\in\EProc_\epsilon,\ \expect\dist^r(X,Y)<\epsilon \}.
\end{equation*}
Then $\EProc$ is tight.
\end{lem}
\preuve
From Jensen inequality, we only need to prove Lemma
\ref{lem:G} for $p=1$.
Indeed, we have $\expect\dist(X,Y)\leq (\expect\dist^r(X,Y))^{1/r}$
thus
$$\EProc\subset\{X\in \randvar{(\espmet)}\tq \exists
Y\in\EProc_\epsilon,\ \expect\dist(X,Y)<\epsilon^{1/r}\}.$$
Let $\Blop{\dist}$ be the set of all mappings $f :\,\espmet\ra[0,1]$ wich are
1--Lischitz with respect to $\dist$.
Let $\beta$ denote the Dudley distance on $\laws{\espmet}$,
that is, for all $\mu,\nu\in\laws{\espmet}$,
\begin{equation*}
\beta(\mu,\nu)=\sup_{f\in\Blop{\dist}}\mu(f)-\nu(f).
\end{equation*}
It is well known that the narrow topology on $\laws{\espmet}$ is induced
by $\beta$ and that $\beta$ is complete, see \eg~\cite{dudley02book}.
For every $X\in\EProc$, there exists $Y\in\EProc_\epsilon$ such that
$\expect\dist(X,Y)<\epsilon$, which implies
$\beta(\law{X},\law{Y})<\epsilon$.
We thus have
\begin{equation*}
\accol{\law{X}\tq X\in\EProc}
\subset \cap_{\epsilon>0}\accol{\law{X}\tq X\in\EProc_\epsilon}^{[\epsilon]},
\end{equation*}
where, for any $\Xi\subset\laws{\espmet}$ and any $\epsilon>0$,
$\Xi^{[\epsilon]}=\{\mu\in\laws{\espmet}\tq \exists \nu\in\Xi,\
\beta(\mu,\nu)<\epsilon\}$.
This proves that $\{\laws{X}\tq X\in\EProc\}$ is totally bounded for
$\beta$, thus relatively compact in the narrow topology.
\fin
\begin{lem}\label{lem:Zero}
Let $\EProc$ be a set of continuous adapted processes on
$\esp$. Assume that each element of $\EProc$ is in
$\Ellp{p}(\esprob\times\Times;\esp)$ and
that
that $\EProc$, considered
as a set of $\skoroC(\Times;\esp)$--valued random variables, is tight.
Assume furthermore that $G$ satisfies Hypothesis
\HyFG-\qref{hyp:growth}.
Let $\Soll$ be the set of processes $\soll$ of the form
$$\soll(t)=\int_0^t\semigr{t-s}g(s)\,d\wien(s)$$
where $g$ is $\predict$--measurable and
$g(s)\in G(s,\proc(s))$ a.e.~for some $\proc\in\EProc$.
The set $\Soll$ is a tight set of $\skoroC(\Times;\esp)$--valued random
variables.
\end{lem}
\preuve
We will prove Lemma \ref{lem:Zero} through a series of reductions.
{\em First step:} {\em We can assume \wlg\ that there exists a compact
subset $\comppC$ of $\skoroC(\Times;\esp)$ such that, for each
$\proc\in\EProc$ and for each $\omega\in\esprob$,
$\proc(\omega,.)\in\comppC$.}
Indeed, assume that Lemma \ref{lem:Zero} is true under this additional hypothesis.
Let $\epsilon>0$. There exists a compact subset
$\comppC_\epsilon$ of $\skoroC(\Times;\esp)$ such that, for each
$\proc\in\EProc$, $\pr(\proc\in\comppC_\epsilon)\geq 1-\epsilon$.
For every $\proc\in\EProc$, there exists a measurable subset
$\esprob_\proc$ of $\esprob$ such that $\pr(\esprob_\proc)\geq
1-\epsilon$ and, for every $\omega\in\esprob_\proc$,
$\proc(\omega)\in\comppC_\epsilon$.
For each $\proc\in\EProc$ and each $t\in\Times$, let us denote
$$\proc^\epsilon(t)=\begin{cases}
\proc(t\wedge \tau)&\text{ if }\tau<+\infty\\
\proc(0)&\text{ if }\tau=+\infty,
\end{cases}
$$
where $\tau(\omega)$ is the infimum of all $s\in[0,\Time]$ such that
there exists $u\in\comppC$ which coincides with $\proc(\omega,.)$ on
$[0,s]$
(we take $\inf\emptyset=+\infty$).
The process $\proc^\epsilon$ is continuous and adapted, thus predictable.
Furthermore,
for every $\omega\in\esprob_\proc$, we
have $\proc(\omega,.)=\proc^\epsilon(\omega,.)$.
Let $\EProc^\epsilon=\{\proc^\epsilon\tq\proc\in\EProc\}$.
The set $\EProc^\epsilon$ is tight,
thus, from our hypothesis, the set
$\Soll^\epsilon$, obtained by replacing $\EProc$ by $\EProc^\epsilon$
in the definition of $\Soll$, is tight.
There exists a compact subset $\compppC_\epsilon$ of $\skoroC(\Times;\esp)$ such that, for each
$\procc^\epsilon\in\Soll^\epsilon$,
$\pr(\procc^\epsilon\in\compppC_\epsilon)\geq 1-\epsilon$.
Let $\procc\in\Soll$. The process $\procc$ has the form
$$\procc(t)=\int_0^t\semigr{t-s}g(s)\,d\wien(s)$$
where $g$ is $\predict$--measurable and
$g(s)\in G(s,\proc(s))$ a.e.~for some
$\proc\in\EProc$.
The set $\{\proc\not=\proc^\epsilon\}$ is predictable, thus there
exists a predictable process $g^\epsilon$ such that
$g^\epsilon(s)\in G(s,\proc^\epsilon(s))$
a.e.~ and $g^\epsilon(t)=g(t)$
for $t\leq \tau$ (we can construct $g^\epsilon$ as a selection of the
predictable multifunction $H$ defined by
$H(t)=\{g(t)\}$ if $t\leq\tau$ and
$H(t)=G(t,\proc^\epsilon(t))$ otherwise).
Let $\procc^\epsilon\in\Soll^\epsilon$ be defined by
$$\procc^\epsilon(t)=\int_0^t\semigr{t-s}g^\epsilon(s)\,d\wien(s).$$
We have
\begin{align*}
\procc(t)-\procc^\epsilon(t)
&=\int_{t\wedge\tau}^t \semigr{t-s}(g(s)-g^\epsilon(s))\,d\wien(s)
\end{align*}
We thus have
$\procc=\procc^\epsilon$ on $\esprob_\proc$.
This shows that, for every $\procc\in\Soll$, we have
\begin{align*}
\pr(\procc\not\in\compppC_\epsilon)
&=\pr(\procc=\procc^\epsilon\text{ and
}\procc^\epsilon\not\in\compppC_\epsilon)+
\pr(\procc\not=\procc^\epsilon\text{ and
}\procc\not\in\compppC_\epsilon)\\
&\leq
\pr(\procc^\epsilon\not\in\compppC_\epsilon)+\pr(\procc^\epsilon\not=\procc)
\leq 2\epsilon.
\end{align*}
Thus $\Soll$ is tight.
{\em Second step: We can furthermore assume \wlg\ that there exists a
compact subset $\comp$ of $\HSz$ such that
$G(t,\proc(\omega,t))\subset\compC$ a.e.~for all $\proc\in\EProc$.}
Assume that Lemma \ref{lem:Zero} holds under this hypothesis.
For any adapted process $g\in\Ellp{p}(\esprob\times\Times;\HSz)$,
we denote by $\soll^{(g)}$ the process
defined by
$$\soll^{(g)}(t)=\int_0^t\semigr{t-s}g(\omega,s)\,d\wien(s). $$
We denote by $\Gamma$ the set of
$\predict$--measurable $\HSz$--valued processes $g$ such that
$g(\omega,t)\in G(s,\proc(t))$ for every $t\in\Times$ a.e.~for some
$\proc\in\EProc$.
Let $\comppp_0$ be a compact subset of $\esp$ such that
$\comppC(t)\subset \comppp_0$ for all $t\in\Times$
(where $\comppC$ is as in Step 1 and
$\comppC(t)=\{u(t)\tq u\in\comppC\}$).
The multifunction
$t\mt G(t,\comppp_0)$ is measurable and has compact values in $\HSz$.
Let $\epsilon>0$.
There exists a compact subset ${\compppK}_\epsilon$ of
$\compacts(\HSz)$ and a measurable subset $I_\epsilon$ of $\Times$
such that, denoting by $\lebesgue$ the Lebesgue measure on $\Times$,
\begin{gather}
\lebesgue\CCO{\Times\setminus I_\epsilon}\leq \epsilon,
\begin{portemanteau}ad\text{ and }\begin{portemanteau}ad
\foreach{t\in I_\epsilon}
G(t,\comppp_0)\in{\compppK}_\epsilon. \label{eq:Iepsilon}
\end{gather}
Now, from a well known characterization of the compact subsets of
$\compactspaconv(\HSz)$
(\cite[Theorem 2.5.2]{michael51topol}, see also
\cite[Theorem {3.1}]{christensen74book} for the converse
implication), the set
$$\comp_\epsilon=\bigcup_{K\in{\compppK}_\epsilon}K$$
is a compact subset of $\HSz$.
Let us define a multifunction
$$G_\epsilon :\,\,\left\{
\begin{array}{lcl}
\Times\times\esp&\ra&\compacts(\HSz)\\
(t,x)&\mt&\un{I_\epsilon}(t)G(t,x).
\end{array}
\right.$$
For every
$g\in\Gamma$ and every $t\in\Times$, let us set
$$
g_\epsilon(t)=\un{I_\epsilon}(t)g(t).
$$
The process $g_\epsilon$ is $\predict$--measurable and satisfies, for some
$\proc\in\EProc$,
\begin{equation*}
g_\epsilon(t)\in G_\epsilon(t,\proc(t))\subset\comp_\epsilon \
\text{ a.e.~for every $t\in\Times$}
\end{equation*}
(we assume \wlg\ that $0\in\comppp_0$).
Thus, from our hypothesis, the set
\begin{gather*}
\Soll_\epsilon=\{\soll^{g_\epsilon}\tq g\in \Gamma \}
\end{gather*}
is tight. Let $R_0=\sup_{x\in\comppp_0}\norm{x}_\esp$.
By \HyFG-\qref{hyp:growth}, we have, for every $g\in\Gamma$ and every $t\in\Times$,
\begin{equation*}
\norm{g(t)-g_\epsilon(t)}\,\begin{cases}
=0&\text{if }t\in I_\epsilon\\
=\norm{g(t)}\leq
\growthconstant\,(1+R_0)&\text{if }t\not\in I_\epsilon.
\end{cases}
\end{equation*}
Using \eqref{stochconvcondense}, we thus have
\begin{align*}
\Hausd[\BTC{\Time}{\esp}]\CCO{\Soll,\Soll_\epsilon}
&\leq \sup_{g\in\Gamma}\,\CteConv\,\Time^{(p/2)-1}\,
\expect\int_0^t\norm{g(s)-g_\epsilon(s)}_{\HSz}^p\,ds \\
&\leq \CteConv\,\Time^{(p/2)-1}
\int_0^t \bigl(\growthconstant\,(1+R_0)\un{\Times\setminus
I_\epsilon(s)}\bigr)^p\,ds\\
&\leq \CteConv\,\Time^{(p/2)-1}\,\growthconstant^p\,(1+R_0)^p\,\epsilon^p.
\end{align*}
From Lemma \ref{lem:G}, we deduce that $\Soll$ is tight.
{\em Third step: We can also assume \wlg\ that $A$ is a bounded
operator on $\esp$.}
Let $A_n$ ($n>\sgcoef$) be the Yosida approximations of $A$.
We are going to prove that
\begin{equation}\label{eq:step3}
\sup_{g\in \Gamma}\expect\sup_{0\leq t\leq \Time}
\norm{
\int_0^t \CCO{\semigr{t-s}-e^{(t-s)A_n}}g(s)\,d\wien(s)
}^p\ra 0\ \text{ as }n\ra+\infty.
\end{equation}
Let $D(A)$ be the domain of $A$. We have $\overline{D(A)}=\esp$ thus,
for each $\epsilon>0$, we can find a
finite $\epsilon$--net $\comp_\epsilon$ of $\comp$ which lies in
$D(A)$. Then, for each $g\in\Gamma$, we can define a predictable
$\comp_\epsilon$--valued process $g_\epsilon$ such that
$\norm{g-g_\epsilon}_\infty\leq \epsilon$.
For each $n$, the semigroup $e^{tA_n}$ satisfies an inequality similar
to \eqref{stochconvcondense} with same constant $\CteConv$, because
$\CteConv$ depends only on the parameters $\sgcontract$ and
$\sgcoef$ in Hypothesis \HySg. We thus have
\begin{align*}
\sup_{g\in \Gamma}\expect\sup_{0\leq t\leq \Time}
\norm{
\int_0^t \semigr{t-s}\CCO{g(s)-g_\epsilon(s)}\,d\wien(s)
}^p
&\leq \CteConv\Time^{p/2}\,\epsilon^p\\
\sup_{g\in \Gamma}\expect\sup_{0\leq t\leq \Time}
\norm{
\int_0^t e^{(t-s)A_n}\CCO{g(s)-g_\epsilon(s)}\,d\wien(s)
}^p&\leq \CteConv\Time^{p/2}\,\epsilon^p.
\end{align*}
Therefore, we only need to prove \eqref{eq:step3} in the case when
$\comp\subset D(A)$.
For every $x\in D(A)$ and every integer $n>\sgcoef$, we have
\begin{align*}
\CCO{\semigr{t-s}-e^{(t-s)A_n}}x
&=\croche{\semigr{t-s-r}\,e^{r
A_n}}_0^{t-s}\,x
=\int_0^{t-s}
\semigr{t-s-r}\,e^{\tau A_n}\CCO{A_n-A}x\,dr.
\end{align*}
Thus, assuming that $\comp\subset D(A)$, and
denoting by $C$ a constant which may difer from line to line,
we have, for every
$g\in\Gamma$, using the stochastic Fubini theorem (see
\cite{dapratozabczyk92book}) and the convolution inequality
\eqref{stochconvcondense},
\begin{multline*}
\expect\sup_{0\leq t\leq \Time}
\norm{
\int_0^t \CCO{\semigr{t-s}-e^{(t-s)A_n}}g(s)\,d\wien(s)
}^p\\
\begin{aligned}
&=\expect\sup_{0\leq t\leq \Time}
\norm{
\int_0^t \int_0^{t-s} \semigr{t-s-r}\,e^{r A_n}\CCO{A_n-A}g(s)\,dr\,d\wien(s)
}^p\\
&=\expect\sup_{0\leq t\leq \Time}
\norm{
\int_0^t e^{r A_n}\,\int_0^{t-r} \semigr{t-s-r}\,e^{r A_n}\CCO{A_n-A}g(s)\,d\wien(s) \,dr
}^p\\
&\leq C\expect\sup_{0\leq t\leq \Time}
\int_0^t \norm{
\int_0^{t-r} \semigr{t-s-r}\,\CCO{A_n-A}g(s)\,d\wien(s)
}^p dr \\
&\leq C\int_0^\Time\expect\sup_{0\leq t\leq \Time}
\norm{
\int_0^{t-r} \semigr{t-s-r}\,\CCO{A_n-A}g(s)\,d\wien(s)
}^p dr \\
&\leq C\int_0^\Time\int_0^\Time\expect
\norm{
\CCO{A_n-A}g(s)}^p
ds\,dr.
\end{aligned}
\end{multline*}
But, for every $x\in D(A)$, we have $(A_n-A)x\ra 0$. So,
using the compactness of $\comp$ and
Lebesgue's dominated convergence theorem, we obtain \eqref{eq:step3}.
From Lemma \ref{lem:G}, we conclude that we only need to check
Lemma \ref{lem:Zero} for the semigroups $e^{tA_n}$ ($n>\sgcoef$), which amounts to
check Lemma \ref{lem:Zero} in the
case when $D(A)=\esp$.
In this case, $S(t)$ is the exponential $e^{tA}$ in the
usual sense.
{\em Fourth step: We can assume \wlg\ that $\esp$ is finite
dimensional.}
Let $(e_n)$ be an orthonormal basis of $\esp$. For each $n$, let
$\esp_n=\Span\CCO{e_1,\dots,e_n}$ and let $\proj_n$ be the orthogonal
projection from $\esp$ onto $\esp_n$.
Let $\Gamma$ be any contour around the spectrum of $A$, say $\Gamma$
is a circle $C(0,\rho)$.
Denoting by $R$ the resolvent operator, we have, for any $g\in\Gamma$,
\begin{multline*}
\int_0^t \CCO{e^{(t-s)A}-e^{(t-s)P_nA}}P_ng(s)\,d\wien(s)
\\
\begin{aligned}
&=\int_0^t\frac{1}{2\pi i}\int_\Gamma
e^{\lambda(t-s)}\,\CCO{R(\lambda,A)-R(\lambda,P_nA)}\,d\lambda\,
P_ng(s)\,d\wien(s)\\
&=\frac{1}{2\pi i}\int_\Gamma\int_0^t
e^{\lambda(t-s)}\,\CCO{R(\lambda,A)-R(\lambda,P_nA)}\,
P_ng(s)\,d\wien(s) \,d\lambda\\
&=\frac{\rho}{2\pi}\int_0^{2\pi}\int_0^t
e^{\rho\, e^{i\theta}\,(t-s)}\,\CCO{R(\rho\, e^{i\theta},A)-R(\rho\, e^{i\theta},P_nA)}\,
P_ng(s)\,d\wien(s) \,d\theta
\end{aligned}
\end{multline*}
by the stochastic Fubini theorem.
Denoting again by $C$ a constant which may change from line to line,
we thus have
\begin{multline*}
\expect\sup_{0\leq t\leq \Time}
\norm{
\int_0^t \CCO{e^{(t-s)A}-e^{(t-s)P_nA}}P_ng(s)\,d\wien(s)
}^p\\
\begin{aligned}
&\leq C \expect\int_0^{2\pi}\sup_{0\leq t\leq \Time}
\int_0^t\norm{
e^{\rho\, e^{i\theta}\,(t-s)}\,\CCO{R(\rho\, e^{i\theta},A)-R(\rho\, e^{i\theta},P_nA)}\,
P_ng(s)\,d\wien(s)
}^p d\theta\\
&\leq C \expect\int_0^{2\pi}
\sup_{0\leq t\leq \Time}\int_0^t\norm{
e^{\rho\, e^{i\theta}\,(t-s)}\,\CCO{R(\rho\, e^{i\theta},A)-R(\rho\, e^{i\theta},P_nA)}\,
P_ng(s)\,d\wien(s)
}^p d\theta\\
&\leq C \int_0^{2\pi}\int_0^\Time
\expect\norm{
\CCO{R(\rho\, e^{i\theta},A)-R(\rho\, e^{i\theta},P_nA)}\,
P_ng(s)\,d\wien(s)
}^p d\theta\\
\end{aligned}
\end{multline*}
using the convolution inequality for the semigroup
$t\mt e^{\rho\, e^{i\theta}\,t}$.
From the compactness of $\comp$ and
Lebesgue's
dominated convergence theorem, we get
$$
\sup_{g\in\Gamma}\expect\sup_{0\leq t\leq \Time}
\norm{
\int_0^t \CCO{e^{(t-s)A}-e^{(t-s)P_nA}}P_ng(s)\,d\wien(s)
}^p
$$
and we conclude as in Step 3.
{\em Fifth step: Assuming all
preceding reductions, we now prove Lemma \ref{lem:Zero}.}
Recall that $R_1=\sup_{x\in\comp}\norm{x}_\HSz$.
We have, for any $\epsilon>0$ and $R>0$,
\begin{align*}
\pr\biggl\{\sup_{0\leq t\leq\Time}\norm{\soll(t)}\leq R\biggr\}
&\leq \dfrac{4}{R^2}\expect\norm{\soll(t)}^2 \\
&= \dfrac{4}{R^2}\expect\int_0^\Time \norm{e^{(\Time-s)A}g(s)}^2\,ds \\
&\leq \dfrac{4}{R^2}\Time e^{\Time\norm{A}} R_1^2
\end{align*}
Taking $R$ large enough, we get
\begin{equation}
\foreach{\epsilon>0}
\thereis{R>0}
\pr\biggl\{\sup_{0\leq t\leq\Time}\norm{\soll(t)}\leq R\biggr\}
\leq \epsilon.\label{eq:aldousI}
\end{equation}
Now, let $\tas$ be the set of stopping times $\tau$ such that
$0\leq\tau\leq\Time$. If $\sigma,\tau\in\tas$ with
$0<\tau-\sigma\leq\delta$ for some $\delta>0$,
we have, for any $\soll\in\Soll$ of the form $\soll=\soll^{(g)}$,
with $g\in G$, and for any $\eta>0$,
\begin{align*}
\pr\accol{\norm{\soll(\tau)-\soll(\sigma)}>\eta}
&\leq \dfrac{1}{\eta^2}\expect\norm{\soll(\tau)-\soll(\sigma)}^2\\
&\leq \dfrac{2}{\eta^2}
\begin{aligned}[t]
\biggl(
\expect\int_0^\sigma
\norm{\bigl(e^{(\tau-s)A}-e^{(\sigma-s)A}\bigr)g(s)}^2\,ds \phantom{\biggr)}&\\
\phantom{\leq \dfrac{2}{\eta^2}\biggl(}
+\expect\int_\sigma^\tau \norm{e^{(\tau-s)A}g(s)}^2\,ds
\biggr)&
\end{aligned}\\
&\leq \dfrac{2}{\eta^2}\CCO{\Time\delta e^{2\Time\norm{A}}\norm{A}^2
R_1^2
+\delta e^{\delta{\norm{A}}R_1^2}}\\
&\leq\delta\, C(\eta)
\end{align*}
{with}
$C(\eta)=\dfrac{2e^{2\Time\norm{A}}\,R_1^2\,(1+\Time)}{\eta^2}$.
Taking $\delta$ small enough, we get
\begin{equation}\label{eq:aldousII}
\foreach{\epsilon>0}
\foreach{\eta>0}
\thereis{\delta>0}
\foreach{\soll\in\Soll}
\sup_{\substack{\sigma,\tau\in\tas\\0< \tau-\sigma\leq\delta}}
\pr\accol{\norm{\soll(\tau)-\soll(\sigma)}>\eta}
\leq\epsilon.
\end{equation}
From \eqref{eq:aldousI} and
\eqref{eq:aldousII}, we conclude, by a criterion of
Aldous \cite{aldous78stopping,jacod85limite},
that $\Soll$ is tight.
\fin
\paragraph{The multivalued operator $\Phi$}
Let us denote by
$\Phi$ the mapping which, with every continuous adapted $\esp$--valued process
$\sol$ such that $\expect{\int_0^\Time\norm{\sol(s)}^p\,ds}<+\infty$,
associates the set of all processes of the form
$$\semigr{t}\initial+\int_0^t
\semigr{t-s}f(s)\,ds+\int_0^t
\semigr{t-s} g(s)\,d\wien(s),$$
where $f$ and $g$ are predictable selections of
$(\omega,t)\mt \allowbreak F(t,\allowbreak \sol(\omega,t))$ and
$(\omega,t)\mt \allowbreak G(t,\allowbreak \sol(\omega,t))$ respectively.
Lemma \ref{lem:Zero} will be used through the following corollary:
\begin{cor}\label{cor:Zero}
\titre{The operator $\Phi$ maps tight sets into tight sets}
Let $\EProc$ be a set of continuous adapted processes on
$\esp$. Assume that each element of $\EProc$ is in
$\Ellp{p}(\esprob\times\Times;\esp)$ and that $\EProc$, considered
as a set of $\skoroC(\Times;\esp)$--valued random variables, is tight.
Assume furthermore that Hypothesis \HySg\ and \HyF\ are
satisfied.
Then
$\Phi\circ\EProc:=\cup_{\proc\in\EProc}\Phi(\proc)$
is a tight set of $\skoroC(\Times;\esp)$--valued random
variables.
\end{cor}
\preuve
Let us denote by $\Phi_F$ the mapping which,
with every continuous adapted $\esp$--valued process
$\sol$ such that $\expect{\int_0^\Time\norm{\sol(s)}^p\,ds}<+\infty$,
associates the set of all processes of the form
$$
\int_0^t \semigr{t-s}f(s)\,ds,
$$
where $f$ is a predictable selection of
$(\omega,t)\mt \allowbreak F(t,\allowbreak \sol(t))$.
By Ascoli's theorem (see the details in the proof of \cite[Lemma 4.2.1]{koz01book}),
the mapping
$$ f\mt \int_0^{\displaystyle .}
\semigr{.-s}f(s)\,ds $$
maps all measurable functions
$f :\,\Times\ra \esp$ with values in a given compact subset of
$\esp$ into a compact subset of $\skoroC(\Times;\esp)$.
Thus, following the same lines as in Steps 1 and 2 of the proof of
Lemma \ref{lem:Zero}, $\Phi_F$
maps tight bounded subsets of
$\Ellp{p}(\esprob\times\Times;\esp)$ into tight sets of
$\skoroC(\Times;\esp)$--valued random variables.
The conclusion immediately follows from Lemma
\ref{lem:Zero}.
\fin
\begin{lem}\label{lem:Aprime}
\titre{The operator $\Phi$ and the measure of noncompactness $\mncp$}
Let $\EProc$ be a bounded subset of $\BTC{\Time}{\esp}$. Assume
Hypothesis
\HyFG.
We then have
\begin{equation*}
{\mncp}^p(\Phi\circ\EProc)(t)
\leq \Lcte
\int_0^t \LL\CCO{s,{\mncp}^p(\EProc)}(s)\,ds
\end{equation*}
for some constant $\Lcte$ which depends only on
$\Time$, $p$, $\CteCauchy{\Time}$,and $\CteConv$.
\end{lem}
\preuve
The main arguments are inspired from \cite[Lemma 4.2.6]{akprs92book}.
For simplicity, the space $\BTC{t}{\esp}$ will be denoted by
$\BTCsimple{t}$.
Let $\epsilon>0$.
The function $t\mt\mncp(\EProc)(t)$ is increasing, thus there exist at
most a finite number of points $0\leq t_1\leq\dots\leq t_n\leq\Time$
for which $\mncp(\EProc)$ makes a jump greater than $\epsilon$.
Let us choose $\delta_1>0$ such that
$i\not=j\Rightarrow ]t_i-\delta_1,t_i+\delta_1[\cap
]t_j-\delta_1,t_j+\delta_1[=\emptyset$.
Using points $\beta_j$, $j=1,\dots,m$,
we divide the remaining part
$[0,\Time]\setminus \CCO{
]t_1-\delta_1,t_1+\delta_1[\cup\dots\cup]t_n-\delta_1,t_n+\delta_1[}$
into disjoint intervals in such
a way that, for each $j$,
\begin{equation}
\label{eq:diameter-range}
\sup_{s,t\in[\beta_{j-1},\beta_j]}\abs{\mncp(\EProc)(s)-\mncp(\EProc)(t)}<\epsilon.
\end{equation}
Let us then choose $\delta_2>0$ such that
$j\not=k\Rightarrow ]\beta_j-\delta_2,\beta_j+\delta_2[\cap
]\beta_k-\delta_2,\beta_k+\delta_2[=\emptyset$.
Now we start the construction of a tight net.
For each $j=1,\dots,m$, we choose a tight
$\CCO{{\mncp}(\EProc)(\beta_j)+\epsilon}$--net $N_j$ of $\EProc$
in $\BTCsimple{\beta_j}$.
(As $\BTCsimple{\beta_j}$ is separable, we can take a countable net
$N_j$, but this fact will not be used here.)
We obtain a (countable) family $\ZZ$ in $\BTCsimple{\Time}$ by
taking all continuous processes which coincide on each
$]\beta_{j-1}+\delta_2,\beta_j-\delta_2[$ ($1\leq j\leq m$) with some
element of $N_j$ and which have affine trajectories on the
complementary segments.
The set $\ZZ$ is tight and bounded in
$\Ellp{p}(\esprob\times\Times;\esp)$,
thus, by Corollary \ref{cor:Zero},
$\Phi(\ZZ)$ is tight.
Consider a fixed $\sol\in\EProc$.
We can find an element $Z$ of $\ZZ$ such that, for each $j=1,\dots,m$,
\begin{equation*}
\norm{\sol-Z}_{\BTCsimple{\beta_j}}
\leq
{{\mncp}(\EProc)(\beta_j)+\epsilon}.
\end{equation*}
For $t\in]\beta_{j-1}+\delta_2,\beta_j-\delta_2[$, we have, using
\eqref{eq:diameter-range},
\begin{align}
\expect\norm{\sol(t)-Z(t)}_\esp^p
&\leq \expect \sup_{\beta_{j-1}+\delta_2\leq s\leq \beta_j+\delta_2}
\norm{\sol(s)-Z(s)}_\esp^p \notag \\
&\leq \norm{\sol-Z}_{\BTCsimple{\beta_j}}^p\notag \\
&\leq \CCO{{\mncp}(\EProc)(\beta_j)+\epsilon}^p\notag \\
&\leq \CCO{{\mncp}(\EProc)(t)+2\epsilon}^p \label{eq:ptitsurI}
\end{align}
Let $\soll\in\Phi(\sol)$, say
$$\soll(t)=\semigr{t}\sol(0)+\int_0^t
\semigr{t-s}f(s)\,ds+\int_0^t
\semigr{t-s} g(s)\,d\wien(s),$$
where $f$ and $g$ are predictable selections of
$(\omega,t)\mt \allowbreak F(t,\allowbreak \sol(\omega,t))$ and
$(\omega,t)\mt \allowbreak G(t,\allowbreak \sol(\omega,t))$
respectively.
We can find predictable selections
$\tilde{f}$ and $\tilde{g}$ of $(\omega,t)\mt \allowbreak F(t,\allowbreak Z(\omega,t))$ and
$(\omega,t)\mt \allowbreak G(t,\allowbreak Z(\omega,t))$
respectively, such that, for every $t\in\Times$,
\begin{align}
\norm{\tilde{f}(t)-f(t)}^p
&\leq 2\Hausd^p\bigl(F(t,\sol(t)),F(t,Z(t))\bigr)\notag\\
&\leq 2\LL\bigl(t,\norm{\sol(t)-Z(t)}^p\bigr)
\label{eq:fftildeL}\\
\intertext{and}
\norm{\tilde{g}(t)-g(t)}^p
&\leq 2\Hausd^p\bigl(G(t,\sol(t)),G(t,Z(t))\bigr)\notag\\
&\leq 2\LL\bigl(t,\norm{\sol(t)-Z(t)}^p\bigr).
\label{eq:ggtildeL}
\end{align}
Let $d_{\BTCsimple{t}}$ be the distance in $\BTCsimple{t}$ associated
with $\norm{.}_{\BTCsimple{t}}$.
We have, using \HyFG\ and the convexity of
$x\mt x^p$,
\begin{align*}
d^p_{\BTCsimple{t}}\CCO{\soll,\Phi(Z)}
&\leq \expect \sup_{0\leq \tau\leq t}\,
\bigl\Vert
\int_0^\tau \semigr{t-s}(f(s)-\tilde{f}(s))\,ds\\
&\phantom{\leq \expect \sup_{0\leq \tau\leq t}\,\bigl\Vert}
+\int_0^\tau \semigr{t-s}(g(s)-\tilde{g}(s))\,d\wien(s)
\bigr\Vert^p\\
&\leq 2^{p-1}\expect\,\sup_{0\leq \tau\leq t}\,
\bigl\Vert \int_0^\tau \semigr{t-s}(f(s)-\tilde{f}(s))\,ds
\bigr\Vert^p\\
&\phantom{\leq \expect \sup_{0\leq \tau\leq t}\,\bigl\Vert}
+\expect\,\sup_{0\leq \tau\leq t}\,
\Vert \int_0^\tau \semigr{t-s}(g(s)-\tilde{g}(s))\,d\wien(s)
\bigr\Vert^p\\
&\leq 2^{p-1}\bigl(
\CteCauchy{t}\int_0^t\expect \norm{f(s)-\tilde{f}(s)}^p\,ds\\
&\phantom{\leq \expect \sup_{0\leq \tau\leq t}\,\bigl\Vert}
+\CteConv\,\Time^{p/2-1}\int_0^t\expect \norm{g(s)-\tilde{g}(s)}^p\,ds
\bigr)
\end{align*}
(recall that $\CteCauchy{t}$ is defined in \HySg).
{Let us denote}
\begin{gather}
\Lcte=2^{p}\max\CCO{\CteCauchy{\Time},\CteConv\,\Time^{p/2-1}},\label{eq:def-de-k}\\
J(t)=[0,t]\cap
\biggl((\cup_{1\leq i\leq n}]t_i-\delta_1,t_i+\delta_1[)
\cup(\cup_{1\leq j\leq
m}]\beta_j-\delta_2,\beta_j+\delta_2[)\biggr),\notag \\
I(t)=[0,t]\setminus J(t).\notag
\end{gather}
As $\EProc$ and $\ZZ$ are bounded in
$\Ellp{p}(\esprob\times\Times;\esp)$, using \HyFG\
\eqref{eq:fftildeL} and \eqref{eq:ggtildeL},
and taking
$\delta_1$ and $\delta_2$ sufficiently small, we get
\begin{align*}
d^p_{\BTCsimple{t}}\CCO{\soll,\Phi(Z)}
&\leq
\Lcte
\bigl(\int_{I(t)} \expect \LL(s,\norm{\sol(s)-Z(s)}^p)\,ds
+\epsilon \bigr).
\end{align*}
Then, using the convexity of
$\LL(t,.)$, and \eqref{eq:ptitsurI},
\begin{align*}
d^p_{\BTCsimple{t}}\CCO{\soll,\Phi(Z)}
&\leq \Lcte\bigl( \int_0^t\LL\CCO{s,\CCO{{\mncp}(\EProc)(s)+2\epsilon}^p}\,ds
+\epsilon\bigr).
\end{align*}
As $\epsilon$, $\sol$ and $\soll$ are arbitrary, the result follows.
\fin
Here is a easy variant for the case when $F$ and $G$ are single-valued:
\begin{lem}\label{lem:compact-single}
Assume that $F$ and $G$ are single-valued. Assume furthermore Hypothesis
\HyFG.
Let $\mncpp$ be the measure of noncompactness on $\BTC{\Time}{\esp}$
defined by
$$ \mncpp(\EProc)(s):=\inf\accol{\epsilon>0\tq
\text{$\EProc\restr{s}$ has a finite $\epsilon$--net in
$\BTC{s}{\esp}$}}
\begin{portemanteau}ad (0\leq s\leq\Time).$$
Let $\EProc$ be a bounded subset of
$\BTC{\Time}{\esp}$.
We have
\begin{equation*}
{\mncpp}^p(\Phi\circ\EProc)(t)
\leq \Lcte'
\int_0^t \LL\CCO{s,{\mncpp}^p(\EProc)}(s)\,ds
\end{equation*}
for some constant $\Lcte'$ which depends only on
$\Time$, $p$, $\CteCauchy{\Time}$,and $\CteConv$.
\end{lem}
\preuve
We only need to repeat the proof of Lemma \ref{lem:Aprime}, but we take for
each $j$ a finite $\CCO{{\mncpp}(\EProc)(\beta_j)+\epsilon}$--net
$N_j$. Then $\ZZ$ is finite, thus $\Phi(\ZZ)$ is compact in
$\BTC{\Time}{\esp}$. The rest of the proof goes as in the proof of
Lemma \ref{lem:Aprime}.
\fin
In the proof of Theorem \ref{theo:main}, we shall consider a variant
of the operator $\Phi$.
For each $n\geq 1$, let $\Phi_n$
be the mapping which, with every continuous adapted $\esp$--valued process
$\sol$ such that $\expect{\int_0^\Time\norm{\sol(s)}^p\,ds}<+\infty$,
associates the set of all processes of the form
$$\semigr{t-1/n}\initial+\int_0^{t-1/n}
\semigr{t-s}f(s)\,ds+\int_0^{t-1/n}
\semigr{t-s} g(s)\,d\wien(s),$$
where $f$ and $g$ are predictable selections of
$(\omega,t)\mt F(t,\sol(\omega,t))$ and
$(\omega,t)\mt G(t,\sol(\omega,t))$.
The following lemma links the tightness properties of $\Phi$ and
$\Phi_n$.
\begin{lem}\label{lem:C}
Let $(\sol_n)$ be a sequence of continuous adapted $\esp$--valued
processes, which is bounded in $\Ellp{p}(\esprob\times\Times;\esp)$.
We then have
\begin{equation*}
\mncp\CCO{\cup_{n}\Phi_n(\sol_n)}
\leq \CteCauchy{\Time}\mncp\CCO{\cup_n\Phi(\sol_n)},
\end{equation*}
where $\CteCauchy{\Time}$ has been defined in \HySg.
\end{lem}
\preuve
First, in the definition of $\Phi_n$ and in that of $\Phi$, we can
assume that $\initial=0$, because $\initial$ does not change the
values of $\mncp\CCO{\cup_{n}\Phi_n(\sol_n)}$ and
$\mncp\CCO{\cup_n\Phi(\sol_n)}$.
We thus have
\begin{align*}
\Phi_n(\sol_n)(t)
&=\biggr\{\begin{aligned}[t]
\semigr{{1}/{n}}\int_0^{t-1/n}&\semigr{(t-1/n)-s}f(s)\,ds\\
&+\semigr{{1}/{n}}\int_0^{t-1/n}\semigr{(t-1/n)-s}g(s)\,d\wien(s)
\tq \\
&\text{ $f$ and $g$ are predictable selections of $F\circ\sol$ and $G\circ\sol$}
\biggl\}\end{aligned}\\
&=\semigr{{1}/{n}}\Phi(\sol_n)(t-1/n).
\end{align*}
Let $\tau_n :,\skoroC(\Times;\esp)\ra\skoroC(\Times;\esp)$ be defined by
$$\tau_n(u)(t)=
\begin{cases}
u(t-1/n)& \text{ if }t\geq 1/n\\
u(0) & \text{ if }t\leq 1/n.
\end{cases}
$$
For any set $\EProc$ of continuous processes, we have
$$\mncp(\tau_n(\EProc))\leq \mncp(\EProc)$$
because, if $\Xi$ is a tight $\epsilon$--net of $\EProc$,
then $\tau_n(\Xi)$ is a tight $\epsilon$--net of $\tau_n(\EProc)$.
We thus have
\begin{multline*}
\rule{1cm}{0em}
\mncp\CCO{\cup_{n}\Phi_n(\sol_n)}
=\mncp\CCO{\semigr{{1}/{n}}\cup_n\tau_n(\Phi(\sol_n))}\\
\leq \CteCauchy{\Time} \mncp\CCO{\cup_n\tau_n(\Phi(\sol_n))}
\leq \CteCauchy{\Time} \mncp\CCO{\cup_n\Phi(\sol_n)}.
\rule{1cm}{0em}
\end{multline*}
\fin
\section{Proof of the main results}
\label{sect:proofs}
\preuvof{Theorem \ref{theo:main}}
{\em First step: construction of a tight sequence of
approximating solutions through Tonelli's scheme}.
For each integer $n\geq 1$, we can easily define a
process $\ssol_n$ on [-1,\Time] by
$\ssol_n(t)=0$ if $t\leq 0$
and, for $t\geq 0$,
\begin{align*}
\ssol_n(t)
&= \semigr{t}\initial+\int_0^t\semigr{t-(s-\frac{1}{n})}f_n(s)\,ds
+\int_0^t\semigr{t-(s-\frac{1}{n})}g_n(s)\,d\wien(s),
\end{align*}
where $f_n :\,\esprob\times\Times\times\ra\esp$ and
$g_n :\,\esprob\times\Times\times\ra\HSz$ are predictable and
\begin{align*}
f_n(s)&\in F(s,\ssol_n(s-1/n)) \text{ a.e.}\begin{portemanteau}ad\text{ and }\begin{portemanteau}ad
g_n(s)\in G(s,\ssol_n(s-1/n)) \text{ a.e.}\
\end{align*}
We then set, for $t\in[1/n,\Time]$,
\begin{align*}
\sol_n(t)
&= \ssol_n(t-1/n)\\
&= \semigr{t-1/n}\initial+\int_0^{t-1/n}\semigr{t-s}f_n(s)\,ds
+\int_0^{t-1/n}\semigr{t-s}g_n(s)\,d\wien(s).
\end{align*}
For $t\leq 1/n$, we set $\sol_n(t)=\initial$.
Let us show that $(\sol_n)$ is bounded in
$\Ellp{p}(\esprob\times\Times;\esp)$.
By convexity of $t\mt\abs{t}^p$, we have
\begin{equation}
\norm{a_1+\dots+a_m}^p\leq m^{p-1}\CCO{\norm{a_1}^p+\dots+\norm{a_m}^p}
\end{equation}
for any finite sequence $a_1,\dots,a_m$ in any normed space.
Recall also that $\CteCauchy{\Time}$ and $\growthconstant$ have been
defined in Hypothesis \HySg\ and \HyF-\qref{hyp:growth} and
$\CteConv$ is the constant of stochastic convolution defined in
\eqref{stochconvcondense}.
For every $n\geq 1$, we have the following chain of inequalities,
where the supremum is taken over all predictable
selections $(f,g)$ of $(\omega,t)\mt F(t,\sol_n(t))\times
G(t,\sol_n(t))$:
\begin{align*}
\expect{\norm{\sol_n(t)}^p}
&\leq \sup_{f,g}
3^{p-1}\begin{aligned}[t]\biggl(
\expect\norm{\semigr{t-1/n}\initial}
&+\expect\int_0^{t-1/n}\norm{\semigr{t-s}f(s)}^p\,ds\\
&+\expect\int_0^{t-1/n}\norm{\semigr{t-s}g(s)}^p\,d\wien(s)
\biggr)\end{aligned}\\
&\leq \sup_g 3^{p-1}\begin{aligned}[t]\biggl(
\CteCauchy{\Time}^p\expect\norm{\initial}^p
&+\CteCauchy{\Time}^p\growthconstant^p\expect\int_0^{t}\CCO{1+\norm{\sol_n}(s)}^p\,ds\\
&+\CteCauchy{\Time}^p\CteConv\expect\int_0^{t-1/n}\norm{g(s)}^p\,ds
\biggr)\end{aligned}\\
&\leq 3^{p-1}\CteCauchy{\Time}^p\begin{aligned}[t]\biggl(
\expect\norm{\initial}^p
&+\growthconstant^p(1+\CteConv)\expect\int_0^t\CCO{1+\norm{\sol_n(s)}}^p\,ds
\biggr).\end{aligned}\\
&\leq 3^{p-1}\CteCauchy{\Time}^p\begin{aligned}[t]\biggl(
\expect\norm{\initial}^p
&+\growthconstant^p(1+\CteConv)2^{p-1}\expect\int_0^t 1+\norm{\sol_n(s)}^p\,ds
\biggr).\end{aligned}
\end{align*}
Let
$\grosscte
=3^{p-1}\CteCauchy{\Time}^p
\bigl(\expect\norm{\initial}^p+2^{p-1}\growthconstant^p(1+\CteConv)\Time\bigr)$.
We have
$$
\expect{\norm{\sol_n(t)}^p}
\leq\grosscte +\grosscte\int_0^t\expect\norm{\sol_n(s)}^p\,ds,
$$
thus, by Gronwall Lemma,
$$
\expect{\norm{\sol_n(t)}^p}\leq \grosscte e^{\grosscte t}$$
which provides the boundedness condition
\begin{equation}\label{eq:xn-bounded}
\sup_n\expect\int_0^\Time \norm{\sol_n(t)}^p\,dt
\leq \Time\grosscte e^{\grosscte \Time}.
\end{equation}
Let us now show that $(\sol_n)$ is a tight sequence of
$\skoroC(\Times;\esp)$--valued random variables.
As $(\sol_n)$ is bounded in
$\Ellp{p}(\esprob\times\Times;\esp)$, we can apply Lemma \ref{lem:C}
and Lemma \ref{lem:Aprime}.
Using
the fact that $\sol_n\in\Phi_n(\sol_n)$ for each $n$ , we get, for every
$t\in\Times$,
\begin{align*}
\CCO{\mncp(\cup_n\{\sol_n\})(t)}^p
&\leq \CCO{\mncp(\cup_n\Phi_n(\sol_n)(t)}^p
\leq \CteCauchy{\Time}^p\mncp\CCO{\cup_n\Phi(\sol_n)(t)}^p\\
&\leq \CteCauchy{\Time}^p\,\Lcte
\int_0^t \LL\CCO{s,{\mncp}^p(\cup_n\{\sol_n\})(s)}\,ds
\end{align*}
(where $\Lcte$ is the constant we obtained in Lemma \ref{lem:Aprime},
see \eqref{eq:def-de-k}).
Thus, by \eqref{eq:hyp:L} in Hypothesis \HyF-\qref{hyp:L},
we have
${\mncp(\cup_n\{\sol_n\})(t)}=0$ for each $t$, which, by Lemma
\ref{lem:G}, implies that $(\sol_n)$ is tight.
{\em Second step: construction of a weak solution}.
By Prohorov's compactness criterion for Young measures,
we can extract a subsequence of $(X_n)$ which converges
{stably}
to a Young measure $\mu\in\youngs(\esprob,\tribu,\pr;\skoroC(\Times;\esp))$.
For simplicity, we denote this extracted sequence by $(X_n)$.
It will be convenient to represent the limiting Young measure $\mu$ as a
random variable defined on an extended probability space.
Let $\Ctribu$ be the Borel $\sigma$-algebra of $\skoroC(\Times;\esp)$
and, for each $t\in\Times$, let
$\Ctribu_t$ be the sub-$\sigma$--algebra of $\Ctribu$ generated by
$\skoroC([0,t];\esp)$.
We define a stochastic basis $(\esprobb,\tribuu,(\tribuu_t)_t,\mu)$ by
\begin{equation*}
\esprobb=\esprob\times\skoroC(\Times;\esp),
\begin{portemanteau}ad
\tribuu=\tribu\otimes\Ctribu
\begin{portemanteau}ad
\tribuu_t=\tribu_t\otimes\Ctribu_t
\end{equation*}
and we define
$X_\infty$ on $\esprobb$ by
$$X_\infty(\omega,u)=u.$$
Clearly, $\law{X_\infty}=\mu$ and $X_\infty$ is
$(\tribuu_t)$--adapted.
Now, the random variables $X_n$ can be seen as random elements defined
on $\esprobb$, using the notation
$$X_n(\omega,u):=X_n(\omega)\begin{portemanteau}ad(n\in\N).$$
Furthermore, $X_n$ is $(\tribuu_t)$--adapted for each $n$.
The $\sigma$--algebra $\tribu$ can be identified with the
sub--$\sigma$--algebra
$\{A\times\skoroC(\Times;\esp)\tq A\in\tribu\}$
of $\tribuu$.
We thus have:
\begin{equation}\label{eq:F-stable}
\foreach{A\in\tribu}
\foreach{f\in\Cb{\esp}}
\lim_n\int_A f(X_n)\,d\mu
=\int_A f(X_\infty)\,d\mu.
\end{equation}
To express \eqref{eq:F-stable}, we say that
$(X_n)$ converges to $X_\infty$ {\em $\tribu$--stably}.
Let us show that $\wien$ is an $(\tribuu_t)$--Wiener process under the
probability $\mu$.
Clearly, $\wien$ is $(\tribuu_t)$--adapted,
so we only need to prove
that $\wien$ has independent increments.
By a result of Balder \cite{balder89Prohorov,balder90new},
each subsequence of $(\ydirac{X_n})$ contains a further subsequence
$(\ydirac{{X}'_n})$
which K--converges to $\mu$, that is, for each subsequence
$(\ydirac{{X}''_n})$ of $(\ydirac{{X}'_n})$, we have
$$\lim_n\frac{1}{n}\sum_{i=1}^n\dirac{{X}''_n(\omega)}=\mu_\omega\ \text{a.e.}$$
This entails that, for every $A\in\Ctribu_t$, the mapping $\omega\mt\mu_\omega(A)$
is $\tribu_t$--measurable\footnotemark.
\footnotetext{From \cite[Lemma 2.17]{jacod-memin81weakstrong},
this means that $(\esprobb,\tribuu,(\tribuu_t)_t,\mu)$
is a {\em very good extension}
of $(\esprob,\tribu,(\tribu_t)_t,\pr)$
in the sense of
\cite{jacod-memin81weakstrong}, that is,
every martingale on $(\esprob,\tribu,(\tribu_t)_t,\pr)$ remains a martingale
on $(\esprobb,\tribuu,(\tribuu_t)_t,\mu)$.}
Let $t\in\Times$ and let $s>0$ such that $t+s\in\Times$.
Let us prove that, for
any $A\in\tribuu_t$ and any Borel subset $C$ of $\hilbw$, we have
\begin{equation}\label{eq:independence}
\mu\CCO{A\cap \{\omega\in\esprob\tq \wien(t+s)-\wien(t)\in C\}}
=\mu(A)\, \mu\{\omega\in\esprob\tq \wien(t+s)-\wien(t)\in C\}.
\end{equation}
Let $B=\{\omega\in\esprob\tq \wien(t+s)-\wien(t)\in C\}$.
We have
\begin{align*}
\mu\CCO{A\cap (B\times\skoroC(\Times;\esp))}
&=\int_{\Omega\times\skoroC(\Times;\esp)}
\un{A}(\omega,u))\un{B}(\omega)\,d\mu(\omega,u) \\
&=\int_\Omega\mu_\omega(\un{A}(\omega,.))\un{B}(\omega)\,d\pr(\omega) \\
&=\int_\Omega\mu_\omega(\un{A}(\omega,.))\,d\pr(\omega)\,\pr(B) \\
&=\mu(A)\, \mu(B\times\skoroC(\Times;\esp)),
\end{align*}
which proves \eqref{eq:independence}. Thus
$\wien(t+s)-\wien(t)$ is independent of $\tribuu_t$.
Now, there remains to prove that $X=X_\infty$ satisfies \eqref{eq:generale}.
Note that the first step of the proof of Theorem \ref{theo:main}
is valid for any choice of the selections $f_n$ and
$g_n$. In this second part, we will use a particular choice.
As $F$ and $G$ are globally measurable, they have measurable graphs
(see \cite[Proposition III.13]{castaingvaladier77book}). As
furthermore they are
continuous in the second
variable, they admit Carath\'eodory selections
\cite[Corollary 1 of the Main Lemma]{kucia98carath},
that is, there exist globally
measurable mappings
$\carathf:\,\Times\times\esp\rightarrow \esp $ and
$\carathg :\,\Times\times\esp\rightarrow \HSz $
such that $\carathf(t,.)$ and $\carathg(t,.)$ are continuous for every
$t\in\Times$ and $\carathf(t,x)\in F(t,x)$ and $\carathg(t,x)\in
G(t,x)$ for all
$(t,x)\in\Times\times\esp$.
We denote by $\NN$ the one-point compactification of $\N$ and we set,
for every $n\in\NN$ and every
$s\in\Times$,
\begin{alignat*}{2}
f_n(s)&=\carathf(s,X_n(s))\\
g_n(s)&=\carathg(s,X_n(s)).
\end{alignat*}
The sequence $(f_n)$ converges in law to $f_\infty$ in the space
$\Ellp{p}\CCO{\Times;\esp}$.
Indeed,
by the growth condition \HyF-\qref{hyp:growth} and Lebesgue's
convergence theorem,
the mapping
$$\supercarath:\,\left\{\begin{array}{lcl}
\skoroC(\Times;\esp])&\rightarrow\Ellp{p}\CCO{\Times;\esp}\\
u&\mapsto \carathf(.,(u(.)))
\end{array}\right.$$
is well defined and continuous, and we have
$f_n=\supercarath\circ X_n$.
Similarly, $(g_n)$ converges in law to
$g_\infty$ in $\Ellp{p}\CCO{\Times;\HSz}$.
Actually, we have more: the sequence
$(X_n,\wien,f_n,g_n)$ converges in law to
$(X_\infty,\wien,f_\infty,g_\infty)$ in
$\skoroC(\Times;\esp)$\allowbreak$\times$\allowbreak
$\skoroC(\Times;\hilbw)$\allowbreak$\times$\allowbreak
$\Ellp{p}\CCO{\Times;\esp}$\allowbreak$\times$\allowbreak$\Ellp{p}\CCO{\Times;\HSz}$.
In particular,
as $p>2$, the convergence in law of $(\wien,g_n)$ to $(\wien,g_\infty)$
implies, for every $t\in\Times$, the convergence in law of the stochastic
integrals
$
\int_0^t\semigr{t-s}g_n(s)\,d\wien(s)
$
to
$
\int_0^t\semigr{t-s}g_\infty(s)\,d\wien(s).
$
To see this, one can e.g.~use a Skorokhod representation $(\wien'_n,g'_n)$ which
converges a.e.~to $(\wien',g_\infty')$, then, by Vitali's convergence
theorem, $(\wien'n,g'_n)$ converges in
$\Ellp{2}\CCO{\esprob\times\Times;\hilbw\times\HSz}$ thus, for every
$t\in\Times$ the stochastic integrals
$
\int_0^.\semigr{t-s}g'_n(s)\,d\wien'(s)
$
converge in
$\Ellp{2}\CCO{\esprob\times[0,t];\esp}$.
Let $t\in\Times$ be fixed and
let us set, for every $n\in\NN$,
$$Z_n(t)=-X_n(t)+\semigr{t}\initial
+\int_0^{t-1/n}\semigr{t-s}f_n(s)\,ds
+\int_0^{t-1/n}\semigr{t-s}g_n(s)\,d\wien(s)$$
(with $1/\infty:=0$).
For every $t\in\Times$, the sequence $(Z_n(t))$ converges in law
to $Z_\infty(t)$.
Now, from the definition of $X_n$ ($n<+\infty$), we have
\begin{multline*}
\foreach{n<+\infty}
Z_n(t)=\CCO{\semigr{t}-\semigr{t-1/n}}\initial\\
+\int_{t-1/n}^t\semigr{t-s}f_n(s)\,ds
+\int_{t-1/n}^t\semigr{t-s}g_n(s)\,d\wien(s).
\end{multline*}
But $\CCO{\semigr{t}-\semigr{t-1/n}}\initial$ converges a.e.~to 0 and
we have, using the growth condition
\HyF-\qref{hyp:growth} and the boundedness property \eqref{eq:xn-bounded},
\begin{align*}
\expect\norm{\int_{t-1/n}^t\semigr{t-s}f_n(s)\,ds}
&\leq \sgcontract e^{\sgcoef \Time}\int_{0}^t\norm{f_n(s)\un{[t-1/n,t]}(s)}\,ds\\
&\leq \sgcontract e^{\sgcoef \Time}
\expect\CCO{\int_{0}^t\norm{f_n(s)}^p\,ds}^{1/p}\CCO{\frac{1}{n}}^{p/(p-1)}\\
&\leq \sgcontract e^{\sgcoef
\Time}\CCO{2^{p-1}\expect\int_0^t(1+\norm{X_n(s)})^p\,ds}^{1/p}\CCO{\frac{1}{n}}^{p/(p-1)}\\
&\rightarrow 0 \text{ as }n\rightarrow \infty
\end{align*}
and, for any $q$ such that $2<q<p$,
\begin{multline*}
\expect\norm{\int_{t-1/n}^t\semigr{t-s}g_n(s)\,d\wien(s)}^q\\
\begin{aligned}
&=\expect\norm{\int_{0}^t\semigr{t-s}g_n(s)\un{[t-1/n,t]}(s)\,d\wien(s)}^q\\
&\leq \CteConv'\Time^{q/2}\expect\int_0^t
\norm{g_n(s)\un{[t-1/n,t]}(s)}q\,ds\\
&\leq \CteConv'\Time^{q/2}\CCO{\expect\int_0^t
\norm{g_n(s)}^p\,ds}^{q/p}\CCO{\frac{1}{n}}^{(p-q)/p}\\
&\leq \CteConv'\Time^{q/2}\CCO{2^{q-1}\expect\int_0^t
(1+\norm{g_n(s)}^p)\,ds}^{q/p}\CCO{\frac{1}{n}}^{(p-q)/p} \\
&\rightarrow 0 \text{ as }n\rightarrow \infty,
\end{aligned}
\end{multline*}
where $\CteConv'$ is the constant of stochastic convolution associated
with $q$.
Thus $Z_n(t)$ converges to 0 in probability.
Thus, for every $t\in\Times$, we have $Z_\infty(t)=0$ a.s.
As $Z_\infty$ is continuous, this means that $Z_\infty=0$ a.s.
Thus $X_\infty$ is a weak mild solution to \eqref{eq:generale}.
\fin
\preuvof{Proposition \ref{prop:single-valued}}
Taking into account Lemma \ref{lem:compact-single},
and following the reasoning of the first part of the proof of Theorem
\ref{theo:main}, we see that the sequence $(X_n)$ provided by the
Tonelli scheme is relatively compact in $\BTC{\Time}{\esp}$.
Moreover, the limit of any convergent
subsequence of $(X_n)$ is a strong mild solution to \eqref{eq:generale}.
\fin
If $\espgene$ is a Banach space,
let us call {\em selection of $\compacts(\espgene)$} every mapping
$\steiner :\,\compacts(\espgene)\rightarrow\espgene$ such that $\steiner(K)\in
K$ for every $K\in\compacts(\espgene)$.
\preuvof{Proposition \ref{cor:finitedim}}
If $\HSz$ is finite dimensional, it is well-known
that there exists a selection $\steiner_\HSz$ of $\compacts(\HSz)$
which is Lipschitz with respect to the Hausdorff distance.
One such mapping is the Steiner
point (see
e.g.~\cite{schneider71steiner,saint-pierre85steiner,vitale85steiner,przeslawski96centres}).
In this case, we can define the Carath\'eodory mapping $\carathg$ of
the proof of Theorem \ref{theo:main} by
$\carathg(s,x)=\steiner_\HSz(G(s,x))$.
Then the selection $\carathg$ satisfies all hypothesis satisfied by $G$,
in particular \HyF-\qref{hyp:FL}.
Similarly, if $\esp$ is finite dimensional, we can define
$\carathf$ as the Steiner point of $F$.
So, the proof of Corollary \ref{cor:finitedim}
reduces to the case
when $F$ and $G$ are single valued, and, in this case, from Proposition
\ref{prop:single-valued}, there exists a strong mild solution to
\eqref{eq:generale}.
\fin
\begin{rem}
It is well known that, if $\HSz$ is infinite dimensional,
there exists no Lipschitz selection of $\compacts(\HSz)$: see
\cite[Theorem 4]{posicelski71lipschitzian},
where the reasoning given for the set of convex
bounded sets also applies to the set of convex compact sets,
and see also \cite{przeslawski-yost89continuity}.
\end{rem}
\noindent{\bf Aknowledgements }
We thank Professor Jan Seidler for several constructive remarks
and for making us know the papers
\cite{yamada81successive,manthey90convergence,taniguchi92successive,barbu98local-global,barbu-bocsan02approx}.
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\end{document} |
\begin{document}
\title{Single machine scheduling problems with uncertain parameters and the OWA criterion}
\author{Adam Kasperski\\
{\small \textit{Institute of Industrial}}\\
{\small \textit{Engineering and Management,}}\\
{\small \textit{Wroc{\l}aw University of Technology,}}\\
{\small \textit{Wybrze{\.z}e Wyspia{\'n}skiego 27,}}\\
{\small \textit{50-370 Wroc{\l}aw, Poland}}\\
{\small \textit{adam.kasperski@pwr.edu.pl}}
\and
Pawe{\l} Zieli{\'n}ski\\
{\small \textit{Institute of Mathematics}}\\
{\small \textit{and Computer Science}}\\
{\small \textit{Wroc{\l}aw University of Technology,}}\\
{\small \textit{Wybrze{\.z}e Wyspia{\'n}skiego 27,}}\\
{\small \textit{50-370 Wroc{\l}aw, Poland}}\\
{\small \textit{pawel.zielinski@pwr.edu.pl}}}
\maketitle
\begin{abstract}
In this paper a class of single machine scheduling problems is discussed. It is assumed that job parameters, such as processing times, due dates, or weights are uncertain and their values are specified in the form of a discrete scenario set. The Ordered Weighted Averaging (OWA) aggregation operator is used to choose an optimal schedule. The OWA operator generalizes traditional criteria used in decision making under uncertainty, such as the maximum, average, median or Hurwicz criterion. It also allows us to extend the robust approach to scheduling by taking into account various attitudes of decision makers towards a risk. In this paper a general framework for solving single machine scheduling problems with the OWA criterion is proposed and some positive and negative computational results for two basic single machine scheduling problems are provided.
\end{abstract}
\begin{keyword}
scheduling, single machine, robust optimization, OWA criterion
\end{keyword}
\section{Introduction}
Scheduling under uncertainty is an important and extensively studied area of operations research and discrete optimization. The importance of this research direction results from the fact that in many real-world problems the precise values of parameters in scheduling models are not known in advance. Thus, instead of possessing the exact values of the parameters, decision makers have rather a set of all their possible realizations, called a \emph{scenario set}. In some cases an additional information with this scenario set is available. If a probability distribution in the scenario set is known, then
\emph{stochastic approach} can be used, which typically consists in minimizing the expected solution cost (see, e.g.~\cite{P02}). The unknown probability distribution can be upper bounded by
a \emph{possibility distribution}, which leads to \emph{possibilistic} (fuzzy) scheduling problems
(see, e.g~\cite{KZ11f}). Finally, if no additional information with scenario set is provided,
then \emph{robust approach} is usually used (see, e.g.~\cite{KY97}). In the robust optimization, we seek a solution minimizing a cost in the worst case, which usually leads to applying the
\emph{minmax} or \emph{minmax regret} criterion for choosing a solution.
The robust approach to decision making is often regarded as too conservative or pessimistic. It follows from the fact, that the minmax criterion takes only the worst-case scenarios into account, ignoring the information connected with the remaining scenarios. This criterion also assumes that decision makers are very risk averse, which is not always true. These drawbacks of the minmax criterion are well known in decision theory, and a detailed discussion on this topic can be found, for example, in~\cite{LR57}.
In this paper we will assume that a scenario set associated with scheduling problem is specified by enumerating all possible scenarios. Such a representation of scenario sets is called a \emph{discrete uncertainty representation} and has been described, for instance, in~\cite{KY97}.
Our goal is to generalize the minmax approach to scheduling problems under uncertainty by using the \emph{Ordered Weighted Averaging} aggregation operator (OWA for short) introduced by Yager in~\cite{YA88}.
The OWA operator is widely applied to aggregate criteria in multiobjective decision problems (see, e.g.,~\cite{GS12, YKB11, OS03}) but it can also be applied to choose a solution under the discrete uncertainty representation by identifying scenarios with objectives in a natural way.
The OWA operator generalizes the classical criteria used in decision making under uncertainty such as the maximum, minimum, average, median, or Hurwicz criterion~\cite{LR57}. So, by using OWA we can extend the minmax approach, typically used in the
robust optimization. Furthermore, the weights used in the OWA operator allows us to model various attitudes of decision makers towards a risk.
Since we generalize the minmax approach to single machine scheduling problems under the
discrete uncertainty representation, let us briefly recall the known results in this area (see also~\cite{KZ14} for a survey).
The minmax version of the single machine scheduling problem with the total flow time criterion has been studied in~\cite{YY02}, where it has been shown that the problem is NP-hard even for two processing time scenarios and strongly NP-hard when the number of processing time scenarios is a part of the
input (the unbounded case). A generalization of this problem, with the weighted sum of completion times criterion, has been recently discussed in~\cite{MNO13, FA10} where, in particular, several inapproximability results for that problem have been established. We will describe these results in more detail later in this paper. In~\cite{AC08} the minmax version of the single machine scheduling problem with the maximum weighted tardiness criterion has been discussed, where it has been shown that some special cases of the problem are polynomially solvable. In this paper, we generalize and extend the algorithms proposed in~\cite{AC08}. In~\cite{ AAK11,AC08} the minmax version of the single machine scheduling problem with the number of late jobs criterion has been investigated. It has been shown in~\cite{AC08} that the problem is NP-hard for deterministic due dates and two processing time scenarios. On the other hand, it has been shown in~\cite{AAK11} that the problem with unit processing times and
the number of due date scenarios being a part of the input is strongly NP-hard and hard to approximate within a factor less than~2.
In a more general version of this problem the weighted sum of late jobs is minimized. This problem is known to be NP-hard for two weight scenarios~\cite{AV01}, strongly NP-hard and hard to approximate within any constant factor if the number of weight scenarios is a part of the input~\cite{KZ13}.
This paper is organized as follows. Section~\ref{sec1} presents
a formulation of the general problem under consideration as well as some its special cases.
The next two sections discuss two basic single machine scheduling problems. Namely, Section~\ref{sec2} explores the problem with the maximum weighted tardiness cost function and Section~\ref{sec3} investigates the problem in which the cost function is the weighted sum of completion times. We show that both problems have various computational properties which depend on the weight distribution in the OWA operator. For some weight distributions the problems are polynomially solvable, while for other ones they become strongly NP-hard and are also hard to approximate.
\section{Problem formulation}
\label{sec1}
Let $J=\{J_1,\dots,J_n\}$ be a set of jobs which must be processed on a single machine. For simplicity of notations, we will identify job $J_j$ with its index~$j$. The set of jobs may be partially ordered by some precedence constraints. The notation $i\rightarrow j$ means that processing of job $j$ cannot start before processing of job $i$ is completed (job $j$ is called a \emph{successor} of job $i$).
For each job $j$ the following parameters may be specified: a nonnegative \emph{processing time} $p_j$, a nonnegative \emph{due date} $d_j$ and a nonnegative \emph{weight} $w_j$. The due date $d_j$ expresses a desired completion time of $j$ and the weight $w_j$ expresses the importance of job~$j$ relative to the other jobs in the system. In all scheduling models discussed in this paper we assume that all the jobs are ready for processing at time~0, in other words, each job has a release date equal to~0. We also assume that each job must be processed without any interruptions, so we consider only nonpreemptive models. Under these assumptions we can define a \emph{schedule} $\pi$ as a feasible permutation of the jobs, in which the precedence constraints among the jobs are preserved. The set of all feasible schedules will be denoted by~$\Pi$.
Let us denote by $C_j(\pi)$ the completion time of job $j$ in schedule $\pi$. We will use $f(\pi)$ to denote a cost of schedule $\pi$. The value of $f(\pi)$ depends on job completion times and may also depend on job due dates or weights. In this paper we will investigate two
basic scheduling problems, in which the cost function is the maximum weighted tardiness, i.e. $f(\pi)=\max_{j\in J} w_j[C_j(\pi)-d_j]^+$ (we use the notation $[x]^+=\max\{0,x\}$) and the weighted sum of completion times, i.e. $f(\pi)=\sum_{j\in J} w_j C_j(\pi)$. In the deterministic case, we wish to find a feasible schedule which minimizes
the cost $f(\pi)$, that is:
$$\mathcal{P}:\; \min_{\pi \in \Pi} f(\pi).$$
We now study a situation in which some or all problem parameters are ill-known.
Let~$S$ be a vector of the problem parameters which may occur. The vector $S$ is called a \emph{scenario}. We will use $p_j(S)$, $d_j(S)$ and $w_j(S)$ to denote the processing time, due date, and weight of job $j$ under scenario $S$. A parameter is deterministic (precisely known) if its value is the same under each scenario.
Let a \emph{scenario set} $\Gamma=\{S_1,\dots,S_K\}$ contain all possible scenarios, where $K> 1$.
In this paper, we
distinguish the \emph{bounded case}, where~$K$ is bounded by a constant and
the \emph{unbounded case}, where~$K$ is a part of the input.
Now, the completion time of job~$j$ in $\pi$ and the cost of~$\pi$ depend on scenario $S_i \in \Gamma$ and will be denoted by~$C_j(\pi,S_i)$ and $f(\pi,S_i)$, respectively.
Since scenario set $\Gamma$ contains more than one scenario, an additional criterion is required to choose a reasonable solution.
In this paper we suggest to use the \emph{Ordered Weighted Averaging} aggregation operator
(OWA for short) proposed by Yager in~\cite{YA88}. We now describe this criterion. Let $(f_1,\dots,f_K)$ be a vector of real numbers. Let us introduce a vector of weights $\pmb{v}=(v_1,\dots,v_K)$ such that $v_j\in [0,1]$ for all $j\in [K]$ ($[K]$ stands for the set $\{1,\dots,K\}$) and $v_1+\dots+v_K=1$. Let $\sigma$ be a permutation of $[K]$ such that $f_{\sigma(1)}\geq f_{\sigma(2)}\geq \dots \geq f_{\sigma(K)}$. The OWA operator is defined as follows:
$${\rm owa}_{\pmb{v}}(f_1,\dots,f_K)=\sum_{i\in[K]} v_i f_{\sigma(i)}.$$
The OWA operator has several natural properties which follow directly from its definition (see, e.g.~\cite{YKB11}). Since it is a \emph{convex combination} of the cost functions,
$\min(f_1,\dots,f_K) \leq \mathrm{owa}_{\pmb{v}}(f_1,\dots,f_K) \leq \max(f_1,\dots,f_K)$. It is also \emph{monotonic}, i.e. if $f_j \geq g_j$ for all $j\in [K]$, then
$\mathrm{owa}_{\pmb{v}}(f_1,\dots,f_K)\geq \mathrm{owa}_{\pmb{v}}(g_1,\dots,g_K)$,
\emph{idempotent}, i.e. if $f_1=\dots=f_k=a$, then $\mathrm{owa}_{\pmb{v}}(f_1,\dots,f_K)=a$ and
\emph{symmetric}, i.e. its value does not depend on the order of the values $f_j$, $j\in [K]$. The OWA operator generalizes some important criteria used in decision making under uncertainty. If $v_1=1$ and $v_j=0$ for $j=2,\dots,K$, then OWA becomes the maximum. If $v_K=1$ and $v_j=0$ for $j=1,\dots,K-1$, then OWA becomes the minimum. In general, if $v_k=1$ and $v_j=0$ for $j\in [K]\setminus\{k\}$, then OWA is the $k$th largest element among $f_1,\dots,f_K$. In particular, when $k=\lfloor K/2 \rfloor +1$, the $k$th element is the median.
If $v_j=1/K$ for all $j\in [K]$, i.e. when the weights are \emph{uniform}, then OWA is the average (or the Laplace criterion). Finally, if $v_1=\alpha$ and $v_K=1-\alpha$ for some fixed $\alpha\in [0,1]$ and $v_j=0$ for the remaining weights, then we get the Hurwicz pessimism-optimism criterion.
We now use the OWA operator to aggregate the costs of a given schedule $\pi$ under scenarios in ~$\Gamma$. Let us define
$$\mathrm{OWA}(\pi)={\rm owa}_{\pmb{v}}(f(\pi,S_1),\dots,f(\pi,S_K))=\sum_{i\in [K]} v_i f(\pi,S_{\sigma(i)}),$$
where $\sigma$ is a permutation of $[K]$ such that $f(\pi,S_{\sigma(1)})\geq \dots\geq f(\pi,S_{\sigma(K)})$. In this paper we examine the following optimization problem:
$$\textsc{Min-Owa}~\mathcal{P}: \min_{\pi\in \Pi} \mathrm{OWA} (\pi).$$
We will also investigate the special cases of the problem, which are listed in Table~\ref{tabsc}.
\begin{table}[ht]
\caption{Special cases of \textsc{Min-Owa}~$\mathcal{P}$.} \label{tabsc}
\begin{tabular}{ll}
\hline
Name of the problem & Weight distribution \\ \hline
\textsc{Min-Max}~$\mathcal{P}$ & $v_1=1$ and $v_j=0$ for $j=2,\dots,K$ \\
\textsc{Min-Min}~$\mathcal{P}$ & $v_K=1$ and $v_j=0$ for $j=1,\dots,K-1$ \\
\textsc{Min-Average}~$\mathcal{P}$ & $v_j=1/K$ for $j\in [K]$ \\
\textsc{Min-Quant}$(k)$~$\mathcal{P}$ & $v_k=1$ and $v_j=0$ for $j\in [K]\setminus \{k\}$ \\
\textsc{Min-Median}~$\mathcal{P}$ & $v_{\lfloor K/2 \rfloor +1}=1$ and $v_j=0$ for $j\in [K] \setminus \{\lfloor K/2 \rfloor +1\}$\\
\textsc{Min-Hurwicz}~$\mathcal{P}$ & $v_1=\alpha$, $v_K=1-\alpha$, $\alpha\in [0,1]$ and $v_j=0$ for $j\in [K]\setminus\{1,K\}$ \\ \hline
\end{tabular}
\end{table}
Notice that \textsc{Min-Owa}~$\mathcal{P}$ can be consistent with
a concept of robustness. Namely, the risk averse decision makers should choose nonincreasing weights, i.e. such that $v_1\geq v_2\geq \dots \geq v_K$. In the extreme case, this leads to the maximum criterion and the \textsc{Min-Max}~$\mathcal{P}$ problem. However, the OWA operator allows us to weaken the maximum criterion by taking more scenarios into account
As we will see in the next sections, the complexity of \textsc{Min-Owa}~$\mathcal{P}$ depends on the properties of the underlying deterministic problem~$\mathcal{P}$ and the weights $v_1,\dots, v_K$. One general and easy observation can be made.
Namely, if $\mathcal{P}$ is solvable in $T(n)$ time, then \textsc{Min-Min}~$\mathcal{P}$ is solvable in $O(K\cdot T(n))$ time.
Indeed, in order to solve the \textsc{Min-Min}~$\mathcal{P}$ problem it is enough to compute an optimal schedule $\pi_k$ under each scenario $S_k$, $k\in [K]$, and choose the one which has the minimum value of~$f(\pi_k, S_k)$, $k\in [K]$. For the remaining problems listed in Table~\ref{tabsc} no such general result can be established and their complexity depends on a structure of the deterministic problem~$\mathcal{P}$.
\section{The maximum weighted tardiness cost function}
\label{sec2}
Let $T_j(\pi,S_i)=[C_j(\pi,S_i)-d_j(S_i)]^+$ be the \emph{tardiness} of job $j$ in $\pi$ under scenario $S_i$, $i\in [K]$. The cost of schedule $\pi$ under $S_i$ is the \emph{maximum weighted tardiness} under $S_i$, i.e. $f(\pi,S_i)=\max_{j\in J} w_j T_j(\pi,S_i)$. The underlying deterministic problem~$\mathcal{P}$ is denoted by
$1|prec|\max w_jT_j$ in
Graham's notation~\cite{GLLK79}.
In this section we will also discuss a special case of this problem, denoted by $1||T_{\max}$, with
unit job weights and
no precedence constraints between the jobs. The deterministic $1|prec|\max w_j T_j$ problem can be solved in $O(n^2)$ time by the well known algorithm designed by Lawler~\cite{LA73}. It follows directly from the Lawler's algorithm that $1||T_{\max}$ can be solved in $O(n\log n)$ time by applying the EDD rule, i.e. by ordering the jobs with respect to nondecreasing due dates.
This section contains the following results.
We will consider first the case when $K$ is unbounded ($K$ is a part of the input).
We will show that the problems of minimizing the average cost or median of the costs are then strongly NP-hard and also hard to approximate. On the other hand, we will prove that the problems of minimizing the maximum cost or the Hurwicz criterion are solvable in polynomial time. We will consider next the problem with a constant $K$. It turns out that in this case the general problem of minimizing the OWA criterion can be solved in pseudopolynomial time. Finally, we will propose an approximation algorithm, which can be efficiently applied to some particular weight distributions in the OWA criterion.
\subsection{Hardness of the problem}
\label{sec1_1}
The following theorem characterizes the complexity of the problem:
\begin{thm}
\label{thm1}
If the number of scenarios is unbounded, then
\begin{enumerate}
\item[(i)] \textsc{Min-Average}~$1||T_{\max}$ is strongly NP-hard and not approximable within $7/6-\epsilon$ for any $\epsilon>0$ unless P=NP,
\item[(ii)] \textsc{Min-Median}~$1|| T_{\max}$ is strongly NP-hard and not at all approximable unless P=NP.
\end{enumerate}
Furthermore, both assertions remain true even for jobs with unit processing times under all scenarios.
\end{thm}
\begin{proof}
We show a polynomial time approximation preserving reduction from the \textsc{Min $k$-Sat} problem, which is defined as follows. We are given boolean variables $x_1,\dots,x_n$ and a collection of clauses $C_1,\dots, C_m$, where each clause is a disjunction of at most $k$ literals (variables or their negations). We ask if there is an assignment to the variables which satisfies at most $L<m$ clauses. This problem is strongly NP-hard even for $k=2$ (see~\cite{AZ02, KM94, MR96}) and its optimization (minimization) version is hard to approximate within $7/6-\epsilon$ for any $\epsilon>0$ when $k=3$ (see~\cite{AZ02}).
\begin{table}[ht]
\centering
\caption{The due date scenarios for the formula $(x_1\vee \overline{x}_2 \vee \overline{x}_3)\wedge (\overline{x}_2 \vee \overline{x}_3 \vee x_4) \wedge (\overline{x}_1 \vee x_2 \vee \overline{x}_4) \wedge (x_1 \vee x_2 \vee x_3) \wedge (x_1 \vee x_3 \vee \overline{x}_4)$.} \label{tab1}
\begin{tabular}{l|lllll}
& $S_1$ & $S_2$ & $S_3$ & $S_4$ & $S_5$ \\ \hline
$J_{x_1}$ & 1 & 2 & 2 & 1 & 1\\
$J_{\overline{x}_1}$ & 2 & 2 & 1 & 2 & 2 \\ \hline
$J_{x_2}$ & 4 & 4 & 3 & 3 & 4\\
$J_{\overline{x}_2}$ & 3 & 3 & 4 & 4 & 4 \\ \hline
$J_{x_3}$ & 6 & 6 & 6 & 5 & 5\\
$J_{\overline{x}_3}$ & 5 & 5 & 6 & 6 & 6 \\ \hline
$J_{x_4}$ & 8 & 7 & 8 & 8 & 8\\
$J_{\overline{x}_4}$ & 8 & 8 & 7 & 8 & 7 \\ \hline
\end{tabular}
\end{table}
We first consider assertion~(i).
Given an instance of \textsc{Min 3-Sat}, we construct the corresponding instance of \textsc{Min-Average}~$1||T_{\max}$ in the following way.
We create two jobs $J_{x_i}$ and $J_{\overline{x}_i}$ for each variable $x_i$,
$i\in [n]$.
The processing times and weights of all the jobs under all scenarios are equal to~1.
The due dates of $J_{x_i}$ and $J_{\overline{x}_i}$ depend on scenario and will take the value of
either $2i-1$ or $2i$. Set $K=m$ and
form
$K$ scenario set~$\Gamma$ in the following way. Scenario~$S_k$ corresponds to clause $C_k=(l_1 \vee l_2 \vee l_3)$. For each $q=1,2,3$, if $l_q=x_i$, then the due date of $J_{x_i}$ is $2i-1$ and the due date of $J_{\overline{x}_i}$ is $2i$; if $l_q=\overline{x}_i$, then the due date of $J_{x_i}$ is $2i$ and the due date of $J_{\overline{x}_i}$ is $2i-1$; if neither $x_i$ nor $\overline{x}_i$ appears in $C_k$, then the due dates of $J_{x_i}$ and $J_{\overline{x}_i}$ are set to $2i$.
A sample reduction is shown in Table~\ref{tab1}.
Finally, we fix $v_k=1/m$ for all $k\in [K]$.
Let us define a subset of the schedules $\Pi'\subseteq \Pi$ such that each schedule $\pi\in \Pi'$ is of the form $\pi=(J_1,J_1',J_2,J_2',\dots,J_n,J_n')$, where $J_i,J_i'\in\{J_{x_i},J_{\overline{x}_i}\}$ for $i\in [n]$. Observe that $\Pi'$ contains exactly $2^n$ schedules and each such a schedule defines an assignment to the variables such that $x_i=0$ if $J_{x_i}$ is processed before $J_{\overline{x}_i}$ and $x_i=1$ otherwise.
Assume that the answer to \textsc{Min 3-Sat} is yes. So, there is an assignment to the variables which satisfies at most $L$ clauses. Choose schedule~$\pi\in \Pi'$ which corresponds to this assignment. It is easily seen that if clause~$C_k$ is not satisfied, then all jobs in~$\pi$ under~$S_k$ are on-time and the maximum tardiness in~$\pi$ under~$S_k$ is~0. On the other hand, if clause~$C_k$ is satisfied, then the maximum tardiness of~$\pi$ under~$S_k$ is~1. In consequence
$\frac{1}{K}\sum_{k \in [K]} f(\pi,S_k)\leq L/m$.
Assume now that there is a schedule $\pi$ such that $\frac{1}{K}\sum_{k \in [K]} f(\pi,S_k)\leq L/m$. Notice that $L/m < 1$ by the nonrestrictive assumption that $L<m$. We first show that $\pi$ must belong to $\Pi'$. Suppose that $\pi \notin \Pi'$ and let $J_i$ ($J_i')$ be the last job in $\pi$ which is not placed properly, i.e. $J_i,(J_i')\notin\{J_{x_i},J_{\overline{x}_i}\}$. Then $J_i$ ($J_i'$) is at least one unit late under all scenarios and $\frac{1}{K}\sum_{k \in [K]} f(\pi,S_k)\geq 1$, a contradiction. Since $\pi\in \Pi'$ and all processing times are equal to~1 it follows that $f(\pi,S_k)\in \{0,1\}$ for all $k\in [K]$. Consequently, the maximum tardiness in~$\pi$ is equal to~1 under at most~$L$ scenarios and the assignment corresponding to $\pi$ satisfies at most $L$ clauses.
The above
reduction is approximation-preserving and the inapproximability result immediately holds.
In order to prove assertion~(ii), it suffices to modify the previous reduction.
Assume first that $L<\lfloor m/2 \rfloor$. We then add to scenario set~$\Gamma$
additional $m-2L$ scenarios with the due dates equal to~0 for all the jobs. So the number of scenarios~$K$ is $2m-2L$.
We fix $v_{m-L+1}=1$ and $v_k=0$ for the remaining scenarios.
Now, the answer to \textsc{Min 3-Sat} is yes, if and only if there is a schedule~$\pi$
whose maximum tardiness is positive under at most $L+m-2L=m-L$ scenarios. According to the definition of the weights $\mathrm{OWA}(\pi)=0$.
Assume that $L>\lfloor m/2 \rfloor$. We then add to~$\Gamma$
additional $2L-m$ scenarios with the due dates to~$n$ for all the jobs.
The number of scenarios~$K$ is then $2L$. We fix $v_{L+1}=1$ and $v_k=0$ for all the remaining scenarios.
Now, the answer to \textsc{Min 3-Sat} is yes, if and only if there is a schedule~$\pi$ whose cost is positive under at most $L$ scenarios. According to the definition of the weights $\mathrm{OWA}(\pi)=0$. We thus can see that it is NP-hard to check whether there is a schedule~$\pi$ such that ${\rm OWA}(\pi)\leq 0$ and the theorem follows.
\end{proof}
The next theorem characterizes the problem complexity when job processing times and due dates are deterministic and only job weights are uncertain.
\begin{thm}
\label{thm2}
If the number of scenarios is unbounded, then
\begin{enumerate}
\item[(i)] \textsc{Min-Average}~$1||\max w_j T_j$ is strongly NP-hard.
\item[(ii)] \textsc{Min-Median}~$1||\max w_j T_j$ is strongly NP-hard and not at all approximable unless P=NP.
\end{enumerate}
Furthermore, both assertions are true when all jobs have unit processing times under all scenarios and all job due dates are deterministic.
\end{thm}
\begin{proof}
As in the proof of Theorem~\ref{thm1}, we show a polynomial time reduction from the \textsc{Min 3-Sat} problem. We start by proving assertion (i). We create two jobs $J_{x_i}$ and $J_{\overline{x}_i}$ for each variable $x_i$. The processing times of these jobs under all scenarios are 1 and their due dates are equal to $2i-1$.
Now for each clause $C_k=(l_1 \vee l_2 \vee l_3)$ we form the weight scenario $S_k$ as follows: for each $q=1,2,3$, if $l_q=x_i$, then the weight of $J_{x_i}$ is $1$ and the weight of $J_{\overline{x}_i}$ is $0$; if $l_q=\overline{x}_i$, then the weight of $J_{\overline{x}_i}$ is $1$ and the weight of $J_{x_i}$ is $0$; if neither $x_i$ nor $\overline{x}_i$ appears in $C_k$, then the weights of $J_{x_i}$ and $J_{\overline{x}_i}$ are 0. We also add one additional scenario $S_{m+1}$ under which the weight of each job is equal to $m$.
We set $K=m+1$ and
fix $v_k=1/(m+1)$ for each $k\in [K]$. We define the subset of schedules $\Pi'\subseteq \Pi$ as in the proof of Theorem~\ref{thm1}.
We will show that the answer to \textsc{Min 3-Sat} is yes if and only if there is a schedule $\pi$ such that ${\rm OWA}(\pi)\leq (m+L)/(m+1)$. Assume that
the answer to \textsc{Min 3-Sat} is yes. Let $\pi\in \Pi'$ be the schedule corresponding to the assignment which satisfies at most $L$ clauses (see the proof of Theorem~\ref{thm1}). It is easy
to verify that $f(\pi,S_k)=0$ if $C_k$ is not satisfied and $f(\pi,S_k)=1$ if $C_k$ is satisfied. Furthermore, $f(\pi,S_{m+1})=m$.
Hence ${\rm OWA}(\pi)\leq (m+L)/(m+1)$. Assume now that ${\rm OWA}(\pi)\leq (m+L)/(m+1)$. Then $\pi$ must belong to $\Pi'$ since otherwise $f(\pi, S_{m+1})\geq 2m$ and ${\rm OWA}(\pi)\geq 2m/(m+1)$, which contradicts the assumption that $L<m$. It must hold $f(\pi, S_{m+1})=m$ and $f(\pi, S_i)\in \{0,1\}$ for each $i\in [K]$.
Consequently $f(\pi,S_i)=1$ under at most $L$ scenarios, which means that the assignment corresponding to $\pi$ satisfies at most $L$ clauses and
the answer to \textsc{Min 3-Sat} is yes.
The proof of assertion~(ii) is very similar to the corresponding proof in Theorem~\ref{thm1}.
\end{proof}
\subsection{Polynomially and pseudopolynomially solvable cases}
In this section we identify some special cases of the
\textsc{Min-Owa}~$1|prec|\max w_j T_j$
problem which are polynomially or pseudopolynomially solvable.
\subsubsection{The maximum criterion}
It has been shown in~\cite{AC08} that \textsc{Min-Max}~$1|prec|T_{\max}$ is solvable in $O(Kn^2)$ time. In this section, we will show that more general version of the problem with arbitrary nonnegative job weights, \textsc{Min-Max}~$1|prec|\max w_j T_j$,
is solvable in $O(Kn^2)$ time as well.
In the construction of the algorithm, we will use some ideas from~\cite{K05, VD10}. Furthermore, the algorithm with some minor modifications will be a basis for solving other special cases of
\textsc{Min-Owa}~$1|prec|\max w_j T_j$.
In this section the OWA operator is the maximum, so
${\rm OWA}(\pi)=\max_{i\in [K]} f(\pi,S_i)$. By interchanging
the maximum operators and some easy transformations,
we can express the value of $\mathrm{OWA}(\pi)$ as follows:
\begin{equation}
\label{defTF}
\mathrm{OWA}(\pi) = \max_{j\in J}\max_{i\in [K]} [w_j(S_i)(C_j(\pi,S_i)-d_j(S_i))]^+.
\end{equation}
Fix a nonempty subset of jobs $D\subseteq J$ and define
\begin{equation}
\label{defFj}
F_j(D)=\max_{i\in [K]} [w_j(S_i) (\sum_{k\in D} p_k(S_i)-d_j(S_i))]^+.\\
\end{equation}
The following proposition immediately follows from the fact that all job processing times and weights are nonnegative:
\begin{prop}
\label{prop2}
If $D_2\subseteq D_1$, then for any $j\in J$ it holds $F_j(D_1)\geq F_j(D_2)$.
\end{prop}
Let $pred(\pi,j)$ be the set of jobs containing job $j$ and all the jobs that precede $j$ in $\pi$.
Since $C_j(\pi,S_i)=\sum_{k\in pred(\pi,j)} p_k(S_i)$,
the maximum cost of~$\pi$ over $\Gamma$ can be expressed as follows (see~(\ref{defTF}) and~(\ref{defFj})):
\begin{equation}
\label{defFtmax}
{\rm OWA}(\pi)=\max_{j\in J} F_j(pred(\pi,j)).
\end{equation}
Consider the algorithm shown in the form of Algorithm~\ref{alg1}.
\begin{algorithm}
\begin{algorithmic}[1]
\STATE $D := \{1,\dots,n\}$
\FORALL {$i\in[K]$} \label{alg1li2}
\STATE $p(S_i):= \sum_{k\in D} p_k(S_i)$ \label{alg1li3}
\ENDFOR \label{alg1li4}
\FOR {$r:= n$ $\bf downto$ $1$}
\STATE Find $j \in D$, which has no successor in $D$ and has the minimum value of $F_j(D)=\max_{i\in[K]}[w_j(S_i)(p(S_i)-d_j(S_i))]^+$\label{alg1l3}
\STATE $\pi(r):= j$
\STATE $D := D\setminus \{j\}$
\FORALL {$i\in[K]$} \label{alg1li9}
\STATE $p(S_i):= p(S_i)- p_j(S_i)$
\ENDFOR \label{alg1li11}
\ENDFOR
\STATE \begin{bf} return \end{bf} $\pi$
\end{algorithmic}
\caption{Algorithm for solving \textsc{Min-Max}~$1|prec|\max w_j T_j$.} \label{alg1}
\end{algorithm}
\begin{thm}
\label{thm3}
Algorithm~\ref{alg1} computes an optimal schedule for \textsc{Min-Max}~$1|prec|\max w_j T_j$ in $O(Kn^2)$ time.
\end{thm}
\begin{proof}
Let $\pi$ be the schedule returned by the algorithm. It is clear that $\pi$ is feasible. Let us renumber the jobs so that $\pi=(1,2,\dots,n)$. Let $\sigma$ be an optimal minmax schedule. Assume that $\sigma(j)=j$ for $j=k+1,\dots,n$, where $k$ is the smallest position among all the optimal minmax schedules. If $k=0$, then we are done, because $\pi=\sigma$ is optimal. Assume that $k>0$, and so $k\neq \sigma(k)=i$. Let us move the job $k$ just after $i$ in $\sigma$ and denote the resulting schedule as $\sigma'$
(see
Figure~\ref{figtmax}).
Schedule $\sigma'$ is feasible, because $\pi$ is feasible.
\begin{figure}
\caption{Illustration of the proof of Theorem~\ref{thm1}
\label{figtmax}
\end{figure}
We need only consider three cases:
\begin{enumerate}
\item If $j \in P \cup R$, then $pred(\sigma',j)=pred(\sigma,j)$ and $F_j(pred(\sigma',j))=F_j(pred(\sigma,j))$.
\item If $j \in Q\cup\{i\}$, then $pred(\sigma',j)\subseteq pred(\sigma,j)$ and, by Proposition~\ref{prop2}, $F_j(pred(\sigma',j))\leq F_j(pred(\sigma,j))$.
\item If $j=k$, then $F_j(D)\leq F_i(D)$ from the construction of Algorithm~1.
Since $pred(\sigma,i)=pred(\sigma',j)=D$, we have $F_j(pred(\sigma',j))\leq F_i(pred(\sigma,i))$.
\end{enumerate}
From the above three cases and equality~(\ref{defFtmax}), we conclude that
$${\rm OWA}(\sigma')=\max_{j\in J} F_j(pred(\sigma',j))\leq \max_{j\in J} F_j(pred(\sigma,j))={\rm OWA}(\sigma),$$
so $\sigma'$ is also optimal, which contradicts the minimality of $k$. Computing $F_j(D)$ for a given $j\in D$ in line~\ref{alg1l3}
requires $O(K)$ time
(note that $p(S_i)$, $i\in [K]$, store the values of $ \sum_{k\in D} p_k(S_i)$
that have been computed in lines~\ref{alg1li2}-\ref{alg1li4}
and they are updated in lines~\ref{alg1li9}-\ref{alg1li11}),
and thus line~\ref{alg1l3} can be executed in $O(Kn)$ time. Consequently,
the overall running time of the algorithm is $O(Kn^2)$.
\end{proof}
\subsubsection{The Hurwicz criterion}
In this section we explore the problem with the Hurwicz criterion. We will examine the case in
which $\alpha \in (0,1)$ as the boundary cases with $\alpha$ equal to 0 (the minimum criterion) or 1 (the maximum criterion) are solvable in $O(Kn^2)$ time.
\begin{thm}
\textsc{Min-Hurwicz}~$1|prec|\max w_j T_j$ is solvable in $O(K^2 n^4 )$ time.
\end{thm}
\begin{proof}
The Hurwicz criterion can be expressed as follows:
$$\textrm{OWA}(\pi)=\alpha\max_{i\in [K]}f(\pi,S_i) +(1-\alpha)\min_{i\in [K]} f(\pi,S_i).$$
Let us define
$$H_k(\pi)=\alpha\max_{i\in [K]}f(\pi,S_i) +(1-\alpha) f(\pi,S_k).$$
Hence
\[
\min_{\pi \in \Pi} \textrm{OWA}(\pi)=\min_{k\in [K]} \min_{\pi \in \Pi} H_k(\pi),
\]
and the problem of minimizing the Hurwicz criterion reduces to solving $K$ auxiliary problems consisting in minimizing $H_k(\pi)$ for a fixed $k\in [K]$.
Let us fix $k\in [K]$ and $t\geq 0$, and define $\Pi_k(t)=\{\pi\in \Pi\,:\,f(\pi,S_k)\leq t\}\subseteq \Pi$ as the set of feasible schedules whose cost under $S_k$ is at most $t$.
Define
$$\Psi_k(t)= \min_{\pi \in \Pi_k(t)}\max_{i\in [K]} f(\pi,S_i).$$
Hence
\begin{equation}
\label{defphi}
\min_{\pi \in \Pi} H_k(\pi)=\min_{t\in [\underline{t}, \overline{t}]} \alpha \Psi_k(t)+(1-\alpha) t,
\end{equation}
where $\underline{t}=\min_{\pi \in \Pi} f(\pi, S_k)$ (for $t<\underline{t}$ it holds $\Pi_k(t)=\emptyset$),
and $\overline{t}=\min_{\pi \in \Pi}\max_{i\in [K]} f(\pi,S_i)$, which
is due to the fact that $\max_{i\in [K]} f(\pi,S_i)\geq f(\pi,S_k)$.
Computing the value of~$ \Psi_k(t)$ for a given $t\in [\underline{t}, \overline{t}]$ can be done
by a slightly modified Algorithm~\ref{alg1}.
It is enough to replace line~\ref{alg1l3} of Algorithm~\ref{alg1} with
the following line:
\[
\ref{alg1l3}': \text{find $j \in D_k(t)$, which has no successor in $D$, and has a minimum value of $F_j(D)$},
\]
where $D_k(t)=\{j\in D: [w_j(S_k)(p(S_k)-d_j(S_k))]^+\leq t\}$.
The proof of the correctness of the modified algorithm is almost the same as the proof of Theorem~\ref{thm3}. It is sufficient to define a feasible schedule~$\pi$ as the one satisfying the precedence constraints and the additional constraint $f(\pi,S_k)\leq t$. Hence, if the algorithm returns a feasible schedule, then it must be optimal. The algorithm fails to compute a feasible schedule when $D_k(t)=\emptyset$ in line~\ref{alg1l3}'. In this case, at least one job in $D\neq\emptyset$ must be completed not earlier than $p(S_k)=\sum_{j\in D} p_j(S_k)$ and $f(\pi,S_k)>t$ for all schedules $\pi\in \Pi$, which means that $\Pi_k(t)=\emptyset$. Clearly, the modified algorithm has the same $O(Kn^2)$ running time.
Note that $\Psi_k$ is a nonincreasing step function on $[\underline{t},\infty)$, i.e.
a constant function on subintervals $[\underline{t}_1,\overline{t}_1)\cup[\underline{t}_2,\overline{t}_2)\cup
\cdots\cup [\underline{t}_l,\infty)$, $\overline{t}_{v-1}=\underline{t}_v$, $v=2,\ldots,l$, $\underline{t}_1=\underline{t}$.
Thus, $ \alpha \Psi_k(t)+(1-\alpha) t$, $\alpha\in(0,1)$, is a piecewise linear function on $[\underline{t},\infty)$,
a linear increasing function on each subinterval
$[\underline{t}_v,\overline{t}_v)$, $v\in [l]$, and attains minimum at one of the points $\underline{t}_1,\dots, \underline{t}_l$. The functions $\Psi_k(t)$ and $\alpha\Psi_k(t)+(1-\alpha)t$ for $k=3$ are depicted in the example shown in
Figure~\ref{figex}. We have $\underline{t}_1=18$, $\underline{t}_2=26$, $\underline{t}_3=60$ and the function $\alpha \Psi_3(t)+(1-\alpha)t$ is minimized for $t=26$. Since $\pi_2=(1,4,2,5,3)$ is an optimal solution to~$\Psi_3(26)$, we conclude that $\pi_2$ minimizes $H_3(\pi)$.
\begin{figure}
\caption{The functions $\Psi_3(t)$ (the dotted line) and
$0.5\Psi_3(t)+0.5t$, $t\in [18,60]$ (the solid line), for a sample problem (there are no precedence constraints between the jobs). The function $H_3(\pi)$ is minimized for $\pi_2=(1,4,2,5,3)$ and $H_3(\pi_2)=51.5$.}
\label{figtmax1}
\label{figex}
\end{figure}
Observe that the value of $t$ minimizing $\alpha \Psi_k(t) + (1-\alpha) t$ can be found in pseudopolynomial time by trying all integers in the interval $[\underline{t},\overline{t}]$.
We now show how to find the optimal value of $t$ in polynomial time.
We first compute $\underline{t}_1=\min_{\pi\in\Pi} f(\pi,S_k)$, and the value of $\Psi_k(\underline{t}_1)$ by the modified Algorithm~\ref{alg1}.
Let us denote by~$\pi_1$ the resulting optimal schedule, $\pi_1\in \Pi_k(\underline{t}_1)$. In the sample problem shown in Figure~\ref{figex}, $\underline{t}_1=18$, $\pi_1=(2,4,5,3,1)$, and $\Psi_3(\underline{t}_1)=91$. Our goal now is to compute the value of $\underline{t}_2$.
Choose the iteration of the
modified
Algorithm~\ref{alg1}, in which the position of job $j$ is fixed in $\pi_1$. The job $j$ satisfies the condition stated in line~\ref{alg1l3}'. We can now compute the smallest value of $t$, $t>\underline{t}_1$, for which job $j$ violates this condition and must be replaced by some other job in $D_k(t)$. In order to do this it suffices to try all values $t_i=w_i(S_k)[p(S_k)-d_i(S_k)]^+$ for
$i\in D\setminus\{j\}$ and fix $t^*_j$ as the smallest among them which violates
the condition in line~\ref{alg1l3}' (if the condition holds for all $t_i$, then $t^*_j=\infty$). Repeating this procedure for each job we get the set of values $t^*_1,\dots, t^*_n$ and $\underline{t}_2$ is the smallest value among them. Consider again the sample problem presented in Figure~\ref{figex}. When job~1 is placed at position 5 in $\pi_1$, it satisfies the condition in line~\ref{alg1l3}' for $t=18$. In fact, it holds $D_3(\underline{t}_1)=\{1\}$. Since $D=\{1,2,3,4,5\}$, we now try the values $t_2=91$, $t_3=26$, $t_4=126$, and $t_5=60$. The condition in line~\ref{alg1l3}' is violated for $t=t_3=26$ as $D_3(26)=\{1,3\}$ and $F_3(D)<F_1(D)$. Hence $t^*_1=26$. In the same way we compute the remaining values $t_2^*,\dots, t^*_5$. It turns out that $t_1^*=26$ is the smallest among them, thus $\underline{t}_2=26$.
The value of $\underline{t}_3$ can be found in the same way. We compute an optimal
schedule~$\pi_2$ corresponding to $\Psi_k(\underline{t}_2)$ and repeat the previous procedure.
Consider the sequence of schedules $\pi_l, \pi_{l-1},\dots,\pi_1$, where $\pi_v$ minimizes $\Psi_k(\underline{t}_v)$. Schedule $\pi_{v-1}$ can be obtained from $\pi_v$ by moving the position of at least one job in $\pi_v$, say $j$, whose current position becomes infeasible as $t$ decreases, to the left. Furthermore the position of $j$ cannot increase in all the subsequent schedules $\pi_{v-2}, \dots, \pi_1$, because the function $f(\pi,S_k)$ is nondecreasing (if $j$ cannot be placed at $i$th position, then it also cannot be placed at positions $i+1,\dots,n$). Hence, if $\pi_l$ is the last schedule, then the position of job $\pi_l(i)$ can be decreased at most $i-1$ times which implies $l=O(n^2)$. Hence problem~(\ref{defphi}) can be solved in $O(Kn^4)$ time and \textsc{Min-Hurwicz}~$1|prec|\max w_j T_j$ is solvable in $O(K^2 n^4 )$ time.
\end{proof}
\subsubsection{The $k$th largest cost criterion}
In this section we investigate the \textsc{Min-Quant($k$)}~$1|prec|\max w_j T_j$ problem. Thus our goal is to minimize the $k$th largest schedule cost. It is clear that this problem is polynomially solvable when $k=1$ or $k=K$. It is, however, strongly NP-hard and not at all approximable when $k$ is a function of $K$, in particular, when the median of the costs is minimized (see Theorem~\ref{thm1}). We now explore the case when $k$ is constant.
\begin{thm}
\label{thm4a}
\textsc{Min-Quant($k$)}~$1|prec|\max w_j T_j$ is solvable in $O\left(\binom{K}{k-1} (K-k+1) n^2 \right)$ time, which is polynomial when $k$ is constant.
\end{thm}
\begin{proof}
The algorithm works as follows. We enumerate all the subsets of scenarios of size $k-1$. For each such a subset, say $C$, we solve \textsc{Min-Max}~$1|prec|\max w_j T_j$ for the scenario set $\Gamma\setminus C$, using Algorithm~\ref{alg1}, obtaining a schedule $\pi_C$. Among the schedules computed we return~$\pi_C$ for which the maximum cost over $\Gamma\setminus C$ is minimal. It is straightforward to verify that this schedule must be optimal. The number of subsets which have to be enumerated is $\binom{K}{k-1}$. For each such a subset we solve
\textsc{Min-Max}~$1|prec|\max w_j T_j$ with scenarios set~$\Gamma \setminus C$, which requires $O((K-k+1) n^2)$ time and the theorem follows.
\end{proof}
The algorithm suggested in the proof of Theorem~\ref{thm4a} is efficient when $k$ is close to~1 or close to $K$. When $k$ is a function of $K$, then this running times becomes exponential and may be prohibitive in practice. In Section~\ref{sappal1}, we will use this algorithm to construct an approximation algorithm for the general \textsc{Min-Owa}~$1|prec|\max w_j T_j$ problem.
\subsubsection{The OWA criterion - the bounded case}
\label{sec1_2}
In Section~\ref{sec1_1}, we have shown that for the unbounded case
\textsc{Min-Owa}~$1|prec|\max w_j T_j$ is strongly NP-hard and not at all approximable unless P=NP. In this section we investigate the case when $K$ is constant.
Without loss of generality we can assume
that all the parameters are nonnegative integers.
Let $f_{\max}$ be an upper bound on the maximum weighted tardiness of any job under any scenario.
By Proposition~\ref{prop2} and equality~(\ref{defFtmax}) we can fix $f_{\max}=\max_{j\in J} F_j(J)$.
\begin{thm}
\textsc{Min-Owa}~$1|prec|\max w_j T_j$ is solvable in $O(f_{\max}^K Kn^2)$ time, which is pseudopolynomial if $K$ is constant.
\label{thm4}
\end{thm}
\begin{proof}
Let $\pmb{t}=(t_1,\dots,t_K)$ be a vector of nonnegative integers.
Let $\Pi(\pmb{t})\subseteq \Pi$ be a subset of the set of feasible schedules such that $\pi \in \Pi(\pmb{t})$ if $f(\pi,S_i)\leq t_i$ for all $i\in [K]$, i.e. the maximum weighted tardiness in $\pi$ under $S_i$ does not exceed $t_i$. Consider the following auxiliary problem. Given a vector $\pmb{t}$, check if $\Pi(\pmb{t})$ is not empty and if so, return any schedule $\pi_{\pmb{t}}\in \Pi(\pmb{t})$. We now show that this auxiliary problem can be solved in polynomial time. Given $\pmb{t}$, we first form scenario set $\Gamma'$ by specifying the following parameters for each $S_i \in \Gamma$ and $j\in J$:
\begin{itemize}
\item $p_j(S_i')=p_j(S_i)$,
\item $ \displaystyle d_j(S_i')=\max\{C\geq 0\,:\,w_j(S_i)(C-d_j(S_i))\leq t_i\}=t_i/w_j(S_i)+d_j(S_i)$,
\item $w_j(S_i')=1$.
\end{itemize}
The scenario set $\Gamma'$ can be determined in $O(Kn)$ time.
We solve \textsc{Min-Max}~$1|prec|\max w_j T_j$ with the scenario set $\Gamma'$ by
Algorithm~\ref{alg1} obtaining schedule $\pi$. If the maximum cost of $\pi$ over $\Gamma'$ is 0, then $\pi_{\pmb{t}}=\pi$; otherwise $\Pi(\pmb{t})$ is empty. Since
\textsc{Min-Max}~$1|prec|\max w_j T_j$
is solvable in $O(Kn^2)$ time, the auxiliary problem is solvable in $O(Kn^2)$ time as well.
We now show that there exists a vector $\pmb{t}^*=(t_1^*,\dots, t^*_K)$, where $t_i^*\in \{0,\dots,f_{\max}\}$, $i\in [K]$, such that each $\pi_{\pmb{t}^*}\in \Pi(\pmb{t}^*)$ minimizes
$\mathrm{OWA(\pi)}$. Let $\pi^*$ be an optimal schedule and let $\pmb{t^*}=(t^*_1,\dots,t^*_K)$ be a vector such that $t^*_i=f(\pi^*,S_i)$ for $i\in [K]$. Clearly, $t^*_i\in \{0,\dots,f_{\max}\}$ for each $i\in [K]$ and $\pi^* \in \Pi(\pmb{t^*})$. By the definition of $\pmb{t^*}$, it holds $\mathrm{owa}_{\pmb{v}}(\pmb{t^*})=\mathrm{OWA}(\pi^*)$. For any $\pi\in \Pi(\pmb{t^*})$ it holds $f(\pi,S_i)\leq t^*_i= f(\pi^*,S_i)$, $i\in [K]$. From the monotonicity of OWA we conclude that each $\pi \in \Pi(\pmb{t^*})$ must be optimal.
The algorithm enumerates all possible vectors $\pmb{t}$ and computes $\pi_{\pmb{t}}\in \Pi(\pmb{t})$ if $\Pi(\pmb{t})$ is nonempty. A schedule~$\pi_{\pmb{t}}$ with the minimum value
of $\mathrm{owa}_{\pmb{v}}(\pmb{t})$ is returned. The number of vectors $\pmb{t}$ which must be enumerated is at most $f_{\max}^K$. Hence the problem is solvable in pseudopolynomial time provided that $K$ is constant and the running time of the algorithm is $O(f_{\max}^K Kn^2)$.
\end{proof}
\subsection{Approximation algorithm}
\label{sappal1}
When $K$ is a part of the input, i.e. in the unbounded case, then the exact algorithm proposed in Section~\ref{sec1_2} may be inefficient. Notice, that due to Theorem~\ref{thm1}, no efficient approximation algorithm can exist for
\textsc{Min-Owa}~$1|prec|\max w_j T_j$ in this case
unless P=NP. We now prove the following result, which can be used to obtain an approximate solution in some special cases of the weight distributions in the OWA operator.
\begin{thm}
\label{thm7}
Suppose that $v_1=\dots =v_{k-1}=0$ and $v_k>0$, $k\in [K]$.
Let $\hat{\pi}$ be an optimal solution to the \textsc{Min-Quant}$(k)$~$1|prec|\max w_j T_j$ problem. Then for each $\pi\in \Pi$, it holds ${\rm OWA}(\hat{\pi})\leq (1/v_k){\rm OWA}(\pi)$ and the bound is tight.
\end{thm}
\begin{proof}
Let $\sigma$ be a sequence of $[K]$ such that $f(\hat{\pi},S_{\sigma(1)})\geq \dots \geq f(\hat{\pi},S_{\sigma(K)})$ and $\rho$ be a sequence of $[K]$ such that $f(\pi,S_{\rho(1)})\geq \dots \geq f(\pi,S_{\rho(K)})$. It holds:
$${\rm OWA}(\hat{\pi})=\sum_{i=k}^K v_i f(\hat{\pi},S_{\sigma(i)})\leq f(\hat{\pi},S_{\sigma(k)}).$$
From the definition of $\hat{\pi}$ and the assumption that $v_k>0$ we get
$$f(\hat{\pi},S_{\sigma(k)})\leq f(\pi,S_{\rho(k)})\leq \frac{1}{v_k}\sum_{i \in [K]} v_i f(\pi,S_{\rho(i)})=\frac{1}{v_k}{\rm OWA}(\pi).$$
Hence ${\rm OWA}(\hat{\pi})\leq (1/v_k){\rm OWA}(\pi)$.
To see that the bound is tight consider
an instance of
the problem with $K$ scenarios and $2K$ jobs. The job processing times and weights are equal to~1 under all scenarios. The job due dates are shown in Table~\ref{tab2a}. We fix $v_i=(1/K)$ for each $i\in [K]$.
\begin{table}[ht]
\centering
\caption{An example of due date scenario set for which the approximation algorithm achieves a ratio of $1/v_k$.} \label{tab2a}
\begin{tabular}{l|ccccccccccc}
& $S_1$ & $S_2$ & $S_3$ & $\dots$ & $S_K$ \\ \hline
$J_1$ & 1 & 1 & 1 & $\dots$ & 1 \\
$J_2$ & 2 & 2 & 2 & $\dots$ & 1 \\
$J_3$ & 3 & 3 & 3 & $\dots$ & 3\\
$J_4$ & 4 & 4 & 4 & $\dots$ & 3\\
$\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \\
$J_{2K-1}$ & $2K-1$ & $2K-1$ & $2K-1$ & $\dots$ & $2K-1$\\
$J_{2K}$ & $2K$ & $2K$ & $2K$ & $\dots$ & $2K-1$
\end{tabular}
\end{table}
Since $v_1>0$, we solve \textsc{Min-Max}~$1|prec|\max w_j T_j$. As a result we can obtain the schedule $\pi=(J_2,J_1,J_4,J_3,\dots,J_{2K},J_{2K-1})$ whose average cost over all scenarios is $1$. But the average cost of the optimal schedule $\pi^*=(J_1,J_2,J_3,J_4,\dots,J_{2K-1},J_{2K})$ is $1/K$.
\end{proof}
We now show several consequences of Theorem~\ref{thm7}. Observe first that if $v_1>0$, then we can use Algorithm~\ref{alg1} to obtain the approximate schedule in polynomial time.
\begin{cor}
If $v_1>0$, then \textsc{Min-Owa}~$1|prec|\max w_j T_j$ is approximable within $1/v_1$.
\end{cor}
Consider now the case of nondecreasing weights, i.e. $v_1\geq v_2\geq \dots \geq v_K$. Recall that nondecreasing weights are used when the idea of robust optimization is adopted. Namely, larger weights are assigned to larger schedule costs. Since in this case the inequality $v_1\geq 1/K$ must hold,
we get the following result:
\begin{cor}
If the weights are nonincreasing, then \textsc{Min-Owa}~$1|prec|\max w_j T_j$ is approximable within $1/v_1\leq K$.
\end{cor}
Finally, the following corollary is an immediate consequence of the previous corollary:
\begin{cor}
\textsc{Min-Average}~$1|prec|\max w_j T_j$ is approximable within~$K$.
\end{cor}
\section{The weighted sum of completion times cost function}
\label{sec3}
Let the cost of schedule~$\pi$ under scenario~$S_i$ be the weighted sum of completion times in $S_i$, i.e.
$f(\pi,S_i)=\sum_{j\in J} w_j(S_i) C_j(\pi,S_i)$. Using the Graham's notation, the deterministic version of the problem is denoted
by $1|prec|\sum w_j C_j$.
We will also examine the special cases of this problem with no precedence constraints between the jobs, i.e. $1||\sum w_j C_j$ and all job weights equal to~1, i.e. $1||\sum C_j$. It is well known that $1|prec|\sum C_j$ is strongly NP-hard for arbitrary precedence constraints~\cite{LK78}. It is, however, polynomially solvable for some special cases of the precedence constraints such as in-tree, out-tree or sp-graph (see, e.g.~\cite{B07}). If there are no precedence constraints between the jobs, then an optimal schedule can be obtained by ordering the jobs with respect to nondecreasing ratios $p_j/w_j$, which reduces to the SPT rule when all job weights are equal to~1.
In this section, we will show that if the number of scenarios is a part of the input, then \textsc{Min-Owa}~$1||\sum w_j C_j$ is strongly NP-hard and not at all approximable. This is the case when the weights in the OWA criterion are nondecreasing, or OWA is the median. We then propose several approximation algorithms which will be valid for nonincreasing weights and the Hurwicz criterion.
\subsection{Hardness of the problem}
The \textsc{Min-Max}~$1||\sum w_j C_j$ and \textsc{Min-Max}~$1||\sum C_j$ problems have
been recently investigated in literature,
and the following results have been established:
\begin{thm}[\cite{YY02}]
\label{thmc1}
\textsc{Min-Max}~$~1||\sum C_j$ is NP-hard even for $K=2$.
\end{thm}
\begin{thm}[\cite{MNO13}]
\label{thmc2}
If the number of scenarios is unbounded, then
\begin{enumerate}
\item[(i)] \textsc{Min-max}~$1||\sum w_j C_j$
is strongly NP-hard and not approximable within $O(\log^{1-\varepsilon}n)$ for any $\varepsilon>0$ unless the problems in NP have quasi-polynomial time algorithms.
\item[(ii)] \textsc{Min-max}~$1||\sum C_j$ and
\textsc{Min-max}~$1|p_j=1|\sum w_j C_j$
are strongly NP-hard and not approximable within $6/5-\varepsilon$ for any $\varepsilon>0$ unless P=NP.
\end{enumerate}
\end{thm}
We now show that the general case is much more complex.
\begin{thm}
\label{thmcc}
If the number of scenarios is unbounded, then \textsc{Min-Owa}~$1||\sum w_j C_j$ is strongly NP-hard and not at all approximable
unless P=NP.
\end{thm}
\begin{proof}
We show a polynomial time reduction from the \textsc{Min 2-Sat} problem which is known to be strongly NP-hard
(see the proof of Theorem~\ref{thm1}).
Given an instance of \textsc{Min 2-Sat}, we construct
the corresponding instance of \textsc{Min-Owa}~$1||\sum w_j C_j$ in the following way.
We associate
two jobs $J_{x_i}$ and $J_{\overline{x}_i}$ with each variable~$x_i$,
$i\in [n]$.
We then set $K=m$ and form scenario set~$\Gamma$
in the following way.
Scenario $S_k$ corresponds to clause $C_k=(l_1 \vee l_2)$. For $q=1,2$, if $l_q=x_i$, then the processing time of $J_{x_i}$ is $0$, the weight of $J_{x_i}$ is~1,
the processing time of $J_{\overline{x}_i}$ is $1$, and the weight of $J_{\overline{x}_i}$ is~0;
if $l_q=\overline{x}_i$, then the processing time of $J_{x_i}$ is $1$, the weight of $J_{x_i}$ is 0,
the processing time of $J_{\overline{x}_i}$ is $0$ and the weight of $J_{\overline{x}_i}$ is~1.
If neither $x_i$ nor $\overline{x}_i$ appears in $C_k$, then both processing times and weights of $J_{x_i}$ and
$J_{\overline{x}_i}$ are set to 0.
We complete the reduction by fixing $v_1=v_2=\dots =v_L=0$ and $v_{L+1}=\dots v_K=1/(m-L)$. A sample reduction is presented in Table~\ref{tab2}.
\begin{table}[ht]
\centering
\caption{Processing times and weights $(p_j(S_i), w_j(S_i))$ corresponding to
the formula $(x_1\vee \overline{x}_2)\wedge (\overline{x}_2 \vee \overline{x}_3) \wedge
(\overline{x}_1 \vee \overline{x}_4) \wedge (x_1 \vee x_3) \wedge (x_1 \vee \overline{x}_4)$.} \label{tab2}
\begin{tabular}{l|lllll}
& $S_1$ & $S_2$ & $S_3$ & $S_4$ & $S_5$ \\ \hline
$J_{x_1}$ & $(0,1)$ & $(0,0)$ & $(1,0)$ & $(0,1)$ & $(0,1)$\\
$J_{\overline{x}_1}$ & $(1,0)$ & $(0,0)$ & $(0,1)$ & $(1,0)$ & $(1,0)$ \\ \hline
$J_{x_2}$ & $(1,0)$ & $(1,0)$ & $(0,0)$ & $(0,0)$ & $(0,0)$\\
$J_{\overline{x}_2}$ & $(0,1)$ & $(0,1)$ & (0,0) & $(0,0)$ & $(0,0)$ \\ \hline
$J_{x_3}$ & (0,0) & $(1,0)$ & (0,0) & $(0,1)$ & $(0,0)$\\
$J_{\overline{x}_3}$ & (0,0) & $(0,1)$ & $(0,0)$ & $(1,0)$ & $(0,0)$ \\ \hline
$J_{x_4}$ & (0,0) & $(0,0)$ & $(1,0)$ & $(0,0)$ & $(1,0)$\\
$J_{\overline{x}_4}$ & (0,0) & $(0,0)$ & $(0,1)$ & $(0,0)$ & $(0,1)$ \\ \hline
\end{tabular}
\end{table}
We now show that there is an assignment to the variables which satisfies at most~$L$ clauses
if and only if there is a schedule~$\pi$ such that $\mathrm{OWA}(\pi)=0$.
Assume that there is an assignment $x_i$, $i\in [n]$, that satisfies at most~$L$ clauses.
According to this assignment we build a schedule~$\pi$ as follows.
We first process $n$ jobs $J_{z_i}$, $z_i\in\{x_i, \overline{x}_i\}$,
which correspond to false literals~$z_i$, $i\in[n]$, in any order and
then the rest $n$ jobs that
correspond to true literals~$z_i$, $i\in[n]$,
in any order.
Choose a clause $C_k=(l_1 \vee l_2)$ which is not satisfied. It is easy to check that the cost of
the schedule~$\pi$ under scenario $S_k$ is~0.
Consequently, there are at most~$L$ scenarios under which the cost of~$\pi$ is
positive and, according to the definition of the weights in the OWA operator,
we get $\mathrm{OWA}(\pi)=0$.
Suppose now that there is a schedule~$\pi$ such that
$\mathrm{OWA}(\pi)=0$.
We construct an assignment to the variables by setting $x_i=0$ if $J_{x_i}$ appears before
$J_{\overline{x}_i}$ in $\pi$ and $x_i=1$ otherwise. Since $\mathrm{OWA}(\pi)=0$,
the cost of~$\pi$ must be~0 under at least $m-L$ scenarios. If the cost of $\pi$ is~0 under
scenario~$S_k$ corresponding to the clause $C_k$, then the assignment does not satisfy $C_k$. Hence,
there is at least $m-L$ clauses that are not satisfied and, consequently, at most $L$ satisfiable clauses.
\end{proof}
\begin{cor}
\textsc{Min-Median}~$1||\sum w_j C_j$ is strongly NP-hard and not at all approximable unless P=NP.
\end{cor}
\begin{proof}
The proof is similar to the proof of Theorem~\ref{thm1} and consists in adding some additional scenarios to
an instance of problem constructed in Theorem~\ref{thmcc}.
\end{proof}
\subsection{Approximation algorithms}
In this section we show several approximation algorithms for \textsc{Min-Owa}~$1|prec|\sum w_j C_j$.
We will explore the case in which the weights in the OWA criterion are
nonincreasing, i.e. $v_1\geq v_2 \geq \dots \geq v_K$. We will then apply the obtained results to the Hurwicz criterion. Observe, that the case with nondecreasing weights, i.e. $v_1\leq v_2\leq \dots\leq v_K$, is not at all approximable (see the proof of Theorem~\ref{thmcc}).
We first recall a well known property
(see, e.g.~\cite{MNO13}) which states
that each problem with uncertain processing times and deterministic weights can be transformed into an equivalent problem with uncertain weights and deterministic processing times (and vice versa). This transformation is cost preserving and works as follows. Under each scenario $S_i$, $i\in [K]$, we invert the role of processing times and weights obtaining scenario $S'_i$. The new scenario set $\Gamma'$ contains scenario $S'_i$ for each $i\in [K]$. We also invert the precedence constraints, i.e. if $i\rightarrow j$ in the original problem, then $j\rightarrow i$ in the new one. It can be easily shown that the cost of schedule $\pi$ under $S$ is equal to the cost of the inverted schedule $\pi'=(\pi(n),\dots,\pi(1))$ under $S'$. Consequently $\mathrm{OWA}(\pi)$ under $\Gamma$ equals $\mathrm{OWA}(\pi')$ under $\Gamma'$.
Notice that if the processing times are deterministic in the original problem, then the weights become deterministic in the new one (and vice versa).
Let $w_{\max}, w_{\min}, p_{\max}, p_{\min}$ be the largest (smallest) weight (processing time) in the input instance. We first consider the case then both processing times an weights can be uncertain. We prove the following result:
\begin{thm}
If $v_1\geq v_2\geq \dots \geq v_K$ and the deterministic $1|prec|\sum w_j C_j$ problem is polynomially solvable, then \textsc{Min-Owa}~$1|prec|\sum w_j C_j$ is approximable within $K\cdot\min\{\frac{w_{\max}}{w_{\min}},\frac{p_{\max}}{p_{\min}}\}$.
\end{thm}
\begin{proof}
Let
$\hat{p}_j=\sum_{i \in [K]} p_j(S_i)$, $\hat{w}_j={\rm owa}_{\pmb{v}}(w_j(S_1),\dots,w_j(S_K))$, $\hat{C}_j(\pi)=\sum_{i\in [K]} C_j(\pi,S_i)$, and $\hat{f}(\pi)=\sum_{j\in J} \hat{w}_j\hat{C}_j(\pi)$. Let $\hat{\pi}\in \Pi$ minimize $\hat{f}(\pi)$. Of course, $\hat{\pi}$ can be computed in polynomial time provided that the deterministic counterpart of the problem is polynomially solvable.
Let $\sigma$ be a sequence of $[K]$ such that $f(\hat{\pi},S_{\sigma(1)})\geq \dots\geq f(\hat{\pi},S_{\sigma(K)})$.
It holds
\begin{equation}
\label{cappr0}
\begin{array}{ll}
{\rm OWA}(\hat{\pi})= & \displaystyle \sum_{i\in [K]} v_i\sum_{j\in J} w_j(S_{\sigma(i)})C_j(\hat{\pi},S_{\sigma(i)})\leq\sum_{j\in J} \sum_{i\in [K]} v_iw_j(S_{\sigma(i)})\hat{C}_j(\hat{\pi})= \\
& \displaystyle=\sum_{j\in J}\hat{C}_j(\hat{\pi}) \sum_{i\in [K]} v_iw_j(S_{\sigma(i)})\leq \sum_{j \in J} \hat{w}_j \hat{C}_j(\hat{\pi})=\hat{f}(\hat{\pi}),
\end{array}
\end{equation}
where the inequality $\hat{w}_j\geq \sum_{i\in [K]} v_i w_j(S_{\sigma(i)})$ follows from the assumption that $v_1\geq v_2\geq \dots \geq v_K$. We also get for any $\pi \in \Pi$
\begin{equation}
\label{cappr1}
\begin{array}{ll}
\displaystyle \hat{f}(\hat{\pi})\leq & \displaystyle\hat{f}(\pi)=\sum_{j \in J} \hat{w}_j \hat{C}_j(\pi)=\sum_{j \in J} \hat{w}_j \sum_{i\in [K]} C_j(\pi,S_i)\leq \frac{w_{\max}}{w_{\min}}\sum_{j\in J}\sum_{i\in[K]}w_j(S_i)C_j(\pi,S_i)= \\
& \displaystyle=\frac{w_{\max}}{w_{\min}}\sum_{i\in[K]}\sum_{j\in J}w_j(S_i)C_j(\pi,S_i),
\end{array}
\end{equation}
where the second inequality follows from the fact that $\hat{w_j}\leq w_{\max}\leq (w_{\max}/w_{\min})w_j(S_i)$ for each $j\in J$, $i\in [K]$.
Again, from the assumption that $v_1\geq v_2\geq \dots \geq v_K$ we have
\begin{equation}
\label{cappr2}
(1/K)\sum_{i\in[K]}\sum_{j\in J}w_j(S_i)C_j(\pi,S_i)\leq {\rm OWA}(\pi).
\end{equation}
From~(\ref{cappr0}), (\ref{cappr1}) and (\ref{cappr2}) we get
${\rm OWA}(\hat{\pi})\leq K\cdot \frac{w_{\max}}{w_{\min}}{\rm OWA}(\pi).$
Since the role of job processing times and weights can be inverted we also get ${\rm OWA}(\hat{\pi})\leq K\cdot \frac{p_{\max}}{p_{\min}}{\rm OWA}(\pi)$ and the theorem follows.
\end{proof}
In~\cite{MNO13} a 2-approximation algorithm for \textsc{Min-Max}~$1|prec|\sum w_j C_j$ has been recently proposed, provided that either job processing times or job weights are deterministic
(they do not vary among scenarios). In this section we will show that this algorithm can be extended to \textsc{Min-Owa}~$1|prec|\sum w_jC_j$ under the additional assumption that the weights in the OWA operator are nonincreasing, i.e. $v_1\geq v_2\geq \dots \geq v_K$.
The idea of the approximation algorithm is to design a mixed integer programming formulation for the problem, solve its linear relaxation and construct an approximate schedule based on the optimal solution to this relaxation.
Assume now that job processing times are deterministic and equal to $p_j$ under each scenario $S_i$, $i\in [K]$. Let $\delta_{ij}\in \{0,1\}$, $i,j\in [n]$, be binary variables such that $\delta_{ij}=1$ if job $i$ is processed before job $j$ in
a schedule constructed. The vectors of all feasible job completion times $(C_1,\dots, C_n)$ can be described by the following system of constraints~\cite{POT80}:
\begin{equation}
\label{cCC2}
\begin {array}{llll}
VC: & C_j=p_j+\sum_{i\in J\setminus\{j\}} \delta_{ij} p_i & j\in J\\
&\delta_{ij}+\delta_{ji}=1 & i,j\in J, i\neq j \\
&\delta_{ij}+\delta_{jk}+\delta_{ki} \geq 1 & i,j,k \in J\\
&\delta_{ij}=1 & i\rightarrow j\\
&\delta_{ij}\in \{0,1\}& i,j \in J
\end{array}
\end{equation}
Let us denote by $VC'$ the relaxation of $VC$, in which the constraints $\delta_{ij}\in \{0,1\}$ are replaced with $0\leq \delta_{ij}\leq 1$. It has been proved in~\cite{SH96a, SH96b} (see also~\cite{HA97}) that each vector $(C_1,\dots, C_n)$ that satisfies $VC'$ also satisfies the following inequalities:
\begin{equation}
\label{Schin}
\sum_{j\in I} p_jC_j\geq \frac{1}{2}\left((\sum_{j\in I} p_j)^2+\sum_{j\in I} p_j^2\right) \text{ for all } I \subseteq J
\end{equation}
In order to build a MIP formulation for the problem, we will use the idea of a deviation model introduced in~\cite{OS03}.
Let $\sigma$ be a permutation of $[K]$ such that $f(\pi,S_{\sigma(1)})\geq \dots \geq f(\pi,S_{\sigma(K)})$ and
let $\theta_k(\pi)=\sum_{i=1}^k f(\pi,S_{\sigma(i)})$ be the cumulative cost of schedule $\pi$. Define $v'_i=v_i-v_{i+1}$ for $i=1,\dots,K-1$ and $v'_K=v_K$. An easy verification shows that
\begin{equation}
\label{mipc00}
{\rm OWA}(\pi)=\sum_{k=1}^K v'_k \theta_k(\pi).
\end{equation}
\begin{lem}
Given $\pi$, the value of $\theta_k(\pi)$ can be obtained by solving the following linear programming problem:
\begin{equation}
\label{mipc0}
\begin{array}{lllll}
\min & \sum_{i=1}^K u_{i} - (K-k) r \\
& r\leq u_{i} & i\in [K] \\
& u_{i} \geq f(\pi,S_i) & i\in[K] \\
& u_i \geq 0 & i\in [K]\\
& r\geq 0
\end{array}
\end{equation}
\end{lem}
\begin{proof}
Consider the following linear programming problem:
\begin{equation}
\label{mipc1}
\begin{array}{lllll}
\max & \sum_{i=1}^K \beta_i f(\pi,S_i) \\
& \alpha_i + \beta_i \leq 1 & i\in [K] \\
& \sum_{i=1}^K \alpha_i \geq (K-k)\\
& \alpha_i, \beta_i \geq 0 & i\in [K]
\end{array}
\end{equation}
It is easy to see that an optimal solution to~(\ref{mipc1}) can be obtained by setting $\beta_{\sigma(i)}=1$ and $\alpha_{\sigma(i)}=0$ for $i=1\dots k$, $\beta_{\sigma(i)}=0$ and $\alpha_{\sigma(i)}=1$ for $i=k+1,\dots K$,
where $\sigma$ is such that $f(\pi,S_{\sigma(1)})\geq \dots \geq f(\pi,S_{\sigma(K)})$.
This gives us the maximum objective function value equal to $\theta_k(\pi)$. To complete the proof it is enough to observe that~(\ref{mipc0}) is the dual linear program to~(\ref{mipc1}).
\end{proof}
If $v_1\geq v_2\geq\dots\geq v_K$, then $v'_i\geq 0$ and~(\ref{cCC2}), (\ref{mipc00}), (\ref{mipc0}) lead to the following mixed integer programming formulation for the problem:
\begin{equation}
\label{mipc2}
\begin{array}{lllll}
\min & \sum_{k=1}^K v'_k (\sum_{i=1}^K u_{ik} - (K-k) r_k) \\
& \text{Constraints } VC \\
& r_k\leq u_{ik} & i,k\in[K] \\
& u_{ik} \geq \sum_{j\in J} C_j w_j(S_i) & i,k\in[K] \\
& u_{ik} \geq 0 & i,k\in[K]\\
& r_k\geq 0 & k\in[K] \\
\end{array}
\end{equation}
In order to construct the approximation algorithm we will also need the following easy observation:
\begin{obs}
\label{lemowa1}
Let $(f_1,\dots, f_K)$ and $(g_1,\dots,g_K)$ be two
nonnegative real vectors
such that $f_i\leq \gamma g_i$ for some constant $\gamma>0$. Then,
$\mathrm{owa}_{\pmb{v}}(f_1,\dots, f_k)\leq \gamma
\mathrm{owa}_{\pmb{v}}(g_1,\dots, g_K)$ for each $\pmb{v}$.
\end{obs}
\begin{proof}
From the monotonicity of the OWA operator and the assumption $\gamma>0$, it follows that ${\rm owa}_{\pmb{v}}(f_1,\dots,f_K)\leq {\rm owa}_{\pmb{v}}(\gamma g_1,\dots,\gamma g_K)=\gamma {\rm owa}_{\pmb{v}}(g_1,\dots,g_K)$.
\end{proof}
The approximation algorithm works as follows. We first solve the linear relaxation of~(\ref{mipc2}) in which $VC$ is replaced with $VC'$ . Clearly, this relaxation can be solved in polynomial time. Let $(C_1^*, \dots, C_n^*)$ be the relaxed optimal job completion times and $z^*$ be the optimal value of the relaxation. It holds $z^*={\rm owa}_{\pmb{v}}(\sum_{j\in J} C^*_j w_j(S_1), \dots, \sum_{j\in J} C^*_j w_j(S_K))$.
We now relabel the jobs so that $C^*_1\leq C^*_2\leq \dots\ C_n^*$ and form schedule $\pi=(1,2,\dots,n)$.
Since the vector $(C_j^*)$ satisfies $VC'$ it must also satisfy~(\ref{Schin}). Hence, by setting $I=\{1,\dots,j\}$, we get
$$\sum_{i=1}^j p_iC^*_i\geq \frac{1}{2}\left((\sum_{i=1}^j p_i)^2+\sum_{i=1}^j p_i^2\right)\geq \frac{1}{2} (\sum_{i=1}^j p_i)^2.
$$
Since $C^*_j\geq C_i^*$ for each $i\in \{1\dots j\}$, we get
$C^*_j\sum_{i=1}^j p_i\geq \sum_{i=1}^j p_iC^*_i \geq \frac{1}{2}(\sum_{i=1}^j p_i)^2$
and, finally $C_j=\sum_{i=1}^j p_i \leq 2 C^*_j$ for each $j\in J$ -- this reasoning is
the same as in~\cite{SH96b}.
For each scenario $S_i$, $i\in [K]$, it holds
$f(\pi,S_i)=\sum_{j\in J} C_j w_j(S_i)\leq 2 \sum_{j\in J} C^*_j w_j(S_i)$, and Observation~\ref{lemowa1} implies
$${\rm OWA}(\pi)={\rm owa}_{\pmb{v}}(\sum_{j\in J} C_j w_j(S_1), \dots, \sum_{j\in J} C_j w_j(S_K))\leq 2z^*.$$
Since $z^*$ is a lower bound on the value of an optimal solution, $\pi$ is a 2-approximate schedule. Let us summarize the obtained result.
\begin{thm}
\label{thmcappr1}
If $v_1\geq v_2\geq \dots\geq v_K$, and job processing times (or weights) are deterministic, then \textsc{Min-Owa}~$1|prec|\sum w_j C_j$ is approximable within~2.
\end{thm}
We now use Theorem~\ref{thmcappr1} to prove the following result:
\begin{thm}
\textsc{Min-Hurwicz}~$1|prec|\sum w_j C_j$ is approximable within~2, if job processing times (or weights) are deterministic.
\end{thm}
\begin{proof}
Assume that job processing times are deterministic (the reasoning for deterministic processing times is the same).
The problem with the Hurwicz criterion can be rewritten as follows:
$$ \min_{\pi \in \Pi} {\rm OWA}(\pi)= \min_{\pi \in \Pi}\min_{k\in [K]} H_k(\pi),$$
where
$$H_k(\pi)=\alpha \max_{i \in [K]} \sum_{j\in J} w_j(S_i)C_j(\pi) + (1-\alpha) \sum_{j\in J} w_j(S_k)C_j(\pi)=$$
$$=\max_{i \in [K]}(\alpha \sum_{j\in J} w_j(S_i)C_j(\pi) + (1-\alpha) \sum_{j\in J} w_j(S_k)C_j(\pi))=\max_{i\in [K]} \sum_{j\in J} \hat{w}_j(S_i)C_j(\pi),$$
where $\hat{w}_j(S_i)=\alpha w_j(S_i)+(1-\alpha)w_j(S_k)$. Hence the problem reduces to solving $K$ auxiliary
\textsc{Min-Max}~$1|prec|\sum w_j C_j$ problems. Since
\textsc{Min-Max}~$1|prec|\sum w_j C_j$ is approximable within~2 (see~\cite{MNO13}, or Theorem~\ref{thmcappr1}),
the theorem follows.
\end{proof}
\section{Conclusion and open problems}
In this paper we have proposed a new approach to scheduling problems with uncertain parameters. The key idea is to use the OWA operator to aggregate all possible values of the schedule cost. The weights in OWA allows decision makers to take their attitude towards a risk into account. In consequence, the main advantage of the proposed approach is to weaken the very conservative minmax criterion, traditionally used in robust optimization. Apart from proposing a general framework, we have discussed the computational properties of two basic single machine scheduling problems. We have shown that they have various computational and approximation properties, which make their analysis very challenging. However, there is still a number of open problems regarding the considered cases. For the problem with the maximum weighted tardiness criterion we do not know if the problem is weakly NP-hard when the number of scenarios is constant (the bounded case). It may be also the case that the pseudopolynomial algorithm designed for a fixed $K$ can be converted into a polynomial one by using a similar idea as for the Hurwicz criterion. We also do not know if the problem with the average criterion admits an approximation algorithm with a constant worst-case ratio (we only know that it is approximable within $K$ and not approximable within a ratio less than 7/6). For the problem with the weighted sum of completion times criterion the complexity of $\textsc{Min-Average}~1||\sum w_jC_j$ with uncertain processing times and weights is open.
The framework proposed in this paper can also be applied to other scheduling problems, for example to the single machine scheduling problem with the sum of late jobs criterion (the minmax version of this problem was discussed
in~\cite{AAK11,AC08}).
\end{document} |
\betaegin{eqnarray}gin{document}
\title{Cold Trapped Ions as Quantum Information Processors}
\alphauthor{Marek \v Sa\v sura and Vladim{\'\i}r Bu\v zek}
\alphaddress{
Research Center for Quantum Information,
Slovak Academy of Sciences,
D\'{u}bravsk\'{a} cesta 9, Bratislava 842 28, Slovakia
}
\date{September 13, 2001}
\betaegin{eqnarray}gin{abstract}
In this tutorial we review physical implementation of quantum computing
using a system of cold trapped ions. We discuss systematically all the aspects for
making the implementation possible.
Firstly, we go through the loading and confining
of atomic ions in the linear Paul trap, then we describe
the collective vibrational
motion of trapped ions. Further, we discuss interactions of the ions with
a laser beam. We treat the interactions in the travelling-wave and
standing-wave configuration for dipole and quadrupole transitions. We
review different types of laser cooling techniques associated with trapped
ions. We address Doppler cooling, sideband cooling in and beyond
the Lamb-Dicke limit, sympathetic cooling and laser cooling using
electromagnetically induced transparency. After that we discuss the problem
of state detection using the electron shelving method. Then
quantum gates are described. We introduce single-qubit rotations,
two-qubit controlled-NOT and multi-qubit controlled-NOT gates. We also
comment on more advanced multi-qubit logic gates. We describe how
quantum logic networks may be
used for the synthesis of arbitrary pure quantum states. Finally, we discuss
the speed of quantum gates and we also give some numerical estimations for
them. A discussion of dynamics on off-resonant transitions associated with
a qualitative estimation of the weak coupling regime and of the Lamb-Dicke
regime is included in Appendix.\\ \\
PACS numbers: 03.65.Ud, 03.67.Lx, 32.80.Pj, 32.80.Ys
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
\langleabel{impl}
Although trapped ions have found many applications in physics
\cite{NIST}, they caused a turning point in the evolution of quantum
computing when the paper entitled {\it Quantum computation with cold trapped
ions} was published by Cirac and Zoller in 1995 \cite{95-5}. This proposal
launched also an avalanche of other physical realizations of quantum
computing using different physical systems, from high finesse cavities
to widely manufactured
semiconductors \cite{sam}. Through the years we have learnt a lot, but also
revealed many peculiarities, about the physical realization of quantum
computing which has led to many discussions concerning the conditions under
which we could in principle implement quantum computing in
certain quantum systems.
Before we give the list of requirements for the physical implementation
of quantum computing we will introduce the fundamental terminology
to appear throughout this paper.
We will follow the definitions in Ref. \cite{00-4}.
\betaegin{eqnarray}gin{itemize}
\item A {\it qubit} is a quantum system in which the logical Boolean
states 0 and 1 are represented by a prescribed pair of normalized and
mutually orthogonal quantum states labelled as $|0\rangle$ and $|1\rangle$. These two
states form a computational basis and any other (pure) state of the
qubit can be written as a superposition
\betaegin{eqnarray}
\langleabel{q1}
|\psi\rangle=\alpha|0\rangle+\beta|1\rangle
\end{eqnarray}
for some $\alpha$ and $\beta$ such that $|\alpha|^2+|\beta|^2=1$.
It can be shown that we may choose $\alpha=\hat{a}^{\dag}s\vartheta$ and
$\beta=e^{i\varphi}\sin\vartheta$. A~qubit is typically a
microscopic system, such as an atom, a nuclear spin or a polarized photon,
etc. In quantum optics a two-level atom with a selected ground $|g\rangle$
and excited $|e\rangle$ state represents a qubit. Hence the notation $|g\rangle$ and
$|e\rangle$ is used for the computational basis instead of $|0\rangle$ and $|1\rangle$.
For instance,
some qubits can serve for logic operations or the storage of information.
Then we refer to {\it logic qubits}. Some others can be used especially
for sympathetic cooling of logic qubits and we may call them {\it cooling qubits}.
Some further qubits can be used as a quantum channel for
transferring the information between distinct
logic qubits and then we refer to them as to a {\it quantum data bus}.
\item A {\it quantum register} of size $N$ refers to a collection
of $N$ qubits.
\item A {\it quantum gate} is a device which performs a fixed unitary
operation on selected qubits in a fixed period of time.
\item A {\it quantum network} is a device consisting of quantum gates
whose computational steps are synchronized in time.
\item A {\it quantum computer (processor)} can be viewed
as a quantum network or a~family of quantum networks.
\item A {\it quantum computation (computing)}
is defined as a unitary evolution associated
with a set of networks which takes a initial quantum state (input)
into a final quantum state (output) and can be interpreted in terms of
the theory of information processing.
\end{itemize}
For the moment we presume that the following five requirements
(termed {\it DiVincenzo's checklist}) should be met in
order to realize quantum information processing on a quantum system
\cite{divin}. Actually, there are two more requirements for the case of the
transmission of qubits in space {\it (flying qubits)}. However, it appears
that all these requirements are necessary but not sufficient
for successful experimental realization of a quantum processor
\cite{loss}.
\betaegin{eqnarray}gin{itemize}
\item[(1)] The system must provide a well characterized qubit and
the possibility to be scalable in order to create a quantum register.
\item[(2)] We must be able to initialize a simple initial state
of the quantum register.
\item[(3)] Quantum gate operation times must be much shorter than
decoherence times. The quantum gate operation time is the period
required to perform a certain quantum gate on a single qubit or on a set of
qubits. The decoherence time approximately corresponds to the duration of
the transformation which turns a pure state of the qubit
$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ into a mixture
$\hat{\rangleho}=|\alpha|^2|0\rangle\langle 0|+|\beta|^2|1\rangle\langle 1|$.
\item[(4)] We need a set of quantum gates, to perform any unitary
evolution operation that can be realized on the quantum system.
It has been shown that any unitary evolution can be
decomposed into a sequence of single qubit rotations and two-qubit
controlled-NOT (CNOT) gates \cite{95-8}.
\item[(5)] The result of a quantum computational process must be efficiently
read out, i.e. the ability to measure distinct qubits is required.
\end{itemize}
Now we introduce briefly the physical system under consideration.
{\it Cold trapped ions} is a quantum system
of $N$ atomic ions confined
in a linear trap. We assume an anisotropic and harmonic trapping
potential. The ions are laser cooled to a very low temperature, beyond
the Doppler cooling limit, reaching the recoil cooling limit \cite{russia}.
Hence the term cold trapped ions.
The ions form a linear crystal and oscillate in vibrational
collective motional modes
around their equilibrium positions. In their internal structure, depending on
the choice of atomic species, we distinguish distinct atomic levels. The ions
are individually addressed with a laser or a set of lasers
in the travelling-wave or standing-wave configuration. We can detect
the internal state of ions using optical detection devices.
Further, we address briefly the requirements for the physical implementation of
quantum computing (mentioned above) using cold trapped ions.
\betaegin{eqnarray}gin{itemize}
\item[(1)] The qubit is represented by a selected pair of internal atomic
states denoted as $|g\rangle$ and $|e\rangle$.
This selection is discussed in detail in Sec.\,\rangleef{lii}.
The quantum register is realized by $N$ ions forming the ion
string in the linear trap, namely the linear Paul trap, which is reviewed in
Sec.\,\rangleef{ital}. A selected collective vibrational motional mode
(normal mode) is used as the quantum data bus. The vibrational motion of the
ions is treated in Sec.\,\rangleef{vmoti}.
\item[(2)] Different laser cooling techniques can be used
for the proper initialization of the motional state of the ions.
They are described in Sec.\,\rangleef{lc}.
The initial internal state where all the ions are in the state $|g\rangle$ can be
reached by optical pumping to atomic states fast decaying to the ground
state $|g\rangle$ (Sec.\,\rangleef{lc} and \rangleef{es}).
\item[(3)] The influence of the decoherence on the motional state
of the ions is suppressed by laser cooling to ground motional states
of the normal modes. The internal levels of the ions representing the qubit
states $|0\rangle$ and $|1\rangle$ are selected such that they form slow transitions
with excited states of long lifetimes. A very detailed discussion
of the decoherence bounds of trapped atomic ions can be
found in Ref. \cite{98-5}.
\item[(4)] Single-qubit quantum rotations can be realized on any ion and
two-qubit controlled-NOT and multi-qubit
controlled-NOT quantum gates can be applied between chosen ions due to the possibility
of individual addressing with laser beams. The implementation of quantum
gates is discussed in Sec.\,\rangleef{qg}.
\item[(5)] The result of a computational process on cold trapped ions is
encoded into the final state of the internal atomic states. This information
can be very efficiently read out using the electron shelving method
addressed in Sec.\,\rangleef{es}.
\end{itemize}
\section{Ion trapping}
\langleabel{ital}
Due to the charge of atomic ions, we can confine them by particular
arrangements of electromagnetic fields. For studies of ions at low energy
two types of traps are used.
(i) {\it Penning trap} uses a combination of static electric and
magnetic fields and
(ii) {\it Paul trap} confines ions by oscillating electric fields.
Paul was awarded the Nobel Prize in 1990 for his work on trapping
particles in electromagnetic fields \cite{90-1}.
The operation of different ion traps is discussed in detail in Ref. \cite{ghosh}.
For the~purpose considered in this paper we will
discuss only one trap configuration: the {\it linear Paul trap}
(FIG.\,\rangleef{trap}).
We will follow Ref. \cite{ghosh} and \cite{LesH} for the mathematical treatment.
\betaegin{eqnarray}gin{figure}[htb]
\centerline{\epsfig{width=9cm,file=ions31.eps}}
\caption{{\footnotesize
Linear Paul trap in the configuration with two ring electrodes spaced by
$2z_0$. The~diagonal distance between a pair of rod electrodes is $2r_0$.
Seven ions are confined at the~trap axis. The potential (\rangleef{trap1}) is
applied between two diagonally opposite rod electrodes.
The other two are grounded. Ideally, equal static potential
is applied on both ring electrodes.
Used by kind permission of Rainer Blatt~\cite{innsbruck}.}}
\langleabel{trap}
\end{figure}
The linear Paul trap is basically a quadrupole mass filter, which is plugged
at the ends with static electric potentials. An electric potential
\betaegin{eqnarray}
\langleabel{trap1}
\phi_0=U_0+V_0\hat{a}^{\dag}s(\Omega t)
\end{eqnarray}
oscillating with the radiofrequency $\Omega$
is applied between two diagonally opposite rod electrodes. The electrodes
are coupled together with capacitors so that the
potential (\rangleef{trap1}) is constant as a function of the $z$ coordinate.
The~other two rod electrodes are grounded. The resulting potential at the trap
axis (parallel with the $z$ direction) has the form
\betaegin{eqnarray}
\langleabel{trap2}
\phi=\frac{\phi_0}{2r_0^2}(x^2-y^2)=
\frac{U_0+V_0\hat{a}^{\dag}s(\Omega t)}{2r_0^2}(x^2-y^2)\,,
\end{eqnarray}
where $r_0$ is the distance from the trap centre to the electrode surface.
In this field the (classical) equations of motion for an ion of the mass $m$
and charge $q$ are
\betaegin{eqnarray}
\langleabel{trap3}
m\ddot{{\betaf r}}=q{\betaf E}=-q{\betaf {\nabla}}\phi\,,\qquad
{\betaf r}=(x,y,z)\,,
\end{eqnarray}
or rewritten in the components
\betaegin{eqnarray}
\langleabel{trap4}
\ddot{x}+\frac{q}{mr^2_0}\betaigg[U_0+V_0\hat{a}^{\dag}s(\Omega t)\betaigg]x&=&0\,,\\
\langleabel{trap5}
\ddot{y}-\frac{q}{mr^2_0}\betaigg[U_0+V_0\hat{a}^{\dag}s(\Omega t)\betaigg]y&=&0\,,\\
\langleabel{trap6}
\ddot{z}&=&0\,.
\end{eqnarray}
After the substitution
\betaegin{eqnarray}
\langleabel{trap6.1}
a=\frac{4qU_0}{mr^2_0\Omega^2},\qquad
b=\frac{2qV_0}{mr^2_0\Omega^2},\qquad
\zeta=\frac{\Omega t}{2}\,,
\end{eqnarray}
Eq. (\rangleef{trap4}) and (\rangleef{trap5}) take the form of
the~{\it Mathieu equation}
\betaegin{eqnarray}
\langleabel{trap7}
\frac{d^2x}{d\zeta^2}+\betaigg[a+2b\hat{a}^{\dag}s(2\zeta)\betaigg]x&=&0\,,\\
\langleabel{trap8}
\frac{d^2y}{d\zeta^2}-\betaigg[a+2b\hat{a}^{\dag}s(2\zeta)\betaigg]y&=&0\,.
\end{eqnarray}
The Mathieu equation can be solved, in general, using
the~{\it Floquet solution}.
However, typically we have $a\langlel b^2\langlel 1$, then the approximate stable
solution of Eq. (\rangleef{trap7}) and (\rangleef{trap8}) are
\betaegin{eqnarray}
\langleabel{trap9}
x(t)&\alphapprox&x_0\langleeft[1+\frac{b}{2}\hat{a}^{\dag}s(\Omega t)\rangleight]
\hat{a}^{\dag}s(\omega_x t+\varphi_x)\,,\\
\langleabel{trap10}
y(t)&\alphapprox&y_0\langleeft[1-\frac{b}{2}\hat{a}^{\dag}s(\Omega t)\rangleight]
\hat{a}^{\dag}s(\omega_y t+\varphi_y)\,,
\end{eqnarray}
where
\betaegin{eqnarray}
\langleabel{trap11}
\omega_x=\frac{\Omega}{2}\sqrt{\frac{b^2}{2}+a}\,,\qquad
\omega_y=\frac{\Omega}{2}\sqrt{\frac{b^2}{2}-a}
\end{eqnarray}
and $x_0$, $y_0$, $\varphi_x$, $\varphi_y$ are constants determined by initial
conditions. We see from Eq. (\rangleef{trap9}) and (\rangleef{trap10}) that the motion of
a single trapped ion in the radial direction is harmonic with the amplitude
modulated with the frequency $\Omega$. The harmonic oscillation
corresponding to the frequencies $\omega_x$ and $\omega_y$
is called the {\it secular motion}, whereas the small contribution oscillating
at $\Omega$ is termed the {\it micromotion} \cite{nagerl, roos}.
We can eliminate the micromotion under certain conditions \cite{roos}. For
instance, well chosen voltages on additional compensation electrodes
(not shown in FIG.\,\rangleef{trap}) null the micromotion.
Then the ion behaves as if it was confined
in a harmonic pseudopotential $\psi_{2D}$
in the radial direction given by
\betaegin{eqnarray}
\langleabel{trap12}
q\psi_{2D}=\frac{m}{2}\langleeft(\omega_x^2x^2+\omega_y^2y^2\rangleight)\,.
\end{eqnarray}
Typically, $U_0=0\,\mbox{V}$ and hence $a=0$, so the radial frequencies
$\omega_x$ and $\omega_y$
are degenerated. Then Eq. (\rangleef{trap12}) reduces to
\betaegin{eqnarray}
\langleabel{trap13}
q\psi_{2D}=\frac{m\omega_r^2}{2}\langleeft(x^2+y^2\rangleight)\,,
\end{eqnarray}
where the {\it radial trapping frequency} $\omega_r$ is given by
\betaegin{eqnarray}
\langleabel{trap14}
\omega_r=\frac{\Omega b}{2\sqrt{2}}=\frac{qV_0}{mr_0^2\Omega\sqrt{2}}\,.
\end{eqnarray}
In experiments \cite{nagerl, roos, 00-3, blatt1, blatt2},
typical operating parameters
are $V_0\simeq 300-800\,\mbox{V}$,
$\Omega/2\pi\simeq 16-18\,\mbox{MHz}$, $r_0=1.2\,\mbox{mm}$,
so we achieve the radial frequency
$\omega_r/2\pi\simeq 1.4-2\,\mbox{MHz}$ for
Calcium ions $^{40}\mbox{Ca}^+$. In nature, $97\%$
of Calcium consists of this isotope.
To provide confinement along the $z$ direction, static potentials $U_1$ and
$U_2$ are applied on the ring electrodes. Ideally, $U_1=U_2=U_{12}$.
Numerical calculations show that the potential near the trap centre at the
trap axis is harmonic with the~approximate {\it axial trapping frequency}
$\omega_z$ given by
\betaegin{eqnarray}
\langleabel{trap15}
\frac{1}{2}m\omega_z^2z_0^2\alphapprox{\xi}qU_{12}\,,
\end{eqnarray}
where $z_0$ is the distance from the trap centre to the ring electrode and
$\xi$ is a geometric factor describing how much of
the static field from the ring electrodes is present along the trap axis
\cite{nagerl}.
Typical parameters are $\omega_z/2\pi\simeq 500-700\mbox{kHz}$ for
$U_{12}\simeq 2000\,\mbox{V}$ and $z_0=5\,\mbox{mm}$ \cite{blatt1,blatt2}.
The resulting pseudopotential for ions confined
in the linear Paul trap in all three directions takes the form
\betaegin{eqnarray}
\langleabel{trap16}
q\psi_{3D}=\frac{m\omega_r^2}{2}\langleeft(x^2+y^2\rangleight)+\frac{m\omega_z^2z^2}{2}\,,
\end{eqnarray}
where the radial trapping frequency $\omega_r$ is given by Eq. (\rangleef{trap14})
and the axial trapping frequency $\omega_z$ is defined by Eq. (\rangleef{trap15}).
For values of experimental parameters given above, we can calculate the depth
of the potential well in the axial direction
($\omega_z/2\pi\simeq 700\,\mbox{kHz}$)
\betaegin{eqnarray}
\langleabel{trap17}
V_z=\frac{m\omega_z^2z_0^2}{2}\simeq 100\,\mbox{eV}
\end{eqnarray}
and in the radial direction ($\omega_r/2\pi\simeq 2\,\mbox{MHz}$)
\betaegin{eqnarray}
\langleabel{trap18}
V_r=\frac{m\omega_r^2r_0^2}{2}\simeq 820\,\mbox{eV}\,.
\end{eqnarray}
The potential well in the radial direction is almost several times deeper than
along the~trap axis, i.e. there is a strong binding in the radial direction.
Therefore we will not take into account radial oscillations of the ions
in our further considerations.
Finally we briefly mention how ions are loaded into the trap. We will
follow the account of practical procedures in Ref. \cite{roos}.
Before starting the loading process, the trapping potentials are
turned off for a while
in order to get rid of any unwanted trapped residual ions.
The atomic oven producing Calcium atoms
is switched on and heats up. This takes about a~minute.
Then we turn on the electron gun ionizing neutral Calcium atoms
directly in the trapping volume. Cooling lasers are directed on the ion
cloud containing several hundreds of ions
with a diameter of about 200\,$\mu$m. The~ion
cloud gradually relaxes into a steady state where the radiofrequency heating
(from the electrodes) is balanced by laser cooling. The number of trapped
ions is reduced by turning off the cooling. At low ion numbers, the ions
undergo a phase transition and form a linear crystal structure. Therefore,
we refer to the {\it ion crystal} or to the {\it ion string} or eventually
to the {\it ion chain}. The loading process itself takes normally about a minute.
\section{Collective vibrational motion}
\langleabel{vmoti}
\subsection{Equilibrium positions}
We have learnt that the ions form a linear crystal structure in the linear
Paul trap after the loading process. We will assume a string of $N$
trapped ions. Due to the strong binding we can neglect the radial
oscillations. However, if a large number of ions is confined in the trap,
the radial vibrations become unstable and the ions undergo a phase
transition from a linear shape to an unstable zig-zag configuration.
The relation
\betaegin{eqnarray}
\langleabel{zig-zag}
\alpha_{crit}=cN^{\beta}
\end{eqnarray}
determines a critical value for the ratio of the trapping frequencies
$\alpha=(\omega_z/\omega_r)^2$ for a given number of trapped ions
$N$. When $\alpha$ exceeds the critical value $\alpha_{crit}$, the ions are exposed
to a zig-zag motion. The experimental values of the constants in
Eq. (\rangleef{zig-zag}) are $c\simeq 3.23$ and $\beta\simeq -1.83$. For experimental
details and the theoretical treatment we refer to Ref. \cite{zig-zag}.
Further, we describe the collective vibrational motion of the ions.
We will follow the treatment given by James in Ref. \cite{98-7}.
The ions are exposed to the harmonic potential (\rangleef{trap16}) due to the
trap electrodes and also to the repulsive Coulomb force from each other.
Taking into account all the assumptions given above, the potential energy of $N$
ions confined in the linear Paul trap is given by the expression
\betaegin{eqnarray}
\langleabel{vibr1}
V=\sum_{i=1}^N\frac{m\omega_z^2z_i^2(t)}{2}+
\sum_{i,j=1\alphatop i<j}^N
\frac{q^2}{4\pi\varepsilon_0}\frac{1}{|z_i(t)-z_j(t)|}\,,
\end{eqnarray}
where $z_i(t)$ is the position of the $i$th ion numbering them from left to
right with the~origin in the trap centre, $m$ is the mass of the ion with
the charge $q$, $\omega_z$ is the~axial trapping frequency (\rangleef{trap15}) and
$\varepsilon_0$ is the permitivity of the vacuum.
Assuming that the ions are cold enough, we can write for the position
of the $i$th ion
\betaegin{eqnarray}
\langleabel{vibr2}
z_i(t)=\betaar{z}_i+\Delta_i(t)\,,
\end{eqnarray}
where $\betaar{z}_i$ is the equilibrium position and $\Delta_i(t)$ expresses
small vibrations around $\betaar{z}_i$.
The ions placed in the equilibrium positions
minimize the potential energy. Hence these positions are determined by the
condition
\betaegin{eqnarray}
\langleabel{vibr4}
\langleeft[\ \frac{\partial V}{\partial z_i}\ \rangleight]_{{\betaf z}=\betaar{{\betaf
z}}}=0\,,\qquad i=1,\dots,N\,,
\end{eqnarray}
where ${\betaf z}=(z_1,\dots,z_N)$ and
$\betaar{{\betaf z}}=(\betaar{z}_1,\dots,\betaar{z}_N)$.
We introduce a scaling factor $\gamma$ by the relation
\betaegin{eqnarray}
\langleabel{vibr5}
\gamma^3={\frac{q^2}{4\pi\varepsilon_0m\omega_z^2}}
\end{eqnarray}
and the dimensionless equilibrium position as ${\cal Z}_i=\betaar{z}_i/\gamma$.
Then one can rewrite Eq. (\rangleef{vibr4}) to the form
\betaegin{eqnarray}
\langleabel{vibr6}
{\cal Z}_i-
\sum_{j=1}^{i-1}
\frac{1}{({\cal Z}_i-{\cal Z}_j)^2}+
\sum_{j=i+1}^N
\frac{1}{({\cal Z}_i-{\cal Z}_j)^2}=0\,,\qquad i=1,\dots,N\,.
\end{eqnarray}
$N=1$ is a trivial case (${\cal Z}_1=0$).
We can find the analytical solution of Eq. (\rangleef{vibr6}) for two and three
ions:
\betaegin{eqnarray}
\langleabel{vibr7}
&N=2\,,&\quad {\cal Z}_1=-\sqrt[3]{1/4}\,,\quad {\cal Z}_2=\sqrt[3]{1/4}\,,\\
&N=3\,,&\quad {\cal Z}_1=-\sqrt[3]{5/4}\,,\quad {\cal Z}_2=0\,,\quad
{\cal Z}_3=\sqrt[3]{5/4}\,.\nonumber
\end{eqnarray}
Numerical calculations are necessary for $N\geq 4$. For the~Calcium ions
$^{40}\mbox{Ca}^+$ and the~trap frequency $\omega_z/2\pi\simeq 700\,\mbox{kHz}$,
we may calculate the equilibrium positions as
\betaegin{eqnarray}
\langleabel{vibr8}
&N=2\,,&\quad\Delta z_{min}\simeq 7.7\,\mu\mbox{m}\,,\\
&N=3\,,&\quad\Delta z_{min}\simeq 6.1\,\mu\mbox{m}\,.\nonumber
\end{eqnarray}
The minimum value $\Delta z_{min}$ of the distance between two neighbouring
ions in the trap occurs at the centre of the ion crystal, because
the outer ions push the inner ions closer together. It has been
calculated from numerical data that this minimum distance is given
approximately by the relation \cite{oxf, 98-7}
\betaegin{eqnarray}
\langleabel{vibr9}
\Delta z_{min}(N)\alphapprox\frac{2.018}{N^{0.559}}\gamma\,.
\end{eqnarray}
However, slightly different numerical results may be found in Ref. \cite{98-5}.
The relation (\rangleef{vibr9}) happens to be important when one considers
individual ion addressing with a laser beam.
Quantum statistics of the ion ensemble is not considered here because
the spatial spread of the zeropoint wavefunctions of the individual ions is of
the order of 10\,nm and the wavefunction overlap is then negligible
\cite{oxf}.
\subsection{Normal modes}
\langleabel{nm}
The (classical) Lagrangian of the ions in the trap is given by the formula
\betaegin{eqnarray}
\langleabel{vibr10}
L\alphapprox\frac{m}{2}\sum_{k=1}^N\dot{\Delta}^2_k-\frac{1}{2}\sum_{k,l=1}^N
\langleeft[\ \frac{\partial^2V}{\partial z_k\partial z_l}\ \rangleight]_
{{\betaf z}=\betaar{{\betaf z}}}\Delta_k\Delta_l\,,
\end{eqnarray}
where we have expanded the potential energy (\rangleef{vibr1}) in a~Taylor
series about the equilibrium positions.
In the expansion we have omitted the constant term and the~linear
term which is zero [see Eq. (\rangleef{vibr4})].
Higher order terms ${\cal O}(\Delta^3_k)$ have been also neglected.
However, they may cause a cross-coupling between different vibrational
modes which becomes a source of decoherence \cite{98-5}.
The~partial derivatives in Eq. (\rangleef{vibr10}) can be
calculated explicitly and we obtain the expression
\betaegin{eqnarray}
\langleabel{vibr11}
L=\frac{m}{2}\langleeft(\sum_{k=1}^N\dot{\Delta}_k^2-
\omega_z^2\sum_{k,l=1}^NV_{kl}\Delta_k\Delta_l\rangleight)\,,
\end{eqnarray}
where
\betaegin{eqnarray}
\langleabel{vibr12}
V_{kl}=\frac{1}{m\omega_z^2}
\langleeft[\ \frac{\partial^2V}{\partial z_k\partial z_l}\ \rangleight]_
{{\betaf z}=\betaar{{\betaf z}}}=\langleeft\{
\betaegin{eqnarray}gin{array}{l}
1+\sum\langleimits_{j=1\alphatop j\neq k}^N
\frac{2}{|{\cal Z}_k-{\cal Z}_j|^3}\,,\
k=l\,,\\ \\
-\frac{2}{|{\cal Z}_k-{\cal Z}_l|^3}\,,\
k\neq l\,.
\end{array}
\rangleight.
\end{eqnarray}
It follows from Eq. (\rangleef{vibr12}) that $V_{kl}=V_{lk}$.
The values of ${\cal Z}_j$ are given by Eq. (\rangleef{vibr7})
for $N=2$ and for $N=3$,
whereas they have to be calculated numerically for $N\geq 4$.
The dynamics of the trapped ions is governed by the Lagrange equations
\betaegin{eqnarray}
\langleabel{vibr13}
\frac{d}{dt}\frac{\partial L}{\partial\dot{\Delta}_k}-
\frac{\partial L}{\partial\Delta_k}=0\,,\qquad k=1,\dots,N\,,
\end{eqnarray}
with the Lagrangian given by Eq. (\rangleef{vibr11}).
We will search for a particular solution of Eq. (\rangleef{vibr13}) in the form
\betaegin{eqnarray}
\langleabel{vibr14}
\Delta_k(t)=C_ke^{-i\nu t}\,,\qquad k=1,\dots,N\,,
\end{eqnarray}
where $C_k$ are constants. Substituting Eq. (\rangleef{vibr14}) into
(\rangleef{vibr13}) we get the condition for $\nu$ in the form
\betaegin{eqnarray}
\langleabel{vibr15}
\langleeft\|\omega_z^2V_{kl}-\nu^2\delta_{kl}\rangleight\|=0\,,
\end{eqnarray}
where $\delta_{kl}$ is the Kronecker symbol and $\|...\|$ denotes the
determinant. The equation (\rangleef{vibr15}) has in general up to $N$ real and
nonnegative solutions $\nu_{\alpha}$.
The frequencies $\nu_{\alpha}$ are
characteristic parameters of the system. They depend only on its physical
features (not on initial conditions). A general solution of Eq. (\rangleef{vibr13})
is a superposition of particular solutions (\rangleef{vibr14}) and we may write
\betaegin{eqnarray}
\langleabel{vibr16}
\Delta_k(t)=\sum_{\alpha=1}^ND_k^{(\alpha)}\,Q_{\alpha}(t)\,,
\qquad k=1,\dots,N\,,
\end{eqnarray}
where
\betaegin{eqnarray}
\langleabel{vibr17}
Q_{\alpha}(t)=C_{\alpha}e^{-i\nu_{\alpha}t}
\end{eqnarray}
By definition we will require the vectors
\betaegin{eqnarray}
\langleabel{18.0}
{\betaf D}^{(\alpha)}=\langleeft(D_1^{(\alpha)},\dots,D_N^{(\alpha)}\rangleight)\,,
\qquad \alpha=1,\dots,N\,,
\end{eqnarray}
to be the eigenvectors of the matrix $V_{kl}$ defined in Eq. (\rangleef{vibr12}),
i.e.
\betaegin{eqnarray}
\langleabel{vibr18}
\sum_{k=1}^NV_{kl}\,D_k^{(\alpha)}=\mu_{\alpha}D_l^{(\alpha)}\,,
\qquad l,\alpha=1,\dots,N\,,
\end{eqnarray}
and also to be orthogonal and properly normalized
\betaegin{eqnarray}
\langleabel{vibr19}
\sum_{k=1}^ND_k^{(\alpha)}D_k^{(\beta)}=\delta_{\alpha\beta}\,,
\qquad \alpha,\beta=1,\dots,N\,.
\end{eqnarray}
We will number the eigenvectors in order of the increasing
eigenvalues $\mu_{\alpha}$. It can be shown that the first two eigenvectors
$(\alpha=1,2)$ always have the form
\betaegin{eqnarray}
\langleabel{com}
{\betaf D}^{(1)}&=&\frac{1}{\sqrt{N}}(1,1,\dots,1)\,,\qquad \mu_{1}=1\,,\\
\langleabel{breath}
{\betaf D}^{(2)}&=&\frac{1}{\sqrt{\sum_{k=1}^N{\cal Z}_k^2}}
({\cal Z}_1,{\cal Z}_2,\dots,{\cal Z}_N)\,,\qquad \mu_{2}=3\,.
\end{eqnarray}
We should emphasize that Eq. (\rangleef{com}) and (\rangleef{breath})
(they characterize two basic collective motional modes)
are not dependent on the number $N$ of the ions in the trap.
Next eigenvectors $(\alpha\geq 3)$ must be, in general, calculated numerically.
Substituting Eq. (\rangleef{com}) into (\rangleef{vibr19}) we get the
relation
\betaegin{eqnarray}
\langleabel{vibr19.1}
\sum_{k=1}^ND^{(\alpha)}_k=0\,,\qquad \alpha=2,\dots,N\,.
\end{eqnarray}
We can determine analytically the eigensystem for two and three ions:
\betaegin{eqnarray}
\langleabel{vibr19.2}
N=2\,,\qquad {\betaf D}^{(1)}&=&\frac{1}{\sqrt{2}}(1,1)\,,\qquad \mu_{1}=1\,,\\
{\betaf D}^{(2)}&=&\frac{1}{\sqrt{2}}(-1,1)\,,\qquad \mu_{2}=3\,,\nonumber\\
N=3\,,\qquad {\betaf D}^{(1)}&=&\frac{1}{\sqrt{3}}(1,1,1)\,,\qquad \mu_{1}=1\,,\\
{\betaf D}^{(2)}&=&\frac{1}{\sqrt{2}}(-1,0,1)\,,\qquad \mu_{2}=3\,,\nonumber\\
{\betaf D}^{(3)}&=&\frac{1}{\sqrt{6}}(1,-2,1)\,,\qquad \mu_{3}=29/5\,.
\end{eqnarray}
For larger $N$, the eigenvectors and eigenvalues must be
computed numerically. The numerical values for up to ten ions can
be found in Ref. \cite{98-7}.
Substituting Eq. (\rangleef{vibr16}) into (\rangleef{vibr11})
we get a new expression for the Lagrangian
\betaegin{eqnarray}
\langleabel{vibr20}
L=\frac{m}{2}\sum_{\alpha=1}^N\langleeft(\dot{Q}^2_{\alpha}-\nu^2_{\alpha}Q^2_{\alpha}\rangleight)\,,
\end{eqnarray}
where
\betaegin{eqnarray}
\langleabel{vibr21}
\nu_{\alpha}=\omega_z\sqrt{\mu_{\alpha}}\,.
\end{eqnarray}
The Lagrangian (\rangleef{vibr20}) has split into $N$ uncoupled terms, where $Q_{\alpha}$
[Eq. (\rangleef{vibr17})] refer to the {\it normal modes} and
$\nu_{\alpha}$ defined in Eq. (\rangleef{vibr21}) are termed
the~{\it normal frequencies}.
Finally, the position of the $i$th ion in the trap can be
rewritten in terms of Eq. (\rangleef{vibr16}) using (\rangleef{vibr2}) to the~form
\betaegin{eqnarray}
\langleabel{vibr22}
z_i(t)=\betaar{z}_i+
{\cal R}e\langleeft\{
\sum_{\alpha=1}^NC_{\alpha}D^{(\alpha)}_i\,e^{-i\,\nu_{\alpha}t}
\rangleight\}\,,\qquad i=1,\dots,N\,,
\end{eqnarray}
where ${\cal R}e\{...\}$ denotes the real part and
$C_{\alpha}$ are constants given by initial conditions.
The collective vibrational motion of trapped ions determined by the
eigenvector ${\betaf D}^{(1)}$ [Eq. (\rangleef{com})] refers to
the normal mode called the {\it center-of-mass (COM) mode}
\betaegin{eqnarray}
\langleabel{vibr23}
z^{(1)}_i(t)=\betaar{z}_i+
{\cal R}e\langleeft\{
\frac{1}{\sqrt{N}}C_1\,e^{-i\omega_zt}
\rangleight\}\,,\qquad i=1,\dots,N\,,
\end{eqnarray}
and corresponds to all of the~ions oscillating back and forth
as if they were a rigid body.
The motion determined by the next eigenvector ${\betaf D}^{(2)}$,
[Eq. (\rangleef{breath}] refers to the {\it breathing mode}
\betaegin{eqnarray}
\langleabel{vibr24}
z_i^{(2)}(t)=\betaar{z}_i+
{\cal R}e\langleeft\{
\frac{\betaar{z}_i}{\sqrt{\sum_{k=1}^N\betaar{z}^2_k}}C_2\,
e^{-i(\omega_z\sqrt{3})t}
\rangleight\}\,,\qquad i=1,\dots,N\,.
\end{eqnarray}
It corresponds to each ion oscillating with the amplitude proportional
to its equilibrium distance from the trap center.
The COM motional mode can be excited in experiments by applying
an additional AC voltage on one of the ring electrodes. For exciting
the breathing motional mode, a 300-times higher voltage must be applied
\cite{00-3}. Higher motional modes require gradient field excitation due
to the nontrivial configuration of the ions in the ion string.
However, in the limit of large ion trap dimension in comparison
with the ion crystal dimension, the electrode electric fields are almost uniform
across the ion crystal and the COM mode is very susceptible to heating due
to these fields. Therefore, it seems to be more advantageous
to use rather the breathing mode,
which is much less influenced by uniform fields, as the quantum data bus.
This will be discussed in more detail later on in the section
on sympathetic cooling (Sec.\,\rangleef{sympcool}).
On the other hand, the ions can be easily addressed
with a laser beam in the COM mode, while higher modes require accurate
bookkeeping when addressing distinct ions in the ion crystal
\cite{98-7,symp1}.
\subsection{Quantized vibrational motion}
The normal modes $Q_{\alpha}$ are uncoupled in Eq. (\rangleef{vibr20}), so the
corresponding canonical momentum conjugated to $Q_{\alpha}$ is
$P_{\alpha}=m\dot{Q_{\alpha}}$ and one may write the (classical) Hamiltonian
\betaegin{eqnarray}
\langleabel{vibr25}
H=\frac{1}{2m}\sum_{\alpha=1}^NP_{\alpha}^2+
\frac{m}{2}\sum_{\alpha=1}^N\nu_{\alpha}^2Q_{\alpha}^2\,.
\end{eqnarray}
The quantum motion of the ions can be considered by introducing the operators
\betaegin{eqnarray}
\langleabel{vibr26}
Q_{\alpha}\ \rangleightarrow\ \hat{Q}_{\alpha}=\sqrt{\frac{\hbar}{2m\nu_{\alpha}}}
\langleeft(\hat{a}^{\dag}_{\alpha}+\alphao_{\alpha}\rangleight)\,,\\
\langleabel{vibr27}
P_{\alpha}\ \rangleightarrow\ \hat{P}_{\alpha}=i\sqrt{\frac{\hbar m\nu_{\alpha}}{2}}
\langleeft(\hat{a}^{\dag}_{\alpha}-\alphao_{\alpha}\rangleight)
\end{eqnarray}
with the corresponding commutation relations
\betaegin{eqnarray}
\langleabel{vibr27.1}
[\hat{Q}_{\alpha},\hat{P}_{\beta}]=i\hbar\delta_{\alpha\beta}\,,\qquad
[\alphao_{a},\hat{a}^{\dag}_{\beta}]=\delta_{\alpha\beta}\,.
\end{eqnarray}
The Hamiltonian operator associated with the external (vibrational)
degrees of freedom of the trapped ions is then expressed as follows
$(H\rangleightarrow\hat{H}_{ext})$
\betaegin{eqnarray}
\langleabel{vibr28}
\hat{H}_{ext}=\sum_{\alpha=1}^N\hbar\nu_{\alpha}\langleeft(\hat{a}^{\dag}_{\alpha}\alphao_{\alpha}+1/2\rangleight)\,,
\end{eqnarray}
where $\alphao_{\alpha}$ and $\hat{a}^{\dag}_{\alpha}$ are the usual annihilation and creation
operators referring to the $\alpha$th normal mode. We use the standard notation
for the number states associated with the collective vibrational
motion of the ions
\betaegin{eqnarray}
\langleabel{vibr-pom1}
\hat{a}^{\dag}_{\alpha}\alphao_{\alpha}|n_{\alpha}\rangle=n_{\alpha}|n_{\alpha}\rangle\,,
\end{eqnarray}
where $|n_{\alpha}\rangle$ refers to the state of the $\alpha$th normal mode and $n_{\alpha}$
denotes the number of vibrational phonons in this mode. The states
$\{|n_{\alpha}\rangle\}$ form the complete and orthonormal basis
\betaegin{eqnarray}
\langleabel{vibr-pom2}
\langle m_{\alpha}|n_{\beta}\rangle&=&\delta_{\alpha\beta}\, \delta_{mn}\,.
\end{eqnarray}
We can quantize the motion of the ions by applying Eq. (\rangleef{vibr26})
to the relation (\rangleef{vibr16}) and expressing the displacement operator
of the $i$th ion in the time-independent picture
\betaegin{eqnarray}
\langleabel{vibr29}
\hat{\Delta}_i&=&\sum_{\alpha=1}^ND^{(\alpha)}_i\sqrt{\frac{\hbar}{2m\nu_{\alpha}}}
(\hat{a}^{\dag}_{\alpha}+\alphao_{\alpha})=\sum_{\alpha=1}^N{\cal K}_i^{(\alpha)}z_0(\hat{a}^{\dag}_{\alpha}+\alphao_{\alpha})\,,
\qquad i=1,\dots,N\,,
\end{eqnarray}
where [see Eq. (\rangleef{vibr21})]
\betaegin{eqnarray}
\langleabel{vibr30}
{\cal K}_i^{(\alpha)}=\frac{D^{(\alpha)}_i}{\sqrt[4]{\mu_{\alpha}}}\,,\qquad
z_0=\sqrt{\frac{\hbar}{2m\omega_z}}\,.
\end{eqnarray}
We can easily calculate from Eq. (\rangleef{com}) that for the COM mode applies
\betaegin{eqnarray}
\langleabel{vibr31}
{\cal K}_i^{(1)}=\frac{1}{\sqrt{N}}
\end{eqnarray}
and for the breathing mode [Eq. (\rangleef{breath})]
\betaegin{eqnarray}
\langleabel{vibr31}
{\cal K}_i^{(2)}=
\frac{\betaar{{\cal Z}}_i}{\sqrt{\sum_{l=1}^N\betaar{{\cal Z}}_l^2}}
\frac{1}{\sqrt[4]{3}}=
\frac{\betaar{z}_i}{\sqrt{\sum_{l=1}^N\betaar{z}_l^2}}
\frac{1}{\sqrt[4]{3}}\,.
\end{eqnarray}
Although we have not considered the radial vibrations due
to the strong binding of the ions in the radial direction, a detailed treatment
of the ion motion in the trap would require the extension to all three
dimensions. Then Eq. (\rangleef{vibr2}) has to be replaced with
\betaegin{eqnarray}
\langleabel{3D-1}
{\betaf q}_i=\betaar{{\betaf q}}_i+{\Delta}{\betaf q}_i\,,\quad i=1,\dots,N\,,
\end{eqnarray}
where $\betaar{{\betaf q}}_i$ denotes the equilibrium position of the $i$th ion
in the 3D space and
${\Delta}{\betaf q}_i$ is its displacement from the equilibrium
position. We can write
\betaegin{eqnarray}
\langleabel{3D-2}
\betaar{{\betaf q}}_i&=&
\betaar{x}_i\,{\betaf x}+\betaar{y}_i\,{\betaf y}+\betaar{z}_i\,{\betaf z}\,,\nonumber\\
\langleabel{3D-3}
{\Delta}{\betaf q}_i&=&
\Delta_i\,{\betaf x}+\Delta_{N+i}\,{\betaf y}+\Delta_{2N+i}\,{\betaf z},
\quad i=1,2,\dots,N\,,
\end{eqnarray}
where $\betaar{x}_i$, $\betaar{y}_i$, $\betaar{z}_i$ are the equilibrium positions
of the $i$th ion and {\betaf x}, {\betaf y}, {\betaf z} are unit vectors
in the 3D space. The free Hamiltonian associated with
the vibrational motion in the 3D space reads
\betaegin{eqnarray}
\langleabel{3D-4}
\hat{H}_{ext}^{(3\!D)}=
\sum_{\alpha=1}^{3N}\hbar\nu_{\alpha}\langleeft(\hat{a}^{\dag}_{\alpha}\alphao_{\alpha}+1/2\rangleight)
\end{eqnarray}
and the displacement operators in Eq. (\rangleef{3D-3}) are given as follows
\betaegin{eqnarray}
\langleabel{3D-5}
\hat{\Delta}_i=\sum_{\alpha=1}^{3N}{\cal K}_i^{(\alpha)}z_0(\hat{a}^{\dag}_{\alpha}+\alphao_{\alpha})\,,
\qquad i=1,\dots,3N\,,
\end{eqnarray}
where the numerical factors ${\cal K}_i^{(\alpha)}$ in general
have to be determined numerically.
\section{Laser-ion interactions}
\langleabel{lii}
Information is encoded in internal (atomic) states, while
it is transferred via external (motional) states of the ions.
We can manipulate these states due to laser-ion interactions.
It can be accomplished in the travelling-wave and standing-wave
configurations. We will address in detail both approaches in what follows.
However, we should first comment on the selection of the two internal
atomic levels to form the qubit.
There are three possibilities \cite{98-7}:
\betaegin{eqnarray}gin{itemize}
\item We can employ a ground and metastable fine structure excited state.
This applies for ions with zero nuclear angular momentum
[FIG.\,\rangleef{3types}(a)]. In this
case we refer to the {\it single beam scheme} and we can drive transitions
on optical frequencies. This configuration is used, for example, by the
group in Innsbruck using Calcium ions $^{40}\mbox{Ca}^{+}$
\cite{innsbruck, 00-11}.
\item We can also choose two sublevels of a ground state within the
hyperfine structure (ions with nonzero nuclear angular momentum)
[FIG.\,\rangleef{3types}(b)].
The spacing of such two sublevels is in the range of GHz. Thus, a two-beam
{\it Raman scheme} via a~third virtual level is required in
order to resolve the individual sublevels. Experiments in this configuration with
Beryllium ions $^9\mbox{Be}^+$ were performed in Boulder
\cite{NIST, 98-5, exp}.
\item It also possible to apply a magnetic field and consider two
Zeeman sublevels of the ground state [FIG.\,\rangleef{3types}(c)].
This scheme also requires Raman excitation.
In this class, we can mention, for example, Magnesium ions
$^{24}\mbox{Mg}^+$ used by the group in Garching \cite{garching}.
\end{itemize}
We have to mention also other active groups running experiments towards quantum
logic with trapped ions.
For instance (in alphabetical order)
IBM Almaden using $^{138}\mbox{Ba}^+$ \cite{IBM},
Imperial College ($^{40}\mbox{Ca}^+$, $^{199}\mbox{Hg}^+$) \cite{imperial},
JPL in Los~Angeles ($^{199}\mbox{Hg}^+$) \cite{JPL},
Los Alamos National Laboratory ($^{40}\mbox{Ca}^+$) \cite{LANL, lanl},
Oxford University ($^{40}\mbox{Ca}^+$) \cite{oxf, oxford},
University of Aarhus ($^{40}\mbox{Ca}^+$) \cite{aarhus},
University of Hamburg ($^{138}\mbox{Ba}^+$, $^{171}\mbox{Yb}^+$) \cite{hamburg}
and University of Mainz ($^{40}\mbox{Ca}^+$) \cite{mainz}.
We can use {\it dipole} and {\it quadrupole} transitions.
Theoretically, the difference is only in the interaction constants
as we will see later on in this section.
On the other hand, in experiments
quadrupole transitions have much longer lifetimes (one second for Calcium ions)
comparing to fast decaying dipole transitions ($10^{-8}\,\mbox{s}$).
Experiments on an {\it octupole}
transition in an Ytterbium ion has also been realized.
The predicted theoretical lifetime in this system is
of the order of $10^8\,\mbox{s}$ \cite{oct}. However, in this case one deals
with very weak transitions with very stringent demands on the laser sources
used in the experiment (although they are of major interest as potential ion
trap {\it clocks}). Moreover, weak transitions have to be driven with
a very intense laser which enhances the possibility for off-resonant
excitations.
From now on we will describe in this paper all experimental procedures
for Calcium ions $^{40}\mbox{Ca}^+$ (FIG.\,\rangleef{Ca}).
\betaegin{eqnarray}gin{figure}[h!]
\centerline{\epsfig{width=14cm,file=ions51.eps}}
\caption{{\footnotesize Three possible choices for two internal atomic states
representing the qubit:
(a) a ground and a metastable excited state, (b) sublevels of a ground state and
(c) Zeeman sublevels of a ground state, where $\omega_L$, $\omega_{L1}$ and
$\omega_{L2}$ refer to the laser frequencies \cite{zeil}.}}
\langleabel{3types}
\end{figure}
In the following we will deal with the single beam scheme, i.e. transitions
being driven by a single laser beam. We will not treat here the Raman
scheme. The derivation of the Hamiltonian in this scheme can be found in
Ref. \cite{raman}.
We just mention that the final Hamiltonian in the Raman scheme
has the same form as the one in the single beam scheme, except for differences
in coupling constants and for atomic frequencies which are Stark light shifted.
In the Raman scheme the resulting effective light field has the direction
(frequency) determined by the difference of the wavevectors (frequencies) of
the two participating laser beams, where each beam is represented
(in a semiclassical approach) with a monochromatic travelling wave. Finally,
the single beam scheme requires a very high laser frequency stability,
while in the Raman scheme we only need to control the relative
frequency stability between the two laser beams which is technically less
demanding. With the Raman scheme we can also ensure the relative wavevector
of the two beams to be parallel to the trap axis which suppresses the coupling to
radial motional modes. On the other hand, the Raman scheme can introduce
significant Stark light shifts \cite{98-5}.
\betaegin{eqnarray}gin{figure}[htb]
\centerline{\epsfig{width=10cm,file=ions52.eps}}
\caption{{\footnotesize
Five lowest available atomic levels of the Calcium ion $^{40}\mbox{Ca}^+$
\cite{roos}. All transitions are accessible with solid state diode lasers.
The nuclear spin of this isotope is zero, i.e. there is no hyperfine
structure. For the spectroscopic notation of the levels see the text.}}
\langleabel{Ca}
\end{figure}
In the rest of the paper
we will use the standard atomic level notation $n\,^{2S+1}L_J$, where $n$ is
the principal quantum number, $S$ is the spin angular momentum, $L$ is the
orbital angular momentum and $J$ is the total angular momentum of electrons.
For the fine structure case the notation is $n\,^{2S+1}L_J(m_J)$ where $m_J$ is
the projection of $J$ onto the quantization axis. In the case of
the hyperfine structure we denote $n\,^{2S+1}L_J(F,m_F)$ where $F$ is the
total angular momentum of the atom (electrons + nucleus) and $m_F$ is the
projection of $F$ onto the quantization axis.
Let us consider that the ion has two internal levels, denoted $|g\rangle$ (lower)
and $|e\rangle$ (upper) with corresponding energies $E_g$ and $E_e$, where the
transition frequency is $\omega_0=(E_e-E_g)/\hbar$. Then the free
Hamiltonian associated with the internal degrees of freedom is given by
\betaegin{eqnarray}
\langleabel{int0}
\hat{H}_{int}=E_e|e\rangle\langle e|+E_g|g\rangle\langle g|=
\frac{\hbar\omega_0}{2}\sigma_z+\frac{E_e+E_g}{2}\openone_{int}\,,
\end{eqnarray}
where $\sigma_z=|e\rangle\langle e|-|g\rangle\langle g|$ and
$\openone_{int}=|e\rangle\langle e|+|g\rangle\langle g|$.
Finally, we can write the total free Hamiltonian for the $j$th ion of $N$
ions confined in the trap communicating via one of the collective
vibrational modes [see Eq. (\rangleef{vibr28})]
\betaegin{eqnarray}
\langleabel{int1}
\hat{H}_{0j}=\hat{H}_{int}+\hat{H}_{ext}=
\frac{\hbar\omega_0}{2}\sigma_{zj}+\hbar\nu\hat{a}^{\dag}\alphao\,,
\end{eqnarray}
where we have omitted constant terms $(E_e+E_g)/2$, $\hbar\nu/2$ and
dropped down the index $\alpha$ denoting a vibrational mode.
The motional mode used for manipulations (especially quantum logic
operations) with the ions is called the {\it quantum data bus} because, as
we will see later, it serves to transfer the information between
distinct ions within the ion crystal (representing a quantum register).
We will consider for this purpose only the COM mode $(\nu=\omega_z)$ or the
breathing mode $(\nu=\omega_z\sqrt{3})$.
Further, we assume a powerful laser, i.e. the interaction with the ions has
no influence on the laser photon statistics. Therefore, we will employ
a semiclassical description of the laser beam. We will consider the laser beam
in the~(i)~{\it travelling-wave} and (ii)~{\it standing-wave configuration}.
\subsection{Travelling-wave configuration}
There are two different ways for addressing the ions.
We can set the laser beam at a fixed position and shift the ion
string by a very slight variation of the DC voltage on the ring electrode.
On the other hand, we can fix the ion string and scan the laser
across the string. In this case an acousto-optical modulator is used for
laser beam deflection \cite{00-3}.
Let us approximate the laser beam as a monochromatic travelling
wave (FIG.\,\rangleef{trav}). We can write
\betaegin{eqnarray}
\langleabel{int2}
{\betaf E}&=&E_0\betaoldsymbol{\epsilon}
\hat{a}^{\dag}s\betaig(\omega_Lt-\betaoldsymbol{\kappa}\cdot{\betaf q}+\phi\betaig)\,,\\
&=&\frac{E_0\betaoldsymbol{\epsilon}}{2}\langleeft[
e^{-i(\omega_Lt-\betaoldsymbol{\kappa}\cdot{\betaf q}+\phi)}+
e^{i(\omega_Lt-\betaoldsymbol{\kappa}\cdot{\betaf q}+\phi)}
\rangleight]\,,\nonumber
\end{eqnarray}
where $E_0$ is the real amplitude, $\betaoldsymbol{\epsilon}$
is the polarization vector with $|\betaoldsymbol{\epsilon}|=1$,
$\omega_L$ is the laser frequency,
$\betaoldsymbol{\kappa}=\kappa{\betaf n}=(\omega_L/c){\betaf n}$ is
the wavevector with $|{\betaf n}|=1$,
$c$ is the speed of light, ${\betaf q}$ is the position
vector and $\phi$ is the phase factor.
The full Hamiltonian for the $j$th ion is given by
\betaegin{eqnarray}
\langleabel{int3}
\hat{H}_j=\hat{H}_{0j}+\hat{V}_j\,,
\end{eqnarray}
where the interaction Hamiltonian
(assuming a hydrogen-like atomic configuration)
expanded to second order
(neglecting magnetic dipole interaction) has only two terms
\betaegin{eqnarray}
\langleabel{int3.1}
\hat{V}_j=\hat{V}_j^{D\!P}+\hat{V}_j^{Q\!D}\,.
\end{eqnarray}
The electric dipole (DP) term is defined as follows
\betaegin{eqnarray}
\langleabel{int4}
\hat{V}^{D\!P}_j=
-q_e\sum_{a}(\hat{{\betaf r}}_j)_{a}\,E_{a}(t,\hat{{\betaf R}}_j)=
-q_e\hat{{\betaf r}}_j\cdot {\betaf E}(t,\hat{{\betaf R}}_j)\,,
\end{eqnarray}
summing over $a=x,y,z$. We refer to Eq. (\rangleef{int4})
as the {\it dipole approximation}.
The electric quadrupole (QD) term reads
\betaegin{eqnarray}
\langleabel{int4.1}
\hat{V}^{Q\!D}_j=
-\frac{q_e}{2}\sum_{a,b}
(\hat{{\betaf r}}_j)_{a}(\hat{{\betaf r}}_j)_{b}
\frac{\partial E_{b}(t,\hat{{\betaf R}}_j)}{\partial q_{a}}\,,
\end{eqnarray}
where the sum is applied over $a,b=x,y,z$ and we refer
to Eq. (\rangleef{int4.1}) as the {\it quadrupole approximation}. We denote
$q_e$ to be the electron charge,
$\hat{{\betaf r}}_j$ is the internal position operator associated with
the position of the valence electron in the $j$th ion and
$\hat{{\betaf R}}_j=(0,0,\hat{z}_j)$ is the external position operator
corresponding to the position of the $j$th ion in the trap.
\betaegin{eqnarray}gin{figure}[htb]
\centerline{\epsfig{width=10cm,file=ions53.eps}}
\caption{{\footnotesize
The travelling-wave configuration corresponds to illuminating
the $j$th ion in the ion string with the laser beam of the frequency
$\omega_L$ at the angle $\vartheta$ to the trap axis.}}
\langleabel{trav}
\end{figure}
For the present we will
consider only the dipole term (\rangleef{int4}), regarding to the situation
when the dipole interaction is present and the quadrupole contribution
(\rangleef{int4.1}) is then negligible.
Later we will also comment on the quadrupole interaction.
If we consider consider only a single motional mode, we get from
Eq.~(\rangleef{vibr29}) for the external position operator of the $j$ ion
\betaegin{eqnarray}
\langleabel{int5.0}
\hat{z}_j=\betaar{z}_j+{\cal K}_jz_0(\hat{a}^{\dag}+\alphao)\,.
\end{eqnarray}
Then we can sandwich the internal position operator
$\hat{{\betaf r}}_j$ with the unity
operator $\openone_j=|e_j\rangle\langle e_j|+|g_j\rangle\langle g_j|$ and rewrite Eq. (\rangleef{int4})
to the form
\betaegin{eqnarray}
\langleabel{int5}
\hat{V}_j=
-q_e\betaigg[
({\betaf r}_{eg})_j\hat{\sigma}_{+j}+({\betaf r}_{eg})_j^*\hat{\sigma}_{-j}
\betaigg]
\cdot\frac{E_0\betaoldsymbol{\epsilon}}{2}
\langleeft\{e^{-i\langleeft[\omega_Lt-\eta_j(\hat{a}^{\dag}+\alphao)+\phi_j\rangleight]}
+\mbox{H.c.}\rangleight\}\,,
\end{eqnarray}
where
$({\betaf r}_{eg})_j=\langle e_j|\hat{{\betaf r}}_j|g_j\rangle$,
$\hat{\sigma}_{+j}=|e_j\rangle\langle g_j|$,
$\hat{\sigma}_{-j}=|g_j\rangle\langle e_j|$,
$\kappa=\omega_L/c$,
$\eta_j={\cal K}_j\betaar{\eta}$,
$\betaar{\eta}=\kappa_{\vartheta}z_0$,
$\kappa_{\vartheta}=\kappa\hat{a}^{\dag}s\vartheta$,
$\phi_j=\phi-\kappa_{\vartheta}\betaar{z}_j$
with ${\cal K}_j$ and $z_0$ defined by Eq. (\rangleef{vibr30}).
In Eq. (\rangleef{int5}) we consider that
$\langle e_j|\hat{{\betaf r}}_j|e_j\rangle=\langle g_j|\hat{{\betaf r}}_j|g_j\rangle=0$, because
we assume spatial symmetry of the wavefunctions associated
with the internal atomic
states $|g_j\rangle$ and $|e_j\rangle$.
The schematic configuration is depicted in FIG.\,\rangleef{trav}.
It is useful to transform to the interaction picture defined by the
prescription
\betaegin{eqnarray}
\langleabel{int6}
i\hbar\frac{\partial}{\partial t}|\Psi\rangle=\hat{H}|\Psi\rangle
\quad &\langleongrightarrow&\quad
i\hbar\frac{\partial}{\partial t}|\psi\rangle=\hat{{\cal H}}|\psi\rangle\,,\qquad
|\psi\rangle=\hat{U}_0^{\dag}|\Psi\rangle\,,
\nonumber\\ \nonumber\\
\hat{H}=\hat{H}_0+\hat{V}
\quad &\langleongrightarrow&\quad
\hat{{\cal H}}=\hat{U}_0^{\dag}\,\hat{V}\,\hat{U}_0\,,\qquad
\hat{U}_0=\exp\langleeft(-\frac{i\hat{H}_0t}{\hbar}\rangleight)\,.
\end{eqnarray}
The Hamiltonian (\rangleef{int5}) after the transformation to the interaction
picture (\rangleef{int6}) reads
\betaegin{eqnarray}
\langleabel{int7}
\hat{{\cal H}}_j=
\frac{\hbar\langleambda_j}{2}\hat{\sigma}_{+j}
\exp\langleeft[
i\eta_j\betaigg(\hat{a}^{\dag} e^{i\nu t}+\alphao e^{-i\nu t}\betaigg)
\rangleight]e^{-i\delta t}
+\mbox{H.c.}\,,
\end{eqnarray}
where $\delta=\omega_L-\omega_0$ and we have neglected rapidly oscillating
terms at the frequency $\omega_L+\omega_0$ compared with low-frequency
terms at $\omega_L-\omega_0$. In practice $\omega_L\simeq\omega_0$,
therefore to a good degree of approximation for times of interest,
high-frequency terms average to zero \cite{louisell}.
This approximation is called
the {\it rotating wave approximation (RWA)}.
In Eq. (\rangleef{int7}) we substitute $\eta_j=\betaar{\eta}/\sqrt{N}$ for the COM
mode or $\eta_j=\betaar{\eta}\betaar{z}_j/(\sqrt[4]{3}\sum_{l=1}^N\betaar{z}_l^2)$
for the breathing mode.
The laser coupling constant $\langleambda_j$ introduced
in Eq. (\rangleef{int7}) is defined by the~relation
\betaegin{eqnarray}
\langleabel{int8}
\langleambda^{D\!P}_j=
-\frac{q_eE_{0}}{\hbar}
\betaigg[\sum_{a}\langle e_j|(\hat{{\betaf r}}_j)_{a}|g_j\rangle\epsilon_{a}\betaigg]\,
e^{-i\phi_j}=
-\frac{q_eE_{0}}{\hbar}
\betaigg[({\betaf r}_{eg})_j\cdot\betaoldsymbol{\epsilon}\betaigg]\,e^{-i\phi_j}\,.
\end{eqnarray}
However, for a dipole forbidden transition when
$\langle e_j|\hat{{\betaf r}}_j|g_j\rangle=0$,
the dipole term (\rangleef{int4})
does not contribute ($\langleambda^{D\!P}_j=0$) and the key role
is played by the weaker quadrupole interaction.
In that case the laser coupling constant in the Hamiltonian (\rangleef{int7}) reads
\betaegin{eqnarray}
\langleabel{int8.1}
\langleambda^{Q\!D}_j=
-\frac{iq_eE_0\omega_L}{2\hbar c}
\betaigg[\sum_{a,b}
\langle e_j|(\hat{{\betaf r}}_j)_{a}(\hat{{\betaf r}}_j)_{b}|g_j\rangle n_{a}\epsilon_{b}
\betaigg]e^{-i\phi_j}\,,
\end{eqnarray}
where all parameters are defined in Eq. (\rangleef{int2}).
Next, let us assume the detuning $\delta$ of the laser frequency $\omega_L$
from the atomic frequency $\omega_0$ for the vibrational frequency $\nu$ in
the form
\betaegin{eqnarray}
\langleabel{int9.0}
\delta=\omega_L-\omega_0=k\nu\,,\qquad k=0,\pm 1,\pm 2,\dots\,,
\end{eqnarray}
and apply the Baker-Campbell-Hausdorff theorem \cite{louisell}
$e^{\hat{A}+\hat{B}}=e^{\hat{A}}e^{\hat{B}}e^{-\frac{1}{2}[\hat{A},\hat{B}]}$
Eq. (\rangleef{int7}). Then we can write
\betaegin{eqnarray}
\langleabel{int9}
\hat{{\cal H}}_j=
\frac{\hbar\langleambda_j}{2}\hat{\sigma}_{+j}\,e^{-(\eta_j^2/2)}
\sum_{\alpha,\beta=0}^{\infty}\langleeft(i\eta_j\rangleight)^{\alpha+\beta}
\frac{(\hat{a}^{\dag})^{\alpha}}{\alpha!}\frac{\alphao^{\beta}}{\beta!}\,
e^{i\nu t(\alpha-\beta-k)}+\mbox{H.c.}\,.
\end{eqnarray}
If the laser is tuned at the frequency $\omega_L$ such that $k>0$,
the spectral line is
termed the~$k$th {\it blue sideband}. For $k=0$ the line is called
the {\it carrier} and for $k<0$ refers the $k$th {\it red sideband}
because the laser is red (blue) detuned from the atomic
frequency $\omega_0$ (FIG.\,\rangleef{3freq}).
\betaegin{eqnarray}gin{figure}[bht]
\centerline{\epsfig{width=13cm,file=ions54.eps}}
\caption{{\footnotesize
Scheme for the transition driven on the carrier (a), on the first red
sideband (b) and on the first blue sideband (c), where $\omega_L$ denotes
the laser frequency, $\omega_0$ is the atomic frequency and $\nu$ is
the vibrational frequency of the respective motional mode.
The parameter $\delta$ is the detuning defined as $\delta=\omega_L-\omega_0$.}}
\langleabel{3freq}
\end{figure}
When the constant $\langleambda_j$ is sufficiently small
we can assume that there are no excitations on
off-resonant transitions ({\it weak coupling regime}).
Then the level structure of the ion can be
considered as a series of isolated two-level systems \cite{97-2}.
Precisely what is meant by sufficiently small is detailed in
Appendix \rangleef{Off}. Assuming the weak coupling regime,
we can neglect off-resonant terms ($\alpha-\beta-k\neq 0$) and
rewrite Eq. (\rangleef{int9}) for $k\geq 0$ to the~form
\betaegin{eqnarray}
\langleabel{int10}
\hat{{\cal H}}_j^{(+)}=
\frac{\hbar\langleambda_j}{2}\,\hat{\sigma}_{+j}\,(\hat{a}^{\dag})^{|k|}\,
{\cal F}_{k}(\hat{a}^{\dag}\alphao)+
\frac{\hbar\langleambda_j^*}{2}\,\hat{\sigma}_{-j}\,
{\cal F}_{k}^{\dag}(\hat{a}^{\dag}\alphao)\,\alphao^{|k|}
\end{eqnarray}
and for $k<0$
\betaegin{eqnarray}
\langleabel{int11}
\hat{{\cal H}}_j^{(-)}=
\frac{\hbar\langleambda_j}{2}\,\hat{\sigma}_{+j}\,
{\cal F}_{k}(\hat{a}^{\dag}\alphao)\,\alphao^{|k|}+
\frac{\hbar\langleambda_j^*}{2}\,\hat{\sigma}_{-j}\,(\hat{a}^{\dag})^{|k|}\,
{\cal F}_{k}^{\dag}(\hat{a}^{\dag}\alphao)
\,.
\end{eqnarray}
In the last relation we introduce the operator function
\betaegin{eqnarray}
\langleabel{int12}
{\cal F}_{k}(\hat{a}^{\dag}\alphao)=e^{-(\eta_j^2/2)}
\langleeft(i\eta_j\rangleight)^{|k|}
\sum_{\alpha=0}^{\infty}(-\eta_j^2)^{\alpha}
\frac{(\hat{a}^{\dag}\alphao)^{\alpha}}{\alpha!(\alpha+|k|)!}\,.
\end{eqnarray}
Although we allow the parameter $k$ to be positive or negative, we
keep writing its absolute value $|k|$ in both cases in order to avoid the tricky
notation of form $(\hat{a}^{\dag})^{-k}$ and $\alphao^{-k}$ in the relation
(\rangleef{int11}) and also in what follows next.
The final form of the Hamiltonian is given by
\betaegin{eqnarray}
\langleabel{int13}
\hat{{\cal H}}_j^{(+)}=\hbar
\sum_{n=0}^{\infty}\langleeft[
\frac{\Omega_{j}^{n,k}}{2}\betaigg(|e_j\rangle\langle g_j|\otimes |n+|k|\rangle\langle n|\betaigg)+
\frac{(\Omega_{j}^{n,k})^*}{2}\betaigg(|g_j\rangle\langle e_j|\otimes |n\rangle\langle n+|k||
\betaigg)
\rangleight]\nonumber\\
\
\end{eqnarray}
and
\betaegin{eqnarray}
\langleabel{int14}
\hat{{\cal H}}_j^{(-)}=\hbar
\sum_{n=0}^{\infty}\langleeft[
\frac{\Omega_{j}^{n,k}}{2}\betaigg(|e_j\rangle\langle g_j|\otimes |n\rangle\langle n+|k||\betaigg)+
\frac{(\Omega_{j}^{n,k})^*}{2}\betaigg(|g_j\rangle\langle e_j|\otimes |n+|k|\rangle\langle n|
\betaigg)\rangleight]\,.\nonumber\\
\
\end{eqnarray}
We have defined a new coupling constant
\betaegin{eqnarray}
\langleabel{int15}
\Omega_j^{n,k}=
\langleambda_j\,e^{-(\eta_j^2/2)}
\langleeft(i\eta_j\rangleight)^{|k|}
\sqrt{\frac{n!}{(n+|k|)!}}\,L_n^{|k|}\langleeft(\eta_j^2\rangleight)\,,
\end{eqnarray}
where
\betaegin{eqnarray}
\langleabel{int16}
L_n^{a}(x)=\sum_{m=0}^n(-1)^m\,\frac{x^m}{m!}
\langleeft({n+a\alphatop n-m}\rangleight)
\end{eqnarray}
is the generalized Laguerre polynomial and
$\nad {n+a}.{n-m}=\frac{(n+a)!}{(n-m)!(a+m)!}$.
Finally, we may write the unitary evolution operator for time-independent
Hamiltonians (\rangleef{int13}) and (\rangleef{int14})
\betaegin{eqnarray}
\langleabel{int17}
\hat{{\cal U}}_j^{(\pm)}=
\exp\langleeft(-\frac{i\hat{{\cal H}}_j^{(\pm)}t}{\hbar}\rangleight)\,,
\end{eqnarray}
which is given for $k\geq 0$ by the formula
\betaegin{eqnarray}
\langleabel{int18}
\hat{{\cal U}}_j^{(+)}&=&
\sum_{n=0}^{\infty}\hat{a}^{\dag}s\langleeft(\frac{|\Omega_{j}^{n,k}|t}{2}\rangleight)
\Bigg[\betaigg(
|e_j\rangle\langle e_j|\otimes |n+|k|\rangle\langle n+|k||\betaigg)+
\betaigg(|g_j\rangle\langle g_j|\otimes |n\rangle\langle n|\betaigg)\Bigg]
\nonumber\\
&-&i\sum_{n=0}^{\infty}\sin\langleeft(\frac{|\Omega_{j}^{n,k}|t}{2}\rangleight)
\Bigg[
\betaigg(|e_j\rangle\langle g_j|\otimes |n+|k|\rangle\langle n|\betaigg)e^{-i\tilde{\phi}_j}+
\betaigg(|g_j\rangle\langle e_j|\otimes |n\rangle\langle n+|k||\betaigg)e^{i\tilde{\phi}_j}
\Bigg]
\nonumber\\
&+&\sum_{n=0}^{|k|-1}|e_j\rangle\langle e_j|\otimes |n\rangle\langle n|
\end{eqnarray}
and for ${k<0}$
\betaegin{eqnarray}
\langleabel{int19}
\hat{{\cal U}}_j^{(-)}&=&
\sum_{n=0}^{\infty}\hat{a}^{\dag}s\langleeft(\frac{|\Omega_{j}^{n,k}|t}{2}\rangleight)
\Bigg[\betaigg(
|e_j\rangle\langle e_j|\otimes |n\rangle\langle n|\betaigg)+
\betaigg(|g_j\rangle\langle g_j|\otimes |n+|k|\rangle\langle n+|k||
\betaigg)\Bigg]
\nonumber\\
&-&i\sum_{n=0}^{\infty}\sin\langleeft(\frac{|\Omega_{j}^{n,k}|t}{2}\rangleight)
\Bigg[
\betaigg(|e_j\rangle\langle g_j|\otimes |n\rangle\langle n+|k||\betaigg)e^{-i\tilde{\phi}_j}+
\betaigg(|g_j\rangle\langle e_j|\otimes |n+|k|\rangle\langle n|\betaigg)e^{i\tilde{\phi}_j}
\Bigg]
\nonumber\\
&+&\sum_{n=0}^{|k|-1}|g_j\rangle\langle g_j|\otimes |n\rangle\langle n|\,.
\end{eqnarray}
We have denoted $\tilde{\phi}_j=\phi_j-\frac{\pi}{2}|k|$.
\langleabel{phase}
For each value of $k$ the phase factor $\tilde{\phi}_j$ can be chosen
arbitrarily for the first application of $\hat{{\cal U}}_j^{(\pm)}$.
However, once chosen, it must be kept track of if subsequent applications of
$\hat{{\cal U}}^{(\pm)}_{j}$ are performed on the $j$th ion \cite{98-5}.
The real parameter $|\Omega_{j}^{n,k}|$ is called
the {\it Rabi frequency} of the transition
$|e_j\rangle|n\rangle\langleeftrightarrow|g_j\rangle|n+|k|\rangle$ or
$|e_j\rangle|n+|k|\rangle\langleeftrightarrow|g_j\rangle|n\rangle$, respectively. This term comes
originally from the field of nuclear magnetic resonance (NMR),
where it refers to the
periodic flipping of a nuclear spin in the magnetic field.
It follows that:
\betaegin{eqnarray}gin{itemize}
\item A $4\pi$-pulse ($|\Omega_{j}^{n,k}|t=4\pi$)
returns the system back to its initial state. For example
\betaegin{eqnarray}
|e_j\rangle|n\rangle\stackrel{4\pi}{\langleongrightarrow}|e_j\rangle|n\rangle\,.
\end{eqnarray}
\item A $2\pi$-pulse ($|\Omega_{j}^{n,k}|t=2\pi$) changes the sign of the
state. For instance
\betaegin{eqnarray}
|g_j\rangle|n+|k|\rangle\stackrel{2\pi}{\langleongrightarrow}-|g_j\rangle|n+|k|\rangle\,.
\end{eqnarray}
\item A $\pi$-pulse ($|\Omega_{j}^{n,k}|t=\pi$) implies that
\betaegin{eqnarray}
|e_j\rangle|n+|k|\rangle\stackrel{\pi}{\langleongrightarrow}|g_j\rangle|n\rangle\,,
\end{eqnarray}
where we set the phase factor $\tilde{\phi}_j$ to be zero.
Other cases may be easily calculated from the formulas
(\rangleef{int18}) and (\rangleef{int19}) given above.
\end{itemize}
In what follows we will assume that all motional modes are
in the {\it Lamb-Dicke regime} characterized by the {\it Lamb-Dicke limit}
(Appendix \rangleef{ldl}). Hence $\eta_j$ introduced in Eq. (\rangleef{int5}) is called
the {\it Lamb-Dicke parameter}. The Lamb-Dicke regime facilitates the ground
state cooling (Sec.\,\rangleef{sideband}) and enables to maintain the contrast of
Rabi oscillations on a longer time scale [see Eq.\,(\rangleef{eit2})].
Then the coupling constant (\rangleef{int15}) simplifies to the form
\betaegin{eqnarray}
\langleabel{int20}
\Omega_j^{n,k}\alphapprox\langleambda_j
\langleeft(i\eta_j\rangleight)^{|k|}
\sqrt{\frac{(n+|k|)!}{n!}}\,\frac{1}{|k|!}\,.
\end{eqnarray}
For the purpose of coherent manipulations with internal states of cold
trapped ions we will be primarily interested in the interaction on the carrier
($k=0$) and on the first red sideband ($k=-1$) which will be used
for the~construction of a wide class of quantum logic
gates. The corresponding unitary evolution operators in the Lamb-Dicke
regime for the transition on the carrier $(\hat{{\cal A}})$
and on the first red sideband $(\hat{{\cal B}})$
may be determined from Eq. (\rangleef{int18}) and (\rangleef{int19}) as follows
\betaegin{eqnarray}
\langleabel{int21}
\hat{{\cal A}}_j&=&
\sum_{n=0}^{\infty}\hat{a}^{\dag}s\langleeft(\frac{|A_j^n|t}{2}\rangleight)
\Bigg[
\betaigg(|e_j\rangle\langle e_j|\otimes |n\rangle\langle n|\betaigg)+
\betaigg(|g_j\rangle\langle g_j|\otimes |n\rangle\langle n|\betaigg)
\Bigg]\\
&-&i\sum_{n=0}^{\infty}\sin\langleeft(\frac{|A_j^n|t}{2}\rangleight)
\Bigg[
\betaigg(|e_j\rangle\langle g_j|\otimes |n\rangle\langle n|\betaigg)e^{-i\phi_j}+
\betaigg(|g_j\rangle\langle e_j|\otimes |n\rangle\langle n|\betaigg)e^{i\phi_j}
\Bigg]
\nonumber
\end{eqnarray}
and
\betaegin{eqnarray}
\langleabel{int22}
\hat{{\cal B}}_j&=&
\sum_{n=0}^{\infty}\hat{a}^{\dag}s\langleeft(\frac{|B_j^n|t}{2}\rangleight)
\Bigg[
\betaigg(|e_j\rangle\langle e_j|\otimes |n\rangle\langle n|\betaigg)+
\betaigg(|g_j\rangle\langle g_j|\otimes |n+1\rangle\langle n+1|\betaigg)
\Bigg]\\
&-&i\sum_{n=0}^{\infty}\sin\langleeft(\frac{|B_j^n|t}{2}\rangleight)
\Bigg[
\betaigg(|e_j\rangle\langle g_j|\otimes |n\rangle\langle n+1|\betaigg)e^{-i\tilde{\phi}_j}+
\betaigg(|g_j\rangle\langle e_j|\otimes |n+1\rangle\langle n|\betaigg)e^{i\tilde{\phi}_j}
\Bigg]
\nonumber\\
&+&|g_j\rangle\langle g_j|\otimes |0\rangle\langle 0|\,.\nonumber
\end{eqnarray}
The respective Rabi frequencies in the Lamb-Dicke limit [Eq. (\rangleef{int20})]
for $k=0$ and $k=-1$ are given by
\betaegin{eqnarray}
\langleabel{conA}
|A_j^n|&=&|\langleambda_j|\,,\\
\langleabel{int24}
\langleabel{conB}
|B_j^n|&=&|\langleambda_j|\eta_j\sqrt{n+1}\,.
\end{eqnarray}
We could by analogy obtain evolution operators for other
sideband transitions.
\subsection{Standing-wave configuration}
As an alternative approach to the laser-ion interactions we could choose
a standing light field (FIG.\,\rangleef{stan}).
One can place a mirror in the setup and let the
laser beam reflect from it. The counter propagating waves interfere and create
a standing-wave configuration with nodes and antinodes. However, it is
experimentally very demanding to place an ion precisely
to a node or an antinode. Let us approximate
the incident laser beam as a monochromatic travelling wave
\betaegin{eqnarray}
\langleabel{int25}
{\betaf E}_i=E_0\betaoldsymbol{\epsilon}
\hat{a}^{\dag}s\betaig(\omega_Lt-\betaoldsymbol{\kappa}\cdot{\betaf q}+\phi\betaig)
\end{eqnarray}
and the reflected beam as a counter propagating travelling wave
\betaegin{eqnarray}
\langleabel{int26}
{\betaf E}_r=E_0\betaoldsymbol{\epsilon}
\hat{a}^{\dag}s\betaig(\omega_Lt+\betaoldsymbol{\kappa}\cdot{\betaf q}+\phi-\pi\betaig)\,,
\end{eqnarray}
where the reflected wave acquires an additional phase $\pi$ on the reflection
at the perfect lossless mirror. Then we can write for the resulting
standing wave
\betaegin{eqnarray}
\langleabel{int27}
{\betaf E}={\betaf E}_i+{\betaf E}_r=2E_0\betaoldsymbol{\epsilon}
\sin\betaig(\omega_Lt+\phi\betaig)\sin\betaig(\betaoldsymbol{\kappa}\cdot{\betaf q}\betaig)\,,
\end{eqnarray}
where the notation is adopted from Eq. (\rangleef{int2}).
Following Eq. (\rangleef{int4}) and (\rangleef{int4.1})
we can write the corresponding relations for the standing wave
in the semiclassical representation
\betaegin{eqnarray}
\langleabel{int28}
{\betaf E}(t,\hat{{\betaf R}}_j)=-iE_0\betaoldsymbol{\epsilon}
\langleeft[e^{i(\omega_Lt+\phi)}-e^{-i(\omega_Lt+\phi)}\rangleight]
\sin\langleeft[\chi_j+\eta_j(\hat{a}^{\dag}+\alphao)\rangleight]\,,
\end{eqnarray}
and
\betaegin{eqnarray}
\langleabel{int29}
\frac{\partial E_{b}(t,\hat{{\betaf R}}_j)}{\partial q_{a}}=
-iE_0\kappa_{a}\epsilon_{b}
\langleeft[e^{i(\omega_Lt+\phi)}-e^{-i(\omega_Lt+\phi)}\rangleight]
\hat{a}^{\dag}s\langleeft[\chi_j+\eta_j(\hat{a}^{\dag}+\alphao)\rangleight]\,,
\end{eqnarray}
where the new parameter $\chi_j=\kappa_{\vartheta}\betaar{z}_j$
determines the position of the $j$th ion in the standing wave
and $\kappa_{a}=(\omega_L/c)n_{a}$. The notation is adopted from
Eq. (\rangleef{int2}) and (\rangleef{int5}).
The condition $\chi_j=0$ refers to the $j$th ion placed in the node, whereas
$\chi_j=\pi/2$ refers to the ion positioned in the antinode of the
standing wave.
\betaegin{eqnarray}gin{figure}[htb]
\centerline{\epsfig{width=10cm,file=ions56.eps}}
\caption{{\footnotesize
The standing-wave configuration corresponds to illuminating
the $j$th ion in the ion crystal with the laser beam of the frequency
$\omega_L$ at the angle $\vartheta$ to the trap axis.}}
\langleabel{stan}
\end{figure}
Following the derivation for the travelling-wave configuration,
one can easily derive the Hamiltonian in
the interaction picture for the standing-wave configuration. It takes the
form of the expressions (\rangleef{int10}) and (\rangleef{int11}), except that
the laser coupling constant $\langleambda_j$ and
the operator function ${\cal F}_k$ are
replaced with $\tilde{\langleambda}_j$ and $\tilde{\cal F}_k$. They are given
in the dipole approximation (assuming a dipole allowed transition) by
\betaegin{eqnarray}
\langleabel{int32}
\tilde{\langleambda}^{D\!P}_j&=&
-\frac{i2q_eE_{0}}{\hbar}
\betaigg[({\betaf r}_{eg})_j\cdot\betaoldsymbol{\epsilon}\betaigg]\,e^{-i\phi}\,,\\
\langleabel{int33}
\tilde{{\cal F}}_{k}^{D\!P}(\hat{a}^{\dag}\alphao)&=&e^{-(\eta_j^2/2)}
\sin\betaigg(\chi_j+\frac{\pi}{2}|k|\betaigg)
\eta_j^{|k|}
\sum_{\alpha=0}^{\infty}
\langleeft(-\eta_j^2\rangleight)^{\alpha}
\frac{(\hat{a}^{\dag}\alphao)^{\alpha}}{\alpha!(\alpha+|k|)!}
\end{eqnarray}
and in the quadrupole approximation
(assuming a dipole forbidden transition) by
\betaegin{eqnarray}
\langleabel{int34}
\tilde{\langleambda}^{Q\!D}_j&=&
-\frac{iq_eE_0\omega_L}{\hbar c}
\betaigg[\sum_{a,b}
\langle e_j|(\hat{{\betaf r}}_j)_{a}(\hat{{\betaf r}}_j)_{b}|g_j\rangle n_{a}\epsilon_{b}
\betaigg]e^{-i\phi}\,,\\
\langleabel{int35}
\tilde{{\cal F}}_{k}^{Q\!D}(\hat{a}^{\dag}\alphao)&=&e^{-(\eta_j^2/2)}
\hat{a}^{\dag}s\betaigg(\chi_j+\frac{\pi}{2}|k|\betaigg)
\eta_j^{|k|}
\sum_{\alpha=0}^{\infty}
\langleeft(-\eta_j^2\rangleight)^{\alpha}
\frac{(\hat{a}^{\dag}\alphao)^{\alpha}}{\alpha!(\alpha+|k|)!}\,.
\end{eqnarray}
Comparing the expressions for the coupling constant in the travelling-wave
configuration [Eq. (\rangleef{int8}) and (\rangleef{int8.1})] with those ones for the
standing-wave configuration [Eq. (\rangleef{int32}) and (\rangleef{int34})],
we find out that $|\tilde{\langleambda}_j^{D\!P}|=2|{\langleambda}_j^{D\!P}|$ and
$|\tilde{\langleambda}_j^{Q\!D}|=2|{\langleambda}_j^{Q\!D}|$. The factor 2 arises
from the expression of the standing wave (\rangleef{int27}) where we have
superposed two travelling waves with equal amplitudes.
Finally, the Hamiltonian can be written in the form given by Eq. (\rangleef{int13})
and (\rangleef{int14}) with the coupling constant
in the dipole approximation
\betaegin{eqnarray}
\langleabel{int36}
(\tilde{\Omega}_j^{n,k})^{D\!P}=
\langleambda_j^{D\!P}\,e^{-(\eta_j^2/2)}
\sin\betaigg(\chi_j+\frac{\pi}{2}|k|\betaigg)
\eta_j^{|k|}
\sqrt{\frac{n!}{(n+|k|)!}}\,L_n^{|k|}(\eta_j^2)
\end{eqnarray}
and in the quadrupole approximation
\betaegin{eqnarray}
\langleabel{int37}
(\tilde{\Omega}_j^{n,k})^{Q\!D}=
\langleambda_j^{Q\!D}\,e^{-(\eta_j^2/2)}
\hat{a}^{\dag}s\betaigg(\chi_j+\frac{\pi}{2}|k|\betaigg)
\eta_j^{|k|}
\sqrt{\frac{n!}{(n+|k|)!}}\,L_n^{|k|}(\eta_j^2)\,.
\end{eqnarray}
It follows from Eq. (\rangleef{int36}) that for the $j$th ion
in the dipole approximation
placed in the node of the standing wave ($\chi_j=0$)
only transitions on odd sidebands ($|k|=2p+1$) are present.
For the same ion in the antinode
($\chi_j=\pi/2$) only even sidebands ($|k|=2p$) are present,
where $p$ is an integer or the zero.
In the quadrupole approximation
the statements above are valid in the opposite order
[compare Eq. (\rangleef{int36}) with (\rangleef{int37})].
The reason for missing transitions in the standing-wave
configuration comes from the destructive interference between the two
counter propagating travelling waves in the standing-wave field.
We could easily write the coupling constant $\tilde{\Omega}^{n,k}_j$ in the
Lamb-Dicke limit [see Eq. (\rangleef{int20})]. We could also write
the unitary evolution operator for the standing-wave configuration. However,
it differs from the evolution operator in the travelling-wave configuration
[Eq. (\rangleef{int18}) and (\rangleef{int19})]
only in the coupling constant and in the phase
factor, but it produces no fundamental problem for further applications.
Therefore, in what follows we will
consider the expressions and formulas for the travelling-wave configuration
keeping in mind the way how to convert to a standing-wave configuration.
\section{Laser cooling}
\langleabel{lc}
{\it Laser cooling} is the process in which the kinetic energy of atoms is
reduced through the action of one or more laser beams. The last decade brought
rapid progress in this research field and this effort culminated in 1997
with the award of the Nobel Prize in physics for laser cooling and trapping
of atoms \cite{98-8,98-9,98-10}. A recent review of different experimental
techniques for laser cooling can be found in Ref. \cite{metcalf}.
One of the requirements for the practical implementation of quantum
computing is the ability to prepare well defined initial states
of the qubits \cite{divin}. In our case the qubits are represented by
trapped ions with vibrational (external) and atomic (internal) degrees of
freedom. {\it Laser cooling} enables the preparation of well defined
initial states of motion and {\it electron shelving} serves for
the proper initialization of the ion register. We will describe
this method later on in Sec.\,\rangleef{es}.
Laser cooling of trapped ions with the axial trapping frequency $\omega_z$
has two stages depending on the linewidth $\Gamma$
of the cooling transition \cite{nagerl,roos}:
\betaegin{eqnarray}gin{itemize}
\item {\it Doppler cooling} is applied when the vibrational frequency of the
ions is smaller than the linewidth of a transition used for cooling
($\omega_z\langleeq\Gamma$). In other words, this means that the velocity
of the ion due to the trapping potential changes on a longer time scale
than the time it takes the ion to absorb or emit a photon (strong laser
driving is assumed). Therefore, we can assume that these processes
change the momentum of the ion instantaneously. For $\omega_z\langleeq\Gamma$ we
refer to the weak confinement regime (in the sense of weak binding of
the~ions to the ion trap).
\item {\it Sideband cooling} is used for further cooling below the Doppler
cooling limit and requires the vibrational frequency to be much bigger
than the linewidth ($\omega_z\gg\Gamma$). Under this condition the
ion develops well resolved sidebands and cooling to a lowest vibrational
state is realized through driving a lower sideband. For $\omega_z\gg\Gamma$
we refer to the strong confinement regime. One can use instead a novel
technique called {\it laser cooling using electromagnetically induced
transparency}.
\end{itemize}
\subsection{Doppler cooling}
This stage of laser cooling is based on the Doppler effect.
The technique is
based on the fact that moving atoms absorb photons from a counter propagating
red detuned laser beam (tuned slightly below the atomic
frequency) and emit spontaneously in a random direction. After several
such cooling cycles (absorption followed by spontaneous emission) we
can write for the total momentum {\betaf p} of atoms
\betaegin{eqnarray}
\langleabel{dopp1}
{\betaf p}={\betaf p}_0+
\sum_j\hbar{\betaf k}_j^{(abs)}+\sum_j\hbar{\betaf k}_j^{(em)}\,,
\end{eqnarray}
where ${\betaf p}_0$ is the initial momentum of atoms, ${\betaf k}_j^{(abs)}$ and
${\betaf k}_j^{(em)}$ denotes the wavevectors of the absorbed and emitted
photons in the $j$th cooling cycle.
We usually use a fast decaying dipole transition for the Doppler
cooling, therefore the spontaneous emission is much faster than
stimulated emission. The average total momentum of atoms after many cooling
cycles takes the form
\betaegin{eqnarray}
\langleabel{dopp2}
\langle {\betaf p}\rangle={\betaf p}_0+\hbar{\betaf k}_L\langle n\rangle\,,
\end{eqnarray}
where $\langle n\rangle$ is the average number of absorption and emission events
(typically $\langle n\rangle\simeq 10^3-10^4$) and by the definition
${\betaf k}_L={\betaf k}_j^{(abs)}$ (${\betaf k}_L$ is the wavevector associated with
the laser light). The spontaneous contribution averages to zero because it
is randomly distributed over the solid angle $4\pi$. If the laser is red
detuned and counter propagating to the motion of atoms
(${\betaf p}_0\uparrow\downarrow{\betaf k}_L$), then the velocity of the atoms is
significantly decreased [see Eq. (\rangleef{dopp2})]. For more details see
ref. \cite{LesH, metcalf}.
The discussion above is valid for free atoms but it also applies for trapped
ions \cite{86-1}, where the motion towards the laser is provided by the
periodic vibrations. The Doppler cooling limit corresponds to the final
temperature $T_{dopp}=\hbar\Gamma/2k$, where $\Gamma$ is
the linewidth of the cooling transition and $k$ is the Boltzmann constant. This
temperature is typically of the order of mK \cite{zeil}.
However, the Doppler cooling limit can be also translated into
the minimum average phonon number in the axial
direction \cite{LesH}
\betaegin{eqnarray}
\langleabel{dopp3}
\langle n_z\rangle_{min}=\frac{\Gamma}{\omega_z}
\langleeft(\frac{1+\alpha}{4}\rangleight)
\langleeft(\frac{\Gamma}{\delta}+\frac{\delta}{\Gamma}\rangleight)-\frac{1}{2}\,,
\end{eqnarray}
where $\Gamma$ is the natural linewidth of the cooling transition,
$\alpha$ is determined from the angular distribution of the emitted radiation
and $\delta$ is the laser detuning from the atomic frequency.
For a dipole radiation pattern we get $\alpha=2/5$.
The cooling is optimal for
the detuning $\delta=\Gamma\gg\omega_z$. Concerning this condition
we can rewrite
Eq. (\rangleef{dopp3}) for a dipole transition to the form
\betaegin{eqnarray}
\langleabel{dopp4}
\langle n_z\rangle_{min}\simeq\frac{7}{10}\frac{\Gamma}{\omega_z}\,.
\end{eqnarray}
We have omitted the factor 1/2 corresponding to
the zero point energy because it has a negligible contribution.
The~Doppler cooling limit is associated with the recoil of the atoms
at the spontaneous emission.
\betaegin{eqnarray}gin{figure}[htb]
\centerline{\epsfig{width=8cm,file=ions61.eps}}
\caption{{\footnotesize
Doppler cooling. The $S_{1/2}\langleeftrightarrow P_{1/2}$ is
the~cooling transition. The~cooling laser on 397\,nm is red detuned by
10\,MHz. The ion spontaneously decays from the $P_{1/2}$
level. There is a $6\%$ probability of the decay to the metastable
state $D_{3/2}$, therefore the pumping laser on 866\,nm is switched on.}}
\langleabel{dopp}
\end{figure}
The relation (\rangleef{dopp4})
leads us to a discussion of how to choose the axial trapping frequency
$\omega_z$. In order to be able to address individually each ion with
a single laser beam, the minimum spacing between the ions
[see Eq. (\rangleef{vibr9})] has to be large enough, which requires small
$\omega_z$. On the other hand,
we do not want the frequency $\omega_z$ to be too small
in order to make the result of Doppler cooling as efficient as possible
[Eq. (\rangleef{dopp4})]. Thus, the design of ion traps is also determined
by the trade-off between these two options.
For Calcium ions $^{40}\mbox{Ca}^+$ the $S_{1/2}\langleeftrightarrow P_{1/2}$
transition with the natural linewidth
$\Gamma=20\,\mbox{MHz}$ is used for the Doppler cooling.
The lifetime of the $P_{1/2}$ level is about 7\,ns and
this level decays with $6\%$ probability to the metastable $D_{3/2}$
level (FIG.\,\rangleef{dopp}).
Therefore, optical pumping between the $P_{1/2}$ and $D_{3/2}$
levels is present on 866\,nm. The laser on the cooling transition
$S_{1/2}\langleeftrightarrow P_{1/2}$ is red detuned by
$\Gamma/2\simeq 10\mbox{MHz}$.
On the other hand, the pumping laser at 866nm is kept on the resonance in
order to prevent population trapping in the superposition of the
$S_{1/2}$ and $D_{3/2}$ levels \cite{roos}. For $\alpha=2.5$,
$\omega_z/2\pi\simeq 700\,\mbox{kHz}$ and $\delta=\Gamma/2$ we can
calculate from Eq. (\rangleef{dopp3}) the minimum average phonon number to be
$\langle n_z\rangle_{min}\simeq 3.5$. This number differs from
experimentally measured values (which are bigger)
because Eq. (\rangleef{dopp3}) has been
derived for a two-level system while the experimental realization of Doppler
cooling involves a three-level system. Nevertheless,
$\langle n_z\rangle_{min}\simeq 3.5$ is still not $\langle n_z\rangle_{min}=0$
required for proper operation of the quantum processor with cold trapped
ions. Therefore a second cooling stage must be launched.
\subsection{Sideband cooling}
\langleabel{sideband}
Doppler cooling represents the precooling stage
in experiments with trapped
ions. The final stage can be realized by the sideband cooling technique which
may prepare the ions to the ground motional state, i.e. a well defined
initial quantum state. Firstly, we address the basic idea of sideband
cooling. Then we illustrate this cooling technique on two trapped ions in
and outside the Lamb-Dicke regime.
Finally, we describe how sideband cooling is realized experimentally.
In the strong confinement
regime ($\omega_z\gg\Gamma$) a single trapped ion exhibits in
its absorption spectrum well resolved sidebands at
$\omega_0\pm k\omega_z$ ($k$ is an integer) spaced on both sides of
the carrier on the atomic frequency $\omega_0$. Sideband cooling occurs
when the cooling laser is tuned to a lower sideband at
$\omega_L=\omega_0-k\omega_z$. In the Lamb-Dicke limit cooling works
efficiently with the laser tuned on the first red sideband at
$\omega_L=\omega_0-\omega_z$. Then the ion absorbs photons of the energy
$\hbar(\omega_0-\omega_z)$ and spontaneously emitted photons of the average
energy $\hbar\omega_0-E_r$ bring the ion back to its initial internal state
(see Appendix \rangleef{ldl}). In every cooling cycle (absorption + emission) the
motional energy of the ion is damped by one vibrational quantum if
$\hbar\omega_z\gg E_r$. This condition implies that in this form
the sideband cooling requires the ion to be in the Lamb-Dicke limit. The whole
process consists of cooling cycles in which the absorption is followed by
the spontaneous emission until the ion reaches the ground motional state
$|n=0\rangle$ and decouples from the cooling laser.
For the sideband cooling of the single trapped ion initially
in the internal state $|g\rangle$ and in the
motional state $|n\rangle$, where $a$ denotes the absorption and $e$ stays for
the spontaneous emission, we may schematically write
\betaegin{eqnarray}
|g\rangle|n\rangle\ \stackrel{a}{\rangleightarrow}\ |e\rangle|n-1\rangle\ \stackrel{e}{\rangleightarrow}
|g\rangle|n-1\rangle\ \stackrel{a}{\rangleightarrow}\ \dots\ \stackrel{e}{\rangleightarrow}
|g\rangle|1\rangle\ \stackrel{a}{\rangleightarrow}|e\rangle|0\rangle\ \stackrel{e}{\rangleightarrow}
|g\rangle|0\rangle\,.\nonumber
\end{eqnarray}
The minimum average phonon number in the axial direction that can be
reached by sideband cooling is then given by \cite{LesH}
\betaegin{eqnarray}
\langleabel{side1}
\langle n_z\rangle_{min}=\langleeft(\frac{\Gamma}{\omega_z}\rangleight)^2
\langleeft(\alpha+\frac{1}{2}\rangleight)\,,
\end{eqnarray}
where the parameter $\alpha$ has been defined in Eq. (\rangleef{dopp1}).
It is evident that now one can
achieve efficient cooling to the ground motional state, i.e.
$\langle n_z\rangle_{min}\simeq 0$, assuming the strong confinement regime
($\omega_z\gg\Gamma$). The limit for the sideband cooling
[Eq. (\rangleef{side1})]
is constrained by the recoil of the ion and is determined by the equilibrium
between cooling and heating processes. Heating is caused mainly by
off-resonant excitations on the carrier ($|g\rangle|n\rangle\langleeftrightarrow|e\rangle|n\rangle$) and
on the first blue sideband ($|g\rangle|n\rangle\langleeftrightarrow|e\rangle|n+1\rangle$).
The sideband cooling of a single ion beyond the Lamb-Dicke limit also exists and is
based on the creation of a dark state in the energy level structure
\cite{morigi0}.
A single trapped Mercury $^{198}\mbox{Hg}^+$
ion was firstly cooled to the ground motional state in 1989 in Boulder
\cite{89}, while sideband cooling of a single Beryllium ion $^9\mbox{Be}^+$
in all three dimensions was firstly reported in 1995 also
by the group in Boulder \cite{95-11}.
\subsubsection{Sideband cooling of two ions}
The key difference between one and more ions lies in the energy spectrum
\cite{oxf, 98-7}.
While single ions have discrete energy levels (three motional degrees of
freedom), a chain of $N$ oscillating ions ($3N$ motional degrees of freedom)
exhibits a quasicontinuous energy spectrum due to the incommensurate
frequencies of the motional modes.
For instance, in the axial direction we have the frequencies
$\omega_z,\,\omega_z\sqrt{3},\,\omega_z\sqrt{5.8},\,\omega_z\sqrt{9.3}$,
etc.
\betaegin{eqnarray}gin{figure}[htb]
\centerline{\epsfig{width=10cm,file=ions55.eps}}
\caption{{\footnotesize
Absorption spectrum $I(\delta)$ of two ions for a thermal distribution
$P({\betaf n})$ in the Lamb-Dicke regime (a) and outside
the Lamb-Dicke regime (b) as a function of detuning $\delta$ (in units of the
axial vibrational frequency $\omega_z$) where $I(\delta)$ is defined
in Eq.~(\rangleef{abs}) and $\delta$
in Eq.~(\rangleef{int9.0}).
Used by kind permission of Giovanna Morigi and
J\"{u}rgen Eschner \cite{morigi}.}}
\langleabel{LDL}
\end{figure}
We will discuss in detail the case of two trapped ions to illustrate
the situation of sideband cooling of more than a single ion \cite{morigi}.
The state of each ion ($j=1,2$) will be expressed in the basis
$\{|g_j\rangle|{\betaf n}\rangle, |e_j\rangle|{\betaf n}\rangle\}$, where
${\betaf n}=(n_1,n_2)$, $n_1$ is the vibrational number associated with the COM
mode ($\nu_1=\omega_z$) and $n_2$ with the breathing mode
($\nu_2=\omega_z\sqrt{3}$). The absorption spectrum of the $j$th ion
will be considered in the form \cite{comm1}
\betaegin{eqnarray}
\langleabel{abs}
I_j(\delta)=\sum_{E_{\betaf n}-E_{\betaf m}=\delta}
|\langle {\betaf n}|\exp{(i\Delta_j\kappa_{\vartheta})|{\betaf m}\rangle|^2P({\betaf n})}\,,
\end{eqnarray}
where $E_{\betaf n}=\hbar\omega_zn_1+\hbar\omega_z\sqrt{3}n_2$, $\delta$ is
the detuning [Eq. (\rangleef{int9.0})], $\kappa_{\vartheta}$ is defined
by Eq. (\rangleef{int5})
and $\Delta_j$ is the displacement operator of the $j$th ion
[Eq. (\rangleef{vibr29})].
$P({\betaf n})=P(n_1,n_2)$ is a probability distribution associated with
the vibrational motion of the ions.
The Lamb-Dicke parameter distinguishes between two very different
regimes of sideband cooling of more ions:
\betaegin{eqnarray}gin{itemize}
\item
In the Lamb-Dicke regime (Appendix \rangleef{ldl}) and
in the strong confinement regime ($\omega_z\gg\Gamma$)
only the first sidebands of the motional modes
at $\omega_0\pm\omega_z$ and $\omega_0\pm\omega_z\sqrt{3}$ appear around the
significant carrier peak at $\omega_0$ in the absorption spectrum
[FIG.\,\rangleef{LDL}(a)].
The higher sidebands are suppressed due to their strength being proportional to
higher powers in the Lamb-Dicke parameter than $\eta$ denoted as
${\cal O}(\eta^2)$. Tuning the laser on the first red sideband of the COM
mode ($\delta=-\omega_z$) we can reach its ground state $|n_1=0\rangle$
at the same cooling rate as for a single ion \cite{morigi}. However,
in the case of two ions the breathing mode is decoupled from the COM mode
and its cooling is almost frozen.
Simultaneous cooling of more modes requires the modes to be coupled to
the cooling laser and then the requirement of the
strong coupling regime ($\omega_z\gg\Gamma$) has to be reconsidered or one
has to use alternative techniques.
\item
Outside the Lamb-Dicke regime higher sidebands
with the strength proportional to ${\cal O}(\eta^2)$ also
contribute and the absorption
spectrum exhibits the structure with many overlapping sidebands
at $\omega_0\pm k\omega_z\pm l\omega_z\sqrt{3}$ where $k, l$
are integers [FIG.\,\rangleef{LDL}(b)].
In this situation the laser tuned on a lower sideband
(it does not have to be strictly the first red sideband of the COM mode)
excites simultaneously all sideband transitions around this lower sideband
in the interval of the linewidth $\Gamma$. Then the COM and the breathing mode
are coupled and cooled at once.
However, the cooling process is much slower
in comparison to cooling of a single ion beyond the Lamb-Dicke limit. It is
partly caused by (i) the increasement of the number of the motional modes
but also by (ii) the appearance of dark states \cite{morigi}. The dark
states are almost decoupled from a resonantly excited state because their
motional wave function after the absorption overlaps with the motional
wavefunction of the excited state only a little. Thus, the ions may be
trapped in these dark states and it slows the cooling process down. This
problem can be solved by escaping from the strong confinement regime
($\omega_z\gg\Gamma$), i.e. by increasing the linewidth $\Gamma$.
It will cause that a single level will be coupled to more levels (more
sidebands are in the resonance) and the ions will be cooled more efficiently
due to more cooling channels. As a result the dark states will
disappear because more channels provide more ways for the ion to escape from
dark (population trapping) states. Moreover, the rate of the cooling
cycles (absorption + emission) is proportional to the linewidth $\Gamma$.
Summarizing both effects we can conclude that the total cooling time beyond
the Lamb-Dicke limit can be shortened significantly for $\Gamma\simeq\omega_z$.
\end{itemize}
\subsubsection{Experimental sideband cooling}
Two ions were cooled for the first time to the ground motional state in 1998
in Boulder.
It was achieved on Beryllium ions $^9\mbox{Be}^+$ illuminating both
ions at once \cite{king}. However, it is sufficient to illuminate only one
ion from the entire ion string because other ions are cooled sympathetically
due to the strong Coulomb coupling. Although we need only one motional mode
(COM or breathing axial mode) as the quantum data bus, which has to be in
the ground motional state, we require also other
modes to be cooled close to the ground state. Uncooled motional modes with
thermal phonon distributions significantly affect the Rabi frequency in the
data mode and spoil the fidelity of the coherent state manipulation
[see Eq. (\rangleef{eit2})].
\betaegin{eqnarray}gin{figure}[htb]
\centerline{\epsfig{width=6.5cm,file=ions62.eps}}
\caption{{\footnotesize
Sideband cooling with cooling and pumping transitions.
The blue transitions correspond to the cooling cycle.
The~laser at 729\,nm is tuned on the first red sideband. The laser at 854\,nm
couples the metastable $D_{5/2}$ level with the fast decaying $P_{3/2}$
level. The cooling cycle is closed by the spontaneous emission
on the~fast $S_{1/2}\langleeftrightarrow P_{3/2}$ dipole transition.
The $P_{3/2}$ level may
decay to the state $D_{3/2}$. The~ion is recycled to the state
$P_{1/2}$ by the laser at 866\,nm followed by the spontaneous emission back
to the state $S_{1/2}$. Decay to the sublevels $S_{1/2}$($m$=+1/2) is
counteracted by driving the~$S_{1/2}\langleeftrightarrow P_{1/2}$ transition with
the $\sigma^-$ polarized laser at 397\,nm.}}
\langleabel{side}
\end{figure}
The group in Innsbruck has realized different approaches in sideband cooling
of two Calcium ions \cite{blatt1, blatt2}. If they cool only one motional
mode, while the other modes are left in the thermal states, they achieve the
ground state population greater than 95\% ($\langle n\rangle\simeq 0.05$) in the
respective mode. However, they can cool sequentially all motional modes
close to the ground state. For this purpose they use a small modification
in the sideband cooling scheme. The laser frequency and laser power has
to be set
sequentially for the respective first red sideband of the given motional
mode. After sequential cooling all modes the corresponding average
phonon numbers are from $\langle n\rangle\simeq 0.05$ to $\langle n\rangle\simeq 2.3$ because
the recoil energy from the spontaneous emission in the cooling process of
one motional mode reheat other modes.
For Calcium ions $^{40}\mbox{Ca}^+$ the quadrupole transition between
the two Zeeman sublevels $S_{1/2} (m_J=-1/2)$ and $D_{5/2} (m_J=-5/2)$ is used
for the sideband cooling (FIG.\,\rangleef{side}).
The laser on 729\,nm is tuned on the first red sideband of the respective
motional mode and a weak magnetic field is applied for Zeeman splitting
of energy levels. The lifetime of the metastable $D_{5/2}$
level is about a second,
therefore there is the pumping on 854\,nm to the fast decaying
$P_{3/2} (m_J=-3/2)$ level in order to decrease the duration of one cooling
cycle, i.e. to increase the cooling rate. The $P_{3/2} (m_J=-3/2)$ level
decays spontaneously to the initial state $S_{1/2} (m_J=-1/2)$ and closes
the cooling cycle. However, the $P_{3/2}$ level may decay with a small
probability to the $D_{3/2}$ level, therefore the pumping laser on 866\,nm
recycles the population to the $P_{1/2}$ state which decays to the $S_{1/2}$
state. The $S_{1/2}\langleeftrightarrow P_{1/2}$ transition is driven with
the $\sigma^-$ polarized laser on 397\,nm to counteract the population of
the $S_{1/2} (m_J=+1/2)$ level \cite{roos}.
\subsection{Sympathetic cooling}
\langleabel{sympcool}
In the previous section we have mentioned that it is sufficient to
illuminate with cooling lasers only one ion from the ion string because
the other ions are cooled sympathetically due to the Coulomb interaction
between them. Hence the term {\it sympathetic cooling}. However, instead of
identical ions one can consider different atomic species (eventually
isotopes) in the ion crystal \cite{symp2}.
Then the addressing of {\it cooling} ions avoids the disturbance
of internal states of {\it logic} ions which store the information
\cite{dfs}.
\betaegin{eqnarray}gin{figure}[h!]
\centerline{\epsfig{width=10.5cm,file=ions63.eps}}
\caption{{\footnotesize
Normalized axial frequencies (\rangleef{vibr21}) as a function of $\zeta$
for (a) three, (b) five, (c) seven and (d) nine ions.
Used by kind permission of David Wineland and David Kielpinski \cite{symp1}.}}
\langleabel{symp1}
\end{figure}
Electric fields from the trap electrodes are one of the sources of
the motional decoherence of the ion crystal due to heating of collective
vibrational motional modes (normal modes).
If one assumes the dimension of the ion trap to be much larger
than the dimension of the ion crystal, then we can expect the electrode
electric fields to be nearly uniform across the ion crystal. Such uniform
fields influence and heat only collective motional modes
involving the centre-of-mass (COM) motion of the ion crystal.
Uniform electric fields can directly heat up the normal mode used for quantum
logic as the quantum data bus. We can overcome this constrain by selecting
a specific normal mode for quantum logic which is decoupled from heating.
However, not all motional modes are prevented from heating. In what follows
we will discuss this point following Ref. \cite{symp1}.
Let us consider
the ion crystal with an odd number $N$ of ions which consists of $N-1$ ions
of mass $m$ and of a central ion of the mass $M$ defining the ratio
$\zeta=M/m$.
Now we can follow the lines in Sec.\,\rangleef{vmoti}
and find the normal modes and frequencies of the ion string with unequal ions.
We find out that (i) there are $(N-1)/2$ axial normal modes
for which the central ion
does not move and corresponding eigenvectors ${\betaf D}^{(\alpha)}$ and eigenfrequencies
$\nu_{\alpha}=\omega_z\sqrt{\mu_{\alpha}}$ do not depend on the parameter
$\zeta$. Moreover, these modes do not have a component associated with the
axial COM motion.
(ii) There are also
other $(N+1)/2$ modes having a component of the COM motion and coupling
to any uniform electric field which causes their heating.
For very small or very large values of $\zeta$ the motional modes
become degenerate and pair up (FIG.\,\rangleef{symp1}).
From this point of view the value $\zeta\simeq 1$ seems to be
suitable. It has also been calculated that those modes having the central
ion at rest (neglecting gradient electric fields) do not heat at all
(we refer to these as {\it cold modes}), while all other motional
modes heat to some extent depending on the value of $\zeta$.
Their heating rate (average number of phonons gained per second) drops
rapidly for $\zeta\to 1$ (FIG.\,\rangleef{symp2}).
We refer to these modes as {\it hot modes}.
Thus, it seems that the optimal choice
is $\zeta\simeq 1$. That means the central (cooling) ion should be chosen
such that it is identical to $N-1$ other (logic) ions or is an isotope
of logic ions. For $\zeta\simeq 1$ we can choose the lowest cold mode
(the second lowest motional mode called breathing mode)
to be used for quantum logic as the quantum data bus because in the case of
$\zeta\simeq 1$ only the lowest motional mode
(corresponding to the COM mode for equal ions) will heat significantly and
can be cooled via the central cooling ion. If the value of $\zeta$ differs
very much from 1, we have to cool all $(N+1)/2$ hot modes via the central ion.
\betaegin{eqnarray}gin{figure}[h!]
\centerline{\epsfig{width=10.5cm,file=ions64.eps}}
\caption{{\footnotesize
Normalized heating rates for hot axial modes as a function of $\zeta$
for (a) three, (b) five, (c) seven and (d) nine ions.
Used by kind permission of David Wineland and David Kielpinski \cite{symp1}.}}
\langleabel{symp2}
\end{figure}
The group in Garching runs experiments where the ion string
consists of Indium ($^{115}\mbox{In}^+$) and Magnesium
($^{25}\mbox{Mg}^+$) ions. The numerical analysis for the ion
crystal containing these two atomic species ordered in different
configuration can be found in Ref. \cite{symp2}.
The mass ratio is $\zeta=4.6$ which is not within the optimal range
discussed above ($\zeta\simeq 1$).
On the other hand, it can quite advantageous because the heavy
ion fulfills the Lamb-Dicke limit (Appendix \rangleef{ldl}) easier
than the light ion. Distinct atomic species can also have
a very different atomic spectrum which may be found convenient when
laser addressing closely spaced ions. However,
for the heavy central cooling ion we pay the price in the form of
heating rates of higher motional modes.
By all means, Indium ions can be efficiently cooled to
the ground motional state \cite{In} and Magnesium ions can serve for quantum
logic operations and storing the information.
Finally, we have to mention that the demonstration of sympathetic
cooling using two different atomic species is very demanding on current
experimental technology due to problems of
loading the ion trap with distinct atoms in a~desired configuration.
\subsection{Laser cooling using electromagnetically induced transparency}
Quantum computing with cold trapped ions requires one of the motional modes
(the one used as the quantum data bus) to be cooled to the motional ground
state and other modes to be inside the Lamb-Dicke regime. For this purpose
one could eventually use Doppler cooling assuming the axial trapping
frequency $\omega_z$ comparable with the linewidth of the cooling transition
$\Gamma$ [Eq.~(\rangleef{dopp4})]. However, it would cause very close spacing of
the ions in the trap [Eq.~(\rangleef{vibr9})] with difficulties at individual
addressing with the laser beam and optical resolving. On the other hand, we
can use sequential sideband cooling of the motional modes described in
Sec.\,\rangleef{sideband}. However, cooling of one motional mode causes heating of
the other modes. Moreover, sideband cooling requires a very narrow bandwidth
to excite the first red sideband of the respective motional mode only.
Otherwise, off-resonant transitions (especially carriers) are also
driven what causes heating as well \cite{blatt1}.
\betaegin{eqnarray}gin{figure}[htb]
\centerline{\epsfig{width=8cm,file=ions65.eps}}
\caption{{\footnotesize
(a) Levels and transitions of the cooling technique using electromagnetically
induced transparency (EIT). The inset shows the absorption on
$|g\rangle\langleeftrightarrow|e\rangle$ while strongly driving the transition
$|r\rangle\langleeftrightarrow|e\rangle$. (b) Absorption on the carrier
($|n\rangle\rangleightarrow|n\rangle$) and on the first sidebands
($|n\rangle\rangleightarrow|n\pm1\rangle$). Used by kind permission of Giovanna Morigi and
J\"{u}rgen Eschner \cite{EIT1}.}}
\langleabel{EIT}
\end{figure}
A novel cooling technique was developed in 2000 with a lower cooling limit
than Doppler cooling and with a wider cooling bandwidth than sideband
cooling. It was named {\it laser cooling using electromagnetically induced
transparency (EIT)} \cite{EIT1, EIT2}. It is based on a quantum interference
effect called EIT or {\it coherent population trapping} or also {\it dark
resonance} \cite{EIT3}. It employes a three-level system with a ground state
$|g\rangle$, a stable or metastable state $|r\rangle$ and an excited state $|e\rangle$
[FIG.\,\rangleef{EIT}(a)]. The transition $|r\rangle\langleeftrightarrow|e\rangle$ is driven
with a laser beam of the intensity $I_r$ blue detuned by $\Delta_r$. The
transition $|g\rangle\langleeftrightarrow|e\rangle$ is coupled by a weak laser with the
intensity $I_g$ (where $I_r/I_g\simeq 100$) also blue detuned by $\Delta_g$.
The intense laser with $|\Omega_r|^2\propto I_r$ introduces a significant
Stark light shift $\Delta\omega$ where \cite{comm2}
\betaegin{eqnarray}
\langleabel{eit1}
\Delta\omega=(\sqrt{\Delta_r^2+|\Omega_r|^2}-|\Delta_r|)/2\,.
\end{eqnarray}
Thus, the laser on the transition $|r\rangle\langleeftrightarrow|e\rangle$ designs the
absorption spectrum seen by the weak laser on $|r\rangle\langleeftrightarrow|e\rangle$ via
the level $|e\rangle$. Then there is a broad resonance at $\Delta_g\simeq 0$,
a dark resonance (EIT) at $\Delta_g=\Delta_r$ and a bright narrow resonance
at $\Delta_g=\Delta_r+\Delta\omega$ [see the inset in FIG.\,\rangleef{EIT}(a)].
Therefore, (i) taking into account also the motional degrees of freedom,
(ii) setting the detunings such that $\Delta_g=\Delta_r$ and (iii) setting
the Stark light shift equal to the vibrational frequency
($\Delta\omega\simeq\nu$) we obtain the absorption spectrum depicted in
FIG.\,\rangleef{EIT}(b). We see that the absorption on the first red sideband (cooling
transition) is enhanced while the absorption on the carrier (heating
transition) is eliminated.
The bright resonance width can be wide enough to cover several motional
modes which can be consecutively cooled at once. It was experimentally
demonstrated in Innsbruck on two motional modes separated in the frequency
by 1.73\,MHz. The modes were cooled to their ground motional states with 74\%
($\langle n\rangle\simeq 0.35$) and 58\% ($\langle n\rangle\simeq 0.72$) occupation \cite{EIT2}.
A great improvement of these results should possible for a rebuilt apparatus
allowing optimal access of laser beams to the ions \cite{danny}.
Following the advantages of laser cooling using EIT it has been estimated
that all $3N$ motional modes can be cooled to a mean phonon number
$\langle n\rangle<1$ for $\omega_z/2\pi=700\,\mbox{kHz}$ and $N=10$
\cite{EIT3}. It is very important to cool $3N-1$ spectator motional modes to
the Lamb-Dicke regime. Otherwise, thermally excited spectator modes cause
the fractional fluctuations (blurring) in the Rabi frequency of the mode
used as the quantum data bus for quantum logic operations \cite{98-5}. These
fluctuations in the Rabi frequency of the $\alphalpha$th mode can be estimated as
\cite{comm3}
\betaegin{eqnarray}
\langleabel{eit2}
\langleeft(\frac{\Delta\Omega}{\Omega}\rangleight)_{\alphalpha}\alphapprox
\sqrt{\sum_{\betaegin{eqnarray}ta\neq\alphalpha}
\eta_{\betaegin{eqnarray}ta}^4\,\langle n_{\betaegin{eqnarray}ta}\rangle\,(\,\langle n_{\betaegin{eqnarray}ta}\rangle+1)}\,,
\end{eqnarray}
where $\eta_{\betaegin{eqnarray}ta}$ is the Lamb-Dicke parameter of the $\betaegin{eqnarray}ta$th motional
mode and $\langle n_{\betaegin{eqnarray}ta}\rangle$ is the respective average phonon number. The ratio
$(\Omega/\Delta\Omega)$ determines the maximal number of Rabi cycles
\cite{blatt2}. A detailed description of experimental laser cooling using
EIT on Calcium ions can be found in Ref. \cite{EIT3}.
\section{Electron shelving}
\langleabel{es}
{\it Electron shelving} is the experimental method for the discrimination
between two electronic levels with an efficiency approaching $100\%$. It was
firstly demonstrated in 1986 \cite{86-2}.
Let us assume a three-level atom consisting of a ground level $|g\rangle$,
a metastable excited state $|e\rangle$ and an auxiliary excited fast
decaying state $|r\rangle$ (FIG.\,\rangleef{shel}).
\betaegin{eqnarray}gin{figure}[htb]
\centerline{\epsfig{width=8cm,file=ions71.eps}}
\caption{{\footnotesize
Electron shelving. The $|g\rangle\langleeftrightarrow |e\rangle$
transition is coupled with a laser pulse forming the superposition
$\alpha|g\rangle+\beta|e\rangle$. The $|g\rangle\langleeftrightarrow |r\rangle$ transition is driven with a
strong laser. The fluorescence detection signal is collected
on the $|g\rangle\langleeftrightarrow |r\rangle$ transition if the ion collapses
to the state $|g\rangle$. If the ion is shelved in the dark state $|e\rangle$, no
fluorescence is observed because the $|e\rangle$ state is a metastable state.}}
\langleabel{shel}
\end{figure}
The $|g\rangle\langleeftrightarrow|e\rangle$ transition is coupled by a weak laser
forming a superposition $\alpha|g\rangle+\betaegin{eqnarray}ta|e\rangle$, while
the $|g\rangle\langleeftrightarrow|r\rangle$ transition is driven with a strong
laser. If the atom collapses to the $|g\rangle$ state during the measurement,
a strong fluorescence signal is collected on the fast transition
$|g\rangle\langleeftrightarrow|r\rangle$, i.e. the atom is excited from $|g\rangle$ to $|r\rangle$
and spontaneously decays back to the $|g\rangle$ state what is observed as the
fluorescence. However, if the atom stays shelved in the metastable excited
state $|e\rangle$, no fluorescence can be observed on the driven
$|g\rangle\langleeftrightarrow|r\rangle$ transition. Hence the name electron shelving.
Even though the detection efficiency is
low, we can keep exciting the measuring transition $|g\rangle\langleeftrightarrow|r\rangle$
and detect some spontaneously emitted photons. Thus, we are able
to discriminate the $|g\rangle$ and $|e\rangle$ states with almost $100\%$ efficiency.
We can also obtain the occupation probability $|\alpha|^2$ for
the $|g\rangle$ state and $|\betaegin{eqnarray}ta|^2$ for the $|e\rangle$ state averaging
over many repetitions of the same experiment \cite{00-3}.
In the case of Calcium ions, the ground state $|g=S_{1/2}\rangle$ and
and the metastable excited state $|e=D_{5/2}\rangle$ form the qubit \cite{roos}.
The auxiliary state corresponds to the $P_{1/2}$ level.
The $S_{1/2}\langleeftrightarrow D_{5/2}$ transition is illuminated with a weak laser
pulse on 729\,nm and the $S_{1/2}\langleeftrightarrow P_{1/2}$ transition
is driven with a strong laser at 397\,nm. However, the ion can decay from the
$P_{1/2}$ level with a small probability to the $D_{3/2}$ level. Therefore,
there is pumping on the $P_{1/2}\langleeftrightarrow D_{3/2}$ transition at
866\,nm (see FIG.\,\rangleef{dopp}).
To clarify the efficiency of the electron shelving method
we report briefly some results measured in
Innsbruck \cite{00-3}. When the ion is found in the $S_{1/2}$
state it scatters about 2000 photons in 100\,ms to the detector.
However, for the ion
in the dark $D_{5/2}$ state the number of events drops to only about 150
photons in 100\,ms. These 150 photons appear due to dark counts
of the photomultiplier and some scattered light from the laser at 397\,nm.
The ion string in the linear ion trap may represent a quantum register,
where the internal state of each ion, i.e. the state of the qubit, can be
detected using a CCD camera. Then the ion in the $|g=S_{1/2}\rangle$ state
appears as a bright spot or a dark spot if the ion is found in the
$|e=D_{5/2}\rangle$ state \cite{nagerl, roos, 00-3}.
\section{Quantum gates}
\langleabel{qg}
One of the requirements for the physical implementation of quantum computing
in
a certain quantum system is a set of quantum gates that can be realized in
the quantum system under consideration. It has been shown that any unitary
operation can be composed of single-qubit rotations and two-qubit
controlled-NOT gates \cite{95-8}. In what follows we will describe how
these and some more complex quantum gates can be implemented on cold trapped
ions. We will use the notation $|g\rangle$ and $|e\rangle$ of the
logical states for the qubit rather than $|0\rangle$ and $|1\rangle$ due
to the representation of the qubit by the internal states of the ion.
\subsection{Single-qubit rotations}
A general {\it single-qubit gate} corresponds to a unitary evolution
operator that acts on a single qubit and is represented
in the basis $\{|g\rangle,|e\rangle\}$ by the matrix
\betaegin{eqnarray}
\langleabel{qg1}
W=\langleeft(
\betaegin{eqnarray}gin{array}{ll}
W_{gg} & W_{ge}\\
W_{eg} & W_{ee}
\end{array}
\rangleight)
\end{eqnarray}
A special case of the single-qubit gates
is a {\it single-qubit rotation} (FIG.\,\rangleef{rot}).
Its parameterization depends on the choice of the coordinates on the Bloch
sphere. We will define it in the matrix form in the basis
$\{|g\rangle,|e\rangle\}$ as follows
\betaegin{eqnarray}
\langleabel{qg2}
{\cal R}(\theta,\phi)=
\langleeft(
\betaegin{eqnarray}gin{array}{cc}
{\cal R}_{gg} & {\cal R}_{ge}\\
{\cal R}_{eg} & {\cal R}_{ee}
\end{array}
\rangleight)=
\langleeft(
\betaegin{eqnarray}gin{array}{cc}
\hat{a}^{\dag}s(\theta/2) & e^{i\phi}\sin(\theta/2)\\
-e^{-i\phi}\sin(\theta/2) & \hat{a}^{\dag}s(\theta/2)
\end{array}
\rangleight)\,,
\end{eqnarray}
where $\theta$ refers to the rotation and $\phi$ to the relative phase shift
of the states $|g\rangle$ and $|e\rangle$ in the corresponding Hilbert space.
\betaegin{eqnarray}gin{figure}[htb]
\centerline{\epsfig{width=2.5cm,file=ions81.eps}}
\caption{{\footnotesize
Schematical representation
of a single-qubit rotation. $R$ is defined by
Eq. (\rangleef{qg2}) in the basis $\{|g\rangle,|e\rangle\}$.}}
\langleabel{rot}
\end{figure}
The single-qubit rotation can be performed on a selected ion
from the ion string in the Lamb-Dicke regime by applying the unitary evolution
operator (\rangleef{int21}). We may rewrite this operator to the form
\betaegin{eqnarray}
\langleabel{qg3}
\hat{{\cal A}}_j^{\ell}(\phi_j)&=&
\sum_{n=0}^{\infty}\hat{a}^{\dag}s (\ell\pi/2)
\Bigg[
\betaigg(|e_j\rangle\langle e_j|\otimes |n\rangle\langle n|\betaigg)+
\betaigg(|g_j\rangle\langle g_j|\otimes |n\rangle\langle n|\betaigg)
\Bigg]\\
&+&\sum_{n=0}^{\infty}\sin (\ell\pi/2)
\Bigg[
-\betaigg(|e_j\rangle\langle g_j|\otimes |n\rangle\langle n|\betaigg)e^{-i\phi_j}+
\betaigg(|g_j\rangle\langle e_j|\otimes |n\rangle\langle n|\betaigg)e^{i\phi_j}
\Bigg]\,,
\nonumber
\end{eqnarray}
where we have applied the arbitrary choice of the phase factor
($\phi_j\rangleightarrow\phi_j+\pi/2$) with respect to the remark below
Eq. (\rangleef{int19}). The operator (\rangleef{qg3}) corresponds to the
$j$th ion illuminated with the laser beam
on the carrier ($\omega_L=\omega_0$) with the laser pulse duration
$t=\ell\pi/|\langleambda_j|$, where the laser coupling constant $\langleambda_j$
depends on the type of (i) the driven transition and (ii) the laser
configuration (see Sec.\,\rangleef{lii}). We will refer to the operation expressed
by Eq. (\rangleef{qg3}) as the $\ell\pi$-pulse on the carrier.
\subsection{Two-qubit controlled-NOT gates}
\langleabel{CNOT}
A two-qubit controlled-NOT (CNOT or XOR) gate acts on two qubits denoted
as a control and a target qubit (FIG.\,\rangleef{2-CNOT}).
If the control qubit ($m_1$) is in the state $|e\rangle$, then the state
of the target qubit ($m_2$) is flipped.
Otherwise, the gate acts trivially, i.e. as the unity operator $\openone$.
We may characterize this gate with the help of the following truth table
\betaegin{eqnarray}
\langleabel{qg4}
\betaegin{eqnarray}gin{array}{lll}
|g_{m_1}\rangle|g_{m_2}\rangle & \langleongrightarrow & |g_{m_1}\rangle|g_{m_2}\rangle\,,\\
|g_{m_1}\rangle|e_{m_2}\rangle & \langleongrightarrow & |g_{m_1}\rangle|e_{m_2}\rangle\,,\\
|e_{m_1}\rangle|g_{m_2}\rangle & \langleongrightarrow & |e_{m_1}\rangle|e_{m_2}\rangle\,,\\
|e_{m_1}\rangle|e_{m_2}\rangle & \langleongrightarrow & |e_{m_1}\rangle|g_{m_2}\rangle\,.
\end{array}
\end{eqnarray}
The implementation of the two-qubit CNOT gate on two selected ions in the
ion string requires the introduction of a third auxiliary internal level
$|r\rangle$. In the original proposal \cite{95-5} the selective
excitation of two sublevels of the $|e\rangle$ level is used instead.
The selection depends on the laser polarization, where the $|e,p=0\rangle$ and
$|e,p=1\rangle$ sublevels are considered. There have also appeared proposals how
to avoid the establishment of the auxiliary internal level to the scheme
\cite{chuang, monroe, jon}. Two of them will be discussed later on in his
section.
\betaegin{eqnarray}gin{figure}[htb]
\centerline{\epsfig{width=2.5cm,file=ions82.eps}}
\caption{{\footnotesize
Schematical representation
of a two-qubit controlled-NOT (CNOT) quantum gate.
The $m_1$ ($m_2$) qubit is control (target).
The gate is defined by the truth table (\rangleef{qg4}).}}
\langleabel{2-CNOT}
\end{figure}
Now we are ready to write two unitary evolution operators
corresponding to laser pulses driven on the first red sideband in the
Lamb-Dicke regime on the $j$th ion [Eq. (\rangleef{int22})] between the internal
levels $|g\rangle\langleeftrightarrow|e\rangle$ with the atomic frequency
$\omega_0^{eg}=(E_e-E_g)/\hbar$
and for $|g\rangle\langleeftrightarrow|r\rangle$ with $\omega_0^{rg}=(E_r-E_g)/\hbar$.
They are given by
\betaegin{eqnarray}
\langleabel{qg5}
\hat{{\cal B}}_j^{\ell,I}(\phi_j)&=&
\hat{a}^{\dag}s(\ell\pi/2)
\Bigg[
\betaigg(|e_j\rangle\langle e_j|\otimes |0\rangle\langle 0|\betaigg)+
\betaigg(|g_j\rangle\langle g_j|\otimes |1\rangle\langle 1|\betaigg)
\Bigg]\nonumber\\
&-&i\sin(\ell\pi/2)
\Bigg[
\betaigg(|e_j\rangle\langle g_j|\otimes |0\rangle\langle 1|\betaigg)e^{-i{\phi}_j}+
\betaigg(|g_j\rangle\langle e_j|\otimes |1\rangle\langle 0|\betaigg)e^{i{\phi}_j}
\Bigg]\nonumber\\
&+&|g_j\rangle\langle g_j|\otimes |0\rangle\langle 0|+{\cal O}
\end{eqnarray}
and
\betaegin{eqnarray}
\langleabel{qg6}
\hat{{\cal B}}_j^{\ell,II}(\phi_j)&=&
\hat{a}^{\dag}s(\ell\pi/2)
\Bigg[
\betaigg(|r_j\rangle\langle r_j|\otimes |0\rangle\langle 0|\betaigg)+
\betaigg(|g_j\rangle\langle g_j|\otimes |1\rangle\langle 1|\betaigg)
\Bigg]\nonumber\\
&-&i\sin(\ell\pi/2)
\Bigg[
\betaigg(|r_j\rangle\langle g_j|\otimes |0\rangle\langle 1|\betaigg)e^{-i{\phi}_j}+
\betaigg(|g_j\rangle\langle r_j|\otimes |1\rangle\langle 0|\betaigg)e^{i{\phi}_j}
\Bigg]\nonumber\\
&+&|g_j\rangle\langle g_j|\otimes |0\rangle\langle 0|+{\cal O}\,,
\end{eqnarray}
where we have applied again the arbitrary choice of the phase factor
($\tilde{\phi}_j=\phi_j-\pi/2\rangleightarrow\phi_j$). The symbol ${\cal O}$
in Eq.~(\rangleef{qg5}) and (\rangleef{qg6})
correspond to the terms in Eq. (\rangleef{int22}) associated with the dynamics on
higher vibrational levels for $n\geq 2$. We do not have to consider them because
the ions are assumed to be cooled to the ground motional state $|n=0\rangle$. We use
the Hilbert space spanned only by the motional states $|n=0\rangle$ and $|n=1\rangle$
forming an auxiliary qubit used as the quantum data bus.
The operators (\rangleef{qg5}) and (\rangleef{qg6}) correspond
to $k\pi$-pulses on the first red sideband
($\omega_L=\omega_0^{eg}-\nu$ and $\omega_L=\omega_0^{rg}-\nu$)
for $n=0$ with the laser pulse duration $t=\ell\pi/|\langleambda_j|\eta_j$.
Finally, the two-qubit CNOT gate on two ions corresponds to the evolution
operator sequence (acting from right to left) \cite{95-5}
\betaegin{eqnarray}
\langleabel{qg7}
\hat{{\cal A}}_{m_2}^{1/2}(\pi)\,
\hat{{\cal B}}_{m_1}^{1,I}\,
\hat{{\cal B}}_{m_2}^{2,II}\,
\hat{{\cal B}}_{m_1}^{1,I}\,
\hat{{\cal A}}_{m_2}^{1/2}(0)\,,
\end{eqnarray}
where $\hat{{\cal A}}_{m_2}^{1/2}(0)$ and $\hat{{\cal A}}_{m_2}^{1/2}(\pi)$
are given by Eq. (\rangleef{qg3}) and stand for the $\pi/2$-pulses on
the carrier ($\omega_L=\omega_0^{eg}$)
on the $m_2$th ion with the phase $\phi_j=0$ and $\phi_j=\pi$,
respectively. The operator $\hat{{\cal B}}_{m_1}^{1,I}$ is defined by
Eq. (\rangleef{qg5}) and represents the $\pi$-pulse on the first red
sideband ($\omega_L=\omega_0^{eg}-\omega_z$) on the $m_1$th ion with the
phase factor $\phi_j=0$. The operator $\hat{{\cal B}}_{m_2}^{2,II}$ defined
in Eq. (\rangleef{qg6}) stands for the $2\pi$-pulse on the first red sideband
($\omega_L=\omega_0^{rg}-\omega_z$) on the $m_2$th ion with $\phi_j=0$.
The middle sequence
$\hat{{\cal B}}_{m_1}^{1,I}\,
\hat{{\cal B}}_{m_2}^{2,II}\,
\hat{{\cal B}}_{m_1}^{1,I}$
in the evolution operator (\rangleef{qg7}) can be schematically represented as
follows
\betaegin{eqnarray}
\langleabel{qg8}
\betaegin{eqnarray}gin{array}{rcrcrcr}
& \hat{{\cal B}}_{m_1}^{1,I} & & \hat{{\cal B}}_{m_2}^{2,II} & &
\hat{{\cal B}}_{m_1}^{1,I}\\
|g_{m_1}\rangle|g_{m_2}\rangle|0\rangle & \langleongrightarrow &
|g_{m_1}\rangle|g_{m_2}\rangle|0\rangle & \langleongrightarrow &
|g_{m_1}\rangle|g_{m_2}\rangle|0\rangle & \langleongrightarrow &
|g_{m_1}\rangle|g_{m_2}\rangle|0\rangle\,,\\
|g_{m_1}\rangle|e_{m_2}\rangle|0\rangle & \langleongrightarrow &
|g_{m_1}\rangle|e_{m_2}\rangle|0\rangle & \langleongrightarrow &
|g_{m_1}\rangle|e_{m_2}\rangle|0\rangle & \langleongrightarrow &
|g_{m_1}\rangle|e_{m_2}\rangle|0\rangle\,,\\
|e_{m_1}\rangle|g_{m_2}\rangle|0\rangle & \langleongrightarrow &
-i|g_{m_1}\rangle|g_{m_2}\rangle|1\rangle & \langleongrightarrow &
i|g_{m_1}\rangle|g_{m_2}\rangle|1\rangle & \langleongrightarrow &
|e_{m_1}\rangle|g_{m_2}\rangle|0\rangle\,,\\
|e_{m_1}\rangle|e_{m_2}\rangle|0\rangle & \langleongrightarrow &
-i|g_{m_1}\rangle|e_{m_2}\rangle|1\rangle & \langleongrightarrow &
-i|g_{m_1}\rangle|e_{m_2}\rangle|1\rangle & \langleongrightarrow &
-|e_{m_1}\rangle|e_{m_2}\rangle|0\rangle\,.
\end{array}
\end{eqnarray}
Finally, the evolution operator (\rangleef{qg7}) refers
to the transformation
\betaegin{eqnarray}
\langleabel{qg9}
\betaegin{eqnarray}gin{array}{rcr}
|g_{m_1}\rangle|g_{m_2}\rangle|0\rangle & \langleongrightarrow & |g_{m_1}\rangle|g_{m_2}\rangle|0\rangle\,,\\
|g_{m_1}\rangle|e_{m_2}\rangle|0\rangle & \langleongrightarrow & |g_{m_1}\rangle|e_{m_2}\rangle|0\rangle\,,\\
|e_{m_1}\rangle|g_{m_2}\rangle|0\rangle & \langleongrightarrow & |e_{m_1}\rangle|e_{m_2}\rangle|0\rangle\,,\\
|e_{m_1}\rangle|e_{m_2}\rangle|0\rangle & \langleongrightarrow & |e_{m_1}\rangle|g_{m_2}\rangle|0\rangle\,,
\end{array}
\end{eqnarray}
on two selected ions labelled as $m_1$ and $m_2$ in the string of $N$ ions.
It is evident from the discussion above that this realization of
the CNOT logic gate on the ion system requires the ions to be cooled
to the ground motional state $|n=0\rangle$ in order to maintain the fidelity of
the computational process. Otherwise, as the ions heat up,
higher terms ${\cal O}$ in Eq. (\rangleef{qg5}) and (\rangleef{qg6})
also contribute and introduce
significant imperfections into the implementation of the quantum gate.
The two-qubit CNOT was firstly demonstrated in Boulder in 1995
\cite{95-11}. A single Beryllium ion was used, where the control qubit was
stored into two lowest vibrational states $|n=0\rangle$ and $|n=1\rangle$ and
the target qubit was represented by two hyperfine levels
$|g=S_{1/2}(F=2,m_F=2)\rangle$ and $|e=S_{1/2}(F=1,m_F=1)\rangle$.
\subsection{Alternative implementation of two-qubit controlled-NOT gates}
\subsubsection{Simplified quantum logic}
\langleabel{monroe}
Monroe et al. have proposed the realization of the two-qubit quantum logic
gate based on the precise setting of the Lamb-Dicke parameter \cite{monroe}.
The control qubit is assumed to be encoded into two lowest
vibrational states $|n=0\rangle$ and $|n=1\rangle$ of the considered collective
vibrational mode, while the target qubit is represented by two internal
levels $|g_j\rangle$ and $|e_j\rangle$ of the $j$th ion from the string of $N$ ions
in the linear Paul trap. The CNOT gate under consideration
is then described by the truth table
\betaegin{eqnarray}
\langleabel{mon1}
\betaegin{eqnarray}gin{array}{rcr}
|0\rangle|g_{j}\rangle & \langleongrightarrow & |0\rangle|g_{j}\rangle\,,\\
|0\rangle|e_{j}\rangle & \langleongrightarrow & |0\rangle|e_{j}\rangle\,,\\
|1\rangle|g_{j}\rangle & \langleongrightarrow & |1\rangle|e_{j}\rangle\,,\\
|1\rangle|e_{j}\rangle & \langleongrightarrow & |1\rangle|g_{j}\rangle\,.
\end{array}
\end{eqnarray}
Further, we adopt the main idea of the original proposal \cite{monroe}.
Driving the $j$th ion with the laser on the carrier ($\omega_L=\omega_0$)
is described by the evolution operator (\rangleef{int18}).
The coupling constant $\Omega_j^{n,k}$ is
introduced by the expression (\rangleef{int15}).
Then for $n=0$ and $n=1$ with $k=0$ we get
\betaegin{eqnarray}
\langleabel{mon2}
\Omega_j^{0,0}&=&\langleambda_je^{-{\eta_j}^2/2}\,,\\
\langleabel{mon3}
\Omega_j^{1,0}&=&\langleambda_je^{-{\eta_j}^2/2}(1-{\eta_j}^2)\,.
\end{eqnarray}
Let us set the Lamb-Dicke parameter such that
\betaegin{eqnarray}
\langleabel{mon4}
{\eta_j}^2=\frac{1}{2p}\,,
\end{eqnarray}
where $p$ is an integer. The realization of the transformation (\rangleef{mon1})
requires driving the carrier transition on the $j$th ion with the duration
$t$ such that
\betaegin{eqnarray}
\langleabel{mon4.1}
\Omega_j^{0,0}t=2p\pi\,.
\end{eqnarray}
We can calculate using Eq. (\rangleef{mon4}) that
\betaegin{eqnarray}
\langleabel{mon4.2}
\Omega_j^{1,0}t=(2p-1)\pi\,.
\end{eqnarray}
Substituting Eq. (\rangleef{mon4.1}) and (\rangleef{mon4.2})
into Eq. (\rangleef{int18}) we find out that the internal
state of the $j$th ion is flipped only if the collective vibrational state
is $|n=1\rangle$. We can write
\betaegin{eqnarray}
\langleabel{mon7}
\betaegin{eqnarray}gin{array}{rcl}
|0\rangle|g_j\rangle & \langleongrightarrow & |0\rangle|g_j\rangle\,,\\
|0\rangle|e_j\rangle & \langleongrightarrow & |0\rangle|e_j\rangle\,,\\
|1\rangle|g_j\rangle & \langleongrightarrow & ie^{-i\phi_j}|1\rangle|e_j\rangle\,,\\
|1\rangle|e_j\rangle & \langleongrightarrow & ie^{+i\phi_j}|1\rangle|g_j\rangle\,.
\end{array}
\end{eqnarray}
This transformation corresponds to the CNOT gate (\rangleef{mon1}) apart from
the phase factor $\phi_j$ which can be eliminated by the appropriate phase
settings of subsequent operations.
The CNOT gate between distinct ions representing two logic qubits can be
implemented (using the proposal being discussed) by two additional laser
pulses on the first red sideband ($\omega_L=\omega_0-\nu$). We have on
mind the CNOT gate given by the truth table (\rangleef{qg9}). Firstly, we apply
a $\pi$-pulse on the first red sideband on the $m_1$th ion (corresponding
to the evolution operator (\rangleef{int19}) with $\Omega_j^{0,1}t=\pi$)
mapping the internal state of this ion onto the collective
vibrational state. We can write
\betaegin{eqnarray}
\langleabel{mon8}
\betaegin{eqnarray}gin{array}{rcl}
|g_{m_1}\rangle|0\rangle & \langleongrightarrow & |g_{m_1}\rangle|0\rangle\,,\\
|e_{m_1}\rangle|0\rangle & \langleongrightarrow & -ie^{+i\tilde{\phi}_{m_1}}|g_{m_1}\rangle|1\rangle\,,\\
|g_{m_1}\rangle|1\rangle & \langleongrightarrow & -ie^{-i\tilde{\phi}_{m_1}}|e_{m_1}\rangle|0\rangle\,.
\end{array}
\end{eqnarray}
Secondly, we apply a laser pulse on the carrier on the $m_2$th ion
representing the {\it reduced} CNOT gate (\rangleef{mon1}) and finally, we map
back the collective vibrational state onto the internal state of the $m_1$th
ion by reapplying a $\pi$-pulse on the first red sideband on this ion.
This sequence of three laser pulses corresponds to the {\it complete} CNOT
gate (\rangleef{qg9}) between two distinct ions with the appropriate choice of
the phase factors.
Comparing this scheme to the original proposal of Cirac and Zoller
(\rangleef{qg7}), we need fewer laser pulses to realize a two-qubit CNOT gates on
trapped ions and there is no need for a third internal auxiliary level.
However, more important is the overall time needed to complete the gate and
the sensitivity to imprecisions. The main limitation of this Monroe scheme
is that it is slow compared with other methods at the same level of
infidelity (caused by off-resonant transitions) and it is
rather sensitive to imprecision in the laser intensity
($|\Omega|^2\propto I$) \cite{stn}.
\subsubsection{Fast quantum gates}
Jonathan et al. have proposed another alternative realization of two-qubit
quantum gates on cold trapped ions scalable on $N$ ions \cite{jon}. It is
based on (i) using carrier transitions and (ii) taking into
account Stark light shifts of atomic levels.
Following Ref. \cite{jon} the basic idea of fast quantum gates is that
the resonant driving of a carrier transition
($|g\rangle|n\rangle\langleeftrightarrow |e\rangle|n\rangle$) with an intense laser causes
the splitting of dressed states $|\pm\rangle=1/\sqrt{2}(|g\rangle\pm|e\rangle)$ in the
interaction picture by amount $2\hbar\Omega$, where the coupling constant
$\Omega$ is proportional to the laser intensity $I$ [FIG.\,\rangleef{jon}(a)].
\betaegin{eqnarray}gin{figure}[h!]
\centerline{\epsfig{width=12cm,file=ions83.eps}}
\caption{{\footnotesize
Fast quantum gates on cold trapped ions are based: (a) on the splitting of
the dressed states $|\pm\rangle=1/\sqrt{2}(|g\rangle\pm|e\rangle)$
by amount proportional to the laser intensity [$\Omega=\Omega(I)$]
when the laser is tuned on the carrier transition ($\omega_L=\omega_0$)
and (b) on setting the splitting such that it is proportional
to one motional quantum ($\nu=2\Omega$).}}
\langleabel{jon}
\end{figure}
When we set the laser intensity such that the splitting
of the dressed states $|-\rangle$
and $|+\rangle$ is equal to one motional quantum $\hbar\nu$.
Then Rabi oscillations appear between the state $|+\rangle|0\rangle$
and $|-\rangle|1\rangle$ with $|0\rangle$ and $|1\rangle$ referring to the lowest collective
vibrational states of the ions [FIG. \rangleef{jon}(b)]. Using this swapping
between the $|+\rangle|0\rangle$ and $|-\rangle|1\rangle$ states one can construct a CNOT gate
between two distinct ions following the truth table (\rangleef{qg9}).
Quantum gates using this idea are faster than standard quantum gates
on trapped ions (discussed in Sec.\,\rangleef{CNOT}) approximately
by the factor of $1/{\eta}$ assuming the Lamb-Dicke regime.
The speed of quantum gates will be discussed in Sec.\,\rangleef{speed}.
\subsection{Multi-qubit controlled-NOT gates}
\langleabel{mqcnot}
A multi-qubit controlled-NOT gate is defined by analogy to the two-qubit CNOT
gate. The only difference is the number of control
qubits (FIG.\,\rangleef{multi-CNOT}).
The multi-qubit (controlled)$^q$-NOT gate acts on $q+1$ qubits with $q$ control
qubits $(m_1,\dots,m_q$) and the $m_{q+1}$th qubit is target. If all control
qubits are in the state $|e\rangle$, then the state of the target qubits is
flipped. Otherwise, the gate acts as the unity operator $\openone$. The
truth table of the multi-qubit (controlled)$^q$-NOT gate acting on
$m_1,\dots,m_{q+1}$ qubits is
\betaegin{eqnarray}
\langleabel{qg20}
\betaegin{eqnarray}gin{array}{llll}
|\Psi_{no}\rangle|g_{m_{q+1}}\rangle & \langleongrightarrow &
\quad |\Psi_{no}\rangle|g_{m_{q+1}}\rangle\,, &
\quad |\Psi_{no}\rangle\neq\prod\langleimits_{j=1}^q\otimes|e_{m_j}\rangle\,,\\
|\Psi_{no}\rangle|e_{m_{q+1}}\rangle & \langleongrightarrow &
\quad |\Psi_{no}\rangle|e_{m_{q+1}}\rangle\,, & \\
|\Psi_{yes}\rangle|g_{m_{q+1}}\rangle & \langleongrightarrow &
\quad |\Psi_{yes}\rangle|e_{m_{q+1}}\rangle\,, &
\quad |\Psi_{yes}\rangle=\prod\langleimits_{j=1}^q\otimes|e_{m_j}\rangle\,,\\
|\Psi_{yes}\rangle|e_{m_{q+1}}\rangle & \langleongrightarrow &
\quad |\Psi_{yes}\rangle|g_{m_{q+1}}\rangle\,. &
\end{array}
\end{eqnarray}
The multi-qubit (controlled)$^q$-NOT gate acting on $q+1$ ions ($m_1,\dots,m_q$
ions represent the control qubits, while the $m_{q+1}$th ion stands for the
target qubit) can be realized by applying the evolution operator (acting
from right to left)
\betaegin{eqnarray}
\langleabel{qg21}
\hat{{\cal A}}_{m_{q+1}}^{1/2}(\pi)\,
\hat{{\cal B}}_{m_1}^{1,I}\,
\langleeft[\prod_{j=2}^q\hat{{\cal B}}_{m_j}^{1,II}\rangleight]\,
\hat{{\cal B}}_{m_{q+1}}^{2,II}\,
\langleeft[\prod_{j=q}^2\hat{{\cal B}}_{m_j}^{1,II}\rangleight]\,
\hat{{\cal B}}_{m_1}^{1,I}\,
\hat{{\cal A}}_{m_{q+1}}^{1/2}(0)\,,
\end{eqnarray}
where the $\hat{{\cal B}}$ operators are taken for the value $\phi=0$.
However, this choice of the phase factor has no fundamental importance.
Eq. (\rangleef{qg21}) applies for three and more ions and
the scheme requires again the auxiliary qubit encoded into two lowest
levels $|n=0\rangle$ and $|n=1\rangle$ of the collective vibrational mode used as the
quantum data bus.
\betaegin{eqnarray}gin{figure}[htb]
\centerline{\epsfig{width=2.75cm,file=ions84.eps}}
\caption{{\footnotesize
Schematical representation
of a multi-bit $(\mbox{controlled})^q$-NOT gate acting on $q+1$ qubits
with $q$ control qubits and $m_{q+1}$th qubit to be target.
The gate is defined by the truth table (\rangleef{qg20}).}}
\langleabel{multi-CNOT}
\end{figure}
Now we verify whether the evolution operator (\rangleef{qg21})
corresponds to the truth table of the multi-qubit CNOT gate given
by Eq.~(\rangleef{qg20}). At first we consider only the $\hat{{\cal B}}$
operators and then we comment on the action of the $\hat{{\cal A}}$ operators.
It holds for $q+1$ ions involved in the multi-qubit CNOT gate that:
\betaegin{eqnarray}gin{itemize}
\item If the $m_1$th ion is in the ground state $|g_{m_1}\rangle$, then the
action of the $\hat{{\cal B}}$ operators in Eq. (\rangleef{qg21}) corresponds to
the unity operator.
\item If the $m_1$th ion is excited with all other ions
in the ground state $|e_{m_1}\rangle|g\rangle^q$, we get
\betaegin{eqnarray}
\langleabel{*1}
|e_{m_1}\rangle|g\rangle^q|0\rangle
\stackrel{\hat{{\cal B}}^{1,I}_{m_1}}{\langleongrightarrow}
-i|g\rangle^{q+1}|1\rangle
\stackrel{\hat{{\cal B}}^{1,II}_{m_2}}{\langleongrightarrow}
-|r_{m_2}\rangle|g\rangle^q|0\rangle
\stackrel{\hat{{\cal B}}^{1,II}_{m_2}}{\langleongrightarrow}
i|g\rangle^{q+1}|1\rangle
\stackrel{\hat{{\cal B}}^{1,I}_{m_1}}{\langleongrightarrow}
|e_{m_1}\rangle|g\rangle^q|0\rangle\,.
\end{eqnarray}
Thus, the transformation is performed on the $m_1$th ion and then
on the first next ion in the ground state. The state of
all other ions in the ground state is not transformed.
If more ions (besides the $m_1$th one) are excited
(except if they all are excited),
their state does not change because the $\hat{{\cal B}}^{\ell, II}_{m_j}$
operator acts only in the Hilbert space spanned by
$\{|g_{m_j}\rangle, |r_{m_j}\rangle\}$ [see Eq. (\rangleef{qg6})].
\item If all the ions are excited, i.e. $|e\rangle^{q+1}$, it follows that
\betaegin{eqnarray}
\langleabel{*2}
|e\rangle^{q+1}|0\rangle
\stackrel{\hat{{\cal B}}^{1,I}_{m_1}}{\langleongrightarrow}
-i|g_{m_1}\rangle|e\rangle^q|1\rangle
\stackrel{\hat{{\cal B}}^{1,I}_{m_1}}{\langleongrightarrow}
-|e\rangle^{q+1}|0\rangle\,.
\end{eqnarray}
\end{itemize}
Finally, the $\hat{{\cal A}}$ operators complete
the operation (\rangleef{qg21}) such that it corresponds
to the transformation (\rangleef{qg20})
by analogy to Eq. (\rangleef{qg8}) and (\rangleef{qg9}).
\subsection{Multi-qubit controlled-$R$ gates}
\langleabel{mqcrot}
A multi-qubit (controlled)$^q$-$R$ gate acts again on $q+1$ qubits. However, it
performs a single-qubit operation (\rangleef{qg2}) on the $m_{q+1}$th (target)
qubit if all $m_1,\dots,m_q$ control qubits
are in the state $|e\rangle$. Otherwise, it acts trivially (FIG.\,\rangleef{CROT}).
Speaking precisely, if all control qubits are in the state
$|e\rangle$, then the rotation
$R=R_1^{\dag}\,\sigma\,R_2^{\dag}\,\sigma\,R_2\,R_1$ is applied (from right
to left) on the target qubit. In the basis of the target qubit
$\{|g\rangle_{m_{q+1}},|e\rangle_{m_{q+1}}\}$ we introduce the matrices
\betaegin{eqnarray}
\langleabel{qg22}
\betaegin{eqnarray}gin{array}{cc}
R=
\langleeft(
\betaegin{eqnarray}gin{array}{cc}
\hat{a}^{\dag}s\theta & e^{i2\phi}\sin\theta\\
-e^{-i2\phi}\sin\theta & \hat{a}^{\dag}s\theta
\end{array}
\rangleight)\,,
&
\sigma=
\langleeft(
\betaegin{eqnarray}gin{array}{cc}
0 & 1 \\
1 & 0
\end{array}
\rangleight)\,,
\\ \\
R_1= \langleeft(
\betaegin{eqnarray}gin{array}{cc}
0 & e^{i\phi}\\
-e^{-i\phi} & 0
\end{array}
\rangleight)\,,
&
R_1^{\dag}=
\langleeft(
\betaegin{eqnarray}gin{array}{cc}
0 & -e^{i\phi} \\
e^{-i\phi} & 0
\end{array}
\rangleight)\,,
\\ \\
R_2=
\langleeft(
\betaegin{eqnarray}gin{array}{rc}
\hat{a}^{\dag}s(\theta/2) & \sin(\theta/2) \\
-\sin(\theta/2) & \hat{a}^{\dag}s(\theta/2)
\end{array}
\rangleight)\,,
&
R_2^{\dag}=
\langleeft(
\betaegin{eqnarray}gin{array}{cr}
\hat{a}^{\dag}s(\theta/2) & -\sin(\theta/2) \\
\sin(\theta/2) & \hat{a}^{\dag}s(\theta/2)
\end{array}
\rangleight)\,,
\end{array}
\end{eqnarray}
where
$R_1={\cal R}(\pi,\phi)$, $R_1^{\dag}={\cal R}^{\dag}(\pi,\phi)$,$R_2={\cal R}(\theta,0)$ and
$R_2^{\dag}={\cal R}(\theta,0)$. The rotation ${\cal R}(\theta,\phi)$ is defined by
Eq. (\rangleef{qg2}). The matrix $\sigma$ denotes the NOT operation.
If not all control qubits are in the state $|e\rangle$,
then the gate performs on the target qubit the unity operator
$\openone=R_1^{\dag}\,\openone\,R_2^{\dag}\,\openone\,R_2\,R_1$.
Finally, we may write the truth table of the multi-qubit
$(\mbox{controlled})^q$-$R$ gate as follows
\betaegin{eqnarray}
\langleabel{qg23}
\betaegin{eqnarray}gin{array}{lll}
|\Psi_{no}\rangle|g_{m_{q+1}}\rangle & \langleongrightarrow &
\quad |\Psi_{no}\rangle|g_{m_{q+1}}\rangle\,,\\ \\
|\Psi_{no}\rangle|e_{m_{q+1}}\rangle & \langleongrightarrow &
\quad |\Psi_{no}\rangle|e_{m_{q+1}}\rangle\,,\\ \\
|\Psi_{yes}\rangle|g_{m_{q+1}}\rangle & \langleongrightarrow &
\quad |\Psi_{yes}\rangle
\betaig(\hat{a}^{\dag}s\theta\,|g_{m_{q+1}}\rangle-e^{-i2\phi}\sin\theta\,|e_{m_{q+1}}\rangle\betaig)\,,
\\ \\
|\Psi_{yes}\rangle|e_{m_{q+1}}\rangle & \langleongrightarrow &
\quad |\Psi_{yes}\rangle
\betaig(e^{i2\phi}\sin\theta\,|g_{m_{q+1}}\rangle+\hat{a}^{\dag}s\theta\,|e_{m_{q+1}}\rangle\betaig)\,,
\end{array}
\end{eqnarray}
where $|\Psi_{no}\rangle$ and $|\Psi_{yes}\rangle$ are defined in Eq. (\rangleef{qg20}).
The multi-qubit controlled-R (CROT) gate (FIG.\,\rangleef{CROT}) is performed
on cold trapped ions by applying the evolution operator (\rangleef{qg21}) for
the multi-qubit CNOT gates and the corresponding operator for the
single-qubit rotations [Eq. (\rangleef{qg3})].
\betaegin{eqnarray}gin{figure}[htb]
\centerline{\epsfig{width=10cm,file=ions85.eps}}
\caption{{\footnotesize
Scheme of a multi-qubit $(\mbox{controlled})^q$-$R$ quantum gate. The gate acts
on $q+1$ qubits with $q$ control qubits and the $m_{q+1}$th qubit is target.
$R$ is defined by Eq. (\rangleef{qg22}) in the~basis
$\{|g\rangle_{m_{q+1}},|e\rangle_{m_{q+1}}\}$ of the target qubit. $R_1$,
$R_1^{\dag}$, $R_2$ and $R_2^{\dag}$
are also defined by Eq. (\rangleef{qg22}) in the same basis.}}
\langleabel{CROT}
\end{figure}
If the preparation of a particular class of quantum states does not
require the introduction of a relative phase shift $\phi$ between the basis
states $|g\rangle$ and $|e\rangle$, then a reduced quantum logic network is
sufficient (FIG.\,\rangleef{CROTr}).
In particular, the rotation $\tilde{R}=\sigma\,R_2^{\dag}\,\sigma\,R_2$ on
the target qubit conditioned by the state of control qubits
can be realized according to the following truth table
\betaegin{eqnarray}
\langleabel{qg24}
\betaegin{eqnarray}gin{array}{lll}
|\Psi_{no}\rangle|g_{m_{q+1}}\rangle & \langleongrightarrow &
\quad |\Psi_{no}\rangle|g_{m_{q+1}}\rangle\,,\\ \\
|\Psi_{no}\rangle|e_{m_{q+1}}\rangle & \langleongrightarrow &
\quad |\Psi_{no}\rangle|e_{m_{q+1}}\rangle\,,\\ \\
|\Psi_{yes}\rangle|g_{m_{q+1}}\rangle & \langleongrightarrow &
\quad |\Psi_{yes}\rangle
\betaig(\hat{a}^{\dag}s\theta\,|g_{m_{q+1}}\rangle-\sin\theta\,|e_{m_{q+1}}\rangle\betaig)\,,\\ \\
|\Psi_{yes}\rangle|e_{m_{q+1}}\rangle & \langleongrightarrow &
\quad |\Psi_{yes}\rangle
\betaig(\sin\theta\,|g_{m_{q+1}}\rangle+\hat{a}^{\dag}s\theta\,|e_{m_{q+1}}\rangle\betaig)\,.
\end{array}
\end{eqnarray}
The results for the multi-qubit controlled-$R$ gates are
compatible with the scheme proposed in Ref.~\cite{95-8}, where
the decomposition of multi-qubit CNOT gates into
the network of two-qubit CNOT gates has been presented as well.
However, this decomposition may require many elementary operations in a
particular realization of quantum logic gates.
It seems to be more appropriate for some practical implementations
of quantum computing to implement directly multi-qubit CNOT gates
(see Sec.\,\rangleef{speed}).
\betaegin{eqnarray}gin{figure}[htb]
\centerline{\epsfig{width=8cm,file=ions86.eps}}
\caption{{\footnotesize
Scheme of a reduced multi-qubit $(\mbox{controlled})^q$-$\tilde{R}$ quantum gate.
The gate acts
on $q+1$ qubits with $q$ control qubits and the $m_{q+1}$th qubit is target.
$\tilde{R}$ is defined by Eq. (\rangleef{qg24}). $R_2$ and
$R_2^{\dag}$ are defined by Eq. (\rangleef{qg22}) in the~basis
$\{|g\rangle_{m_{q+1}},|e\rangle_{m_{q+1}}\}$ of the target qubit.}}
\langleabel{CROTr}
\end{figure}
\section{Quantum logic networks}
\langleabel{networks}
In this section we present quantum logic networks as effective
tools for the synthesis of quantum coherent superpositions of internal atomic
states. We provide two particular networks, where both of them
apply to an arbitrary register of qubits. The networks
consist of single-qubit rotations, multi-qubit controlled-NOT and
multi-qubit controlled-$R$ gates. Their implementation on cold trapped
ions is described in detail in Sec.\,\rangleef{mqcnot} and \rangleef{mqcrot}.
The generation
of nonclassical motional states of a trapped ion experimentally
is described in Ref. \cite{gen}.
We keep the notation $|g\rangle$ and $|e\rangle$ for the logical states of the qubit
also in this section.
Firstly, let us introduce the network for the preparation of a totally
symmetric state (with respect to the permutations) of $N$ qubits, such that all
qubits except one are in the excited state
\betaegin{eqnarray}
\langleabel{net1}
|\Psi\rangle
=\frac{1}{\sqrt{N}}\betaigg(
|gee\dots e\rangle+|ege\dots e\rangle+|eeg\dots e\rangle+\dots+|eee\dots g\rangle
\betaigg)\,.
\end{eqnarray}
It has been shown that the maximal degree of bipartite entanglement
measured in the concurrence \cite{conc}
is equal to $2/N$ and is achieved when a system
of $N$ qubits is prepared just in the state (\rangleef{net1}). The synthesis of
this state realizes the network in FIG.\,\rangleef{fnet1} assuming all qubits to
be initially prepared in the state $|e\rangle$. The rotations $Q_j$ are given as
follows
\betaegin{eqnarray}
\langleabel{net2}
Q_j=\langleeft(
\betaegin{eqnarray}gin{array}{cc}
\sqrt{\frac{N-j}{N-j+1}} & \frac{1}{\sqrt{N-j+1}}\\
-\frac{1}{\sqrt{N-j+1}} & \sqrt{\frac{N-j}{N-j+1}}
\end{array}
\rangleight)\,,\quad
j=1,\dots,N-1\,.
\end{eqnarray}
For more details we refer to our original paper \cite{ms}.
\betaegin{eqnarray}gin{figure}[htb]
\centerline{\epsfig{width=7.5cm, file=ions91.eps}}
\caption{{\footnotesize
The network for the synthesis of the symmetric entangled
state (\rangleef{net1}). The rotations $Q_j$ are given
by Eq.~(\rangleef{net2}). $N$ qubits are assumed to be initially prepared
in the state $|eee...e\rangle$.}}
\langleabel{fnet1}
\end{figure}
\betaegin{eqnarray}gin{figure}[htb]
\centerline{\epsfig{width=11cm, file=ions92.eps}}
\caption{{\footnotesize
An array of networks for the synthesis of an arbitrary pure
quantum state (\rangleef{net3}) on three qubits. The initial state is $|ggg\rangle$ and
the rotations $U_j$ are given by Eq.~(\rangleef{net4})--(\rangleef{net6}).
The networks (a)--(g) generate gradually the respective terms
in the superposition (\rangleef{net3}).}}
\langleabel{fnet2}
\end{figure}
\betaegin{eqnarray}gin{figure}[htb]
\centerline{\epsfig{width=11cm, file=ions93.eps}}
\caption{{\footnotesize
A compact form of the array of the networks
shown in FIG.\,\rangleef{fnet2}.}}
\langleabel{fnet3}
\end{figure}
Secondly, we propose an array of quantum logic networks
for the synthesis of an arbitrary pure quantum state for illustration
depicted on three qubits. However, the scheme is quite easily scalable on $N$
qubits \cite{ms}. Let us assume a general state of three qubits
in the form
\betaegin{eqnarray}
\langleabel{net3}
|\psi\rangle&=&
\alpha_0|ggg\rangle+e^{i\varphi_1}\alpha_1|gge\rangle+e^{i\varphi_2}\alpha_2|geg\rangle+
e^{i\varphi_3}\alpha_3|egg\rangle\nonumber\\
&+&e^{i\varphi_4}\alpha_4|gee\rangle+e^{i\varphi_5}\alpha_5|ege\rangle+
e^{i\varphi_6}\alpha_6|eeg\rangle+e^{i\varphi_7}\alpha_7|eee\rangle\,,
\end{eqnarray}
The state (\rangleef{net3}) can be realized by applying
the array of the networks in FIG.\,\rangleef{fnet2} (shown in a more compact form in
FIG.\,\rangleef{fnet3}) on the initial state $|ggg\rangle$, i.e. all three
qubits in the state $|g\rangle$. We have denoted the rotations $U_j$ as
follows
\betaegin{eqnarray}
\langleabel{net4}
U_j=\langleeft(
\betaegin{eqnarray}gin{array}{cc}
a_j & e^{i2\phi_j}b_j\\
-e^{-i2\phi_j}b_j & a_j
\end{array}
\rangleight)\,,\qquad j=0,\dots,6\,,
\end{eqnarray}
where $a_j=\hat{a}^{\dag}s\theta_j$ and $b_j=\sin\theta_j$. The state (\rangleef{net3}) is
given by 14 real parameters and the network in FIG.\,\rangleef{fnet3} preparing
this state is also determined by 14 parameters (seven rotations), where
\betaegin{eqnarray}
\langleabel{net5}
\phi_0=\frac{1}{2}(\pi-\varphi_7)\,, \qquad
\phi_j=\frac{1}{2}(\varphi_j-\varphi_7)\,,\qquad j=1,\dots,6
\end{eqnarray}
and
\betaegin{eqnarray}
\langleabel{net6}
b_0=\sqrt{1-\alphalpha_0^2}\,,\qquad
b_j=\frac{\alphalpha_j}{\sqrt{1-\sum\langleimits_{k=0}^{j-1}\alphalpha_k^2}}\,,\qquad
j=1,\dots,6\,.
\end{eqnarray}
Thus, the mapping between the state under preparation (\rangleef{net3})
and the network (FIG. \rangleef{fnet3}) is clearly defined.
\section{Speed of quantum gates}
\langleabel{speed}
One of the most important requirements for the implementation of quantum
logic (Sec.\,\rangleef{impl}) on a particular candidate quantum system
is the physical realization of quantum gates on time scales which are
much shorter compared to time
scales associated with decoherence effects.
We have represented quantum gates on cold trapped ions with unitary
evolution operators (\rangleef{qg3}), (\rangleef{qg5}) and (\rangleef{qg6}) associated with
laser pulses on the carrier and on the first red sideband. However, these
operators are valid only in the Lamb-Dicke and weak coupling regime. Taking
into account the complete Hamiltonian (\rangleef{int9})
we have to deal with resonant and off-resonant transitions
accompanied with Stark light shifts of the energy levels (Appendix \rangleef{Off}).
A detailed treatment of this problem was presented by Steane et al.
in Ref. \cite{speed} and we adopt some of their results in this section.
The analysis of the speed of gate operations in ion traps was firstly
discussed by Plenio and Knight in Ref. \cite{plen1, plen2}.
At first we discuss the speed of single-qubit rotations and two-qubit CNOT
gates. Then we include some estimations for the speed of multi-qubit CNOT
gates with cold trapped ions.
\betaegin{eqnarray}gin{itemize}
\item
The single-qubit rotations are associated with the transition on the carrier
(\rangleef{qg3}) with the duration $T_{{\cal A}}=\ell\pi/|\langleambda|$,
i.e. a $\ell\pi$-pulse on the carrier applied on a given ion. However,
we have to consider rather the evolution operator corresponding to the
Hamiltonian (\rangleef{int9})
when we want to discuss unwanted off-resonant transitions. Directing
the laser beam such that it is perpendicular to the $z$ axis,
the Lamb-Dicke parameter $\eta$ becomes equal to zero and off-resonant
transitions $|e\rangle|n\rangle\langleeftrightarrow |g\rangle|n\pm |k|\rangle$ for $k\neq 0$ do
not appear in the dynamics. Therefore, one can make the laser coupling
constant $|\langleambda|$ large without the restriction on the weak coupling
regime characterized by the condition $|\langleambda|\langlel\nu$
(Appendix \rangleef{Off}).
We can assume $|\langleambda|/2\pi\simeq 300\,\mbox{kHz}$, then we get typically
for a $\pi/2$-pulse on the carrier $T_{{\cal A}}\simeq 1\,\mu\mbox{s}$.
\item
The two-qubit CNOT gate (\rangleef{qg7}) is realized by two $\pi/2$-pulses on the
carrier [$\hat{{\cal A}}_{m_2}^{1/2}(0)$, $\hat{{\cal A}}_{m_2}^{1/2}(\pi)$],
two $\pi$-pulses ($\hat{{\cal B}}_{m_1}^{1,I}$) and a single $2\pi$-pulse
($\hat{{\cal B}}_{m_2}^{2,II}$) on the first red sideband.
A pulse on the first red sideband is represented with
the unitary evolution operator (\rangleef{int19}) for $k=-1$. However, it was derived
for an ideal case when off-resonant transitions and Stark light shifts were
not considered. We can correct for the light shifts
by tuning the laser on the frequency
$\omega_L=\omega_0-\omega_z+\Delta\omega$, where $\Delta\omega$
corresponds to the light shifts caused by the presence of the carrier
transitions. The imprecision caused by the excitation of off-resonant
transitions can be corrected by applying a correction laser pulse
with a correspondingly adjusted phase. This was accomplished in Ref. \cite{speed}
and the limit for the duration of the operation corresponding
to the $\pi$-pulse on the first red sideband is given as
\betaegin{eqnarray}
\langleabel{sp1}
\frac{1}{T_{{\cal B}}}\langleeq
2\sqrt{2}\epsilon\,\sqrt{\frac{E_r}{Nh}\frac{\omega_z}{2\pi}}\,,
\end{eqnarray}
where $\epsilon=\sqrt{1-F}$ is the imprecision defined via the fidelity $F$,
$E_r=\hbar^2\kappa_{\vartheta}^2/2m$ is the recoil energy of a single ion of
the mass $m$, $\kappa=(2\pi/\Lambda)\hat{a}^{\dag}s\vartheta$,
$\Lambda$ is the laser wavelength,
$h=2\pi\hbar$ and $\omega_z$ is the axial trapping frequency.
The limit for the duration of the operation
corresponding to the $2\pi$-pulse on the first red sideband
is the double of the expression given by Eq. (\rangleef{sp1}).
\end{itemize}
In TABLE \rangleef{tab} we give the estimations for Calcium ions $^{40}\mbox{Ca}^+$.
We assume the angle between the laser beam and the $z$ axis to be
$\vartheta=60^{\circ}$,
the laser wavelength is $\Lambda=729\,\mbox{nm}$ and the axial trapping frequency
is $\omega_z/2\pi\simeq 700\,\mbox{kHz}$. Then we get the recoil frequency
$E_r/h\simeq 2.33\,\mbox{kHz}$ and $\eta=\sqrt{E_r/\hbar\omega_z}\simeq 0.06$.
\betaegin{eqnarray}gin{table}[htb]
\betaegin{eqnarray}gin{center}
\betaegin{eqnarray}gin{tabular}{|c|cc|cc|}
\hline
&
\multicolumn{2}{|c|}{$T_{{\cal B}}\ [\mu\mbox{s}]$} &
\multicolumn{2}{|c|}{$T\ [\mbox{ms}]$}\\[0.75mm]
$\ \ N\ \ $ &
$\ F=99\%\ $ &
$\ F=75\%\ $ &
$\ F=99\%\ $ &
$\ F=75\%\ $\\[0.75mm]
\hline\hline
2 & 124 & 24.8 & 0.50 & 0.10\\[0.75mm]\hline
3 & 152 & 30.3 & 0.91 & 0.18\\[0.75mm]\hline
6 & 214 & 42.9 & 2.58 & 0.52\\[0.75mm]\hline
9 & 263 & 52.5 & 4.74 & 0.98\\[0.75mm]\hline
10 & 277 & 55.4 & 5.55 & 1.12\\[0.75mm]\hline
\end{tabular}
\end{center}
\caption{{\footnotesize
$N$ is the total number of the ions confined in the trap and also the number
of the ions involved in the realization of the multi-qubit CNOT gate,
${T}_{{\cal B}}$ is the duration of the operation corresponding to
the $\pi$-pulse on the first red sideband given by Eq.~(\rangleef{sp1})
calculated for two values of the fidelity
($F=99\%, F=75\%$) and $T$ is the total minimal time
[Eq.~(\rangleef{sp2})] for the realization of the multi-qubit CNOT gate on $N$ ions
[see Eq.~(\rangleef{qg21})] evaluated for two different fidelities.}}
\langleabel{tab}
\end{table}
The multi-qubit CNOT on $Q$ ions (\rangleef{qg21}) differs
from the two-qubit
CNOT gate only in the number of laser pulses required for its realization.
The multi-qubit CNOT gate corresponds to two $\pi/2$-pulses on the carrier
[$\hat{{\cal A}}_{m_{q+1}}^{1/2}(0)$, $\hat{{\cal A}}_{m_{q+1}}^{1/2}(\pi)$],
a single $2\pi$-pulse on the first red sideband
($\hat{{\cal B}}_{m_{q+1}}^{2,II}$) and $(2Q-2)$ $\pi$-pulses also on the
first red sideband
($\hat{{\cal B}}_{m_1}^{1,I}$, $\hat{{\cal B}}_{m_2}^{1,II}$, $\dots$,
$\hat{{\cal B}}_{m_{q}}^{1,II}$). Thus, it requires all together $2Q+1$
laser pulses, where $Q=q+1$ refers to the number of the ions
involved in the gate ($q$ control ions, one target ion).
Then, in the spirit of the previous discussion
the minimal total time for the realization of the multi-qubit CNOT gate
on $Q$ ions reads
\betaegin{eqnarray}
\langleabel{sp2}
T=2(T_{{\cal A}}+QT_{{\cal B}})\,,
\end{eqnarray}
where we assume $T_{{\cal A}}=5\,\mu\mbox{s}$ and $T_{{\cal B}}$ is given by
Eq. (\rangleef{sp1}) for a total number $N$ ($Q\langleeq N$) of the ions confined
in the trap. We give some estimations for the realization of
multi-qubit CNOT gates in TABLE \rangleef{tab}, where the number~$Q$ of the ions
involved in the gate and the total number~$N$ of the ions in the
trap are equal ($N=Q$). We stress this point because there is a difference if
we realize a two-qubit CNOT gate ($Q=2$) having just two ions in the trap
($N=2$), then we get for the total time of the gate
\betaegin{eqnarray}
\langleabel{sp3}
T_1=2(T_{{\cal A}}+2T_{{\cal B}}^{N=2})\simeq 0.5\,\mbox{ms}\,,
\end{eqnarray}
or having a larger register of ten ions in the trap ($N=10$), what gives
\betaegin{eqnarray}
\langleabel{sp4}
T_2=2(T_{{\cal A}}+2T_{{\cal B}}^{N=10})\simeq 1.1\,\mbox{ms}
\end{eqnarray}
at the same fidelity $F=99\%$.
Any multi-qubit gate on a register of size $N$ can be decomposed into a network
of single-qubit rotations and two-qubit CNOT gates \cite{95-8}. However,
there might be a possibility to realize this multi-qubit directly, if
a given physical system allows it. For instance, the multi-qubit CNOT gate on six
qubits can be decomposed into the network of 12 two-qubit CNOT gates
including three additional auxiliary qubits \cite{95-8}.
In the case of cold trapped
ions it requires the total time for the realization of the whole network
($Q=2$, $N=9$)
\betaegin{eqnarray}
\langleabel{sp5}
T_3=12\times 2(T_{{\cal A}}+2T_{{\cal B}}^{N=9})\simeq 12.7\,\mbox{ms}\,.
\end{eqnarray}
However, the direct implementation (\rangleef{qg21}) would take only ($Q=N=6$)
\betaegin{eqnarray}
\langleabel{sp6}
T_4=2(T_{{\cal A}}+6T_{{\cal B}}^{N=6})\simeq 2.6\,\mbox{ms}\,,
\end{eqnarray}
which is about five-times less than the former case.
We have again assumed almost
the perfect fidelity $F=99\%$ of the operation. We conclude that
there can be quantum systems that may support a direct implementation of
multi-qubit gates, rather than their decomposition into fundamental gates,
what may bring advantages at experimental realization as well as in
quantum state synthesis \cite{ms}.
\section{Discussion}
\subsection{Decoherence}
Throughout the paper we have discussed many aspects of cold trapped ions for
quantum computing but we have not dealt with the decoherence which appears
to be a main obstacle in achievements of experimental quantum computing.
The reason was that our main goal has been to give a basic review and the
discussion on decoherence sources and effects would refer more to an
advanced study \cite{plen1, plen2, garg, hughes, dfv, res, dfqm}.
Nevertheless, for the sake of completeness we would like to
mention on this place some decoherence aspects met in the lab. We will
follow a detailed study of experimental issues in quantum manipulations with
trapped ions given by Wineland et al. \cite{98-5}. The decoherence will be
met in a more general usage of this term. Thus, by the {\it decoherence}
we mean any effect that limits the fidelity (the match between
desired and achieved realization). Further, we will distinguish three
categories.
\betaegin{eqnarray}gin{itemize}
\item
{\it Motional state decoherence} is the most troublesome source of
the decoherence in ion trap experiments and refers to the relaxation of two
vibrational states $|n=0\rangle$ and $|n=1\rangle$ of a given motional mode used as
the quantum data bus. The ions are cooled to the ground motional state
$|n=0\rangle$ and the excitation to the $|n=1\rangle$ state is used for the transfer
of information on a distinct ion. However, this scenario is not ideal
for several reasons:
\betaegin{eqnarray}gin{itemize}
\item[$\circ$] Instability of trap parameters. We are simply not able to
control all voltages as they undergo fluctuations and dephasing.
\item[$\circ$] We also have to count on (i) the micromotion, (ii) the
Coulomb repulsion between the ions making the motional modes
(except the COM mode) anharmonic in reality and (iii) stray electrode fields
causing possible excitations of the ion motion.
\item[$\circ$] We have considered just a single motional mode in our
approach, but there are also other $3N-1$ modes present and the
cross-coupling between the modes appears. If spectator $3N-1$ modes are not
cooled to their ground motional states, the energy can be transferred to the
mode of interest. This happens because the trapping potential is anharmonic
in real and these higher anharmonic terms are responsible for the
cross-coupling.
\item[$\circ$] We should mention also inelastic and elastic collisions with the
background gas, even though experiments are carried out in an excellent
environment ($p\simeq 10^{-8}\,\mbox{Pa}$).
\end{itemize}
\item
{\it Internal state decoherence} corresponds to the evolution when
a pure state of the ion $|\psi\rangle=\alpha|g\rangle+\beta|e\rangle$ transforms into a mixture
$\hat{\rangleho}=|\alpha|^2|g\rangle\langle g|+|\beta|^2|e\rangle\langle e|$. The ions demonstrate internal
decoherence times of the order of seconds (Calcium) up to minutes and hours
(Beryllium). The type of the decoherence discussed here can be eliminated by
a proper choice of metastable excited states with long lifetimes.
\item
{\it Operational decoherence} refers to the precision of coherent laser-ion
manipulations. There are several aspects that we have to consider:
\betaegin{eqnarray}gin{itemize}
\item[$\circ$]
When the ion is illuminated with a laser beam, one has to
control the pulse duration and the phase adjustment in order to avoid the
preparation of unwanted states.
\item[$\circ$]
Due to the laser spatial intensity
profile, there is a probability (if the ions are spaced too closely) that
the state of a neighbouring ion will be affected.
\item[$\circ$]
If we consider the standing-wave configuration we have to take care of
the precise position of the ion in the node or the antinode of the standing
wave what seems to be very troublesome.
\item[$\circ$]
Finally, off-resonant transitions are always present and we have to control
the laser power very carefully to avoid their excitations.
\end{itemize}
\end{itemize}
However, there is a way to eliminate the effect of the decoherence.
We can encode information into a~decoherence-free subspace whose states are
invariant under coupling to the environment \cite{dfs, knight1, knight2}.
\subsection{Ion trap systems}
We have been discussing cold trapped ions so far. {\it Cold} refers to the
fact that all motional modes have to be cooled to their ground motional
states because the dynamics assumes the precise control over the motional
state. However, there have appeared other proposals referring to {\it warm}
or {\it hot} trapped ions which assume an arbitrary motional state.
\betaegin{eqnarray}gin{itemize}
\item Poyatos et al.\,\cite{poyatos} proposed a scheme for the realization of
two-qubit CNOT gates between two trapped ions using ideas from the atomic
interferometry. They split the wavepacket of the control ion into two
directions depending on its internal state with a laser pulse.
Then they address one of the
wavepackets of the target ion changing conditionally its internal state and
finally they bring together the wavepackets of the ions using another laser
pulse. The ions communicate through the Coulomb repulsion and under ideal
conditions the scheme is independent on the motional state of the ions.
\item Milburn et al.\,\cite{milburn} described two schemes for manipulations
with warm trapped ions. Firstly, they use the adiabatic passage for
the conditional phase shift, i.e. the phase of the ion is flipped if the
motional mode is in the superposition of odd number states and the ion is
excited. The COM mode in an arbitrary vibrational state is used for quantum
logic but all other motional modes are assumed to be cooled to their ground
motional states. Secondly, they apply the idea of the collective spin
\cite{collspin} for faster gates. This idea has been also used for
the introduction of multi-qubit gates for quantum computing \cite{coll}.
\item S{\o}rensen and M{\o}lmer \cite{hot1, hot2}
proposed a novel scheme based on the idea of
bichromatic light ($\omega_1, \omega_2$).
Realizing the two-qubit CNOT gate they illuminate two
ions with the bichromatic light coupling the states $|gg\rangle|n\rangle$ and
$|ee\rangle|n\rangle$. They choose detunings far enough from the resonance with
the first red and blue sidebands such that the intermediate states
$|eg\rangle|n\pm 1\rangle$ and $|ge\rangle|n\pm 1\rangle$ are not populated in the process.
The scheme is not sensitive on fluctuations of the number of phonons in the
relevant motional mode.
It is also possible to illuminate with the bichromatic light more ions and
generate a multiparticle entangled state. Actually, these experiments were
already realized in NIST \cite{sackett}
and they generated the GHZ state with two ions
$|\psi_2\rangle=1/\sqrt{2}(|gg\rangle-i|ee\rangle)$ with a fidelity $F=83\%$ and also the
GHZ state with four ions $|\psi_4\rangle=1/\sqrt{2}(|gggg\rangle+i|eeee\rangle)$
with a fidelity $F=57\%$ using Beryllium ions and the Raman scheme.
Jonathan and Plenio proposed light shift induced quantum gates for trapped
ions insensitive on phonon number in motional modes (thermal motion)
\cite{jon2}.
\item Finally, there has appeared a proposal of a scalable quantum computer
with the ions in an array of microtraps by Cirac and Zoller \cite{micro1}
detailed in Ref. \cite{micro2}. The ions are placed in a 2D array of
independent ion microtraps \cite{devoe}
and there is another ion (head) that moves above
this plane. If we position the head above a particular ion from the array
and switch on the laser in the perpendicular direction, we can realize a
two-qubit gate. This operation allows us to swap the state of the ion
to the head which can be moved immediately above a distinct ion in the array
and transfer the information onto it. The ions oscillating in the microtraps
are not assumed to be cooled to their ground motional states. However,
their motion can couple to the environment. It becomes relevant during
the time when the ion interacts with the head but not in the case
when the head moves.
\end{itemize}
\section{Conclusion}
In this paper we have tried to review achievements accomplished in the field
of cold trapped ions. In the first part we have discussed in detail
the ion loading and trapping process, the collective vibrational motion
of the ions. We have also given a detailed
derivation of the Hamiltonian governing the dynamics of the system including
the discussion of weak coupling and Lamb-Dicke regime. Further,
we have reviewed laser cooling techniques and a detection process with
experimental illustrations on Calcium ions. In the second part we have
discussed the implementation of quantum computing using cold trapped ions.
In particular,
we have described how to realize single-qubit, two-qubit and multi-qubit quantum
logic gates. Finally, we have estimated the speed of quantum gates
with cold trapped ions. The aim of this
paper is to give an introduction to this field with many references on
relevant papers and studies.
\section*{Acknowledgments}
We would like to thank
Peter Knight, Danny Segal, Martin Plenio, Andrew Steane
and Miloslav Du\v{s}ek
for their helpful comments and suggestions.
We are also grateful to Rainer Blatt, Giovanna Morigi and David
Kielpinski for sending us original files of their figures.
This work was supported by the European Union projects
QUBITS (IST-1999-13021) and QUEST (HPRN-CT-2000-00121).
We acknowledge the special support from
the Slovak Academy of Sciences.
One of us (M.\v{S}.) is thankful for the support from the ESF via the
{\em Programme on Quantum information and quantum computation}.
\alphappendix
\section{Weak coupling regime}
\langleabel{Off}
The expression (\rangleef{int9}) corresponds to the complete Hamiltonian
in the sense that it includes also off-resonant transitions. For instance,
even though a sufficiently intense laser is tuned on the carrier,
the off-resonant transitions on the sidebands are also present and they cause
imprecisions and perturbations in the dynamics. This can be avoided by
setting the laser intensity $I\propto |\langleambda|^2$,
i.e. the laser coupling constant [Eq.~(\rangleef{int8}) or (\rangleef{int8.1})],
sufficiently small.
In what follows we
will determine conditions (characterizing the {\it weak coupling regime})
under which we can neglect off-resonant transitions.
\subsection{Off-resonant transitions}
The implementation of quantum gates on cold trapped ions requires laser
pulses on the carrier [FIG.\,\rangleef{3freq}(a)] and on the first red sideband
[FIG.\,\rangleef{3freq}(b)]. Therefore, we will discuss the dynamics on these two
spectral lines.
At first, let us assume that the laser is tuned on the carrier $(\delta=0)$.
The closest off-resonant transitions (detuned by the frequency~$\nu$) are on
the first blue and on the first red sideband [FIG.\,\rangleef{off}(a)].
We will consider only the respective terms in the Hamiltonian
(\rangleef{int9}) and drop down the index $j$. In the Lamb-Dicke limit we can write
\betaegin{eqnarray}
\langleabel{off1}
\hat{{\cal H}}_1=
\frac{\hbar\langleambda}{2}\hat{\sigma}_++
\frac{\hbar\langleambda}{2}(i\eta)\hat{\sigma}_+\hat{a}^{\dag}\,e^{i\nu t}+
\frac{\hbar\langleambda}{2}(i\eta)\hat{\sigma}_+\alphao\,e^{-i\nu t}+
\mbox{H.c.}\,.
\end{eqnarray}
Further, we assume the ion to be initially in the state $|g\rangle|n\rangle$, we apply
the Hamiltonian (\rangleef{off1}) and calculate the probability of
off-resonant transitions. The dynamics governed by a time-dependent
Hamiltonian $\hat{{\cal H}}(t)$ is described to the first order by the
unitary evolution operator
\betaegin{eqnarray}
\langleabel{off3}
\hat{U}(t,t_0)\alphapprox
\openone-
\frac{i}{\hbar}\int_{t_0}^t\hat{{\cal H}}(t^{\prime})dt^{\prime}\,.
\end{eqnarray}
Then the probability to find the ion (initially prepared in the state
$|g\rangle|n\rangle$) in the state $|e\rangle|n+1\rangle$ (i.e. undergoing the off-resonant
transition on the first blue sideband) is
\betaegin{eqnarray}
\langleabel{off4}
P_B=|\,\langle e|\langle n+1|\,\hat{U}_1\,|g\rangle|n\rangle\,|^2=
\frac{|\langleambda|^2\eta^2(n+1)}{\nu^2}
\sin^2\langleeft[\frac{\nu(t-t_0)}{2}\rangleight]\,,
\end{eqnarray}
where $\hat{U}_1$ is given by Eq. (\rangleef{off3}) for the Hamiltonian
(\rangleef{off1}). The probability to find the ion in the state $|e\rangle|n-1\rangle$
corresponding to the off-resonant transition on the first red sideband is
given as
\betaegin{eqnarray}
\langleabel{off5}
P_R=|\,\langle e|\langle n-1|\,\hat{U}_1\,|g\rangle|n\rangle\,|^2=
\frac{|\langleambda|^2\eta^2n}{\nu^2}
\sin^2\langleeft[\frac{\nu(t-t_0)}{2}\rangleight]\,.
\end{eqnarray}
If there is no population transferred via the off-resonant transitions to
the states $|e\rangle|n+1\rangle$ and $|e\rangle|n-1\rangle$, i.e. $P_B\langlel 1$ and $P_R\langlel 1$ at
any time $t$, we can neglect these off-resonant transitions. Then we get the
conditions of the weak coupling regime for the transition on the carrier
in the form
\betaegin{eqnarray}
\langleabel{off6}
|\langleambda|\eta\sqrt{n+1}\langlel\nu
\end{eqnarray}
and
\betaegin{eqnarray}
\langleabel{off7}
|\langleambda|\eta\sqrt{n}\langlel\nu\,.
\end{eqnarray}
We can also avoid the off-resonant transitions by setting the laser beam
perpendicular to the $z$ axis ($\vartheta=\pi/2$). Then the Lamb-Dicke
parameter [see Eq. (\rangleef{int5})] is equal to zero
($\eta\simeq\kappa\hat{a}^{\dag}s\vartheta z_0$) and the coupling on the off-resonant
transitions vanishes.
\betaegin{eqnarray}gin{figure}[htb]
\centerline{\epsfig{width=11cm,file=ions100.eps}}
\caption{{\footnotesize
(a) The laser is tuned on the carrier ($\delta=0$) with the coupling
constant $\langleambda$. The off-resonant transitions are present on the first
blue and red sideband with the coupling constant $\eta\langleambda$.
(b) The laser is tuned on the first red sideband ($\delta=-\nu$) with
the coupling constant $\eta\langleambda$. The off-resonant transitions are
present on the carrier with the coupling constant $\langleambda$ and on the weak
second red sideband ($\eta^2\langleambda$).}}
\langleabel{off}
\end{figure}
Analogically, we can assume the laser to be tuned on the first red sideband
($\delta=-\nu$) and the closest off-resonant transitions are on the carrier
and on the second red sideband [FIG.\,\rangleef{off}(b)]. However, the strength
of the second red sideband is of the order of $\eta^2$ and we can omit it in
the Lamb-Dicke limit.
Then the respective Hamiltonian is given as
\betaegin{eqnarray}
\langleabel{off8}
\hat{{\cal H}}_2=
\frac{\hbar\langleambda}{2}(i\eta)\hat{\sigma}_+\alphao+
\frac{\hbar\langleambda}{2}\hat{\sigma}_+e^{i\nu t}+
\mbox{H.c.}\,.
\end{eqnarray}
The probability to find the ion (initially in the state $|g\rangle|n\rangle$) in the
state $|e\rangle|n\rangle$ (after the off-resonant transition on the carrier) can be
calculated as
\betaegin{eqnarray}
\langleabel{off9}
P_C=|\,\langle e|\langle n|\,\hat{U}_2\,|g\rangle|n\rangle\,|^2=
\frac{|\langleambda|^2}{\nu^2}\sin^2\langleeft[\frac{\nu (t-t_0)}{2}\rangleight]\,,
\end{eqnarray}
where $\hat{U}_2$ is given by Eq. (\rangleef{off3}) for the Hamiltonian
(\rangleef{off8}).
We can neglect the off-resonant dynamics if $P_C\langlel 1$ at any time $t$.
Then for the weak coupling regime on the first red sideband applies
\betaegin{eqnarray}
\langleabel{off11}
|\langleambda|\langlel\nu\,.
\end{eqnarray}
Even though the transition on the carrier is off-resonant,
it has stronger coupling in the Lamb-Dicke limit than the first red sideband.
Therefore, it is very important to follow in the experiment the constraint given
by Eq. (\rangleef{off11}).
\subsection{Stark light shifts}
Besides the population of the off-resonant levels there is another
source of imprecisions in the state manipulation.
However, it is weaker and it doesn't
require any special constraints on physical parameters except those for the
weak coupling regime. When the laser drives a transition between two levels
there appears a frequency shift called {\it Stark light shift}
caused by the presence of other spectator levels. Therefore, in the experiment we
have to consider the detuning (\rangleef{int9.0}) rather in the form
\betaegin{eqnarray}
\langleabel{off20}
\tilde{\delta}=\omega_L-\omega_0-\Delta\omega\,,
\end{eqnarray}
where
$\Delta\omega$ corresponds to the Stark light shift. The higher the laser
intensity, the more significant the light shift is. We can correct for this
effect by shifting the laser frequency
($\omega_L\rangleightarrow\omega_L+\Delta\omega$) which tunes the transition back
to the resonance. Further, we will estimate the frequency shift
$\Delta\omega$.
Let us consider the Hamiltonian (\rangleef{int3}) and transform it to the
interaction picture (\rangleef{int6}) with $\hat{U}_0=\exp(-i\hat{H}_0t/\hbar)$, where
$\hat{H}_0=(\hbar\omega_L/2)\hat{\sigma}_z$. Then we get
\betaegin{eqnarray}
\langleabel{off21}
\hat{{\cal H}}=\hat{{\cal H}}_0+\hat{{\cal V}}=
\langleeft(-\frac{\hbar\delta}{2}\hat{\sigma}_z+\hbar\nu\hat{a}^{\dag}\alphao\rangleight)+
\langleeft(
\frac{\hbar\langleambda}{2}\hat{\sigma}_+\,e^{i\eta(\alphao+\hat{a}^{\dag})}+\mbox{H.c.}
\rangleight)\,,
\end{eqnarray}
where in the Lamb-Dicke limit the interaction term reduces to
\betaegin{eqnarray}
\langleabel{off22}
\hat{{\cal V}}\alphapprox
\frac{\hbar\langleambda}{2}\hat{\sigma}_++
\frac{\hbar\langleambda}{2}(i\eta)\hat{\sigma}_+\hat{a}^{\dag}+
\frac{\hbar\langleambda}{2}(i\eta)\hat{\sigma}_+\alphao+
\mbox{H.c.}
\end{eqnarray}
with the first term corresponding to the transition on the carrier, the
second term to the first blue sideband and the last one to the transition on
the first red sideband.
In the second order of the time-independent perturbation theory (the first
order gives no contribution) we can write for the shift of the energy levels
\betaegin{eqnarray}
\langleabel{off23}
\Delta E_{|g\rangle|n\rangle}&=&
\sum_{m\alphatop \delta\neq\nu (m-n)}
\frac{\langle g|\langle n|\hat{{\cal V}}|e\rangle|m\rangle\langle e|\langle m|\hat{{\cal V}}|g\rangle|n\rangle}
{\hbar\delta+\hbar\nu (n-m)}
\end{eqnarray}
and
\betaegin{eqnarray}
\langleabel{off24}
\Delta E_{|e\rangle|n\rangle}&=&
\sum_{m\alphatop \delta\neq\nu (n-m)}
\frac{\langle e|\langle n|\hat{{\cal V}}|g\rangle|m\rangle\langle g|\langle m|\hat{{\cal V}}|e\rangle|n\rangle}
{-\hbar\delta+\hbar\nu (n-m)}\,.
\end{eqnarray}
For instance, for the transition on the first red sideband ($\delta=-\nu$)
we can calculate
\betaegin{eqnarray}
\langleabel{off25}
\Delta E_{|g\rangle|n\rangle}\alphapprox -\frac{\hbar |\langleambda|^2}{4\nu}\,,\qquad
\Delta E_{|e\rangle|n-1\rangle}\alphapprox\frac{\hbar |\langleambda|^2}{4\nu}
\end{eqnarray}
and the corresponding light shift can be estimated as
\betaegin{eqnarray}
\langleabel{off26}
\Delta\omega=
\betaigg(\Delta E_{|e\rangle|n-1\rangle}-\Delta E_{|g\rangle|n\rangle}\betaigg)/\hbar\alphapprox
\frac{|\langleambda|^2}{2\nu}\,.
\end{eqnarray}
If we choose $|\langleambda|/2\pi=50\,\mbox{kHz}$ and
$\nu/2\pi=700\,\mbox{kHz}$ so
that the condition (\rangleef{off11}) holds, then
$\Delta\omega/2\pi\simeq 1.8\,\mbox{kHz}$.
\section{Lamb-Dicke regime}
\langleabel{ldl}
In the relation for the coupling constant (\rangleef{int15})
we can expand the exponential function to the Taylor series
about the value $\eta_j=0$ and use the expression (\rangleef{int16})
for the Laguerre polynomial. Then we get
\betaegin{eqnarray}
\langleabel{ldl1}
\Omega_j^{n,k}(\eta)&=&
\langleambda_j
\langleeft(i\eta_j\rangleight)^{|k|}
\sqrt{\frac{n!}{(n+|k|)!}}
\langleeft[1-\frac{\eta_j^2}{2}+{\cal O}(\eta_j^4)\rangleight]\\
&\times&
\langleeft[\langleeft({n+|k|\alphatop n}\rangleight)
-\eta_j^2\langleeft({n+|k|\alphatop n-1}\rangleight)+{\cal O}(\eta_j^4)\rangleight]
\nonumber\,,
\end{eqnarray}
where ${\cal O}(\eta_j^4)$ denotes the terms proportional to the fourth and
higher powers of $\eta_j$.
In the {\it Lamb-Dicke regime}
we consider the dependence of $\Omega_j^{n,k}$ on the parameter $\eta_j$
only to its lowest order, i.e. in Eq. (\rangleef{ldl1})
we neglect all terms with any higher
power than $\eta_j^{|k|}$ and we get the coupling constant
$\Omega_j^{n,k}$ given by the expression (\rangleef{int20}).
This approximation can be done only if the conditions
\betaegin{eqnarray}
\langleabel{ldl2}
\frac{\eta_j^2}{2}\langlel 1
\end{eqnarray}
and
\betaegin{eqnarray}
\langleabel{ldl3}
\eta_j^2
\langleeft({n+|k|\alphatop n-1}\rangleight)\langlel
\langleeft({n+|k|\alphatop n}\rangleight)\qquad{\cal R}ightarrow\qquad
\eta_j^2\frac{n}{|k|+1}\langlel 1
\end{eqnarray}
are satisfied.
We will refer to the condition ($|k|=0$)
\betaegin{eqnarray}
\langleabel{ldl4}
\eta_j\sqrt{\langle n \rangle+\frac{1}{2}}\langlel 1
\end{eqnarray}
as the {\it Lamb-Dicke limit}, where $\langle n \rangle$ is the average number of
phonons in the respective vibrational mode \cite{jonathan}.
\betaegin{eqnarray}gin{itemize}
\item
The Lamb-Dicke limit corresponds physically to the situation where
the spatial extent of the vibrational motion of the ion $z_0$
is much smaller than the wavelength $\Lambda$ of the laser,
where $\eta_j\simeq\kappa z_0$ and
$\kappa=2\pi/\Lambda$ [see def. in Eq.~(\rangleef{int5})].
\item
The Lamb-Dicke limit can be physically interpreted also from a different
point of view. We may rewrite the Lamb-Dicke parameter of the $j$ ion of $N$
ions in the COM mode to the form $\eta_j^2=E_r/\hbar\omega_z$,
where $E_r=\hbar^2\kappa^2/2mN$ is the recoil energy.
It can be shown \cite{tan2} that the trapped ion emits spontaneously photons
of the average energy
$\hbar\omega_0-E_r$, where $\omega_0$ is for the atomic frequency. Taking
into account the Lamb-Dicke limit ($E_r\langlel\hbar\omega_z$) we may say that
during the spontaneous emission the change in the vibrational state of the
ion is very unlikely. In other words, the trapped ion in the Lamb-Dicke
regime decays spontaneously mostly on the carrier
$(\omega_0\alphapprox\omega_0-E_r/\hbar)$.
\item
The Lamb-Dicke parameter for a single trapped ion equals to $\betaar{\eta}$
and for an ion from the string of $N$ ions in the COM mode is given as
$\eta_j=\betaar{\eta}/\sqrt{N}$. It means that we can reach the Lamb-Dicke
limit (\rangleef{ldl4})
for $N$ ions even if the limit is not fulfilled for single ions
\cite{morigi}.
\end{itemize}
\thebibliography{99}
\betaibitem{NIST} {\tt http://www.bldrdoc.gov/timefreq/ion}
\betaibitem{95-5} J.I. Cirac and P. Zoller,
{\it Quantum computation with cold trapped ions},
Phys. Rev. Lett. {\betaf 74}, 4091 (1995)
\betaibitem{sam} Fortschritte der Physik {\betaf 48}, Number 9 -- 11 (2000)
\betaibitem{00-4} A. Ekert, P. Hayden, H. Inamori
{\it Basic concepts in quantum computation},
{\tt quant-ph/0011013} (2000)
\betaibitem{divin} D.P. DiVincenzo,
{\it The physical implementation of quantum computation}
Fortschritte der Physik {\betaf 48}, 771 (2000)
\betaibitem{loss} Private communication with Daniel Loss.
\betaibitem{95-8}
A. Barenco, Ch. Bennett, R. Cleve, D.P. DiVincenzo, N. Margolus, P. Shor, T.
Sleator, J.A. Smolin and H. Weinfurter,
{\it Elementary gates for quantum computation},
Phys. Rev. {\betaf A52}, 3457 (1995)
\betaibitem{russia} V.I. Balykin, V.G. Minogin and V.S. Letokhov,
{\it Electromagnetic trapping of cold atoms},
Rep. Prog. Phys. {\betaf 63}, 1429 (2000)
\betaibitem{90-1} W. Paul,
{\it Electromagnetic traps for charged and neutral particles}
Reviews of Modern Physics {\betaf 62}, 531 (1990)
\betaibitem{ghosh} P.K. Ghosh,
{\it Ion traps} (Clarendon Press, Oxford 1995)
\betaibitem{LesH} edited by J. Dalibard et al.,
{\it Fundamental systems in quantum optics} (Elsevier, Amsterdam 1992)
\betaibitem{innsbruck} {\tt http://heart-c704.uibk.ac.at}
\betaibitem{nagerl} H.Ch. N\"{a}gerl,
{\it PhD thesis} (Innsbruck 1998)
\betaibitem{roos} Ch.F. Roos,
{\it PhD thesis} (Innsbruck 2000)
\betaibitem{00-3} H.Ch. N\"{a}gerl, C. Roos, H. Rohde, D. Leibfried, J. Eschner,
F. Schmidt-Kaler and R. Blatt,
{\it Addressing and cooling of single ions in Paul traps},
Fortschritte der Physik {\betaf 48}, 623 (2000)
\betaibitem{blatt1} D. Leibfried, C. Roos, P. Barton, H. Rohde, S. Gulde,
A. Mundt, G. Reymond, M. Lederbauer, F. Schmidt-Kaler, J. Eschner
and R. Blatt,
{\it Experiments towards quantum information with trapped Calcium ions},
{\tt quant-ph/0009105} (2000)
\betaibitem{blatt2} H. Rohde, S.T. Gulde, C.F. Roos, P.A. Barton, D. Leibfried,
J. Eschner, F. Schmidt-Kaler and R. Blatt,
{\it Sympathetic ground state cooling and coherent manipulation with
two-ion crystals},
{\tt quant-ph/0009031} (2000)
\betaibitem{zig-zag}
D.G. Enzer, M.M. Schauer, J.J. Gomez, M.S. Gulley, M.H. Holzscheiter,
P.G. Kwiat, S.K. Lamoreaux, C.G. Peterson, V.D. Sandberg, D. Tupa,
A.G. White, R.J. Hughes and D.F.V. James,
{\it Observation of power-law scaling for phase transitions in linear
trapped ion crystals},
Phys. Rev. Lett. {\betaf 85}, 2466 (2000)
\betaibitem{oxf} A. Steane,
{\it The Ion Trap Quantum Information Processor},
Appl. Phys. {\betaf B64}, 623 (1997)
\betaibitem{98-7} D.F.V. James,
{\it Quantum dynamics of cold trapped ions with application to quantum
computation},
App. Phys. {\betaf B66}, 181 (1998)
\betaibitem{louisell} W.H. Louisell,
{\it Quantum statistical properties of radiation}
(John Wiley \& Sons, New York 1973
\betaibitem{98-5} D.J. Wineland, C. Monroe, W.M. Itano,
D. Leibfried, B.E. King BE and D.M. Meekhof,
{\it Experimental issues in coherent quantum-state manipulation
of trapped atomic ions},
Journal of Research of the National Institute of Standards and Technology
{\betaf 103}, 259 (1998), (see also {\tt quant-ph/9710025})
\betaibitem{99-2} H.Ch. N\"{a}gerl, D. Leibfried, H. Rohde, G. Thalhammer,
J. Eschner, F. Schmidt-Kaler and R. Blatt,
{\it Laser addressing of individual ions in a linear ion trap},
Phys. Rev. {\betaf A60}, 145 (1999)
\betaibitem{99-7} D. Kielpinski, B.E. King, C.J. Myatt, C.A. Sackett,
Q.A. Turchette, W.M. Itano, C. Monroe, D.J. Wineland and W.H. Zurek,
{\it Quantum Logic Using Sympathetically Cooled Ions},
{\tt quant-ph/9909035} (1999)
\betaibitem{00-11} F. Schmidt-Kaler, C. Roos C, H.Ch. N\"{a}gerl, H. Rohde,
S. Gulde, A. Mundt, M. Lederbauer, G. Thalhammer, T. Zeiger, P. Barton,
L. Hornekaer, G. Reymond, D. Leibfried, J. Eschner and R. Blatt,
{\it Ground state cooling, quantum state engineering and study of
decoherence of ions in Paul traps},
Journal of Modern Optics {\betaf 47}, 2573 (2000)
\betaibitem{exp} D.J. Wineland, C. Monroe, W.M. Itano, B.E. King, D. Leibfried,
D.M. Meekhof, C. Myatt and C. Wood,
{\it Experimental primer on the trapped ion quantum computer},
Fortschritte der Physik {\betaf 46}, 363 (1998)
\betaibitem{garching} {\tt http://www.mpq.mpg.de/laserphysics.html}
\betaibitem{IBM} {\tt http://researchweb.watson.ibm.com/quantuminfo}
\betaibitem{imperial} {\tt http://www.lsr.ph.ic.ac.uk/iontrap}
\betaibitem{JPL} {\tt http://horology.jpl.nasa.gov}
\betaibitem{LANL} {\tt http://p23.lanl.gov/Quantum}
\betaibitem{oxford} {\tt http://www.qubit.org/research/IonTrap}
\betaibitem{aarhus} {\tt http://www.ifa.au.dk/iontrapgroup}
\betaibitem{hamburg}
{\tt http://www.physnet.uni-hamburg.de/ilp/english/research.html}
\betaibitem{mainz} {\tt http://www.physik.uni-mainz.de/werth}
\betaibitem{lanl} R.J. Hughes, D.F.V. James, J.J. Gomez, M.S. Gulley,
M.H. Holzscheiter, P.G. Kwiat, S.K. Lamoreaux, C.G. Peterson, M.M. Sandberg,
V.D. Schauer, C.M. Simmons, C.E. Thorburn, D. Tupa, P.Z. Wang and A.G. White,
{\it The Los Alamos trapped ion quantum computer experiment},
Fortschritte der Physik {\betaf 46}, 329 (1998)
\betaibitem{oct} M. Roberts, P. Taylor, G.P. Barwood, P. Gill,
H.A. Klein and W.R.C. Rowley,
{\it Observation of an Electric Octupole Transition in a Single Ion},
Phys. Rev. Lett. {\betaf 78}, 1876 (1997)
\betaibitem{plen1} M.B. Plenio and P.L. Knight,
{\it Realistic lower bounds for the factorization time of large numbers
on a quantum computer},
Phys. Rev. {\betaf A53}, 2986 (1996)
\betaibitem{plen2} M.B. Plenio and P.L. Knight,
{\it Decoherene limits to quantum computation using trapped ions},
Proc. R. Soc. Lond. {\betaf A453}, 2017 (1997)
\betaibitem{raman} J. Steinbach, J. Twamley and P.L. Knight,
{\it Engineering two-mode interactions in ion traps},
Phys. Rev. {\betaf A56}, 4815 (1997)
\betaibitem{97-2} S.A. Gardiner, J.I. Cirac and P. Zoller,
{\it Nonclassical states and measurement of general motional
observables of a trapped ion},
Phys. Rev. {\betaf A55}, 1683 (1997)
\betaibitem{98-8} S. Chu,
{\it The manipulation of neutral particles},
Reviews of Modern Physics {\betaf 70}, 685 (1998)
\betaibitem{98-9} C.N. Cohen-Tannoudji,
{\it Manipulating atoms with photons},
Reviews of Modern Physics {\betaf 70}, 707 (1998)
\betaibitem{98-10} W.D. Phillips,
{\it Laser cooling and trapping of neutral atoms},
Reviews of Modern Physics {\betaf 70}, 721 (1998)
\betaibitem{metcalf} H.J. Metcalf and P. van der Straten,
{\it Laser cooling and trapping} (Springer-Verlag, New York 1999)
\betaibitem{86-1} S. Stenholm,
{\it The semiclassical theory of laser cooling},
Reviews of Modern Physics {\betaf 58}, 699 (1986)
\betaibitem{zeil} D. Bouwmeester, A. Ekert and A. Zeilinger,
{\it The Physics of Quantum Information} (Springer-Verlag, Berlin 2000)
\betaibitem{morigi0} G. Morigi, J.I. Cirac, K. Ellinger and P. Zoller,
{\it Laser cooling of trapped atoms to the ground state: A dark state in
position space},
Phys. Rev. {\betaf A57}, 2909 (1998)
\betaibitem{89} F. Diedrich, J.C. Bergquist, W.M. Itano and D.J. Wineland,
{\it Laser cooling to the zero-point energy of motion},
Phys. Rev. Lett. {\betaf 62}, 403 (1989)
\betaibitem{king} B.E. King, C.S. Wood, C.J. Myatt, Q.A. Turchette,
D. Leibfried, W.M. Itano, C. Monroe and D.J. Wineland,
{\it Cooling the Collective Motion of Trapped Ions to Initialize a Quantum
Register},
Phys. Rev. Lett. {\betaf 81}, 1525 (1998)
\betaibitem{95-11} C. Monroe, D.M. Meekhof, B.E. King, W.M. Itano
and D.J. Wineland,
{\it Demonstration of a fundamental quantum logic gate},
Phys. Rev. Lett. {\betaf 75}, 4714 (1995)
\betaibitem{jonathan} D. Jonathan,
{\it PhD thesis} (Imperial College, London 2000)
\betaibitem{tan2} C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg,
{\it Atom-photon interactions: Basic processes and applications}
(John Wiley \& Sons, New York 1992)
\betaibitem{morigi} G. Morigi, J. Eschner, J.I. Cirac and P. Zoller,
{\it Laser cooling of two trapped ions: Sideband cooling beyond
the Lamb-Dicke limit},
Phys. Rev. {\betaf A59}, 3797 (1999)
\betaibitem{comm1} Eq. (14) in Ref. \cite{morigi}
\betaibitem{98-16} B.E. King, C.S. Wood, C.J. Myatt, Q.A. Turchette, D. Leibfried,
W.M. Itano, C. Monroe and D.J. Wineland,
{\it Cooling the Collective Motion of Trapped Ions to Initialize a Quantum
Register},
Phys. Rev. Lett. {\betaf 81}, 1525 (1998)
\betaibitem{symp2} G. Morigi and H. Walther,
{\it Low temperature dynamics and laser-cooling of two-species Coulomb
chains for quantum logic},
{\tt quant-ph/0005082} (2000)
\betaibitem{dfs} D. Kielpinski, A. Ben-Kish, J. Britton, V. Meyer, M.A. Rowe,
C.A. Sackett, W.M. Itano, C. Monroe and D.J. Wineland,
{\it Recent Results in Trapped-Ion Quantum Computing},
{\tt quant-ph/0102086} (2001)
\betaibitem{symp1} D. Kielpinski, B.E. King, C.J. Myatt, C.A. Sackett,
Q.A. Turchette, W.M. Itano, C. Monroe, D.J. Wineland and W.H. Zurek,
{\it Sympathetic cooling of trapped ions for quantum logic},
Phys. Rev. {\betaf A61}, 032310 (2000)
\betaibitem{In} E. Peik, J. Abel, Th. Becker, J. von Zanthier and H. Walther,
{\it Sideband cooling of ions in radio-frequency traps},
Phys. Rev. {\betaf A60}, 439 (1999)
\betaibitem{EIT1} G. Morigi, J. Eschner and Ch. H. Keitel,
{\it Ground State Laser Cooling Using Electromagnetically Induced
Transparency},
Phys. Rev. Lett. {\betaf 85}, 4458 (2000)
\betaibitem{EIT2} C.F. Roos, D. Leibfried, A. Mundt, F. Schmidt-Kaler,
J. Eschner and R. Blatt,
{\it Experimental Demonstration of Ground State Laser Cooling
with Electromagnetically Induced Transparency},
Phys. Rev. Lett. {\betaf 85}, 5547 (2000)
\betaibitem{EIT3} F. Schmidt-Kaler, J. Eschner, G. Morigi, C.F. Roos,
D. Leibfried, A. Mundt and R. Blatt,
{\it Laser cooling with electromagnetically induced transparency:
Application to trapped samples of ions or neutral atoms},
{\tt quant-ph/0107087} (2001)
\betaibitem{comm2} Eq. (4) in Ref. \cite{EIT3}
\betaibitem{danny} Private communication with Danny Segal.
\betaibitem{comm3} Eq. (126) in Ref. \cite{98-5}
\betaibitem{86-2} W. Nagourney, J. Sandberg and H. Dehmelt,
{\it Shelved optical electron amplifier: Observation of quantum jumps},
Phys. Rev. Lett. {\betaf 56}, 2797 (1986)
\betaibitem{chuang} A.M. Childs and I.L. Chuang,
{\it Universal quantum computation with two-level trapped ions},
{\tt quant-ph/0008065} (2000)
\betaibitem{monroe} C. Monroe, D. Leibfried, B.E. King, D.M. Meekhof,
W.M. Itano and D.J. Wineland,
{\it Simplified quantum logic with trapped ions},
Phys. Rev. {\betaf A55}, R2489 (1997)
\betaibitem{stn} Private communication with Andrew Steane.
\betaibitem{jon} D. Jonathan, M.B. Plenio and P.L. Knight,
{\it Fast quantum gates for cold trapped ions},
Phys. Rev. {\betaf A62}, 42307 (2000)
\betaibitem{speed} A. Steane, Ch.F. Roos, D. Stevens, A. Mundt, D. Leibfried,
F. Schmidt-Kaler and R. Blatt,
{\it Speed of ion-trap quantum-information processors},
Phys. Rev. {\betaf A62}, 042305 (2000)
\betaibitem{gen} D.M. Meekhof, C. Monroe, B.E. King, W.M. Itano
and D.J. Wineland,
{\it Generation of Nonclassical Motional States of a Trapped Atom},
Phys. Rev. Lett. {\betaf 76}, 1796 (1996)
\betaibitem{conc} W.K. Wootters,
{\it Entanglement of Formation of an Arbitrary State of Two Qubits},
Phys. Rev. Lett. {\betaf 80}, 2245 (1998)
\betaibitem{ms} M. \v{S}a\v{s}ura and V. Bu\v{z}ek,
{\it Multiparticle entanglement with quantum logic networks:
Application to cold trapped ions},
Phys. Rev. {\betaf A64}, 012305 (2001)
\betaibitem{garg} A. Garg,
{\it Decoherence in Ion Trap Quantum Computers},
Phys. Rev. Lett. {\betaf 77}, 964 (1996)
\betaibitem{hughes} R.J. Hughes, D.F.V. James, E.H. Knill, R. Laflamme
and A.G. Petschek,
{\it Decoherence Bounds on Quantum Computation with Trapped Ions},
Phys. Rev. Lett. {\betaf 77}, 3240 (1996)
\betaibitem{dfv} D.F.V. James,
{\it Theory of Heating of the Quantum Ground State of Trapped Ions},
Phys. Rev. Lett. {\betaf 81}, 317 (1998)
\betaibitem{res} Q.A. Turchette, C.J. Myatt, B.E. King, C.A. Sackett,
D. Kielpinski, W.M. Itano, C. Monroe and D.J. Wineland,
{\it Decoherence and decay of motional quantum states of a trapped atom
coupled to engineered reservoirs},
Phys. Rev. {\betaf A62}, 053807 (2000)
\betaibitem{dfqm} D. Kielpinski, V. Meyer, M.A. Rowe, C.A. Sackett, W.M. Itano,
C. Monroe and D.J. Wineland,
{\it A decoherence-free quantum memory using trapped ions},
Science {\betaf 291}, 1013 (2001)
\betaibitem{knight1} M.B. Plenio, S.F. Huelga, A. Beige and P.L. Knight,
{\it Cavity-loss-induced generation of entangled atoms},
Phys. Rev. {\betaf A59}, 2468 (1999)
\betaibitem{knight2} A. Beige, S. Bose, D. Braun, S.F. Huelga, P.L. Knight,
M.B. Plenio, V. Vedral,
{\it Entangling atoms and ions in dissipative environments},
Journal of Modern Optics {\betaf 47}, 2583 (2000)
\betaibitem{poyatos} J.F. Poyatos, J.I. Cirac and P. Zoller,
{\it Quantum Gates with "Hot" Trapped Ions},
Phys. Rev. Lett. {\betaf 81}, 1322 (1998)
\betaibitem{milburn} G.J. Milburn, S. Schneider and D.F.V. James,
{\it Ion trap quantum computing with warm ions},
Fortschritte der Physik {\betaf 48}, 801 (2000)
\betaibitem{collspin} G.J. Milburn,
{\it Simulating nonlinear spin models in an ion trap},
{\tt quant-ph/9908037} (1999)
\betaibitem{coll} X. Wang, A. S{\o}rensen and Klaus M{\o}lmer,
{\it Multibit Gates for Quantum Computing},
Phys. Rev. Lett. {\betaf 86}, 3907 (2001)
\betaibitem{hot1} K. M{\o}lmer and A. S{\o}rensen,
{\it Multiparticle Entanglement of Hot Trapped Ions},
Phys. Rev. Lett. {\betaf 82}, 1835 (1998)
\betaibitem{hot2} A. S{\o}rensen and A. S{\o}rensen,
{\it Quantum Computation with Ions in Thermal Motion},
Phys. Rev. Lett. {\betaf 82}, 1971 (1998)
\betaibitem{jon2} D. Jonathan and M.B. Plenio,
{\it Light-shift-induced quantum gates for ions in thermal motion},
{\tt quant-ph/0103140} (2001)
\betaibitem{sackett} C.A. Sackett, D. Kielpinski, B.E. King, C. Langer, V. Meyer,
C.J. Myatt, M. Rowe, Q.A. Turchette, W.M. Itano, D.J. Wineland and I.C. Monroe,
{\it Experimental entanglement of four particles},
Nature {\betaf 404}, 256 (2000)
\betaibitem{micro1} J.I. Cirac and P. Zoller,
{\it A scalable quantum computer with ions in an array of microtraps},
Nature {\betaf 404}, 579 (2000)
\betaibitem{micro2} T. Calarco, H.J. Briegel, D. Jaksch, J.I. Cirac and P. Zoller,
{\it Quantum computing with trapped particles in microscopic potentials},
Fortschritte der Physik {\betaf 48}, 945 (2000),
(see also {\tt quant-ph/0010105})
\betaibitem{devoe} R.G. DeVoe,
{\it Elliptical ion traps and trap arrays for quantum computation},
Phys. Rev. {\betaf A58}, 910 (1998)
\end{document} |
\begin{document}
\title{Quantum Algorithm for the Collision Problem}
\thispagestyle{empty}
\begin{abstract}
In this note, we give a quantum algorithm that finds collisions in
arbi\-trary \mbox{$r$-to-one} functions after only
\mbox{$O(\sqrt[3]{\mbox{\vphantom{$N$}\smash{$N/r$}}}\,)$}
expected evaluations of the function.
\mbox{Assuming} the function is given by a black box, this is
more efficient than the best possible classical algorithm,
even allowing probabilism.
We~also give a similar algorithm for finding claws in pairs of
functions.
Furthermore, we~exhibit a space-time tradeoff for our technique.
Our approach uses Grover's quantum searching algorithm in a novel~way.
\end{abstract}
\section{Introduction}
A {\em collision\/} for function~\mbox{$F:X \rightarrow Y$} consists of
two
distinct elements \mbox{$x_0,x_1 \in X$} such that
\mbox{$F(x_0)=F(x_1)$}.
The~{\em collision problem\/} is to find a collision in~$F$ under the
promise that there is~one.
This problem is of particular interest for cryptology because some
functions known as {\em hash functions\/} are used in various
cryptographic protocols. The secu\-rity of these protocols depends
crucially on the presumed difficulty of finding collisions in such
functions.
A~related question is to find so-called {\em claws\/} in pairs of
functions;
our quantum algorithm extends to this task. This has consequences for
the
security of classical signature and bit commitment schemes. We~refer
the
interested reader to~\cite{Stinson95} for general background on
cryptography,
which is not required for understanding our new collision-finding
algorithm.
A~function $F$ is said to be {\em $r$-to-one\/} if every element in its
image has exactly $r$ distinct preimages.
We~assume throughout this note that function~$F$ is given as a
black~box, so that it is not possible to obtain knowledge about it by
any
other means than evaluating it on points in its domain.
When $F$ is \mbox{two-to-one},
the most efficient classical algorithm possible for the collision
problem
requires an expected $\Theta(\sqrt{N}\,)$ evaluations of~$F$, where
\mbox{$N=|X|$} denotes the cardinality of the domain. This classical
algorithm, which uses a principle reminiscent of the birthday paradox,
is reviewed in the next section.
Recently, at a talk held at AT\&T, Eric Rains~\cite{Rains97} asked if it
is possible to do better on a quantum computer. In~this note, we give a
positive answer to this question by providing a quantum algorithm that
finds
a collision in an arbitrary \mbox{two-to-one} function $F$ after only
$O(\sqrt[3]{N}\,)$ expected evaluations.
Earlier, Simon~\cite{Simon94} addressed the {\em {\sc xor}-mask
problem\/}
defined as follows. Consider integers \mbox{$m \ge n$}. We~are given a
function \mbox{$F: \{0,1\}^n \rightarrow \{0,1\}^m$} and promised that
either~$F$ is \mbox{one-to-one} or it is \mbox{two-to-one} and there
exists
an~\mbox{$s \in \{0,1\}^n$} such that \mbox{$F(x_0)=F(x_1)$}
if and only if \mbox{$x_0 \oplus x_1 = s$}, for all distinct
\mbox{$x_0, x_1 \in \{0,1\}^n$}, where~$\oplus$ denotes the bitwise
\mbox{exclusive-or}. Simon's problem is to decide which of these two
conditions holds, and to find~$s$ in the latter case.
Note that finding $s$ is equivalent to finding a collision in the
case that $F$ is \mbox{two-to-one}.
Simon gave a quantum algorithm to solve his problem in
expected time polynomial in~$n$ and in the time required to compute~$F$.
The~running time required for this task on a quantum computer was
recently
improved to being worst-case (rather than expected) polynomial time
thanks to
a more sophisticated algorithm~\cite{BH97}. Simon's algorithm is
interesting
from a theoretical point of view because any classical algorithm that
uses only
sub-exponentially (in~$n$) many evaluations of~$F$ cannot hope to
distinguish
between the two types of functions significantly better than simply by
tossing
a coin, assuming equal {\em a~priori} probabilities~\cite{Simon94,BH97}.
Unfortunately, the {\sc xor}-mask constraint when $F$ is
\mbox{two-to-one}
is so restrictive that Simon's algorithm has not yet found a practical
application.
More recently, Grover~\cite{Grover96} discovered a quantum algorithm for
a
different searching problem. We~are given a function
\mbox{$F:X \rightarrow \{0,1\}$} with the promise that there exists a
unique
\mbox{$x_0 \in X$} so that \mbox{$F(x_0)=1$}, and we are asked to
find~$x_0$.
Provided the domain of the function is of cardinality a power of two
(\mbox{$N=2^n$}), Grover gave a quantum algorithm that finds the unknown
$x_0$ with probability at least~$1/2$ after only $\Theta(\sqrt{N}\,)$
evaluations of~$F$.
A~natural generalization of this searching problem occurs when
\mbox{$F:X \rightarrow Y$} is an arbitrary function. Given some
\mbox{$y_0 \in Y$}, we are asked to find an \mbox{$x \in X$} such that
\mbox{$F(x)=y_0$}, provided such an~$x$ exists.
If~\mbox{$t=|\{x \in X \, | \, F(x)=y_0\}|$} denotes the number of
different
solutions, \cite{BBHT96}~gives a generalization of Grover's algorithm
that
can find a solution whenever it exists (\mbox{$t \geq 1$}) after an
expected number of $O(\sqrt{\mbox{\vphantom{$N$}\smash{$N/t$}}}\,)$ evaluations of~$F$.
Although the algorithm does not need to know the value of $t$ ahead of
time,
it is more efficient (in~terms of the hidden constant in the $O$
notation)
when $t$ is known, which will be the case for most
algorithms given here. From now on, we refer to this generalization of
Grover's algorithm as~$\mbox{\bf Grover}(F,y_0)$.
Note that the number of evaluations of~$F$ is not polynomially bounded
in~$\log N$ when~\mbox{$t \ll N$}; nevertheless Grover's algorithm is
considerably more efficient than classical brute-force searching.
In~the next section, we give our new quantum algorithm for solving the
collision problem for \mbox{two-to-one} functions. We~then discuss a
straightforward generalization to \mbox{$r$-to-one} functions and even
to arbitrary functions whose image is sufficiently smaller than their
domain. A~natural space-time tradeoff emerges for our technique.
Finally, we give applications to finding claws in pairs of functions.
\section{Algorithms for the collision problem}
We~first state two simple algorithms for the collision problem, one
classical and one quantum. Both of these algorithms use an expected
number of $O(\sqrt{N}\,)$ evaluations of the given function, but the
quantum algorithm is more space efficient.
We~derive our improved algorithm from these two simple solutions.
The first solution is a well-known classical probabilistic
algorithm, here stated in slightly different terms than traditionally.
The algorithm consists of three steps. First, it selects a random
subset
\mbox{$K \subseteq X$} of cardinality \mbox{$k = c \sqrt{N}$} for an
appropriate constant~$c$. Then, it computes the pair~\mbox{$(x,F(x))$}
for
each \mbox{$x \in K$} and sorts these pairs according to the second
entry.
Finally, it outputs a collision in~$K$ if there is one, and otherwise
reports
that none has been found. Based on the birthday paradox, it is not
difficult
to show that if~$F$ is \mbox{two-to-one} then this algorithm returns a
collision with probability at least~$1/2$ provided $c$ is sufficiently
large
(\mbox{$c \approx 1.18$} will~do).
If~we take a pair~\mbox{$(x,F(x))$} as unit of space then the algorithm
can be
implemented in space $\Theta(\sqrt{N}\,)$, and $\Theta(\sqrt{N}\,)$
evaluations of~$F$ suffice to succeed with probability~$1/2$.
If~we care about running time rather than simply the number of
evaluations
of~$F$, it may be preferable to resort to universal
hashing~\cite{CarterWegman}
rather than sorting to find a collision in~$K$. This would avoid
spending
\mbox{$\Theta(\sqrt{N} \log N)$} time sorting the table, making possible
a
$\Theta(\sqrt{N}\,)$ overall expected running time if we assume that
each
evaluation of $F$ takes constant time. We~stick to the sorting paradigm
for simplicity and because it is not clear if the benefits of universal
hashing carry over to quantum parallelism situations such as~ours.
We~come back to this issue in Section~\ref{sect:disc}.
The simple quantum algorithm for two-to-one functions also consists of
three
steps. First, it picks an arbitrary element \mbox{$x_0 \in X$}. Then,
it
computes \mbox{$x_1= \textbf{Grover}(H,1)$} where
\mbox{$H:X \rightarrow \{0,1\}$} denotes the function defined by
\mbox{$H(x)=1$} if and only if \mbox{$x \neq x_0$} and
\mbox{$F(x)=F(x_0)$}.
Finally, it outputs the collision \mbox{$\{x_0,x_1\}$}. There is
exactly one
\mbox{$x \in X$} that satisfies \mbox{$H(x)=1$} so~\mbox{$t=1$}, and
thus the
expected number of evaluations of~$F$ is also \mbox{$O(\sqrt{N}\,)$},
still to succeed with probability~$1/2$, but constant space suffices.
Our new algorithm, denoted {\bf Collision} and given below, can be
thought of as the logical union of the two algorithms above. The main
idea is to select a subset~$K$ of~$X$ and then use {\bf Grover} to find
a collision \mbox{$\{x_0,x_1\}$} with \mbox{$x_0 \in K$} and
\mbox{$x_1 \in X \setminus K$}. The~expected number of evaluations
of~$F$ and
the space used by the algorithm are determined by the parameter
\mbox{$k=|K|$}, the cardinality of~$K$.
\noindent $\textbf{Collision}(F,k)$
\begin{enumerate}
\item \label{step:one}
Pick an arbitrary subset \mbox{$K \subseteq X$} of
cardinality~$k$.
Construct a table~$L$ of size~$k$ where each item in~$L$ holds a
distinct pair \mbox{$(x,F(x))$} with \mbox{$x \in K$}.
\item Sort~$L$ according to the second entry in each item of~$L$.
\item \label{step:check} Check if $L$ contains a collision, that is,
check if there exist distinct elements
\mbox{$(x_0,F(x_0)), (x_1,F(x_1)) \in L$} for which
\mbox{$F(x_0) = F(x_1)$}.
If~so, goto step~\ref{step:last}.
\item \label{step:grover} Compute \mbox{$x_1 = \textbf{Grover}(H,1)$}
where
\mbox{$H:X \rightarrow \{0,1\}$} denotes the function defined by
\mbox{$H(x)=1$} if and only if there exists \mbox{$x_0 \in K$} so
that
\mbox{$(x_0,F(x)) \in L$} but \mbox{$x \not= x_0$}.
(Note that $x_0$ is unique if it exists since we already checked
that there are no collisions in~$L$.)
\item Find \mbox{$(x_0,F(x_1)) \in L$}.
\item Output the collision \mbox{$\{x_0,x_1\}$}. \label{step:last}
\end{enumerate}
\begin{theorem}\label{thm:twotoone}
Given a two-to-one function \mbox{$F:X \rightarrow Y$} with
\mbox{$N=|X|$}
and an integer \mbox{$1 \leq k \leq N$}, algorithm
\mbox{$\textnormal{\bf Collision}(F,k)$} returns a collision after an
expected
number of \mbox{$O(k+\sqrt{N/k}\,)$} evaluations of~$F$ and uses
space~$\Theta(k)$. In~particular, when \mbox{$k=\sqrt[3]{N}$} then
\mbox{$\textnormal{\bf Collision}(F,k)$} evaluates~$F$ an expected
number of
\mbox{$O(\sqrt[3]{N}\,)$} times and uses space
\mbox{$\Theta(\sqrt[3]{N}\,)$}.
\end{theorem}
\begin{proof}
The correctness of the algorithm follows easily from the definition
of~$H$ and the construction of $\textbf{Grover}(H,1)$.
We~now count the number of evaluations of~$F$. In~the first step, the
algorithm uses~$k$ such evaluations. Set $t=|\{x \in X \, | \,
H(x)=1\}|$.
By~the previous section, subroutine {\bf Grover} in
step~\ref{step:grover} uses an expected number of $O(\sqrt{\mbox{\vphantom{$N$}\smash{$N/t$}}}\,)$
evaluations of the function~$H$ to find one of the $t$ solutions. Each
evaluation of $H$ can be implemented by using only one evaluation
of~$F$. Finally, our algorithm evaluates~$F$ once in the penultimate
step,
giving a total expected number of $k+O(\sqrt{\mbox{\vphantom{$N$}\smash{$N/t$}}}\,)+1$ evaluations
of~$F$.
Since $F$ is two-to-one, $t$ equals the cardinality of~$K$, that is,
$t=k$, and the first part of the theorem follows. The second part is
immediate.
\end{proof}
In~a nutshell, the improvement of our algorithm over
the simple quantum algorithm is achieved by trading time for space.
Suppose~the cardinality of set~$K$ is large. Then the expected
number of evaluations of~$H$ used by subroutine
$\textbf{Grover}(H,1)$ is small, but on the other hand more space is
needed to store table~$L$. Analogously, if~$K$ is small then the
space requirements are less but also $\textbf{Grover}(H,1)$ runs slower.
Suppose now that we apply algorithm {\bf Collision}, not necessarily on
a
\mbox{two-to-one} function, but on an arbitrary \mbox{$r$-to-one}
function
where \mbox{$r \geq 2$}. Then we have the following theorem, whose
proof
is essentially the same as that of Theorem~\ref{thm:twotoone}.
\begin{theorem}\label{thm:rtoone}
Given an \mbox{$r$-to-one} function \mbox{$F:X \rightarrow Y$} with
\mbox{$r \geq 2$} and an integer \mbox{$1 \leq k \leq N=|X|$},
algorithm \mbox{$\textnormal{\bf Collision}(F,k)$} returns a collision
after
an expected number of \mbox{$O(k+\sqrt{N/(rk)}\,)$} evaluations of~$F$
and uses
space~$\Theta(k)$. In~particular, when \mbox{$k=\sqrt[3]{\mbox{\vphantom{$N$}\smash{$N/r$}}}$} then
\mbox{$\textnormal{\bf Collision}(F,k)$} uses an expected number of
\mbox{$O(\sqrt[3]{\mbox{\vphantom{$N$}\smash{$N/r$}}}\,)$} evaluations of~$F$ and space
\mbox{$\Theta(\sqrt[3]{\mbox{\vphantom{$N$}\smash{$N/r$}}}\,)$}.
\end{theorem}
Note that algorithm \mbox{$\textnormal{\bf Collision}(F,k)$}
can also be applied on an arbitrary function
\mbox{$F:X \rightarrow Y$} for which \mbox{$|X| \ge r |Y|$}
for some \mbox{$r > 1$}, even if $F$ is not \mbox{$r$-to-one}.
However, the algorithm must be modified in two ways for the general
case.
First of all, the subset \mbox{$K \subseteq X$} of cardinality~$k$ must
be
picked at random, rather than arbitrarily, at step~\ref{step:one}.
Furthermore, the~fully generalized version of Grover's \mbox{algorithm}
given
in~\cite{BBHT96} must be used at step~\ref{step:grover} because the
number
of solutions for \mbox{$\textbf{Grover}(H,1)$} is no longer known in
advance
to be exactly \mbox{$t=(r-1)k$}.
By~varying~$k$ in Theorem~\ref{thm:rtoone}, the following space-time
tradeoff emerges.
\begin{corollary}
There exists a quantum algorithm that can find a collision in an
arbitrary
\mbox{$r$-to-one} function \mbox{$F:X \rightarrow Y$},
for any \mbox{$r \geq 2$}, using space $S$ and an expected number of
$O(T)$
evaluations of~$F$ for every $1 \leq S \leq T$ subject to
\[S T^2 \geq |F(X)| \]
where $F(X)$ denotes the image of~$F$.
\end{corollary}
Consider now two functions \mbox{$F:X \rightarrow Z$}
and \mbox{$G:Y \rightarrow Z$} that have the same codomain.
By~definition, a {\em claw\/} is a pair \mbox{$x \in X$},
\mbox{$y \in Y$} such that \mbox{$F(x)=G(y)$}.
Many cryptographic protocols are based on the assumption
that there are efficiently-computable functions $F$ and $G$
for which claws cannot be found efficiently even though
they exist in large number.
The simplest case arises when both $F$ and $G$ are bijections,
which is the usual situation when such functions are used to create
unconditionally-concealing bit commitment schemes~\cite{BCC88}.
If~\mbox{$N=|X|=|Y|=|Z|$}, algorithm {\bf Collision} is easily
modified as follows.
\noindent $\textbf{Claw}(F,G,k)$
\begin{enumerate}
\item \label{claw:one}
Pick an arbitrary subset \mbox{$K \subseteq X$} of
cardinality~$k$.
Construct a table~$L$ of size~$k$ where each item in~$L$ holds a
distinct pair \mbox{$(x,F(x))$} with \mbox{$x \in K$}.
\item Sort~$L$ according to the second entry in each item of~$L$.
\item \label{claw:grover} Compute \mbox{$y_0 = \textbf{Grover}(H,1)$}
where
\mbox{$H:Y \rightarrow \{0,1\}$} denotes the function defined by
\mbox{$H(y)=1$} if and only if a pair \mbox{$(x,G(y))$} appears
in~$L$ for some arbitrary~\mbox{$x \in K$}.
\item Find \mbox{$(x_0,G(y_0)) \in L$}.
\item Output the claw \mbox{$(x_0,y_0)$}. \label{claw:last}
\end{enumerate}
\begin{theorem}\label{thm:clawtwotoone}
Given two one-to-one functions \mbox{$F:X \rightarrow Z$}
and \mbox{$G:Y \rightarrow Z$} with \mbox{$N=|X|=|Y|=|Z|$} and an
integer \mbox{$1 \leq k \leq N$}, algorithm $\textnormal{\bf
Claw}(F,G,k)$
\mbox{returns} a claw after $k$ evaluations of~$F$
and \mbox{$O(\sqrt{N/k}\,)$} evaluations of~$G$, and uses
space~$\Theta(k)$. In~particular, when \mbox{$k=\sqrt[3]{N}$} then
$\textnormal{\bf Claw}(F,G,k)$ evaluates~$F$ and $G$ an
expected number of
\mbox{$O(\sqrt[3]{N}\,)$} times and uses space
\mbox{$\Theta(\sqrt[3]{N}\,)$}.
\end{theorem}
\begin{proof}
Similar to the proof of Theorem~\ref{thm:twotoone}.
\end{proof}
The case in which both $F$ and $G$ are \mbox{$r$-to-one} for some
\mbox{$r \ge 2$} and \mbox{$N=|X|=|Y|=r|Z|$} is handled similarly.
However, it becomes necessary in step~\ref{claw:one} of algorithm
{\bf Claw} to select the elements of $K$ so that no two of them are
mapped to the same point by~$F$. This will ensure that the call on
\mbox{$\textbf{Grover}(H,1)$} at step~\ref{claw:grover} has exactly
$kr$ solutions to choose from. The~simplest way to choose~$K$ is
to pick random elements in~$X$ until \mbox{$|F(K)|=k$}.
As~long as \mbox{$k \le |Z|/2$}, this requires trying less than
$2k$ random elements of~$X$, except with vanishing probability.
The~proof of the following theorem is again essentially
as before.
\begin{theorem}\label{thm:clawrtoone}
Given two \mbox{$r$-to-one} functions \mbox{$F:X \rightarrow Z$}
and \mbox{$G:Y \rightarrow Z$} with \mbox{$N=|X|=|Y|=r|Z|$} and an
integer \mbox{$1 \leq k \leq N/2r$}, modified algorithm
$\textnormal{\bf Claw}(F,G,k)$ returns a claw after an
expected number of
$\Theta(k)$ evaluations of~$F$ and
\mbox{$O(\sqrt{N/(rk)}\,)$} evaluations of~$G$, and uses
space~$\Theta(k)$. In~particular, when \mbox{$k=\sqrt[3]{\mbox{\vphantom{$N$}\smash{$N/r$}}}$}
then $\textnormal{\bf Claw}(F,G,k)$ evaluates~$F$ and $G$ an
expected number of
\mbox{$O(\sqrt[3]{\mbox{\vphantom{$N$}\smash{$N/r$}}}\,)$} times and uses space
\mbox{$\Theta(\sqrt[3]{\mbox{\vphantom{$N$}\smash{$N/r$}}}\,)$}.
\end{theorem}
\section{Discussion}\label{sect:disc}
When we say that our quantum algorithms require $\Theta(k)$ space
to hold table~$L$, this corresponds unfortunately to the amount of
{\em quantum\/} memory, a~rather scarce resource with current
technology.
Note however that this table is built classically in the initial
steps of algorithms {\bf Collision} and {\bf Claw}: it needs to
live in quantum memory for read purposes only.
In~practice, it may be easier to build large read-only
quantum memories than general read/write memories.
We considered only the number of evaluations of~$F$
in the analysis of algorithm {\bf Collision}.
The~time spent sorting~$L$ and doing
binary search in~$L$ should also be taken into account if we
wanted to analyse
the running time of our algorithm. If~we assume that it
takes time~$T$ to compute the function (rather than assuming
that it is given as a black box),
then it is straightforward to show that the algorithm given by
Theorem~\ref{thm:rtoone} runs in expected time
\[O((k+\sqrt{N/(kr)}\,) (T+\log k))\,.\]
Thus, the time spent sorting is negligible only if it takes
$\Omega(\log k)$ time to compute~$F$.
Similar considerations apply to algorithm {\bf Claw}.
It~is tempting to try using universal hashing to bypass the
need for sorting, as in the simple classical algorithm,
but it is not clear that this approach saves time here because
our use of quantum parallelism when we apply Grover's algorithm
will take a time that is given by the {\em maximum\/} time taken
for all requests to the table, which is unlikely to be constant
even though the expected {\em average\/} time is constant.
\end{document} |
\begin{document}
\begin{frontmatter}
\title{Monochromatic factorisations of words and periodicity}
\author[label1]{Ca\"ius Wojcik}
\ead{caius.wojcik@gmail.com}
\author[label1]{Luca Q. Zamboni}
\ead{zamboni@math.univ-lyon1.fr}
\address[label1]{Universit\'e de Lyon,
Universit\'e Lyon 1, CNRS UMR 5208,
Institut Camille Jordan,
43 boulevard du 11 novembre 1918,
F69622 Villeurbanne Cedex, France}
\begin{abstract}
In 2006 T. Brown asked the following question in the spirit of Ramsey theory: Given a non-periodic infinite word $x=x_1x_2x_3\cdots$ with values in a non-empty set $\mathbb{A},$ does there exist a finite colouring $\varphi: \mathbb{A}^+\rightarrow C$ relative to which $x$ does not admit a $\varphi$-monochromatic factorisation, i.e., a factorisation of the form $x=u_1u_2u_3\cdots$ with $\varphi(u_i)=\varphi(u_j)$ for all $i,j\geq 1$? This question belongs to the class of Ramsey type problems in which one asks whether some abstract form of Ramsey's theorem holds in a certain generalised setting.
Various partial results in support of an affirmative answer to this question have appeared in the literature in recent years. In particular it is known that the question admits an affirmative answer for all non-uniformly recurrent words and hence for almost all words relative to the standard Bernoulli measure on $A^{\mathbb N}.$ This question also has a positive answer for various classes of uniformly recurrent words including Sturmian words and fixed points of strongly recognizable primitive substitutions. In this paper we give a complete and optimal affirmative answer to this question by showing that if $x=x_1x_2x_3\cdots$ is an infinite non-periodic word with values in a non-empty set $\mathbb{A},$ then there exists a $2$-colouring $\varphi: \mathbb{A}^+\rightarrow \{0,1\}$
such that for any factorisation $x=u_1u_2u_3\cdots$ we have $\varphi(u_i)\neq \varphi(u_j)$ for some $i\neq j.$ In fact this condition gives a characterization of non-periodic words. It may be reformulated in the language of ultrafilters as follows: Let $\beta \mathbb{A}^+$ denote the Stone-\v Cech compactification of the discrete semigroup $\mathbb{A}^+$ which we regard as the set of all ultrafilters on $\mathbb{A}^+.$ Then an infinite word $x=x_1x_2x_3\cdots$ with values in $\mathbb{A}$ is periodic if and only if there exists $p\in \beta \mathbb{A}^+$ such that for each $A\in p$ there exists a factorisation $x=u_1u_2u_3\cdots $ with each $u_i \in A.$ Moreover $p$ may be taken to be an idempotent element of $\beta \mathbb{A}^+.$
\end{abstract}
\begin{keyword}Combinatorics on words, Ramsey theory.
\MSC[2010] 68R15, 05C55, 05D10.
\end{keyword}
\journal{}
\end{frontmatter}
\section{Introduction}
Give a non-empty (not necessarily finite) set $\mathbb{A},$ let $\mathbb{A}^+$ denote the free semigroup generated by $\mathbb{A}$ consisting of all finite words $u_1u_2\cdots u_n$ with $u_i \in \mathbb{A},$ and let $\mathbb{A}^{\mathbb N}$ denote the set of all infinite words $x=x_1x_2x_3\cdots$ with $x_i\in \mathbb{A}.$
We say $x\in \mathbb{A}^{\mathbb N}$ is {\it periodic} if $x=u^\omega=uuu\cdots $ for some $u\in \mathbb{A}^+.$
The following question was independently posed by T. Brown in \cite{BTC} and by the second author in \cite{LQZ}\footnote{The original formulation of the question was stated in terms of finite colourings of the set of all factors of $x.$}:
\begin{question}\label{conj} Let $x\in \mathbb{A}^{\mathbb N}$ be non-periodic. Does there exist a finite colouring $\varphi: \mathbb{A}^+\rightarrow C$ relative to which $x$ does not admit a $\varphi$-monochromatic factorisation, i.e., a factorisation of the form $x=u_1u_2u_3\cdots$ with $\varphi(u_i)=\varphi(u_j)$ for all $i,j\geq 1$?
\end{question}
A finite colouring $\varphi: \mathbb{A}^+\rightarrow C$ is called a {\it separating colouring} for $x$ (or a separating $|C|$-colouring for $x)$ if for all factorisations $x=u_1u_2u_3\cdots$ there exist $i,j\geq 1$ such that $\varphi(u_i)\neq \varphi(u_j).$ While T. Brown originally stated it as a question, Question~\ref{conj} has evolved into a conjecture which states that every non-periodic word admits a separating colouring.
We begin by illustrating Question~\ref{conj} with an example: Consider the {\it Thue-Morse} infinite word \[x=011010011001011010010\cdots\] where the $n$th term of $x$ (starting from $n=0)$ is defined as the sum modulo $2$ of the digits in the binary expansion of
$n.$ The origins of this word go back to the beginning of the last century with the works of A. Thue \cite{Th1, Th2} in which he proves amongst other things that $x$ is {\it overlap-free} i.e., $x$ contains no factor of the form $uuu'$ where $u'$ is a non-empty prefix of $u.$
Consider the $3$ colouring $\varphi : \{0,1\}^+\rightarrow \{0,1,2\}$ defined by
$$ \varphi(u) = \begin{cases}
0 & \text{if $u$ is a prefix of $x$ ending with $0;$} \\ 1 & \text{if $u$ is a prefix of $x$ ending with $1;$}\\
2& \text{if $u$ is not a prefix of $x.$}
\end{cases} $$
We claim that no factorisation of $x$ is $\varphi $-monochromatic. In fact, suppose to the contrary that $x=u_1u_2u_3\cdots$ is a $\varphi $-monochromatic factorisation of $x.$ Since $u_1$ is a prefix of $x$, it follows that $\varphi(u_1)\in \{0,1\},$ i.e., there exists $a\in \{0,1\}$ such that each $u_i$ is a prefix of $x$ terminating with $a.$ Pick $i\geq 2$ such that $|u_i|\leq |u_{i+1}|.$ Then as each $u_i$ is a prefix of $x,$ it follows that $u_i$ is a prefix of $u_{i+1}$and hence $au_iu_i$ is a factor of $x.$ Writing
$u_i=va,$ (with $v$ empty or in $\{0,1\}^+),$ we have $au_iu_i=avava $ which is an overlap, contradicting that $x$ is overlap-free.
This proves that there exists a separating $3$-colouring for the Thue-Morse word. Recently S. Avgustinovich and O. Parshina~\cite{AP} proved that it is possible to colour $\{0,1\}^+$ using only $2$ colours in such a way that no factorisation of the Thue-Morse word is monochromatic.
Let us remark in the example above that since the Thue-Morse word $x$ is not periodic, it follows that each proper suffix $x'$ of $x$ begins in some factor $u$ which is not a prefix of $x.$ This means that $x'$ may be written as an infinite concatenation $x'=u_1u_2u_3\cdots$ where $\varphi(u_i)=2$ for all $i\geq 1.$ So while $x$ itself does not admit a $\varphi$-monochromatic factorisation, where $\varphi$ is the $3$-colouring of $\{0,1\}^+$ defined above, it turns out that every proper suffix of $x$ does admit a $\varphi$-monochromatic factorisation. Moreover, this monochromatic factorisation of $x'$ has an even stronger monochromatic property: The set $\{u_n: n\geq 1\}^+$ is $\varphi$-monochromatic (each element has $\varphi$ colour equal to $2).$ This is because each element of $\{u_n: n\geq 1\}$ is a non-prefix of $x$ and hence the same is true of any concatenation formed by elements from this set. It turns out that a weaker version of this phenomenon is true in greater generality: Given any $x\in A^{\mathbb N}$ and any finite colouring $ \varphi: \mathbb{A} ^+ \rightarrow C ,$ one can always find a suffix $x'$ of $x$ which admits a factorisation $x'=u_1u_2u_3\cdots$ where $\varphi(u_i\cdots u_j)=\varphi(u_1)$ for all $1\leq i\leq j.$ This fact may be obtained via a straightforward application of the infinite Ramsey theorem \cite{Ram} (see \cite{BTC, DZ1} or see \cite{schutz} for a proof by M. P. Sch\"utzenberger which does not use Ramsey's theorem).
Thus Question~\ref{conj} belongs to the class of Ramsey type problems in which one tries to show that some abstract form of Ramsey's theorem does not hold in certain settings. For instance, the infinite version of Ramsey's theorem \cite{Ram} (for colouring of pairs) states that whenever the set $\Sigma_2({\mathbb N})$ of all $2$-element subsets of ${\mathbb N}$ is finitely coloured, there exists an infinite set $X\subseteq {\mathbb N}$ with $\Sigma_2(X)$ monochromatic. Hence the same applies when ${\mathbb N}$ is replaced by ${\mathbb R}.$ On the other hand, W. Sierpi\'nski \cite{Serp} showed that there exists a finite colouring of $\Sigma_2({\mathbb R})$ such that there does not exist an uncountable set $X$ with $\Sigma_2(X)$ monochromatic. In other words, Ramsey's theorem does not extend to the uncountable setting in ${\mathbb R}.$ Similarly, by a straightforward application of Ramsey's theorem, one deduces that given any finite colouring of ${\mathbb N},$ there exists an infinite $X\subseteq {\mathbb N}$ all of whose pairwise sums $\{n+m: n,m\in X, n\neq m\}$ is monochromatic. Again it follows the same is true with ${\mathbb N}$ replaced by ${\mathbb R}.$ On the other hand, N. Hindman, I. Leader and D. Strauss \cite{HLS} recently exhibited (using the Continuum Hypothesis CH) the existence of a finite colouring of ${\mathbb R}$ such that there does not exist an uncountable set with all its pairwise sums monochromatic. In other words, this additive formulation of Ramsey's Theorem also fails in the uncountable setting in ${\mathbb R}.$ A related question in these sorts of problems concerns the least number of colours necessary to avoid the presence of monochromatic subsets of a certain kind. For instance N. Hindman \cite{H} showed that there exists a $3$-colouring of ${\mathbb N}$ such that there does not exist an infinite subset $X$ with $X+X$ monochromatic, and it is an open question of Owings \cite{Ow} whether the same result may be obtained with only $2$ colours. Again, using CH and a few more colours ($288$ to be precise), it is possible to extend Hindman's result to the reals (see Theorem 2.8 in \cite{HLS}). But again it is not known whether the same result can be obtained with only $2$ colours~\cite{Leader}.
Thus a stronger version of Question~\ref{conj} would read: Does every non-periodic word admit a separating $2$-colouring?
Various partial results in support of an affirmative answer to Question~\ref{conj} were obtained in \cite{BPS, DPZ, DZ1, DZ2, ST}. For instance, in \cite{DPZ}, it is shown that Question~\ref{conj} admits an affirmative answer for all non-uniformly recurrent words and various classes of uniformly recurrent words including Sturmian words. In \cite{ST}, V. Salo and I. T\"orm\"a prove that for every aperiodic linearly recurrent word $x\in \mathbb{A}^{\mathbb N}$ there exists a finite colouring of $\mathbb{A}^+$ relative to which $x$ does not admit a monochromatic factorisation into factors of increasing lengths. And recently A. Bernardino, R. Pacheco and M. Silva \cite{BPS} prove that Question~\ref{conj} admits an affirmative answer for all fixed points of primitive strongly recognizable substitutions. In addition to the fact that these partial results concern only restricted classes of non-periodic words (e.g., Sturmian words or certain fixed points of primitive substitutions), in most cases the number of colours required to colour $\mathbb{A}^+$ in order to avoid a monochromatic factorisation of $x$ is found to be quite high. For instance in \cite{ST}, the authors prove that if $x\in \mathbb{A}^{\mathbb N}$ is an aperiodic linearly recurrent word, then there exists a constant $K\geq 2$ and a colouring $\varphi: \mathbb{A}^+\rightarrow C$ with ${\rm Card}(C)=2+\sum_{i=0}^{K^5-1}2K^i(K+1)^{2i},$ such that no factorisation of $x=u_1u_2u_3\cdots ,$ verifying the additional constraint that $|u_i|\leq |u_{i+1}|$ for each $i\geq 1,$ is $\varphi$-monochromatic.
The constant $K$ above is chosen such that for every factor $u$ of $x,$ every first return $w$ to $u$ satisfies $|w|\leq K|u|$ (see for instance \cite{DHS}). A similar large bound depending on the recognizability index of a substitution is obtained in \cite{BPS} in the context of fixed points of strongly recognizable substitutions. In contrast, it is shown in \cite{DPZ} that every Sturmian word admits a separating $3$-colouring.
In this paper we give a complete and optimal affirmative answer to Question~\ref{conj} by showing that for every non-periodic word $x=x_1x_2x_3\cdots \in \mathbb{A}^{\mathbb N},$ there exists a $2$-colouring $\varphi: \mathbb{A}^+\rightarrow \{0,1\}$ relative to which no factorisation of $x$ is $\varphi$-monochromatic. Moreover, this is a characterization of periodicity of infinite words:
\begin{thm}\label{M} Let $x=x_1x_2x_3\cdots \in \mathbb{A}^{\mathbb N}$ be an infinite word. Then $x$ is periodic if and only if for every $2$-colouring
$\varphi: \mathbb{A}^+\rightarrow \{0,1\}$ there exists a $\varphi$-monochromatic factorisation of $x,$ i.e., a factorisation $x=u_1u_2u_3\cdots$ such that $\varphi(u_i)=\varphi(u_j)$ for all $i,j\geq 1.$
\end{thm}
Theorem~\ref{M} has several nice immediate consequences: For instance, fix a symbol $a\in \mathbb{A}$ and suppose an infinite word $x\in \mathbb{A}^{\mathbb N}$ admits a factorisation $x=u_1u_2u_3\cdots$ where each $u_i$ is a prefix of $x$ rich in the symbol $a\in \mathbb{A},$ meaning
that for each $i\geq 1$ and for each factor $v$ of $x$ with $|v|=|u_i|$ we have $f_a(u_i)\geq f_a(v)$ where $f_a(u_i)$ denotes the frequency of $a$ in $u_i.$ Then $\{n\in {\mathbb N}: x_n=a\}$ is a finite union of arithmetic progressions. As another application of Theorem~\ref{M} we show that if $u_1,u_2,u_3,\ldots, u_{2k+1}\in A^+$ (with $k$ a positive integer), then any $x\in \mathbb{A}^{\mathbb N}$ belonging to $\{u_1,u_2\}^{\mathbb N} \cap \{u_2,u_3\}^{\mathbb N}\cap\cdots \cap \{u_{2k},u_{2k+1}\}^{\mathbb N}\cap \{u_{2k+1},u_1\}^{\mathbb N}$ is periodic.
Theorem~\ref{M} may be reformulated in the language of ultrafilters as follows: Let $\beta \mathbb{A}^+$ denote the Stone-\v Cech compactification of the discrete semigroup $\mathbb{A}^+$ which we regard as the set of all ultrafilters on $\mathbb{A}^+$, identifying the points of
$\mathbb{A}^+$ with the principal ultrafilters. As is well known, the operation of concatenation on $\mathbb{A}^+$ extends uniquely to $\beta \mathbb{A}^+$ making $\beta \mathbb{A}^+$ a compact right topological semigroup with $\mathbb{A}^+$ contained in its topological center (see for instance \cite{HS}). In particular by the Ellis-Numakura lemma, $\beta\mathbb{A}^+$ contains an idempotent element i.e., an element $p$ verifying $p\cdot p=p.$ We show that an infinite word $x=x_1x_2x_3\cdots \in \mathbb{A}^{\mathbb N}$ is periodic if and only if there exists $p\in \beta \mathbb{A}^+$ such that for each $A\in p$ there exists a factorisation $x=u_1u_2u_3\cdots $ with each $u_i \in A.$ Moreover $p$ can be taken to be an idempotent element of $\beta \mathbb{A}^+.$
\section{Proof of Theorem~\ref{M} \& Applications}
Before embarking on the proof of Theorem~\ref{M}, we introduce some notation which will be relevant in what follows.
Let $\mathbb{A}$ be a non-empty set (the {\it alphabet}). We do not assume that the cardinality of $\mathbb{A}$ is finite. Let $\mathbb{A}^*$ denote the set of all finite words $u=u_1u_2\cdots u_{n}$ with
$u_i\in \mathbb{A}$. We call $n$ the {\it length} of $u$ and denote it $|u|$. The empty word is denoted $\varepsilon$ and by convention
$|\varepsilon|=0$. We put $\mathbb{A}^+=\mathbb{A}^*\setminus \{\varepsilon\}$. For $u\in \mathbb{A}^+$ and $a\in \mathbb{A},$ we let $|u|_a$ denote the number of occurrences of $a$ in $u.$
Let $\mathbb{A}^{\mathbb N}$ denote the set of all right sided infinite words $x=x_1x_2x_3\cdots$ with values in $\mathbb{A}.$
More generally, for $x\in \mathbb{A}^{\mathbb N}$ and $A\subseteq \mathbb{A}^+,$ we write $x\in A^{\mathbb N}$ if $x=u_1u_2u_3\cdots $ with $u_i\in A$ for all $i\geq 1,$ that is in case $x$ factors over the set $A.$ We say $x$ is {\it periodic} if $x\in \{u\}^{\mathbb N}$ for some $u\in \mathbb{A}^+.$
\begin{proof}[Proof of Theorem~\ref{M}] First assume $x=x_0x_1x_2\cdots \in \mathbb{A}^{\mathbb N}$ is periodic, i.e., $x\in \{u\}^{\mathbb N}$ for some $u\in \mathbb{A}^+.$ Then the factorisation $x=u_1u_2u_3\cdots$ with each $u_i=u$ is $\varphi$-monochromatic for any choice of $\varphi: \mathbb{A}^+\rightarrow \{0,1\}.$ Next assume $x$ is not periodic and we will define a $2$-colouring $\varphi: \mathbb{A}^+\rightarrow \{0,1\}$ with the property that no factorisation of $x$ is $\varphi$-monochromatic. Pick any total order on the set $\mathbb{A}$ and let $\rhorec$ denote the induced lexicographic order on $\mathbb{A}^+$ and $\mathbb{A}^{\mathbb N}.$ For $u,v\in \mathbb{A}^+$ with $|u|=|v|,$ we write $u\rhoreccurlyeq v$ if either $u\rhorec v$ or $u=v.$ For each $n\geq 1,$ let $P_x(n)$ denote the prefix of $x$ of length $n,$ and for each $y \in \mathbb{A}^{\mathbb N},$ let $x\wedge y$ denote the longest common prefix of $x$ and $y.$
\noindent For $u \in A^+, $ set
\begin{equation}\label{E} \varphi(u) = \begin{cases}
0 & \text{if $u \rhorec P_x(|u|)$ or if $u=P_x(|u|)$ and $x\rhorec y$ where $x=uy;$} \\ 1 & \text{if $P_x(|u|) \rhorec u$ or if $u=P_x(|u|)$ and $y\rhorec x$ where $x=uy.$}
\end{cases}\end{equation}
\noindent As $x$ is not periodic, for every proper suffix $y$ of $x$ we have either $x\rhorec y$ or $y\rhorec x,$ whence $\varphi: \mathbb{A}^+\rightarrow \{0,1\}$ is well defined.
We claim that no factorisation of $x$ is $\varphi$-monochromatic. To see this, fix a factorisation $x=u_1u_2u_3\cdots .$
For each $k\geq 0$ put $y_k=u_{k+1}u_ {k+2}u_{k+3}\cdots \in \mathbb{A}^{\mathbb N}$ and let $w_k=x \wedge y_k.$ We note that
$y_0=x=w_0.$ For each $k\geq 0$ let
\[S_k=|u_1| + |u_2|+\cdots + |u_k| + |w_k|\]
so that $S_0=|w_0|=+\infty.$ Clearly $S_k\geq k$ for each $k\geq 0$ and hence $\lim S_k=+\infty.$
\begin{lemma}\label{L1} Assume that $\varphi (u_i)=0$ for all $i\geq 1.$ Then $\varphi(u_1u_2\cdots u_k)=0$ for all $k\geq 1.$
\end{lemma}
\begin{proof} We proceed by induction on $k.$ The result is clear for $k=1$ since $\varphi(u_1)=0. $ For the inductive step, fix $k\geq 1$ and suppose $\varphi(u_1\cdots u_k)=0.$ As $u_1\cdots u_k$ is a prefix of $x$ and $\varphi(u_1\cdots u_k)=0,$ we can write $x=w_kay'$ and $x=u_1u_2\cdots u_kw_kby''$ for some $y', y'' \in \mathbb{A}^{\mathbb N}$ and $a,b\in \mathbb{A}$ with $a\rhorec b.$
It follows that $|u_{k+1}|\leq |w_k|$ for otherwise $w_kb$ is a prefix of $u_{k+1}$ and we would have $P_x(|u_{k+1}|)\rhorec u_{k+1}$ whence $\varphi(u_{k+1})=1,$ a contradiction. Thus we can write $w_k=u_{k+1}v$ for some $v\in \mathbb{A}^*$ so that $x=u_{k+1}vay'$ and $x=u_1u_2\cdots u_ku_{k+1}vby''.$ Since $u_{k+1}$ is a prefix of $x$ and $\varphi(u_{k+1})=0,$ it follows that $P_x(|vb|)=P_x(|va|)\rhoreccurlyeq va \rhorec vb$ which combined with the fact that $u_1\cdots u_{k+1}$ is a prefix of $x$ implies that $\varphi(u_1\cdots u_{k+1})=0$ as required. \end{proof}
\begin{lemma}\label{L2} Assume that $\varphi (u_i)=0$ for all $i\geq 1.$ Then $S_k\leq S_{k-1}$ for all $k\geq 1.$
\end{lemma}
\begin{proof} We first note that $S_1\leq S_0$ since $S_0=+\infty.$ Now fix $k\geq 1;$ we will show that $S_{k+1}\leq S_k.$ By the previous lemma we have that $\varphi(u_1u_2\cdots u_k)=0.$ As in the proof of the previous lemma, we can write $x=u_{k+1}vay'$ and $x=u_1u_2\cdots u_ku_{k+1}vby'' $ for some $y', y'' \in \mathbb{A}^{\mathbb N},$ $v\in \mathbb{A}^*$ and $a,b\in \mathbb{A}$ with $a\rhorec b.$ We claim that $|w_{k+1}|\leq |v|.$ In fact, if $|w_{k+1}|>|v|,$ then $vb$ would be a prefix of $x,$ and hence $va\rhorec vb =P_x(|va|)$ which in turn implies
that $\varphi(u_{k+1})=1,$ a contradiction. Thus $|w_{k+1}|\leq |v|=|w_k|-|u_{k+1}|$ or equivalently $|u_{k+1}| + |w_{k+1}| \leq |w_k|$ from which it follows that $S_{k+1}\leq S_k$ as required.\end{proof}
Returning to the proof of Theorem~\ref{M}, we must show that the factorisation $x=u_1u_2u_3\cdots $ is not $\varphi$-monochromatic.
Suppose to the contrary that $\varphi(u_i)=\varphi(u_j)$ for all $i,j\geq 0.$ If $\varphi(u_i)=0$ for all $i\geq 1,$ then by the previous lemma we have that $S_k\leq S_{k-1}$ for all $k\geq 1$ contradicting that $\lim S_k=+\infty.$ If on the other hand $\varphi(u_i)=1$ for all $i\geq 1,$ then by replacing the original order on $A$ by the reverse order we would have $\varphi(u_i)=0$ for all $i\geq 1,$ and hence as above the previous lemma yields the desired contradiction. This concludes our proof of Theorem~\ref{M}. \end{proof}
We end this section with some applications of Theorem~\ref{M} or its proof. An infinite word $x\in \mathbb{A}^{\mathbb N}$ is called {\it Lyndon} if there exists an order on $\mathbb{A}$ relative to which $x$ is strictly smaller than each of its proper suffixes. Our first application is the following result originally proved in \cite{DPZ}:
\begin{corollary}\label{C1} A Lyndon word $x\in \mathbb{A}^{\mathbb N}$ does not admit a prefixal factorisation, i.e., a factorisation of the form $x=u_1u_2u_3\cdots$ where each $u_i$ is a prefix of $x.$
\end{corollary}
\begin{proof} Suppose to the contrary that $x=u_1u_2u_3\cdots$ is a prefixal factorisation of $x.$ Let $\varphi: \mathbb{A}^+ \rightarrow \{0,1\}$ be the separating $2$-colouring for $x$ defined in (\ref{E}). Note that $x$ being Lyndon is not-periodic. Since $x$ is Lyndon we have that $\varphi(u_i)=0$ for all $i\geq 1,$ contradicting that $\varphi$ is a separating $2$-colouring for $x.$ \end{proof}
Let $x\in \mathbb{A}^{\mathbb N}$ and $a\in \mathbb{A}.$ A factor $u$ of $x$ is said to be {\it rich} in $a$ if $|u|_a\geq |v|_a$ for all factors $v$ of $x$ with $|v|=|u|.$ Theorem~6.7 in \cite{DPZ} states that a Sturmian word $x\in \{0,1\}^{\mathbb N}$ does not admit a factorisation of the form $x=u_1u_2u_3\cdots$ where each $u_i$ is a prefix of $x$ rich in the same letter $a\in \{0,1\}.$ The following generalises this result to all binary non-periodic words:
\begin{corollary}\label{C2}Let $x\in \{0,1\}^{\mathbb N}$ and $a\in \{0,1\}.$ Suppose $x$ admits a prefixal factorisation $x=u_1u_2u_3\cdots$ with each $u_i$ rich in $a.$ Then $x$ is periodic.
\end{corollary}
\begin{proof}Suppose to the contrary that $x$ is not periodic. Let $\varphi:\{0,1\}^+\rightarrow \{0,1\}$ be the separating $2$-colouring for $x$ defined in (\ref{E}) relative to the order on $\{0,1\}$ where $a$ is taken to be the least element. For each $i\geq 1,$ writing
$x=u_iy_i$ with $y_i\in \{0,1\}^{\mathbb N},$ we claim that $x\rhorec y_i.$ Otherwise if $y_i\rhorec x,$ then we can write
$x=zbx'=u_izay'$ for some $z\in \{0,1\}^*,$ $x',y'\in \{0,1\}^{\mathbb N}$ and where $\{a,b\}=\{0,1\}.$ But then the factor $u_i'$ of length $|u_i|$ immediately preceding the suffix $y'$ of $x$ would contain one more occurrence of the symbol $a$ than $u_i,$ contradicting that
$u_i$ was rich in $a.$ Having established that $x\rhorec y_i,$ it follows that $\varphi(u_i)=0$ for all $i \geq 1$ contradicting that $\varphi$ is a separating $2$-colouring for $x.$ \end{proof}
\begin{corollary}\label{C3}Let $\mathbb{A}$ be an arbitrary non-empty set, $x\in \mathbb{A}^{\mathbb N}$ and $a\in \mathbb{A}.$ Suppose $x$ admits a prefixal factorisation $x=u_1u_2u_3\cdots$ with each $u_i$ rich in $a.$ Then $\{n\in {\mathbb N}: x_n=a\}$ is a finite union of (infinite) arithmetic progressions.
\end{corollary}
\begin{proof} Consider the morphism $\rhohi: \mathbb{A}\rightarrow \{0,1\}$ given by $\rhohi(a)=1$ and $\rhohi(b)=0$ for all $ b\in \mathbb{A}\setminus \{a\}.$
Then applying $\rhohi$ to the prefixal factorisation $x=u_1u_2u_3\cdots$ gives a prefixal factorisation of $\rhohi(x)$ in which each $\rhohi(u_i)$ is rich in $1.$ Hence by Corollary~\ref{C2} we have that $\rhohi(x)$ is periodic and hence the set of occurrences of $1$ in $\rhohi(x),$ which is equal to the set of occurrences of $a$ in $x,$ is a finite union of arithmetic progressions. \end{proof}
\begin{corollary}\label{C4} Let $x=x_1x_2x_3\cdots \in \mathbb{A}^{\mathbb N}$ and $k$ be a positive integer. Let $B\subseteq \mathbb{A}^+$ with ${\rm Card}(B)\geq 2k-1.$ Suppose that $x\in A^{\mathbb N}$ for every $k$-element subset $A$ of $B.$ Then $x$ is periodic. \end{corollary}
\begin{proof} Let us assume to the contrary that $x$ is not periodic. By Theorem~\ref{M} there exists a $2$-colouring $\varphi:\mathbb{A}^+\rightarrow \{0,1\}$ relative to which no factorisation of $x$ is $\varphi$-monochromatic. Let $\Sigma_k(B)$ denote the set of all $k$-element subsets of $B.$ By assumption $x$ factors over each $A\in \Sigma_k(B).$ On the other hand, since ${\rm Card}(B)\geq 2k-1,$ it follows that there exists a $\varphi$-monochromatic subset $A\in \Sigma_k(B).$ This gives rise to a $\varphi$-monochromatic factorisation of $x,$ a contradiction.
\end{proof}
\begin{corollary}\label{C5} Let $x=x_1x_2x_3\cdots \in \mathbb{A}^{\mathbb N}$ and $k$ be a positive integer. Let $u_1,u_2,u_3,\ldots, u_{2k+1}\in A^+$ and suppose $x\in \{u_1,u_2\}^{\mathbb N} \cap \{u_2,u_3\}^{\mathbb N}\cap\cdots \cap \{u_{2k},u_{2k+1}\}^{\mathbb N}\cap \{u_{2k+1},u_1\}^{\mathbb N}.$ Then $x$ is periodic. \end{corollary}
\begin{proof} Suppose to the contrary that $x$ is not periodic. Pick any separating $2$-colouring $\varphi: \mathbb{A}^+\rightarrow \{0,1\}$ for $x.$ Then $\varphi(1)=\varphi(2i+1)$ for each $1\leq i\leq k.$ Thus $x\notin \{u_{2k+1},u_1\}^{\mathbb N},$ a contradiction.
\end{proof}
\section{A reformulation of Theorem~\ref{M} in the language of ultrafilters}
In this section give an equivalent reformulation of Theorem~\ref{M} in terms of ultrafilters and the Stone-\v Cech compactification of the discrete semigroup $\mathbb{A}^+.$
We begin by recalling some basic facts. For more information we refer the reader to \cite{HS}.
Let $S$ be a non-empty set and let $\mathscr{P}(S)$ denote the set of all subsets of $S.$ A set $p\subseteq \mathscr{P}(S)$ is called a {\it filter} on $S$ if
\begin{itemize}
\item $S\in p$ and $\emptyset \notin p$
\item If $A\in p$ and $B\in p$ then $A\cap B\in p$
\item If $A\subseteq B$ and $A\in p$ then $B\in p.$
\end{itemize}
A filter $p$ on $S$ is called an {\it ultrafilter} if for all $A\in \mathscr{P}(S)$ either $A\in p$ or $A^c\in p$ where $A^c$ denotes the complement of $A,$ i.e., $A^c=S\setminus A.$ Equivalently, a filter $p$ is an ultrafilter if for each $A\in p$ whenever $A=A_1\cup \cdots \cup A_n$ we have that at least one $A_i\in p.$ Each $x\in S$ determines an ultrafilter $e(x)$ on $S$ defined by $e(x)=\{A\subseteq S: x\in A\}.$ An ultrafilter $p$ on $S$ is called {\it principal} if
$p=e(x)$ for some $x\in S.$ Otherwise $p$ is said to be {\it free}.
Let $\beta S$ denote the collection of all ultrafilters $p$ on $S.$ By identifying each $x\in S$ with the principal ultrafilter $e(x),$ we regard $S\subseteq \beta S.$
If $S$ is infinite, then a straightforward application of Zorn's lemma guarantees the existence of free ultrafilters on $S.$
Given $A\subseteq S$, we put $\overline A=\{p\in\beta S:A\in p\}$. Then $\{\overline A:A\subseteq S\}$
defines a basis for a topology on $\beta S$ relative to which $\beta S$ is both compact and Hausdorff and the mapping
$x\mapsto e(x)$ defines an injection $S\hookrightarrow \beta S$ whose image is dense in $\beta S.$ In fact, if $S$ is given the discrete topology, then $\beta S$ is identified with the Stone-\v Cech compactification of $S:$ Any continuous mapping from $f: S\rightarrow K,$ where $K$ is a compact Hausdorff space, lifts uniquely to a continuous mapping $\beta f: \beta S \rightarrow K.$ Of special interest is the case in which $S$ is a discrete semigroup. In this case the operation on $S$ extends uniquely to $\beta S$ making $\beta S$ a right topological semigroup with $S$ contained in its topological center. This means that
$\rho_p:\beta S\rightarrow \beta S,$ defined by $\rho_p(q)=q\cdot p,$ is continuous for each $p\in\beta S$ and $\lambda_s:\beta S\rightarrow \beta S,$ defined by $\lambda_s(q)=s\cdot q,$ is continuous for each $s\in S.$
The operation $\cdot$ on $\beta S$ is defined as follows: For $p,q\in \beta S$
\[p\cdot q=\{A\subseteq S: \{s\in S:s^{-1}A\in q\}\in p\},\] where $s^{-1}A=\{t\in S:st\in A\}.$
As a consequence of the Ellis-Numakura lemma, $\beta S$ contains an idempotent element i.e., an element $p$ verifying $p\cdot p=p$ (see for instance \cite{HS}). Subsets $A\subseteq S$ belonging to idempotents in $\beta S$ have rich combinatorial structures: Let $\rm{Fin}({\mathbb N})$ denote the set of all finite subsets of ${\mathbb N}.$ Given an infinite sequence $\langle s_n\rangle_{n\in {\mathbb N}}$ in $S,$
let
\[FP\left(\langle s_n\rangle_{n\in {\mathbb N}}\right)=\{\rhorod_{n\in F} s_n: F\in \rm{Fin}({\mathbb N})\}\]
where for each $F\in \rm{Fin}({\mathbb N}),$ the product $\rhorod_{n\in F} s_n$ is taken in increasing order of indices. A subset $A$ of $S$ is called an {\it IP-set} if $A$ contains $FP\left(\langle s_n\rangle_{n\in {\mathbb N}}\right)$ for some infinite sequence $\langle s_n\rangle_{n\in {\mathbb N}}$ in $S.$ IP-sets are characterised as belonging to idempotent elements: $A\subseteq S$ is an IP-set if and only if $A$ belongs to some idempotent element of $\beta S$ (see for instance Theorem~5.12 in \cite{HS}).
\noindent The following is a reformulation of Theorem~\ref{M} in the language of ultrafilters:
\begin{thm}\label{UF} Let $x=x_1x_2x_3\cdots \in \mathbb{A}^{\mathbb N}$ be an infinite word. Then $x$ is periodic if and only if there exists $p\in \beta \mathbb{A}^+$ such that for each $A\in p$ there exists a factorisation $x=u_1u_2u_3\cdots $ with each $u_i \in A.$
\end{thm}
\begin{proof} Suppose $x$ is periodic, i.e., $x\in \{u\}^{\mathbb N}$ for some $u\in \mathbb{A}^+. $ Then the principal ultrafilter $e(u)=\{A\subseteq \mathbb{A}^+: u\in A\}$ obviously verifies the required condition of Theorem~\ref{UF}. Conversely, suppose there exists $p\in \beta \mathbb{A}^+$ such that for each $A\in p$ there exists a factorisation $x=u_1u_2u_3\cdots $ with each $u_i \in A.$ Let $\varphi: \mathbb{A}^+\rightarrow \{0,1\}$ be any $2$-colouring of $\mathbb{A}^+.$ We will show that $x$ admits a $\varphi$-monochromatic factorisation. The result then follows from Theorem~\ref{M}. Consider the partition $\mathbb{A}^+=\varphi^{-1}(0)\cup \varphi^{-1}(1).$ Since $\mathbb{A}^+\in p,$ it follows that $\varphi^{-1}(a)\in p$ for some $a\in \{0,1\}.$ Thus there exists a factorisation $x=u_1u_2u_3\cdots $ with each $u_i\in \varphi^{-1}(a).$ In other words, $x$ admits a $\varphi$-monochromatic factorisation.\end{proof}
\begin{remark}{\rm It can be shown that Theorem~\ref{UF} is actually an equivalent reformulation of Theorem~\ref{M}. More precisely, given an infinite word $x=x_1x_2x_3\cdots \in \mathbb{A}^{\mathbb N},$ the following statements are equivalent:
\begin{enumerate}
\item[a)] There exists $p\in \beta \mathbb{A}^+$ such that for each $A\in p$ we can write $x=u_1u_2u_3\cdots $ with each $u_i \in A.$
\item[b)] For each finite colouring $\varphi:\mathbb{A}^+\rightarrow C$ there exists a $\varphi$-monochromatic factorisation of $x.$
\end{enumerate}
To see that $a)\mathcal{L}ongrightarrow b),$ pick $p$ as in a) and let $\varphi:\mathbb{A}^+\rightarrow C$ be any finite colouring of $\mathbb{A}^+.$ Consider the partition $\mathbb{A}^+=\bigcup_{c\in C}\varphi^{-1}(c).$ Then there exists some $c\in C$ such that $\varphi^{-1}(c)\in p.$ This implies that we can write $x=u_1u_2u_3\cdots$ with each $u_i\in \varphi^{-1}(c).$ In other words, $x$ admits a $\varphi$-monochromatic factorisation. Conversely to see that $b)\mathcal{L}ongrightarrow a),$ assume b) and define $\rhohi: \mathscr{P}(\mathbb{A}^+)\rightarrow \{0,1\}$ by $\rhohi(A)=1$ if and only if for every finite partition $A=A_1\cup A_2\cup \cdots \cup A_n$ there exists $1\leq i\leq n$ with $A_i\in \mathscr{F}(x).$ Then $\rhohi$ verifies the following three conditions for all subsets $A,B \in \mathscr{P}(\mathbb{A}^+):$ \item (1) $\rhohi(\mathbb{A}^+)=1$ (this is a consequence of b). \item (2) $A\subseteq B$ and $\rhohi(A)=1$ then $\rhohi(B)=1.$ \item (3) If $\rhohi (A)=\rhohi(B)=0$ then $\rhohi (A\cup B)=0.$
\noindent By a standard application of Zorn's lemma, it is shown that there exists $p\in \beta \mathbb{A}^+$ such that $\rhohi(A)=1$ for all $A\in p$ (see Theorem~3.11 in \cite{HS}).
Finally for each $A\in p,$ by considering the trivial partition $A=A,$ we deduce that $A\in \mathscr{F}(x)$ as required.}
\end{remark}
We next show that $p$ in Theorem~\ref{UF} may be taken to be an idempotent element of $\beta \mathbb{A}^+.$ For $x=x_1x_2x_3\cdots \in \mathbb{A}^{\mathbb N},$ we set \[\mathscr{F}(x)=\{A\subseteq \mathbb{A}^+: x\in A^{\mathbb N}\}\] and \[\mathcal{U}(x)=\{p\in \beta \mathbb{A}^+: p\subseteq \mathscr{F}(x)\}.\]
Thus $p\in \mathcal{U}(x)$ if and only if $x$ factors over every $A\subseteq \mathbb{A}^+$ belonging to $p.$ Thus Theorem~\ref{UF} states that $\mathcal{U}(x)\neq \emptyset$ if and only if $x$ is periodic.
\begin{thm}\label{UF2} Let $x=x_0x_1x_2\cdots\in \mathbb{A}^{\mathbb N}.$ Then the following are equivalent:
\begin{enumerate}
\item[i)] $x$ is periodic.
\item[ii)] $\mathcal{U}(x)$ is a closed sub-semigroup of $\beta \mathbb{A}^+.$
\item[iii)] $\mathcal{U}(x)$ contains an idempotent element.
\item[iv)] The set $\rm{Pref}(x)$ consisting of all prefixes of $x$ is an IP-set
\end{enumerate}
\end{thm}
\begin{proof} We first note that for each infinite word $x,$ we have that $\mathcal{U}(x)$ is a closed subset of $\beta \mathbb{A}^+.$ In fact, suppose $p\in \beta \mathbb{A}^+\setminus \mathcal{U}(x),$ then there exists $A\in p$ with $A\notin \mathscr{F}(x).$ Then $\overline A$ is an open neighbourhood of $p$ and any $q\in \overline A$ contains the set $A$ and hence is not in $\mathcal{U}(x).$ To see that $i)\mathcal{L}ongrightarrow ii),$ suppose that $x$ is periodic. Pick the shortest $u\in \mathbb{A}^+$ such that $x\in \{u\}^{\mathbb N}.$ Then the principal ultrafilter $e(u)=\{A\subseteq \mathbb{A}^+: u\in A\}$ clearly belongs to $\mathcal{U}(x).$ Thus $\mathcal{U}(x)\neq \emptyset.$ We note that if $x$ is periodic, then $\mathcal{U}(x)$ also contains free ultrafilters. In fact, for each $i\geq 1,$ let $A_i=\{u^j: j\geq i\}.$ Then $p\in \mathcal{U}(x)$ if and only if $A_1\in p$ and any $p\in \beta \mathbb{A}^+$ containing $\{A_i: i\geq 1\}$ is a free ultrafilter belonging to $\mathcal{U}(x).$ It remains to show that $p\cdot q\in \mathcal{U}(x)$ whenever $p,q\in \mathcal{U}(x).$ Let $A\in p\cdot q$ with $p,q\in \mathcal{U}(x).$ Then $\{s\in \mathbb{A}^+:s^{-1}A\in q\}\in p$ and since $p\in \mathcal{U}(x)$ it follows that $u^n\in \{s\in \mathbb{A}^+:s^{-1}A\in q\}$ for some $n\in {\mathbb N}.$ Thus $\{t\in \mathbb{A}^+: u^nt\in A\}\in q$ and since $q\in \mathcal{U}(x)$ it follows that $u^m \in \{t\in \mathbb{A}^+: u^nt\in A\}$ for some $m\in {\mathbb N}.$ In other words $u^{n+m}\in A$ and hence $x$ factors over $A.$ Thus $p\cdot q\in \mathcal{U}(x).$
The implication $ii)\mathcal{L}ongrightarrow iii)$ follows from the Ellis-Numakura lemma. To see $iii)\mathcal{L}ongrightarrow iv)$ pick an idempotent element $p\in \mathcal{U}(x).$ We note that $\rm{Pref}(x)$ belongs to every $q\in \mathcal{U}(x).$ In fact, suppose that $\rm{Pref}(x)\notin q,$ for some $q\in \mathcal{U}(x).$ Then $\mathbb{A}^+\setminus \rm{Pref}(x)\in q$ which implies that $x$ factors over $\mathbb{A}^+\setminus \rm{Pref}(x).$ But this is a contradiction since in any factorisation of $x,$ the first term occurring in the factorisation belongs to $\rm{Pref}(x).$ Thus in particular $\rm{Pref}(x)\in p.$ Since $p$ is an idempotent, it follows that $\rm{Pref}(x)$ is an IP-set. Finally, to see that $iv)\mathcal{L}ongrightarrow i)$ assume that $\rm{Pref}(x)$ is an IP-set. Then $\rm{Pref}(x)$ contains $FP\left(\langle s_n\rangle_{n\in {\mathbb N}}\right)$ for some infinite sequence $\langle s_n\rangle_{n\in {\mathbb N}}$ of prefixes of $x.$ This means that for each $n\geq 2,$ both $s_1s_2\cdots s_n$ and $s_2\cdots s_n$ are prefixes of $x.$ Thus $x=s_1x$ which implies that $x$ is periodic. \end{proof}
\small
\end{document} |
\begin{document}
\title{The Boosted Double-Proximal Subgradient Algorithm \ for Nonconvex Optimization hanks{Submitted to the editors oday.
unding{F. J. Arag\'on-Artacho and D. Torregrosa-Bel\'en were partially supported by the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund (ERDF) of the European Commission, Grant PGC2018-097960-B-C22, and by the Generalitat Valenciana (AICO/2021/165).
P. P\'erez-Aros was supported by Centro de Modelamiento Matem\'atico
(CMM), ACE210010 and FB210005, BASAL
funds for center of excellence and ANID-Chile grant: Fondecyt Regular 1200283 and Fondecyt Regular 1220886 and Fondecyt Exploración 13220097.
D. Torregrosa-Bel\'en was supported by MINECO and European Social Fund (PRE2019-090751) under the program ``Ayudas para contratos predoctorales para la formaci\'{o}
\begin{abstract}
In this paper we introduce the Boosted Double-proximal Subgradient Algorithm (BDSA), a novel splitting algorithm designed to address general structured nonsmooth and nonconvex mathematical programs expressed as sums and differences of composite functions. BDSA exploits the combined nature of subgradients from the data and proximal steps, and integrates a line-search procedure to enhance its performance. While BDSA encompasses existing schemes proposed in the literature, it extends its applicability to more diverse problem domains. We establish the convergence of BDSA under the Kurdyka--{\L}ojasiewicz property and provide an analysis of its convergence rate.
To evaluate the effectiveness of BDSA, we introduce a novel family of challenging test functions with an abundance of critical points. We conduct comparative evaluations demonstrating its ability to effectively escape non-optimal critical points. Additionally, we present two practical applications of BDSA for testing its efficacy, namely, a constrained minimum-sum-of-squares clustering problem and a nonconvex generalization of Heron's problem.
\end{abstract}
\begin{keywords}
nonconvex optimization, difference programming, Kurdyka--{\L}ojasiewicz property, descent lemma, minimum-sum of squares clustering, Heron's problem
\end{keywords}
\begin{MSCcodes}
49J53, 90C26, 90C30, 65K05, 90C46
\end{MSCcodes}
\section{Introduction}
In this paper we are concerned with the structured nonconvex optimization problem
\begin{equation}\label{eq:P1}
\min_{x\in\mathbb{R}^n} f(x) +g(x)-\sum_{i =1 }^p h_i(\Psi_i (x)), \tag{P}
\end{equation}
where the function $f:\mathbb{R}^n\to\mathbb{R}$ is locally Lipschitz and satisfies the descent lemma, $g:\mathbb{R}^n\to{]-\infty,+\infty]}$ is lower-semicontinuous and prox-bounded, $h_i:\mathbb{R}^{m_i}\to\mathbb{R}$ are convex continuous functions and $\Psi_i:\mathbb{R}^n\to\mathbb{R}^{m_i}$ are differentiable functions with Lipschitz continuous gradients, for $i=1,\ldots,p$ (see Assumption~\ref{assump:1} for more specific details).
Problems in this form appear in a broad variety of fields such as machine learning, image recovery or signal processing~\cite{wen2018survey, lou2015weighted,zhang2009some}.
Numerous algorithms have been developed to handle simpler instances of~\eqref{eq:P1} (see, e.g., \cite{an2017convergence, bot2019doubleprox, doi:10.1137/060655183, sun2003proximal}), but as far as we are aware of, not for the more general problem that we address in this work.
We present a new splitting algorithm, named \emph{Boosted Double-proximal Subgradient Algorithm} (BDSA), that makes use of the inherent structure of problem~\eqref{eq:P1}. More specifically, the method employs the subdifferential of $f$, the gradients of $\Psi_i$ and the \emph{proximal point operators} of the functions $g$ and $h_i$, for $i=1,\ldots,p$. Subsequently, an optional linesearch can be performed to obtain the final update, which intends to steer (or \emph{``boost''}) the iteration to a point with a reduced value of the objective function, in a similar manner than the linesearch introduced in~\cite{aragon2018accelerating, boostedDCA} permits to accelerate the \emph{Difference of Convex functions Algorithm} (DCA)~\cite{Oliveira2020,TAO1986249, pham2014recent}.
In our numerical tests, we show that the addition of the linesearch provides major improvements in the performance of the method. On the one hand, it \emph{accelerates} the algorithm, significantly reducing both the number of iterations and the time that it needs to converge. On the other hand, it may help the sequence to converge to \emph{better solutions}. Indeed, note that the algorithms employed for tackling this class of nonconvex problems usually converge to critical points (see Section~\ref{sect:BDSA}). Being a critical point is a necessary condition for local optimality, but not sufficient. Therefore, the algorithms often converge to critical points which are not local minima. We show that the linesearch introduced in our scheme helps the method to converge to local minima by avoiding some of the non-desirable critical points.
The paper is structured as follows. We begin by introducing the notation and some basic notions of variational analysis in Section~\ref{sect:preliminaries}. In Section~\ref{sect:BDSA}, we present our algorithm and study its convergence properties in Subsection~\ref{subsec:convergence}. In Subsection~\ref{subsec:KL} we make use of the Kurdyka--{\L}ojasiewicz property to prove its global convergence and deduce some convergence rates.
Section~\ref{sect:numerical} contains multiple numerical experiments demonstrating the good performance of BDSA. The benefit of the linesearch for escaping non-minimal critical points is demonstrated with the introduction of a new family of challenging test functions in Subsection~\ref{subsect:avoidingcriticalpoints}. The reduction in time and iterations is illustrated with an application of the \emph{minimum sum-of-squares clustering problem} and a generalization of the classical \emph{Heron problem} in Subsections~\ref{sect:numerical2} and~\ref{sect:numerical3}, respectively. We finish with some concluding remarks in Section~\ref{sec:conc}.
\section{Preliminaries and Notational Conventions}\label{sect:preliminaries}
Throughout this paper, we use $\mathbb{R}^n$ to denote the Euclidean space of dimension $n$, while we denote the extended-real-valued line by $\overline{\mathbb{R}} := \mathbb{R}\cup\{-\infty,+\infty\}$, and set $1/{0}=+\infty$ by convention. The notations $\|\cdot\|$ and $\langle \cdot, \cdot\rangle$ represent the Euclidean norm and inner product in $\mathbb{R}^n$, respectively. We use $\mathbb{B}_\epsilon(\bar{x})$ to denote the closed ball of radius $\epsilon>0$ centered at $\bar{x}$.
Given $p$ positive integers $m_1,\ldots, m_p$, the inner product of the product space $ \mathbb{R}^{m_1}\times\cdots\times\mathbb{R}^{m_p}$ is defined by
\begin{equation*}
\langle (x_1,\ldots,x_p),(y_1,\ldots,y_p)\rangle := \sum_{i=1}^p \langle x_i,y_i\rangle \quad \text{for all } (x_1,\ldots,x_p),(y_1,\ldots,y_p)\in\mathbb{R}nm,
\end{equation*}
with $m := \sum_{i=1}^p m_i$, and its induced norm is denoted by $\|(x_1,\ldots,x_p)\|$.
Vectors in product spaces will be marked with bold, e.g., $\mathbf{x} = (x_1,\ldots,x_p)\in\mathbb{R}^m$.
\subsection{Notions of Variational Analysis}
Given some constant $L\geq 0$, a vector-valued function $F: C\subseteq \mathbb{R}^n \to \mathbb{R}^m$ is said to be $L$-\emph{Lipschitz continuous} on $C$ if
\begin{equation*}
\|F(x)-F(y)\| \leq L \|x-y\|\quad \text{for all } x,y\in C,
\end{equation*}
and \emph{locally Lipschitz continuous} around $\bar x\in C$ if it is Lipschitz continuous in some neighborhood of $\bar x$.
The \emph{domain} of a function $f:\mathbb{R}^n\to\overline{\mathbb{R}}$ is defined as $\dom f := \{ x \in\mathbb{R}^n : f(x) < +\infty\}$, and we say that $f$ is \emph{proper} if it does not attain the value $-\infty$ and $ \dom f\neq \emptyset$. The function $f$ is \emph{lower-semicontinuous} (\emph{l.s.c.}) at some point $\bar{x}\in\mathbb{R}^n$ if $\liminf_{x\to\bar{x}} f(x) \geq f(\bar{x})$. The \emph{Dini upper directional derivative} of $f$ at some point $\bar{x}\in\dom f$ in the direction $d\in\mathbb{R}^n$ is defined as
$$d^{+} f(\bar{x} ; d):=\limsup_{t \downarrow 0} \frac{f(\bar{x}+t d)-f(\bar{x})}{t}.$$
Given $\bar{x}\in f^{-1}(\mathbb{R})$, the \emph{regular subdifferential} of the function $f$ at $\bar{x}\in\mathbb{R}^n$ is the closed and convex set of \emph{regular subgradients}
\begin{equation*}
\hat{\partial} f(\bar{x}) := \left\{ v\in\mathbb{R}^n : f(x) \geq f(\bar{x}) + \langle v, x-\bar{x}\rangle + o(\|x-\bar{x}\|) \right\}.
\end{equation*}
We say that $v\in\mathbb{R}^n$ is a (\emph{basic}) \emph{subgradient} of $f$ at $\bar{x}$ if there exists sequences $(x^k)_{k\in\mathbb{N}}$ and $(v^k)_{k\in\mathbb{N}}$, with $v^k\in\hat{\partial}f(x^k)$ for all $k\in\mathbb{N}$, such that $x^k\to\bar{x}$, $f(x^k)\to f(\bar{x})$ and $v^k\to v$, as $k \to \infty$. The (\emph{basic}) \emph{subdifferential} of $f$ at $\bar{x}$, denoted by $\partial f(\bar{x})$, is the set of all subgradients of $f$ at $\bar{x}$. We use the convention $ \hat{\partial} f(\bar{x})={\partial} f(\bar{x}):=\emptyset$ if $|f(\bar x) |= +\infty$. We say that $f$ is \emph{lower regular} at $\bar{x}\in\dom f$ if $\partial f(\bar{x})=\hat{\partial} f(\bar{x})$.
Let us recall that the regular and basic subdifferentials coincide for convex functions and are equal to the convex subdifferential, which in an abuse of notation we denote by
\begin{equation*}
\partial f(\bar{x}) := \left\{ v \in\mathbb{R}^n : f(x) \geq f(\bar{x}) + \langle v, x-\bar{x} \rangle \right\}.
\end{equation*}
If a function $f:\mathbb{R}^n\to\mathbb{R}$ is strictly differentiable at a point $\bar{x}\in\mathbb{R}^n$ then all its subdifferentials are singletons and coincide with its gradient, which is denoted by $\nabla f(\bar{x})$. We also use the same symbol to denote the transpose of the Jacobian matrix of a multivariable function $F=(F_1,\ldots,F_m):\mathbb{R}^n\to\mathbb{R}^m$, i.e., the matrix of gradients
$ \nabla F(x) = \left( \nabla F_1(x), \ldots, \nabla F_m(x)\right)$.
Given a function $f:\mathbb{R}^n\to\overline{\mathbb{R}}$, its convex conjugate is the function $f^*:\mathbb{R}^n\to\overline{\mathbb{R}}$ given by
\begin{equation*}
f^*(x) := \sup_{u\in\mathbb{R}^n} \left\{ \langle u,x\rangle -f(u) \right\}.
\end{equation*}
If $f$ is proper, the \emph{Fenchel--Moreau} theorem states that $f$ is convex and l.s.c. if and only if $f=f^{**}$, where $f^{**}:=(f^*)^*$ denotes the biconjugate of $f$.
Given a proper l.s.c. function $f:\mathbb{R}^n\to\overline{\mathbb{R}}$ and some constant $\gamma>0$, the \emph{proximal mapping} (also known as the \emph{proximity operator} or \emph{proximal point operator}) is the multifunction $\prox_{\gamma f}:\mathbb{R}^n\rightrightarrows\mathbb{R}^n$ defined as the solution set of the optimization problem
\begin{equation*}\label{eq:argmin}
\prox_{\gamma f}(x) := \argmin_{u\in\mathbb{R}^n} \left\{ f(u) + \frac{1}{2 \gamma} \|u-x\|^2\right\}.
\end{equation*}
A function $f:\mathbb{R}^n \to \overline{\mathbb{R}}$ is said to be \emph{prox-bounded} if there exists some ${\gamma} >0$ such that $f(\cdot)+\frac{1}{2{\gamma}} \|\cdot-x\|^2$ is bounded from below for all $x\in\mathbb{R}^n$. The supremum of the set of all such constants $\gamma$ is called the \emph{prox-boundedness threshold} of $f$ and is denoted by~${\gamma}^f$. For a proper, l.s.c. and prox-bounded function, the proximal mapping $\prox_{\gamma f}$ has full domain for any $\gamma\in{]0,{\gamma}^f[}$, but it may not be single-valued (see, e.g., \cite[Theorem~1.25]{MR1491362}). It is worth noting that all proper, convex and l.s.c. functions are prox-bounded, but the class is much larger. Any proper l.s.c. function $f:\mathbb{R}^n\to \overline{\mathbb{R}}$ that is bounded from below by an affine function has a threshold of prox-boundedness $\gamma^f=+\infty$ (see~\cite[Example 3.28]{MR1491362}). For example, the indicator function $\iota_C$ of a nonempty and closed set $C\subseteq\mathbb{R}^n$ is prox-bounded with threshold ${\gamma}^{\iota_C} = +\infty$.
\subsection{The Family of Upper-$\mathcal{C}^2$ Functions}\label{subsec:upperC2}
The class of functions with a Lipschitz continuous gradient is very important in optimization. Its relevance relies on the fact that these functions verify the so-called \emph{descent lemma} (see, e.g.,~\cite[Lemma A.11]{MR3289054}). Namely, given a differentiable function $f:\mathbb{R}^n\to\mathbb{R}$ with $L_f$-Lipschitz continuous gradient, then the following inequality holds
\begin{equation}\label{eq:descentlemmaineq}
f(y) \leq f(x) + \langle \nabla f(x),y-x\rangle + \frac{L_f}{2}\|y-x\|^2\quad \text{for all } x,y\in\mathbb{R}^n.
\end{equation}
Different authors have identified a larger family of functions that also satisfies an inequality similar to~\eqref{eq:descentlemmaineq}, preserving thus the same nice properties for optimization (see, e.g. \cite[Theorem~5.1]{MR1363364} and~\cite[Definition~10.29]{MR1491362}). We extend our analysis to this broader class of functions, which we present next.
\begin{definition}
Let $V \subseteq \mathbb{R}^n$ be an open, convex and bounded set and let $f :\mathbb{R}^n \to \overline{\mathbb{R}}$ be Lipschitz continuous on $V$. We say that $f$ is $\kappa$-\emph{upper-$\mathcal{C}^2$} on $V$ for some $\kappa\geq0$ if there exist a compact set $S$ (in some topological space) and some continuous functions $b: S \to \mathbb{R}^n$ and $c: S \to \mathbb{R}$ such that
\begin{equation}\label{eq:upperC2}
f(x)= \min\limits_{ s\in S} \left\{ \kappa\| x\|^2 - \langle b(s),x\rangle - c(s) \right\}\quad \text{for all } x\in V.
\end{equation}
\end{definition}
From~\eqref{eq:upperC2} it directly follows that $\kappa$-upper-$\mathcal{C}^2$ functions are DC (difference of convex) functions with the specific DC decomposition
$$f(x)=\kappa \|x\|^2 - \max_{s\in S} \left\{\langle b(s), x\rangle+c(s)\right\},$$
since $x\mapsto \max_{s\in S} \left\{\langle b(s), x\rangle+c(s)\right\}$ is a convex continuous function.
The following result, based on~\cite[Theorem~5.1]{MR1363364}, establishes the relationship between upper-$\mathcal{C}^2$ functions and the descent lemma. The convex hull of a set is denoted by ``$\conv$''.
\begin{proposition}\label{p:deslemma}
Let $U$ be an open and convex set such that $f:\mathbb{R}^n\to\overline{\mathbb{R}}$ is locally Lipschitz on $U$. Then, the following assertions are equivalent for a parameter $\kappa\geq 0$:
\begin{enumerate}[(i)]
\item\label{it:p25-i} $f$ is $\kappa$-upper-$\mathcal{C}^2$ on every bounded subset of $U$;
\item\label{it:p25-iii} for all $x\in U$ and $\xi \in \conv{\partial f(x)}$, it holds
\begin{align}\label{equpperdesc}
f(y) \leq f(x) + \langle \xi , y-x\rangle + \kappa \| y -x\|^2\quad\text{for all } y \in U;
\end{align}
\item\label{it:p25-ii} for each $x\in U$, there exits $\xi \in \mathbb{R}^n$ such that \eqref{equpperdesc} holds.
\end{enumerate}
\end{proposition}
\begin{proof}
The case of $\kappa=0$ can be straightforwardly established. Therefore, for the remainder of the proof, we assume $\kappa>0$.
(\ref{it:p25-i})$\mathbb{R}ightarrow$(\ref{it:p25-iii}) Let $x,y\in U$ and $\xi\in\conv{\partial f(x)}$. Let $V$ be an open, convex and bounded subset of $U$ that contains both $x$ and $y$. By~\cite[Theorem~5.1(a)$\mathbb{R}ightarrow$(c) and Theorem~5.2]{MR1363364} applied to $-f$ and~$V$, it holds
\begin{equation}\label{eq:Psub}
f(w) \leq f(x) + \langle -\zeta, w-x\rangle + \kappa \|w-x\|^2,\quad\forall w\in V
\end{equation}
for all $\zeta \in\partial(-f)(x)=\hat{\partial}( -f)(x)$. By convexity of the regular subdifferential,
\begin{equation}\label{eq:subinclusions}
\partial (-f)(x)=\conv{ \partial (-f)(x)} =-\conv{\partial f(x)}.
\end{equation}
Since $-\xi\in-\conv{\partial f(x)}= \partial (-f)(x)$, we conclude that~\eqref{eq:Psub} holds for $\zeta:=-\xi$ and $w:=y\in V$, which implies~\eqref{equpperdesc}.
(\ref{it:p25-iii})$\mathbb{R}ightarrow$(\ref{it:p25-ii}) This is straightforward: $\partial f(x)\neq\emptyset$ since $f$ is locally Lipschitz around $x$ (see, e.g., \cite[Theorem 1.22]{MR3823783}).
(\ref{it:p25-ii})$\mathbb{R}ightarrow$(\ref{it:p25-i}) If $V$ is an open, convex and bounded subset of $U$, then~\eqref{equpperdesc} holds on $V$, so \cite[Theorem~5.1 (b)$\mathbb{R}ightarrow$(a)]{MR1363364} implies that $f$ is $\kappa$-upper-$\mathcal{C}^2$ on V. This completes the proof.
\end{proof}
We conclude this section with some examples of upper-$\mathcal{C}^2$ functions, the first of which was considered in \cite[Lemma~5.2]{ferreira2021boosted}. A local variation of this result can be found in~\cite[Lemma~3.6]{aragonartacho2023coderivativebased}.
\begin{example}[Difference of an $L$-smooth function and a convex function]\label{example01}
Let $f_1:\mathbb{R}^n\to\mathbb{R}$ be a differentiable function whose gradient is $L_{f_1}$-Lipschitz continuous and let $f_2:\mathbb{R}^n\to\mathbb{R}$ be a convex and l.s.c. function. Then, the function $f:= f_1-f_2$ is $L_{f_1}/2$-upper-$\mathcal{C}^2$ on every bounded set of $\mathbb{R}^n$. Indeed, observe first that~\eqref{eq:descentlemmaineq} holds for $f_1$. Moreover, for any $v\in\partial f_2(x)$, by definition of the convex subdifferential, we have
\begin{equation*}
-f_2(y) \leq -f_2(x) + \langle -v,y-x\rangle, \text{ for all } y\in\mathbb{R}^n.
\end{equation*}
Adding together~\eqref{eq:descentlemmaineq} for $f_1$ and the above inequality, we get that
\begin{equation*}
f(y) \leq f(x) + \langle \nabla f_1(x)-v,y-x\rangle + \frac{L_{f_1}}{2}\|y-x\|^2.
\end{equation*}
Hence, the assertion follows by Proposition~\ref{p:deslemma}.
\end{example}
\begin{example}[Squared distance to a nonconvex set]\label{prop:Asplund}
Consider a (possibly nonconvex) nonempty closed set $C \subseteq \mathbb{R}^s$ and a matrix $Q \in \mathbb{R}^{s\times n}$. Then the mapping $x\to \frac{1}{2}d^2(Qx,C)$ is upper $C^2$ on $\mathbb{R}^n$. Indeed, let us notice that
\begin{align}\label{rep_01}
\frac{1}{2} d^2(Qx,C) =\frac{1}{2} \| Qx\|^2 - \Asp{C}(Qx),
\end{align}
where $\Asp{C}$ is the \emph{Asplund function} associated to the set $C$ given by
\begin{align*}
\Asp{C}(w):=\sup_{ y \in C} \left\{\langle y, w\rangle - \frac{1}{2} \| y\|^2\right\}=\left(\iota_C+\frac{1}{2}\|\cdot\|^2\right)^*(w).
\end{align*}
By representation \eqref{rep_01} and Example \ref{example01}, we have that $x\mapsto \frac{1}{2} d^2(Qx,C)$ is $\frac{\|Q\|^2}{2}$-upper-$\mathcal{C}^2$ on $\mathbb{R}^n$.
\end{example}
\section{The Boosted Double-Proximal Subgradient Algorithm}\label{sect:BDSA}
In this section, we design an algorithm for solving the nonconvex optimization problem~\eqref{eq:P1}. The algorithm utilizes subgradients of $f$, the gradients of~$\Psi_i$, and proximal steps of $g$ and $h_i^*$, for $i=1,\ldots,p$. Additionally, it incorporates a linesearch step, leading us to name it the \emph{Boosted Double-proximal Subgradient Algorithm} (in short BDSA).
Le us define the function $\varphi: \mathbb{R}^n \to \overline{\mathbb{R}}$ as follows:
\begin{equation}\label{Def_varphi}
\varphi(x) := f(x) +g(x)-\sum_{i =1 }^p h_i(\Psi_i (x)).
\end{equation}
The following assumptions are made throughout the paper.
\begin{assumption}\label{assump:1}
Let $U\subseteq \mathbb{R}^n$ be an open convex set such that $\dom g \subseteq U$. Suppose that $\inf_{x\in\mathbb{R}^n} \varphi(x) > -\infty$ and that the functions in~\eqref{eq:P1} satisfy:
\begin{enumerate}[(i)]
\item $f:\mathbb{R}^n\to\overline{\mathbb{R}}$ is locally Lipschitz on $U$ and $\kappa$-upper-$\mathcal{C}^2$ on every bounded subset of $U$;
\item $g:\mathbb{R}^n\to\overline{\mathbb{R}}$ is proper, l.s.c. and prox-bounded for some ${\gamma}^g >0$;
\item $h_i:\mathbb{R}^{m_i}\to\mathbb{R}$ are convex and continuous functions for all $i=1,\ldots,p$;
\item $\Psi_i:\mathbb{R}^n\to\mathbb{R}^{m_i}$ are differentiable functions with $L_i$-Lipschitz continuous gradients on $U$ for all $i=1,\ldots,p$.
\end{enumerate}
\end{assumption}
\begin{remark} Despite the fact that the class of upper-$\mathcal{C}^2$ functions is large, as demonstrated by the examples presented in Subsection~\ref{subsec:upperC2}, it is worth noting that, in general, one cannot subsume the function $-\sum_{i =1 }^p h_i(\Psi_i (x))$ into the function $f$ in~\eqref{Def_varphi}, since $-h_i(\Psi_i(x))$ may not belong to the class of upper-$\mathcal{C}^2$ functions (e.g., if $h_i(y)=y^2$ and $\Psi_i(x)=x^2$, then $-h_i(\Psi_i(x))=-x^4$ does not satisfy~\eqref{equpperdesc} on $U=\mathbb{R}$ for any fixed $\kappa\geq0$).
\end{remark}
Instead of directly addressing problem~\eqref{eq:P1}, which consists in the minimization of the function $\varphi$ in~\eqref{Def_varphi}, we consider the primal-dual formulation
\begin{equation}\label{eq:P2}
\min_{(x,\mathbf{y}) \in\mathbb{R}^n \times \mathbb{R}nm}\Phi(x,\mathbf{y}), \tag{PD}
\end{equation}
where $\Phi:\mathbb{R}^n\times\mathbb{R}nm\to\overline{\mathbb{R}}$ is given by
\begin{equation}\label{Def_PHI}
\Phi(x,\mathbf{y}) := f(x)+g(x)+\sum_{i=1}^p \left(h_i^*(y_i)-\langle \Psi_i(x),y_i\rangle\right),
\end{equation}
with $\mathbf{y}=(y_1,\ldots,y_p)\in\mathbb{R}^{m_1}\times\cdots\times\mathbb{R}^{m_p}=\mathbb{R}nm$. It is easy to check that the optimal values of both problems coincide, that is,
\begin{align}\label{Eq_Optvalues}
\inf_{x\in\mathbb{R}^n} \varphi(x) = \inf_{(x,\mathbf{y}) \in\mathbb{R}^n \times \mathbb{R}nm}\Phi(x,\mathbf{y}).
\end{align}
Indeed, by the Fenchel--Moreau theorem,
\begin{equation*}
\begin{aligned}
\inf_{x\in\mathbb{R}^n} \varphi(x)
&= \inf_{x\in\mathbb{R}^n} \left\{ f(x)+g(x)-\sum_{i=1}^p h_i(\Psi_i(x)) \right\}\\
& = \inf_{x\in\mathbb{R}^n} \left\{f(x)+g(x)-\sum_{i=1}^p \sup_{y_i\in\mathbb{R}^{m_i}} \left\{ \langle \Psi_i (x),y_i\rangle -h_i^*(y_i)\right\} \right\}\\
& = \inf_{x\in\mathbb{R}^n}\inf_{\mathbf{y}\in\mathbb{R}nm}\left\{f(x)+g(x)+\sum_{i=1}^p \left( h_i^*(y_i)-\langle \Psi_i(x),y_i\rangle\right)\right\}\\
&= \inf_{(x,\mathbf{y}) \in\mathbb{R}^n \times \mathbb{R}nm}\Phi(x,\mathbf{y}).
\end{aligned}
\end{equation*}
By the generalized Fermat rule, a necessary condition for a point $(\bar{x},\bar{\mathbf{y}})\in\mathbb{R}^n\times\mathbb{R}^m$ to be a local minimum of $\Phi$ is that $0\in\partial \Phi(\bar{x},\bar{\mathbf{y}})$. Observe that we can express $\Phi=\Phi_1+\Phi_2$, with $\Phi_1(x,\mathbf{y}):=f(x)+g(x)+\sum_{i=1}^p h_i^*(y_i)$ and $\Phi_2(x,\mathbf{y}):=\sum_{i=1}^p \langle \Psi_i(x),y_i\rangle$. Since $\Phi_2$ is~$\mathcal{C}^1$, by the sum rule in~\cite[Proposition~1.30]{MR3823783},
\begin{equation}\label{eq:subdiff_Phi}
\begin{aligned}
\partial \Phi(x,\mathbf{y})&=\partial\Phi_1(x,\mathbf{y})-\nabla \Phi_2(x,\mathbf{y})\\
&=\partial(f+g)(x)\times\left(\bigtimes_{i=1}^p \partial h_i^*(y_i)\right)-\left(\sum_{i=1}^p\nabla \Psi_i(x)y_i,\Psi_1(x),\ldots,\Psi_p(x)\right),
\end{aligned}
\end{equation}
where the second equality holds because $\Phi_1$ has separate variables. Therefore, the necessary condition $0\in\partial \Phi(\bar{x},\bar{\mathbf{y}})$ is equivalent to
\begin{equation}\label{eq:estpoint}
\left\{
\begin{aligned}
&\sum_{i=1}^p \nabla\Psi_i(\bar{x}) \bar{y}_i \in \partial (f+ g)(\bar{x}) ,\\
&\Psi_i( \bar{x}) \in \partial h_i^* (\bar{y}_i), \quad \forall i=1,\ldots,p.
\end{aligned}
\right.
\end{equation}
By~\cite[Corollary~2.20]{MR3823783}, we have $\partial (f+ g)(\bar{x}) \subseteq \partial f(\bar x)+\partial g(\bar x)$. Thus, the inclusions~\eqref{eq:estpoint} imply
\begin{equation}\label{eq:critpoint}
\left\{
\begin{aligned}
&\sum_{i=1}^p \nabla\Psi_i(\bar{x}) \bar{y}_i \in \partial f(\bar x )+\partial g(\bar{x}) ,\\
&\bar{y}_i \in \partial h_i (\Psi_i( \bar{x})), \quad \forall i=1,\ldots,p.
\end{aligned}
\right.
\end{equation}
We refer to a point $(\bar{x},\bar{\mathbf{y}})\in\mathbb{R}^n\times\mathbb{R}^m$ that satisfy~\eqref{eq:critpoint} as a \emph{critical point of~\eqref{eq:P2}}.
On the other hand, given a point $\bar{x}\in\dom{\varphi}$, it is well-known (see, e.g., \cite{MR4228320,correa2023optimality,MR2453095}) that the subdifferential inclusion
\begin{equation}\label{eq:statpoint}
\partial\left( \sum_{i=1}^p h_i \circ \Psi_i\right)(\bar{x}) \subset \partial (f+g)(\bar{x}).
\end{equation}
constitutes a necessary condition for local optimality of problem~\eqref{eq:P1}. A point verifying~\eqref{eq:statpoint} is called a \emph{stationary point} of~\eqref{eq:P1}. In general, finding stationary points is highly challenging, so it is useful to consider relaxed notions. We say that a point $\bar{x}\in\dom{\varphi}$ is a \emph{critical point} of~\eqref{eq:P1} if there exists $\barbf{y}\in\mathbb{R}nm$ such that~\eqref{eq:critpoint} holds. By~\cite[Theorem~10.6]{MR1491362} and~\cite[Corollary~2.21]{MR3823783}, it easily follows that every stationary point of~\eqref{eq:P1} is a critical point, but the converse is not true in general.
Therefore, if $(\bar{x},\bar{\mathbf{y}})\in\mathbb{R}^n\times\mathbb{R}nm$ is a critical point of~\eqref{eq:P2}, then $\bar{x}$ is a critical point of~\eqref{eq:P1}. Conversely, if $\bar{x}\in\mathbb{R}^n$ is a critical point of~\eqref{eq:P1}, then there exists $ \bar{\mathbf{y}}\in\mathbb{R}^m$ such that $(\bar{x},\bar{\mathbf{y}})$ is a critical point of~\eqref{eq:P2}. The next result establishes further connections between critical points and solutions of the two minimization problems \eqref{eq:P1} and \eqref{eq:P2} presented above.
\begin{proposition}\label{p:solphiL}
Let $(\bar{x},\bar{\mathbf{y}})\in\mathbb{R}^n\times\mathbb{R}nm$. Then, the following claims hold.
\begin{enumerate}[(i)]
\item\label{it:psolphiL-3} If $(\bar{x},\bar{\mathbf{y}})$ is a critical point of~\eqref{eq:P2}, then $\Phi(\bar{x},\bar{\mathbf{y}})=\varphi(\bar{x})$.
\item\label{it:psolphiL-4} If $(\bar{x},\bar{\mathbf{y}})$ is a solution of~\eqref{eq:P2}, then $\bar{x}$ is a solution of~\eqref{eq:P1}.
\item\label{it:psolphiL-5} If $\bar{x}$ is a solution of~\eqref{eq:P1}, then there exists $ \bar{\mathbf{y}}\in\mathbb{R}^m$ such that $(\bar{x},\bar{\mathbf{y}})$ is a solution of~\eqref{eq:P2}.
\end{enumerate}
\end{proposition}
\begin{proof}
\eqref{it:psolphiL-3} Let $(\bar{x},\bar{\mathbf{y}})=(\bar{x},\bar{y}_1,\ldots,\bar{y}_p)$ be a critical point of~\eqref{eq:P2}. In particular, $y_i\in\partial h_i(\Psi_i(\bar{x}))$ for all $i=1,\ldots,p$, so for these points the Fenchel--Young inequality becomes an equality (see, e.g.~\cite[Proposition~16.10]{bauschke2017}), and we obtain the following expressions
\begin{equation}\label{F_Y_Eq}
\begin{aligned}
h_i(\Psi_i(\bar{x})) +h_i^*(\bar{y}_i) = \langle \Psi_i(\bar{x}),\bar{y}_i\rangle, \quad \forall i=1,\ldots,p.
\end{aligned}
\end{equation}
This yields the equality
\begin{equation*}\label{eq:phiL}
\begin{aligned}
\varphi(\bar{x}) & = f(\bar{x}) + g(\bar{x}) -\sum_{i=1}^p h_i(\Psi_i (\bar{x})) \\
& = f(\bar{x}) + g(\bar{x}) + \sum_{i=1}^p \left( h_i^*(\bar{y}_i) - \langle \Psi_i(\bar{x}),\bar{y}_i\rangle\right) = \Phi(\bar{x},\bar{y}_1,\ldots,\bar{y}_p).
\end{aligned}
\end{equation*}
\eqref{it:psolphiL-4} If $(\bar{x},\bar{\textbf{y}})$ is a solution of~\eqref{eq:P2}, as argued above, it must be a critical point.
Then, by~\eqref{it:psolphiL-3}, we have that $
\Phi(\bar{x},\bar{\mathbf{y}})=\varphi(\bar{x})
$. Since the optimal values of problems~\eqref{eq:P1} and~\eqref{eq:P2} coincide (recall \eqref{Eq_Optvalues}), the above expression implies that $\bar{x}$ is a solution to~\eqref{eq:P1}.
\eqref{it:psolphiL-5} Finally, let us suppose that $\bar{x}$ is a solution of~\eqref{eq:P1}. Then, $\bar{x}$ is necessarily a critical point of~\eqref{eq:P1}, and thus there exists $\bar{y}_i \in \partial h_i(\Psi_i(\bar x))$ such that \eqref{F_Y_Eq} holds. Once more, this implies that $\varphi(\bar{x})= \Phi(\bar x, \bar{\mathbf{y}}) $, where $\bar{\mathbf{y}}:=(\bar{y}_1,\ldots,\bar{y}_p )$. Therefore, using again the fact that optimal values of problems~\eqref{eq:P1} and~\eqref{eq:P2} coincide, we get that $(\bar x, \bar{\mathbf{y}}) $ is a solution of problem \eqref{eq:P2}.
\end{proof}
Now we present in Algorithm \ref{alg:1} the pseudo-code of the \emph{Boosted Double-proximal Subgradient Algorithm}.
\begin{algorithm}[hb!]
\caption{Boosted Double-proximal Subgradient Algorithm for problem~\eqref{eq:P1}}\label{alg:1}
\begin{algorithmic}[1]
\mathbb{R}equire{$(x^0,\mathbf{y}^0) = (x^0,y^0_1,\ldots,y^0_p)\in\mathbb{R}^n\times\mathbb{R}nm$, $R\geq 0$, $\rho\in{]0,1[}$ and $\alpha\geq 0$. Set $k:=0$.}
\State{Choose $v^{k}\in\partial f(x^k)$.}
\State{Take some positive $\gamma_k<\min\left\{{\gamma}^g,\left(2\kappa + \sum_{i=1 }^p L_i\left\| y_i^k\right\|\right)^{-1}\right\}$ and compute
\begin{equation}\label{eq:algx}
\begin{aligned}
\hat{x}^{k} & \in \prox_{\gamma_k g} \left(x^k+\gamma_k \sum_{i=1}^p \nabla \Psi_i(x^k) y_i^k-\gamma_k v^k\right) .
\end{aligned}
\end{equation}
}
\State{For each $i=1,\ldots,p$, take $\mu_i^k>0$ and compute
\begin{equation}\label{eq:algy}
\begin{aligned}
\hat{y}_i^{k} & = \prox_{\mu^k_i h_i^*}\left(y_i^k+\mu^k_i \Psi_i(\hat{x}^{k}) \right).
\end{aligned}
\end{equation}
}
\State{Choose any $\overline{\lambda}_k\geq 0$. Set $\lambda_{k}:=\overline{\lambda}_{k}$, $r:=0$ and $(d^{k}, \mathbf{e}^{k}):=(\hat{x}^{k}, \hat{\mathbf{y}}^{k}) -(x^{k},\mathbf{y}^k)$.}
\State{\textbf{if} $(d^{k},\mathbf{e}^{k})=0$ \textbf{then} STOP and return $x^k$.}
\State{\textbf{while} $r<R$ and
\begin{align}\label{eq:whilecond}
\Phi\big((\hat{x}^{k},\hat{\mathbf{y}}^{k}) + \lambda_{k}(d^{k}, \mathbf{e}^{k}) \big) > \Phi(\hat{x}^{k}, \hat{\mathbf{y}}^{k})- \alpha \lambda_k^2 \| (d^{k},\mathbf{e}^k)\|^2
\end{align}
\textbf{do }{$r:=r+1$ and $\lambda_k:=\rho^r\overline{\lambda}_k$.}}
\State{\textbf{if }$r=R$ \textbf{then} $\lambda_k:=0$.}
\State{Set $(x^{k+1},\mathbf{y}^{k+1}) :=(\hat{x}^{k},\hat{\mathbf{y}}^k)+ \lambda_{k}(d^{k}, \mathbf{e}^{k})$, $k:=k+1$ and go to Step 1.}
\end{algorithmic}
\end{algorithm}
\begin{remark}\label{r:alg1}
From the definition of the proximity operator,~\eqref{eq:algx} and~\eqref{eq:algy} imply that the sequences generated by Algorithm~\ref{alg:1} verify
\begin{equation}\label{eq:subdif1}
\begin{aligned}
&\frac{x^k-\hat{x}^{k}}{\gamma_k} + \sum_{i=1}^p \nabla \Psi_i(x^k) y_i^k - v^k \in \partial g(\hat{x}^{k}),
\\
&\frac{y_i^k-\hat{y}_i^{k}}{\mu_i^k} + \Psi_i ( \hat{x}^{k}) \in \partial h_i^*(\hat{y}_i^{k}), \quad \forall i= 1,\ldots, p.
\end{aligned}
\end{equation}
In particular, if $(d^k,\mathbf{e}^k)=0$, the above inclusions become~\eqref{eq:critpoint} for $(\bar{x},\barbf{y}):=(x^k,\mathbf{y}^k)$ and hence $\bar{x}$ is a critical point of problem~\eqref{eq:P1}.
Let us also observe that in Algorithm~\ref{alg:1} it is possible to replace $v^k \in \partial f(x^k)$ by $v^k \in \conv{\partial f(x^k)}$. This is justified by Proposition~\ref{p:deslemma}, which shows that such subgradients satisfy the descent inequality~\eqref{equpperdesc}. This straightforward modification in Step~1 of Algorithm~\ref{alg:1} can be advantageous in numerical applications, particularly when the calculus rules for the basic subdifferential only provide an upper estimate rather than an equality. For example, if the objective function $\varphi(x)$ contains a term of the form $-\upsilon(x)$ and $\upsilon:\mathbb{R}^n\to\overline{\mathbb{R}}$ is convex, it is natural to set $f:=-\upsilon$, which is upper-$\mathcal{C}^2$ by Example~\ref{example01}. Algorithm~\ref{alg:1} requires choosing $v^k \in \partial f(x^k)=\partial\left(-\upsilon(x^k)\right)\subset -\partial\upsilon(x^k)$, while the modification $v^k \in \conv{\partial f(x^k)}=-\partial \upsilon(x^k)$ (see~\eqref{eq:subinclusions}) allows choosing $v^k$ from the possibly larger set $-\partial \upsilon(x^k)$.
On the other hand, according to~\eqref{eq:subdif1}, allowing $v^k \in \conv{\partial f(x^k)}$ in Step~1 entails the weaker notion of criticality
\begin{equation*}
\left\{
\begin{aligned}
&\sum_{i=1}^p \nabla\Psi_i(\bar{x}) \bar{y}_i \in \conv{\partial f(\bar x )}+\partial g(\bar{x}) ,\\
&\bar{y}_i \in \partial h_i (\Psi_i( \bar{x})), \quad \forall i=1,\ldots,p,
\end{aligned}
\right.
\end{equation*}
when $(d^k,\mathbf{e}^k)=0$ and $(\bar{x},\barbf{y}):=(x^k,\mathbf{y}^k)$. Thus, the price to pay for having more freedom in the choice of $v^k$ is the possibility of having a larger set of non-optimal critical points to which the algorithm might converge. Nonetheless, a more subtle analysis can be performed to derive stronger notions of criticality, see Remark~\ref{Remark_sub} and Section~\ref{sect:numerical2} for a numerical example.
\end{remark}
\begin{remark}[Particular cases of Algorithm~\ref{alg:1}]\label{remark:particularcases}
Different known algorithms can be obtained as particular cases of Algorithm~\ref{alg:1}.
\begin{enumerate}[(i)]
\item\label{it:particularcases1} Consider the problem of minimizing $\varphi(x):=g_1(x)-g_2(x)$, with $g_1$ and $g_2$ being convex. If we let $f:=-g_2$, $g:=g_1$ and $\Psi_i:=0=:h_i$ for all $i=1,\ldots,p$, then Step~1 of Algorithm~\ref{alg:1} becomes $v^k\in\partial (-g_2)(x^k)$. Since $\partial (-g_2)(x^k)\subseteq -\partial g_2(x^k)$, when $R=0$ one recovers a specific choice for the \emph{Proximal DC Algorithm} of~\cite{sun2003proximal} in which $v^k$ is chosen from a smaller set of subgradients. If $x^{k+1}=x^k=:\bar{x}$, one gets $-v^k\in \partial g_1(\bar{x})\cap\left(-\partial (-g_2)(\bar{x})\right)$. Observe that the more restrictive condition $\partial g_1(\bar{x})\cap\left(-\partial (-g_2)(x)\right)\neq\emptyset$ can serve to discard some points which are not local minima (e.g., if $g_1(x)=x^2$ and $g_2(x)=|x|$, then $\bar{x}:=0$ is a local maximum which satisfies $0\in\nabla g_1(\bar{x})\cap\partial g_2(\bar{x})$, but $0\not\in\nabla g_1(\bar{x})\cap\left(-\partial (-g_2)(\bar{x})\right)$). When $R=\infty$, one obtains the \emph{Boosted Proximal DC Algorithm} introduced in~\cite{alizadeh2022new}, which is a modification of the \emph{Boosted Difference of Convex functions Algorithm} from~\cite{aragon2018accelerating, boostedDCA} that adds a proximal term.
\item Likewise, if $\varphi(x):=g_1(x)-g_2(x)+g_3(x)$ with $g_1$ being proper and l.s.c. with $\inf_{x\in\mathbb{R}^n}g_1>-\infty$, $g_2$ being convex and $g_3$ being $L$-smooth, letting $f:=-g_2+g_3$ and the rest of the functions as in the previous case with $R=0$, we obtain a specific choice for the \emph{Generalized Proximal Point Algorithm} presented in~\cite{an2017convergence}.
\item The \emph{Double-Proximal Gradient Algorithm} proposed in~\cite{bot2019doubleprox} is also recovered by Algorithm~\ref{alg:1} in the case in which $f$ is convex and $L$-smooth, $g$ is convex, $R=0$, $p=1$ and $\Psi_1$ is a linear operator.
\end{enumerate}
\end{remark}
Steps 6-7 of Algorithm~\ref{alg:1} correspond to an optional linesearch step in the direction $(d^{k}, \mathbf{e}^{k})$ with a fixed number $R$ of attempts. On the one hand, note that the computational burden of these steps can be avoided if the user either chooses $R=0$ or $\overline{\lambda}_k=0$, in which case we refer to the resulting algorithm as \emph{Double-proximal Subgradient Algorithm} (abbr. \emph{DSA}). On the other hand, if $R>0$ and $\overline{\lambda}_k>0$, Step~6 allows to achieve a further decrease of the primal-dual objective function~$\Phi$. Step~7 sets $\lambda_k=0$ when the linesearch was not successful.
For general problems satisfying Assumption~\ref{assump:1} there is no guarantee that the vector $(d^{k},\mathbf{e}^{k})$ defined in Step $4$ of BDSA provides a descent direction for the function $\Phi$. The motivation for the linesearch step comes from the case in which the functions $g$ and $h_i^*$ are differentiable at the points $\hat{x}^k$ and $\hat{y}_i^k$, respectively. In this case, $(d^{k},\mathbf{e}^{k})$ is a descent direction for $\Phi$ at $(\hat{x}^{k},\hat{\mathbf{y}}^{k})$, since the upper Dini directional derivative of $\Phi$ at $(\hat{x}^{k},\hat{\mathbf{y}}^{k})$ in the direction $(d^{k},\mathbf{e}^{k})$ is negative. This fact is proved in the next result.
\begin{proposition}\label{l:descentdirection}
Suppose that Assumption~\ref{assump:1} holds and consider the sequences generated by Algorithm~\ref{alg:1} for problem~\eqref{eq:P1}. Assume also that
\begin{enumerate}[(i)]
\item $g$ is differentiable at $\hat{x}^{k}$,
\item $h_i^*$ is differentiable at $\hat{y_i}^{k}$ for all $i = 1,\ldots,p$,
\item\label{it:descentdirection-4} $\gamma_k \in\left] 0, \left(2\kappa +\frac{p}{2}+\sum_{i=1}^p L_i\|y_i^k\|\right)^{-1}\right[$ and $\mu_i^{k} \in{\left]0,2\,\|\nabla\Psi_i(\hat{x}^k)\|^{-2}\right[}$ for all $i=1,\ldots,p$.
\end{enumerate}
Then, for all $k\geq0$,
\begin{equation}\label{eq:direc_deriv}
\begin{aligned}
d^+\Phi\big(&(\hat{x}^{k},\hat{\mathbf{y}}^{k}); (d^{k},\mathbf{e}^{k})\big) \leq\\
&\left(2\kappa +\frac{p}{2}+\sum_{i=1}^p L_i\|y_i^k\| -\frac{1}{\gamma_k}\right) \|d^{k}\|^2
+ \sum_{i=1}^p\left(\frac{1}{2}\|\nabla\Psi_i(\hat{x}^k)\|^2-\frac{1}{\mu_i^k}\right) \|e_i^{k}\|^2.
\end{aligned}
\end{equation}
Consequently, if $(d^{k},\mathbf{e}^{k})\neq 0$, then for every $\alpha>0$ there is some $\delta_k>0$ such that
\begin{equation}\label{eq:linesearch_guarantee}
\Phi\big((\hat{x}^{k},\hat{\mathbf{y}}^{k}) + \lambda(d^{k}, \mathbf{e}^{k}) \big) \leq\Phi(\hat{x}^{k}, \hat{\mathbf{y}}^{k})- \alpha \lambda^2 \|(d^{k},\mathbf{e}^k)\|^2, \quad\text{for all }\lambda\in{[0,\delta_k]}.
\end{equation}
\end{proposition}
\begin{proof}
Indeed, for any $\hat{v}^{k}\in\partial f(\hat{x}^{k})$, we get
{\small
\begin{equation}\label{eq:direc_deriv0}
\begin{aligned}
d^+\Phi\big((\hat{x}^{k},\hat{\mathbf{y}}^{k}); (d^{k},\mathbf{e}^{k})\big) & \leq \limsup_{t\downarrow 0} \frac{f(\hat{x}^{k}+ td^{k}) -f(\hat{x}^{k}) }{t} + \limsup_{t\downarrow 0} \frac{g(\hat{x}^{k}+ td^{k}) -g(\hat{x}^{k}) }{t} \\
& \quad + \sum_{i=1}^p \limsup_{t\downarrow 0} \frac{h_i^*(\hat{y}_i^{k}+te_{i}^{k}) - h_i^*(\hat{y}_i^{k})}{t} \\
& \quad - \sum_{i=1}^p \liminf_{t\downarrow 0}\frac{ \langle\Psi_i(\hat{x}^{k}+td^{k})-\Psi_i(\hat{x}^{k}),\hat{y}_i^{k}\rangle +t\langle\Psi_i(\hat{x}^{k}+td^{k}),e_i^{k} \rangle}{t} \\
& \leq \langle \hat{v}^{k}, d^{k}\rangle + \langle\nabla g(\hat{x}^{k}),d^{k}\rangle + \sum_{i=1}^p \langle\nabla h_i^*(\hat{y}_i^{k}),e_{i}^{k}\rangle \\
& \quad - \sum_{i=1}^p \left(\langle\nabla\Psi_i(\hat{x}^k) \hat{y}_i^{k},d^{k}\rangle + \langle \Psi_i(\hat{x}^{k}), e_i^{k}\rangle \right), \\
\end{aligned}
\end{equation} }
where the second inequality is due to Proposition~\ref{p:deslemma}.
Now, since $g$ and $h_i^*$ are assumed to be differentiable at $\hat{x}^{k}$ and $\hat{y}_i^{k}$, respectively,~\eqref{eq:subdif1} yields
\begin{equation}\label{eq:diff}
\begin{aligned}
&\frac{x^k-\hat{x}^{k}}{\gamma_k} + \sum_{i=1}^p \nabla \Psi_i(x^k) y_i^k - v^k = \nabla g(\hat{x}^{k}),
\\
&\frac{y_i^k-\hat{y}_i^{k}}{\mu_i^k} + \Psi_i ( \hat{x}^{k}) =\nabla h_i^*(\hat{y}_i^{k}), \quad \forall i= 1,\ldots, p.
\end{aligned}
\end{equation}
On the other hand, again making use of Proposition~\ref{p:deslemma}, we get the following inequality by setting $y:=\hat{x}^{k}$, $x:= x^{k}$ and $\xi:=v^k$ in equation~\eqref{equpperdesc}
\begin{equation*}
f(\hat{x}^{k})-f(x^k) - \langle v^k, \hat{x}^{k} - x^k \rangle \leq \kappa \|\hat{x}^{k}-x^{k}\|^2.
\end{equation*}
Likewise, setting $y:=x^k$, $x:=\hat{x}^{k}$ and $\xi:=\hat{v}^{k}$ in~\eqref{equpperdesc} yields
\begin{equation*}
f(x^k) - f(\hat{x}^{k}) + \langle \hat{v}^{k},\hat{x}^{k}-x^k\rangle \leq \kappa \|\hat{x}^{k}-x^{k}\|^2.
\end{equation*}
Summing together these two equations, we get
\begin{equation}\label{eq:subdiffbound}
\langle \hat{v}^{k}-v^{k}, \hat{x}^{k}-x^k \rangle \leq 2 \kappa \|\hat{x}^{k}-x^k\|^2 \quad \text{ for all } v^k\in\partial f(x^k).
\end{equation}
Substituting~\eqref{eq:diff} and~\eqref{eq:subdiffbound} in~\eqref{eq:direc_deriv0}, it becomes
\begin{equation}\label{eq:direc_deriv1}
\begin{aligned}
d^+\Phi\big((\hat{x}^{k},\hat{\mathbf{y}}^{k}); (d^{k},\mathbf{e}^{k})\big) & \leq \langle \hat{v}^{k}-v^k,d^{k}\rangle -\frac{1}{\gamma_k} \langle \hat{x}^{k}-x^k,d^{k}\rangle + \sum_{i=1}^p\langle \nabla \Psi_i(x^k)y_i^k,d^{k}\rangle \\
&\quad -\sum_{i=1}^p \frac{1}{\mu_i^k} \langle \hat{y}_i^{k}-y_i^k,e_i^{k}\rangle +\sum_{i=1}^p \langle \Psi_i(\hat{x}^{k}),e_i^{k}\rangle \\
&\quad - \sum_{i=1}^p \left(\langle\nabla\Psi_i(\hat{x}^k) \hat{y}_i^{k},d^{k}\rangle + \langle \Psi_i(\hat{x}^{k}), e_i^{k}\rangle \right) \\
& \leq \left(2\kappa-\frac{1}{\gamma_k}\right)\|d^{k}\|^2 -\sum_{i=1}^p \frac{1}{\mu_i^k} \|e_i^{k}\|^2 \\
&\quad- \sum_{i=1}^p \langle \nabla\Psi_i(\hat{x}^k) \hat{y}_i^{k}-\nabla \Psi_i(x^k)y_i^k ,d^{k} \rangle.
\end{aligned}
\end{equation}
Finally, using the Cauchy--Schwartz inequality and Young's inequality, the terms in the last summation can be upper bounded as follows:
\begin{equation*}
\begin{aligned}
-\langle \nabla\Psi_i(\hat{x}^k) \hat{y}_i^{k}-\nabla \Psi_i(x^k)y_i^k ,d^{k} \rangle&=\langle \nabla\Psi_i(\hat{x}^k) e_i^k ,d^{k} \rangle+\langle (\nabla\Psi_i(\hat{x}^k)- \nabla \Psi_i(x^k))y_i^k ,d^{k} \rangle\\
&\leq \|\nabla\Psi_i(\hat{x}^k)\| \|e_i^k\|\|d^{k}\| + L_i\|y_i^k\| \|d^k\|^2\\
&\leq \frac{1}{2}\left(\|\nabla\Psi_i(\hat{x}^k)\|^2\|e_i^k\|^2+\|d^k\|^2\right)+ L_i\|y_i^k\| \|d^k\|^2\\
&=\frac{1}{2}\|\nabla\Psi_i(\hat{x}^k)\|^2\|e_i^k\|^2+\left(\frac{1}{2}+L_i\|y_i^k\|\right)\|d^k\|^2,
\end{aligned}
\end{equation*}
for all $i = 1,\ldots,p$. Putting this into~\eqref{eq:direc_deriv1}, we deduce~\eqref{eq:direc_deriv}.
Thanks to assumption~(\ref{it:descentdirection-4}), we have
$$K:=\min_{i=1,\ldots,p}\left\{\frac{1}{\gamma_k}-2\kappa +\frac{p}{2}-\sum_{i=1}^p L_i\|y_i^k\|, \frac{1}{\mu_i^k}-\frac{1}{2}\|\nabla\Psi_i(\hat{x}^k)\|^2\right\}>0.
$$
Thus, if $(d^{k},\mathbf{e}^k)\neq 0$, one has
$$d^+\Phi\big((\hat{x}^{k},\hat{\mathbf{y}}^{k}); (d^{k},\mathbf{e}^{k})\big) \leq -K\| (d^{k},\mathbf{e}^k)\|^2<-\frac{K}{2}\| (d^{k},\mathbf{e}^k)\|^2,$$
so there exist $\tau_k>0$ such that
$$\Phi\big((\hat{x}^{k},\hat{\mathbf{y}}^{k}) + \lambda(d^{k}, \mathbf{e}^{k}) \big) \leq \Phi(\hat{x}^{k}, \hat{\mathbf{y}}^{k})- \lambda\frac{K}{2} \|(d^{k},\mathbf{e}^k)\|^2, \quad\text{for all }\lambda\in{[0,\tau_k]}.$$
Then, given any $\alpha>0$, letting $\delta_k:=\min\{K/(2\alpha),\tau_k\}>0$, we have that $-K/2\leq-\lambda\alpha$ for all $\lambda\in[0,\delta_k]$, so we obtain~\eqref{eq:linesearch_guarantee}.
\end{proof}
The differentiability of $h_i^*$ is guaranteed when $h_i$ is strictly convex. Actually, for proper l.s.c. convex functions, \emph{essential strict convexity} is equivalent to \emph{essential smoothness} of the conjugate function, cf.~\cite[Theorem~26.3]{Rockafellar1970}.
Before moving to the convergence analysis in the next subsection, let us explain the rationale behind Algorithm~\ref{alg:1} in the simplest case in which $h_i=\Psi_i=0$. Thanks to the $\kappa$-upper-$\mathcal{C}^2$ assumption on $f$, since $v^k\in\partial f(x^k)$ (Step~1), it holds
$$f(z)\leq f(x^k)+\langle v^k,z-x^k\rangle+\kappa\|z-x^k\|^2,\quad\forall z\in\mathbb{R}^n.$$
Thus, if $\gamma_k\in{\left]0,\frac{1}{2\kappa}\right[}$ (as in Step~2), one gets
\begin{align*}
\varphi(z)&=f(z)+g(z)\\
&\leq g(z)+f(x^k)+\langle v^k,z-x^k\rangle+\frac{1}{2\gamma_k}\|z-x^k\|^2=:\widetilde{\varphi}_k(z).
\end{align*}
for all $z\in\mathbb{R}^n$. Therefore, the function $\widetilde{\varphi}_k$ provides an upper bound to $\varphi$, so it makes sense to take
$$\hat{x}^k=\argmin_{z\in\mathbb{R}^n}\widetilde{\varphi}_k(z)= \prox_{\gamma_k g} \left(x^k-\gamma_k v^k\right), $$
which coincides with~\eqref{eq:algx}. Finally, when $g$ is differentiable at $\hat{x}^k$, the linesearch condition~\eqref{eq:whilecond} in Step~6 permits to further reduce the original function $\varphi$.
For illustration, consider the function from~\cite[Example~2.4]{boostedDCA} given by $\varphi(x):=x_1+x_2 -\|x\|_1+\|x\|^2$, for $x\in\mathbb{R}^2$. If we let $f(x):=x_1+x_2 -\|x\|_1$ and $g(x):=\|x\|^2$, then $f$ is $\kappa$-upper-$\mathcal{C}^2$ for $\kappa=0$, by Example~\ref{example01}. If we take $x^0:=(0,1)^T$ and $\gamma_0:=1$, we have $v^0:=(2,0)^T\in \partial f(x^0)=\left\{(2,0)^T,(0,0)^T\right\}$ and $\hat{x}^0=\prox_{\gamma_0 g}(x^0-\gamma_0v^0)=\frac{1}{3}(-2,1)^T$, which minimizes the function $\widetilde{\varphi}_0$. In Figure~\ref{fig:example} we represent the sections of $\varphi$ and $\widetilde{\varphi}$ at $\hat{x}^0$ in the direction $d^0=\hat{x}^0-x^0=-\frac{2}{3}(1,1)^T$. Taking for instance $\alpha=0.1$, we can observe how the linesearch step in the direction $d^0$ can help achieving an additional reduction of the objective function.
\begin{figure}
\caption{Sections of the functions $\varphi$ and $\widetilde{\varphi}
\label{fig:example}
\end{figure}
\subsection{Convergence Analysis}\label{subsec:convergence}
The following result shows that the primal-dual functional $\Phi$ of problem~\eqref{eq:P2} evaluated at the sequence $( x^k,\mathbf{y}^k)_{k\in\mathbb{N}}$ generated by BDSA decreases after every iteration of the algorithm.
\begin{proposition}\label{p:des}
Let $\Phi$ be the function defined in~\eqref{Def_PHI} and suppose that Assumption~\ref{assump:1} holds. Given a starting point $(x^0,\mathbf{y}^0)=(x^{0},y^{0}_1,\ldots,y^{0}_p)\in\mathbb{R}^n\times\mathbb{R}nm$, consider the sequence $(x^k,\mathbf{y}^k)_{k\in\mathbb{N}}=(x^k,y^k_1,\ldots,y^k_p)_{k\in\mathbb{N}}$ generated by Algorithm~\ref{alg:1}. Then, for all $k\geq1$,
\begin{equation}\label{eq:des}
\begin{aligned}
\Phi(x^{k+1},&\mathbf{y}^{k+1}) - \Phi(x^{k},\mathbf{y}^k
)
& \leq -a_k \|x^{k+1}-x^k\|^2 - \sum_{i=1}^p b_i^k \|y_i^{k+1}-y_i^k\|^2,
\end{aligned}
\end{equation}
where
\begin{equation*}\label{eq:akbk}
a_k:=\frac{2\alpha \lambda_k^2+\gamma_k^{-1}- 2\kappa -\sum_{i=1}^p L_i\| y_i^k\|}{2(1+\lambda_k)^2}>0\quad\text{and}\quad b^k_i:=\frac{1+\alpha \lambda_k^2\mu_i^k}{\mu_i^k(1+\lambda_k)^2}>0,
\end{equation*}
for $i=1,\ldots,p$.
\end{proposition}
\begin{proof}
First, note that for $k\geq1$ the vector $(x^k,\mathbf{y}^k)$ belongs to $\dom \Phi$. By the definition of the proximal point operator, equation~\eqref{eq:algx} yields the inequality
{\small
\begin{equation*}
g(\hat{x}^{k}) + \frac{1}{2\gamma_k} \left\|\hat{x}^{k}-x^k-\gamma_k \left(\sum_{i=1}^p \nabla\Psi_i(x^k) y_i^k-v^k\right)\right\|^2 \leq g(x^k) + \frac{\gamma_k}{2}\left\|\sum_{i=1}^p \nabla \Psi_i(x^k) y_i^k -v^k\right\|^2.
\end{equation*}}
Rearranging this expression and remembering that $d^k=\hat{x}^k-x^k$ we get
\begin{equation}\label{eq:subg_1}
g(\hat{x}^{k})-g(x^k) \leq \left\langle d^k,\sum_{i=1}^p\nabla\Psi_i(x^k) y_i^k-v^k\right\rangle-\frac{1}{2\gamma_k}\|d^k\|^2.
\end{equation}
Now, let us notice that since the function $x\mapsto -\langle \Psi_i(x),y_i^k\rangle$ is $\mathcal{C}^{1}$ with $L_i\| y_i^k\|$-Lipschitz gradient, then by \eqref{eq:descentlemmaineq} we have
\begin{align*}
\langle \Psi_i(x^k)-\Psi_i(\hat{x}^{k}),y_i^{k}\rangle \leq- \langle \nabla\Psi_i(x^k) y_i^k, d^k \rangle + \frac{L_i\| y_i^k\|}{2} \| d^{k}\|^2.
\end{align*}
Using this expression and~\eqref{eq:subg_1}, we obtain
\begin{equation}\label{eq:Phixk}
\begin{aligned}
\Phi(\hat{x}^{k},\mathbf{y}^k) - \Phi(x^k,\mathbf{y}^k) = & f(\hat{x}^{k})- f(x^{k}) +
g(\hat{x}^{k})- g(x^k)\\
&+\sum_{i=1 }^p \langle \Psi_i(x^k)-\Psi_i(\hat{x}^{k}), y_i^k \rangle \\
\leq & f(\hat{x}^{k})- f(x^{k}) +
g(\hat{x}^{k})- g(x^k)\\& +\sum_{i=1 }^p \left(- \langle \nabla\Psi_i(x^k) y_i^k, d^k \rangle + \frac{L_i\| y_i^k\|}{2} \| d^{k}\|^2 \right) \\
\leq& f(\hat{x}^{k})- f(x^{k}) -\frac{1}{2\gamma_k}\|d^{k}\|^2 - \langle v^k, d^{k} \rangle + \frac{1}{2}\sum_{i=1 }^p L_i\| y_i^k\| \| d^{k}\|^2 \\
\leq & \left( \kappa + \frac{1}{2}\sum_{i=1 }^p L_i\| y_i^k\| - \frac{1}{2\gamma_k}\right) \|d^k\|^2,
\end{aligned}
\end{equation}
where the last inequality is due to Proposition~\ref{p:deslemma}. On the other hand, in equation~\eqref{eq:algy} we are computing the proximity operator of the convex function $h_i^*$, which yields the subgradient inequality
\begin{equation}\label{eq:subh_2}
h_i^*(\hat{y}^{k}) + \left\langle \frac{y_i^k-\hat{y}_i^{k}}{\mu_i^k} + \Psi_i(\hat{x}^{k}),y^k_i-\hat{y}_i^{k} \right\rangle \leq h_i^*(y_i^k), \quad \forall i=1,\ldots, p.
\end{equation}
Making use of this expression and $\mathbf{e}^k=\hat{\mathbf{y}}^k-\mathbf{y}^k$, we obtain
\begin{equation}\label{eq:Phiyk}
\begin{aligned}
\Phi(\hat{x}^{k},\hat{\mathbf{y}}^{k})-\Phi(\hat{x}^{k},\mathbf{y}^k) & =\sum_{i=1}^p \left( h_i^\ast(\hat{y}_i^{k}) - h_i^*(y^{k})-\langle \Psi_i(\hat{x}^{k}),e_i^{k} \rangle \right)
\leq-\sum_{i=1}^p \frac{1}{\mu_i^k} \|e_i^k\|^2.
\end{aligned}
\end{equation}
Then, \eqref{eq:Phiyk} and \eqref{eq:Phixk} give
\begin{equation}\label{eq:hats}
\Phi(\hat{x}^{k},\hat{\mathbf{y}}^{k})\leq \Phi(x^k,\mathbf{y}^k)-\left(\frac{1}{2\gamma_k}- \kappa -\frac{1}{2}\sum_{i=1 }^p L_i\| y_i^k\|\right)\|d^k\|^2 - \sum_{i=1}^p \frac{1}{\mu_i^k} \|e_i^k\|^2.
\end{equation}
Finally, using the linesearch~\eqref{eq:whilecond} and~\eqref{eq:hats}, we get
\begin{align*}
\Phi(x^{k+1},\textbf{y}^{k+1})&\leq \Phi(\hat{x}^{k}, \hat{\mathbf{y}}^{k})-\alpha \lambda_k^2 \| (d^{k},\mathbf{e}^k)\|^2\\
&\leq \Phi({x}^{k}, {\mathbf{y}}^{k})- \left(\alpha \lambda_k^2+\frac{1}{2\gamma_k}- \kappa -\frac{1}{2}\sum_{i=1 }^p L_i\| y_i^k\|\right) \| d^{k}\|^2 \\
&\quad- \sum_{i=1}^p \left(\alpha \lambda_k^2+\frac{1}{\mu_i^k}\right) \| e_i^{k}\|^2\\
&=\Phi({x}^{k}, {\mathbf{y}}^{k})- a_k \| x^{k+1}-x^k\|^2 - \sum_{i=1}^p b_i^k \| y_i^{k+1}-y_i^k\|^2,
\end{align*}
where we note that the first inequality trivially holds when the linesearch procedure was not successful, as in that case $\lambda_k=0$ by Step~7. Therefore,~\eqref{eq:des} holds.
\end{proof}
Next, we present the convergence result of Algorithm~\ref{alg:1}.
\begin{theorem}\label{th:conv}
Suppose that Assumption~\ref{assump:1} holds and consider the functions $\varphi$ and $\Phi$ defined in~\eqref{Def_varphi} and~\eqref{Def_PHI}, respectively. Given $(x^0,\mathbf{y}^0)\in\mathbb{R}^n\times\mathbb{R}nm$ and $\eta\in{]0,1[}$, consider the pair of sequences $(x^k,\mathbf{y}^k)_{k\in\mathbb{N}}$ generated by Algorithm~\ref{alg:1} with $\gamma_k\in{\left]0,\eta \min{\left\{\gamma^g, \left(2\kappa + \sum_{i=1 }^p L_i\left\| y_i^k\right\|\right)^{-1}\right\}}\right]}$ for all $k\in\mathbb{N}$ and $\sup_{ k \in \mathbb{N}, i=1,\ldots, p} \mu_i^k<+\infty$. Then, either Algorithm~\ref{alg:1} stops at a critical point of~\eqref{eq:P2} after a finite number of iterations or it generates an infinite sequence $(x^k,\mathbf{y}^k)_{k\in\mathbb{N}}$ such that the following assertions hold:
\begin{enumerate}[(i)]
\item\label{it:th-1} The sequence $\big(\Phi(x^k,\mathbf{y}^k)\big)_{k\in\mathbb{N}}$ monotonically (strictly) decreases and converges. Moreover, the sequences $(x^k)_{k\in\mathbb{N}}$ and $(\mathbf{y}^k)_{k\in\mathbb{N}}$ verify that
\begin{equation}\label{eq:Ostrowski}
\sum_{k=0}^{\infty} \|x^{k+1}-x^k\|^2 < \infty \text{ and } \sum_{k=0}^{\infty} \|\mathbf{y}^{k+1}-\mathbf{y}^k\|^2 < \infty.
\end{equation}
\item\label{it:th-2} If the sequence $(x^k,\mathbf{y}^k)_{k\in\mathbb{N}}$ is bounded, the set of its accumulation points is nonempty, closed and connected.
\item\label{it:th-3} If $(\bar{x},\barbf{y})\in\mathbb{R}^n\times\mathbb{R}nm$ is an accumulation point of the sequence $(x^k,\mathbf{y}^k)_{k\in\mathbb{N}}$, then there exists $\bar{v}\in\partial f(\bar{x})$ such that~\eqref{eq:critpoint} holds, i.e., $\bar{x}$ is a critical point of~\eqref{eq:P1}. In addition, $\varphi(\bar{x}) = \inf_{k\in\mathbb{N}}\Phi(x^k,y^k)$.
\item\label{it:th-4} If $(x^k,\mathbf{y}^k)_{k\in\mathbb{N}}$ has at least one isolated accumulation point, then the whole sequence $(x^k,\mathbf{y}^k)_{k\in\mathbb{N}}$ converges to a critical point of~\eqref{eq:P2}. Consequently, $(x^k)_{k\in\mathbb{N}}$ converges to a critical point of problem~\eqref{eq:P1}.
\end{enumerate}
\end{theorem}
\begin{proof}
If Algorithm~\ref{alg:1} stops at some iteration $k+1$ with $x^*=x^{k+1}=x^k$ and $\mathbf{y}^{k+1}=\mathbf{y}^k$, then $x^*$ is a critical point of~\eqref{eq:P1}, as shown in Remark~\ref{r:alg1}. Otherwise, Algorithm~\ref{alg:1} generates an infinite sequence $(x^k,\mathbf{y}^k)_{k\in\mathbb{N}}$.
(\ref{it:th-1}) Again, observe that $(x^k,\mathbf{y}^k) \in \dom \Phi$ for all $k\geq 1$. By Proposition~\ref{p:des}, summing~\eqref{eq:des} for all $k\geq 1$, we get
\begin{equation}\label{eq:3}
\begin{aligned}
\Phi(x^1,\mathbf{y}^1) - \inf_{ k\in \mathbb{N} } \Phi(x^k,\mathbf{y}^k)&\geq\sum_{k=1}^{\infty} a_k\|x^{k+1}-x^k\|^2 + \sum_{k=1}^{\infty} \sum_{i=1}^p b_i^k\|\mathbf{y}^{k+1}-\mathbf{y}^k\|^2\\
&\geq C\left(\sum_{k=1}^{\infty} \|x^{k+1}-x^k\|^2 + \sum_{k=1}^{\infty} \|\mathbf{y}^{k+1}-\mathbf{y}^k\|^2 \right),
\end{aligned}
\end{equation}
where $C:= \inf_{k\in\mathbb{N}, i=1,\ldots, p} \left\{ a_k,b_i^k \right\}$. Let us see that $C>0$. Indeed, minimizing the value of $a_k$ with respect to $\lambda_k$, we deduce
$$a_k=\frac{2\alpha \lambda_k^2+\gamma_k^{-1}- 2\kappa -\sum_{i=1 }^p L_i\| y_i^k\|}{2(1+\lambda_k)^2}\geq\frac{(\gamma_k^{-1}- 2\kappa-\sum_{i=1 }^p L_i\| y_i^k\|)\alpha}{\gamma_k^{-1}- 2\kappa-\sum_{i=1 }^p L_i\| y_i^k\|+2\alpha},$$
whose right-hand side, as a function of $\gamma_k$, is strictly decreasing in $\left]0,\eta / \left(2\kappa + \sum_{i=1 }^p L_i\left\| y_i^k\right\|\right)\right]$. Hence,
$$a_k\geq \frac{(1-\eta)\alpha}{1-\eta+2\alpha\eta/(2\kappa+\sum_{i=1 }^p L_i\| y_i^k\|)}\geq\frac{(1-\eta)\alpha}{1-\eta+\alpha\eta/\kappa}>0, \quad \forall k\in\mathbb{N}.$$
Likewise,
\begin{align*}
b^k_i=\frac{1+\alpha \lambda_k^2\mu_i^k}{\mu_i^k(1+\lambda_k)^2}\geq\frac{\alpha}{1+\alpha\mu_i^k}\geq\frac{\alpha}{1+\alpha\sup_{k\in\mathbb{N},i=1,\ldots,p}{\mu_i^k}}>0,\quad \forall k\in\mathbb{N}.
\end{align*}
Therefore, $C>0$ and we obtain from~\eqref{eq:3} that
\begin{equation*}
\begin{aligned}
\sum_{k=1}^{\infty} \|x^{k+1}-x^k\|^2 + \sum_{k=1}^{\infty} \|\mathbf{y}^{k+1}-\mathbf{y}^k\|^2 \leq C^{-1} \left( \Phi(x^1,\mathbf{y}^1) - \inf_{ k\in \mathbb{N} } \Phi(x^k,\mathbf{y}^k) \right).
\end{aligned}
\end{equation*}
By assumption, the right-hand side of the equation is bounded from above, so the sums in the left-hand side are finite, which proves~\eqref{eq:Ostrowski}.
(\ref{it:th-2})~Equation~\eqref{eq:Ostrowski} implies that the sequences $(x^k)_{k\in\mathbb{N}}$ and $(\mathbf{y}^k)_{k\in\mathbb{N}}$ verify the so-called \emph{Ostrowski condition}, that is,
\begin{equation}\label{eq:Ostrowski2}
\lim_{k\to\infty} \|x^{k+1}-x^k\| = 0 \text{ and } \lim_{k\to\infty} \|\mathbf{y}^{k+1}-\mathbf{y}^k\| =0.
\end{equation}
Now, the result directly follows from~\cite[Theorem~8.3.9]{MR1955649}.
(\ref{it:th-3}) Let $(x^{k_j},\mathbf{y}^{k_j})_{j\in\mathbb{N}}$ be a subsequence of $(x^k,\mathbf{y}^k)_{k\in\mathbb{N}}$ such that $(x^{k_j},\mathbf{y}^{k_j})\to(\bar{x},\barbf{y})$. By~\eqref{eq:Ostrowski}, since
$$(x^{k_j+1}-x^{k_j},\mathbf{y}^{k_j+1}-\mathbf{y}^{k_j})=(1+\lambda_{k_j})(\hat{x}^{k_j}-x^{k_j},\hat{\mathbf{y}}^{k_j}-\mathbf{y}^{k_j}),$$
the sequence $(\hat{x}^{k_j},\hat{\mathbf{y}}^{k_j})_{k\in\mathbb{N}}$ also converges to $(\bar{x},\barbf{y})$. Now, we can assume without loss of generality that $\gamma_{k_j} \to \bar{\gamma} \in {]0,\infty[}$. Consider the subsequence $(v^{k_j})_{j\in\mathbb{N}}$ of $(v^k)_{k\in\mathbb{N}}$. In particular, $v^{k_j}\in\partial f(x^{k_j})$ for all $j\in\mathbb{N}$. Since $f$ is locally Lipschitz continuous, $v^{k_j}$ is bounded for all sufficiently large $j\in\mathbb{N}$ (see, e.g., \cite[Proposition~9.13]{MR1491362}). Therefore, we can also assume without loss of generality that $(v^{k_j})_{j\in\mathbb{N}}$ converges to some point $\bar{v}\in\partial f(\bar{x})$. Thus, the sequence
$$\left(x^{k_j}+\gamma_{k_j}\sum_{i=1}^{m}\nabla\Psi_i(x^{k_j})y_i^{k_j}-\gamma_{k_j} v^{k_j},\hat{x}^{k_j}\right)_{j\in\mathbb{N}} \subseteq \gra \prox_{\gamma_{k_j} g}$$ converges to $(\bar{x}+\bar{\gamma}\sum_{i=1}^p\nabla\Psi_i(\bar{x})\bar{y}_i-\bar{\gamma} \bar{v},\bar{x})$ as $j\to\infty$. Hence, by \cite[Theorem 1.25]{MR1491362}), we get that
\begin{equation*}
\bar{x}\in \prox_{\bar \gamma g} \left(\bar{x}+\bar \gamma\sum_{i=1}^p\nabla\Psi_i(\bar{x})\bar{y}_i-\bar{\gamma} \bar{v}\right),
\end{equation*}
which implies that
\begin{equation}\label{eq:thst-1}
\sum_{i=1}^p \nabla\Psi_i(\bar{x}) \bar{y}_i \in\bar{v} + \partial g(\bar{x}) \subseteq \partial f(\bar{x})+\partial g(\bar{x}).
\end{equation}
Further, using~\eqref{eq:subdif1}, we get
\begin{equation*}\label{eq:subdiff_j}
\frac{y^{k_j}_i-\hat{y}^{k_j}_i}{\mu_i^{k_j}} + \Psi_i(\hat{x}^{k_j})\in \partial h_i^*(\hat{y}^{k_j}_i), \quad \forall i=1,\ldots,p.
\end{equation*}
Again, we can assume without loss of generality that $\mu_i^{k_j} \to \bar{\mu}_i \in {]0,\infty[}$ as $j\to\infty$ for all $ i=1,\ldots,p$. Taking limits as $j\to\infty$, the closedness of the subdifferential of convex functions results in
\begin{equation}\label{eq:tiii-cp}
\begin{aligned}
\Psi_i (\bar{x} )\in \partial h_i^*(\bar{y}_i),& \quad \forall i=1,\ldots,p.
\end{aligned}
\end{equation}
Therefore,~\eqref{eq:thst-1} and~\eqref{eq:tiii-cp} imply that $\bar{x}$ is a critical point of~\eqref{eq:P1}.
Let us now prove that $\varphi(\bar{x}) = \inf_{k\in\mathbb{N}}\Phi(x^k,y^k)$.
On the one hand, due to~\eqref{eq:subdif1}, for all $i=1,\ldots,p$, we have
\begin{equation}\label{eq:2}
h_i^*(\hat{y}_i^{k_j}) + \left\langle \frac{y_i^{k_j}-\hat{y}_i^{k_j}}{\mu^{k_j}_i}+ \Psi_i ( \hat{x}^{k_j}), \bar{y}_i - \hat{y}_i^{k_j}\right\rangle \leq h_i^*(\bar{y}_i),
\end{equation}
from the definition of the convex subdifferential.
On the other hand, for all $j\in\mathbb{N}$, we have
{
\begin{equation}\label{eq:gnorm}
\begin{aligned}
g(\hat{x}^{k_j}) & +\frac{1}{2\gamma_{k_j}} \Bigg\| \hat{x}^{k_j}-\left(x^{k_j}+\gamma_{k_j} \sum_{i=1}^p \nabla\Psi_i(x^{k_j}) y_i^{k_j}- \gamma_{k_j} v^{k_j}\right)\Bigg\|^2 \\
& \leq g(\bar{x}) + \frac{1}{2\gamma_{k_j}} \left\| \bar{x}-\left(x^{k_j}+\gamma_{k_j} \sum_{i=1}^p \nabla\Psi_i(x^{k_j}) y_i^{k_j}-\gamma_{k_j}v^{k_j}\right)\right\|^2,
\end{aligned}
\end{equation}}
Thus, we deduce from~\eqref{eq:2} and~\eqref{eq:gnorm} that $\limsup_{j\to\infty} \Phi(\hat{x}^{k_j},\hat{\mathbf{y}}^{k_j}) \leq \Phi(\bar{x},\barbf{y})$.
By~\eqref{eq:Ostrowski2}, we have that $(x^{k_j+1},\mathbf{y}^{k_j+1})_{j\in\mathbb{N}}$ also converges to $(\bar{x},\barbf{y})$, so by lower semicontinuity of the functions defining $\Phi$ in~\eqref{Def_PHI}, we have
\begin{align*}
\liminf_{j\to\infty} \Phi(x^{k_j+1},\textbf{y}^{k_j+1})&\geq \Phi(\bar{x},\bar{\mathbf{y}})\\
&\geq\limsup_{j\to\infty}\Phi(\hat{x}^{k_j},\hat{\mathbf{y}}^{k_j})\\
&\geq\limsup_{j\to\infty} \Phi(x^{k_j+1},\mathbf{y}^{k_j+1}),
\end{align*}
where the last inequality is a consequence of the linesearch~\eqref{eq:whilecond}. Therefore, using Proposition~\ref{p:solphiL}\eqref{it:psolphiL-3} and item~\eqref{it:th-1}, we obtain
$$\varphi(\bar{x})=\Phi(\bar{x},\barbf{y})=\lim_{j\to\infty}\Phi(x^{k_j+1},\mathbf{y}^{k_j+1})=\inf_{k\in\mathbb{N}}\Phi(x^{k},\mathbf{y}^k),$$
which proves the claim.
(\ref{it:th-4}) In this case, by~\cite[Proposition~8.3.10]{MR1955649} the sequence $(x^k,\mathbf{y}^k)_{k\in\mathbb{N}}$ converges to some point $(\bar{x},\barbf{y})$, which is a critical point of problem~\eqref{eq:P2} by~(\ref{it:th-3}).
\end{proof}
\begin{remark}[Possible modification of Algorithm \ref{alg:1}]
It is worth mentioning that it would be possible to replace $\hat{x}^{k}$ in~\eqref{eq:algy} by some interpolation between the points $x^k$ and $\hat{x}^{k}$ of the form $(1-\beta^k_i) \hat{x}^{k} + \beta^k_i x^k$, where $\beta^k_i \in \mathbb{R}$ is arbitrarily chosen at each iteration. In principle, this could allow to improve the overall performance of the algorithm. For example, setting $\beta_i = 1$ for all $i=1,\ldots,p$ would permit to fully run the algorithm in parallel, since only $x^{k}$ and $y_i^k$ would be required to compute $\hat{y}_i^{k}$. This would allow to simultaneously compute $\hat{x}^{k}$ and $\hat{y}_i^{k}$, which could improve the algorithm's overall efficiency when the computation of the proximal mappings is time-consuming. However, it is important to note that the bounds obtained in the accordingly modified version of Proposition~\ref{p:des} would be considerably more technical and difficult to be satisfied in practical applications when $\beta_i\neq 0$. As, in addition, in our experiments we did not find numerical evidence of the benefits to justify the inclusion of such extra linear terms, for conciseness we only consider the case where $\beta_i^k=0$.
\end{remark}
\begin{remark}\label{Remark_sub}
As mentioned in Remark~\ref{r:alg1}, one can allow choosing $v^k \in \conv{\partial f(x^k)}$ in Algorithm~\ref{alg:1}. The only modification in Theorem~\ref{th:conv} is that if $(x^{k_j}, \mathbf{y}^{k_j})_{k\in\mathbb{N}}$ converges to
$(\bar{x},\bar{\mathbf{y}})$, then there exists $\bar{y}_i \in \partial h_i (\Psi_i( \bar{x}))$ for all $ i=1,\ldots,p$, such that
\begin{equation}\label{general_critical_point}
\sum_{i=1}^p \nabla\Psi_i(\bar{x}) \bar{y}_i \in \limsup_{j\to +\infty} \left\{v^{k_j}\right\} +\partial g(\bar{x}),
\end{equation}
where here $\limsup$ refers to the Painlev\'e--Kuratowski upper-limit of the sequence $(v^{k_j})_{k\in \mathbb{N} }$. Furthermore, it is easy to prove that $\limsup_{j\to +\infty} \left\{v^{k_j}\right\} \subseteq \conv \partial f(\bar x)$.
\end{remark}
\subsection{Convergence under the Kurdyka--{\L}ojasiewicz property}\label{subsec:KL}
In this subsection, we establish the global convergence of Algorithm~\ref{alg:1} and some convergence rates. In addition to the assumptions required by Theorem~\ref{th:conv}, we assume that the primal-dual function $\Phi$ satisfies the Kurdyka--{\L}ojasiewicz property at some accumulation point of the sequence generated by Algorithm~\ref{alg:1}. Recall that the Kurdyka--{\L}ojasiewicz property holds for $\phi:\mathbb{R}^n\to\mathbb{R}$ at $\bar{x}\in\mathbb{R}^n$ if there exists $\beta>0$ and a continuous concave function $ \theta:[0,\beta] \to [0,+\infty[$ such that $\theta(0)=0$, $\theta$ is $\mathcal{C}^1$-smooth on $]0,\beta[$ with a strictly positive derivative $\theta'$ and
\begin{align}\label{Kur-Loj}
\theta'\big(\phi(x) - \phi(\bar{x})\big)\,d(0,\partial \phi(x))\geq 1
\end{align}
for all $x\in \mathbb{B}_\beta(\bar{x})$ with $\phi(\bar{x}) < \phi(x) <\phi( \bar{x} ) + \beta$, where $d(\cdot,\Omega)$ stands for the distance function to a set $\Omega$.
\begin{lemma}\label{Lemma_desgrad}
Consider a point $(\bar{x},\bar{\mathbf{y}}) \in \mathbb{R}^n\times \mathbb{R}nm $. In addition to the assumptions of Theorem {\rm\ref{th:conv}}, suppose that $f$ is $\mathcal{C}^{1,+}$ around $\bar{x}$. Then, there exists $r>0$, $\rho>0$ and $\hat{k}\in\mathbb{N}$ such that, for all $(x^k, \mathbf{y}^k) \in \mathbb{B}_r(\bar{x},{\bar{\mathbf y}})$ with $k \geq \hat{k}$, there exists $(u^k,\mathbf{w}^k) \in \partial \Phi(\hat{x}^k,\hat{\mathbf{y}}^k)$ verifying
\begin{align}\label{eq001_Lemma_desgrad}
\| (u^k,\mathbf{w}^k)\| \leq \rho \| (x^{k+1},\mathbf{y}^{k+1}) - (x^{k},\mathbf{y}^{k})\|.
\end{align}
\end{lemma}
\begin{proof}
Let $L_1>0$ and $r>0$ be such that $f$ is continuously differentiable with $L_1$-Lipschitz gradient on $\mathbb{B}_{2r}(\bar{x},\bar{\mathbf{y}})$. Let $\hat{k}\in \mathbb{N}$ be such that $\| (x^{k+1},\mathbf{y}^{k+1}) - (x^{k},\mathbf{y}^{k})\| \leq r$ for all $k\geq \hat{k}$. Now, consider $(x^k, \mathbf{y}^k) \in \mathbb{B}_r(\bar{x},{\bar{\mathbf y}})$ with $k \geq \hat{k}$. It follows that $(\hat{x}^k,\hat{\mathbf{y}}^k)$ belongs to $\mathbb{B}_{2r}(\bar{x},{\bar{\mathbf y}})$. Using \eqref{eq:subdif1}, we get that
\begin{equation*}
\begin{aligned}
&\frac{x^{k}-\hat{x}^{k}}{\gamma_{k}} + \sum_{i=1}^p \nabla \Psi_i(x^{k}) y_i^{k} - \nabla f(x^{k}) \in \partial g(\hat{x}^{k}),
\\
&\frac{y_i^{k}-\hat{y}_i^{k}}{\mu_i^{k}} + \Psi_i ( \hat{x}^{k}) \in \partial h_i^*(\hat{y}_i^{k}), \quad \forall i=1, \ldots,p.
\end{aligned}
\end{equation*}
Let us define $(u^k,\mathbf{w}^k)=(u^k, w_1^k, \ldots,w_p^k)$ by
\begin{align*}
u^k &:= \frac{x^{k}-\hat{x}^{k}}{\gamma_{k}} +\sum_{i=1}^p \left( \nabla \Psi_i(x^{k}) y_i^{k} - \nabla \Psi_i(\hat{x}^{k}) \hat{y}_i^{k} \right)+ \nabla f(\hat{x}^{k})- \nabla f(x^{k}),\\
w_i^k&:= \frac{y_i^{k}-\hat{y}_i^{k}}{\mu_i^{k}}, \quad \forall i=1,\ldots, p.
\end{align*}
It follows from~\eqref{eq:subdiff_Phi} that $(u^k,\mathbf{w}^k) \in \partial \Phi(\hat{x}^k,\hat{\mathbf{y}}^k)$. Furthermore, the function $(x,\mathbf{y}) \mapsto \sum_{i=1}^p \nabla\Psi_i(x)y_i$ is Lipschitz continuous on $\mathbb{B}_{2r}(\bar{x})$, let us say $L_2$-Lipschitz continuous, so we can make the following estimations
\begin{align*}
\| u^k\| &\leq \left(\frac{1}{\gamma_{k}}+L_1\right)\|x^{k}- \hat{x}^{k}\| + L_2 \|(x^{k}- \hat{x}^{k}, \mathbf{y}^k - \hat{\mathbf{y}}^{k})\|, \\
\|w_i^k\| &\leq \frac{1}{\mu_i^{k}}\|y_i^{k}-\hat{y}_i^{k}\|, \quad \forall i=1,\ldots, p.
\end{align*}
Finally, let us notice that by Step~8 of Algorithm~\ref{alg:1} we have that
\begin{equation}\label{eq:ineq_hat}
\begin{aligned}
\|(x^{k+1},\mathbf{y}^{k+1}) - (x^{k},\mathbf{y}^{k})\|&=(1+\lambda_k)\|( \hat{x}^{k},\hat{\mathbf{y}}^{k})- (x^{k},\mathbf{y}^{k})\|\\
&\geq \|( \hat{x}^{k},\hat{\mathbf{y}}^{k})- (x^{k},- \mathbf{y}^{k})\| ,
\end{aligned}
\end{equation}
which yields~\eqref{eq001_Lemma_desgrad} taking $\rho>0$ sufficiently large.
\end{proof}
\begin{theorem}\label{t:KL}
In addition the assumptions of Theorem~{\rm\ref{th:conv}}, suppose that the sequence $(x^k, \mathbf{y}^k)_{k\in\mathbb{N}}$ generated by Algorithm~{\rm\ref{alg:1}} has an accumulation point $(\bar{x},\bar{\mathbf{y}})\in\mathbb{R}^n\times\mathbb{R}nm$ at which the Kurdyka--{\L}ojasiewicz property \eqref{Kur-Loj} holds, assume that $f$ is $\mathcal{C}^{1,+}$ around $\bar{x}$, and $\sup_{k\in\mathbb{N}}\lambda_k<+\infty$. Then $(x^k, \mathbf{y}^k )_{k\in\mathbb{N}}$ converges to $(\bar{x},\bar{\mathbf{y}})$ as $k\to\infty$.
\end{theorem}
\begin{proof}
If Algorithm~{\rm\ref{alg:1}} stops after a finite number of iterations, then the results clearly holds. Otherwise, Algorithm~{\rm\ref{alg:1}} produces an infinite sequence $(x^k, \mathbf{y}^k)_{k\in\mathbb{N}}$. Let $r$, $\rho$ and $\hat{k}$ be the constants given by Lemma~\ref{Lemma_desgrad}, let $\beta$ and $\theta$ be the constant and function in the definition of the Kurdyka--{\L}ojasiewicz property, and let $c_0:=\inf_{ k\in \mathbb{N},i =1,\ldots, p}\left\{\frac{(1-\eta)\kappa}{\eta}, \frac{1}{\mu_i^k}\right\} $, $\lambda_\infty:=\sup_{k\in\mathbb{N}}\lambda_k$ and $\sigma:= \rho(1+\lambda_\infty)^2/c_0$. Consider an arbitrary $\epsilon \in{] 0, \min\{ r, \beta/2\}]}$ and pick $k_0 \geq \hat{k}$ large enough such that the following conditions hold:
\begin{itemize}
\item $\|(x^{k_0}, \mathbf{y}^{k_0}) -(\bar{x},\bar{\mathbf{y}})\| \leq\epsilon/4$,
\item $\| (x^{k+1}, \mathbf{y}^{k+1}) - (x^{k}, \mathbf{y}^{k})\|\leq \epsilon/4$, for all $k \geq k_0$,
\item $\sigma \theta( \Phi(x^{k_0}, \mathbf{y}^{k_0}) - \Phi(\bar{x},\bar{\mathbf{y}})) \leq\epsilon/4$,
\item $\Phi(\bar{x},\bar{\mathbf{y}}) < \Phi(\hat{x}^k, \hat{\mathbf{y}}^k) <\Phi(\bar{x},\bar{\mathbf{y}}) + \beta$, for all $k \geq k_0$,
\end{itemize}
where in the last assertion we have used the fact that $(\Phi(\hat{x}^k,\hat{\mathbf{y}}^k))_{k\in\mathbb{N}}$ also converges to $\Phi(\bar{x},\bar{\mathbf{y}})$, since
$$\Phi(x^{k+1},\mathbf{y}^{k+1})\leq\Phi(\hat{x}^k,\hat{\mathbf{y}}^k)\leq\Phi(x^k,\mathbf{y}^k),$$
by the linesearch~\eqref{eq:whilecond} and~\eqref{eq:hats}.
The rest of the proof is split into three claims.
\noindent\textbf{Claim~1:} \emph{Let $k \geq k_0$ be such that $(x^{k}, \mathbf{y}^{k}) \in \mathbb{B}_\varepsilon(\bar{x},\bar{\mathbf{y}})$. Then, the following estimation holds
\begin{align}\label{Inq_Claim1}
\Delta_{k+1} \leq \left(\sigma \Delta s_k \Delta_{k}\right)^{\frac{1}{2}},
\end{align}
where $\Delta_k:= \| (x^{k+1}, \mathbf{y}^{k+1}) - (x^{k}, \mathbf{y}^{k})\|$, $\Delta s_k:= s_k -s_{k+1}$ and
$s_k:= \theta( \Phi(\hat{x}^{k}, \hat{\mathbf{y}}^{k}) - \Phi(\bar{x},\bar{\mathbf{y}}))$. }
Indeed, let $(u^{k},\mathbf{w}^{k}) \in \partial \Phi(\hat{x}^{k}, \hat{\mathbf{y}}^{k})$ be the vector given by Lemma \ref{Lemma_desgrad} (recall that $(x^{k}, \mathbf{y}^{k}) \in \mathbb{B}_r(\bar{x},\bar{\mathbf{y}})$). Observe that $(\hat{x}^k,\hat{\mathbf{y}}^k)\in\mathbb{B}_\beta(\bar{x},\bar{\mathbf{y}})$, since by~\eqref{eq:ineq_hat}, it holds
\begin{align*}
\|(\hat{x}^k,\hat{\mathbf{y}}^k)-(\bar{x},\bar{\mathbf{y}})\|&\leq \|(\hat{x}^k,\hat{\mathbf{y}}^k)-(x^k,\mathbf{y}^k)\|+\|(x^k,\mathbf{y}^k)-(\bar{x},\bar{\mathbf{y}})\|\\
&\leq\|(x^{k+1},\mathbf{y}^{k+1}) - (x^{k},\mathbf{y}^{k})\|+\epsilon\leq 2\epsilon\leq \beta
\end{align*}
Then, by the concavity of $\theta$, the Kurdyka--{\L}ojasiewicz property \eqref{Kur-Loj} applied to $(\hat{x}^k,\hat{\mathbf{y}}^k)$, inequality \eqref{eq:hats}, and the linesearch~\eqref{eq:whilecond}, we have that
\begin{align*}
\Delta s_k \| (u^k,\mathbf{w}^k)\| &\geq \theta'( \Phi(\hat{x}^{k}, \hat{\mathbf{y}}^{k}) - \Phi(\bar{x},\bar{\mathbf{y}})) ) \left(\Phi(\hat{x}^{k}, \hat{\mathbf{y}}^{k}) - \Phi(\hat{x}^{k+1}, \hat{\mathbf{y}}^{k+1}) \right)\| (u^k,\mathbf{w}^k)\| \\
&\geq \Phi(\hat{x}^{k}, \hat{\mathbf{y}}^{k}) - \Phi(\hat{x}^{k+1}, \hat{\mathbf{y}}^{k+1})\\
&\geq \Phi(\hat{x}^{k}, \hat{\mathbf{y}}^{k}) - \Phi(x^{k+1},\mathbf{y}^{k+1})\\
&\quad+\frac{1}{2}\left(\frac{1}{\gamma_{k+1}}- 2\kappa -\sum_{i=1 }^p L_i\| y_i^{k+1}\|\right)\|d^{k+1}\|^2 + \sum_{i=1}^p \frac{1}{\mu_i^{k+1}} \|e_i^{k+1}\|^2\\
&\geq\frac{(1-\eta)\kappa}{\eta}\|d^{k+1}\|^2 + \sum_{i=1}^p \frac{1}{\mu_i^{k+1}} \|e_i^{k+1}\|^2\\
&\geq c_0\|(\hat{x}^{k+1},\hat{\mathbf{y}}^{k+1})-(x^{k+1},\mathbf{y}^{k+1})\|^2\\
&= \frac{c_0}{(1+\lambda_{k+1})^2}\Delta_{k+1}^2\geq \frac{c_0}{(1+\lambda_\infty)^2}\Delta_{k+1}^2.
\end{align*}
Hence,
\begin{align*}
\Delta_{k+1}&\leq (1+\lambda_\infty)\sqrt{ \frac{ \Delta s_k \| (u^k,\mathbf{w}^k)\| }{c_0}} \leq (1+\lambda_\infty)\sqrt{ \frac{\rho}{c_0} \Delta s_k \Delta_{k}} = \sqrt {\sigma \Delta s_k \Delta_{k}} ,
\end{align*}
which proves the claim.
\noindent\textbf{Claim~2:} \emph{ Let $k\geq k_0$ and assume that $(x^{j}, \mathbf{y}^{j}) \in \mathbb{B}_\varepsilon(\bar{x},\bar{\mathbf{y}})$, for all $j\in\{k_0,\ldots, k\}$. Then
\begin{align}\label{Inq_Claim2_ind}
\Delta_{k+1} \leq \sigma\left( \sum_{j=0}^{k-k_0} \frac{1}{2^{j+1}} \Delta s_{k-j} \right) + \frac{1}{2^{k+1-k_0}}\Delta_{k_0}.
\end{align}
}
Using \eqref{Inq_Claim1} inductively for $j\in\{k_0,\ldots, k\}$, we get
\begin{align*}
\Delta_{k+1} &\leq \left( \prod\limits_{j=0}^{k-k_0}\left( \sigma \Delta s_{k-j}\right)^{\frac{1}{2^{j+1}}} \right)\Delta_{k_0}^{\frac{1}{2^{k+1-k_0}}}.
\end{align*}
Now, let us recall the (generalized) inequality of arithmetic and geometric means (see, e.g.,~\cite[Proposition~3.14]{Aragon2019}), which states that for any nonnegative numbers $b_0, \ldots, b_{\ell+1}$,
\begin{align*}
\prod_{j=0}^{\ell+1} b_j^{\nu_j } \leq \sum_{ j=0}^{\ell+1} \nu_j b_j ,\text{ whenever } \nu_j\geq 0,\text{ with } \sum_{j=0}^{\ell+1} \nu_j = 1.
\end{align*}
Using this inequality with $b_j:= \sigma \Delta s_{k-j}$ and $\nu_j := \frac{1}{2^{j+1}}$ for $j=0,\ldots, k-k_0=:\ell$, and $b_{\ell+1} :=\Delta_{k_0}$ and $\nu_{\ell+1}:=\frac{1}{2^{k+1-k_0}}$, we have that
\begin{align*}
\left( \prod\limits_{j=0}^{k-k_0}\left( \sigma \Delta s_{k-j}\right)^{\frac{1}{2^{j+1}}} \right)\Delta_{k_0}^{\frac{1}{2^{k+1-k_0}}} \leq \sigma\left( \sum_{j=0}^{k-k_0} \frac{1}{2^{j+1}} \Delta s_{k-j} \right) + \frac{1}{2^{k+1-k_0}}\Delta_{k_0}
\end{align*}
which concludes the proof of \eqref{Inq_Claim2_ind}.
\noindent\textbf{Claim~3:} \emph{For all $k\geq k_0$, $(x^{k}, \mathbf{y}^{k}) \in \mathbb{B}_\varepsilon(\bar{x},\bar{\mathbf{y}})$. Therefore, $(x^{k}, \mathbf{y}^{k})_{k\in \mathbb{N}}$ converges to $(\bar{x},\bar{\mathbf{y}})$.
}
We prove by induction that $(x^{k}, \mathbf{y}^{k}) \in \mathbb{B}_\varepsilon(\bar{x},\bar{\mathbf{y}})$ for all $k\geq k_0$. The assertion clearly holds for $k=k_0$ and $k=k_0+1$, so we can assume that there is $k_1> k_0+1$ such that $(x^{k}, \mathbf{y}^{k}) \in \mathbb{B}_\varepsilon(\bar{x},\bar{\mathbf{y}})$ for all $k\in\{k_0,\ldots,k_1\}$. Then \eqref{Inq_Claim2_ind} holds for all $k \in \{k_0,\ldots, k_1\}$, so we get that
\begingroup\allowdisplaybreaks
\begin{align*}
\sum_{k=k_0}^{k_1}\| (x^{k+1}, \mathbf{y}^{k+1}) - (x^{k}, \mathbf{y}^{k})\|&= \Delta_{k_0}+\sum_{k=k_0}^{k_1-1}\Delta_{k+1}\\
& \leq \Delta_{k_0} + \sum_{k=k_0}^{k_1-1}\left( \sigma\left( \sum_{j=0}^{k-k_0} \frac{1}{2^{j+1}} \Delta s_{k-j} \right) + \frac{1}{2^{k+1-k_0}}\Delta_{k_0} \right) \\
&\leq \Delta_{k_0} +\sigma \sum_{k=k_0}^{k_1-1}\left( \sum_{j=0}^{k-k_0} \frac{1}{2^{j+1}} \Delta s_{k-j} \right) + \left(\sum_{k=1}^{\infty} \frac{1}{2^{j}} \right) \Delta_{k_0}\\
&\leq 2\Delta_{k_0} +\sigma \sum_{k=k_0}^{k_1-1}\left( \sum_{j=0}^{k-k_0} \frac{1}{2^{j+1}} \Delta s_{k-j} \right) \\
&= 2\Delta_{k_0} +\sigma \sum_{j=1}^{k_1-k_0}\frac{1}{2^{j}} \left( \sum_{k=k_0}^{k_1-j} \Delta s_{k} \right) \\
&\leq 2\Delta_{k_0} +\sigma \sum_{j=1}^{k_1-k_0}\frac{1}{2^{j}}s_{k_0} \leq 2\Delta_{k_0} + \sigma s_{k_0}.
\end{align*}
\endgroup
Hence,
\begingroup\allowdisplaybreaks
\begin{align*}
\| (x^{k_1+1}, \mathbf{y}^{k_1+1}) - (\bar{x},\bar{\mathbf{y}})\| &\leq \| (x^{{k_0}}, \mathbf{y}^{{k_0}}) - (\bar{x},\bar{\mathbf{y}})\| + \sum_{k=k_0}^{k_1}\| (x^{k+1}, \mathbf{y}^{k+1}) - (x^{k}, \mathbf{y}^{k})\|\\
&\leq \frac{\epsilon}{4} + 2\Delta_{k_0} + \sigma s_{k_0}\leq\frac{\epsilon}{4}+\frac{2\epsilon}{4}+\frac{\epsilon}{4}=\epsilon,
\end{align*}
\endgroup
which demonstrates the assertion for $k=k_1+1$.
Therefore,
\begin{equation}\label{eq:sumDeltak}
\sum_{k=k_0}^{\infty}\| (x^{k+1}, \mathbf{y}^{k+1}) - (x^{k}, \mathbf{y}^{k})\|\leq 2\Delta_{k_0} + \sigma s_{k_0},
\end{equation}
which proves that $(x^{k}, \mathbf{y}^{k})$ is a Cauchy sequence, so it converges to $(\bar{x},\bar{\mathbf{y}})$.
\end{proof}
The next theorem allows to deduce convergence rates of the sequence generated by Algorithm~\ref{alg:1} when the Kurdyka--{\L}ojasiewicz property holds for a specific choice of function~$\theta$.
\begin{theorem}[Convergence rates]
In addition to the assumptions of Theorem~\ref{t:KL}, suppose that the function $\theta$ in the definition of the Kurdyka--{\L}ojasiewicz property is given by $\theta(t) := Mt^{1-\vartheta}$ for some $M>0$ and $0\leq \vartheta <1$. Then, we obtain the following convergence rates:
\begin{enumerate}[(i)]
\item\label{it:convrates-1} If $\vartheta=0$, then the sequence $(x^k,\mathbf{y}^k)_{k\in\mathbb{N}}$ converges in a finite number of steps to $(\bar{x},\barbf{y})$.
\item\label{it:convrates-2} If $\vartheta\in{\left]0,\frac{1}{2}\right]}$, then the sequence $(x^k,\mathbf{y}^k)_{k\in\mathbb{N}}$ converges linearly to $(\bar{x},\barbf{y})$.
\item\label{it:convrates-3} If $\vartheta\in{\left]\frac{1}{2},1\right[}$, then there exists a positive constant $\varrho$ such that for all $k$ large enough
\begin{equation*}
\|(x^k,\mathbf{y}^k)-(\bar{x},\barbf{y})\| \leq \varrho k^{-\frac{1-\vartheta}{2\vartheta-1}}.
\end{equation*}
\end{enumerate}
\end{theorem}
\begin{proof}
For proving~\eqref{it:convrates-1}, let $\vartheta=0$. By~\eqref{Kur-Loj}, \eqref{eq001_Lemma_desgrad} and Claim~3 from previous theorem, we have that
$$
1 \leq M \|(u^k,\mathbf{w}^k)\| \leq \rho \|(x^{k+1},\mathbf{y}^{k+1})-(x^k,\mathbf{y}^k)\|
$$
for all $k$ sufficiently large. Therefore, Theorem~\ref{th:conv} concludes that $(x^k,\mathbf{y}^k)_{k\in\mathbb{N}}$ stops after a finite number of iterations, or otherwise we would enter into contradiction with~\eqref{eq:Ostrowski}.
For the remaining cases, consider the sequence $S_{k}:=\sum_{\ell=k}^{\infty} \|(x^{\ell+1},\mathbf{y}^{\ell+1})-(x^{\ell},\mathbf{y}^{\ell})\|$, which is finite for any $k\geq0$ due to~\eqref{eq:sumDeltak}. The convergence of $(x^{k},\mathbf{y}^{k})_{k\in\mathbb{N}}$ to $(\bar{x},\barbf{y})$ can be studied by means of $S_k$ since $\|(x^{k},\mathbf{y}^{k})-(\bar{x},\barbf{y})\|\leq S_{k}$.
Recall that $\theta^{\prime}(t) = (1-\vartheta)Mt^{-\vartheta}$. Then, for any $k$ large enough \eqref{eq:sumDeltak} implies
\begin{equation}\label{eq:S_k}
\begin{aligned}
S_{k}& = \sum_{\ell=k}^{\infty} \Delta_{\ell} \leq 2\Delta_{k} + \sigma s_{k} \\
&= 2 \Delta_{k} + \sigma M \left(\Phi(\hat{x}^k,\hatbf{y}^k)-\Phi(\bar{x},\barbf{y})\right)^{1-\vartheta} \\
&= 2 \Delta_{k} + \frac{\sigma M^{\frac{1}{\vartheta}}(1-\vartheta)^{\frac{1-\vartheta}{\vartheta}}}{\theta^{\prime}\left(\Phi(\hat{x}^k,\hatbf{y}^k)-\Phi(\bar{x},\barbf{y})\right)^{\frac{1-\vartheta}{\vartheta}}} \\
&\leq 2 \Delta_{k} + \sigma M^{\frac{1}{\vartheta}}(1-\vartheta)^{\frac{1-\vartheta}{\vartheta}}\rho^{\frac{1-\vartheta}{\vartheta}} \Delta_{k}^{\frac{1-\vartheta}{\vartheta}},
\end{aligned}
\end{equation}
where the last inequality is due to~\eqref{Kur-Loj} and~\eqref{eq001_Lemma_desgrad}. If $\vartheta\in{\left]0,\frac{1}{2}\right]}$ , the dominant term in the right hand side of the above equation is the first summand. Therefore, there exists $k_1>0$ and $K_1>0$ such that
$$
S_k \leq K_1 \Delta_k, \quad \text{ for all } k\geq k_1.
$$
This implies~\eqref{it:convrates-2} by resorting to~\cite[Lemma~1(ii)]{aragon2018accelerating}. On the other hand, if $\vartheta\in{\left]\frac{1}{2},1\right[}$, the second term in the right hand side of~\eqref{eq:S_k} would be the dominant one. This yields the existence of some $k_2>0$ and $K_2>0$ such that
$$
S_k^{\frac{\vartheta}{1-\vartheta}} \leq K_2 \Delta_k, \quad \text{ for all } k\geq k_2.
$$
Finally, the conclusion of~\eqref{it:convrates-3} similarly follows from~\cite[Lemma~1(iii)]{aragon2018accelerating}.
\end{proof}
\section{Numerical Experiments}\label{sect:numerical}
In this section we present some computational experiments where we evaluate the performance of Algorithm~\ref{alg:1}. We recall that when $R>0$, so the linesearch in Steps~$6$-$7$ is performed, the resulting algorithm is referred as BDSA; otherwise, Algorithm~\ref{alg:1} without linesearch is named as DSA.
The linesearch of BDSA requires the selection of a number of hyperparameters, namely, the initial stepsize $\overline{\lambda}_k$, the backtracking constant~$\rho$ and the number of trials $R$. This may seem to be a drawback, as each particular problem could require of a specific tuning of all these parameters in order to obtain a good performance of the linesearch. Quite the opposite, the next numerical experiments on very different applications evidence that this is not the case: we ran all instances of BDSA with the same choice of parameters specified below and this general tuning was good enough for BDSA to significantly outperform its counterpart DSA with no linesearch.
\paragraph{Parameter tuning for Algorithm~\ref{alg:1} linesearch}
All the linesearches for BDSA in our numerical experiments were performed with the following choice of parameters: $R=2$, $\rho = 0.5$ and $\alpha=0.1$. The initial stepsize $\overline{\lambda}_k$ was chosen according to the self-adaptive trial stepsize scheme presented in Algorithm~\ref{alg:fortracking} with $\lambda_0=2$ and $\delta=2$.
\begin{algorithm}[H]
\caption{Self-adaptive trial stepsize}\label{alg:fortracking}
\begin{algorithmic}[1]
\mathbb{R}equire{$\delta>1$ and $\lambda_0>0$. Obtain $\lambda_k$ from $\overline{\lambda}_k$ by Steps $4$-$7$ of BDSA (Algorithm~\ref{alg:1}).}
\If{$r=0$}
\State{set $\overline{\lambda}_{k+1}:=\delta\overline{\lambda}_k$;}
\Else
\State{set $\overline{\lambda}_{k+1}:=\max{\{ \lambda_0, \rho^{r}\,\overline{\lambda}_k\}}$.}
\EndIf
\end{algorithmic}
\end{algorithm}
The \emph{self-adaptive} trial stepsize given by Algorithm~\ref{alg:fortracking} is based on the one proposed in~\cite{boostedDCA} for the \emph{Boosted Difference of Convex functions Algorithm} (BDCA). We note that a similar adaptive scheme for the \emph{gradient descend method} was recently introduced in~\cite{truong2021backtracking}. The procedure works as follows.
Algorithm~\ref{alg:fortracking} determines how to choose the starting stepsize $\overline{\lambda}_{k+1}$ of the next iteration of the method. If in the current iteration a decrease of $\Phi$ was achieved in the first attempt of the linesearch (i.e., when $r=0$), then the starting stepsize for the next iteration of the main algorithm is increased by setting $\overline{\lambda}_{k+1}:=\delta \lambda_k$, with $\delta>1$. Otherwise, $\overline{\lambda}_{k+1}$ is set as the maximum between the default initial stepsize and the smallest stepsize accepted in the previous iteration, i.e., $\overline{\lambda}_{k+1} :=\max{\{ \lambda_0, \rho^{r}\,\overline{\lambda}_k\}}$ (observe that $\lambda_k$ could be zero if the linesearch was not successful).
This section is divided into three subsections, each containing a different application. The purpose of the experiments in the first subsection is twofold. First, to demonstrate how the linesearch from the boosting step can help reaching better critical points. Second, to show that the assignment of the terms of the objective function of~\eqref{eq:P1} to each of the functions $f$, $g$, $h_i$ and $\Psi_i$ has a big impact in the success of the resulting scheme derived from Algorithm~\ref{alg:1}. In Subsection~\ref{sect:numerical2} we consider an application with real-data for clustering cities in a region and show how the linesearch of BDSA helps finding better solutions in considerably less time than DSA (which, in this context, coincides with GPPA~\cite{an2017convergence}). Lastly, Subsection~\ref{sect:numerical3} contains a nonconvex generalization of Heron's problem that can be addressed with BDSA. In this case, BDSA is not a particular instance of any other known algorithm.
All the experiments were ran in a computer of Intel Core i7-12700H 2.30 GHz with 16GB RAM, under Windows 11 (64-bit).
\subsection{Avoiding Non-Optimal Critical Points}\label{subsect:avoidingcriticalpoints}
Theorems~\ref{th:conv} and~\ref{t:KL} prove the convergence of Algorithm~\ref{alg:1} to some critical point of~\eqref{eq:P1}. We recall that being a critical point is a necessary (but not sufficient) condition for local optimality of problem~\eqref{eq:P1}. In~\cite[Example~3.3]{boostedDCA} it was shown how the linesearch performed by the BDCA helps prevent the algorithm from being trapped by critical points which are not local minima. In this subsection we illustrate the same phenomenon by considering different known algorithms that can be obtained as particular cases of Algorithm~\ref{alg:1}. We show that its \emph{boosted} version, with the additional linesearch, outperforms the basic methods in avoiding these non-desirable critical points.
To this aim, we introduce a new family of functions which entails a challenge to this class of methods. These functions have a large number of critical points where the algorithm can easily get stuck, but a unique global minimum. Specifically, for any $q\in \mathbb{N}$, we define the functions $\varphi_{q}:\mathbb{R}^n\to\mathbb{R}$ as
\begin{equation}\label{eq:func_phi_q}
\varphi_{q}(x) := \|x\|^2 - \|x\|_{1} - \sum_{j=1}^q \left( \|x-je\|_{1} + \|x+je\|_{1}\right) - \|x-(q+1)e\|_{1},
\end{equation}
where $e$ is the vector of ones in $\mathbb{R}^n$.
It is a simple exercise to check that the function $\varphi_q$ possesses $(2q+3)^n$ critical points, which are given by the set ${\{-(q+1),-q,\ldots,0,\ldots,q,q+1\}}^n$, and a unique local minimum at $x^* := (-(q+1),-(q+1),\ldots,-(q+1))^T$, which corresponds to its global optimum, with optimal value $\varphi_q(x^*)=-n(q^2+3q+2)$.
Note that the function $\varphi_q$ admits different representations as an instance of~\eqref{eq:P1}, and different algorithms are derived from BDSA depending on which terms one assigns to each of the functions $f$, $g$ and $h_i$ (recall Remark~\ref{remark:particularcases}):
\begin{itemize}
\item Setting $f(x):= 0$, $g(x) := \| x\|^2$ and $h_i$, for $i= 1,\ldots, 2q+2$, to be the remaining terms involving the $\ell_1$-norm, the \emph{Double-proximal Gradient Algorithm} (DGA) by Banert--Bo\c{t}~\cite{bot2019doubleprox} is obtained.
\item If we take $f(x):= - \|x\|_{1} - \sum_{j=1}^q \left( \|x-je\|_{1} + \|x+je\|_{1}\right) -\|x-(q+1)e\|_{1}$, $g(x) := \| x\|^2$ and $h(x):= 0$, we recover the particular case of the \emph{Proximal DC Algorithm} (PDCA) discussed in Remark~\ref{remark:particularcases}~\eqref{it:particularcases1}, which would become the \emph{Boosted Proximal DC Algorithm} (BPDCA) from~\cite{alizadeh2022new} when $R=\infty$ (but recall that we take $R=2$ in our experiments). Due to variable separability of the $\ell_1$-norm, it can be proved that the subdifferential of $f$ coincides with the sum of subdifferentials of the $\ell_1$-norm terms. Therefore, for every $k\geq0$, we take $v^k \in \partial f(x^k)$ as a sum of subgradients of the form
\[
v^k = \sum_{s\in I} v^k_{s} \quad \text{ where } \quad v^k_{s}\in\partial\left(-\|\cdot-se\|_1\right) (x^k),
\]
where $I:=\{0,1,-1,\ldots,q,-q,q+1\}$ and every subgradient is componentwise chosen as
\begin{equation*}
(v^k_{s})_i = \left\{
\begin{aligned}
1 & \text{ if } x^k_i\leq s, \\
-1 & \text{ if } x^k_i >s,
\end{aligned}
\right. \quad \text{ for all }i=1,\ldots,n.
\end{equation*}
\end{itemize}
For different combinations of $n$ and $q$, we performed $10\,000$ runs of DGA, PDCA, and their boosted counterparts with linesearch (abbr. as BDGA and BPDCA) all initialized at the same starting points randomly chosen in the interval $[-q-2,q+2]^{n}$, and $[-1,1]^n$ for the dual variables (when necessary), with $\mu=\gamma=1$. We note that the conjugate of the $\ell_1$-norm is the indicator function of $[-1,1]^n$, so this seems a fair set in which to choose the initial dual variables.
We stopped all the algorithms when the norm of the difference between two consecutive iterates is smaller than $n\times 10^{-6}$ and counted how many times each of the methods converged to the optimal solution $x^*$. The results are summarized in Table~\ref{tab:counterexample}.
\begin{table}[ht!]\centering
\begin{tabular}{rrcccc}
\toprule
n & q& DGA& BDGA& PDCA & BPDCA \tabularnewline
\midrule
$2$ & $3$ &273 &1\,202& 410 & 10\,000 \tabularnewline
\midrule
$2$ & $5$ &72 &774& 201 & 10\,000 \tabularnewline
\midrule
$2$ & $10$ &10 &440& 71 & 10\,000 \tabularnewline
\midrule
$2$ & $20$ & 0 &253 & 21 & 10\,000 \tabularnewline
\midrule
$10$& $3$ & 0 &2\,229& 0 & 10\,000 \tabularnewline
\midrule
$20$ & $3$ &0 &2\,076& 0 & 10\,000 \tabularnewline
\bottomrule
\end{tabular}
\caption{For different values of $n$ and $q$, and $10 \, 000$ random starting points, we count the number of instances that each of the algorithms converged to the global minimum $x^*=(-q-1,\ldots,-q-1)^T$ of the function $\varphi_q$ in~\eqref{eq:func_phi_q}. All algorithms are particular instances of BDSA.}\label{tab:counterexample}
\end{table}
The most remarkable fact is that BPDCA converged to the optimal point $x^*$ in every single instance. By contrast, its non accelerated version PDCA very rarely managed to reach the optimum (1.17\% of the overall instances). On the other side, DGA also got trapped very often by non-optimal critical points, only converging to the global minimum in 0.59\% of the instances. Its accelerated version BDGA greatly improved this poor result and converged 11.62\% of the times to the optimal solution.
For illustration, we display in Figure~\ref{fig:counterexample} for $n=2$ and $q=3$ the sequences generated by DGA and PDCA from the starting point $x^0= (1.5,-0.5)^T$, using $y^0= (0,0)^T$ as the dual starting point for DGA. In addition, we show the iterates obtained after accelerating both methods with the boosted linesearch scheme proposed in Algorithm~\ref{alg:1}. We observe that PDCA got caught by $(0,-1)^T$, which is the nearest critical point that it encountered, while DGA converged to the slightly better critical point $(-1,-1)^T$. On the other hand, both BPDCA and BDGA managed to converge to $x^*=(-4,-4)^T$.
\begin{figure}
\caption{Sequence of iterates generated by PDCA, DGA and their boosted versions for the same starting point when they were applied to the function $\varphi_3$ in $\mathbb{R}
\label{fig:counterexample}
\end{figure}
The fact that PDCA and DGA have such a low rate of success in reaching the global minimum is an indicator of how challenging the proposed family of functions is for this type of algorithms. The advantage of the boosted versions of the algorithms for this family is clear. Even so, it is important to mention that although the linesearch in BDGA only succeeded to improve its success rate up to 11.62\%, it consistently improved the objective values. BDGA converged to a point with lower objective value than DGA in 46.61\% of the instances, while both algorithms attained the same value in the remaining in 53.39\%. DGA did not surpass BDGA in any of the 60\,000 instances.
\subsection{Minimum Sum-of-Squares Constrained Clustering Problem}\label{sect:numerical2}
\emph{Clustering analysis} is a widely-employed technique in data science for classifying a collection of objects into groups, called \emph{clusters}, whose elements share similar characteristics. In order to mathematically describe the clustering problem, we can think of our data as a finite set of points $A=\left\{a^1,\ldots,a^q\right\}$ in $\mathbb{R}^s$. Our goal is to group $A$ into $\ell$ disjoint subsets $A^1,\ldots,A^\ell$, based on the minimization of some clustering measure.
In the \emph{minimum sum-of-squares clustering problem}, the groups are determined by the minimization of the squared Euclidean distance of each data to the \emph{centroid} of its cluster. In this way, each cluster $A^j$ is identified by its centroid, which we denote by $x^j\in\mathbb{R}^s$, for $j=1,\ldots,\ell$. Letting $X:=(x^1,\ldots,x^{\ell})\in\mathbb{R}^{s \times \ell}$, this clustering problem can be reformulated as the optimization problem.
\begin{equation}\label{eq:DC-clustering}
\min_{X\in\mathbb{R}^{s\times \ell}} f(X) := \frac{1}{q}\sum_{i=1}^q \omega_i(X),
\end{equation}
where $\omega_i(X):= \min\left\{ \|x^j-a^i\|^2 : j=1,\ldots, \ell \right\}.$ The function $f$ is $1$-upper-$\mathcal{C}^2$, since each of the functions $\omega_i$ is $1$-upper-$\mathcal{C}^2$ (simply by definition).
In~\cite{boostedDCA}, the authors considered the clustering problem \eqref{eq:DC-clustering} with the aim of grouping the 4001 Spanish cities in the peninsula with more than 500 residents.
In this work, we consider a more challenging version of the above problem in which we add a nonconvex constraint on $X$. This is useful for example when the centroids represent facilities (e.g., hospitals or government administrations). In this case, the centroids cannot be located in the sea, or even in certain areas that should be avoided. Therefore, we are interested in solving the problem
\begin{equation}\label{eq:const-clustering}
\min_{X\in C} f(X).
\end{equation}
where $C\subseteq\mathbb{R}^{s\times \ell}$ is the newly introduced (not necessarily convex) constraint. This allows us to make the experiment in~\cite{boostedDCA} more challenging, in the following way:
\begin{itemize}
\item We consider the cities with more than 500 residents in the Spanish peninsula, but also those in the Balearic Islands, which is an archipelago in the Mediterranean Sea. They sum a total of 4049 cities.
\item We exclude a region in the center of Spain as a possible location for centroids, which would be useful if decentralization policies were aimed.
\item We exclude Portugal, which is also contained in the same peninsula as Spain.
\end{itemize}
The resulting closed nonconvex constraint is depicted in Figure~\ref{fig:constraint}.
Now, considering the objective function of problem \eqref{eq:const-clustering} as a large sum of nonsmooth functions, the sum rule for the basic subdifferential only offers an upper estimation rather than an equality. Consequently, it becomes more convenient to compute subgradients of individual functions $\omega_i$ instead of examining the entire function $f$. In this context, the following proposition formally provides the computation of the subdifferential of the functions $\omega_i$, for $i=1,\ldots,q$.
\begin{proposition}
Given $a\in\mathbb{R}^s$, consider the function $\omega :\mathbb{R}^{s\times \ell} \to \mathbb{R}$ given by
\begin{equation*}
\omega(X):= \min\left\{ \|x^j-a\|^2 : j=1,\ldots, \ell \right\},
\end{equation*}
with $X=(x^1,\ldots,x^{\ell})\in\mathbb{R}^{s \times \ell}$. Then, the following formula holds
\begin{align}\label{basic_subd_f_i}
\partial \omega(X)= \left\{ (0,\ldots, 0, \underbrace{2(x^j - a)}_{
j\text{-th position} }, 0 , \ldots,0) : \omega(X) = \| x^j - a\|^2 \right\}, \text{ for all } X\in \mathbb{R}^{s\times \ell }.
\end{align}
\end{proposition}
\begin{proof}
To prove \eqref{basic_subd_f_i} let us notice that the inclusion $\subseteq$ follows from the calculus rule for the minimum function (see, e.g., \cite[Proposition 1.113]{MR2191744}). To prove the opposite inclusion, we recall that by \eqref{eq:subinclusions} the following equality holds \begin{equation}\label{eq_conv}
\conv \partial \omega(X) = -\partial (-\omega)(X).
\end{equation}
Now, since $-\omega$ is a maximum of quadratic forms, we can apply \cite[Theorem 3.46]{MR2191744} to $-\omega$ to conclude that
\begin{align*}
-\partial (-\omega)(X) = \conv B(X),
\end{align*}
where $B(X)$ is the set in the right-hand side of \eqref{basic_subd_f_i}. Finally, since all the points in the set $B(X)$ are linearly independent, we get that $\supset$ must hold in \eqref{basic_subd_f_i}, as otherwise it would contradict \eqref{eq_conv}.
\end{proof}
Based on the aforementioned observation, we are motivated to present Algorithm \ref{alg:3:clustering} as a well-suited variant of the Boosted Double-proximal Subgradient Algorithm for effectively addressing the constrained clustering problem \eqref{eq:DC-clustering}. When $C$ is defined by linear inequality constraints, observe that feasibility of the direction $D^k$ defined in Step~3 can be checked as in~\cite[Algorithm~1]{AragonArtacho2022} (see also Lemma~3.1 there), so the boosting in Step~5 is only run when $D^k$ is in the cone of feasible directions. The set-valued \emph{projector} onto the set $C\subseteq\mathbb{R}^m$ is denoted by $P_C:\mathbb{R}^m\rightrightarrows \mathbb{R}^m$.
\begin{algorithm}
\caption{Boosted proximal Subgradient Algorithm for constrained clustering}\label{alg:3:clustering}
\begin{algorithmic}[1]
\mathbb{R}equire{$X^0 \in\mathbb{R}^{s\times \ell}$, $R\geq 0$, $\rho\in{]0,1[}$ and $\alpha\geq 0$. Set $k:=0$.}
\State{Choose $v_i^{k}\in \partial \omega_i(X^k)$ for $i=1, \ldots,q$ and set $V^k = \frac{1}{q} \sum_{i=1}^q v_i^k$ .}
\State{Take some positive $\gamma_k<\frac{1}{2}$ and compute
\begin{equation*}
\begin{aligned}
\hat{X}^{k} & \in P_C \left(X^k-\gamma_k V^k\right) .
\end{aligned}
\end{equation*}
}
\State{Choose any $\overline{\lambda}_k\geq 0$. Set $\lambda_{k}:=\overline{\lambda}_{k}$, $r:=0$ and $D^{k} :=\hat{X}^{k} -X^{k}$.}
\State{\textbf{if} $D^{k}=0$ \textbf{then} STOP and return $x^k$.}
\State{\textbf{while} $r<R$ and
\begin{align*}
\hat{X}^{k} + \lambda_{k}D^{k} \notin C \text{ or } f\big(\hat{X}^{k} + \lambda_{k}D^{k} \big) > f(\hat{X}^{k})- \alpha \lambda_k^2 \| D^{k}\|^2
\end{align*}
\textbf{do }{$r:=r+1$ and $\lambda_k:=\rho^r\overline{\lambda}_k$.}}
\State{\textbf{if }$r=R$ \textbf{then} $\lambda_k:=0$.}
\State{Set $X^{k+1} :=\hat{X}^{k} + \lambda_{k}D^{k}$, $k:=k+1$ and go to Step 1.}
\end{algorithmic}
\end{algorithm}
Now, in order to present our convergence result for Algorithm \ref{alg:3:clustering} we define a suitable notion of critical point: we say that $\bar X$ is a \emph{critical point} of the constrained clustering problem \eqref{eq:const-clustering} if
\begin{align*}\label{eq:clustering:critical}
0 \in \frac{1}{q} \sum_{ i=1}^q \partial \omega_i(\bar{X}) + N_C(\bar{X}),
\end{align*}
where $N_C$ denotes the (\emph{basic}) \emph{normal cone} to $C$, which coincides with $\partial\iota_C$ (see, e.g.,\cite[Proposition~1.79]{MR2191744}).
\begin{corollary}\label{th:conv:clustering}
Given $ X^0 \in\mathbb{R}^{s\times \ell }$ and $\eta\in{]0,1[}$, consider the sequence $(X^k)_{k\in\mathbb{N}}$ generated by Algorithm~\ref{alg:3:clustering} with $\gamma_k\in{\left]0,\frac{\eta}{2}\right]}$ for all $k\in\mathbb{N}$. Then, either Algorithm~\ref{alg:3:clustering} stops at a critical point of~\eqref{eq:const-clustering} after a finite number of iterations or it generates an infinite sequence $ (X^k)_{k\in\mathbb{N}}$ such that the following assertions hold:
\begin{enumerate}[(i)]
\item The sequence $\big(f(X^k )\big)_{k\in\mathbb{N}}$ monotonically (strictly) decreases and converges, and $X^k \in C$ for all $k \geq 1$. Moreover, the sequences $(X^k)_{k\in\mathbb{N}}$ verifies that
\begin{equation*}\label{eq:Ostrowski:clustering}
\sum_{k=0}^{\infty} \|X^{k+1}-X^k\|^2 < \infty.
\end{equation*}
\item If the sequence $(X^k)_{k\in\mathbb{N}}$ is bounded, the set of its accumulation points is nonempty, closed and connected.
\item If $\bar{X}\in\mathbb{R}^{s\times \ell }$ is an accumulation point of the sequence $(X^k)_{k\in\mathbb{N}}$, then $\bar{X}$ is a critical point of~\eqref{eq:const-clustering}. In addition, $f(\bar{X}) = \inf_{k\in\mathbb{N}}f(X^k)$.
\item If $(X^k)_{k\in\mathbb{N}}$ has at least one isolated accumulation point, then the whole sequence $(X^k)_{k\in\mathbb{N}}$ converges to a critical point of~\eqref{eq:const-clustering}.
\end{enumerate}
\end{corollary}
\begin{proof}
To prove this corollary, let us first notice that every subgradient $V^k $ belongs to $\conv \partial f(X^k)$. Indeed, using \cite[Theorem 3.46(ii)]{MR2191744} we have that each function $-\omega_i$ is lower regular at any point. Hence, using \eqref{eq:subinclusions} and the sum rule for the basic subdifferential we get that
\begin{align*}
\conv \partial f (X) = - \partial (-f) (X) = -\frac{1}{q}\sum_{ i=1}^q \partial (-\omega_i)(X) =\frac{1}{q}\sum_{ i=1}^q \conv \partial \omega_i(X) \supset \frac{1}{q}\sum_{ i=1}^q \partial \omega_i(X).
\end{align*}
Therefore, using Remark~\ref{Remark_sub} we get the result, where the justification that $\bar{X}$ is a critical point of~\eqref{eq:const-clustering} follows from \eqref{general_critical_point}.
\end{proof}
\begin{figure}
\caption{The blue squares represent the 4049 cities of Spain peninsula and Balearic Islands with more than 500 inhabitants. In order to accurately gather all the area of Spain including these cities, we build our constraint $C$ as the union of a finite number of shaded polyhedral sets. Note that the rectangle in the center of Spain is excluded.}
\label{fig:constraint}
\end{figure}
\begin{figure}
\caption{
We ran GPPA and BDSA from the same initial random point, for finding $9$ centroids satisfying the constraints shown in Figure~\ref{fig:constraint}
\label{fig:clustering9}
\end{figure}
In our first experiment, we aim to find a partition into $9$ clusters of the $4049$ cities in consideration. For this experiment, we set $\gamma := 0.9 \times \left( 1/2\right)$ and ran both GPPA and BDSA, starting from the same initial random point, until BDSA reached a relative error in the objective function (i.e., $|f(X^{k+1})-f(X^k)|/f(X^{k+1})$) smaller than $10^{-3}$. This stopping criterion was achieved after $27$ iterations, represented in Figure~\ref{fig:clustering9}, with a value of the objective function of $1.0822$. Simultaneously, GPPA returned an objective value of $1.5138$ for the same number of iterations. In particular, Figure~\ref{fig:clustering9} (left) demonstrates significant progress made by BDSA in moving towards regions with a higher concentration of cities. Meanwhile, GPPA required $116$ iterations before reaching the same relative error. Figure~\ref{fig:clustering_decrease} presents an illustrative example where both algorithms converge to distinct critical points. In this occasion, GPPA converged to a critical point with an objective function value of $3.2670$ after $30$ iterations, while BDSA only needed $16$ iterations to converge to a point with a superior value of $2.1827$. The plots on both Figure~\ref{fig:clustering9} and~\ref{fig:clustering_decrease} give more insight into how the linesearch helps the boosted algorithm to achieve a more significant decrease in the objective value.
\begin{figure}
\caption{We ran GPPA and BDSA from the same initial random point, for finding $5$ centroids satisfying the constraints shown in Figure~\ref{fig:constraint}
\label{fig:clustering_decrease}
\end{figure}
To demonstrate that this is the general trend, we solved the same problem with the Spanish cities for a different number of clusters $\ell\in{\{3,5, 10, 15, 20, 30, 40, 50\}}$. The results are summarized in Figure~\ref{fig:clusteringF}. For each of these values and for $10$ different random starting points, we ran BDSA until the relative error in the objective function was smaller than $10^{-3}$. Then, GPPA was ran from the same starting point until it reached the same objective value than BDSA or until the relative error in the objective function was smaller than $10^{-3}$. In particular, GPPA failed to reach the same value than BDSA in $68$ out of the $80$ runs. In Figure~\ref{fig:clusteringF} we present the iterations ratio (left) and time ratio (right) of both algorithms for all the instances. On average, BDSA was approximately $3$ times faster with respect to the number of iterations than GPPA, and more than $2$ times faster in time. There was only one instance where GPPA was faster than BDSA (both in time and iterations), for $\ell=5$, but the value of $\varphi$ at the stopping point for GPPA was $1.43$ times larger than that of BDSA. Therefore, the increase in the running time and iterations of BDSA was a consequence of converging to a better critical point than GPPA.
\begin{figure}
\caption{Iteration and time ratios between GPPA and BDSA for solving problem~\eqref{eq:const-clustering}
\label{fig:clusteringF}
\end{figure}
\subsection{A Nonconvex Generalization of Heron's Problem}\label{sect:numerical3}
The original formulation of Heron's problem consists in the following: given a straight line in the plane, find a point $x$ in it such that the sum of the distances from $x$ to two given points is minimal. A generalization of Heron's problem to a Euclidean space of arbitrary dimension $\mathbb{R}^n$ was introduced in~\cite{mordukhovich2012solving}, where the line was substituted by a closed convex set $C_0$ and the two given points by a finite family of closed convex sets $\{C_i\}_{i=1}^p$. This convex problem was then solved by means of a projected subgradient algorithm. Lately, different splitting methods have also been employed to tackle this generalization of Heron's problem~\cite{bot2013douglas,campoy2022product}.
In this subsection, we go one step ahead and consider a more general version of the problem. Specifically, we seek to minimize a weighted sum of the squared distance of the images of $x$ by certain differentiable functions $\Psi_i:\mathbb{R}^n\to\mathbb{R}^{m_i}$ with Lipschitz continuous gradients, for $i=1,\ldots,p$. Namely, given some closed (but not necessarily convex) sets $C_0\subseteq \mathbb{R}^n$ and $C_i\subseteq\mathbb{R}^{m_i}$, for $i=1,\ldots,p$, we are interested in solving the following nonconvex generalization of Heron's problem
\begin{align}\label{eq:GQHP}
\min_{x \in C_0} \sum_{i=1}^p \frac{w_i}{2}d^2 (\Psi_i(x),C_i),
\end{align}
where $w_i > 0$ represents a weight associated to the $i$-th constraint.
Problem~\eqref{eq:GQHP} can be easily reformulated as an instance of~\eqref{eq:P1}. Indeed, as shown in Example~\ref{prop:Asplund}, the squared distance function admits the following decomposition:
\begin{align*}
\frac{1}{2} d^2 (\Psi_i(x),C_i)= \frac{1}{2} {\| \Psi_i(x)\|^2 } - \Asp{C_i}{}(\Psi_i(x)).
\end{align*}
Hence, problem~\eqref{eq:GQHP} is equivalent to the unconstrained problem
\begin{equation*}
\min_{x\in\mathbb{R}^n} \iota_{C_0}(x) + \sum_{i=1}^p w_i \left( \frac{1}{2}\|\Psi_i(x)\|^2 - \Asp{C_i}(\Psi_i(x))\right).
\end{equation*}
It is clear that this problem can be expressed in the form of~\eqref{eq:P1} by choosing $f:= \sum_{i=1}^p \frac{w_i}{2}\|\Psi_i(\cdot)\|^2$, $g=\iota_{C_0}$ and $h_i = w_i\Asp{C_i}$, for $i=1,\ldots,p$. Note that although the Asplund function $\Asp{C_i}$ is always convex, $\Asp{C_i} \circ \Psi_i$ may not be convex if $\Psi_i$ is not linear. Therefore, $\frac{w_i}{2} d^2 (\Psi_i(x),C_i)$ is not necessarily upper-$\mathcal{C}^2$.
In the following we analyze the performance of DSA and BDSA for the particular instance of the problem in which $w_i:=1$ for all $i=1,\ldots,p$, $C_0$ is the closed ball of radius $r_{C_0} := 5$ in $\mathbb{R}^n$, and the soft constraints $C_i$ are hypercubes of edge length $2$. To avoid intersections with $C_0$, the centroids of the hypercubes were randomly generated with norm between $7$ and $10$. We set all $\Psi_i := \Psi:\mathbb{R}^n\to \mathbb{R}^m$, with
\begin{equation*}
\Psi(x) := \left( x^TQ_1x, x^TQ_2x, \ldots, x^TQ_mx \right)^T,
\end{equation*}
where, for simplicity, we chose $Q_1,\ldots,Q_m$ as diagonal matrices with randomly generated entries in $]-1,1[$. Note, that the gradient of $\Psi$ is the linear transformation given by
\[
\nabla \Psi(x) = 2(Q_1x,Q_2x,\ldots,Q_mx),
\]
which is Lipschitz continuous with constant $L_{\Psi}:=2 \rho(Q) $, where $\rho(Q)$ denotes the spectral radius of $Q:=(Q_1,Q_2,\ldots,Q_m)$. On the other hand, note that $\nabla f$ is also $L_f$-Lipschitz in the ball $C_0$ for $L_f:=6pr_{C_0}^2\rho(Q)\sqrt{\sum_{i=1}^m\rho(Q_i)^2}$, since
\begin{align*}
\|\nabla f(x)-\nabla f(y)\|&=\|p\left(\nabla\Psi(x)\Psi(x)-\nabla\Psi(y)\Psi(y)\right)\|\\
&\leq p\|\nabla\Psi(x)\Psi(x)-\nabla\Psi(x)\Psi(y)\|+p\|\nabla\Psi(x)\Psi(y)-\nabla\Psi(y)\Psi(y)\|\\
&\leq p\|\nabla\Psi(x)\|\sqrt{\sum_{i=1}^m(x^TQ_ix-y^TQ_iy)^2}+p\|\Psi(y)\|L_{\Psi}\|x-y\|\\
&\leq 2\rho(Q)p\left(\|x\|\sqrt{\sum_{i=1}^m4r_{C_0}^2\rho(Q_i)^2}+\max_{z\in C_0}\|\Psi(z)\|\right)\|x-y\|\\
&\leq 2\rho(Q)p\left(\|x\|2r_{C_0}\sqrt{\sum_{i=1}^m\rho(Q_i)^2}+\max_{z\in C_0}\sqrt{\sum_{i=1}^m\|z\|^4\ \rho(Q_i)^2}\right)\|x-y\|\\
&\leq 6\rho(Q)pr_{C_0}^2\sqrt{\sum_{i=1}^m\rho(Q_i)^2}\|x-y\|=L_f\|x-y\|.
\end{align*}
Therefore, according to~\eqref{eq:descentlemmaineq} and Proposition~\ref{p:deslemma}, the function $f$ is $L_f /2$-upper-$\mathcal{C}^2$.
In order to fairly illustrate the advantages of the linesearch step in BDSA, we initially perform an experiment to find some adequate performing parameters for DSA (i.e., when no linesearch was performed).
In all the experiments in this subsection, we stopped the algorithms when
\begin{equation}\label{eq:stop}
|\Phi(x^{k+1},y^{k+1})-\Phi(x^k,y^k)|< 10^{-6}.
\end{equation}
\paragraph{Tuning the parameters for DSA}
We set $n=3$, $m=4$ and $p=3$, and ran Algorithm~\ref{alg:1} with different choices of parameters for $5$ randomly generated problems and $5$ different starting points for each problem (i.e., 25 instances in total). Having in mind the bounds for the parameters given in Theorem~\ref{th:conv}, we tested the algorithm for every combination of the following choices:
\begin{align*}
\gamma_k&:=\eta\left( L_f + L_{\Psi} \sum_{i=1}^p \|y_i^k\| \right)^{-1},\text{ with }
\eta \in {\{0.1, 0.3, 0.5, 0.7, 0.9, 0.99\}},\\
\mu^k_i &:=\mu\in{\{0.5, 1, 5\}},\text{ for all } k\geq 0.
\end{align*}
For every combination of stepsize parameters the algorithm obtained the same value of $\Phi$ in the last iterate. However, there is a considerable variability in the number of iterations needed for reaching the stopping criterion, which we show in Table~\ref{fig:ParamSplitting}. Note that the parameter~$\gamma_k$ is the only one providing significant differences, being $\eta= 0.99$ the best performing value. The parameter $\mu$ does not seem to have much influence in the results obtained.
\begin{table}[ht!]\centering
\begin{tabular}{lcccccc}
\toprule
& $\eta=0.1$ & $\eta=0.3$ & $\eta=0.5$ & $\eta=0.7$ & $\eta=0.9$ & $\eta=0.99$\tabularnewline
\midrule
$\mu= 0.5$ & 8\,013.0 & 3\,104.8 & 1\,989.0& 1\,486.6 & 1\,201.5& 1\,104.9\tabularnewline
$\mu = 1$ & 8\,012.4 & 3\,103.7 & 1\,987.6 &1\,485.0&1\,199.9 &1\,103.4\tabularnewline
$\mu = 5$ & 8\,011.7 &3\,102.9&1\,986.5& 1\,483.9 & 1\,198.6& 1\,102.1\tabularnewline
\bottomrule
\end{tabular}\caption{Average number of iterations of DSA for $5$ random problems~\eqref{eq:GQHP} and $5$ random starting points for each problem, with $n=3$, $m=4$, $p=3$ and different values of the parameters.}\label{fig:ParamSplitting}
\end{table}
\paragraph{DSA vs. BDSA: Benefit of linesearches}
Now we compare both versions of Algorithm~\ref{alg:1} with the best choice parameter $\eta=0.99$. Since $\mu$ does not have much effect, we set a small value $\mu^k=0.5$, which is more likely to satisfy the bound in Proposition~\ref{l:descentdirection}(iii).
We tested DSA and BDSA for different values of $n$, $m$ and~$p$. Both algorithms obtained similar results regarding the objective values, showing only differences after the second decimal, in favor of BDSA in all but one instance, so we only present the results regarding number of iterations (without counting those needed for the linesearch) and the running time.
The results are summarized in Figures~\ref{fig:SplitVSBoosted} and~\ref{fig:SplitVSBoosted_r}, where we observe that BDSA clearly outperformed DSA in each of the $120$ instances. In particular, BDSA was on average more than $2.5$ times faster than DSA.
\begin{figure}
\caption{For $p=3$, each $n\in{\{5,10,15,20,30,50\}
\label{fig:SplitVSBoosted}
\end{figure}
\begin{figure}
\caption{For $n=20$, $m=16$ and each $p\in{\{3, 5, 7, 10, 15, 20\}
\label{fig:SplitVSBoosted_r}
\end{figure}
\paragraph{The generalized Heron Problem with nonconvex sets} In our last experiment, we consider examples of~\eqref{eq:GQHP} in which the sets $C_i$ are not necessarily convex. Instances of the generalized Heron problem with nonconvex sets have already been studied for example in \cite{mordukhovich2012applications}. In this experiment, we let $\Psi_i$ be a linear mapping of the form $\Psi_i(x)=Qx$, with $Q\in\mathbb{R}^{m\times n}$, for all $i=1,\ldots,p$. We showed in Example~\ref{prop:Asplund} that $x\mapsto\frac{1}{2}d^2(Qx,C)$ is a $\kappa$-upper-$\mathcal{C}^2$ function with $\kappa = \rho(Q)^2/2$. Moreover, note that by \cite[Theorem~5.3~(iii)]{aragonartacho2023coderivativebased} its subdifferential at a point $x\in\mathbb{R}^n$ is given by
\[
\partial \left(\frac{1}{2}d^2(Q(\cdot),C)\right)(x) = Q^T \bigg(Qx-P_{C}(Qx)\bigg).
\]
These two facts allow us to tackle~\eqref{eq:GQHP} when $C_i$ are not necessarily convex as an instance of problem~\eqref{eq:P1} by setting $f:= \sum_{i=1}^p \frac{w_i}{2}d^2(Q(\cdot),C_i)$, $g= \iota_{C_0}$ and $h=0$.
We work with the particular instance of~\eqref{eq:GQHP} in which $p=1$, $w_1= 1$ and $C_1$ is given as the union of $5$ hypercubes which were generated in the same way as in the previous experiment. The entries of $Q$ were randomly generated in the interval $]-1,1[$. As in the previous experiment, up to the authors' knowledge, in this setting BDSA does not recover any method already proposed in the literature. In Figure~\ref{fig:SplitVSBoosted_NC} we show the results of running both DSA and BDSA for $5$ different dimensions and $10$ different randomly generated problems for each dimension. In this case, BDSA reached a better value of the objective function than DSA in $49$ out of $50$ instances. Regarding the comparison in iterations and time, BDSA was again significantly faster: on average, DSA needed around $5$ times more iterations and $2.5$ more running time than BDSA to satisfy the stopping criterion~\eqref{eq:stop}.
\begin{figure}
\caption{Let $n\in{\{50, 100,200,300,500\}
\label{fig:SplitVSBoosted_NC}
\end{figure}
\section{Conclusions and Future Work}\label{sec:conc}
In this paper, we developed a new algorithm for structured nonconvex optimization problems, which we named as \emph{Boosted Double-proximal Subgradient Algorithm} (BDSA). One of the main features of our method is the inclusion of a linesearch at the end of every iteration. If the stepsizes in every iteration of the linesearch are set to be $0$, then algorithms such as the \emph{Proximal Difference of Convex functions Algorithm} (PDCA)~\cite{sun2003proximal}, the \emph{Generalized Proximal Point Algorithm} (GPPA)~\cite{an2017convergence} and the \emph{Double-proximal Gradient Algorithm} (DGA)~\cite{bot2019doubleprox} can be recovered as particular cases of BDSA (see Remark~\ref{remark:particularcases}). Nevertheless, BDSA can also be applied to far more general problems.
The convergence of the sequence generated by BDSA is guaranteed under the usual assumptions required for this class of nonconvex problems. In addition, when the Kurdyka--{\L}ojasiewicz property holds, global convergence and convergence rates can be derived.
We demonstrated the advantages of the additional linesearch included in BDSA with multiple experiments. By recurring to a new family of test functions, we showed that BDSA remarkably improves ``non-boosted'' methods in avoiding non-optimal critical points and achieving convergence to minima. Indeed, in our test problems BDSA had a $100\%$ rate of success, while the PDCA only reached the global minimum in $1.17\%$ of the instances. Further, the boosting step significantly reduced the running time and the number of iterations employed by the algorithm. For example, the GPPA needed approximately $2.5$ more iterations and time for reaching the same accuracy than BDSA for an application of the minimum sum-of-squares clustering problem. For different generalizations of the Heron problem, BDSA also managed to be much faster than its non-accelerated version, both in time and number of iterations.
To conclude, we point out two possible directions for future research.
\paragraph{Considering a nonmonotone linesearch} The authors in~\cite{ferreira2021boosted} have recently proposed a nonmonotone modification of the BDCA~\cite{aragon2018accelerating,AragonArtacho2022}. Their scheme is suitable for handling DC problems where both functions are nondifferentiable by allowing the linesearch to accept steps leading to some controlled growth in the objective function. It would be interesting to study whether a similar linesearch can be considered in Step $6$ of Algorithm~\ref{alg:1}.
\paragraph{Incorporating second-order information} The importance of including second-order information in algorithms for improved performance is widely well-recognized. For some applications, it can be crucial to extend our results by incorporating Hessian information into the data. This becomes particularly significant when dealing with data that is not twice continuously differentiable. Recent studies propose the integration of generalized Hessians (see, e.g., \cite{aragonartacho2023coderivativebased} and the references therein) to enable linesearches in Newton-like methods.
\end{document} |
\begin{document}
\title{Self-testing of binary Pauli measurements requiring neither entanglement nor any dimensional restriction}
\author{Ananda G. Maity}
\email{anandamaity289@gmail.com}
\affiliation{S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 106, India}
\author{Shiladitya Mal}
\email{shiladityamal@hri.res.in}
\affiliation{Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India}
\affiliation{Department of Physics and Center for Quantum Frontiers of Research and Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan}
\author{Chellasamy Jebarathinam}
\email{jebarathinam@cft.edu.pl}
\affiliation{Department of Physics and Center for Quantum Frontiers of Research and Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan}
\affiliation{Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotnik\'ow 32/46, 02-668 Warsaw, Poland}
\author{A. S. Majumdar}
\email{archan@bose.res.in}
\affiliation{S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 106, India}
\begin{abstract}
Characterization of quantum devices received from unknown providers is a significant primary task for any quantum information processing protocol. Self-testing protocols are designed for this purpose of certifying quantum components from the observed statistics under a set of minimal assumptions. Here we propose a self-testing protocol for certifying binary Pauli measurements employing the violation of a Leggett-Garg inequality. The scenario based on temporal correlations does not require entanglement, a costly and fragile resource. Moreover, unlike previously proposed self-testing protocols in the prepare and measure scenario, our approach requires neither dimensional restrictions, nor other stringent assumptions on the type of measurements. We further analyse the robustness of this hitherto unexplored domain of self-testing of measurements.
\end{abstract}
\maketitle
\section{Introduction}
It is hard to overemphasise the role of emerging quantum technology in recent times. Various real life applications, such as quantum key distribution \mathscrite{qkd}, quantum sensing \mathscrite{qsensing}, quantum metrology \mathscrite{metrology}, quantum internet \mathscrite{internet}, and machine learning \mathscrite{mlearning} have been investigated with great prospects. For such prospects to materialize in practice, it becomes utmost important to design experiments which can test whether the quantum components of the required devices function properly. To this end, various certification or verification protocols have been employed, such as those based on tomography and benchmarking \mathscrite{bench}, as well as self-testing protocols based essentially on observed statistics \mathscrite{my'98, my'04}.
Among several certification protocols generic quantum tomography is the most powerful, but at the same time it is the most resource consuming. Randomized benchmarking refers to a collection of methods that aim at reliably estimating the figure of merit of the overlap between the physical quantum process and it's ideal counterpart. For both randomized benchmarking or quantum tomography, trusted mesurements are required in order to certify quantum devices. On the other hand, self-testing of quantum components based on statistics collected from lesser number of measurements and minimal physical assumptions, require lesser trust on the measurement devices and are experimentally more efficient and less resource consuming\mathscrite{bench}.
Self-testing based on the Bell test was proposed initially in the context of quantum cryptography \mathscrite{my'98, my'04}, where it was shown that maximum violation of the Bell-CHSH inequality \mathscrite{Bell'64,chsh'69} implies that the underlying quantum state is maximally entangled and the local measurements performed by two distant parties are anticommuting \mathscrite{popescu'92}. In such self-testing protocols one can uniquely identify quantum states and measurements (upto some local isometries) by observing extreme correlations in measurement statistics. The underlying assumptions are the same as those required for implementation of loophole free Bell tests \mathscrite{hensen'15,knill'15,zeilinger'15,weinfurter'17}. Since then all pure bipartite entangled states and certain multipartite quantum states such as graph states have been self-tested \mathscrite{Col,mayers2003} in this scenario. Self-testing of pure entangled bipartite states has also been investigated \mathscrite{Supic,goswami,Goh,SBK20} employing Einstein-Podolski-Rosen steering \mathscrite{Wiseman}. Self-testing using steering is weaker in the sense that it requires more assumptions compared to that based on Bell tests, but the former is easier to verify experimentally \mathscrite{peng}.
In this regard, it is worthwhile to mention that self-testing based on maximal violation of some nonclassicality revealing criterion is very fragile as idealised situations in the laboratory are rare to occur. Therefore, a self-testing statement is useful for practical purposes if it has reasonable robustness \mathscrite{Scarani2,Yang,Bamps, Kani2}. For instance, there are robust self-testing protocols for which given a certain level of violation of a Bell inequality (but not necessarily maximal), nontrivial lower bound on the fidelity between the initially unknown state and a given target state has been shown. Robust self-testing of higher dimensional state has also been studied \mathscrite{Scarani3,supic2018,Zhang'18}. Apart from various special class of quantum states, different types of measurements including maximally incompatible observables \mathscrite{popescu'92}, entangling Bell-measurement \mathscrite{bancal'18,renou'18}, binary observables \mathscrite{kani'17}, more than two dichotomic observables \mathscrite{Mck,Bowles'18}, and non-maximally anticommuting observables \mathscrite{Pironio} have also been self-tested (See Ref.\mathscrite{supic'19} for a review).
Apart from the above schemes of self-testing of measurements based on the use of quantum correlations in entangled states, some certification methods for self-testing have been proposed that do not require entanglement \mathscrite{chen'11,brunner'15,bourennane'16,Tavakoli}. A pair of anticommuting Pauli observables have been self-tested in the prepare and measure scenario \mathscrite{tavakoli'18}, employing the quantum advantage of $2\mapsto 1$ random access code (RAC) \mathscrite{ambainis'99, tavakoli'15}. Such approaches in the context of self-testing of local states and measurements are more resource efficient and experimentally feasible compared to schemes based on nonlocal quantum correlations since they do not require entanglement, an expensive resource. However, the assumption of an upper bound on the system-dimension has been crucial for such approaches, and hence, the verifier needs to trust the measurement devices more.
Another self-testing protocol has been constructed for three-dimensional states and measurements \mathscrite{kishor'19} based on contextuality of quantum theory \mathscrite{ks'67}, which requires certain restrictive assumptions such as the compatibility between subsequent projective measurements \mathscrite{supic'19}. Recently, an extension of the scheme has been proposed based on the sum of sqaures decomposition of a family of non-contextual inequalities \mathscrite{kcbs}, relaxing some of the above restrictions \mathscrite{saha'20}. Self-testing of arbitrary high-dimensional local quantum systems using contextuality has also been proposed where the assumption of projective measurement is necessary \mathscrite{kishor'19(1)}. The above protocols pertain to local measurements in three and above dimensions. On the other hand, among all measurements in quantum information processing, the measurements with dichotomic settings and outcomes are the most widely used ones. Hence, the design of resource efficient self-testing protocols for such meaurements with minimal assumptions is necessary for practical purposes.
In the present work, we develop a scheme of self-testing binary Pauli measurements in a hitherto unexplored scenario, {\it i.e.}, employing nonclassicality of temporal correlations exhibited via violation of the Leggett-Garg inequality(LGI) \mathscrite{LG'85,emary'14} and predictability\mathscrite{kofler'13,rand'16}, which does not require any entanglement. LGI has been employed to probe macroscopic coherence and to study quantum to classical transition \mathscrite{LG'85,emary'14,LG'02, kofler'07,kofler'13,brukner'04,mal'16, mal'18,mal'19}. Various aspects of temporal correlations have been investigated \mathscrite{fritz'10,budroni'13,rand'16,bm'14,brierley'19,usha'13,alok'17,das'18,ku'18,titas'18} and experimentally realized \mathscrite{knee'12,robens'15,knee'16,ku'19,shayan'19,spee'20}. Here, we employ the violation of LGI, and invoke a minimal set of assumptions which can be easily met in a real experiment, in order to provide a self-testing statement for binary Pauli measurements without any need to restrict the dimension of the Hilbert space accessed by the quantum devices. We further perform the robustness analysis of our self-testing protocol by deriving a lower bound on the fidelity of the measured observables with that of ideal ones.
\section{Description of the scenario}
We start with a brief description of the Leggett-Garg (LG) test (Fig.-\ref{Seq_LGI}) which enables self-testing of Pauli measurements whenever extremal correlations of LGI are observed. In a LG test, a single system is measured sequentially at different instants of time in order to obtain temporal correlations. Two sequential binary measurements are performed (by Alice and Bob, respectively) on an identical initial state prepared by the experimenter in every run of the experiment. In contrast to the sequential measurement scenario of self-testing using contextuality \mathscrite{kishor'19}, here subsequent measurements are not required to commute with each other.
Alice and Bob have two choices of binary measurements, say, $\{A_1,A_2\}$ and $\{B_1,B_2\}$ to perform in each run. The probability of obtaining outcome $a_i$ and $b_j$ is denoted by $P(a_i,b_j \mid A_i,B_j)$, when Alice measures $A_i$ at time $t_m$ and Bob measures $B_j$ at some later instant $t_{m+1}$, with $i,j \in\{1,2\}$ and $a_i, b_j\in\{0,1\}$. Let us denote, $\mathcal{P}_{a_i|A_i}$, $\mathcal{P}_{b_j|B_j}$ as projectors so that $\sum_{a_i}\mathcal{P}_{a_i|A_i}=\mathbb{I}, \sum_{b_j}\mathcal{P}_{b_j|B_j}=\mathbb{I}$. The two-time joint probability can be obtained using Bayes' rule as,
\begin{align}\label{pij}
&P(a_i,b_j \mid A_i,B_j) = P(a_i \mid A_i) P(b_j\mid a_i, A_i,B_j ) \nonumberonumber \\
&= \text{Tr}\left[ \mathcal{P}_{a_i|A_i} \rho_{in}\right] \text{Tr}\left[ \mathcal{P}_{b_j|B_j} \frac{\mathcal{P}_{a_i|A_i}\rho_{in}\mathcal{P}_{a_i|A_i}^{\dagger}}{\text{Tr}\left[ \mathcal{P}_{a_i|A_i}\rho_{in}\mathcal{P}_{a_i|A_i}^{\dagger}\right]}\right] .
\end{align}
The two-time correlation is defined as,
\begin{equation}\label{cij}
\mathcal{C}_{ij} =\sum_{a_i,b_j} (-1)^{a_i\oplus b_j} P(a_i,b_j \mid A_i,B_j),
\end{equation}
where $\oplus$ denotes addition modulo $2$.
The four-term LGI in terms of the above correlators is given by,
\begin{equation}\label{k4}
\mathcal{K}_4 = C_{11} + C_{21} + C_{22} - C_{12} \leq 2.
\end{equation}
\begin{figure}\label{Seq_LGI}
\end{figure}
Any model compatible with classical theory predicts that the maximal value of the above expression is $2$, whereas the quantum theory can violate this inequality up to $2\sqrt{2}$. Suppose, in a single run, Alice performs a projective qubit measurement, $A_1\equiv \hat{a_1}.\vec{\sigma}$ (to make the preliminary discussion simple, but it can be any general measurement too) on an arbitrary input state, $\rho_{in}$ at time $t_1$, and Bob performs a qubit measurement, $B_1\equiv \hat{b_1}.\vec{\sigma}$ at some later time $t_2$, where $\hat{a_1}$ and $\hat{b_1}$ are the Bloch vectors denoting Alice's and Bob's measurement directions respectively and $\vec{\sigma}$ is the vector of Pauli matrices. Then the maximum quantum violation, $2\sqrt{2}$, can be achieved by the following measurement settings,
\begin{eqnarray}\label{mea}
A_1^{\text{ideal}}= \sigma_z, \nonumberonumber \\
A_2^{\text{ideal}}= \sigma_x, \nonumberonumber \\
B_1^{\text{ideal}} = \frac{\sigma_x + \sigma_z}{\sqrt{2}}, \nonumberonumber \\
B_2^{\text{ideal}} = \frac{\sigma_x - \sigma_z}{\sqrt{2}},
\end{eqnarray}
upto some local unitary. In the context of self-testing, all the four measurements are {\it a priori} unknown to Alice and Bob. Our goal is to certify above measurements from the observed statistics, under some suitable and minimal assumptions.
\section{Derivation of Leggett-Garg inequality}
LGI has been earlier derived using the assumptions of non-invasive measurement and realism \mathscrite{brukner'04,mal'16}. However, those assumptions are at ontological level, and hence, cannot be verified individually. Rather a conjunction of them can be verified in an experiment. On the other hand, in \mathscrite{rand'16}, LGI was derived from some operational assumptions called predictability and no signalling in time (NSIT) \mathscrite{kofler'13}. Subsequently, the NSIT condition has been experimentally observed on a variety of input states \mathscrite{expnsit}. Here we present the essential features of the LGI derivation \mathscrite{rand'16}. Let us first state the two assumptions in precise mathematical form.
\textbf{Predictability :} A model is said to be predictable if the joint statistics $P(a_i,b_j \mid A_i,B_j)\in\{0,1\}~ \forall a_i,b_j,A_i,B_j$ \mathscrite{wiseman'12}.
\\
\textbf{NSIT :} NSIT is defined by the condition that measurement statistics is not influenced by the earlier measurements. Mathematically, $P(b_j\mid B_j)=P(b_j\mid A_i, B_j) ~\forall A_i, B_j,b_j$.
\\
In order to derive LGI from the above two assumptions, our aim is to show that
\begin{eqnarray}
NSIT \wedge Predictability \Rightarrow LGI.
\end{eqnarray}
Suppose $\lambda$ denotes some underlying variable at the ontological level, averaging over which we obtain joint probabilities observed in an experiment, {\it i.e.},
\begin{eqnarray}
P(a_i,b_j \mid A_i,B_j)=\int_{\lambda} d\lambda p(\lambda) P(a_i,b_j \mid A_i,B_j,\lambda). \nonumberonumber
\end{eqnarray}
LGI follows in a straightforward way when the joint probability at the ontological level gets factorised, or in other words, we have to show that predictability together with NSIT leads to
\begin{eqnarray}\label{facto}
P(a_i,b_j \mid A_i,B_j,\lambda)=P(a_i\mid A_i,\lambda) P(b_j \mid B_j,\lambda).
\end{eqnarray}
As further conditioning does not change the deterministic probability distribution, we have from the predictability, $P(a_i,b_j \mid A_i,B_j,\lambda)=P(a_i,b_j \mid A_i,B_j)$. Using Bayes' rule one has
\begin{equation}
P(a_i,b_j \mid A_i,B_j)=P(a_i\mid A_i, B_j, b_j) P(b_j\mid A_i, B_j). \nonumberonumber
\end{equation}
The condition of NSIT implies $P(b_j\mid A_i, B_j)=P(b_j\mid B_j)$. Also, as
physically reasonable and broadly accepted, a later measurement cannot influence the past measurement result, and hence $P(a_i\mid A_i, B_j, b_j)=P(a_i\mid A_i)$. With the above implications, one can construct a theory at the ontological level where $P(a_i\mid A_i,\lambda)=P(a_i\mid A_i)$, $P(b_j\mid B_j,\lambda)=P(b_j\mid B_j)$, which lead to Eq. \eqref{facto}. Now, a straightforward calculation leads us to derive the four-term LGI \eqref{k4} which is bounded by $2$.
Despite some structural similarities, the LG-test is essentially different from the Bell-test \mathscrite{clemente}, and also has some loopholes which are different from that of the Bell-test \mathscrite{wilde}. The approach adopted in this paper for the derivation of LGI is based on the conjunction of {\it Predictability} and {\it NSIT}. In the present scenario it is ensured that NSIT condition is satisfied, as we see in the next section. This means that if LGI is violated, then predictability must have to be violated. Thus, the violation of predictability naturally guarantees non-classicality which is required for the purpose of self-testing.
\section{Self-testing of measurements using Leggett-Garg inequality}
The above derivation of LGI with the assumptions of predictability and NSIT helps us to devise a self-testing protocol for binary Pauli measurements \eqref{mea} whenever extremal non-classical temporal correlations are observed. We formulate the self-testing protocol in such a way that the NSIT condition is satisfied so that maximal violation of the LGI will imply the violation of the predictability condition, and hence, no classical strategy is able to reproduce the statistics. We now make a minimal assumption which is again very natural in the sequential measurement scenario, since otherwise, a classical model may simulate the quantum violation of LGI.
\textbf{Assumption :}~\emph{The measurement device of Alice acts only on the input state prepared by the experimenter, and the measurement device of Bob acts only on the state produced by Alice's measurement, with both returning only the respective post-measurement states.}
A similar assumption was also considered in Ref.\mathscrite{saha'20} for the purpose of self-testing of local three dimensional measurements using the sum-of-squares decomposition of a family of non-contextual inequalities \mathscrite{kcbs}. However, in our approach using LGI we are interested here in the self-testing of binary qubit measurements. Employing the maximum violation of LGI along with NSIT condition we can self-test the binary qubit measurements given by Eq. \eqref{mea} upto some local unitaries. Our self-testing protocol does not depend on the input state. However, for the sake of a formal proof of self-testing, we first consider a general qubit state (in Lemma 1). Then, making use of this proof, we extend our result for states in any dimension (Lemma 2). Our formal proof of self-testing is thus accomplished in the following two steps (lemma 1 and 2), and finally we present the proof of isometry in Theorem 1.
\textbf{Lemma 1.}~ \emph{The maximum violation of LGI (i.e, $\mathcal{K}_4^{\text{max}} = 2\sqrt{2}$) implies implementation of the qubit measurement observables given by equation \eqref{mea} upto some local unitaries, satisfying the NSIT condition. }
\begin{proof} Maximum quantum violation of LGI can only be achieved if the measurements are taken to be projective . We maximize the LGI considering the most general positive operator valued measure (POVM) with two outcomes. Maximizing the four-time LGI expression numerically over all the POVM parameters, it is found that the maximum value, $2\sqrt{2}$ can only be achieved if the measurements are taken to be projective (the details of the proof are given in Appendix-\ref{Appa}). Hence, without any loss of generality, we restrict ourselves to projective measurements only.
\\
To obtain the two-time correlation $\mathcal{C}_{ij}$ we have to calculate joint probabilities $P(a_i,b_j \mid A_i, B_j)$ given in Eq.\eqref{pij}. Considering the general qubit input state, $\rho_{in}=\frac{\mathbb{I}+\hat{n}.\vec{\sigma}}{2}$, the first term of Eq.\eqref{pij} can be simplified to $\text{Tr}\left[ \mathcal{P}_{a_i|A_i} \rho_{in}\right] = \frac{1}{2}(1+(-1)^{a_i}\hat{a_i}.\hat{n})$. The post-measurement state after obtaining outcome $a_i$ is given by, $\frac{\mathcal{P}_{a_i|A_i}\rho_{in}\mathcal{P}_{a_i|A_i}^{\dagger}}{\text{Tr}\left[\mathcal{P}_{a_i|A_i}\rho_{in}\mathcal{P}_{a_i|A_i}^{\dagger} \right] } = \frac{1}{2}\left[ \mathbb{I}+ \left( -1\right) ^{a_i}\hat{a_i}.\vec{\sigma}\right]$.
Now, the term, $\text{Tr}\left[\mathcal{P}_{b_j|B_j}\frac{\mathcal{P}_{a_i|A_i}\rho_{in}\mathcal{P}_{a_i|A_i}^{\dagger}}{\text{Tr}\left[ \mathcal{P}_{a_i|A_i}\rho_{in}\mathcal{P}_{a_i|A_i}^{\dagger}\right]}\right]$ can be simplified to $\frac{1}{2}\left[ 1+ (-1)^{a_i+b_j}\hat{a_i}.\hat{b_j}\right]$.
So, the joint probability distribution of getting outcomes $a_i$ and $b_j$ when measurements $A_i$ and $B_j$ are performed respectively, is given by,
\begin{equation}\label{jp}
P(a_i,b_j \mid A_i, B_j)= \frac{1}{4}(1+(-1)^{a_i}\hat{a_i}.\hat{n})( 1+ (-1)^{a_i+b_j}\hat{a_i}.\hat{b_j}).
\end{equation}
After simplification, we get, $\mathcal{C}_{ij}= \hat{a_i} . \hat{b_j}.$ Then, the four-time LGI in terms of four correlators will be,
\begin{eqnarray}
\mathcal{K}_4 &= \hat{a_1} . \hat{b_1}+\hat{a_2} . \hat{b_1}+\hat{a_2} . \hat{b_2} -\hat{a_1} . \hat{b_2}\leq 2.
\end{eqnarray}
It follows that the maximum value of $\mathcal{K}_{4}$ is $2\sqrt{2}$ when all $\vert \hat{a_i}.\hat{b_j}\vert=\frac{1}{\sqrt{2}}$ \mathscrite{fritz'10}, which can be obtained for the settings described in Eq. (4) or their local unitary rotation.
\\
Here, one can verify that the NSIT condition is also satisfied in case of maximum LGI violation when $\vert \hat{a_i}.\hat{b_j}\vert=\frac{1}{\sqrt{2}}$. The NSIT condition on the probabilities, $P(b_j\mid A_i, B_j)=P(b_j\mid B_j)$ $\forall b_j,A_i,B_j$, implies $(-1)^{a_1+b_1}\hat{a_1}.\hat{b_1} = (-1)^{a_2+b_1}\hat{a_2}.\hat{b_1}=(-1)^{a_1+b_2}\hat{a_1}.\hat{b_2}=(-1)^{a_2+b_2}\hat{a_2}.\hat{b_2}$. Thus clearly, when maximum violation of $\mathcal{K}_{4}$ is observed along with the NSIT condition, it is obvious that the predictability condition does not hold anymore.
\end{proof}
We now show how maximal violation of LGI can also be used to self-test the Pauli measurements even if the measurement operators act on a higher dimensional Hilbert space. Let the initial state be $\rho_{in}$ in an arbitrary dimensional Hilbert space, and the measurements $A_i, B_j$ both act on this space.
\textbf{Lemma 2.}~ \emph{The maximum violation of LGI (i.e, $\mathcal{K}_4^{\text{max}} = 2\sqrt{2}$) implies implementation of the block diagonal measurement, i.e., $A_1 = \oplus_i \sigma_z^i, A_2= \oplus_i \sigma_x^i , B_1 = \oplus_j (\sigma_x^j+ \sigma_z^j)/\sqrt{2}, B_2 = \oplus_j (\sigma_x^j- \sigma_z^j)/\sqrt{2}$.}
\begin{proof}
Suppose $\left\lbrace \mathcal{P}_{a_i|A_i}\right\rbrace $ and $\left\lbrace \mathcal{P}_{b_j|B_j}\right\rbrace $ are two dichotomic measurements which act on an arbitrary dimensional Hilbert space. Then, according to Jordan's lemma \mathscrite{Kani2}, $ \mathcal{P}_{a_i|A_i}= \oplus _m \mathcal{P}_{a_i|A_i}^{m} $ and $ \mathcal{P}_{b_j|B_j}= \oplus _n \mathcal{P}_{b_j|B_j}^{n} $, where $\mathcal{P}_{a_i|A_i}^{m} $ and $\mathcal{P}_{b_j|B_j}^{n}$ are projectors on $\mathcal{H}_d$ with $d\leq 2$ (here, in view of our proof of lemma 1, we again stick to projective measurements without any loss of generality). Therefore, one has
\begin{align}
&P(a_i,b_j \mid A_i,B_j) = \nonumberonumber\\
&\sum_{m,n}p_m \text{Tr}\left[ \mathcal{P}_{a_i|A_i}^{m} \rho_{m}\right] \text{Tr}\left[ \mathcal{P}_{b_j|B_j}^{n} \frac{\mathcal{P}_{a_i|A_i}^{m}\rho_{m}\mathcal{P}_{a_i|A_i}^{m \dagger}}{\text{Tr}\left[ \mathcal{P}_{a_i|A_i}^{m}\rho_{m}\mathcal{P}_{a_i|A_i}^{m \dagger}\right]}\right] \nonumberonumber
\end{align}
where $p_m = \text{Tr}(\rho_{in}\Pi_m)$, with $\Pi_m =\sum_{a_i}\mathcal{P}^m_{a_i|A_i}$, and $\rho_m = (\Pi_m \rho_{in}\Pi_m)/p_m$, which is at most a qubit state. with $\rho=\oplus_m p_m \rho_m$, where $\rho_m = \frac{\mathbb{I}+\hat{r}.\vec{\sigma}}{2}$ acts on $\mathcal{H}_d$ with $d\leq 2$. Now, the operators of the above equation acting on $\mathcal{H}_d$ with $d=2$ implies that
\begin{eqnarray}
&&P(a_i,b_j \mid A_i, B_j) = \nonumberonumber \\
&& \sum_{m}p_m \frac{1}{4}(1+(-1)^{a^m_i}\hat{a^m_i}.\hat{r})( 1+ (-1)^{a_i+b_j}\hat{a^m_i} \mathscrdot \hat{b^n_j}).\nonumberonumber
\end{eqnarray}
From the above equation it follows that,
\begin{eqnarray}
C_{ij}= \sum_{m,n}p_m \hat{a^m_i} \mathscrdot \hat{b^n_j}
\end{eqnarray}
The above expectation implies $C_{11} + C_{21} + C_{22} - C_{12} = 2\sqrt{2}$ if and only if $\hat{a_1}^m = \hat{z}, \hat{a_2}^m = \hat{x},\hat{b_1}^n = (\hat{z} + \hat{x})/\sqrt{2},\hat{b_2}^n = (\hat{z} -\hat{x})/\sqrt{2}$ and $\sum_{m}p_m = 1$. This is achieved if and only if $A_1 = \oplus_m \sigma_z^m, A_2= \oplus_m \sigma_x^m , B_1 = \oplus_n (\sigma_x^n+ \sigma_z^n)/\sqrt{2}, B_2 = \oplus_n (\sigma_x^n - \sigma_z^n)/\sqrt{2}$ and $\rho_{in} = \oplus_m \rho_m$.
\end{proof}
The above two lemmas enables us to present the following theorem.
\textbf{Theorem 1.}~ \emph{If $\mathcal{K}_4^{\text{max}} = 2\sqrt{2}$ is observed in LG-test under Assumption 1, with the measurements of Alice, $A_i$ acting on $H_d $, producing the post measurement states $\left\lbrace \frac{\mathcal{P}_{a_i|A_i}\rho_{in}\mathcal{P}_{a_i|A_i}^{\dagger}}{\text{Tr}\left[ \mathcal{P}_{a_i|A_i}\rho_{in}\mathcal{P}_{a_i|A_i}^{\dagger}\right]} \right\rbrace_a$, and the measurements of Bob $B_j$ acting on these post measurement states, then there exists an isometry $\Phi : \mathcal{H}^d \rightarrow \mathcal{C}^2 \otimes \mathcal{H}^d $ such that
\begin{eqnarray}
&&\Phi\left( B_j \frac{\mathcal{P}_{a_i|A_i}\rho_{in}\mathcal{P}_{a_i|A_i}^{\dagger}}{\text{Tr}\left[ \mathcal{P}_{a_i|A_i}\rho_{in}\mathcal{P}_{a_i|A_i}^{\dagger}\right]} \right) \Phi^{\dagger} \nonumberonumber \\
&=&B^{\texttt{ideal}}_j \left|\psi^{\texttt{ideal}}_{a|A_i}\right \rangle \left \langle\psi^{\texttt{ideal}}_{a|A_i} \right |\otimes \left| \texttt{junk} \right \rangle \left \langle \texttt{junk}\right |\nonumberonumber
\end{eqnarray}
where $\left|\psi^{\texttt{ideal}}_{a|A_i}\right \rangle$ are the eigenstates of Alice's ideal measurements and $B^{\texttt{ideal}}_j$ are Bob's ideal measurements given by Eq. (4) respectively, and $\left| \texttt{junk} \right \rangle $ is a junk state acting on $\mathcal{H}^d$.}
\begin{proof}
The details proof are given in the Appendix-\ref{Appb}. Choosing the eigenbasis of $A_1$ as the computational basis, from lemma $1$ and $2$, it follows that the post measurement states of Alice can be written as
$\frac{\mathcal{P}_{a_i|A_i}\rho_{in}\mathcal{P}_{a_i|A_i}^{\dagger}}{\text{Tr}\left[ \mathcal{P}_{a_i|A_i}\rho_{in}\mathcal{P}_{a_i|A_i}^{\dagger}\right]} = \oplus_m p_m \left|\psi^m_{a|A_i}\right \rangle \left \langle \psi^m_{a|A_1}\right |$, with $\left|\psi^m_{0|A_1}\right \rangle= \left| 2m \right \rangle $
and $\left|\psi^m_{1|A_1}\right \rangle= \left| 2m +1 \right \rangle $. We append an ancilla qubit prepared in the state $\left|0 \right \rangle$
and look for an isometry $\Phi$ such that
\begin{eqnarray}
&&\Phi\left( B_j \frac{\mathcal{P}_{a_i|A_i}\rho_{in}\mathcal{P}_{a_i|A_i}^{\dagger}}{\text{Tr}\left[ \mathcal{P}_{a_i|A_i}\rho_{in}\mathcal{P}_{a_i|A_i}^{\dagger}\right]} \otimes \left|0 \right \rangle \left \langle 0 \right | \right) \Phi^{\dagger} \nonumberonumber \\
&=&B^{\texttt{ideal}}_j \left|\psi^{\texttt{ideal}}_{a|A_i}\right \rangle \left \langle\psi^{\texttt{ideal}}_{a|A_i} \right |\otimes \left| \texttt{junk} \right \rangle \left \langle \texttt{junk}\right |\nonumberonumber
\end{eqnarray}
This can be achieved for $\Phi$ defined by the map
\begin{align}
\Phi \left|2m,0 \right \rangle &\rightarrow \left| 2m, 0 \right \rangle \nonumberonumber \\
\Phi \left|2m+1,0 \right \rangle &\rightarrow \left| 2m, 1 \right \rangle \nonumberonumber
\end{align}
\end{proof}
\section{Robust self-testing of measurements}
Robustness of self-testing of measurements is quantified as how the actual observables which are to be self-tested differ from the ideal ones. Hence, characterization of the fidelities between the real measurements and the ideal measurements is warranted. Let us first perform the robustness analysis of the measurements on Alice's side i.e, $\left\lbrace \mathcal{P}_{a_i|A_i}\right\rbrace$. Given an arbitrary set of measurements $\left\lbrace \mathcal{P}_{a_i|A_i}\right\rbrace$, the average fidelity with the ideal measurements are,
\begin{equation}
S(\left\lbrace \mathcal{P}_{a_i|A_i}\right\rbrace) = \text{max}_\Lambda \sum_{i,a_i} F(\mathcal{P}_{a_i|A_i}^{\text{ideal}}, \Lambda[\mathcal{P}_{a_i|A_i}])/4.
\end{equation}
Here $\Lambda$ is a quantum channel and the fidelities are defined as usual, $F(\rho,\sigma)=\text{Tr}(\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}})$. The fidelities between the real and the ideal measurements, $F(\mathcal{P}_{a_i|A_i}^{\text{ideal}}, \Lambda[\mathcal{P}_{a_i|A_i}])$ simplify to $\text{Tr}(\Lambda \left[ \mathcal{P}_{a_i|A_i} \right] \mathcal{P}_{a_i|A_i}^{\text{ideal}})$. The lower bound on the smallest possible value of the average fidelity $S$, given a particular violation of the four-time LGI $\mathcal{K}_4$, can be found by minimizing over all sets of four measurements of $\mathcal{P}_{a_i|A_i}$,
\begin{equation}
\mathcal{F}(\mathcal{K}_4)= \text{min}_{\mathcal{P}_{a_i|A_i}} S(\lbrace \mathcal{P}_{a_i|A_i}\rbrace).
\end{equation}
In order to a lower bound for $\mathcal{F}$ as a function of LGI-violation $\mathcal{K}_4$, we use the operator inequality approach given in Refs. \mathscrite{Kani2,tavakoli'18}. Rewriting $\mathcal{K}_4$ in terms of an operator $W_{ia_i}$ and $\mathcal{P}_{a_i|A_i}$, where $W_{ia_i}= \frac{1}{2} [(-1)^{a_i} (\mathcal{P}_{B_1} + (-1)^{i-1} \mathcal{P}_{B_2})]$, $\mathcal{P}_{B_i}\equiv \mathcal{P}_{0|B_i}-\mathcal{P}_{1|B_i}$, we have,
\begin{equation}
\mathcal{K}_4 = \sum_{i,a_i} \text{Tr}[W_{ia_i}\mathcal{P}_{a_i|A_i}].
\end{equation}
Let us now define another operator $K_{ia_i}(\mathcal{P}_{B_1}, \mathcal{P}_{B_2}) \equiv \Lambda^{\dagger}(\mathcal{P}_{B_1}, \mathcal{P}_{B_2})[\mathcal{P}_{a_i|A_i}^{\text{ideal}}] $.
Considering an operator inequality of the form,
\begin{equation}
K_{ia_i}(\mathcal{P}_{B_1}, \mathcal{P}_{B_2}) \geq s W_{ia_i} + \mu_{ia_i} (\mathcal{P}_{B_1}, \mathcal{P}_{B_2}) \mathbb{I}
\end{equation}
with $s$ and $\mu_{ia_i}(\mathcal{P}_{B_1}, \mathcal{P}_{B_2})$ being real coeficients, the average fidelity $S$ (and hence $\mathcal{F}$) can be lower bounded as,
\begin{eqnarray}
\mathcal{F} \geq S &\geq & \frac{1}{4}\sum_{i,a_i} F(\mathcal{P}_{a_i|A_i}^{\text{ideal}}, \Lambda \left[ \mathcal{P}_{a_i|A_i}\right] \nonumberonumber \\
&=& \frac{1}{4}\sum_{i,a_i} Tr[K_{ia_i}\mathcal{P}_{a_i|A_i}] \nonumberonumber \\
&\geq & \frac{s}{4} \sum_{i,a_i} Tr[W_{ia_i}\mathcal{P}_{a_i|A_i}] +\frac{1}{4} \sum_{i,a_i} \mu_{ia_i} \nonumberonumber \\
&=& \frac{s}{4}\mathcal{K}_4 + \mu
\end{eqnarray}
where $\mu \equiv 1/4 ~min_{\mathcal{P}_{B_1}, \mathcal{P}_{B_2}}\sum_{i,a_i} \mu_{ia_i} (\mathcal{P}_{B_1}, \mathcal{P}_{B_2})$.
Since we can use Jordan's lemma to write the observables of Alice in a block-diagonal form with block of size at most $2 \times 2$, it suffices for our purpose to focus on each block as in Ref. \mathscrite{Kani2} to find the constants $s$ and $\mu$ for lower bounding the fidelity. The goal of the robustness analysis is to obtain a self-testing bound as tight as possible. Following Ref. \mathscrite{tavakoli'18}, adoption of the dephasing channel $\Lambda$, as an extraction map suffices to achieve our aim of finding an optimal self-testing bound.
\begin{equation}
\Lambda_\theta (\rho) = \frac{1+ \xi(\theta)}{2}\rho + \frac{1- \xi(\theta)}{2} \Gamma (\theta) \rho \Gamma(\theta);
\end{equation}
where $\xi(\theta) \in [-1,1] $ and
\begin{align}
\Gamma(\theta)&=\left\{
\begin{array}{lr}
\sigma_z & \quad \text{for} \quad \theta \in \left[0, \frac{\pi}{4}\right] \\
\sigma_x & \quad \text{for} \quad \theta \in \left(\frac{\pi}{4}, \frac{\pi}{2}\right] \\
\end{array} \nonumberonumber
\right.
\end{align}
For the interval $0\leq\theta\leq \pi/4$
\begin{eqnarray}
K_{10} &=& \Lambda^\dagger[\mathcal{P}_{0|A_1}] \nonumberonumber \\
&=& \frac{1+ \xi(\theta)}{2}\left[ \frac{\mathbb{I}+\sigma_z}{2}\right] + \frac{1- \xi(\theta)}{2}\left[ \sigma_z \frac{\mathbb{I}+\sigma_z}{2} \sigma_z \right] \nonumberonumber \\
&=& \frac{\mathbb{I}+\sigma_z}{2} ,\nonumberonumber
\end{eqnarray}
and similarly
$K_{20} = \frac{\mathbb{I}+\xi(\theta)\sigma_x}{2}$ , $K_{21} = \frac{\mathbb{I}-\xi(\theta)\sigma_x}{2}$, $K_{11} = \frac{\mathbb{I}-\sigma_z}{2}$. \\
whereas, for the interval $\pi/4\leq\theta\leq \pi/2$,\\
~~~~~~~~$K_{10}=\frac{\mathbb{I}+\xi(\theta)\sigma_z}{2}$,~~~~~~~~~~~~ $K_{20}= \frac{\mathbb{I}+\sigma_x}{2}$, \\
~~~~~~~~$K_{11}=\frac{\mathbb{I}-\xi(\theta)\sigma_z}{2}$,~~~~~~~~~~~~ $K_{21}= \frac{\mathbb{I}-\sigma_x}{2}$. \\
We can, without loss of generality, represent the two measurements in the x-z plane as,
\begin{eqnarray}
\mathcal{P}_{B_1} &=& \mathscros(\theta) \sigma_z + \sin(\theta) \sigma_x \nonumberonumber \\
\mathcal{P}_{B_2} &=& \mathscros(\theta) \sigma_z - \sin(\theta) \sigma_x
\end{eqnarray}
and therefore,
$W_{10} = \mathscros(\theta) \sigma_z$, $W_{20} = \sin(\theta) \sigma_x$, and
$W_{21} = - \sin(\theta) \sigma_x$, $W_{11} = - \mathscros(\theta) \sigma_z$.
We analyze the operator inequality in the interval $0\leq\theta\leq \pi/4$ and $\pi/4\leq\theta\leq \pi/2$ separately. One can see that the effective number of inequalities may be reduced since there are symmetries in the expression of $K_{ia_i}$ and $W_{ia_i}$, and hence, without loss of generality we can choose $\mu_o\equiv \mu_{20}=\mu_{21}$ and $\mu_e\equiv \mu_{10}=\mu_{11}$.
After simplication we get,
\begin{eqnarray}
\mu_e &\leq & 1-s\mathscros(\theta), \nonumberonumber\\
\mu_e &\leq & s\mathscros(\theta).
\end{eqnarray}
and,
\begin{eqnarray}
\mu_o &\leq & \frac{1}{2} + \frac{1}{2} \xi(\theta)- s \sin(\theta) ,\nonumberonumber \\
\mu_o &\leq & \frac{1}{2} - \frac{1}{2} \xi(\theta)+ s \sin(\theta).
\end{eqnarray}
In order to obtain the strongest bound, we have to choose the largest value of $\mu_o$ and $\mu_e$ consistant with their respective constraints, {\it i.e.}, in the first interval,
\begin{align}
\mu_e &= \text{min}\left\lbrace 1-s\mathscros(\theta), s\mathscros(\theta)\right\rbrace , \nonumberonumber \\
\mu_o &= \text{min}\left\lbrace \frac{1}{2} + \frac{1}{2} \xi(\theta)- s \sin(\theta), \frac{1}{2} - \frac{1}{2} \xi(\theta)+ s \sin(\theta)\right\rbrace .
\end{align}
A similar procedure in the second interval leads to,
\begin{align}
\mu_o &= \text{min}\left\lbrace 1-s\sin(\theta), s\sin(\theta)\right\rbrace , \nonumberonumber \\
\mu_e &= \text{min}\left\lbrace \frac{1}{2} + \frac{1}{2} \xi(\theta)- s \mathscros(\theta), \frac{1}{2} - \frac{1}{2} \xi(\theta)+ s \mathscros(\theta)\right\rbrace .
\end{align}
The expressions in the two intervals are related to each other by the transfomations $\mu_o\leftrightarrow \mu_e$ and $\sin(\theta)\leftrightarrow \mathscros(\theta)$.
Hence, the lower bound in fidelity becomes,
\begin{equation}
\mathcal{F}(\mathcal{K}_4)\geq \frac{s}{4} \mathcal{K}_4 + \text{min}_{\mathcal{P}_{B_1}, \mathcal{P}_{B_2}} \mu(\mathcal{P}_{B_1}, \mathcal{P}_{B_2}).
\end{equation}
where $\mu(\mathcal{P}_{B_1}, \mathcal{P}_{B_2})=(\mu_e+\mu_o)/2$. To compute this value, we fix $s = \frac{1+ \sqrt{2}}{2}$, and choosing the dephasing function as $\xi(\theta) = \text{min}\lbrace 1, 2s \sin(\theta)\rbrace$ in the interval $\theta\in[0,\pi/4]$, and $\xi(\theta) = \text{min}\lbrace 1, 2s \mathscros(\theta)\rbrace$ in the interval $\theta\in (\pi/4,\pi/2]$. After simplification, we get, $\mu = \frac{2-\sqrt {2}}{4}$ which gives the lower bound,
\begin{equation}\label{robustness1}
\mathcal{F}(\mathcal{K}_4)\geq \frac{(1+ \sqrt{2})}{8}\mathcal{K}_4 + \frac{2-\sqrt {2}}{4}.
\end{equation}
This provides robust self-testing of Alice's measurements. Clearly, the maximal quantum violation of the LGI, {\it i.e.}, $\mathcal{K}_4=2\sqrt{2}$ implies $\mathcal{F}(\mathcal{K}_4)=1$, which suggests that the measurements must be the ideal ones.
For $\mathcal{K}_4=2$, $\mathcal{F}(\mathcal{K}_4)\ge 3/4$. This bound can be obtained by $A_1=A_2=B_1=B_2=\sigma_z$. Therefore, we see that our bound is optimal.
We now quantify the average fidelity of the measurements on Bob's side with respect to the ideal ones: $\mathcal{S}^{\prime}(\lbrace \mathcal{P}_{b_i|B_i} \rbrace) = \text{max}_{\Lambda} \sum_{i,b_i} F((\mathcal{P}_{b_i|B_i})^{\text{ideal}}\Lambda [\mathcal{P}_{b_i|B_i}])/4$, where $\Lambda$ must be a unital channel. Now, define,
\begin{equation}
\mathcal{F}^{\prime}(\mathcal{K}_4) = \text{min}_{\lbrace \mathcal{P}_{b_i|B_i} \rbrace} \mathcal{S}^{\prime}(\lbrace \mathcal{P}_{b_i|B_i} \rbrace).
\end{equation}
First, we rewrite $\mathcal{K}_4 = \sum_{i,b_i} Tr[\mathcal{P}_{b_i|B_i} Z_{ib_i}]$ where $Z_{ib_i}= \frac{1}{2} (-1)^{b_i}[\mathcal{P}_{A_1} + (-1)^{i-1}\mathcal{P}_{A_2}] $ and $\mathcal{P}_{A_i} \equiv \mathcal{P}_{0|A_i} - \mathcal{P}_{1|A_i}$.
Let us take an operator inequality of the form,
\begin{equation}
K_{ib_i}(\lbrace \mathcal{P}_{A_1}, \mathcal{P}_{A_2} \rbrace) \geq s Z_{ib_i} + \mu_{ib_i} (\lbrace \mathcal{P}_{A_1}, \mathcal{P}_{A_2}\rbrace) \mathbb{I},
\end{equation}
with $K_{ib_i} = \Lambda ^{\dagger}[\mathcal{P}_{b_i|B_i}^{\text{ideal}}]$. Similar to the previous case,
\begin{equation}
\mathcal{F}^{\prime}(\mathcal{K}_4) \geq \text{min}_{\mathcal{P}_{A_1}, \mathcal{P}_{A_2}}\frac{1}{4}\sum_{i,b_i} Tr[K_{ib_i} \mathcal{P}_{b_i|B_i} ].
\end{equation}
Using the same map and the same technique, we have $\xi(\theta)= \text{min} \lbrace 1, 2s \sin(\theta)\rbrace$ in interval $\theta\in[0,\pi/4]$ and $\xi(\theta) = \text{min}\lbrace 1, 2s \mathscros(\theta)\rbrace$ $\theta\in (\pi/4,\pi/2]$.
If we repeat the same procedure as in Alice's measurements, we get the bound, $\mathcal{F}^{\prime}(\mathcal{K}_4)\geq \frac{(1+ \sqrt{2})}{8}\mathcal{K}_4 + \frac{2-\sqrt {2}}{4}$. This provides robust self-testing of Bob's measurements which is again optimal.
In order to provide an operational analysis of robustness, let us choose a dephasing channel, $\Lambda_\theta (\rho) = \frac{1+ \xi(\theta)}{2}\rho + \frac{1- \xi(\theta)}{2} \sigma_z \rho \sigma_z$. It keeps the measurement, $A_1 = \sigma_z$ as it is and dephase the measurement, $A_2 = \sigma_x$ to $\xi(\theta)\sigma_x$. Let us consider Bob's measurements as, $B_1 = \mathscros(\phi) \sigma_z + \sin(\phi) \sigma_x$ and $B_2 = \mathscros(\phi) \sigma_z - \sin(\phi) \sigma_x.$
Taking $\xi(\theta)= \tan(\phi)$, we get,
\begin{equation}
\mathcal{K}_4 = 2 \sqrt{1+\tan^2(\phi)} ~~~~~~\text{and} ~~~~~~\mathcal{F} = \frac{1}{4}(3+ \tan(\phi)).\nonumberonumber
\end{equation}
The above expressions provide a parametric curve as a function of $\phi \in [0,\pi/4]$, which is the dashed (blue) line in Fig.-\ref{Robust}. The straight (red) line represents the lower bound of fidelity as given by Eq.\eqref{robustness1}. A straightforward calculation shows that the robustness analysis for Bob's measurements($\mathcal{F}^{\prime}$) produces a matching parametric curve.
\begin{figure}\label{Robust}
\end{figure}
\section{Conclusions}
With the rapid development of quantum technologies, it is important to first characterize and certify various quantum devices. Among various certification
protocols, self-testing protocols are designed for the purpose of certifying quantum components from the observed statistics under a set of minimal assumptions. Self-testing is regarded to be more resource-efficient, and requires lesser trust on measurement devices. Previously, self-testing of measurements has been performed mainly employing nonlocal spatial quantum correlations \mathscrite{my'04,Col,supic2018,supic'19,bancal'18,renou'18}, as exhibited by the violation of Bell-type inequalities, which require entanglement, a costly resource. Schemes of self-testing measurements without entanglement proposed earlier require either dimensional restrictions \mathscrite{tavakoli'18, tavakoli'15}, or other stringent assumptions like compatibility and projectivity of the measurements \mathscrite{kishor'19,kishor'19(1)} in case of approaches based on contextuality, though it may be possible to relax some of these assumptions for local measurements in three and higher dimensions \mathscrite{saha'20}.
In this work our purpose is to self-test binary measurements with dichotomic settings and outcomes without using entangled states. To this end we have exploited another fundamental property of quantum mechanics, {\it viz.}, temporal quantum correlations exhibited by violation of Leggett-Garg inequalities together with violation of predictability \mathscrite{LG'85, emary'14, kofler'13, brukner'04, rand'16}. We have presented a scheme of self-testing of binary Pauli measurements through LGI violation using a minimal assumption that is inevitable in the present context. We have shown how the maximum quantum violation of the four-term LGI along with the NSIT condition can be used to provide a self-testing statement of binary measurements without entanglement, which requires neither compatibility or projectivity, nor any dimensional restriction. Moreover, we have formulated the robustness bound of our self-testing protocol, and analysed it with an example of dephasing noise.
Before concluding, we would like to highlight certain salient features of our analysis. First, it is evident from our analysis that the proposed self-testing protocol for binary qubit measurements does not depend on the input state. Secondly, though in the present work we have not discussed self-testing of quantum states, once the measurement devices are characterized through our approach of self-testing through the LGI framework, any subsequent certification of quantum states using such devices may reduce essentially to the task of decoherence control. Finally, the NSIT condition used here is experimentally implementable, as shown recently, for a variety of states \mathscrite{expnsit}. Experimental viability of our protocol is ensured in the backdrop of several recent LG tests \mathscrite{knee'12,robens'15,knee'16,expnsit, ku'19,shayan'19,spee'20}.
\begin{acknowledgments}
AGM would like to thank Debarshi Das and Debashis Saha for helpful comments. SM acknowledges the Ministry of Science and Technology in Taiwan (Grant no. 110-2811-M-006 -501). CJ thanks Huan-Yu Ku for discussions and acknowledges the financial support from the Ministry of Science and Technology, Taiwan (Grant No. MOST 108-2811-M-006-516) and the Foundation for Polish Science through the First Team project (No First TEAM/2017-4/31). ASM acknowledges support from the project DST/ICPS/QuEST/Q98 from the Department of Science and Technology, India.
\end{acknowledgments}
\appendix
\section{Maximal violation of the LGI with POVMs.}\label{Appa}
Here we obtain the maximal of the Leggett-Garg inequality for POVMs \mathscrite{busch'91,das'18} which are a set of positive operators that add to identity, {\it i. e.}, $E \equiv \{ E_i | \sum_i E_i = \mathbb{I}, 0 <E_i \leq \mathbb{I} \}$. Each of the operators $E_i$, called effect operator determines the probability $\text{Tr}[\rho E_{i}]$ of obtaining the $i^{\text{th}}$ outcome when applied on the state $\rho$. The most general POVM with two outcomes are defined by two parameters,-- sharpness parameter and biasedness parameter. Let us consider that $\gamma_{a}, \gamma_{b}$ are the biasedness parameter and $\lambda_i$ and $\mu_j$ are the sharpness parameters for Alice's $i$-th and Bob's $j$-th measurements, respectively. The most general effect operators with two outcomes can be written as,
\begin{eqnarray}
&&E^{\lambda_i}_{a_i} = \lambda_i \mathcal{P}_{a_i|A_i} + (1\pm \gamma_{a} -\lambda_i) \frac{\mathbb{I}_2}{2}, \nonumberonumber\\
&&E^{\mu_j}_{b_j} = \mu_j \mathcal{P}_{b_j|B_j} + (1 \pm \gamma_{b}-\mu_j) \frac{\mathbb{I}_2}{2}
\end{eqnarray}
The post measurement state can be derived using generalized von Neumann-L\"{u}ders transformation rule, $\dfrac{\sqrt{E^{\lambda_i}_{a_i}} \rho \sqrt{E^{\lambda_i}_{a_i}}}{\text{Tr}[E^{\lambda_i}_{a_i} \rho]}$.
\\
Now, in order to derive the LGI we shall follow the same scenario and procedure as discussed in the main text. The two-time joint probability is obtained using Baye's rule,
\begin{eqnarray}
&&P(a_i,b_j \mid A_i,B_j) = P(a_i \mid A_i) P(b_j\mid a_i, A_i,B_j ) \nonumberonumber \\
&=& Tr\left[ E^{\lambda_i}_{a_i} \rho_{in}\right] Tr\left[ E^{\mu_j}_{b_j} \dfrac{\sqrt{E^{\lambda_i}_{a_i}} \rho \sqrt{E^{\lambda_i}_{a_i}}}{\text{Tr}[E^{\lambda_i}_{a_i} \rho]}\right] .
\end{eqnarray}
The two-time correlation is defined as,
\begin{eqnarray}
\mathcal{C}_{ij} =\sum_{a_i,b_j} (-1)^{a_i\oplus b_j} P(a_i,b_j \mid A_i,B_j),
\end{eqnarray}
where $\oplus$ denotes addition modulo $2$.
The four-term LGI in terms of the above correlators is given by,
\begin{equation}
\mathcal{K}_4 = C_{11} + C_{21} + C_{22} - C_{12} \leq 2
\end{equation}
For a general input state $\rho_{in}=\frac{\mathbb{I}+\hat{n}.\vec{\sigma}}{2}$, the first term (probability obtained by Alice) reduces to,
\begin{equation}
Tr\left[ E^{\lambda_i}_{a_i} \rho_{in}\right] = \frac{1}{2}((1\pm \gamma_a) + (-1)^{a_i}\lambda_i\hat{a_i}.\hat{n}) \nonumberonumber
\end{equation}
Similarly, one can calcultate $Tr\left[ E^{\mu_j}_{b_j} \dfrac{\sqrt{E^{\lambda_i}_{a_i}} \rho \sqrt{E^{\lambda_i}_{a_i}}}{\text{Tr}[E^{\lambda_i}_{a_i} \rho]}\right]$
\iffalse
\begin{widetext}
\begin{eqnarray}
Tr\left[ E^{\mu_j}_{b_j} \dfrac{\sqrt{E^{\lambda_i}_{a_i}} \rho \sqrt{E^{\lambda_i}_{a_i}}}{\text{Tr}[E^{\lambda_i}_{a_i} \rho]}\right] =\frac{1}{2(1\pm \gamma_a)}\left[ (1\pm \gamma_a)(1\pm \gamma_b)+ (-1)^{a_i+b_j} \lambda_i\mu_j\hat{a_i}.\hat{b_j}\right]. \nonumberonumber
\end{eqnarray}
\end{widetext}
So, the joint probability distribution is given by,
\begin{equation}
P(a_i,b_j \mid A_i, B_j)= \frac{1}{4}[ (1\pm \gamma_a)(1\pm \gamma_b)+ (-1)^{a_i+b_j}\lambda_i\mu_j\hat{a_i}.\hat{b_j}].
\end{equation}
Substitution of the above expression in the two-time correlation $\mathcal{C}_{ij}$, leads to,
$\mathcal{C}_{ij}= \lambda_i\mu_j\hat{a_i} . \hat{b_j}.$
\fi
One can now calculate $\mathcal{K}_{4}$ and maximize it numerically. It can be found that the maximum quantum value of $\mathcal{K}_{4}$ is $2\sqrt{2}$ when all $\vert \hat{a_i}.\hat{b_j}\vert=\frac{1}{\sqrt{2}}$ along with $\lambda_i=\mu_j=1$ and $\gamma_{a}=\gamma_{b}=0$. Hence, the maximum value of LGI can only be achieved if Alice and Bob perform sharp-projective measurements.
\section{Details of proof of Theorem 1.}\label{Appb}
Appending an ancilla qubit prepared in the state $\left|0 \right \rangle$, let us show that there exists an isometry $\Phi$ defined by the map,
\begin{align}
\Phi \left|2m,0 \right \rangle &\rightarrow \left| 2m, 0 \right \rangle \nonumberonumber \\
\Phi \left|2m+1,0 \right \rangle &\rightarrow \left| 2m, 1 \right \rangle
\end{align}
such that
\begin{eqnarray}
&&\Phi\left( B_j \frac{\mathcal{P}_{a_i|A_i}\rho_{in}\mathcal{P}_{a_i|A_i}^{\dagger}}{\text{Tr}\left[ \mathcal{P}_{a_i|A_i}\rho_{in}\mathcal{P}_{a_i|A_i}^{\dagger}\right]} \otimes \left|0 \right \rangle \left \langle 0 \right | \right) \Phi^{\dagger} \nonumberonumber \\
&=&B^{\texttt{ideal}}_j \left|\psi^{\texttt{ideal}}_{a|A_i}\right \rangle \left \langle\psi^{\texttt{ideal}}_{a|A_i} \right |\otimes \left| \texttt{junk} \right \rangle \left \langle \texttt{junk}\right |\nonumberonumber
\end{eqnarray}
holds.
\\
We present here the calculation for one term (say, $\Phi\left( B_1 \frac{\mathcal{P}_{a_1|A_1}\rho_{in}\mathcal{P}_{a_1|A_1}^{\dagger}}{\text{Tr}\left[ \mathcal{P}_{a_1|A_1}\rho_{in}\mathcal{P}_{a_1|A_1}^{\dagger}\right]} \otimes \left|0 \right \rangle \left \langle 0 \right | \right) \Phi^{\dagger}$) explicitly. Other terms can be calculated in a similar fashion.
\\
Choosing the eigenbasis of $A_1$ as the computational basis, from lemma $1$ and $2$, it follows that the post-measurement states of Alice can be written as
$\frac{\mathcal{P}_{a_i|A_i}\rho_{in}\mathcal{P}_{a_i|A_i}^{\dagger}}{\text{Tr}\left[ \mathcal{P}_{a_i|A_i}\rho_{in}\mathcal{P}_{a_i|A_i}^{\dagger}\right]} = \oplus_m p_m \left|\psi^m_{a|A_1}\right \rangle \left \langle \psi^m_{a|A_1}\right | $, with $\left|\psi^m_{0|A_1}\right \rangle= \left| 2m \right \rangle $
and $\left|\psi^m_{1|A_1}\right \rangle= \left| 2m +1 \right \rangle $.
The measurement $B_1$ on Bob's side is $ \oplus_n (\sigma_x^n+ \sigma_z^n)/\sqrt{2} $.
Expanding the sum for $a=0$ leads to
\begin{widetext}
\begin{align}
& \oplus_n \frac{\sigma_x^n+ \sigma_z^n}{\sqrt{2}} \oplus_m p_m \left| 2m \right \rangle \left \langle 2m \right | \nonumberonumber \\
&= \bigg( \frac{\ketbra{0}{0}-\ketbra{1}{1}+\ketbra{0}{1}+\ketbra{1}{0}}{\sqrt{2}}+\frac{\ketbra{2}{2}-\ketbra{3}{3}+\ketbra{2}{3}+\ketbra{3}{2}}{\sqrt{2}}+\mathscrdots
\nonumberonumber \\
&+\frac{\ketbra{2m}{2m}-\ketbra{2m+1}{2m+1}+\ketbra{2m}{2m+1}+\ketbra{2m+1}{2m}}{\sqrt{2}} \bigg) \mathscrdot \left( p_0 \ket{0}\bra{0} + p_1 \ket{2}\bra{2} + \mathscrdots + p_m \ket{2m}\bra{2m} \right) \nonumberonumber \\
&=\bigg( p_0 \frac{\ket{0}+\ket{1}}{\sqrt{2}}\bra{0}+ p_1 \frac{\ket{2}+\ket{3}}{\sqrt{2}}\bra{2}+\mathscrdots +p_m \frac{\ket{2m}+\ket{2m+1}}{\sqrt{2}}\bra{2m} \bigg)
\end{align}
Now, for the map defined above,
\begin{align}
&\Phi \left(\oplus_n \frac{\sigma_x^n+ \sigma_z^n}{\sqrt{2}} \oplus_m p_m \left| 2m \right \rangle \left \langle 2m\right | \otimes \ket{0}\bra{0} \right) \Phi^{\dagger}\nonumberonumber \\
&=\frac{\ket{0}+\ket{1}}{\sqrt{2}}\bra{0} \otimes \oplus_m p_m \left| 2m \right \rangle\left \langle 2m\right | \nonumberonumber \\
&=B^{\texttt{ideal}}_1 \left|\psi^{\texttt{ideal}}_{0|A_1}\right \rangle \left \langle\psi^{\texttt{ideal}}_{0|A_1} \right |\otimes \left| \texttt{junk} \right \rangle \left \langle \texttt{junk}\right |
\end{align}
with $\left| \texttt{junk} \right \rangle \left \langle \texttt{junk}\right |= \sum_m p_m \left| 2m \right \rangle \left \langle 2m\right |$.
\\
Next, for $a=1$, we have
\begin{align}
& \oplus_n \frac{\sigma_x^n+ \sigma_z^n}{\sqrt{2}} \oplus_m p_m \left| 2m +1 \right \rangle \left \langle 2m+1 \right | \nonumberonumber \\
&=\bigg( p_0 \frac{\ket{0}-\ket{1}}{\sqrt{2}}\bra{0}+ p_1 \frac{\ket{2}-\ket{3}}{\sqrt{2}}\bra{2}+\mathscrdots +p_m \frac{\ket{2m}-\ket{2m+1}}{\sqrt{2}}\bra{2m} \bigg)
\end{align}
Hence, for the map defined above,
\begin{align}
&\Phi \left(\oplus_n \frac{\sigma_x^n+ \sigma_z^n}{\sqrt{2}} \oplus_m p_m \left| 2m+1 \right \rangle\left \langle 2m+1 \right | \otimes \ket{0}\bra{0} \right) \Phi^{\dagger}\nonumberonumber \\
&=\frac{\ket{0}-\ket{1}}{\sqrt{2}}\bra{0} \otimes \oplus_m p_m \left| 2m \right \rangle\left \langle 2m \right | \nonumberonumber \\
&=B^{\texttt{ideal}}_1 \left|\psi^{\texttt{ideal}}_{1|A_1}\right \rangle \left \langle\psi^{\texttt{ideal}}_{1|A_1} \right |\otimes \left| \texttt{junk} \right \rangle \left \langle \texttt{junk}\right | .
\end{align}
\end{widetext}
\end{document} |
\begin{document}
\title{\bf A Generalization of Fermat's Principle for Classical and Quantum Systems}
\author{Tarek A. Elsayed}
\email{T.Elsayed@thphys.uni-heidelberg.de}
\address{Institute of Theoretical Physics, University of Heidelberg, Philosophenweg 19, 69120 Heidelberg, Germany}
\begin{abstract}
The analogy between dynamics and optics had a great influence on the development of the foundations of classical and quantum mechanics. We take this analogy one step further and investigate the validity of Fermat's principle in many-dimensional spaces describing dynamical systems (i.e., the quantum Hilbert space and the classical phase and configuration space). We propose that if the notion of a metric distance is well defined in that space and the velocity of the representative point of the system is an invariant of motion, then a generalized version of Fermat's principle will hold. We substantiate this conjecture for time-independent quantum systems and for a classical system consisting of coupled harmonic oscillators. An exception to this principle is the configuration space of a charged particle in a constant magnetic field; in this case the principle is valid in a frame rotating by half the Larmor frequency, not the stationary lab frame.
\end{abstract}
\maketitle
\section{Introduction}
An important lesson that has been emphasized throughout the history of physics is that illuminating new aspects of the interwoven connections between geometry and physics leads to paradigm shifts in physics. Typically, novel geometric considerations of physical quantities lead to new variational principles which assign the natural evolution of physical systems with an extremum of some functional or a geodesic curve in some hyperspace. The oldest of these variational principles is the Fermat principle of least time, which became a fundamental principle in geometric optics. The principle was introduced by Fermat, who also called it \emph{ the principle of natural economy} \cite{basdevant}, and it states that light rays travel in a general medium along the path that minimizes the time taken to travel between the initial and final destinations. The concept of natural economy inspired Maupertuis to introduce the principle of least action in analytical mechanics, which later evolved through the work of Euler, Lagrange, Hamilton, and Jacobi to become a fundamental concept in classical mechanics. By 1887, it had become clear that the least action is a universal concept in physics when Helmholtz expanded its domain of validity by applying it to two regimes beyond the standard problems of classical mechanics, namely, thermodynamics and electrodynamics \cite{helmholtz}. Since then, the pursuit of new variational principles in physics has not relented \cite{novikov2004}.
The mathematical formulation of Fermat's principle states that the time functional $\mathcal{T}$, defined as
\begin{equation}
\mathcal{T}=\int\frac{ds}{\nu(s)},
\label{T}
\end{equation}
where $ \nu(s) $ is the speed of light and $ ds $ is the distance element along the light trajectory, is minimized \cite{mandelstam}. In other words, if $ \mathcal{T} $ is computed along all possible trajectories between fixed initial and final positions, $ \mathcal{T} $ will always be minimum along the actual path traveled by the light rays (the physical path). The modern version of Fermat's principle is written in terms of the index of refraction $ n(s) =\frac{c}{\nu(s)} $, where c is the speed of light in free space, and states that the optical path length $\int n(s)ds$ is a minimum. In that sense, Fermat's principle is the optical analog of Jacobi's principle of least action \cite{lanczos}, which states that for a conservative classical system at energy $ E $, with potential function $V$ between its constituent particles, the action functional \begin{equation}
I=\int\sqrt{E-V(s)}ds
\label{jac}
\end{equation}
is an extremum.
The remarkable property of this action that distinguishes it from other variational principles in analytical mechanics, i.e., Hamilton and Lagrange's variational principles, is that it represents a purely geometric quantity. This quantity is computed along different trajectories in the configuration space between fixed points without referring to any time evolution. Therefore, Eq. (\ref{jac}) can be used to define a new Riemannian space, whose metric $ ds'=\sqrt{E-V(s)}ds $, where the natural evolution of the representative point of the system is along geodesic curves.
In this work, we set out to seek how far the analogy between dynamics and optics applies as far as the Fermat principle is concerned. In particular, we investigate the validity of Fermat's principle for a generic many-body classical and quantum system, and pose the following question: If the state of a conservative dynamical system is represented in some metric space $ \mathbb{S} $ by a point, and the velocity field $ \nu(s) $ is computed everywhere in $ \mathbb{S} $ from the equations of motion using the proper metric of that space, will the motion of this point be along a path that extremizes the time functional $\mathcal{T}$?
\section{The Generalized Fermat Principle}
We answer the question posed above by proposing the \emph{generalized fermat principle} (GFP): {Whenever the speed of the representative point of a conservative dynamical system, $\nu(s)$, is an integral of motion in a metric space $ \mathbb{S} $, the path followed during the dynamical evolution of that system in $ \mathbb{S} $ between fixed initial and final states makes the time functional $\mathcal{T}$ stationary against small variations of the path. } In contrast to light rays, where $\mathcal{T}$ is an extremum even when the speed of light is not constant (i.e., in an inhomogeneous medium), this conjecture considers only the case when $\nu(s)$ is invariant during the time evolution. A corollary that follows from this conjecture is that the length of the physical path $\int ds$ is stationary (e.g., the path can be a geodesic) on the sub-manifold of a given value of $\nu(s)$ embedded in $\mathbb{S} $ when the above condition is fulfilled.
Mathematically speaking, the GFP states that if $\nu(s)$ is a constant of motion along the physical path (not necessarily in the whole space), then among all possible trajectories between initial and final states, only those which make $ \mathcal{T} $ invariant under an infinitesimal variation of the path, i.e.,
\begin{equation}
\delta \mathcal{T} =0,
\label{var}
\end{equation}
are possible candidates for the dynamical evolution. The value of $ \mathcal{T} $ corresponding to the physical path is not necessarily the global minimum between all paths connecting the initial and final states. We emphasize here that we are not aiming to derive the equations of motion from the time action, because we have to use them to find $ \nu(s) $ in the first place. We rather propose that they necessarily lead to a stationary time action when $\nu(s)$ is an integral of motion. Unlike the original Fermat principle, not every pair of states are connected by a physical path. Rather, the principle proposed here gives a geometrically appealing argument to explain why the evolution of the system followed a certain trajectory between a given pair of initial and final states which we know a priori are connected by some physical path.
We investigate the validity of this conjecture by considering three cases: (i) The evolution of quantum systems in the projective Hilbert space $\mathbb{P}$, where wavefunctions are defined up to an overall phase factor. (ii) The evolution of a system of coupled harmonic oscillators in the phase space consisting of coordinates and momenta and equipped with an Euclidean metric. (iii) The motion of a charged particle in a constant magnetic field, which turns out to be an exception. Similar to the Jacobi's principle, the principle proposed here represents a geometric variational principle in the phase and Hilbert space. In both cases, the velocity field $\nu(s)$ is defined completely by the Hamiltonian of the problem, and is obtained from the equations of motion of the system that will drive its evolution along the physical path (i.e., Schr\"odinger equation in quantum systems and Hamilton's equations of motion in classical systems).
\subsection{Generalized Fermat Principle in Hilbert Space}
The development of the concept of geometric phase in quantum mechanics triggered the interest of many physicists to look for more connections between quantum mechanics and geometry \cite{ashtekar}. Anandan and Aharonov investigated the nature of the geometry of quantum evolution in the projective Hilbert space $\mathbb{P}$ through a series of papers in the late 80s \cite{aharonov87,aharonov90,aharonov88}. They have shown \cite{aharonov90} that the speed of quantum evolution in $\mathbb{P}$ is related to the energy uncertainty $ \Delta E=\left ( \left \langle H^2 \right \rangle - \left \langle H \right \rangle^2\right )^{1/2} $ via
\begin{equation}
ds= \Delta E\ dt/\hbar,
\end{equation}
where $ ds $ is the infinitesimal distance in $\mathbb{P}$ given by the Fubini-Study (FS) metric $ ds^2= \frac{\left \langle \delta \psi |\delta \psi \right \rangle}{\left \langle \psi |\psi \right \rangle} - \frac{|\left \langle \delta \psi | \psi \right \rangle|^2}{\left \langle \psi |\psi \right \rangle^2}$. On the unit sphere, $ds= \langle \delta \psi |1-\hat{P}|\delta \psi\rangle^\frac{1}{2}$, where $ \hat{P} $ is the projection operator $|\psi\rangle \langle \psi| $. The trajectory traversed by a ray in $\mathbb{P}$ under unitary evolution is generally not a geodesic, i.e., $ \delta\int ds\neq 0 $. This can be easily conceived by considering a system composed of a single quantum spin-1/2. In this case, $\mathbb{P}$ is simply the Bloch sphere and the precession motion of the spin on Bloch sphere off the equator is not a geodesic.
Several attempts \cite{kryukov,grigorenko} have been made to find new formulations where the quantum evolution is a geodesic flow. Noticing that the speed of quantum evolution $ \Delta E/\hbar$ is invariant for time-independent Hamiltonians, the simple answer to this problem suggested by the present paper is to consider $ \mathcal{T}=\int \frac{ds}{\Delta E} $ as a geodesic quantity, i.e., \emph{Fermat's principle in Hilbert space}. This issue should be distinguished from the quantum brachistochrone problem \cite{optimal}, where the Hamiltonian that leads to optimal time evolution between an initial and final state is sought. The above proposition, however, states that the unitary evolution generated by any time-independent Hamiltonian is optimal, with respect to all other possible trajectories connecting the initial and final states (Fig. 1-a).
To show that $\mathcal{T}$ is stationary along the physical path through $\mathbb{P}$, let us parameterize the evolution along any path connecting the initial and final states $ | \psi_i \rangle $ and $ | \psi_f \rangle $ by some arbitrary parameter $ \tau $. We can write Eq. (\ref{T}) as
\begin{equation}
\mathcal{T}=\int_{|\psi_i \rangle }^{|\psi_f \rangle} d\tau\frac{\langle\dot{\psi}|1-\hat{P}|\dot{\psi}\rangle^{\frac{1}{2}}}{\left ( \left \langle \psi| H^2 |\psi\right \rangle - \left \langle\psi| H|\psi \right \rangle^2\right )^{\frac{1}{2}}},
\label{TQ}
\end{equation}
where $ | \dot{\psi} \rangle =\frac{|\delta \psi \rangle}{\delta \tau} $.
Taking the variational derivative of $ \mathcal{T}$ with respect to $ \langle \delta \psi| $ subject to the constraints of normalization and fixed initial and final states, we arrive at the Euler-Lagrange (EL) equation,
\begin{equation}
\frac{\delta L}{\langle \delta \psi |}-\frac{d}{d\tau}\frac{\delta L}{\langle \delta \dot{\psi}|}=0,
\label{EL}
\end{equation}
where $L$ is the integrand in Eq. (\ref{TQ}) added to the Lagrange multiplier term $\lambda(\tau) (\langle \psi |\psi \rangle-1)$. Although we have already imposed the normalization condition in the Fubini-Study metric, we use the Lagrange-multiplier method here to ensure that the variations of the path will respect the conservation of the norm of $|\psi \rangle$.
Calling the numerator and denominator in Eq. (\ref{TQ}), $A$ and $B$ respectively, Eq. (\ref{EL}) reads
\begin{widetext}
\footnotesize
\begin{equation}
\begin{split}
&\left [ \frac{-1}{2AB}\langle\dot{\psi}|\psi\rangle |\dot \psi\rangle - \frac{A}{2B^3}\left ( H^2 |\psi\rangle -2\langle H\rangle H | \psi \rangle \right ) \right ] -\frac{1}{2AB}\left [ |\ddot\psi \rangle - \langle\psi |\dot\psi \rangle |\dot\psi \rangle -\left ( \langle \dot \psi |\dot \psi \rangle + \langle \psi|\ddot \psi \rangle \right )| \psi \rangle\right ] -\frac{1}{2}\left ( |\dot\psi \rangle - \langle\psi |\dot\psi \rangle |\psi \rangle \right)*\\ & \left [ \frac{-1}{2AB^3}\left ( \langle \psi H^2 | \dot \psi \rangle + \langle \dot \psi | H^2| \psi \rangle -2 \langle H \rangle\left [ \langle \dot \psi | H |\psi \rangle + \langle \psi |H|\dot\psi \rangle \right ] \right ) - \frac{1}{2A^3B}\left ( \langle \ddot \psi |\dot \psi \rangle - \langle \psi | \dot \psi \rangle \left ( \langle \ddot \psi | \psi \rangle +\langle \dot \psi | \dot \psi \rangle\right ) +c.c \right )\right ] +\\ &\lambda(\tau) |\psi \rangle =0.
\end{split}
\label{long}
\end{equation}
\normalsize
\end{widetext}
Although Eq. (\ref{long}) is a highly nonlinear equation, it is easy to verify that the Schr\"odinger equation $ |\dot{\psi}\rangle=\pm iH|\psi\rangle $ satisfies this equation with a vanishing Lagrange multiplier, and therefore makes the time functional stationary when $\tau$ equals the real time $t$. The sign ambiguity can be considered a reminiscence of the non-unique mapping between $\tau$ and $t$. In cases where quantum ergodicity applies, i.e., when ``all states within a given energy range can be reached from all other states within the range" \cite{haar}, the opposite sign can be related to the other route to reach $|\psi_f \rangle $ starting from $|\psi_i \rangle $, i.e., backward in time.
\twocolumngrid
The above discussion provokes several interesting issues. First, it is intriguing to explore whether there is a nonlinear Schr\"odinger equation that would satisfy Eq. (\ref{long}) with a vanishing Lagrange multiplier and fulfill the Fermat principle as a possible extension to quantum mechanics. It is unlikely that such an equation exists since any equation that satisfies Eq. (\ref{long}) and keeps the norm of $|\psi\rangle$ conserved should also satisfy
\begin{equation}
\frac{\langle\dot{\psi}|\dot{\psi}\rangle}{|\langle\psi|\dot{\psi}\rangle|^2}= \frac{\left \langle \psi| H^2 |\psi\right \rangle}{\left \langle\psi| H|\psi \right \rangle^2}.
\end{equation}
This equation results after taking the inner product of Eq. (\ref{long}) with $\langle\psi|$ and making use of the normalization condition of the wavefunction. Second, it has to be emphasized that Eq. (\ref{EL}) is satisfied only for time-independent Hamiltonians. A very interesting problem is how to generalize this concept to time-dependent Hamiltonians as we shall do in the classical domain below. Finally, we expect the Fermat principle to be equally valid for the unitary evolution of a density matrix with the FS metric replaced by the Hilbert-Schmidt metric. It would be interesting, though, to investigate whether the non-unitary evolution of the density matrix of an open quantum system described by a master equation follows a Fermat principle. We present no further details on these issues in the present paper.
\begin{figure*}
\caption{ \label{fig}
\label{fig}
\end{figure*}
\normalsize
\subsection{Generalized Fermat Principle in Phase Space}
In optics, the Euler-Lagrange equation for the functional $\frac{1}{c}\int n(s)ds$ reduces to the ray equation (also called the eikonal equation) \cite{johns2011}
\begin{equation}
\frac{d}{ds} (n(s) \mathbf{\hat{t}}) -\mathbf{\nabla} n(s)=0,
\label{EL-n}
\end{equation}
where $\mathbf{\hat{t}}$ is a unit tangent vector defined in terms of the position vector $\mathbf{r}$ as $\mathbf{\hat{t}}=d\vec {\mathbf{r}}/ds$ (the length of the path $s$ plays the role of time in this derivation). When $n(s)$ is constant along the path (i.e., independent of $s$), Eq. (\ref{EL-n}) can be rewritten as
\begin{equation}
\frac{d}{ds} ( \mathbf{\hat{t}})=-\frac{\mathbf{\nabla}\nu(s)}{\nu(s)}.
\label{curv}
\end{equation}
The left hand side of this equation represents a curvature vector $ \vec{\mathbf{\kappa}} $ whose magnitude equals the curvature of the path and direction is orthogonal to the direction of motion. Therefore, for the case of an Euclidean space $\mathbb{S}$, the GFP is equivalent to stating that $ \vec{\mathbf{\kappa}} $ equals $-\mathbf{\nabla}\nu(s)/ \nu(s)$ for a dynamical system that has invariable speed $\nu(s)$ along its evolution in $ \mathbb{S} $. On the other hand, since $ \vec{\mathbf{\kappa}} $ is the acceleration vector of the representative point of the system, we can regard the RHS of Eq. (\ref{curv}) as a force that drives its evolution. The potential function responsible for this force is $\log(\nu(s))$.
We now consider a generic conservative classical system composed of $ N $ particles described by a set of generalized coordinates $ \{q_i,p_i\} $ that have the same units and Hamiltonian $H$ (Fig. 1-b). Let the distance element in the phase space be described by the Euclidean metric $ ds^2=\sum_{i=1}^{N}dq_i^2+dp_i^2 $. The speed $ \nu(s) $ along the physical path generated by the Hamiltonian flow equals $ \sqrt{\sum_{i=1}^N \left( \frac{\partial H}{\partial q_i}\right)^2+\left(\frac{\partial H}{\partial p_i}\right)^2} $ \cite{haar}. We therefore express the time functional $\mathcal{T}$ as
\begin{equation}
\mathcal{T}=\int_{\vec{\mathbf{x}}_i}^{\vec{\mathbf{x}}_f}\frac{ds}{\sqrt{\sum_i \left( \frac{\partial H}{\partial x_i}\right)^2}},
\end{equation}
where $x_i$ denotes any of the generalized coordinates $ {q_i,p_i} $ treated on an equal footing and $ \vec{\mathbf{x}}$ denotes $\{ x_1,x_2,...x_{2N} \}$.
If we parameterize any arbitrary trajectory connecting $ \vec{\mathbf{x}}_i $ and $ \vec{\mathbf{x}}_f $ by $ \tau $, then the EL equation which satisfies the variational principle $\delta \mathcal{T} =0$ is
\begin{equation}
\frac{\delta L}{\delta x }-\frac{d}{d\tau}\frac{\delta L}{\delta \dot{x} }=0,
\label{ELPS}
\end{equation}
where $ L= FG$, $ F=\frac{1}{\sqrt{\sum_i\left(\frac{\partial H}{\partial x_i}\right)^2}} $, $G=\sqrt{\sum_i \dot{x_i}^2}$ and $x$ denotes any of the $2N$ coordinates. The explicit form of Eq. (\ref{ELPS}) for coordinate $x_i$ is
\begin{equation}
G^2\frac{\partial F}{\partial x_i}-\frac{G^2\left(\ddot{x_i}F+\dot{x_i}\sum_{j} \frac{\partial F}{\partial x_j}\dot{x_j}\right )-\dot{x_i}F\sum_j\dot{x_j}\ddot{x_j}}{G^2}=0.
\label{fullELPS}
\end{equation}
This equation is not generally satisfied for an arbitrary choice of the generalized coordinates ${q_i,p_i}$. The only system known to the author for which a metric space is defined and $\nu(s)$ is a constant of motion is the harmonic oscillator. It is easy to verify numerically that Eq. (\ref{fullELPS}) is satisfied for a system of $N$ coupled harmonic oscillators with equal masses and coupling constants with Hamiltonian $H=\sum_{i} \frac{p_i^2}{2m}+\sum_{i<j}\frac{1}{2}m\omega^2(q_i-q_j)^2$ if we work in the phase space $\{P_i,Q_i\}$, where $P_i=p_i$ and $Q_i=m\omega\sqrt{N} q_i$. The speed of evolution $\nu(s)$ in this case is proportional to the energy. In this example, we had to perform a simple scale transformation in order to obtain a new set of coordinates that have the same units, leading to a metric space in which Fermat's principle is fulfilled. In other examples, one might need to look for more complicated transformations that would render $\nu(s)$ an integral of motion in the new phase space.
For externally driven systems whose time-dependent Hamiltonian is $H(p,q,t)$, we define the new Hamiltonian, $H'=H(q,p,\tau) +\eta$, where $(\tau,\eta)$ is an extra degree of freedom that satisfies $\dot{\tau}=\partial H'/\partial \eta=1$ and $\dot{\eta}=-\partial H'/\partial \tau=-\partial H/\partial \tau$ \cite{hossein}. The above discussion can be generalized to the extended phase space which consists of the original phase space supplemented by the new coordinates $(\tau,\eta)$.
\subsection{Generalized Fermat Principle in Configuration Space}
An obvious example where GFP is valid in configuration space is the free motion of a single particle constrained to a curved surface following a geodesic trajectory. Similarly, for a system of free particles, GFP holds in the configuration space $\{q_i\}$ (Fig. 1-c), while for a system of interacting particles it holds in the Riemannian manifold defined by the metric $ds'$ introduced in the introduction. Another example is the bound states of the two-body Kepler problem. The 3D motion of the reduced one-body problem can be mapped onto the motion of a free particle on the inner surface of an $S^3$ sphere embedded in a 4D space \cite{moser,guillemin}. In the rest of this section, we discuss the applicability of Fermat's principle in the configuration space of another system, namely a charge $q$ with mass $m$ moving in a constant uniform magnetic field $\mathbf{B}$.
The dynamics of the charge controlled by the Lorentz Force $m \frac{d\mathbf{v}}{dt}=q\mathbf{v}\times \mathbf{B}$ guarantees that the magnitude of the charge velocity is constant and GFP indicates that the time functional should be stationary in the configuration space of the charge. However, in this case, a stationary time functional $\mathcal{T}$ indicates that the length of the path of the charge in space is an extremum, which we know is not the case; the circular path of the charge in a constant magnetic field is not an extremal path. Therefore, we consider this case a clear counter-example to the GFP if we do not restrict it to velocity-independent potentials. However, we note that even in this case, there is another frame of reference where GFP is valid. What we will show next is that in a frame rotating with frequency $\boldsymbol{\omega}=-q/2m \mathbf{B}$ the time functional is a stationary quantity and GFP restores its validity.
The time derivative of the charge position vector in the rotating frame $\frac{d \mathbf{r} }{d\tau}$ is related to its time derivative in the lab frame $\frac{d \mathbf{r} }{dt}$ by $\frac{d \mathbf{r} }{d\tau} = \frac{d \mathbf{r} }{dt} - \boldsymbol{\omega} \times \mathbf{r}$ \cite{abragam}. By working in the Coulomb gauge, $\mathbf{A}=-\mathbf{r}\times \mathbf{B}/2$, it can be easily shown that $\frac{d \mathbf{r} }{d\tau}=\frac{d \mathbf{r} }{dt}+q\mathbf{A}/m$. The RHS of this equation is nothing but the canonical momentum $\mathbf{P}=m\mathbf{v}+q\mathbf{A}$ divided by the mass of the charge. Let us call the velocities in the rotating frame and the lab frame $\mathbf{u}$ and $\mathbf{v}$ respectively. The time functional in the rotating frame is then $\mathcal{T}=\int \frac{ds}{u}$. If we parameterize the path in terms of the lab frame time coordinate $t$, the time functional transforms into $\mathcal{T}=\int \frac{v dt}{u}$. Assuming the magnetic field in the $z$ direction and the charge initially having a horizontal velocity, we find that $\mathcal{T}=\int \frac{ \sqrt{\dot x^2 + \dot y^2 } dt }{\sqrt{(\dot x-\omega y)^2 + (\dot y+\omega x)^2 } }$ where $\omega=|\boldsymbol{\omega}|$. The author has verified numerically that the EL equations are verified for this action and hence $\mathcal{T}$ is a stationary quantity in the rotating frame.
\section{Discussion} Having demonstrated the validity of the Fermat's principle for quantum Hilbert spaces and the phase space of a system of harmonic oscillators, we need to emphasize that the validity of Eq. \ref{var} is not a trivial consequence of the invariance of $\nu(s)$ along the physical path, but is rather attributed to the underlying dynamics, or more precisely the differential equations governing those dynamics. Not every functional of the form $\int F(s)ds$ is an extremum along the path of constant $F$. On the opposite side, a dynamical system can have an extremum action $\int F(s)ds$ even when $F(s)$ is not a constant of motion as in Eq. \ref{jac}. Moreover, the validity of Eq. \ref{var} in both Hilbert and phase spaces may not be related to the fact that the speed of evolution is directly connected with the energy in both cases since we can easily check that the functional $\int \langle H \rangle ds$ in the first case and $\int H(P_i,Q_i) ds$ in the second case are not stationary along the physical path. Satisfying Eq. \ref{var} is not synonymous to the conservation of energy, and we did not have to impose the constraint of constant energy to verify it. Moreover, on the energy manifolds in Hilbert and phase spaces, there is a multitude of trajectories connecting the initial and final states, but not all of them satisfy $\delta \mathcal{T} =0$.
While it is true that one can find a variational principle that describes the solution of almost every differential equation, the conceptual significance of the GFP is the link it makes with an established and a well known principle in optics. The principle advertised in this paper is obviously not as practical as the original Fermat's principle or the Lagrange principle of least action for example since we cannot easily solve Eqs. \ref{fullELPS} and \ref{long} and find the physical path. However, it illustrates an interesting property of the evolution in Hilbert and phase spaces that we believe is worthy to be highlighted.
As a leap of faith based on the previous examples, we claim that GFP will be universally true in any space where the dynamical system evolves with a constant speed. This implies that if the state of the system is projected to a sub-manifold of the full space $ \mathbb{S} $, where a metric is defined and $\nu(s)$ of the projected state is an integral of motion, Fermat's principle will hold in that space as well. An important exception of this generalization is the motion of a charge in a magnetic field; in this case we found a new frame of reference where GFP is valid in its configuration space.
The ability to find the manifold or the transformation that renders $\nu(s)$ an integral of motion relies on our ability to find the integrals of motion of the given dynamical system, not a trivial task in many cases. Therefore, the generalized Fermat principle is more likely to be relevant in integrable classical systems than in non-integrable systems. We recall that the original Fermat's principle is a wave phenomenon. The minimization of the time functional $\mathcal{T}$ occurs along the path which leaves the phase of the light wave stationary with respect to small variations of the path (see \cite{science} and references therein.) The notion of periodic phase variations is found naturally in two types of dynamical systems: completely integrable classical systems and quantum systems. The description of the system in terms of the action-angle variables in the first case and the eigenbasis of the Hamiltonian in the second case offers an intuitive representation of the dynamics as a superposition of {\it waves}.
On the other hand, even for a completely integrable system, it is not a trivial task to find the metric space where $\nu(s)$ is a constant of motion and it is not even clear whether such a space exists for every integrable system. It is therefore instructive to ask: if we abandon the notion of single trajectories in the classical domain and consider a semiclassical system described by a quasi-probability distribution in the phase space, can a generalized form of Fermat's principle hold in this representation? It seems plausible to expect interesting phenomena to occur in this case due to the quantum corrections to the classical Liouville's equation or because quantum quasi-probability wavepackets can be thought of as superpositions of many classical trajectories that interfere with each other \cite{martens}. We hope that this discussion will trigger more research efforts into this direction.
\section{Conclusion}
In conclusion, we have introduced a conjecture that generalizes Fermat's principle in Hilbert and phase spaces of quantum and classical systems respectively and illuminates new aspects of Fermat's surmise of \emph{natural economy}. The generalized Fermat principle provides a new geometric variational principle satisfied naturally by the Schr\"odinger and Hamilton's equations of motion in the proper space that has an associated metric distance when the speed of evolution is an integral of motion. This principle may have implications for the protein folding problem, one of the major challenges of biological sciences today \cite{dill}. The mechanism followed by a protein to evolve from its unfolded structure to the native structure that has a minimum free energy is not fully understood. The puzzle lies in the ultrashort time the protein takes to fold with respect to the astronomical number of intermediate conformations. Although the folding protein is an open system, we may gain some insight into this problem by searching for the proper space $ \mathbb{S} $ where the generalized Fermat principle is fulfilled.
The author would like to thank Chris Gray for his comments and Boris Fine for several fruitful discussions and useful comments on the manuscript.
\end{document} |
\begin{document}
\title{
With a little help from your friends:
semi-cooperative games via Joker moves
}
\pagestyle{plain}
\pagenumbering{arabic}
\begin{abstract}
This paper coins the notion of Joker games where Player 2 is not strictly adversarial:
Player 1 gets help from Player 2 by playing a Joker.
We formalize these games as cost games, and study their theoretical
properties.
Finally, we illustrate their use in model-based testing.
\end{abstract}
\section{Introduction}
\myparagraph{Winning strategies.} We study 2 player concurrent games played on a game graph with reachability objectives, i.e., the goal for Player 1 is to reach a set of goal states $R$. A central notion in such games is the concept of a {\em winning strategy}, which assigns ---in states where this is possible
---moves to Player 1 so that she always reaches the goal $R$, no matter how Player 2 plays.
Concurrent reachability games have abundant applications, e.g. controller synthesis \cite{bertsekas},
assume/guarantee reasoning \cite{ChatterjeeH07-assume-guarantee}, interface theory \cite{DBLP:journals/entcs/AlfaroS04},
security \cite{conf-gamesec/2016} and test case optimization \cite{DBLP:conf/fortest/HesselLMNPS08,realtimegames}.
However, it is also widely acknowledged
\cite{Berwanger07,Faella09,ChatterjeeH07-assume-guarantee}
that the concept of a winning strategy is often too strong in these applications: unlike games like chess, the opponent does not try to make Player 1's life as hard as possible, but rather, the behaviour of Player 2 is unknown.
Moreover, winning strategies do not prescribe any move in states where Player 1 cannot win. This is a major drawback: even if Player 1 cannot win, some strategies are better than others.
Several solutions have been proposed to remedy this problem, via {\em best effort strategies}, i.e. strategies that determine which move to play in losing states. For example, Berwanger \cite{Berwanger07} coins the notion of strategy dominance, where one strategy is better than another if it wins against more opponent strategies.
The maximal elements in the lattice of Player-1 strategies, called {\em admissible} strategies, are proposed as best strategies in losing states.
Further, specific solutions have been proposed to refine the notion of winning in specific areas, especially in controller synthesis: \cite{ChatterjeeH07-assume-guarantee} refines the synthesis problem
into a co-synthesis problem, where
components
compete conditionally: their first aim is to satisfy their own specification, and their second aim is to violate the other component's specification.
In \cite{DBLP:conf/atva/BloemEK15}, a hierarchy of collaboration levels is defined, formalizing how cooperative a controller for an LTL specification can be.
\myparagraph{Our approach: Joker games.}
The above notions are all qualitative, i.e., they refine the notion of winning strategy, but do not quantify how much collaboration from the opponent is needed. To fill this gap, we introduce {\em Joker games}.
By fruitfully building upon robust strategy synthesis results by Neider et al. \cite{synthesisresilient,synthesissafety}, a \emph{Joker strategy} provides a best effort strategy in cases where Player 1 cannot win. Concretely,
we allow Player 1, in addition to her own moves, to play a so-called {\em Joker move}.
With such a Joker move, Player 1 can choose her own move \emph{and} the opponent's move, so that Player 2 helps Player 1 reaching the goal.
Then, we minimize the number of Jokers needed to win the game,
thereby minimizing the help needed from Player 2 Player to win. We formalize such Joker-minimal strategies as cost-minimal strategies in a priced game.
While the construction for Joker strategies closely follows the construction in \cite{synthesisresilient,synthesissafety} (and later in \cite{BARTOCCI2021229}), our games extend these results in several ways: First, the games in \cite{synthesisresilient,synthesissafety,BARTOCCI2021229} are turn-based and deterministic, whereas ours are concurrent and nondeterministic. That is, outcome of the moves may lead to several next states. Nondeterminism is essential in game-based testing, to model a faithful interaction between the tester and the system-under-test \cite{mbtgames}.
An important difference is that Neider et al. focus on finding optimal strategies in the form of attractor strategies, while we present our Joker strategies as general cost-minimal strategies.
In fact, we show that all attractor strategies are cost-minimal, but not all cost minimal strategies are attractor strategies. In particular, non-attractor strategies may outperform attractor strategies with respect to other objectives like the number of moves needed to reach the goal.
Furthermore, we establish several new results for Joker games: (1) While concurrent games require randomized strategies to win a game, this is not true for the specific Joker moves: these can be played deterministically. (2) If we play a strategy with $n$ Jokers, each play contains exactly $n$ Joker moves.
(3)
Even with deterministic strategies, Joker games are determined. (4) The classes of Joker strategies and admissible strategies do not coincide.
Finally, we illustrate how Joker strategies can be applied in practice, by extracting test cases from Joker strategies. Here we use techniques from previous work \cite{mbtgames}, to translate game strategies to test cases. We refer to \cite{mbtgames} for related work on game-based approaches for testing. In the experiments of this paper we show on four classes of case studies that obtained test cases outperform the standard testing approach of random testing. Specifically,
Joker-inspired test cases reach the goal more often than random ones, and require fewer steps to do so.
\myparagraph{Contributions.} Summarizing, the main contributions of this paper are:
\begin{itemize}
\item We formalize the minimum help Player 1 needs from Player 2 to win as cost-minimal strategies in a Joker game.
\item We establish several properties: the minimum number of Jokers equals minimum cost, each play of a Joker strategy uses $n$ Jokers, Joker game determinacy, Jokers can be played deterministically, in randomized setting, and admissible strategies do not coincide with cost-minimal strategies.
\item We refine our Joker approach with second objective of the number of moves.
\item We illustrate the benefits of our approach for test case generation.
\end{itemize}
\myparagraph{Paper organization.} Section~\ref{sec:games} recapitulates concurrent games. Section~\ref{sec:joker-games} introduces Joker games, and Section~\ref{sec:properties-joker-games} investigates their properties. In Sect~\ref{sec:short} we study multi-objective Joker strategies, and in \sectionref{sec:admissible} admissible strategies. In Section ~\ref{sec:experiments} we apply Joker strategies to test case generation. Section~\ref{sec:concl} concludes the paper. The artefact of our experimental results of \sectionref{sec:experiments} is provided in \cite{artefact}.
\section{Concurrent Games}
\label{sec:games}
We consider concurrent games played by two players on a game graph.
In each state, Player 1 and 2 \emph{concurrently} choose an action, leading the game in one of the (nondeterministically chosen) next states.
\begin{definition}\label{def:gamearena}
A \emph{concurrent game} is a tuple ${G}=(Q,q^0,\textrm{Act}_1,\textrm{Act}_2,\allowbreak\Gamma_1,\allowbreak\Gamma_2, \allowbreak\textit{Moves})$ where:
\begin{itemize}
\item $Q$ is a finite set of states,
\item $q^0 \in Q$ is the initial state,
\item For $i\in \{1,2\}$, $\textrm{Act}_i$ is a finite and non-empty set of Player $i$ actions,
\item For $i\in \{1,2\}$, $\Gamma_i: Q\to 2^{\textrm{Act}_i} \setminus \emptyset$
is an enabling condition, which assigns to each state $q$ a non-empty set $\Gamma_i(q)$ of actions available to Player $i$ in $q$,
\item $\textit{Moves}: Q \times \textrm{Act}_1 \times \textrm{Act}_2 \rightarrow 2^Q$ is a function that
given the actions of Player 1 and 2 determines the set of next states $Q' \subseteq Q$ the game can be in.
We require that $\textit{Moves}(q,a,x) = \emptyset$ iff $a\not\in\Gamma_1(q) \vee x\not\in\Gamma_2(q)$.
\end{itemize}
\end{definition}
For the rest of the paper, we fix concurrent game ${G} = (Q,q^0,\allowbreak\textrm{Act}_1,\textrm{Act}_2,\allowbreak\Gamma_1,\allowbreak\Gamma_2, \textit{Moves})$.
Next, we define a \emph{play} as a
sequence of states and actions.
\begin{definition} \label{def:play}
An \emph{infinite play} is an infinite sequence\\
$\pi = q_0 \langle a_0,x_0 \rangle q_1 \langle a_1, x_1 \rangle q_2 \dots $
with $a_j\in\Gamma_1(q_j)$,
$x_j\in\Gamma_2(q_j)$, and
$q_{j+1} \in \textit{Moves}({q_j},\allowbreak{a_j},{x_j})$ for all $j \in \mathbb{N}$.
We write $\pi_j^q = q_j$, $\pi_j^a = a_j$, and $\pi_j^x = x_j$ for the $j$-th state, Player 1 action, and Player 2 action respectively.
The set of infinite plays with $\pi_0^q = q$ is denoted $\Pi^\infty(q)$.
We define $\Pi^\infty({G}) = \Pi^\infty(q^0)$.
A \emph{play} $\pi_{0:j} = q_0\langle a_0,x_0 \rangle q_1 \dots \langle a_{j-1},x_{j-1} \rangle q_j$ is a (finite) play prefix of infinite play $\pi$.
We write $\pi^a_{end}=a_{j-1}$, $\pi^x_{end}=x_{j-1}$, and $\pi^q_{end}=q_j$ for the last Player 1 action, last Player 2 action and last state of a finite play $\pi$.
All states in $\pi_{0:j}$ are collected in $\textit{States}(\pi_{0:j}) = \{\pi^q_k \mid 0\leq k \leq j \}$.
The set of all (finite) plays of a set of infinite plays $P \subseteq \Pi^\infty(q)$ is denoted ${\textit{Pref}}(P) = \{\pi_{0:j} \mid \pi \in P, j \in \mathbb{N}\}$.
We define $\textnormal{play?}pref({G}) = {\textit{Pref}}(\Pi^\infty({G}))$, and for any $q \in Q$: $\Pi(q) = {\textit{Pref}}(\Pi^\infty(q))$.
\end{definition}
\noindent
We consider plays where Player 1 wins, i.e. reaches a state in $R \subseteq Q$.
\begin{definition} \label{def:winning}
A play $\pi \in \Pi^\infty(q) \cup \Pi(q)$ for $q \in Q$ is {\em winning} for reachability goal $R$, if there exist a $j \in \mathbb{N}$ such that $\pi_j^q \in R$.
The \emph{winning index} of a play $\pi \in \Pi^\infty(q) \cup \Pi({G})$ is:
$\textit{WinInd}(\pi,R) = \min\{j \in \mathbb{N} \mid \pi^q_j \in R\}$, where $\min{\emptyset} = \infty$.
\end{definition}
Given a play prefix, players choose their actions according to a strategy.
A strategy is positional, if the choice for an action only depends on the last state of the play.
The game outcomes are the possible plays of the game when using the strategy.
We define deterministic strategies here;
see Section~\ref{sec:properties-joker-games} for randomization.
\begin{definition} \label{def:strategy}
A \emph{strategy} for Player $i \in \{1,2\}$ starting in state $q \in Q$ is a function $\sigma_i: \Pi(q) \rightarrow \textrm{Act}_i$, such that $\sigma_i(\pi) \in \Gamma_i(\pi^q_{end})$ for any $\pi \in \Pi(q)$.
We write $\Sigma_i(q)$ for the set of all Player $i$ strategies starting in $q$, and set $\Sigma_i({G}) = \Sigma_i(q_0)$.
A strategy $\sigma_i \in \Sigma_i(q)$ is \emph{positional} if for all plays
$\forall \pi,\tau \in \Pi(q)$ we have $\pi^q_{end} = \tau^q_{end} \implies \sigma_i(\pi) = \sigma_i(\tau).$
The \emph{outcome} of a Player 1 strategy $\sigma_1 \in \Sigma_1(q)$ is the set of infinite plays that occur if Player 1 plays according to $\sigma_1$:
\[\textit{Outc}(\sigma_1) = \{ \pi \in \Pi^\infty(q) \mid \forall j \in \mathbb{N}: \
\sigma_1(\pi_{0:j}) = \pi_{j+1}^a \}\]
\end{definition}
A Player 1 strategy is winning if all its game outcomes are winning.
Player 1 can win the game by using a winning strategy from a state of its winning region.
\begin{definition} \label{def:winstrat}
Let $q \in Q$ be a game state.
A Player 1 strategy $\sigma_1 \in \Sigma(q)$ is \emph{winning}, if all plays from $\textit{Outc}(\sigma_1)$ are winning.
The Player 1 \emph{winning region} $\textit{WinReg}({G},R)$
for game ${G}$ and goal $R$ is the set of all states $Q' \subseteq Q$ such that for each $q \in Q'$, Player $1$ has a winning strategy $\sigma_1 \in \Sigma_1(q)$.
\end{definition}
\section{Joker games}
\label{sec:joker-games}
We formalize the notion of help by Player 2 by associating
to each concurrent game ${G}$ a Joker game $\textit{JAttr}(\game,\finally)ame$.
In $\textit{JAttr}(\game,\finally)ame$, Player 1 can always get control, i.e. choose any enabled move, at the cost of using a Joker. Thus, in any state $q$ of the game, Player 1 may either choose a regular Player 1 action $a \in \Gamma_1(q)$, or a Joker action $(a,x,q') \in \Gamma_1(q) \times \Gamma_2(q) \times Q$ with $q' \in \textit{Moves}(q,a,x)$. In this way, a Joker action encodes getting help from Player 2 and `the system' (the nondeterministic choice of a state from $\textit{Moves}(q,a,x)$).
We are interested in strategies using the minimum number of Jokers that Player 1 needs to win, because we will use this later
in model-based testing settings, where the
test execution is neither cooperative nor adversarial.
Thus, we set up a cost game where Joker actions have cost 1 and regular Player 1 actions cost 0. We use $\varspadesuit$ to define the parts of a Joker game that differ from those of a regular game.
With $\varspadesuitreal$, we specifically refer to states, moves, etc. where Jokers are actually played.
\begin{definition} \label{def:costgame}
We associate to each concurrent game ${G}$ a Joker game
$\textit{JAttr}(\game,\finally)ame=(Q,q^0,\textrm{Act}_1 \cup (\textrm{Act}_1 \times \textrm{Act}_2 \times Q),\textrm{Act}_2,\Gamma_1^\varspadesuit,\Gamma_2, \textit{Moves}^\varspadesuit,\textit{Cost}(\textit{JAttr}(\game,\finally)ame,\finally))$ where:
\begin{align*}
&\Gamma_1^\varspadesuitreal(q) = \{(a,x,q') \in \Gamma_1(q) \times\Gamma_2(q) \times Q \mid q' \in \textit{Moves}(q,a,x) \}\\
&\Gamma_1^\varspadesuit(q) = \Gamma_1(q) \cup \Gamma_1^\varspadesuitreal(q)\\
&\textit{Moves}^\varspadesuit (q,a,x) = \begin{cases}
\{q'\} &\text{if } a=(a',x',q') \in \Gamma_1^\varspadesuitreal(q)\\
\textit{Moves}(q,a,x) &\text{otherwise}
\end{cases}\\
&\textit{Cost}(\textit{JAttr}(\game,\finally)ame,\finally)(q,a) = \begin{cases}
1 &\text{if } a \in \Gamma_1^\varspadesuitreal(q)\\
0 &\text{otherwise}\\
\end{cases}
\end{align*}
\end{definition}
For the rest of the paper we fix Joker game $\textit{JAttr}(\game,\finally)ame$. We will write $\Sigma_1^\varspadesuit(q)$ for the set of strategies in $\textit{JAttr}(\game,\finally)ame$ starting at state $q$. Note that the states of both games are the same, and that all plays of ${G}$ are also plays of $\textit{JAttr}(\game,\finally)ame$.
\definitionref{def:costfunction} defines the cost of plays, strategies, and states.
The cost of a play $\pi$ arises by adding the costs of all moves until the goal $R$ is reached. If $R$ is never reached, the cost of $\pi$ is $\infty$.
The cost of a strategy $\sigma_1$ considers the worst case resolution of Player 2 actions and nondeterministic choices.
The cost of a state $q$ is the cost of Player 1's cost-minimal strategy from $q$.
\begin{definition} \label{def:costfunction}
Let $q \in Q$ be a state, $\pi \in \Pi^\infty(q) \cup \Pi(q)$ a play, and $\sigma_1 \in \Sigma_1^\varspadesuit(q)$ a strategy in Joker game $\textit{JAttr}(\game,\finally)ame$. For goal states $R$, define their cost as follows:
\begin{align*}
\textit{Cost}(\textit{JAttr}(\game,\finally)ame,\finally)(\pi) & =
\begin{cases}
\Sigma^{{\textit{WinInd}(\pi,R)}-1}_{j=0} \textit{Cost}(\textit{JAttr}(\game,\finally)ame,\finally)(q_j,a_j) & \text{if } \pi \text{is winning}\\
\infty & \text{otherwise}\\
\end{cases}\\
\textit{Cost}(\textit{JAttr}(\game,\finally)ame,\finally)(\sigma_1) & =
\sup_{\pi \in \textit{Outc}(\sigma_1)} \textit{Cost}(\textit{JAttr}(\game,\finally)ame,\finally)(\pi)\\
\textit{Cost}(\textit{JAttr}(\game,\finally)ame,\finally)(q) & = \inf_{\sigma_1 \in \Sigma_1^\varspadesuit(q) } \textit{Cost}(\textit{JAttr}(\game,\finally)ame,\finally)(\sigma_1)
\end{align*}
A strategy $\sigma \in \Sigma_1^\varspadesuit(q)$ is \emph{cost-minimal} if
$\sigma \in \underset{\sigma_1 \in \Sigma_1^\varspadesuit(q)}{\mathrm{arg inf}} \{\textit{Cost}(\textit{JAttr}(\game,\finally)ame,\finally)(\sigma_1) \}$
\end{definition}
\myparagraph{Winning states in Joker games.}
Normally, the value of a cost game is computed by a fixed point computation.
\begin{align*}
v_0(q) & = 0 \text{ if $q\in R$} \\
v_0(q) & = \infty \text{ if $q\notin R$}\\
v_{k+1}(q) & = \min_ {a\in\Gamma_1(q)}\max_{x\in\Gamma_2(q)}\max_{q'\in\textit{Moves}(q,a,x)} \textit{Cost}(\textit{JAttr}(\game,\finally)ame,\finally)(q,a) + v_k (q')
\end{align*}
\newcommand{{\textit{Distr}}}{{\textit{Distr}}}
Joker games allow this computation to be simplified, by exploiting their specific structure with cost 0 for competitive and cost 1 for cooperative moves.
We adapt
the classical attractor construction (see e.g., \cite{zielonka}) on the original game ${G}$.
The construction relies on two concepts:
the predecessor $\textit{Pre}(Q')$ contains all states with some move into $Q'$;
the controllable predecessor contains those states where Player 1 can force the game into $Q'$, no matter how Player 2 plays and how the nondeterminism is resolved.
We note that $\textit{Pre}(Q')$ can be equivalently defined as the states $q \in Q$ with $(a,x,q') \in \Gamma_1^\varspadesuitreal(q)$ (for $q'\in Q'$).
\begin{definition}
\label{def:predecessors}
Let $Q' \subseteq Q$ be a set of states.
The \emph{predecessor} $\textit{Pre}(Q')$ of $Q'$, and the \emph{controllable predecessor} $\textit{CPre}_1(Q')$ of $Q'$ are:
\begin{align*}
\textit{Pre}(Q') & = \{q \in Q \mid \exists a \in \Gamma_1(q), \exists x \in \Gamma_2(q), \exists q' \in \textit{Moves}(q,a,x): q' \in Q'\}\\
\textit{CPre}_1(Q') & = \{q \in Q \mid \exists a \in \Gamma_1(q), \forall x \in \Gamma_2(q): \textit{Moves}(q,a,x) \subseteq Q'\}
\end{align*}
\end{definition}
The classical \emph{attractor} is the set of states from which Player 1 can force the game to reach $R$, winning the game.
It is constructed by expanding the goal states via $\textit{CPre}_1$, until a fixed point is reached \cite{dAHK98};
the rank $k$ indicates in which computation step a state was added \cite{zielonka}.
Thus, the lower $k$, the fewer moves Player 1 needs to reach its goal.
\begin{definition} \label{def:attr}
The Player 1 \emph{attractor} is $\textit{Attr}(\game,\finally)$, where:
\begin{align*}
&\textit{Attr}(\game,\finally)i{0} = R\\
&\textit{Attr}(\game,\finally)i{k+1} = \textit{Attr}(\game,\finally)i{k}\ \cup \textit{CPre}_1(\textit{Attr}(\game,\finally)i{k})\\
&\textit{Attr}(\game,\finally) = \bigcup_{k \in \mathbb{N}} \textit{Attr}(\game,\finally)i{k}
\end{align*}
The function $\arankop(\game,\finally): Q \rightarrow \mathbb{N}$ associates to each state $q\in Q$ a rank $\arankop(\game,\finally)(q) = \min \{k \in \mathbb{N} \mid q \in \textit{Attr}(\game,\finally)i{k}\}$.
Recall that $\min \emptyset = \infty$.
\end{definition}
The cost of state $q$ in ${G}^\varspadesuit$ is obtained by interleaving the attractor and predecessor operators, computing sets of states that we call the \emph{Joker attractor}.
See \figureref{fig:ui} for an illustration of the computation.
Since $\textit{Attr}$ and $\textit{Pre}$ only use elements from ${G}$, the computation is performed in ${G}$. We will see that for defining the Joker attractor strategy (\definitionref{def:jokerstrat}) we do need Joker game $\textit{JAttr}(\game,\finally)ame$.\
In states of set $\joker{k}$, the game can be won with at most $k$ Jokers, i.e., with cost $k$.
Clearly, states that can be won by Player 1 in ${G}$ have cost 0, so these fall into $\joker{0}$.
Similarly, if all states in $Q'$ can be won with at most $k$ jokers, then so can states in $\textit{Attr}(Q')$.
In Joker states, Player 1 can only win from any opponent if she uses a Joker. By playing a Joker, the game moves to a state in $\joker{k}$. Joker states are the predecessors of $\joker{k}$.
\begin{figure}
\caption{Illustration of the Joker attractor computation: it starts with states in R initially and then adds states using the attractor and Joker attractor operations, until a fixpoint is reached.}
\label{fig:ui}
\end{figure}
\begin{definition} \label{def:jokers}
The Player 1 Joker attractor is $\textit{JAttr}(\game,\finally)$, where:
\begin{align*}
\joker{0} &= \textit{Attr}(\game,\finally)\\
\jokerreal{k+1} &= \joker{k} \cup \textit{Pre}(\joker{k})\\
\joker{k+1} &= \textit{Attr}(G,\jokerreal{k+1})\\
\textit{JAttr}(\game,\finally) &= \bigcup_{k \in \mathbb{N}} \joker{k}
\end{align*}
We call $\textit{JAttr}(\game,\finally)$ the Joker attractor of ${G}$. The Joker states are
$\jokerop_\varspadesuitreal(\game,\finally) = \bigcup_{k \in \mathbb{N}} \jokerreal{k+1}\setminus\joker{k}$.
To each Joker attractor $\textit{JAttr}(\game,\finally)$ we associate a Joker rank function $\jrankop(\game,\finally) : Q \rightarrow \mathbb{N}$, where for each state $q \in Q$ we define $\jrankop(\game,\finally)(q) = \min \{k \in \mathbb{N} \mid q \in \joker{k}\}$.
\end{definition}
\begin{example} \label{exmp:jokerstrat}
We compute the Joker attractor for goal \smiley\ in for concurrent game ${G}_{a\vee b}$ of \figureref{fig:jokergame}, together with the $\jrankop(\game,\finally)$. We indicate whether the state is a Joker state or not, i.e., whether Player 1 plays a Joker.
\begin{minipage}{0.6\textwidth}
\footnotesize
\begin{align*}
\textit{JAttr}^0({G}_{a\vee b},\smiley) &= \{\smiley\}\\
\jokerreal{1}({G}_{a\vee b},\smiley) &= \{1,2,\smiley\}\\
\textit{JAttr}^{1}({G}_{a\vee b},\smiley) &= \{1,2,4,\smiley\}\\
\jokerreal{2}({G}_{a\vee b},\smiley) &= \\%\{1,2,3,4,\smiley\}\\
\textit{JAttr}^{2}({G}_{a\vee b},\smiley) = \textit{JAttr}^{3}({G}_{a\vee b},\smiley) &= \{1,2,3,4,\smiley\}
\end{align*}
\end{minipage}
\quad
\begin{minipage}{0.3\textwidth}
\centering
\footnotesize
\begin{tabular}{c|c|c}
$q \in Q$ & $J\rankop
$ & Joker state\\\hline
1 & 1 & yes\\
2 & 1 & yes \\
3 & 2 & yes\\
4 & 1 & no\\
\smiley & 0 & no \\
\frownie & $\infty$ & no\\
\end{tabular}\\
\noindent
\end{minipage}
\end{example}
\noindent
\begin{figure}
\caption{Left: concurrent game ${G}
\label{fig:jokergame}
\end{figure}
\theoremref{thm:costgameisjokergameNew}
states the correctness of equations in
\definitionref{def:jokers}:
the number of Jokers needed to reach $R$, i.e. $\jrankop(\game,\finally)(q)$, equals $\textit{Cost}(\textit{JAttr}(\game,\finally)ame,\finally)(q)$.
\begin{restatable}{theorem}{jokeriscost}
\label{thm:costgameisjokergameNew}
For all $q \in Q$, we have
$\jrankop(\game,\finally)(q)=\textit{Cost}(\textit{JAttr}(\game,\finally)ame,\finally)(q)$.
\end{restatable}
\myparagraph{Winning strategies in Joker games.}
The attractor construction cannot only be used to compute the states of the Joker attractor, but also to construct the corresponding winning strategy. To do so, we need to update the definitions of the (controllable) predecessor.
This requires some extra administration, recording which actions
used in $\textit{Pre}$ and $\textit{CPre}_1$ are \emph{witnesses} for moving to states $Q'$.
\begin{align*}
w\textit{Pre}(Q') & = \{(q,a,x) \in Q \times \textrm{Act}_1 \times \textrm{Act}_2 \mid a \in \Gamma_1(q) \wedge x \in \Gamma_2(q)\\
&\qquad\qquad\qquad\qquad\qquad\qquad\quad\wedge (\exists q' \in \textit{Moves}(q,a,x): q' \in Q')\}\\
w\textit{CPre}_1(Q') & = \{(q,a) \in Q \times \textrm{Act}_1 \mid a \in \Gamma_1(q) \wedge (\forall x \in \Gamma_2(q): \textit{Moves}(q,a,x) \subseteq Q')\}
\end{align*}
Now, we obtain the sets of winning actions, by replacing, in the construction of $\textit{JAttr}(\game,\finally)$, $\textit{Pre}$ and $\textit{CPre}$ by
$w\textit{Pre}$ and $w\textit{CPre}_1$, respectively.
With $w\textit{Pre}$ and $w\textit{CPre}_1$ we select actions for moving from the states that are newly added to the $k+1$-th attractor, or the $k+1$-th Joker attractor set, respectivly, to move to states of the $k$-th attractor, or $k$-th Joker attractor set, respectively.
\begin{definition} \label{def:witness}
We define the \emph{witnessed} attractor $w\textit{Attr}(\game,\finally)$ and Joker attractor $w\textit{JAttr}(\game,\finally)$:
\begin{align*}
w\textit{Attr}(\game,\finally)i{0} &= \emptyset\\
w\textit{Attr}(\game,\finally)i{k+1} &= w\textit{Attr}(\game,\finally)i{k} \cup \{(q,a) \in w\textit{CPre}_1(\textit{Attr}(\game,\finally)i{k}) \mid q \notin \textit{Attr}(\game,\finally)i{k}\}\\
w\textit{Attr}(\game,\finally) &= \bigcup_{k \in \mathbb{N}}w\textit{Attr}(\game,\finally)i{k}\\
w\joker{0}&= \emptyset\\
w\joker{k+1} &= w\joker{k} \cup \{ (q,a,x) \in w\textit{Pre}(\joker{k}) \mid q \notin \joker{k} \}\\
w\textit{JAttr}(\game,\finally) &= \bigcup_{k \in \mathbb{N}}w\joker{k}
\end{align*}
\end{definition}
A {\em Joker attractor strategy} in $\textit{JAttr}(\game,\finally)ame$ is a strategy that
plays according to the witnesses: in Joker states, a Joker action from $w\joker{k}$ is played. In non-Joker states $q$, the strategy takes its action from $w\textit{Attr}(\game,\finally)$ if the state has $\jrankop(\game,\finally)(q)$=0, and from $w\textit{Attr}(\jokerreal{k})$ if the state has $\jrankop(\game,\finally)(q) = k$ for some $k > 0$.
\begin{definition}\label{def:jokerstrat}
A strategy $\sigma_1 \in \Sigma_1^\varspadesuit(q)$ in $\textit{JAttr}(\game,\finally)ame$ is a \emph{Joker attractor strategy}, if for any $\pi \in \textnormal{play?}pref(\textit{JAttr}(\game,\finally)ame)$ with $\pi^q_{end} \in \textit{JAttr}(\game,\finally)$ and $\jrankop(\game,\finally)(\pi^q_{end}) = k$ we have:
\begin{align*}
&(\pi^q_{end} \in \jokerop_\varspadesuitreal(\game,\finally) \implies \sigma_1(\pi) \in w\textit{JAttr}(\game,\finally))\qquad \wedge\\
&(k = 0 \wedge \pi^q_{end} \notin R \implies (\pi^q_{end},\sigma_1(\pi)) \in w\textit{Attr}(\game,\finally))\qquad \wedge\\
&(k>0 \wedge \pi^q_{end} \notin \jokerop_\varspadesuitreal(\game,\finally)\implies (\pi^q_{end},\sigma_1(\pi)) \in w\textit{Attr}(\jokerreal{k}))
\end{align*}
\end{definition}
\section{Properties of Joker games}
\label{sec:properties-joker-games}
We establish five fundamental properties of Joker games.
\myparagraph{1. All outcomes use same number of Jokers.}
A first, and perhaps surprising result is that, all outcomes in Joker attractor strategy $\sigma$, use exactly the same number of Joker actions, namely $\jrankop(\game,\finally)(q)$ Jokers, if $\sigma$ starts in state
$q$. This follows from the union-like computation illustrated in \figureref{fig:ui}.
This is unlike cost-minimal strategies in general cost games, where some outcomes may use lower costs.
This is also unlike cost-minimal strategies in $\textit{JAttr}(\game,\finally)$ that are not obtained via the attactor construction, see \figureref{fig:costless}.
\begin{restatable}{theorem}{nrjokermoves}
\label{thm:nrjokermoves}
Let $q \in \textit{JAttr}(\game,\finally)$.
Then:
\begin{enumerate}
\item Let $\sigma^J_1 \in \Sigma_1^\varspadesuit(q)$ be a Joker attractor strategy in $\textit{JAttr}(\game,\finally)ame$.
Then any play $\pi \in \textit{Outc}(\sigma^J_1)$ has \emph{exactly} $\jrankop(\game,\finally)(q)$ Joker actions in winning prefix $\pi_{0:\textit{WinInd}(\pi)}$.
\item Let $\sigma_1 \in \Sigma_1^\varspadesuit(q)$ be a cost-minimal strategy in $\textit{JAttr}(\game,\finally)ame$.
Then any play $\pi \in \textit{Outc}(\sigma_1)$ has \emph{at most} $\jrankop(\game,\finally)(q)$ Joker actions in the winning prefix $\pi_{0:\textit{WinInd}(\pi)}$.
\end{enumerate}
\end{restatable}
\begin{figure}
\caption{A cost-minimal strategy is depicted by the dashed edges. It plays a cost-0 $a$ action in state 1, and hopes Player 2 plays $x$ to arrive in {\smiley}
\label{fig:costless}
\end{figure}
The intuition of the proof of \theoremref{thm:nrjokermoves} can be derived from \figureref{fig:costless}: a Joker is always played in a Joker state. By construction of the Joker attractor strategy, the Joker action moves the game from $q$ to a state $q'$ with $\jrankop(\game,\finally)(q') + 1 = \jrankop(\game,\finally)(q)$. In non-Joker states no Joker is used to reach a next Joker state.
\myparagraph{2. Characterization of winning states.} For the set of states $\textit{JAttr}(\game,\finally)$, we establish that:
(1) for every goal $R$, the winning region in Joker game $\textit{JAttr}(\game,\finally)ame$ (i.e., states where Player 1 can win with any strategy, not necessarily cost-minimal) coincides with the Joker attractor $\textit{JAttr}(\game,\finally)$, and
(2) the set of all states having a play reaching a state from $R$ coincide with Joker attractor $\textit{JAttr}(\game,\finally)$.
\begin{restatable}{theorem}{jokerwinreach}
\label{thm:jokerwinreach}
Let ${\reachop(\game,\finally)} =
\{q \in Q \mid q \text{ can reach a state } q'\in R\}$.
Then
\begin{align*}
\textit{WinReg}(\textit{JAttr}(\game,\finally)ame,R) = \textit{JAttr}(\game,\finally) = {\reachop(\game,\finally)}
\end{align*}
\end{restatable}
\myparagraph{3. Joker attractor strategies and minimality.}
We show some fundamental results relating Joker attractor strategies to Joker-minimal strategies: \theoremref{thm:jokerstratwinning} states correctness of Joker attractor strategies: they are indeed cost-minimal.
The converse, cost-minimal strategies are Joker attractor strategies, is not true, as shown by \figureref{fig:costless} and \ref{fig:costminnojokerstrat}. The game of
\figureref{fig:costminnojokerstrat} also shows that Joker attractor strategies need not take the shortest route to the goal (see also \sectionref{sec:short}).
\begin{SCfigure}[9]
\begin{tikzpicture}[shorten >=1pt,node distance=1.1cm,>=stealth']
\tikzstyle{every state}=[draw=black,text=black,inner sep=1pt,minimum size=11pt,initial text=]
\node[state,fill=lightgray,initial,initial where=above] (1) {1};
\node[state,fill=lightgray] (2) [right of=1] {2};
\node[state,fill=lightgray] (3) [left of=1] {3};
\node[state,fill=lightgray] (4) [below of=2] {4};
\node[state,fill=lightgray] (5) [below of=3] {5};
\node[state,fill=lightgray] (6) [below of=5] {6};
\node[circle,inner sep=-2pt,fill=lightgray] (7) [below of=4] {\LARGE\smiley};
\node[circle,inner sep=-2pt,fill=lightgray] (8) [below of=1] {\LARGE\frownie};
\path[->]
(1) edge [dashed] node [above] {$a$ $x$} (2)
(1) edge [dotted] node [above] {$b$ $x$} (3)
(2) edge [dashed] node [left] {$a$ $x$} (4)
(3) edge [dotted] node [left] {$a$ $x$} (5)
(3) edge node [above right=-1mm and -1mm] {$a$ $y$} (8)
(4) edge node [below] {$a$ $y$} (8)
(4) edge [dashed] node [left] {$a$ $x$} (7)
(5) edge [dotted] node [left] {$a$ $x$} (6)
(6) edge [dotted] node [above] {$a$ $x$} (7)
;
\end{tikzpicture}
\caption{This game ${G}_{cost}$ has a cost-minimal strategy (dashed; cost 1 for the Joker used in state 4) that is not a Joker attractor strategy. The unique Joker attractor strategy (dotted) selects Player 1 action $b$ from state 1 (to Joker state 3), since $1,2 \in \textit{Attr}^1({G}_{cost},\textit{JAttr}^1_\varspadesuitreal({G}_{cost},\{\smiley\}))$.
The dashed strategy requires fewer moves to reach {\smiley} (3 vs 4).}
\label{fig:costminnojokerstrat}
\end{SCfigure}
\begin{restatable}{theorem}{jokerstratcostminimal}
\label{thm:jokerstratwinning}
Any Joker attractor strategy is cost-minimal.
\end{restatable}
\myparagraph{4. Determinacy.}
Unlike arbitrary concurrent games with deterministic strategies, Joker games are \emph{determined}. That is, in each state of the Joker game, either Player 1 has a winning strategy or Player 2 can keep the game outside $R$ forever. Determinacy (\theoremref{thm:jokerdetermined}) follows from \theoremref{thm:jokerwinreach}: by $\textit{WinReg}(\textit{JAttr}(\game,\finally)ame,R) = \textit{JAttr}(\game,\finally)$ we have a winning strategy in any state from $\textit{JAttr}(\game,\finally)$, and by $\textit{JAttr}(\game,\finally) = {\reachop(\game,\finally)}$ we have that states not in $\textit{JAttr}(\game,\finally)$ have no play to reach $R$, so Player 2 wins with any strategy.
\begin{restatable}{theorem}{jokerdetermined}
\label{thm:jokerdetermined}
Joker games are determined.
\end{restatable}
\section{Joker games with randomized strategies}
\label{sec:randstrats}
A distinguishing feature of concurrent games is that, unlike turn-based games, randomized strategies may help to win a game. A classical example is the Penny Matching Game of
\figureref{fig:penny}.
We show that randomization is not needed for winning Joker games, but does help to reduce the number of Joker moves. However, randomization in Joker states is never needed. We first set up the required machinery, following \cite{dAHK98}.
\begin{SCfigure}[8]
\centering
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,>=stealth']
\tikzstyle{every state}=[draw=black,text=black,inner sep=1pt,minimum
size=11pt,initial text=]
\node[state,initial,initial where=above, fill=lightgray] (1) {1};
\node[circle,inner sep=-2pt, fill=lightgray] (3) [right of=1] {\LARGE\smiley};
\path[->]
(1) edge [loop left] node [left] {H T} (1)
(1) edge [loop below] node [below] {T H} (1)
(1) edge node [above] {H H} (3)
(1) edge [bend right] node [below] {T T} (3)
;
\end{tikzpicture}
\caption{Penny Matching Game: Each Player chooses a side of a coin. If they both choose heads, or both tails, then Player 1 wins, otherwise they play again. Player 1 wins this game with probability 1, by flipping the coin (each side has probability $\frac 12$). }
\label{fig:penny}
\end{SCfigure}
\begin{definition}
\label{def:probstrat}
A \emph{(probability) distribution} over a finite set $X$ is a function $\nu:X\to [0,1]$ such that $\sum_{x\in X}\nu(x)=1$. The set of all probability distributions over $X$ is denoted ${\textit{Distr}}(X)$. Let $q \in Q$. A randomized Player $i$ strategy from $q$ is a function $\sigma_i: {\textit{Pref}}(q) \rightarrow {\textit{Distr}}(\textrm{Act}_i)$, such that $\sigma_i(\pi)(a)>0$ implies $a \in \Gamma_i(\pi^q_{end})$.
Let $\Sigma_i^r(q)$ denote the randomized strategies for Player $i \in \{1,2\}$ from $q$.
\end{definition}
To define the probability of an outcome, we must not only know how player Players 1 and 2 play, but also how the nonderminism is resolved. Given moves $a$ and $x$ by Player 1 and 2, the Player 3 strategy $\sigma_3(\pi,a,x)$ chooses, probabilistically, one of the next states in $\textit{Moves}(\pi^q_{end},a,x)$.
\begin{definition} \label{def:probplayerthree}
A randomized Player 3 strategy is a function $\sigma_3: \Pi(q) \times \textrm{Act}_1 \times \textrm{Act}_2 \rightarrow {\textit{Distr}}(Q)$, such that $\sigma_3(\pi,a,x)(q')>0$ implies $ q'\in \textit{Moves}(\pi^q_{end},a,x)$ for all $\pi \in \Pi(q)$.
We write $\Sigma_3^r(q)$ for the set of all randomized Player $3$ strategies from $q$.
The \emph{outcome} $\textit{Outc}(\sigma_1,\sigma_2,\sigma_3)$ of randomized strategies $\sigma_1 \in \Sigma_1^r(q)$, $\sigma_2 \in \Sigma_2^r(q)$, and $\sigma_3 \in \Sigma_3^r(q)$ are the plays $\pi \in \Pi^\infty(q)$ such that for all $j \in \mathbb{N}$:
$
\sigma_1(\pi_{0:j})(\pi_{j}^a) > 0 \ \wedge\ \sigma_2(\pi_{0:j})(\pi_{j}^x) > 0 \ \wedge\ \sigma_3(\pi_{0:j},\sigma_1(\pi_{0:j}),\sigma_2(\pi_{0:j}))( \pi_{j+1}^q) > 0
$
\end{definition}
Given randomized strategies for Player 1, Player 2, and Player 3, \definitionref{def:proboutc} defines the probability of a play prefix of the game. This probability is computed as the multiplication of the probabilities given by the strategies for the play prefix. A Player 1 strategy is then an \emph{almost sure winning strategy} if it can win from any Player 2 and Player 3 strategy with probability 1.
\begin{definition}
\label{def:proboutc}
Let $\sigma_1 \in \Sigma_1^r(q_0)$, $\sigma_2 \in \Sigma_2^r(q_0)$, and $\sigma_3 \in \Sigma_3^r(q_0)$ be randomized strategies, and let
$\pi = q_0\langle a_0,x_0 \rangle q_1 \dots \langle a_{j-1},x_{j-1} \rangle q_j$ be a finite play prefix.
We define its probability as:
\begin{align*}
P(\pi) = \prod_{i=0}^{j-1} \sigma_1(\pi_{0:i})(a_i)\cdot \sigma_2(\pi_{0:i})(x_i)\cdot
\sigma_3\left (\pi_{0:i},\sigma_1(\pi_{0:i}),\sigma_2(\pi_{0:i})\right)(q_{i+1})
\end{align*}
The strategies $\bar{\sigma} = (\sigma_1, \sigma_2, \sigma_3)$ define a probability space
$(\Omega, \mathcal{F}, \mathcal{P}^{\bar{\sigma}})$ over the set of outcomes.
A Player 1 strategy $\sigma_1 \in \Sigma_1^r(q)$ is {\em almost sure winning} for reachability goal $R$ if for all $\sigma_2 \in \Sigma_2^r(q)$, and $\sigma_3 \in \Sigma_3^r(q)$,
we have:
$\mathcal{P}^{\bar{\sigma}}[\{\pi\in \textit{Outc}(\sigma_1,\sigma_2,\sigma_3) \mid \pi \text{ is winning}\} ]=1$. Strategy $\sigma_1$ is \emph{sure winning} if for all $\sigma_2 \in \Sigma_2^r(q)$, and $\sigma_3 \in \Sigma_3^r(q)$, and for any $\pi \in \textit{Outc}(\sigma_1,\sigma_2,\sigma_3)$, $\pi$ is winning.
\end{definition}
\myparagraph{Properties of Joker games with randomization.}
In \sectionref{sec:properties-joker-games}, several fundamental properties were given, which hold for randomized strategies too (\theoremref{thm:randomjokerproperties}).
By replacing the standard attractor in the definition of the Joker attractor (\definitionref{def:jokers}), by the probabilistic atractor \cite{dAHK98}, we obtain the definition of the probabilistic Joker attractor.
Next, by using this probabilistic Joker attractor instead of the Joker attractor, the definitions for the witnissed probablistic Joker attractor (\definitionref{def:witness}), and probablistic Joker strategies (\definitionref{def:jokerstrat}) can be reformulated.
From these probablistic definitions, \theoremref{thm:randomjokerproperties} then follows.
\begin{restatable}{theorem}{randomjokerproperties}
\label{thm:randomjokerproperties}
\theoremref{thm:nrjokermoves}, \ref{thm:jokerwinreach}, \ref{thm:jokerstratwinning} and \ref{thm:jokerdetermined} hold for Joker games with randomized strategies.
\end{restatable}
\myparagraph{The (non-)benefits of randomization in Joker games.}
We show that Joker games do not need randomized strategies:
if Player 1 can win a Joker game with a randomized strategy, then she can win this game with a deterministic strategy.
This result is less surprising than it may seem (\theoremref{thm:jokerrandom}(1)), since Joker actions are very powerful:
they can determine the next state of the game to be any state reachable in one move.
In the Penny Matching Game, Player 1 may in state 1 just take the Joker move $(H,H,\smiley)$ and reach the state {\smiley} immediately. With randomization, Player 1 can win this game without using Joker moves.
We note however that we only need to use the power of Jokers in Joker states (\theoremref{thm:jokerrandom}(2)). We can use a probabilistic attractor \cite{dAHK98} to attract to Joker states with probability 1, and then use a Joker move in this Joker state, where even using randomization, the game cannot be won with probability 1. The Jokers used in these Joker states then only needs to be played deterministically, i.e. with probability 1 (\theoremref{thm:jokerrandom}(3)). The intuition for this last statement is that a Joker move determines the next state completely, so by choosing the `best' next state there is no need to include a chance for reaching any other state.
\begin{restatable}{theorem}{jokernotrandom}
\label{thm:jokerrandom}
If a state $q\in Q$ of Joker game $\textit{JAttr}(\game,\finally)ame$ has an almost sure winning strategy $\sigma_1^r \in \Sigma_1^{r,\varspadesuit}(q)$, then
\begin{enumerate}
\item she also has a winning deterministic strategy.
\item she also has an almost sure winning strategy that only uses Jokers in Joker states.
\item she also has an almost sure winning strategy that only uses Jokers in Joker states, such that these Jokers can all be played with probability 1.
\end{enumerate}
\end{restatable}
\section{Better help from your friends: multi-objective and admissible strategies}
\label{sec:short}
\myparagraph{Short Joker strategies.}
Although multiple Joker attractor strategies may be constructed via \definitionref{def:jokerstrat},
their number of moves may not be minimal.
The cause for this is that Joker attractor strategies have to reduce the $\arankop(\game,\finally)$ or $\jrankop(\game,\finally)$ by 1 each step. \figureref{fig:costminnojokerstrat} shows that this is not always beneficial for the total number of moves taken towards the goal: Joker attractor strategies may need more moves in total than other cost-minimal strategies.
To take a shortest path while spending the minimum number of Jokers, we use the structure of the Joker attractor sets to compute a distance function.
\definitionref{def:short} defines distance for the following four types of states: goal states $R$, Joker states $\jokerop_\varspadesuitreal(\game,\finally)$, non-Joker states part of some attractor $\jokerop_{\not\varspadesuit}(\game,\finally)$, and unreachable states (states not in $\textit{JAttr}(\game,\finally)$).
In a goal state, the distance is 0, and in an unreachable state it is infinite.
In a Joker state, we use the Joker action to the state of the next Joker attractor set that has the smallest distance to the goal. In a non-Joker state from some attractor, we choose a Player 1 action that results in reaching a state within the current Joker attractor set, such that this action minimizes the distance to the goal from the reached state.
In the latter case we assume that Player 2 and 3 cooperate with decreasing the distance, by using $\min$ in the definition.
We chose this because we use the definition in \theoremref{thm:shortcostminimal}, where we consider the situation that Player 2 and 3 cooperate.
Less cooperation can be assumed by e.g. replacing $\min$ by $\max$.
\definitionref{def:postrestr} states auxilary definitions used in \definitionref{def:short}. Most importantly it defines the Joker restricted enabling condition $\Gamma_1^{J}(q)$ that returns Player 1 actions that surely lead to states within the current Joker attractor set.
\begin{definition}
\label{def:postrestr}
Let $q \in Q$, and $Q' \subseteq Q$. Define the states reachable in one move $\textit{Post}(q)$, the $k$-th Joker states $\jokerop_\varspadesuitreal(\game,\finally)k{k}$, the non-Joker, attractor states $\jokerop_{\not\varspadesuit}(\game,\finally)$, the $k$-th non-Joker states, the restricted enabling condition $\Gamma_1(Q')$, and the Joker restricted enabling condition $\Gamma_1^{J}$ as:
\begin{align*}
\textit{Post}(q) &= \{q' \in Q \mid \exists a \in \Gamma_1(q), \exists x \in \Gamma_2(q): q' \in \textit{Moves}(q,a,x)\}\\
\jokerop_\varspadesuitreal(\game,\finally)k{k} &= \jokerreal{k}\setminus\joker{k-1}\\
\jokerop_{\not\varspadesuit}(\game,\finally) &= \textit{JAttr}(\game,\finally)\setminus\jokerop_\varspadesuitreal(\game,\finally)\\
\jokerop_{\not\varspadesuit}(\game,\finally)k{k} & = \joker{k}\setminus\jokerreal{k}\\
\Gamma_1(Q')(q) &= \{a \in \Gamma_1(q) \mid \forall x \in \Gamma_2(q): \textit{Moves}(q,a,x) \subseteq Q'\}\\
\Gamma_1^{J}(q) &=
\begin{cases}
\Gamma_1(\textit{Attr}(\game,\finally))(q) &\text{if }q \in \textit{Attr}(\game,\finally)\\
\Gamma_1(\joker{k+1}\setminus\joker{k})(q) &\text{if }q \in \jokerop_{\not\varspadesuit}(\game,\finally)k{k+1}\\
\emptyset &\text{otherwise}
\end{cases}
\end{align*}
\end{definition}
\begin{definition} \label{def:short}
We define the distance function $d : Q \rightarrow \mathbb{N}$ as follows:
\begin{align*}
d(q) =
\begin{cases}
0 &q \in R\\
1 + \min_{q'\in \textit{Post}(q) \cap \joker{k}}{d(q')} &q \in \jokerop_\varspadesuitreal(\game,\finally)k{k+1}\\
1 + \min_{a \in \Gamma_1^{J}(q)}{\min_{x \in \Gamma_2(q)}{\min_{q'\in\textit{Moves}(q,a,x)}{d(q')}}} &q \in \jokerop_{\not\varspadesuit}(\game,\finally)\\
\infty &q \notin \textit{JAttr}(\game,\finally)
\end{cases}
\end{align*}
\end{definition}
A short Joker strategy (\definitionref{def:diststrat}) minimizes the distance while using the minimum number of Joker actions. The construction of such a strategy from the distance function is straightforward:
we use a distance minimizing Player 1 action in a non-Joker state, and a distance minimizing Joker action in a Joker state. This distance minimization follows the structure of \definitionref{def:short}.
\begin{definition}
\label{def:diststrat}
A strategy $\sigma \in \Sigma_1(q)$ in Joker game $\textit{JAttr}(\game,\finally)ame$ is a \emph{short Joker strategy} from $q$, if for any play $\pi \in \textit{Outc}(\sigma)$ with $q' = \pi_{end}$ the following formulas hold:
\begin{align*}
q' \in \jokerop_\varspadesuitreal(\game,\finally) &\implies \sigma(\pi) \in \argmin_{(a,x,q'') \in \Gamma_1^\varspadesuitreal(q')} \{d(q'') \}\\
q' \in \jokerop_{\not\varspadesuit}(\game,\finally) &\implies \sigma(\pi) \in \argmin_{a \in \Gamma_1^J(q')} \{ d(q'') \mid x \in \Gamma_2(q'), q'' \in \textit{Moves}(q',a,x)\}
\end{align*}
\end{definition}
Unfolding \definitionref{def:short} for state 1 of \figureref{fig:costminnojokerstrat} results in $d(1) = 3$, as expected. With \definitionref{def:diststrat} we then obtain a short Joker strategy that is cost-minimal (it uses 1 Joker action), and uses the minimum number of moves (namely 3).
\theoremref{thm:shortcostminimal} states that \definitionref{def:diststrat} indeed defines cost-minimal strategies using the minimum number of moves for the number of used Joker actions, when helped by Player 2 and 3 (i.e. there is a play). A short Joker strategy prefers the minimum number of Joker actions over the minimum number of moves, so it will not take a shorter route if that costs more than the minimum number of Jokers.
\begin{restatable}{theorem}{shortcostminimal}
\label{thm:shortcostminimal}
A short Joker strategy $\sigma$ is cost-minimal (1), and has a play $\pi \in \textit{Outc}(\sigma)$ using the minimum number of moves to win while using the minimum number of Joker actions (2).
\end{restatable}
\myparagraph{Admissible strategies.} \label{sec:admissible}
Various papers \cite{admissibilityingraphs,nonzerosumgames,Faella09} advocate that a player should play \emph{admissible} strategies if there is no winning strategy.
Admissible strategies (\definitionref{def:dominance}) are the maximal elements in the lattice of all strategies equipped with the dominance order $<_d$. Here, a Player 1 strategy $\sigma_1$ dominates strategy $\sigma'_1$, denoted $\sigma'_1 <_d \sigma_1$,
iff whenever $\sigma'_1$ wins from opponent strategy $\rho$, so does $\sigma_1'$.
In \definitionref{def:dominance}, the set of winning strategies is defined as the pairs of Player 2 \'and Player 3 strategies, since the nondeterministic choice for the next state affects whether Player 1 wins. \definitionref{def:playerthree} defines non-randomized Player 3 strategies, similar to its randomized variant in \definitionref{def:probplayerthree}.
\begin{definition} \label{def:playerthree}
A Player 3 strategy is a function $\sigma_3: \Pi(q) \times \textrm{Act}_1 \times \textrm{Act}_2 \rightarrow Q$, such that $\sigma_3(\pi,a,x) \in \textit{Moves}(\pi^q_{end},a,x)$ for all $\pi \in \Pi(q)$.
We write $\Sigma_3(q)$ for the set of all Player $3$ strategies from $q$.
The \emph{outcome} $\textit{Outc}(\sigma_1,\sigma_2,\sigma_3)$ of strategies $\sigma_1 \in \Sigma_1(q)$, $\sigma_2 \in \Sigma_2(q)$, and $\sigma_3 \in \Sigma_3(q)$ is the play $\pi \in \Pi^\infty(q)$ such that:
\[\forall j \in \mathbb{N}: \
\sigma_1(\pi_{0:j}) = \pi_{j}^a \wedge \sigma_2(\pi_{0:j}) = \pi_{j}^x \wedge \sigma_3(\pi_{0:j},\sigma_1(\pi_{0:j}),\sigma_2(\pi_{0:j})) = \pi_{j+1}^q\]
\end{definition}
\iffalse
In this section, we consider ordinary concurrent games again, without Joker actions for Player 1.
According to several papers \cite{admissibilityingraphs,nonzerosumgames,Faella09}, a player should prefer \emph{admissible} strategies in such games. There is no strategy that dominates, i.e.\ is strictly `better', than such an admissible strategy, because it can win from more opponent strategies.
In particular, Faella \cite{Faella09} concludes that a player should prefer admissible strategies when the game is in a state outside the player's winning region, i.e. states from $\textit{JAttr}(\game,\finally)\setminus\textit{Attr}(\game,\finally)$. In these states we defined our Joker attractor strategies. This section therefore investigates the relation between Joker attractor strategies and admissible strategies.
To make this comparison, we translate Joker attractor strategies in the Joker game $\textit{JAttr}(\game,\finally)ame$ to ordinary strategies in the concurrent game ${G}$, by using the Player 1 action of any Joker action. We call the resulting strategies \emph{Joker inspired strategies}.
\fi
\begin{definition} \label{def:dominance}
Let ${G}$ be a concurrent game and $R$ a reachability goal.
Define the Player 2 and 3 strategy pairs that are winning for a strategy $\sigma_1 \in \Sigma_1({G})$ as:
\begin{align*}
{\textit{Win$\Sigma$}}_{2,3}(\sigma_1,R) = \{(\sigma_2,\sigma_3) \in \Sigma_2({G}) \times \Sigma_3({G}) \mid \textit{Outc}(\sigma_1,\sigma_2,\sigma_3) \in {\textit{Win$\Pi$}}({G},R) \}
\end{align*}
For any $q \in Q$, a strategy $\sigma_1 \in \Sigma_1(q)$ is \emph{dominated by} a strategy $\sigma_1' \in \Sigma_1(q)$, denoted $\sigma_1 <_d \sigma_1'$, if ${\textit{Win$\Sigma$}}_{2,3}(\sigma_1,R) \subset {\textit{Win$\Sigma$}}_{2,3}(\sigma_1',R)$.
Strategy $\sigma_1 \in \Sigma_1(q)$ is \emph{admissible} if there is no strategy $\sigma_1' \in \Sigma_1(q)$ with $\sigma_1 <_d \sigma_1'$.
\end{definition}
To compare Joker attractor strategies and admissible strategies, we note that Joker attractor strategies are played in Joker games, where Joker actions have full control over the opponent. Admissible strategies, however, are played in regular concurrent games, without Joker actions.
To make the comparison, \definitionref{def:insp} therefore
associates to any Player 1 strategy $\sigma$ in $\textit{JAttr}(\game,\finally)ame$, a Joker-inspired strategy $\sigma_{insp}$ in $G$: if $\sigma$ chooses Joker action $(a,x,q)$, then $\sigma_{insp}$ plays Player 1 action $a$.
\begin{definition} \label{def:insp}
Let $\sigma \in \Sigma(q)$ be a strategy in $\textit{JAttr}(\game,\finally)ame$. Define the \emph{Joker-inspired strategy} $\sigma_{insp}$ of $\sigma$ for any $\pi \in \Pi({G})$ as:
\begin{align*}
\sigma_{insp}(\pi) = \begin{cases}
a &\text{ if } \sigma(\pi) = (a,x,q)\\
\sigma(\pi) &\text{ otherwise}
\end{cases}
\end{align*}
\end{definition}
\begin{figure}
\caption{
Left: the dashed cost-minimal strategy uses bad action $b$ in state 2 instead of good action $g$, because using Joker ($a$,$h$,\smiley) in state 1 will make Player 1 win the game. Hence, the Joker-inspired strategy of this winning cost-minimal strategy is not admissible, as it is dominated by strategies choosing $g$ in state 2.
Middle: the dotted strategy uses 2 Jokers, and is admissible, since it wins from strategy $\sigma_2^a$ while the dashed cost-minimal strategy (using 1 Joker) does not.
Right: the dashed cost-minimal strategy dominates dotted cost-minimal strategy, because dashed strategy wins from $\sigma_2^{uh}
\label{fig:admcounter}
\end{figure}
It turns out that Joker-inspired strategies of cost-minimal strategies need not be admissible, for a rather trivial reason: a cost-minimal strategy that chooses a losing move in a non-visited state is not admissible. See \figureref{fig:admcounter}(left). Therefore, \theoremref{thm:adm} relates admissible and \emph{global} cost-minimal strategies (\definitionref{def:globcostmin}): strategies that are cost-minimal for any initial state of the Joker game. Then still, the classes of admissible and of the Joker-inspired variants of global cost-minimal strategies do not coincide. \figureref{fig:admcounter}(middle,right) shows two examples, with Player 2 strategies using memory, for both parts of \theoremref{thm:adm}: (1) an admissible strategy need not be cost-minimal, and (2) vice versa.
We conjecture that the converse of \theoremref{thm:adm} holds for memoryless Player 2 and 3 strategies.
\begin{definition} \label{def:globcostmin}
Let $\sigma \in \Sigma_1^\varspadesuit(q)$ be a strategy in Joker game $\textit{JAttr}(\game,\finally)ame$. Then $\sigma$ is \emph{global cost-minimal} if $\sigma$ is cost-minimal from any $q' \in {\reachop(\game,\finally)}$.
\end{definition}
\begin{restatable}{theorem}{admjoker}
\label{thm:adm}\text{ }
\begin{enumerate}
\item A Joker-inspired strategy of a global cost-minimal strategy is not always admissible.
\item A admissble strategy is not always a Joker-inspired strategy of a global cost-minimal strategy.
\end{enumerate}
\end{restatable}
\section{Experiments}
\label{sec:experiments}
We illustrate the application of Joker games in model-based testing.
\myparagraph{Testing as a game.}
We translate input-output labeled transition systems describing the desired behaviour of the System Under Test (SUT) to concurrent games, as described in \cite{mbtgames}.
In each game state,
Player 1/Tester has 3 options: stop testing, provide one of the inputs to the SUT, or observe the behaviour of the SUT. Player 2/SUT has 2 options: provide an output to the SUT, or do nothing.
Hence Player 1 and 2 decide concurrently what their next action is.
The next state is then determined as follows: if the Tester provides an input, and the SUT does nothing, the input is processed. If the Tester observes the SUT (virtually also doing no action of its own), the output is processed. If the Tester provides an input, while the SUT provides an output, then several regimes are possible, such as input-eager, output-eager \cite{mbtgames}. We opt for the {\em nondeterministic} regime, where picked action is chosen nondeterministically (this corresponds with having a set of states $\textit{Moves}(q,a,x)$).
In this set up, we investigate the effectiveness of Joker-based testing through a comparison with randomized testing. In Joker-based testing we take the Joker-inspired strategies of Joker attractor strategies, to allow for a fair and realistic (no use of Joker actions) comparison.
\myparagraph{Case studies.} We applied our experiments on four case studies from \cite{VandenBosVaandragerJournal}: the opening and closing behaviour of the TCP protocol (26 states, 53 transitions) \cite{tcp}, an elaborate vending machine for drinks (269 states,687 transitions) \cite{vendmach,modelsward17}
the Echo Algorithm for leader election (105 states, 275 transitions) \cite{fokkink2013distributed}, and
a tool for file storage in the cloud (752 states, 1520 transitions) \cite{hughesdropbox,tretmansdropbox}.
\noindent
\textbf{Experimental setup.}
For each case study, we randomly selected the different goal states: 5 for the (relatively small) TCP case study and 15 for the other cases.
For each of these goals, we extract a Joker attractor strategy, and translate it via its corresponding Joker-inspired strategy, without Joker actions, to a test case, as described in \cite{mbtgames}.
We run this Joker test case 10.000 times. Also, we run 10.000 random test cases. A random test case chooses an action uniformly at random in any state.
Test case execution is done according to the standard model-against-model testing approach \cite{VandenBosVaandragerJournal}. In this setup, an impartial SUT is simulated from the model, by making the simulation choose any action with equal probability.
From the 10.000 test case runs for each goal of each case study, we compute:
(1) the number of runs that reach the goal state, and
(2) the average number of actions to reach the goal state, if a goal state was reached.
\myparagraph{Results.}
\figureref{fig:exp} shows the experimental results. In the left graph, a point represents one goal state of a case study, and compares the averages of the Joker test case, and random test case. For the right graph, for each goal, the number of actions needed for reaching the goal for Random testing has been divided by the number of actions for the Joker test case. Results for all goals of one case study are shown with one bar. We omitted results for 5 goal states of the Vending Machine case study, as these states were not reached in any of the 10.000 runs of random testing, while there were $> 3500$ successful Joker runs for each of those goals.
Clearly, the graphs show that Joker test cases outperform random testing.
The experimental results can be reproduced with the artefact of this paper \cite{artefact}.
\begin{figure}
\caption{Experimental results: Joker-based versus randomized testing.}
\label{fig:exp}
\end{figure}
\section{Conclusions}
\label{sec:concl}
We introduced the notion of Joker games, showed that its attractor-based Joker strategies use a minimum number of Jokers, and proved properties on determinacy, randomization, and admissible strategies in the context of Joker games.
In future work, we would like to extend the experimental evaluation of Joker strategies on applications, prove (or disprove) that, against memoryless Player 2 and 3 strategies, admissible strategies are global cost-minimal and vice versa, and investigate other multi-objective Joker strategies, a.o. to quantify the `badness' of adversarial moves \cite{synthesissafety}, where a bad move is e.g. a move that takes the game to a state where many Jokers need to be spend to reach the goal again.
\appendix
\appendix
\section*{Appendix of paper: ``With a little help from your friends: semi-cooperative games via Joker moves''}
\section{Computing randomized strategies with Jokers.}
To compute almost sure strategies that use a minimal number of Joker moves,
we slightly modify the recursive equations for computing the Joker attractor in \definitionref{def:jokers}:
we replace the (deterministic) attractor $\textit{Attr}(X)$, which always attracts the game to a set $X$, by the almost-sure attractor \cite{dAHK98}, that attracts the game to set $X$ with probability one.
This way, we obtain a strategy that corresponds to \theoremref{thm:jokerrandom}(3).
We formally define this in \definitionref{def:pattr}. This definition restates the probabilistic attractor \cite{dAHK98}.
\definitionref{def:pjokerstrat} then defines the probabilistic Joker attractor strategy. \theoremref{thm:probjokerstrat} concludes then that such strategies are sure winning strategies that use Jokers in Joker states with probability 1, similar to \theoremref{thm:jokerrandom}(3).
\section{Definitions}
\definitionref{def:pattr}
restates the definition of probabilistic attractor \cite{dAHK98}. The states $\mathit{Safe}_1(Q')$, and $\textit{CPre}_2(Q',W')$ are the states where Player 1 and Player 2 can confine the game to, respectively. The Player 1 probabilistic attractor $\textit{pAttr}_1(\game,\finally)$ is computed applying $\mathit{Safe}_1$ iteratively to find the set of states that Player 1 can visit forever. The starting point for computing $\textit{pAttr}_1(\game,\finally)$ are all the states, where Player 2 cannot prevent Player 1 from ever reaching $R$, the states $\textit{pAttr}_2(\game,\finally)i{0}$. When removing states from $\textit{pAttr}_1(\game,\finally)i{k}$ to compute $\textit{pAttr}_1(\game,\finally)i{k+1}$, again we need to remove states where Player 2 can make sure Player 1 does not reach $R$.
\begin{definition} \label{def:pattr}
\begin{align*}
&\mathit{Safe}^{0}_1(Q') = Q'\\
&\mathit{Safe}^{k+1}_1(Q') = Q'\ \cap \textit{CPre}_1(\mathit{Safe}_1^{k}(Q'))\\
&\mathit{Safe}_1(Q') = \bigcap_{k \in \mathbb{N}} \mathit{Safe}^{k}_1(Q')
\end{align*}
\begin{align*}
&\textit{CPre}_2(Q',W') = \{\ q \in Q \mid \exists x\in\Gamma_2(q), \forall (q',a) \in W': q = q' \wedge \textit{Moves}(q,a,x) \subseteq Q'\}\\
&\mathit{Safe}^{0}_2(Q',W') = Q'\\
&\mathit{Safe}^{k+1}_2(Q',W') = Q'\ \cap \textit{CPre}_2(\mathit{Safe}_2^k(Q',W'),W')\\
&\mathit{Safe}_2(Q',W') = \bigcap_{k \in \mathbb{N}} \mathit{Safe}^{k}_2(Q',W')
\end{align*}
\begin{align*}
&\textit{pAttr}_1(\game,\finally)i{0} = Q\\
&w\textit{pAttr}_1(\game,\finally)i{k} = \textit{CPre}_1^w(\textit{pAttr}_1(\game,\finally)i{k})\\
&\textit{pAttr}_2(\game,\finally)i{k} = \mathit{Safe}_2(\textit{pAttr}_1(\game,\finally)i{k} \setminus R, w\textit{pAttr}_1(\game,\finally)i{k})\\
&\textit{pAttr}_1(\game,\finally)i{k+1} = \mathit{Safe}_1(\textit{pAttr}_1(\game,\finally)i{k} \setminus \textit{pAttr}_2(\game,\finally)i{k})\\
&\textit{pAttr}_1(\game,\finally) = \bigcap_{k \in \mathbb{N}} \textit{pAttr}_1(\game,\finally)i{k}\\
&w\textit{pAttr}_1(\game,\finally) = \min_{k \in \mathbb{N}}\{w\textit{pAttr}_1(\game,\finally)i{k}\mid \textit{pAttr}_1(\game,\finally)i{k} = \textit{pAttr}_1(\game,\finally)i{k+1}\}\\
&\textit{pAttr}_2(\game,\finally) = \bigcap_{k \in \mathbb{N}} \textit{pAttr}_2(\game,\finally)i{k}
\end{align*}
The function $\parankop(\game,\finally): Q \times 2^Q \rightarrow \mathbb{N}$ associates to each state $q\in Q$ a rank $\parankop(\game,\finally)(q, R) = \min \{k \in \mathbb{N} \mid q \in \textit{pAttr}_1(\game,\finally)i{k}\}$.
A probabilistic attractor strategy $\sigma \in \Sigma_1^r(q)$ has $\sigma(\pi)(a) > 0 \iff (\pi_{end},a) \in w\textit{pAttr}_1(\game,\finally)$.
\end{definition}
Below we formulate the probabilistic Joker attractor and probablistic Joker attractor strategies. These definitions basically substitute $\textit{Attr}(\game,\finally)$ by $\textit{pAttr}_1(\game,\finally)$.
\begin{definition} \label{def:pjokers}
The Player 1 probablistic Joker attractor is $p\jokerop(\game,\finally)$, where:
\begin{align*}
\pjoker{0} &= \textit{pAttr}_1(\game,\finally)\\
\pjokerreal{k+1} &= \textit{Pre}(\pjoker{k})\\
\pjoker{k+1} &= \pjoker{k}\ \cup \textit{pAttr}(\pjokerreal{k+1})\\
p\jokerop(\game,\finally) &= \bigcup_{k \in \mathbb{N}} \pjoker{k}
\end{align*}
We call $p\jokerop(\game,\finally)$ the probabilistic Joker attractor of ${G}$. The probabilistic Joker states are
$\pjokerop_\varspadesuitreal(\game,\finally) = \bigcup_{k \in \mathbb{N}} \pjokerreal{k}\setminus\pjoker{k}$.
To each probabilistic Joker attractor $p\jokerop(\game,\finally)$ we associate a probabilistic Joker rank function $\pjrankop(\game,\finally) : Q \times 2^Q \rightarrow \mathbb{N}$, where for each state $q \in Q$ we define $\pjrankop(\game,\finally)(q) = \min \{k \in \mathbb{N} \mid q \in \pjoker{k}\}$.
\end{definition}
\begin{definition} \label{def:pwitness}
We define the \emph{witnessed} probabilistic Joker attractor $w\textit{JAttr}(\game,\finally)$:
\begin{align*}
w\pjoker{0}&= \emptyset\\
w\pjoker{k+1} &= w\pjoker{k} \cup \{ (q,a,x) \in w\textit{Pre}(\pjoker{k}) \mid q \notin \pjoker{k} \}\\
wp\jokerop(\game,\finally) &= \bigcup_{k \in \mathbb{N}}w\pjoker{k}
\end{align*}
\end{definition}
\begin{definition}\label{def:pjokerstrat}
A strategy $\sigma_1 \in \Sigma_1^{\varspadesuit(q),r}$ in $\textit{JAttr}(\game,\finally)ame$ is a \emph{probabilistic Joker attractor strategy}, if for any $\pi \in \textnormal{play?}pref(\textit{JAttr}(\game,\finally)ame)$ with $\pi^q_{end} \in p\jokerop(\game,\finally)$ and $\pjrankop(\game,\finally)(\pi^q_{end}) = k$ we have:
\begin{align*}
&(\pi^q_{end} \in \pjokerop_\varspadesuitreal(\game,\finally) \implies \sigma_1(\pi) \in wp\jokerop(\game,\finally))\qquad \wedge\\
&(k = 0 \wedge \pi^q_{end} \notin R \implies (\pi^q_{end},\sigma_1(\pi)) \in w\textit{pAttr}_1(\game,\finally))\qquad \wedge\\
&(k>0 \wedge \pi^q_{end} \notin \jokerop_\varspadesuitreal(\game,\finally)\implies (\pi^q_{end},\sigma_1(\pi)) \in w\textit{Attr}(\jokerreal{k}))
\end{align*}
\end{definition}
\section{Theorems with randomized strategies}
\label{sec:app:thms}
This section restates the theorems of \sectionref{sec:properties-joker-games} for randomized strategies (\definitionref{def:probstrat}) and probabilistic Joker attractors (\definitionref{def:pjokers}).
\begin{restatable}{theorem}{probnrjokermoves}
\label{thm:probnrjokermoves}
Let $q \in p\jokerop(\game,\finally)$.
Then:
\begin{enumerate}
\item Let $\sigma^J_1 \in \Sigma_1^{\varspadesuit,r}(q)$ be a probabilistic Joker attractor strategy in $\textit{JAttr}(\game,\finally)ame$.
Then any play $\pi \in \textit{Outc}(\sigma^J_1)$ has \emph{exactly} $\pjrankop(\game,\finally)(q)$ Joker actions in winning prefix $\pi_{0:\textit{WinInd}(\pi)}$.
\item Let $\sigma_1 \in \Sigma_1^{\varspadesuit(q),r}$ be a randomized, cost-minimal strategy in $\textit{JAttr}(\game,\finally)ame$.
Then any play $\pi \in \textit{Outc}(\sigma_1)$ has \emph{at most} $\pjrankop(\game,\finally)(q)$ Joker actions in the winning prefix $\pi_{0:\textit{WinInd}(\pi)}$.
\end{enumerate}
\end{restatable}
\begin{restatable}{theorem}{probjokerwinreach}
\label{thm:probjokerwinreach}
Let ${\reachop(\game,\finally)} =
\{q \in Q \mid q \text{ can reach a state } q'\in R\}$.
Then
\begin{align*}
\textit{WinReg}(\textit{JAttr}(\game,\finally)ame,R) = p\jokerop(\game,\finally) = {\reachop(\game,\finally)}
\end{align*}
\end{restatable}
\begin{restatable}{theorem}{probjokerstratwinning}
\label{thm:probjokerstratwinning}
Any probabilistic Joker attractor strategy is cost-minimal (in a Joker game with randomized strategies).
\end{restatable}
\begin{restatable}{theorem}{probjokerdetermined}
\label{thm:probjokerdetermined}
Joker games are determined (in Joker games with randomized strategies).
\end{restatable}
\begin{restatable}{theorem}{probjokerstrat}
\label{thm:probjokerstrat}
Any strategy from \definitionref{def:pjokerstrat} satisfies \theoremref{thm:jokerrandom}(3).
\end{restatable}
\section{Proofs}
\jokeriscost*
\noindent
\textit{Proof.}
First consider case $q \notin \textit{JAttr}(\game,\finally)$. By \definitionref{def:jokers} we then have that $\jrankop(\game,\finally)(q) = \infty$.
By \theoremref{thm:jokerwinreach} $q \notin {\reachop(\game,\finally)}$.
Hence $q$ cannot reach goal $R$ with any play. By \definitionref{def:costfunction}, we then have that any play has cost $\infty$, so any strategy has cost $\infty$, and then $\textit{Cost}(\textit{JAttr}(\game,\finally)ame,\finally)(q) = \infty.$
Hence, $\jrankop(\game,\finally)(q) = \textit{Cost}(\textit{JAttr}(\game,\finally)ame,\finally)(q)$.
Now consider $q \in \textit{JAttr}(\game,\finally)$, so $q \in {\reachop(\game,\finally)}$. By \definitionref{def:jokerstrat} we then have a Joker attractor strategy. By \theoremref{thm:jokerstratwinning}, this strategy is cost-minimal. By \theoremref{thm:nrjokermoves} it uses $\jrankop(\game,\finally)(q)$ Jokers in any play, so its cost is $\jrankop(\game,\finally)(q)$.
Since $\textit{Cost}(\textit{JAttr}(\game,\finally)ame,\finally)(q)$ is defined as the cost of cost-minimal strategies (\definitionref{def:costfunction}), we have that $\jrankop(\game,\finally)(q) = \textit{Cost}(\textit{JAttr}(\game,\finally)ame,\finally)(q)$.
\nrjokermoves*
\noindent
\textit{Proof.}
\noindent
(1) Let $q'$ be a state in a play of $\textit{Outc}(\sigma_1^J)$. Observe the following three cases:
\begin{itemize}
\item If $q' \in \textit{Attr}(\game,\finally)$, then $\sigma_1^J$ is a classical attractor strategy from $q$ that is winning \cite{dAHK98}, so reaches $R$, without using any Joker actions.
\item If $q' \in \joker{k}\setminus\jokerop_\varspadesuitreal(\game,\finally)$ is a non-Joker state of the $k$-th Joker attractor set, then $\sigma_1^J$ is a classical attractor strategy from $q'$ with goal $\jokerreal{k}$, and will surely reach one of the Joker states $\jokerreal{k}$, without using any Joker actions.
\item If $q' \in \jokerop_\varspadesuitreal(\game,\finally)$, then by the definition of a Joker attractor strategy (\definitionref{def:jokerstrat}), $\sigma_1^J$ uses a Joker action derived from the predecessor $\textit{Pre}$, such that the Joker action causes the game to reach state $q'' \in \joker{k}$ from $q' \notin \joker{k}$. The use of $\textit{Pre}_w$ in the definition of a Joker attractor strategy (\definitionref{def:jokerstrat}) exactly corresponds with the use of $\textit{Pre}$ in the definition of the Joker attractor (\definitionref{def:jokers}), so we have $\jrankop(\game,\finally)(q'') + 1 = \jrankop(\game,\finally)(q')$, and spend 1 Joker action making this move.
\end{itemize}
Consequently, $\sigma_1^J$ is a winning strategy.
Moreover, using 1 Joker action, reduces the rank of the next state by 1. Using the attractor in non-Joker states, costs no Joker actions. Since each of the $\jrankop(\game,\finally)(q)$ encountered Joker states we use 1 Joker action, we spend exactly $\jrankop(\game,\finally)(q)$ Joker actions in total.
\noindent
2) By \theoremref{thm:jokerstratwinning} we have that a Joker attractor strategy is cost-minimal, so any other cost-minimal strategy must have cost $\jrankop(\game,\finally)(q)$. That the cost can be less than $\jrankop(\game,\finally)(q)$ for some plays follows from \figureref{fig:costless}.
\textit{JokerWin}({G})reach*
\textit{Proof.}
To prove ${\reachop(\game,\finally)} \subseteq \textit{WinReg}(\textit{JAttr}(\game,\finally)ame,R)$, we observe that a state $q' \in R$ that can be reached from a state $q$ has a play $q_0 \langle a_0, x_0 \rangle q_1 \dots q_n$ with $q_0 = q$ and $q^n = q'$. This play allows constructing the winning strategy $\sigma \in \Sigma_1^\varspadesuit(q)$ in the Joker game, that only plays Joker actions to imitate the play: $\sigma(q_0 \dots q_k) = (a_k,x_k,q_{k+1})$ for all $0 \le k < n$ in $\textit{JAttr}(\game,\finally)ame$.
We conclude $\textit{WinReg}(\textit{JAttr}(\game,\finally)ame,R) \subseteq {\reachop(\game,\finally)}$ from the three facts that (1)
a state with a winning strategy in $\textit{JAttr}(\game,\finally)ame$ has a play reaching $R$, and (2) $\textit{JAttr}(\game,\finally)ame$ has the same states as ${G}$, and (3) playing a Joker action in $\textit{JAttr}(\game,\finally)ame$ corresponds to a regular move in ${G}$.
From the proof of \theoremref{thm:nrjokermoves}(1) we can derive that any state in $\textit{JAttr}(\game,\finally)$ has a winning strategy in $\textit{JAttr}(\game,\finally)ame$, so $q \in \textit{WinReg}(\textit{JAttr}(\game,\finally)ame,R)$. Consequently we have: $\textit{JAttr}(\game,\finally) \subseteq \textit{WinReg}(\textit{JAttr}(\game,\finally)ame,R)$.
Lastly, to prove that ${\reachop(\game,\finally)} \subseteq \textit{JAttr}(\game,\finally)$, we observe that $q \in (\textit{Pre})^k(\textit{JAttr}(\game,\finally)) \iff q \in \textit{JAttr}(\game,\finally)$ for any $k \in \mathbb(N)$ by the definition of $\textit{JAttr}(\game,\finally)$ (\definitionref{def:jokers}). For any $q\in{\reachop(\game,\finally)}$, we must have $q \in (\textit{Pre})^k(\textit{JAttr}(\game,\finally))$ for some $k \in \mathbb{N}$, as $q$ can reach $R$ in a finite number of steps, because $Q$ is finite. Consequently, we have $q \in \textit{JAttr}(\game,\finally)$ too.
From above $\subseteq$-relations the stated equalities follow trivially.
\jokerstratcostminimal*
\noindent
\textit{Proof.}
Let $q \in Q$ and $\sigma \in \Sigma_1(q)$ a Joker attractor strategy according to \definitionref{def:jokerstrat}.
First suppose that $q \notin \textit{JAttr}(\game,\finally)$. Then $q \notin {\reachop(\game,\finally)}$ by \theoremref{thm:jokerwinreach}, so $q$ has no winning play. Then any strategy -- also $\sigma$ -- has $\textit{Cost}(\textit{JAttr}(\game,\finally)ame,\finally)(\sigma) = \infty$ by \definitionref{def:costfunction}.
Consequently, $\sigma$ is cost-minimal.
Now suppose that $q \in \textit{JAttr}(\game,\finally)$. We prove that $\sigma$ is cost-minimal by induction on $k = \jrankop(\game,\finally)(q)$.
If $k = 0$, then $\sigma$ is a plain attractor strategy that does not spend any Joker actions, so $\textit{Cost}(\textit{JAttr}(\game,\finally)ame,\finally)(\sigma) = 0$. A Joker game has no actions with negative costs, so trivially, $\sigma$ is cost-minimal.
For $k > 0$, we have the induction hypothesis that any Joker strategy $\sigma' \in \Sigma_1(q')$ with $\jrankop(\game,\finally)(q') = k-1$ is cost-minimal.
If $q \in \jokerop_\varspadesuitreal(\game,\finally)$, then $\sigma$ uses a Joker action $(q'',a,x) \in w\textit{JAttr}(\game,\finally)$. By \definitionref{def:witness} we have that $q'' \in \joker{k-1}$, so $\sigma$ is cost-minimal when starting in $q''$. By the Joker attractor construction we know that there is no possibility to reach $q''$ (or any other state in $\joker{k-1}$) from $q$ for sure (i.e. against any Player 2 and 3), because $q$ is not in the controllable predecessor of $\joker{k-1}$. So Player 2 or 3 can prevent Player 1 from winning if she uses no Joker action, which would yield a path, and corresponding strategy, with cost $\infty$ . Hence, using one Joker action is the minimum cost for reaching a state in $\joker{k-1}$. Consequently, $\sigma$ is cost-minimal in this case.
If $q \notin \jokerop_\varspadesuitreal(\game,\finally)$, then $\sigma$ is a plain attractor strategy attracting to a Joker state of $\joker{k}$. These plain attractor actions have cost 0. Upon reaching a Joker state, the same reasoning as above applies: we need to spend one Joker action. Hence, $
\sigma$ is also cost-minimal in this case.
\jokerdetermined*
\textit{Proof.} This follows from \theoremref{thm:jokerwinreach}:
\begin{itemize}
\item In any state $q \in \textit{JAttr}(\game,\finally)$ we have a winning Player 1 strategy.
\item From any $q \notin \textit{JAttr}(\game,\finally)$, no state of $R$ can be reached, so Player 2 wins with any strategy.
\end{itemize}
\jokernotrandom*
\textit{Proof.}
Let $\sigma_1 \in \Sigma_1^r(q^0)$ be an almost sure winning randomized strategy. \\
1) Then there is a winning play in $\textit{Outc}(\sigma_1)$. As in the proof of \theoremref{thm:jokerwinreach}, we can use this play for constructing a strategy that in plays the Joker actions matching the play.\\
2) Construct the Joker attractor $\textit{JAttr}(\game,\finally)$ (\definitionref{def:jokers}), and witnesses (\definitionref{def:witness}), using probabilistic attractor $\textit{pAttr}_1(\game,\finally)$ instead of $\textit{Attr}(\game,\finally)$. This is defined formally in \definitionref{def:pattr}-\ref{def:pjokerstrat}. By \definitionref{def:pjokerstrat} we have that, in each Joker attractor set, we can attract with probability 1 to a Joker state. In a Joker state $q$ we choose Joker action $(a,x,q')$ such that $\pjrankop(\game,\finally)(q) > \pjrankop(\game,\finally)(q')$ with probability 1. By choosing such a move in a Joker state, the game moves to a state with a smaller Joker rank with probability 1. Decreasing the Joker rank in each Joker state leads to reaching a goal state. Because a next Joker state is reached with probability 1 each time, the overall probability of the resulting strategy is 1 too, so it is almost sure winning.\\
3) See the proof of 2).\\
\probjokerstrat*
\textit{Proof.}
This is exactly what we use in the proof of \theoremref{thm:jokerrandom}(2).
\randomjokerproperties*
\textit{Proof.} We prove each of the adapted theorems below.
\probnrjokermoves*
\noindent
\textit{Proof.}
\noindent
(1) We can follow the original proof of \theoremref{thm:nrjokermoves}(1), and only make the following two changes to use it in the setting of randomized strategies:
\begin{itemize}
\item In case $q' \in \textit{Attr}(\game,\finally)$, we have that $\sigma_1^J$ is a \emph{probablistic} attractor strategy from $q$ that is \emph{almost} sure winning, i.e. reach R \emph{with probability 1}, without using any Joker actions.
\item In case $ q'\in \joker{k}\setminus\jokerop_\varspadesuitreal(\game,\finally)$, we similarly have a \emph{probablistic} attractor strategy now, that \emph{almost} surely reaches one of the Joker states without using any Joker actions.
\end{itemize}
\noindent
(2) follows from \theoremref{thm:probjokerstratwinning} as in the original proof of \theoremref{thm:nrjokermoves}(2).
\probjokerwinreach*
\noindent
\textit{Proof.}
We follow the proof of \theoremref{thm:jokerwinreach} with the following changes:
\begin{itemize}
\item In the proof of case ${\reachop(\game,\finally)} \subseteq \textit{WinReg}(\textit{JAttr}(\game,\finally)ame,R)$ we adapt the strategy $\sigma$ to play the proposed Joker actions with probability 1.
\item In the proof of case $\textit{WinReg}(\textit{JAttr}(\game,\finally)ame,R) \subseteq {\reachop(\game,\finally)}$, we use for fact (1) that there is an almost sure winning strategy, and in fact (3) we use a Joker action that is played with probability 1 .
\item The proof of case $p\jokerop(\game,\finally) \subseteq \textit{WinReg}(\textit{JAttr}(\game,\finally)ame,R)$ needs to be adapted to use the proof of \theoremref{thm:nrjokermoves}(1) for randomized strategies (part of this theorem).
\item In the proof of case ${\reachop(\game,\finally)} \subseteq p\jokerop(\game,\finally)$, we need no changes, as the definition of $p\jokerop(\game,\finally)$ uses $\textit{Pre}$ also when $\textit{pAttr}_1(\game,\finally)$ is used instead of $\textit{Attr}(\game,\finally)$.
\end{itemize}
\probjokerstratwinning*
\noindent
\textit{Proof.} We follow the proof of \theoremref{thm:jokerstratwinning}, but use the probabilistic definitions at all places in this proof. Hence we adapt the proof as follows:
\begin{itemize}
\item We define $\sigma \in \Sigma_1^r(q)$ as a probabilistic Joker attractor strategy.
\item In the first case we suppose that $q \notin p\jokerop(\game,\finally)$, and use \theoremref{thm:probjokerwinreach} instead of \theoremref{thm:jokerwinreach}.
\item In the second case we take $q \in p\jokerop(\game,\finally)$ and prove that $\sigma$ is cost-minimal by induction on $k = \pjrankop(\game,\finally)(q)$.
\begin{itemize}
\item In case $k=0$ we use a probabilistic attractor strategy (\definitionref{def:pattr}) instead of a plain attractor strategy, and similarly use no Joker actions, so $\sigma$ is cost-minimal.
\item In case $k > 0$ we have the induction hypothesis that any $\sigma' \in \Sigma_1^r(q')$ with $\pjrankop(\game,\finally)(q') = k-1$ is cost-minimal.
\begin{itemize}
\item If $q \in \pjokerop_\varspadesuitreal(\game,\finally)$, then $\sigma$ uses a Joker action $(q'',a,x) \in wp\jokerop(\game,\finally)$. By \definitionref{def:pjokers} we have that $ q'' \in \pjoker{k-1}$, so $\sigma$ is cost-minimal when starting in $q''$. By the probabilistic Joker attractor construction we know that there is no possibility to reach $ q''$ from $q$ with probability 1 (...), because $q$ has been excluded by $\mathit{Safe}_1$ at some point. For same reasons as in the proof of \theoremref{thm:jokerstratwinning} it is minimum cost to spend 1 Joker action, so $\sigma$ is cost-minimal in this case.
\item If $q \notin \pjokerop_\varspadesuitreal(\game,\finally)$ then $\sigma$ is a probabilistic attractor strategy to a Joker state of $\pjoker{k}$, using 0 cost actions only. Upon reaching a Joker state we follow the above reasoning. Hence $\sigma$ is also cost-minimal in this case.
\end{itemize}
\end{itemize}
\end{itemize}
\probjokerdetermined*
\noindent
\textit{Proof.}
Similar to the proof of \theoremref{thm:jokerdetermined} this follows from \theoremref{thm:probjokerwinreach}.
\shortcostminimal*
\noindent
\textit{Proof.}\\
(1)
By \definitionref{def:diststrat} Joker actions are only played in Joker states.
In non-Joker states $\jokerop_{\not\varspadesuit}(\game,\finally)$ a non-Joker Player 1 action is played, chosen using the Joker enabling condition $\Gamma_1^\varspadesuit$. By definition, $\Gamma_1^\varspadesuit(q)$ is the set of states from the same Joker attractor set $k = \jrankop(\game,\finally)(q)$. Hence, to reach a state with a lower $J\rankop$, this has to happen due to playing a Joker action in a Joker state. Because distance is decreased by at least 1 every step, this indeed is the case: if the game would stay in states of the same Joker attractor set forever, distance does not decrease at some point due to the finite number of states in the set. Hence, a short Joker strategy uses $\jrankop(\game,\finally)(q^0)$ Joker actions from initial state $q^0$. A plain Joker attractor strategy also uses $\jrankop(\game,\finally)(q^0)$ actions (by \theoremref{thm:nrjokermoves}) and is cost-minimal by \theoremref{thm:jokerstratwinning}. Consequently, a short Joker strategy is cost-minimal.
\noindent
(2) It follows from (1) that a minimum number of Joker actions is used in any case.
The distance function \definitionref{def:short} computes the minimum number of moves to win for this number of Joker actions for the following reasons:
\begin{itemize}
\item In all goal states the distance is 0, as the game has been won.
\item In Joker states, we take the Joker action to a state with minimum distance. As noted in the proof of (1), this is a state of the next Joker attractor set.
\item In non-Joker states from a Joker attractor set, we choose a Player 1 action that minimizes distance for at least some Player 2 and 3 actions.
\item In states $Q \setminus \textit{JAttr}(\game,\finally)$, it is impossible to win and hence reach a goal state (\theoremref{thm:jokerwinreach}), so the minimum number of moves indeed is $\infty$.
\end{itemize}
This minimum number of moves is realized by \definitionref{def:diststrat}:
\begin{itemize}
\item In Joker states, it chooses a Joker action that has a next state with minimum distance, in accordance with the distance function.
\item In non-Joker states from the Joker attractor, it chooses a Player 1 action such that Player 2 and 3 can help to indeed minimize distance. The case(s) where Player 2 and 3 help will be represented in plays that are part of the outcome of the short Joker strategy.
\item In goal states winning has happend, and in unreachable states, nothing can be done. That a short Joker strategy is undefined in these cases doesn't influence the minimum number of moves to win.
\end{itemize}
Hence, the short Joker strategy will have a play using minimum number of moves for the minimum number of Joker actions.
\admjoker*
\noindent
\textit{Proof.}\\
See the examples in \figureref{fig:admcounter}(middle,right).
\end{document} |
\begin{document}
\baselineskip 18pt
\hfuzz=6pt
\newtheorem{theorem}{Theorem}[section]
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{cor}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\rightarrow}{\rightarrow}
\renewcommand{\theequation}
{\thesection.\arabic{equation}}
\newcommand{{\mathcal C}}{{\mathcal C}}
\newcommand{1\hspace{-4.5pt}1}{1\hspace{-4.5pt}1}
\def \Gg {\widetilde{{\mathcal G}}_{L}}
\def \GG {{\mathcal G}_{L}}
\def \SL {\sqrt{L}}
\def \GL{G_{\mu,\Phi}^{\ast}}
\def \gL{g_{\mu,\Psi}^{\ast}}
\def \RN {\mathbb{R}^n}
\def\mathbb R{\mathbb R}
\newcommand{\infty}{\infty}
\allowdisplaybreaks
\title[Weighted $L^p$ estimates for the area integral]
{ Weighted $L^p$ estimates for the area integral\\ [2pt] associated to self-adjoint operators}
\author{Ruming Gong \ and \ Lixin Yan}
\footnotetext[1]{{\it {\rm 2000} Mathematics Subject Classification:}
42B20, 42B25, 46E35, 47F05.}
\footnotetext[1]{{\it Key words and phrase:} Weighted norm inequalities, area integral, self-adjoint operators, heat kernel,
semigroup, Whitney decomposition, $A_p$ weighs.}
\address{
Ruming Gong, Department of Mathematics, Sun Yat-sen (Zhongshan) University, Guangzhou, 510275, P.R. China}
\email{
gongruming@163.com}
\address{
Lixin Yan, Department of Mathematics, Sun Yat-sen (Zhongshan) University, Guangzhou, 510275, P.R. China}
\email{
mcsylx@mail.sysu.edu.cn
}
\begin{abstract} This article is concerned with some weighted norm inequalities
for the so-called horizontal (i.e. involving time derivatives) area integrals associated to a non-negative self-adjoint operator
satisfying a pointwise Gaussian estimate for its heat kernel,
as well as the corresponding vertical (i.e. involving space derivatives) area integrals associated to a non-negative self-adjoint operator
satisfying in addition a pointwise upper bounds for the gradient of the heat kernel.
As applications, we obtain sharp estimates for the operator norm of the area integrals on $L^p(\RN)$ as $p$ becomes large,
and the growth of the $A_p$ constant on estimates of the area integrals
on the weighted $L^p$ spaces.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction }
\setcounter{equation}{0}
\noindent
{\bf 1.1. Background.}\
Let $\varphi\in C_0^{\infty}({\RN})$ with $\int \varphi =0.$
Let $\varphi_t(x)=t^{-n}\varphi(x/t), t>0$, and define the Lusin area integral by
\begin{eqnarray}\label{e1.1}
S_{\varphi}(f)(x)=\bigg(\int_{|x-y|<t}
\big|f\ast \varphi_t(y)\big|^2 {dy \, dt\over t^{n+1}}\bigg)^{1/2}.
\end{eqnarray}
\noindent
A celebrated result of
Chang-Wilson-Wolff (\cite{CWW}) says that for all $w\geq 0$, $w\in L_{\rm loc}^{1}(\RN)$ and all $f\in \mathcal{S}(\RN)$,
there is a constant $C=C(n, \varphi)$ independent of $w$ and $f$ such that
\begin{eqnarray}\label{e1.2}
\int_{\RN}S^2_{\varphi}(f)w\, dx\leq C\int_{\RN}|f|^2Mw\, dx,
\end{eqnarray}
\noindent
where $Mw$ denotes the Hardy-Littlewood maximal operator of $w$.
The fact that $\varphi$ has compact support is crucial in the proof of Chang, Wilson and Wolff.
In \cite{CW}, Chanillo and Wheeden overcame this difficulty, and they obtained weighted $L^p$ inequalities for
$1<p<\infty$ of
the area integral, even when $\varphi$
does not have compact support, including the classical area function defined by means of the Poisson kernel.
From the theorem of Chang, Wilson and Wolff, it was already observed in \cite{FP} that R. Fefferman and Pipher obtained
sharp estimates for the operator norm of a classical Calder\'on-Zygmund
singular integral, or the classical area integral for $p$ tending to infinity, e.g.,
\begin{eqnarray}\label{e1.3}
\big\|S_{\varphi}(f)\big\|_{L^p(\RN)}\leq C p^{1/2} \big\|f\big\|_{L^p(\RN)}
\end{eqnarray}
\noindent
as $p\rightarrow \infty.$\\
\noindent
{\bf 1.2. Assumptions, notation and definitions.}\, In this article, our main goal is
to provide an extension of the result of
Chang-Wilson-Wolff to study some weighted norm inequalities
for the area integrals associated to non-negative self-adjoint operators,
whose kernels are not smooth enough
to fall under the scope of \cite{CWW, CW, W}.
The relevant classes of operators is determined by the following condition:
\noindent
{\bf Assumption $(H_1)$.}\,
Assume that $L$ is a non-negative self-adjoint operator on $L^2({\mathbb R}^{n}),$
the semigroup $e^{-tL}$, generated by $-L$ on $L^2(\RN)$, has the kernel $p_t(x,y)$
which satisfies
the following Gaussian upper bound if there exist $C$ and $c$ such that
for all $x,y\in {\mathbb R}^{n}, t>0,$
$$
|p_{t}(x,y)| \leq \frac{C}{t^{n/2} } \exp\Big(-{|x-y|^2\over
c\,t}\Big).
\leqno{(GE)}
$$
\noindent
Such estimates are typical for elliptic or sub-elliptic differential operators of second
order (see for instance, \cite{Da} and \cite{DOS}).\\
For $f\in {\mathcal S}(\RN)$, define the (so called vertical) area functions $S_{P}$ and $S_{H}$
by
\begin{eqnarray}\label{e1.4}
S_{P}f(x)&=&\bigg(\int_{|x-y|<t}
|t\nabla_y e^{-t\sqrt{L}} f(y)|^2 {dy dt\over t^{n+1}}\bigg)^{1/2},\\
S_{H}f(x)&=&\bigg(\int_{|x-y|<t}
|t\nabla_y e^{-t^2L} f(y)|^2 {dy dt\over t^{n+1}}\bigg)^{1/2}, \label{e1.5}
\end{eqnarray}
\noindent
as well as the (so-called horizontal) area functions $s_{p}$ and $s_{h}$
by
\begin{eqnarray}\label{e1.6}
s_{p}f(x)&=&\bigg(\int_{|x-y|<t}
|t\sqrt{L} e^{-t\sqrt{L}} f(y)|^2 {dy dt\over t^{n+1}}\bigg)^{1/2},\\
s_{h}f(x)&=&\bigg(\int_{|x-y|<t}
|t^2L e^{-t^2L} f(y)|^2 {dy dt\over t^{n+1}}\bigg)^{1/2}.\label{e1.7}
\end{eqnarray}
It is well known (cf. e.g. \cite{St, G}) that when $L=\Delta$ is the Laplacian on $\RN$, the classical
area functions $S_{P}, S_{H}, s_{p}$ and $s_{h}$ are all bounded on $L^p(\RN), 1<p<\infty.$
For a general non-negative self-adjoint operator $L$, $L^p$-boundedness of
the area functions $S_{P}, S_{H}, s_{p}$ and $s_{h}$ associated to $L$ has been
studied extensively -- see for examples \cite{A}, \cite{ACDH}, \cite{ADM}, \cite{CDL}, \cite{St1} and \cite{Y}, and the references therein.
\noindent
{\bf 1.3. Statement of the main results.}
Firstly, we have the following weighted $L^p$ estimates for the area functions
$s_{p}$ and $s_{h}.$
\begin{theorem}\label{th1.1} \ Let $L$ be a non-negative self-adjoint operator such that the corresponding
heat kernels satisfy Gaussian bounds $(GE)$.
If $w\geq 0$, $w\in L_{\rm loc}^{1}(\RN)$ and $f\in \mathcal{S}(\RN)$, then
\begin{eqnarray*}
\hspace{-2.5cm}
&{\rm (a)}& \hspace{0.1cm}
\int_{\RN}s_h(f)^pw\, dx\leq C(n,p)\int_{\RN}|f|^pMw\, dx,\ \ \ 1<p\leq 2,\\
\hspace{-2.5cm}&{\rm (b)}& \hspace{0.1cm}\int_{\{s_h(f)>\lambda\}} wdx\leq {C(n)\over \lambda}\int_{\RN}|f| Mwdx,\ \ \ \lambda>0,
\\
\hspace{-2.5cm}&{\rm (c)}& \hspace{0.1cm} \int_{\RN} s_h(f)^pwdx\leq C(n,p)\int_{\RN}|f|^p(Mw)^{p/2}w^{-(p/2-1)}dx, \ \ \ 2<p<\infty.
\end{eqnarray*}
Also, estimates (a), (b) and (c) hold for the operator $s_p.$
\end{theorem}
To study weighted $L^p$-boundedness of the (so-called vertical) area integrals $S_{P}$ and $S_{H}$,
one assumes in addition the following condition:
\noindent
{\bf Assumption $(H_2)$.}\, Assume that
the semigroup $e^{-tL}$, generated by $-L$ on $L^2(\RN)$, has the kernel $p_t(x,y)$
which satisfies a pointwise upper bound for the gradient of the heat kernel. That is, there exist $C$ and $c$ such that for all $x,y\in{\RN}, t>0$,
$$
\big|\nabla_x p_t(x,y)\big|\leq {C\over t^{(n+1)/2}} \exp\Big(-{|x-y|^2\over
c\,t}\Big).
\leqno{(G)}
$$
Then the following result holds.
\begin{theorem}\label{th1.2} \ Let $L$ be a non-negative self-adjoint operator such that the corresponding
heat kernels satisfy conditions $(GE)$ and $(G)$.
\noindent
If $w\geq 0$, $w\in L_{\rm loc}^{1}(\RN)$ and $f\in \mathcal{S}(\RN)$, then $(a), (b)$ and $(c)$ of Theorem~\ref{th1.1}
hold for the area functions $S_{P}$ and $S_{H}$.
\end{theorem}
Let us now recall a definition. We say that a weight $w$ is in the the Muckenhoupt class $A_p, 1<p<\infty,$ if
\begin{eqnarray*}
\|w\|_{A_p}\equiv \sup_Q \bigg({1\over |Q|}\int_Q w(x)dx \bigg)\bigg({1\over |Q|}\int_Q w(x)^{-1/(p-1)}dx \bigg)^{p-1}<\infty.
\end{eqnarray*}
\noindent
$\|w\|_{A_p}$ is usually called the $A_p$ constant (or characterization or norm) of the weight. The case $p=1$
is understand by replacing the right hand side by $(\inf_Q w)^{-1}$ which is equivalent to the one defined above.
Observe the duality relation:
$$
\|w\|_{A_p}=\|w^{1-p'}\|^{p-1}_{A_{p'}}.
$$
Following the R. Fefferman-Pipher's method, we can use Theorems~\ref{th1.1} and ~\ref{th1.2} to
establish the $L^p$ estimates of the area integrals as $p$ becomes large.
\begin{theorem}\label{th1.3} Let $T$ be of the area functions $s_h$, $s_p$, $S_{P}$ and $S_{H}.$
Under assumptions of Theorems~\ref{th1.1} and ~\ref{th1.2}, there exists a constant $C$ such that
for all $w\in A_1$, the following
estimate holds:
\begin{eqnarray}\label{e1.8}
\|Tf\|_{L^2_w(\RN)}\leq C\|w\|_{A_1}^{1/2}\|f\|_{L^2_w(\RN)}.
\end{eqnarray}
\noindent This inequality implies that as $p\rightarrow \infty$
\begin{eqnarray}\label{e1.9}
\|Tf\|_{L^p(\RN)}\leq Cp^{1/2}\|f\|_{L^p(\RN)}.
\end{eqnarray}
\end{theorem}
The next result we will prove is the following.
\begin{theorem}\label{th1.4} Let $T$ be of the area functions $s_h$, $s_p$, $S_{P}$ and $S_{H}.$
Under assumptions of Theorems~\ref{th1.1} and ~\ref{th1.2}, there exists a constant $C$ such that
for all $w\in A_p$, the following
estimate holds for all $f\in L^p_w(\RN), 1<p<\infty$:
\begin{eqnarray}\label{e1.10}
\|T f\|_{L^p_w(\RN)}\leq C\|w\|_{A_p}^{\beta_p+1/(p-1)} \|f\|_{L^p_w(\RN)} \ \ (1<p<\infty),
\end{eqnarray}
\noindent where $\beta_p=\max\{1/2,1/(p-1)\}.$
\end{theorem}
We should mention that
Theorems~\ref{th1.1} and ~\ref{th1.2} are of some independent of interest, and they provide an immediate
proof of weighted $L^p$ estimates of
the area functions $s_h$, $s_p$, $S_{P}$ and $S_{H}$ on $L^p_w(\RN), 1<p<\infty $ and $ w\in A_p $
(see Lemma~\ref{le5.1} below). In the proofs of Theorems~\ref{th1.1} and ~\ref{th1.2}, the main tool is that
each area integral is controlled by $\gL$ pointwise:
\begin{eqnarray}\label{e3.2}
Tf(x) \leq C\gL(f)(x),\ \ \ x\in\RN,
\end{eqnarray}
\noindent
where $T$ is of $S_P, S_H, s_p$ and $s_H$, and $\gL$ is defined by
\begin{eqnarray}\label{e3.1} \hspace{1cm}
\gL(f)(x)=\Bigg(\iint_{\mathbb R_{+}^{n+1}}
\bigg({t\over t+|x-y|}\bigg)^{n\mu}
|\Psi(t\sqrt{L})f(y)|^2{dydt\over t^{n+1}}\Bigg)^{1/2}, \ \ \mu>1
\end{eqnarray}
\noindent
with some $\Psi\in{\mathcal S}(\RN)$. The idea of using $\gL$ to control the area integrals is due
to Calder\'on and Torchinsky \cite{CT} (see also
\cite{CW} and \cite{W}). Note that the singular integral $\gL$ does not satisfy the standard regularity condition
of a so-called Calder\'on-Zygmund operator, thus standard techniques of Calder\'on-Zugmund
theory (\cite{CW, W}) are not applicable. The lacking of smoothness of the kernel was indeed the main
obstacle and it was overcome by
using the method developed in \cite{ CD, DM}, together with some estimates on heat kernel bounds, finite propagation speed of solutions
to the wave equations and spectral theory of non-negative self-adjoint operators.
The layout of the paper is as follows. In Section 2 we recall some basic properties of
heat kernels and finite propagation speed for the wave equation, and build the necessary kernel estimates
for functions of an operator, which is useful in
the proof of weak-type $(1,1)$ estimate for the area integrals.
In Section 3 we will prove that the area integral is controlled by $\gL$ pointwise,
which implies Theorems~\ref{th1.1} and \ref{th1.2} for $p=2$, and then we employ
the R. Fefferman-Pipher's method to obtain sharp estimates for the operator norm of
the area integrals on $L^p(\RN)$ as $p$ becomes large. In Section 4, we will
give the proofs of Theorems~\ref{th1.1} and \ref{th1.2}.
Finally, in Section 5 we will prove our Theorem~\ref{th1.4}, which gives
the growth of the $A_p$ constant on estimates
on the weighted $L^p$ spaces.
Throughout, the letter ``$c$" and ``$C$" will denote (possibly different)
constants that are independent of the essential variables.
\section{Notation and preliminaries}
\setcounter{equation}{0}
Let us
recall that, if $L$ is a self-adjoint positive definite operator acting
on $L^2({\mathbb R}^n)$, then it admits a spectral resolution
\begin{eqnarray*}
L=\int_0^{\infty} \lambda dE(\lambda).
\end{eqnarray*}
\noindent
For every bounded
Borel function $F:[0,\infty)\to{\mathbb{C}}$,
by using the spectral theorem we can define the operator
\begin{eqnarray}\label{e2.1}
F(L):=\int_0^{\infty}F(\lambda)\,dE_{L}(\lambda).
\end{eqnarray}
This is of course, bounded on $L^2({\mathbb R}^n)$.
In particular, the operator $\cos(t\sqrt{L})$ is then well-defined and bounded
on $L^2({\mathbb R}^{n})$. Moreover, it follows from Theorem 3 of \cite{CS}
that if the corresponding
heat kernels $p_{t}(x,y)$ of $e^{-tL}$ satisfy Gaussian bounds $(GE)$, then there exists a finite,
positive constant $c_0$ with the property that the Schwartz
kernel $K_{\cos(t\sqrt{L})}$ of $\cos(t\sqrt{L})$ satisfies
\begin{eqnarray}\label{e2.2} \hspace{1cm}
{\rm supp}K_{\cos(t\sqrt{L})}\subseteq
\big\{(x,y)\in {\mathbb R}^{n}\times {\mathbb R}^{n}: |x-y|\leq c_0 t\big\}.
\end{eqnarray}
\noindent
See also \cite{CGT} and \cite{Si}. The precise value of $c_0$ is inessential and throughout the article we will choose $c_0=1$.
By the Fourier inversion
formula, whenever $F$ is an even, bounded, Borel function with its Fourier transform
$\hat{F}\in L^1(\mathbb{R})$, we can write $F(\sqrt{L})$ in terms of
$\cos(t\sqrt{L})$. More specifically, we have
\begin{eqnarray}\label{e2.3}
F(\sqrt{L})=(2\pi)^{-1}\int_{-\infty}^{\infty}{\hat F}(t)\cos(t\sqrt{L})\,dt,
\end{eqnarray}
which, when combined with (\ref{e2.2}), gives
\begin{eqnarray}\label{e2.4} \hspace{1cm}
K_{F(\sqrt{L})}(x,y)=(2\pi)^{-1}\int_{|t|\geq |x-y|}{\hat F}(t)
K_{\cos(t\sqrt{L})}(x,y)\,dt,\qquad \forall\,x,y\in{\mathbb R}^{n}.
\end{eqnarray}
The following result
is useful for certain estimates later.
\begin{lemma}\label{le2.1}\, Let $\varphi\in C^{\infty}_0(\mathbb R)$ be
even, $\mbox{supp}\,\varphi \subset (-1, 1)$. Let $\Phi$ denote the Fourier transform of
$\varphi$. Then for every $\kappa=0,1,2,\dots$, and for every $t>0$,
the kernel $K_{(t^2L)^{\kappa}\Phi(t\sqrt{L})}(x,y)$ of the operator
$(t^2L)^{\kappa}\Phi(t\sqrt{L})$ which was defined by the spectral theory, satisfies
\begin{eqnarray}\label{e2.5}
{\rm supp}\ \! K_{(t^2L)^{\kappa}\Phi(t\sqrt{L})}
\subseteq \big\{(x,y)\in \RN\times \RN: |x-y|\leq t\big\}
\end{eqnarray}
\noindent
and
\begin{eqnarray}\label{e2.6}
\big|K_{(t^2L)^{\kappa}\Phi(t\sqrt{L})}(x,y)\big|
\leq C \, t^{-n}
\end{eqnarray}
\noindent
for all $t>0$ and $x,y\in \RN.$
\end{lemma}
\begin{proof} The proof of this lemma is standard (see \cite{SW} and \cite{HLMMY}). We give a brief argument
of this proof for completeness and convenience for the reader.
For every $\kappa=0,1,2,\dots$, we set
$\Psi_{\kappa, t}(\zeta):=(t\zeta)^{2\kappa}\Phi(t\zeta)$.
Using the definition of the Fourier transform, it can be verified that
$$
\widehat{\Psi_{\kappa,t}}(s)=(-1)^{\kappa}
{1\over t}\psi_{\kappa}({s\over t}),
$$
where we have set
$\psi_{\kappa} (s)={d^{2\kappa}\over ds^{2\kappa}}\varphi(s)$.
Observe that for every $\kappa=0,1,2,\dots$, the function
$\Psi_{\kappa,t}\in{\mathcal S}(\mathbb R)$ is an even function.
It follows from formula (\ref{e2.4}) that
\begin{equation}\label{e2.7}
K_{(t^2L)^{\kappa}\Phi(t\sqrt{L})}(x,y)
=(-1)^{\kappa}{1\over 2\pi}\int_{|st|\geq |x-y|}
{d^{2\kappa }\over ds^{2\kappa}}\varphi({s})K_{\cos(st\sqrt{L})}(x,y)\,ds.
\end{equation}
\noindent Since $\varphi\in C^{\infty}_0(\mathbb R)$ and
$\mbox{supp}\,\varphi \subset(-1, 1)$, (\ref{e2.5})
follows readily from this.
Note that for any $m\in {\Bbb N}$ and $t>0$, we have the relationship
$$
(I+tL)^{-m}={1\over (m-1)!} \int\limits_{0}^{\infty}e^{-tsL}e^{-s} s^{m-1} ds
$$
\noindent
and so when $m>n/4$,
\begin{eqnarray*}
\big\| (I+tL)^{-m} \big\|_{L^2\rightarrow L^{\infty}}\leq {1\over (m-1)!} \int\limits_{0}^{\infty}
\big\| e^{-tsL}\big\|_{L^2\rightarrow L^{\infty}} e^{-s} s^{m-1} ds\leq C t^{-n/4}
\end{eqnarray*}
\noindent for all $t>0.$ Now $ \big\| (I+tL)^{-m} \big\|_{L^1\rightarrow L^{2}}=\big\| (I+tL)^{-m} \big\|_{L^2\rightarrow L^{\infty}}\leq
C t^{-n/4}$, and so
\begin{eqnarray*}
\big\|(t^2L)^{\kappa}\Phi(t\sqrt{L})\big\|_{L^1\rightarrow L^{\infty}} \leq
\big\| (I+t^2L)^{2m}(t^2L)^{\kappa}\Phi(t\sqrt{L}) \big\|_{L^2\rightarrow L^2} \big\| (I+t^2L)^{-m} \big\|^2_{L^2\rightarrow L^{\infty}}.
\end{eqnarray*}
\noindent
The $L^2$ operator norm of the last term is equal to the $L^{\infty}$
norm of the function $(1+t^2|s|)^{2m} (t^2|s|)^{\kappa}\Phi(t\sqrt{|s|})$ which is uniformly
bounded in $t>0$. This implies that (\ref{e2.6}) holds. The proof of this lemma is concluded.
\end{proof}
\begin{lemma}\label{le2.2}\, Let $\varphi\in C^{\infty}_0(\mathbb R)$ be
even function with $\int \varphi =1$, $\mbox{supp}\,\varphi \subset (-1/10, 1/10)$.
Let $\Phi$ denote the Fourier transform of
$\varphi$ and let $\Psi(s)=s^{2n+2}\Phi^3(s)$. Then there exists a positive constant $C=C({n,\Phi})$
such that the kernel $K_{\Psi(t\sqrt{L})(1-\Phi(r\sqrt{L}))}(x,y)$ of $ \Psi(t\sqrt{L})(1-\Phi(r\sqrt{L}))$
satisfies
\begin{eqnarray}\label{e2.8}
\big|K_{\Psi(t\sqrt{L})(1-\Phi(r\sqrt{L}))}(x,y)\big|
\leq C\, { r\over t^{n+1}}\Big(1+{|x-y|^2\over t^2}\Big)^{-(n+1)/2}
\end{eqnarray}
\noindent
for all $t>0, r>0$ and $x, y\in \RN$.
\end{lemma}
\begin{proof} By rescaling, it is enough to
show that
\begin{eqnarray}\label{e2.9}
|K_{\Psi(\sqrt{L})(1-\Phi(r\sqrt{L}))}(x,y)|
\leq C{r}\big(1+{|x-y|^2}\big)^{-(n+1)/2}.
\end{eqnarray}
Let us prove (\ref{e2.9}). One writes $\Psi(s)=\Psi_1(s)\Phi^2(s)$, where $\Psi_1(s)=s^{2n+2}\Phi(s)$. Then we have
$\Psi(\sqrt{L})=\Psi_1(\sqrt{L})\Phi^2(\sqrt{L})$. It follows from Lemma 2.1 that $|K_{\Phi(\sqrt{L}) }(z,y)|\leq C$
and $K_{\Phi(\sqrt{L}) }(z,y)=0$ when $|z-y|\geq 1.$ Note that if $|z-y|\leq 1$, then
$\big(1+|x-y|\big) \leq 2(1+|x-z|)$. Hence,
\begin{eqnarray*}
&&\hspace{-1cm}\Big|\big(1+|x-y|\big)^{n+1}K_{\Psi(\sqrt{L})(1-\Phi(r\sqrt{L}))}(x,y)\Big|\\
&&= \big(1+|x-y|\big)^{n+1}\Big|\int_{\RN}
K_{\Psi_1(\sqrt{L})(1-\Phi(r\sqrt{L}))\Phi(\sqrt{L})}(x,z) K_{\Phi(\sqrt{L}) }(z,y) dz\Big|\\
&&\leq C\int_{\RN}\big|K_{\Psi_1(\sqrt{L})(1-\Phi(r\sqrt{L}))\Phi(\sqrt{L})}(x,z)\big|\big(1+|x-z| \big)^{n+1}dz.
\end{eqnarray*}
\noindent
By symmetry, we will be done if we show that
\begin{eqnarray}\label{e2.10}
\int_{\RN}\big|K_{\Psi_1(\sqrt{L})(1-\Phi(r\sqrt{L}))\Phi(\sqrt{L})}(x,z)\big|\big(1+|x-z| \big)^{n+1}dx\leq Cr.
\end{eqnarray}
\noindent Let $G_r(s)=\Psi_1(s)(1-\Phi(rs))$. Since $G_r(s)$ is an even function, apart from a $(2\pi)^{-1}$ factor we
can write
$$G_r(s)=\int^{+\infty}_{-\infty}\widehat{G_r}(\xi){\rm cos}(s\xi)d\xi,
$$
\noindent and by (\ref{e2.3}),
\begin{eqnarray}\label{e2.11}\Psi_1(\sqrt{L})(1-\Phi(r\sqrt{L}))\Phi(\sqrt{L})
=\int^{+\infty}_{-\infty}\widehat{G_r}(\xi){\rm cos}(\xi\sqrt{L})\Phi(\sqrt{L})d\xi.
\end{eqnarray}
\noindent
By Lemma 2.1 again, it can be seen that $K_{{\rm cos}(\xi\sqrt{L})\Phi(\sqrt{L})}(x,z)=0$ if $|x-z|\geq 1+|\xi|.$
Using the unitarity of $\cos(\xi\sqrt{L})$, estimates (\ref{e2.5}) and (\ref{e2.6}), we have
\begin{eqnarray*}
\int_{\RN} \big|K_{{\rm cos}(\xi\sqrt{L})\Phi(\sqrt{L})}(x,z)\big|dx &=&
\int_{\RN} \big|\cos(\xi\sqrt{L}) \big(K_{\Phi(\sqrt{L})}(\cdot\ , z)\big)(x)\big|dx\nonumber\\
&\leq& (1+|\xi|)^{n/2} \big\| \cos(\xi\sqrt{L}) \big(K_{\Phi(\sqrt{L})}(\cdot\ , z)\big)\big\|_{L^2(\RN)} \nonumber\\
&\leq& (1+|\xi|)^{n/2} \big\| K_{\Phi(\sqrt{L})}(\cdot\ , z) \big\|_{L^2(\RN)} \nonumber\\
&\leq& (1+|\xi|)^{n/2}.
\end{eqnarray*}
\noindent
This,
in combination with (\ref{e2.11}), gives
\begin{eqnarray}\label{e2.12}
{\rm LHS\ \ of \ \ } (\ref{e2.10}) \ &\leq& C\int^{+\infty}_{-\infty} |\widehat{G_r}(\xi)| \, (1+|\xi|)^{2n+1}\, d\xi\nonumber\\
&\leq& C\Big(\int^{+\infty}_{-\infty} |\widehat{G_r}(\xi)|^2 \, (1+|\xi|)^{4n+4}\, d\xi\Big)^{1/2}\nonumber\\
&\leq& C\big\|G_r\|_{W^{2n+2, \,2}(\RN)}.
\end{eqnarray}
\noindent Next we estimate the term $\big\|G_r\|_{W^{2n+2, \,2}(\RN)}$.
Note that $G_r(s)=\Psi_1(s)(1-\Phi(rs)), \Phi(0)=\widehat{\varphi}(0)=\int \varphi =1$ and
$\Phi=\widehat{\varphi}\in \mathcal{S}(\mathbb R),$ also $\Psi_1(s)=s^{2n+2}\Phi(s).$ We have
\begin{eqnarray}\label{e2.13} \hspace{1cm}
\|G_r\|_{L^2}^2= \int_{\mathbb R} |\Psi_1(s)|^2|1-\Phi(rs)|^2ds
\leq C \|\Phi'\|^2_{L^\infty} \int_{\mathbb R} |\Psi_1(s)|^2\, (rs)^2\,ds \leq C r^2.
\end{eqnarray}
\noindent Moreover, observe that for any $k\in{\mathbb N}$,
$\big|{d^k\over ds^k}\big(1-\Phi(rs)\big)\big|=r^k|\Phi^{(k)}(rs)|\leq Crs^{1-k}.$
By Leibniz's rule, we obtain
\begin{eqnarray}\label{e2.14}
\Big\|{d^{2n+2}\over ds^{2n+2}}G_r(s)\Big\|_{L^2}&=& \Big\|{d^{2n+2}\over ds^{2n+2}}\Big(\Psi_1(s)\big(1-\Phi(rs)\big)\Big)\Big\|_{L^2(\RN)}\nonumber\\
&\leq&\sum_{m+k=2n+2}\Big\|{d^{m}\over ds^{m}}\Big(s^{2n+2}\Phi\Big) {d^{k}\over ds^{k}}\Big(1-\Phi(rs)\Big)\Big\|_{L^2(\RN)}\nonumber\\
&\leq&Cr \sum_{m=0}^{2n+2}\Big\|s^{m-(2n+1)}{d^{m}\over ds^{m}}\Big(s^{2n+2}\Phi\Big) \Big\|_{L^2(\RN)}\nonumber\\
&\leq&Cr.
\end{eqnarray}
\noindent From estimates (\ref{e2.13}) and (\ref{e2.14}),
it follows that $\big\|G_r\|_{W^{2n+2, \,2}(\RN)}\leq Cr$. This, in combination with (\ref{e2.12}),
shows that the desired estimate (\ref{e2.10}) holds, and concludes the proof of Lemma 2.2.
\end{proof}
Finally, for $s>0$, we define
$$
{\Bbb F}(s):=\Big\{\psi:{\Bbb C}\to{\Bbb C}\ {\rm measurable}: \ \
|\psi(z)|\leq C {|z|^s\over ({1+|z|^{2s}})}\Big\}.
$$
Then for any non-zero function $\psi\in {\Bbb F}(s)$, we have that
$\{\int_0^{\infty}|{\psi}(t)|^2\frac{dt}{t}\}^{1/2}<\infty$.
Denote by $\psi_t(z)=\psi(tz)$. It follows from the spectral theory
in \cite{Yo} that for any $f\in L^2(\RN)$,
\begin{eqnarray}
\Big\{\int_0^{\infty}\|\psi(t\sqrt{L})f\|_{L^2(\RN)}^2{dt\over t}\Big\}^{1/2}
&=&\Big\{\int_0^{\infty}\big\langle\,\overline{ \psi}(t\sqrt{L})\,
\psi(t\sqrt{L})f, f\big\rightarrowngle {dt\over t}\Big\}^{1/2}\nonumber\\
&=&\Big\{\big\langle \int_0^{\infty}|\psi|^2(t\sqrt{L}) {dt\over t}f,
f\big\rightarrowngle\Big\}^{1/2}\nonumber\\
&=& \kappa \|f\|_{L^2(\RN)},
\label{e2.15}
\end{eqnarray}
\noindent where $\kappa=\big\{\int_0^{\infty}|{\psi}(t)|^2 {dt/t}\big\}^{1/2},$
an estimate which will be often used in the sequel.
\section{An auxiliary $\gL$ function}
\setcounter{equation}{0}
\subsection{The $\gL$ function}
Let $\varphi\in C^{\infty}_0(\mathbb R)$ be
even function with $\int \varphi =1$, $\mbox{supp}\,\varphi \subset (-1/10, 1/10)$.
Let $\Phi$ denote the Fourier transform of
$\varphi$ and let $\Psi(s)=s^{2n+2}\Phi^3(s)$ (see Lemma 2.2 above).
We define the $\gL$ function by
\begin{eqnarray}\label{e3.1} \hspace{1cm}
\gL(f)(x)=\Bigg(\iint_{\mathbb R_{+}^{n+1}}
\bigg({t\over t+|x-y|}\bigg)^{n\mu}
|\Psi(t\sqrt{L})f(y)|^2{dydt\over t^{n+1}}\Bigg)^{1/2}, \ \ \mu>1.
\end{eqnarray}
In this section, we will show that the area integrals $s_p$, $s_h $, $s_H$ and $ S_H$
are all controlled by $\gL$ pointwise. To achieve this, we need some results on the kernel estimates
of the semigroup. Firstly, we note that the Gaussian upper bounds for $p_t(x,y)$ are further
inherited by the time derivatives of $p_{t}(x,y)$. That is, for each
$k\in{\mathbb N}$, there exist two positive constants $c_k$ and $C_k$ such
that
\begin{eqnarray}\label{e3.0}
\Big|{\partial^k \over\partial t^k} p_{t}(x,y) \Big|\leq
\frac{C_k}{ t^{n /2+k} } \exp\Big(-{|x-y|^2\over
c_k\,t}\Big)
\end{eqnarray}
\noindent for all $t>0$, and $x, y\in {\mathbb R}^{n}$. For
the proof of (\ref{e3.0}), see \cite{Da}
and \cite{Ou}, Theorem~6.17.
Note that in the absence of regularity on space variables of $p_t(x,y)$, estimate (\ref{e3.0})
plays an important role in our theory.
\begin{lemma} \label{le3.1} \,
Let $L$ be a non-negative self-adjoint operator such that the corresponding
heat kernels $p_t(x,y)$ of the semigroup $e^{-tL}$ satisfy Gaussian bounds $(GE)$. Then
for every $\kappa=0,1, ..., $
the operator $ (t\sqrt{L})^{2\kappa} e^{-t\sqrt{L}}$ satisfies
\begin{eqnarray}\label{e3.00}\hspace{1cm}
\big|K_{(t\sqrt{L})^{2\kappa} e^{-t\sqrt{L}}}(x,y)\big|\leq C_\kappa t^{-n}\Big(1+{ |x-y|\over t}\Big)^{-(n+2\kappa+1)}, \ \ \forall t>0
\end{eqnarray}
\noindent
for almost every $x,y\in \RN.$
\end{lemma}
\begin{proof}
The proof of (\ref{e3.00}) is simple. Indeed, the subordination formula
$$e^{-t\sqrt{L}}={1\over \sqrt{\pi}}\int^\infty_0e^{-u}u^{-1/2}e^{-{t^2\over 4u}L}du$$
allows us to estimate
\begin{eqnarray*}
\big|K_{(t\sqrt{L})^{2\kappa} e^{-t\sqrt{L}}}(x,y)\big|&\leq&
C_\kappa\int_0^{\infty}{e^{-u}\over\sqrt{u}}\Big({t^2\over u}\Big)^{-n/2}\exp \Big(-{u|x-y|^2\over c t^2}\Big)u^\kappa du\\
&\leq&C_\kappa t^{-n}\int_0^{\infty}e^{-u}u^{n/2+\kappa-1/2} \exp \Big(-{u|x-y|^2\over c t^2}\Big)\, du\\
&\leq&C_\kappa t^{-n}\Big(1+{|x-y| \over t }\Big)^{-(n+2\kappa+1)}
\end{eqnarray*}
\noindent for every $t>0$ and almost every $x,y\in \RN$.
\end{proof}
\begin{lemma} \label{le3.2} \,
Let $L$ be a non-negative self-adjoint operator such that the corresponding
heat kernels $p_t(x,y)$ of the semigroup $e^{-tL}$ satisfy $(GE)$ and ($G$). Then
for every $\kappa=0,1, ..., $
the operator $ t^{2\kappa+1}\nabla (L^\kappa e^{-t^2L} )$ satisfies
\begin{eqnarray*}
\Big|K_{t^{2\kappa+1}\nabla (L^\kappa e^{-t^2L} )}(x,y)\Big|
&\leq& C t^{-n} \exp\Big(-{|x-y|^2\over
c \,t^2}\Big), \ \ \ \forall t>0
\end{eqnarray*}
\noindent
for almost every $x,y\in \RN.$
\end{lemma}
\begin{proof} Note that $ t^{2\kappa+1}\nabla (L^\kappa e^{-t^2L} )= t\nabla e^{-{t^2\over 2}L} \circ (t^2L)^\kappa e^{-{t^2\over 2}L}.$
Using (\ref{e3.0}) and the pointwise gradient estimate ($G$) of heat kernel $p_t(x,y)$, we have
\begin{eqnarray*}
\Big|K_{t^{2\kappa+1}\nabla (L^\kappa e^{-t^2L} )}(x,y)\Big|&=& \Big| \int_{\RN} K_{t\nabla e^{-{t^2\over 2}L}}(x,z) K_{(t^2L)^\kappa e^{-{t^2\over 2}L}} (z,y)dz\Big|\nonumber\\
&\leq& C t^{-2n}\int_{\RN} \exp\Big(-{|x-z|^2\over
c \,t^2}\Big) \exp\Big(-{|z-y|^2\over
c \,t^2}\Big) dz\nonumber\\
&\leq& C t^{-n} \exp\Big(-{|x-y|^2\over
c \,t^2}\Big)
\end{eqnarray*}
\noindent for every $t>0$ and almost every $x,y\in \RN$.
\end{proof}
Now we start to prove the following Propositions~\ref{prop3.3} and ~\ref{prop3.4}.
\begin{prop}\label{prop3.3} Let $L$ be a non-negative self-adjoint operator such that the corresponding
heat kernels satisfy condition $(GE)$.
Then for $f\in {\mathcal S}(\RN),$ there
exists a constant $C=C_{n,\mu,\Psi}$ such that the area integral $s_p$ satisfies the pointwise estimate:
\begin{eqnarray}\label{e3.4}
s_pf(x) \leq C\gL(f)(x).
\end{eqnarray}
Estimate (\ref{e3.4}) also holds for the area integral $s_h$.
\end{prop}
\begin{prop}\label{prop3.4} Let $L$ be a non-negative self-adjoint operator such that the corresponding
heat kernels satisfy conditions $(GE)$ and $(G)$.
Then for $f\in {\mathcal S}(\RN),$ there
exists a constant $C=C_{n,\mu,\Psi}$ such that the area integral $S_P$ satisfies the pointwise estimate:
\begin{eqnarray}\label{e3.2}
S_Pf(x) \leq C\gL(f)(x).
\end{eqnarray}
Estimate (\ref{e3.2}) also holds for the area integral $S_H$.
\end{prop}
\begin{proof} [Proofs of Propositions~\ref{prop3.3} and ~\ref{prop3.4}]\
Let us begin to prove (\ref{e3.2}).
By the spectral theory (\cite{Yo}),
for every $f\in {\mathcal S}(\RN)$ and every $\kappa \in{\mathbb N}$,
\noindent\begin{eqnarray*}
f =C_\Psi\int^\infty_0
(t^2L)^{\kappa}e^{-t^2{L}}\Psi(t\sqrt{L}) f {dt\over t}
\end{eqnarray*}
\noindent
with $C^{-1}_{\Psi}=\int^\infty_0 t^{2\kappa}
e^{-t^2} \Psi(t ){dt/t}$, and the integral converges in $L^2(\RN)$.
Recall the subordination formula:
$$
e^{-t\sqrt{L}} ={1\over \sqrt{\pi}}\int^\infty_0 e^{-u} {u}^{-1/2} e^{-{t^2\over 4u}L} du.
$$
\noindent
One writes
\begin{eqnarray} \label{e3.666}
\hspace{-1cm} s\nabla e^{-s\SL}f(y)
&=& {1\over \sqrt{\pi}}\int^\infty_0e^{-u} {u}^{-1/2}s\nabla e^{-{s^2\over 4u}L}f(y)du \nonumber\\
&=&{ C_{\Psi} \over \sqrt{\pi}}\int^{\infty}_{0}\int^\infty_0e^{-u} {u}^{-1/2} st^{2\kappa}\nabla\big(L^\kappa
e^{-({s^2\over 4u}+t^2)L}\big)\Psi(t\sqrt{L})f (y){dt\, du\over t}.
\end{eqnarray}
\noindent
Fix $\kappa=[{n(\mu-1)\over 2}]+1$.
Using Lemma~\ref{le3.2} and the H\"older inequality, we can estimate (\ref{e3.666}) as follows:
\begin{eqnarray*}
&&\hspace{-1cm}|s\nabla e^{-s\SL}f(y)|\\
&\leq&C \int^{\infty}_0\int^{\infty}_0\int_{\RN}e^{-u}u^{-1/2}st^{2\kappa}\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa-n/2}e^{-|y-z|^2/c({s^2\over 4u}+{t^2 })}
|\Psi(t\SL)f(z)|{dzdtdu\over t}\\
&\leq&C A\cdot B,
\end{eqnarray*}
\noindent
where
\begin{eqnarray*}
A^2 &= & \int^{\infty}_0\int^{\infty}_0\int_{\RN}|\Psi(t\SL)f(z)|^2e^{-u} {u}^{-1/2}s t^{2\kappa}
\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa-n/2}e^{-|y-z|^2/c({s^2\over 4u}+{t^2 })}
{dzdtdu\over t} \end{eqnarray*}
\noindent
and
\begin{eqnarray*}
B^2
&=& \int^{\infty}_0\int^{\infty}_0\int_{\RN}e^{-u}u^{-1/2}st^{2\kappa}\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa-n/2}e^{-|y-z|^2/c({s^2\over 4u}+{t^2 })}
{dzdtdu\over t}\\
&=&C \int^{\infty}_0\int^{\infty}_0\int_{0}^{\infty}e^{-u}u^{-1/2}st^{2\kappa}\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa}e^{-r^2 }
r^{n-1}{drdtdu\over t}\\
&\leq& C \int^{\infty}_0\int^{\infty}_0 e^{-u} \, v^{2\kappa}\Big(1+{v^2 }\Big)^{-1/2-\kappa} { du dv\over v} \\
&\leq& C.
\end{eqnarray*}
\noindent
Note that in the first equality of the above term $B$, we have changed variables $|y-z|\rightarrow r({s^2\over 4u}+{t^2 })^{1/2}$ and $t\rightarrow v ({s^2/u})^{1/2}$.
Hence,
\begin{eqnarray*}
&&\hspace{-0.3cm}|s\nabla e^{-s\SL}f(y)|^2\\
&&\leq C
\int^{\infty}_0\int^{\infty}_0\int_{\RN}|\Psi(t\SL)f(z)|^2e^{-u}u^{-1/2}st^{2\kappa}\Big({s^2\over 4u}
+{t^2 }\Big)^{-1/2-\kappa-n/2}e^{-|y-z|^2/c({s^2\over 4u}+{t^2 })}
{dzdtdu\over t}.
\end{eqnarray*}
\noindent Therefore, we put it into the definition of $S_p$ to obtain
\begin{eqnarray*}
&&\hspace{-0.6cm}S^2_P(f)(x) = \int^{\infty}_0\int_{|x-y|<s}|s\nabla e^{-s\SL}f(y)|^2{dyds\over s^{n+1}}\\
&&\leq C\iint_{\mathbb R^{n+1}_{+}}|\Psi(t\SL)f(z)|^2\\
&&\hspace{0.2cm}\times\Bigg(\int^{\infty}_0\int^{\infty}_0
\int_{|x-y|<s}e^{-u}u^{-1/2}st^{2\kappa+n}\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa-n/2}e^{-|y-z|^2/c({s^2\over 4u}+{t^2 })}
{dydsdu\over s^{n+1}}\Bigg)
{dzdt\over t^{n+1}}.
\end{eqnarray*}
\noindent
We will be done if we show that
\begin{eqnarray}\label{e3.3}
&&
\hspace{-1.2cm}
\int^{\infty}_0\int^{\infty}_0\int_{|v-y|<s}e^{-u}u^{-1/2}st^{2\kappa+n}\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa-n/2}e^{-|y|^2/c({s^2\over 4u}+{t^2 })}
{dydsdu\over s^{n+1}}\\
&\leq& C\Big({t\over t+|v|}\Big)^{n\mu},\nonumber
\end{eqnarray}
\noindent
where we set $x-z=v,$ and we will prove estimate (\ref{e3.3}) by considering the following two cases.
\noindent
{\it Case 1.} $|v|\leq t.$ \ In this case, it is easy to show that
\begin{eqnarray*}
{\rm LHS\ of \ (\ref{e3.3})}
&&\leq C \int^{\infty}_0\int^{\infty}_0 e^{-u}u^{-1/2}st^{2\kappa+n}\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa-n/2}
{ dsdu\over s }\\
&&\leq C \int^{\infty}_0\int^{\infty}_0 e^{-u}u^{-1/2}\sqrt{u}s\big({s^2}+1\big)^{-1/2-\kappa-n/2}
{ dsdu\over s }\\
&&\leq C.
\end{eqnarray*}
\noindent But $|v|\leq t$, so
$$\Big({t\over t+|v|}\Big)^{n\mu}\geq C_{n,\mu}.$$
\noindent This implies that (\ref{e3.3}) holds when $|v|\leq t.$
\noindent
{\it Case 2.} $|v|> t$. In this case, we break the integral into two pieces:
\begin{eqnarray*}
\int^\infty_0\int^{|v|/2}_0\int_{|v-y|<s}\cdots +\int^\infty_0\int^{\infty}_{|v|/2}\int_{|v-y|<s}\cdots =:I+{\it II}.
\end{eqnarray*}
\noindent For the first term, note that $|y|\geq |v|-|v-y|>|v|/2.$ This yields
\begin{eqnarray*}
I&\leq&\int^{\infty}_0\int^{|v|/2}_0\int_{|v-y|<s}e^{-u}u^{-1/2}st^{2\kappa+n}
\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa-n/2}e^{-|v|^2/c({s^2\over 4u}+{t^2 })}
{dydsdu\over s^{n+1}}\\
&\leq&C\int^{\infty}_0\int^{\infty}_0e^{-u}u^{-1/2}st^{2\kappa+n}
\Big({s^2\over 4u}+{t^2}\Big)^{-1/2-\kappa-n/2} \Big({s^2\over 4u}+{t^2 }\Big)^{n\mu/2}/|v|^{n\mu}
{ dsdu\over s }\\
&\leq&C\Big({t\over |v|}\Big)^{n\mu}\int^{\infty}_0\int^{\infty}_0
e^{-u}u^{-1/2}st^{2\kappa+n-n\mu}\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa-n/2+n\mu/2}
{ dsdu\over s }\\
&\leq&C\Big({t\over |v|}\Big)^{n\mu}\int^{\infty}_0\int^{\infty}_0e^{-u} s \big({s^2+1}\big)^{-1/2-\kappa-n/2+n\mu/2}
{ dsdu\over s }\\
&\leq&C\Big({t\over |v|}\Big)^{n\mu},
\end{eqnarray*}
\noindent where we used condition $\kappa=[{n(\mu-1)\over2}]+1$ in the last inequality.
Since $|v|>t$,
so $I\leq C_{n,\mu}\Big({t \over t+|v|}\Big)^{n\mu}.$
\noindent
For the term $\it{II}$, we have
\begin{eqnarray*}
\it{II}
&\leq&C\int^{\infty}_0\int_{|v|/2}^{\infty}\int_{0}^{\infty}e^{-u}
u^{-1/2}st^{2\kappa+n}\big({s^2\over 4u}+{t^2 }\big)^{-1/2-\kappa-n/2}e^{-r^2/ ({s^2\over 4u}+{t^2 })}r^{n-1}
{drdsdu\over s^{n+1}}\\
&\leq&C\int^{\infty}_0\int_{|v|/2}^{\infty} e^{-u}u^{-1/2}st^{2\kappa+n}\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa}
{ dsdu\over s^{n+1}}\\
&\leq&C\int^{\infty}_0\int_{|v|/(4\sqrt{u}t)}^{\infty} e^{-u}u^{-1/2}(\sqrt{u})^{1-n}
\Big({s^2+1}\Big)^{-1/2-\kappa}
{ dsdu\over s^{n}}\\
&\leq&C\int^{\infty}_0 e^{-u}u^{-1/2}(\sqrt{u})^{1-n}
\Big({\sqrt{u}\, t\over |v|}\Big)^{2\kappa+n}
du\\&\leq&C\Big({t\over |v|}\Big)^{2\kappa+n}\\
&\leq&
C \Big({t\over t+|v|}\Big)^{n\mu},
\end{eqnarray*}
\noindent since $\kappa=[{n(\mu-1)\over2}]+1$ and $|v|>t$.
From the above {\it Cases 1} and {\it 2}, we have obtained estimate (\ref{e3.3}), and then the proof of
estimate (\ref{e3.2}) is complete.
The similar argument as above gives estimate (\ref{e3.2}) since for the area integral $S_H$, and this completes
the proof of Proposition~\ref{prop3.4}.
For the area functions $s_P$ and $s_h$, we can use a similar argument to show Proposition~\ref{prop3.3}
by using either estimate (\ref{e3.0}) or Lemma~\ref{le3.1} instead of Lemma~\ref{le3.2} in the proof
of estimate (\ref{e3.2}), and we skip it here.
\end{proof}
\subsection{Weighted $L^2$ estimate of $\gL$.}
\begin{theorem}\label{th3.5}
Let $\mu>1$. Then there exists a constant $C=C_{n,\mu,\Phi}$ such that
for all $w\geq 0$ in $L_{loc}^{1}(\RN)$ and all $f\in {\mathcal S}(\RN)$, we
have
\begin{eqnarray}\label{e3.6}
\int_{\RN}\gL (f)^2wdx\leq C\int_{\RN}|f|^2Mwdx.
\end{eqnarray}
\end{theorem}
\begin{proof} The proof essentially follows from \cite{CWW} and \cite{CW} for the classical area function.
Note that by Lemma~\ref{le2.1}, the kernel $K_{\Psi(t\sqrt{L})}$ of the operator
$\Psi(t\sqrt{L})$ satisfies supp $K_{\Psi(t\sqrt{L})}\subseteq \big\{ (x,y)\in \RN\times \RN: |x-y|\leq t\big\}.$
By (\ref{e3.1}), one writes
\begin{eqnarray}\label{e3.7}\hspace{1cm}
\int_{\RN}\gL (f)^2wdx
&=& \int_{\RN} \int_{\mathbb R_{+}^{n+1}}
\bigg({t\over t+|x-y|}\bigg)^{n\mu}
|\Psi(t\sqrt{L})f(y)|^2{dydt\over t^{n+1}} dx\nonumber\\
&=&
\int_{\mathbb R^{n+1}_{+}}|\Psi(t\sqrt{L})f(y)|^2\bigg({1\over t^n}\int_{\RN}w(x)
\Big({t\over t+|x-y|}\Big)^{n\mu}dx\bigg){dydt\over t}.
\end{eqnarray}
\noindent For $k$ an integer, set
$$A_k=\Big\{(y,t):\ 2^{k-1}<{1\over t^n}\int_{\RN}w(x)
\Big({t\over t+|x-y|}\Big)^{n\mu}dx\leq 2^k\Big\}.$$
\noindent Then
\begin{eqnarray}\label{e3.8}
{\rm RHS \ of \ (\ref{e3.7})} \ \leq \sum_{k\in \mathbb{Z}}2^k
\int_{\mathbb R^{n+1}_{+}}|\Psi(t\sqrt{L})f(y)|^2\chi_{A_k}(y,t){dydt\over t}.
\end{eqnarray}
\noindent We note that if $(y,t)\in A_k$, then since $\mu>1$,
$$2^{k-1}\leq {1\over t^n}\int_{\RN}w(x)
\Big({t\over t+|x-y|}\Big)^{n\mu}dx\leq CMw(y).$$
\noindent Now if $|y-z|<t$, then $t+|x-y|\approx t+|x-z|$. Thus if
$|y-z|<t$ and $(y,t)\in A_k,$
$$2^{k-1}\leq {C\over t^n}\int_{\RN}w(x)
\Big({t\over t+|x-z|}\Big)^{n\mu}dx\leq CMw(z).$$
\noindent In particular, if $(y,t)\in A_k$ and $|y-z|<t$, then
$z\in E_k=\{z:\ Mw(z)\geq C2^k\}$. Now since ${\rm supp}\ \! K_{ \Psi(t\sqrt{L})}(y,z)
\subseteq \big\{(y,z)\in \RN\times \RN: |y-z|\leq t\big\}$, for
$(y,t)\in A_k$,
$$\Psi(t\sqrt{L})f(y)=\int_{|y-z|<t}K_{ \Psi(t\sqrt{L})}(y,z)f(z)dz
=\int_{\RN}K_{ \Psi(t\sqrt{L})}(y,z)f(z)\chi_{E_k}(z)dz.$$
\noindent Therefore,
\begin{eqnarray*}
{\rm RHS \ of \ (\ref{e3.7})} \ &\leq& \sum_{k\in \mathbb{Z}}2^k
\int_{\mathbb R^{n+1}_{+}}|\Psi(t\sqrt{L})(f\chi_{E_k})(y)|^2\chi_{A_k}(y,t){dydt\over t}\\
&\leq& \sum_{k\in \mathbb{Z}}2^k
\int_{\mathbb R^{n+1}_{+}}|\Psi(t\sqrt{L})(f\chi_{E_k})(y)|^2{dydt\over t}\\
&=& \sum_{k\in \mathbb{Z}}2^k
\int_0^{\infty}\|\Psi(t\sqrt{L})(f\chi_{E_k})\|_{L^2(\RN)}^2{ dt\over t}\\
&=& C_{\Psi} \sum_{k\in \mathbb{Z}}2^k \| f\chi_{E_k} \|_{L^2(\RN)}^{2}
\end{eqnarray*}
\noindent with $C_{\Psi}=\int^\infty_0|\Psi(t)|^2{dt/t}<\infty,$ and the last inequality follows from the
spectral theory (see \cite{Yo}). By interchanging the
order of summation and integration, we have
\begin{eqnarray*}
\int_{\RN}\gL (f)^2wdx&\leq& C \sum_{k\in \mathbb{Z}}2^k\int_{\RN}|f|^2\chi_{E_k}dx\\
&\leq& C\int_{\RN}|f|^2\Big(\sum_{k\in \mathbb{Z}}2^k\chi_{E_k}\Big)dx\\
&\leq& C \int_{\RN}|f|^2Mwdx.
\end{eqnarray*}
\noindent This concludes the proof of the theorem.
\end{proof}
As a consequence of Propositions~\ref{prop3.3}, ~\ref{prop3.4} and Theorem~\ref{th3.5}, we have the following
analogy for the area function of the result of Chang, Wilson and Wolff.
\begin{cor}\label{c3.3} Let $T$ be of the area integrals $s_h$, $s_p$, $S_{P}$ and $S_{H}.$
Under assumptions of Theorems~\ref{th1.1} and ~\ref{th1.2}, there exists a constant $C$ such that
for all $w\geq 0$ in $L_{loc}^{1}(\RN)$ and all $f\in {\mathcal S}(\RN)$,
\begin{eqnarray*}
\int_{\RN}|Tf|^2wdx\leq C\int_{\RN}|f|^2Mwdx.
\end{eqnarray*}
\end{cor}
\subsection{Proof of Theorem 1.3}
Let $T$ be of the area functions $s_h$, $s_p$, $S_{P}$ and $S_{H}.$
For $w\in A_1$, we have $Mw(x)\leq \|w\|_{A_1}w(x)$ for a.e. $x\in \RN$.
According to Corollary \ref{c3.3},
\begin{eqnarray*}
\int_{\RN}T(f)^2wdx\leq C\int_{\RN}|f|^2Mwdx\leq C\|w\|_{A_1}\int_{\RN}|f|^2wdx.
\end{eqnarray*}
\noindent This implies (\ref{e1.8}) holds.
For (\ref{e1.9}), we follow the
method of Cordoba and Rubio de Francia (see pages 356-357, \cite{FP}). Let $p>2$ and take $f\in L^p(\RN).$ Then from duality, we know that
there exist some $\varphi\in L^{(p/2)'}(\RN),$ with $\varphi\geq0,$ $\|\varphi\|_{L^{(p/2)'}(\RN)}=1$, such that
$$\|Tf\|^2_{L^p(\RN)}\leq \int_{\RN}|Tf|^2\varphi dx.$$
\noindent Set
$$v=\varphi +{M\varphi\over 2\|M\|_{L^{(p/2)'}(\RN)}}+{M^2\varphi\over (2\|M\|_{L^{(p/2)'}(\RN)})^2}+\cdots$$
\noindent following Rubio de Francia's familiar method (Here $\|M\|_{L^{(p/2)'}(\RN)}$
denotes the
operator norm of the Hardy-Littlewood maximal operator on $L^{(p/2)'}(\RN)$). Then
$\|v\|_{L^{(p/2)'}(\RN)}\leq 2$ and $\|v\|_{A_1}\leq 2\|M\|_{L^{(p/2)'}(\RN)}\equiv O(p)$ as $p\to \infty.$
Therefore
\begin{eqnarray*}
\|Tf\|^2_{L^p(\RN)}&\leq& \int_{\RN}|Tf|^2\varphi dx\\
&\leq&\int_{\RN}|Tf|^2v dx\\
&\leq&C\|v\|_{A_1}\int_{\RN}|f|^2v dx\\
&\leq&Cp\|f\|^2_{L^p(\RN)}.
\end{eqnarray*}
\noindent This proves (\ref{e1.9}), and then the proof of this theorem is complete.
{}$\Box$
Note that in Theorem~\ref{e1.3}, when $L=-\Delta$ is the Laplacian on $\RN$, it is well known
that estimate (\ref{e1.9})
of the classical area integral on $L^p(\RN)$ is sharp, in general (see, e.g., \cite{FP}).
\section{Proofs of Theorems 1.1 and 1.2}
\setcounter{equation}{0}
Note that from Propositions~\ref{prop3.3} and ~\ref{prop3.4}, the area functions $S_H, S_P, s_H$ and $s_p$ are all controlled by the $\gL$ function.
In order to prove Theorems 1.1 and 1.2, it suffices to show the following result.
\begin{theorem}\label{th4.1} \ Let $L$ be a non-negative self-adjoint operator such that the corresponding
heat kernels satisfy Gaussian bounds $(GE)$. Let $\mu>3$.
If $w\geq 0$, $w\in L_{\rm loc}^{1}(\RN)$ and $f\in {\mathcal S}(\RN)$, then
\begin{eqnarray*}
\hspace{-2.5cm}&{\rm (a)}& \hspace{0.1cm}\int_{\{\gL (f)>\lambda\}} wdx\leq {c(n)\over \lambda}\int_{\RN}|f| Mwdx,\ \ \ \lambda>0,\\
\hspace{-2.5cm}
&{\rm (b)}& \hspace{0.1cm}
\int_{\RN}\gL (f)^pw\, dx\leq c(n,p)\int_{\RN}|f|^pMw\, dx,\ \ \ 1<p\leq 2,
\\
\hspace{-2.5cm}&{\rm (c)}& \hspace{0.1cm} \int_{\RN} \gL (f)^pwdx\leq c(n,p)\int_{\RN}|f|^p(Mw)^{p/2}w^{-(p/2-1)}dx, \ \ \ 2<p<\infty.
\end{eqnarray*}
\end{theorem}
\subsection{Weak-type $(1,1)$ estimate}\
We first state a Whitney decomposition. For its proof, we refer to Chapter 6, \cite{St}.
\begin{lemma} \label{le4.2}
Let $F$ be a non-empty closed set in $\RN$. Then its complement
$\Omega$ is the union of a sequence of cubes $Q_k$, whose sides are parallel to the axes,
whose interiors are mutually disjoint, and whose diameters are
approximately proportional to their distances from $F$. More explicitly:
\noindent
(i) $\Omega=\RN\setminus F=\bigcup\limits_{k=1}^{\infty}Q_k.$
\noindent
(ii) $Q_j\bigcap Q_k=\varnothing$ if $j\neq k$.
\noindent
(iii) There exist two constants $c_1, c_2 > 0$, (we can take $c_1 = 1$, and
$c_2 = 4$), so that
$$c_1 {\rm diam}(Q_k)\leq {\rm dist}(Q_k,\ F)\leq c_2 {\rm diam}(Q_k).$$
\end{lemma}
Note that if $\Omega$ is an open set with $\Omega=\bigcup\limits_{k=1}^{\infty}Q_k$
a Whitney decomposition, then for every $\varepsilon:\ 0<\varepsilon<1/4,$ there exists $N\in \mathbb{N}$
such that no point in $\Omega$ belongs to more than $N$ of the cubes $Q_{k}^{\ast}$, where
$Q_{k}^{\ast}=(1+\varepsilon)Q_k.$
\begin{proof}[Proof of (a) of Theorem~\ref{th4.1}]
Since $g^{\ast}_{\mu',\Psi}(f)\leq \gL(f)$ whenever $\mu'\geq \mu$, it is enough
to prove ($a$) of Theorem~\ref{th4.1} for $3<\mu<4.$ Since $\gL$ is subadditive, we may assume that $f\geq0$
in the proof (if not we only need to consider the positive part and the negative part of $f$).
For $\lambda>0,$ we set $\Omega=\{x\in\RN:\ Mf(x)>\lambda\}$. By \cite{FS} it follows that
\begin{eqnarray}\label{e4.2}
\int_{\Omega}wdx\leq {C\over \lambda}\int_{\RN}|f|Mwdx.
\end{eqnarray}
\noindent Let $\Omega=\cup Q_j$ be a Whitney decomposition, and define
\begin{eqnarray*}
h(x)&=&\left\{
\begin{array}{ll}
f(x), & x\notin \Omega \\ [12pt]
{1\over |Q_j|}\int_{Q_j}f(x)dx, & x\in Q_j
\end{array}
\right.\\[12pt]
b_j(x)&=&\left\{
\begin{array}{ll}
f(x)-{1\over |Q_j|}\int_{Q_j}f(x)dx, & x\in Q_j \\ [12pt]
0, & x\notin Q_j.
\end{array}
\right.
\end{eqnarray*}
\noindent Then $f=h+\sum_jb_j$, and we set $b=\sum_jb_j.$ As in \cite{St}, we have
$|h|\leq C\lambda$ a.e. By (\ref{e4.2}), it suffices to show
\begin{eqnarray}\label{e4.3}
w\{x\notin\Omega:\ \gL(f)(x)>\lambda\}\leq {C\over \lambda}\int_{\RN}|f|Mwdx.
\end{eqnarray}
\noindent By Chebychev's inequality and Theorem~\ref{th3.5},
\begin{eqnarray*}
w\{x\notin\Omega:\ \gL(h)(x)>\lambda\}&\leq& {1\over \lambda^2}
\int_{\RN}\gL(h)^2(w\chi_{\RN\setminus\Omega})dx\\
&\leq& {C\over \lambda^2}
\int_{\RN}|h|^2 M(w\chi_{\RN\setminus\Omega})dx\\
&\leq&{C\over \lambda }
\int_{\RN}|h| M(w\chi_{\RN\setminus\Omega})dx
\end{eqnarray*}
\noindent since $|h|\leq C\lambda$ a.e. By definition of $h$, the last expression is at most
\begin{eqnarray}\label{e4.4}
{C\over \lambda }
\int_{\RN}|f| Mwdx +\sum_j{C\over \lambda }
\int_{Q_j}\Big({1\over |Q_j|}\int_{Q_j}|f(z)|dz\Big) M(w\chi_{\RN\setminus\Omega})(x)dx.
\end{eqnarray}
\noindent From the property (iii) of Lemma \ref{le4.2}, we know that for $x,z\in Q_j$ there is a constant
$C$ depending only on $n$ so that $M(w\chi_{\RN\setminus\Omega})(x)\leq CM(w\chi_{\RN\setminus\Omega})(z)$.
Thus (\ref{e4.4}) is less than
$${C\over \lambda }
\int_{\RN}|f| Mwdx +\sum_j{C\over \lambda }
\int_{Q_j}\Big({1\over |Q_j|}\int_{Q_j}|f(z)|Mw(z)dz\Big) dx\leq {C\over \lambda }
\int_{\RN}|f| Mwdx.$$
\noindent This gives
$$w\{x\notin\Omega:\ \gL(h)(x)>\lambda\}\leq{C\over \lambda }
\int_{\RN}|f| Mwdx.$$
\noindent Therefore, estimate (\ref{e4.3}) will follow if we show that
\begin{eqnarray}\label{eb}
w\{x\notin\Omega:\ \gL(b)(x)>\lambda\}\leq{C\over \lambda }
\int_{\RN}|f| Mwdx.
\end{eqnarray}
\noindent To prove (\ref{eb}), we follow an idea of \cite{DM} to decompose $b=\sum_jb_j=\sum_j\Phi_j(\sqrt{L})b_j + \sum_j\big(1-\Phi_j(\sqrt{L})\big)b_j,$
where $\Phi_j(\sqrt{L})=\Phi\Big({\ell(Q_j)\over32}\sqrt{L}\Big),$ $\Phi$ is the function as in Lemma~\ref{le2.2}
and $\ell(Q_j)$ is the side length of the cube $Q_j.$ See also \cite{CD}.
So, it reduces to
show that
\begin{eqnarray}\label{ei1}
w\{x\notin\Omega:\ \gL\Big(\sum_j\Phi_j(\sqrt{L})b_j\Big)(x)>\lambda\}\leq{C\over \lambda }
\int_{\RN}|f| Mwdx
\end{eqnarray}
\noindent
and
\begin{eqnarray}\label{ei2}
w\{x\notin\Omega:\ \gL\Big(\sum_j\big(1-\Phi_j(\sqrt{L})\big)b_j\Big)(x)>\lambda\}\leq{C\over \lambda }
\int_{\RN}|f| Mwdx.
\end{eqnarray}
\noindent By Chebychev's inequality and Theorem \ref{th3.5} again, we have
\begin{eqnarray*}
{\rm LHS\ \ of \ \ (\ref{ei1})}\ &\leq&{C\over \lambda^2}\int_{\RN}\Big|\gL\Big(\sum_j\Phi_j(\sqrt{L})b_j\Big)\Big|^2
(w\chi_{\RN\setminus\Omega})dx\\
&\leq&{C\over \lambda^2}\int_{\RN}\Big| \sum_j\Phi_j(\sqrt{L})b_j \Big|^2
M(w\chi_{\RN\setminus\Omega})dx.
\end{eqnarray*}
\noindent Note that $\Phi_j(\sqrt{L})=\Phi\Big({\ell(Q_j)\over32}\sqrt{L}\Big),$ it follows from Lemma~\ref{le2.1}
that ${\rm supp}\ \Phi_j(\sqrt{L})b_j\subset {{17 }Q_j/16} $ and $\big|K_{\Phi_j(\sqrt{L})}(x,y)\big|\leq C/\ell(Q_j)$.
Hence, the above inequality is at most
\begin{eqnarray*}
{C\over \lambda^2}\sum_j \int_{\RN}\Big| \Phi_j(\sqrt{L})b_j \Big|^2
M(w\chi_{\RN\setminus\Omega})dx.
\end{eqnarray*}
\noindent
This, together with Lemma~\ref{le2.1} and the definition of $b$,
yields
\begin{eqnarray*}
{\rm LHS\ \ of \ \ (\ref{ei1})}\
&\leq& {C\over \lambda^2}\sum_j\int_{{17 }Q_j/16}\Big({\ell(Q_j)^{-n}}\int_{Q_j} |b(y)|dy \Big)^2
M(w\chi_{\RN\setminus\Omega})(x)dx\\
&\leq& {C\over \lambda^2}\sum_j\int_{{17 }Q_j/16}\Big( {1\over|Q_j|}\int_{Q_j}|f(y)| dy \Big)^2
M(w\chi_{\RN\setminus\Omega})(x)dx\\
&\leq& {C\over \lambda }\sum_j\int_{{17 }Q_j/16}\Big( {1\over|Q_j|}\int_{ Q_j}|f(y)| dy \Big)
M(w\chi_{\RN\setminus\Omega})(x)dx\\
&\leq& {C\over \lambda }\sum_j{1\over|Q_j|}\int_{{17 }Q_j/16} \int_{ Q_j}|f(y)|M(w\chi_{\RN\setminus\Omega})(y) dy
dx\\
&\leq&{C\over \lambda }\int_{\RN}|f |Mwdy.
\end{eqnarray*}
\noindent This proves the desired estimate (\ref{ei1}).
Next we turn to estimate (\ref{ei2}). It suffices to show that
$$\sum_j\int_{\RN\setminus\Omega}\gL\Big( \big(1-\Phi_j(\sqrt{L})\big)b_j\Big)wdx
\leq C\int_{\RN}|f|Mwdx.$$
\noindent
Further, the above inequality reduces to prove the following result:
\begin{eqnarray}\label{ei22}
\int_{\RN\setminus\Omega}\gL\Big( \big(1-\Phi_j(\sqrt{L})\big)b_j\Big)wdx
\leq C\int_{Q_j}|f|Mwdx.
\end{eqnarray}
\noindent Let $x_j$ denote the center of $Q_j$. Let us estimate
$\Psi(t\sqrt{L})\big(1-\Phi_j(\sqrt{L})\big)b_j(y)=:\Psi_{jt}(\sqrt{L})b_j(y)$ by considering
two cases: $t\leq \ell(Q_j)/4$ and $t>\ell(Q_j)/4$.
\noindent
{\it Case 1. $t\leq \ell(Q_j)/4$}. \ In this case, we use Lemma \ref{le2.1} to obtain
\begin{eqnarray*}
\big|\Psi_{jt}(\sqrt{L})b_j(y)\big|&\leq& |\Psi(t\sqrt{L})b_j(y)|+\Big|\Psi(t\sqrt{L})\Phi_j(\sqrt{L})b_j(y)\Big|\\
&\leq& \Big|\int_{Q_j}K_{\Psi(t\sqrt{L})}(y,z)b(z)dz\Big|\\
&&+\Big|\int_{{17\over16}Q_j}
K_{\Psi(t\sqrt{L})}(y,z)\Big(\int_{Q_j}K_{\Phi_j(\sqrt{L})}(z,x)b(x)dx\Big)dz\Big|\\
&\leq& C\|b_j\|_{ 1}t^{-n}.
\end{eqnarray*}
\noindent
{\it Case 2. $t> \ell(Q_j)/4$}. Using Lemma \ref{le2.2}, we have
\begin{eqnarray*}
\big|\Psi_{jt}(\sqrt{L})b_j(y)\big|
& \leq&\int_{\RN}\Big|K_{\Psi(t\sqrt{L})\big(1-\Phi_j(\sqrt{L})\big)}(y,z)\Big||b_j(z)|dz\\
& \leq& C \|b_j\|_{ 1}\ell(Q_j)t^{-n-1}.
\end{eqnarray*}
From the property (iii) of Lemma \ref{le4.2}, we know that if $x\notin\Omega,$
then $|x-x_j|> (\sqrt{n}+1/2)\ell(Q_j)$. By Lemma \ref{le2.1}, we have
$\Psi(t\sqrt{L})\big(1-\Phi_j(\sqrt{L})\big)b_j(y)=0$ unless $|y-x_j|\leq t+(1/32+\sqrt{n}/2)\ell(Q_j).$
Note that for $x\notin\Omega$, $0<t\leq \ell(Q_j)/4$ and $|y-x_j|\leq t+(1/32+\sqrt{n}/2)\ell(Q_j)$,
$|x-y|\geq|x-x_j|-|y-x_j|>{\sqrt{n}/2+7/32\over \sqrt{n}+1/2}|x-x_j|$.
Denote $F_j=:\{y:\ |y-x_j|<(9/32+\sqrt{n}/2)\ell(Q_j)\}$.
Then for $x\notin\Omega$ and $\mu>3$, we have
\begin{eqnarray*}
&&\hspace{-1cm}\bigg(\int^{\ell(Q_j)/4}_{0}\int_{F_j}
\big|\Psi_{jt}(\sqrt{L})b_j(y)\big|^2
\Big({t\over t+|x-y|}\Big)^{n\mu}{dydt\over t^{n+1}}\bigg)^{1/2}\\
&&\leq C {\|b_j\|_1\ell(Q_j)^{n/2}\over |x-x_j|^{n\mu/2}}
\bigg(\int^{\ell(Q_j)/4}_{0}
t^{n\mu-2n-n-1}dt\bigg)^{1/2}\\
&&\leq C {\|b_j\|_1\ell(Q_j)^{n/2}\over |x-x_j|^{n\mu/2}}\ell(Q_j)^{(n\mu-3n)/2}\\
&&\leq C {\|b_j\|_1\ell(Q_j)^{-n}\over (1+|x-x_j|/\ell(Q_j))^{n\mu/2}} \\
&&\leq C |f\chi_{Q_j}|\ast \tau_{\ell(Q_j)}(x),
\end{eqnarray*}
\noindent where $\tau_{\ell(Q_j)}(x)=1/(1+|x|)^{n\mu/2}\in L^1(\RN)$.
For the next part of the integral we consider two cases: $n=1$ and $n>1$.
Note that for $x\notin\Omega$, $\ell(Q_j)/4<t\leq |x-x_j|/4$ and $y\in E_{jt}=:\{y:\ |y-x_j|\leq t+(1/32+\sqrt{n}/2)\ell(Q_j)\}$,
$|x-y|\geq|x-x_j|-|y-x_j|>{\sqrt{n}/4+11/32\over \sqrt{n}+1/2}|x-x_j|$. Thus for $3<\mu<4$, if $n=1$,
\begin{eqnarray*}
&&\hspace{-1cm}\bigg(\int_{\ell(Q_j)/4}^{|x-x_j|/4}\int_{E_{jt}}
\big|\Psi_{jt}(\sqrt{L})b_j(y)\big|^2
\Big({t\over t+|x-y|}\Big)^{n\mu}{dydt\over t^{n+1}}\bigg)^{1/2}\\[3pt]
&&\leq C {\|b_j\|_1\ell(Q_j)\over |x-x_j|^{ \mu/2}}
\bigg(\int_{\ell(Q_j)/4}^{|x-x_j|/4}
t^{-4+1+\mu-2}dt\bigg)^{1/2}\\[3pt]
&&\leq C {\|b_j\|_1\ell(Q_j) \over |x-x_j|^{ \mu/2}}\ell(Q_j)^{( \mu-4)/2}\\[3pt]
&&\leq C {\|b_j\|_1\ell(Q_j)^{-1}\over (1+|x-x_j|/\ell(Q_j))^{ \mu/2}} \\[3pt]
&&\leq C |f\chi_{Q_j}|\ast \sigma_{\ell(Q_j)}(x),
\end{eqnarray*}
\noindent where $\sigma_{\ell(Q_j)}(x)=1/(1+|x|)^{ \mu/2} $. On the other hand, for
$3<\mu<4$, if $n>1$,
\begin{eqnarray*}
&&\hspace{-1cm} \bigg(\int_{\ell(Q_j)/4}^{|x-x_j|/4}\int_{E_{jt}}
\big|\Psi_{jt}(\sqrt{L})b_j(y)\big|^2
\Big({t\over t+|x-y|}\Big)^{n\mu}{dydt\over t^{n+1}}\bigg)^{1/2}\\
&&\leq C {\|b_j\|_1\ell(Q_j)\over |x-x_j|^{ n\mu/2}}
\bigg(\int_{\ell(Q_j)/4}^{|x-x_j|/4}
t^{-2n-2+n\mu+n-n-1}dt\bigg)^{1/2}\\
&&\leq C {\|b_j\|_1\ell(Q_j) \over |x-x_j|^{ n\mu/2}}\ell(Q_j)^{( n\mu-2n-2)/2}\\
&&\leq C {\|b_j\|_1\ell(Q_j)^{-n}\over (1+|x-x_j|/\ell(Q_j))^{ n+1}} \\
&&\leq C |f\chi_{Q_j}|\ast P_{\ell(Q_j)}(x),
\end{eqnarray*}
\noindent where $P_{\ell(Q_j)}(x)=1/(1+|x|)^{ n+1} $.
Finally, since $t/(t+|x-y|)\leq 1$, so
\begin{eqnarray*}
&&\hspace{-1cm}\bigg(\int^{\infty}_{|x-x_j|/4}\int_{E_{jt}}
\big|\Psi_{jt}(\sqrt{L})b_j(y)\big|^2
\Big({t\over t+|x-y|}\Big)^{n\mu}{dydt\over t^{n+1}}\bigg)^{1/2}\\
&&\leq C {\|b_j\|_1\ell(Q_j) }
\bigg(\int^{\infty}_{|x-x_j|/4}
t^{-2n-3}dt\bigg)^{1/2}\\
&&\leq C |f\chi_{Q_j}|\ast P_{\ell(Q_j)}(x).
\end{eqnarray*}
\noindent Therefore, if $x\notin\Omega$, and $n>1$, then
$\gL\Big(\big(1-\Phi_j(\sqrt{L})\big)b_j\Big)(x)\leq C |f\chi_{Q_j}|\ast P_{\ell(Q_j)}(x).$
And
\begin{eqnarray*}
\int_{\RN\setminus\Omega}\gL\Big( \big(1-\Phi_j(\sqrt{L})\big)b_j\Big)wdx
&\leq& C \int_{\RN\setminus\Omega}|f\chi_{Q_j}|\ast P_{\ell(Q_j)}wdx\\
&\leq& C \int_{Q_j}|f|( P_{\ell(Q_j)}\ast w)dx\\
&\leq& C \int_{Q_j}|f|Mwdx.
\end{eqnarray*}
\noindent If $n=1$ we get the same thing, but with $P$ replaced by $\sigma.$ This concludes the proof of
(\ref{e4.3}). And the proof of this theorem is complete.
\end{proof}
\subsection{Estimate for $2<p<\infty$}
We proceed by duality. If $h(x)\geq0$ and $h\in L^{(p/2)'}(wdx)$, then
\begin{eqnarray*}
\int_{\RN}\gL(f)^2hwdx=\int_{\mathbb R^{n+1}_{+}}|\Psi(t\sqrt{L})f(y)|^2{1\over t}
\bigg({1\over t^n}\int_{\RN}h(x)w(x)\Big({t\over t+|x-y|}\Big)^{n\mu}dx\bigg)dydt.
\end{eqnarray*}
\noindent Set
$$E_k=\Big\{(y,t):\ {1\over t^n}\int_{\RN}h(x)w(x)\Big({t\over t+|x-y|}\Big)^{n\mu}dx\sim2^k\Big\}.$$
\noindent Note that if $|y-z|<t$, then $t+|x-y|\sim t+|x-z|$. Thus, if $(y,t)\in E_k$ and
$|y-z|<t$, then
\begin{eqnarray*}
2^k<{1\over t^n}\int_{\RN}h(x)w(x)\Big({t\over t+|x-y|}\Big)^{n\mu}dx\sim
{1\over t^n}\int_{\RN}h(x)w(x)\Big({t\over t+|x-z|}\Big)^{n\mu}dx.
\end{eqnarray*}
\noindent The last expression is at most
\begin{eqnarray*}
&&\hspace{-1.2cm}C\sum_{j=0}^{\infty}{1\over2^{jn\mu}}{1\over t^n}\int_{B(z,2^jt)}hwdx\\
&=&C\sum_{j=0}^{\infty}{1\over2^{jn(\mu-1)}}
{w(B(z,2^jt))\over (2^jt)^n}{1\over w(B(z,2^jt))}\int_{B(z,2^jt)}hwdx\\
&\leq&C\sum_{j=0}^{\infty}{1\over2^{jn(\mu-1)}}Mw(z)M_w(h)(z)\\
&\leq&C Mw(z)M_w(h)(z),
\end{eqnarray*}
\noindent where
$$M_w(h)(z)=\sup_{t>0}\Big({1\over w(B(z,t))}\int_{B(z,t)}hwdx\Big).$$
Recall that ${\rm supp}\ K_{\Psi(t\sqrt{L})}(y,\cdot)\subset B(y,t)$. Since for
$(y,t)\in E_k$ and $|y-z|<t$ we have $z\in A_k=\{z:\ Mw(z)M_w(h)(z)\geq C_{n,\mu}2^k\},$ it
follows that for $(y,t)\in E_k$,
$\Psi(t\sqrt{L})f(y)=\Psi(t\sqrt{L})(f\chi_{A_k})(y).$
Thus,
\begin{eqnarray*}
&&\hspace{-1.2cm}\int_{\mathbb R^{n+1}_{+}}|\Psi(t\sqrt{L})f(y)|^2{1\over t}
\bigg({1\over t^n}\int_{\RN}h(x)w(x)\Big({t\over t+|x-y|}\Big)^{n\mu}\bigg)dydt\\
&&\leq \sum_k 2^{k+1}\int_{E_k}|\Psi(t\sqrt{L})f(y)|^2{dydt\over t}\\
&&= \sum_k 2^{k+1}\int_{E_k}|\Psi(t\sqrt{L})(f\chi_{A_k})(y)|^2{dydt\over t}\\
&&\leq C \sum_k 2^{k+1}\int_{\RN}| f|^2\chi_{A_k}{dy }\\
&&\leq C \int_{\RN}| f|^2 {MwM_w(h) }{dy }.
\end{eqnarray*}
\noindent Applying the H\"older inequality with exponents $p/2$ and $(p/2)'$, we obtain the bound
$$ C \bigg(\int_{\RN}| f|^p (Mw)^{p/2}w^{-(p/2-1)}{dy }\Big)^{2/p}
\Big(\int_{\RN}M_w(h)^{(p/2)'}w{dy }\bigg)^{(p-2)/p}.$$
However, since $M_w$ is the centered maximal function, we have
$$\int_{\RN}M_w(h)^{(p/2)'}wdx\leq C_{n,p}\int_{\RN}h^{(p/2)'}wdx,$$
by a standard argument based on the Besicovitch covering lemma. Since
$h$ is arbitrary, we obtain our result.
\section{Proof of Theorem~\ref{th1.4} }
\setcounter{equation}{0}
In order to prove Theorem \ref{th1.4}, we first prove the following result.
\begin{lemma}\label{le5.1}
Let $T$ be of the area functions $s_h$, $s_p$, $S_{P}$, $S_{H}$ and $\gL$ with $\mu>3$.
Under assumptions of Theorems~\ref{th1.1}, ~\ref{th1.2} and ~\ref{th4.1}, for $w\in A_p,\
1<p<\infty$, we have
\begin{eqnarray}
\label{e5.1}\|Tf\|_{L^p_w(\RN)}\leq C \|f\|_{L^p_w(\RN)}
\end{eqnarray}
\noindent
where
constant $C$ depends only on
$p$, $n$ and $w$.
\end{lemma}
\begin{proof} Let $T$ be
of the area functions $s_h$, $s_p$, $S_{P}$, $S_{H}$ and $\gL$ with $\mu>3$.
Note that if $w\in A_1$, then $Mw\leq Cw$ a.e. By
Theorems~\ref{th1.1}, ~\ref{th1.2} and ~\ref{th4.1}, $T$ is bounded on
$L^p_w(\RN), 1<p<\infty,$ for any $w\in A_1,$ i.e.,
$$\|Tf\|_{L^p_w(\RN)}\leq C \|f\|_{L^p_w(\RN)}.
$$
By extrapolation theorem, these operators are all bounded on
$L^p_w(\RN), 1<p<\infty,$ for any $w\in A_p,$
and estimate (\ref{e5.1}) holds. For the detail, we refer the reader
to pages 141-142, Theorem 7.8, \cite{D}.
\end{proof}
Going further, we introduce some definitions. Given a weight $w$, set $w(E)=\int_E w(x)dx$. The non-increasing
rearrangement of a measurable function $f$ with respect to a weight
$w$ is defined by (cf. \cite{CR})
\begin{eqnarray*}
f^\ast_w(t)=\sup_{w(E)=t} \inf_{x\in E}|f(x)|\ \ \ \ (0<t<w(\RN)).
\end{eqnarray*}
\noindent If $w\equiv1$, we use the notation $f^\ast(t)$.
Given a measurable function $f$, the local sharp maximal function $M^\sharp_{\lambda}f$ is defined by
\begin{eqnarray*}
M^\sharp_{\lambda}f(x)=\sup_{Q\ni x}\inf_{c}\big((f-c)\chi_Q\big)^\ast(\lambda|Q|)\ \ \ (0<\lambda<1).
\end{eqnarray*}
\noindent This function was introduced by Str\"omberg \cite{S}, and
motivated by an alternate characterization of the space $BMO$ given
by John \cite{J}.
\begin{lemma}\label{le5.4}
For any $w\in A_p$ and for any locally integrable function $f$ with
$f^\ast_w(+\infty)=0$ we have
\begin{eqnarray}\label{e5.2}
\|Mf\|_{L^p_w(\RN)}\leq C\|w\|_{A_p}^{\gamma_{p,q}}\cdot\|M^\sharp_{\lambda_n}
(|f|^q)\|^{1/q}_{L^{p/q}_w(\RN )}\ \ \ (1<p<\infty,1\leq q<\infty),
\end{eqnarray}
\noindent where $\gamma_{p,q}=\max\{1/q,1/(p-1)\}$, $C$ depends only on $p,q$ and on the
underlying dimension $n$, and $\lambda_n$ depends only on $n$.
\end{lemma}
For the proof of this lemma, see Theorem 3.1 in \cite{L1}.
\begin{prop}\label{pro5.5} Let $\gL$ be a function with $\mu>3$ in (\ref{e3.1}).
Then for any $f\in C^\infty_0(\RN)$ and for all
$x\in \RN,$
$$M^\sharp_{\lambda}\big(\gL(f)^2\big)(x)\leq CMf(x)^2,$$
where $C$ depends on $\lambda,\mu,\Psi$ and $n$.
\end{prop}
\begin{proof}
Given a cube $Q$, let $T(Q)=\{(y,t):\ y\in Q,0<t<l(Q)\}$, where $\ell(Q)$ denotes the
side length of $Q$. For $(y,t)\in T(Q)$, using (\ref{e2.5}) of Lemma \ref{le2.1} we have
\begin{eqnarray}\label{e5.3}
\Psi(t\sqrt{L})f(y)=\Psi(t\sqrt{L})(f\chi_{3Q})(y).
\end{eqnarray}
Now, fix a cube $Q$ containing $x$. For any $z\in Q$ we decompose
$\gL(f)^2$ into the sum of
$$I_1(z)=\iint_{T(2Q)}|\Psi(t\sqrt{L})f(y)|^2\Big({t\over t+|z-y|}\Big)^{n\mu}{dydt\over t^{n+1}}$$
and
$$I_2(z)=\iint_{\mathbb R^{n+1}_{+} \setminus T(2Q)}|\Psi(t\sqrt{L})f(y)|^2\Big({t\over t+|z-y|}\Big)^{n\mu}{dydt\over t^{n+1}}.$$
\noindent
From Theorem \ref{th4.1}, we know that for $\mu>3$, $\gL(f)$ is of weak type $(1, 1)$. Then using
(\ref{e5.3}), we have
\begin{eqnarray}\label{e5.4}
(I_1)^\ast(\lambda|Q|)&\leq& \Big(\gL(f\chi_{6Q})\Big)^{\ast}(\lambda|Q|)^2\\
&\leq&\Big({C\over \lambda|Q|}\int_{6Q}|f|\Big)^2\leq CMf(x)^2.\nonumber
\end{eqnarray}
\noindent
Further, for any $z_0 \in Q$ and $(y, t)\notin T (2Q)$, by the Mean
Value Theorem,
$$(t+|z-y|)^{-n\mu}-(t+|z_0-y|)^{-n\mu}\leq C\ell(Q)(t+|z-y|)^{-n\mu-1}.$$
From this and (\ref{e5.3}), using Lemma \ref{le2.1} again and
$\mu>3$, we have
\begin{eqnarray*}
&&\hspace{-1.2cm}|I_2(z)-I_2(z_0)|\\
&&\leq C\ell(Q)\iint_{\mathbb R^{n+1}_{+}\setminus
T(2Q)}t^{n\mu}|\Psi(t\sqrt{L})f(y)|^2\Big({1\over
t+|z-y|}\Big)^{n\mu+1}{dydt\over t^{n+1}}\\
&&\leq C\sum^\infty_{k=1}{1\over 2^k}{1\over (2^k\ell(Q))^{n\mu}}
\iint_{T(2^{k+1}Q)\setminus
T(2^{k}Q)}t^{n\mu}|\Psi(t\sqrt{L})f(y)|^2{dydt\over t^{n+1}}\\
&&\leq C\sum^\infty_{k=1}{1\over 2^k}{|2^{k+1}Q|\over
(2^k\ell(Q))^{n\mu}}\Big(\int^{2^{k+1}\ell(Q)}_0t^{n\mu-3n-1}dt\Big)
\Big(\int_{6\cdot2^kQ}|f|\Big)^2\\
&&\leq C\sum^\infty_{k=1}{1\over 2^k}\Big({1\over
|2^{k+1}Q|}\int_{6\cdot2^kQ}|f|\Big)^2\leq CMf(x)^2.
\end{eqnarray*}
Combining this estimate with (\ref{e5.4}) yields
\begin{eqnarray*}
\inf_{c}\Big((\gL(f)^2-c)\chi_Q\Big)^\ast(\lambda|Q|)&\leq&
\big((I_1+I_2-I_2(z_0))\chi_Q\big)^\ast(\lambda|Q|)\\
&\leq&(I_1)^\ast(\lambda|Q|)+CMf(x)^2\\
&\leq& CMf(x)^2,
\end{eqnarray*}
which proves the desired result.
\end{proof}
Then we have the following result.
\begin{theorem}\label{th5.4}
Let $T$ be of the area functions $s_h$, $s_p$, $S_{P}$,$S_{H}$ and $\gL$ with $ \mu>3.$
Under assumptions of Theorems~\ref{th1.1}, ~\ref{th1.2} and ~\ref{th4.1}, for $w\in A_p,\
1<p<\infty$, if $\|f\|_{L^p_w(\RN )}<\infty$, then
\begin{eqnarray}
\label{e5.5}
\Big(\int_{\RN}\big(M(Tf)\big)^pwdx\Big)^{1/p}\leq C\|w\|_{A_p}^{\beta_p}\Big(\int_{\RN}
\big(M( f)\big)^pwdx\Big)^{1/p},
\end{eqnarray}
\noindent
where
$\beta_p=\max\{1/2,1/(p-1)\}$, and a constant $C$ depends only on
$p$ and $n$.
\end{theorem}
\begin{proof}
Suppose $T=\gL.$ From Lemma~\ref{le5.1}, we know that
$\gL$ is bounded on $L^p_w(\RN ) $ when $w\in A_p$. Therefore, assuming that
$\|f\|_{L^p_w(\RN)}$ is finite, we clearly obtain that
$(\gL)^\ast_w(+\infty)=0 $. Letting $\gL(f)$ instead of $f$ in
(\ref{e5.2}) with $q=2$ and applying Proposition \ref{pro5.5}, we get
$$\Big(\int_{\RN}\big(M(\gL(f))\big)^pwdx\Big)^{1/p}\leq C\|w\|_{A_p}^{\beta_p}\Big(\int_{\RN}
\big(M( f)\big)^pwdx\Big)^{1/p}.$$
Under assumptions of Theorems~\ref{th1.1} and ~\ref{th1.2}, it follows that the area functions
$s_h$, $s_p$, $S_{P}$ and $S_{H}$ are all controlled by $\gL$ pointwise.
So we have the estimate (\ref{e5.5}) for $s_h$, $s_p$, $S_{P}$ and $S_{H}$. Then the
proof of this theorem is complete.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th1.4}]
In \cite{B}, Buckley proved that for the Hardy-Littlewood maximal
operator,
\begin{eqnarray}\label{m}
\|M\|_{L^p_w(\RN)}\leq C\|w\|_{A_p}^{1/(p-1)}\ \ (1<p<\infty),
\end{eqnarray}
\noindent and this result is sharp.
From (\ref{m}) and Theorem~\ref{th5.4}, there exists a constant $C=C(T, n, p)$ such that
for all $w\in A_p$,
\begin{eqnarray}\label{5.7}
\|T\|_{L^p_w(\RN)}\leq C\|w\|_{A_p}^{ {1\over p-1}+ \, \max\big\{{1\over 2},\, {1\over p-1}\big\}} \ \ \ \ \ \ (1<p<\infty),
\end{eqnarray}
\noindent where $T$ is of the area functions $s_h$, $s_p$, $S_{P}$ and $S_{H}.$
This proves Theorem \ref{th1.4}.
\end{proof}
\noindent
{\bf Remarks.}\
(i) \, Note
that when $L=-\Delta$ is the Laplacian on $\RN$, it is well known
that the exponents $\beta_p$ of (\ref{e5.5}) in Theorem~\ref{th5.4} is best possible,
in general (see, e.g., Theorem 1.5, \cite{L1}).
(ii)\, For the classical
area function $S_{\varphi}$ in (\ref{e1.1}), the result of Theorem~\ref{th1.4} was recently improved by A. Lerner in \cite{L2}, i.e.,
there exists a constant $C=C(S_{\varphi}, n, p)$ such that
for all $w\in A_p, 1<p<\infty$,
\begin{eqnarray}\label{e5.8}
\|S_{\varphi}\|_{L^p_w(\RN)}\leq C\|w\|_{A_p}^{ \, \max\big\{{1\over 2},\, {1\over p-1}\big\}},
\end{eqnarray}
\noindent
and the estimate (\ref{e5.8})
is the best possible for all $1<p<\infty.$ However, we do not know whether one can deduce the same
bounds (\ref{e5.8}) for the $L^p_w$ operator norms of the area functions $s_h$, $s_p$, $S_{P}$ and $S_{H},$
and they are of interest in their own right.
Note that sharp weighted optimal bounds for singular integrals has been
studied extensively, see for examples, \cite{CMP, HLRSUV, LOP1, LOP2, P} and the references therein.
(iii)\, Finally, for $f\in {\mathcal S}(\RN)$, we define the (so called vertical) Littlewood-Paley-Stein
functions ${\mathcal G}_P $ and $ {\mathcal G}_H$
by
\begin{eqnarray*}
{\mathcal G}_P(f)(x)&=&\bigg(\int_0^{\infty}
|t\nabla_x e^{-t\sqrt{L}} f(x)|^2{ dt\over t }\bigg)^{1/2},\\
{\mathcal G}_H(f)(x)&=&\bigg(\int_0^{\infty}
|t\nabla_x e^{-t^2L} f(x)|^2 { dt\over t }\bigg)^{1/2},
\end{eqnarray*}
\noindent
as well as the (so-called horizontal) Littlewood-Paley-Stein functions $ g_p$ and $g_h$
by
\begin{eqnarray*}
g_p(f)(x)&=&\bigg(\int_0^{\infty}
|t\sqrt{L} e^{-t\sqrt{L}} f(x)|^2 { dt\over t }\bigg)^{1/2},\\
g_h(f)(x)&=&\bigg(\int_0^{\infty}
|t^2L e^{-t^2L} f(x)|^2 { dt\over t }\bigg)^{1/2}.
\end{eqnarray*}
One then has the analogous statement as in Theorems 1.1, 1.2, 1.3 and 1.4 replacing
$s_p, s_h, S_P, S_H $ by $g_p, g_h, {\mathcal G}_P,
{\mathcal G}_H$, respectively.
\vskip 1cm
\noindent
{\bf Acknowledgment}:\ The research of Lixin Yan is supported by NNSF of China (Grant No. 10771221)
and National
Science Foundation for Distinguished
Young Scholars of China (Grant No. 10925106).
\end{document} |
\begin{document}
\begin{nouppercase}
\maketitle
\end{nouppercase}
\begin{abstract}
Consider the Fukaya category associated to a Lefschetz fibration. It turns out that the Floer cohomology of the monodromy around $\infty$ gives rise to natural transformations from the Serre functor to the identity functor, in that category. We pay particular attention to the implications of that idea for Lefschetz pencils.
\end{abstract}
\section{Introduction}
This paper is part of an investigation of the Floer-theoretic structures arising from Lefschetz fibrations. Its purpose is to introduce a new piece of that puzzle; and also, to set up a wider algebraic framework into which that piece should (conjecturally) fit, centered on a notion of noncommutative pencil.
\subsection{Symplectic geometry}
It is well-known that there is a version of the Fukaya category tailored to a Lefschetz fibration (the idea is originally due to Kontsevich). In \cite{seidel06}, it was pointed out that these categories always come with some added structure: a distinguished natural transformation from the Serre functor to the identity functor (see \cite{seidel08, bourgeois-ekholm-eliashberg09, seidel12b, abouzaid-seidel13} for related theoretical developments, and \cite{maydanskiy09, maydanskiy-seidel09, abouzaid-seidel10} for applications). Concretely, let's fix a symplectic Lefschetz fibration
\begin{equation} \label{eq:lefschetz-pencil}
\pi: E^{2n} \longrightarrow {\mathbb C}
\end{equation}
with fibre $M^{2n-2}$. For technical simplicity, we will impose an exactness condition on the symplectic form, hence assume that the fibre is a Liouville domain. Choose a basis of Lefschetz thimbles, and let $\EuScript A$ be the associated (directed) Fukaya $A_\infty$-algebra, defined over a coefficient field ${\mathbb K}$. In general, $\EuScript A$ is ${\mathbb Z}/2$-graded, and this lifts to a ${\mathbb Z}$-grading if $c_1(E) = 0$. Natural transformations of degree $n$ from the Serre functor to the identity functor (in the triangulated envelope, or equivalently derived $A_\infty$-category, of $\EuScript A$) can be described as maps of $A_\infty$-bimodules
\begin{equation} \label{eq:natural-transformation}
\EuScript A^\vee[-n] \longrightarrow \EuScript A.
\end{equation}
Here $\EuScript A$ is the diagonal bimodule; and $\EuScript A^\vee$ is its dual, to which we have applied an upwards shift by $n$. To be more precise, $A_\infty$-bimodules form a dg category, denoted here by $[\EuScript A,\EuScript A]$. By a bimodule map \eqref{eq:natural-transformation}, we mean an element of
\begin{equation} \label{eq:bimodule-morphisms}
H^0(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript A^\vee[-n],\EuScript A)) \cong H^n(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript A^\vee,\EuScript A)).
\end{equation}
In these terms, the story so far can be summarized as:
\begin{lemma} \label{th:old}
$\EuScript A$ always comes with a distinguished element of \eqref{eq:bimodule-morphisms}, denoted here by $\rho$.
\end{lemma}
The definition in \cite{seidel12b} uses the geometry and Floer cohomology of Lefschetz thimbles in $E$. However, there is another description (equivalent up to automorphisms of $\EuScript A^\vee$), in terms of the functor from $\EuScript A$ to the standard Fukaya category $\EuScript F(M)$ of the fibre (this second description was the one originally proposed in \cite{seidel06}; the relation between the two is \cite[Corollary 7.1]{seidel12b}).
The specific contribution of this paper is to introduce another geometric source of maps \eqref{eq:natural-transformation}. This uses fixed point Floer cohomology for symplectic automorphisms $\phi$ of $M$ (which are exact and equal the identity near the boundary). To define it, we make an auxiliary choice of perturbation, by the time $\epsilon$ map of the Reeb flow near the boundary, and write the outcome as $\mathit{HF}^*(\phi,\epsilon)$ (for small $|\epsilon|$, this appears in \cite{seidel00b, mclean12}; for the general case, see \cite{uljarevic14} or the exposition in the body of this paper).
\begin{theorem} \label{th:main}
Let $\mu: M \rightarrow M$ be the monodromy of the Lefschetz fibration around a large circle. For sufficiently small $\epsilon>0$, there is a canonical map
\begin{equation} \label{eq:fix-to-aa}
\mathit{HF}^{*+2}(\mu,\epsilon) \longrightarrow H^*(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript A^\vee[-n],\EuScript A)).
\end{equation}
\end{theorem}
This is the outcome of a Floer cohomology computation in $E$ which follows \cite{mclean12}, together with an application of a suitable open-closed string map, similar to those in \cite{abouzaid-ganatra14}. Generally, both sides of \eqref{eq:fix-to-aa} are ${\mathbb Z}/2$-graded; when $c_1(E) = 0$, this lifts to ${\mathbb Z}$-gradings, and the map between them will respect those gradings.
An important special class of Lefschetz fibrations comes from anticanonical Lefschetz pencils: by which we mean, on a monotone closed symplectic manifold $E^{\mathit{cl}}$, a symplectic Lefschetz pencil of hypersurfaces representing the first Chern class. One obtains $E$ from $E^{\mathit{cl}}$ by removing a suitable neighbourhood of one such hypersurface; hence, $c_1(E) = 0$ always holds in this case. The Reeb flow on $\partial M$ is periodic (say with period $1$), and the monodromy is the boundary twist associated to that periodic flow:
\begin{equation} \label{eq:mu-tau}
\mu = \tau_{\partial M}.
\end{equation}
This satisfies
\begin{equation} \label{eq:trivial-monodromy}
\mathit{HF}^{*+2}(\tau_{\partial M},\epsilon) \cong
\begin{cases} H^*(M,\partial M) & \epsilon \in (0,1), \\
H^*(M) & \epsilon \in (1,2).
\end{cases}
\end{equation}
Theorem \ref{th:main} applies only to the first case of \eqref{eq:trivial-monodromy}. However, by a more precise analysis, one can show that in the second case, an analogue of the map \eqref{eq:fix-to-aa} can be defined in the lowest degree $\ast = 0$. Hence, the element of $\mathit{HF}^2(\tau_{\partial M},\epsilon)$, $\epsilon \in (1,2)$, corresponding to $1 \in H^0(M)$ still gives rise to a map \eqref{eq:bimodule-morphisms}, which we denote by $\sigma$. We summarize the consequence:
\begin{theorem} \label{th:fano}
For a Lefschetz fibration arising from an anticanonical Lefschetz pencil, there is a distinguished pair $(\rho,\sigma)$ of maps \eqref{eq:natural-transformation} (obtained, respectively, by Lemma \ref{th:old}, and a modified version of the argument from Theorem \ref{th:main}).
\end{theorem}
\begin{remark} \label{th:fractional-cy-1}
More generally, one can consider a Lefschetz pencil such that $c_1(E^{\mathit{cl}})$ is $(1+m)$ times the class of the hypersurfaces in the pencil, for some $m \in {\mathbb Z}$. One then still has $c_1(E) = 0$, but the appropriate generalization of \eqref{eq:trivial-monodromy} is
\begin{equation} \label{eq:trivial-monodromy-2}
\mathit{HF}^{*+2}(\tau_{\partial M},\epsilon) \cong
\begin{cases} H^{*+2m}(M,\partial M) & \epsilon \in (0,1), \\
H^{*+2m}(M) & \epsilon \in (1,2).
\end{cases}
\end{equation}
While $\rho$ still has degree $0$, our construction now leads to a $\sigma$ which has degree $-2m$.
\end{remark}
\subsection{Noncommutative geometry\label{subsec:nc-geometry}}
The starting point for our algebraic framework is the notion of {\em noncommutative divisor}. While the terminology is new, the concept already appeared in \cite{seidel08}, and a version was considered in \cite{seidel12b, kontsevich-vlassopoulos13}. The last two references include a cyclic symmetry condition, which corresponds more specifically to a {\em noncommutative anticanonical divisor} (see Remark \ref{th:cyclic}); we have chosen not to impose that condition here, for the sake of simplicity.
While precise definitions will be given later on, it makes sense to give an outline now. Let $\EuScript A$ be an $A_\infty$-algebra. A noncommutative divisor on $\EuScript A$ consists of an $A_\infty$-bimodule $\EuScript P$ which is invertible (with respect to tensor product), together with an $A_\infty$-algebra structure on
\begin{equation} \label{eq:b-space}
\EuScript B = \EuScript A \oplus \EuScript P[1],
\end{equation}
which extends the given one on $\EuScript A$ as well as the $\EuScript A$-bimodule structure of $\EuScript P$. If one then considers $\EuScript B$ as an $\EuScript A$-bimodule, it fits into a short exact sequence
\begin{equation} \label{eq:a-b-sequence}
0 \rightarrow \EuScript A \longrightarrow \EuScript B \longrightarrow \EuScript P[1] \rightarrow 0.
\end{equation}
As boundary homomorphism of that sequence, one gets a bimodule map $\theta: \EuScript P \rightarrow \EuScript A$, which we call the {\em section} associated to the noncommutative divisor. We extend the algebro-geometric language to related structures:
\begin{equation} \label{eq:list-of-categories-1}
\left\{\!\!\!\!\!\!
\parbox{35em}{
\begin{itemize} \itemsep.5em
\item[(i)] The $A_\infty$-algebra $\EuScript B$ is thought of as describing the {\em divisor by itself} (independently of its relationship with the {\em ambient space} $\EuScript A$).
\item[(ii)] There is a localization process, in which one makes $\theta$ invertible by passing to a quotient $A_\infty$-algebra. We consider this to be the abstract analogue of taking the {\em complement of the divisor} (see \cite[Section 1]{seidel08} for an explanation), and write the outcome accordingly as $\EuScript A \setminus \EuScript B$. This depends only on the section associated to the divisor.
\end{itemize}
} \right.
\end{equation}
Generalizing the previous concept, we will introduce {\em noncommutative pencils}. As in classical algebraic geometry, such a pencil has an associated pair of bimodule maps (sections, in our geometrically inspired terminology)
\begin{equation}
\rho,\sigma: \EuScript P \longrightarrow \EuScript A.
\end{equation}
This leads to enhancements of the previously described constructions:
\begin{equation} \label{eq:list-of-categories-2}
\left\{\!\!\!\!\!\!
\parbox{35em}{
\begin{itemize} \itemsep.5em
\item[(i)] A noncommutative pencil determines a family of noncommutative divisors, parametrized by $z \in {\mathbb P}^1 = {\mathbb K} \cup \{\infty\}$. The sections associated to those divisors are (at least up to a scalar multiple)
\[
\begin{aligned}
& \theta_z = \sigma + z\rho \quad \text{for $z \in {\mathbb K}$}, \\
& \theta_\infty = \rho.
\end{aligned}
\]
In particular, one gets an $A_\infty$-algebra $\EuScript B_\infty$. More interestingly, one can work with varying $z$. For instance, taking $z = 1/q$, where $q$ is a formal variable, leads to a formal deformation of $\EuScript B_\infty$, which we denote by $\hat\EuScript B_\infty$.
\item[(ii)] $\EuScript A \setminus \EuScript B_\infty$ also acquires a distinguished (curved) formal deformation, whose deformation parameter has degree $2$. Let's call this the {\em noncommutative Landau-Ginzburg model}, and denote it by $\EuScript L\EuScript G$.
\end{itemize}
}\right.
\end{equation}
\begin{remark}
This by no means exhausts the noncommutative geometry structures which arise in this context. For one thing, from a purely algebraic perspective, there is no particular reason to single out the point $z = \infty$, which means that there are more general versions of the constructions listed above. Two other notions, the {\em graph} and {\em base locus} of a noncommutative pencil, will be mentioned briefly in Remark \ref{th:graph}. Different kinds of algebraic structures, notably ones involving Hochschild and cyclic homology (such as, the Gau\ss-Manin connection for the periodic cyclic homology of the fibres of the noncommutative pencil), are entirely beyond the scope of our discussion.
\end{remark}
\subsection{Speculations}
The $A_\infty$-algebra $\EuScript A$ associated to a Lefschetz fibration comes with a natural structure of a noncommutative divisor, with associated bimodule
\begin{equation} \label{eq:serre-n}
\EuScript P = \EuScript A^\vee[-n].
\end{equation}
The algebraic constructions introduced above have the following meaning:
\begin{equation} \label{eq:list-of-categories-3}
\left\{\!\!\!\!\!\!
\parbox{35em}{
\begin{itemize} \itemsep.5em
\item[(i)] The divisor by itself, $\EuScript B$, is the full subcategory of $\EuScript F(M)$ consisting of the vanishing cycles in our basis. In fact, this is how the noncommutative divisor was constructed in \cite{seidel06,seidel08}, by identifying $\EuScript A$ with the directed $A_\infty$-subcategory of $\EuScript B$.
\item[(ii)] The complement of the divisor, $\EuScript A \setminus \EuScript B$, yields a full subcategory of the wrapped Fukaya category of the total space, $\EuScript W(E)$. This is the main result of \cite{abouzaid-seidel13}.
\end{itemize}
} \right.
\end{equation}
We will now propose a continuation of this line of thought, to the point where it can include Theorem \ref{th:fano}. In spite of its tentative nature, this direction seems worth while exploring, because of the potential implications for symplectic geometry and mirror symmetry.
\begin{conjecture} \label{th:conjecture}
Take an exact symplectic Lefschetz fibration coming from an anticanonical Lefschetz pencil. Then, the $A_\infty$-algebra $\EuScript A$ carries a canonical structure of a noncommutative pencil, satisfying \eqref{eq:serre-n}, whose associated sections are those described in Theorem \ref{th:fano}.
\end{conjecture}
The appeal of the noncommutative pencil structure in this context is that it can serve as an intermediate object between various other Fukaya $A_\infty$-structures. Consider:
\begin{equation} \label{eq:list-of-categories-4}
\left\{\!\!\!\!\!\!
\parbox{35em}{
\begin{itemize} \itemsep.5em
\item[(i)] $\EuScript B_\infty$ should be a full subcategory of $\EuScript F(M)$ (formally, this is the same statement as in \eqref{eq:list-of-categories-3}(i); but the level of difficulty of proving it depends on how one would go about constructing the noncommutative pencil structure). Now, $M$ admits a natural closure $M^{\mathit{cl}}$, obtained by gluing in a disc bundle over the base locus of the original pencil. Associated to that is the relative Fukaya category \cite{seidel02, seidel03b, sheridan11b}, which is a formal deformation of $\EuScript F(M)$. More generally, one can modify the deformation by including a suitable bulk term, as in \cite{fukaya-oh-ohta-ono11}. The conjecture is that, for a specific choice of bulk term, and after a change in the deformation parameter, this deformation is $\hat\EuScript B_\infty$. This is made more precise in \cite{seidel15}.
\item[(ii)]
$E$ itself embeds into a closed manifold $E^{\mathit{cl}}$, the manifold on which the original Lefschetz pencil was defined. This leads to another relative Fukaya category, but where now (because of monotonicity) the deformation parameter has degree $2$. At least on the basis of formal analogy, one expects this to be related (by a cohomologically full embedding) to the Landau-Ginzburg category $\EuScript L\EuScript G$ from \eqref{eq:list-of-categories-2}(ii).
\end{itemize}
} \right.
\end{equation}
\begin{remark} \label{th:fractional-cy-2}
We have limited ourselves to anticanonical Lefschetz pencils (the setup of Theorem \ref{th:fano}) for the sake of concreteness, but other cases are also of interest. For instance, along the same lines as in Conjecture \ref{th:conjecture}, the situation from Remark \ref{th:fractional-cy-1} should lead to a ${\mathbb Z}/2 m$-graded noncommutative pencil. The residual information given by the ${\mathbb Z}$-grading of $\EuScript A$ means that this pencil is homogeneous with respect to a circle action on ${\mathbb P}^1$ of weight $2m$.
\end{remark}
\subsection{Homological Mirror Symmetry\label{subsec:mirror}}
The mirror ($B$-model) counterpart of Conjecture \ref{th:conjecture} is far more straightforward, and amounts to embedding ordinary algebraic geometry into its noncommutative cousin. Suppose that we have a smooth algebraic variety $A$, a line bundle $P$ on it, and a section $t \in \Gamma(A,P^{-1})$ of its dual. Let $\EuScript A$ be an $A_\infty$-algebra underlying the bounded derived category $D^b\mathit{Coh}(A)$. Then, $P$ gives rise to an invertible bimodule $\EuScript P$, and $t$ to a bimodule map $\theta: \EuScript P \rightarrow \EuScript A$. In fact, there is a noncommutative divisor structure on $\EuScript A$ of which $\theta$ is part. This fits in with mirror symmetry in the formulation of \cite{auroux07}, where the crucial ingredient is a choice of anticanonical divisor (see \cite[Section 6]{seidel08}, where the connection is made explicit in a simple example). Similarly, suppose that for $A$ and $P$ as before, we have two elements of $\Gamma(A,P^{-1})$; this yields the structure of a noncommutative pencil on $\EuScript A$.
Homological Mirror Symmetry can now be thought of as operating on the level of pencils, schematically like this:
\begin{equation}
{\begin{tabular}{|c|} \hline Symplectic \\ geometry \\($A$-model) \\ \hline \it Symplectic \\ \it pencil \\ \hline \end{tabular}}
\longrightarrow
{\begin{tabular}{|c|} \hline Homological algebra \\ (Noncommutative geometry) \\ \hline \it Noncommutative pencil \\ \hline \end{tabular}}
\longleftarrow
{\begin{tabular}{|c|} \hline Algebraic \\ geometry \\ ($B$-model) \\ \hline \it Algebraic \\ \it pencil\\ \hline \end{tabular}}
\end{equation}
As a consequence, one gets a unified approach to the associated equivalences of categories. By \eqref{eq:list-of-categories-4}(ii), this picture includes counts of holomorphic discs, following the model of \cite{auroux07}.
{\em Acknowledgments.} Conversations with Anatoly Preygel were useful at an early stage of this project. Mohammed Abouzaid and Maxim Kontsevich provided very valuable comments about \eqref{eq:list-of-categories-4}(i), which are reflected in its current formulation. The definition \eqref{eq:katz1} was suggested by Ludmil Katzarkov. I am grateful to the anonymous referee for keeping me on the straight and narrow. Partial support was provided by: NSF grant DMS-1005288; the Simons Foundation, through a Simons Investigator award; and the Radcliffe Institute for Advanced Study at Harvard University, through a Radcliffe Fellowship. I would also like to thank Boston College, where the first version of this paper was written, for its hospitality.
\section{Homological algebra\label{sec:algebra}}
This section is of an elementary algebraic nature. To make the exposition more self-contained, we recall the relation between Hochschild homology and bimodule maps. We then discuss noncommutative divisors, largely following \cite[Section 3]{seidel08} except for the terminology. Finally, we adapt the same ideas to noncommutative pencils.
\subsection{$A_\infty$-bimodules}
Let $\EuScript A$ be a graded vector space over a field ${\mathbb K}$. We denote by $T(\EuScript A[1])$ the tensor algebra over the (downwards) shifted space $\EuScript A[1]$. The structure of an $A_\infty$-algebra on $\EuScript A$ is given by a map
\begin{equation} \label{eq:a}
\mu_{\EuScript A}: T(\EuScript A[1]) \longrightarrow \EuScript A[2],
\end{equation}
which vanishes on the constants ${\mathbb K} \subset T(\EuScript A[1])$. The components of \eqref{eq:a} are operations $\mu_{\EuScript A}^d: \EuScript A^{\otimes d} \rightarrow \EuScript A[2\!-\!d]$ (for $d \geq 1$). We will assume that $\EuScript A$ is strictly unital, denoting the unit by $e_{\EuScript A}$, and write $\bar{\EuScript A} = \EuScript A/{\mathbb K}\,e_{\EuScript A}$.
An $A_\infty$-bimodule over $\EuScript A$ consists of a graded vector space $\EuScript P$ and a map
\begin{equation}
\mu_{\EuScript P}: T(\EuScript A[1]) \otimes \EuScript P \otimes T(\EuScript A[1]) \longrightarrow \EuScript P[1],
\end{equation}
whose components we write as $\mu_{\EuScript P}^{s;1;r}: \EuScript A^{\otimes s} \otimes \EuScript P \otimes \EuScript A^{\otimes r} \rightarrow \EuScript P[1\!-\!r\!-\!s]$ (for $r,s \geq 0$). All $A_\infty$-bimodules will be assumed to be strictly unital. A basic example is the diagonal bimodule $\EuScript P = \EuScript A$, which has
\begin{equation}
\mu_{\EuScript A}^{s;1;r}(a'_s,\dots,a'_1;a;a_r,\dots,a_1) = (-1)^{\|a_1\| + \cdots + \|a_r\|+1}
\mu_{\EuScript A}^{r+1+s}(a'_s,\dots,a'_1,a,a_r,\dots,a_1).
\end{equation}
Here and later on, $\|a\| = |a|-1$ stands for the reduced degree.
$A_\infty$-bimodules over $\EuScript A$ form a dg category, which we denote by $[\EuScript A,\EuScript A]$. An element of $\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P,\EuScript Q)$ of degree $d$ is given by a map
\begin{equation} \label{eq:bimodule-map}
\phi: T(\EuScript A[1]) \otimes \EuScript P \otimes T(\EuScript A[1]) \longrightarrow \EuScript Q[d],
\end{equation}
which factors through the projection to $T(\bar{\EuScript A}[1])$ on both sides. We denote the components of \eqref{eq:bimodule-map} again by $\phi^{s;1;r}$. The cocycles in $\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P,\EuScript Q)$ are solutions of
\begin{equation} \label{eq:bimodule-d}
\begin{aligned}
& \;\;\; \sum_{i,j} (-1)^{|\phi| (\|a_1\|+\cdots+\|a_i\|)} \; \mu_{\EuScript Q}^{s-j;1;i}(a'_s,\dots;
\phi^{j;1;r-i}(a'_j,\dots,a'_1;p;a_r,\dots,a_{i+1});\dots,a_1) \\
= & \;\;\;\sum_{i,j} (-1)^{|\phi|+\|a_1\|+\cdots+\|a_i\|} \;
\phi^{s-j;1;i}(a'_s,\dots;\mu^{j;1;r-i}_{\EuScript P}(a'_j,\dots,a'_1;p;a_r,\dots,a_{i+1});\dots,a_1) \\
& \!+ \sum_{i,j} (-1)^{|\phi|+\|a_1\|+\cdots+\|a_i\|} \;
\phi^{s;1;r-j+1}(a'_s,\dots;p;a_r,\dots,\mu_{\EuScript A}^j(a_{i+j},\dots,a_{i+1}),\dots,a_1) \\
& \!+ \sum_{i,j} (-1)^{|\phi|+\|a_1\|+\cdots+\|a_r\|+|p|+\|a'_1\|+\cdots+\|a'_i\|}
\phi^{s-j+1;1;r}(a'_s,\dots, \\[-1.5em]
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \mu_{\EuScript A}^j(a_{i+j}',\dots,a_{i+1}'),\dots;p;a_r,\dots,a_1).
\end{aligned}
\end{equation}
For instance, the identity endomorphism $\mathit{id}_{\EuScript P}$ has only one nonzero component, $\mathit{id}_{\EuScript P}^{0;1;0}(p) = p$.
To relate these structures to their (classical) cohomology level counterparts, on equips $H(\EuScript A)$ with the graded associative algebra structure given by
\begin{equation}
[a_2] \cdot [a_1] = (-1)^{|a_1|} [\mu^2_{\EuScript A}(a_2,a_1)],
\end{equation}
and makes $H(\EuScript P)$ into a graded bimodule over this algebra by setting
\begin{equation}
\begin{aligned}
& [a'] \cdot [p] = -(-1)^{|p|} [\mu^{1;1;0}_{\EuScript P}(a';p)],
\\
& [p] \cdot [a] = [\mu^{0;1;1}_{\EuScript P}(p;a)].
\end{aligned}
\end{equation}
If $\phi: \EuScript P \rightarrow \EuScript Q$ is as in \eqref{eq:bimodule-d}, the associated cohomology level bimodule map is
\begin{equation}
\begin{aligned}
& H(\phi): H^*(\EuScript P) \longrightarrow H^{*+|\phi|}(\EuScript Q), \\
& [p] \longmapsto (-1)^{|\phi|\,|p|} [\phi^{0;1;0}(p)].
\end{aligned}
\end{equation}
\begin{conventions}
Our sign conventions for $A_\infty$-algebras follow \cite{seidel04}. The sign conventions for $A_\infty$-bimodules follow \cite{seidel08} except that, for greater compatibility with \cite{seidel04}, we reverse the ordering of the entries. This applies in particular to \eqref{eq:bimodule-d}, which agrees with \cite{seidel08} up to ordering (but differs from the convention for $A_\infty$-modules in \cite{seidel04}).
The following observation may help address some sign issues. Given any $A_\infty$-bimodule structure, the sign change
\begin{equation} \label{eq:tilde-p}
\begin{aligned}
& \mu_{\EuScript P}^{s;1;r}(a_s',\dots,a_1';p;a_r,\dots,a_1) \longmapsto \\ & \qquad (-1)^{\|a_1\|+\cdots+\|a_r\|+\|a_1'\|+\cdots+\|a_s'\|+1} \mu_{\EuScript P}^{s;1;r}(a_s',\dots,a_1'; p; a_r,\dots,a_1)
\end{aligned}
\end{equation}
yields another $A_\infty$-bimodule structure. The two structures are generally distinct, but isomorphic (by an isomorphism that acts by $\pm 1$ on each graded piece). Hence, whenever we define some construction of $A_\infty$-bimodules, there are potentially at least two equivalent versions, which differ by \eqref{eq:tilde-p}.
\end{conventions}
We need to recall a few specific operations on $A_\infty$-bimodules:
\begin{itemize} \itemsep1em
\item The shifted space $\EuScript P[1]$ becomes an $A_\infty$-bimodule, with
\begin{equation}
\mu_{\EuScript P[1]}^{s;1;r}(a_s',\dots,a_1';p;a_r,\dots,a_1) = (-1)^{\|a_1\|+\cdots+\|a_r\|+1}
\mu_{\EuScript P}^{s;1;r}(a_s',\dots,a_1';p;a_r,\dots,a_1).
\end{equation}
\item
The dual is $\EuScript P^\vee = \mathit{Hom}(\EuScript P,{\mathbb K})$, with
\begin{equation}
\langle \mu_{\EuScript P^\vee}^{s;1;r}(a_s,\dots,a_1;\pi;a'_r,\dots,a'_1), p \rangle =
(-1)^{|p|+1} \langle \pi, \mu_{\EuScript P}^{r;1;s}(a'_r,\dots,a'_1;p;a_s,\dots,a_1) \rangle.
\end{equation}
\item Given two $A_\infty$-bimodules $\EuScript Q$ and $\EuScript P$, one defines the tensor product $\EuScript Q \otimes_{\EuScript A} \EuScript P$ to be the graded vector space $\EuScript Q \otimes T(\bar\EuScript A[1]) \otimes \EuScript P$, with the differential
\begin{equation}
\begin{aligned}
& \mu^{0;1;0}_{\EuScript Q \otimes_{\EuScript A} \EuScript P}(q \otimes a'_t \otimes \cdots \otimes a'_1 \otimes p) = \\
& \qquad
\sum_i (-1)^{|p|+\|a_1'\|+\cdots+\|a_{i}'\|} \mu_{\EuScript Q}^{0;1;t-i}(q;a'_t,\dots,a'_{i+1}) \otimes \cdots \otimes a'_1 \otimes p
\\
& \quad + \sum_{i,j} (-1)^{|p|+\|a_1'\|+\cdots+\|a_i'\|} q \otimes a'_t \otimes \cdots \otimes \mu_{\EuScript A}^j(a'_{i+j},\dots,a'_{i+1}) \otimes \cdots \otimes a'_1 \otimes p \\
& \quad + \sum_i q \otimes a'_t \otimes \cdots \otimes \mu_{\EuScript P}^{i;1;0}(a'_i,\dots,a'_1;p);
\end{aligned}
\end{equation}
the operations (for $r>0$ or $s>0$)
\begin{equation}
\begin{aligned}
& \mu^{s;1;0}_{\EuScript Q \otimes_{\EuScript A} \EuScript P}(a''_s,\dots,a''_1; q \otimes a'_t \otimes \cdots \otimes a'_1 \otimes p) = \\
& \quad \sum_i (-1)^{|p|+\|a_1'\|+\cdots+\|a_{i}'\|} \mu_{\EuScript Q}^{s;1;t-i}(a''_s,\dots,a''_1;q;a'_t,\dots,a'_{i+1})
\otimes \cdots \otimes a'_1 \otimes p,
\\
& \mu^{0;1;r}_{\EuScript Q \otimes_{\EuScript A} \EuScript P}(q \otimes a'_t \otimes \cdots \otimes a'_1 \otimes p; a_r,\dots, a_1) = \\
& \quad \sum_i q \otimes a'_t \otimes \cdots \otimes \mu_{\EuScript P}^{i;1;r}(a'_i,\dots,a'_1;p;a_r,\dots,a_1);
\end{aligned}
\end{equation}
and with $\mu_{\EuScript Q \otimes_{\EuScript A} \EuScript P}^{s;1;r} = 0$ if both $r$ and $s$ are positive.
\end{itemize}
Tensor product with the diagonal bimodule is essentially a trivial operation. More precisely, there are canonical quasi-isomorphisms \cite[Eqn.~(2.21)--(2.24)]{seidel08}
\begin{equation} \label{eq:neutral}
\EuScript A \otimes_{\EuScript A} \EuScript P \simeq \EuScript P \simeq \EuScript P \otimes_{\EuScript A} \EuScript A.
\end{equation}
For arbitrary $\EuScript A$-bimodules $\EuScript P$, $\EuScript Q$, $\EuScript R$, there are canonical isomorphisms
\begin{align}
&
\label{eq:2-rotation}
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P,\EuScript Q^\vee) \cong \mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript Q,\EuScript P^\vee),
\\
&
\label{eq:3-rotation}
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P \otimes_{\EuScript A} \EuScript Q, \EuScript R^\vee) \cong \mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript Q \otimes_{\EuScript A} \EuScript R, \EuScript P^\vee) \cong
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript R \otimes_{\EuScript A} \EuScript P, \EuScript Q^\vee).
\end{align}
As an application of \eqref{eq:neutral} and \eqref{eq:3-rotation}, one gets
\begin{align}
& \label{eq:dual-rotate-1}
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^\vee,\EuScript P^\vee)
\simeq \mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript A \otimes_{\EuScript A} \EuScript P^\vee, \EuScript P^\vee)
\cong \mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^\vee \otimes_{\EuScript A} \EuScript P, \EuScript A^\vee),
\\
& \label{eq:dual-rotate-2}
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^\vee,\EuScript P^\vee) \simeq \mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^\vee \otimes_{\EuScript A} \EuScript A, \EuScript P^\vee) \cong \mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P \otimes_{\EuScript A} \EuScript P^\vee, \EuScript A^\vee).
\end{align}
By starting with $\mathit{id}_{\EuScript P^\vee}$ and going through \eqref{eq:dual-rotate-1}, \eqref{eq:dual-rotate-2}, one obtains canonical maps
\begin{align} \label{eq:left-pi}
& \lambda_{\mathit{left}}: \EuScript P^\vee \otimes_{\EuScript A} \EuScript P \longrightarrow \EuScript A^\vee, \\
\label{eq:right-pi}
& \lambda_{\mathit{right}}: \EuScript P \otimes_{\EuScript A} \EuScript P^\vee \longrightarrow \EuScript A^\vee.
\end{align}
There are explicit formulae for \eqref{eq:neutral}--\eqref{eq:right-pi}, but we prefer to omit them. For more foundational material on $A_\infty$-bimodules, see \cite{tradler01, lefevre, kontsevich-soibelman06, lyubashenko-manzyuk08, seidel08}.
\subsection{Invertible bimodules}
A bimodule $\EuScript P$ is called invertible if there is another bimodule $\EuScript P^{-1}$ and quasi-isomorphisms
\begin{equation}
\EuScript P^{-1} \otimes_{\EuScript A} \EuScript P \simeq \EuScript A \simeq \EuScript P \otimes_{\EuScript A} \EuScript P^{-1}.
\end{equation}
In that case, taking the tensor product with $\EuScript P$ (on objects, and the tensor product with $\mathit{id}_{\EuScript P}$ on morphisms) is an automorphism, or more rigorously a quasi-equivalence, of the dg category $[\EuScript A,\EuScript A]$.
\begin{lemma}
If $\EuScript P$ is invertible, $\lambda_{\mathit{left}}$ and $\lambda_{\mathit{right}}$ are quasi-isomorphisms.
\end{lemma}
\begin{proof}
Consider the chain of quasi-isomorphisms
\begin{equation} \label{eq:quasi-shift}
\begin{aligned}
& \mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript Q,\EuScript P^\vee) \simeq \mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript A \otimes_{\EuScript A} \EuScript Q, \EuScript P^\vee) \\ & \qquad \cong \mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript Q \otimes_{\EuScript A} \EuScript P, \EuScript A^\vee) \simeq
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript Q, \EuScript A^\vee \otimes_{\EuScript A} \EuScript P^{-1}).
\end{aligned}
\end{equation}
This is functorial in $\EuScript Q$, hence (by the Yoneda Lemma) arises from a quasi-isomorphism
\begin{equation} \label{eq:v-inverse-quasi-iso}
\EuScript P^\vee \simeq \EuScript A^\vee \otimes_{\EuScript A} \EuScript P^{-1}.
\end{equation}
More concretely, one obtains that quasi-isomorphism by setting $\EuScript Q = \EuScript P^\vee$ in \eqref{eq:quasi-shift}, and then taking the identity in the leftmost morphism group. By comparing this with \eqref{eq:dual-rotate-1}, one sees that the quasi-isomorphism is in fact $\lambda_{\mathit{left}} \otimes_{\EuScript A} \mathit{id}_{\EuScript P^{-1}}$. By tensoring with $\mathit{id}_{\EuScript P}$, it follows that $\lambda_{\mathit{left}}$ itself is a quasi-isomorphism. The other case is similar.
\end{proof}
Note that, by combining \eqref{eq:v-inverse-quasi-iso} with its counterpart for $\EuScript P^{-1} \otimes_{\EuScript A} \EuScript A^\vee$, we can get a canonical quasi-isomorphism (for invertible $\EuScript P$)
\begin{equation} \label{eq:central-tensor}
\EuScript A^\vee \otimes_{\EuScript A} \EuScript P^{-1} \simeq \EuScript P^{-1} \otimes_{\EuScript A} \EuScript A^\vee.
\end{equation}
\subsection{Hochschild homology\label{subsec:hochschild-homology}}
The Hochschild homology of $\EuScript A$ with coefficients in a bimodule $\EuScript P$, written as $\mathit{HH}_*(\EuScript A,\EuScript P)$, is the homology of $\mathit{CC}_*(\EuScript A,\EuScript P) = T(\bar\EuScript A[1]) \otimes \EuScript P$, with differential
\begin{equation}
\begin{aligned}
\partial(a_d \otimes \cdots \otimes a_1 \otimes p) =
& \sum_{i,j} (-1)^{|p|+\|a_1\|+\cdots+\|a_i\|} a_d \otimes \cdots \otimes \mu_{\EuScript A}^j(a_{i+j},\dots,a_{i+1}) \otimes \cdots \otimes a_1 \otimes p \\[-1em]
& \qquad + \sum_{i,j} (-1)^\ast a_{d-i} \otimes \cdots \otimes \mu_{\EuScript P}^{j;1;i}(a_j,\dots,a_1;p;a_d,\dots,a_{d-i+1}),
\end{aligned}
\end{equation}
where $\ast = (\|a_{d-i+1}\| + \cdots + \|a_d\|)(|p|+\|a_1\| + \cdots + \|a_{d-i}\|)$. In spite of the subscript notation, the grading on $\mathit{CC}_*(\EuScript A,\EuScript P)$ is cohomological, meaning that $\partial$ has degree $1$.
There is a canonical isomorphism
\begin{equation} \label{eq:general-hochschild}
\mathit{CC}_*(\EuScript A, \EuScript Q \otimes_{\EuScript A} \EuScript P)^\vee \cong
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P,\EuScript Q^\vee).
\end{equation}
In particular,
\begin{equation} \label{eq:second-hochschild}
\mathit{CC}_*(\EuScript A,\EuScript P)^\vee \simeq
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript A,\EuScript P^\vee).
\end{equation}
By spelling out \eqref{eq:second-hochschild}, one obtains the following:
\begin{lemma} \label{th:recognize-diagonal}
Suppose that we have a cocycle $\xi \in \mathit{CC}_0(\EuScript A,\EuScript P)^\vee$, whose constant term $\xi^0 \in \EuScript P^\vee$ has the property that
\begin{equation} \label{eq:zero-order-map}
\begin{aligned}
& H^*(\EuScript A) \longrightarrow H^*(\EuScript P)^\vee, \\
& [a] \longmapsto [\xi^0(\mu_{\EuScript P}^{1;1;0}(a; \cdot))]
\end{aligned}
\end{equation}
is an isomorphism. Then, under \eqref{eq:second-hochschild}, $\xi$ gives rise to a quasi-isomorphism $\EuScript A \simeq \EuScript P^\vee$.
\end{lemma}
If $\EuScript A$ is proper (has finite-dimensional cohomology), the natural map $\EuScript A \rightarrow (\EuScript A^\vee)^\vee$ is a quasi-isomorphism. In that case, \eqref{eq:general-hochschild} implies that
\begin{equation} \label{eq:first-hochschild}
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P,\EuScript A) \simeq
\mathit{CC}_*(\EuScript A, \EuScript A^\vee \otimes_{\EuScript A} \EuScript P)^\vee.
\end{equation}
Hochschild homology is cyclically invariant, which means that there is a canonical isomorphism of chain complexes
\begin{equation} \label{eq:cyclic-hh}
\mathit{CC}_*(\EuScript A,\EuScript Q \otimes_{\EuScript A} \EuScript P) \cong \mathit{CC}_*(\EuScript A,\EuScript P \otimes_{\EuScript A} \EuScript Q).
\end{equation}
The dual of this isomorphism, using \eqref{eq:general-hochschild}, is \eqref{eq:2-rotation}. Similarly, \eqref{eq:cyclic-hh} implies that
\begin{equation}
\mathit{CC}_*(\EuScript A,\EuScript R \otimes_{\EuScript A} \EuScript Q \otimes_{\EuScript A} \EuScript P) \cong
\mathit{CC}_*(\EuScript A,\EuScript P \otimes_{\EuScript A} \EuScript R \otimes_{\EuScript A} \EuScript Q),
\end{equation}
and the dual of that is \eqref{eq:3-rotation}. One can apply the same idea to the tensor powers of a single bimodule, leading to an action of the cyclic group ${\mathbb Z}/i$ on $\mathit{CC}_*(\EuScript A,\EuScript P^{\otimes_{\EuScript A} i})$. By applying \eqref{eq:first-hochschild}, one gets a ${\mathbb Z}/i$-action on $H^*(\mathit{hom}_{[\EuScript A,\EuScript A]}( (\EuScript A^\vee)^{\otimes_{\EuScript A} i-1}, \EuScript A))$, assuming that $\EuScript A$ is proper.
\subsection{Hochschild cohomology\label{subsec:hochschild-coho}}
Even though we have emphasized Hochschild homology, it makes sense to also mention the Hochschild cohomology $\mathit{HH}^*(\EuScript A,\EuScript P)$, whose underlying chain complex is $\mathit{CC}^*(\EuScript A,\EuScript P) = \mathit{Hom}(T(\bar\EuScript A[1]), \EuScript P)$. It is well-known that
\begin{equation} \label{eq:hh-diag}
\mathit{CC}^*(\EuScript A,\EuScript P) \simeq \mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript A,\EuScript P).
\end{equation}
One can interpret \eqref{eq:second-hochschild} and Lemma \ref{th:recognize-diagonal} in those terms, given that
\begin{equation}
\mathit{CC}_*(\EuScript A,\EuScript P)^\vee = \mathit{CC}^*(\EuScript A,\EuScript P^\vee).
\end{equation}
There is another, less straightforward, connection between the two Hochschild theories. Namely, suppose that $\EuScript A$ is proper and homologically smooth, in which case $\EuScript A^\vee$ is always invertible (see e.g.\ \cite[Theorem 4.5]{shklyarov07b}). Then (see e.g.\ \cite[Remark 8.2.4]{kontsevich-soibelman06})
\begin{equation} \label{eq:serre-hochschild}
\mathit{CC}_*(\EuScript A,\EuScript P) \simeq
\mathit{hom}_{[\EuScript A,\EuScript A]}((\EuScript A^\vee)^{-1},\EuScript P) \simeq
\mathit{CC}^*(\EuScript A,\EuScript A^\vee \otimes_{\EuScript A} \EuScript P)
\end{equation}
If one specializes this to $\EuScript P = \EuScript A$, the outcome is a quasi-isomorphism between $\mathit{CC}_*(\EuScript A,\EuScript A)$ and its dual, which gives rise to a nondegenerate pairing on $\mathit{HH}_*(\EuScript A,\EuScript A)$ (see \cite{shklyarov07b} for an extended discussion).
\subsection{Self-conjugation}
Fix an invertible bimodule $\EuScript P$, and some $i \geq 1$. We define an endomorphism of $\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i}, \EuScript A)$, unique up to chain homotopy, through the homotopy commutative diagram
\begin{equation} \label{eq:k-diagram}
\xymatrix{
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i}, \EuScript A)
\ar[d]_-{\mathit{id}_{\EuScript P} \otimes_{\EuScript A} \cdot}^-{\simeq}
\ar@{-->}[rr]
&&
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i}, \EuScript A)
\ar[d]^-{\cdot \otimes_{\EuScript A} \mathit{id}_{\EuScript P}}_-{\simeq}
\\
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i+1},\EuScript P \otimes_{\EuScript A} \EuScript A)
\ar[r]^-{\simeq}
&
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i+1},\EuScript P)
&
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i+1},\EuScript A \otimes_{\EuScript A} \EuScript P).
\ar[l]_-{\simeq}
}
\end{equation}
Again up to homotopy, these maps are compatible with the ring structure induced by the tensor product
\begin{equation}
\begin{aligned}
& \mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} j}, \EuScript A) \otimes
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i}, \EuScript A) \longrightarrow
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i+j}, \EuScript A \otimes_{\EuScript A} \EuScript A)
\\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\stackrel{\simeq}{\longrightarrow} \mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i+j}, \EuScript A).
\end{aligned}
\end{equation}
We call the induced cohomology automorphisms {\em self-conjugation maps}, and denote them by
\begin{equation} \label{eq:c-automorphism}
\xymatrix{
\ar@(dl,ul)[0,0]^-{K_i} & \hspace{-3em}
H^*(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i},\EuScript A)).
}
\end{equation}
\begin{example} \label{th:exterior-algebra}
Let $A = \Lambda^*({\mathbb K})$ be the exterior algebra in one variable (denoted by $x$, and given degree $1$). Given any $\lambda \in {\mathbb K}^\times$, one has an invertible $A$-bimodule $P_{\lambda}$, whose underlying graded vector space is the same as $A$, but where the action of $x$ on the right is multiplied by $\lambda$ (while that on the left remains the same). Consider the bimodule map (of degree $1$) $t: P_{\lambda} \rightarrow A$ which sends $1$ to $x$. Then
\begin{equation}
\begin{cases}
P_{\lambda} \otimes_A P_{\lambda} \xrightarrow{t \otimes_A \mathit{id}_{P_\lambda}} A \otimes_A P_{\lambda} \cong P_{\lambda}
& \text{maps $1 \otimes 1 \mapsto x$,} \\
P_{\lambda} \otimes_A P_{\lambda} \xrightarrow{\mathit{id}_{P_\lambda} \otimes_A t} P_{\lambda} \otimes_A A \cong P_{\lambda}
& \text{maps $1 \otimes 1 \mapsto \lambda x$.}
\end{cases}
\end{equation}
This translates straightforwardly into the $A_\infty$-world, and provides an example where \eqref{eq:c-automorphism}, for $i = 1$, is not the identity.
\end{example}
For any invertible bimodule $\EuScript P$, there is a homotopy commutative diagram
\begin{equation} \label{eq:top-diagram}
\xymatrix{
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript A^\vee, \EuScript A^\vee \otimes_{\EuScript A} \EuScript P^{-1})
\ar[dd]_-{\eqref{eq:central-tensor}}
&&
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript A, \EuScript P^{-1})
\ar[ll]_-{\mathit{id}_{\EuScript A^\vee} \otimes_{\EuScript A} \cdot}
\ar[d]_-{\cdot \otimes_{\EuScript A} \mathit{id}_{\EuScript P}}
\\
&
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript A^\vee,\EuScript P^\vee)
\ar[ul]
\ar[dl]
&
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P,\EuScript A)
\ar[l]_-{\text{dualize}}
\\
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript A^\vee,\EuScript P^{-1} \otimes_{\EuScript A} \EuScript A^\vee)
&&
\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript A,\EuScript P^{-1})
\ar[ll]^-{\cdot \otimes_{\EuScript A} \mathit{id}_{\EuScript A^\vee}}
\ar[u]^-{\mathit{id}_{\EuScript P} \otimes_{\EuScript A} \cdot}
}
\end{equation}
The diagonal arrows are \eqref{eq:v-inverse-quasi-iso} and its counterpart; hence, the left triangle in the diagram commutes by definition of \eqref{eq:central-tensor}. Proving the commutativity of the remaining parts requires one to go back to \eqref{eq:quasi-shift}, and we will not explain the details here.
Going vertically down the right column of \eqref{eq:top-diagram} gives an automorphism of $H^*(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript A,\EuScript P^{-1}))$, which is a form of $K_1$. Suppose that $\EuScript A^\vee$ is an invertible bimodule. Then, by going around the diagram the other way, we get another description of that automorphism, which now involves the interaction of $\EuScript P$ and $\EuScript A^\vee$.
\begin{example}
Suppose that $\EuScript A$ is weakly Calabi-Yau of dimension $n$, by which we mean that it comes with a quasi-isomorphism
\begin{equation} \label{eq:weak-cy-structure}
\EuScript A^\vee[-n] \simeq \EuScript A.
\end{equation}
By inserting that quasi-isomorphism on both sides of \eqref{eq:central-tensor}, one gets a distinguished automorphism of $\EuScript P^{-1}$ (and correspondingly of $\EuScript P$), which describes the failure of the tensor product with that bimodule to be compatible with the Calabi-Yau structure. It then follows from \eqref{eq:top-diagram} that $K_1$ is the composition with that automorphism. This agrees with Example \ref{th:exterior-algebra}.
\end{example}
\begin{lemma} \label{th:z-action}
Suppose that $\EuScript A$ is proper. Suppose also that $\EuScript A^\vee$ is invertible (which holds if $\EuScript A$ is smooth). Then, the generator of the ${\mathbb Z}/(i+1)$-action on $H^*(\mathit{hom}_{[\EuScript A,\EuScript A]}((\EuScript A^\vee)^{\otimes_{\EuScript A} i},\EuScript A))$ equals the self-conjugation map $K_i$, for the bimodule $\EuScript A^\vee$.
\end{lemma}
We will only explain the counterpart of this in classical algebra. Given a finite-dimensional algebra $A$, a bimodule map
\begin{equation} \label{eq:classical-bimodule}
(A^\vee)^{\otimes_A i} \longrightarrow A
\end{equation}
is given by an element
\begin{equation} \label{eq:tensor-element}
\sum_j a^j_{i+1} \otimes \cdots \otimes a^j_1 \in A^{\otimes_{{\mathbb K}} i+1}
\end{equation}
satisfying the equation
\begin{equation} \label{eq:alpha-eq}
\sum_j a^j_{i+1}a \otimes a^j_i \otimes \cdots \otimes a^j_1 = \sum_j
a^j_{i+1} \otimes aa^j_i \otimes \cdots \otimes a^j_1
\end{equation}
for $a \in A$, as well as all its cyclic permutations. The bimodule map associated to \eqref{eq:tensor-element} is
\begin{equation} \label{eq:cyclic-relations}
\alpha_1 \otimes \cdots \otimes \alpha_i \longmapsto
\sum_j \alpha_1(a^j_1) \cdots \alpha_i(a^j_i) a^j_{i+1} \in A.
\end{equation}
If one tensors this with the identity map of $A^\vee$ on the right or left, the outcome are bimodule maps $(A^\vee)^{\otimes_A i+1} \rightarrow A^\vee$ given by, respectively,
\begin{align}
& \alpha_1 \otimes \cdots \otimes \alpha_{i+1} \longmapsto \sum_j \alpha_1(a^j_1) \cdots \alpha_i(a^j_i) \alpha_{i+1}(\cdot\, a^j_{i+1}), \label{eq:lambda-1-map} \\
& \alpha_1 \otimes \cdots \otimes \alpha_{i+1} \longmapsto \sum_j \alpha_2(a_1^j) \cdots \alpha_{i+1}(a^j_i) \alpha_1(a^j_{i+1}\, \cdot). \label{eq:lambda-2-map}
\end{align}
Again using \eqref{eq:alpha-eq}, one can write the right hand side of \eqref{eq:lambda-2-map} as
\begin{equation}
\sum_j \alpha_2(\cdot \, a^j_1) \alpha_3(a_2^j) \cdots \alpha_1(a_{i+1}^j),
\end{equation}
which is indeed obtained from \eqref{eq:lambda-1-map} by cyclically permuting the tensor factors in \eqref{eq:tensor-element}.
\begin{example}
For some $i \geq 1$, consider a graded algebra $A = \bigoplus_{k \in {\mathbb Z}/(i+1)} {\mathbb K} e_k \oplus {\mathbb K} x_k$. We draw it in quiver form as follows:
\begin{equation}
\xymatrix{
\stackrel{e_0}{\bullet} \ar[r]^-{x_1} &
\stackrel{e_1}{\bullet} \ar[r]^-{x_2} & \cdots \ar[r]^-{x_i} & \stackrel{e_i}{\bullet}
\ar@/^1pc/[lll]^-{x_0}
}
\end{equation}
The vertices give rise to mutually orthogonal idempotents $e_k$, and the $x_k$ satisfy
\begin{equation}
\begin{aligned}
& e_{k+1}x_k e_k = x_k, \\
& x_{k+1}x_k = 0.
\end{aligned}
\end{equation}
The $e_k$ have degree zero, and we choose the degrees of the $x_k$ so that they add up to $i-1$. Following \eqref{eq:tensor-element}, each of the expressions
\begin{equation} \label{eq:cyclic-expressions}
x_k \otimes x_{k-1} \otimes \cdots \otimes x_{k-i}
\end{equation}
gives rise to a bimodule map \eqref{eq:classical-bimodule} of degree $i-1$.
If one thinks of $A$ (in the straightforward way) as an $A_\infty$-algebra $\EuScript A$, the elements \eqref{eq:cyclic-expressions} induce $A_\infty$-bimodule maps $(\EuScript A^\vee)^{\otimes_{\EuScript A} i} \rightarrow \EuScript A[i-1]$, which are cyclically exchanged by $K^i$ (up to signs).
It is a familiar fact that $\EuScript A$ is ``virtually Calabi-Yau of dimension $(i-1)/(i+1)$'', meaning that
\begin{equation} \label{eq:virtual-cy}
(\EuScript A^\vee)^{\otimes_{\EuScript A} i+1} \simeq \EuScript A[i-1].
\end{equation}
Hence,
\begin{equation} \label{eq:i-plus-1-morphism}
H^{i-1}(\mathit{hom}_{[\EuScript A,\EuScript A]}((\EuScript A^\vee)^{\otimes_{\EuScript A} i},\EuScript A)) \cong
H^0(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript A,\EuScript A^\vee)) \cong \mathit{HH}_0(\EuScript A,\EuScript A)^\vee.
\end{equation}
One easily computes that
\begin{equation} \label{eq:am-hochschild}
\mathit{HH}_*(\EuScript A,\EuScript A) = \begin{cases} {\mathbb K}^{i+1} & \ast = 0, \\
{\mathbb K} & \ast < 0, \\
0 & \ast > 0.
\end{cases}
\end{equation}
The elements of \eqref{eq:i-plus-1-morphism} which we have constructed are a dual basis of $\mathit{HH}_0(\EuScript A,\EuScript A)$.
\end{example}
\begin{example}
Start with the graded algebra from the previous example (with $i>1$), but now let $\EuScript A$ be an $A_\infty$-deformation of it, in which $\mu_{\EuScript A}^{i+1}(x_k,\dots,x_{k-i})$ is a nonzero multiple of $e_k$ (if this is the case for one $k$, it must be true for all $k$, since these operations are related to each other by the $A_\infty$-associativity equations; in fact, $\EuScript A$ is unique up to $A_\infty$-isomorphism). The counterpart of \eqref{eq:am-hochschild} is that
\begin{equation}
\mathit{HH}_*(\EuScript A,\EuScript A) = \begin{cases} {\mathbb K}^i & \ast = 0, \\ 0 & \ast \neq 0.
\end{cases}
\end{equation}
This deformation still satisfies \eqref{eq:virtual-cy}, and hence \eqref{eq:i-plus-1-morphism}. The ${\mathbb Z}/(i+1)$-action on \eqref{eq:i-plus-1-morphism} generated by $K_i$ is the action on ${\mathbb K}^i$ obtained by taking the standard cyclic permutation representation, and dividing by the one-dimensional diagonal subspace.
\end{example}
\subsection{Noncommutative divisors\label{subsec:div}}
We now have all the ingredients needed to flesh out the outline previously given in Section \ref{subsec:nc-geometry}.
\begin{definition}
A noncommutative divisor on $\EuScript A$, with underlying invertible bimodule $\EuScript P$, is an $A_\infty$-algebra structure $\mu_{\EuScript B}$ on the graded vector space \eqref{eq:b-space}, with the following properties:
\begin{itemize} \itemsep.5em
\item[(i)] $\EuScript A \subset \EuScript B$ is an $A_\infty$-subalgebra.
\item[(ii)] Consider $\EuScript B$ as an $\EuScript A$-bimodule (by restriction of the diagonal bimodule $\EuScript B$ to the subalgebra $\EuScript A$). Then, the induced $\EuScript A$-bimodule structure on $\EuScript B/\EuScript A = \EuScript P[1]$ agrees with the previously given one.
\end{itemize}
Two noncommutative divisors (with the same underlying bimodule) are isomorphic if there is an isomorphism of the associated $A_\infty$-algebras $\EuScript B$ which restricts to the identity map on $\EuScript A$, and induces the identity map on the $\EuScript A$-bimodule $\EuScript B/\EuScript A$.
\end{definition}
\begin{example}
There is a trivial special case, denoted by $\EuScript B^{\mathit{triv}}$, which is the trivial extension algebra obtained from $\EuScript A$ and $\EuScript P$. In that case, the only nonzero components of $\mu_{\EuScript B}$ are those dictated by (i) and (ii) above.
\end{example}
What does this mean concretely? If we restrict $\mu_{\EuScript B}$ to $T(\EuScript A[1]) \subset T(\EuScript B[1])$, it takes values in $\EuScript A$ and agrees with $\mu_{\EuScript A}$.
Next, if we restrict $\mu_{\EuScript B}$ to $T(\EuScript A[1])\otimes \EuScript P[2] \otimes T(\EuScript A[1]) \subset T(\EuScript B[1])$, it has the form
\begin{equation} \label{eq:reduces-to-p}
\begin{aligned}
\mu_{\EuScript B}^{r+s+1}(a'_s,\dots,a'_1,p,a_r,\dots,a_1) = & \; \mu_{\EuScript P}^{s;1;r}(a'_s,\dots,a'_1;p;a_r,\dots,a_1) \\ & + \theta^{s;1;r}(a'_s,\dots,a'_1;p;a_r,\dots,a_1)
\end{aligned}
\end{equation}
for some
\begin{equation}
\theta: T(\EuScript A[1]) \otimes \EuScript P \otimes T(\EuScript A[1]) \longrightarrow \EuScript A,
\end{equation}
which is the first new piece of information (not described by $\EuScript A$ and $\EuScript P$) in the noncommutative divisor. The $A_\infty$-associativity equation for $\mu_{\EuScript B}$ implies that $\theta$ is a bimodule map $\EuScript P \rightarrow \EuScript A$, meaning that it satisfies \eqref{eq:bimodule-d}. We call its cohomology class
\begin{equation} \label{eq:leading-order-part}
[\theta] \in H^0(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P,\EuScript A))
\end{equation}
the section associated to the noncommutative divisor. It is an isomorphism invariant, which classifies the bimodule extension \eqref{eq:a-b-sequence}. For instance, $\EuScript B^{\mathit{triv}}$ has $\theta = 0$. Hence, \eqref{eq:leading-order-part} is an obstruction (not the only one, in general) to the triviality of a noncommutative divisor.
The next piece of the structure of a noncommutative divisor is obtained by restricting $\mu_{\EuScript B}$ to $T(\EuScript A[1]) \otimes \EuScript P[2] \otimes T(\EuScript A[1]) \otimes \EuScript P[2] \otimes T(\EuScript A[1])$, and then projecting the outcome to $\EuScript P[3]$. This can be viewed as an element of $\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P \otimes_{\EuScript A} \EuScript P, \EuScript P)$ of degree $-1$. It is not a cocycle; instead, as shown in \cite[Lemma 3.2]{seidel08}, it provides a homotopy (in the dg category of bimodules) between the two sides of the diagram
\begin{equation} \label{eq:h-commutative}
\xymatrix{
& \EuScript P \otimes_{\EuScript A} \EuScript P
\ar[dl]_-{\mathit{id}_{\EuScript P} \otimes_{\EuScript A} \theta} \ar[dr]^-{\theta \otimes_{\EuScript A} \mathit{id}_{\EuScript P}}
\\
\EuScript P \otimes_{\EuScript A} \EuScript A
\ar[dr]_{\simeq} &&
\EuScript A \otimes_{\EuScript A} \EuScript P
\ar[dl]^{\simeq} \\
& \EuScript P.
}
\end{equation}
In terms of \eqref{eq:c-automorphism}, this means that
\begin{equation} \label{eq:k-fixed}
K_1([\theta]) = [\theta].
\end{equation}
One can put the previous discussion on a more systematic footing, as follows. Consider the Hochschild cochain complex of $\EuScript B^{\mathit{triv}}$. We recall (from Section \ref{subsec:hochschild-coho}) that this is
\begin{equation} \label{eq:cc}
\mathit{CC}^*(\EuScript B^{\mathit{triv}}, \EuScript B^{\mathit{triv}})[1] = \mathit{Hom}\big(T(\bar\EuScript B^{\mathit{triv}}[1]),\EuScript B^{\mathit{triv}}[1]\big),
\end{equation}
with a differential given by $\mu_{\EuScript B^{\mathit{triv}}}$, which means by $\mu_{\EuScript A}$ and $\mu_{\EuScript P}$. Let's equip $\EuScript B^{\mathit{triv}}$ with an additional grading (called {\em weight grading} to distinguish it from the usual one), in which $\EuScript A$ has weight $0$ and $\EuScript P[1]$ has weight $-1$. For any $i > 0$, let $F^i\mathfrak{g}$ be the subspace of \eqref{eq:cc} consisting of maps which increase the weight by (exactly) $i$. Write $\mathfrak{g}$ for the (bigraded) space which combines all the $F^i\mathfrak{g}$. The Hochschild differential maps each $F^i \mathfrak{g}$ to itself. Moreover, its cohomology forms a long exact sequence
\begin{multline} \label{eq:g-les}
\cdots \rightarrow H^{*-2i}\big(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i+1},\EuScript P)\big) \longrightarrow H^*(F^i\frak g) \\ \longrightarrow
H^{*-2i+1}\big(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i}, \EuScript A )\big) \longrightarrow
H^{*-2i+1}\big(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i+1}, \EuScript P)\big) \rightarrow \cdots
\end{multline}
where the last map is the difference between $\phi \mapsto \mathit{id}_{\EuScript P} \otimes_{\EuScript A} \phi$ and $\phi \mapsto \phi \otimes_{\EuScript A} \mathit{id}_{\EuScript P}$. Since either map is an isomorphism, one can identify $H^*(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i+1},\EuScript P)) \cong H^*(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i},\EuScript A))$, and then \eqref{eq:g-les} becomes
\begin{multline} \label{eq:g-les-2}
\cdots \rightarrow H^{*-2i}\big(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i},\EuScript A)\big) \longrightarrow H^*(F^i\frak g) \\ \longrightarrow
H^{*-2i+1}\big(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i}, \EuScript A )\big) \xrightarrow{K_i-\mathit{id}}
H^{*-2i+1}\big(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i}, \EuScript A)\big) \rightarrow \cdots
\end{multline}
where $K_i$ is \eqref{eq:c-automorphism}. From this point of view, \eqref{eq:k-fixed} holds because the structure of a noncommutative divisor specifies a lift of $[\theta]$ to $H^1(F^1\mathfrak{g})$.
\begin{remark} \label{th:cyclic}
In our applications, $\EuScript A$ is proper and smooth (hence $\EuScript A^\vee$ is invertible), and
\begin{equation} \label{eq:p-dual}
\EuScript P = \EuScript A^\vee[-n].
\end{equation}
By Lemma \ref{th:z-action}, the kernel of the boundary map in \eqref{eq:g-les} is $H^*(\mathit{hom}_{[\EuScript A,\EuScript A]}((\EuScript A^\vee)^{\otimes_{\EuScript A} i}, \EuScript A))^{{\mathbb Z}/(i+1)}$. This hints at the existence of a more refined notion, considered in \cite{seidel12b, kontsevich-vlassopoulos13}. Let's suppose for simplicity that $\EuScript A$ is finite-dimensional. More importantly, we assume that our coefficient field ${\mathbb K}$ has characteristic $0$. Then, a {\em noncommutative anticanonical divisor} is one with $\EuScript P$ as in \eqref{eq:p-dual}, and such that the $A_\infty$-structure on $\mu_{\EuScript B}$ is cyclic with respect to the obvious inner product on $\EuScript B = \EuScript A \oplus \EuScript A^\vee[1-n]$. The counterparts $F^i\mathfrak{g}^{\mathit{cyc}}$ of the spaces $F^i\mathfrak{g}$ satisfy
\begin{equation}
H^*(F^i\mathfrak{g}^{\mathit{cyc}}) \cong H^{*+(n-2)i + 1}\big(\mathit{hom}_{[\EuScript A,\EuScript A]}((\EuScript A^\vee)^{\otimes_{\EuScript A} i}, \EuScript A)^{{\mathbb Z}/(i+1)}\big).
\end{equation}
\end{remark}
We keep the assumption that the coefficient field ${\mathbb K}$ has characteristic $0$. The complex \eqref{eq:cc} carries the structure of dg Lie algebra, and since the bracket respects weight gradings, we get the structure of a (bigraded) dg Lie algebra on $\mathfrak{g}$. A noncommutative divisor is given by a solution of the Maurer-Cartan equation \cite{goldman-millson88,manetti04}
\begin{equation} \label{eq:mc-deformation-theory}
\beta \in \mathfrak{g}^1, \;\; d_{\mathfrak{g}} \beta + \half [\beta,\beta] = 0
\end{equation}
(the trivial solution $\beta = 0$ corresponds to $\EuScript B^{\mathit{triv}}$). One can decompose \eqref{eq:mc-deformation-theory} into a series of equations for the weight components $(\beta^1,\beta^2,\dots)$, which are
\begin{equation} \label{eq:weighted-maurer-cartan}
d_{\mathfrak{g}} \beta^i + \half \sum_j [\beta^j,\beta^{i-j}] = 0.
\end{equation}
The notion of isomorphism for noncommutative divisors reduces to the usual gauge equivalence for solutions of \eqref{eq:mc-deformation-theory}. Standard obstruction theory yields:
\begin{lemma} \label{th:obstructions}
Suppose that
\begin{equation} \label{eq:no-negative-degree}
H^*\big(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P^{\otimes_{\EuScript A} i}, \EuScript A)\big) = 0 \quad \text{for all $i\geq 1$ and $\ast<0$.}
\end{equation}
Then, the structure of a noncommutative divisor is determined (up to isomorphism) by \eqref{eq:leading-order-part}, which moreover must satisfy \eqref{eq:k-fixed}. \qed
\end{lemma}
Here is another equivalent formulation, which will be convenient when considering generalizations. Take the one-dimensional vector space ${\mathbb K}[-1]$, placed in degree $1$ and weight $1$, and considered as a dg Lie algebra with trivial differential and bracket. A solution of \eqref{eq:weighted-maurer-cartan} is the same as an $L_\infty$-homomorphism
\begin{equation} \label{eq:k-map}
{\mathbb K}[-1] \longrightarrow \mathfrak{g}
\end{equation}
which preserves the weight grading (more precisely, the image of $1^{\otimes i}$ under the $i$-linear part of \eqref{eq:k-map} should be divided by $i!$ to get $\beta^i$; this is the usual action of $L_\infty$-homomorphisms on Maurer-Cartan elements \cite[Section 4.3]{kontsevich97}).
Following up on \eqref{eq:list-of-categories-1}, we have:
\begin{equation} \label{eq:curved-d-algebra}
\left\{\!\!\!\!\!\!
\parbox{35em}{
\begin{itemize}\itemsep0.5em
\item[(i)] Of course, $\EuScript B$ itself is an $A_\infty$-algebra.
\item[(ii)]
Take $\EuScript B[[u]]$, where the formal variable $u$ has degree $2$. Let
\[
\EuScript D = \EuScript A \oplus \prod_{i \geq 1} u^i\EuScript B \subset =
\EuScript A[[u]] \oplus u(\EuScript P[1])[[u]] \subset \EuScript B[[u]]
\]
be the space of $\EuScript B$-valued formal series in $u$ whose constant term lies in $\EuScript A$. Following \cite{seidel06}, we make $\EuScript D$ into a curved $A_\infty$-algebra by extending $\mu_{\EuScript B}$ $u$-linearly, and then adding a curvature term $\mu^0_{\EuScript D} = u\, e_{\EuScript A}$.
Then, $\EuScript A$ is a (right) $A_\infty$-module over $\EuScript D$, by pullback under the projection $\EuScript D \rightarrow \EuScript A$. One can then define $\EuScript A \setminus \EuScript B$ as the endomorphism ring of that module (it is important that we consider $\EuScript D$ to be defined over ${\mathbb K}$, but equipped with the $u$-adic topology). Explicitly, the underlying vector space \cite[Equation (4.7)]{seidel08} is
\[
\begin{aligned}
\EuScript A \setminus \EuScript B = & \bigoplus_{\substack{k \geq 0 \\ i_1, \dots, i_k > 0}}
\mathit{Hom}\big(\EuScript A \otimes T(\bar{\EuScript A}[1]) \otimes u^{i_k}\EuScript B[1] \otimes T(\bar{\EuScript A[1]})
\otimes \cdots
\\[-2em] & \qquad \qquad\qquad \qquad \qquad \qquad \cdots
\otimes u^{i_1}\EuScript B[1] \otimes T(\bar{\EuScript A}[1]), \EuScript A\big).
\end{aligned}
\]
The identification of this construction with a suitable categorical localization (or quotient) is provided by \cite[Theorem 4.1]{seidel08}.
\end{itemize}
}
\right.
\end{equation}
\subsection{Noncommutative pencils\label{subsec:pencil}}
We continue in the same framework as before, including the assumption that ${\mathbb K}$ has characteristic $0$. Fix a two-dimensional vector space $V$, which will be given weight grading $-1$ (and ordinary grading $0$). Denote the symmetric algebra by $\mathit{Sym}^*(V)$, and the dual vector space by $W = V^\vee$.
\begin{definition}
A noncommutative pencil on $\EuScript A$, with underlying bimodule $\EuScript P$, is a map
\begin{equation} \label{eq:wp}
\wp: T(\EuScript B[1]) \longrightarrow \EuScript B[2] \otimes \mathit{Sym}^*(V)
\end{equation}
which preserves both the ordinary grading and the one by weight, and with the following property: for any $w \in W$, we can use the associated evaluation map $\mathit{Sym}^*(V) \longrightarrow {\mathbb K}$ to specialize \eqref{eq:wp} to $\mu_{\EuScript B,w}: T(\EuScript B[1]) \longrightarrow \EuScript B[2]$; and all these should be noncommutative divisors.
\end{definition}
As before, this definition lends itself to piecewise analysis. The first term not determined by $\EuScript A$ and $\EuScript P$ has the form (after dualizing)
\begin{equation} \label{eq:w1}
W \otimes T(\EuScript A[1]) \otimes \EuScript P \otimes T(\EuScript A[1]) \longrightarrow \EuScript A.
\end{equation}
This is a family of bimodule maps $\EuScript P \rightarrow \EuScript A$, depending linearly on $W$. To make things even more concrete, suppose that we identify $V$ and $W$ with ${\mathbb K}^2$ by choosing dual bases. Then, \eqref{eq:w1} consists of two bimodule maps $\rho$ and $\sigma$, corresponding to $w = (1,0)$ resp.\ $w = (0,1)$, which we have called the sections associated to the noncommutative pencil.
Let's return to the bigraded dg Lie algebra $\mathfrak{g}$. Take $W[-1]$, as a vector space placed in degree $1$ and weight $1$, and equip it with the trivial dg Lie structure. A noncommutative pencil is the same as an $L_\infty$-homomorphism, preserving the weight grading,
\begin{equation}
W[-1] \longrightarrow \mathfrak{g}.
\end{equation}
There is a standard obstruction theory for such homomophisms, which in particular yields an analogue of Lemma \ref{th:obstructions}:
\begin{lemma} \label{th:obstructions-2}
If \eqref{eq:no-negative-degree} holds, the structure of a noncommutative pencil is determined up to isomorphism by the cohomology classes $[\rho],\, [\sigma]$ (each of which must be fixed by $K_1$). \qed
\end{lemma}
By definition, a noncommutative pencil yields a noncommutative divisor for each $w \in W$. For $w = 0$, this reduces to the trivial extension algebra $\EuScript B^{\mathit{triv}}$. Assuming as before that $W = {\mathbb K}^2$, we write $\EuScript B_\infty$ for the noncommutative divisor associated to $w = (1,0)$, and $\EuScript B_z$ for that associated to $w = (z,1)$. The associated sections are those given in \eqref{eq:list-of-categories-2}. This essentially exhausts all the possibilities, since the noncommutative divisors associated to $w$ and $\lambda w$, for $\lambda \in {\mathbb K}^\times$, are related by an automorphism of $\EuScript B$ (which acts trivially on $\EuScript A$ and rescales $\EuScript P$ by $\lambda$).
Let's spell out the other structures mentioned in \eqref{eq:list-of-categories-2}:
\begin{equation}
\left\{\!\!\!\!\!\!
\parbox{35em}{
\begin{itemize} \itemsep.5em
\item[(i)] Set $w = (1,q)$, where $q$ is a formal variable. The corresponding specialization of \eqref{eq:wp} is a map $T(\EuScript B[1]) \longrightarrow (\EuScript B[2])[[q]]$. This defines a ${\mathbb C}[[q]]$-linear $A_\infty$-structure on $\EuScript B[[q]]$, which is a deformation of $\EuScript B_\infty$. This is what we denoted by $\hat\EuScript B_{\infty}$.
\item[(ii)] Using the basis $r = (1,0)$ and $s = (0,1)$ of $V$, write $\wp^d = \sum_{i,j} \wp^d_{i,j} \otimes r^i s^j$. Because $\wp^d_{i,j}$ increases weights by $i+j$, it is nontrivial only if $d \geq i+j$. Moreover, if we extend it $u$-linearly to a map $\EuScript D^{\otimes d} \rightarrow \EuScript D$, then it is divisible by $u^{i+j}$. Hence, the expression
\[
\sum_{i,j} \wp^d_{i,j} u^{-j} h^j
\]
makes sense, as a family of maps $\EuScript D^{\otimes d} \rightarrow \EuScript D$ depending on an auxiliary formal variable $h$ of degree $2$. We add to it a curvature term as in \eqref{eq:curved-d-algebra}, and get a formal deformation of our previous curved $A_\infty$-structure on $\EuScript D$, with parameter $h$.
This induces a deformation of $\EuScript A \setminus \EuScript B_\infty$ (with a curvature term, which vanishes if we set $h = 0$). We write it as $\EuScript L\EuScript G$.
\end{itemize}
}\right.
\end{equation}
\begin{remark} \label{th:graph}
(i) A pencil in ordinary algebraic geometry has a (compactified) graph, see e.g.\ \cite[p.~154]{tibar07}. Here is one possible approach towards defining its noncommutative analogue: over the projective line $\mathbb{P}^1 = \mathbb{P}(W)$, consider the graded quasi-coherent sheaf
\begin{equation}
{\mathcal A} = \EuScript A \otimes {\mathcal O}_{\mathbb P^1} \oplus \EuScript P[1] \otimes {\mathcal O}_{\mathbb P^1}(-1).
\end{equation}
Then, \eqref{eq:wp} makes this into a sheaf of $A_\infty$-algebras. Using e.g.\ Cech resolutions, one can associate to this a single $A_\infty$-algebra, whose underlying cohomology is $H^*(\mathbb{P}^1, {\mathcal A} \otimes \mathit{End}({\mathcal E}))$ for some generator ${\mathcal E}$ of the derived category of $\mathbb{P}^1$, for instance ${\mathcal E} = {\mathcal O}_{\mathbb P^1} \oplus {\mathcal O}_{\mathbb P^1}(-1)$.
(ii)
A pencil in classical algebraic geometry also has a base locus (or axis). A first idea for its noncommutative analogue is
\begin{equation} \label{eq:katz1}
\EuScript B_0 \otimes_{\EuScript A} \EuScript B_\infty,
\end{equation}
but this is not a priori an $A_\infty$-algebra (only a bimodule over $\EuScript A$, and as such, does not use the full structure of the noncommutative pencil). However, even at this level, the notion has geometric significance: we conjecture that for, the noncommutative pencils obtained from anticanonical Lefschetz pencils as in Conjecture \ref{th:conjecture}, \eqref{eq:katz1} is trivial, meaning quasi-isomorphic to zero (this property would distinguish them from Lefschetz fibrations with singular fibre at $\infty$).
In classical algebraic geometry, (i) and (ii) are related, since (under suitable smoothness assumptions) the graph is obtained by blowing up the base locus. Noncommutative counterparts of this relation would be of interest, because they arise when considering the relation between the Fukaya category of an ample hypersurface and the wrapped Fukaya category of its complement. Concretely, in the case where \eqref{eq:katz1} is trivial, the Fukaya category of the ample hypersurface should appear as a formal completion of the wrapped Fukaya category of the complement.
\end{remark}
\subsection{Discussion of the assumptions}
Throughout this section, we have required that all $A_\infty$-structures should be ${\mathbb Z}$-graded. One can actually work with ${\mathbb Z}/2$-gradings throughout, with the same results, except for Lemmas \ref{th:obstructions} and \ref{th:obstructions-2}.
\begin{remark}
This is not quite the end of this topic, as illustrated by the intermediate situation mentioned in Remark \ref{th:fractional-cy-2}. There, $\EuScript A$ and $\EuScript P$ are ${\mathbb Z}$-graded, and so is $\wp$ provided that we take $V = {\mathbb K} \oplus {\mathbb K}[2m]$. The associated sections $\rho$ and $\sigma$ have degrees $0$ and $2m$, respectively. While $\EuScript B_\infty$ is still ${\mathbb Z}$-graded, the other $\EuScript B_z$, $z \in {\mathbb K}$, are only ${\mathbb Z}/2$-graded. The original ${\mathbb Z}$-grading leads to an isomorphism $\EuScript B_z \rightarrow \EuScript B_{\lambda^{2m}z}$, for any $\lambda \in {\mathbb K}^\times$. Similarly, the formal deformation $\hat\EuScript B_\infty$ is ${\mathbb Z}$-graded if we assign degree $-2m$ to the deformation variable $q$.
\end{remark}
The other assumption we have imposed, starting with \eqref{eq:mc-deformation-theory}, is that the coefficient field ${\mathbb K}$ has characteristic $0$. This can be dropped as well, at the cost of making some formulations a little more complicated. Maurer-Cartan theory, which means the classification theory of solutions of \eqref{eq:mc-deformation-theory}, does not apply in positive characteristic. However, a simpler obstruction theory argument suffices to prove Lemma \ref{th:obstructions} (and its variant, Lemma \ref{th:obstructions-2}), and that does not require any assumption on ${\mathbb K}$. The second case where we have used the assumption on ${\mathbb K}$ is in the definition of noncommutative pencil, which was formulated by specializing two-variable polynomials, appearing as elements of $\mathit{Sym}(V)$, to one-dimensional linear subspaces. In positive characteristic, such specializations may fail to recover the original polynomial ($f(x,y) = x^py - xy^p$ vanishes on every line, for ${\mathbb K} = {\mathbb F}_p$). Hence, the definition of noncommutative pencil would have to be reformulated as a condition on \eqref{eq:wp} itself.
Finally, there is a variation on our general setup which will be important for applications. Namely, one can replace the ground field ${\mathbb K}$ by the semisimple ring
\begin{equation} \label{eq:semisimple}
R = {\mathbb K}^m = {\mathbb K} e_1 \oplus \cdots \oplus {\mathbb K} e_m.
\end{equation}
An $A_\infty$-algebra over $R$ consists of a graded $R$-bimodule $\EuScript A$, together with $R$-bimodule maps
\begin{equation} \label{eq:r-bilinear}
\mu_{\EuScript A}^d: \EuScript A \otimes_R \EuScript A \otimes_R \cdots \otimes_R \EuScript A \longrightarrow \EuScript A[2\!-\!d].
\end{equation}
This is in fact the same as an $A_\infty$-category with objects labeled by $\{1,\dots,m\}$. The strict unit $e_{\EuScript A}$ must lie in the diagonal part of the $R$-bimodule $\EuScript A$, which means that it can be written as $e_{\EuScript A} = e_{\EuScript A,1} + \cdots + e_{\EuScript A,m}$ with $e_{\EuScript A,i} \in e_i \EuScript A e_i$. Similarly, an $\EuScript A$-bimodule consists of a graded $R$-bimodule $\EuScript P$, with structure maps that are linear over $R$ as in \eqref{eq:r-bilinear}. The same applies to the definition of morphisms in $[\EuScript A,\EuScript A]$ and of Hochschild homology. All definitions and results in this section then carry over without any other significant modifications.
\section{Hamiltonian Floer cohomology}
This section recalls the most familiar version of Floer cohomology \cite{floer88}, adapted to Liouville domains following \cite{viterbo97a}. The only non-standard aspect of the exposition is our extensive use of the BV (Batalin-Vilkovisky or loop rotation) operator \cite{seidel07, seidel-solomon10, bourgeois-oancea09b}. As a specific example, we will consider Liouville domains whose boundary is a contact circle bundle. Their Floer cohomology is very well understood in relation to Symplectic Field Theory, thanks to \cite{bourgeois-oancea09, diogo12} and ongoing work of Diogo-Lisi; but our limited needs can be met by a more elementary approach.
\subsection{Basic definitions}
We will work in the following setting, which benefits from strong exactness assumptions:
\begin{setup} \label{th:hamiltonian-setup}
(i) Let $M^{2n}$ be a Liouville domain. This means that it is a compact manifold with boundary, equipped with an exact symplectic structure $\omega_M = d\theta_M$, such that the Liouville vector field $Z_M$ dual to $\theta_M$ points strictly outwards along the boundary. The Liouville flow provides a canonical collar embedding $(-\infty,0] \times \partial M \hookrightarrow M$. The exponential of the time variable then gives a function $\rho_M$ (defined on the image of the embedding in $M$) such that
\begin{equation} \label{eq:h-function}
\left\{\begin{aligned}
& \rho_M|\partial M = 1, \\
& Z_M.\rho_M = \rho_M.
\end{aligned}
\right.
\end{equation}
We denote by $R_M$ the Hamiltonian vector field of $\rho_M$. This is the natural extension of the Reeb vector field $R_{\partial M}$ associated to the contact one-form $\theta_M|\partial M$.
(ii) During part of our argument, we will additionally assume that $M$ comes with a symplectic Calabi-Yau structure (a distinguished homotopy class of trivializations of its canonical bundle $K_M = \Lambda^n_{{\mathbb C}}(TM)^\vee$, for some compatible almost complex structure).
(iii) We consider functions $H \in C^\infty(M,{\mathbb R})$ with
\begin{equation} \label{eq:hamiltonian}
H = \epsilon \rho_M \quad \text{near $\partial M$}
\end{equation}
for some $\epsilon \in {\mathbb R}$. Throughout, it is assumed that
\begin{equation} \label{eq:no-reeb}
\text{$\epsilon R_{\partial M}$ has no $1$-periodic orbits;}
\end{equation}
equivalently, $\epsilon \neq 0$ and there are no Reeb orbits on $\partial M$ whose period is $|\epsilon|/k$, for any positive integer $k$.
(iii) We use compatible almost complex structures $J$ such that
\begin{equation} \label{eq:j-convex}
\theta_M \circ J = d\rho_M \quad \text{near $\partial M$;}
\end{equation}
equivalently, $J Z_M = R_M$.
\end{setup}
\begin{example} \label{th:contact-circle-bundle}
(i) The most important case for us is when $\partial M$ is a {\em contact circle bundle}, by which we mean that the Reeb flow is a free $S^1$-action (for concreteness, we assume that this has period $1$). In that case, $M$ can be embedded into a closed symplectic manifold $M^{\mathit{cl}}$ by attaching a disc bundle over $\partial M/S^1$. The zero-section $\partial M/S^1 \hookrightarrow M^{\mathit{cl}}$ represents a positive multiple of the symplectic class $[\omega_{M^{\mathit{cl}}}] \in H^2(M^{\mathit{cl}};{\mathbb R})$.
(ii) In the same situation as before, suppose additionally that
\begin{equation}
c_1(M^{\mathit{cl}}) = m\, [\partial M/S^1] \in H^2(M^{\mathit{cl}};{\mathbb Z}), \quad \text{for some $m \in {\mathbb Z}$.}
\end{equation}
For $m = 0$, this means that $K_{M^{\mathit{cl}}}$ is trivial; one can then choose a symplectic Calabi-Yau structure on $M^{\mathit{cl}}$, which then obviously restricts to one on $M$. For general $m$, one can still find a section of $K_{M^{\mathit{cl}}}$ which has an order $m$ ``pole'' along $\partial M/S^1$, and is nonzero elsewhere. By restricting that to $M$, one again obtains a symplectic Calabi-Yau structure.
\end{example}
In its most basic form, the Floer cohomology $\mathit{HF}^*(M,\epsilon)$ is a ${\mathbb Z}/2$-graded vector space over a field ${\mathbb K}$ of characteristic $2$, which depends on $M$ and a choice of constant $\epsilon$ satisfying \eqref{eq:no-reeb}. To define it, take a family of functions $H = (H_t)$ as in \eqref{eq:hamiltonian}, as well as a family of almost complex structures $J = (J_t)$ as in \eqref{eq:j-convex}, both parametrized by $t \in S^1 = {\mathbb R}/{\mathbb Z}$. On the free loop space $\EuScript L = C^\infty(S^1, M)$, consider the $H$-perturbed action functional:
\begin{equation} \label{eq:action}
A_H(x) = \int_{S^1} -x^*\theta_M + H_t(x(t)) \, dt.
\end{equation}
Its critical points are solutions $x \in \EuScript L$ of
\begin{equation} \label{eq:periodic-y}
\dot{x} = X_{H,t},
\end{equation}
where $X_H$ is the time-dependent Hamiltonian vector field of $H$. Such $x$ correspond bijectively to fixed points $x(1)$ of $\phi_H^1$, where $(\phi_H^t)$ is the Hamiltonian isotopy generated by $H$. By assumption on $\epsilon$, all such $x$ are contained in the interior of $M$. For a generic choice of $H$, all $x$ are nondegenerate; assume from now on that this is the case. One defines the Floer cochain space $\mathit{CF}^*(M,H)$ to be the ${\mathbb Z}/2$-graded vector space with one summand ${\mathbb K}$ for each $x$ (the ${\mathbb Z}/2$-grading encodes the local Lefschetz number). Consider solutions of Floer's gradient flow equation for \eqref{eq:action}. These ``Floer trajectories'' are maps $u: {\mathbb R} \times S^1 \rightarrow M$, satisfying
\begin{equation} \label{eq:floer}
\partial_s u + J_t (\partial_t u - X_{H,t}) = 0,
\end{equation}
with limits
\begin{equation} \label{eq:floer-limits}
\textstyle \lim_{s \rightarrow \pm\infty} u(s,\cdot) = x_{\pm}
\end{equation}
as in \eqref{eq:periodic-y}. While $M$ is not closed, a standard maximum principle argument shows that Floer trajectories can never reach $\partial M$. We consider the moduli space of solutions to \eqref{eq:floer}, \eqref{eq:floer-limits} (more precisely, solutions that are not constant in $s$, modulo translation in that direction). For generic choice of $J$, these moduli spaces are regular. One counts isolated points in them to obtain the coefficients of the differential $d$ on $\mathit{CF}^*(M,H)$, of which $\mathit{HF}^*(M,\epsilon)$ is the cohomology.
The free loop space admits an obvious circle action (loop rotation). In classical cohomology, a circle action gives rise to an operator of degree $-1$, dual to moving cycles around orbits. The analogue for Floer cohomology is the BV operator
\begin{equation} \label{eq:bv}
\Delta: \mathit{HF}^*(M,\epsilon) \longrightarrow \mathit{HF}^{*-1}(M,\epsilon).
\end{equation}
To define it, suppose that we have fixed $(H,J)$ as necessary to define the Floer cochain complex. The rotated versions, for $r \in S^1$, are
\begin{equation}
(H^{(r)}_t,J^{(r)}_t) = (H_{t-r},J_{t-r}).
\end{equation}
Similarly, if $x$ is as in \eqref{eq:periodic-y}, we write $x^{(r)}(t) = x(t-r)$. Choose a family of functions and almost complex structures $(H_{r,s,t}, J_{r,s,t})$, depending on $(r,s,t) \in S^1 \times {\mathbb R} \times S^1$, which lie in the same class as before and satisfy
\begin{equation} \label{eq:bv-data}
(H_{r,s,t}, J_{r,s,t}) = \begin{cases} (H_t^{(r)}, J_t^{(r)}) & s \leq -1, \\
(H_t,J_t) & s \geq 1.
\end{cases}
\end{equation}
One considers the parametrized moduli space of pairs $(r,u)$, consisting of $r \in S^1$ and a solution $u: {\mathbb R} \times S^1 \rightarrow M$ of the $r$-dependent equation
\begin{equation} \label{eq:bv-equation}
\partial_s u + J_{r,s,t} (\partial_t u - X_{H,r,s,t}) = 0,
\end{equation}
with limits
\begin{equation} \label{eq:bv-limits}
\left\{
\begin{aligned}
& \textstyle \lim_{s \rightarrow -\infty} \; u(s,\cdot) = x_-^{(r)}, \\
& \textstyle \lim_{s \rightarrow +\infty} \, u(s,\cdot) = x_+.
\end{aligned}
\right.
\end{equation}
The same maximum principle argument as before applies to solutions of \eqref{eq:bv-equation}, \eqref{eq:bv-limits}, ensuring that they can never reach $\partial M$. For suitably generic choices, counting isolated points in the parametrized moduli space yields a degree $-1$ chain map
\begin{equation} \label{eq:bv-chain}
\delta: \mathit{CF}^*(M,H) \longrightarrow \mathit{CF}^{*-1}(M,H),
\end{equation}
which induces \eqref{eq:bv} on cohomology. Unlike the differential $d$, the operator $\delta$ is not always compatible with the action filtration. More precisely, given a solution of \eqref{eq:bv-equation}, \eqref{eq:bv-limits}, one has
\begin{equation} \label{eq:energy-cont}
E(u) = \int_{{\mathbb R} \times S^1} |\partial_s u|^2\, \mathit{ds} \wedge \mathit{dt} \leq
A_H(x_-) - A_H(x_+) + \int_{{\mathbb R} \times S^1} \|\partial_s H_{r,s,t}\|_{\infty} \, \mathit{ds} \wedge \mathit{dt}.
\end{equation}
If $H$ was autonomous (independent of $t$) one could choose $H_{r,s,t} = H$ for all $(r,s,t)$, and then the last term in \eqref{eq:energy-cont} would vanish. In general, this interferes with the nondegeneracy of $1$-periodic orbits; but one can always choose $H$ to be a small perturbation of an autonomous function, and thereby make that term as small as needed. For such choices, $\delta$ will decrease actions only by a small amount.
\begin{remark} \label{th:signs-and-grading}
By counting Floer trajectories with appropriate signs, one can define Floer cohomology over an arbitrary coefficient field, as already pointed out in \cite{floer88}. A symplectic Calabi-Yau structure on $M$ determines a lift of the ${\mathbb Z}/2$-grading on $\mathit{HF}^*(M,\epsilon)$ to a ${\mathbb Z}$-grading \cite{salamon-zehnder92}. Since these are standard additions to the basic construction, we will continue our exposition in the simplest setup ($\mathrm{char}({\mathbb K}) = 2$ and ${\mathbb Z}/2$-gradings), but take care that all formulae remain correct when signs and ${\mathbb Z}$-gradings are added. For instance, in the ${\mathbb Z}$-graded case, the BV operator has degree $-1$, as indicated in \eqref{eq:bv}.
\end{remark}
Floer cohomology has a Poincar{\'e} type duality, obtained by reversing loops in $\EuScript L$:
\begin{equation} \label{eq:id-poincare-duality}
\mathit{HF}^*(M,-\epsilon) \cong \mathit{HF}^{2n-*}(M,\epsilon)^\vee.
\end{equation}
This comes from an isomorphism of the underlying chain complexes, assuming that the choices of $(H,J)$ have been suitably coordinated. For equally straightforward reasons, it is compatible with BV operators.
As is implicit in the notation, Floer cohomology is independent (up to canonical isomorphism) of all the auxiliary choices made in its construction, and the same holds for the BV operator. It is maybe useful to elaborate on this slightly. Suppose that we have two possible choices, giving rise to Floer complexes $\mathit{CF}^*(M,H)$ and $\mathit{CF}^*(M,\tilde{H}$), with their differentials $d$, $\tilde{d}$ and chain level BV operators $\delta$, $\tilde{\delta}$. The continuation map method \cite{salamon-zehnder92} provides maps
\begin{equation} \label{eq:cont-maps}
\begin{aligned}
& k: \mathit{CF}^*(M,H) \longrightarrow \mathit{CF}^*(M,\tilde{H}), \\
& \kappa: \mathit{CF}^*(M,H) \longrightarrow \mathit{CF}^{*-2}(M,\tilde{H}).
\end{aligned}
\end{equation}
The first one is a chain map (and actually a quasi-isomorphism), and the second one satisfies
\begin{equation} \label{eq:compatibility-with-delta}
\tilde{d} \kappa - \kappa d = \tilde{\delta} k - k \delta.
\end{equation}
Moreover, the maps \eqref{eq:cont-maps} are themselves unique in a suitable homotopical sense.
\subsection{Reeb orbits}
We now consider the dependence on $\epsilon$. This usually appears as part of the construction of symplectic cohomology (in the sense of \cite{viterbo97a}).
\begin{lemma} \label{th:no-wall}
Suppose that all $\epsilon \in [\epsilon_-,\epsilon_+]$ satisfy \eqref{eq:no-reeb}. Then we have an isomorphism, compatible with BV operators,
\begin{equation}
\mathit{HF}^*(M,\epsilon_-) \cong \mathit{HF}^*(M,\epsilon_+).
\end{equation}
\end{lemma}
\begin{proof}
Even though that is not absolutely necessary, we find it convenient to introduce finite enlargements of our Liouville domain. Such an enlargement, by an amount $C>1$, is
\begin{equation} \label{eq:attach-cone}
\hat{M} = M \cup_{\partial M} ([1,C] \times \partial M).
\end{equation}
The conical part which we have added carries the one-form $\theta_{\hat{M}} = r (\theta_M|\partial M)$ ($r$ being the coordinate in $[1,C]$). One extends $\rho_M$ to $\hat{M}$ by setting $\rho_{\hat{M}}(r,x) = r$. Choose a function
\begin{equation} \label{eq:turn}
\left\{
\begin{aligned}
& h_+: (0,1] \longrightarrow {\mathbb R}, \\
& h_+(a) = \epsilon_- a \quad \text{for $a$ sufficiently small,} \\
& h_+'(a) = \epsilon_+ \quad \text{for $a$ close to $1$,} \\
& h_+''(a) \geq 0 \quad \text{everywhere}, \\
& h_+''(a) > 0 \quad \text{for all $a$ such that $\epsilon_- < h_+'(a) < \epsilon_+$.}
\end{aligned}
\right.
\end{equation}
Given the $H_-$ used to define $\mathit{HF}^*(M,\epsilon_-)$, we extend it to a time-dependent function $\hat{H}_+$ on $\hat{M}$ by setting
\begin{equation} \label{eq:extend-hamiltonian}
\hat{H}_{+,t}(r,x) =C\, h_+(C^{-1}r).
\end{equation}
This makes sense provided that $C$ is large (so that $h_+(a) = \epsilon_- a$ near $a = C^{-1}$). This extension does not quite belong to the class \eqref{eq:hamiltonian} for the manifold $\hat{M}$ and constant $\epsilon_+$, but that could be remedied by adding a constant, which has no effect on our construction. Hence, $\hat{H}_+$ can be used to define $\mathit{HF}^*(\hat{M},\epsilon_+)$.
Note that on $[1,C] \times \partial M$,
\begin{equation}
X_{\hat{H}_+,t} = h'_+(C^{-1}r)\, R_{\partial M}.
\end{equation}
Because of the assumption, it follows that all $1$-periodic orbits of \eqref{eq:extend-hamiltonian} are actually contained in $M$. A general almost complex structure on $M$ does not naturally extend to $\hat{M}$, so there we proceed in the other direction: we choose $\hat{J} = (\hat{J}_t)$ on $\hat{M}$ which satisfy the analogue of \eqref{eq:j-convex} in a neighbourhood of $[1,C] \times \partial M$, and then take $J = (J_t)$ to be the restriction to $M$. A maximum principle argument shows that Floer trajectories in $\hat{M}$ are in fact all contained in $M$. As a consequence, we get an isomorphism (of complexes, and hence of Floer cohomology groups)
\begin{equation}
\mathit{HF}^*(\hat{M},\epsilon_+) \cong \mathit{HF}^*(M,\epsilon_-).
\end{equation}
For similar reasons, this is compatible with the BV operators. A repetition of the same argument, with a linear function instead of $h_+$, shows that
\begin{equation}
\mathit{HF}^*(\hat{M},\epsilon_+) \cong \mathit{HF}^*(M,\epsilon_+).
\end{equation}
Combining the two isomorphisms yields the desired result.
\end{proof}
\begin{lemma} \label{th:wall}
Take $\epsilon_- < \epsilon_+$, both of which satisfy \eqref{eq:no-reeb}, and suppose that there is exactly one $\epsilon \in (\epsilon_-,\epsilon_+)$ for which \eqref{eq:no-reeb} fails. Then there is a long exact sequence
\begin{equation} \label{eq:les-f}
\cdots \rightarrow \mathit{HF}^*(M,\epsilon_-) \longrightarrow \mathit{HF}^*(M,\epsilon_+) \longrightarrow H^*(Q) \rightarrow \cdots
\end{equation}
where $H^*(Q)$ depends only on the local geometry near the $1$-periodic orbits of $\epsilon R_{\partial M}$. Moreover, $H^*(Q)$ carries an endomorphism of degree $-1$, with the same locality property, and which fits in with \eqref{eq:les-f} and the BV operators on $\mathit{HF}^*(M,\epsilon_{\pm})$.
\end{lemma}
\begin{proof}
Let's use the same extension $\hat{H}_+$ as in the proof of Lemma \ref{th:no-wall}. This time, there are additional $1$-periodic orbits in the conical part. Their actions are
\begin{equation} \label{eq:h-hat-x-action}
C (h_+(a_*)-ah'_+(a_*)), \quad \text{for the unique $a_*$ such that $h'_+(a_*) = \epsilon$.}
\end{equation}
The convexity properties of $h_+$ ensure that $h_+(a_*) - a_*h'_+(a_*) < 0$. Therefore, \eqref{eq:h-hat-x-action} goes to $-\infty$ as we make $C$ large. In particular, we can choose $C$ so that \eqref{eq:h-hat-x-action} is smaller than the actions of any $1$-periodic orbits of $H_-$. The $1$-periodic orbits lying in the conical part will be degenerate, but one can remedy that by a small time-dependent perturbation of $\hat{H}_+$ (concentrated near those orbits), and the statement about actions will continue to hold. Hence, the $1$-periodic orbits lying in $M$ form a subcomplex of $\mathit{CF}^*(\hat{M},\hat{H}_+)$. Using the integrated maximum principle (see \cite{abouzaid-seidel07}), one can show that the differential on this subcomplex agrees with the Floer differential in $M$ (alternatively, one can arrive at the same conclusion by taking $C$ very large, and using the Monotonicity Lemma; or by the standard maximum principle and a Gromov compactness argument, applied to the limit where the perturbation of $\hat{H}_+$ goes to $0$). In other words, one has an inclusion of chain complexes
\begin{equation} \label{eq:small-h-complex}
\mathit{CF}^*(M,H_-) \subset \mathit{CF}^*(\hat{M},\hat{H}_+).
\end{equation}
Defining $Q^*$ to be the quotient, one obviously gets a long exact sequence
\begin{equation} \label{eq:reeb-les}
\cdots \rightarrow \mathit{HF}^*(M,\epsilon_-) \longrightarrow \mathit{HF}^*(\hat{M},\epsilon_+) \longrightarrow H^*(Q) \rightarrow \cdots
\end{equation}
In the original situation \eqref{eq:h-hat-x-action}, all $1$-periodic orbits in the conical part had the same action. The perturbation will destroy that, but only by a small amount. Hence, $H^*(Q)$ is still a local Floer cohomology group, in the sense of \cite{pozniak, ginzburg06b}. As in Lemma \ref{th:no-wall}, one can replace $\mathit{HF}^*(\hat{M},\epsilon_+)$ by $\mathit{HF}^*(M,\epsilon_+)$ in \eqref{eq:reeb-les}, which yields \eqref{eq:les-f}.
The corresponding statements concerning BV operators are slightly more tricky. Let's choose all our Hamiltonians to be small perturbations of autonomous ones. In that case, it follows from \eqref{eq:h-hat-x-action} and \eqref{eq:energy-cont} that the chain level BV map on $\mathit{CF}^*(\hat{M},\hat{H}_+)$ will preserve the subcomplex \eqref{eq:small-h-complex}. A Gromov compactness argument (where one decreases the size of the perturbations, as briefly mentioned before) implies that the induced endomorphism of $Q^*$ is again local in nature. To show that its restriction to $\mathit{CF}^*(M,H_-)$ counts only solutions lying in $M$, one combines the same idea of Gromov compactness with the methods we've used for the differential (integrated maximum principle, or Monotonicity Lemma).
\end{proof}
\begin{lemma} \label{th:wall-2}
In the situation of Lemma \ref{th:wall}, assume additionally that the $1$-periodic orbits of $\epsilon R_{\partial M}$ form a Morse-Bott nondegenerate connected manifold $F$. Then,
\begin{equation}
H^*(Q) \cong H^{*+k}(F;\xi),
\end{equation}
for some $k$ and local ${\mathbb K}$-coefficient system $\xi \rightarrow F$ (with holonomy $\pm 1$). That coefficient system is equivariant with respect to the $S^1$-action given by loop rotation on $F$. Hence, $H^*(F;\xi)$ comes with an endomorphism of degree $-1$, which fits in with the BV operators and \eqref{eq:les-f}.
\end{lemma}
The connectedness assumption is for notational convenience only (otherwise, each component of $F$ comes with its own index offset $k$, and one has to take the direct sum of their contributions).
\begin{example}
Suppose that $\epsilon R_{\partial M}$ has exactly one $1$-periodic orbit (up to loop rotation), which moreover is transversally nondegenerate. This means that $F \cong S^1$. The local coefficient system $\xi$ is trivial if the orbit is ``good'', and has holonomy $-1$ if it is ``bad'' (in the terminology from Symplectic Field Theory; for an explanation in a framework close to ours, see \cite[\S 5]{bourgeois-mohnke04} or \cite[\S 4.4]{bourgeois-oancea09c}).
\end{example}
Lemma \ref{th:wall-2} can be proved by using the Morse-Bott formalism for Floer cohomology \cite{austin-braam95, bourgeois02}. While we will not give a complete proof, it makes sense to describe that formalism in the appropriate form, since we'll return to it later on.
We start with a given $H_-$ and extend that to $\hat{H}_+$ as in \eqref{eq:extend-hamiltonian}, but do not perturb this extension further. As usual, $C$ is assumed to be large, to get the necessary action inequalities. Fix a Morse-Smale pair on $F$, consisting of a Morse function $f$ and a metric $g$. The Morse-Bott-Floer complex $\mathit{CF}^*(\hat{M},\hat{H}_+,f)$ has two kinds of generators: ones coming from the $1$-periodic orbits in $M$, and ones coming from the critical points of $f$ (the constant $k$ and local coefficient system $\xi$ appear when we take gradings and signs into account). Let's write this as
\begin{equation} \label{eq:morse-bott-c}
\mathit{CF}^*(\hat{M},\hat{H}_+,f) = \mathit{CF}^*(M,H_-) \oplus \mathit{CM}^{*+k}(f;\xi).
\end{equation}
The differential on \eqref{eq:morse-bott-c} is of the form
\begin{equation} \label{eq:morse-bott-d}
d = \begin{pmatrix}
d_{\mathit{Floer}} & d_{\mathit{mixed}} \\
0 & d_{\mathit{Morse}}
\end{pmatrix},
\end{equation}
where the various parts are defined as follows:
\begin{itemize} \itemsep.5em
\item[(i)] If $x_-$ and $x_+$ are $1$-periodic orbits in $M$, one counts Floer trajectories as before, and this yields $d_{\mathit{Floer}}$.
\item[(ii)] If $x_-$ and $x_+$ are both critical points of $f$, one counts trajectories of $-\nabla_g f$ going from $x_-$ to $x_+$, which yields $d_{\mathit{Morse}}$.
\item[(iii)] Suppose that $x_-$ lies in $M$, and $x_+$ is a critical point of $f$. In that case, one considers a mixed moduli space consisting of pairs $(u,v)$, where $u$ is a Floer trajectory, and $v: [0,\infty) \rightarrow F$ is a positive half flow line of $-\nabla_g f$, with
\begin{equation}
\left\{
\begin{aligned}
& \textstyle
\lim_{s \rightarrow -\infty} u(s,\cdot) = x_-, \\ & \textstyle
\lim_{s \rightarrow +\infty} u(s,\cdot) = v(0), \\ & \textstyle
\lim_{s \rightarrow +\infty} v(s) = x_+.
\end{aligned}
\right.
\label{eq:matching-conditions}
\end{equation}
Counting solutions of this equation yields $d_{\mathit{mixed}}$.
\end{itemize}
Assume that the original $H_-$ was a small perturbation of an autonomous Hamiltonian. We then choose a family of Hamiltonians $H_{-,r,s,t}$ on $M$ as in \eqref{eq:bv-data}, which are perturbations of the same autonomous Hamiltonian, and extend them to $\hat{H}_{+,r,s,t}$ on $\hat{M}$ as in \eqref{eq:extend-hamiltonian}. The main property, established by similar arguments as in the proof of Lemma \ref{th:wall}, is that any solution of \eqref{eq:bv-equation} whose left-hand limit $x_-^{(r)}$ lies in $F$ must be stationary, which means $u(s,t) = x_-^{(r)}(t) = x_+(t)$. Then, the Morse-Bott version of \eqref{eq:bv-chain} takes on a form parallel to \eqref{eq:morse-bott-d}:
\begin{equation} \label{eq:morse-bott-delta}
\delta = \begin{pmatrix}
\delta_{\mathit{Floer}} & \delta_{\mathit{left}} + \delta_{\mathit{right}} \\
0 & \delta_{\mathit{Morse}}
\end{pmatrix}.
\end{equation}
The definition is a more elaborate version of the previous one:
\begin{itemize} \itemsep.5em
\item[(i)] $\delta_{\mathit{Floer}}$ is defined by considering solutions of \eqref{eq:bv-equation} whose limits both lie in $M$.
\item[(ii)]
Loop rotation restricts to an $S^1$-action on $F$. Denote by $f^{(r)}$ and $g^{(r)}$ the images of $f$ and $g$ under that action, for time $r$. Choose generic families $f_{r,s}$ and $g_{r,s}$, depending on $(r,s) \in S^1 \times {\mathbb R}$, which satisfy the analogue of \eqref{eq:bv-data}:
\begin{equation}
(f_{r,s},g_{r,s}) = \begin{cases} (f^{(r)},g^{(r)}) & s \leq -1, \\ (f,g) & s \geq 1. \end{cases}
\end{equation}
The counterpart of \eqref{eq:bv-equation}, \eqref{eq:bv-limits} is an equation for $r \in S^1$ and $v: {\mathbb R} \rightarrow F$:
\begin{equation} \label{eq:morse-bv}
\left\{
\begin{aligned}
& \partial_s v + \nabla_{g_{r,s}} f_{r,s} = 0, \\
& \textstyle \lim_{s \rightarrow -\infty} v(s) = x_-^{(r)}, \\
& \textstyle \lim_{s \rightarrow +\infty} v(s) = x_+,
\end{aligned}
\right.
\end{equation}
where $x_\pm$ are critical points of $f$. By counting isolated solutions of this equation, one defines $\delta_{\mathit{Morse}}$.
\item[(iii)]
Suppose that $x_-$ lies in $M$, and $x_+$ is a critical point of $f$. Consider triples $(r,u,v)$, where $(r,u)$ is a solution of the same parametrized equation as in (i), and $v$ is a half flow line of $-\nabla_g f$, with asymptotic conditions
\begin{equation}
\left\{\begin{aligned}
& \textstyle
\lim_{s \rightarrow -\infty} u(s,\cdot) = x_-^{(r)}, \\
& \textstyle
\lim_{s \rightarrow +\infty} u(s,\cdot) = v(0), \\
& \textstyle
\lim_{s \rightarrow +\infty} v(s) = x_+.
\label{eq:matching-conditions-2}
\end{aligned}
\right.
\end{equation}
This defines $\delta_{\mathit{left}}$, which satisfies
\begin{equation} \label{eq:delta-left}
\delta_{\mathit{left}} d_{\mathit{Morse}} + \delta_{\mathit{Floer}} d_{\mathit{mixed}} + d_{\mathit{Floer}} \delta_{\mathit{left}} + b = 0.
\end{equation}
The two last terms in \eqref{eq:delta-left} correspond to limits where a Floer trajectory ``breaks off'' from a sequence of solutions $u$ on the left hand side $s \ll 0$. If the limit of the $u$ themselves is non-stationary, such broken solutions are accounted for by $d_{\mathit{Floer}} \delta_{\mathit{left}}$ in the standard way. The other term $b$ counts the remaining limiting configurations. It is defined using a parametrized moduli space of triples $(r,u,v)$, where: $u$ is a solution of Floer's equation; $v$ is a half flow line; and
\begin{equation} \label{eq:matching-conditions-3}
\left\{\begin{aligned}
& \textstyle
\lim_{s \rightarrow -\infty} u(s,\cdot) = x_-, \\
& \textstyle
\lim_{s \rightarrow +\infty} u(s,\cdot) = v(0)^{(-r)}, \\
& \textstyle
\lim_{s \rightarrow +\infty} v(s) = x_+.
\end{aligned}
\right.
\end{equation}
\item[(iv)]
For $x_{\pm}$ as in (iii), we consider an equation for pairs $(u,v)$ with additional parameters $(q,r) \in [0,\infty) \times S^1$. Asymptotic conditions are as in \eqref{eq:matching-conditions-3}, and $u$ is always a solution of Floer's equation. The $q$-dependence lies entirely in the equation satisfied by $v$. For $q \gg 0$, that equation should be that which defines $\delta_{\mathit{Morse}}$, but applied to the positive half-line only, and with the $s$-parameter shifted by $q$:
\begin{equation} \label{eq:morse-bv-shifted}
\partial_s v + \nabla_{g_{r,s-q}} f_{r,s-q} = 0.
\end{equation}
In contrast, for $q = 0$ the equation should be the gradient flow equation for $-\nabla_g f$ (and one connects those two behaviours by choosing some intermediate data). The outcome is that
\begin{equation} \label{eq:delta-right}
d_{\mathit{Floer}} \delta_{\mathit{right}} + \delta_{\mathit{right}} d_{\mathit{Morse}} + d_{\mathit{mixed}} \delta_{\mathit{Morse}} - b = 0,
\end{equation}
where the last two terms correspond to $q \rightarrow \infty$ and $q = 0$, respectively.
\end{itemize}
Lemma \ref{th:wall-2} follows directly once one establishes that the Morse-Bott approach yields the same Floer cohomology (and BV operator) as the original definition. There are several ways of doing that, the most natural one being a suitable generalization of continuation maps; but we will not explain the details.
\subsection{Autonomous Hamiltonians}
The relation between Floer cohomology and ordinary cohomology is established by the following classical result:
\begin{proposition} \label{th:bv-vanish}
For sufficiently small $\epsilon > 0$, we have $\mathit{HF}^*(M,\epsilon) \cong H^*(M)$. Moreover, the BV operator $\Delta$ vanishes.
\end{proposition}
The most natural proof is a version of \cite{piunikhin-salamon-schwarz94}, but here we choose instead to follow \cite{floer-hofer-salamon94, hofer-salamon95}, which is more elementary. Take a time-independent $H$ and $J$, such that $H$ is Morse and the metric associated to $J$ is Morse-Smale. Then, after possibly multiplying $H$ with a small positive constant, we have:
\begin{equation} \label{eq:time-independent}
\left\{\!\!\!\!\!\! \parbox{35em}{
\begin{itemize} \itemsep0.5em
\item[(i)]
all one-periodic orbits of $X_H$ are constant (at the critical points of $H$);
\item[(ii)]
any solution of Floer's equation is constant in $t$, hence a negative gradient flow line for $H$ \cite[Lemma 7.1]{hofer-salamon95};
\item[(iii)]
for the linearization of Floer's equation (at a gradient flow line), all solutions are also constant in $t$ \cite[Proposition 4.2]{salamon-zehnder92}; together with an easy index computation, this shows that the moduli spaces are all regular.
\end{itemize}
}
\right.
\end{equation}
Hence, $\mathit{CF}^*(M,H)$ is isomorphic to the Morse complex of $H$ (this continues to hold when signs are taken into account). Moreover, when defining the BV operator, one may choose $(H_{r,s,t},J_{r,s,t}) = (H,J)$, in which case there are no isolated solutions of \eqref{eq:bv-equation}, since $r$ can be changed freely. Hence, $\delta$ itself vanishes.
We have proved that the BV operator is zero on the chain level, for a specific kind of Hamiltonian and almost complex structure. One can be slightly more precise about the meaning of this vanishing result; for that, it is convenient to introduce some simple algebraic language.
\begin{setup} \label{th:null}
(i) Suppose that we have a chain complex $(C^*,d)$ together with a chain endomorphism $\delta$ of degree $-1$. Recall that a nullhomotopy for $\delta$ is a map $\chi: C^* \rightarrow C^{*-2}$ satisfying $d\chi - \chi d = \delta$. Two nullhomotopies are called equivalent if the difference between them is a nullhomotopic chain map of degree $-2$. Equivalence classes are called {\em homotopy trivializations} of $\delta$.
(ii) Take two chain complexes $(C,d)$ and $(\tilde{C},\tilde{d})$, with endomorphisms $\delta$ and $\tilde\delta$. Suppose that we have maps
\begin{equation}
\begin{aligned}
& k: C^* \longrightarrow \tilde{C}^*, \\
& \kappa: C^* \longrightarrow \tilde{C}^{*-2},
\end{aligned}
\end{equation}
of which the first one is a chain map, and the second satisfies \eqref{eq:compatibility-with-delta}. Given homotopy trivializations $[\chi]$ and $[\tilde{\chi}]$ on our complexes, we say that they are {\em compatible with $(k,\kappa)$} if there is a map
\begin{equation} \label{eq:ggg}
g: C^* \longrightarrow \tilde{C}^{*-3}
\end{equation}
such that
\begin{equation}
\tilde{\chi} k - k \chi - \kappa = \tilde{d} g + g d.
\end{equation}
It is easy to see that compatibility depends only on the equivalence classes of $\chi$ and $\tilde{\chi}$. Moreover, if $k$ is a quasi-isomorphism, every homotopy trivialization on one complex determines a unique compatible homotopy trivialization on the other one.
\end{setup}
In this terminology, the desired statement is that the chain complex underlying $\mathit{HF}^*(M,\epsilon)$ (still in the situation of Proposition \ref{th:bv-vanish}, but now with $(H,J)$ arbitrary) comes with a canonical homotopy trivialization of the BV operator, which is compatible with \eqref{eq:cont-maps}. To prove that, one takes two choices $(H,J)$ and $(\tilde{H},\tilde{J})$ to which the proof of Proposition \ref{th:bv-vanish} applies. The associated BV operators vanish on the chain level, hence have trivial nullhomotopies. From \eqref{eq:compatibility-with-delta}, one gets a degree $-2$ chain map between these Floer complexes (itself canonical up to chain homotopy). One can show that, within a slightly more precisely defined class of Hamiltonians and almost complex structures, those chain maps are always nullhomotopic. Nullhomotopies for them provide the desired maps \eqref{eq:ggg}, which show compatibility of the homotopy trivializations. After that, one uses continuation maps to extend to arbitrary $(H,J)$. We omit the details.
\begin{proposition} \label{th:bv-vanish-2}
Suppose that $\partial M$ is a contact circle bundle. Take $\epsilon \in (1,2)$. Then there is a long exact sequence (of ${\mathbb Z}/2$-graded spaces)
\begin{equation} \label{eq:morse-bott-sequence}
\cdots \rightarrow H^*(M) \longrightarrow \mathit{HF}^*(M,\epsilon) \longrightarrow H^*(\partial M) \rightarrow \cdots
\end{equation}
The connecting map in \eqref{eq:morse-bott-sequence}, $\mathit{H}^*(\partial M) \rightarrow H^{*+1}(M)$, vanishes on $H^0(\partial M)$. Moreover, the BV operator vanishes on the preimage of $H^0(\partial M)$ inside $\mathit{HF}^*(M,\epsilon)$. Finally, suppose that $M$ carries a symplectic Calabi-Yau structure constructed as in Example \ref{th:contact-circle-bundle}(ii). Then, \eqref{eq:morse-bott-sequence} becomes ${\mathbb Z}$-graded if we take the rightmost term to be $H^{*+2m-2}(\partial M)$.
\end{proposition}
The existence of such an exact sequence is a special case of Lemma \ref{th:wall-2}, combined with Proposition \ref{th:bv-vanish}; what's new are the additional properties (of the connecting map and BV operator).
\begin{proof}
Choose $\epsilon_- > 0$ small, and define $\mathit{HF}^*(M,\epsilon_-)$ using time-independent $H_-$ (and the same for the almost complex structure), as in Proposition \ref{th:bv-vanish}. For $\epsilon_+ = \epsilon \in (1,2)$, extend $H_-$ to $\hat{H}_+$ as in \eqref{eq:extend-hamiltonian}. This enlargement produces a Morse-Bott-nondegenerate manifold of $1$-periodic orbits $F$, which is a copy of $\partial M$ (and one can check that it carries a trivial local system $\xi$). Floer trajectories whose limits $x_{\pm}$ lie in $M$ remain entirely within that subset, and have the same description as in Proposition \ref{th:bv-vanish}. The remaining point is to ensure that Floer trajectories with mixed limits ($x_-$ lies in $M$, and $x_+$ in $F$) can be made regular. Because $x_+$ is a simple periodic orbit, such trajectories are themselves simple, in the sense of \cite[p.~279]{floer-hofer-salamon94}. Then, \cite[Theorem 7.4]{floer-hofer-salamon94} ensures that one can choose $J$ (by varying it near $x_-$) so that regularity holds generically. Three technical remarks are appropriate. First, transversality theory needs to be carried out in an analytic formalism suitable for the Morse-Bott case. Secondly, while the results in \cite{floer-hofer-salamon94} are stated for trajectories with both limits being constant orbits, only one such limit is actually necessary for the argument to go through (and, since the constant orbits are still nondegenerate in our situation, their treatment does not need to modified). Finally, we need a technical condition on the Hessian of $H_-$ at its critical points \cite[Definition 7.1]{floer-hofer-salamon94}, but it is unproblematic to arrange that it holds.
Given that, we can use this same almost complex structure everywhere in the definition of \eqref{eq:morse-bott-d} and \eqref{eq:morse-bott-delta}. Then, as in Proposition \ref{th:bv-vanish} (which means using the fact that $r$ is a free parameter), we get
\begin{align} \label{eq:1-vanish}
& \delta_{\mathit{Floer}} = 0, \\
& \delta_{\mathit{left}} = 0.
\end{align}
Now let $x_+$ be a local minimum (Morse index $0$ critical point) of $f$, so that for degree reasons,
\begin{equation}
\delta_{\mathit{Morse}}(x_+) = 0.
\end{equation}
There is an open subset of $F$ consisting of points $v(0)$ which flow to $x_+$ under $-\nabla_g f$. The Floer trajectories $u$ such that $\lim_{s \rightarrow +\infty} u(s,\cdot)$ lies in that open subset form a space of dimension $\geq 1$ (since one can rotate them a little in the $t$-variable). The same applies if one replaces the gradient flow equation by \eqref{eq:morse-bv-shifted}. As a consequence,
\begin{align}
& d_{\mathit{mixed}}(x_+) = 0, \\
& \delta_{\mathit{right}}(x_+) = 0. \label{eq:5-vanish}
\end{align}
The desired properties of \eqref{eq:morse-bott-sequence} and $\Delta$ follow directly from this. The observation about ${\mathbb Z}$-gradings is a standard Conley-Zehnder index computation.
\end{proof}
As before, it can be useful to encode this observation in a more abstract framework, which also allows one to formulate a uniqueness property.
\begin{setup} \label{th:truncation}
(i) In the situation of Setup \ref{th:null}(i), fix some integer $j$. Let $C^{\leq j} \subset C^*$ be the subcomplex consisting of all cocycles of degree $j$, together with all cochains of degree $<j$. Note that this is automatically preserved by $\delta$. A {\em homotopy trivialization of $\delta$ in degrees $\leq j$} is a homotopy trivialization of $\delta^{\leq j} = \delta|C^{\leq j}$, in the previously defined sense. The existence of such a trivialization implies that $[\delta]: H^*(C) \rightarrow H^{*-1}(C)$ vanishes in degrees $\leq j$, since the map $H^*(C^{\leq j}) \rightarrow H^*(C)$ is an isomorphism in those degrees.
(ii) In the situation of Setup \ref{th:null}(ii), suppose that we have homotopy trivializations of $C$ and $\tilde{C}$ in degrees $\leq j$. We say that these are compatible if there exists a map \eqref{eq:ggg} defined on $C^{\leq j}$, with the same properties as before.
\end{setup}
Suppose that we are in the situation of Proposition \ref{th:bv-vanish-2}, with a symplectic Calabi-Yau structure constructed as in Example \ref{th:contact-circle-bundle}(ii). Then, the proof of that Proposition yields (for a very special choice of Morse-Bott complex underlying Floer cohomology) a homotopy trivialization of the BV operator in degrees $\leq 2m-2$. The relevant uniqueness result would say that this is compatible with continuation maps. Proving this requires an analogue of Proposition \ref{th:bv-vanish-2} for continuation maps, which we will not explain here.
\begin{remark}
The statement of Proposition \ref{th:bv-vanish-2} is actually not optimal. One can show that the connecting map vanishes on $\mathit{im}(H^*(\partial M/S^1) \rightarrow H^*(\partial M))$, and that the BV operator is zero on the preimage of that subspace inside $\mathit{HF}^*(M,\epsilon)$. To do that, one has to use a chain level model for $H^*(\partial M)$ which is more closely adapted to the circle action than what we've done (one possibility is to perturb the Reeb flow on $\partial M$ so that the one-periodic orbits become transversally nondegenerate).
Alternatively, one can use the $S^1$-equivariant version of Hamiltonian Floer cohomology (see e.g.\ \cite{seidel07}), which is a ${\mathbb K}[[u]]$-module $\mathit{HF}^*_{S^1}(M,\epsilon)$. In the situation of Proposition \ref{th:bv-vanish-2}, it fits into a long exact sequence of such modules,
\begin{equation} \label{eq:s1-equi-les}
\cdots \rightarrow H^*(M)[[u]] \longrightarrow \mathit{HF}^*_{S^1}(M,\epsilon) \longrightarrow H^*(\partial M/S^1) \rightarrow \cdots
\end{equation}
where $u$ acts on $H^*(\partial M/S^1)$ by cup product with the first Chern class of the circle bundle $\partial M \rightarrow \partial M/S^1$. Since that action is nilpotent, the boundary operator of \eqref{eq:s1-equi-les} is necessarily zero. Now, \eqref{eq:s1-equi-les} and \eqref{eq:morse-bott-sequence} fit into a commutative diagram whose rows and columns are long exact sequences:
\begin{equation}
\xymatrix{
&
\vdots
\ar[d]^-{\text{zero}}
&
\vdots
\ar[d]
&
\vdots
\ar[d]
&
\\
\cdots
\ar[r]^-{\text{zero}}
&
H^*(M)[[u]]
\ar[r] \ar[d]^-{u}
&
\mathit{HF}^*_{S^1}(M,\epsilon)
\ar[r] \ar[d]^-{u}
&
H^*(\partial M/S^1)
\ar[r]^-{\text{zero}} \ar[d]^-{u}
&
\cdots
\\
\cdots
\ar[r]^-{\text{zero}}
&
H^*(M)[[u]]
\ar[r] \ar[d]
&
\mathit{HF}^*_{S^1}(M,\epsilon)
\ar[r] \ar[d]
&
H^*(\partial M/S^1)
\ar[r]^-{\text{zero}} \ar[d]
&
\cdots
\\
\cdots
\ar[r]
&
H^*(M)
\ar[r]
\ar[d]^-{\text{zero}}
&
\mathit{HF}^*(M,\epsilon)
\ar[r]
\ar[d]
&
H^*(\partial M)
\ar[r]
\ar[d]
&
\cdots
\\
&
\vdots
&
\vdots
&
\vdots
&
}
\end{equation}
The right hand column is the standard Gysin sequence; and the BV operator is the composition of two vertical arrows (going from $\mathit{HF}^*(M,\epsilon)$ to $\mathit{HF}^*_{S^1}(M,\epsilon)$ and then back). Diagram-chasing yields the desired result: if a class in $\mathit{HF}^*(M,\epsilon)$ has the property that its image in $H^*(\partial M)$ is the pullback of a class in $H^*(\partial M/S^1)$, then it necessarily comes from a class in $\mathit{HF}^*_{S^1}(M,\epsilon)$, hence is killed by the BV operator.
\end{remark}
\section{Fixed point Floer cohomology}
The generalisation of Floer cohomology from Hamiltonian to general symplectic automorphisms was introduced in \cite{dostoglou-salamon93, dostoglou-salamon94}. Here, we use a version for Liouville domains, as in \cite[Section 4]{seidel00b}, \cite{mclean12}, or \cite{uljarevic14}.
\subsection{Definition}
For $M$ as before, we will consider the following situation.
\begin{setup} \label{th:auto-setup}
(i) Let $\phi$ be a symplectic automorphism of $M$ which is equal to the identity near $\partial M$ and exact. The latter condition means that there is a function $G_\phi$ (necessarily locally constant near $\partial M$) such that $\phi^*\theta_M - \theta_M = dG_\phi$.
(ii) If $M$ carries a symplectic Calabi-Yau structure, we will assume that $\phi$ is compatible with it, and in fact comes with a choice of grading (making it a graded symplectic automorphism \cite{seidel99}).
\end{setup}
\begin{example} \label{th:boundary-twist}
Suppose that $\partial M$ is a contact circle bundle. Fix a function $F$ on $M$ which agrees with $\rho_M$ near the boundary, and let $(\phi^t_F)$ be its flow. By construction, $\phi_F^{-1}$ is the identity near the boundary. We call it the {\em boundary twist} of $M$ (see \cite[Section 4]{seidel99} for a general discussion of such symplectic automorphisms), and denote it by $\tau_{\partial M}$.
Let's now assume that $M$ carries a symplectic Calabi-Yau structure as in Example \ref{th:contact-circle-bundle}(ii). Then, $\tau_{\partial M}$ can be equipped with the structure of a graded symplectic automorphism. In fact, there are two reasonable choices (which coincide for $m = 1$):
(i) One can take the trivial grading of the identity, and extend it continuously over the isotopy $(\phi_F^t)$ to get a grading of $\tau_{\partial M}$. Near the boundary, that grading is a shift $[2-2m]$.
(ii) Alternatively, by changing the previous grading by a constant $2-2m$, one can get the unique grading of $\tau_{\partial M}$ which is trivial near the boundary.
\end{example}
Take families of functions $H = (H_t)$ and $J = (J_t)$, parametrized by $t \in {\mathbb R}$. Each of those should be as in \eqref{eq:hamiltonian} and \eqref{eq:j-convex}, but now with $\phi$-twisted periodicity properties:
\begin{equation} \label{eq:phi-periodicity}
\left\{
\begin{aligned}
& H_{t+1}(x) = H_t(\phi(x)), \\
& J_{t+1} = \phi^*J_t.
\end{aligned}
\right.
\end{equation}
Take the space $\EuScript L_\phi = \{x: {\mathbb R} \rightarrow M \,:\, x(t) = \phi(x(t+1))\}$, with action functional
\begin{equation} \label{eq:phi-action}
A_{\phi,H}(x) = \Big( \int_0^1 -x^*\theta_M + H_t(x(t)) \, dt \Big) - G_\phi(x(1)).
\end{equation}
Its critical points are solutions $x \in \EuScript L_\phi$ of \eqref{eq:periodic-y}, and correspond bijectively to fixed points $x(1)$ of $\phi_H^1 \circ \phi$. For a generic choice of $H$, these will be nondegenerate, and we use them as generators of a ${\mathbb Z}/2$-graded ${\mathbb K}$-vector space $\mathit{CF}^*(\phi,H)$. To define the differential, one considers solutions $u: {\mathbb R}^2 \rightarrow M$ of \eqref{eq:floer}, but now satisfying
\begin{equation} \label{eq:floer-2}
u(s,t) = \phi(u(s,t+1)). \\
\end{equation}
The resulting Floer cohomology is denoted by $\mathit{HF}^*(\phi,\epsilon)$, with the previous $\mathit{HF}^*(M,\epsilon)$ being the special case $\phi = \mathit{id}_M$. Remark \ref{th:signs-and-grading} carries over to this more general situation; except that to make $\mathit{HF}^*(\phi,\epsilon)$ ${\mathbb Z}$-graded, one needs to impose conditions on $\phi$ as well as on $M$, as in Setup \ref{th:auto-setup}(ii).
\begin{remark} \label{th:circle-and-discrete-actions}
In general, $\mathit{HF}^*(\phi,\epsilon)$ does not carry a BV operator. The exception is when $\phi$ can be written as the time-one map of a Hamiltonian flow $(\phi^t)$. Here, the general $\phi^t$ do not have to be equal to the identity near $\partial M$, but they have to preserve $\theta_M$ near $\partial M$ (one example of this would be the flow that leads to $\phi = \tau_{\partial M}$). One then defines a circle action on $\EuScript L_\phi$ by mapping a twisted loop $x$ to
\begin{equation}
x^{(r)}(t) = \phi^t(x(t-r)).
\end{equation}
There is a related situation where fixed point Floer cohomology admits a discrete symmetry, which was extensively studied in \cite{seidel14c, polterovich-shelukhin15}. Namely, suppose that $\phi$ admits a $k$-th root $\phi^{1/k}$ (such that $(\phi^{1/k})^*\theta_M = \theta_M$ near $\partial M$). This gives rise to an action of ${\mathbb Z}/k$ on $\EuScript L_\phi$, whose generator maps $x$ to
\begin{equation}
x^{(1/k)}(t) = \phi^{1/k}(x(t-1/k)).
\end{equation}
One gets an induced action of ${\mathbb Z}/k$ on $\mathit{HF}^*(\phi)$ (in the previously considered case of Hamiltonian flows, $\phi$ has $k$-th roots for any $k$; but the induced ${\mathbb Z}/k$-actions on Floer cohomology are trivial, which is intuitively clear since they embed into a continuous symmetry).
\end{remark}
Floer cohomology is invariant under isotopies of $\phi$ (within the class of symplectic automorphisms that are allowed). One has a straightforward counterpart of \eqref{eq:id-poincare-duality}:
\begin{equation}\label{eq:phi-poincare-duality}
\mathit{HF}^*(\phi^{-1},-\epsilon) \cong \mathit{HF}^{2n-*}(\phi,\epsilon)^\vee.
\end{equation}
As for the dependence on $\epsilon$, Lemmas \ref{th:no-wall}--\ref{th:wall-2}, with the parts about BV operators omitted, carry over with essentially the same proofs.
\begin{lemma} \label{th:tau-shift}
Suppose that $\partial M$ is a contact circle bundle, and let $\tau_{\partial M}$ be the boundary twist. For any $\phi$ and $\epsilon$,
\begin{equation} \label{eq:shift-isomorphism}
\mathit{HF}^*(\tau_{\partial M}^{-1} \circ \phi, \epsilon - 1) \cong \mathit{HF}^*(\phi,\epsilon).
\end{equation}
This isomorphism is compatible with ${\mathbb Z}$-gradings if we use option (i) from Example \ref{th:boundary-twist} as a grading for $\tau_{\partial M}$; if instead we use option (ii), the right hand side of \eqref{eq:shift-isomorphism} should be replaced with $\mathit{HF}^{*+2-2m}(\phi,\epsilon)$.
\end{lemma}
\begin{proof}
When defining $\tau_{\partial M}$, one can choose the flow $(\phi^t_F)$ to be supported arbitrarily close to $\partial M$, so that it commutes with $\phi$. Consider the diffeomorphism
\begin{equation} \label{eq:change-loop}
\begin{aligned}
& \EuScript F: \EuScript L_{\tau_{\partial M}^{-1} \circ \phi} \longrightarrow \EuScript L_{\phi}, \\
& (\EuScript F x)(t) = \phi^t_F(x(t)),
\end{aligned}
\end{equation}
which satisfies
\begin{equation}
\begin{aligned}
& \EuScript F^* A_{\phi,H} = A_{\tau_{\partial M}^{-1} \circ \phi, \tilde{H}}, \\
& \tilde{H}_t(x) = H_t(\phi^t_F(x)) - F(x).
\end{aligned}
\end{equation}
Note that close to $\partial M$, $\tilde{H}_t = \epsilon\rho_M - \rho_M$. Given suitable choices of almost complex structures, \eqref{eq:change-loop} gives rise to an isomorphism of Floer cochain complexes, which induces \eqref{eq:shift-isomorphism}.
\end{proof}
\begin{example} \label{th:grading-of-boundary-twist}
Assuming choice (ii) for the grading,
$\mathit{HF}^{*+2-2m}(\tau_{\partial M}, \epsilon) \cong \mathit{HF}^*(M,\epsilon-1)$. In particular, using Proposition \ref{th:bv-vanish} and \eqref{eq:id-poincare-duality}, one obtains \eqref{eq:trivial-monodromy} (for $m = 0$) and its generalization \eqref{eq:trivial-monodromy-2} (for arbitrary $m$).
\end{example}
\subsection{Symplectic mapping tori\label{subsec:mapping-tori}}
As usual, the mapping torus construction relates discrete dynamics (symplectic automorphisms) and its continuous counterpart (Hamiltonian flows).
\begin{setup} \label{th:mapping-torus-setup}
(i) Let $M$ be a Liouville domain, and $\mu$ an exact symplectic automorphism (as in Setup \ref{th:auto-setup}, with associated function $G_\mu$). Its symplectic mapping torus is the fibration
\begin{equation} \label{eq:mapping-cone-projection}
E = \frac{{\mathbb R}^2 \times M}{(p,q,x) \sim (p,q-1,\mu(x))}
\xrightarrow{\pi(p,q,x) = (p,q)}
{\mathbb R} \times S^1.
\end{equation}
The symplectic form is $\omega_E = dp \wedge dq + \omega_M$. To obtain a primitive, one chooses a function $G = G(q,x)$ such that $G_\mu(x) = G(q,x) - G(q-1,\mu(x))$, and sets $\theta_E = p\,\mathit{dq} + \theta_M + dG$.
(ii) If $M$ has a symplectic Calabi-Yau structure, and $\mu$ is a graded symplectic automorphism, the mapping torus inherits a symplectic Calabi-Yau structure.
(iii) We use the class of functions $H$ on $E$ of the following form. Fix constants $\epsilon$, as in \eqref{eq:no-reeb}, and $\gamma_{\pm} \in {\mathbb R} \setminus {\mathbb Z}$. Let $H_M$ be a function on the fibre, which equals $\epsilon \rho_M$ near the boundary, and which is invariant under $\mu$ (such functions can easily be constructed as cutoffs of $\epsilon \rho_M$). Let $H_{{\mathbb R} \times S^1}$ be a function on the base, such that $H_{{\mathbb R} \times S^1}(p,q) = \gamma_{\pm} p$ if $\pm p \gg 0$. Then, there should be an open subset $U \subset E$, which contains $\partial E$ and has compact complement, such that
\begin{equation} \label{eq:mapping-torus-h}
H(p,q,x) = H_{{\mathbb R} \times S^1}(p,q) + H_M(x) \quad \text{for $(p,q,x) \in U$.}
\end{equation}
We stress that if we have several different $H$, the associated $H_{{\mathbb R} \times S^1}$ and $H_M$ can also be different (they do not have to be fixed once and for all).
(iv) We will use almost complex structures $J$ on $E$ such that: outside a compact subset, $\pi$ is $J$-holomorphic (with respect to the standard complex structure $i$ on the base); and near the boundary, $J = J_{{\mathbb R} \times S^1} \times J_{M,p,q}$, where $J_{{\mathbb R} \times S^1}$ is an almost complex structure on the base which is standard outside a compact subset, and the $J_{M,p,q}$ are as in Setup \ref{th:hamiltonian-setup}.
\end{setup}
Given time-dependent $H = (H_t)$ and $J = (J_t)$, one can build a Floer complex $\mathit{CF}^*(E,H)$ as before. Of course, one has to check that solutions of Floer's equation $u: {\mathbb R} \times S^1 \rightarrow E$ remain inside a compact subset of $E \setminus \partial E$. To see that this is the case, note that on the subset where the $|p|$-coordinate of $u$ is large, $v = \pi(u)$ will itself be a solution of
\begin{equation}
\partial_s v + i(\partial_t v - \gamma_{\pm} \partial_q) = 0,
\end{equation}
($\partial_q$ stands for the unit vector field in $q$-direction), hence its $p$-component is harmonic. On the other hand, where $u$ is close to $\partial E$, one can consider its fibre component alone, and then apply the maximum principle in the same way as when constructing Floer cohomology inside $M$. The resulting groups $\mathit{HF}^*(E,\gamma_-,\gamma_+,\epsilon)$ depend only on the constants involved (and we can change those, without affecting Floer cohomology, as long as no ``forbidden values'' are crossed, in parallel with Lemma \ref{th:no-wall}). They also carry a BV operator.
\begin{lemma} \label{th:mclean-example}
Suppose that $\gamma_- \in (0,1)$ and $\gamma_+ \in (1,2)$. Then
\begin{equation} \label{eq:s1-floer}
\mathit{HF}^*(E,\gamma_-,\gamma_+,\epsilon) \cong H^*(S^1) \otimes \mathit{HF}^*(\mu,\epsilon).
\end{equation}
The BV operator is given by rotation on $S^1$, tensored with the identity map on $\mathit{HF}^*(\mu,\epsilon)$.
\end{lemma}
The Floer cohomology computation \eqref{eq:s1-floer} is \cite[Theorem 1.3]{mclean12}, which can be proved relatively straightforwardly by a Morse-Bott approach. A similar argument determines the BV operator.
\section{Rotations at infinity\label{subsec:rotate}}
The next step is to apply Hamiltonian Floer cohomology to total spaces of Lefschetz fibrations. Besides making the necessary adjustments to the construction of Floer cohomology, we will consider some specific computations. Those follow \cite{mclean12} fairly closely, but with the BV operator as an added ingredient.
\subsection{Target spaces}
The Lefschetz (complex nondegeneracy) condition is not important yet, and we will allow considerably more freedom for the local geometry. On the other hand, we impose quite strict conditions on the behaviour near infinity.
\begin{setup} \label{th:setup-e}
(i) An {\em exact symplectic fibration with singularities} is a $2n$-dimensional manifold with boundary $E$, together with an exact symplectic form $\omega_E = d\theta_E$, and a proper map
\begin{equation} \label{eq:symplectic-fibration}
\pi: E \longrightarrow {\mathbb C},
\end{equation}
subject to the following conditions.
At any $x \in E$, define the horizontal subspace $TE^h_x \subset TE$ as the $\omega_E$-orthogonal complement of $TE^v_x = \mathit{ker}(D\pi_x)$. We ask that there should be an open subset $U \subset E$ which contains $\partial E$ and has compact complement, such that at each $x \in U$, one has: $TE_x^v$ is a codimension $2$ symplectic subspace of $TE$ (hence, $x$ is a regular point of $\pi$, and $D\pi_x|TE^h_x$ is an isomorphism); and
\begin{equation} \label{eq:flat}
(\omega_E-\pi^*\omega_{{\mathbb C}})\,|\,TE^h_x = 0,
\end{equation}
where $\omega_{{\mathbb C}} = d\mathrm{re}(y) \wedge d\mathrm{im}(y)$ is the standard symplectic form on the base.
Next, at each point $x \in \partial E$, $TE_x^h$ should lie inside $T_x(\partial E)$.
Finally, there should be a $y_* > 0$ such that: all fibres $E_y = \pi^{-1}(y)$ with $|y| \geq y_*$ lie inside the previously introduced subset $U$; and $M = E_{y_*}$, with its induced exact symplectic structure, is a Liouville domain (in all subsequent developments, we will assume that such a $y_*$ has been fixed).
(ii) For part of our considerations, we will assume that $E$ comes with a symplectic Calabi-Yau structure. This induces the same kind of structure on $M$.
\end{setup}
Let's consider the implications of these conditions. We have symplectic parallel transport maps defined in a neighbourhood of $\partial E$. More precisely, let $W \subset M$ be a small open neighbourhood of $\partial M$. Then, parallel transport yields a canonical embedding, whose image is an open neighbourhood of $\partial E \subset E$:
\begin{equation} \label{eq:first-trivialization}
\left\{
\begin{aligned}
& \Theta: {\mathbb C} \times W \longrightarrow E, \\
& \Theta({\mathbb C} \times \partial M) = \partial E, \\
& \pi(\Theta(y,x)) = y, \\
& \Theta(y_*,x) = x, \\
& \Theta^*\omega_E = \omega_{\mathbb C} + \omega_M.
\end{aligned}
\right.
\end{equation}
To see why that is the case, note that \eqref{eq:flat} implies that the symplectic connection is flat near $\partial E$. Hence, the relevant part of parallel transport is independent of the choice of path, and $\Theta^*\omega_E - \omega_M$ must be locally the pullback of some two-form on ${\mathbb C}$; to determine that two-form, one again appeals to \eqref{eq:flat}.
We also have the same flatness property outside a compact subset, with the following consequence. Let $\mu$ be the monodromy around the circle of radius $y_*$, which is an exact symplectic automorphism of $M$ (and restricts to the identity on $W$).
Then, for $p_* = \log(y_*)$, there is a unique covering map
\begin{equation} \label{eq:second-trivialization}
\left\{
\begin{aligned}
& \Psi: [p_*,\infty) \times {\mathbb R} \times M \longrightarrow \{|\pi(x)| \geq y_*\} \subset E, \\
& \pi(\Psi(p,q,x)) = e^{p+iq}, \\
& \Psi(p_*,0,x) = x, \\
& \Psi(p,q+2\pi,x) = \Psi(p,q,\mu(x)), \\
& \Psi^*\,\omega_E = e^{2p} dp \wedge dq + \omega_M.
\end{aligned}
\right.
\end{equation}
The partial trivializations \eqref{eq:first-trivialization} and \eqref{eq:second-trivialization} are compatible, in the sense that
\begin{equation}
\Theta(e^{p+iq},x) = \Psi(p,q,x) \quad \text{for $p \geq p_*$ and $x \in W$.}
\end{equation}
In words, what we have observed is that \eqref{eq:symplectic-fibration} is symplectically trivial near $\partial E$, whereas outside the preimage of a disc, it is like a symplectic mapping torus for $\mu$ (but with a different symplectic structure on the base than before). Finally, in the situation of Setup \ref{th:setup-e}(ii), $\mu$ is a graded symplectic automorphism (in a preferred way, such that the grading is trivial near $\partial M$).
\begin{example} \label{th:anticanonical-lefschetz-pencil}
The most important examples for us are those obtained from anticanonical Lefschetz pencils, or more generally, from Lefschetz pencils satisfying the condition from Remark \ref{th:fractional-cy-1}. In that case, the boundary of the fibre is a contact circle bundle; $E$ carries a symplectic Calabi-Yau structure, such that the induced structure on the fibre is as in Example \ref{th:contact-circle-bundle}(ii); and as already stated in \eqref{eq:mu-tau}, the monodromy is the associated boundary twist.
\end{example}
\begin{setup} \label{th:setup-rotation}
(i) We consider functions $H \in C^\infty(E,{\mathbb R})$ such that
\begin{equation}
\left\{
\begin{aligned}
& H(\Theta(y,x)) = H_{{\mathbb C}}(y) + \epsilon \rho_M(x) \quad \text{for $x$ near $\partial M$}, \\
& H(\Psi(p,q,x)) = \textstyle \gamma e^{2p}/2 + H_M(x) \quad \text{for $p \gg 0$.}
\end{aligned}
\right.
\end{equation}
Here, the constant $\epsilon$ is as in \eqref{eq:no-reeb}, and $\gamma \in {\mathbb R} \setminus 2\pi{\mathbb Z}$. $H_{{\mathbb C}}$ is a function on the base such that $H_{{\mathbb C}}(y) = \gamma |y|^2/2$ for $|y| \gg 0$, and $H_M$ is as in Setup \ref{th:mapping-torus-setup}.
(ii) We use compatible almost complex structures $J$ on $E$ satisfying the following conditions. Outside a compact subset, $\pi$ is $J$-holomorphic, with respect to the standard complex structure $i$ on the base; and at points $(y,x) \in {\mathbb C} \times M$ with $x$ sufficiently close to $\partial M$,
\begin{equation} \label{eq:theta-star-j}
\Theta^*J = J_{{\mathbb C}} \times J_{M,y}.
\end{equation}
Here, $J_{{\mathbb C}}$ is an almost complex structure on ${\mathbb C}$ which is standard outside a compact subset, and each $J_{M,y}$ is an almost complex structure on $M$ as in Setup \ref{th:hamiltonian-setup}.
\end{setup}
\subsection{Floer cohomology and its properties}
The construction of Hamiltonian Floer cohomology in this context proceeds pretty much as in Section \ref{subsec:mapping-tori}. One gets chain complexes $\mathit{CF}^*(E,H)$ and cohomology groups $\mathit{HF}^*(E,\gamma,\epsilon)$, together with a BV operator. A suitable analogue of Lemma \ref{th:no-wall} holds. There are also counterparts of Lemma \ref{th:wall} for changing either $\gamma$ or $\epsilon$. We will not consider them in full generality, but one important special case is this:
\begin{lemma} \label{th:gamma-wall}
For $\gamma_- \in (0,2\pi)$, $\gamma_+ \in (2\pi,4\pi)$, and any $\epsilon$, we have a long exact sequence
\begin{equation} \label{eq:2pi-sequence}
\cdots \rightarrow \mathit{HF}^*(E,\gamma_-,\epsilon) \longrightarrow
\mathit{HF}^*(E,\gamma_+,\epsilon) \longrightarrow H^*(S^1) \otimes \mathit{HF}^{*+2}(\mu,\epsilon) \rightarrow \cdots
\end{equation}
where $\mu$ is the monodromy. This is compatible with BV operators, where the operator on the rightmost group is as in Lemma \ref{th:mclean-example}. Moreover, if $E$ has a symplectic Calabi-Yau structure, \eqref{eq:2pi-sequence} is compatible with ${\mathbb Z}$-gradings.
\end{lemma}
\begin{proof}
To simplify the notation, we assume that one can take $y_* = 1$. Fix a function $H_M$ which is supported in a small neighbourhood of $\partial M$ (hence is invariant under $\mu$), and which equals $\epsilon\rho_M$ close to $\partial M$. Via \eqref{eq:first-trivialization}, there is an obvious extension of this function to all of $E$, which we again denote by $H_M$. Choose a function
\begin{equation} \label{eq:rotation-function}
\left\{
\begin{aligned}
& h_+: [0,\infty) \longrightarrow {\mathbb R}, \\
& h_+'(a) = 0 \quad \text{if and only if $a \leq 1/2$,} \\
& h_+'(a) = \gamma_- \quad \text{for $a$ close to $2$,} \\
& h_+(a) = \gamma_+ a \quad \text{for $a \gg 0$}, \\
& h_+''(a) \geq 0 \quad \text{everywhere}, \\
& h_+''(a) > 0 \quad \text{whenever $h'_+(a) \in (\gamma_-,\gamma_+)$.}
\end{aligned}
\right.
\end{equation}
Define $H_+: E \rightarrow {\mathbb R}$ by
\begin{equation}
H_+(x) = h_+(|\pi(x)|^2/2) + H_M(x).
\end{equation}
This has two kinds of $1$-periodic orbits:
\begin{itemize} \itemsep.5em
\item[(i)] The first kind are contained in a fibre $\pi^{-1}(y)$, $|y| \leq 1$. They are either constant, or else nontrivial $1$-periodic orbits of $H_M$ (lying close to $\partial M$).
\item[(ii)] The second kind of $1$-periodic orbit is fibered over the circle $|y|^2/2 = a_*$, where $a_* > 2$ is the unique value such that $h'_+(a_*) = 2\pi$. In each fibre, they correspond to fixed points of $\phi_{H_M}^1 \circ \mu$.
\end{itemize}
Let's perturb $H_+$ slightly (in a time-dependent way, and without changing the notation), so as to make the $1$-periodic points nondegenerate; we will assume that the perturbation is trivial near $\partial E$, as well as outside a compact subset. The outcome lies in the class of Hamiltonians which can be used to construct $\mathit{HF}^*(E,\gamma_+,\epsilon)$. As long as the perturbation is small, one can still maintain the distinction between type (i) and (ii) orbits, even though they no longer have exactly the same properties as before.
We now consider Floer trajectories. When perturbing $H_+$, we can assume that the perturbation is trivial near the preimage of the circle $Z = \{|y| = 2\}$. We will also use almost complex structures $J = (J_t)$ such that $\pi$ becomes pseudo-holomorphic near that same preimage. As a consequence, if $u$ is a Floer trajectory, then $v = \pi(u)$ satisfies
\begin{equation} \label{eq:v-projection}
\partial_s v + i (\partial_t v - \gamma_- iv) = 0 \quad \text{whenever $v$ is close to $Z$.}
\end{equation}
We exploit that by introducing a degenerate version of the action functional (this is essentially what \cite{seidel12b} called a ``barrier argument''). On the base, choose a two-form $\bar{\omega}_{{\mathbb C}}$ which is: rotationally invariant; everywhere nonnegative; supported in a small neighbourhood of $Z$; and positive at every point of $Z$. Then,
\begin{equation} \label{eq:barrier}
0 \leq \int_{{\mathbb R} \times S^1} \bar\omega_{{\mathbb C}}(\partial_s v, \partial_tv - \gamma_- iv) =
\bar{A}(y_-) - \bar{A}(y_+),
\end{equation}
where $\bar{A}(y_{\pm})$ depends only on the limits $y_{\pm}$ of $v$. To compute it explicitly, fix a primitive $\bar{\omega}_{{\mathbb C}} = d\bar{\theta}_{{\mathbb C}}$. Also, consider the function $\bar{H}_{{\mathbb C}}$ such that $d\bar{H}_{{\mathbb C}} = \bar{\omega}_{{\mathbb C}}(\cdot, \gamma_- i y)$, normalized by asking that it should vanish near the origin. Then,
\begin{equation}
\bar{A}(y) = \int_{S^1} -y^*\bar{\theta}_{{\mathbb C}} + \bar{H}_{{\mathbb C}}(y(t))\, \mathit{dt}.
\end{equation}
Concretely,
\begin{equation}
\bar{A}(y) = \begin{cases} 0 & \text{if $y = \pi(x)$, with $x$ of type (i) as listed above,} \\
\textstyle (\frac{\gamma_-}{2\pi} - 1) \textstyle \int_{{\mathbb C}} \bar{\omega}_{{\mathbb C}} < 0 & \text{if $x$ is of type (ii).}
\end{cases}
\end{equation}
Finally, note that equality in \eqref{eq:barrier} holds only if $v^{-1}(Z) = \emptyset$. It follows that there are only three kinds of Floer trajectories:
\begin{itemize} \itemsep.5em
\item[(i, i)]
If both limits $x_{\pm}$ are of type (i), the Floer trajectory remains entirely on the inside of $\pi^{-1}(Z)$;
\item[(ii, ii)]
If both limits $x_{\pm}$ are of type (ii), the Floer trajectoy remains entirely on the outside of $\pi^{-1}(Z)$, since $\bar{A}(y_-) = \bar{A}(y_+)$ (the same purpose was achieved by the ``minimum principle argument'' in \cite[Lemma 6.1]{mclean12});
\item[(i, ii)]
Finally, $x_-$ could be of type (i), and $x_+$ of type (ii).
\end{itemize}
Take a function $h_-: [0,\infty) \rightarrow {\mathbb R}$ which agrees with $h$ on $[0,2]$, and satisfies $h'_-(a) = \gamma_-$ for $a \geq 2$. Let $H_-$ be a (time-dependent) function which agrees with $H_+$ over the preimage of $\{|y| \leq 2\}$, and satisfies $H_-(x) = h_-(|\pi(x)|^2/2) + H_M(x)$ elsewhere. This has only the type (i) periodic orbits, and the associated chain complex $\mathit{CF}^*(E,H_-)$ has cohomology $\mathit{HF}^*(E, \gamma_-,\epsilon)$. From the argument above, we get (assuming suitable choices of almost complex structures) a short exact sequence of chain complexes
\begin{equation} \label{eq:gamma-gamma}
0 \rightarrow \mathit{CF}^*(E,H_-) \longrightarrow \mathit{CF}^*(E,H_+) \longrightarrow Q^* \rightarrow 0.
\end{equation}
The differential on the quotient $Q^*$ counts type (ii, ii) trajectories only, hence defines a (slightly modified) version of Floer cohomology inside the mapping torus of $\mu$, which is as computed in Lemma \ref{th:mclean-example}. This shows that \eqref{eq:gamma-gamma} induces the desired long exact sequence \eqref{eq:2pi-sequence}. The argument about BV operators is parallel, since one can arrange that the relevant equation \eqref{eq:bv-equation} has the same property \eqref{eq:v-projection}. The statement about gradings is straightforward (the shift by $2$ reflects the Conley-Zehnder index of the circle $|y|^2/2 = a_*$, as a $1$-periodic orbit of the Hamiltonian $h_+(|y|^2/2)$ on ${\mathbb C}$).
\end{proof}
\begin{proposition} \label{th:bv-vanish-3}
For $\gamma \in (0,2\pi)$ and sufficiently small $\epsilon>0$,
\begin{equation} \label{eq:e-pss}
\mathit{HF}^*(E,\gamma,\epsilon) \cong H^*(E).
\end{equation}
Moreover, the BV operator vanishes.
\end{proposition}
This is an analogue of Proposition \ref{th:bv-vanish}, and has the same proof. We will now dig a little deeper into the consequences. In the situation of Lemma \ref{th:gamma-wall}, choose a class $z \in \mathit{HF}^{*+2}(\mu,\epsilon)$. Using notation as in \eqref{eq:gamma-gamma}, take $[\mathit{point}] \otimes z \in H^1(S^1) \otimes \mathit{HF}^{*+2}(\mu,\epsilon) \subset H^{*+1}(Q)$, represent it by a cocycle in $Q^*$, and lift that cocycle to a cochain
\begin{equation} \label{eq:x-cochain}
x \in \mathit{CF}^{*+1}(E,H_+), \quad dx \in \mathit{CF}^{*+2}(E,H_-).
\end{equation}
Assume that $\gamma_-$ and $\epsilon$ are small. Using the proof of Proposition \ref{th:bv-vanish} (as applied to our situation in Lemma \ref{th:bv-vanish-3}), one can arrange that the chain level BV operator vanishes on $\mathit{CF}^*(E,H_-) \subset \mathit{CF}^*(E,H_+)$. As a result, we get a cocycle
\begin{equation} \label{eq:bar-x}
\bar{x} = \delta x \in \mathit{CF}^{*}(E,H_+),
\end{equation}
since $d \bar{x} = -\delta (dx) = 0$. Clearly, $\bar{x}$ is independent of the choice of lift $x$. Moreover, if the original cohomology class was trivial, we would have $x - dy \in \mathit{CF}^{*+1}(E,H_-)$ for some $y$, and hence
\begin{equation}
\bar{x} = \delta(dy) = -d(\delta y).
\end{equation}
This shows that $[\bar{x}] \in \mathit{HF}^*(E,\gamma,\epsilon_+)$ depends only on $z$. Finally, if we project $\bar{x}$ to $Q^*$, it represents $1 \otimes z \in H^0(S^1) \otimes \mathit{HF}^{*+2}(\mu,\epsilon) \subset H^*(Q)$ (by the description of the BV operator in Lemmas \ref{th:mclean-example} and \ref{th:gamma-wall}). The outcome of this argument can be summarized as follows:
\begin{proposition} \label{th:partial-splitting}
Take the situation from Lemma \ref{th:gamma-wall}, with $\epsilon>0$ small. Then the map
\begin{equation} \label{eq:proj-s1}
\mathit{HF}^*(E,\gamma_+,\epsilon) \longrightarrow H^*(S^1) \otimes \mathit{HF}^{*+2}(\mu,\epsilon)
\end{equation}
from \eqref{eq:2pi-sequence} has a partial splitting, defined on $H^0(S^1) \otimes \mathit{HF}^{*+2}(\mu,\epsilon)$. \qed
\end{proposition}
For the benefit of readers concerned by the apparent ad hoc nature of the previous argument, we can explain how it fits into an appropriate formal framework.
\begin{setup} \label{th:split-setup}
(i) Take a chain complex $C^*_+$ together with an endomorphism $\delta$ of degree $-1$. Let $C^*_-$ be a subcomplex such that $\delta(C^*_-) \subset C^*-$. Then, a homotopy trivialization $[\chi]$ of $\delta|C^*_-$ induces a lift
\begin{equation} \label{eq:lift-bv}
\xymatrix{
&& H^{*-1}(C_+) \ar[d] \\
H^*(Q) \ar@{-->}[urr] \ar[rr] && H^{*-1}(Q)
}
\end{equation}
where the horizontal arrow is the operation induced by $\delta$ on the quotient complex $Q^* = C^*_+/C^*_-$, and the vertical arrow is the projection map. The lift in \eqref{eq:lift-bv} is defined as follows: given a cocycle in $Q^*$, lift it to a cochain $x$ in $C^*_+$, and then map that to the cocycle
\begin{equation} \label{eq:corrected-bar}
\bar{x} = \delta x + \chi(dx).
\end{equation}
One sees immediately that this induces a map on cohomology, which depends only on the equivalence class $[\chi]$.
(ii) Take two such complexes $C^*_+$ and $\tilde{C}^*_+$, each with the same structure as in (i) (including subcomplexes $C^*_-$ and $\tilde{C}^*_-$). Suppose that we have maps between them as in Setup \ref{th:null}(ii), which preserve the subcomplexes. If we are given mutually compatible homotopy trivializations $[\chi]$ and $[\tilde{\chi}]$ on those subcomplexes, which are compatible in the sense of Setup \ref{th:null}(ii), then (by a straightforward computation) the lifts from \eqref{eq:lift-bv} fit into a commutative diagram
\begin{equation} \label{eq:lift-lift}
\xymatrix{
H^*(C_+) \ar[rr] && H^*(\tilde{C}_+)
\\
\ar@{-->}[u] H^*(Q) \ar[rr] && H^*(\tilde{Q}). \ar@{-->}[u]
}
\end{equation}
\end{setup}
Mapping this abstract framework onto our specific situation is fairly straightforward. We had $H^*(Q) \cong H^*(S^1) \otimes \mathit{HF}^{*+2}(\mu,\epsilon)$, with the BV operator on the quotient as described in Lemma \ref{th:mclean-example}. We arranged that $\delta|C^*_-$ vanished, and made the trivial choice $\chi = 0$. The partial splitting of \eqref{eq:proj-s1} is the map which fits into the commutative diagram
\begin{equation}
\xymatrix{
\mathit{HF}^*(E,\gamma_+,\epsilon) \\
\ar[u]
H^1(S^1) \otimes \mathit{HF}^{*}(\mu,\epsilon) \ar[rr]^-{\cong} &&
H^0(S^1) \otimes \mathit{HF}^{*}(\mu,\epsilon) \ar@{-->}[ull]
}
\end{equation}
where the vertical arrow is the restriction of the lift \eqref{eq:lift-bv}, and the horizontal arrow is the BV operator. Inspection shows that this indeed recovers the formula \eqref{eq:bar-x}. As in the discussion following Setup \ref{th:null}, one can use the fact that the homotopy trivialization is canonical to show that the partial splitting is independent of all auxiliary choices.
We will also need a modified version of Proposition \ref{th:partial-splitting}, whose proof combines much of the technical material about Floer cohomology that has been mentioned so far:
\begin{proposition} \label{th:partial-splitting-2}
Take the situation from Lemma \ref{th:gamma-wall}. Assume that $\partial M$ is a contact circle bundle, and that $\epsilon \in (1,2)$. Additionally, assume that $E$ has a symplectic Calabi-Yau structure, such that the induced symplectic Calabi-Yau structure on $M$ is as in Example \ref{th:contact-circle-bundle}(ii). Then \eqref{eq:proj-s1} has a partial splitting, defined on the subspace of $H^0(S^1) \otimes \mathit{HF}^{*+2}(\mu,\epsilon)$ where $\ast \leq -2m$.
\end{proposition}
\begin{proof}
One can set up a Morse-Bott chain complex which computes $\mathit{HF}^*(E,\gamma_-,\epsilon)$ exactly as in Proposition \ref{th:bv-vanish-2}. Namely, generators consist of constant orbits, together with a Morse-Bott nondegenerate submanifold $F$ which is a copy of $\partial M$ (located in some fibre of $\pi$). Moreover, the Hamiltonian and almost complex structure can be taken to be time-independent. Then, the argument used there shows that the Morse-Bott analogue of the BV operator vanishes on all cochains of degree $\leq 2-2m$.
What does this mean for our original complex $\mathit{CF}^*(E,H_-)$? It is related to the Morse-Bott version by a chain homotopy equivalence, which is a version of a continuation map, hence comes with a secondary version as in \eqref{eq:cont-maps}, expressing its compatibility with BV operators. Using Setup \ref{th:truncation}(ii), we obtain a nullhomotopy for the BV operator on the correspondingly truncated subcomplex:
\begin{equation} \label{eq:chi-kill}
\chi: \mathit{CF}^{*}(E,H_-)^{\leq 2-2m} \longrightarrow \mathit{CF}^{*-2}(E,H_-)^{\leq 2-2m}.
\end{equation}
Now take \eqref{eq:x-cochain}, where $\ast \leq -2m$, and replace \eqref{eq:bar-x} with \eqref{eq:corrected-bar}, which provides the desired splitting; this makes sense because $dx$ is a cocycle in $\mathit{CF}^*(E,H_-)$ of degree $\ast+2 \leq 2-2m$, hence lies in the domain of definition of \eqref{eq:chi-kill}.
\end{proof}
As before, one can interpret this as an instance of a more general setup:
\begin{setup} \label{th:final-setup}
(i) Consider a variation on the situation of Setup \ref{th:split-setup}(i), where the homotopy trivialization exists only on $(C^*_-)^{\leq j}$, for some $j$. Then, \eqref{eq:corrected-bar} defines a partial lift, as in \eqref{eq:lift-bv} but defined only on $H^*(Q)$ for $\ast \leq j-1$.
(ii) There is a corresponding variation of Setup \ref{th:split-setup}(ii), with two homotopy trivializations defined on $(C^*_-)^{\leq j}$ respectively $(\tilde{C}^*_-)^{\leq j}$. Then, one gets a diagram \eqref{eq:lift-lift} in degrees $\ast \leq j-1$.
\end{setup}
To apply this to our case, we first used a Morse-Bott formalism to put ourselves in a situation where part of the BV operator can be explicitly computed (allowing an obvious choice of partial homotopy trivialization), and then used continuation maps to carry over that trivialization to the general case. In principle, part (ii) of Setup \ref{th:final-setup} can be used to show that the partial splitting from Proposition \ref{th:partial-splitting} is canonical, but that would require additional arguments not carried out here.
\begin{example} \label{th:lift-1}
In the situation of Example \ref{th:anticanonical-lefschetz-pencil}, we can use \eqref{eq:trivial-monodromy-2} for $\epsilon \in (1,2)$. The splitting from Proposition \ref{th:partial-splitting-2} then takes $1 \in H^0(M) \cong \mathit{HF}^{2-2m}(\mu,\epsilon)$ to an element of $\mathit{HF}^{-2m}(E,\gamma_+,\epsilon)$.
\end{example}
\section{Translations at infinity\label{subsec:translations}}
We remain in the same class of manifolds (Setup \ref{th:setup-e}), but now set up Floer cohomology differently. First of all, we follow the version for symplectic automorphisms, rather than the Hamiltonian one. More importantly, the perturbations involved have different behaviour at infinity, following \cite{seidel12b}.
\subsection{Geometric data}
As usual, we begin by assembling the basic geometric ingredients, this time starting with the class of symplectic automorphisms that are allowed.
\begin{setup} \label{th:setup-t}
(i) We consider exact symplectic automorphisms $\phi: E \rightarrow E$ with the following properties. There is an open subset $U \subset E$ of the same kind as in Setup \ref{th:setup-e}, such that
\begin{equation} \label{eq:fibrewise-phi}
\pi(\phi(x)) = \phi_{{\mathbb C}}(\pi(x)) \quad \text{for $x \in U$.}
\end{equation}
Here, $\phi_{{\mathbb C}}$ is a symplectic automorphism of the base, which is the identity outside a compact subset. Moreover, if we restrict $\phi$ to a fibre $E_y$, $|y| \gg 0$, then it should be the identity near $\partial E_y$.
(ii) As usual, if $E$ has a symplectic Calabi-Yau structure, we assume that $\phi$ is a graded symplectic automorphism (then, the same holds for its restriction to any fibre $E_y$, $|y| \gg 0$).
(iii) We use functions $H \in C^\infty(E,{\mathbb R})$ such that
\begin{equation} \label{eq:h-translation}
\left\{
\begin{aligned}
& H(\Theta(y,x)) = H_{{\mathbb C}}(y) + \epsilon\rho_M(x) \quad \text{for $x$ near $\partial M$,} \\
& H(\Psi(p,q,x)) = \delta \,\mathrm{re}(e^{p+iq}) + H_M(x) \quad \text{for $p \gg 0$.}
\end{aligned}
\right.
\end{equation}
Here, $\epsilon$ is as in \eqref{eq:no-reeb}, and $\delta \in {\mathbb R} \setminus \{0\}$. $H_{{\mathbb C}}$ is a function on the base such that $H_{{\mathbb C}}(y) = \delta\, \mathrm{re}(y)$ outside a compact subset, and $H_M$ is as in Setup \ref{th:mapping-torus-setup}.
(iv) We use compatible almost complex structures $J$ on $E$ of the following kind. Near the boundary, we impose the same condition \eqref{eq:theta-star-j} as before. At points $(p,q,x)$, where $p \gg 0$ and $-\pi \leq q \leq \pi$,
\begin{equation} \label{eq:two-families}
\Psi^*J = \begin{cases}
i \times J_{M,\mathrm{re}(e^{p+iq})}^+ & \text{for $0 \leq q \leq \pi$}, \\
i \times J_{M,\mathrm{re}(e^{p+iq})}^- & \text{for $-\pi \leq q \leq 0$.}
\end{cases}
\end{equation}
Here, $J_{M,r}^\pm$ are two auxiliary families of compatible almost complex structures on the fibre, both parametrized by $r \in {\mathbb R}$ and belonging to the class \eqref{eq:j-convex}, and additionally satisfying
\begin{equation}
\begin{cases}
J_{M,r}^- = J_{M,r}^+ & \text{if $r \gg 0$,} \\
J_{M,r}^- = \mu_*J_{M,r}^+ & \text{if $r \ll 0$.}
\end{cases}
\end{equation}
\end{setup}
The assumption \eqref{eq:fibrewise-phi} is quite restrictive, since it implies that $\phi$ must commute with symplectic parallel transport between the fibres. In combination with the other condition, this implies that
\begin{equation} \label{eq:phi-first-trivialization}
\phi(\Theta(y,x)) = \Theta(\phi_{{\mathbb C}}(y),x) \quad \text{for $x$ close to $\partial M$.}
\end{equation}
Similarly, we have
\begin{equation} \label{eq:conjugate-mu}
\phi(\Psi(p,q,x)) = \Psi(p,q,\phi_M(x)) \quad \text{if $p \gg 0$},
\end{equation}
where $\phi_M$ is an exact symplectic automorphism of $M$, which is the identity near $\partial M$, and which must commute with $\mu$.
\begin{example} \label{th:global-monodromy}
Fix a function on ${\mathbb C}$, which equals $\pi |y|^2$ outside a compact subset, and pull it back to $E$. Then, its flow for time $1$ gives a symplectic automorphism within the class defined above. We denote it by $\nu$, and call it the {\em global monodromy}. The associated automorphism of the fibre is the monodromy $\mu$.
\end{example}
The class of almost complex structures introduced above has the property that $\pi$ is $J$-holomorphic outside a compact subset. Moreover, because of \eqref{eq:two-families}, the induced almost complex structures on the fibres $E_y$, $|y| \gg 0$, are locally constant under parallel transport in imaginary direction (this is sketched in Figure \ref{fig:j}).
\begin{figure}
\caption{\label{fig:j}
\label{fig:j}
\end{figure}
\subsection{The compactness argument}
Given $\phi$, choose $H = (H_t)$ and $J = (J_t)$, with the same periodicity condition as in \eqref{eq:phi-periodicity} (this makes sense because the relevant classes of Hamiltonians and almost complex structures are invariant under $\phi$). The solutions of \eqref{eq:periodic-y} in $\EuScript L_\phi$, or equivalently the fixed points of $\phi_H^1 \circ \phi$, will always be contained in a compact subset of $E \setminus \partial E$. For generic choice of $(H_t)$, these fixed points will also be nondegenerate; assume from now on that this is the case. Consider solutions $u: {\mathbb R}^2 \rightarrow E$ of Floer's equation \eqref{eq:floer} with periodicity conditions \eqref{eq:floer-2}, and with limits $x_{\pm}$ as usual.
\begin{lemma} \label{th:b-energy}
There is a compact subset of $E$, such that for any solution $u$ and any $s$ such that $u(s,1)$ lies outside that subset,
\begin{equation}
\int_{[0,1]} |\partial_s u(s,t)|^2 \, \mathit{dt} \geq \delta^2.
\end{equation}
\end{lemma}
\begin{proof}
For $x$ outside a compact subset,
\begin{equation} \label{eq:no-asymptotic-fixed-points}
\mathrm{dist}(x, (\phi_H^1 \circ \phi)(x)) \geq \delta,
\end{equation}
where the distance is with respect to the metric associated to any almost complex structure in our class (recall that the fibres of $\pi$ are compact, so ``outside a compact subset'' means going to infinity in base direction). For essentially the same reason, $\int_{[0,1]} |\partial_s u(s,t)| \mathit{dt} \geq \delta$.
\end{proof}
\begin{lemma} \label{th:base-convexity}
There is a constant $B>0$, such that $|\mathrm{re}(\pi(u))| \leq B$ for all solutions $u$.
\end{lemma}
\begin{proof}
Write $v = \pi(u)$. On the subset where $\mathrm{re}(v)$ is large, it satisfies the equation
\begin{equation}
\left\{
\begin{aligned}
& \partial_s v + i(\partial_t v - i \delta) = 0, \\
& v(s,t+1) = v(s,t) + i\delta.
\end{aligned}
\right.
\end{equation}
Hence, $\mathrm{re}(v)$ is harmonic, so that we can apply the maximum principle.
\end{proof}
\begin{lemma} \label{th:fibre-convexity}
There is a neighbourhood $V \subset M$ of $\partial M$, such that the image of any solution $u$ is disjoint from $\Theta({\mathbb C} \times V)$.
\end{lemma}
This is again a maximum principle argument, but now used in fibre direction.
\begin{lemma} \label{th:gromov}
There is a constant $C$ such that $\|du\|_{\infty} \leq C$ for any Floer trajectory.
\end{lemma}
\begin{proof}
This is a simple Gromov compactness argument, following \cite[Proposition 5.1]{seidel12b}, and we will only give limited details. Suppose that the result is false. After passing to a subsequence, $\|\partial_s u_k\|$ goes to infinity. Take points $z_k = (s_k,t_k)$ where $|\partial_s u_k|$ reaches its maximum.
{\em If the $u_k(z_k)$ remain inside a compact subset of $E$}, one can rescale locally near $z_k$, and get a non-constant pseudo-holomorphic plane as a limit. Since we have a priori bounds on the energy, this plane has finite energy, hence extends to a pseudo-holomorphic sphere, in contradiction to exactness. {\em Suppose on the other hand that the $u_k(z_k)$ do not remain inside a compact subset of $E$.} In view of Lemma \ref{th:base-convexity}, one can pass to a subsequence and then assume that $\mathrm{im}(\pi(u_k(z_k)))$ converges to either $+\infty$ or $-\infty$. Then, after a translation in imaginary direction over the base, the same argument as before applies, except that the limit lies in ${\mathbb C} \times M$, and is pseudo-holomorphic for an almost complex structure $i \times J_{M,\mathrm{re}(y)}^{\pm}$, where $y$ is the coordinate on ${\mathbb C}$ (it is here that we use the specific property of our almost complex structures indicated in Figure \ref{fig:j}).
\end{proof}
\begin{proposition} \label{th:floer-bound}
There is a compact subset of $E \setminus \partial E$ which contains all Floer trajectories.
\end{proposition}
\begin{proof}
Assume that this is not true. By Lemmas \ref{th:base-convexity} and \ref{th:fibre-convexity}, there must be a sequence $(u_k)$ of solutions, such that the maximal value of $|\mathrm{im}(\pi(u_k))|$ goes to infinity. Because we have an absolute bound on $|\partial_su_k|$ from Lemma \ref{th:gromov}, $u_k(s,1)$ has to spend an increasingly large interval (in $s$) inside the region where Lemma \ref{th:b-energy} applies. But that contradicts the a priori bound on the energy.
\end{proof}
\subsection{Floer cohomology and its properties}
Having obtained Proposition \ref{th:floer-bound}, it is now a familiar process to set up Floer complexes $\mathit{CF}^*(\phi,H)$, whose cohomology we denote by $\mathit{HF}^*(\phi,\delta,\epsilon)$. The appropriate version of \eqref{eq:phi-poincare-duality} is
\begin{equation} \label{eq:translation-duality}
\mathit{HF}^*(\phi^{-1},-\delta,-\epsilon) \cong \mathit{HF}^{2n-*}(\phi,\delta,\epsilon)^\vee.
\end{equation}
\begin{lemma} \label{th:vanishing-homology}
For small $\delta > 0$ and $\epsilon > 0$, $\mathit{HF}^*(\mathit{id},\delta,\epsilon) \cong H^*(E,\{\mathrm{re}(\pi) \ll 0\})$.
\end{lemma}
We will not explain the proof of this, which can be done by reduction to Morse theory as in Proposition \ref{th:bv-vanish} (if $\pi$ is a Lefschetz fibration, $H^*(E,\{\mathrm{re}(\pi) \ll 0\})$ is concentrated in degree $n$, and has one generator for each critical point; one can in fact arrange that the underlying chain complex has the same property, and thereby give an elementary proof of Lemma \ref{th:vanishing-homology} for that special case).
The dependence of Floer cohomology on the parameters $(\delta,\epsilon)$ is a more interesting issue than before. One can show (for instance, using parametrized moduli spaces) that only the sign of $\delta$ matters. In fact, one also has isomorphisms
\begin{equation} \label{eq:reverse-delta}
\mathit{HF}^*(\phi,-\delta,\epsilon) \cong \mathit{HF}^*(\phi,\delta,\epsilon),
\end{equation}
but not canonical ones. To see that, note that the sign of $\delta$ depends on our identification of the base with the standard complex plane ${\mathbb C}$. Reversing that identification ($y \mapsto -y$) takes $\delta$ to $-\delta$. To construct \eqref{eq:reverse-delta}, one has to rotate the plane by some amount in $\pi + 2\pi{\mathbb Z}$, and different choices yield different maps \eqref{eq:reverse-delta}.
\begin{remark} \label{th:canonical-automorphism}
One way to think of the ambiguity in \eqref{eq:reverse-delta} is as follows. $\mathit{HF}^*(\phi,\delta,\epsilon)$ carries a canonical automorphism, induced by a full rotation of the plane (whose angle is thought of as an additional parameter), or equivalently by conjugation with the global monodromy. Then, \eqref{eq:reverse-delta} is unique up to composition with powers of that automorphism.
Another version of the same explanation goes as follows: on can allow translations over the base in any direction, corresponding to a parameter $\delta \in {\mathbb C}^*$ (for compatibility with our previous notation, the translation would have to be by $i\delta$). For fixed $\epsilon$, these more general Floer cohomology groups $\mathit{HF}^*(\phi,\delta,\epsilon)$ would be canonical locally trivial in $\delta$, which means that they would form a local system over ${\mathbb C}^*$. The previously mentioned automorphism is just the holonomy of the local system, and \eqref{eq:reverse-delta} would be a parallel transport map between two different fibres.
\end{remark}
\begin{lemma} \label{th:independence-of-epsilon}
For any $\epsilon_- < \epsilon_+$ which satisfy \eqref{eq:no-reeb}, one has $\mathit{HF}^*(\phi,\delta,\epsilon_-) \cong \mathit{HF}^*(\phi,\delta,\epsilon_+)$.
\end{lemma}
\begin{proof}
Our strategy follows that of Lemma \ref{th:no-wall}. We enlarge the given $E$ by attaching a conical piece to the boundary of each fibre, as in \eqref{eq:attach-cone}:
\begin{equation} \label{eq:attach-cone-2}
\hat{E} = E \cup_{\partial E} ({\mathbb C} \times [1,C] \times \partial M),
\end{equation}
where $\partial E$ is identified with ${\mathbb C} \times \partial M$ using parallel transport, which means \eqref{eq:first-trivialization}. Extend the given $\phi$ to an automorphism $\hat\phi$ of $\hat{E}$ by setting it equal to $\phi_{{\mathbb C}} \times \mathit{id}_{[1,C] \times \partial M}$ on the conical part, where $\phi_{{\mathbb C}}$ is as in \eqref{eq:phi-first-trivialization}.
Suppose that $H_-$ is the function used to define the chain complex $\mathit{CF}^*(\phi,H_-)$ underlying $\mathit{HF}^*(\phi,\delta,\epsilon_-)$. We extend it to a function $\hat{H}_+$ on $\hat{E}$ by a fibrewise version of \eqref{eq:extend-hamiltonian}:
\begin{equation}
\hat{H}_{t,+}(r,y,x) = H_{{\mathbb C},t}(y) + C h(C^{-1}r)
\end{equation}
where $h$ is as in \eqref{eq:turn}, and the $H_{{\mathbb C},t}$ are functions as in \eqref{eq:h-translation}. It is unproblematic to show that $\mathit{CF}^*(\hat\phi,\hat{H}_+)$ computes $\mathit{HF}^*(\phi,\delta,\epsilon_+)$. At this point, we want to be more specific about the choice of function for the original Floer cohomology group. Using the fact that the group of compactly supported symplectic automorphisms of ${\mathbb C}$ is connected, one can find a time-dependent $H_{{\mathbb C}}$ such that
\begin{equation}
(\phi_{H_{{\mathbb C}}}^1 \circ \phi_{{\mathbb C}})(y) = y + i\delta
\end{equation}
is simply a translation. In that case, $\mathit{CF}^*(\hat\phi,\hat{H}_+)$ has the same generators as $\mathit{CF}^*(\phi,H_-)$, and an easy maximum principle argument (in fibre direction) shows that the differentials also coincide.
\end{proof}
\begin{remark}
Alternatively, one can avoid the use of specific choices of $H_{{\mathbb C}}$, and argue as in Lemma \ref{th:wall}. Namely, suppose that there is only one $\epsilon \in (\epsilon_-,\epsilon_+)$ such that $\epsilon R_{\partial M}$ has $1$-periodic orbits. Then, there is a long exact sequence
\begin{equation} \label{eq:les-sequence-2}
\cdots \rightarrow \mathit{HF}^*(\phi,\delta,\epsilon_-) \longrightarrow \mathit{HF}^*(\phi,\delta,\epsilon_+) \longrightarrow H(Q^*) \rightarrow \cdots
\end{equation}
One can arrange that $Q^* \cong Q^*_{{\mathbb C}} \otimes Q^*_{M}$, where the first factor is a version of the Floer chain complex for $\phi^1_{H_{{\mathbb C}}} \circ \phi_{{\mathbb C}}$, which is then shown to be acyclic (this last step would again use the connectedness of the group of compactly supported symplectic automorphisms of ${\mathbb C}$).
\end{remark}
Our final topic is the relation between the two versions of Floer cohomology on $E$. Because of the different classes of perturbations used, this is not quite straightforward. We only need a partial result:
\begin{lemma} \label{th:rotation-translation}
Take $\gamma_- \in (2\pi(k-1),2\pi k)$, and arbitrary $\delta,\epsilon$. Let $\nu$ be the global monodromy. Then there is a canonical map
\begin{equation} \label{eq:rotation-translation}
\mathit{HF}^*(E,\gamma_-,\epsilon) \longrightarrow \mathit{HF}^*(\nu^k,\delta,\epsilon).
\end{equation}
\end{lemma}
To clarify the grading conventions: the source in \eqref{eq:rotation-translation} carries its natural ${\mathbb Z}$-grading (as a form of Hamiltonian Floer cohomology, on a manifold with a symplectic Calabi-Yau structure). For the target, we use the structure of the global monodromy $\nu$ as a graded symplectic automorphism which comes from its original construction by a flow (see Example \ref{th:global-monodromy}; this is parallel to (i) in Example \ref{th:boundary-twist}).
\begin{proof}
The argument follows Lemma \ref{th:gamma-wall} closely, and we will only give a few details. Set $\gamma_+ = 2\pi k$, and choose a function $h_+$ as in \eqref{eq:rotation-function}. We can assume that $\nu^k$ is defined as the time-one map of the function $F(x) = h_+(|\pi(x)|^2/2)$. When choosing a perturbation $H_+$ to be used to define $\mathit{CF}^*(\nu^k,H_+)$, we can assume that $H_{t,+}(x) = H_M(x)$ close to $Z = \{|\pi(x)| = 2\}$.
A suitable ``barrier'' argument shows that (for small $|\delta|$) the generators corresponding to fixed points lying on the inside of $Z$ form a subcomplex of $\mathit{CF}^*(\nu^k,H_+)$. Moreover, that subcomplex can be identified with $\mathit{CF}^*(E,H_-)$, where
\begin{equation} \label{eq:minusham}
H_{t,-}(x) =
\begin{cases}
H_{t,+}(\phi_F^{-t}(x)) + F(x) & |\pi(x)| \leq 2, \\
H_M(x) + h_-(|\pi(x)|^2/2) & |\pi(x)| \geq 2.
\end{cases}
\end{equation}
Here, $h_-$ is the function that agrees with $h_+$ on $[0,2]$, and satisfies $h_-'(a) = \gamma_-$ for all $a \geq 2$. But \eqref{eq:minusham} defines $\mathit{HF}^*(E,\gamma_-,\epsilon)$.
\end{proof}
\subsection{Additional remarks}
One can show that the map \eqref{eq:rotation-translation} fits into a long exact sequence
\begin{equation} \label{eq:r-t-1}
\cdots \rightarrow \mathit{HF}^*(E,\gamma_-,\epsilon) \longrightarrow \mathit{HF}^*(\nu^k,\delta,\epsilon) \longrightarrow \mathit{HF}^{*+2k}(\mu^{k+1},\epsilon) \rightarrow \cdots
\end{equation}
It seems likely (but we have not checked the details) that there is a similar long exact sequence
\begin{equation} \label{eq:r-t-2}
\cdots \rightarrow \mathit{HF}^*(\nu^k,\delta,\epsilon) \longrightarrow \mathit{HF}^*(E,\gamma_+,\epsilon) \longrightarrow \mathit{HF}^{*+2k+1}(\mu^{k+1},\epsilon) \rightarrow \cdots
\end{equation}
where $\gamma_+ \in (2\pi k, 2\pi (k+1))$. One piece of supporting evidence is that the combination of the two sequences above (in the appropriate order) is compatible with \eqref{eq:2pi-sequence}. Repeated use of those two sequences gives a step-by-step ``decomposition'' of all the Floer cohomology groups $\mathit{HF}^*(E,\gamma,\epsilon)$ and $\mathit{HF}^*(\nu^k,\delta,\epsilon)$. Alternatively, one can approach the same idea through spectral sequences:
\begin{lemma} \label{th:rotate-ss}
For any $\epsilon$ and any $\gamma \in (2\pi k, 2\pi(k+1))$, $k \geq 0$, there is a spectral sequence converging to $\mathit{HF}^*(E,\gamma,\epsilon)$, whose starting page is
\begin{equation} \label{eq:rotate-ss}
E_1^{pq} = \begin{cases}
H^{q+1}(E,\{\mathrm{re}(\pi) \ll 0\}) & p = 1, \\
\mathit{HF}^q(\mu^{-\lfloor p/2 \rfloor},\epsilon) & -2k \leq p \leq 0, \\
0 & \text{otherwise.}
\end{cases}
\end{equation}
\end{lemma}
\begin{lemma} \label{th:nu-ss}
For any $k \geq 0$, $\delta>0$, and $\epsilon$, there is a spectral sequence converging to $\mathit{HF}^*(\nu^k,\delta,\epsilon)$, whose starting page is
\begin{equation} \label{eq:nu-ss}
E_1^{pq} = \begin{cases}
H^{q+1}(E,\{\mathrm{re}(\pi) \ll 0\}) & p = 1, \\
\mathit{HF}^q(\mu^{-\lfloor p/2 \rfloor},\epsilon) & -2k+1 \leq p \leq 0, \\
0 & \text{otherwise.}
\end{cases}
\end{equation}
\end{lemma}
In fact, these spectral sequences are more straightforward to prove than \eqref{eq:r-t-1} or \eqref{eq:r-t-2}. The first one is a weaker version of that in \cite{mclean12}, and the second one is not substantially different. We will not explain them further, but we do want to show one application.
\begin{example} \label{th:negative-degrees}
Take the situation arising from an anticanonical Lefschetz pencil (see Examples \ref{th:contact-circle-bundle}(ii) and \ref{th:anticanonical-lefschetz-pencil}, but where $m = 0$). By repeatedly applying Lemma \ref{th:tau-shift}, we get
\begin{equation} \label{eq:r-shift}
\mathit{HF}^*(\mu^j, \epsilon) \cong \mathit{HF}^{*-2j}(M,\epsilon-j).
\end{equation}
Moreover, for any $\epsilon>0$, there is a Morse-Bott spectral sequence converging to $\mathit{HF}^*(M,\epsilon)$ (generalizing Proposition \ref{th:bv-vanish-2}), with
\begin{equation} \label{eq:bottish}
E_1^{pq} = \begin{cases} H^{p+q}(M) & p = 0, \\
H^{q-p}(\partial M) & -\lfloor \epsilon \rfloor \leq p < 0, \\
0 & \text{otherwise.}
\end{cases}
\end{equation}
In particular, $\mathit{HF}^*(M,\epsilon)$ is always concentrated in degrees $\ast \geq 0$.
Choose some $k>0$, and take $\epsilon > k$. From \eqref{eq:r-shift} and \eqref{eq:bottish}, it follows that
\begin{equation}
\left\{
\begin{aligned}
& \text{$\mathit{HF}^*(M, \epsilon)$ is concentrated in degrees $\ast \geq 0$,} \\
& \text{$\mathit{HF}^*(\mu,\epsilon) \cong \mathit{HF}^{*-2}(M, \epsilon-1)$ is concentrated
in degrees $\ast \geq 2$,} \\
& \dots \\
& \text{$\mathit{HF}^*(\mu^k,\epsilon) \cong \mathit{HF}^{*-2k}(M,\epsilon-k)$ is concentrated in degrees $\ast \geq 2k$.}
\end{aligned}
\right.
\end{equation}
By feeding that into \eqref{eq:rotate-ss} and \eqref{eq:nu-ss}, it follows that $\mathit{HF}^*(E,\gamma,\epsilon)$ and $\mathit{HF}^*(\nu^k,\delta,\epsilon)$ are all concentrated in nonnegative degrees
(moreover, all the contributions coming from closed Reeb orbits on $\partial M$ land in degrees $\geq 2$).
\end{example}
To summarize, we have two infinite sequences of ``closed string'' Floer cohomology groups associated to any Lefschetz fibration. The first of these sequences, $\mathit{HF}^*(E,\gamma,\epsilon)$ for $\gamma \in (2\pi k, 2\pi(k+1))$, comes with an additional dependence on $\epsilon$ (one can remove that dependence by passing to the direct limit $\epsilon \rightarrow \infty$, as in the definition of symplectic cohomology; the price to pay is that the resulting groups will typically be infinite-dimensional). Moreover, these groups carry BV operators. The second sequence is $\mathit{HF}^*(\nu^k,\delta,\epsilon)$. These groups are independent of $\epsilon$, do not have BV operators, but come with canonical automorphisms (mentioned in Remark \ref{th:canonical-automorphism}).
\begin{remark}
It is possible to interpret this situation in terms of mirror symmetry. Consider a smooth projective variety $A$, together with a section $r$ of its anticanonical bundle, which gives rise to a smooth divisor $B = r^{-1}(0)$. We'll discuss the analogues of the two Floer cohomology groups mentioned above, but in reverse order.
Take the sheaves $\Omega^i_A(kB)$ of algebraic $i$-forms with poles of order at most $k$ along $B$, and form
\begin{equation} \label{eq:hh-b}
\bigoplus_i H^{*+i}(A,\Omega^i_A(kB))
\end{equation}
For $k = 0$, this is the Hochschild homology of $A$. It admits a BV type operator, induced by the de Rham differential, but that is known to vanish. Derived autoequivalences act on Hochschild homology, and in particular, one gets a distinguished automorphism from the action of the Serre functor. More concretely, this automorphism is given by multiplying with the exponential of the class of the canonical bundle in $H^1(A,\Omega^1_A)$. For $k>0$, \eqref{eq:hh-b} does not carry a natural BV type operator, since the de Rham differential increases pole order (but one can still define a canonical automorphism, as before).
Instead, consider the subsheaves
\begin{equation} \label{eq:log}
\Omega^i_A((k-1)B +\log B) \subset \Omega^i_A(kB)
\end{equation}
of those differential forms $\alpha$ such that both $\alpha$ and $d\alpha$ have poles of order $\leq k$ along $B$. This gives rise to another sequence of graded groups,
\begin{equation} \label{eq:coherent-or-not}
\bigoplus_i H^{*+i}(A,\Omega^i_A((k-1)B + \log B)),
\end{equation}
which do carry BV operators. To be more precise, the situation we have just considered (with a smooth $B$) is mirror to working with a Lefschetz fibration which has closed fibres; hence, there is no parameter corresponding to our $\epsilon$ here.
\end{remark}
We also would like to consider a variant of Lemma \ref{th:nu-ss}, where some of the columns in the $E_1$ page have already been combined, as in \eqref{eq:r-t-1}. Again, the proof is omitted.
\begin{lemma} \label{th:3e}
There is a spectral sequence converging to $\mathit{HF}^*(\nu^2,\delta,\epsilon)$, with
\begin{equation} \label{eq:3cols}
E_1^{pq} = \begin{cases} \mathit{HF}^q(E,\gamma,\epsilon) & p = 0, \text{ where $\gamma>0$ is small,} \\
\mathit{HF}^{q+1}(\mu,\epsilon) \oplus \mathit{HF}^{q+2}(\mu,\epsilon) & p = -1, \\
\mathit{HF}^{q+1}(\mu^2,\epsilon) & p = -2, \\
0 & \text{otherwise.}
\end{cases}
\end{equation}
This spectral sequence is compatible with the ${\mathbb Z}/2$-action on $\mathit{HF}^*(\nu^2,\delta,\epsilon)$ from Remark \ref{th:circle-and-discrete-actions}: the induced ${\mathbb Z}/2$-action on \eqref{eq:3cols} is trivial except in the $p = -2$ column, where it is the corresponding action on $\mathit{HF}^*(\mu^2,\epsilon)$.
\end{lemma}
\begin{example} \label{th:trivial-involution}
Take the situation from Example \ref{th:negative-degrees} with $\epsilon>2$. Then the $p = -2$ column contributes only in positive degrees. Hence,
the ${\mathbb Z}/2$-action on $\mathit{HF}^0(\nu^2,\delta,\epsilon)$ is trivial, at least if we assume that $\mathrm{char}({\mathbb K}) \neq 2$.
\end{example}
\section{Open-closed string maps}
We now combine fixed point Floer cohomology, in the version considered in Section \ref{subsec:translations}, with its counterpart for Lagrangian submanifolds. The relation between the two theories, together with the preceding Floer cohomology computations, leads directly to our main results (Theorems \ref{th:main} and \ref{th:fano}).
\subsection{Lagrangian Floer cohomology}
We will continue to work in the situation of Setups \ref{th:setup-e} and \ref{th:setup-t}, with the following additional geometric ingredient.
\begin{setup} \label{th:setup-l}
(i) We consider oriented exact Lagrangian submanifolds $L \subset E \setminus \partial E$ such that $\pi|L$ is proper, and
\begin{equation} \label{eq:lambda}
\pi(L) = \{\text{compact subset}\} \cup \{\mathrm{re}(y) \gg 0, \;\; \mathrm{im}(y) = o\} \subset {\mathbb C}
\quad\text{for some $o \in {\mathbb R}$.}
\end{equation}
(ii) If $E$ comes with a Calabi-Yau structure, we will assume that $L$ is a graded Lagrangian submanifold.
(iii) Independently, one may want to assume that $L$ comes with a {\em Spin} structure.
\end{setup}
At any point $x \in L$ outside a compact subset, we know that $TE_x^v$ is a symplectic subspace, and hence that
\begin{equation} \label{eq:n-1}
\mathrm{rank}(D\pi_x|TL_x) = n - \mathrm{dim}(TL_x \cap \mathit{TE}_x^v) \geq 1.
\end{equation}
By combining this with \eqref{eq:lambda}, one sees that equality holds in \eqref{eq:n-1}, hence that $D\pi_x|TL_x: TL_x \rightarrow {\mathbb R}$ is onto. This implies that there is a unique closed Lagrangian submanifold $L_M \subset M \setminus \partial M$ such that, if $y = e^{p+iq}$ with $q \in (-\pi/2,\pi/2)$, $\mathrm{re}(y) \gg 0$ and $\mathrm{im}(y) = o$, then
\begin{equation} \label{eq:l-m}
L \cap \pi^{-1}(y) = \Psi((p,q) \times L_M).
\end{equation}
In other words, at infinity $L$ is fibered over a horizontal half-infinite path, with each fibre being equal to $L_M$. Note that $L_M$ is exact, and inherits an orientation (as well as the other structure mentioned in (ii) and (iii) above, whenever that exists on $L$).
Given two such submanifolds $L_0,L_1$, such that the corresponding numbers \eqref{eq:lambda} satisfy $o_1 - o_0 \neq 0$, there is a well-defined Floer cohomology $\mathit{HF}^*(L_0,L_1)$. To make the setup formally parallel to fixed point Floer cohomology, we will also introduce a perturbed version $\mathit{HF}^*(L_0,L_1,\delta,\epsilon)$, which reduces to the previous one for $\delta = \epsilon = 0$, and is defined under the assumption that
\begin{equation} \label{eq:delta-inequality}
o_1-o_0 - \delta \neq 0.
\end{equation}
Floer cohomology is invariant under automorphisms of $E$,
\begin{equation}
\mathit{HF}^*(\phi(L_0),\phi(L_1),\delta,\epsilon) \cong \mathit{HF}^*(L_0,L_1,\delta,\epsilon).
\end{equation}
The analogue of \eqref{eq:phi-poincare-duality} says that
\begin{equation} \label{eq:lagrangian-floer-duality}
\mathit{HF}^*(L_1,L_0,-\delta,-\epsilon) \cong \mathit{HF}^{n-*}(L_0,L_1,\delta,\epsilon)^\vee.
\end{equation}
\begin{lemma} \label{th:lag-1}
Suppose that $\delta_{\pm}$ are such that $o_1 - o_0 - \delta_{\pm}$ have the same sign. Then $\mathit{HF}^*(L_0,L_1,\delta_-,\epsilon) \cong \mathit{HF}^*(L_0,L_1,\delta_+,\epsilon)$.
\end{lemma}
\begin{lemma} \label{th:lag-2}
For any $\epsilon_{\pm}$, $\mathit{HF}^*(L_0,L_1,\delta,\epsilon_-) \cong \mathit{HF}^*(L_0,L_1,\delta,\epsilon_+)$.
\end{lemma}
The two Lemmas above are analogues of Lemma \ref{th:no-wall}. Note that this time, there are no ``forbidden values'' of $\epsilon$. The passage through the unique ``forbidden value'' of $\delta$ is described by the following result, whose proof we omit:
\begin{lemma} \label{th:lag-3}
Take $\delta_- < o_1-o_0 < \delta_+$. Then there is a long exact sequence
\begin{equation}
\cdots \rightarrow \mathit{HF}^*(L_0,L_1,\delta_-,\epsilon) \longrightarrow
\mathit{HF}^*(L_0,L_1,\delta_+,\epsilon) \longrightarrow \mathit{HF}^*(L_{0,M},L_{1,M}) \rightarrow \cdots
\end{equation}
where the third group is the Floer cohomology of the associated closed Lagrangian submanifolds \eqref{eq:l-m} in the fibre $M$.
\end{lemma}
Finally, we have an analogue of Proposition \ref{th:bv-vanish}, also given here without proof:
\begin{lemma} \label{th:albers}
For $\delta>0$, $\mathit{HF}^*(L,L,\delta,\epsilon) \cong H^*(L)$.
\end{lemma}
\begin{remark} \label{th:spin}
In general, $\mathit{HF}^*(L_0,L_1,\delta,\epsilon)$ is a ${\mathbb Z}/2$-graded space over a coefficient field ${\mathbb K}$ of characteristic $2$. In the situation from Setup \ref{th:setup-l}(ii), one can obtain ${\mathbb Z}$-graded Floer cohomology groups; and in that of Setup \ref{th:setup-l}(iii), an arbitrary coefficient field ${\mathbb K}$ can be allowed (see \cite[Chapter 8]{fooo} or \cite[Section 12]{seidel04}).
\end{remark}
To define the chain complex $\mathit{CF}^*(L_0,L_1,H)$ underlying $\mathit{HF}^*(L_0,L_1,\delta,\epsilon)$, one chooses functions $H = (H_t)$ and almost complex structures $J = (J_t)$ as in Setup \ref{th:setup-t}, but this time parametrized by $t \in [0,1]$. On the path space
\begin{equation} \label{eq:path-space}
\EuScript L_{L_0,L_1} = \{x: [0,1] \rightarrow E \,:\, x(0) \in L_0, \; x(1) \in L_1\},
\end{equation}
one has a counterpart of \eqref{eq:phi-action}:
\begin{equation} \label{eq:lagrangian-action}
A_{L_0,L_1,H}(x) = \Big( \int_0^1 -x^*\theta_E + H_t(x(t)) \, dt \Big) + G_{L_1}(x(1)) - G_{L_0}(x(0)),
\end{equation}
where the $G_{L_k}$ are functions such that $dG_{L_k} = \theta_E|L_k$. The critical points are solutions $x$ of \eqref{eq:periodic-y} in \eqref{eq:path-space}, hence correspond to points of $\phi_H^1(L_0) \cap L_1$. The condition \eqref{eq:delta-inequality} implies that all such $x$ are contained in a compact subset of $E$. Moreover, for generic choice of $H$, the $x$ will be nondegenerate. Assuming this to be the case, one considers solutions $u: {\mathbb R} \times [0,1] \rightarrow E$ of \eqref{eq:floer}, with boundary conditions
\begin{equation} \label{eq:floer-boundary}
u(s,0) \in L_0, \;\; u(s,1) \in L_1.
\end{equation}
There is an analogue of Proposition \ref{th:floer-bound} in this context, with similar strategy of proof. The rest of the construction of $\mathit{HF}^*(L_0,L_1,\delta,\epsilon)$ follows the classical theory for closed Lagrangian submanifolds \cite{floer88c}.
\begin{remark} \label{th:referee}
It may make sense to mention the one point where the situation here differs from that in Proposition \ref{th:floer-bound} (by being easier, in fact). There is no analogue of \eqref{eq:no-asymptotic-fixed-points} in the present context, but that is in fact unnecessary: because of \eqref{eq:floer-boundary}, a bound on $\|du_\infty\|$ (obtained as in Lemma \ref{th:gromov}) implies that the image of $u$ is contained in a region $\{\mathrm{im}(\pi) \leq D\}$. (The reader may also want to consult the proof of \cite[Proposition 5.1]{seidel12b}, even though that result itself does not apply here, because of the different choice of inhomogeneous terms.)
\end{remark}
\subsection{The formalism\label{subsec:formalism}}
We will now introduce the additional structures on Floer cohomology which underlie our main argument (for the moment, only the formal aspects will be considered; discussion of their actual construction will take place later on, in Section \ref{subsec:cr}).
Given Lagrangian submanifolds $(L_0,L_1,L_2)$ and constants $(\delta_0,\delta_1)$, $(\epsilon_0,\epsilon_1)$ such that all Floer cohomology groups involved are well-defined, the triangle product (due to Donaldson) is a map
\begin{equation} \label{eq:triangle-product}
\mathit{HF}^*(L_1,L_2,\delta_1,\epsilon_1) \otimes \mathit{HF}^*(L_0,L_1,\delta_0,\epsilon_0) \longrightarrow \mathit{HF}^*(L_0,L_2,\delta_0+\delta_1,\epsilon_0+\epsilon_1).
\end{equation}
This satisfies an appropriate associativity condition. As an application, fix Lagrangian submanifolds $L_1,\dots,L_m$ with pairwise different constants $o_1, \dots, o_m \in {\mathbb R}$. To these, one can associate an algebra $A$ over \eqref{eq:semisimple}, namely
\begin{equation} \label{eq:directed-1}
A = R \oplus \bigoplus_{i<j} \mathit{HF}^*(L_i,L_j),
\end{equation}
or equivalently
\begin{equation} \label{eq:directed-2}
e_j A e_i = \begin{cases} \mathit{HF}^*(L_i,L_j) & i<j, \\ {\mathbb K} & i=j, \\ 0 & i>j.
\end{cases}
\end{equation}
Recall that here, the Floer cohomology groups under discussion are $\mathit{HF}^*(L_i,L_j,\delta,\epsilon)$ with $\delta = \epsilon = 0$ (in view of Lemmas \ref{th:lag-1} and \ref{th:lag-2}, one could equivalently use any $\delta$ which lies on the same side of $o_j-o_i$ as the origin, and any $\epsilon$, but that it not helpful for thinking about the products). The nontrivial part of the algebra structure of $A$ consists of the products $\mathit{HF}^*(L_j,L_k) \otimes \mathit{HF}^*(L_i,L_j) \rightarrow \mathit{HF}^*(L_i,L_k)$ for $i<j<k$, which are special cases of \eqref{eq:triangle-product}. Next, take an automorphism $\phi$ as in Setup \ref{th:setup-t}, and constants $\delta$, $\epsilon$ satisfying
\begin{equation} \label{eq:lambda-difference}
\delta \neq o_i - o_j \quad \text{for all $i,j \in \{1,\dots,m\}$ (including $i = j$, which means $\delta \neq 0$).}
\end{equation}
Generalizing \cite[Section 6.3]{seidel12b}, one associates to this a bimodule $P_{\phi,\delta,\epsilon}$ over $A$, namely:
\begin{equation}
P_{\phi,\delta,\epsilon} = \bigoplus_{i,j} \mathit{HF}^*(\phi(L_i),L_j,\delta,\epsilon).
\end{equation}
These structures have cochain level refinements: an $A_\infty$-algebra $\EuScript A$ (following Fukaya), and $A_\infty$-bimodules $\EuScript P_{\phi,\delta,\epsilon}$ over $\EuScript A$. There are corresponding refinements of the properties of Lagrangian Floer cohomology mentioned above. Namely, there is a quasi-isomorphism (in fact, if the choices are suitably coordinated, an isomorphism)
\begin{equation} \label{eq:dual-bimodule}
\EuScript P_{\phi,\delta,\epsilon} \simeq \EuScript P_{\phi^{-1},-\delta,-\epsilon}^\vee[-n].
\end{equation}
\begin{lemma}
The quasi-isomorphism type of $\EuScript P_{\phi,\delta,\epsilon}$ is independent of $\epsilon$. It is also locally constant in $\delta$, within the allowed range \eqref{eq:lambda-difference}.
\end{lemma}
This is a refinement of the previously mentioned invariance properties of Lagrangian Floer cohomology (Lemmas \ref{th:lag-1}, \ref{th:lag-2}). We omit the proof.
\begin{example}
In our eventual application, $(L_1,\dots,L_m)$ will be a basis of Lefschetz thimbles. In that case, if one takes $\delta \gg 0$, then $\EuScript P_{\mathit{id},\delta,\epsilon}$ is quasi-isomorphic to the diagonal bimodule $\EuScript A$ \cite[Corollary 6.1]{seidel12b}; and if one takes $\delta \ll 0$, $\EuScript P_{\mathit{id},\delta,\epsilon}$ is quasi-isomorphic to the shifted dual diagonal bimodule $\EuScript A^\vee[-n]$ \cite[Corollary 6.2]{seidel12b}.
\end{example}
For $\phi$ as before, take $(\delta,\epsilon)$ so that $\mathit{HF}^*(\phi,\delta,\epsilon)$ is well-defined. Given any $L$, one has a canonical map relating the two kinds of Floer cohomology groups:
\begin{equation} \label{eq:oc0}
\mathit{HF}^*(\phi(L_i),L_i,\delta,\epsilon) \longrightarrow \mathit{HF}^{*+n}(\phi,\delta,\epsilon).
\end{equation}
After applying \eqref{eq:translation-duality}, \eqref{eq:lagrangian-floer-duality} and changing notation from $(\phi,\delta,\epsilon)$ to $(\phi^{-1},-\delta,\epsilon)$, the map dual to \eqref{eq:oc0} can be written as
\begin{equation} \label{eq:dual-oc0}
\mathit{HF}^*(\phi,\delta,\epsilon) \longrightarrow \mathit{HF}^*(\phi(L_i),L_i,\delta,\epsilon).
\end{equation}
\begin{example}
Let's specialize to $\phi = \mathit{id}$ and $\delta >0$. Applying Lemmas \ref{th:albers} and \ref{th:vanishing-homology}, one sees that \eqref{eq:oc0} reduces to a map
\begin{equation} \label{eq:i-1}
H^*(L_i) \longrightarrow H^{*+n}(E,\{\mathrm{re}(\pi) \ll 0\}),
\end{equation}
Let's instead take $\delta < 0$. Then, because of \eqref{eq:lagrangian-floer-duality} and \eqref{eq:translation-duality}, the map \eqref{eq:oc0} looks like this:
\begin{equation} \label{eq:i-2}
H_*(L_i) \longrightarrow H_*(E,\{\mathrm{re}(\pi) \ll 0\}),
\end{equation}
and its dual \eqref{eq:dual-oc0} is correspondingly
\begin{equation} \label{eq:i-3}
H^*(E,\{\mathrm{re}(\pi) \ll 0\}) \longrightarrow H^*(L_i).
\end{equation}
Unsurprisingly, \eqref{eq:i-1} and \eqref{eq:i-3} can be identified with the ordinary pushforward and restriction maps in cohomology; in particular, \eqref{eq:i-3} factors through $H^*(E)$.
\end{example}
As before, \eqref{eq:oc0} is part of a more complicated structure, the {\em open-closed string map}
\begin{equation} \label{eq:open-closed-string-map}
\mathit{HH}_*(\EuScript A,\EuScript P_{\phi,\delta,\epsilon}) \longrightarrow \mathit{HF}^{*+n}(\phi,\delta,\epsilon),
\end{equation}
which is defined for all $(\delta,\epsilon)$ such that both sides make sense. Let's use the same Floer-theoretic duality as before, as well as \eqref{eq:second-hochschild} and \eqref{eq:dual-bimodule}. After applying that, and suitably adjusting notation, one can write the dual of \eqref{eq:open-closed-string-map} as
\begin{equation} \label{eq:dual-open-closed-string-map}
\mathit{HF}^*(\phi,\delta,\epsilon) \longrightarrow
\mathit{HH}^*(\EuScript A,\EuScript P_{\phi,\delta,\epsilon}) = H^*(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript A, \EuScript P_{\phi,\delta,\epsilon})).
\end{equation}
\begin{remark}
The composition of \eqref{eq:open-closed-string-map} and \eqref{eq:dual-open-closed-string-map} yields a map from Hochschild homology to Hochschild cohomology (of degree $n$), for any bimodule $\EuScript P_{\phi,\delta,\epsilon}$. Taking into account \eqref{eq:serre-hochschild} (since $\EuScript A$ is directed, it is homologically smooth), one can write that as
\begin{equation} \label{eq:our-map}
H^*(\mathit{hom}_{[\EuScript A,\EuScript A]}((\EuScript A^\vee)^{-1}, \EuScript P_{\phi,\delta,\epsilon}))
\longrightarrow
H^{*+n}(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript A,\EuScript P_{\phi,\delta,\epsilon})).
\end{equation}
Generally speaking, one natural source of such homomorphisms are bimodule maps $\EuScript A \rightarrow (\EuScript A^\vee)^{-1}$ of degree $n$, or equivalently bimodule maps $\EuScript A^\vee[-n] \rightarrow \EuScript A$. At least in the case of Lefschetz fibrations, to be discussed later (and assuming $\delta \gg 0$), it seems likely that \eqref{eq:our-map} is induced by the bimodule map we have called $\rho$ (Lemma \ref{th:old}).
\end{remark}
We will also need a variation of \eqref{eq:open-closed-string-map} which involves two automorphisms $\phi_k$ ($k = 0,1$), with their associated bimodules $\EuScript P_{\phi_k,\delta_k,\epsilon_k}$. For simplicity, we assume that $\epsilon_0/\delta_0 = \epsilon_1/\delta_1$. Set
\begin{equation} \label{eq:product-phi}
\left\{
\begin{aligned}
& \phi = \phi_1\,\phi_0, \\
& \delta = \delta_0 + \delta_1, \\
& \epsilon = \epsilon_0 + \epsilon_1.
\end{aligned}
\right.
\end{equation}
We suppose that $\mathit{HF}^*(\phi,\delta,\epsilon)$ is defined. Then, the new version of the open-closed string map has the form
\begin{equation} \label{eq:double-open-closed}
\mathit{HH}_*(\EuScript A, \EuScript P_{\phi_1,\delta_1,\epsilon_1} \otimes_{\EuScript A} \EuScript P_{\phi_0,\delta_0,\epsilon_0}) \longrightarrow \mathit{HF}^{*+n}(\phi,\delta,\epsilon).
\end{equation}
This is the same kind of construction as the ``two-pointed open-closed string maps'' of \cite[Section 5.6]{ganatra13}. In parallel with \eqref{eq:dual-open-closed-string-map}, but this time using \eqref{eq:general-hochschild}, one can write the dual of \eqref{eq:double-open-closed} as
\begin{equation} \label{eq:dual-double}
\mathit{HF}^*(\phi,\delta,\epsilon) \longrightarrow H^*\big(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P_{\phi_0^{-1},-\delta_0,-\epsilon_0}, \EuScript P_{\phi_1,\delta_1,\epsilon_1})\big).
\end{equation}
\begin{remark} \label{th:consider-double}
One can think of \eqref{eq:double-open-closed} as follows. Write $\EuScript P_k = \EuScript P_{\phi_k,\delta_k,\epsilon_k}$ and $\EuScript P = \EuScript P_{\phi,\delta,\epsilon}$, assuming that the latter is defined. The triangle products
\begin{equation}
\begin{aligned} &
\mathit{HF}^*(\phi_1(L_j),L_k,\delta_1,\epsilon_1) \otimes \mathit{HF}^*(\phi_0(L_i),L_j,\delta_0,\epsilon_0) \\
& \quad \cong \mathit{HF}^*(\phi_1(L_j),L_k,\delta_1,\epsilon_1) \otimes
\mathit{HF}^*(\phi(L_i),\phi_1(L_j),\delta_0,\epsilon_0)
\longrightarrow \mathit{HF}^*(\phi(L_i),L_k,\delta,\epsilon)
\end{aligned}
\end{equation}
can be lifted to an $A_\infty$-bimodule homomorphism
\begin{equation} \label{eq:bimodule-tensor}
\EuScript P_1 \otimes_{\EuScript A} \EuScript P_0 \longrightarrow \EuScript P.
\end{equation}
One can reduce \eqref{eq:double-open-closed} to \eqref{eq:open-closed-string-map} for $\phi$, by writing it as the composition
\begin{equation}
\mathit{HH}_*(\EuScript A, \EuScript P_1 \otimes_{\EuScript A} \EuScript P_0) \longrightarrow
\mathit{HH}_*(\EuScript A,\EuScript P) \longrightarrow \mathit{HF}^{*+n}(\phi,\delta,\epsilon),
\end{equation}
where the first map is induced by \eqref{eq:bimodule-tensor} (\cite{ganatra13} uses the same idea to relate ``two-pointed open-closed string maps'' to ordinary ``one-pointed'' ones). However, we will prefer a direct definition of \eqref{eq:double-open-closed}, since that is a little simpler, and we have no use for the maps \eqref{eq:bimodule-tensor} by themselves.
\end{remark}
\begin{remark} \label{th:swap-factors}
Let's slightly change notation, and assume that $(\phi,\delta,\epsilon)$ is given, with the property that $\EuScript P_{\phi,\delta,\epsilon}$ and $\mathit{HF}^*(\phi^2,2\delta,2\epsilon)$ are well-defined. Then, \eqref{eq:double-open-closed} specializes to yield a map
\begin{equation} \label{eq:square-map}
\mathit{HH}_*(\EuScript A,\EuScript P_{\phi,\delta,\epsilon}^{\otimes_{\EuScript A} 2}) \longrightarrow \mathit{HF}^{*+n}(\phi^2,2\delta,2\epsilon).
\end{equation}
The left hand side carries a natural ${\mathbb Z}/2$-action, as discussed in Section \ref{subsec:hochschild-homology}; so does the right hand side, because it is the Floer cohomology of a square (Remark \ref{th:circle-and-discrete-actions}); and \eqref{eq:square-map} is compatible with those actions (essentially, because Figure \ref{fig:disc2} is rotationally symmetric).
\end{remark}
\subsection{Cauchy-Riemann equations\label{subsec:cr}}
The previously mentioned algebraic structures belong to a kind of TCFT (Topological Conformal Field Theory), which means a general framework of algebraic operations parametrized by Riemann surfaces. We will now outline some of the ingredients of this construction. Compared to similar ideas in the literature, the main difference is that we allow nontrivial monodromy around interior punctures on the Riemann surfaces. In other respects, the exposition here is actually somewhat restrictive: for instance, unlike \cite{abouzaid-seidel07}, we only allow closed one-forms on our Riemann surfaces.
\begin{setup} \label{th:setup-s}
(i) Fix constants $(\epsilon,\delta)$. A {\em worldsheet} is a connected non-compact Riemann surface $S$, possibly with boundary, with the following properties and additional data.
We assume that there is a compactification $\bar{S} = S \sqcup \Sigma$, obtained by adding a finite set of points. We divide the points of this finite set $\Sigma$ into closed string (interior) and open string (boundary) ones, and (independently, and arbitrarily) into inputs and outputs, denoting the respective subsets by $\Sigma^{\mathit{cl}/\mathit{op},\mathit{in}/\mathit{out}}$. These should come with local holomorphic coordinates (tubular and strip-like ends), of the form
\begin{equation} \label{eq:ends}
\begin{cases}
\epsilon_\zeta: (-\infty,0] \times S^1 \longrightarrow S, & \zeta \in \Sigma^{\mathit{cl},\mathit{out}}, \\
\epsilon_\zeta: [0,\infty) \times S^1 \longrightarrow S, & \zeta \in \Sigma^{\mathit{cl},\mathit{in}}, \\
\epsilon_\zeta: (-\infty,0] \times [0,1] \longrightarrow S, & \zeta \in \Sigma^{\mathit{op},\mathit{out}}, \\
\epsilon_\zeta: [0,\infty) \times [0,1] \longrightarrow S, &
\zeta \in \Sigma^{\mathit{op},\mathit{in}}.
\end{cases}
\end{equation}
Let $\tilde{S} \rightarrow S$ be the universal covering. We fix lifts of \eqref{eq:ends}, of the form
\begin{equation}
\begin{cases}
\tilde\epsilon_\zeta: (-\infty,0] \times {\mathbb R} \longrightarrow \tilde{S}, & \zeta \in \Sigma^{\mathit{cl},\mathit{out}}, \\
\tilde\epsilon_\zeta: [0,\infty) \times {\mathbb R} \longrightarrow \tilde{S}, & \zeta \in \Sigma^{\mathit{cl},\mathit{in}}, \\
\tilde\epsilon_\zeta: (-\infty,0] \times [0,1] \longrightarrow \tilde{S}, & \zeta \in \Sigma^{\mathit{op},\mathit{out}}, \\
\tilde\epsilon_\zeta: [0,\infty) \times [0,1] \longrightarrow \tilde{S}, &
\zeta \in \Sigma^{\mathit{op},\mathit{in}}.
\end{cases}
\end{equation}
Write $\Gamma \cong \pi_1(S)$ for the covering group of $\tilde{S} \rightarrow S$. We want to have a homomorphism
\begin{equation} \label{eq:monodromy}
\Phi: \Gamma \longrightarrow \mathit{Symp}(E),
\end{equation}
which takes values in the subgroup of those symplectic automorphisms of $E$ described in Setup \ref{th:setup-t}. In particular, given $\zeta \in \Sigma^{\mathit{cl}}$, passing from $\tilde\epsilon_\zeta(s,t+1)$ to $\tilde\epsilon_\zeta(s,t)$ amounts to acting by an element of $\Gamma$, which determines an automorphism $\phi_\zeta$ by \eqref{eq:monodromy}.
To each point $\tilde{z} \in \partial \tilde{S}$ we want to associate a Lagrangian submanifold $L_{\tilde{z}}$ as in Setup \ref{th:setup-l}, in a way which is locally constant in $z$, and compatible with \eqref{eq:monodromy}:
\begin{equation} \label{eq:equivariant-lagrangian}
L_{\gamma(\tilde{z})} = \Phi(\gamma)(L_{\tilde{z}}).
\end{equation}
In particular, given $\zeta \in \Sigma^{\mathit{op}}$, we have distinguished Lagrangian submanifolds $(L_{\zeta,0}, L_{\zeta,1})$ which are associated to the points $(\tilde{\epsilon}_{\zeta}(s,0), \tilde\epsilon_\zeta(s,1))$. By definition, these come with real numbers \eqref{eq:lambda}, which we denote by $(o_{\zeta,0},o_{\zeta,1})$.
Finally, we want $S$ to carry a closed one-form $\beta_S$ with $\beta_S|\partial S = 0$, and which over each end \eqref{eq:ends} satisfies
\begin{equation}
\epsilon_\zeta^* \beta_S = \beta_\zeta \mathit{dt},
\end{equation}
where the $\beta_\zeta$ are constants. Set $\delta_\zeta = \beta_\zeta \delta$, $\epsilon_\zeta = \beta_\zeta \epsilon$. We require that for $\zeta \in \Sigma^{\mathit{cl}}$, $\epsilon_\zeta$ should satisfy \eqref{eq:no-reeb}, and $\delta_\zeta \neq 0$; while for $\zeta \in \Sigma^{\mathit{op}}$, the pair $(L_{\zeta,0},L_{\zeta,1})$ and the number $\delta_\zeta$ must satisfy \eqref{eq:delta-inequality}.
(ii) Given $S$, we consider families $J_S = (J_{S,\tilde{z}})$ of almost complex structures in the class from Setup \ref{th:setup-t}, para\-me\-trized by $\tilde{z} \in \tilde{S}$. These should be equivariant with respect to \eqref{eq:monodromy}, and over the strip-like ends they should be invariant under translation in the first variable. Concretely, this means that there are families $J_\zeta = (J_{\zeta,t})$ such that
\begin{equation} \label{eq:j-zeta}
J_{S,\tilde\epsilon_\zeta(s,t)} = J_{\zeta,t};
\end{equation}
and if $\zeta \in \Sigma^{\mathit{cl}}$, the associated family $J_\zeta$ satisfies a periodicity condition as in \eqref{eq:phi-periodicity} with respect to $\phi_\zeta$.
(iii) We will equip $S$ with an {\em inhomogeneous term} $K_S$, which is a one-form on $\tilde{S}$ with values in the space of functions on $E$. More precisely, for any tangent vector $\xi$, $K_S(\xi) \in C^\infty(E,{\mathbb R})$ belongs to the class from Setup \ref{th:setup-t} for the constants $(\beta_S(\xi) \delta, \beta_S(\xi)\epsilon )$. This family must be equivariant with respect to \eqref{eq:monodromy}, and if $\xi$ is tangent to $\partial S$, then $K_S(\xi)|L_{\tilde{z}} = 0$. Finally, over each end, $\epsilon_\zeta^*K_S$ is invariant under translation in the first variable, and satisfies $(\epsilon_\zeta^* K_S)(\partial_s) = 0$. In analogy with \eqref{eq:j-zeta}, one can therefore write
\begin{equation} \label{eq:k-zeta}
\tilde\epsilon_\zeta^* K_S = H_{\zeta,t} \, \mathit{dt};
\end{equation}
and if $\zeta \in \Sigma^{\mathit{cl}}$, $H_\zeta$ again satisfies \eqref{eq:phi-periodicity}.
\end{setup}
We usually refer to a pair $(J_S,K_S)$ as a {\em perturbation datum}, since it specifies a particular pertubed Cauchy-Riemann equation on $S$. Concretely, this is an equation for maps
\begin{equation} \label{eq:tilde-u}
\left\{\begin{aligned}
& \tilde{u}: \tilde{S} \longrightarrow E, \\
& \tilde{u}(\gamma(\tilde{z})) = \Phi(\gamma)(\tilde{u}(\tilde{z})), \\
& \tilde{u}(\tilde{z}) \in L_{\tilde{z}} \quad \text{for $\tilde{z} \in \partial \tilde{S}$},
\end{aligned}
\right.
\end{equation}
namely:
\begin{equation} \label{eq:generalized-floer}
(d\tilde{u} - Y_{K_S})^{0,1} = 0.
\end{equation}
Here, $Y_{K_S}$ is the one-form on $\tilde{S}$ with values in $C^\infty(TE)$ associated to $K_S$ (by passing from functions to Hamiltonian vector fields). We use the complex structure on $S$, and the almost complex structures $J_{S,\tilde{z}}$, to form the $(0,1)$-part of the linear map $d\tilde{u} - Y_{K_S}: TS_{\tilde{z}} \longrightarrow TE_{\tilde{u}(\tilde{z})}$. One checks readily that \eqref{eq:generalized-floer} is invariant under the twisted $\Gamma$-periodicity condition from \eqref{eq:tilde-u}. Over the ends, which means for $u_\zeta(s,t) = \tilde{u}(\tilde{\epsilon}_\zeta(s,t))$, \eqref{eq:generalized-floer} reduces to equations for suitable Floer trajectories:
\begin{equation} \label{eq:end-floer}
\left\{
\begin{aligned}
& u_\zeta(s,k) \in L_{\zeta,k} \quad \text{for $k = 0,1$, if $\zeta \in \Sigma^{\mathit{op}}$}, \\
& u_\zeta(s,t) = \phi_\zeta(u_\zeta(s,t+1)) \quad \text{if $\zeta \in \Sigma^{\mathit{cl}}$}, \\
& \partial_s u_\zeta + J_{\zeta,t}(\partial_t u_\zeta - X_{H_\zeta,t}) = 0.
\end{aligned}
\right.
\end{equation}
It therefore makes sense to impose convergence conditions
\begin{equation} \label{eq:generalized-limit}
\textstyle \lim_{s \rightarrow \pm\infty} u_\zeta(s,\cdot) = x_\zeta.
\end{equation}
The necessary compactness argument for the moduli space of solutions of \eqref{eq:generalized-floer} combines: an overall energy bound; a maximum principle argument in horizontal direction over the base (as in Lemma \ref{th:base-convexity}); and the following idea. Let's fix a metric on $S$ which is standard over the ends. Then there is a constant $c$ such that each point of $S$ either (i) lies within distance $c$ of $\partial S$, or else (ii) lies on a tubular end (the image of $\epsilon_\zeta$ for some $\zeta \in \Sigma^{\mathit{cl}}$). For preimages $\tilde{z}$ of points as in (i), one can bound $|\pi(\tilde{u}(\tilde{z}))|$ as in Lemma \ref{th:gromov}; whereas in case (ii), one uses an argument as in Lemma \ref{th:b-energy}. Taken together, these ingredients yield an analogue of Proposition \ref{th:floer-bound}. Transversality is straightforward, given the freedom to choose $J_S$ and $K_S$.
Counting solutions of \eqref{eq:generalized-floer}, \eqref{eq:generalized-limit} for a single Riemann surface $S$ yields a map between Floer cochain spaces, which induces a cohomology level map of degree $n(-\chi(\bar{S}) + |\Sigma^{\mathit{out,op}}| + 2|\Sigma^{\mathit{out,cl}}|)$:
\begin{equation} \label{eq:tqft}
\xymatrix{
\displaystyle
\bigotimes_{\zeta \in \Sigma^{\mathit{in},\mathit{op}}} \mathit{HF}^*(L_{\zeta,0},L_{\zeta,1},\delta_\zeta,\epsilon_\zeta) \otimes \bigotimes_{\zeta \in \Sigma^{\mathit{in},\mathit{cl}}} \mathit{HF}^*(\phi_\zeta,\delta_\zeta,\epsilon_\zeta)
\ar[d] \\
\displaystyle
\bigotimes_{\zeta \in \Sigma^{\mathit{out},\mathit{op}}} \mathit{HF}^*(L_{\zeta,0},L_{\zeta,1},\delta_\zeta,\epsilon_\zeta) \otimes
\bigotimes_{\zeta \in \Sigma^{\mathit{out},\mathit{cl}}}
\mathit{HF}^*(\phi_\zeta,\delta_\zeta,\epsilon_\zeta).
}
\end{equation}
\begin{remark}
A suitable analogue of Remarks \ref{th:signs-and-grading} and \ref{th:spin} applies. In general, \eqref{eq:tqft} will be a ${\mathbb Z}/2$-graded map, and defined over a coefficient field with $\mathrm{char}({\mathbb K}) = 2$. However, if one assumes that: $E$ carries a symplectic Calabi-Yau structure; that \eqref{eq:monodromy} comes with a lift to the graded symplectic automorphism group; and that all boundary conditions are graded Lagrangian submanifolds; then \eqref{eq:tqft} will have the specified degree with respect to the ${\mathbb Z}$-gradings of the Floer cohomology groups involved. Moreover, if the Lagrangian submanifolds are {\em Spin}, arbitrary ${\mathbb K}$ are allowed.
\end{remark}
So far, what we have explained is a version of the TQFT formalism for Floer theory. For the extension to a TCFT, which is relevant for us, one has to include families of Riemann surfaces. Strictly speaking, a TCFT framework would have to allow a class of such families which is large enough to form a chain level model for the homology of the relevant compactified moduli spaces of Riemann surfaces, and to allow appropriate composition (operad) structures. For our purpose, only certain specific families are needed; there, the underlying analysis remains the same as for the TQFT case (except that the gluing theory has to be carried out in a parametrized sense, which is something that can be regarded as well-understood).
\subsection{Moduli spaces}
As before, we start with $(L_1,\dots,L_m)$ having pairwise different constants $(o_1,\dots,o_m)$. For $1 \leq i < j \leq m$, fix the additional data (functions $H_{ij}$, and almost complex structures $J_{ij}$) needed to define the Floer cochain complexes $\mathit{CF}^*(L_i,L_j,H_{ij})$. To obtain an $A_\infty$-algebra structure, consider worldsheets $S$ which are discs with $d+1 \geq 3$ boundary punctures. We equip the components of $\partial S$ with boundary conditions $L_{i_0},\dots,L_{i_d}$, for $i_0 < \cdots < i_d$ (in positive order around $\partial \bar{S}$). The one-forms $\beta_S$ are taken to be zero throughout. The perturbation data $(J_S,K_S)$ need to be chosen so that their restriction to the strip-like ends, as in \eqref{eq:j-zeta} and \eqref{eq:k-zeta}, reduces to the choices previously made to define Floer theory. Finally, the choices need to depend smoothly on the moduli of $S$, and there are conditions about their behaviour as this surface degenerates. We will not give any details, referring instead to \cite[Section 9]{seidel04}. The outcome are maps
\begin{equation}
\mathit{CF}^*(L_{i_{d-1}},L_{i_d},H_{i_{d-1}i_d}) \otimes \cdots
\otimes \mathit{CF}^*(L_{i_0},L_{i_1},H_{i_0i_1}) \longrightarrow
\mathit{CF}^{*+2-d}(L_{i_0},L_{i_d},H_{i_0i_d}),
\end{equation}
which satisfy the $A_\infty$-associativity equations. To define the cochain level structure underlying \eqref{eq:directed-1}, one extends these operations in the unique way to a strictly unital $A_\infty$-structure on
\begin{equation}
\EuScript A = R \oplus \bigoplus_{i<j} \mathit{CF}^*(L_i,L_j,H_{ij}).
\end{equation}
Fix an automorphism $\phi$. To define $\EuScript P = \EuScript P_{\phi,\delta,\epsilon}$, one chooses Hamiltonians $H_{\phi,ij}$ and almost complex structures $J_{\phi,ij}$ for all $(i,j)$, and then sets
\begin{equation} \label{eq:p-space}
\EuScript P = \bigoplus_{ij} \mathit{CF}^*(\phi(L_i),L_j,H_{\phi,ij}),
\end{equation}
with $\mu^{0;1;0}_{\EuScript P}$ the direct sum of Floer differentials. Fundamentally, the moduli spaces of Riemann surfaces which define the higher operations $\mu^{s;1;r}_{\EuScript P}$ are the same (Stasheff polyhedra) as for $\mu^{r+1+s}_{\EuScript A}$. However, the other data they carry are different, and we find it convenient to think of the Riemann surfaces themselves in a slightly different way, namely to write them as
\begin{equation} \label{eq:s-strip}
S = T \setminus \{\zeta_1,\dots,\zeta_r,\zeta_1',\dots,\zeta_s'\},
\end{equation}
where: $T = {\mathbb R} \times [0,1]$; the $\zeta_i \in {\mathbb R} \times \{0\} \subset \partial T$ are increasing; and the $\zeta_i' \in {\mathbb R} \times \{1\} \subset \partial T$ are decreasing. The one-forms $\beta_S$ should be equal to $\mathit{dt}$ on the region where $|s| \gg 0$, and should vanish sufficiently close to the $\zeta_i$ and $\zeta_i'$. The boundary conditions are
\begin{equation} \label{eq:boundary-0}
(\phi(L_{i_0}),\dots,\phi(L_{i_r})) \quad \text{along ${\mathbb R} \times \{0\}$}
\end{equation}
(as $s$ increases, and where $i_0 < \cdots < i_r$), respectively
\begin{equation} \label{eq:boundary-1}
(L_{i_0'},\dots,L_{i_s'}) \quad \text{along ${\mathbb R} \times \{1\}$}
\end{equation}
(as $s$ decreases, and where again $i_0' < \cdots < i_s'$). The inhomogeneous terms and almost complex structures are determined by those chosen for \eqref{eq:p-space} over the ends where $|s| \gg 0$, and by the choices made for $\EuScript A$ on the remaining ends. As we vary over all possible $S$, the outcome are operations
\begin{equation}
\begin{CD}
\mathit{CF}^*(L_{i_{s-1}'},L_{i_s'},H_{i_{s-1}'i_s'}) \otimes \cdots \otimes \mathit{CF}^*(L_{i_0'},L_{i_1'},H_{i_0'i_1'}) \\
\otimes \, \mathit{CF}^*(\phi(L_{i_r}),L_{i_0'},H_{\phi,i_r i_0'})\, \otimes \\
\mathit{CF}^*(L_{i_{r-1}},L_{i_r},H_{i_{r-1}i_r}) \otimes \cdots \otimes
\mathit{CF}^*(L_{i_0},L_{i_1},H_{i_0i_1})
\\
@VVV
\\
\mathit{CF}^{*+1-r-s}(\phi(L_{i_0}),L_{i_s'},H_{\phi,i_0 i_s'}),
\end{CD}
\end{equation}
which (extended over the units in $\EuScript A$ in the obvious way) constitute $\mu_{\EuScript P}^{s;1;r}$. It may be convenient to think of \eqref{eq:s-strip} as glued together from two half-strips $S_{\pm} = \{(s,t) \in S \;:\; \pm t \leq \half\}$, each of which carries boundary conditions taken from the $L_k$, but where the gluing identifies the two target spaces $E$ using $\phi$. This amounts to rewriting the relevant equation \eqref{eq:generalized-floer} as two parts $u_\pm: S_\pm \rightarrow E$, joined by a ``seam'' $u_+(s,\half) = \phi(u_-(s,\half))$. Mathematically, this does not change anything, but it can be useful as an aid to the intuition, since it makes the connection with quilted Floer cohomology \cite{wehrheim-woodward10} (Figure \ref{fig:half-strips}).
\begin{figure}
\caption{\label{fig:half-strips}
\label{fig:half-strips}
\end{figure}
Next, consider surfaces of the form
\begin{equation} \label{eq:h-punctured}
S = H \setminus \{\zeta_*,\zeta_1,\dots,\zeta_d\}
\end{equation}
where $H \subset {\mathbb C}$ is the closed upper half-plane, from which we remove a fixed interior point $\zeta_*$ (say $\zeta_* = i$) as well as boundary points $\zeta_1 < \cdots < \zeta_d$. We consider the $\zeta_k$ as inputs, and $\zeta_*$ as an output. The one-form $\beta_S$ should vanish near the $\zeta_k$, and its integral along a small (counterclockwise) loop around $\zeta_*$ should be equal to $1$ (in other words, $\beta_{\zeta_k} = 0$ and $\beta_{\zeta_*} = 1$). The representation \eqref{eq:monodromy} maps that same loop to $\phi$ (Figure \ref{fig:disc1} shows two equivalent pictures of this; in the right-hand one, we have drawn $H$ itself as a disc with one boundary point removed). Over $\partial H \subset H \setminus \{\zeta_*\}$, we can choose a section of the universal cover. Along that section we place the boundary conditions $L_{i_0},\dots,L_{i_d}$ with $i_0 < \cdots < i_d$, and then extend that to all of $\partial \tilde{S}$ in the unique way which satisfies \eqref{eq:equivariant-lagrangian}. The outcome of this construction are maps
\begin{equation} \label{eq:hochschild-maps}
\begin{CD}
\mathit{CF}^*(L_{i_{d-1}},L_{i_d},H_{i_{d-1}i_d}) \otimes \cdots \otimes \mathit{CF}^*(L_{i_0},L_{i_1},H_{i_0i_1}) \otimes \mathit{CF}^*(\phi(L_{i_d}),L_{i_0},H_{\phi,i_di_0}) \\
@VVV \\
\mathit{CF}^{*+n-d}(\phi,H_{\phi}),
\end{CD}
\end{equation}
where $H_\phi$ is a new Hamiltonian, chosen (together with a corresponding $J_\phi$) to form $\mathit{HF}^*(\phi,\delta,\epsilon)$. The \eqref{eq:hochschild-maps} are the components of a chain map, which induces \eqref{eq:open-closed-string-map}.
\begin{figure}
\caption{\label{fig:disc1}
\label{fig:disc1}
\end{figure}
The construction of \eqref{eq:double-open-closed} is a mixture of the previous two. One considers surfaces
\begin{equation} \label{eq:s-strip-2}
S = T \setminus \{\zeta_*,\zeta_1,\dots,\zeta_r,\zeta_1',\dots,\zeta_s'\},
\end{equation}
where $\zeta_*$ is a fixed interior point of $T$, say $\zeta_* = (0,1/2)$, and the other points are as in \eqref{eq:s-strip}. Note however that this time, both ends $s \rightarrow \pm \infty$ will be considered as inputs. The map \eqref{eq:monodromy} has monodromy $\phi = \phi_1\phi_0$ around $\zeta_*$. This structure, as well as the choice of boundary conditions, is indicated in Figure \ref{fig:disc2}. One chooses one-forms $\beta_S$ such that
\begin{equation}
\beta_S = \begin{cases} -(\epsilon_0/\epsilon)\, \mathit{dt} & s \ll 0, \\
(\epsilon_1/\epsilon)\, \mathit{dt} & s \gg 0,
\end{cases}
\end{equation}
and which vanish near the $\zeta_k, \zeta_k'$. As a consequence, the integral of $\beta_S$ along a loop around $\zeta_*$ is necessarily $\epsilon_0/\epsilon+\epsilon_1/\epsilon = 1$. The outcome are maps
\begin{equation} \label{eq:double-open-closed-components}
\begin{CD}
\mathit{CF}^*(L_{i_{s-1}'},L_{i_s'},H_{i_{s-1}'i_s'}) \otimes \cdots \otimes \mathit{CF}^*(L_{i_0'},L_{i_1'},H_{i_0'i_1'}) \\
\otimes \, \mathit{CF}^*(\phi_1(L_{i_r}),L_{i_0'},H_{\phi_0,i_r i_0'})\, \otimes \\
\mathit{CF}^*(L_{i_{r-1}},L_{i_r},H_{i_{r-1}i_r}) \otimes \cdots \otimes
\mathit{CF}^*(L_{i_0},L_{i_1},H_{i_0i_1}) \\
\; \otimes \, \mathit{CF}^*(\phi_0(L_{i_s'}),L_{i_0},H_{\phi_1,i_{s'} i_0})
\\
@VVV
\\
\mathit{CF}^{*+n-r-s}(\phi,H_{\phi}).
\end{CD}
\end{equation}
These are the components of a chain map, which induces \eqref{eq:double-open-closed}.
\begin{figure}
\caption{\label{fig:disc2}
\label{fig:disc2}
\end{figure}
\subsection{The Lefschetz condition}
What's been missing so far is a natural source of Lagrangian submanifolds, which would make the construction of $\EuScript A$ meaningful. We will provide that now, in the form of Lefschetz thimbles.
\begin{setup} \label{th:setup-lefschetz}
(i) Take an exact symplectic fibration with singularities. We call it an {\em exact symplectic Lefschetz fibration} if it satisfies the following additional conditions. First of all, $TE^v_x = \mathit{ker}(D\pi_x) \subset TE_x$ is a symplectic subspace at every regular point $x$. Secondly, near each critical point, there is an (integrable and $\omega_E$-compatible) complex structure $I_E$, such that $\pi$ is $I_E$-holomorphic, and the (complex) Hessian at the critical point is nondegenerate. Together with the previous conditions, this implies that there are only finitely many critical points. For practical bookkeeping purposes, we also impose the additional condition that there should be at most one critical point in each fibre.
(ii) Suppose that we have an exact symplectic Lefschetz fibration. A {\em basis of vanishing paths} (see Figure \ref{fig:basis}) is a collection of properly embedded half-infinite paths $l_1,\dots,l_m \subset {\mathbb C}$, whose endpoints are precisely the critical values of $\pi$, and with the following properties. There is a half-plane $\{\mathrm{re}(y) \leq C\}$ which contains all critical values of $\pi$, and such that the parts of the $l_k$ lying in that half-plane are pairwise disjoint. Outside that half-plane, we have
\begin{equation}
l_k \cap \{\mathrm{re}(y) \geq C\} = \{\mathrm{im}(y) = \iota_k(\mathrm{re}(y)) \}.
\end{equation}
Here, the functions $\iota_1,\dots,\iota_m: [C,\infty] \rightarrow {\mathbb R}$ are constant near infinity, let's say $\iota_k(r) = o_k$ for $r \gg 0$, and
\begin{equation}
\left\{
\begin{aligned}
& \iota_1(C) < \cdots < \iota_m(C), \\
& o_1 > \cdots > o_m.
\end{aligned}
\right.
\end{equation}
The corresponding {\em basis of Lefschetz thimbles} consists of the unique Lagrangian submanifolds $L_1,\dots,L_m \subset E$ such that $\pi(L_k) = l_k$. We choose orientations arbitrarily (since the Lefschetz thimbles are diffeomorphic to ${\mathbb R}^n$, they carry unique {\em Spin} structures; and if $E$ has a symplectic Calabi-Yau structure, they can be equipped with gradings).
\end{setup}
\begin{figure}
\caption{\label{fig:basis}
\label{fig:basis}
\end{figure}
\begin{conjecture} \label{th:decomposition-of-diagonal}
Let's use a basis of vanishing cycles to define the $A_\infty$-algebra $\EuScript A$ and bimodules $\EuScript P_{\phi,\delta,\epsilon}$. Then, for any automorphism $\phi$ and $\delta \gg 0$, the open-closed string map \eqref{eq:open-closed-string-map} is an isomorphism.
\end{conjecture}
This is a kind of ``decomposition of the diagonal'' statement, which is of general interest since it would give a way of computing fixed point Floer cohomology in terms of open string (Fukaya category) data. It is plausible that it could be approached by the methods from \cite{abouzaid-ganatra14}, but we will not discuss that possibility further here.
\begin{lemma} \label{th:li-floer}
Let $\nu$ be the global monodromy. For any $\delta < 0$,
$\mathit{HF}^*(\nu(L_i),L_i,\delta,\epsilon)$ is one-dimensional and concentrated in degree $0$. Moreover, for any $i<j$ and any $\delta < o_j - o_i < 0$, the triangle product
\begin{equation} \label{eq:isomorphic-triangle-product}
\mathit{HF}^*(L_i,L_j) \otimes \mathit{HF}^*(\nu(L_i),L_i,\delta,\epsilon) \longrightarrow \mathit{HF}^*(\nu(L_i),L_j,\delta,\epsilon)
\end{equation}
is an isomorphism.
\end{lemma}
\begin{proof}[Sketch of proof]
Let $H$ be the Hamiltonian used to define $\mathit{HF}^*(\nu(L_i),L_i,\delta,\epsilon)$. The generators of the underlying chain complex correspond to points of $(\phi_H^1 \circ \nu)(L_i) \cap L_i$. After a compactly supported isotopy (see Figure \ref{fig:triangle}), these two Lagrangian submanifolds will intersect in a single point, which is a critical point of $\pi$; and one easily computes that its Maslov index is $0$. For a suitable choice of auxiliary data, the product \eqref{eq:isomorphic-triangle-product} is obtained by counting holomorphic triangles in $E$ which project to the triangle in the base shaded in Figure \ref{fig:triangle}. One can in principle determine those directly, but it is easier to apply a further isotopy as indicated in Figure \ref{fig:triangle2}, after which the triangle in the base can be shrunk to a point, making the computation straightforward (the same trick is used in \cite[Figure 6]{maydanskiy-seidel09}).
\end{proof}
\begin{figure}
\caption{\label{fig:triangle}
\label{fig:triangle}
\end{figure}
\begin{figure}
\caption{\label{fig:triangle2}
\label{fig:triangle2}
\end{figure}
Take $\gamma \in (0,2\pi)$, $\delta > 0$, and a small $\epsilon>0$. Consider the map
\begin{equation} \label{eq:combined-oc}
H^*(E) \cong \mathit{HF}^*(E,\gamma,\epsilon) \longrightarrow \mathit{HF}^*(\nu,\delta,\epsilon) \longrightarrow \mathit{HF}^*(\nu(L_i),L_i,\delta,\epsilon).
\end{equation}
Here, the first isomorphism is Lemma \ref{th:bv-vanish-3}; the map after that comes from Lemma \ref{th:rotation-translation}; and the final one is \eqref{eq:dual-oc0}. Let $u_i$ be the image of $1 \in H^0(E)$ under \eqref{eq:combined-oc}.
\begin{lemma} \label{th:last-lemma}
$u_i$ is nontrivial.
\end{lemma}
\begin{proof}[Sketch of proof]
One can define a version of Lagrangian Floer homology perturbed by a rotational Hamiltonian (in parallel with Section \ref{subsec:rotate}), which we will denote by $\mathit{HF}^*(L_i,L_i,\gamma,\epsilon)$. Then, the map \eqref{eq:rotation-translation} has a Lagrangian counterpart, which fits into a commutative diagram of the form including all the maps in \eqref{eq:combined-oc}:
\begin{equation}
\xymatrix{
H^*(E) \ar[d] \ar[r] & \mathit{HF}^*(E,\gamma,\epsilon) \ar[r] \ar[d] & \mathit{HF}^*(\nu,\delta,\epsilon) \ar[d] \\
H^*(L) \ar[r] & \mathit{HF}^*(L_i,L_i,\gamma,\epsilon) \ar[r] & \mathit{HF}^*(\nu(L_i),L_i,\delta,\epsilon).
}
\end{equation}
The leftmost $\downarrow$ is the standard restriction map; and all the $\rightarrow$ in the bottom row are isomorphisms (of one-dimensional vector spaces, concentrated in degree $0$).
\end{proof}
\begin{lemma}
Suppose that $\epsilon>0$ is small, and that $\delta < o_m - o_1$. Take the element of $\mathit{HF}^0(\nu,\delta,\epsilon)$ produced from $1 \in H^0(E)$ as in \eqref{eq:combined-oc}. Applying the open-closed string map in its dual form \eqref{eq:dual-open-closed-string-map} to that element yields a quasi-isomorphism $\EuScript A \rightarrow \EuScript P_{\nu,\delta,\epsilon}$.
\end{lemma}
\begin{proof}
This is a direct application of Lemma \ref{th:recognize-diagonal} (with $\EuScript P = \EuScript P_{\nu^{-1},-\delta,-\epsilon}[n] \cong \EuScript P_{\nu,\delta,\epsilon}^\vee$). To use that criterion, one has to consider the maps
\begin{equation}
e_j H^*(\EuScript A) e_i \longrightarrow e_j H^*(\EuScript P_{\nu,\delta,\epsilon}) e_i = \mathit{HF}^*(\nu(L_i),L_j,\delta,\epsilon)
\end{equation}
given by
\begin{equation}
\left\{
\begin{aligned}
& \text{the product with $u_i$, if $i<j$;} \\
& \text{taking the unit $e_i$ to $u_i$, if $i = j$;} \\
& \text{zero, if $i>j$.}
\end{aligned}
\right.
\end{equation}
The first two cases yield an isomorphism, by Lemmas \ref{th:li-floer} and \ref{th:last-lemma}. As for the last case, it is straightforward to show that $\mathit{HF}^*(\nu(L_i),L_j,\delta,\epsilon) = 0$ for $i>j$.
\end{proof}
To summarize, we now know that
\begin{equation} \label{eq:p-dual-p}
\text{for $\delta \ll 0$,}\quad \left\{
\begin{aligned}
& \EuScript P_{\nu,\delta,\epsilon} \simeq \EuScript A, \\
& \EuScript P_{\nu^{-1},-\delta,-\epsilon} \simeq \EuScript A^\vee[-n],
\end{aligned}
\right.
\end{equation}
where the second part of the statement follows from the first one by \eqref{eq:dual-bimodule}. Note that $\epsilon$ can be arbitrary, since the bimodules are independent of $\epsilon$ up to quasi-isomorphism. Applying \eqref{eq:dual-double}, one therefore gets a map
\begin{equation} \label{eq:omg}
\begin{aligned}
\mathit{HF}^*(\nu^2,\delta,\epsilon) \longrightarrow & H^*\big(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript P_{\nu^{-1},-\delta/2,-\epsilon/2}, \EuScript P_{\nu,\delta/2,\epsilon/2})\big) \\ & \quad \cong H^*(\mathit{hom}_{[\EuScript A,\EuScript A]}(\EuScript A^\vee[-n],\EuScript A)).
\end{aligned}
\end{equation}
\begin{proof}[Proof of Theorem \ref{th:main}]
For small $\epsilon>0$, Proposition \ref{th:partial-splitting} and Lemma \ref{th:rotation-translation} yield a map
\begin{equation}
\mathit{HF}^{*+2}(\mu,\epsilon) \longrightarrow \mathit{HF}^*(\nu^2,\delta,\epsilon).
\end{equation}
One combines this with \eqref{eq:omg} to get the desired construction.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th:fano}]
This is the same argument as before, but instead of Proposition \ref{th:partial-splitting}, one uses Proposition \ref{th:partial-splitting-2} (and $1<\epsilon<2$).
\end{proof}
\begin{remark}
By comparing \eqref{eq:g-les} and Conjecture \ref{th:conjecture}, one sees that for the conjecture to hold, the bimodule map $\EuScript A^\vee \rightarrow \EuScript A$ constructed in the proof of Theorem \ref{th:fano} must necessarily be invariant under self-conjugation \eqref{eq:c-automorphism}. Equivalently by Lemma \ref{th:z-action}, if one thinks of that map as an element of $\mathit{HH}_*(\EuScript A,\EuScript A^\vee \otimes_{\EuScript A} \EuScript A^\vee)^\vee$, it must be invariant under the ${\mathbb Z}/2$-action which exchanges the two tensor factors. Going back to the geometric definition, which means using \eqref{eq:p-dual-p}, the desired statement is that the image of a specific element under the map
\begin{equation} \label{eq:zz}
\mathit{HF}^0(\nu^2,\delta,\epsilon) \longrightarrow
\mathit{HH}_0(\EuScript A, \EuScript P_{\nu^{-1},-\delta/2,-\epsilon/2} \otimes_{\EuScript A} \EuScript P_{\nu^{-1},-\delta/2,-\epsilon/2})^\vee
\end{equation}
is ${\mathbb Z}/2$-invariant. Assuming $\mathrm{char}({\mathbb K}) \neq 2$, it follows from Remark \ref{th:swap-factors} and Example \ref{th:trivial-involution} that the entire image of \eqref{eq:zz} is ${\mathbb Z}/2$-invariant.
\end{remark}
\begin{remark}
As a generalization of Conjecture \ref{th:decomposition-of-diagonal}, one could consider maps \eqref{eq:double-open-closed}, but with an arbitrary number of bimodules involved. We will be interested only in one special case, written in dual form as in \eqref{eq:dual-double}:
\begin{equation} \label{eq:i-power}
\begin{aligned}
\mathit{HF}^*(\nu^k,\delta,\epsilon) \longrightarrow & H^*\big(\mathit{hom}_{[\EuScript A,\EuScript A]}(
\EuScript P_{\nu^{-1},-\delta/k,-\epsilon/k}^{\otimes_{\EuScript A} k-1}, \EuScript P_{\nu,\delta/k,\epsilon/k})\big)
\\ & \quad \cong H^*(\mathit{hom}_{[\EuScript A,\EuScript A]}((\EuScript A^\vee)^{\otimes_{\EuScript A} k-1}, \EuScript A)),
\quad \delta \ll 0.
\end{aligned}
\end{equation}
Suppose that \eqref{eq:i-power} is an isomorphism. Let's specialize to Lefschetz fibrations coming from anticanonical Lefschetz pencils. In that case, we have seen in Example \ref{th:negative-degrees} that, for geometric reasons, the Floer cohomology groups of $\nu^k$ are concentrated in nonnegative degrees. It would then follow that \eqref{eq:no-negative-degree} is satisfied, so that Lemmas \ref{th:obstructions} and \ref{th:obstructions-2} would become applicable, leading to a proof of Conjecture \ref{th:conjecture} (however, this approach would not apply to other situations, such as that of Remark \ref{th:fractional-cy-2}).
\end{remark}
\end{document} |
\begin{document}
\title{\sc {A linear-time algorithm for the strong
chromatic index of Halin graphs}
\begin{abstract}
We show that there exists a linear-time algorithm that computes the
strong chromatic index of Halin graphs.
\end{abstract}
\section{Introduction}
\begin{definition}
Let $G=(V,E)$ be a graph. A {\em strong edge coloring\/} of $G$ is a
proper edge coloring such that no edge is adjacent to two edges of the
same color.
\end{definition}
Equivalently, a strong edge coloring of $G$ is a vertex coloring
of $L(G)^2$, the square of the linegraph of $G$.
The strong chromatic index of $G$ is the minimal integer $k$ such
that $G$ has a strong edge coloring with $k$ colors. We denote the
strong chromatic index of $G$ by $s\chi^{\prime}(G)$.
Recently it was shown that the strong chromatic index
is bounded by
\[(2-\epsilon)\Delta^2\] for some $\epsilon >0$, where
$\Delta$ is the maximal degree of the graph~\cite{kn:molloy}.
\footnote{In their paper Molloy and Reed state that $\epsilon \geq 0.002$
when $\Delta$ is sufficiently large.}
Earlier, Andersen showed that the strong chromatic index of a cubic graph
is at most ten~\cite{kn:andersen}.
Let $\mathcal{G}$ be the class of chordal graphs or the class of
cocomparability graphs. If $G \in \mathcal{G}$ then also
$L(G)^2 \in \mathcal{G}$ and it follows that the strong chromatic index
can be computed in polynomial time for these classes.
Also for graphs of bounded treewidth
there exists a polynomial time algorithm
that computes the strong chromatic
index~\cite{kn:salavatipour}.
\footnote{This algorithm checks in $O(n(s+1)^t)$ time whether a
partial $k$-tree has a strong edge coloring that uses at most $s$ colors.
Here, the exponent $t=2^{4(k+1)+1}$.}
\begin{definition}
Let $T$ be a tree without vertices of degree two. Consider a plane embedding
of $T$ and connect the leaves of $T$ by a cycle that crosses no edges of $T$.
A graph that is constructed in this way is called a {\em Halin graph\/}.
\end{definition}
Halin graphs have treewidth at most three. Furthermore, if $G$
is a Halin graph of bounded degree, then also $L(G)^2$ has
bounded treewidth
and thus the strong chromatic index of $G$ can be computed in
linear time. Recently, Ko-Wei Lih, {\em et al.\/},
proved that a cubic Halin graph
other than one of the two `necklaces' $Ne_2$ (the complement of
$C_6$) and $Ne_4$, has strong
chromatic index at most 7. The two exceptions have
strong chromatic index 9 and 8, respectively. If $T$ is the underlying
tree of the Halin graph, and if $G \neq Ne_2$ and $G$ is
not a wheel $W_n$
with $n \neq 0 \bmod{3}$, then Ping-Ying Tsai, {\em et al.\/},
show that the strong chromatic index is bounded by
$s\chi^{\prime}(T)+3$.
(See~\cite{kn:shiu2,kn:shiu} for earlier results that
appeared in regular papers.
\footnote{The results of Ko-Wei Lih and Ping-Ying Tsai, {\em et al.\/}, were
presented at the Sixth Cross-Strait Conference on Graph Theory and
Combinatorics which was held at the National Chiao Tung University
in Taiwan in 2011.})
If $G$ is a Halin graph then $L(G)^2$
has bounded rankwidth.
In~\cite{kn:ganian} it is shown that there exists a
polynomial algorithm that computes the chromatic number
of graphs with bounded rankwidth, thus the strong chromatic
index of Halin graphs can be computed in polynomial time.
In passing, let us mention the following result.
A class of graphs $\mathcal{G}$ is
$\chi$-bounded if there exists a function $f$ such that
$\chi(G) \leq f(\omega(G))$ for $G \in \mathcal{G}$. Here
$\chi(G)$ is the chromatic number of $G$ and $\omega(G)$ is the
clique number of $G$.
Recently, Dvo\v{r}\'ak and Kr\'al showed that for every $k$,
the class of graphs with rankwidth at most $k$
is $\chi$-bounded~\cite{kn:dvorak}.
Obviously, the graphs $L(G)^2$ have a uniform $\chi$-bound for
graphs $G$ in the class of Halin graphs.
In this note we show that there exists a linear-time algorithm
that computes the strong chromatic index of Halin graphs.
\section{The strong chromatic index of Halin graphs}
The following lemma is easy to check.
\begin{lemma}[Ping-Ying Tsai]
\label{basics}
Let $C_n$ be the cycle with $n$ vertices and let $W_n$ be the wheel
with $n$ vertices in the cycle.
Then
\[s\chi^{\prime}(C_n) = \begin{cases}
3 & \quad \text{if $n= 0 \bmod{3}$} \\
5 & \quad \text{if $n=5$} \\
4 & \quad \text{otherwise}
\end{cases}
\quad
s\chi^{\prime}(W_n)= \begin{cases}
n+3 & \quad \text{if $n=0 \bmod{3}$}\\
n+5 & \quad \text{if $n=5$}\\
n+4 & \quad \text{otherwise.}
\end{cases}\]
\end{lemma}
A double wheel is a Halin graph in which the tree
$T$ has exactly two vertices
that are not leaves.
\begin{lemma}[Ping-Ying Tsai]
\label{basic2}
Let $W$ be a double wheel where $x$ and $y$ are
the vertices of $T$ that are not leaves.
Then $s\chi^{\prime}(T)=d(x)+d(y)-1$ where $d(x)$ and
$d(y)$ are the degrees of $x$ and $y$. Furthermore,
\[s\chi^{\prime}(W)=
\begin{cases}
s\chi^{\prime}(T)+4=9 & \quad \text{if $d(x)=d(y)=3$, {\em i.e.\/}, if
$W=\Bar{C_6}$} \\
s\chi^{\prime}(T)+2=d(y)+4 & \quad \text{if $d(y) > d(x)=3$} \\
s\chi^{\prime}(T)+1=d(x)+d(y) & \quad \text{if $d(y) \geq d(x) >3$.}
\end{cases}\]
\end{lemma}
Let $G$ be a Halin graph with tree $T$ and cycle $C$.
Then obviously,
\begin{equation}
\label{eq1}
s\chi^{\prime}(G) \leq s\chi^{\prime}(T) + s\chi^{\prime}(C).
\end{equation}
The linegraph of a tree is a claw-free blockgraph. Since a sun
$S_r$ with $r > 3$ has a claw,
$L(T)$ has no induced sun $S_r$ with $r > 3$. It follows
that $L(T)^2$ is a chordal graph~\cite{kn:laskar} (see
also~\cite{kn:cameron}; in this paper Cameron proves
that $L(G)^2$ is chordal for any chordal graph $G$). Notice that
\begin{equation}
\label{bound}
s\chi^{\prime}(T)=\chi(L(T)^2)=\omega(L(T)^2) \leq 2\Delta(G)-1
\quad\Rightarrow\quad
s\chi^{\prime}(G) \leq 2\Delta(G)+4.
\end{equation}
\subsection{Cubic Halin graphs}
In this subsection we outline a simple linear-time algorithm for
the cubic Halin graphs.
\begin{theorem}
\label{cubic case}
There exists a linear-time algorithm that computes the
strong chromatic index of cubic Halin graphs.
\end{theorem}
\begin{proof}
Let $G$ be a cubic Halin graph with plane tree $T$ and cycle $C$.
Let $k$ be a natural number. We describe a linear-time
algorithm that
checks if $G$ has a strong edge coloring with at most $k$ colors.
By Equation (\ref{bound})
we may assume that $k$ is at most 10. Thus the
correctness of this algorithm proves the theorem.
Root the tree $T$ at an arbitrary leaf $r$ of $T$.
Consider a vertex $x$ in $T$. There is a unique path $P$
in $T$ from $r$ to $x$ in $T$. Define the subtree $T_x$ at $x$
as the maximal connected subtree of $T$ that does not contain
an edge of $P$. If $x=r$ then $T_x=T$.
Let $H(x)$ be the subgraph of $G$ induced by the vertices of
$T_x$.
Notice that,
if $x \neq r$ then the edges of $H(x)$ that are
not in $T$ form a path $Q(x)$ of edges in $C$.
For $x \neq r$ define the boundary $B(x)$
of $H(x)$ as the following set
of edges.
\begin{enumerate}[\rm (a)]
\item
\label{i1}
The unique edge of $P$ that
is incident with $x$.
\item
\label{i2}
The two edges of $C$ that connect
the path $Q(x)$ of $C$ with the rest of $C$.
\item Consider the endpoints of the
edges mentioned in (\ref{i1}) and (\ref{i2})
that are in $T_x$.
Add the remaining two edges that are incident with each of
these endpoints to $B(x)$.
\end{enumerate}
Thus the boundary $B(x)$ consists of at most 9 edges.
The following claim is easy to check. It proves
the correctness of the algorithm described below.
Let $e$ be an edge of $H(x)$. Let $f$ be an edge
of $G$ that is not an edge of $H(x)$. If $e$ and $f$ are
at distance at most 1 in $G$ then $e$ or $f$ is in
$B(x)$.\footnote{Two edges in $G$ are at distance at most one
if the subgraph induced by their endpoints is either $P_3$, or
$K_3$ or $P_4$. We assume that it can be checked in constant time
if two edges $e$ and $f$ are at distance at most one. This can be
achieved by a suitable data structure.}
Consider all possible colorings of the edges in $B(x)$.
Since $B(x)$ contains at most 9 edges and since there are at
most $k$ different colors for each edge, there are at most
\[k^9 \leq 10^9\]
different colorings of the edges in $B(x)$.
The algorithm now fills a table which gives a boolean
value for each coloring of the boundary $B(x)$. This boolean value is
{\tt TRUE} if and only if
the coloring of the edges in $B(x)$ extends to an
edge coloring of the union of the sets of edges in $B(x)$
and in $H(x)$ with at most $k$ colors, such that any pair of
edges in this set that are
at distance at most one in $G$, have different colors.
These boolean values are computed as follows. We prove the correctness
by induction on the size of the subtree at $x$.
First consider the case where the subtree at $x$ consists of the
single vertex $x$. Then $x \neq r$ and $x$ is a leaf of $T$.
In this case
$B(x)$ consists
of three edges, namely the three edges that are incident with $x$.
These are two edges of $C$ and one edge of $T$.
If the colors of these three edges in
$B$
are different
then the boolean value is set to
{\tt TRUE}. Otherwise it is set to {\tt FALSE}.
Obviously, this is a correct assignment.
Next consider the case where $x$ is an internal vertex of $T$.
Then $x$ has two children in the subtree at $x$.
Let $y$ and $z$ be the two children and consider the two
subtrees rooted at $y$ and $z$.
The algorithm that computes the tables for each vertex $x$
processes the subtrees in order of increasing number of vertices.
(Thus the roots of the subtrees are visited in postorder).
We now assume that the tables at $y$ and $z$ are
computed correctly and show how the
table for $x$ is computed correctly and in constant time.
That is, we prove that the algorithm described below
computes the table at $x$ such that it contains a
coloring of $B(x)$ with a value {\tt TRUE} if and only
if there
exists an extension of this coloring to the
edges of $H(x)$ and $B(x)$ such that any two
different edges $e$ and $f$
at distance at most one in $G$, each one in
$H(x)$ or in $B(x)$, have different colors.
Consider a coloring of the edges in the
boundary $B(x)$.
The boolean value in the table of $x$
for this coloring is computed as follows.
Notice that
\begin{enumerate}[\rm (i)]
\item $B(y) \cap B(z)$ consists of one edge and this
edge is not in $B(x)$, and
\item $B(x) \cap B(y)$ consists
of at most four edges, namely the edge $(x,y)$ and the three
edges of $B(y)$ that are incident with one vertex of
$C \cap H(y)$.
Likewise, $B(x) \cap B(z)$ consists of at most four edges.
\end{enumerate}
The algorithm varies the possible colorings of
the edge in $B(y) \cap B(z)$.
Colorings of $B(x)$, $B(y)$ and $B(z)$ are
consistent if the intersections are the same color and the pairs
of edges in
\[B(x) \cup B(y) \cup B(z)\]
that are at distance at most one in $G$ have different colors.
A coloring of $B(x)$ is assigned the value {\tt TRUE}
if there exist colorings of $B(y)$ and $B(z)$ such that the
three colorings are consistent and $B(y)$ and $B(z)$ are assigned the
value {\tt TRUE} in the tables at $y$ and at $z$ respectively.
Notice that the table at $x$ is built in constant time.
Consider a coloring of $B(x)$ that is assigned the value {\tt TRUE}.
Consider colorings of the edges of $B(y)$ and $B(z)$ that are
consistent with $B(x)$ and that are assigned the value
{\tt TRUE} in the tables at $y$ and $z$. By induction, there
exist extensions of the colorings of $B(y)$ and $B(z)$
to the edges of $H(y)$ and $H(z)$. The union of these
extensions provides a $k$-coloring of the edges in $H(x)$.
Consider two edges $e$ and $f$ in $B(x) \cup B(y) \cup B(z)$.
If their distance
is at most one then they have different colors since
the coloring of $B(x) \cup B(y) \cup B(z)$ is
consistent. Let $e$ and $f$ be a pair of edges in $H(x)$.
If they are both in $H(y)$ or both in $H(z)$ then they have
different colors.
Assume that $e$ is in $H(y)$ and assume that $f$ is not in
$H(y)$. If $e$ and $f$ are at distance at most one,
then $e$ or $f$ is in $B(y)$. If they are both in $B(y)$,
then they have different colors, due to the consistency.
Otherwise, by the induction hypothesis, they have different colors.
This proves the claim on the correctness.
Finally, consider the table for the vertex $x$ which is
the unique neighbor of $r$ in $T$.
By the induction hypothesis,
and the fact that every edge in $G$ is either in
$B(x)$ or in $H(x)$,
$G$ has a strong edge coloring with at most
$k$ colors if and only if the table at $x$ contains a coloring
of $B(x)$ with three different colors for which the boolean is
set to {\tt TRUE}.
This proves the theorem.
\qed\end{proof}
\begin{remark}
The involved constants in this algorithm are improved considerably
by the recent results of Ko-Wei Lih, Ping-Ying Tsai, {\em et al.\/}.
\end{remark}
\subsection{Halin graphs of general degree}
\begin{theorem}
\label{general Halin graphs}
There exists a linear-time algorithm that computes the strong
chromatic index of Halin graphs.
\end{theorem}
\begin{proof}
The algorithm is similar to the algorithm for
the cubic case.
Let $G$ be a Halin graph, let $T$ be the underlying plane tree,
and let $C$ be the cycle that connects the leaves of $T$.
Since $L(T)^2$ is chordal the chromatic number of $L(T)^2$
is equal to the
clique number of $L(T)^2$, which is
\[s\chi^{\prime}(T)= \max \;\{\;d(u)+d(v)-1\;|\; (u,v) \in E(T)\;\},\]
where $d(u)$ is the degree of $u$ in the tree $T$.
By Formula~(\ref{eq1}) and Lemma~\ref{basics}
the strong chromatic index of $G$ is one of
the six possible values
\footnote{Actually, according to the recent results of
Ping-Ying Tsai, {\em et al.\/},
the strong chromatic index of $G$ is at most $s\chi^{\prime}(T)+3$
except when $G$ is a wheel or $\Bar{C_6}$.}
\[s\chi^{\prime}(T), s\chi^{\prime}(T)+1, \ldots, s\chi^{\prime}(T)+5.\]
Root the tree at some leaf $r$ and consider a subtree $T_x$ at a
node $x$ of $T$.
Let $H(x)$ be the subgraph of $G$ induced by the vertices of $T_x$.
Let $y$ and $z$ be the two boundary vertices of $H(x)$ in $C$.
We distinguish the following six types of edges corresponding
to $H(x)$.
\begin{enumerate}[\rm 1.]
\item The set of edges in $T_x$ that are adjacent to $x$.
\item The edge that connects $x$ to its parent in $T$.
\item The edge that connects $y$ to its neighbor in $C$ that is
not in $T_x$.
\item The set of edges in $H(x)$ that have endpoint $y$.
\item The edge that connects $z$ to its neighbor in $C$ that is
not in $T_x$.
\item The set of edges in $H(x)$ that have endpoint $z$.
\end{enumerate}
Notice that the set of edges of every type has bounded
cardinality, except the first type.
Consider a $0/1$-matrix $M$ with rows indexed by the six
types of edges and columns indexed by the colors.
A matrix entry $M_{ij}$ is 1 if there is an edge of the row-type
$i$ that is colored with the color $j$ and otherwise this
entry is 0. Since $M$ has only 6 rows, the rank over
$GF[2]$ of $M$ is at most 6.
Two colorings are equivalent if there is a
permutation of the colors that maps one coloring to the other one.
Let $S \subseteq \{1,\ldots,6\}$ and let $W(S)$ be the
set of colors that are used by
edges of type $i$ for all $i \in S$.
A class of equivalent colorings is fixed by the set
of cardinalities
\[\{\; |W(S)| \;|\; S \subseteq \{1,\ldots,6\} \;\}.\]
We claim that the number of equivalence classes is
constant. The number of ones in the row of the first type
is the degree of $x$ in $H(x)$. Every other row has at most
3 ones. This proves the claim.
Consider the union of two subtrees, say at $x$ and $x^{\prime}$.
The algorithm considers all equivalence classes of colorings of the
union, and checks, by table look-up, whether it decomposes into
valid colorings of $H(x)$ and $H(x^{\prime})$. An easy way to do this
is as follows. First double the number of types, by distinguishing
the edges of $H(x)$ and $H(x^{\prime})$. Then enumerate
all equivalence classes of colorings. Each equivalence class
is fixed by a sequence of $2^{12}$ numbers, as above.
By table look-up,
check if an equivalence class restricts to a valid coloring
for each of $H(x)$ and $H(x^{\prime})$.
Since this takes constant time, the algorithm runs in linear time.
This proves the theorem.
\qed\end{proof}
\end{document} |
\begin{document}
\begin{frontmatter}
\title{Instrumental Variables: An~Econometrician's Perspective\thanksref{T1}}
\relateddois{T1}{Discussed in \relateddoi{d}{10.1214/14-STS494},
\relateddoi{d}{10.1214/14-STS485}, \relateddoi{d}{10.1214/14-STS488},
\relateddoi{d}{10.1214/14-STS491}; rejoinder at \relateddoi{r}{10.1214/14-STS496}.}
\runtitle{Instrumental Variables}
\begin{aug}
\author[a]{\fnms{Guido W.}~\snm{Imbens}\corref{}\ead[label=e1]{imbens@stanford.edu}
\ead[label=u1,url]{http://www.gsb.stanford.edu/users/imbens}}
\runauthor{G. W. Imbens}
\affiliation{Stanford University}
\address[a]{Guido W. Imbens is the Applied Econometrics Professor and Professor of Economics,
Graduate School of Business, Stanford University, Stanford, California 94305, USA and
NBER \operatorname{pr}intead{e1,u1}.}
\end{aug}
\begin{abstract}
I review recent work in the statistics literature on instrumental
variables methods from an econometrics perspective. I discuss some of
the older, economic, applications including supply and demand models
and relate them to the recent applications in settings of randomized
experiments with noncompliance. I discuss the assumptions underlying
instrumental variables methods and in what settings these may be
plausible. By providing context to the current applications, a better
understanding of the applicability of these methods may arise.
\end{abstract}
\begin{keyword}
\kwd{Simultaneous equations models}
\kwd{randomized experiments}
\kwd{potential outcomes}
\kwd{noncompliance}
\kwd{selection models}
\end{keyword}
\end{frontmatter}
\setcounter{footnote}{1}
\section{Introduction}
Instrumental Variables (IV) refers to a set of methods developed in
econometrics starting in the 1920s to draw causal inferences in
settings where the treatment of interest cannot be credibly viewed as
randomly assigned, even after conditioning on additional covariates,
that is, settings where the assumption of no unmeasured confounders
does not hold.\footnote{There is another literature in econometrics
using instrumental variables methods also to deal with classical
measurement error (where explanatory variables are measured with error
that is independent of the true values). My remarks in the current
paper do not directly reflect on the use of instrumental variables to
deal with measurement error. See \citet{Sar58} for a classical paper,
and Hillier (\citeyear{Hil90}) and \citet{Are02} for more recent discussions.}
In the last two decades, these methods have attracted considerable
attention in the statistics literature. Although this recent statistics
literature builds
on the earlier econometric literature, there are nevertheless
important differences.
First, the recent statistics literature
primarily focuses on the binary treatment case. Second, the recent
literature explicitly allows for treatment effect heterogeneity. Third,
the recent instrumental variables literature (starting with \cite*{ImbAng94}; \cite*{AngImbRub}; \cite*{Hec}; \cite*{Man90}; and \cite*{Rob86}) explicitly uses the potential outcome framework
used by Neyman for randomized experiments and generalized to
observational studies by
Rubin (\citeyear{Rub74}, \citeyear{Rub78}, \citeyear{Rub90}).
Fourth, in the
applications this literature has concentrated on, including randomized
experiments with noncompliance, the intention-to-treat or reduced-form
estimates are often of greater interest than they are in the
traditional econometric simultaneous equations applications.
Partly the recent statistics literature has been motivated by the earlier
econometric literature on instrumental variables, starting with \citet{Wri28} (see the discussion on the origins of instrumental variables in
\cite*{StoTre03}). However, there are also other antecedents,
outside of the traditional econometric instrumental variables
literature, notably the work by Zelen on encouragement designs (Zelen, \citeyear{Zel79}, \citeyear{Zel90}).
Early papers in the recent statistics literature include \citet{AngImbRub}, \citet{Rob} and
\citet{McCNew94}.
Recent reviews include \citet{Ros10}, \citet{Vanetal11} and
\citet{HerRob06}. Although these reviews include
many references to the earlier economics literature, it might still be
useful to discuss the econometric literature in more detail to provide
some \mbox{background} and perspective on the applicability of instrumental
variables methods in other fields. In this discussion, I~will do so.
Instrumental variables methods have been a central part of the
econometrics canon since the first half of the twentieth century,
and\vadjust{\goodbreak}
continue to be an integral part of most graduate and undergraduate
textbooks (e.g., Angrist and Pischke, \citeyear{AngPis09}; \cite*{BowTur84}; \cite*{Gre11}; \cite*{Hay00}; \cite*{Man95}; \cite*{StoWat10}; Wooldridge, \citeyear{Woo10}, \citeyear{Woo08}).
Like the statisticians Fisher
and Neyman (\cite*{Fis25}; Splawa-Neyman, \citeyear{Spl90}),
early econometricians such as \citet{Wri28}, \citet{Wor27},
\citet{Tin30} and\break \citet{Haa43} were interested in drawing causal
inferences, in their case about the effect of economic policies on
economic behavior. However, in sharp contrast to the statistical
literature on causal inference,
the starting point for these econometricians was \textit{not} the
randomized experiment. From the outset, there was a recognition that in
the settings they studied, the causes, or treatments, were not {assigned} to passive units (economic agents in their setting, such as
individuals, households, firms or countries). Instead the economic
agents actively influence, or even explicitly choose, the level of the
treatment they receive. Choice, rather than chance, was the starting
point for thinking about the assignment mechanism in the econometrics
literature. In this perspective, units receiving the active treatment
are different from those receiving the control treatment not just
because of the receipt of the treatment: they (choose to) receive the
active treatment because they are different to begin with. This makes
the treatment potentially \textit{endogenous}, and creates what is
sometimes in the econometrics literature referred to as the \textit{selection problem} (\cite{Hec79}).
The early econometrics literature on instrumental variables did not
have much impact
on thinking in the statistics community. Although some of the technical
work on large sample properties of various estimators did get published
in statistics journals (e.g., the still influential Anderson and
Rubin, \citeyear{A49} paper), applications by noneconomists were rare. It is not
clear exactly what the reasons for this are. One possibility is the
fact that the early literature on instrumental variables was closely
tied to substantive economic questions (e.g., interventions in
markets), using theoretical economic concepts that may have appeared
irrelevant or difficult to translate to other fields (e.g.,
supply and demand). This may have suggested to noneconomists that the
instrumental variables methods in general had limited applicability
outside of economics. The use of economic concepts was not entirely
unavoidable, as the critical assumptions underlying instrumental
variables methods are substantive and require subtle subject matter knowledge.
A second reason may be that although the early work by Tinbergen and
Haavelmo used a notation that is very similar to what \citet{Rub74}
later called the potential outcome notation, quickly the literature
settled on a notation only involving realized or observed outcomes; see
for a historial perspective \citet{autokey71} and \citet{Imb97}.
This realized-outcome notation that remains common in the econometric
textbooks obscures the connections between the Fisher and Neyman work
on randomized experiments and the instrumental variables literature. It
is only in the 1990s that econometricians returned to the potential
outcome notation for causal questions (e.g., \cite*{Hec}; \cite*{Man90}; \cite*{ImbAng94}), facilitating and initiating a
dialogue with statisticians on instrumental variable methods.
The main theme of the current paper is that the early work in
econometrics is helpful in understanding the modern instrumental
variables literature, and furthermore, is potentially useful in
improving applications of these methods and identifying potential instruments.
These methods may in fact be useful in many settings statisticians
study. Exposure to treatment is rarely solely a matter of chance or
solely a matter of choice. Both aspects are important and help to
understand when causal inferences are credible and when they are not.
In order to make these points, I will discuss some of the early work
and put it in a modern framework and notation. In doing so, I will
address some of the concerns that have been raised about the
applicability of instrumental variables methods in statistics.
I will also discuss some areas where the recent statistics literature
has extended and improved our understanding of instrumental variables
methods. Finally, I will review some of the econometric terminology and
relate it to the statistical literature to remove some of the semantic
barriers that continue to separate the literatures.
I should emphasize that many of the topics discussed in this review
continue to be active research areas, about which there is considerable
controversy both inside and outside of econometrics.
The remainder of the paper is organized as follows.
In Section~\ref{section:choice}, I will discuss the distinction
between the statistics literature on causality with its primary focus
on chance, arising from its origins in the experimental literature, and
the econometrics or economics literature with its emphasis on choice.
The next two sections discuss in detail two classes of examples.
In Section~\ref{section:supplydemand}, I discuss the canonical example
of instrumental variables in economics, the estimation of supply and
demand functions.
In Section~\ref{section:modern}, I discuss a modern class of examples,
randomized experiments with noncompliance.
In
Section~\ref{section:content}, I discuss the substantive content of
the critical assumptions, and in Section~\ref{section:textbook}, I
link the current literature to the older textbook discussions.
In Section~\ref{section:extensions}, I discuss some of the recent
extensions of traditional instrumental variables methods.
Section~\ref{section:conclusion} concludes.
\section{Choice versus Chance in Treatment~Assignment}
\label{section:choice}
Although the objectives of causal analyses in statistics and
econometrics are very similar, traditionally statisticians and
economists have approached these questions very differently.
A
key difference in the approaches taken in the statistical and
econometric literatures is the focus on different assignment
mechanisms, those with an emphasis on chance versus those with an
emphasis on choice.
Although in practice in many observational studies assignment
mechanisms have elements of both chance and choice, the traditional
starting points in the two literatures are very different, and it is
only recently that these literatures have discovered how much they have
in common.\footnote{In both literatures, it is typically assumed that
there is no interference between units. In the statistics literature,
this is often referred to as the \textit{Stable Unit Treatment Value
Assumption} (SUTVA, \cite*{Rub78}). In economics, there are many cases
where this is not a reasonable assumption because there are \textit{general equilibrium} effects. In an interesting recent experiment,
\citet{Creetal}
varied the scale of experimental interventions (job training programs
in their case) in different labor markets and found that the scale
substantially affected the average effects of the interventions. There
is also a growing literature on settings directly modeling interactions.
In this discussion, I will largely ignore the complications arising
from interference between units. See, for example, Manski (\citeyear{Man00N1}).}
\subsection{The Statistics Literature: The Focus on Chance}
The starting point in the statistics literature, going back to \citet{Fis25} and
Splawa-Neyman (\citeyear{Spl90}), is the randomized experiment, with both
Fisher and Neyman motivated by agricultural applications where the
units of analysis are plots of land. To be specific, suppose we are
interested in the average causal effect of a binary treatment or
intervention, say fertilizer $A$ or fertilizer $B$, on plot yields. In the
modern notation and language originating with \citet{Rub74}, the unit
(plot) level causal effect is a comparison
between the two potential outcomes, $Y_i(A)$ and $Y_i(B)$ [e.g.,
the difference
$\tau_i=Y_i(B)-Y_i(A)$], where $Y_i(A)$ is the potential outcome given
fertilizer $A$ and $Y_i(B)$ is the potential outcome given fertilizer
$B$, both for plot $i$. In a completely randomized experiment with $N$
plots, we select $M$ (with $M\in\{1,\ldots,N-1\}$) plots at random to
receive fertilizer $B$, with the remaining $N-M$ plots assigned to
fertilizer $A$.
Thus, the
treatment assignment,
denoted by $X_i\in\{A,B\}$ for plot $i$, is by design independent of
the potential outcomes.\footnote{To facilitate comparisons with the
econometrics literature, I will follow the notation that is common in
econometrics, denoting the endogenous regressors, here the treatment of
interest, by $X_i$, and later the instruments by $Z_i$. Additional
(exogenous) regressors will be denoted by $V_i$. In the statistics
literature, the treatments of interested are often denoted by $W_i$,
the instruments by $Z_i$, with $X_i$ denoting additional regressors or
attributes.}
In this specific setting, the work by Fisher and Neyman shows how one
can draw exact causal inferences. Fisher focused on calculating exact
$p$-values for sharp null hypotheses, typically the null hypothesis of no
effect whatsoever, $Y_i(A)=Y_i(B)$ for all plots. Neyman focused on
developing unbiased estimators for the average treatment effect $\sum_i(Y_i(A)-Y_i(B))/N$ and the variance of those estimators.
The subsequent literature in statistics, much of it associated with the
work by Rubin and coauthors (\cite*{Coc68}; \cite*{CocRub73};
Rubin, \citeyear{Rub74}, \citeyear{Rub90}, \citeyear{Rub06}; Rosenbaum and Rubin, \citeyear{RosRub83};
\cite*{RubTho92}; Rosenbaum, \citeyear{Ros02}, \citeyear{Ros10}; \cite*{Hol86}) has focused on
extending and generalizing the Fisher and Neyman results that were
derived explicitly for randomized experiments to the more general
setting of observational studies. A large part of this literature
focuses on the case where the researcher has additional background
information available about the units in the study. The additional
information is in the form of pretreatment variables or covariates not
affected by the treatment. Let $V_i$ denote these covariates. A key
assumption in this literature is that conditional on these pretreatment
variables the assignment to treatment is independent of the treatment
assignment. Formally,
\[
X_i \perp \bigl(Y_i(A), Y_i(B)\bigr) \vert
V_i \quad \mbox{(unconfoundedness)}.
\]
Following \citet{Rub90}, I refer to this assumption as \textit{unconfoundedness given} $V_i$, also known as \textit{no unmeasured
confounders}. This assumption, in combination with the auxiliary
assumption that for all values of the covariates the probability of
being assigned to each level of the treatment is strictly positive is
referred to as
\textit{strong ignorability} (Rosenbaum and Rubin, \citeyear{RosRub83}). If we assume
only that $X_i\perp Y_i(A)|V_i$ and $X_i\perp Y_i(B)|V_i$ rather than
jointly, the assumption is referred to as \textit{weak unconfoundedness}
(\cite{Imb00}), and the combination as \textit{weak ignorability}.
Substantively, it is not clear that there are cases in the setting with
binary treatments where the weak version is plausible but not the
strong version, although the difference between the two assumptions has
some content in the multivalued treatment case (\cite{Imb00}).
In the econometric literature, closely related assumptions are referred
to as
\textit{selection-on-observables} (\cite{BarCaiGol80}) or
\textit{exogeneity}.
Under weak ignorability (and thus also under strong ignorability), it
is possible to estimate precisely the average effect of the treatment in large
samples. In other words, the average effect of the treatment is \textit{identified}.
Various specific methods have been proposed, including matching,
subclassification and regression. See \citet{Ros10}, \citet{Rub06},
Imbens (\citeyear{Imb04}, \citeyear{ImbN2}), \citet{GelHil06}, \citet{ImbRub} and \citet{AngPis09} for general discussions and surveys.
Robins and coauthors (\cite*{Rob86}; \cite*{GilRob01}; \cite*{RicRob}; Van der Laan and Robins, \citeyear{vanRob03}) have extended this
approach to settings with sequential treatments.
\subsection{The Econometrics Literature: The~Focus~on~Choice}
In contrast to the statistics literature whose point of departure was
the randomized experiment, the starting point in the economics and
econometrics literatures for studying causal effects emphasizes the
choices that led to the treatment received. Unlike the original
applications in statistics where the units are passive, for example,
plots of land, with no influence over their treatment exposure, units
in economic analyses are typically economic agents, for example,
individuals, families, firms or administrations. These are agents with
objectives and the ability to pursue these objectives within constraints.
The objectives are typically closely related to the outcomes under the
various treatments.
The constraints may be legal, financial or information-based.
The starting point of economic science is to model these agents as
behaving optimally. More specifically, this implies that economists
think of everyone of these agents as choosing the level of the
treatment to most efficiently pursue their objectives given the
constraints they face.\footnote{In principle, these objectives may
include the effort it takes to find the optimal strategy, although it
is rare that these costs are taken into account.}
In practice, of course, there is often evidence that not all
agents behave optimally. Nevertheless, the starting point is the
presumption that optimal behavior is a reasonable approximation to
actual behavior, and the models economists take to the data often
reflect this.
\subsection{Some Examples}
Let us contrast the statistical and econometric approaches in a highly
stylized example.
\citet{Roy51} studies the problem of occupational choice and the
implications for the observed distribution of earnings. He focuses on
an example where individuals can choose between two occupations,
hunting and fishing. Each individual has a level of productivity
associated with each occupation, say, the total value of the catch per day.
For individual $i$, the two productivity levels
are $Y_i(h)$ and $Y_i(f)$, for the productivity level if hunting and
fishing, respectively.\footnote{In this example, the no-interference
(SUTVA) assumption that there are no effects of other individual's
choices and, therefore, that the individual level potential outcomes
are well defined is tenuous---if one hunter is successful that will
reduce the number of animals available to other hunters---but I will
ignore these issues here.}
Suppose the researcher is interested in the average difference in
productivity in these two occupations, $\tau=\mathbb{E}[Y_i(f)-Y_i(h)]$, where the averaging is over the population of
individuals.\footnote{That is not actually the goal of Roy's original
study, but that is beside the point here.}
The researcher observes for all units in the sample the occupation they
chose ($X_i$, equal to $h$ for hunters and $f$ for fishermen) and the
productivity in their chosen occupation,
\[
Y_i^{\mathrm{obs}}=Y_i(X_i)=
\cases{Y_i(h) & $\mbox{if } X_i=h$,\vspace*{2pt}
\cr
Y_i(f) & $\mbox{if } X_i=f$.}
\]
In the Fisher--Neyman--Rubin statistics tradition, one might start by
estimating $\tau$ by comparing productivity levels by occupation:
\[
\hat{\tau}=\overline{Y}^{\mathrm{obs}}_f-\overline{Y}^{\mathrm{obs}}_h,
\]
where
\begin{eqnarray*}
\overline{Y}^{\mathrm{obs}}_f &=& \frac{1}{N_f}\sum
_{i:X_i=f} Y^{\mathrm{obs}}_i,\quad
\overline{Y}^{\mathrm{obs}}_h=\frac{1}{N_h}\sum
_{i:X_i=h} Y^{\mathrm{obs}}_i,
\\
N_f &=& \sum_{i=1}^N
\mathbf{1}_{X_i=f} \quad \mbox{and} \quad N_h=N-N_f.
\end{eqnarray*}
If there is concern that these unadjusted differences are not credible
as estimates of the average causal effect, the next step in this
approach would be to
adjust for observed individual characteristics such as education levels
or family background. This would be justified if individuals can be
thought of as choosing, at least within homogenous groups defined by
covariates, randomly which occupation to engage in.
Roy, in the economics tradition, starts from a very different place.
Instead of assuming that individuals choose their occupation (possibly
after conditioning on covariates) randomly, he assumes that each
individual chooses her occupation optimally, that is, the occupation
that maximizes her productivity:
\[
X_i= \cases{ f & $\mbox{if }Y_i(f)\geq
Y_i(h)$,
\cr
h & $\mbox{otherwise}$.}
\]
There need not be a solution in all cases, especially if there is
interference, and thus there are general equilibrium effects, but I
will assume here that such a solution exists.
If this assumption about the occupation choice were strictly true, it
would be difficult to learn much about $\tau$ from data on occupations
and earnings.
In the spirit of research by Manski (\citeyear{Man90}, \citeyear{Man00N2}, \citeyear{Man01}),
Manski and Pepper (\citeyear{ManPep00}), and Manski et al. (\citeyear{Manetal92N1}),
one can
derive bounds on $\tau$, exploiting the fact that if $X_i=f$,
then the unobserved $Y_i(h)$ must satisfy $Y_i(h)\leq Y_i(f)$, with
$Y_i(f)$ observed. For the Roy model, the specific calculations have
been reported in Manski (\citeyear{Man95}), Section~2.6. Without additional
information or restrictions, these bounds might be fairly wide, and often one
would not learn much about $\tau$. However, the original version of
the Roy model, where individuals know ex ante the exact value of
the potential outcomes and choose the level of the treatment
corresponding to the maximum of those, is ultimately not plausible in
practice. It is likely that individuals face uncertainty regarding their future productivity,
and thus may not be able to choose the ex post optimal
occupation; see for bounds under that scenario \citet{ManNag98}.
Alternatively, and this is emphasized in \citet{AthSte98},
individuals may have more complex objective functions taking into
account heterogenous costs or nonmonetary benefits associated with each
occupation. This creates a wedge between the outcomes that the
researcher focuses on and the outcomes that the agent optimizes over.
What is key here in relation to the statistics literature is that under
the Roy model and its generalizations the very fact that two
individuals have different occupations is seen as indicative that they
have different potential outcomes, thus fundamentally calling into
question the unconfoundedness assumption that individuals with similar
pretreatment variables but different treatment levels are comparable.
This concern about differences between individuals with the same values
for pretreatment variables but different treatment levels underlies
many econometric analyses of causal effects, specifically in the
literature on selection models. See \citet{HecRob} for a
general discussion.
Let me discuss two additional examples. There is a large literature in
economics concerned with estimating the causal effect of educational
achievement (measured as years of education) on earnings; see for
general discussions \citet{Gri77} and \citet{Car01}. One starting
point, and in fact the basis of a large empirical literature, is to
compare earnings for individuals who look similar in terms of
background characteristics, but who differ in terms of educational
achievement. The concern in an equally large literature is that those
individuals who choose to acquire higher levels of education did so
precisely because they expected their returns to additional years of
education to be higher than individuals who choose not to acquire
higher levels of education expected their returns to be. In the
terminology of the returns-to-education literature, the individuals
choosing higher levels of education may have higher levels of ability,
which lead to higher earnings for given levels of education.
Another canonical example is that of voluntary job training programs.
One approach to estimate the causal effect of training programs on
subsequent earnings would be to compare earnings for those
participating in the program with earnings for those who did not. Again
the concern would be that those who choose to participate did so
because they expected bigger benefits (financial or otherwise) from
doing so than individuals who chose not to participate.
These issues also arise in the missing data literature.
The statistics literature (Rubin, \citeyear{Rub76}, \citeyear{Rub87}, \citeyear{Rub96}; Little and Rubin, \citeyear{LitRub87})
has primarily
focused on models that assume that units with item nonresponse are
comparable to units with complete response, conditional on covariates
that are always observed. The econometrics literature (Heckman, \citeyear{Hec76},
\citeyear{Hec79}) has focused more heavily on models that interpret the nonresponse
as the \mbox{result} of systematic differences between units.
Philipson (\citeyear{Phi97N1}, \citeyear{Phi97N2}), Philipson and DeSimone
(\citeyear{PhiDeS97}), and Philipson and Hedges (\citeyear{PhiHed98}) take this
even further, viewing survey response as
a market transaction, where individuals not responding the survey do so
deliberately because the costs of responding outweighs the benefits to
these nonrespondents.
The Heckman-style selection models often assume strong parametric
alternatives to the Little and Rubin missing-at-random or ignorability
condition. This has often in turn led to estimators that are sensitive
to small changes in the data
generating process. See Little (\citeyear{L85}).
These issues of nonrandom selection are of course not special to
economics. Outside of randomized experiments, the exposure to treatment
is typically also chosen to achieve some objectives, rather than
randomly within homogenous populations. For example, physicians
presumably choose treatments for their patients optimally, given their
knowledge and given other constraints (e.g., financial).
Similarly, in economics and other social sciences one may view
individuals as making optimal decisions, but these are typically made
given incomplete information, leading to errors that may make the
ultimate decisions appear as good as random within homogenous
subpopulations. What is important is that the starting point is
different in the two disciplines, and this has led to the development
of substantially different methods for causal inference.
\subsection{Instrumental Variables}
How do instrumental variables methods address the type of selection
issues the Roy model raises? At the core, instrumental variables change
the incentives for agents to choose a particular level of the
treatment, without affecting the potential outcomes associated with
these treatment levels. Consider a job training program example where
the researcher is interested in the average effect of the training
program on earnings.
Each individual is characterized by two potential earnings outcomes,
earnings given the training and earnings in the absence of the
training. Each individual chooses to participate or not based on their
perceived net benefits from doing so.
As pointed out in \citet{AthSte98}, it is important that these
net benefits that enter into the individual's decision differ from the
earnings that are the primary outcome of interest to the researcher.
They do so
by the costs associated with participating in that regime.
Suppose that there is variation in the costs individuals incur with
participation in the training program.
The costs are broadly defined, and may include travel time to the
program facilities, or the effort required to become informed about the program.
Furthermore, suppose that these costs are independent of the potential
outcomes. This is a strong assumption, often made more plausible by
conditioning on covariates. Measures of the participation cost may then
serve as instrument variables and aid in the identification of the
causal effects of the program. Ultimately, we compare earnings for
individuals with low costs of participation in the program with those
for individuals with high costs of participation and attribute the
difference in average earnings to the increased rate of participation
in the program among the two groups.
In almost all cases, the assumption that there is no direct effect of
the change in incentives on the potential outcomes is controversial,
and it needs to be assessed at a case-by-case level. The second part of
the assumption, that the costs are independent of the potential
outcomes, possibly after conditioning on covariates, is qualitatively
very different. In some cases, it is satisfied by design, for example,
if the incentives are randomized. In observational studies, it is a
substantive, unconfoundedness-type, assumption, that may be more
plausible or at least approximately hold after conditioning on
covariates. For example, in a number of studies researchers have used
physical distance to facilities as instruments for exposure to
treatments available at such facilities. Such studies include \citet{McCNew94} and
\citet{Baietal10}
who use distance to hospitals with particular capabilities as an
instrument for treatments associated with those capabilities, after
conditioning on distance to the nearest medical facility, and \citet{Car95}, who uses distance to colleges as an instrument for attending college.
\section{The Classic Example: Supply~and~Demand}
\label{section:supplydemand}
In this section, I will discuss the classic example of instrumental
variables methods in econometrics, that is, simultaneous equations.
Simultaneous equations models are both at the core of the econometrics
canon and at the core of the confusion concerning instrumental
variables methods in the statistics literature.
More precisely, in this section I will look at supply and demand models
that motivated the original research into instrumental variables. Here,
the \textit{endogeneity}, that is, the violation of unconfoundedness,
arises from an equilibrium condition.
I will discuss the model in a very specific example to make the issues
clear, as I think that perhaps the level of abstraction used in the
older econometric text books has hampered communication with
researchers in other fields.
\begin{figure*}
\caption{Scatterplot of log prices and log quantities.}
\label{f1}
\end{figure*}
\subsection{Discussions in the Statistics Literature}
To show the level of frustration and confusion in the statistics
literature with these models, let me present some quotes.
In a comment on Pratt and Schlaifer (\citeyear{Pra}), \citet{Daw84} writes
``I despair of ever understanding the logic of simultaneous
equations well enough to tackle them,'' (page 24).
\citet{Cox92}
writes in a discussion on causality ``it seems
reasonable that models should be
specified in a way that would allow direct computer simulation
of the data$\ldots$\,. This, for example, precludes the use of $y_2$ as an
explanatory variable for $y_1$ if at the same time $y_1$ is an
explanatory variable for $y_2$'' (page 294).
This restriction appears to rule out the first model Haavelmo
considers, that is, equations (1.1) and (1.2) (\cite*{Haa43}, page 2):
\[
Y=aX+\epsilon_1, \quad X=bY+\epsilon_2
\]
(see also Haavelmo, \citeyear{Haa44}). In fact, the comment by Cox
appears to rule out all simultaneous
equations models of the type studied by economists.
\citet{Hol}, in comment on structural equation methods in econometrics,
writes ``why should [this disturbance] be
independent of [the instrument]$\ldots$ when the very
definition of [this disturbance] involves [the
instrument],''
(page 460).
Freedman writes
``Additionally, some
variables are taken to be exogenous (independent of the disturbance terms)
and some endogenous (dependent on the disturbance terms). The rationale
is seldom clear, because---among other things---there is seldom any very
clear description of what the disturbance terms mean, or where they come
from'' (\cite*{Fre06}, page 699).
\subsection{The Market for Fish}
The specific example I will use in this section is the market for
whiting (a particular white fish, often used in fish sticks) traded at
the Fulton fish market in New York City. Whiting was sold at the Fulton
fish market at the time by a small number of dealers to a large number
of buyers. Kathryn Graddy collected data on quantities and prices of
whiting sold by a particular trader at the Fulton fish market on 111
days between December 2, 1991, and May 8, 1992 (Graddy, \citeyear{Gra95}, \citeyear{Gra96};
\cite*{AngGraImb00}). I will take as the unit of analysis
a day, and interchangeably refer to this as a market. Each day, or
market, during the period covered in this data set, indexed by
$t=1,\ldots,111$, a number of pounds of whiting are sold by this
particular trader, denoted by $Q_t^{\mathrm{obs}}$. Not every transaction on the
same day involves the same price, but to focus on the essentials I will
aggregate the total amount of whiting sold and the total amount of
money it was sold for, and calculate a price per pound (in cents) for
each of the 111 days, denoted by $P_t^{\mathrm{obs}}$.
Figure~\ref{f1} presents a scatterplot of the observed log price and log
quantity data.
The average quantity sold over the 111 days was 6335 pounds, with a
standard deviation of 4040 pounds, for an average of the average
within-day prices of 88 cts per pound and a standard deviation of 34 cts.
For example, on the first day of this period 8058 pounds were sold for
an average of 65 cents, and the next day 2224 pounds were sold for an
average of 100 cents.
Table~\ref{summ_stats_fish} presents averages of log prices and log
quantities for the fish data.
\begin{table*}[b]
\tablewidth=13.5cm
\caption{Fulton fish market data ($N=111$)}
\label{summ_stats_fish}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}ld{3.0}d{2.2}ccc@{}}
\hline
& & \multicolumn{2}{c}{\textbf{Logarithm of price}} & \multicolumn {2}{c@{}}{\textbf{Logarithm of quantity}}
\\[-4pt]
&&\multicolumn{2}{l}{\rule{4.14cm}{1pt}} &
\multicolumn{2}{l@{}}{\rule{4.14cm}{1pt}} \\
&\multicolumn{1}{c}{\multirow{2}{55pt}[10pt]{\textbf{Number of observations}}} & \multicolumn{1}{c}{\textbf{Average}} & \textbf{Standard deviation} & \textbf{Average}
& \textbf{Standard deviation} \\
\hline
All & 111 & -0.19 & (0.38) & 8.52 & (0.74) \\[3pt]
Stormy & 32 & 0.04 & (0.35) & 8.27 & (0.71) \\
Not-stormy & 79 & -0.29 & (0.35) & 8.63 & (0.73) \\[3pt]
Stormy & 32 & 0.04 & (0.35) & 8.27 & (0.71) \\
Mixed & 34 & -0.16 & (0.35) & 8.51 & (0.77) \\
Fair & 45 & -0.39 & (0.37) & 8.71 & (0.69) \\
\hline
\end{tabular*}
\end{table*}
Now suppose we are interested in predicting the effect of a tax in this
market. To be specific, suppose the government is considering imposing
a $100\times r$\% tax (e.g., a 10\% tax) on all whiting sold,
but before doing so it wishes to predict the average percentage change
in the quantity sold as a result of the tax. We may formalize that by
looking at the average effect on the logarithm of the quantity, $\tau
=\mathbb{E}[\ln Q_t(r)-\ln Q_t(0)]$, where $Q_t(r)$ is the quantity
traded in market/day $t$ if the tax rate were set at $r$. The problem,
substantially worse than in the standard causal inference setting where
for some units we observe one of the two potential outcomes and for
other units we observe the other potential outcome, is that in all 111
markets we observe the quantity traded at tax rate $0$, $Q_t^\mathrm{obs}=Q_t(0)$, and we never see the quantity traded at the tax rate
contemplated by the government, $Q_t(r)$.
Because only $\mathbb{E}[\ln Q_t(0)]$ is directly estimable from data
on the quantities we observe, the question is how to draw inferences
about $\mathbb{E}[\ln Q_t(r)]$.
A naive approach would be to assume that a tax increase by 10\% would
simply raise prices by 10\%.
If one additionally is willing to make the unconfoundedness assumption
that prices can be viewed as set independently of market conditions on
a particular day, it follows that those markets after the introduction
of the tax where the price net of taxes is \$1.00 would on average be
like those markets prior to the introduction of the 10\% tax where the
price was \$1.10. Formally, this approach assumes that
\begin{eqnarray}
\label{model1} && \mathbb{E}\bigl[\ln Q_t(r)|P^{\mathrm{obs}}
_t=p\bigr]
\nonumber
\\[-8pt]
\\[-8pt]
&& \quad =\mathbb{E}\bigl[\ln Q_t(0)|P^{\mathrm{obs}}_t=(1+r)
\times p\bigr],
\nonumber
\end{eqnarray}
implying that
\begin{eqnarray*}
&& \mathbb{E}\bigl[\ln Q_t(r)-\ln Q_t(0)|P^{\mathrm{obs}}_t=p
\bigr]
\\
&& \quad =\mathbb{E}\bigl[\ln Q_t^{\mathrm{obs}}
|P^{\mathrm{obs}}_t=(1+r)\times p\bigr]
\\
&& \qquad {}-\mathbb{E}\bigl[\ln Q_t^{\mathrm{obs}}|P^{\mathrm{obs}}_t=
p\bigr]
\\
&& \quad \approx\mathbb{E}\bigl[\ln Q_t^{\mathrm{obs}}|\ln
P^{\mathrm{obs}}_t=r+\ln p\bigr]
\\
&& \qquad {}-\mathbb{E}\bigl[\ln Q_t^{\mathrm{obs}}|\ln
P^{\mathrm{obs}}_t= \ln p\bigr].
\end{eqnarray*}
The last quantity is often estimated using linear regression methods.
Typically, the regression function is assumed to be
linear in logarithms with constant coefficients,
\begin{equation}
\label{regression} \ln Q_t^{\mathrm{obs}}= \alpha^{\mathrm{ls}}+
\beta^{\mathrm{ls}}\times \ln P_t^{\mathrm{obs}}+
\varepsilon_t.
\end{equation}
Ordinary least squares estimation with the Fulton fish market data
collected by Graddy leads to
\[
\begin{array} {ccccc} \widehat{\ln Q_t^{\mathrm{obs}}}=&
8.42&-&0.54& \times \ \ln P_t^{\mathrm{obs}}.
\\
& (0.08)&& (0.18)\end{array}
\]
The estimated regression line is also plotted in Figure~\ref{f1}.
Interestingly, this is what \citet{Wor27} calls the ``statistical `demand curve',''
as opposed to the concept of a demand curve in economic theory. This
simple regression, in combination with the assumption embodied in (\ref{model1}), suggests that the quantity traded would go down, on average,
by 5.4\% in response to a 10\% tax.
\[
\hat\tau=-0.054 \quad (\mbox{s.e. }0.018).
\]
Why does this answer, or at least the method in which it was derived,
not make any sense to an economist? The answer assumes that prices can
be viewed as independent of the potential quantities traded, or, in
other words, unconfounded.
This assignment mechanism is unrealistic. In reality, it is likely the
markets/days, prior to the introduction of the tax, when the price was
\$1.10 were systematically different from those where the price was \$1.00. From an economists' perspective, the fact that the price was
\$1.10 rather than \$1.00 implies that market conditions \textit{must} have
been different, and it is likely that these differences are directly
related to the potential quantities traded. For example, on days where the price
was high there may have been more buyers, or buyers may have been
interested in buying larger quantities, or there may have been less
fish brought ashore.
In order to predict the effect of the tax, we need to think about the
responses of both buyers and sellers to changes in prices, and about
the determination of prices. This is where economic theory comes in.
\subsection{The Supply of and Demand for Fish}
So, how do economists go about analyzing questions such as this one if
not by regressing quantities on prices? The starting point for
economists is to think of an economic model for the determination of
prices (the treatment assignment mechanism in Rubin's potential outcome
terminology). The first part of the simplest model an economist would
consider for this type of setting is a pair of functions, the demand
and supply functions. Think of the buyers coming to the Fulton
fishmarket on a given market/day (say, day $t$) with a demand function
$Q^d_t(p)$. This function tells us, for that particular morning, how
much fish all buyers combined would be willing to buy if the price on
that day were $p$, for any value of $p$. This function is conceptually
exactly like the potential outcomes set up commonly used in causal
inference in the modern literature. It is more complicated than the
binary treatment case with two potential outcomes, because there is a
potential outcome for each value of the price, with more or less a
continuum of possible price values, but it is in line with continuous
treatment extensions such as those in \citet{GilRob01}. Common
sense, and economic theory, suggests that this demand function is a
downward sloping function: buyers would likely be willing to buy more
pounds of whiting if it were cheaper. Traditionally, the demand
function is specified parametrically, for example, linear in logarithms:
\begin{equation}
\label{demand} \ln Q^d_t(p)=\alpha^d+
\beta^d\times \ln p+\varepsilon^d_t,
\end{equation}
where $\beta^d$ is the price elasticity of demand.
This equation is \textit{not} a regression function like (\ref
{regression}). It is interpreted as a \textit{structural equation} or
behavioral equation, and in the treatment effect literature
terminology, it is a model for the potential outcomes.
Part of the confusion between the model for the potential outcomes in
(\ref{demand}) and the regression function in (\ref{regression}) may
stem from the traditional notation in the econometrics literature where
the same symbol (e.g.,\vspace*{1pt} $Q_t$) would be used for the observed
outcomes ($Q^{\mathrm{obs}}_t$ in our notation) and the potential outcome
function [$Q^d_t(p)$ in our notation], and the same symbol (e.g.,
$P_t$) would be used for the observed value of the treatment ($P^{\mathrm{obs}}_t$ in our notation) and the argument in the potential\vadjust{\goodbreak} outcome function
($p$~in our notation). Interestingly, the pioneers in this literature,
Tinbergen (\citeyear{Tin30}) and \citet{Haa43}, \textit{did} distinguish between
these concepts in their notation, but the subsequent literature on
simultaneous equations dropped that distinction and adopted a notation
that did not distinguish between observed and
potential outcomes. For a historical perspective see
Christ (\citeyear{Chr94}) and Stock and Trebbi (\citeyear{StoTre03}). My view
is that dropping this distinction was merely incidental, and that
implicitly the interpretation of the simultaneous equations models
remained that in terms of potential outcomes.\footnote{As a reviewer
pointed out, once one views simultaneous equations in terms of
potential outcomes, there is a natural normalization of the equations.
This suggests that perhaps the discussions of issues concerning
normalizations of equations in simultaneous equations models (e.g.,
Basmann, \citeyear{Bas63N1}, \citeyear{Bas63N2}, \citeyear{Bas65}; \cite*{Hil90}) implicitly rely on a
different interpretation, for example, thinking of the endogeneity
arising from measurement error. Throughout this discussion, I will
interpret simultaneous equations in terms of potential outcomes,
viewing the realized outcome notation simply as obscuring that.}
Implicit (by the lack of a subscript on the coefficients) in the
specification of the demand function in (\ref{demand}) is the strong
assumption that the effect of a unit change in the logarithm of the price
(equal to $\beta^d$)
is the same for all values of the price, and that the effect is the
same in all markets. This is clearly a very strong assumption, and the
modern literature on simultaneous equations (see \cite*{Mat07} for an
overview) has developed less restrictive specifications allowing for
nonlinear and nonadditive effects while maintaining identification.
The unobserved component in the demand function, denoted by
$\varepsilon^d_t$, represents unobserved determinants of the demand on
any given day/market: a particular buyer may be sick on a particular
day and not go to the market, or may be expecting a client wanting to
purchase a large quantity of whiting.
We can normalize this unobserved component to have expectation zero,
where the expectation is taken over all markets or days:
\[
\mathbb{E}\bigl[\ln Q^d_t(p)\bigr]=
\alpha^d+\beta^d\times\ln p.
\]
The interpretation of this expectation is subtle, and again it is part
of the confusion that sometimes arises. Consider the expected demand at
$p=1$, $\mathbb{E}[\ln Q^d_t(1)]$, under the linear
specification in (\ref{demand}) equal to $\alpha^d+\beta^d\cdot\ln
(1)=\alpha^d$. This $\alpha^d$ is the average of all demand
functions, evaluated at price equal to \$1.00, irrespective of what the
actual price in the market is, where the expectation is taken over \textit{all} markets. It is \textit{not}, and this is key, the {conditional}
expectation of the observed quantity in markets where the observed
price is equal to \$1.00 (or which is the same the demand function at 1
in those markets), which is $\mathbb{E}[\ln Q^{\mathrm{obs}}_t|\mbox{$P^{\mathrm{obs}}
_t=1$}]=\mathbb{E}[\ln Q^d_t(1)|P^{\mathrm{obs}}_t=1]$.
The original Tinbergen and Haavelmo notation and the modern potential
outcome version is helpful in making this distinction, compared to the
sixties econometrics textbook notation.\footnote{Other notations have
been recently proposed to stress the difference between the conditional
expectation of the observed outcome and the expectation of the
potential outcome.
\citet{Pea00} writes the
expected demand when the price is \textit{set} to \$1.00 as $\mathbb
{E}[\ln Q^d_t|\operatorname{do}(P_t=1)]$, rather than conditional
on the price being observed to be \$1.00.
\citet{HerRob06} write this average potential outcome as
$\mathbb{E}[\ln Q^d_t(P_t=1)]$, whereas
\citet{LauRic02} write it as $\mathbb{E}[\ln Q^{\mathrm{obs}}
_t\parallel P^{\mathrm{obs}}_t=1]$ where the double $\parallel$ implies conditioning by
intervention. }
Similar to the demand function, the supply function $Q^s_t(p)$
represents the quantity of whiting the sellers collectively are willing
to sell at any given price $p$, on day $t$. Here, common sense would
suggest that this function is sloping upward: the higher the price, the
more the sellers are willing to sell.
As with the demand function, the supply function is typically specified
parametrically with constant coefficients:
\begin{equation}
\label{supply} \ln Q^s_t(p)=\alpha^s+
\beta^s\times \ln p+\varepsilon^s_t,
\end{equation}
where $\beta^s$ is the price elasticity of supply.
Again we can normalize the expectation of $\varepsilon^s_t$ to zero
(where the expectation is taken over markets), and write
\[
\mathbb{E}\bigl[\ln Q^s_t(p)\bigr]=
\alpha^s+\beta^s\times\ln p.
\]
Note that the $\varepsilon^d_t$ and $\varepsilon^s_t$ are not assumed
to be independent in general, although in some applications that may be
a reasonable assumption. In this specific example, $\varepsilon_t^d$
may represent random variation in the set or number of buyers coming to
the market on a particular day, and $\varepsilon^s_t$ may represent
random variation in suppliers showing up at the market and in their
ability to catch whiting during the preceding days. These components
may well be uncorrelated, but there may be common components, for
example, in traffic conditions around the market that make it difficult
for both suppliers and buyers to come to the market.
\subsection{Market Equilibrium}
Now comes the second part of the simple economic model, the
determination of the price, or, in the terminology of the treatment
effect literature, the assignment mechanism.
The conventional\vadjust{\goodbreak} assumption in this type of market is that the price
that is observed, that is, the price at which the fish is traded in
market/day $t$, is the (unique) market clearing price at which demand
and supply are equal. In other words, this is the price at which the
market is in \textit{equilibrium}, denoted by $P^{\mathrm{obs}}_t$. This
equilibrium price solves
\begin{equation}
\label{equilibrium} Q^d_t\bigl(P_t^{\mathrm{obs}}
\bigr)=Q^s_t\bigl(P_t^{\mathrm{obs}}\bigr).
\end{equation}
The observed quantity on that day, that is the quantity actually
traded, denoted by $Q^{\mathrm{obs}}_t$, is then equal to the demand function at
the equilibrium price (or, equivalently, because of the equilibrium
assumption, the supply function at that price):
\begin{equation}
\label{q_equi} Q^{\mathrm{obs}}_t=Q^d_t
\bigl(P_t^{\mathrm{obs}}\bigr)=Q^s_t
\bigl(P_t^{\mathrm{obs}}\bigr).
\end{equation}
Assuming that the demand function does slope downward and the supply
function does slope upward, and both are linear in logarithms, the
equilibrium price exists and is unique, and we can solve for the
observed price and quantities in terms of the parameters of the model
and the unobserved components:
\begin{eqnarray*}
\ln P^{\mathrm{obs}}_t &=& \frac{\alpha^d-\alpha^s}{\beta^s-\beta^d}+ \frac{\varepsilon^d_t-\varepsilon^s_t}{\beta^s-\beta^d}
\quad \mbox{and}
\\
\ln Q^{\mathrm{obs}}_t &=& \frac{\beta^s\cdot\alpha^d-\beta^d\cdot\alpha
^s}{\beta^s-\beta^d}+ \frac{\beta^s\cdot\varepsilon^d_t-\beta^d\cdot\varepsilon
^s_t}{\beta^s-\beta^d}.
\end{eqnarray*}
For economists, this is a more plausible model for the determination of
realized prices and quantities than the model that assumes prices are
independent of market conditions. It is not without its problems
though. Chief among these from our perspective is the complication
that, just as in the Roy model, we cannot necessarily infer the values
of the unknown parameters in this model even if we have data on
equilibrium prices and quantities $P^{\mathrm{obs}}_t$ and $Q^{\mathrm{obs}}_t$ for many markets.
Another issue is how buyers and sellers arrive at the equilibrium price.
There is a theoretical economic literature addressing this question.
Often the idea is that there is a sequential process of buyers making
bids, and suppliers responding with offers of quantities at those
prices, with this process repeating itself until it arrives at a price
at which supply and demand are equal. In practice, economists often
refrain from specifying the details of this process and simply assume
that the market is in equilibrium. If the process is fast enough, it
may be reasonable to ignore the fact the specifics of the process and
analyze the data as if equilibrium was instantaneous.\footnote{See
\citet{ShaShu77} and \citet{Gir03}, and for some experimental
evidence, Plott and Smith (\citeyear{PloSmi87}) and \citet{Smi}.}
A related issue is whether this model with an equilibrium prices that
equates supply and demand is a reasonable approximation to the actual
process that determines prices and quantities. In fact, Graddy's data
contains information showing that the seller would trade at different
prices on the same day, so strictly speaking this model does not hold.
There is a long tradition in economics, however, of using such models
as approximations to price determination and we will do so here.
Finally, let me connect this to the textbook discussion of supply and
demand models. In many textbooks, the demand and supply equations would
be written directly in terms of the observed (equilibrium) quantities
and prices as
\begin{eqnarray}
\label{wold1} Q^{\mathrm{obs}}_t &=& \alpha^s+
\beta^s\times\ln P^{\mathrm{obs}}_t+\varepsilon^s_t,
\\
\label{wold2} Q^{\mathrm{obs}}_t &=& \alpha^d+
\beta^d\times\ln P^{\mathrm{obs}}_t+\varepsilon^d_t.
\end{eqnarray}
This representation leaves out much of the structure that gives the
demand and supply function their meaning, that is, the demand equation
(\ref{demand}), the supply equation (\ref{supply}) and the
equilibrium condition (\ref{equilibrium}). As \citet{StrWol60}
write, ``Those who write such systems [(\ref{wold2}) and (\ref{wold2})] do not, however, really mean what they write, but introduce
an ellipsis which is familiar to economists'' (page 425), with the
ellipsis referring to the market equilibrium condition
that is left out. See also Strotz (\citeyear{Str60}), Strotz and Wold (\citeyear{StrWol65}),
and Wold (\citeyear{Wol60})
\subsection{The Statistical Demand Curve}
Given this set up, let me discuss two issues. First,
let us explore, under this model, the interpretation of what \citet{Wor27} called the ``statistical demand curve.'' The covariance between
observed (equilibrium) log quantities and log prices is
$ \operatorname{cov} (\ln Q_t^{\mathrm{obs}},\ln P^{{\mathrm{obs}}}_t )=(\beta
^s\cdot\sigma^2_{d}+
\beta^d\cdot\sigma^2_{s}-\rho\cdot\sigma_d\cdot\sigma_s\cdot
(\beta^d+\beta^s))/( (\beta^s-\beta^d )^2)$,
where $\sigma_d$ and $\sigma_s$ are the standard deviations of
$\varepsilon_t^d$ and $\varepsilon_t^s$, respectively, and $\rho$ is
their correlation.
Because the variance of $\ln P^{{\mathrm{obs}}}_t$ is $(\sigma_s^2+\sigma
^2_d-2\cdot\rho\cdot\sigma_d\cdot\sigma_s)/(\beta^s-\beta
^d)^2$, it follows that
the regression coefficient in the regression of log quantities on log
prices is
\begin{eqnarray*}
&& \frac{\operatorname{cov} (\ln Q_t^{{\mathrm{obs}}},\ln P^{{\mathrm{obs}}}_t )}{\operatorname{var} (\ln P^{{\mathrm{obs}}}_t )}
\\
&& \quad =\frac{\beta^s\cdot\sigma^2_{d}+
\beta^d\cdot\sigma^2_{s}-\rho\cdot\sigma_d\cdot\sigma_s\cdot
(\beta^d+\beta^s)}{\sigma_s^2+\sigma^2_d-2\cdot\rho\cdot\sigma
_d\cdot\sigma_s}.
\end{eqnarray*}
Working focuses on the interpretation of this relation between
equilibrium quantities and prices. Suppose that
the correlation between $\varepsilon^d_t$ and $\varepsilon^s_t$,
denoted by $\rho$, is zero. Then the regression coefficient is a
weighted average of the two slope coefficients of the supply and demand
function, weighted by the variances of the residuals:
\[
\frac{\operatorname{cov} (\ln Q_t^{{\mathrm{obs}}},\ln P^{{\mathrm{obs}}}_t )}{\operatorname{var} (\ln P^{{\mathrm{obs}}}_t )}
=\beta^s\cdot
\frac{\sigma^2_{d}}{\sigma_s^2+\sigma^2_d} +\beta^d\cdot\frac{
\sigma^2_{s}}{\sigma_s^2+\sigma^2_d}.
\]
If $\sigma_d^2$ is small relative to $\sigma^2_s$, then we estimate
something close to the slope of the demand function, and if $\sigma
_s^2$ is small relative to $\sigma^2_d$, then we estimate something
close to the slope of the supply function. In general, however, as
Working stresses, the ``statistical demand curve'' is not informative
about the demand function (or about the supply function); see also
\citet{LeaN1}.
\subsection{The Effect of a Tax Increase}
The
second question is how this model with supply and demand functions and
a market clearing price helps us answer the substantive question of
interest. The specific question considered is
the effect of the tax increase on the average quantity traded. In a
given market, let $p$ be the price sellers receive per pound of
whiting, and let $\tilde{p}=p\times(1+r)$ the price buyers pay after
the tax has been imposed. The key assumption is that the only way
buyers and sellers respond to the tax is through the effect of the tax
on prices: they do not change how much they would be willing to buy or
sell at any given price, and the process that determines the
equilibrium price does not change. The technical econometric term for
this is that the demand and supply functions are \textit{structural} or
\textit{invariant} in the sense that they are not affected by changes in
the treatment, taxes in this case. This may not be a perfect
assumption, but certainly in many cases it is reasonable: if I have to
pay \$1.10 per pound of whiting, I probably do not care whether 10 cts
of that goes to the government and \$1 to the seller, or all of it goes
to the seller. If we are willing to make that assumption, we can solve
for the new equilibrium price and quantity. Let $P_t(r)$ be the new
equilibrium price [net of taxes, that is, the price sellers receive,
with $(1+r)\cdot P_t(r)$ the price buyers pay], given a tax rate $r$,
with in our example $r=0.1$. This price solves
\[
Q^d_t\bigl(P_t(r)\times(1+r)
\bigr)=Q^s_t\bigl(P_t(r)\bigr).
\]
Given the log linear specification for the demand and supply functions,
this leads to
\[
\ln P_t(r)=\frac{\alpha^d-\alpha^s}{\beta^s- \beta^d}+ \frac{\beta^d\times\ln(1+r)}{\beta^s-\beta^d}+
\frac{\varepsilon^d_t-\varepsilon^s_t}{\beta^s- \beta^d}.
\]
The result of the tax is that the average of the logarithm of the price
that sellers receive with a positive tax rate $r$ is less than what
they would have received in the absence of the tax rate:
\begin{eqnarray*}
\mathbb{E}\bigl[\ln P_t(r)\bigr] & =& \frac{\alpha^d-\alpha^s}{\beta
^s-\beta^d}+
\frac{\beta^d\times\ln(1+r)}{\beta^s-\beta^d}
\\
& \leq & \frac{\alpha^d-\alpha^s}{\beta^s- \beta^d}= \mathbb{E}\bigl[\ln P_t(0)\bigr].
\end{eqnarray*}
(Note that $\beta^d<0$.)
On the other hand,
the buyers will pay more on average:
\begin{eqnarray*}
\mathbb{E}\bigl[\ln\bigl((1+r)\cdot P_t(r)\bigr)\bigr] &=&
\frac{\alpha
^d-\alpha^s}{\beta^s- \beta^d}+\frac{\beta^s\times\ln
(1+r)}{\beta^s-\beta^d}
\\
& \geq &
\mathbb{E}\bigl[\ln P_t(0)\bigr].
\end{eqnarray*}
The quantity traded after the tax increase is
\begin{eqnarray*}
\ln Q_t(r) &= & \frac{\beta^s\cdot\alpha^d-\beta^d\cdot\alpha
^s}{\beta^s-\beta^d} +\frac{\beta^s\cdot\beta^d\cdot\ln(1+r)}{\beta^s-\beta^d}
\\
&&{}+ \frac{\beta^s\cdot\varepsilon^d_t-\beta^d\cdot\varepsilon
^s_t}{\beta^s-\beta^d},
\end{eqnarray*}
which is less than the quantity that would be traded in the absence of
the tax increase. The
causal effect is
\[
\ln Q_t(r)-\ln Q_t(0)= \frac{\beta^s\cdot\beta^d\cdot\ln(1+r)}{\beta^s-\beta^d},
\]
the same in all markets, and
proportional to the supply and demand elasticities and, for small $r$,
proportional to the tax.
What should we take away from this discussion? There are three points.
First, the regression coefficient in the regression of log quantity on
log prices does not tell us much about the effect of new tax. The sign
of this regression coefficient is ambiguous, depending on the variances
and covariance of the unobserved determinants of supply and demand.
Second, in order to predict the magnitude of the effect of a new tax we
need to learn about the demand and supply functions separately, or in
the econometrics terminology, \textit{identify} the supply and demand
functions. Third, observations on equilibrium prices and quantities by
themselves do not identify these functions.
\subsection{Identification with Instrumental Variables}
Given this identification problem, how \textit{do} we identify the demand
and supply functions? This is where instrumental variables enter the discussion.
To identify the demand function, we look for determinants of the
supply of whiting that do not affect the demand for whiting, and,
similarly, to identify the supply function we look for determinants of
the demand for whiting that do not affect the supply. In this specific
case, Graddy (\citeyear{Gra95}, \citeyear{Gra96}) assumes that weather conditions at sea on the
days prior to market $t$, denoted by $Z_t$, affect supply but do not
affect demand. Certainly, it appears reasonable to think that weather
is a direct determinant of supply: having high waves and strong winds
makes it harder to catch fish.
On the other hand, there does not seem to be any reason why demand on
day $t$, at a given price $p$, would be correlated with wave height or
wind speed on previous days. This assumption may be made more plausible
by conditioning on covariates. For example, if one is concerned that
weather conditions on land affect demand, one may wish to condition on
those, and only look at variation in weather conditions at sea given
similar weather conditions on land as an instrument.
Formally, the key assumptions are that
\[
Q_t^d(p) \perp Z_t \quad \mbox{and}\quad
Q_t^s(p) \not\perp Z_t,
\]
possibly conditional on covariates.
If both of these conditions hold,
we can use weather conditions as an instrument.
How do we exploit these assumptions? The traditional approach is to
generalize the functional form of the supply function to explicitly
incorporate the effect of the instrument on the supply of whiting. In
our notation,
\[
\ln Q^s_t(p,z)=\alpha^s+\beta^s
\times\ln p+\gamma^s\times z+\varepsilon^s_t.
\]
The demand function remains unchanged, capturing the fact that demand
is not affected by the instrument:
\[
\ln Q^d_t(p,z)=\alpha^d+\beta^d
\times\ln p+\varepsilon^d_t.
\]
We assume
that the unobserved components of supply and demand are independent of
(or at least uncorrelated with) the weather conditions:
\[
\bigl(\varepsilon_t^d,\varepsilon^s_t
\bigr) \perp Z_t.
\]
The equilibrium price $P^{\mathrm{obs}}_t$ is the solution for $p$ in the equation
\[
Q^d(p,Z_t)=Q^s_t(p,Z_t),
\]
which, in combination with the log linear specification for the demand
and supply functions, leads to
\[
\ln P^{{\mathrm{obs}}}_t=\frac{\alpha^d-\alpha^s}{\beta^s-\beta^d}+ \frac{\varepsilon^d_t-\varepsilon^s_t}{\beta^s-\beta^d}-
\frac{\gamma^s\cdot Z_t}{\beta^s-\beta^d}
\]
and
\begin{eqnarray*}
\ln Q^{{\mathrm{obs}}}_t &=& \frac{\beta^s\cdot\alpha^d-\beta^d\cdot\alpha
^s}{\beta^s-\beta^d}+ \frac{\beta^s\cdot\varepsilon^d_t-\beta^d\cdot\varepsilon
^s_t}{\beta^s-\beta^d}
\\
&& {}- \frac{\gamma^s\cdot\beta^d\cdot Z_t}{\beta^s-\beta^d}.
\end{eqnarray*}
Now consider the expected value of the equilibrium price and quantity
given the weather conditions:
\begin{eqnarray}
\label{eq:reduced1}
&& {\mathbb{E}}\bigl[ \ln Q^{{\mathrm{obs}}
}_t|Z_t=z
\bigr]
\nonumber
\\[-8pt]
\\[-8pt]
&& \quad =\frac{\beta^s\cdot\alpha^d-\beta^d\cdot
\alpha^s}{\beta^s-\beta^d} - \frac{\gamma^s\cdot\beta^d}{\beta^s-\beta^d}\cdot z
\nonumber
\end{eqnarray}
and
\begin{equation}
\label{eq:reduced2} \qquad\quad{\mathbb{E}}\bigl[ \ln P^{{\mathrm{obs}}}_t
|Z_t=z\bigr]=\frac{\alpha
^d-\alpha^s}{\beta^s-\beta^d}- \frac{\gamma^s}{\beta^s-\beta^d}\cdot z.
\end{equation}
Equations (\ref{eq:reduced1}) and
(\ref{eq:reduced2}) are what is called in econometrics the \textit{reduced form} of the simultaneous equations model.
It expresses the
\textit{endogenous} variables (those variables whose values are determined inside
the model, price and quantity in this example) in terms of the \textit{exogenous} variables (those variables whose values are not determined within the
model, weather conditions in this example).
The slope coefficients on the instrument in these reduced form
equations are what in randomized experiments with noncompliance would
be called the \textit{intention-to-treat} effects.
One can estimate the coefficients in the reduced form by least squares
methods. The key insight is that the ratio of the coefficients on the
weather conditions in the two regression functions,
$\gamma^s\cdot\beta^d/(\beta^s-\beta^d)$ in the quantity
regression and $\gamma^s/(\beta^s-\beta^d)$ in the price regression,
is equal to the slope coefficient in the demand function.
\begin{figure*}
\caption{Scatterplot of log prices and log quantities by weather conditions.}
\label{fig2}
\end{figure*}
For some purposes, the reduced-form or intention-to-treat effects may
be of substantive interest. In the Fulton fish market example, people
attempting to predict prices and quantities under the current regime
may find these estimates of interest. They are of less interest to
policy makers contemplating the introduction of a new tax.
In simultaneous equations settings, the demand and supply functions are
viewed as \textit{structural} in the sense that they are not affected by
interventions in the market such as new taxes. As such they, and not
the reduced-form regression functions, are the key components of
predictions of market outcomes under new regimes.
This is somewhat different in many of the recent applications of
instrumental variables methods in the statistics literature in the
context of randomized experiments with noncompliance where the
intention-to-treat effects are traditionally of primary interest.
Let me illustrate this with the Fulton Fish Market data collected by
Graddy. For ease of illustration, let me simplify the instrument to a
binary one: the weather conditions are good for catching fish ($Z_t=0$,
fair weather, corresponding to low wind speed and low wave height) or
stormy ($Z_t=1$, corresponding to relatively strong winds and high
waves).\footnote{The formal definition I use, following \citet{AngGraImb00} is that stormy is defined as wind speed
greater than 18 knots in combination with wave height more than 4.5 ft,
and fair weather is anything else.} The price is the average daily
price in cents for one dealer, and the quantity is the daily quantity
in pounds. The two estimated reduced forms are
\[
\begin{array} {ccccc}\widehat{\ln Q}^{\mathrm{obs}}_t=&
8.63&-&0.36&\times Z_t
\\
& (0.08)&& (0.15) \end{array}
\]
and
\[
\begin{array}{ccccc}
\widehat{\ln P}^{\mathrm{obs}}_t=& \!\!\!
-0.29&+&0.34&\times Z_t.
\\
& \!\!\!\hphantom{-}(0.04)&& (0.07) \end{array}
\]
Hence, the instrumental variables estimate of the slope of the demand
function is
\[
\hat\beta^d=\frac{-0.36}{0.34}=-1.08\quad (\mbox{s.e. }0.46).
\]
Another, perhaps more intuitive way of looking at these estimates is to
consider the location of the average log quantity and average log price
separately by weather conditions. Figure~\ref{fig2} presents the scatter plot of
log quantity and log prices, with the stars indicating stormy days and
the plus signs indicating calm days.
On fair weather days the average log price is $-0.29$, and the average
log quantity is 8.6. On stormy days, the average log price is 0.04, and
the average log quantity is 8.3. These two loci are marked
by circles in Figure~\ref{fig2}.
On stormy days, the price is higher and the quantity traded is lower
than on fair weather days. This is used to estimate the slope of the
demand function.
The figure also includes the estimated demand function based on using
the indicator for stormy days as an instrument for the price: the
estimated demand function goes through the two points defined by the
average of the log price and log quantity for stormy and fair weather days.
With the data collected by Graddy, it is more difficult to point
identify the supply curve. The traditional route toward identifying the
supply curve would rely on finding an instrument that shifts demand
without directly affecting supply. Without such an instrument, we
cannot point identify the effect of the introduction of the tax on
quantity and prices. It is possible under weaker assumptions to find
bounds on these estimands (e.g., \cite*{LeaN1}; \cite*{Man03}), but
we do not pursue this here.
\subsection{Recent Research on Simultaneous Equations~Models}
The traditional econometric literature on simultaneous equations models
is surveyed in \citet{Hau83}. Compared to the discussion in the
preceding sections, this literature focuses on a more general case,
allowing for multiple endogenous variables and multiple instruments.
The modern econometric literature, starting in the 1980s, has relaxed
the linearity and additivity assumptions in specification (\ref
{demand}) substantially.
Key references to this literature are \citet{Bro83}, \citet{Roe88},
Newey and Powell (\citeyear{N03}), Chesher (\citeyear{Che03}, \citeyear{Che10}), \citet{BenBer06}, Matzkin (\citeyear{Mat03}, \citeyear{Mat07}),
\citet{AltMat05}, \citet{ImbNew09}, \citet{HodMam07},
\citet{Hor11} and \citet{HorLee07}.
\citet{Mat07} provides a recent survey of this technically demanding
literature.
This literature has continued to use the observed outcome notation,
making it more difficult to connect to the statistical literature.
Here, I briefly review some of this literature.
The starting point is a structural equation, in the potential outcome notation,
\[
Y_i(x)=\alpha+\beta\cdot x+\varepsilon_i
\]
and an instrument $Z_i$ that satisfies
\[
Z_i\perp\varepsilon_i\quad \mbox{and}\quad
Z_i\not\perp X_i.
\]
The traditional econometric literature would formulate this in the
observed outcome notation as
\[
Y_i=\alpha+\beta\cdot X_i+\varepsilon_i,
\quad Z_i\perp\varepsilon_i \quad \mbox{and}\quad
Z_i\not\perp X_i.
\]
There are a number of generalizations considered in the modern
literature. First, instead of assuming independence of the unobserved
component and the instrument, part of the current literature assumes
only that the conditional mean of the unobserved component given the
instrument is free of dependence on the instrument, allowing the
variance and other distributional aspects to depend on the value of the
instrument; see \citet{Hor11}. Another generalization of the linear
model allows for general nonlinear function forms of the type
\[
Y_i=g(X_i)+\varepsilon_i,\quad
Z_i\perp\varepsilon_i \quad \mbox{and} \quad
Z_i\not\perp X_i,
\]
where the focus is on nonparametric identification and estimation of
$g(x)$; see \citet{Bro83}, \citet{Roe88}, \citet{BenBer06}.
Allowing for even more generality, researchers have studied nonadditive
versions of these models with
\[
Y_i=g(X_i,\varepsilon_i),\quad
Z_i\perp\varepsilon_i \quad \mbox{and} \quad
Z_i\not\perp X_i,
\]
with $g(x,\varepsilon)$ strictly monotone in a scalar unobserved
component $\varepsilon$. In these settings, point identification often
requires strong assumptions on the support of the instrument and its
relation to the endogenous regressor and, therefore, researchers have
also explored bounds. See Matzkin (\citeyear{Mat03}, \citeyear{Mat07}, \citeyear{Mat08}) and \citet{ImbNew09}.
\section{A Modern Example: Randomized Experiments with Noncompliance
and Heterogenous Treatment Effects}
\label{section:modern}
In this section, I will discuss part of the modern literature on
instrumental variables methods that has evolved simultaneously in the
statistics and econometrics literature. I will do so in the context of
a second example.
On the one hand, concern arose in the econometric literature about the
restrictiveness of the functional form assumptions in the traditional
instrumental variables methods and in particular with the constant
treatment effect assumption that were commonly used in the so-called
selection models (\cite{Hec79}; \cite{HecRob}).
The initial results in this literature demonstrated the difficulties in
establishing point identification (\cite{Hec}; \cite{Man90}),
leading to the bounds approach developed by Manski (\citeyear{Man95}, \citeyear{Man03}). At the
same time, statisticians analyzed the complications arising from
noncompliance in randomized experiments (\cite{Rob}) and the merits
of encouragement designs (Zelen, \citeyear{Zel79}, \citeyear{Zel90}).
By adopting a common framework and notation in \citet{ImbAng94} and \citet{AngImbRub}, these literatures have
become closely connected and influenced each other substantially.
\subsection{\texorpdfstring{The McDonald, Hiu and Tierney (\citeyear{McDHiuTie92}) Data}{The McDonald, Hiu and Tierney (1992) Data}}
The canonical example in this literature is that of a randomized
experiment with noncompliance.
To illustrate the issues,
I will use here data previously analyzed in Hirano et al. (\citeyear{Hiretal00})
and
McDonald, Hiu and Tierney (\citeyear{McDHiuTie92}).
McDonald, Hiu and Tierney (\citeyear{McDHiuTie92}) carried out a randomized experiment to
evaluate the effect of an influenza vaccination on flu-related hospital
visits. Instead of randomly assigning individuals to receive the
vaccination, the researchers randomly assigned physicians to receive
letters reminding them of the upcoming flu season and encouraging them
to vaccinate their patients. This is what Zelen (\citeyear{Zel79}, \citeyear{Zel90}) refers to
as an \textit{encouragement design}. I discuss this using the potential
outcome notation used for this particular set up in \citet{AngImbRub}, and in general sometimes referred to as the Rubin Causal
Model (\cite{Hol86}), although there are important antecedents in
Splawa-Neyman (\citeyear{Spl90}). I consider two distinct treatments: the first the
receipt of the letter, and second the receipt of the influenza
vaccination. Let $Z_i\in\{0,1\}$ be the indicator for the receipt of
the letter, and let $X_i\in\{0,1\}$ be the indicator for the receipt
of the vaccination. We start by postulating the existence of four
potential outcomes. Let $Y_i(z,x)$ be the potential outcome
corresponding to the receipt of letter equal to $Z_i=z$, and the
receipt of vaccination equal to $X_i=x$, for $z=0,1$ and $x=0,1$.
In addition, we postulate the existence of two potential outcomes
corresponding to the receipt of the vaccination as a function of the
receipt of the letter, $X_i(z)$, for $z=0,1$.
We observe for each unit in a population of size $N=2861$ the value of
the assignment, $Z_i$, the treatment actually received, $X_i^{\mathrm{obs}}
=X_i(Z_i)$ and the potential outcome corresponding to the assignment
and treatment received, $Y^{\mathrm{obs}}_i=Y_i(Z_i,X_i(Z_i))$.
Table~\ref{summ_stats_flu} presents the number of individuals for each
of the eight values of the triple $(Z_i,X^{\mathrm{obs}}_i,Y^{\mathrm{obs}}_i)$ in the
McDonald, Hiu and Tierney data set.
It should be noted that the randomization in this experiment is at the
physician level. I do not have physician indicators and, therefore,
ignore the clustering. This will tend to lead to underestimation of the
standard errors.
\begin{table}
\caption{Influenza data ($N=2861$)}\label{summ_stats_flu}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}lccd{4.0}@{}}
\hline
\textbf{Hospitalized for} & \multicolumn{1}{c}{\textbf{Influenza}} & &
\\
\textbf{flu-related reasons} & \textbf{vaccine} & \textbf{Letter} & \multicolumn{1}{c@{}}{\textbf{Number of}} \\
$\boldsymbol{Y}_{\bolds{i}}^{\mathbf{obs}}$ & $\bolds{X}^{\mathbf{obs}}_{\bolds{i}}$ & $\bolds{Z}_{\bolds{i}}$ & \multicolumn{1}{c@{}}{\textbf{individuals}} \\
\hline
No & No & No & 1027 \\
No & No & Yes & 935 \\
No & Yes & No & 233 \\
No & Yes & Yes & 422 \\
Yes & No & No & 99 \\
Yes & No & Yes & 84 \\
Yes & Yes & No & 30 \\
Yes & Yes & Yes & 31 \\
\hline
\end{tabular*}
\end{table}
\subsection{Instrumental Variables Assumptions}
\label{assumptions}
There are four key of assumptions underlying instrumental variables
methods beyond the no-interference assumption or SUTVA, with different
versions for some of them.
I will introduce these assumptions in this section, and in Section~\ref{section:content} discuss their substantive content in the context of
some examples.
The first assumption concerns the assignment to the instrument $Z_i$,
in the flu example the receipt of the letter by the physician. The
assumption requires that the instrument is as good as randomly assigned:
\begin{eqnarray}
\label{eq:een}
&& Z_i \perp \bigl( Y_i(0,0),Y_i(0,1),Y_i(1,0),
\nonumber
\\
&&\hspace{48pt} Y_i(1,1),X_i(0),X_i(1) \bigr)
\\
\eqntext{\mbox{(random assignment).}}
\end{eqnarray}
This assumption is often satisfied by design: if the assignment is
physically randomized, as the letter in the flu example and as in many
of the applications in the statistics literature (e.g., see the
discussion in \cite*{Rob}), it is automatically satisfied. In other
applications with observational data, common in the econometrics
literature, this assumption is more controversial. It can in those
cases be relaxed by requiring it to hold only within subpopulations
defined by covariates $V_i$, assuming the assignment of the instrument
is unconfounded:
\begin{eqnarray}
&& Z_i \perp \bigl( Y_i(0,0),Y_i(0,1),Y_i(1,0),
\nonumber\\
&&\hspace{48pt} Y_i(1,1),X_i(0),X_i(1) \bigr) |
V_i
\nonumber\\
\eqntext{(\mbox{unconfounded assignment given }V_i).}
\end{eqnarray}
This is identical to the generalization from random assignment to
unconfounded assignment in observational studies.
Either version of this assumption justifies the causal interpretation
of \textit{Intention-To-Treat} (ITT) effects, the comparison of outcomes
by assignment to the treatment. In many cases, these ITT effects are
only of limited interest, however, and this motivates the consideration
of additional assumptions that do allow the researcher to make
statements about the causal effects of the treatment of interest.
It should be stressed, however, that in order to draw inferences beyond
ITT effects, additional assumptions will be used; whether the resulting
inferences are credible will depend on the credibility of these assumptions.
The second class of assumptions limits or rules out completely direct
effects of the assignment (the receipt of the letter in the flu
example) on the outcome, other than through the effect of the
assignment on the receipt of the treatment of interest (the receipt of
the vaccine). This is the most critical, and typically most
controversial assumption underlying instrumental variables methods,
sometimes viewed as the defining characteristic of instruments. One way
of formulating this assumption is as
\begin{eqnarray}
&& Y_i(0,x)=Y_i(1,x)\quad \mbox{for } x=0,1, \mbox{for
all } i
\nonumber\\
\eqntext{\mbox{(exclusion restriction).}}
\end{eqnarray}
\citet{Rob} formulates a similar assumption as requiring that the
instrument is ``not an independent causal risk factor'' (\cite*{Rob},
page 119).
Under this assumption, we can drop the $z$ argument of the potential
outcomes and write the potential outcomes without ambiguity as $Y_i(x)$.
This assumption is typically a substantive one. In the flu example, one
might be concerned that the physician, in response to the receipt of
the letter, takes actions that affect the likelihood of the patient
getting infected with the flu other than simply administering the flu
vaccine. In randomized experiments with noncompliance, the exclusion
restriction is sometimes made implicitly by indexing the potential
outcomes only by the treatment $x$ and not the instrument $z$ (e.g., \cite*{Zel90}).
There are other, weaker versions of this assumption. Hirano et al. (\citeyear{Hiretal00}) use a stochastic version of the exclusion
restriction that only requires that the distribution of $Y_i(0,x)$ is
the same as the distribution of $Y_i(1,x)$. \citet{Man90} uses a weaker
restriction that he calls a \textit{level set restriction}, which requires
that the average value of $Y_i(0,x)$ is equal to the average value of
$Y_i(1,x)$. In another approach, Manski and Pepper (\citeyear{ManPep00}) consider
monotonicity assumptions that restrict the sign of $Y_i(1,x)-Y_i(0,x)$
across individuals without requiring that the effects are completely absent.
\citet{ImbAng94} combine the random assignment assumption and
the exclusion restriction by postulating the existence of a pair of
potential outcomes $Y_i(x)$, for $x=0,1$, and directly assuming that
\[
Z_i \perp \bigl(Y_i(0),Y_i(1) \bigr).
\]
A disadvantage of this formulation is that it becomes less clear
exactly what role randomization of the instrument plays.
Another version of this combination of the exclusion restriction and
random assignment assumption
does not require full independence, but assumes that the conditional
mean of $Y_i(0)$ and $Y_i(1)$ given the instrument is free of
dependence on the instrument. A~concern with such assumptions is that
they are functional form dependent: if they hold in levels, they do not
hold in logarithms unless full independence holds.
A third assumption that is often used, labeled \textit{monotonicity} by
\citet{ImbAng94}, requires that
\[
X_i(1)\geq X_i(0)\quad \mbox{for all } i \quad
\mbox{(monotonicity)},
\]
for all units. This assumption rules out the presence of units who
always do the opposite of their assignment [units with $X_i(0)=1$ and
$X_i(1)=0$], and is therefore also referred to as the \textit{no-defiance}
assumption (\cite{autokey15}).
It is implicit in the latent index models often used in econometric
evaluation models (e.g., Heckman and Robb, \citeyear{HecRob}).
In the randomized experiments such as the flu example, this assumption
is often plausible. There it requires that in response to the receipt
of the letter by their physician, no patient
is less likely to get the vaccine. \citet{Rob} makes this assumption
in the context of a randomized trial for the effect of AZT on AIDS, and
describes the assumption as ``often, but not always, reasonable''
(\cite*{Rob}, page~122).
Finally, we need the instrument to be correlated with the treatment, or
the instrument to be \textit{relevant} in the terminology of \citet{Phi89} and \citet{StaSto97}:
\[
X_i\not\perp Z_i.
\]
In practice, we need the correlation to be substantial in order to draw
precise inferences. A recent literature on \textit{weak instruments} is
concerned with credible inference in settings where this correlation
between the instrument and the treatment is weak; see \citet{StaSto97} and \citet{AndSto}.
The random assignment assumption and the exclusion restriction are
conveniently captured by the graphical model below, although the
monotonicity assumption does not fit in as easily.
The unobserved component $U$ has a direct effect on both the treatment
$X$ and the outcome $Y$ (captured by arrows from $U$ to $X$ and to~$Y$). The instrument $Z$ is not related to the unobserved component $U$
(captured by the absence of a link between $U$ and $Z$), and is only
related to the outcome $Y$ through the treatment $X$ (as captured by
the arrow from $Z$ to $X$ and an arrow from $X$ to $Y$, and the absence
of an arrow between $Z$ and $Y$).
I will primarily focus on the case with all four assumptions
maintained, random assignment, the exclusion restriction, monotonicity
and instrument relevance, without additional covariates, because this
case has been the focus of, or a special case of the focus of, many
studies, allowing me to compare different approaches. Methodological
studies considering essentially this set of assumptions, sometimes
without explicitly stating instrument relevance, and sometimes adding
additional assumptions, include \citet{Rob}, \citet{Hec}, \citet{Man90},
\citet{ImbAng94}, \citet{AngImbRub}, \citet{RobGre96}, Balke and Pearl (\citeyear{autokey15}, \citeyear{BalPea97}), \citet{Gre}, Hern\'
an and Robins (\citeyear{HerRob06}), \citet{Rob94}, \citet{RobRot04},
\citet{VanGoe03}, Vansteelandt et al. (\citeyear{Vanetal11}), Hirano et al. (\citeyear{Hiretal00}),
Tan (\citeyear{Tan06}, \citeyear{Tan10}), Abadie (\citeyear{Aba02}, \citeyear{Aba03}), Duflo,
Glennester and Kremer (\citeyear{DufGleKre07}), Brookhart et al. (\citeyear{Broetal06}),
Martens et al. (\citeyear{Maretal06}), Morgan and
Winship (\citeyear{MorWin07}), and others.
Many more studies make the same assumptions in combination with a
constant treatment effect assumption.
The modern literature analyzed this setting from a number of different
approaches. Initially, the literature focused on the inability, under
these four assumptions, to identify the average effect of the
treatment. Some researchers, including prominently \citet{Man90}, \citet{autokey15} and \citet{Rob}, showed that although one could not
point-identify the average effect under these assumptions, there was
information about the average effect in the data under these
assumptions and they derived bounds for it. Another strand of the
literature, starting with \citet{ImbAng94} and \citet{AngImbRub} abandoned the effort to do inference for the overall
average effect, and focused on subpopulations for which the average
effect could be identified, the so-called compliers, leading to the
local average treatment effect. We discuss the bounds approach in the
next section (Section~\ref{bounds}) and the local average treatment
effect approach in Sections~\ref{types}--\ref{section:dowecare}.
\subsection{Point Identification {versus} Bounds}
\label{bounds}
In a number of studies, the primary estimand is the average effect of
the treatment, or the average effect for the treated:
\begin{eqnarray}
\label{ate} \tau & = &\mathbb{E}\bigl[Y_i(1)-Y_i(0)
\bigr]\quad \mbox{and}
\nonumber
\\[-8pt]
\\[-8pt]
\tau_t &=& \mathbb{E}\bigl[Y_i(1)-Y_i(0)|X_i=1
\bigr].
\nonumber
\end{eqnarray}
With only the four assumptions, random assignment, the exclusion
restriction, monotonicity, and instrument relevance \citet{Rob},
\citet{Man90} and \citet{autokey15} established that the
average treatment effect can often not be consistently estimated even
in large samples. In other words, that it is often \textit{not point-identified}.
Following this result, a number of different approaches have been taken.
\citet{Hec} showed that if the instrument takes on values such that
the probability of treatment given the instrument can be arbitrarily
close to zero and one, then the average effect is identified. This is
sometimes referred to as \textit{identification at infinity}.
\citet{Rob} also formulates assumptions that allow for point
identification, focusing on the average effect for the treated, $\tau
_t$. These assumptions restrict the average value of the potential
outcomes when not observed in terms of average outcomes that are
observed. For example, Robins formulates the condition that
\begin{eqnarray*}
&& \mathbb{E}\bigl[Y_i(1)-Y_i(0)|Z_i=1,X_i=1
\bigr]
\\
&& \quad =\mathbb{E}\bigl[Y_i(1)-Y_i(0)|Z_i=0,X_i=1
\bigr],
\end{eqnarray*}
which,
in combination with the random assignment and the exclusion
restriction, this allows for point identification of the average effect
for the treated. Robins also formulates two other assumptions,
including one where the effects are proportional to survival rates
$\mathbb{E}[Y_i(1)|Z_i=1,X_i=1]$ and $\mathbb{E}[Y_i(1)|Z_i=0,X_i=1]$
respectively, that also point-identifies the average effect for the
treated. However, Robins questions the applicability of these results
by commenting that ``it would be hard to imagine that there is
sufficient understanding of the biological mechanism$\ldots$ to have strong
beliefs that any of the three conditions$\ldots$ is more likely to hold
than either of the other two'' (\cite*{Rob}, page~122).
As an alternative to adding assumptions, \citet{Rob}, \citet{Man90}
and \citet{autokey15}, focused on the question what can be learned
about $\tau$ or $\tau_t$ given these four assumptions that do not
allow for point identification. Here, I focus on the case where the
three assumptions, random assignment, the exclusion restriction and
monotonicity
are maintained
(without necessarily instrument relevance holding), although \citet{Rob} and \citet{Man90} also consider other combinations of
assumptions. For ease of exposition, I focus on the bounds for the
average treatment effect $\tau$ under these assumptions, in the case
where $Y_i(0)$ and $Y_i(1)$ are binary. Then
\begin{eqnarray*}
&& \mathbb{E}\bigl[Y_i(1)-Y_i(0)\bigr]
\\
&& \quad \in \bigl[-\bigl(1-{\mathbb{E}}[X_i|Z_i=1]\bigr)\cdot
\mathbb{E}[Y_i|Z_i=1,X_i=0]
\\
&&\qquad \hspace*{2pt} {}+ {\mathbb{E}}[Y_i|Z_i=1]-
{\mathbb{E}}[Y_i|Z_i=0]
\\
&&\qquad \hspace*{2pt} {}+{\mathbb{E}}[X_i|Z_i=0]\cdot \bigl(
\mathbb{E}[Y_i|Z_i=0,X_i=1]-1\bigr),
\\
&& \qquad \hspace*{2pt}\bigl(1-{\mathbb{E}}[X_i|Z_i=1]\bigr)
\\
&&\qquad \hspace*{2pt} {}\cdot\bigl(1- \mathbb{E}[Y_i|Z_i=1,X_i=0]
\bigr)
\\
&&\qquad \hspace*{2pt} {}+ {\mathbb{E}}[Y_i|Z_i=1]-
{\mathbb{E}}[Y_i|Z_i=0]
\\
&&\qquad \hspace*{32pt} {}+{\mathbb{E}}[X_i|Z_i=0]\cdot
\mathbb{E}[Y_i|Z_i=0,X_i=1] \bigr],
\end{eqnarray*}
which are known at the \textit{natural bounds}.
In this simple setting, this is a straightforward calculation. Work by
Manski (\citeyear{Man95}, \citeyear{Man03}, \citeyear{Man05}, \citeyear{Man07}),
\citet{Rob} and \citet{HerRob06} extends the partial identification approach to substantially
more complex settings.
For the McDonald--Hiu--Tierney flu data, the estimated identified set for
the population average treatment effect is
\[
\mathbb{E}\bigl[Y_i(1)-Y_i(0)\bigr]\in [ -0.24, 0.64].
\]
There is a growing literature developing methods for establishing
confidence intervals for parameters in settings with partial
identification taking sampling uncertainty into account; see Imbens and
\citet{Man04} and \citet{CheHonTam07}.
\subsection{Compliance Types}
\label{types}
\citet{ImbAng94} and \citet{AngImbRub} take a
different approach. Rather than focusing on the average effect for the
population that is not identified under the three assumptions given in
Section~\ref{assumptions}, they focus on different average causal effects.
A first key step in the Angrist--Imbens--Rubin set up is that we can
think of four different compliance types defined by the pair of values
of $(X_i(0),X_i(1))$, that is, defined by how individuals would respond
to different assignments in terms of receipt of the treatment:\footnote
{\citet{FraRub02} generalize this notion of subpopulations
whose membership is not completely observed into their \textit{principal
stratification} approach; see also Section~\ref{sec72}.}
\[
T_i= \cases{ n \ (\mbox{never-taker}) & $\mbox{if } X_i(0)=X_i(1)=0$,
\cr
c \ (\mbox{complier}) & $\mbox{if } X_i(0)=0, X_i(1)=1$,
\cr
d \ (
\mbox{defier}) & $\mbox{if } X_i(0)=1,X_i(1)=0$,
\cr
a \ (\mbox{always-taker}) &
$\mbox{if } X_i(0)=X_i(1)=1$.}
\]
Given the existence of deterministic potential outcomes this
partitioning of the population into four subpopulations is simply a
definition.\footnote{Outside of this framework, the existence of these
four subpopulations would be an assumption.} It clarifies immediately
that it will be difficult to identify the average effect of the primary
treatment (the receipt of the vaccine) for the entire population:
never-takers and always-takers can only be observed exposed to a single
level of the treatment of interest, and thus for these groups any point
estimates of the causal effect of the treatment must be based on extrapolation.
We cannot infer without additional assumptions the compliance type of
any unit: for each unit we observe $X_i(Z_i)$, but the data contain no
information about the value of $X_i(1-Z_i)$. For each unit, there are
therefore two compliance types consistent with the observed behavior.
We can also not identify the proportion of
individuals of each compliance type without additional restrictions.
The monotonicity assumption implies that there are no defiers. This, in
combination with random assignment, implies that we can identify the
population shares of the remaining three compliance types. The
proportion of always-takers and never-takers are
\begin{eqnarray*}
\pi_a &=& \operatorname{pr}(T_i=a)=\operatorname{pr}(X_i=1|Z_i=0)
\quad \mbox{and}
\\
\pi_n &=& \operatorname{pr}(T_i=n)=\operatorname{pr}(X_i=0|Z_i=1),
\end{eqnarray*}
respectively,
and the proportion of compliers is the remainder:
\[
\pi_c=\operatorname{pr}(T_i=c)=1-\pi_a-
\pi_n.
\]
For the McDonald--Hiu--Tierney data these shares are estimated to be
\[
\hat\pi_a=0.189,\quad \hat\pi_n=0.692,\quad \hat
\pi_c=0.119,
\]
although, as I discuss in
Section~\ref{section:exclusion}, these shares may not be consistent
with the exclusion restriction.
\subsection{Local Average Treatment Effects}
\label{section_late}
If, in addition to monotonicity, we also assume that the exclusion
restriction holds, \citet{ImbAng94} and \citet{AngImbRub} show that the
\textit{local average treatment effect} or \textit{complier average causal
effect} is identified:
\begin{eqnarray}
\label{eq:late} \tau_{\mathrm{late}} &=& \mathbb{E}\bigl[ Y_i(1)-Y_i(0)
|T_i= c \bigr]
\nonumber
\\[-8pt]
\\[-8pt]
&=& \frac{\mathbb{E}[ Y_i|Z_i=1]-\mathbb{E}[
Y_i|Z_i=0]}{\mathbb{E}[ X_i|Z_i=1]-\mathbb
{E}[ X_i|Z_i=0]}.
\nonumber
\end{eqnarray}
The components of the right-hand side of this expression can be
estimated consistently from a random sample $(Z_i,X_i,Y_i)_{i=1}^N$.
For the McDonald--Hiu--Tierney data, this leads to
\[
\hat{\tau}_{\mathrm{late}}= -0.125\quad (\mbox{s.e. }0.090).
\]
Note that just as in the supply and demand example, the causal estimand
is the ratio of the intention-to-treat effects of the letter on
hospitalization and of the letter on the receipt of the vaccine. These
intention-to-treat effects
are
\begin{eqnarray*}
\widehat{\mbox{ITT}}_Y & =& -0.015 \quad (\mbox{s.e. } 0.011),
\\
\widehat{\mbox{ITT}}_X &=& \hat\pi_c= 0.119 \quad (
\mbox{s.e. }0.016),
\end{eqnarray*}
with the latter equal to the estimated proportion of compliers in the
population.
Without the monotonicity assumption, but maintaining the random
assignment assumption and the exclusion restriction, the ratio of ITT
effects still has a clear interpretation. In that case, it
is equal to a linear combination average of the effect of the treatment
for compliers and defiers:
\begin{eqnarray}
\label{eq:late1} \quad&& \frac{\mathbb{E}[ Y_i|Z_i=1]-\mathbb{E}[
Y_i|Z_i=0]}{\mathbb{E}[ X_i|Z_i=1]-\mathbb
{E}[ X_i|Z_i=0]}
\nonumber
\\
&& \quad =\frac{\operatorname{pr}(T_i=c)}{\operatorname{pr}(T_i=c)-\operatorname{pr}(T_i=d)}
\nonumber
\\
&& \qquad {}\cdot \mathbb{E}\bigl[ Y_i(1)-Y_i(0)
|T_i=c\bigr]
\\
&& \qquad {}- \frac{\operatorname{pr}(T_i=d)}{\operatorname{pr}(T_i=c)-\operatorname{pr}(T_i=d)}
\nonumber
\\
&& \quad\qquad {}\cdot \mathbb{E}\bigl[ Y_i(1)-Y_i(0)
|T_i=d\bigr].
\nonumber
\end{eqnarray}
This estimand has a clear interpretation if the treatment effect is
constant across all units, but if there is heterogeneity in the
treatment effects it is a weighted average with some weights negative.
This representation shows that if the monotonicity assumption is
violated, but the proportion of defiers is small relative to that of
compliers, the interpretation of the instrumental variables estimand is
not severely impacted.
\subsection{Do We Care About the Local Average Treatment Effect?}
\label{section:dowecare}
The local average treatment effect is an unusual estimand. It is an
average effect of the treatment for a subpopulation that cannot be
identified in the sense that there are no units whom we know for sure
to belong to this subpopulation, although there are some units whom we
know do not belong to it.
A more conventional approach is to start an analysis by clearly
articulating the object of interest, say the average effect of a
treatment for a well-defined population. There may be challenges in
obtaining credible estimates of this object of interest, and along the
way one may make more or less credible assumptions, but typically the
focus remains squarely on the originally specified object of interest.
Here, the approach appears to be quite different. We started off by
defining unit-level treatment effects for all units. We did not
articulate explicitly what the target estimand was. In the
McDonald--Hiu--Tierney influenza-vaccine application a natural estimand
might be the population average effect of the vaccine. Then, apparently
more or less by accident, the definition of the compliance types led us
to focus on the average effects for compliers. In this example, the
compliers were defined by the response in terms of the receipt of the
vaccine to the receipt of the letter. It appears difficult to argue
that this is a substantially interesting group, and in fact no attempt
was made to do so.
This type of example has led distinguished researchers both in
economics and in statistics to question whether and why one should care
about the local average treatment effect. The economist Deaton writes
``I find it hard to make any sense of the LATE [local average treatment
effect]''
(\cite*{Dea10}, page 430). Pearl similarly wonders ``Realizing that the
population averaged treatment effect
(ATE) is not identifiable in experiments marred by noncompliance, they have
shifted attention to a specific response type (i.e., compliers) for
which the causal
effect was identifiable, and presented the latter [the local average
treatment effect] as an approximation for ATE. $\ldots$ However, most
authors in this category do not state
explicitly whether their focus on a specific stratum is motivated by
mathematical
convenience, mathematical necessity (to achieve identification) or a genuine
interest in the stratum under \mbox{analysis}'' (\cite*{Pea11}, page 3).
Freedman writes ``In
many circumstances, the instrumental-variables estimator turns out to be
estimating some data-dependent average of structural parameters, whose
meaning would have to be elucidated'' (\cite*{Fre06}, pages 700--701).
Let me attempt to clear up this
confusion. See also Imbens (\citeyear{Imb10}). An instrumental
variables
analysis is an analysis in a second-best setting. It would have been
preferable if one had been able to carry out a well-designed randomized
experiment. However, such an experiment was not carried out, and we
have noncompliance. As a result, we cannot answer all the questions we
might have wanted to ask. Specifically, if the noncompliance is
substantial, we are limited in the questions we can answer credibly and
precisely. Ultimately, there is only one subpopulation we can credibly
(point-)identify the average effect of the treatment for, namely, the compliers.
It may be useful to draw an analogy. Suppose a researcher is interested
in evaluating a medical treatment and suppose a randomized experiment
had been carried out to estimate the average effect of this new
treatment. However, the population of the randomized experiment
included only men, and the researcher is interested in the average
effect for the entire population, including both men and women. What
should the researcher do? I would argue that the researcher should
report the results for the men, and acknowledge the limitation of the
results for the original question of interest. Similarly, in the
instrumental variables I see the limitation of the results to the
compliers as one that was unintended, but driven by the lack of
identification for other subpopulations given the design of the study.
This limitation should be acknowledged, but one should not drop the
analysis simply because the original estimand cannot be identified.
Note that our case with instrumental variables is slightly worse than
in the gender example, because we cannot actually identify all
individuals with certainty as compliers.
There are alternatives to this view. One approach is to focus solely or
primarily on intention-to-treat effects. The strongest argument for
that is in the context of randomized experiments with noncompliance.
The causal interpretation of intention-to-treat effects is justified by
the randomization. As Freedman writes, ``Experimental data should
therefore be analyzed first by comparing rates or
averages, following the intention-to-treat principle. Such comparisons are
justified because the treatment and control groups are balanced, within the
limits of chance variation, by randomization'' (\cite*{Fre06}, page 701).
Even in that case one may wish to also report estimates of the local
average treatment effects because they may correspond more closely to
the object of ultimate interest. The argument for focusing on
intention-to-treat or reduced-form estimates is weaker in other
settings. For example, in the Fulton Fish Market demand and supply
application, the intention-to-treat effects are the effects of weather
conditions on prices and quantities. These effects may be of little
substantive interest to policy makers interested in tax policy. The
substantive interest for these policy makers is almost exclusively in
the \textit{structural} effects of price changes on demand and supply, and
reduced form effects are only of interest in sofar as they are
informative about those structural effects. Of course, one should bear
in mind that the reduced form or intention-to-treat effects rely on
fewer assumptions.
A second alternative is associated with the partial identification
approach by
Manski (\citeyear{Man90}, \citeyear{Man02}, \citeyear{Man03}, \citeyear{Man07});
see also \citet{Rob} and \citet{LeaN1} for antecedents. In this
setting that suggests maintaining the focus on the original estimand,
say the overall average effect, we cannot estimate that accurately
because we cannot estimate the average value of $Y_i(0)$ for
always-takers or the average value of $Y_i(1)$ for nevertakers, but we
can {bound} the average effect of interest because we know a priori that the average value of $Y_i(0)$ for always-takers and the
average value of $Y_i(0)$ for nevertakers is restricted to lie in the
unit interval. Manski's is a principled and coherent approach. One
concern with the approach is that it has often focused on reporting
solely these bounds, leading researchers to miss relevant information
that is available given the maintained assumptions. Two different data
sets may lead to the same bounds even though in one case we may know
that the average effect for one subpopulation (the compliers) is
positive and statistically significantly different from zero whereas in
the other case there need not be any evidence of a nonzero effect for
any subpopulation. It would appear to be useful to distinguish between
such cases by reporting estimates of both the local average treatment
effect and the bounds.
\section{The Substantive Content of the Instrumental Variables Assumptions}
\label{section:content}
In this section, I will discuss the substantive content of the three
key assumptions, random assignment, the exclusion restriction and the
monotonicity assumption. I will not discuss here the fourth assumption,
instrument relevance. In practice, the main issue with that assumption
concerns the quality of inferences when the assumption is close to
being violated. See Section~\ref{section_weak} for more discussion, and \citet{StaSto97} for a detailed study.
\subsection{Unconfoundedness of the Instrument}
First, consider the random assignment or unconfoundedness assumption.
In a slightly different setting, this is a very familiar assumption.
Matching methods often rely on random assignment, either
unconditionally or conditionally, for their justification.
In some of the leading applications of instrumental variables methods,
this assumption is satisfied by design, when the instrument is
physically randomized.
For example, in the draft lottery example (Angrist, \citeyear{Ang90}), draft
priority is used as an instrument for veteran status in an evaluation
of the causal effect of veteran status on mortality and earnings. In
that case, the instrument, the draft priority number was assigned by
randomization. Similarly, in the flu example (Hirano et al., \citeyear{Hiretal00}), the instrument for influenza vaccinations, the letter to
the physician, was randomly assigned.
In other cases, the conditional version of this assumption is more
plausible. In the \citet{McCNew94} study, proximity of an
individual to a hospital with particular facilities is used as an
instrument for the receipt of intensive
treatment of acute myocardial infarction. This proximity measure is not
randomly assigned, and McClellan and Newhouse use covariates to make
the unconfoundedness assumption more plausible. For example, they worry
about differences between individuals living in rural versus urban
areas. To adjust for such differences, they use as one of the
covariates the distance to the nearest hospital (regardless of the
facilities at the nearest hospital).
A key issue is that although on its own this random assignment or
unconfoundedness assumption justifies a causal interpretation of the
intention-to-treat effects, it is \textit{not} sufficient for a causal
interpretation of the instrumental variables estimand, the ratio of the
ITT effects for outcome and treatment.
\subsection{The Exclusion Restriction}
\label{section:exclusion}
Second, consider the exclusion restriction. This is the most critical
and typically most controversial assumption underlying instrumental
variables methods.
First of all, it has some testable implications; see \citet{BalPea97} and the recent discussions in
\citet{Kit09} and \citet{RamLau11}. This testable restriction can be seen most easily in
a binary outcome setting. Under the three assumptions, random
assignment, the exclusion restriction and monotonicity, the
intention-to-treatment effect of the assignment on the outcome is the
product of two causal effects. First, the average effect of the
assignment on the outcome for compliers, and second, the
intention-to-treat effect of the assignment on receipt of the
treatment, which is equal to the population proportion of compliers. If
the outcome is binary, the first factor is between $-1$ and 1. Hence, the
intention-to-treat effect of the assignment on the outcome has to be
bounded in absolute value by the intention-to-treat effect of the
assignment on the receipt of the treatment. This is a testable
restriction. If the outcomes are multivalued, there is in fact a range
of restrictions implied by the assumptions. However, there exist no
consistent tests that will reject the null hypothesis with probability
going to one as the sample size increases in all scenarios where the
null hypothesis is wrong.
Let us assess these restrictions in the flu example. Because
\begin{eqnarray*}
&& \operatorname{pr}(Y_i=1,X_i=0|Z_i=1)
\\
&& \quad =\operatorname{pr}\bigl(Y_i(0)=1|T_i=n\bigr)
\cdot \operatorname{pr}(T_i=n)
\end{eqnarray*}
and
\begin{eqnarray*}
&& \operatorname{pr}(Y_i=1,X_i=0|Z_i=0)
\\
&& \quad =\operatorname{pr}\bigl(Y_i(0)=1| T_i=n\mbox{ or } c\bigr)
\\
&& \qquad {}\cdot\operatorname{pr}(T_i=n \mbox{ or }c)
\\
&& \quad =\operatorname{pr}\bigl(Y_i(0)=1|T_i=n\bigr)
\cdot\operatorname{pr}(T_i=n)
\\
&& \qquad {}+\operatorname{pr}\bigl(Y_i(0)=1|T_i=c\bigr)
\cdot\operatorname{pr}(T_i=c)
\end{eqnarray*}
it follows that
\begin{eqnarray}
\label{test} && \operatorname{pr}(Y_i=1,X_i=0|Z_i=1)
\nonumber
\\[-8pt]
\\[-8pt]
&& \quad \leq\operatorname{pr}(Y_i=1,X_i=0|Z_i=0).
\nonumber
\end{eqnarray}
There are three more restrictions in this setting with a binary
outcome, binary treatment and binary instrument; see \citet{ImbRub97N2}, \citet{BalPea97} and \citet{RicEvaRob11} for details.
For the flu data, the simple frequency estimator for the left-hand side
of (\ref{test}) is
$30/1389=0.0216$, and the right-hand side is $31/72=0.0211$, leading to
a slight violation as pointed out in \citet{RicEvaRob11} and Imbens and Rubin (\citeyear{ImbRub}). Although not statistically
significant, it shows that these restrictions have content in practice.
To assess the plausibility of the exclusion restriction, it is often
helpful to do so separately in subpopulations defined by compliance status.
Let us first consider the exclusion restriction for
always-takers,
who would receive
the influenza vaccine irrespective of the receipt of the
letter by their physician.
Presumably, such patients are generally at higher risk for the flu.
Why would such patients be
affected by a letter warning their physicians about the upcoming
flu season when they will get inoculated irrespective of this warning?
It may be that the letter led the physician to
take other actions beyond giving the flu vaccine,
such as encouraging the patient to avoid exposure.
These other actions may
affect health outcomes, in which case the exclusion restriction would
be violated.
The exclusion restriction for never-takers has different content.
These patients would not receive the vaccine in any case.
If their physicians
did not regard the risk of flu as sufficiently high to encourage their
patients to have the vaccination,
presumably the physician would not take other actions either. For these
patients, the exclusion restriction may therefore be reasonable.
Consider the draft lottery example. In that case, the always-takers are
individuals who volunteer for military service irrespective of their
draft priority number. It seems plausible that the draft priority
number has no causal effect on their outcomes. never-takers are
individuals who do not serve in the military irrespective of their
draft priority number. If this is for medical reasons, or more
generally reasons that make them ineligible to serve, this seems
plausible. If, on the other hand these are individuals fit but
unwilling to serve, they may have had to take actions to stay out of
the military that could have affected their subsequent civilian labor
market careers. Such actions may include extending their educational
career, or temporarily leaving the country.
Note that these issues are not addressed by the random assignment of
the instrument.
In general, the concern is that the instrument creates incentives not
only to receive the treatment, but also to take additional actions that
may affect the outcome of interest. The nature of these actions may
well differ by compliance type.
Most important is to keep in mind that this assumption is typically a
substantive assumption, not satisfied by design outside of
double-blind, single-dose placebo control randomized experiments with
noncompliance.
\subsection{Monotonicity}
Finally, consider the monotonicity or no-defiers assumption. Even
though this assumption is often the least controversial of the three
instrumental variables assumptions, it is still sometimes viewed with
suspicion. For example, whereas Robins views the assumption as ``often,
but not always reasonable'' (\cite*{Rob}, page 122), \citet{Fre06} wonders:
``The identifying restriction for the
instrumental-variables estimator
is troublesome: just why are there no defiers?''
(\cite*{Fre06}, page 700). In many applications, it is perfectly clear
why there should be no or at most few defiers.
The instrument plays the role of an \textit{incentive} for the individual
to choose the active treatment by either making it more attractive to
take the active treatment or less attractive to take the control
treatment. As long as individuals do not respond perversely to this
incentive, monotonicity is plausible with either no or a negligible
proportion of defiers in the population.
The term incentive is used broadly here: it may be a financial
incentive, or the provision of information, or an imperfectly monitored
legal requirement, but in all cases something that makes it more
likely, at the individual level, that the individual participates in
the treatment.
Let us consider some examples.
If noncompliance is one-sided, and those assigned to the control group
are effectively embargoed from receiving the treatment, monotonicity is
automatically satisfied. In that case $X_i(0)=0$, and there are no
always-takers or defiers. The example discussed in \citet{SomZeg91},
Imbens and Rubin (\citeyear{ImbRub97N1}) and \citet{Gre} fits this set up.
In the
flu application introduced in Section~\ref{section:modern}, the letter
to the physician creates an additional incentive for the physician to
provide the flu vaccine to a patient, something beyond any incentives
the physician may have had already to provide the vaccine.
Some individuals may already be committed to the vaccine, irrespective
of the letter (the always-takers), and some may not be swayed by the
receipt of the letter (the never-takers), and that is consistent with
this assumption. Monotonicity only requires that there is no patient,
who, if their physician receives the letter, would not take the
vaccine, whereas they would have taken the vaccine in the absence of
the letter.
Consider a second example, the influential draft lottery application by
\citet{Ang90} (see also \cite*{HeaNewHul}). Angrist is
interested in evaluating the effect of military service on subsequent
civilian earnings, using the draft priority established by the draft
lottery as an instrument. Monotonicity requires that assigning an
individual priority for the draft rather than not, may induce them to
serve in the military, or may not affect them, but cannot induce them
to switch from serving to not serving in the military.
Again that seems plausible. Having high priority for the draft
increases the cost of staying out of the military: that may not be
enough to change behavior, but it would be unusual if the increased
cost of staying out of the military induced an individual to switch
from serving in the military to not serving.
As a third example, consider
the \citet{PerHeb89} study of the effect of smoking on birthweight.
Permutt and Hebel use the random assignment to a smoking-cessation
program as an instrument for the amount of smoking.
In this case, the monotonicity assumption requires that there are no
individuals who as a causal effect of the assignment to the
smoking-cessation program end up smoking more. There may be individuals
who continue to smoke as much under either assignment and individuals
who reduce smoking as a result of the assignment, but the assumption is
that there is nobody who increases their smoking as a result of the
smoking-cessation program. In all these examples, monotonicity requires
individuals not to respond perversely to changes in incentives.
Systematic and major violations in such settings seem unlikely.
In other settings, the assumption is less attractive.
Suppose a program has assignment criteria that are checked by two
administrators. Individuals entering the assignment process are
assigned randomly to one of
the two administrators. The assignment criteria may be interpreted
slightly differently by the two administrators, with on average
administrator A being more strict than administrator B. Monotonicity
requires that anyone admitted by administrator A would also be admitted
by administrator B, or {vice-versa}. In this type of setting,
monotonicity does not appear to be as plausible as it is in the
settings where the instrument can be viewed as creating an incentive to
participate in the treatment.
For example, in an analysis of the effect of prison time on recidivism,
Aizer and Doyle (\citeyear{AizDoy13}) use
random assignment of cases to judges, and in an
analysis of the effect of bankruptcy,
Dobbie and Song (\citeyear{DobSon13}) use
random
assignment of bankruptcy applications to judges.
The discussion in this section focuses primarily on the case with a
binary treatment and a binary instrument. In cases with multivalued
treatments, the monotonicity can be generalized in two different ways.
In both cases, it may be less plausible than in the binary case. Let
$X_i(z)$ be the potential treatment level associated with the
assignment $z$. One can generalize the monotonicity assumption for the
binary instrument case to this case as
\begin{eqnarray}
&& X_i(z) \mbox{ is nondecreasing in } z \quad \mbox{for all } i
\nonumber\\
\eqntext{\mbox{(monotonicity in instrument)}.}
\end{eqnarray}
This generalization is used in \citet{AngImb95}. It is
consistent with the view of the instrument as changing the incentive to
participate in the treatment: increasing the incentive cannot decrease
the level of the treatment received.
Angrist and Imbens show that this assumption has testable implications.
An alternative generalization is
\begin{eqnarray}
&& \mbox{if } X_i(z)> X_j(z)\quad
\nonumber\\
&& \quad \mbox{then } X_i\bigl(z'\bigr)\geq
X_j\bigl(z'\bigr) \quad \mbox{for all }
z,z',i,j
\nonumber\\
\eqntext{\mbox{(monotonicity in unobservables).}}
\end{eqnarray}
This assumption, referred to as \textit{rank preservation} in \citet{Rob86}, implicitly ranks all units in terms of some unobservables
(\cite{ImbN1}). It assumes this ranking is invariant to the level of
the instrument. It implies that if $X_i(z)>X_j(z)$, then it cannot be
that $X_j(z')>X_i(z')$. It is equivalent to the ``continuous
prescribing preference'' in \citet{HerRob06}.
In both cases, the special case with a binary treatment is identical to
the previously stated monotonicity. In settings with multivalued
treatments, these assumptions are more restrictive than in the binary
treatment case. In the demand and supply example in Section~\ref{section:supplydemand} with linear supply and demand functions, both
the monotonicity in the instrument and monotonicity in the
unobservables conditions are satisfied.
\section{The Link to the Textbook Discussions of Instrumental Variables}
\label{section:textbook}
Most textbook discussions of instrumental variables use a framework
that is quite different at first sight from the potential outcome set
up used in Sections~\ref{section:modern} and~\ref{section:content}. These textbook discussions
(graduate texts include Wooldridge, \citeyear{Woo10}; \cite*{AngPis09}; \cite*{Gre11}; and \cite*{Hay00}, and introductory undergraduate
textbooks include \cite*{Woo08}; and \cite*{StoWat10})
are often closer to the simultaneous equations example from Section~\ref{section:supplydemand}. An exception is \citet{Man07} who uses the
potential outcome set up used in this discussion. In this section I
will discuss the standard textbook set up and relate it to the
potential outcome framework and the simultaneous equations set up.
The textbook version of instrumental variables does not explicitly
define the potential outcomes. Instead the starting point is a linear
regression function describing the relation between the realized
(observed) outcome $Y_i$, the endogenous regressor of interest $X_i$
and other regressors $V_i$:
\begin{equation}
\label{eq:classical} Y^{\mathrm{obs}}_i=\beta_0+
\beta_1 X_i+\beta_2'V_i+
\varepsilon_i.
\end{equation}
These other regressors a well as the instruments are often referred to
in the econometric literature as \textit{exogenous} variables. Although
this term does not have a well-defined meaning, informally it includes
variables that \citet{Cox92} called \textit{attributes}, as well as potential
causes whose assignment is unconfounded.
This set up covers both the demand function setting and the randomized
experiment example.
Although this equation looks like a standard regression function, that
similarity is misleading. Equation (\ref{eq:classical}) is not an
ordinary regression function in the sense that the first part does \textit{not} represent the conditional expectation of the outcome $Y_i$ given
the right-hand side variables $X_i$ and $V_i$.
Instead it is what is sometimes called a \textit{structural equation}
representing the causal response to changes in the input~$X_i$.
The key assumption in this formulation is that the unobserved component
$\varepsilon_i$ in this regression function is independent of the
exogenous regressors $V_i$ and the
instruments $Z_i$, or, formally
\begin{equation}
\label{eq:class_ass} \varepsilon_i \perp ( Z_i,V_i
).
\end{equation}
The unobserved component is \textit{not} independent of the endogenous
regressor $X_i$ though.
The value of the regressor $X_i$ may be partly chosen by individual $i$
to optimize some objection function as in the noncompliance example, or
the result of an equilibrium condition as in the supply and demand
model. The precise relation between $X_i$ and $\varepsilon_i$ is often
not fully specified.
How does this set up relate to the earlier discussion involving
potential outcomes?
Implicitly, there is in the background of this set up a causal,
unit-level response function. In the potential outcome notation, let
$Y_i(x)$ denote this causal response function for unit $i$, describing
for each value of $x$ the potential outcome corresponding to that level
of the treatment for that unit. Suppose the conditional expectation of
this causal response function is linear in $x$ and some exogenous covariates:
\begin{equation}
\label{lineareq} \mathbb{E}\bigl[Y_i(x)|V_i\bigr]=\beta
_0+\beta_1\cdot x+\beta_2^{\operatorname{pr}ime}V_i.
\end{equation}
Moreover, let us make the (strong) assumption that the difference
between the response function $Y_i(x)$ and its conditional expectation
does not depend on $x$, so we can define
the residual unambiguously as
\[
\varepsilon_i=Y_i(x)- \bigl(\beta_0+
\beta_1\cdot x+\beta _2'V_i
\bigr),
\]
with the equality holding for all $x$.
The residual $\varepsilon_i$ is now uncorrelated with $V_i$ by
definition. We will assume that it is in fact independent of $V_i$. Now
suppose we have an instrument $Z_i$ such that
\[
Y_i(x)\perp Z_i | V_i.
\]
This assumption is, given the linear representation for $Y_i(x)$,
equivalent to
\[
\varepsilon_i\perp Z_i | V_i.
\]
In combination with the assumption that $\varepsilon_i\perp V_i$, this
gives us the textbook version of the assumption given in (\ref{eq:class_ass}).
We observe $V_i$, $X_i$, the instrument $Z_i$, and the realized outcome
\[
Y^{\mathrm{obs}}_i=Y_i(X_i)=
\beta_0+\beta_1 X_i+\beta_2'V_i+
\varepsilon_i,
\]
which is the starting point in the econometric textbook discussion
(\ref{eq:classical}).
This set up is more restrictive than it needs to be. For example, the
assumption that the
difference between the response function $Y_i(x)$ and its conditional
expectation does not depend on $x$ can be relaxed to allow for
variation in the slope coefficient,
\[
Y_i(x)-Y_i(0)=\beta_1\cdot x+
\eta_i\cdot x,
\]
as long as the $\eta_i$ satisfies conditions similar to those on
$\varepsilon_i$.
The modern literature (e.g., \cite*{Mat07}) discusses such models
in more detail.
One key feature of the textbook version is that there is no separate
role for the monotonicity assumption. Because the linear model
implicitly assumes that the per-unit causal effect is constant across
units and levels of the treatment, violations of the monotonicity
assumption do not affect the interpretation of the estimand.
A second feature of the textbook version is that the exclusion
restriction and the random assignment assumption are combined in (\ref
{eq:class_ass}). Implicitly, the exclusion restriction is captured by
the absence of $Z_i$ in the equation (\ref{eq:classical}), and the
(conditional) random assignment is captured by (\ref{eq:class_ass}).
\section{Extensions and Generalizations}\label{section:extensions}
In this section, I will briefly review some of other approaches taken
in the instrumental variables literature. Some of these originate in
the statistics literature, some in the econometrics literature. They
reflect different concerns with the traditional instrumental variables
methods, sometimes because of different applications, sometimes because
of different traditions in econometrics and statistics. This discussion
is not exhaustive. I~will focus on highlighting the most interesting
developments and provide some references to the relevant literature.
\subsection{Model-based Approaches to Estimation and Inference}
Traditionally, instrumental variables analyses relied on linear
regression methods. Additional explanatory variables are incorporated
linearly in the regression function. The recent work in the statistics
literature has explored more flexible approaches to include covariates.
These approaches often involve modeling the conditional distribution of
the endogenous regressor given the instruments and the exogenous
variables. This is in contrast to the traditional econometric
literature which has focused on settings and methods that do not rely
on such models.
Robins (\citeyear{Rob}, \citeyear{Rob94}), \citet{HerRob06}, \citet{Gre},
Robins and Rotnitzky (\citeyear{RobRot04}) and \citet{Tan10} developed an approach that
allow for identification of average treatment effect by adding
parametric modelling assumptions. This approach starts with the
specification of what they call the \textit{structural mean}, the
expectation of $Y_i(x)$. This structural mean can be the conditional
mean given covariates, or the marginal mean, labeled the \textit{marginal
structural mean}. The specification for this expectation is typically
parametric. Then estimating equations for the parameters of these
models are developed. In the simple setting considered here, this would
typically lead to the same estimators considered already. An important
virtue of the method is that it has been extended to much more general
settings, in particular with time-varying covariates and dynamic
treatment regimes in a series of papers. In other settings, it has also
led to the development of doubly robust estimators (\cite{RobRot04}).
A~key feature of the models is that the models are robust in a
particular sense. Specifically, the estimators for the average
treatment effects are consistent irrespective of the misspecification
of the model, in the absence of intention-to-treat effects (what they
call the conditional ITT null).
\citet{ImbRub97N1} and
Hirano et al. (\citeyear{Hiretal00}) propose building a parametric
model for the compliance status in terms of additional covariates,
combined with models for the potential outcomes conditional on
compliance status and covariates.
Given the monotonicity assumption, there are three compliance types:
never-takers, always-takers and compliers. A natural model for
compliance status given individual characteristics $V_i$ is therefore a
trinomial logit model:
\begin{eqnarray*}
\operatorname{pr}(T_i=n|V_i=v) &=& \frac{\exp(v'\gamma_n)}{1+\exp(v'\gamma_n)+\exp
(v'\gamma_n)},
\\
\operatorname{pr}(T_i=a|V_i=v) &=& \frac{\exp(v'\gamma_a)}{1+\exp(v'\gamma_n)+\exp
(v'\gamma_n)}
\end{eqnarray*}
and
\[
\operatorname{pr}(T_i=c|V_i=v)=\frac{1}{1+\exp(v'\gamma_n)+\exp(v'\gamma_n)}.
\]
With continuous outcomes, the conditional outcome distributions given
compliance status and covariates may be normal:
\[
Y_i(x)|T_i=t, \quad V_i=v\sim \mathcal{N}
\bigl(\beta_{tx}' v,\sigma^2_{tx}
\bigr),
\]
for $(t,x)=(n,0),(a,1),(c,0),(c,1)$.
With binary outcomes, one may wish to use logistic regression models here.
This specification defines the likelihood function.
Hirano et al. (\citeyear{Hiretal00}) apply this to the flu data
discussed before.
Simulations in Richardson, Evans and Robins (\citeyear{RicEvaRob11}) suggest that the
modeling of the compliance status here is key. Specifically, they point
out that even in the absence of ITT effects there can be biases if the
model of the compliance status is misspecified.
Like Hirano et al. (\citeyear{Hiretal00}), \citet{RicEvaRob11} build parametric model only for the identified
distributions. They use them to estimate the bounds so that the
parametric assumptions do not contain identifying information.
Little and Yau (\citeyear{LitYau98}) and Yau and Little (\citeyear{YauLit01})
similarly
model the conditional expectation of
the outcome given compliance status and covariates. In their
application, there are no always-takers, only never-takers and
compliers. Their specification specifies parametric forms for the
conditional means given the compliance types and the treatment status:
\begin{eqnarray*}
\mathbb{E}\bigl[Y_i(0)|T_i=n,V_i=v\bigr] &=&
\beta_{n0}+\beta_{n1}'v,
\\
\mathbb{E}\bigl[Y_i(0)|T_i=c,V_i=v\bigr] &=&
\beta_{c00}+\beta_{c01}'v
\end{eqnarray*}
and
\[
\mathbb{E}\bigl[Y_i(1)|T_i=c,V_i=v\bigr]=
\beta_{c00}+\beta_{c11}'v.
\]
\subsection{Principal Stratification}\label{sec72}
\citet{FraRub02} generalize the latent compliance type
approach to instrumental variables in an important and novel way. Their
focus is on the causal effect of a binary treatment on some outcome.
However, it is not the average effect of the treatment they are
interested in, but the average within a subpopulation.
It is the way this subpopulation is defined that creates the
complications as well as the connection to instrumental variables.
There is a post-treatment variable that may be affected by the
treatment. Frangakis and Rubin postulate the existence of a pair of
potential outcomes for this post-treatment variable. The subpopulation
of interest is then defined by the values for the pair of potential
outcomes for this post-treatment variables.
Let us consider two examples: first, the randomized experiment with
noncompliance. The treatment here is the random assignment. The
post-treatment variable is the actual receipt of the treatment. The
pair of potential outcomes for this post-treatment variable captures
the compliance status. The subpopulation of interest is the
subpopulation of compliers.
The second example shows how principal stratification generalizes the
instrumental variables set up to other cases.
Examples of this type are considered in \citet{ZhaRubMea09},
Frumento et al. (\citeyear{Fruetal12}) and \citet{Rob86}.
Suppose we have a randomized experiment with perfect compliance. The
primary outcome is survival after one year. For patients who survive, a
quality of life measure is observed. We may be interested in the effect
of the treatment on quality of life. This is only defined for patients
who survive up to one year. The principal stratification approach
suggests focusing on the subpopulation or \textit{principal stratum} of
patients who survive irrespective of the treatment assignment.
Membership in this stratum is not observed, and so we cannot directly
estimate the average effect of the treatment on quality of life for
individuals in this stratum, but the data are generally still
informative about such effects, particularly under monotonicity assumptions.
\subsection{Randomization Inference with Instrumental~Variables}
Most of the work on inference in instrumental variables settings is
model-based. After specifying a model relating the treatment to the
outcome, the conditional distribution or conditional mean of outcomes
given instruments is derived. The resulting inferences are conditional
on the values of the instruments. A very different approach is taken in
\citet{Ros96} and Imbens and Rosenbaum (\citeyear{ImbRos05}).
Rosenbaum focuses on the distribution for statistics generated by the
random assignment of the instruments. In the spirit of the work by
\citet{Fis25} confidence intervals for the parameter of interest,
$\beta_1$ in equation (\ref{lineareq}) are based on this randomization
distribution.
Similar to confidence intervals for treatment effects based on
inverting conventional Fisher $p$-values, these intervals have exact
coverage under the stated assumptions.
However, these results rely on arguably restrictive constant treatment
effect assumptions.
\subsection{Matching and Instrumental Variables}
\label{section_matchinginstruments}
In many observational studies using instrumental variables approaches,
the instruments are not randomly assigned. In that case, adjustment for
additional pretreatment variables can sometimes make causal inferences
more credible. Even if the instrument is \mbox{randomly} assigned, such
adjustments can make the inferences more precise. Traditionally, in
econometrics these adjustments are based on regression methods.
Recently, in the statistics literature matching methods have been
proposed as a way to do the adjustment for pretreatment variables
(Baiocchi et al., \citeyear{Baietal10}).
\subsection{Weak Instruments}
\label{section_weak}
One concern that has arisen in the econometrics literature is about
\textit{weak} instruments. For an instrument to be helpful in estimating
the effect of the treatment, it not only needs to have no direct effect
on the outcome, it also needs to be correlated with the treatment.
Suppose this correlation is very close to zero. In the simple case, the
IV estimator is the ratio of covariances,
\begin{eqnarray*}
\hat\beta_{1,\mathrm{iv}} &=& \frac{\widehat{\operatorname{cov}}(Y_i,Z_i)}{\widehat{\operatorname{cov}}(X_i,Z_i)} \\
& = & \frac{({1}/{N})\sum_{i=1}^N (Y_i-\overline{Y}) (Z_i-\overline{Z})}{
({1}/{N})\sum_{i=1}^N (X_i-\overline{X}) (Z_i-\overline{Z})}.
\end{eqnarray*}
The distribution of this ratio can be approximated by a normal
distribution in large samples, as long as the covariance in the
denominator is nonzero in the population.
If the population value of the covariance in the denominator is exactly
zero, the distribution of the ratio $\hat\beta_{1,\mathrm{iv}}$ is Cauchy in
large samples, rather than normal (\cite{Phi89}; \cite{StaSto97}). The weak instrument literature is concerned with the
construction of confidence intervals in the case the covariance is
close to zero. Interest in this problem rose sharply after a study by
\citet{AngKru91}, which remains the primary empirical
motivation for this literature. Angrist and Krueger were interested in
estimating the causal effect of years of education on \mbox{earnings}. They
exploited variation in educational achievement by quarter of birth
attributed to differences in compulsory schooling laws. These
differences in average years of education by quarter of birth were
small, and they attempted to improve precision of their estimators by
including interactions of the basic instruments, the three quarter of
birth dummies, with indicators for year and state of birth. \citet{BouJaeBak95} showed that the estimates using the
interactions as additional instruments were potentially severely
affected by the weakness of the instruments. In one striking analysis,
they reestimated the Angrist--Krueger regressions using randomly
generated quarter of birth data (uncorrelated with earnings or years of
education). One might have expected, and hoped, that in that case one
would find an imprecisely estimated effect. Surprisingly, \citet{BouJaeBak95} found that the confidence intervals constructed by
Angrist and Krueger suggested precisely estimated effects for the
effect of years of education on earnings. It was subsequently found
that with weak instruments the TSLS estimator, especially with many
instruments, was biased, and that the standard variance estimator led
to confidence intervals with substantial undercoverage (\cite{BouJaeBak95}; \cite{StaSto97}; \cite{ChaImb04}).
Motivated by the Bound--Jaeger--Baker findings, the weak and many
instruments literature focused on point and interval estimators
with better properties in settings with weak instruments.
Starting with
\citet{StaSto97}, a literature developed to construct
confidence intervals for the instrumental variables estimand that
remained valid irrespective of the strength of the instruments. A key
insight was that confidence intervals based on the inversion of
Anderson--Rubin (\citeyear{A49}) statistics have good properties in settings with
weak instruments; see also \citet{Mor03}, \citet{AndSto},
\citet{Kle02} and \citet{AndMorSto06}.
Let us look at the simplest case with a single endogenous
regressor, a single instrument, and no additional regressors and
normally distributed residuals:
\[
Y_i(x)=\beta_0+\beta_1\cdot x+
\varepsilon_i\quad \mbox{with } \varepsilon_i
|Z_i\sim\mathcal{N} \bigl(0,\sigma^2_\varepsilon
\bigr).\vadjust{\goodbreak}
\]
The Anderson--Rubin statistic
is, for a given value of $b$
\begin{eqnarray*}
\operatorname{AR}(b) &=& \Biggl(\frac{1}{\sqrt N}\sum_{i=1}^N
(Z_i-\overline{Z})\cdot (Y_i-b\cdot X_i)
\Biggr)^2 \\
&&{}\bigg/ \Biggl(\frac{1}{N}\sum
_{i=1}^N (Z_i-\overline{Z})^2 \cdot
\hat\sigma ^2_\varepsilon \Biggr),
\end{eqnarray*}
where $\overline{Z}=\sum_{i=1}^N Z_i/N$, and for some estimate of the residual
variance $\sigma^2_\varepsilon$. At the true value $b=\beta_1$, the
AR statistic has in large samples a chi-squared distribution with one
degree of freedom. \citet{StaSto97} propose constructing a
confidence interval by inverting this test statistic:
\[
\mathrm{CI}^{0.95}(\beta_1)= \bigl\{b| \operatorname{AR}(b)\leq3.84 \bigr\}.
\]
The subsequent literature has extended this by allowing for multiple
instruments and developed various alternatives, all with the focus on
methods that remain valid irrespective of the strength of the
instruments; see \citet{AndSto} for an overview of this literature.
\subsection{Many Instruments}
\label{section_many}
Another strand of the literature motivated by the Angrist--Krueger
study focused on settings with many weak instruments. The concern
centered on the Bound, Jaeger and Baker (\citeyear{BouJaeBak95}) finding that in a setting
similar to the Angrist--Krueger setting using TSLS with many randomly
generated instruments led to confidence intervals that had very low
coverage rates.
To analyze this setting,
Bekker (\citeyear{Bek94}) considered the behavior of various estimators under an
asymptotic sequence where the number of instruments increases with the
sample size.
Asymptotic approximations to sampling distributions based on this
sequence turned out to be much more accurate than those based on
conventional asymptotic approximations.
A key finding in Bekker (\citeyear{Bek94}) is that under such sequences one of the
leading estimators, Two-Stage-Least-Squares (TSLS, see the \hyperref[appA]{Appendix} for
details) estimator is no longer consistent, whereas another estimator,
Limited Information Maximum Likelihood (LIML, again see the \hyperref[appA]{Appendix}
for details) estimator remains consistent although the variance under
this asymptotic sequence differs from that under the standard sequence;
see also
Kunitomo (\citeyear{K80}), Morimune (\citeyear{M83}),
Bekker and van der Ploeg (\citeyear{B05}),
\citet{ChaImb04}, \citet{ChaSwa05}, Hahn (\citeyear{H02}),
Hansen, Hausman and Newey (\citeyear{H08}),
Koles\'ar et al. (\citeyear{K13}).
\subsection{Proxies for Instruments}
\citet{HerRob06} and
\citet{Cha11} explores settings where the instrument is not directly
observed. Instead a proxy variable $Z^*_i$ is observed. This proxy
variable is correlated with the underlying instrument $Z_i$, but not
perfectly so.
The potential outcomes $Y_i(z,x)$ are still defined in terms of the
underlying, unobserved instrument $Z_i$.
The unobserved \mbox{instrument} $Z_i$ satisfies the instrumental variables
assumptions, random assignment, the exclusion restriction and the
monotonicity assumption. In addition, the observed proxy $Z_i^*$ satisfies
\begin{eqnarray*}
Z_i^* &\perp & Y_i(0,0),Y_i(0,1),Y_i(1,0),\\
&& {} Y_i(1,1),X_i(0),X_i(1)|Z_i.
\end{eqnarray*}
Chalak shows that the ratio of covariances (now no longer the ratio of
intention-to-treat effects) still has an
interpretation
of an average
causal effect.
\subsection{Regression Discontinuity Designs}
Regression Discontinuity (RD) designs attempt to estimate causal
effects of a binary treatment in settings where the assignment
mechanism is a deterministic function of a pretreatment variable.
In the sharp version of the RD design, the assignment mechanism takes
the form
\[
X_i=\mathbf{1}_{V_i\geq c},
\]
for some fixed threshold $c$: all units with a value for the covariate
$V_i$ exceeding $c$ receive the treatment and all units with a value
for $V_i$ less than $c$ are in the control group. Under smoothness
assumptions, it is possible in such settings to estimate the average
effect of the treatment for units with a value for the pretreatment
variable equal to $V_i\approx c$:
\begin{eqnarray*}
&& \mathbb{E}\bigl[Y_i(1)-Y_i(0)|V_i=c
\bigr] \\
&& \quad =\lim_{w\uparrow c} \mathbb {E}[Y_i|V_i=w]-
\lim_{w\downarrow c} \mathbb{E}[Y_i|V_i=w].
\end{eqnarray*}
These designs were introduced by
\citet{ThiCam60}, and have been used in psychology,
sociology, political science and economics. For example, many
educational programs have eligibility criteria that allow for the
application of RD methods;
see \citet{Coo08} for a recent historical perspective and \citet{ImbWoo09} for a recent review.
A generalization of the sharp RD design is the \textit{Fuzzy Regression
Discontinuity} or FRD design. In this case, the probability of receipt
of the treatment increases discontinuously at the threshold,
but not necessarily from zero to one:
\[
\lim_{w\downarrow c}\operatorname{pr}(X_i=1|V_i=w)\neq\lim
_{w\uparrow c}\operatorname{pr}(X_i=1|V_i=w).
\]
In that case, it is no longer possible to consistently estimate the
average effect of the treatment for all units at the threshold.
Hahn, Todd and Van der Klaauw (\citeyear{HahTodvan00}) demonstrate that there is a close
link to the instrumental variables set up. Specifically Hahn, Todd and
Van der Klaauw show that one can estimate a local average treatment
effect at the threshold. To be precise, one can identify the average
effect of the treatment for those who are on the margin of getting the
treatment:
\begin{eqnarray*}
&& \mathbb{E}\Bigl[Y_i(1)-Y_i(0) \big|\\
&& \quad\hspace*{2pt} {}V_i=c,\lim
_{w\uparrow c} X_i(w)=0, \lim_{w\downarrow c}
X_i(w)=1 \Bigr]\\
&& \quad =
\frac{\lim_{w\uparrow c} \mathbb{E}[Y_i|V_i=w]-\lim_{w\downarrow c} \mathbb{E}[Y_i|V_i=w]}{
\lim_{w\uparrow c} \mathbb{E}[X_i|V_i=w]-\lim_{w\downarrow c}
\mathbb{E}[X_i|V_i=w]}.
\end{eqnarray*}
This estimand can be estimated as the ratio of an estimator for the
discontinuity in the regression function for the outcome and an
estimator for the discontinuity in the regression function for the
treatment of interest.
\section{Conclusion}
\label{section:conclusion}
In this paper, I review the connection between the recent statistics
literature on instrumental variables and the older econometrics
literature. Although the econometric literature on instrumental
variables goes back to the 1920s, until recently it had not made much
of an impact on the statistics literature. The recent statistics
literature has combined some of the older insights from the
econometrics instrumental variables literature with the separate
literature on causality, enriching both in the process.
\begin{appendix}
\section*{Appendix: Estimation and Inference,
Two-Stage-Least-Squares and Other
Traditional Methods}\label{appA}
\subsection{Set up}
In this section, I will discuss the traditional econometric approaches
to estimation and inference in instrumental variables settings. Part of
the aim of this section is to provide easier access to the econometric
literature and terminology on instrumental variables, and to provide a
perspective and context for the recent advances.
The textbook setting is the one discussed in the previous section,
where a scalar outcome $Y_i$ is linearly related to a scalar covariate
of interest
$X_i$. In addition, there may be additional exogenous covariates $V_i$.
The traditional model is
\begin{equation}
\label{eq:outcome}
Y_i=\beta_0+\beta_1
X_i+\beta_2'V_i+
\varepsilon_i.
\end{equation}
In addition, we have a vector of instrumental variables $Z_i$, with
dimension $K$.
An important distinction in the traditional econometric literature is
between the case with a single instrument ($K=1$), and the case with
more than one instrument ($K>1$). More generally, with more than one
endogenous regressor, the distinction is between the case with the
number of instruments equal to the number of endogenous regressors and
the case with the number of instruments larger than the number of
endogenous regressors. In the empirical literature, there are few
credible examples with more than one endogenous regressor, so I focus
here on the case with a single endogenous regressor.
The first case, with a single instrument, is referred to as the \textit{just-identified} case, and the second, with multiple instruments and a
single endogenous regressor, as the \textit{over-identified} case. In the
textbook setting with a linear model and constant coefficients, this
distinction has motivated different estimators and specification tests.
In the modern literature, with its explicit allowance for heterogeneity
in the treatment effects, these tests, and the distinction between the
various estimators, are of less interest. In the recent statistics
literature, little attention has been paid to the over-identified case
with multiple instruments. An exception is \citet{Sma07}.
Obviously, it is often difficult in applications to find even a single
variable that satisfies the conditions for it to be a valid instrument.
This raises the question how relevant the literature focusing on
methods to deal with multiple instruments is for empirical practice.
There are two classes of applications where multiple instruments could
credible arise. First, suppose one has a single continuous (or
multivalued) instrument that satisfies the instrumental variables
assumptions, monotonicity, random assignment and the exclusion
restriction. Then any monotone function of the instruments also
satisfies these assumptions, and one can use multiple monotone
functions of the original instrument as instruments.
Second,
if one has a single instrument in combination with exogenous
covariates, then one can use interactions of the instrument and the
covariates to generate additional instruments.
Consider, for example, the Fulton fish market study by Graddy (\citeyear{Gra95},
\citeyear{Gra96}). Graddy uses weather conditions as an instrument that affects
supply but not demand. Specifically, she measures wind speed and wave
height, giving her two basic instruments. She also constructs functions
of these basic instruments, such as indicators that the wind speed or
wave height exceeds some threshold.
\subsection{The Just-Identified Case with no Additional~Covariates}
The traditional approach to estimation in this case is to use what is
known in the econometrics literature as the instrumental variables
estimator. In the case without additional exogenous covariates, the
most widely used estimator is simply the ratio of two covariances:
\begin{eqnarray*}
\hat{\beta}_{1,\mathrm{iv}} &=& \frac{\widehat{\operatorname{cov}}(Y_i,Z_i)}{\widehat{\operatorname{cov}}(X_i,Z_i)} \\
&=& \frac{({1}/{N})\sum_{i=1}^N (Y_i-\overline{Y}) (Z_i-\overline{Z})}{
({1}/{N})\sum_{i=1}^N (X_i-\overline{X}) (Z_i-\overline{Z})},
\end{eqnarray*}
where $\overline{Y}$, $\overline{Z}$ and $\overline{X}$ denote
sample averages.
If the instrument $Z_i$ is binary, this is also known as the Wald estimator:
\[
\hat\beta_{1,\mathrm{iv}}=\frac{\overline{Y}_1-\overline{Y}_0}{
\overline{X}_1-\overline{X}_0},
\]
where for $z=0,1$
\[
\overline{Y}_z=\frac{1}{N_z}\sum_{i:Z_i=z}
Y_i,\quad \overline{X}_z=\frac{1}{N_z}\sum
_{i:Z_i=z} X_i,
\]
and $N_1=\sum_{i=1}^N Z_i$ and $N_0=\sum_{i=1}^N (1-Z_i)$.
One can interpret this estimator in two different ways. These
interpretations are useful for motivating extensions to settings with
multiple instruments and additional exogenous regressors. First, the
\textit{indirect least squares} interpretation. This relies on first
estimating separately the two \textit{reduced form regressions}, the
regressions of the outcome on the instrument:
\[
Y_i=\pi_{10}+\pi_{11}\cdot Z_i+
\varepsilon_{1i},
\]
and the regression of the endogenous regressor
on the instrument:
\[
X_i=\pi_{20}+\pi_{21}\cdot Z_i+
\varepsilon_{2i}.
\]
The indirect least squares estimator is the ratio of the least squares
estimates of $\pi_{11}$ and $\pi_{21}$, or $\hat\beta_{1,\mathrm{ils}}=\hat\pi_{11}/\hat\pi_{21}$. Note that in the randomized
experiment example where $X_i$ and $Z_i$ are binary, the $\pi_{11}$
and $\pi_{12}$ are the \textit{intention-to-treat} effects,
with
$\hat\pi_{11}=\overline{Y}_1-\overline{Y}_0$ and
$\hat\pi_{12}=\overline{X}_1-\overline{X}_0$.
Second, I discuss the two-stage-least-squares interpretation of the
instrumental variables estimator. First, estimate the
reduced form regression of the treatment on the instruments and the
exogenous covariates. Calculate the predicted value for the endogenous
regressor from this regression:
\[
\hat X_i=\hat\pi_{20}+\hat\pi_{21}\cdot
Z_i.
\]
The estimate the regression of the outcome on the predicted endogenous
regressor and the additional covariates,
\[
Y_i=\beta_0+\beta_1 \hat
X_i+\eta_i,
\]
by least squares to get the TSLS estimator $\hat\beta_{\mathrm{tsls}}$.
In this just-identified setting, the three estimators for $\beta_1$
are numerically identical:
$\hat\beta_{1,\mathrm{iv}}=\hat\beta_{1,\mathrm{ils}}=\hat\beta_{1,\mathrm{tsls}}$.
\subsection{The Just-Identified Case with Additional~Covariates}
In most econometric applications, the instrument is not physically
randomized. There is in those cases no guarantee that the instrument is
independent of the potential outcomes. Often researchers use covariates
to weaken the requirement on the instrument to conditional independence
given the exogenous covariates.
In addition, the additional exogenous covariates can serve to increase
precision.
In that case with additional covariates, the estimation strategy
changes slightly.
The two reduced form regressions now take the form
\[
Y_i=\pi_{10}+\pi_{11}\cdot Z_i+
\pi_{12}' V_i+\varepsilon_{1i},
\]
and the regression of the endogenous regressor
on the instrument:
\[
X_i=\pi_{20}+\pi_{21}\cdot Z_i+
\pi_{22}' V_i+\varepsilon_{2i}.
\]
The indirect least squares estimator is again the ratio of the least
squares estimates of $\pi_{11}$ and $\pi_{21}$, or $\hat\beta
_{1,\mathrm{ils}}=\hat\pi_{11}/\hat\pi_{21}$.
For the two-stage-least-squares estimator, we again first estimate the
regression of the endogenous regressor on the instrument, now also
including the exogenous regressors. The next step is to predict the
endogenous covariate:
\[
\hat{X}_i=\hat\pi_{20}+\hat\pi_{21}\cdot
Z_i+\hat\pi_{22}' V_i .
\]
Finally, the outcome is regressed on the predicted value of the
endogenous regressor and the actual values of the exogenous variables:
\[
Y_i=\beta_0+\beta_1 \hat
X_i+\beta_2'V_i+
\eta_i.
\]
The TSLS estimator is again identical to the ILS estimator.
For inference, the traditional approach is to assume homoscedasticity
of the residuals $Y_i-\beta_0-\beta_1 X_i-\beta_2'V_i$ with variance
$\sigma^2_\varepsilon$.
In large samples, the distribution of the estimator $\hat{\beta}_{\mathrm{iv}}$ is approximately normal, centered around the true value $\beta
_1$. Typically, the variance is estimated as
\[
\widehat{\mathbb{V}}=\hat\sigma^2_\varepsilon\cdot \left( \left(
\begin{array} {@{}c@{}} 1
\\
\hat X_i
\\
V_i \end{array}
\right) \left(
\begin{array} {@{}c@{}} 1
\\
\hat X_i
\\
V_i \end{array}
\right)^{\operatorname{pr}ime} \right)^{-1} .
\]
See the textbook discussion in Wooldridge (\citeyear{Woo10}).
\subsection{The Over-Identified Case}
The second case of interest is the overidentified case. The main
equation remains
\[
Y_i=\beta_0+\beta_1 X_i+
\beta_2'V_i+\varepsilon_i,
\]
but now the instrument $Z_i$ has dimension $K>1$.
We continue to assume that the residuals $\varepsilon_i$ are
independent of the instruments with mean zero and variance $\sigma
^2_\varepsilon$.
This case is the subject of a large literature, and
many estimators have been proposed. I will briefly discuss two. For a
more detailed discussion, see Wooldridge (\citeyear{Woo10}).
\subsection{Two-Stage-Least-Squares}
The TSLS approach extends naturally to the setting with multiple instruments.
First, estimate the reduced form regression of the endogenous variable
$X_i$ on the instruments $Z_i$ and the exogenous variables $V_i$,
\[
X_i=\pi_{20}+\pi_{21}'Z_i+
\pi_{22}'V_i+\varepsilon_{2i},
\]
by least squares. Next, calculate the predicted value,
\[
\hat{X}_i=\hat{\pi}_{20}+\hat{\pi}_{21}'Z_i+
\hat{\pi}_{22}'V_i .
\]
Finally, regress the outcome on the predicted value from this regression:
\[
Y_i=\beta_0+\beta_1 \hat
X_i+\beta_2'V_i+
\eta_i.
\]
The fact that the dimension of the instrument $Z_i$ is greater than one
does not affect the mechanics of the procedure.
To illustrate this, consider the Graddy Fulton Fish Market data.
Instead of simply using the binary indicator stormy/not-stormy as the
instrument, we can use the trivalued weather indicator,
stormy/mixed/fair to generate two instruments. This leads to TSLS
estimates equal to
\[
\hat\beta_{1,\mathrm{tsls}} = -1.014\quad (\mbox{s.e. }0.384).
\]
\subsection{Limited-Information-Maximum-Likelihood}
The second most popular estimator in this over-identified setting is
the limited-information-maximum-likelihood (LIML) estimator, originally
proposed by Anderson and Rubin (\citeyear{A49}) in the statistics literature.
The likelihood is based on joint normality of the joint endogenous
variables, $(Y_i,X_i)'$, given the instruments and exogenous variables
$(Z_i,V_i)$:
\[
\left(
\begin{array} {@{}c@{}} Y_i
\\
X_i \end{array}
\right) \bigg| Z_i,V_i
\sim\mathcal{N} \left( \left(
\begin{array} {@{}c@{}} \vspace*{2pt}\pi_{10}+
\beta_1\pi_{21}'Z_i+
\pi_{12}'V_i
\\
\pi_{20}+\pi_{21}'Z_i+
\pi_{22}'V_i \end{array}
\right),
\Omega \right).
\]
The LIML estimator can be expressed in terms of some eigenvalue
calculations, so that it is computationally fairly simple, though more
complicated than the TSLS estimator which only requires matrix inversion.
Although motivated by a normal-distribution-based likelihood function,
the LIML estimator is consistent under much weaker conditions, as long
as $(\varepsilon_{1i},\varepsilon_{2i})'$ are independent of
$(Z_i,V_i)$ and the model (\ref{eq:outcome}) is correct with
$\varepsilon_i$ independent of $(Z_i,V_i)$.
Both the TSLS and LIML estimators are consistent and asymptotically
normally distributed with the same variance. In the just-identified
case, the two estimators are numerically identical. The variance can be
estimated as in the just-identified case as
\[
\widehat{\mathbb{V}}=\hat\sigma^2_\varepsilon\cdot \left( \left(
\begin{array} {@{}c@{}} 1
\\
\hat X_i
\\
V_i \end{array}
\right) \left(
\begin{array} {@{}c@{}} 1
\\
\hat X_i
\\
V_i \end{array}
\right)' \right)^{-1} .
\]
In practice, there can be substantial differences between the TSLS and
LIML estimators when the instruments are weak (see Section~\ref{section_weak}) or when there are many instruments (see Section~\ref{section_many}), that is, when the degree of overidentification is high.
For the fish data, the LIML estimates are
\[
\hat\beta_{1,\mathrm{liml}} = -1.016\quad (\mbox{s.e. }0.384).
\]
\subsection{Testing the Over-Indentifying Restrictions}
The indirect least squares procedure does not work well in the case
with multiple instruments. The two reduced form regressions are
\[
X_i=\pi_{20}+\pi_{21}'Z_i+
\pi_{22}'V_i+\varepsilon_{2i}
\]
and
\[
Y_i=\pi_{10}+\pi_{11}'Z_i+
\pi_{12}'V_i+\varepsilon_{1i}.
\]
If the model is correctly specified, the $K$-component vector $\pi
_{11}$ should be equal to $\beta_1\cdot\pi_{21}$. However, there is
nothing in the reduced form estimates that imposes proportionality of
the estimates. In principle, we can use any element of the
$K$-component vector or ratios $\hat\pi_{21}/\pi_{11}$ as an
estimator for $\beta_1$.
If the assumption that $\varepsilon_{1i}$ is independent of $Z_i$ is
true for each component of the instrument, all estimators will estimate
the same object, and differences between them should be due to sampling
variation.
Comparisons of these $K$ estimators can therefore be used to test the
assumptions that all instruments are valid.
Although such tests have been popular in the econometrics literature,
they are also sensitive to the other maintained assumptions in the
model, notably linearity in the endogenous regressor and the constant
effect assumption. In the local-average-treatment-effect set up from
Section~\ref{section_late}, differences in estimators based on
different instruments can simply be due to the fact that the different
instruments correspond to different populations of compliers.
\end{appendix}
\section*{Acknowledgments}
Financial support for this research was generously provided
through NSF Grants 0820361 and 0961707. I~am grateful to Joshua
Angrist who got me interested in these topics many years ago, and over
the years, has taught me much about the issues discussed in this
manuscript, the editor of \textit{Statistical Science} for suggesting this
review and three anonymous referees who wrote remarkably thoughtful
reviews.
\begin{thebibliography}{185}
\bibitem[\operatorname{pr}otect\citeauthoryear{Abadie}{2002}]{Aba02}
\begin{barticle}[mr]
\bauthor{\bsnm{Abadie},~\bfnm{Alberto}\binits{A.}}
(\byear{2002}).
\btitle{Bootstrap tests for distributional treatment effects in
instrumental variable models}.
\bjournal{J. Amer. Statist. Assoc.}
\bvolume{97}
\bpages{284--292}.
\bid{doi={10.1198/016214502753479419}, issn={0162-1459}, mr={1947286}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Abadie}{2003}]{Aba03}
\begin{barticle}[mr]
\bauthor{\bsnm{Abadie},~\bfnm{Alberto}\binits{A.}}
(\byear{2003}).
\btitle{Semiparametric instrumental variable estimation of treatment
response models}.
\bjournal{J. Econometrics}
\bvolume{113}
\bpages{231--263}.
\bid{doi={10.1016/S0304-4076(02)00201-4}, issn={0304-4076}, mr={1960380}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Aizer and Doyle}{2013}]{AizDoy13}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Aizer},~\bfnm{A.}\binits{A.}} \AND
\bauthor{\bsnm{Doyle},~\bfnm{J.}\binits{J.}}
(\byear{2013}).
\bhowpublished{Juvenile incarceration, human capital, and future
crime: Evidence from randomly assigned judges. Unpublished working paper,
Dept. Economics, Brown Univ., Providence, RI.}
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Altonji and Matzkin}{2005}]{AltMat05}
\begin{barticle}[mr]
\bauthor{\bsnm{Altonji},~\bfnm{Joseph~G.}\binits{J.~G.}} \AND
\bauthor{\bsnm{Matzkin},~\bfnm{Rosa~L.}\binits{R.~L.}}
(\byear{2005}).
\btitle{Cross section and panel data estimators for nonseparable
models with endogenous regressors}.
\bjournal{Econometrica}
\bvolume{73}
\bpages{1053--1102}.
\bid{doi={10.1111/j.1468-0262.2005.00609.x}, issn={0012-9682}, mr={2149241}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Anderson and Rubin}{1949}]{A49}
\begin{barticle}[mr]
\bauthor{\bsnm{Anderson},~\bfnm{T.~W.}\binits{T.~W.}} \AND
\bauthor{\bsnm{Rubin},~\bfnm{Herman}\binits{H.}}
(\byear{1949}).
\btitle{Estimation of the parameters of a single equation in a complete system of stochastic equations}.
\bjournal{Ann. Math. Statistics}
\bvolume{20}
\bpages{46--63}.
\bid{issn={0003-4851}, mr={0028546}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Andrews, Moreira and
Stock}{2006}]{AndMorSto06}
\begin{barticle}[mr]
\bauthor{\bsnm{Andrews},~\bfnm{Donald~W.~K.}\binits{D.~W.~K.}},
\bauthor{\bsnm{Moreira},~\bfnm{Marcelo~J.}\binits{M.~J.}} \AND
\bauthor{\bsnm{Stock},~\bfnm{James~H.}\binits{J.~H.}}
(\byear{2006}).
\btitle{Optimal two-sided invariant similar tests for instrumental
variables regression}.
\bjournal{Econometrica}
\bvolume{74}
\bpages{715--752}.
\bid{doi={10.1111/j.1468-0262.2006.00680.x}, issn={0012-9682}, mr={2217614}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Andrews and Stock}{2007}]{AndSto}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Andrews},~\bfnm{D.}\binits{D.}} \AND
\bauthor{\bsnm{Stock},~\bfnm{J.}\binits{J.}}
(\byear{2007}).
\bhowpublished{Inference with weak instruments. In \textit{Advances
in Economics and Econometrics\textup{:} Theory and Applications\textup{,} Ninth World Congress\textup{,} Vol. \textup{III}}
(R.~Blundell, W. Newey and T. Persson, eds.) 122--173. Cambridge Univ. Press, Cambridge.}
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Angrist}{1990}]{Ang90}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Angrist},~\bfnm{J.}\binits{J.}}
(\byear{1990}).
\btitle{Lifetime earnings and the Vietnam era draft lottery: Evidence
from social security administrative records}.
\bjournal{American Economic Review}
\bvolume{80}
\bpages{313--335}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Angrist, Graddy and
Imbens}{2000}]{AngGraImb00}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Angrist},~\bfnm{J.}\binits{J.}},
\bauthor{\bsnm{Graddy},~\bfnm{K.}\binits{K.}} \AND
\bauthor{\bsnm{Imbens},~\bfnm{G.}\binits{G.}}
(\byear{2000}).
\btitle{The interpretation of instrumental variables estimators in
simultaneous equations models with an application to the demand for fish}.
\bjournal{Rev. Econom. Stud.}
\bvolume{67}
\bpages{499--527}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Angrist and Imbens}{1995}]{AngImb95}
\begin{barticle}[mr]
\bauthor{\bsnm{Angrist},~\bfnm{Joshua~D.}\binits{J.~D.}} \AND
\bauthor{\bsnm{Imbens},~\bfnm{Guido~W.}\binits{G.~W.}}
(\byear{1995}).
\btitle{Two-stage least squares estimation of average causal effects
in models with variable treatment intensity}.
\bjournal{J. Amer. Statist. Assoc.}
\bvolume{90}
\bpages{431--442}.
\bid{issn={0162-1459}, mr={1340501}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Angrist, Imbens and Rubin}{1996}]{AngImbRub}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Angrist},~\bfnm{J.}\binits{J.}},
\bauthor{\bsnm{Imbens},~\bfnm{G.}\binits{G.}} \AND
\bauthor{\bsnm{Rubin},~\bfnm{D.}\binits{D.}}
(\byear{1996}).
\bhowpublished{Identification of causal effects using instrumental
variables (with discussion). \textit{J.~Amer. Statist. Assoc.}
\textbf{91}
444--472.}
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Angrist and Krueger}{1991}]{AngKru91}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Angrist},~\bfnm{J.}\binits{J.}} \AND
\bauthor{\bsnm{Krueger},~\bfnm{A.}\binits{A.}}
(\byear{1991}).
\btitle{Does compulsory school attendance affect schooling and earnings}.
\bjournal{Quarterly Journal of Economics}
\bvolume{106}
\bpages{979--1014}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Angrist and Pischke}{2009}]{AngPis09}
\begin{bbook}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Angrist},~\bfnm{J.}\binits{J.}} \AND
\bauthor{\bsnm{Pischke},~\bfnm{S.}\binits{S.}}
(\byear{2009}).
\btitle{Mostly Harmless Econometrics}.
\bpublisher{Princeton Univ. Press},
\blocation{Princeton, NJ}.
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Arellano}{2002}]{Are02}
\begin{barticle}[mr]
\bauthor{\bsnm{Arellano},~\bfnm{Manuel}\binits{M.}}
(\byear{2002}).
\btitle{Sargan's instrumental variables estimation and the generalized
method of moments}.
\bjournal{J. Bus. Econom. Statist.}
\bvolume{20}
\bpages{450--459}.
\bid{doi={10.1198/073500102288618595}, issn={0735-0015}, mr={1973797}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Athey and Stern}{1998}]{AthSte98}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Athey},~\bfnm{S.}\binits{S.}} \AND
\bauthor{\bsnm{Stern},~\bfnm{S.}\binits{S.}}
(\byear{1998}).
\bhowpublished{An empirical framework for testing theories about
complementarity in organizational
design.
Working Paper 6600,
NBER}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Baiocchi et~al.}{2010}]{Baietal10}
\begin{barticle}[mr]
\bauthor{\bsnm{Baiocchi},~\bfnm{Mike}\binits{M.}},
\bauthor{\bsnm{Small},~\bfnm{Dylan~S.}\binits{D.~S.}},
\bauthor{\bsnm{Lorch},~\bfnm{Scott}\binits{S.}} \AND
\bauthor{\bsnm{Rosenbaum},~\bfnm{Paul~R.}\binits{P.~R.}}
(\byear{2010}).
\btitle{Building a stronger instrument in an observational study of
perinatal care for premature infants}.
\bjournal{J. Amer. Statist. Assoc.}
\bvolume{105}
\bpages{1285--1296}.
\bid{doi={10.1198/jasa.2010.ap09490}, issn={0162-1459}, mr={2796550}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Balke and Pearl}{1995}]{autokey15}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Balke},~\bfnm{A.}\binits{A.}} \AND
\bauthor{\bsnm{Pearl},~\bfnm{J.}\binits{J.}}
(\byear{1995}).
\bhowpublished{Counterfactuals and policy analysis in structural
models. In \textit{Uncertainty in Artificial Intelligence \textup{11}}
(P. Besnard and
S. Hanks, eds.) 11--18. Morgan Kaufmann, San Francisco, CA}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Balke and Pearl}{1997}]{BalPea97}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Balke},~\bfnm{A.}\binits{A.}} \AND
\bauthor{\bsnm{Pearl},~\bfnm{J.}\binits{J.}}
(\byear{1997}).
\btitle{Bounds on treatment effects from studies with imperfect compliance}.
\bjournal{J. Amer. Statist. Assoc.}
\bvolume{92}
\bpages{1171--1176}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Barnow, Cain and
Goldberger}{1980}]{BarCaiGol80}
\begin{bincollection}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Barnow},~\bfnm{B.~S.}\binits{B.~S.}},
\bauthor{\bsnm{Cain},~\bfnm{G.~G.}\binits{G.~G.}} \AND
\bauthor{\bsnm{Goldberger},~\bfnm{A.~S.}\binits{A.~S.}}
(\byear{1980}).
\btitle{Issues in the analysis of selectivity bias}.
In \bbooktitle{Evaluation Studies, Vol. 5}
(\beditor{\bfnm{E.}\binits{E.}~\bsnm{Stromsdorfer}} \AND
\beditor{\bfnm{G.}\binits{G.}~\bsnm{Farkas}}, eds.).
\bpublisher{Sage},
\blocation{San Francisco, CA}.
\end{bincollection}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Basmann}{1963a}]{Bas63N1}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Basmann},~\bfnm{R.}\binits{R.}}
(\byear{1963}a).
\btitle{The causal interpretation of non-triangular systems of
economic relations}.
\bjournal{Econometrica}
\bvolume{31}
\bpages{439--448}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Basmann}{1963b}]{Bas63N2}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Basmann},~\bfnm{R.}\binits{R.}}
(\byear{1963}b).
\btitle{On the causal interpretation of non-triangular systems of
economic relations: A rejoinder}.
\bjournal{Econometrica}
\bvolume{31}
\bpages{451--453}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Basmann}{1965}]{Bas65}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Basmann},~\bfnm{R.}\binits{R.}}
(\byear{1965}).
\btitle{Causal systems and stability: Reply to R.~W.~Clower}.
\bjournal{Econometrica}
\bvolume{33}
\bpages{242--243}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Bekker}{1994}]{Bek94}
\begin{barticle}[mr]
\bauthor{\bsnm{Bekker},~\bfnm{Paul~A.}\binits{P.~A.}}
(\byear{1994}).
\btitle{Alternative approximations to the distributions of
instrumental variable estimators}.
\bjournal{Econometrica}
\bvolume{62}
\bpages{657--681}.
\bid{doi={10.2307/2951662}, issn={0012-9682}, mr={1281697}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Bekker and van~der Ploeg}{2005}]{B05}
\begin{barticle}[mr]
\bauthor{\bsnm{Bekker},~\bfnm{Paul~A.}\binits{P.~A.}} \AND
\bauthor{\bsnm{van~der Ploeg},~\bfnm{Jan}\binits{J.}}
(\byear{2005}).
\btitle{Instrumental variable estimation based on grouped data}.
\bjournal{Statist. Neerlandica}
\bvolume{59}
\bpages{239--267}.
\bid{doi={10.1111/j.1467-9574.2005.00296.x}, issn={0039-0402}, mr={2189771}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Benkard and Berry}{2006}]{BenBer06}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Benkard},~\bfnm{L.}\binits{L.}} \AND
\bauthor{\bsnm{Berry},~\bfnm{S.}\binits{S.}}
(\byear{2006}).
\btitle{On the nonparametric identification of nonlinear simultaneous
equations models: Comment on Brown (1983) and Roehrig (1988)}.
\bjournal{Econometrica}
\bvolume{74}
\bpages{1429--1440}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Bound, Jaeger and Baker}{1995}]{BouJaeBak95}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Bound},~\bfnm{J.}\binits{J.}},
\bauthor{\bsnm{Jaeger},~\bfnm{D.}\binits{D.}} \AND
\bauthor{\bsnm{Baker},~\bfnm{R.}\binits{R.}}
(\byear{1995}).
\btitle{Problems with instrumental variables estimation when the
correlation between the instruments and the endogenous explanatory
variable is weak}.
\bjournal{J. Amer. Statist. Assoc.}
\bvolume{90}
\bpages{443--450}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Bowden and Turkington}{1984}]{BowTur84}
\begin{bbook}[mr]
\bauthor{\bsnm{Bowden},~\bfnm{Roger~J.}\binits{R.~J.}} \AND
\bauthor{\bsnm{Turkington},~\bfnm{Darrell~A.}\binits{D.~A.}}
(\byear{1984}).
\btitle{Instrumental Variables}.
\bseries{Econometric Society Monographs in Quantitative Economics}
\bvolume{8}.
\bpublisher{Cambridge Univ. Press},
\blocation{Cambridge}.
\bid{mr={0798790}}
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Brookhart et~al.}{2006}]{Broetal06}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Brookhart},~\bfnm{M.}\binits{M.}},
\bauthor{\bsnm{Wang},~\bfnm{P.}\binits{P.}},
\bauthor{\bsnm{Solomon},~\bfnm{D.}\binits{D.}} \AND
\bauthor{\bsnm{Schnee\-weiss},~\bfnm{S.}\binits{S.}}
(\byear{2006}).
\btitle{Evaluating short-term drug effects using a physician-specific
prescribing preference as an instrumental variable}.
\bjournal{Epidemiology}
\bvolume{17}
\bpages{268--275}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Brown}{1983}]{Bro83}
\begin{barticle}[mr]
\bauthor{\bsnm{Brown},~\bfnm{Bryan~W.}\binits{B.~W.}}
(\byear{1983}).
\btitle{The identification problem in systems nonlinear in the variables}.
\bjournal{Econometrica}
\bvolume{51}
\bpages{175--196}.
\bid{doi={10.2307/1912253}, issn={0012-9682}, mr={0694456}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Card}{1995}]{Car95}
\begin{bincollection}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Card},~\bfnm{D.}\binits{D.}}
(\byear{1995}).
\btitle{Using geographic variation in college proximity to estimate
the return to schooling}.
In \bbooktitle{Aspects of Labor Market Behaviour: Essays in Honour of
John Vanderkamp}
(\beditor{\bfnm{L.~N.}\binits{L.~N.}~\bsnm{Christofides}},
\beditor{\bfnm{E.~K.}\binits{E.~K.}~\bsnm{Grant}} \AND
\beditor{\bfnm{R.}\binits{R.}~\bsnm{Swidinsky}}, eds.).
\bpublisher{Univ. Toronto Press},
\blocation{Toronto}.
\end{bincollection}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Card}{2001}]{Car01}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Card},~\bfnm{D.}\binits{D.}}
(\byear{2001}).
\btitle{Estimating the return to schooling: Progress on some
persistent econometric problems}.
\bjournal{Econometrica}
\bvolume{69}
\bpages{1127--1160}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Chalak}{2011}]{Cha11}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Chalak},~\bfnm{K.}\binits{K.}}
(\byear{2011}).
\bhowpublished{Identification of local treatment effects using a
proxy for an instrument. Unpublished
Manuscript, Dept. Economics,
Boston College, Boston, MA}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Chamberlain and Imbens}{2004}]{ChaImb04}
\begin{barticle}[mr]
\bauthor{\bsnm{Chamberlain},~\bfnm{Gary}\binits{G.}} \AND
\bauthor{\bsnm{Imbens},~\bfnm{Guido}\binits{G.}}
(\byear{2004}).
\btitle{Random effects estimators with many instrumental variables}.
\bjournal{Econometrica}
\bvolume{72}
\bpages{295--306}.
\bid{doi={10.1111/j.1468-0262.2004.00485.x}, issn={0012-9682}, mr={2031020}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Chao and Swanson}{2005}]{ChaSwa05}
\begin{barticle}[mr]
\bauthor{\bsnm{Chao},~\bfnm{John~C.}\binits{J.~C.}} \AND
\bauthor{\bsnm{Swanson},~\bfnm{Norman~R.}\binits{N.~R.}}
(\byear{2005}).
\btitle{Consistent estimation with a large number of weak instruments}.
\bjournal{Econometrica}
\bvolume{73}
\bpages{1673--1692}.
\bid{doi={10.1111/j.1468-0262.2005.00632.x}, issn={0012-9682}, mr={2156676}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Chernozhukov, Hong and
Tamer}{2007}]{CheHonTam07}
\begin{barticle}[mr]
\bauthor{\bsnm{Chernozhukov},~\bfnm{Victor}\binits{V.}},
\bauthor{\bsnm{Hong},~\bfnm{Han}\binits{H.}} \AND
\bauthor{\bsnm{Tamer},~\bfnm{Elie}\binits{E.}}
(\byear{2007}).
\btitle{Estimation and confidence regions for parameter sets in
econometric models}.
\bjournal{Econometrica}
\bvolume{75}
\bpages{1243--1284}.
\bid{doi={10.1111/j.1468-0262.2007.00794.x}, issn={0012-9682}, mr={2347346}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Chesher}{2003}]{Che03}
\begin{barticle}[mr]
\bauthor{\bsnm{Chesher},~\bfnm{Andrew}\binits{A.}}
(\byear{2003}).
\btitle{Identification in nonseparable models}.
\bjournal{Econometrica}
\bvolume{71}
\bpages{1405--1441}.
\bid{doi={10.1111/1468-0262.00454}, issn={0012-9682}, mr={2000252}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Chesher}{2010}]{Che10}
\begin{barticle}[mr]
\bauthor{\bsnm{Chesher},~\bfnm{Andrew}\binits{A.}}
(\byear{2010}).
\btitle{Instrumental variable models for discrete outcomes}.
\bjournal{Econometrica}
\bvolume{78}
\bpages{575--601}.
\bid{doi={10.3982/ECTA7315}, issn={0012-9682}, mr={2656640}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Christ}{1994}]{Chr94}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Christ},~\bfnm{C.}\binits{C.}}
(\byear{1994}).
\btitle{The Cowles Commission's contributions to econometrics at
Chicago, 1939--1955}.
\bjournal{Journal of Economic Literature}
\bvolume{32}
\bpages{30--59}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Cochran}{1968}]{Coc68}
\begin{barticle}[mr]
\bauthor{\bsnm{Cochran},~\bfnm{W.~G.}\binits{W.~G.}}
(\byear{1968}).
\btitle{The effectiveness of adjustment by subclassification in
removing bias in observational studies}.
\bjournal{Biometrics}
\bvolume{24}
\bpages{295--313}.
\bid{issn={0006-341X}, mr={0228136}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Cochran and Rubin}{1973}]{CocRub73}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Cochran},~\bfnm{W.}\binits{W.}} \AND
\bauthor{\bsnm{Rubin},~\bfnm{D.}\binits{D.}}
(\byear{1973}).
\btitle{Controlling bias in observational studies: A review}.
\bjournal{Sankhy\=a}
\bvolume{35}
\bpages{417--446}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Cook}{2008}]{Coo08}
\begin{barticle}[mr]
\bauthor{\bsnm{Cook},~\bfnm{Thomas~D.}\binits{T.~D.}}
(\byear{2008}).
\btitle{``{W}aiting for life to arrive'': A history of the
regression-discontinuity design in psychology, statistics and economics}.
\bjournal{J. Econometrics}
\bvolume{142}
\bpages{636--654}.
\bid{doi={10.1016/j.jeconom.2007.05.002}, issn={0304-4076}, mr={2416822}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Cox}{1992}]{Cox92}
\begin{barticle}[mr]
\bauthor{\bsnm{Cox},~\bfnm{D.~R.}\binits{D.~R.}}
(\byear{1992}).
\btitle{Causality: Some statistical aspects}.
\bjournal{J. Roy. Statist. Soc. Ser. A}
\bvolume{155}
\bpages{291--301}.
\bid{doi={10.2307/2982962}, issn={0964-1998}, mr={1157712}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Cr{\'e}pon et~al.}{2012}]{Creetal}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Cr{\'e}pon},~\bfnm{B.}\binits{B.}},
\bauthor{\bsnm{Duflo},~\bfnm{E.}\binits{E.}},
\bauthor{\bsnm{Gurgand},~\bfnm{M.}\binits{M.}},
\bauthor{\bsnm{Rathelot},~\bfnm{M.}\binits{M.}} \AND
\bauthor{\bsnm{Zamoray},~\bfnm{P.}\binits{P.}}
(\byear{2012}).
\bhowpublished{Do labor market policies have displacement effects?
Evidence from a clustered randomized experiment. Unpublished manuscript.}
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Dawid}{1984}]{Daw84}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Dawid},~\bfnm{P.}\binits{P.}}
(\byear{1984}).
\btitle{Causal inference from messy data. Comment on `On the nature and discovery of structure'}.
\bjournal{J. Amer. Statist. Assoc.}
\bvolume{79}
\bpages{22--24}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Deaton}{2010}]{Dea10}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Deaton},~\bfnm{A.}\binits{A.}}
(\byear{2010}).
\btitle{Instruments, randomization, and learning about development}.
\bjournal{Journal of Economic Literature}
\bvolume{48}
\bpages{424--455}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Dobbie and Song}{2013}]{DobSon13}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Dobbie},~\bfnm{W.}\binits{W.}} \AND
\bauthor{\bsnm{Song},~\bfnm{J.}\binits{J.}}
(\byear{2013}).
\bhowpublished{Debt relief and debtor outcomes: Measuring the effects of
consumer bankruptcy protection. Unpublished working paper, Dept. Economics,
Princeton Univ., Princeton, NJ}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Duflo, Glennester and
Kremer}{2007}]{DufGleKre07}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Duflo},~\bfnm{E.}\binits{E.}},
\bauthor{\bsnm{Glennester},~\bfnm{R.}\binits{R.}} \AND
\bauthor{\bsnm{Kremer},~\bfnm{M.}\binits{M.}}
(\byear{2007}).
\bhowpublished{Using randomization in development economics research: A~toolkit.
In \textit{Handbook of Development
Economics}, \textit{Vol.} 4 (T. P. Schultz and J. Strauss, eds.) 3895--3962. North-Holland, Amsterdam.}
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Fisher}{1925}]{Fis25}
\begin{bbook}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Fisher},~\bfnm{R.~A.}\binits{R.~A.}}
(\byear{1925}).
\btitle{The Design of Experiments}, 1st ed.
\bpublisher{Oliver \& Boyd},
\blocation{London}.
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Frangakis and Rubin}{2002}]{FraRub02}
\begin{barticle}[mr]
\bauthor{\bsnm{Frangakis},~\bfnm{Constantine~E.}\binits{C.~E.}}
\AND
\bauthor{\bsnm{Rubin},~\bfnm{Donald~B.}\binits{D.~B.}}
(\byear{2002}).
\btitle{Principal stratification in causal inference}.
\bjournal{Biometrics}
\bvolume{58}
\bpages{21--29}.
\bid{doi={10.1111/j.0006-341X.2002.00021.x}, issn={0006-341X}, mr={1891039}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Freedman}{2006}]{Fre06}
\begin{barticle}[pbm]
\bauthor{\bsnm{Freedman},~\bfnm{David~A.}\binits{D.~A.}}
(\byear{2006}).
\btitle{Statistical models for causation: What inferential leverage do
they provide?}
\bjournal{Eval. Rev.}
\bvolume{30}
\bpages{691--713}.
\bid{doi={10.1177/0193841X06293771}, issn={0193-841X}, pii={30/6/691},
pmid={17093105}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Frumento et~al.}{2012}]{Fruetal12}
\begin{barticle}[mr]
\bauthor{\bsnm{Frumento},~\bfnm{Paolo}\binits{P.}},
\bauthor{\bsnm{Mealli},~\bfnm{Fabrizia}\binits{F.}},
\bauthor{\bsnm{Pacini},~\bfnm{Barbara}\binits{B.}} \AND
\bauthor{\bsnm{Rubin},~\bfnm{Donald~B.}\binits{D.~B.}}
(\byear{2012}).
\btitle{Evaluating the effect of training on wages in the presence of
noncompliance, nonemployment, and missing outcome data}.
\bjournal{J. Amer. Statist. Assoc.}
\bvolume{107}
\bpages{450--466}.
\bid{doi={10.1080/01621459.2011.643719}, issn={0162-1459}, mr={2980057}}
\bptnote{check year}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Gelman and Hill}{2006}]{GelHil06}
\begin{bbook}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Gelman},~\bfnm{A.}\binits{A.}} \AND
\bauthor{\bsnm{Hill},~\bfnm{J.}\binits{J.}}
(\byear{2006}).
\btitle{Data Analysis Using Regression and Multilevel/Hierarchical Models}.
\bpublisher{Cambridge Univ. Press},
\blocation{Cambridge}.
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Gill and Robins}{2001}]{GilRob01}
\begin{barticle}[mr]
\bauthor{\bsnm{Gill},~\bfnm{Richard~D.}\binits{R.~D.}} \AND
\bauthor{\bsnm{Robins},~\bfnm{James~M.}\binits{J.~M.}}
(\byear{2001}).
\btitle{Causal inference for complex longitudinal data: The continuous case}.
\bjournal{Ann. Statist.}
\bvolume{29}
\bpages{1785--1811}.
\bid{doi={10.1214/aos/1015345962}, issn={0090-5364}, mr={1891746}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Giraud}{2003}]{Gir03}
\begin{barticle}[mr]
\bauthor{\bsnm{Giraud},~\bfnm{Ga{\"e}l}\binits{G.}}
(\byear{2003}).
\btitle{Strategic market games: An introduction}.
\bjournal{J.~Math. Econom.}
\bvolume{39}
\bpages{355--375}.
\bid{doi={10.1016/S0304-4068(03)00049-1}, issn={0304-4068}, mr={1996481}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Graddy}{1995}]{Gra95}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Graddy},~\bfnm{K.}\binits{K.}}
(\byear{1995}).
\bhowpublished{Who pays more? Essays on bargaining and price discrimination. Ph.D. thesis,
Dept. Economics, Princeton Univ., Princeton, NJ.}
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Graddy}{1996}]{Gra96}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Graddy},~\bfnm{K.}\binits{K.}}
(\byear{1996}).
\btitle{Testing for imperfect competition at the Fulton fish market}.
\bjournal{RAND Journal of Economics}
\bvolume{26}
\bpages{75--92}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Greene}{2011}]{Gre11}
\begin{bbook}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Greene},~\bfnm{W.}\binits{W.}}
(\byear{2011}).
\btitle{Econometric Analysis},
\bedition{7th} ed.
\bpublisher{Prentice Hall},
\blocation{Upper Saddle River, NJ}.
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Greenland}{2000}]{Gre}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Greenland},~\bfnm{S.}\binits{S.}}
(\byear{2000}).
\bhowpublished{An introducation to instrumental variables for
epidemiologists. \textit{International Journal of Epidemiology}
\textbf{29}
722--729}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Griliches}{1977}]{Gri77}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Griliches},~\bfnm{Z.}\binits{Z.}}
(\byear{1977}).
\btitle{Estimating the returns to schooling: Some econometric problems}.
\bjournal{Econometrica}
\bvolume{45}
\bpages{1--22}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Haavelmo}{1943}]{Haa43}
\begin{barticle}[mr]
\bauthor{\bsnm{Haavelmo},~\bfnm{Trygve}\binits{T.}}
(\byear{1943}).
\btitle{The statistical implications of a system of simultaneous equations}.
\bjournal{Econometrica}
\bvolume{11}
\bpages{1--12}.
\bid{issn={0012-9682}, mr={0007954}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Haavelmo}{1944}]{Haa44}
\begin{barticle}[mr]
\bauthor{\bsnm{Haavelmo},~\bfnm{Trygve}\binits{T.}}
(\byear{1944}).
\btitle{The probability approach in econometrics}.
\bjournal{Econometrica}
\bvolume{12 (Supplement)}
\bpages{118 pages}.
\bid{issn={0012-9682}, mr={0010953}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Hahn}{2002}]{H02}
\begin{barticle}[mr]
\bauthor{\bsnm{Hahn},~\bfnm{Jinyong}\binits{J.}}
(\byear{2002}).
\btitle{Optimal inference with many instruments}.
\bjournal{Econometric Theory}
\bvolume{18}
\bpages{140--168}.
\bid{doi={10.1017/S0266466602181084}, issn={0266-4666}, mr={1885354}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Hahn, Todd and Van~der
Klaauw}{2001}]{HahTodvan00}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Hahn},~\bfnm{J.}\binits{J.}},
\bauthor{\bsnm{Todd},~\bfnm{P.}\binits{P.}} \AND
\bauthor{\bsnm{Van~der Klaauw},~\bfnm{W.}\binits{W.}}
(\byear{2001}).
\btitle{Identification and estimation of treatment effects with a
regression-discontinuity design}.
\bjournal{Econometrica}
\bvolume{69}
\bpages{201--209}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Hansen, Hausman and Newey}{2008}]{H08}
\begin{barticle}[mr]
\bauthor{\bsnm{Hansen},~\bfnm{Christian}\binits{C.}},
\bauthor{\bsnm{Hausman},~\bfnm{Jerry}\binits{J.}} \AND
\bauthor{\bsnm{Newey},~\bfnm{Whitney}\binits{W.}}
(\byear{2008}).
\btitle{Estimation with many instrumental variables}.
\bjournal{J. Bus. Econom. Statist.}
\bvolume{26}
\bpages{398--422}.
\bid{doi={10.1198/073500108000000024}, issn={0735-0015}, mr={2459342}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Hausman}{1983}]{Hau83}
\begin{bincollection}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Hausman},~\bfnm{J.}\binits{J.}}
(\byear{1983}).
\btitle{Specification and estimation of simultaneous equations models}.
In \bseries{Handbook of
Econometrics, Vol. 1} (Z.~Grilliches and M. D. Intrilligator, eds.).
\bpublisher{North-Holland},
\blocation{Amsterdam}.
\end{bincollection}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Hayashi}{2000}]{Hay00}
\begin{bbook}[mr]
\bauthor{\bsnm{Hayashi},~\bfnm{Fumio}\binits{F.}}
(\byear{2000}).
\btitle{Econometrics}.
\bpublisher{Princeton Univ. Press},
\blocation{Princeton, NJ}.
\bid{mr={1881537}}
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Hearst, Newman and Hulley}{1986}]{HeaNewHul}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Hearst},~\bfnm{N.}\binits{N.}},
\bauthor{\bsnm{Newman},~\bfnm{T.}\binits{T.}} \AND
\bauthor{\bsnm{Hulley},~\bfnm{S.}\binits{S.}}
(\byear{1986}).
\bhowpublished{Delayed effects of the military draft on mortality: A
randomized natural experiment. \textit{N.~Engl. J. Med.}
\textbf{314}
620--624}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Heckman}{1976}]{Hec76}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Heckman},~\bfnm{J.}\binits{J.}}
(\byear{1976}).
\btitle{The common structure of statistical models of truncation,
sample selection and limited dependent variables and a simple estimator
for such models}.
\bjournal{Annals of Economic and Social Measurement}
\bvolume{5}
\bpages{475--492}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Heckman}{1979}]{Hec79}
\begin{barticle}[mr]
\bauthor{\bsnm{Heckman},~\bfnm{James}\binits{J.}}
(\byear{1979}).
\btitle{Sample selection bias as a specification error}.
\bjournal{Econometrica}
\bvolume{47}
\bpages{153--161}.
\bid{doi={10.2307/1912352}, issn={0012-9682}, mr={0518832}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Heckman}{1990}]{Hec}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Heckman},~\bfnm{J.}\binits{J.}}
(\byear{1990}).
\bhowpublished{Varieties of selection bias. \textit{American Economic
Review: Papers and Proceedings} \textbf{80}
313--318}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Heckman and Robb}{1985}]{HecRob}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Heckman},~\bfnm{J.}\binits{J.}} \AND
\bauthor{\bsnm{Robb},~\bfnm{R.}\binits{R.}}
(\byear{1985}).
\bhowpublished{Alternative methods for evaluating the impact of
interventions. In \textit{Longitudinal Analysis of Labor Market Data} (J. Heckman and B. Singer, eds.).
Cambridge Univ. Press, Cambridge}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Hendry and Morgan}{1992}]{autokey71}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Hendry},~\bfnm{D.}\binits{D.}} \AND
\bauthor{\bsnm{Morgan},~\bfnm{M.}\binits{M.}}
(\byear{1992}).
\bhowpublished{\textit{The Foundations of Econometric Analysis}.
Cambridge Univ. Press, Cambridge}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Hern{\'a}n and Robins}{2006}]{HerRob06}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Hern{\'a}n},~\bfnm{M.}\binits{M.}} \AND
\bauthor{\bsnm{Robins},~\bfnm{J.}\binits{J.}}
(\byear{2006}).
\btitle{Instruments for causal inference: An epidemiologist's dream?}
\bjournal{Epidemiology}
\bvolume{17}
\bpages{360--372}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Hillier}{1990}]{Hil90}
\begin{barticle}[mr]
\bauthor{\bsnm{Hillier},~\bfnm{Grant~H.}\binits{G.~H.}}
(\byear{1990}).
\btitle{On the normalization of structural equations: Properties of
direction estimators}.
\bjournal{Econometrica}
\bvolume{58}
\bpages{1181--1194}.
\bid{doi={10.2307/2938305}, issn={0012-9682}, mr={1079413}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Hirano et~al.}{2000}]{Hiretal00}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Hirano},~\bfnm{K.}\binits{K.}},
\bauthor{\bsnm{Imbens},~\bfnm{G.}\binits{G.}},
\bauthor{\bsnm{Rubin},~\bfnm{D.}\binits{D.}} \AND
\bauthor{\bsnm{Zhou},~\bfnm{X.}\binits{X.}}
(\byear{2000}).
\btitle{Identification and estimation of local average treatment effects}.
\bjournal{Biostatistics}
\bvolume{1}
\bpages{69--88}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Hoderlein and Mammen}{2007}]{HodMam07}
\begin{barticle}[mr]
\bauthor{\bsnm{Hoderlein},~\bfnm{Stefan}\binits{S.}} \AND
\bauthor{\bsnm{Mammen},~\bfnm{Enno}\binits{E.}}
(\byear{2007}).
\btitle{Identification of marginal effects in nonseparable models
without monotonicity}.
\bjournal{Econometrica}
\bvolume{75}
\bpages{1513--1518}.
\bid{doi={10.1111/j.1468-0262.2007.00801.x}, issn={0012-9682}, mr={2347352}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Holland}{1986}]{Hol86}
\begin{barticle}[mr]
\bauthor{\bsnm{Holland},~\bfnm{Paul~W.}\binits{P.~W.}}
(\byear{1986}).
\btitle{Statistics and causal inference}.
\bjournal{J. Amer. Statist. Assoc.}
\bvolume{81}
\bpages{945--970}.
\bid{issn={0162-1459}, mr={0867618}}
\bptnote{check related}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Holland}{1988}]{Hol}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Holland},~\bfnm{P.}\binits{P.}}
(\byear{1988}).
\bhowpublished{Causal inference, path analysis, and recursive
structural equations models. In
\textit{Sociological Methodology}, Chapter~13. American Sociological
Association, Washington, DC}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Horowitz}{2011}]{Hor11}
\begin{barticle}[mr]
\bauthor{\bsnm{Horowitz},~\bfnm{Joel~L.}\binits{J.~L.}}
(\byear{2011}).
\btitle{Applied nonparametric instrumental variables estimation}.
\bjournal{Econometrica}
\bvolume{79}
\bpages{347--394}.
\bid{doi={10.3982/ECTA8662}, issn={0012-9682}, mr={2809374}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Horowitz and Lee}{2007}]{HorLee07}
\begin{barticle}[mr]
\bauthor{\bsnm{Horowitz},~\bfnm{Joel~L.}\binits{J.~L.}} \AND
\bauthor{\bsnm{Lee},~\bfnm{Sokbae}\binits{S.}}
(\byear{2007}).
\btitle{Nonparametric instrumental variables estimation of a quantile
regression model}.
\bjournal{Econometrica}
\bvolume{75}
\bpages{1191--1208}.
\bid{doi={10.1111/j.1468-0262.2007.00786.x}, issn={0012-9682}, mr={2333498}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Imbens}{1997}]{Imb97}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Imbens},~\bfnm{G.}\binits{G.}}
(\byear{1997}).
\btitle{Book review of `The foundations of econometric analysis,' by
David Hendry and Mary Morgan}.
\bjournal{J. Appl. Econometrics}
\bvolume{12}
\bpages{91--94}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Imbens}{2000}]{Imb00}
\begin{barticle}[mr]
\bauthor{\bsnm{Imbens},~\bfnm{Guido~W.}\binits{G.~W.}}
(\byear{2000}).
\btitle{The role of the propensity score in estimating dose-response
functions}.
\bjournal{Biometrika}
\bvolume{87}
\bpages{706--710}.
\bid{doi={10.1093/biomet/87.3.706}, issn={0006-3444}, mr={1789821}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Imbens}{2004}]{Imb04}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Imbens},~\bfnm{G.}\binits{G.}}
(\byear{2004}).
\btitle{Nonparametric estimation of average treatment effects under
exogeneity: A review}.
\bjournal{Rev. Econom. Statist.}
\bvolume{86}
\bpages{1--29}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Imbens}{2007}]{ImbN1}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Imbens},~\bfnm{G.}\binits{G.}}
(\byear{2007}).
\bhowpublished{Nonadditive models with endogenous regressors. In \textit{Advances in Economics and Econometrics}: \textit{Theory and Applications}, \textit{Ninth
World Congress}, \textit{Vol.} III (R. Blundell, W. Newey and T. Persson, eds.) 17--46. Cambridge Univ. Press, Cambridge}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Imbens}{2010}]{Imb10}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Imbens},~\bfnm{G.}\binits{G.}}
(\byear{2010}).
\btitle{Better LATE than nothing: Some comments on Deaton (2009) and
Heckman and Urzua (2009)}.
\bjournal{Journal of Economic Literature}
\bvolume{48}
\bpages{399--423}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Imbens}{2014}]{ImbN2}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Imbens},~\bfnm{G.}\binits{G.}}
(\byear{2014}).
\bhowpublished{Matching in practice. \textit{Journal of Human
Resources}. To appear.}
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Imbens and Angrist}{1994}]{ImbAng94}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Imbens},~\bfnm{G.}\binits{G.}} \AND
\bauthor{\bsnm{Angrist},~\bfnm{J.}\binits{J.}}
(\byear{1994}).
\btitle{Identification and estimation of local average treatment effects}.
\bjournal{Econometrica}
\bvolume{61}
\bpages{467--476}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Imbens and Newey}{2009}]{ImbNew09}
\begin{barticle}[mr]
\bauthor{\bsnm{Imbens},~\bfnm{Guido~W.}\binits{G.~W.}} \AND
\bauthor{\bsnm{Newey},~\bfnm{Whitney~K.}\binits{W.~K.}}
(\byear{2009}).
\btitle{Identification and estimation of triangular simultaneous
equations models without additivity}.
\bjournal{Econometrica}
\bvolume{77}
\bpages{1481--1512}.
\bid{doi={10.3982/ECTA7108}, issn={0012-9682}, mr={2561069}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Imbens and Rosenbaum}{2005}]{ImbRos05}
\begin{barticle}[mr]
\bauthor{\bsnm{Imbens},~\bfnm{Guido~W.}\binits{G.~W.}} \AND
\bauthor{\bsnm{Rosenbaum},~\bfnm{Paul~R.}\binits{P.~R.}}
(\byear{2005}).
\btitle{Robust, accurate confidence intervals with a weak instrument:
Quarter of birth and education}.
\bjournal{J. Roy. Statist. Soc. Ser. A}
\bvolume{168}
\bpages{109--126}.
\bid{doi={10.1111/j.1467-985X.2004.00339.x}, issn={0964-1998}, mr={2113230}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Imbens and Rubin}{1997a}]{ImbRub97N1}
\begin{barticle}[mr]
\bauthor{\bsnm{Imbens},~\bfnm{Guido~W.}\binits{G.~W.}} \AND
\bauthor{\bsnm{Rubin},~\bfnm{Donald~B.}\binits{D.~B.}}
(\byear{1997}a).
\btitle{Bayesian inference for causal effects in randomized
experiments with noncompliance}.
\bjournal{Ann. Statist.}
\bvolume{25}
\bpages{305--327}.
\bid{doi={10.1214/aos/1034276631}, issn={0090-5364}, mr={1429927}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Imbens and Rubin}{1997b}]{ImbRub97N2}
\begin{barticle}[mr]
\bauthor{\bsnm{Imbens},~\bfnm{Guido~W.}\binits{G.~W.}} \AND
\bauthor{\bsnm{Rubin},~\bfnm{Donald~B.}\binits{D.~B.}}
(\byear{1997}b).
\btitle{Estimating outcome distributions for compliers in instrumental
variables models}.
\bjournal{Rev. Econom. Stud.}
\bvolume{64}
\bpages{555--574}.
\bid{doi={10.2307/2971731}, issn={0034-6527}, mr={1485828}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Imbens and Rubin}{2014}]{ImbRub}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Imbens},~\bfnm{G.}\binits{G.}} \AND
\bauthor{\bsnm{Rubin},~\bfnm{D.}\binits{D.}}
(\byear{2014}).
\bhowpublished{\textit{Causal Inference for Statistics, Social and
Biomedical Sciences: An Introduction}. Cambridge Univ.
Press, Cambridge}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Imbens and Wooldridge}{2009}]{ImbWoo09}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Imbens},~\bfnm{G.}\binits{G.}} \AND
\bauthor{\bsnm{Wooldridge},~\bfnm{J.}\binits{J.}}
(\byear{2009}).
\btitle{Recent developments in the econometrics of program evaluation}.
\bjournal{Journal of Economic Literature}
\bvolume{47}
\bpages{5--86}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Kitagawa}{2009}]{Kit09}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Kitagawa},~\bfnm{T.}\binits{T.}}
(\byear{2009}).
\bhowpublished{Identification region of the potential outcome distributions
under instrument independence. Manuscript, Dept. Economics, Univ. College London}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Kleibergen}{2002}]{Kle02}
\begin{barticle}[mr]
\bauthor{\bsnm{Kleibergen},~\bfnm{Frank}\binits{F.}}
(\byear{2002}).
\btitle{Pivotal statistics for testing structural parameters in
instrumental variables regression}.
\bjournal{Econometrica}
\bvolume{70}
\bpages{1781--1803}.
\bid{doi={10.1111/1468-0262.00353}, issn={0012-9682}, mr={1925156}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Koles\'a et~al.}{2013}]{K13}
\begin{bmisc}[auto:STB|2014/08/04|07:23:14]
\bauthor{\bsnm{Koles\'a},~\bfnm{M.}\binits{M.}},
\bauthor{\bsnm{Chetty},~\bfnm{R.}\binits{R.}},
\bauthor{\bsnm{Friedman},~\bfnm{J.~N.}\binits{J.~N.}},
\bauthor{\bsnm{Glaeser},~\bfnm{E.}\binits{E.}} \AND
\bauthor{\bsnm{Imbens},~\bfnm{G.~W.}\binits{G.~W.}}
(\byear{2013}).
\bhowpublished{Identification and inference with many invalid instruments. Unpublished manuscript.}
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Kunitomo}{1980}]{K80}
\begin{barticle}[mr]
\bauthor{\bsnm{Kunitomo},~\bfnm{Naoto}\binits{N.}}
(\byear{1980}).
\btitle{Asymptotic expansions of the distributions of estimators in a linear functional relationship and simultaneous equations}.
\bjournal{J. Amer. Statist. Assoc.}
\bvolume{75}
\bpages{693--700}.
\bid{issn={0162-1459}, mr={0590703}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Lauritzen and Richardson}{2002}]{LauRic02}
\begin{barticle}[mr]
\bauthor{\bsnm{Lauritzen},~\bfnm{Steffen~L.}\binits{S.~L.}} \AND
\bauthor{\bsnm{Richardson},~\bfnm{Thomas~S.}\binits{T.~S.}}
(\byear{2002}).
\btitle{Chain graph models and their causal interpretations}.
\bjournal{J. R. Stat. Soc. Ser. B Stat. Methodol.}
\bvolume{64}
\bpages{321--361}.
\bid{doi={10.1111/1467-9868.00340}, issn={1369-7412}, mr={1924296}}
\bptnote{check related}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Leamer}{1981}]{LeaN1}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Leamer},~\bfnm{E.}\binits{E.}}
(\byear{1981}).
\bhowpublished{Is it a demand curve, or is it a supply curve?
Partial identification through inequality constraints. \textit{Rev. Econom. Statist.} \textbf{63}
319--327}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Little}{1985}]{L85}
\begin{barticle}[auto:STB|2014/08/04|07:23:14]
\bauthor{\bsnm{Little},~\bfnm{R.}\binits{R.}}
(\byear{1985}).
\btitle{A note about models for selectivity bias}.
\bjournal{Econometrica}
\bvolume{53}
\bpages{1469--1474}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Little and Rubin}{1987}]{LitRub87}
\begin{bbook}[mr]
\bauthor{\bsnm{Little},~\bfnm{Roderick~J.~A.}\binits{R.~J.~A.}}
\AND
\bauthor{\bsnm{Rubin},~\bfnm{Donald~B.}\binits{D.~B.}}
(\byear{1987}).
\btitle{Statistical Analysis with Missing Data}.
\bpublisher{Wiley},
\blocation{New York}.
\bid{mr={0890519}}
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Little and Yau}{1998}]{LitYau98}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Little},~\bfnm{R.}\binits{R.}} \AND
\bauthor{\bsnm{Yau},~\bfnm{L.}\binits{L.}}
(\byear{1998}).
\btitle{Statistical techniques for analyzing data from prevention
trials: Treatment of no-shows using Rubin's causal model}.
\bjournal{Psychological Methods}
\bvolume{3}
\bpages{147--159}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Manski}{1990}]{Man90}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Manski},~\bfnm{C.}\binits{C.}}
(\byear{1990}).
\btitle{Nonparametric bounds on treatment effects}.
\bjournal{American Economic Review: Papers and Proceedings}
\bvolume{80}
\bpages{319--323}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Manski}{1995}]{Man95}
\begin{bbook}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Manski},~\bfnm{C.}\binits{C.}}
(\byear{1995}).
\btitle{Identification Problems in the Social Sciences}.
\bpublisher{Harvard Univ. Press},
\blocation{Cambridge}.
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Manski}{2000a}]{Man00N1}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Manski},~\bfnm{C.}\binits{C.}}
(\byear{2000}a).
\btitle{Economic analysis of social interactions}.
\bjournal{Journal of Economic Perspectives}
\bvolume{14}
\bpages{115--136}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Manski}{2000b}]{Man00N2}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Manski},~\bfnm{C.}\binits{C.}}
(\byear{2000}b).
\btitle{Identification problems and decisions under ambiguity:
Empirical analysis of treatment response and normative analysis of
treatment choice}.
\bjournal{J. Econometrics}
\bvolume{95}
\bpages{415--442}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Manski}{2001}]{Man01}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Manski},~\bfnm{C.}\binits{C.}}
(\byear{2001}).
\btitle{Designing programs for heterogenous populations: The value of
covariate information}.
\bjournal{American Economic Review: Papers and Proceedings}
\bvolume{91}
\bpages{103--106}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Manski}{2002}]{Man02}
\begin{barticle}[mr]
\bauthor{\bsnm{Manski},~\bfnm{Charles~F.}\binits{C.~F.}}
(\byear{2002}).
\btitle{Treatment choice under ambiguity induced by inferential problems}.
\bjournal{J. Statist. Plann. Inference}
\bvolume{105}
\bpages{67--82}.
\bid{doi={10.1016/S0378-3758(01)00204-X}, issn={0378-3758}, mr={1911559}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Manski}{2003}]{Man03}
\begin{bbook}[mr]
\bauthor{\bsnm{Manski},~\bfnm{Charles~F.}\binits{C.~F.}}
(\byear{2003}).
\btitle{Partial Identification of Probability Distributions}.
\bpublisher{Springer},
\blocation{New York}.
\bid{mr={2151380}}
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Manski}{2004}]{Man04}
\begin{barticle}[mr]
\bauthor{\bsnm{Manski},~\bfnm{Charles~F.}\binits{C.~F.}}
(\byear{2004}).
\btitle{Statistical treatment rules for heterogeneous populations}.
\bjournal{Econometrica}
\bvolume{72}
\bpages{1221--1246}.
\bid{doi={10.1111/j.1468-0262.2004.00530.x}, issn={0012-9682}, mr={2064712}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Manski}{2005}]{Man05}
\begin{bbook}[mr]
\bauthor{\bsnm{Manski},~\bfnm{Charles~F.}\binits{C.~F.}}
(\byear{2005}).
\btitle{Social Choice with Partial Knowledge of Treatment Response}.
\bpublisher{Princeton Univ. Press},
\blocation{Princeton, NJ}.
\bid{mr={2178946}}
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Manski}{2007}]{Man07}
\begin{bbook}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Manski},~\bfnm{C.}\binits{C.}}
(\byear{2007}).
\btitle{Identification for Prediction and Decision}.
\bpublisher{Princeton Univ. Press},
\blocation{Princeton, NJ}.
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Manski and Nagin}{1998}]{ManNag98}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Manski},~\bfnm{C.}\binits{C.}} \AND
\bauthor{\bsnm{Nagin},~\bfnm{D.}\binits{D.}}
(\byear{1998}).
\bhowpublished{Bounding disagreements about treatment effects: A case study
of sentencing and recidivism. \textit{Sociological Methodology} \textbf{28} 99--137}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Manski and Pepper}{2000}]{ManPep00}
\begin{barticle}[mr]
\bauthor{\bsnm{Manski},~\bfnm{Charles~F.}\binits{C.~F.}} \AND
\bauthor{\bsnm{Pepper},~\bfnm{John~V.}\binits{J.~V.}}
(\byear{2000}).
\btitle{Monotone instrumental variables: With an application to the
returns to schooling}.
\bjournal{Econometrica}
\bvolume{68}
\bpages{997--1010}.
\bid{doi={10.1111/1468-0262.00144}, issn={0012-9682}, mr={1771587}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Manski et~al.}{1992}]{Manetal92N1}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Manski},~\bfnm{C.}\binits{C.}},
\bauthor{\bsnm{Sandefur},~\bfnm{G.}\binits{G.}},
\bauthor{\bsnm{McLanahan},~\bfnm{S.}\binits{S.}} \AND
\bauthor{\bsnm{Powers},~\bfnm{D.}\binits{D.}}
(\byear{1992}).
\btitle{Alternative estimates of the effect of family structure during
adolescence on high school}.
\bjournal{J. Amer. Statist. Assoc.}
\bvolume{87}
\bpages{25--37}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Martens et~al.}{2006}]{Maretal06}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Martens},~\bfnm{E.}\binits{E.}},
\bauthor{\bsnm{Pestman},~\bfnm{W.}\binits{W.}},
\bauthor{\bparticle{de} \bsnm{Boer},~\bfnm{A.}\binits{A.}},
\bauthor{\bsnm{Belitser},~\bfnm{S.}\binits{S.}} \AND
\bauthor{\bsnm{Klungel},~\bfnm{O.}\binits{O.}}
(\byear{2006}).
\btitle{Instrumental variables: Application and limitations}.
\bjournal{Epidemiology}
\bvolume{17}
\bpages{260--267}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Matzkin}{2003}]{Mat03}
\begin{barticle}[mr]
\bauthor{\bsnm{Matzkin},~\bfnm{Rosa~L.}\binits{R.~L.}}
(\byear{2003}).
\btitle{Nonparametric estimation of nonadditive random functions}.
\bjournal{Econometrica}
\bvolume{71}
\bpages{1339--1375}.
\bid{doi={10.1111/1468-0262.00452}, issn={0012-9682}, mr={2000250}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Matzkin}{2007}]{Mat07}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Matzkin},~\bfnm{R.}\binits{R.}}
(\byear{2007}).
\bhowpublished{Nonparametric identification. In \textit{Handbook of Econometrics} 6B
(J. Heckman and E. Leamer, eds.). North-Holland, Amsterdam.}
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Matzkin}{2008}]{Mat08}
\begin{barticle}[mr]
\bauthor{\bsnm{Matzkin},~\bfnm{Rosa~L.}\binits{R.~L.}}
(\byear{2008}).
\btitle{Identification in nonparametric simultaneous equations models}.
\bjournal{Econometrica}
\bvolume{76}
\bpages{945--978}.
\bid{doi={10.3982/ECTA5940}, issn={0012-9682}, mr={2455118}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{McClellan and Newhouse}{1994}]{McCNew94}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{McClellan},~\bfnm{M.}\binits{M.}} \AND
\bauthor{\bsnm{Newhouse},~\bfnm{J.~P.}\binits{J.~P.}}
(\byear{1994}).
\btitle{Does more intensive treatment of acute myocardial infarction
in the elderly reduce mortality}.
\bjournal{Journal of the American Medical Association}
\bvolume{272}
\bpages{859--866}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{McDonald, Hiu and
Tierney}{1992}]{McDHiuTie92}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{McDonald},~\bfnm{C.}\binits{C.}},
\bauthor{\bsnm{Hiu},~\bfnm{S.}\binits{S.}} \AND
\bauthor{\bsnm{Tierney},~\bfnm{W.}\binits{W.}}
(\byear{1992}).
\btitle{Effects of computer reminders for influenza vaccination on
morbidity during influenza epidemics}.
\bjournal{MD Computing}
\bvolume{9}
\bpages{304--312}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Moreira}{2003}]{Mor03}
\begin{barticle}[mr]
\bauthor{\bsnm{Moreira},~\bfnm{Marcelo~J.}\binits{M.~J.}}
(\byear{2003}).
\btitle{A conditional likelihood ratio test for structural models}.
\bjournal{Econometrica}
\bvolume{71}
\bpages{1027--1048}.
\bid{doi={10.1111/1468-0262.00438}, issn={0012-9682}, mr={1995822}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Morgan and Winship}{2007}]{MorWin07}
\begin{bbook}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Morgan},~\bfnm{S.}\binits{S.}} \AND
\bauthor{\bsnm{Winship},~\bfnm{C.}\binits{C.}}
(\byear{2007}).
\btitle{Counterfactuals and Causal Inference}.
\bpublisher{Cambridge Univ. Press},
\blocation{Cambridge}.
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Morimune}{1983}]{M83}
\begin{barticle}[mr]
\bauthor{\bsnm{Morimune},~\bfnm{Kimio}\binits{K.}}
(\byear{1983}).
\btitle{Approximate distributions of {$k$}-class estimators when the degree of overidentifiability is large compared with the sample size}.
\bjournal{Econometrica}
\bvolume{51}
\bpages{821--841}.
\bid{doi={10.2307/1912160}, issn={0012-9682}, mr={0712372}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Newey and Powell}{2003}]{N03}
\begin{barticle}[mr]
\bauthor{\bsnm{Newey},~\bfnm{Whitney~K.}\binits{W.~K.}} \AND
\bauthor{\bsnm{Powell},~\bfnm{James~L.}\binits{J.~L.}}
(\byear{2003}).
\btitle{Instrumental variable estimation of nonparametric models}.
\bjournal{Econometrica}
\bvolume{71}
\bpages{1565--1578}.
\bid{doi={10.1111/1468-0262.00459}, issn={0012-9682}, mr={2000257}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Pearl}{2000}]{Pea00}
\begin{bbook}[mr]
\bauthor{\bsnm{Pearl},~\bfnm{Judea}\binits{J.}}
(\byear{2000}).
\btitle{Causality: Models, Reasoning, and Inference}.
\bpublisher{Cambridge Univ. Press},
\blocation{Cambridge}.
\bid{mr={1744773}}
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Pearl}{2011}]{Pea11}
\begin{barticle}[mr]
\bauthor{\bsnm{Pearl},~\bfnm{Judea}\binits{J.}}
(\byear{2011}).
\btitle{Principal stratification---A goal or a tool?}
\bjournal{Int. J. Biostat.}
\bvolume{7}
\bpages{Art. 20}.
\bid{doi={10.2202/1557-4679.1322}, issn={1557-4679}, mr={2787410}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Permutt and Hebel}{1989}]{PerHeb89}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Permutt},~\bfnm{T.}\binits{T.}} \AND
\bauthor{\bsnm{Hebel},~\bfnm{J.}\binits{J.}}
(\byear{1989}).
\btitle{Simultaneous-equation estimation in a clinical trial of the
effect of smoking on birth weight}.
\bjournal{Biometrics}
\bvolume{45}
\bpages{619--622}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Philipson}{1997a}]{Phi97N1}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Philipson},~\bfnm{T.}\binits{T.}}
(\byear{1997}a).
\btitle{The evaluation of new health care technology: The labor
economics of statistics}.
\bjournal{J. Econometrics}
\bvolume{76}
\bpages{375--396}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Philipson}{1997b}]{Phi97N2}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Philipson},~\bfnm{T.}\binits{T.}}
(\byear{1997}b).
\btitle{Data markets and the production of surveys}.
\bjournal{Rev. Econom. Stud.}
\bvolume{64}
\bpages{47--73}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Philipson and DeSimone}{1997}]{PhiDeS97}
\begin{barticle}[mr]
\bauthor{\bsnm{Philipson},~\bfnm{Tomas}\binits{T.}} \AND
\bauthor{\bsnm{DeSimone},~\bfnm{Jeffrey}\binits{J.}}
(\byear{1997}).
\btitle{Experiments and subject sampling}.
\bjournal{Biometrika}
\bvolume{84}
\bpages{619--630}.
\bid{doi={10.1093/biomet/84.3.619}, issn={0006-3444}, mr={1603928}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Philipson and Hedges}{1998}]{PhiHed98}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Philipson},~\bfnm{T.}\binits{T.}} \AND
\bauthor{\bsnm{Hedges},~\bfnm{L.}\binits{L.}}
(\byear{1998}).
\btitle{Subject evaluation in social experiments}.
\bjournal{Econometrica}
\bvolume{66}
\bpages{381--408}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Phillips}{1989}]{Phi89}
\begin{barticle}[mr]
\bauthor{\bsnm{Phillips},~\bfnm{P.~C.~B.}\binits{P.~C.~B.}}
(\byear{1989}).
\btitle{Partially identified econometric models}.
\bjournal{Econometric Theory}
\bvolume{5}
\bpages{181--240}.
\bid{doi={10.1017/S0266466600012408}, issn={0266-4666}, mr={1006540}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Plott and Smith}{1987}]{PloSmi87}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Plott},~\bfnm{C.}\binits{C.}} \AND
\bauthor{\bsnm{Smith},~\bfnm{V.}\binits{V.}}
(\byear{1987}).
\btitle{An experimental examination of two exchange institutions}.
\bjournal{Rev. Econom. Stud.}
\bvolume{45}
\bpages{133--153}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Pratt}{1984}]{Pra}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Pratt},~\bfnm{J.}\binits{J.}} \AND
\bauthor{\bsnm{Shlaifer},~\bfnm{R.}\binits{R.}}
(\byear{1984}).
\bhowpublished{On the nature and discovery of structure. \textit{J. Amer. Statist. Assoc.} \textbf{79}
9--21}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Ramsahai and Lauritzen}{2011}]{RamLau11}
\begin{barticle}[mr]
\bauthor{\bsnm{Ramsahai},~\bfnm{R.~R.}\binits{R.~R.}} \AND
\bauthor{\bsnm{Lauritzen},~\bfnm{S.~L.}\binits{S.~L.}}
(\byear{2011}).
\btitle{Likelihood analysis of the binary instrumental variable model}.
\bjournal{Biometrika}
\bvolume{98}
\bpages{987--994}.
\bid{doi={10.1093/biomet/asr040}, issn={0006-3444}, mr={2860338}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Richardson, Evans and
Robins}{2011}]{RicEvaRob11}
\begin{bincollection}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Richardson},~\bfnm{T.}\binits{T.}},
\bauthor{\bsnm{Evans},~\bfnm{R.}\binits{R.}} \AND
\bauthor{\bsnm{Robins},~\bfnm{J.}\binits{J.}}
(\byear{2011}).
\btitle{Transparent parametrizations of models for potential outcomes}.
In \bbooktitle{Bayesian Statistics 9}
(\beditor{\bfnm{Bernardo}\binits{B.}~\bsnm{Bayarri}},
\beditor{\bfnm{Dawid}\binits{D.}~\bsnm{Berger}} \AND
\beditor{\bfnm{SmithWest}\binits{S.}~\bsnm{Heckerman}}, eds.).
\bpublisher{Oxford Univ. Press},
\blocation{Oxford}.
\end{bincollection}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Richardson and Robins}{2013}]{RicRob}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Richardson},~\bfnm{T.}\binits{T.}} \AND
\bauthor{\bsnm{Robins},~\bfnm{J.}\binits{J.}}
(\byear{2013}).
\bhowpublished{Single World Intervention Graphs (SWIGs):
A unification of the counterfactual and graphical approaches to causality.
Working Paper 128, Center for Statistics and the Social Sciences, Univ.
Washington, Seattle, WA}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Robins}{1986}]{Rob86}
\begin{barticle}[mr]
\bauthor{\bsnm{Robins},~\bfnm{James}\binits{J.}}
(\byear{1986}).
\btitle{A new approach to causal inference in mortality studies with a
sustained exposure period---Application to control of the healthy
worker survivor effect}.
\bjournal{Math. Modelling}
\bvolume{7}
\bpages{1393--1512}.
\bid{doi={10.1016/0270-0255(86)90088-6}, issn={0270-0255}, mr={0877758}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Robins}{1989}]{Rob}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Robins},~\bfnm{James~M.}\binits{J.~M.}}
(\byear{1989}).
\bhowpublished{The analysis of randomized and non-randomized AIDS
treatment trials using a new approach to causal inference in
longitudinal studies. In \textit{Health Service Research Methodology: A
Focus on AIDS} (L. Sechrest, H. Freeman and A. Bailey, eds.).
NCHSR, U.S. Public Health
Service, Washington, DC}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Robins}{1994}]{Rob94}
\begin{barticle}[mr]
\bauthor{\bsnm{Robins},~\bfnm{James~M.}\binits{J.~M.}}
(\byear{1994}).
\btitle{Correcting for non-compliance in randomized trials using
structural nested mean models}.
\bjournal{Comm. Statist. Theory Methods}
\bvolume{23}
\bpages{2379--2412}.
\bid{doi={10.1080/03610929408831393}, issn={0361-0926}, mr={1293185}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Robins and Greenland}{1996}]{RobGre96}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Robins},~\bfnm{J.}\binits{J.}} \AND
\bauthor{\bsnm{Greenland},~\bfnm{S.}\binits{S.}}
(\byear{1996}).
\btitle{Comment on: Identification of causal effects using
instrumental variables}.
\bjournal{J. Amer. Statist. Assoc.}
\bvolume{91}
\bpages{456--468}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Robins and Rotnitzky}{2004}]{RobRot04}
\begin{barticle}[mr]
\bauthor{\bsnm{Robins},~\bfnm{James}\binits{J.}} \AND
\bauthor{\bsnm{Rotnitzky},~\bfnm{Andrea}\binits{A.}}
(\byear{2004}).
\btitle{Estimation of treatment effects in randomised trials with
non-compliance and a dichotomous outcome using structural mean models}.
\bjournal{Biometrika}
\bvolume{91}
\bpages{763--783}.
\bid{doi={10.1093/biomet/91.4.763}, issn={0006-3444}, mr={2126032}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Roehrig}{1988}]{Roe88}
\begin{barticle}[mr]
\bauthor{\bsnm{Roehrig},~\bfnm{Charles~S.}\binits{C.~S.}}
(\byear{1988}).
\btitle{Conditions for identification in nonparametric and parametric models}.
\bjournal{Econometrica}
\bvolume{56}
\bpages{433--447}.
\bid{doi={10.2307/1911080}, issn={0012-9682}, mr={0935634}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Rosenbaum}{1996}]{Ros96}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Rosenbaum},~\bfnm{P.}\binits{P.}}
(\byear{1996}).
\btitle{Comment on: Identification of causal effects using
instrumental variables}.
\bjournal{J. Amer. Statist. Assoc.}
\bvolume{91}
\bpages{465--468}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Rosenbaum}{2002}]{Ros02}
\begin{bbook}[mr]
\bauthor{\bsnm{Rosenbaum},~\bfnm{Paul~R.}\binits{P.~R.}}
(\byear{2002}).
\btitle{Observational Studies},
\bedition{2nd} ed.
\bpublisher{Springer},
\blocation{New York}.
\bid{doi={10.1007/978-1-4757-3692-2}, mr={1899138}}
\bptnote{check year}
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Rosenbaum}{2010}]{Ros10}
\begin{bbook}[mr]
\bauthor{\bsnm{Rosenbaum},~\bfnm{Paul~R.}\binits{P.~R.}}
(\byear{2010}).
\btitle{Design of Observational Studies}.
\bpublisher{Springer},
\blocation{New York}.
\bid{doi={10.1007/978-1-4419-1213-8}, mr={2561612}}
\bptnote{check year}
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Rosenbaum and Rubin}{1983}]{RosRub83}
\begin{barticle}[mr]
\bauthor{\bsnm{Rosenbaum},~\bfnm{Paul~R.}\binits{P.~R.}} \AND
\bauthor{\bsnm{Rubin},~\bfnm{Donald~B.}\binits{D.~B.}}
(\byear{1983}).
\btitle{The central role of the propensity score in observational
studies for causal effects}.
\bjournal{Biometrika}
\bvolume{70}
\bpages{41--55}.
\bid{doi={10.1093/biomet/70.1.41}, issn={0006-3444}, mr={0742974}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Roy}{1951}]{Roy51}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Roy},~\bfnm{A.}\binits{A.}}
(\byear{1951}).
\btitle{Some thoughts on the distribution of earnings}.
\bjournal{Oxford Economics Papers}
\bvolume{3}
\bpages{135--146}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Rubin}{1974}]{Rub74}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Rubin},~\bfnm{D.}\binits{D.}}
(\byear{1974}).
\btitle{Estimating causal effects of treatments in randomized and
non-randomized studies}.
\bjournal{Journal of Educational Psychology}
\bvolume{66}
\bpages{688--701}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Rubin}{1976}]{Rub76}
\begin{barticle}[mr]
\bauthor{\bsnm{Rubin},~\bfnm{Donald~B.}\binits{D.~B.}}
(\byear{1976}).
\btitle{Inference and missing data}.
\bjournal{Biometrika}
\bvolume{63}
\bpages{581--592}.
\bid{issn={0006-3444}, mr={0455196}}
\bptnote{check related}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Rubin}{1978}]{Rub78}
\begin{barticle}[mr]
\bauthor{\bsnm{Rubin},~\bfnm{Donald~B.}\binits{D.~B.}}
(\byear{1978}).
\btitle{Bayesian inference for causal effects: The role of randomization}.
\bjournal{Ann. Statist.}
\bvolume{6}
\bpages{34--58}.
\bid{issn={0090-5364}, mr={0472152}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Rubin}{1987}]{Rub87}
\begin{bbook}[mr]
\bauthor{\bsnm{Rubin},~\bfnm{Donald~B.}\binits{D.~B.}}
(\byear{1987}).
\btitle{Multiple Imputation for Nonresponse in Surveys}.
\bpublisher{Wiley},
\blocation{New York}.
\bid{doi={10.1002/9780470316696}, mr={0899519}}
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Rubin}{1990}]{Rub90}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Rubin},~\bfnm{D.~B.}\binits{D.~B.}}
(\byear{1990}).
\btitle{Formal modes of statistical inference for causal effects}.
\bjournal{J. Statist. Plann. Inference}
\bvolume{25}
\bpages{279--292}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Rubin}{1996}]{Rub96}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Rubin},~\bfnm{D.~B.}\binits{D.~B.}}
(\byear{1996}).
\btitle{Multiple imputation after 18$+$ years}.
\bjournal{J.~Amer. Statist. Assoc.}
\bvolume{91}
\bpages{473--489}.
\bnote{-292}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Rubin}{2006}]{Rub06}
\begin{bbook}[mr]
\bauthor{\bsnm{Rubin},~\bfnm{Donald~B.}\binits{D.~B.}}
(\byear{2006}).
\btitle{Matched Sampling for Causal Effects}
(D. B. Rubin, ed.).
\bpublisher{Cambridge Univ. Press},
\blocation{Cambridge}.
\bid{doi={10.1017/CBO9780511810725}, mr={2307965}}
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Rubin and Thomas}{1992}]{RubTho92}
\begin{barticle}[mr]
\bauthor{\bsnm{Rubin},~\bfnm{Donald~B.}\binits{D.~B.}} \AND
\bauthor{\bsnm{Thomas},~\bfnm{Neal}\binits{N.}}
(\byear{1992}).
\btitle{Affinely invariant matching methods with ellipsoidal distributions}.
\bjournal{Ann. Statist.}
\bvolume{20}
\bpages{1079--1093}.
\bid{doi={10.1214/aos/1176348671}, issn={0090-5364}, mr={1165607}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Sargan}{1958}]{Sar58}
\begin{barticle}[mr]
\bauthor{\bsnm{Sargan},~\bfnm{J.~D.}\binits{J.~D.}}
(\byear{1958}).
\btitle{The estimation of economic relationships using instrumental variables.}
\bjournal{Econometrica}
\bvolume{26}
\bpages{393--415}.
\bid{issn={0012-9682}, mr={0110567}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Shapley and Shubik}{1977}]{ShaShu77}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Shapley},~\bfnm{L.}\binits{L.}} \AND
\bauthor{\bsnm{Shubik},~\bfnm{M.}\binits{M.}}
(\byear{1977}).
\btitle{Trade using one commodity as a means of payment}.
\bjournal{Journal of Political Economy}
\bvolume{85}
\bpages{937--968}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Small}{2007}]{Sma07}
\begin{barticle}[mr]
\bauthor{\bsnm{Small},~\bfnm{Dylan~S.}\binits{D.~S.}}
(\byear{2007}).
\btitle{Sensitivity analysis for instrumental variables regression
with overidentifying restrictions}.
\bjournal{J. Amer. Statist. Assoc.}
\bvolume{102}
\bpages{1049--1058}.
\bid{doi={10.1198/016214507000000608}, issn={0162-1459}, mr={2411664}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Smith}{1982}]{Smi}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Smith},~\bfnm{V.}\binits{V.}}
(\byear{1982}).
\bhowpublished{Markets as economizers of information: Experimental
examination of the Hayek hypothesis. \textit{Economic
Inquiry} \textbf{20} 165--179}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Sommer and Zeger}{1991}]{SomZeg91}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Sommer},~\bfnm{A.}\binits{A.}} \AND
\bauthor{\bsnm{Zeger},~\bfnm{S.}\binits{S.}}
(\byear{1991}).
\btitle{On estimating efficacy from clinical trials}.
\bjournal{Stat. Med.}
\bvolume{10}
\bpages{45--52}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Splawa-Neyman}{1990}]{Spl90}
\begin{barticle}[mr]
\bauthor{\bsnm{Splawa-Neyman},~\bfnm{Jerzy}\binits{J.}}
(\byear{1990}).
\btitle{On the application of probability theory to agricultural experiments. {E}ssay on principles. {S}ection~9}.
\bjournal{Statist. Sci.}
\bvolume{5}
\bpages{465--472}.
\bnote{Translated from the Polish and edited by D. M. D{\polhk{a}}browska and T. P. Speed}.
\bid{issn={0883-4237}, mr={1092986}}
\bptnote{check related}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Staiger and Stock}{1997}]{StaSto97}
\begin{barticle}[mr]
\bauthor{\bsnm{Staiger},~\bfnm{Douglas}\binits{D.}} \AND
\bauthor{\bsnm{Stock},~\bfnm{James~H.}\binits{J.~H.}}
(\byear{1997}).
\btitle{Instrumental variables regression with weak instruments}.
\bjournal{Econometrica}
\bvolume{65}
\bpages{557--586}.
\bid{doi={10.2307/2171753}, issn={0012-9682}, mr={1445622}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Stock and Trebbi}{2003}]{StoTre03}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Stock},~\bfnm{J.}\binits{J.}} \AND
\bauthor{\bsnm{Trebbi},~\bfnm{F.}\binits{F.}}
(\byear{2003}).
\btitle{Who invented instrumental variable regression?}
\bjournal{Journal of Economic Perspectives}
\bvolume{17}
\bpages{177--194}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Stock and Watson}{2010}]{StoWat10}
\begin{bbook}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Stock},~\bfnm{J.}\binits{J.}} \AND
\bauthor{\bsnm{Watson},~\bfnm{M.}\binits{M.}}
(\byear{2010}).
\btitle{Introduction to Econometrics},
\bedition{3rd} ed.
\bpublisher{Addison-Wesley},
\blocation{Reading, MA}.
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Strotz}{1960}]{Str60}
\begin{barticle}[mr]
\bauthor{\bsnm{Strotz},~\bfnm{Robert~H.}\binits{R.~H.}}
(\byear{1960}).
\btitle{Interdependence as a specification error}.
\bjournal{Econometrica}
\bvolume{28}
\bpages{428--442}.
\bid{issn={0012-9682}, mr={0120035}}
\bptnote{check year}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Strotz and Wold}{1960}]{StrWol60}
\begin{barticle}[mr]
\bauthor{\bsnm{Strotz},~\bfnm{Robert~H.}\binits{R.~H.}} \AND
\bauthor{\bsnm{Wold},~\bfnm{H.~O.~A.}\binits{H.~O.~A.}}
(\byear{1960}).
\btitle{Recursive vs. nonrecursive systems: An attempt at synthesis}.
\bjournal{Econometrica}
\bvolume{28}
\bpages{417--427}.
\bid{issn={0012-9682}, mr={0120034}}
\bptnote{check year}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Strotz and Wold}{1965}]{StrWol65}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Strotz},~\bfnm{R.}\binits{R.}} \AND
\bauthor{\bsnm{Wold},~\bfnm{H.}\binits{H.}}
(\byear{1965}).
\btitle{The causal interpretability of structural parameters: A reply}.
\bjournal{Econometrica}
\bvolume{31}
\bpages{449--450}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Tan}{2006}]{Tan06}
\begin{barticle}[mr]
\bauthor{\bsnm{Tan},~\bfnm{Zhiqiang}\binits{Z.}}
(\byear{2006}).
\btitle{Regression and weighting methods for causal inference using
instrumental variables}.
\bjournal{J. Amer. Statist. Assoc.}
\bvolume{101}
\bpages{1607--1618}.
\bid{doi={10.1198/016214505000001366}, issn={0162-1459}, mr={2279483}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Tan}{2010}]{Tan10}
\begin{barticle}[mr]
\bauthor{\bsnm{Tan},~\bfnm{Zhiqiang}\binits{Z.}}
(\byear{2010}).
\btitle{Marginal and nested structural models using instrumental variables}.
\bjournal{J. Amer. Statist. Assoc.}
\bvolume{105}
\bpages{157--169}.
\bid{doi={10.1198/jasa.2009.tm08299}, issn={0162-1459}, mr={2757199}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Thistlewaite and Campbell}{1960}]{ThiCam60}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Thistlewaite},~\bfnm{D.}\binits{D.}} \AND
\bauthor{\bsnm{Campbell},~\bfnm{D.}\binits{D.}}
(\byear{1960}).
\btitle{Regression-discontinuity analysis: An alternative to the
ex-post facto experiment}.
\bjournal{Journal of Educational Psychology}
\bvolume{51}
\bpages{309--317}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Tinbergen}{1930}]{Tin30}
\begin{bmisc}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Tinbergen},~\bfnm{J.}\binits{J.}}
(\byear{1930}).
\bhowpublished{Bestimmung und Deuting von Angebots\-kurven. Ein Beispiel.
\textit{Zeitschrift fur Nationalokonomie} \textbf{1} 669--679. Translated
as:
Determination and interpretation of supply curves. An example.
In
\textit{The Foundations of Econometric Analysis}
(D. Hendryand and M. Morgan, eds.) 233--245.
Cambridge Univ. Press, Cambridge}.
\end{bmisc}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Van~der Laan and Robins}{2003}]{vanRob03}
\begin{bbook}[mr]
\bauthor{\bsnm{Van~der Laan},~\bfnm{Mark~J.}\binits{M.~J.}} \AND
\bauthor{\bsnm{Robins},~\bfnm{James~M.}\binits{J.~M.}}
(\byear{2003}).
\btitle{Unified Methods for Censored Longitudinal Data and Causality}.
\bpublisher{Springer},
\blocation{New York}.
\bid{doi={10.1007/978-0-387-21700-0}, mr={1958123}}
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Vansteelandt et~al.}{2011}]{Vanetal11}
\begin{barticle}[mr]
\bauthor{\bsnm{Vansteelandt},~\bfnm{Stijn}\binits{S.}},
\bauthor{\bsnm{Bowden},~\bfnm{Jack}\binits{J.}},
\bauthor{\bsnm{Babanezhad},~\bfnm{Manoochehr}\binits{M.}} \AND
\bauthor{\bsnm{Goetghebeur},~\bfnm{Els}\binits{E.}}
(\byear{2011}).
\btitle{On instrumental variables estimation of causal odds ratios}.
\bjournal{Statist. Sci.}
\bvolume{26}
\bpages{403--422}.
\bid{doi={10.1214/11-STS360}, issn={0883-4237}, mr={2917963}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Vansteelandt and
Goetghebeur}{2003}]{VanGoe03}
\begin{barticle}[mr]
\bauthor{\bsnm{Vansteelandt},~\bfnm{S.}\binits{S.}} \AND
\bauthor{\bsnm{Goetghebeur},~\bfnm{E.}\binits{E.}}
(\byear{2003}).
\btitle{Causal inference with generalized structural mean models}.
\bjournal{J. R. Stat. Soc. Ser. B Stat. Methodol.}
\bvolume{65}
\bpages{817--835}.
\bid{doi={10.1046/j.1369-7412.2003.00417.x}, issn={1369-7412}, mr={2017872}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Wold}{1960}]{Wol60}
\begin{barticle}[mr]
\bauthor{\bsnm{Wold},~\bfnm{H.~O.~A.}\binits{H.~O.~A.}}
(\byear{1960}).
\btitle{A generalization of causal chain models}.
\bjournal{Econometrica}
\bvolume{28}
\bpages{443--463}.
\bid{issn={0012-9682}, mr={0120036}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Wooldridge}{2008}]{Woo08}
\begin{bbook}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Wooldridge},~\bfnm{J.}\binits{J.}}
(\byear{2008}).
\btitle{Introductory Econometrics}.
\bpublisher{South-Western College Pub., New York}.
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Wooldridge}{2010}]{Woo10}
\begin{bbook}[mr]
\bauthor{\bsnm{Wooldridge},~\bfnm{Jeffrey~M.}\binits{J.~M.}}
(\byear{2010}).
\btitle{Econometric Analysis of Cross Section and Panel Data},
\bedition{2nd} ed.
\bpublisher{MIT Press},
\blocation{Cambridge, MA}.
\bid{mr={2768559}}
\bptnote{check year}
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Working}{1927}]{Wor27}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Working},~\bfnm{E.}\binits{E.}}
(\byear{1927}).
\btitle{What do statistical `demand curves' show?}
\bjournal{Quarterly Journal of Economics}
\bvolume{41}
\bpages{212--235}.
\end{barticle}
\bptok{imsref}
\endbibitem\
\bibitem[\operatorname{pr}otect\citeauthoryear{Wright}{1928}]{Wri28}
\begin{bbook}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Wright},~\bfnm{P.}\binits{P.}}
(\byear{1928}).
\btitle{The Tariff on Animal and Vegetable Oils}.
\bpublisher{MacMillan},
\blocation{New York}.
\end{bbook}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Yau and Little}{2001}]{YauLit01}
\begin{barticle}[mr]
\bauthor{\bsnm{Yau},~\bfnm{Linda~H.~Y.}\binits{L.~H.~Y.}} \AND
\bauthor{\bsnm{Little},~\bfnm{Roderick~J.}\binits{R.~J.}}
(\byear{2001}).
\btitle{Inference for the complier-average causal effect from
longitudinal data subject to noncompliance and missing data, with
application to a job training assessment for the unemployed}.
\bjournal{J. Amer. Statist. Assoc.}
\bvolume{96}
\bpages{1232--1244}.
\bid{doi={10.1198/016214501753381887}, issn={0162-1459}, mr={1973667}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Zelen}{1979}]{Zel79}
\begin{barticle}[pbm]
\bauthor{\bsnm{Zelen},~\bfnm{M.}\binits{M.}}
(\byear{1979}).
\btitle{A new design for randomized clinical trials}.
\bjournal{N.~Engl. J. Med.}
\bvolume{300}
\bpages{1242--1245}.
\bid{doi={10.1056/NEJM197905313002203}, issn={0028-4793}, pmid={431682}}
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Zelen}{1990}]{Zel90}
\begin{barticle}[auto:STB|2014/06/18|12:29:53]
\bauthor{\bsnm{Zelen},~\bfnm{M.}\binits{M.}}
(\byear{1990}).
\btitle{Randomized consent designs for clinical trials: An update}.
\bjournal{Stat. Med.}
\bvolume{9}
\bpages{645--656}.
\end{barticle}
\bptok{imsref}
\endbibitem
\bibitem[\operatorname{pr}otect\citeauthoryear{Zhang, Rubin and Mealli}{2009}]{ZhaRubMea09}
\begin{barticle}[mr]
\bauthor{\bsnm{Zhang},~\bfnm{Junni~L.}\binits{J.~L.}},
\bauthor{\bsnm{Rubin},~\bfnm{Donald~B.}\binits{D.~B.}} \AND
\bauthor{\bsnm{Mealli},~\bfnm{Fabrizia}\binits{F.}}
(\byear{2009}).
\btitle{Likelihood-based analysis of causal effects of job-training
programs using principal stratification}.
\bjournal{J. Amer. Statist. Assoc.}
\bvolume{104}
\bpages{166--176}.
\bid{doi={10.1198/jasa.2009.0012}, issn={0162-1459}, mr={2663040}}
\end{barticle}
\bptok{imsref}
\endbibitem
\end{thebibliography}
\end{document} |
\begin{document}
\title{Transitive sensitive subsystems for interval~maps
ootnotetext{2000 Mathematics Subject Classification: 37E05.}
\begin{abstract}
We state that for continuous interval maps the existence of a non empty
closed invariant subset which is transitive and
sensitive to initial conditions is implied by positive topological
entropy and implies chaos in the sense of Li-Yorke, and we exhibit
examples showing that these three notions are distinct.
\end{abstract}
\section{Introduction}
In this paper an {\em interval map} is a topological
dynamical system given by a continuous map $f\colon I\to I$ where $I$
is a compact interval. In the literature much has been said about chaos
for interval maps. The point is that the relations between the various
properties related to chaos are much more numerous for these systems than
for general dynamical systems. As a consequence there is a rather ordered
``scale of chaos'' on the interval. For example, for interval maps
topological weak mixing and topological strong mixing are equivalent
\cite{BM}, and transitivity implies sensitivity to initial conditions
\cite{BM2}, which in turn implies positive topological entropy \cite{Blo3}.
For more
details on this topic see e.g. \cite{BCop},
\cite[\S\S 6-9]{KS} and \cite{R3}.
Among the different definitions of chaos, a well known one is
{\em chaos in the sense of Li-Yorke}. It follows the ideas of \cite{LY}
but was formalised later.
\begin{defi}
Let $T\colon X\to X$ be a continuous map on the metric space $X$,
$d$ denoting the distance. The map $T$ is said {\em chaotic
in the sense of Li-Yorke} if there exists an uncountable set $S\subset X$
such that, for all $x,y\in S$, $x\not=y$, one has
$$
\limsup_{n\to+\infty}d(T^n(x),T^n(y))>0\quad\mbox{and}\quad
\liminf_{n\to+\infty}d(T^n(x),T^n(y))=0.
$$
\end{defi}
Note that in the definition of chaos in the sense of Li-Yorke
some people make the extra assumption that
for all $x\in S$ and all periodic points $z\in X$ one has
$\limsup_{n\to+\infty}d(T^n(x),T^n(z))>0$. This gives an equivalent
definition since this property is satisfied by all but at most one points
of the set $S$ \cite[p 144]{BCop}.
Li and Yorke showed that an interval map with a periodic point of
period $3$ is chaotic in the sense of Li-Yorke \cite{LY}. In
\cite{JS} Jankov\'a and Sm\'{\i}tal generalised this result as follows:
\begin{theo}[Jankov\'a-Sm\'{\i}tal]\label{theo:htop-positive-chaos-LY}
If $f\colon I\to I$ is an interval map of positive entropy, then it is
chaotic in the sense of Li-Yorke.
\end{theo}
Recently, Blanchard, Glasner, Kolyada
and Maass proved that, if \mbox{$T\colon X\to X$} is
a continuous map on the compact metric space $X$ such that the topological
entropy of $T$ is positive, then
the system is chaotic in the sense of Li-Yorke \cite{BGKM}.
The converse of this result is not true, even for interval maps:
Sm\'{\i}tal \cite{Smi} and Xiong
\cite{Xio3} built interval maps of zero entropy which are chaotic in
the sense of Li-Yorke. See also \cite{MSmi} (a correction is given in
\cite{Jim}) or \cite{Du2} for examples of a
$C^{\infty}$ interval map which is chaotic in the sense of Li-Yorke
and has a null entropy.
\par
Recall that
the map $T\colon X\to X$ is {\em transitive} if for all non empty open subsets
$U,V$ there exists an integer $n\geq 0$ such that $T^{-n}(U)\cap V\not=
\emptyset$; if $X$ is compact with no isolated point, $T$ is transitive
if and only if there exists $x\in X$ such that $\omega(x,T)=X$ (where
$\omega(x,T)$ is the set of limit points of $\{T^n(x)\mid n\geq 0\}$).
The map $T$ has {\em sensitive dependence to initial conditions}
(or simply
is {\em sensitive}) if there exists $\delta>0$ such that for all $x\in X$ and
all neighbourhoods $U$ of $x$ there exist $y\in U$ and $n\geq 0$ such that
$d(T^n(x),T^n(y))\geq \delta$. A subset $Y\subset X$ is {\em invariant}
if $T(Y)\subset Y$.
The work of Wiggins \cite{Wig} leads to the following definition
(see, e.g., \cite{FD}).
\begin{defi}
Let $X$ be a metric space. The continuous map $T\colon X\to X$
is said {\em chaotic in the sense of Wiggins} if there exists a non empty
closed invariant subset $Y$ such that the restriction $T|_Y$ is transitive
and sensitive.
\end{defi}
The aim of this paper is to locate this notion with
respect to the other definitions of chaos.
\begin{rem}
A continuous map $T\colon X\to X$ which is transitive and sensitive is
sometimes called {\em chaotic in the sense of Auslander-Yorke} \cite{AY}.
If in addition the periodic points are dense, then it is called
{\em chaotic in the sense of Devaney} \cite{Dev}.
\end{rem}
Transitive sensitive subsystems appear naturally when considering a horseshoe,
that is, two disjoint closed intervals $J,K$ such that
$f(J)\cap f(K)\supset J\cup K$, because the points the orbits of which
never escape from $J\cup K$ form a subset on which $f$ acts almost like
a $2$-shift \cite{Bloc3}. For interval maps, positive entropy is equivalent
to the
existence of a horseshoe for some power of $f$ \cite{Mis2,BGMY}
(see also \cite[chap. VIII]{BCop}), thus one can deduce that
a positive entropy interval map has a transitive, sensitive subsystem.
More precisely, Shihai Li proved the following result \cite{Li}.
\begin{theo}[Shihai Li]\label{theo:htop-wiggins}
Let $f\colon I\to I$ be an interval map.
The topological entropy of $f$ is positive if and only if
there exists a non empty closed invariant
subset $X\subset I$ such that $f|_X$ is transitive, sensitive
to initial conditions and the periodic points are dense in $X$
(in other words, $f|_X$ is Devaney chaotic).
\end{theo}
In the ``if'' part of this theorem one cannot suppress the assumption on
the periodic points. In Section~\ref{sec:wiggins-htop0} we build a
counter-example, which leads to the following theorem.
\begin{theo}\label{theo:wiggins-htop0}
There exists a continuous map $f\colon [0,1]\to [0,1]$
of zero topological entropy which is chaotic in the sense of Wiggins.
\end{theo}
In \cite{Smi} Sm\'{\i}tal built a zero entropy map $f$ which is chaotic in the
sense of Li-Yorke. If one looks at the construction of $f$, it is not
hard to prove that $f|_{\omega(0,f)}$ is transitive and sensitive to
initial conditions. We show next theorem in Section~\ref{sec:wiggins-LY}.
\begin{theo}\label{theo:Wiggins-implies-LY}
Let $f\colon I\to I$ be an interval map.
If $f$ is Wiggins chaotic then it is Li-Yorke chaotic.
\end{theo}
The converse of this theorem is not true, contrary to what one may expect
by considering Sm\'{\i}tal's example. The last and longest section is
devoted to the construction of a counter-example that proves the
following result.
\begin{theo}\label{theo:LY-not-wiggins}
There exists a continuous interval map $g\colon I\to I$ which is
chaotic in the sense of Li-Yorke but not in the sense of Wiggins.
\end{theo}
From Theorems \ref{theo:htop-wiggins}, \ref{theo:wiggins-htop0},
\ref{theo:Wiggins-implies-LY}, \ref{theo:LY-not-wiggins} it follows
that, for interval maps, chaos in the sense of Wiggins is a strictly
intermediate notion between positive entropy and chaos in the sense of
Li-Yorke.
\par
Furthermore the examples of Sections \ref{sec:wiggins-htop0} and
\ref{sec:LY-not-Wiggins} show that the behaviours of zero entropy interval
maps are more varied that one might expect. Let us expose the different
kinds of dynamics exhibited by these maps.
The next result is well known (see, e.g., \cite[p218]{BCop}).
\begin{theo}\label{theo:htop-power-of-2}
Let $f\colon I\to I$ be an interval map. The following properties
are equivalent:
\begin{itemize}
\item the topological entropy of $f$ is zero,
\item every periodic point has a period equal to $2^n$ for some integer
$n\geq 0$.
\end{itemize}
\end{theo}
According to Sharkovskii's Theorem \cite{Sha} the set of periods of periodic
points of a zero entropy interval map is either $\{2^k; 0\leq k\leq n\}$
for some integer $n$ and $f$ is said of type $2^n$, or
$\{2^k;k\geq 0\}$ and $f$ is said of type $2^{\infty}$.
There is little to say about the dynamics of type $2^n$, and some
interval maps of type $2^{\infty}$ share almost the same dynamics \cite{Del}:
every orbit converges to some periodic orbit of period $2^k$;
these maps are never Li-Yorke chaotic.
The interval maps of type $2^{\infty}$
that admit an infinite $\omega$-limit set may be Li-Yorke chaotic or
not, as shown by Sm\'{\i}tal \cite{Smi}. A map $f$ that is not Li-Yorke chaotic
is called ``uniformly non-chaotic'' in \cite{BCop} and it satisfies the
following property: every point $x$ is approximately periodic, that is,
for every $\varepsilon>0$ there exists a periodic point $y$ and an integer $N$
such that $|f^n(x)-f^n(y)[<\varepsilon$ for all $n\geq N$.
The maps built in Sections \ref{sec:wiggins-htop0} and
\ref{sec:LY-not-Wiggins} are both zero entropy and Li-Yorke
chaotic. In the first example there is a transitive sensitive subsystem
which is the core of the dynamics; in particular Li-Yorke chaos can be
read on this subsystem. In the second example this situation does
not occur since there is no transitive sensitive subsystem.
\section{Wiggins chaos implies Li-Yorke chaos}
\label{sec:wiggins-LY}
The following notion of $f$-non separable points was introduced by
Sm\'{\i}tal to give an equivalent condition for chaos in the sense of Li-Yorke
\cite{Smi}. Note that Theorem~\ref{theo:htop0-chaos-LY} was proven to
remain valid for all interval maps by Jankov\'a and Sm\'{\i}tal~\cite{JS}.
\begin{defi}
Let $f\colon I\to I$ be an interval map and $a_0,a_1$ two distinct
points in $I$. The points $a_0, a_1$ are called {\em $f$-separable} if
there exist two disjoint subintervals $J_0,J_1$ and two integers
$n_0,n_1$, such that for $i=0,1$, $a_i\in J_i$, $f^{n_i}(J_i)=J_i$ and
$(f^k(J_i))_{0\leq k<n_i}$ are disjoint. Otherwise they are said {\em
$f$-non separable}.
\end{defi}
\begin{theo}[Sm\'{\i}tal]\label{theo:htop0-chaos-LY}
Let $f\colon I\to I$ be an interval map of zero entropy. The following
properties are equivalent:
\begin{itemize}
\item $f$ is chaotic in the sense of Li-Yorke,
\item there exists $x_0\in I$ such that the set $\omega(x_0,f)$ is
infinite and contains two $f$-non separable points.
\end{itemize}
\end{theo}
In the proof of this theorem,
Sm\'{\i}tal showed the following intermediate
result which describes the structure of an infinite $\omega$-limit set
of a zero entropy map.
\begin{lem}\label{lem:htop0-Lki}
Let $f\colon I\to I$ be an interval map of zero entropy and $x_0\in I$
such that $\omega(x_0,f)$ is infinite. For all $n\geq 0$
and $0\leq i<2^n$, define
$$
I_n^i=[\min \omega(f^i(x_0),f^{2^n}), \max\omega(f^i(x_0),f^{2^n})]
\quad\mbox{and}\quad
L_n^i=\bigcup_{k\geq 0}f^{k2^n}(I_n^i).
$$
Then
$f(L_k^i)=L_k^{i+1\bmod 2^k}$ for all $0\leq i< 2^k$,
and the intervals $(L_k^i)_{0\leq i< 2^k}$ are pairwise disjoint.
\end{lem}
\begin{lem}\label{lem:fixed-point}
Let $f\colon [a,b]\to \mathbb{R}$ be a continuous map. If
$f([a,b])\supset [a,b]$, then $f$ has a fixed point.
\end{lem}
\begin{demo}
There exist $x,y\in [a,b]$ such that
$f(x)\leq a$ and $f(y)\geq b$. One has then
$f(x)-x\leq a-x\leq 0$ and $f(y)-y\geq b-y\geq 0$,
thus there is a point $c\in [x,y]$ such that $f(c)-c=0$.
\end{demo}
\begin{lem}\label{lem:periodic-interval-power-of-2}
Let $f\colon I\to I$ be an interval map of zero entropy. If $J\subset I$ is a
(non necessarily closed) subinterval such that $f^p(J)=J$ and
$(f^i(J))_{0\leq i<p}$ are pairwise disjoint then $p$ is a power of $2$.
\end{lem}
\begin{demo}
If $J$ is reduced to one point then it is a periodic orbit and by
Theorem~\ref{theo:htop-power-of-2} $p$ is a power of $2$. We assume that
$J$ is non degenerate, which implies that
$f^n(J)$ is a non degenerate interval for all $n\geq 0$.
One has $f^p(\overline{J})=\overline{J}$ thus by Lemma~\ref{lem:fixed-point}
there exists $x\in \overline{J}$ such that $f^p(x)=x$. According to
Theorem~\ref{theo:htop-power-of-2} the period of $x$ is equal to $2^k$
for some $k$; write $p=m2^k$. If $x\in J$ then $(f^i(x))_{0\leq i<p}$ are
distinct and $p=2^k$.
Suppose that $m\geq 3$. Then $x\in \partial J$; we assume
that $x=\sup J$, the case with $x=\inf J$ being
symmetric. One has $x=f^{2^k}(x)\in f^{2^k}(\overline{J})$ and
$f^{2^k}(J)\cap J=\emptyset$ thus $x=\inf f^{2^k}(J)$. But one also has
$x\in f^{2^{k+1}}(\overline{J})$, which contradicts the fact that
$J,f^{2^k}(J), f^{2^{k+1}}(J)$ are pairwise disjoint non degenerate intervals.
Therefore $m=1$ or $2$ and $p$ is a power of $2$.
\end{demo}
The following result is the key tool in the proof of
Theorem~\ref{theo:Wiggins-implies-LY}. A rather similar result
can be found in a paper of Fedorenko, Sharkovskii and Sm{\'\i}tal~\cite{FSS}.
\begin{lem}\label{lem:LY-sensitive}
Let $f\colon I\to I$ be an interval map of zero entropy and $x_0$ in $I$
such that $\omega(x_0,f)$ is infinite and does not contain two $f$-non
separable points. Then
for all $\varepsilon>0$ there exists $\delta>0$ such that, if $x,y\in
\omega(x_0,f)$, $|x-y|<\delta$, then $|f^n(x)-f^n(y)|<\varepsilon$ for all
$n\geq 0$.
\end{lem}
\begin{demo}
Let $X=\omega(x_0,f)$. For all integers $n\geq 0$
and $0\leq i<2^n$ define $a_n^i=\min\omega(f^i(x_0),f^{2^n})$
and $b_n^i=\max\omega(f^i(x_0),f^{2^n})$. Define $I_n^i$ and $L_n^i$
as in Lemma~\ref{lem:htop0-Lki}; one has $I_n^i=[a_n^i,b_n^i]$. The
points $a_n^i, b_n^i$ belong to $X$ and
\begin{equation}\label{eq:nested-In}
I_{n+1}^i\cup I_{n+1}^{i+2^n}\subset I_n^i\ \mbox{ for all }\ 0\leq i<2^n.
\end{equation}
Suppose that there exists $\varepsilon>0$ such that
\begin{equation}
\label{eq:In>eps}
\mbox{for all }n\geq 0\mbox{ there is }0\leq i<2^n\mbox{ with }
|I_n^i|\geq\varepsilon.
\end{equation}
Using Equation~(\ref{eq:nested-In}) we can build a sequence $(i_n)_{n\geq
0}$ such that $I_{n+1}^{i_{n+1}}\subset I_n^{i_n}$ and
$|I_n^{i_n}|\geq \varepsilon$ for all $n\geq 0$. Define $I_{\infty}=\bigcap_{n\geq 1}
I_n^{i_n}$. It is a decreasing intersection of compact
intervals thus $I_{\infty}$ is
a closed interval and $|I_{\infty}|\geq \varepsilon$. Write $I_{\infty}=[a,b]$; then
$$
a=\lim_{n\to+\infty} a_n^{i_n}\quad\mbox{and}\quad
b=\lim_{n\to+\infty}b_n^{i_n},
$$
thus $a,b\in X$.
One has $a,b\in L_n^{i_n}$ for all $n\geq 0$.
By Lemma~\ref{lem:htop0-Lki}
the intervals $L_{n+2}^{i_{n+2}}$, $f^{2^n}(L_{n+2}^{i_{n+2}})$,
$f^{2^{n+1}}(L_{n+2}^{i_{n+2}})$ are pairwise disjoint thus
$\{a,f^{2^n}(a),f^{2^{n+1}}(a)\}$ are distinct.
One has $a\not=b$ and by assumption
$a,b$ are $f$-separable, thus there exist an interval
$J$ and an integer $p\geq 1$ such that $a\in J$, $b\not\in J$,
$f^p(J)=J$ and $(f^i(J))_{0\leq i<p}$ are pairwise disjoint.
By Lemma~\ref{lem:periodic-interval-power-of-2}
$p$ is a power of $2$; write $p=2^k$.
Define the interval $K=L_k^{i_k}\cap J$; $K$ contains $a$ and
$f^{2^k}(K)\subset K$ because $f^{2^k}(L_k^{i_k})=L_k^{i_k}$ by
Lemma~\ref{lem:htop0-Lki}. Thus $K$ contains $\{a,f^{2^k}(a),
f^{2^{k+1}}(a)\}$. These three points belong to $\omega(x_0,f)$ and
are distinct, so one of them belongs to $\Int{K}$
and there exists an integer $n$ such that
$f^n(x_0)\in K$. We have then
$$
X=\omega(x_0,f)\subset \bigcup_{j=0}^{2^k-1} f^j(\overline{K}).
$$
Let $b'\in X$ such that $f^{2^{k+1}}(b')=b$ and
let $0\leq j<2^k$ such that $b'\in f^j(\overline{K})$. The points
$b', f^{2^k}(b')$ and $f^{2^{k+1}}(b')$ belong to $f^j(\overline{K})$ and
they are distinct (same proof as for $a$)
thus one of them belongs to $f^j(K)$, which implies that
$b\in f^j(K)$. One has $j\not=0$ because $b\not\in J$ and
$K\subset J$. But on the
other hand $b\in f^j(L_k^{i_k})\cap L_k^{i_k}$ which is empty by
Lemma~\ref{lem:htop0-Lki}, thus we get a contradiction.
We deduce that Equation~(\ref{eq:In>eps}) is false.
Let $\varepsilon>0$; the negation of Equation~(\ref{eq:In>eps}) implies that
there exists $n\geq 0$ such
that $|I_n^i|<\varepsilon$ for all $0\leq i<2^n$.
Let $\delta>0$ be the minimal distance
between two distinct intervals among $(I_n^i)_{0\leq i<2^n}$. If
$x,y\in X$ with $|x-y|<\delta$ then there exists $0\leq i<2^n$ such
that $x,y\in I_n^i\cap \omega(x_0,f)=\omega(f^i(x_0),f^{2^n})$,
thus for all $k\geq 0$
one has $f^k(x),f^k(y)\in \omega(f^{i+k}(x_0),f^{2^n})\subset
I_n^{i+k\bmod 2^n}$, which implies that $|f^k(x)-f^k(y)|<\varepsilon$.
\end{demo}
Now we are ready to prove
\par
\noindent{\bf Theorem \ref{theo:Wiggins-implies-LY} }
Let $f\colon I\to I$ be an interval map.
If $f$ is Wiggins chaotic then it is Li-Yorke chaotic.
\par
\begin{demo}
We show the result by refutation.
Suppose that $f$ is not chaotic in the sense of Li-Yorke. By
Theorem~\ref{theo:htop-positive-chaos-LY} one has $h_{top}(f)=0$.
Consider a closed invariant subset $Y\subset I$ such that $f|_Y$ is
transitive. If $Y$ is finite or has an isolated point then
$f|_Y$ is not sensitive. If $Y$ is infinite with no isolated point,
there exists $x_0\in Y$ such that
$\omega(x_0,f)=Y$. By Theorem~\ref{theo:htop0-chaos-LY} $Y$ does not
contain two $f$-non separable points, thus
by Lemma~\ref{lem:LY-sensitive} $f|_Y$ is not
sensitive.
\end{demo}
\section{Wiggins chaos does not imply positive entropy}
\label{sec:wiggins-htop0}
We are going to build an interval map of zero entropy which is chaotic in
the sense of Wiggins. It resembles the maps built by Sm\'{\i}tal
(map $f$ in \cite{Smi}) and Delahaye (map $g$ in \cite{Del}),
however we give full details because this construction will
be used as a basis for the next example.
\par
\noindent{\bf Notation. } If $I$ is an interval, let
${\rm mid}(I)$ denote the middle of $I$. If $f$ is a linear map, let
${\rm slope}(f)$ denote its constant slope. We write $\uparrow$ (resp.
$\downarrow$) for ``increasing'' (resp. ``decreasing'').
\par
Let $(a_n)_{n\geq 0}$ be an increasing sequence of numbers less
that $1$ such that $a_0=0$. Define
$I_0^1=[a_0,1]$ and, for all $n\geq 1$,
$$
I_n^0=[a_{2n-2},a_{2n-1}],\ L_n=[a_{2n-1},a_{2n}],\ I_n^1=[a_{2n},1].
$$
One has $I_n^0\cup L_n\cup I_n^1=I_{n-1}^1$.
We fix $(a_n)_{n\geq 0}$ such that the lengths of the intervals
$I_n^0, I_n^1$ satisfy:
\begin{itemize}
\item if $n$ is odd, $|I_n^0|=\frac{1}{3^n}|I_{n-1}^1|$
and $|I_n^1|=\left(1-\frac{2}{3^n}\right)|I_{n-1}^1|$,
\item if $n$ is even, $|I_n^0|=\left(1-\frac{2}{3^n}\right)|I_{n-1}^1|$ and
$|I_n^1|=\frac{1}{3^n}|I_{n-1}^1|$.
\end{itemize}
This implies that $|L_n|=\frac{1}{3^n}|I_{n-1}^1|$ for all $n\geq 1$.
Note that $|I_n^1|\to 0$, that is, $\lim_{n\to +\infty} a_n=1$; hence
$\bigcup_{n\geq 1}(I_n^0\cup L_n)=[0,1)$.
For all $n\geq 1$, let $\varphi_n\colon I_n^0\to I_n^1$ be the increasing linear
homeomorphism mapping $I_n^0$ onto $I_n^1$; the slope of $\varphi_n$ is
${\rm slope}(\varphi_n)=\frac{|I_n^1|}{|I_n^0|}$. Define the map $f\colon
[0,1]\to [0,1]$ such that $f$ is continuous on $[0,1)$ and
\begin{eqnarray*}
&& f(x)=\varphi_1^{-1}\circ
\varphi_2^{-1}\circ\cdots\circ\varphi_{n-1}^{-1}\circ \varphi_n(x)\mbox{ for all }
x\in I_n^0,\ n\geq 1,\\
&& f|_{L_n} \mbox{ is linear for all }n\geq
1,\\
&&f(1)=0.
\end{eqnarray*}
Note that $f|_{I_n^0}$ is linear $\uparrow$.
We will show below that $f$ is continuous at $1$.
Let us explain the underlying construction. At step $n=1$ the
interval $I_1^0$ is sent linearly onto $I_1^1$ (hence
$f|_{I_1^0}=\varphi_1$) and we decide that $f(I_1^1)\subset I_1^0$ (grey
area on Figure~\ref{fig:wiggins-htop0-partial-construction}). Then we do the same
kind of construction in the grey area with respect to
$I_2^0,I_2^1\subset I_1^1$: we rescale $I_2^0,I_2^1$ as
$\varphi_1^{-1}(I_2^0),\varphi_1^{-1}(I_2^1) \subset I_1^0$ (on the vertical
axis) then we send linearly $I_2^0$ onto $\varphi_1^{-1}(I_2^1)$; in this
way $f|_{I_2^0}=\varphi_1^{-1}\circ\varphi_2$. We repeat this construction on
$I_2^1$ (black area), and so on. Finally we fill the gaps in a linear
way and we get the whole map, which is
pictured on the right side of
Figure~\ref{fig:wiggins-htop0-partial-construction}.
\begin{figure}
\caption{The first steps of the construction of $f$ (left) and
the graph of $f$ (right). This map has a zero entropy and
the invariant set $\omega(0,f)$ is transitive and sensitive.
\label{fig:wiggins-htop0-partial-construction}
\label{fig:wiggins-htop0-partial-construction}
\end{figure}
\par
Let $J_0^0=[0,1]$ and for all $n\geq 1$ define the subintervals
$J_n^0,J_n^1\subset J_{n-1}^0$ by
$\min J_n^0=0$, $\max J_n^1=\max J_{n-1}^0$ and
$\displaystyle \frac{|J_n^i|}{|J_{n-1}^0|}=\frac{|I_n^i|}{|I_{n-1}^1|}$ for $i=0,1$.
\par
To show that $f$ is continuous at $1$, it is enough to prove that
$\max (f|_{I_n^1})$ tends to $0$ when $n$ goes to infinity.
For all $n\geq 1$ one has
\begin{eqnarray*}
\varphi_n(\max I_n^0)&=&\max I_n^1=1=\min I_{n-1}^1+|I_{n-1}^1|\\
\varphi_{n-1}^{-1}\circ \varphi_n(\max I_n^0)&=&
\min I_{n-1}^0+|I_{n-1}^1|{\rm slope} (\varphi_{n-1}^{-1})\\
&=&\min I_{n-2}^1+|I_{n-1}^1|{\rm slope} (\varphi_{n-1}^{-1})\\
\varphi_{n-2}^{-1}\circ \varphi_{n-1}^{-1}\circ\varphi_n(\max I_n^0)&=&
\min I_{n-2}^0+|I_{n-1}^1|{\rm slope} (\varphi_{n-2}^{-1}){\rm slope} (\varphi_{n-1}^{-1})\\
&\vdots &
\end{eqnarray*}
\begin{eqnarray*}
\varphi_1^{-1}\circ \varphi_2^{-1}\circ\cdots\circ
\varphi_{n-1}^{-1}\circ\varphi_n(\max I_n^0)&=&
\displaystyle \min I_1^0+|I_{n-1}^1|\prod_{i=1}^{n-1}{\rm slope}(\varphi_i^{-1})\\
&=&\prod_{i=1}^{n-1}\frac{|I_i^0|}{|I_{i-1}^1|}=|J_{n-1}^0|
\end{eqnarray*}
Consequently,
\begin{equation}\label{eq:wiggins-htop0-max-f(In0)}
f(\max I_n^0)=|J_{n-1}^0|=\max J_{n-1}^0.
\end{equation}
According to the definition of $f$, one has $\max (f|_{I_{n-1}^1})=
f(\max I_n^0)$, thus $\max (f|_{I_{n-1}^1})=|J_{n-1}^0|$.
By definition, $|J_{n-1}^0|\leq\frac{1}{3^{n-2}}$, which tends to $0$,
therefore $f$ is continuous at $1$.
\par
Next Lemma describes the action of $f$ on the intervals $(J_n^i)$ and
$(I_n^i)$ and collects the properties that we will use later.
The interval $I_n^1$ is periodic
of period $2^n$ and the map $f^{2^{n-1}}$ swaps $I_n^0$ and $I_n^1$,
however we prefer to deal with $J_n^0=f(I_n^1)$; this will simplify
the proofs because $f|_{I_n^1}$
is not monotone whereas $f^i|_{J_n^0}$ is linear for all
$1\leq i\leq 2^n-1$.
\begin{lem}\label{lem:wiggins-htop0-summary-Jn0}
Let $f$ be the map defined above. Then, for all
$n\geq 1$,
\begin{enumerate}
\item $f(I_n^1)=J_n^0$,
\item $f(I_n^0)=J_n^1$,
\item $f^i|_{J_n^0}$ is linear $\uparrow$ for all $1\leq i\leq 2^n-1$,
\item $f^{2^{n-1}-1}(J_n^0)=I_n^0$ and $f^{2^n-1}(J_n^0)=I_n^1$,
\item $f^i(J_n^0)\subset \bigcup_{1\leq k\leq n} I_k^0$ for all
$0\leq i\leq 2^n-2$,
\item $\left(f^i(J_n^0)\right)_{0\leq i<2^n}$ are pairwise disjoint;
\end{enumerate}
and the previous points also imply
\begin{enumerate}
\addtocounter{enumi}{6}
\item $f^{2^{n-1}}(J_n^0)=J_n^1$,
\item $f^{2^n}(J_n^0)=J_n^0$,
\item $f^{2^{n-1}}|_{I_n^0}$ is linear $\uparrow$ and
$f^{2^{n-1}}(I_n^0)=I_n^1$,
\item $f^{2^{n-1}}(I_n^1)=I_n^0$,
\item $(f^i(I_n^0))_{0\leq i<2^n}$ are pairwise disjoint and
$f^{2^n}(I_n^1)=I_n^1$.
\end{enumerate}
\end{lem}
\begin{demo}
According to Equation~(\ref{eq:wiggins-htop0-max-f(In0)}),
$\max (f|_{I_n^1})=f(\max I_{n+1}^0)=\max J_n^0$; moreover $f(1)=0=\min
J_n^0$. Thus $f(I_n^1)=J_n^0$ by continuity; this is the point~(i).
According to the definition of $f$,
\begin{eqnarray*}
|f(I_n^0)|&=&
|I_n^0| {\rm slope}(\varphi_n) \prod_{i=1}^{n-1}{\rm slope}(\varphi_i^{-1})\\
&= &|I_n^1|\prod_{i=1}^{n-1}\frac{|I_i^0|}{|I_i^1|}
=\frac{|I_n^1|}{|I_{n-1}^1|}\prod_{i=1}^{n-1}\frac{|I_i^0|}{|I_{i-1}^1|}\\
&=&|J_n^1|
\end{eqnarray*}
Moreover $f(\max I_n^0)=\max J_{n-1}^0=\max J_n^1$ according to
Equation~(\ref{eq:wiggins-htop0-max-f(In0)}),
thus $f(I_n^0)=J_n^1$. This gives the point~(ii).
\par
We show by induction on $n$ that the points (iii) and (iv) are satisfied.
\begin{itemize}
\item
This is true for $n=1$ because $J_1^0=I_1^0$, $J_1^1=I_1^1$,
$f|_{I_1^0}=\varphi_1$ is linear $\uparrow$ and $f(I_1^0)=I_1^1$.
\item
Suppose that the induction hypothesis is true for $n$. Since
$J_{n+1}^0\subset J_n^0$, the map $f^i|_{J_{n+1}^0}$ is linear $\uparrow$
for all $1\leq i\leq 2^n-1$ and $f^{2^n-1}(J_{n+1}^0)\subset I_n^1$;
moreover the linearity implies that
$$
\min f^{2^n-1}(J_{n+1}^0)=\min f^{2^n-1}(J_n^0)=\min I_n^1=\min I_{n+1}^0
$$
and
$$
\frac{|f^{2^n-1}(J_{n+1}^0)|}{|I_n^1|}=\frac{|J_{n+1}^0|}{|J_n^0|}
=\frac{|I_{n+1}^0|}{|I_n^1|}.
$$
Therefore $f^{2^n-1}(J_{n+1}^0)=I_{n+1}^0$.
Then $f^{2^n}(J_{n+1}^0)=J_{n+1}^1$ by the point~(ii). Since $J_{n+1}^1
\subset J_n^0$, the induction hypothesis applies: $f^i|_{J_{n+1}^1}$
is linear $\uparrow$ for all $1\leq i\leq 2^n-1$, $f^{2^n-1}(J_{n+1}^1)
\subset I_n^1$, and by linearity
$$
\max f^{2^n-1}(J_{n+1}^1)=\max f^{2^n-1}(J_n^0)=1=\max I_{n+1}^1
$$
and $|f^{2^n-1}(J_{n+1}^1)|=|I_{n+1}^1|$, hence $f^{2^{n+1}-1}(J_{n+1}^0)\!=\!
f^{2^n-1}(J_{n+1}^1)\!=\!I_{n+1}^1$. This gives the points (iii) and (iv)
for $n+1$.
\end{itemize}
Now we prove the point (v) by induction on $n$.
\begin{itemize}
\item
This is true for $n=1$ because $J_1^0=I_1^0$.
\item
Suppose that the induction hypothesis is true for $n$. Since $J_{n+1}^0
\subset J_n^0$ we have that $f^i(J_{n+1}^0)\subset \bigcup_{1\leq k\leq n}
I_k^n$ for all $0\leq i<2^n-1$. Moreover $f^{2^n-1}(J_{n+1}^0)=
I_{n+1}^0$ by the point~(iv) and $f^{2^n}(J_{n+1}^0)=f(I_{n+1}^0)=
J_{n+1}^1$ by the point~(ii). Since $J_{n+1}^1\subset J_n^0$ we can use
the induction hypothesis again and we get that
$f^{2^n+i}(J_{n+1}^0)\subset \bigcup_{1\leq k\leq n} I_k^n$ for all
$0\leq i<2^n-1$.
This gives the point~(v) for $n+1$.
\end{itemize}
Next we prove the point~(vi).
Suppose that $f^i(J_n^0)\cap f^j(J_n^0)\not=\emptyset$ for some
$0\leq i<j<2^n$. Then $f^{2^n-1-j}(f^i(J_n^0))\cap f^{2^n-1-j}(f^j(J_n^0))
\not=\emptyset$. But $f^{2^n-1}(J_n^0)=I_n^1$ by the point~(iv)
and $f^{2^n-1-(j-i)}(J_n^0)\subset [0,\max I_n^0]$ by the point~(v),
thus these two sets are disjoint, which is a contradiction.
We deduce that $(f^i(J_n^0))_{0\leq i<2^n}$ are pairwise disjoint.
\par
Finally we indicate how to obtain the other points from the previous ones.
The points (vii) and (viii) are implied respectively by (iv)+(ii) and
(iv)+(i). The point (ix) is implied by (iii)+(iv). The point (x)
is given by (i)+(iv). The point (xi) is given by the combination of
(i), (iv) and (vi).
\end{demo}
Define $K_n=\bigcup_{i\geq 0}f^i(I_n^1)$ for all $n\geq 0$ and $K=\bigcap_{n\geq 0} K_n$.
According to Lemma~\ref{lem:wiggins-htop0-summary-Jn0},
$K_n$ is the disjoint union of the intervals
$\left(f^i(J_n^0)\right)_{0\leq i\leq 2^n-1}$. The set $K$ has a
Cantor-like construction: at each step a middle part of every connected
component of $K_n$ is removed to get $K_{n+1}$. However $K$ is not a
Cantor set because its interior
is not empty (see Proposition~\ref{prop:wiggins-htop0-omega(0,f)}).
In Proposition~\ref{prop:wiggins-htop0-2-infty}
we state that the entropy of $f$ is null. Next we show
in Proposition~\ref{prop:wiggins-htop0-omega(0,f)} that the set $\omega(0,f)$
contains $\partial K$. Then we prove that $\omega(0,f)$ is
transitive and sensitive to initial conditions.
\begin{prop}\label{prop:wiggins-htop0-2-infty}
Let $f$ be the map defined above. Then $h_{top}(f)=0$.
\end{prop}
\begin{demo}
By definition the map $f|_{L_n}$ is linear decreasing thus
$f(L_n)$ is included in $[0,f(\max I_n^0)]$. Moreover $f(\max I_n^0)=\max J_{n-1}^0$
by Equation~(\ref{eq:wiggins-htop0-max-f(In0)}), thus $f(L_n)\subset J_{n-1}^0$.
Then Lemma~\ref{lem:wiggins-htop0-summary-Jn0}(iii) implies that
$f^{2^{n-1}}|_{L_n}$ is linear decreasing.
The map $f^{2^{n-1}}|_{I_n^0}$
is linear increasing and $f^{2^{n-1}}(I_n^0)=I_n^1$ by
Lemma~\ref{lem:wiggins-htop0-summary-Jn0}(ix),
thus $f^{2^{n-1}}(\min L_n)=\max I_n^1=1$;
moreover $f^{2^{n-1}}(I_n^1)=I_n^0$ by Lemma~\ref{lem:wiggins-htop0-summary-Jn0}(x),
thus $f^{2^{n-1}}(\max L_n)\in I_n^0$. We deduce that
$f^{2^{n-1}}(L_n)$ contains $L_n\cup I_n^1$, thus by Lemma~\ref{lem:fixed-point}
there exists $z_n\in L_n$ such that $f^{2^{n-1}}(z_n)=z_n$. The period of
$z_n$ is exactly $2^{n-1}$ because $L_n\subset I_{n-1}^1$ and the
intervals $(f^i(I_{n-1}^1))_{0\leq i<2^n}$ are pairwise disjoint
by Lemma~\ref{lem:wiggins-htop0-summary-Jn0}(xi).
By definition $|L_n|\leq |I_n^1|$ thus
${\rm slope}(f^{2^{n-1}}|_{L_n})\leq -2$.
If the points $x, f^{2^{n-1}}(x),\ldots, f^{k2^{n-1}}(x)$ belong to $L_n$ then
$|f^{(k+1)2^{n-1}}(x)-z_n|\geq 2^k|x-z_n|$ thus, for all $x\in L_n$,
$x\not=z_n$, there exists $k\geq 1$ such that
$f^{k2^{n-1}}(x)\not\in L_n$. Since $I_{n-1}^1=I_n^0\cup L_n\cup I_n^1$ and
$f^{2^{n-1}}(I_{n-1}^1)=I_{n-1}^1$
by Lemma~\ref{lem:wiggins-htop0-summary-Jn0}(xi),
this implies that $f^{k2^{n-1}}(x)\in I_n^0\cup I_n^1$. In addition
$f^{2^{n-1}}(I_n^0)=I_n^1$ by Lemma~\ref{lem:wiggins-htop0-summary-Jn0}(ix)
thus
$$
\forall x\in I_{n-1}^1,\ x\not=z_n,\ \exists k\geq 0,\
f^k(x)\in I_n^1.
$$
Starting with $I_0^1=[0,1]$, a straightforward induction shows that,
for all $x\in [0,1]$, if the orbit of $x$ does not meet
$\{z_n\mid n\geq 1\}$ then for all integers $n\geq 1$ there exists
$k\geq 0$ such that $f^k(x)\in I_n^1$; in particular
$\omega(x,f)\subset K$.
According to Lemma~\ref{lem:wiggins-htop0-summary-Jn0}(xi)
the set $K$ contains no periodic point because $K\subset \bigcup_{i\geq 0}f^i(I_n^1)$ for
all $n\geq 1$, thus
every periodic point is in the orbit of some $z_n$, therefore its
period is a power of $2$.
Finally, $h_{top}(f)=0$ by Theorem~\ref{theo:htop-power-of-2}.
\end{demo}
The orbit of $0$ obviously enters $f^i(J_n^0)$ for all $n\geq 0$ and
$0\leq i<2^n$, thus $\omega(0,f)$ meets all connected components of $K$.
We show in
next Lemma that $\omega(0,f)$ contains $\partial K$; the proof relies
on the idea that the smaller interval among $J_{n+1}^0$ and $J_{n+1}^1$
contains alternatively either $\min J_n^0$ or $\max J_n^0$ when $n$
varies, so that both endpoints of a connected component of $K$ can be
approximated by small intervals of the form $f^i(J_n^0)$.
\begin{prop}\label{prop:wiggins-htop0-omega(0,f)}
Let $f$ and $K$ be as defined above. Then
$\partial K\subset \omega(0,f)$. In particular $\omega(0,f)$ is
infinite and contains $0$, and $f|_{\omega(0,f)}$ is transitive.
\end{prop}
\begin{demo}
According to the definition of $K$, the connected components of $K$ are
exactly the non empty sets of the form $\bigcap_{n\geq 0}f^{j_n}(J_n^0)$
with $0\leq j_n<2^n$.
Let $y$ be a point in
$\partial K$. For all $n\geq 0$ there exists $0\leq j_n<2^n$
such that $y\in f^{j_n}(J_n^0)$, and there exists a sequence $(y_n)_{n\geq 0}$
such that $y=\lim_{n\to+\infty}y_n$ and $y_n\in\partial f^{j_n}(J_n^0)=
\{\min f^{j_n}(J_n^0),\max f^{j_n}(J_n^0)\}$. Let $\varepsilon>0$ and $N\geq 0$.
Let $n$ be an even integer such that $\frac{1}{3^{n+1}}<\varepsilon$ and
$|y_n-y|<\varepsilon$, and let $k\geq 0$ such that $k2^{n+1}\geq N$.
Firstly suppose that $y_n=\min f^{j_n}(J_n^0)$. The point $0$ belongs
to $J_{n+1}^0$ and $f^{2^{n+1}}(J_{n+1}^0)=J_{n+1}^0$ by
Lemma~\ref{lem:wiggins-htop0-summary-Jn0}(viii)
thus $f^{k2^{n+1}+j_n}(0)$ belongs to $f^{j_n}(J_{n+1}^0)$. By
Lemma~\ref{lem:wiggins-htop0-summary-Jn0}(iii) one has
$\min f^{j_n}(J_{n+1}^0)=\min f^{j_n}(J_n^0)=y_n$ and
$$
\frac{|f^{j_n}(J_{n+1}^0)|}{|f^{j_n}(J_n^0)|}=\frac{|J_{n+1}^0|}{|J_n^0|}=
\frac{1}{3^{n+1}}<\varepsilon
$$
thus $|f^{k2^{n+1}+j_n}(0)-y_n|<\varepsilon|f^{j_n}(J_n^0)|\leq \varepsilon$.
Secondly suppose that $y_n=\max f^{j_n}(J_n^0)$. The point
$f^{k2^{n+2}}(0)$ belongs to $J_{n+2}^0$ and $f^{2^{n+1}}(J_{n+2}^0)=
J_{n+2}^1$ by Lemma~\ref{lem:wiggins-htop0-summary-Jn0}(vii) thus
$$
f^{k2^{n+2}+2^{n+1}+2^n+j_n}(0)\in f^{2^n+j_n}(J_{n+2}^1).
$$
According to Lemma~\ref{lem:wiggins-htop0-summary-Jn0}(iii)-(vii) one has
$$
\max f^{2^n+j_n}(J_{n+2}^1)=\max f^{2^n+j_n}(J_{n+1}^0)
=\max f^{j_n}(J_{n+1}^1)
=\max f^{j_n}(J_n^0)=y_n
$$
Moreover
$$
f^{2^n}(J_{n+2}^1)\subset f^{2^n}(J_{n+1}^0)=J_{n+1}^1\subset J_n^0.
$$
Thus
$$
\frac{|f^{j_n+2^n}(J_{n+2}^1)|}{|f^{j_n}(J_n^0)|}=
\frac{|f^{2^n}(J_{n+2}^1)|}{|J_n^0|}
=\frac{|f^{2^n}(J_{n+2}^1)|}{|f^{2^n}(J_{n+1}^0)|}\times
\frac{|J_{n+1}^1|}{|J_n^0|}
=\frac{1}{3^{n+2}}\left(1-\frac{2}{3^{n+1}}\right).
$$
Consequently we get that
$|f^{k2^{n+2}+2^{n+1}+2^n+j_n}(0)-y_n|\leq |f^{j_n+2^n}(J_{n+2}^1)|<\varepsilon$.
In both cases there exists $p\geq N$ such that $|f^p(0)-y_n|<\varepsilon$,
thus $|f^p(0)-y|<2\varepsilon$. This means that $y\in \omega(0,f)$, that is,
$\partial K\subset \omega(0,f)$. The point $\{0\}=\bigcap_{n\geq 0}J_n^0$
belongs to $\partial K$ thus $0\in \omega(0,f)$ and $f|_{\omega(0,f)}$ is
transitive.
Finally, $K_n$ has $2^n$ connected components, each of which containing
$2$ connected components of $K_{n+1}$, thus $K$ has an infinite number
of connected components, which implies that $\partial K$ is infinite.
\end{demo}
In the proof of next proposition, we first show that
$K$ contains a non degenerate connected component $B$.
\begin{prop}\label{prop:wiggins-htop0-non-separable-K}
Let $f$ be the map defined above. Then
$f|_{\omega(0,f)}$ is sensitive to initial condition.
\end{prop}
\begin{demo}
First we define by induction a sequence of intervals
$B_n=f^{i_n}(J_n^0)$ for some $0\leq i_n<2^n$ such that $B_n\subset
B_{n-1}$ and $|B_n|=\left(1-\frac{2}{3^n}\right)|B_{n-1}|$ for all $n\geq 1$.
\begin{itemize}
\item
Take $B_0=J_0=[0,1]$.
\item
Suppose that $B_{n-1}=f^{i_{n-1}}(J_{n-1}^0)$ is already built.
If $n$ is even take $i_n=i_{n-1}$ and $B_n=f^{i_n}(J_n^0)$. The map
$f^{i_{n-1}}|_{J_{n-1}^0}$ is linear $\uparrow$ by
Lemma~\ref{lem:wiggins-htop0-summary-Jn0}(iii) and $J_n^0\subset J_{n-1}^0$ thus
$$
\frac{|B_n|}{|B_{n-1}|}=\frac{|J_n^0|}{|J_{n-1}^0|}=1-\frac{2}{3^n}.
$$
If $n$ is odd take $i_n=i_{n-1}+2^{n-1}$ and $B_n=f^{i_n}(J_n^0)$.
According to Lemma~\ref{lem:wiggins-htop0-summary-Jn0}(vii)-(iii)
one has $B_n=f^{i_{n-1}}(J_n^1)$
and $f^{i_{n-1}}|_{J_{n-1}^0}$ is linear $\uparrow$ thus
$$
\frac{|B_n|}{|B_{n-1}|}=\frac{|J_n^1|}{|J_{n-1}^0|}=1-\frac{2}{3^n}.
$$
\end{itemize}
\par
Let $B=\bigcap_{n\geq 0}B_n$. This is a compact interval and it is
non degenerate because
$$
\log |B|=\log |B_0|+\sum_{n\geq 1}\log \left(1-\frac{2}{3^n}\right)
>-\infty.
$$
Moreover $B$ is a connected component of $K$, thus $\partial B\subset
\partial K$. Let $b_0=\min B$ and $b_1=\max B$; one has $b_0, b_1\in
\omega(0,f)$ by Proposition~\ref{prop:wiggins-htop0-omega(0,f)}.
The set $\omega(0,f)$ is included in the periodic orbit of $J_n^0$,
consequently
$f^k(J_n^0\cap \omega(0,f))=f^k(J_n^0)\cap\omega(0,f)$ for all $k\geq 0$.
Let $\varepsilon>0$ and $k\geq 1$.
There exists $n\geq 0$ such that $|J_n^0|<\varepsilon$ and there exist $x_0,x_1
\in J_n^0\cap \omega(0,f)$ such that $f^{i_n+k2^n}(x_0)=b_0$ and
$f^{i_n+k2^n}(x_1)=b_1$. Let $\delta=|b_1-b_0|/4$.
The triangular inequality implies that either
$|f^{i_n+k2^n}(0)-b_0)|\geq 2\delta$ or
$|f^{i_n+k2^n}(0)-b_1|\geq 2\delta$.
In other words, for all $\varepsilon>0$ and $k\geq 1$ there exist $x\in [0,\varepsilon]
\cap \omega(0,f)$ and $i\geq k$ such that $|f^i(0)-f^i(x)|\geq 2\delta$.
Let $y\in \omega(0,f)$ and $\varepsilon>0$; there exists $k\geq 0$ such that
$|f^k(0)-y|<\varepsilon/2$. By continuity of $f^k$ there is $\eta>0$ such that
$f^k([0,\eta])\subset [y-\varepsilon,y+\varepsilon]$. What precedes shows that there
exists $x\in [0,\eta]cap \omega(0,f)$
and $i>k$ such that $|f^i(0)-f^i(x)|\geq 2\delta$
thus, if $z=f^k(x)$, $z'=f^k(0)$ and $j=i-n$ we get that $z,z'\in
[y-\varepsilon, y+\varepsilon]$ and $|f^j(z)-f^j(z')|\geq 2\delta$. Then the triangular
inequality implies that either $|f^j(y)-f^j(z)|\geq \delta$ or
$|f^j(y)-f^j('z)|\geq \delta$. We conclude that $f|_{\omega(0,f)}$ is
sensitive to initial conditions.
\end{demo}
Finally Propositions \ref{prop:wiggins-htop0-2-infty},
\ref{prop:wiggins-htop0-omega(0,f)} and
\ref{prop:wiggins-htop0-non-separable-K} give Theorem~\ref{theo:wiggins-htop0}.
\begin{rem}
According to Theorem \ref{theo:Wiggins-implies-LY} and
\ref{prop:wiggins-htop0-non-separable-K}, the map $f$ is chaotic in the
sense of Li-Yorke. It can be proven directly that $b_0,b_1$ are
$f$-non separable thus Theorem~\ref{theo:htop0-chaos-LY} applies.
\end{rem}
\section{Li-Yorke chaos does not imply Wiggins chaos}\label{sec:LY-not-Wiggins}
The aim of this section is to exhibit an interval map which is
chaotic in the sense of Li-Yorke but has no transitive sensitive subsystem.
This map resembles the one of Section~\ref{sec:wiggins-htop0}:
the construction on the set $\bigcup I_n^0$ is the same except that the
lengths of the intervals differ; the dynamics on $L_n$ is different.
\subsection{Definition of the map {\boldmath $g$}}
\label{subsec:LY-Wiggins-def}
We are going to build a continuous map $g\colon [0,3/2]\to
[0,3/2]$. Let $(a_n)_{n\geq 0}$ be an increasing sequence of numbers
less than $1$ such that $a_0=0$. Define $I_0^1=[a_0,1]$ and for
all $n\geq 1$
$$
I_n^0=[a_{2n-2},a_{2n-1}],\ L_n=[a_{2n-1},a_{2n}],\ I_n^1=[a_{2n},1].
$$
One has $I_n^0\cup L_n\cup I_n^1=I_{n-1}^1$.
Fix $(a_n)_{n\geq 0}$ such that the lengths of the intervals
satisfy
$$
\forall n\geq 1,\ |I_n^0|=|L_n|=\frac{1}{3^n}|I_{n-1}^1|\mbox{ and }
|I_n^1|=\left(1-\frac{2}{3^n}\right)|I_{n-1}^1|.
$$
Let $a=\lim_{n\to +\infty}a_n$.
One has $\bigcup_{n\geq 1}(I_n^0\cup L_n)=[0,a)$ and $a<1$ because
$$
\log (1-a)=\sum_{n=1}^{+\infty}\log \left(1-\frac{2}{3^n}\right)>-\infty.
$$
For all $n\geq 1$,
let $\varphi_n\colon I_n^0\to I_n^1$ be the increasing linear homeomorphism
mapping $I_n^0$ onto $I_n^1$.
Define the map $g\colon [0,3/2]\to [0,3/2]$ such that $g$ is
continuous on $[0,3/2]\setminus\{a\}$ and
\begin{eqnarray*}
&& g(x)=\varphi_1^{-1}\circ \varphi_2^{-2}\circ\cdots\circ
\varphi_{n-1}^{-1}\circ\varphi_n(x)\mbox{ for all } x\in I_n^0,\ n\geq 1,\\
&& g\mbox{ is linear $\uparrow$ of slope }\lambda_n\mbox{ on } [\min L_n,{\rm mid}(L_n)]
\mbox{ for all }n\geq 1,\\
&&g \mbox{ is linear $\downarrow$ on }
[{\rm mid}(L_n),\max L_n]\mbox{ for all }n\geq 1,\\
&&g(x)=0 \mbox{ for all }x\in [a,1],\\
&&g(x)=x-1 \mbox{ for all } x\in [1,3/2],
\end{eqnarray*}
where the slopes $(\lambda_n)$ will be defined below. We will also show below
that $g$ is continuous at $a$. The map $g$ is pictured on
Figure~\ref{fig:LY-not-wiggins}.
\begin{figure}
\caption{The graph of $g$; this map is Li-Yorke chaotic
but not Wiggins chaotic. \label{fig:LY-not-wiggins}
\label{fig:LY-not-wiggins}
\end{figure}
Let $J_0^0=[0,1]$ and for all $n\geq 1$ define the subintervals $J_n^0,
J_n^1\subset J_{n-1}^0$ such that
$\min J_n^0=0$, $\max J_n^1=\max J_{n-1}^0$ and
$\frac{|J_n^i|}{|J_{n-1}^0|}=\frac{|I_n^i|}{|I_{n-1}^1|}\mbox{ for }i=0,1$;
let $M_n=[\max J_n^0,\min J_n^1]$.
Note that on the set $\bigcup_{n\geq 1}I_n^0$ the map $g$ is defined
similarly to the map $f$ in Section~\ref{sec:wiggins-htop0}, thus
the assertions of Lemma~\ref{lem:wiggins-htop0-summary-Jn0} remain valid for
$g$, except the point (i) and its derived results (viii), (x), (xi).
\begin{lem}\label{lem:wiggins-LY-summary-Jn0}
Let $g$ be the map defined above.
Then for all $n\geq 1$ one has
\begin{enumerate}
\item $g(I_n^0)=J_n^1$,
\item $g^i|_{J_n^0}$ is linear $\uparrow$ for all $0\leq i\leq 2^n-1$,
\item $g^{2^{n-1}-1}(J_n^0)=I_n^0$ and $g^{2^n-1}(J_n^0)=I_n^1$,
\item $g^i(J_n^0)\subset \bigcup_{1\leq k\leq n}I_k^0$ for all
$0\leq i\leq 2^n-2$,
\item $(g^i(J_n^0))_{0\leq i<2^n}$ are pairwise disjoint.
\item $g^{2^{n-1}}|_{I_n^0}$ is linear $\uparrow$ and
$g^{2^{n-1}}(I_n^0)=I_n^1$,
\item $g^{2^{n-1}-1}|_{M_n}$ is linear $\uparrow$ and
$g^{2^{n-1}-1}(M_n)=L_n$,
\item $g(\min L_n)=\min M_{n-1}$,
\item $g^{2^{n-2}}(\min L_n)=\min L_{n-1}$.
\end{enumerate}
\end{lem}
\begin{demo}
For the points (i) to (vi) see the proof of
Lemma~\ref{lem:wiggins-htop0-summary-Jn0}.
According to the point (ii), the map $g^{2^{n-1}-1}|_{M_n}$ is
linear $\uparrow$ because $M_n$ is included in $J_{n-1}^0$.
Since $M_n=[\max J_n^0,\min J_n^1]$ and
$L_n=[\max I_n^0,\min I_n^1]$, the points (i), (ii) and (iii) imply that
$g^{2^{n-1}-1}(M_n)=L_n$, which is the point~(vii).
The map $g|_{I_n^0}$ is increasing and $\min L_n=\max I_n^0$ thus,
according to the point~(i), one has $g(\min L_n)=\max J_n^1=\max J_{n-1}^0=
\min M_{n-1}$; this is the point~(viii).
Finally, the points (vii) and (viii) imply the point (ix).
\end{demo}
For all $n\geq 0$, define $x_n={\rm mid}(M_{n+1})$, that is,
$x_n=\frac{3}{2}\prod_{i=1}^{n+1}\frac{1}{3^i}$.
It is a decreasing
sequence and $x_0=1/2$ thus for all $n\geq 0$, $g(1+x_n)$ is well
defined and is equal to $x_n$.
For all $n\geq 0$ let $t_n={\rm slope}\left(g^{2^n-1}|_{J_n^0}\right)$;
by convention $g^0$ is the identity map so $t_0=1$. Fix $\lambda_1=
\frac{2x_1}{|L_1|}$ and for all $n\geq 2$ define
inductively $\lambda_n$ such that
\begin{equation}
\label{eq:defi-lambda-n}
\frac{|L_n|}{2} \prod_{i=1}^n \lambda_i\prod_{i=0}^{n-2}t_i=x_n.
\end{equation}
By convention an empty product is equal to $1$,
so way Equation~(\ref{eq:defi-lambda-n}) is satisfied for $n=1$.
The slopes $(\lambda_n)_{n\geq 1}$ have been fixed such that
$g^{2^{n-1}}([\min L_n,{\rm mid}(L_n)])=[1,1+x_n]$, as
proven in next lemma. This means that under the action of $g^{2^{n-1}}$
the image of $L_n$ falls outside of $[0,1]$ but remains close to $1$.
We also list properties of $g$ on
the intervals $L_n$, $I_n^1$ and $[1,1+x_n]$.
\begin{lem}\label{lem:wiggins-LY-summary2}
Let $g$ be the map defined above. Then one has
\begin{enumerate}
\item
$g^{2^n}|_{[1,1+x_n]}$ is linear $\uparrow$ and
$g^{2^n}([1,1+x_n])=[\min I_{n+1}^0,{\rm mid}(L_{n+1})]$ for all $n\geq 0$,
\item
$g^{2^{n-1}}|_{[\min L_n,{\rm mid}(L_n)]}$ is linear $\uparrow$
and $g^{2^{n-1}}([\min L_n,{\rm mid}(L_n)])\!=\![1,1+x_n]$ for all $n\geq 1$,
\item
$g^{2^{n+1}}|_{[1,1+x_n]}$ is $\uparrow$ and
$g^{2^{n+1}}([1,1+x_n])=I_{n+1}^1\cup [1,1+x_{n+1}]$ for
all $n\geq 1$,
\item
$g(I_n^1)\subset [0,{\rm mid}(M_n)]$ for all $n\geq 1$,
\item
$g^{2^n}([\min I_n^1,1+x_n])\subset [\min I_n^1,1+x_n]$ and
$g^i([\min I_n^1,1+x_n])\subset [0,1]$ for all $1\leq i\leq 2^n-1$, $n\geq 1$.
\end{enumerate}
\end{lem}
\begin{demo}
The map $g|_{[1,1+x_n]}$ is linear $\uparrow$ and $g([1,1+x_n])=
[0,{\rm mid}(M_{n+1})]\subset J_n^0$, thus $g^{2^n}|_{[1,1+x_n]}$ is linear
$\uparrow$ by Lemma~\ref{lem:wiggins-LY-summary-Jn0}(ii). Moreover
$g^{2^n-1}(0)=\min I_{n+1}^0$ and $g^{2^n-1}({\rm mid}(M_{n+1}))=
{\rm mid}(L_{n+1})$ by Lemma~\ref{lem:wiggins-LY-summary-Jn0}(iii)+(iv),
which gives the point~(i).
\par
Before proving the point~(ii) we show some intermediate results.
Let $n\geq 2$ and $2\leq k\leq n$. One has
\begin{eqnarray*}
\lambda_n\ldots \lambda_k\cdot t_{n-2}\ldots t_{k-2}&=&
\frac{\prod_{i=1}^n\lambda_i\prod_{i=0}^{n-2}t_i}{
\prod_{i=1}^{k-1}\lambda_i\prod_{i=0}^{k-3}t_i}\\
&=&\frac{x_n/|L_n|}{x_{k-1}/|L_{k-1}|}\quad\mbox{by
Equation~(\ref{eq:defi-lambda-n})}\\
&=&\prod_{i=k+1}^{n+1}\frac{1}{3^i}
\prod_{i=k-1}^{n-1}\frac{1}{1-2/3^i}\times\frac{3^n}{3^{k-1}}\\
&=&\frac{1}{3^{n-k+1}}\prod_{i=k-1}^{n-1}\frac{1}{3^i-2}
\end{eqnarray*}
thus
\begin{equation}\label{eq:wiggins-LY-summary2:1}
\lambda_n\ldots \lambda_k\cdot t_{n-2}\ldots t_{k-2} <1.
\end{equation}
\par
By definition, $g({\rm mid}(L_n))=g(\min L_n)+\lambda_n \frac{|L_n|}{2}$, and
by Equation~(\ref{eq:defi-lambda-n}),
\begin{eqnarray*}
\lambda_n \frac{|L_n|}{2}
&=&\frac{x_n}{t_{n-2}\prod_{i=1}^{n-1}\lambda_i\prod_{i=0}^{n-3}t_i}\\
&=& \frac{x_n|L_{n-1}|}{2x_{n-1}t_{n-2}}\\
&=&\frac{1}{3^{n+1}}\frac{|M_{n-1}|}{2}\ \mbox{ because }
t_{n-2}=\frac{|L_{n-1}|}{|M_{n-1}|}\mbox{ by
Lemma~\ref{lem:wiggins-LY-summary-Jn0}(vii)}\\
&<&\frac{|M_{n-1}|}{2}
\end{eqnarray*}
Moreover $g(\min L_n)=\min M_{n-1}$ by
Lemma~\ref{lem:wiggins-LY-summary-Jn0}(viii) thus
\begin{equation}\label{eq:wiggins-LY-summary2:2}
g([\min L_n,{\rm mid}(L_n)])\subset [\min M_{n-1},{\rm mid}(M_{n-1})]
\mbox{ for all }n\geq 2.
\end{equation}
\par
We show by induction on $k=n,\ldots,2$ that\\
-- the map
$g^{2^{n-2}+2^{n-3}+\cdots+2^{k-2}}$
is linear $\uparrow$ of slope $\lambda_n\ldots\lambda_k t_{n-2}\ldots t_{k-2}$
on $[\min L_n,{\rm mid}(L_n)]$ and
maps $\min L_n$ to $\min L_{k-1}$,\\
-- $g^i([\min L_n,{\rm mid}(L_n)])\subset [0,1]$ for all
$0\leq i\leq 2^{n-2}+2^{n-3}+\cdots+2^{k-2}$.
\begin{itemize}
\item
By Equation~(\ref{eq:wiggins-LY-summary2:2})
one has $g([\min L_n,{\rm mid}(L_n)])\subset M_{n-1}\subset J_{n-2}^0$, thus
the map $g^{2^{n-2}}|_{[\min L_n,{\rm mid}(L_n)]}$ is linear $\uparrow$ of slope
$\lambda_n t_{n-2}$.
According to Lemma~\ref{lem:wiggins-LY-summary-Jn0}(ix) one has
$g^{2^{n-2}}(\min L_n)=\min L_{n-1}$.
Equation~(\ref{eq:wiggins-LY-summary2:2}) and
Lemma~\ref{lem:wiggins-LY-summary-Jn0}(iii)+(iv) imply that
$g^i([\min L_n,{\rm mid}(L_n)])\subset [0,1]$
for all $1\leq i\leq 2^{n-2}$. This is the induction property at rank
$k=n$.
\item
Suppose that the induction property is true for $k$ with $3\leq k\leq n$.
By Equation~(\ref{eq:wiggins-LY-summary2:1}) one has
$\lambda_n\ldots\lambda_k\cdot t_{n-2}\ldots t_{k-2}\frac{|L_n|}{2}
\leq \frac{|L_{k-1}|}{2}$ thus
$g^{2^{n-2}+2^{n-3}+\cdots+2^{k-2}}([\min L_n,{\rm mid}(L_n)])\subset
[\min L_{k-1},{\rm mid}(L_{k-1})]$. The map $g$ is of slope $\lambda_{k-1}$
on this interval, $g(\min L_{k-1})=\min M_{k-2}$ by
Lemma~\ref{lem:wiggins-LY-summary-Jn0}(viii) and
$g([\min L_{k-1},{\rm mid}(L_{k-1})])\subset
M_{k-2}$ by Equation~(\ref{eq:wiggins-LY-summary2:2}).
Since $M_{k-2}\subset J_{n-1}^0$, the map
$g^{2^{n-2}+2^{n-3}+\cdots+2^{k-2}+2^{k-3}}$ is linear $\uparrow$ of slope
$\lambda_n\ldots\lambda_{k-1}\cdot t_{n-2}\ldots t_{k-3}$ on
$[\min L_n,{\rm mid}(L_n)]$, and it maps $\min L_n$ to $\min L_{k-2}$
by Lemma~\ref{lem:wiggins-LY-summary-Jn0}(ix).
Moreover $g^i([\min L_n,{\rm mid}(L_n)])\subset [0,1]$ for all
$0\leq i\leq 2^{n-2}+2^{n-3}+\cdots+2^{k-2}+2^{k-3}$ by
Lemma~\ref{lem:wiggins-LY-summary-Jn0}(iv) and the induction hypothesis.
This is the property at rank $k-1$.
\end{itemize}
For $k=2$ we finally get that
$g^{2^{n-2}+\cdots +2^0}=g^{2^{n-1}-1}$ is linear $\uparrow$ of slope
$\prod_{i=2}^n\lambda_i\prod_{i=0}^{n-2}t_i$ on $[\min L_n,{\rm mid}(L_n)]$,
with $g^{2^{n-1}-1}(\min L_n)=\min L_1$ and $g^{2^{n-1}-1}
([\min L_n,{\rm mid}(L_n)])\subset [\min L_1,{\rm mid}(L_1)]$.
The map $g$ is of slope $\lambda_1$ on this interval thus,
according to the definition of $\lambda_n$,
the point~(ii) holds for all $n\geq 2$; it trivially holds for $n=1$ too.
The induction property for $k=2$ also gives that
\begin{equation}\label{eq:wiggins-LY-summary2:5}
g^i([\min L_n,{\rm mid}(L_n)])\subset [0,1]\mbox{ for all }
0\leq i\leq2^{n-1}-1,\ n\geq 1.
\end{equation}
\par
The points (i) and (ii) and
Lemma~\ref{lem:wiggins-LY-summary-Jn0}(vi) imply the point (iii).
\par
One has $I_n^1=\bigcup_{k\geq n+1}(I_k^0\cup L_k)\cup [a,1]$.
One can see from the definition of $g$ that
$$
\max\{g(x)\mid x\in I_k^0\cup
L_k\}=g({\rm mid}(L_k)),
$$
thus $g(I_k^0\cup L_k)\subset [0,{\rm mid}(M_{k-1})]$
by Equation~(\ref{eq:wiggins-LY-summary2:2}). Hence
$$
g(I_n^1)\subset [0,{\rm mid}(M_n)]=J_n^0\cup [\min M_n,{\rm mid}(M_n)],
$$
which is the point~(iv).
According to Lemma~\ref{lem:wiggins-LY-summary-Jn0}(iii)+(vii), one has that
$g^{2^n-1}(J_n^0)=I_n^1$ and $g^{2^{n-1}-1}([\min M_n,{\rm mid}(M_n)])=[\min L_n,
{\rm mid}(L_n)]$, and by the point~(ii)
$g^{2^{n-1}}([\min L_n,{\rm mid}(L_n)])=[1,1+x_n]$.
Combined with the point~(iv) we get that
\begin{equation}\label{eq:g2n}
g^{2^n}(I_n^1)\subset I_n^1\cup [1,1+x_n].
\end{equation}
Moreover $g^i(J_n^0)\subset [0,1]$ for all $0\leq i\leq 2^n-2$
and $g^i([\min M_n,{\rm mid}(M_n)])\subset [0,1]$ for all $0\leq i\leq 2^{n-1}-2$
according to Lemma~\ref{lem:wiggins-LY-summary-Jn0}(iv). In addition,
$g^{2^{n-1}+i-1}([\min M_n,{\rm mid}(M_n)])=g^i([\min L_n, {\rm mid}(L_n)])\subset
[0,1]$ for all \mbox{$0\leq i\leq 2^{n-1}-1$} by
Equation~(\ref{eq:wiggins-LY-summary2:5}).
Therefore
\begin{equation}\label{eq:wiggins-LY-summary2:8}
g^i(I_n^1)\subset [0,1]\mbox{ for all }0\leq i <2^n.
\end{equation}
Finally, $g([1,1+x_n])=[0,{\rm mid}(M_{n+1})]\subset J_n^0$ and
the point~(i) implies that $g^{2^n}([1,1+x_n])\subset I_n^1$. Combined
with Equation~(\ref{eq:wiggins-LY-summary2:8}) and (\ref{eq:g2n}) and
Lemma~\ref{lem:wiggins-LY-summary-Jn0}(iv), this gives the point~(v).
\end{demo}
Now we show that
$g$ is continuous at point $a$ as claimed at the beginning of the section.
\begin{lem}
The map $g$ defined above is continuous.
\end{lem}
\begin{demo}
We just have to show the continuity at $a$.
It is clear from the definition that $g$ is continuous at $a^+$.
According to Lemma~\ref{lem:wiggins-LY-summary2}(iv) one has
$g(I_n^1)\subset J_{n-1}^0$. This implies that $g$ is continuous
at $a^-$ because $\displaystyle\lim_{n\to+\infty} \max J_n^0=0$.
\end{demo}
To end this subsection, let us explain the main underlying ideas of the
construction of $g$ by comparing it with the map $f$ built in
Section~\ref{sec:wiggins-htop0}. The map $g$ and $f$ are similar on
the set $\bigcup_{n\geq 1} I_n^0$ -- which is the core of the dynamics
of $f$ -- the only difference is the length of the intervals.
For $f$ we showed that $K=\bigcap_{n\geq 0}\bigcup_{i=0}^{2^n-1}
f^i(J_n^0)$ has a non degenerate connected component $B$ and it can be proven
that the endpoints of $B$ are
$f$-non separable. The same remains true for $g$ with
$B=[a,1]=\bigcap_{n\geq 0}I_n^1$ (the fact that $a,1$ are $g$-non
separable will be proven in Proposition~\ref{prop:wiggins-LY-g-LY-chaotic}).
For $f$ we proved that $\partial K\subset \omega(0,f)$ hence
$\partial B\subset \omega(0,f)$;
for $g$ it is not true that $\{a,1\}\subset \omega(0,g)$ because
the orbit of $0$ stays in $[0,a]$.
The construction of $g$ on the intervals $L_n$ allows to approach $1$ from
outside of $[0,1]$: we will see in
Proposition~\ref{prop:wiggins-LY-g-LY-chaotic} that $\omega(1+x_0,g)$
contains both $a$ and $1$, which implies chaos in the sense of Li-Yorke.
On the other hand the proof showing that $f|_{\omega(0,f)}$
is transitive and sensitive
fails for $g$ because $\omega(0,g)$ does not contain
$\{a,1\}$ and $\omega(1+x_0,g)$ is not transitive.
We will see in Proposition~\ref{prop:wiggins-LY-g-not-wiggins-chaotic}
that $g$ has no transitive sensitive subsystem at all.
\subsection{{\boldmath $g$} is chaotic in the sense of Li-Yorke}
\begin{prop}\label{prop:wiggins-LY-g-LY-chaotic}
Let $g$ be the map defined in
Section~\ref{subsec:LY-Wiggins-def}. Then the set $\omega(1+x_0,g)$ is
infinite and contains the points $a,1$, which are $g$-non separable.
Consequently the map $g$ is chaotic in the sense of Li-Yorke.
\end{prop}
\begin{demo}
Lemma~\ref{lem:wiggins-LY-summary2}(iii) implies that
$g^{2^{n+1}}(1+x_n)=1+x_{n+1}$ for all $n\geq 0$.
Since $x_n\to 0$ when $n$ goes to infinity, this implies that
$1\in \omega(1+x_0,g)$.
Moreover Lemma~\ref{lem:wiggins-LY-summary2}(i) implies that
$g^{2^n}(1)=\min I_{n+1}^0=a_{2n}$ for all $n\geq 1$,
hence $a\in \omega(1,g)\subset \omega(1+x_0,g)$.
Suppose that $A_1,A_2$ are two
periodic intervals such that $a\in A_1$ and $1\in A_2$,
and let $p$ be a common multiple of their periods. One has $g(a)=g(1)=0$
thus $g^p(a)=g^p(1)\in A_1\cap A_2$, and $A_1,A_2$ are not disjoint.
This means that $a,1$ are $g$-non separable.
It is well known that a finite $\omega$-limit set is cyclic.
Therefore, if $y_0,y_1$ are two distinct points in
a finite $\omega$-set, the degenerate intervals $\{y_0\}$, $\{y_1\}$
are periodic and $y_0,y_1$ are $g$-separable. This implies that
$\omega(1+x_0,g)$ is infinite.
We deduce that the map $g$ is chaotic in the sense
of Li-Yorke by Theorem~\ref{theo:htop0-chaos-LY}.
\end{demo}
\subsection{{\boldmath $g$} is not chaotic in the sense of Wiggins}
The main result of this subsection is
Proposition~\ref{prop:wiggins-LY-g-not-wiggins-chaotic} stating that
$g$ has no transitive sensitive subsystem.
Next lemma is about the location of transitive subsystems.
\begin{lem}\label{lem:wiggins-LY-transitive-set}
Let $g$ be the map defined in
Section~\ref{subsec:LY-Wiggins-def} and $Y\subset [0,3/2]$ a closed
invariant subset with no isolated point such that $g|_Y$ is transitive.
Then
\begin{enumerate}
\item $Y\subset [0,a]$,
\item $\displaystyle Y\subset \bigcup_{i=0}^{2^n-1}g^i(J_n^0)$ for all $n\geq 1$,
\item $g^i(J_n^0\cap Y)=g^i(J_n^0)\cap Y=g^{i\bmod 2^n}(J_n^0)\cap Y$ for all
$i\geq 0$, $n\geq 0$.
\end{enumerate}
\end{lem}
\begin{demo}
By transitivity there exists $y_0\in Y$ such that $\omega(y_0,g)=Y$;
in particular the set $Y'=\{g^k(y_0),k\geq 0\}$
is dense in $Y$ and $y\in\omega(y,g)$ for all $y\in Y'$.
Let $n\geq 0$. By Lemma~\ref{lem:wiggins-LY-summary2}(iii),
$g^{2^{n+1}}([1,1+x_n])=I_{n+1}^1\cup [1,1+x_{n+1}]$
thus, according to Lemma~\ref{lem:wiggins-LY-summary2}(v), one obtains
that for all integers $k\geq 1$,
$g^{k2^{n+1}}([1,1+x_n])\subset I_{n+1}^1\cup [1,1+x_{n+1}]$
and $g^i([1,1+x_n])\subset [0,1]$ for all $i>2^{n+1}$, $i\not\in 2^{n+1}\mathbb{N}$.
This implies that
$$
\mbox{for all }i\geq 2^{n+1},\ g^i((1+x_{n+1},1+x_n])\subset
[0,1+x_{n+1}].
$$
Consequently there is no $y\in (1,3/2]=\bigcup_{n\geq 0}
(1+x_{n+1},1+x_n]$ such that $y\in \omega(y,g)$, hence
$Y'\cap (1,3/2]=\emptyset$ and by density $Y\cap (1,3/2]=\emptyset$.
One has $g^{2^n-1}(0)=a_{2n}$ by
Lemma~\ref{lem:wiggins-LY-summary-Jn0}(ii)+(iii)
thus the point $0$ is not periodic, hence $\forall k\geq 1$,
$g^k(0)\not\in [a,1]$.
If $y\in (a,1)$ then $g(y)=0$ and $g^k(y)\not\in [a,1]$ for all
$k\geq 1$, which implies that $y\not\in \omega(y,g)$. Consequently,
$Y\cap (a,1)=\emptyset$. We get that $Y\subset [0,a]\cup\{1\}$, and
$1\not\in Y$ because $Y$ has no isolated point; this gives the point~(i).
\par
Let $n\geq 1$. One has $\min L_n=\max I_n^0$ and $\max L_n=\min I_{n+1}^0$
thus $g(\min L_n)=\max J_n^1$ and $g(\max L_n)=\min J_{n+1}^1$ by
Lemma~\ref{lem:wiggins-LY-summary-Jn0}(i). Moreover
$g|_{[\min L_n,{\rm mid}(L_n)]}$ is $\uparrow$ and
$g|_{[{\rm mid}(L_n),\max L_n]}$ is linear $\downarrow$ thus there exists
$c_n$ in $[{\rm mid}(L_n),\max L_n]$ such that $g(c_n)=g(\min L_n)$.
Since $g([c_n,\max L_n])=[\min J_{n+1}^1,\max J_n^1]$ is included
in $J_{n-1}^0$,
the map $g^{2^{n-1}}|_{[c_n,\max L_n]}$ is linear $\downarrow$ by
Lemma~\ref{lem:wiggins-LY-summary-Jn0}(ii). Moreover $M_n\subset
g([c_n,\max L_n])$ thus $g^{2^{n-1}}([c_n,\max L_n])$ contains $L_n$ by
Lemma~\ref{lem:wiggins-LY-summary-Jn0}(vii). Consequently
there exists $z_n\in [c_n,\max L_n]$ such that $g^{2^{n-1}}(z_n)=z_n$
(Lemma~\ref{lem:fixed-point}) and
${\rm slope}(g^{2^{n-1}}|_{[c_n,\max L_n]})\leq -2$.
Then for every $x\in [c_n,\max L_n]$,
$x\not=z_n$, there exists $k\geq 1$ such that $g^{k2^{n-1}}(x)\not\in
[c_n,\max L_n]$. By Lemma~\ref{lem:wiggins-LY-summary2}(v) one has
$g^{2^{n-1}}(I_{n-1}^1\cup [1,1+x_{n-1}])\subset I_{n-1}^1\cup [1,1+x_{n-1}]$
thus
\begin{eqnarray}
\lefteqn{\forall x\in [c_n,\max L_n], x\not=z_n,}\nonumber\\
&\exists k\geq 1,\
g^{k2^{n-1}}(x)\in I_n^0\cup [\min L_n, c_n]\cup I_n^1\cup [1,1+x_{n-1}].&
\label{eq:out-of-[cn,maxLn]}
\end{eqnarray}
We show by induction on $n$ that
\begin{equation}\label{eq:Y-In1}
\forall n\geq 0,\ Y'\cap I_n^1\not=\emptyset.
\end{equation}
This is true for $n=0$ because $Y\subset [0,1]=I_0^1$ by the point~(i).
Suppose that there exists $y\in Y'\cap I_{n-1}^1$. Write
$I_{n-1}=I_{n-1}^1=I_n^0\cup L_n\cup I_n^1$; to prove that
$Y'\cap I_n^1\not=\emptyset$ we split into four cases.
\begin{itemize}
\item If $y\in I_n^1$ there is nothing to do.
\item If $y\in I_n^0$ then $g^{2^{n-1}}(y)\in I_n^1$ by
Lemma~\ref{lem:wiggins-LY-summary-Jn0}(vi) and $g^{2^{n-1}}(y)\in Y'$.
\item If $y\in [\min L_n,c_n]$ then $g(y)\in g([\min L_n,{\rm mid}(L_n)]$
and $g^{2^{n-1}}(y)\in [1,1+x_n]$ by Lemma~\ref{lem:wiggins-LY-summary2}(ii),
which is impossible because $Y\subset [0,a]$ by the point~(i).
\item If $y\in [c_n,\max L_n]$ then $y\not=z_n$ because $Y$ is infinite.
In addition $g^j(y)\in [0,1]$ for all $j\geq 0$ according to the
point~(i). Then
Equation~(\ref{eq:out-of-[cn,maxLn]}) says that there exists $j\geq 1$
such that $g^j(y')$ belongs to $I_n^0\cup [\min L_n,c_n]\cup I_n^1$ and one of the
first three cases applies.
\end{itemize}
One has $g(I_n^1)\subset J_n^0\cup [\min M_n,{\rm mid}(M_n)]$ by
Lemma~\ref{lem:wiggins-LY-summary2}(iv) and
$g^{2^n-1}([\min M_n,{\rm mid}(M_n)])=g^{2^{n-1}}([\min L_n,{\rm mid}(L_n)])
=[1,1+x_n]$ by Lemmas \ref{lem:wiggins-LY-summary-Jn0}(vii) and
\ref{lem:wiggins-LY-summary2}(ii) respectively. Together with the point~(i)
this implies that
\begin{equation}\label{eq:Y-Jn0}
g(Y\cap I_n^1)\subset J_n^0.
\end{equation}
Equations (\ref{eq:Y-In1}) and (\ref{eq:Y-Jn0}) combined with
Lemma~\ref{lem:wiggins-LY-summary-Jn0}(i)+(iii) imply that
$$
Y\subset \bigcup_{i=0}^{2^n-1}g^i(J_n^0) \mbox{ for all }n\geq 1,
$$
which is the point~(ii); furthermore
$Y\cap g^i(J_n^0)=Y\cap g^{i\bmod 2^n}(J_n^0)$ for all $i\geq 0$.
Since $g(Y)=Y$ it is clear that $g^i(J_n^0\cap Y)\subset
g^i(J_n^0)\cap Y$ and that $g^{2^n}(g^i(J_n^0)\cap Y)\subset
g^{2^n+i}(J_n^0)\cap Y$, thus
$$
g^i(J_n^0\cap Y)=g^i(J_n^0)\cap Y=g^{i\bmod 2^n}(J_n^0)\cap Y \mbox{ for all }
i\geq 0,
$$
which concludes the proof of the Lemma.
\end{demo}
Next lemma is the key tool in the proof of
Proposition~\ref{prop:wiggins-LY-g-not-wiggins-chaotic}.
It relies on the knowledge of the precise location of $g^i(J_n^0)$ in
$\bigcup_{1\leq k\leq n}I_n^0$.
\begin{lem}\label{lem:slope}
Let $g$ be the map defined in
Section~\ref{subsec:LY-Wiggins-def}.
For all $n\geq 1$ and all $0\leq k\leq 2^n-1$ one has
${\rm slope}\!\left(g^{2^n-1-k}|_{g^k(J_n^0)}\right)\geq 1$.
\end{lem}
\begin{demo}
A {\em (finite) word} $B$ is an element of $\mathbb{N}^n$ for some $n\in \mathbb{N}$.
If $B,B'$ are two words, $BB'$ denotes their concatenation and
$|B|=n$ is the length of $B$.
We define inductively a sequence of words $(B_n)_{n\geq 1}$ by:
\begin{itemize}
\item $B_1=1$,
\item $B_n=nB_1B_2\ldots B_{n-1}$.
\end{itemize}
and we define the infinite word $\omega=(\omega(i))_{i\geq 1}$ by
concatenating the $B_n$'s: $\omega=B_1B_2B_3\ldots B_n\ldots$
A straightforward induction shows that
$|B_n|=2^{n-1}$ thus $|B_1|+|B_2|+\cdots+|B_k|=2^k-1$ and
the word $B_{k+1}$ begins at $\omega(2^k)$, which gives
\begin{equation}
\label{eq:omega(2k)}
\omega(2^k)=k+1,
\end{equation}
and
\begin{equation}
\label{eq:omega-repeat}
\omega(2^k+1)\ldots\omega(2^{k+1}-1)=B_1\ldots B_k=
\omega(1)\ldots\omega(2^k-1).
\end{equation}
We prove by induction on $k\geq 1$ that
\begin{equation}
\label{eq:gi(Jn)-omega}
g^{i-1}(J_n^0)\subset I_{\omega(i)}^0\mbox{ for all } n\geq k,\
1\leq i\leq 2^k-1.
\end{equation}
\begin{itemize}
\item
Case $k=1$: $J_n^0\subset I_1^0=I_{\omega(1)}^0$ for all $n\geq 1$.
\item
Suppose that Equation~(\ref{eq:gi(Jn)-omega}) holds for $k$ and let
$n\geq k+1$. Since $J_n^0$ is included in $J_{k+1}^0$,
Lemma~\ref{lem:wiggins-LY-summary-Jn0}(iii) implies that
$g^{2^k-1}(J_n^0)\subset I_{k+1}^0$, thus
$g^{2^k}(J_n^0)\subset J_k^0$ by
Lemma~\ref{lem:wiggins-LY-summary-Jn0}(i). By
induction one has $g^{i-1}(J_k^0)\subset I_{\omega(i)}^0$ for all
$1\leq i\leq 1^k-1$, and by Equation~(\ref{eq:omega-repeat}) , one has
$\omega(i)=\omega(2^k+i)$ for all $1\leq i\leq 2^k-1$. Consequently
$g^{2^k+i-1}(J_n^0)\subset I_{\omega(2^k+i)}^0$ for all $1\leq i\leq 2^k-1$.
Together with the induction hypothesis this gives
Equation~(\ref{eq:gi(Jn)-omega}) for $k+1$.
\end{itemize}
\par
Let $\mu_n={\rm slope}(g|_{I_n^0})$. By definition of $g$ one has
$$
\mu_n=\frac{{\rm slope}(\varphi_n)}{\prod_{i=1}^{n-1}{\rm slope}(\varphi_i)}.
$$
It is straightforward from Equation~(\ref{eq:gi(Jn)-omega}) that
for all $2\leq k\leq 2^n-1$
\begin{equation}\label{eq:slope-gk}
{\rm slope} (g^{k-1}|_{J_n^0})=\prod_{i=1}^{k-1} \mu_{\omega(i)}.
\end{equation}
By Lemma~\ref{lem:wiggins-LY-summary-Jn0}(ii)+(iii)
the map $g^{2^n-1}|_{J_n^0}$
is linear and $g^{2^n-1}(J_n^0)=I_n^1$, thus
$$
{\rm slope}(g^{2^n-1}|_{J_n^0})=\frac{|I_n^1|}{|J_n^0|}
=\prod_{i=1}^n\frac{1-2/3^i}{1/3^i}.
$$
Since ${\rm slope}(\varphi_i)=\frac{|I_i^1|}{|I_i^0|}=\frac{1-2/3^i}{1/3^i}$, we
get
\begin{equation}
\label{eq:slope-g2n-1}
{\rm slope}(g^{2^n-1}|_{J_n^0})=\prod_{i=1}^{2^n-1}\mu_{\omega(i)}=
\prod_{i=1}^n {\rm slope}(\varphi_i).
\end{equation}
We show by induction on $n\geq 1$ that for all $1\leq k\leq 2^n-1$
\begin{equation}
\label{eq:prod-mui}
\prod_{i=1}^{k}\mu_{\omega(i)}=\prod_{i=1}^n{\rm slope}(\varphi_i)^{\varepsilon_i}
\mbox{ for some }\varepsilon_i=\varepsilon(i,k,n)\in\{0,1\}.
\end{equation}
\begin{itemize}
\item
$\mu_{\omega(1)}=\mu_1={\rm slope}(\varphi_1)$; this gives the case $n=1$.
\item
Suppose that the induction hypothesis is true for
$n$. One has
\begin{eqnarray*}
\prod_{i=1}^{2^n}\mu_{\omega(i)}&=&\prod_{i=1}^{2^n-1}\mu_{\omega(i)}\times
\mu_{n+1}\mbox{ by Equation~(\ref{eq:omega(2k)})}\\
&=&\displaystyle \prod_{i=1}^n{\rm slope}(\varphi_i)
\frac{{\rm slope}(\varphi_{n+1})}{\prod_{i=1}^n{\rm slope}(\varphi_i)}
\mbox{ by Equation~(\ref{eq:slope-g2n-1})}\\
&=&{\rm slope}(\varphi_{n+1})
\end{eqnarray*}
This is Equation~(\ref{eq:prod-mui}) for $n+1$ and $k=2^n$ with
$\varepsilon(i,k,n1)=0$ for $1\leq i\leq n$ and $\varepsilon(n+1,2^n,n+1)=1$.
Next, $\omega(2^n+1)\ldots\omega(2^{n+1}-1)=\omega(1)\ldots
\omega(2^n-1)$ by Equation~(\ref{eq:omega-repeat}) thus, if
$2^n+1\leq k\leq 2^{n+1}-1$ one has
\begin{eqnarray*}
\prod_{i=1}^k\mu_{\omega(i)}&=&
\prod_{i=1}^{2^n}\mu_{\omega(i)}\prod_{i=2^n+1}^k\mu_{\omega(i)}\\
&=&{\rm slope}(\varphi_{n+1})\prod_{i=1}^{k-2^n}\mu_{\omega(i)}\\
&=&{\rm slope}(\varphi_{n+1})\prod_{i=1}^n{\rm slope}(\varphi_i)^{\varepsilon_=(i,k_2^n,n)}
\end{eqnarray*}
That is, Equation~(\ref{eq:prod-mui}) holds with
$\varepsilon(i,k,n+1)=\varepsilon(i,k-2^n,n)$ for all $1\leq i\leq n$ and
$\varepsilon(n+1,k,n+1)=1$. This concludes the induction.
\end{itemize}
Equations (\ref{eq:slope-gk}) and (\ref{eq:prod-mui}) show that for all
$1\leq k\leq 2^n-1$
\begin{equation}
\label{eq:slope-gk-bis}
{\rm slope}(g^k|_{J_n})=\prod_{i=1}^{k+1}\mu_{\omega(i)}=
\prod_{i=1}^{n}{\rm slope}(\varphi_i)^{\varepsilon_i}\mbox{ for some }
\varepsilon_i\in\{0,1\}.
\end{equation}
Since ${\rm slope}\!\left(g^{2^n-1-k}|_{g^k(J_n^0)}\right)=
\frac{{\rm slope}(g^{2^n-1}|_{J_n^0})}{{\rm slope}(g^k|_{J_n^0})}$, Equations
(\ref{eq:slope-g2n-1}) and (\ref{eq:slope-gk-bis})
imply that ${\rm slope}\!\left(g^{2^n-1-k}|_{g^k(J_n)}\right)$ is a
product of at most $n$ terms of the form ${\rm slope}(\varphi_i)$. This
concludes the proof of the
lemma because ${\rm slope}(\varphi_i)\geq 1$ for all $i\geq 1$.
\end{demo}
\begin{prop}\label{prop:wiggins-LY-g-not-wiggins-chaotic}
The map $g$ defined in
Section~\ref{subsec:LY-Wiggins-def} is not Wiggins chaotic.
\end{prop}
\begin{demo}
Let $Y\subset [0,3/2]$ be a closed invariant subset such that $g|_Y$ is
transitive. We assume that $Y$ has no isolated point, otherwise $g|_Y$
is not sensitive.
The sets $\left(g^i(J_n^0\cap Y)\right)_{0\leq i\leq 2^n-1}$ are closed and
by Lemma~\ref{lem:wiggins-LY-summary-Jn0}(v)
they are pairwise disjoint; let $\delta_n>0$
be the minimal distance between two of these sets.
If $x,x'\in Y$ and $|x-x'|<\delta_n$ then there is $0\leq i\leq 2^n-1$ such
that $x,x'\in g^i(J_n^0)$ and for all $k\geq 0$ one has
$g^k(x),g^k(x')\in g^{i+k\bmod{2^n}}(J_n^0)$ by
Lemma~\ref{lem:wiggins-LY-transitive-set}(ii)+(iii). Let
$$
\varepsilon_n=\max\{\diam{g^i(J_n^0)\cap Y}\mid 0\leq i<2^n\}.
$$
By Lemma~\ref{lem:slope}, we get that $\diam{g^k(J_n^0)\cap Y}
\leq \diam{g^{2^n-1}(J_n^0)\cap Y}$ for all $0\leq k\leq 2^n-1$.
By Lemma~\ref{lem:wiggins-LY-summary-Jn0}(iii) one has
$ g^{2^n-1}(J_n^0)=I_n^1$ and by
Lemma~\ref{lem:wiggins-LY-transitive-set}(i) one has
$I_n^1\cap Y\subset [a_{2n},a]$, thus $\varepsilon_n\leq \diam{I_n^1\cap Y}\leq
a-a_{2n}$, which implies that
$$
\lim_{n\to+\infty}\varepsilon_n=0.
$$
This implies that $g|_Y$ is not sensitive.
\end{demo}
At last this example is completed.
Theorem~\ref{theo:LY-not-wiggins} is given by
Propositions \ref{prop:wiggins-LY-g-LY-chaotic} and
\ref{prop:wiggins-LY-g-not-wiggins-chaotic}.
\noindent
Laboratoire de Math\'ematiques --
Topologie et Dynamique -- B\^atiment 425~-- Universit\'e Paris-Sud --
F-91405 Orsay cedex -- France.\\
E-mail: {\tt Sylvie.Ruette@math.u-psud.fr}
\end{document} |
\begin{document}
\begin{abstract}
We study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schrödinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for the solution of the linear nonhomogenenous problem by using the Fokas method (also known as the \emph{unified transform method}). We use this representation formula to prove space and time estimates on the solutions of the linear model in fractional Sobolev spaces by using Fourier analysis. Secondly, we consider the nonlinear model with a power type nonlinearity and prove the local wellposedness by means of a classical contraction argument. We obtain Strichartz estimates to treat the low regularity case by using the oscillatory integral theory directly on the representation formula provided by the Fokas method. Global wellposedness of the defocusing model is established up to cubic nonlinearities by using the multiplier technique and proving hidden trace regularities.
\end{abstract}
\keywords{fourth-order Schrödinger equation; biharmonic Schrödinger equation; Fokas method; unified transform method; local wellposedness; global wellposedness; space estimates; time estimates; Strichartz estimates; \and inhomogeneous boundary data}
\subjclass[2010]{35Q55, 35C15, 35A07, 35A22, 35G15, 35G30}
\maketitle
\tableofcontents
\section{Introduction}
This article studies the local and global wellposedness of the initial\,-\,(inhomogeneous) boundary value problem for the biharmonic nonlinear Schrödinger equation (NLS) which is posed on the right half-line:
\begin{align}
&iq_{t}+\partial_x^{4}q=f(q), \quad (x,t)\in\mathbb{R}_+\times (0,T),\label{4th.1}\\
&q(x,0)=q_{0}(x), \quad x\in\mathbb{R}_+,\label{4th.2}\\
&q(0,t)=g_{0}(t), \quad t\in(0,T),\label{4th.3} \\
&q_{x}(0,t)=g_{1}(t), \quad t\in(0,T),\label{4th.4}
\end{align} where $f(q)=\kappa|q|^pq$, $\kappa\in\mathbb{C}$, $T,p>0$, and $q$ is a complex valued function. The analysis here is carried out in the $L^2-$based fractional Sobolev space $H^s(\mathbb{R}_+)$ at the spatial level, where throughout the paper (without any restatement) we will assume the following in order to work with a sufficiently nice nonlinearity:
\begin{itemize}
\item[(a1)] if $s$ is integer, then $p\ge s$ if $p$ is an odd integer and $\floor{p}\ge s-1$ if $p$ is non-integer,
\item[(a2)] if $s$ is non-integer, then $p>s$ if $p$ is an odd integer and $\floor{p}\ge \floor{s}$ if $p$ is non-integer.
\end{itemize}
The fourth-order NLS, in the form
\begin{equation}\label{mixeddisp}iu_{t}+\Delta u+\gamma \Delta^2u+|u|^pu=0,\,x\in \mathbb{R}^n,\,t\in\mathbb{R}\end{equation}
was introduced by \cite{Karpman96}-\cite{Karpman2000} to study the stabilizing role of the higher-order dispersive effects. It was shown that the solutions are stable if $\gamma<0$, $\displaystyle p\le\frac{4}{n}$ or if $\gamma\ll-1$, $\displaystyle p\in \left(\frac{4}{n},\frac{8}{n}\right)$. Moreover, solutions were found to be unstable for $\displaystyle\gamma<0$, $\displaystyle p\ge\frac{8}{n}$ in which case solutions may cease to exist globally.
In the absence of the Laplacian, the fourth order NLS takes the form \begin{equation}\label{biharmonic}iu_{t}+\gamma \Delta^2u+|u|^pu=0,\,x\in \mathbb{R}^n,\,t\in\mathbb{R}\end{equation} and it is called the biharmonic NLS. It was shown by \cite{Fin2002} (see also \cite{Fin2011} and the references therein) that all solutions of the biharmonic NLS are global if $\gamma>0$. Moreover, it was found that $\displaystyle p=\frac{8}{n}$ is the critical exponent for singularity formation if $\gamma<0$, and smallness in the mean-square sense is sufficient for global existence if $\displaystyle p=\frac{8}{n}$. The biharmonic NLS $$iu_{t}+\gamma \Delta^2u+\kappa|u|^pu=0,$$ with $\gamma,\kappa\in\mathbb{R}$, is said to be focusing (resp. defocusing) if $\gamma\kappa<0$ (resp. $\gamma\kappa>0$).
The rigourous analysis of the solutions of the fourth order Schrödinger equation started with the proof of sharp space-time decay properties for the linear group associated to the operator $i\partial_t+\lambda \Delta+\Delta^2$, where $\lambda\in \{-1,0,1\}$ \cite{Ben2000}. One can actually use these properties to obtain Strichartz estimates, which gives the local wellposedness at $H^2$-level.
Local well-posedness of the nonlinear fourth order Schrödinger equations in one space dimension was studied in \cite{Seg04}, \cite{Hao2006}, and \cite{Zheng11}. Global well-posedness in one dimensional case with small initial data was proved with various nonlinearities in \cite{Hayashi15}, \cite{Hayashi15-2}, \cite{Hayashi15-3}, \cite{Hayashi15-4}, and \cite{aoki16}. Local well-posedness in the muti-dimensional case was treated in \cite{Hao2007} and the global well-posedness was studied in \cite{cui2007}, \cite{Pausader10}, \cite{guo2010}, \cite{guo12}, and \cite{Zhang2010}. Global well-posedness at the $H^2-$level in the energy-critical case with power-type nonlinearities was shown by \cite{Pausader07} for radial initial data. Global well-posedness and ill-posedness of the cubic defocusing biharmonic NLS was studied in \cite{Pausader09}. It turns out that the cubic defocusing problem is ill-posed in dimensions $n\ge 9$, and well-posed in dimensions $n\le 8$, while the scattering holds true for dimensions $5\le n\le 8$. Other scattering results in the one dimensional scenario were obtained by \cite{Seg06} and \cite{Seg06-2}, while the high dimensional scattering problems were studied in \cite{Pausader09}, \cite{Miao2009}, \cite{Miao2015}, \cite{Wang2012}, \cite{Ruz16}, \cite{Pausader13}, and \cite{Pausader07-2}. The last paper in particular proves the Levandosky-Strauss conjecture in the defocusing case. The blow-up phenomenon for the biharmonic NLS was studied in \cite{Fin2011}, \cite{Zhul2010}, \cite{Zhu11}, \cite{Zhu11-2}, \cite{dinh17}, and \cite{boul17}.
The references given above studied the fourth order Schrödinger equation in the whole Euclidean space, namely the spatial domain was assumed to be equal to $\mathbb{R}^n$. The absence of the boundary in these studies simplified the mathematical and physical analysis of the problem to some extent. However, in order to boost the physical reality, it is common to assume that the evolution takes place in a region with boundary, and what happens at the boundary influences the nature of the solutions. This is especially important for a control scientist, since boundary can be used as a control point, particularly when it is difficult or impossible to access the medium of the evolution. This idea motivated some of the recent studies related with the controllability of the the solutions of the linear fourth order Schrödinger equation. For instance, \cite{Wen16-2}, \cite{Wen14}, and \cite{Wen16} studied the well-posedness and exact controllability of the linear biharmonic Schrödinger equation on a bounded domain $\Omega\subset \mathbb{R}^n$. Most recently, \cite{Aksas17} studied the stabilization of the linear biharmonic Schrödinger equation on a bounded domain with a locally supported internal damping.
From the physical point of view, the model under consideration in this paper corresponds to a situation in which the wave is generated from a fixed source such that the wave train moves into the medium in one specific direction. Wellposedness of similar inhomogenenous initial boundary value problems on the half-line were recently considered for the classical Schrödinger equation; see for example \cite{Carroll}, \cite{bu92}, \cite{bona}, \cite{holmer}, and \cite{fokas}. We prove the corresponding wellposedness theorems for the biharmonic Schrödinger equations, and as far as we know this is the first treatment of the fourth order Schrödinger equations subject to inhomogeneous boundary conditions.
\subsection{Main results}
In this paper, attention is given only to the biharmonic NLS. More general fourth order Schrödinger equations with mixed dispersion as in \eqref{mixeddisp} will be taken into consideration in a further study. Our first main result is the local well-posedness of solutions in fractional Sobolev spaces. More precisely, we prove the following theorem.
\begin{thm}[Local wellposedness I] Let $T>0$, $s\in \left(\frac{1}{2},\frac{9}{2}\right)$, $s\neq \frac{3}{2}$, $p>0$, $q_0\in H^s(\mathbb{R_+})$, $\displaystyle g_0\in H^{\frac{2s+3}{8}}(0,T)$, $\displaystyle g_1\in H^{\frac{2s+1}{8}}(0,T)$, $q_0(0)=g_0(0)$, (also $q_0'(0)=g_1(0)$ if $s>\frac{3}{2}$). Then, \eqref{4th.1}-\eqref{4th.4} has the following local wellposedness properties:
\begin{itemize}
\item[(i)] Local existence and uniqueness: there exists a unique local solution $q\in C([0,T_0];H^s(\mathbb{R_+}))$ for some ${T_0\in (0,T]}$,
\item[(ii)] Continuous dependence: if $B$ is a bounded subset of $H^s(\mathbb{R}_+)\times H^{\frac{2s+3}{8}}(0,T_0)\times H^{\frac{2s+1}{8}}(0,T_0)$, then there is $T_0>0$ such that the flow $(q_0,g_0,g_1)\rightarrow q$ is Lipschitz continuous from $B$ into $C([0,T_0];H^s(\mathbb{R_+}))$,
\item[(iii)] Blow-up alternative: Let $S$ be the set of all $T_0\in (0,T]$ such that there exists a unique local solution in $C([0,T_0];H^s(\mathbb{R_+}))$. If $\displaystyle T_{max}:=\sup_{T_0\in S}T_0<T$, then $\displaystyle\lim_{t\uparrow T_{max}}\|q(t)\|_{H^s(\mathbb{R}_+)}=\infty.$
\end{itemize}
\end{thm}
\begin{rem}
The proof of the above theorem is based on the Fokas method (\cite{fokas1}, \cite{fokasb}) combined with classical contraction arguments. The Fokas method is a unified approach for solving initial-(inhomogeneous) boundary value problems for a general class of linear evolution equations. It has significant advantages over traditional methods. One of these advantages is that the solution formula obtained via the Fokas method is uniformly convergent at the boundary points. This is an important property for numerical studies.
Another remarkable feature of the Fokas method is that one can obtain the necessary space and time estimates for the corresponding linear evolution operator directly from the representation formula by using Fourier analysis. This allows one to easily study the wellposedness of the initial-boundary value problem for corresponding nonlinear models. The study of rigorous wellposedness analysis of nonlinear initial-boundary value problems using the Fokas method was initiated by \cite{fokasKdV}, \cite{fokas}. In these studies, the authors determine the fractional Sobolev spaces for initial and boundary data for which the local Hadamard well-posedness holds true. Similar results have been obtained for other models such as the \emph{good} Boussinesq equation \cite{himonas15}, and two dimensional nonlinear Schrödinger \cite{himarx1} and reaction diffusion equations \cite{himarx2}.
\end{rem}
Second, we prove the local existence and uniqueness at the low regularity setting $s<1/2$.
\begin{thm}[Low regularity]\label{thmlowreg} Let $T>0$, $s\in [0,\frac{1}{2})$, $p\in (0,\frac{8}{1-2s}]$, $q_0\in H^s(\mathbb{R_+})$, $\displaystyle g_0\in H^{\frac{2s+3}{8}}(0,T)$, $\displaystyle g_1\in H^{\frac{2s+1}{8}}(0,T)$, $\lambda=\frac{8(p+2)}{p(1-2s)}$, and $r=\frac{p+2}{1+sp}$. Then, \eqref{4th.1}-\eqref{4th.4} has a unique solution $q\in C([0,T_0];H^s(\mathbb{R_+}))\cap L^\lambda(0,T_0;W^{s,r}(\mathbb{R}_+))$ for some ${T_0\in (0,T]}$.
\end{thm}
Finally, we prove the global wellposedness of weak (more precisely $H^2$) solutions for the defocusing problem. We are able to prove the global wellposedness for $p\le 2$, as opposed to $p>0$, which is the case for the problem posed with homogeneous boundary conditions.
\begin{thm}[Global wellposedness] Let $\kappa\in \mathbb{R}_{-}$ (defocusing nonlinearity), $T>0$, $p\le 2$, $q_0\in H^2(\mathbb{R_+})$, $g_0\in H^1(0,T)$, $g_1\in H^1(0,T)$, $q_0(0)=g_0(0)$, $q_0'(0)=g_1(0)$. Then, the corresponding local solution $q\in C([0,T];H^2(\mathbb{R}_+))$ is global. Moreover, the global solution $q$ satisfies the hidden trace regularities given by $q_{xx}(0,\cdot), q_{xxx}(0,\cdot)\in L^2(0,T)$.
\end{thm}
\begin{rem}
The global wellposedness problem for the nonlinear biharmonic Schrödinger equation is a nontrivial problem in the presence of inhomogeneous boundary conditions as opposed to the case of homogeneous boundary conditions. The main difficulty lies in the fact that one looses all energy conservation and control properties once $g_0$ and $g_1$ are non-zero. For instance, the most basic energy identiy, namely the $L^2$-energy, satisfies an equality given by $$\frac{1}{2}\frac{d}{dt}\int_0^\infty|q|^2dx=\text{Im}\left[q_{xxx}(0,t)\bar{g}_0(t)\right]-\text{Im}\left[q_{xx}(0,t)\bar{g}_1(t)\right].$$ This identity involves the unknown traces $q_{xxx}(0,t)$ and $q_{xx}(0,t)$. Higher order energy estimates are even more complicated.
The extra regularity result $q_{xx}(0,\cdot), q_{xxx}(0,\cdot)\in L^2(0,T)$ proved in the above theorem is called a hidden trace regularity since in general for an arbitrary $H^2$ function, these traces do not need to be well-defined in the sense of Sobolev trace theory. Hence, this shows that the biharmonic Schrödinger operator has a hidden regularizing property in the sense of traces.
\end{rem}
\subsection{Orientation}
We prove the main results in several steps:
\subsubsection*{Step 1 - Representation formula.} We use the Fokas method (also known as the \emph{unified transform method}) to obtain a representation formula for the solution of the linear biharmonic Schrödinger equation with interior force and inhomogenenous boundary inputs. The derivation of the representation formula is more complicated than the classical Schrödinger equation due to the higher order nature of the evolution operator.
\subsubsection*{Step 2 - Cauchy problem.} Secondly, we study the Cauchy problem on the spatial domain $\mathbb{R}$. We obtain the necessary space and times estimates on the solutions of the Cauchy problem. These estimates are later used to study the half-line problem with zero boundary inputs by extending the given initial datum from half-line to the whole line.
\subsubsection*{Step 3 - Half line problem.} We use the representation formula obtained in Step 1 to study the half-line problem with zero initial data and inhomogeneous Dirichlet-Neumann boundary inputs. We obtain space and time estimates on the solutions of the half-line problem. The analysis poses more challenges than the Schrödinger equation with only Dirichlet input, because the inhomogeneous Neumann input here requires us to deal with integrals that involve singular integrands. Singularities are treated with cut-off functions.
\subsubsection*{Step 4 - Operator theoretic formula.} In this step, we express the representation formula of the linear problem with internal force in operator theoretic form. Then, we replace the internal source with the given nonlinearity and applying the contraction argument to this form of the representation formula to obtain the local wellposedness.
\subsubsection*{Step 5 - Strichartz estimates.} We treat the low regularity case $s<\frac{1}{2}$ for the nonlinear model by proving Strichartz estimates by using the oscillatory integral theory.
\subsubsection*{Step 6 - Global wellposedness.} Finally we use the multiplier method to obtain useful energy estimates that involve information on the unknown boundary traces. The classical multipliers for the biharmonic Schrödinger equations are not sufficient. Therefore, we use also the control theoretic multipliers to prove hidden trace regularities for the second and third order boundary traces of the solution. We combine these with $H^2$ energy identities to deduce the global wellposedness of solutions up to cubic powers. The results obtained here are quite interesting compared to solutions of the homogeneous boundary value problem, whose solutions can be shown to be global for all powers in the defocusing case.
\section{Linear model}
In this section, we consider the linear biharmonic Schrödinger equation on the right half-line with inhomogeneous Dirichlet-Neumann data in $L^2-$based fractional Sobolev spaces:
\begin{align}
&iq_{t}+\partial_x^4q=f, \quad (x,t)\in\mathbb{R}_+\times (0,T),\label{1.1}\\
&q(x,0)=q_{0}(x), \quad x\in\mathbb{R}_+,\label{1.2}\\
&q(0,t)=g_{0}(t), \quad t\in(0,T),\label{1.3} \\
&q_{x}(0,t)=g_{1}(t), \quad t\in(0,T).\label{1.4}
\end{align}
\subsection{Representation formula} We will obtain a representation formula for the solution of \eqref{1.1}-\eqref{1.4}. In order to do this, we will use the Fokas (unified transform) method. To this end, let $\hat{q}(k,t)$ denote the spatial Fourier transform of the solution of \eqref{1.1}-\eqref{1.4} on the right half-line:
\begin{equation}\label{qhatkt}
\hat{q}(k,t) \equiv \int_0^\infty e^{-ikx}q(x,t)dx, \quad \text{Im}\,k\le 0.
\end{equation} Note that the condition $\text{Im}\,k\le 0$ is enforced so that the right hand side of the formula \eqref{qhatkt} will in general converge. Applying this transform to the problem \eqref{1.1}-\eqref{1.4}, after some computations we get
\begin{multline}\label{qthatkt}
i\hat{q}_t(k,t)+k^4\hat{q}(k,t)-q_{xxx}(0,t)-ikq_{xx}(0,t)+k^2q_x(0,t)+ik^3q(0,t)=\hat{f}(k,t), \quad \text{Im}\,k\le 0.
\end{multline}
Integrating the above equation in the temporal variable, we obtain
\begin{multline}\label{Intqthatkt}
e^{-ik^4t}\hat{q}(k,t)=\hat{q}_0(k)\\
-\int_0^te^{-ik^4s}\left[iq_{xxx}(0,s)-kq_{xx}(0,s)-ik^2q_x(0,s)+k^3q(0,s)+i\hat{f}(k,s)\right]ds, \quad \text{Im}\,k\le 0.
\end{multline} Introducing the formula \begin{equation}\label{qtildej}
\tilde{g_{j}}(k,t)=\int_{0}^{t}e^{ks}\partial_x^jq(0,s)ds,\quad k\in\mathbb{C}, \,j= \overline{0,3},
\end{equation} we can rewrite \eqref{Intqthatkt} as
\begin{multline}\label{IntqthatktRe}
e^{-ik^4t}\hat{q}(k,t)=\hat{q}_0(k)-i\tilde{g}_3(-ik^4,t)+k\tilde{g}_2(-ik^4,t)+ik^2\tilde{g}_1(-ik^4,t)-k^3\tilde{g}_0(-ik^4,t)\\
-i\int_0^te^{-ik^4s}\hat{f}(k,s)ds, \quad \text{Im}\,k\le 0.
\end{multline}
Multiplying both sides of \eqref{IntqthatktRe} by $e^{ik^4t}$ and then taking the inverse Fourier transform, we get
\begin{multline}\label{qxt}
{q}(x,t)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{ikx+ik^4t}\hat{q}_0(k)dk\\
-\frac{1}{2\pi}\int_{-\infty}^\infty e^{ikx+ik^4t}\left[i\tilde{g}_3(-ik^4,t)-k\tilde{g}_2(-ik^4,t)-ik^2\tilde{g}_1(-ik^4,t)+k^3\tilde{g}_0(-ik^4,t)\right]dk\\
-\frac{i}{2\pi}\int_{-\infty}^\infty e^{ikx+ik^4t}\left[\int_0^te^{-ik^4s}\hat{f}(k,s)ds\right]dk, 0<x<\infty, t>0.
\end{multline}
The above formula consists of two unknown boundary traces, namely $q_{xxx}(0,t)$ and $q_{xx}(0,t)$. The idea of the uniform transform method is based on eliminating these unknown quantities from the solution formula. This is achieved in two steps:
\begin{enumerate}
\item[(1)] We deform the contour of integration involving unknown quantities from $(-\infty,\infty)$ to another appropriate contour.
\item[(2)] We take advantage of the invariant properties of equation \eqref{IntqthatktRe} satisfied by $\hat{q}(k,t)$.
\end{enumerate}
To this end, we first consider the region $D$ described by
\begin{equation} D\equiv \{k\in\mathbb{C}\,|\,\text{Re}\,(-ik^4)<0\}.
\end{equation} It is easy to show that the above region can also be written by
\begin{equation} D\equiv \left\{k\in\mathbb{C}\,|\,\text{Arg}\,k\in \bigcup_{m=0}^3\left(\frac{(2m+1)\pi}{4},\frac{(m+1)\pi}{2}\right)\right\},
\end{equation} where the principle argument of a complex number is assumed to be defined in the interval $[0,2\pi).$
Now, we split $D$ in two disjoint parts depending on whether $k\in D$ is in the upper or lower half-plane: $D^+\equiv D\cap \mathbb{C}^+$ and $D^-\equiv D\cap \mathbb{C}^-$. See Figure \ref{D+D-} below.
\begin{figure}
\caption{The region $D=D^+\cup D^-$}
\label{D+D-}
\end{figure}
The following lemma follows from the the unified theory given in \cite[Proposition 1.1, Chapter 1]{fokasb}:
\begin{lem}[Deformation]
Let $q$ be a solution of \eqref{1.1} on $\Omega\equiv \mathbb{R}_+\times (0,T)$ such that $q$ is sufficiently smooth up to the boundary of $\Omega$ and decays sufficiently fast as $x\rightarrow \infty$, uniformly in $[0,T]$. Then, $q(x,t)$ can be represented by
\begin{multline}\label{qDeformed}
{q}(x,t)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{ikx+ik^4t}\hat{q}_0(k)dk
-\frac{i}{2\pi}\int_{-\infty}^\infty e^{ikx+ik^4t}\left[\int_0^te^{-ik^4s}\hat{f}(k,s)ds\right]dk\\
-\frac{1}{2\pi}\int_{\partial D^+} e^{ikx+ik^4t}\tilde{g}(k)dk,
\end{multline} where $\tilde{g}(k)=i\tilde{g}_3(-ik^4,T)-k\tilde{g}_2(-ik^4,T)-ik^2\tilde{g}_1(-ik^4,T)+k^3\tilde{g}_0(-ik^4,T)$ and $\partial D^+$ is oriented in such a way that $D^+$ is to the left of $\partial D^+$.
\end{lem}
Now, we will use the invariant properties of the equation \eqref{IntqthatktRe}. Replacing $k$ by $-k$ in this equation keeps $\tilde{g}_j(-ik^4,T)$ invariant for $j = \overline{0,3}$, and one obtains
\begin{multline}\label{Intqthatkt-b}
e^{-ik^4T}\hat{q}(-k,T)=\hat{q}_0(-k)-i\tilde{g}_3(-ik^4,T)-k\tilde{g}_2(-ik^4,T)+ik^2\tilde{g}_1(-ik^4,T)+k^3\tilde{g}_0(-ik^4,T)\\
-i\int_0^Te^{-ik^4s}\hat{f}(-k,s)ds, \quad \text{Im}\,k\ge 0.
\end{multline} Similarly, replacing $k$ by $ik$ and $-ik$, one can keep $\tilde{g}_j(-ik^4,T)$ invariant for $j = \overline{0,3}$. Moreover, we have the identities
\begin{multline}\label{Intqthatkt-c}
e^{-ik^4T}\hat{q}(ik,T)=\hat{q}_0(ik)-i\tilde{g}_3(-ik^4,T)+ik\tilde{g}_2(-ik^4,T)-ik^2\tilde{g}_1(-ik^4,T)+ik^3\tilde{g}_0(-ik^4,T)\\
-i\int_0^Te^{-ik^4s}\hat{f}(ik,s)ds, \quad \text{Re}\,k\le 0,
\end{multline} and
\begin{multline}\label{Intqthatkt-d}
e^{-ik^4T}\hat{q}(-ik,T)=\hat{q}_0(-ik)-i\tilde{g}_3(-ik^4,T)-ik\tilde{g}_2(-ik^4,T)-ik^2\tilde{g}_1(-ik^4,T)-ik^3\tilde{g}_0(-ik^4,T)\\
-i\int_0^Te^{-ik^4s}\hat{f}(-ik,s)ds, \quad \text{Re}\,k\ge 0,
\end{multline}respectively. Let us define two subregions of $D^+$ (See Figure \ref{D+D-}) by $$D_1^+:=D^+\cap \{\text{Re}\,k\ge 0\}\text{ and }D_2^+:=D^+\cap \{\text{Re}\,k\le 0\}$$ and let us also set the following transformation for the given boundary data: \begin{equation}\label{Gj}G_j(k,t):=\int_0^te^{ks}g_j(s)ds,\,k\in\mathbb{C},j=0,1.\end{equation}
\begin{figure}
\caption{The region $D_1^+$ and $D_2^+$}
\label{D1+D2+}
\end{figure}
The identities \eqref{Intqthatkt-b} and \eqref{Intqthatkt-d} are valid in $D_1^+$. Solving these identities for $\tilde{g}_2$ and $\tilde{g}_3$ and using the fact that $G_j=\tilde{g}_j$ for $j=0,1$, we obtain
\begin{multline}\label{g2tilded1}\tilde{g}_2(-ik^4,T)=\frac{e^{-ik^4T}}{k(1-i)}\left(\hat{q}(-ik,T)-\hat{q}(-k,T)\right)+\frac{\hat{q}_{0}(-k)-\hat{q}_{0}(-ik)}{k(1-i)}\\
+\frac{k^{2}(1+i)}{1-i}G_0(-ik^{4},T)+\frac{2ik}{1-i}G_1(-ik^{4},T)+\frac{i-1}{2k}\int_0^Te^{-ik^4s}\left[\hat{f}(-ik,s)-\hat{f}(-k,s)\right]ds\end{multline} and
\begin{multline}\label{g3tilded1}\tilde{g}_3(-ik^4,T)=-\frac{e^{-ik^4T}}{1+i}\left(\hat{q}(-ik,T)-i\hat{q}(-k,T)\right)+\frac{-i\hat{q}_{0}(-k)+\hat{q}_{0}(-ik)}{1+i}\\
-\frac{2ik^{3}}{1+i}G_0(-ik^{4},T)+\frac{k^2(1-i)}{1+i}G_1(-ik^{4},T)+\frac{i-1}{2}\int_0^Te^{-ik^4s}\left[i\hat{f}(-ik,s)+\hat{f}(-k,s)\right]ds\end{multline} for $k\in D_1^+$. Similarly, by using the identities \eqref{Intqthatkt-b} and \eqref{Intqthatkt-c}, which are valid on $D_2^+$, we have
\begin{multline}\label{g2tilded2}\tilde{g}_2(-ik^4,T)=\frac{e^{-ik^4T}}{k(1+i)}\left(\hat{q}(ik,T)-\hat{q}(-k,T)\right)+\frac{\hat{q}_{0}(-k)-\hat{q}_{0}(ik)}{k(1+i)}\\
+\frac{k^{2}(1-i)}{1+i}G_0(-ik^{4},T)+\frac{2ik}{1+i}G_1(-ik^{4},T)+\frac{1+i}{2k}\int_0^Te^{-ik^4s}\left[\hat{f}(ik,s)-\hat{f}(-k,s)\right]ds\end{multline} and
\begin{multline}\label{g3tilded2}\tilde{g}_3(-ik^4,T)=\frac{e^{-ik^4T}}{1-i}\left(\hat{q}(ik,T)+i\hat{q}(-k,T)\right)-\frac{i\hat{q}_{0}(-k)+\hat{q}_{0}(ik)}{1-i}\\
-\frac{2ik^{3}}{1-i}G_0(-ik^{4},T)+\frac{k^2(1+i)}{1-i}G_1(-ik^{4},T)+\frac{1+i}{2}\int_0^Te^{-ik^4s}\left[i\hat{f}(ik,s)-\hat{f}(-k,s)\right]ds\end{multline} for $k\in D_2^+$. Using \eqref{g2tilded1}-\eqref{g3tilded2} in \eqref{qDeformed} together with the fact that $G_j=\tilde{g}_j$ for $j=0,1$, we deduce the following identity:
\begin{multline}\label{qDeformednew}
{q}(x,t)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{ikx+ik^4t}\hat{q}_0(k)dk-\frac{1}{2\pi}\int_{\partial D_1^+} e^{ikx+ik^4(t-T)}\left[-(1+i)\hat{q}(-ik,T)+i\hat{q}(-k,T)\right]dk\\
-\frac{i}{2\pi}\int_{-\infty}^\infty e^{ikx+ik^4t}\left[\int_0^te^{-ik^4s}\hat{f}(k,s)ds\right]dk-\frac{1}{2\pi}\int_{\partial D_1^+} e^{ikx+ik^4t}\left[(1+i)\hat{q}_0(-ik)-i\hat{q}_0(-k)\right]dk\\
-\frac{1}{2\pi}\int_{\partial D_1^+} e^{ikx+ik^4t}\left[2k^3(1-i)G_0(-ik^4,T)+2k^2\left(1-i\right)G_1(-ik^4,T)\right]dk\\
-\frac{1}{2\pi}\int_{\partial D_1^+}e^{ikx+ik^4t}\left[\int_0^Te^{-ik^4s}\left((1-i)\hat{f}(-ik,s)-\hat{f}(-k,s)\right)ds\right]dk\\
-\frac{1}{2\pi}\int_{\partial D_2^+} e^{ikx+ik^4(t-T)}\left[(i-1)\hat{q}(ik,T)-i\hat{q}(-k,T)\right]dk\\
-\frac{1}{2\pi}\int_{\partial D_2^+} e^{ikx+ik^4t}\left[(1-i)\hat{q}_0(ik)+i\hat{q}_0(-k)\right]dk\\
-\frac{1}{2\pi}\int_{\partial D_2^+} e^{ikx+ik^4t}\left[2k^3(1+i)G_0(-ik^4,T)-2k^2(1+i)G_1(-ik^4,T)\right]dk\\
-\frac{1}{2\pi}\int_{\partial D_2^+}e^{ikx+ik^4t}\left[\int_0^Te^{-ik^4s}\left(\hat{f}(-k,s)-(1+i)\hat{f}(ik,s)\right)ds\right]dk.
\end{multline}
The second and the seventh integrals at the right hand side of \eqref{qDeformednew} vanish by Cauchy's theorem and one obtains the following representation formula where the right hand side includes information coming only from the prescribed initial-boundary-interior data:
\begin{multline}\label{qDeformednew2}
{q}(x,t)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{ikx+ik^4t}\hat{q}_0(k)dk-\frac{i}{2\pi}\int_{-\infty}^\infty e^{ikx+ik^4t}\left[\int_0^te^{-ik^4s}\hat{f}(k,s)ds\right]dk\\-\frac{1}{2\pi}\int_{\partial D_1^+} e^{ikx+ik^4t}\left[(1+i)\hat{q}_0(-ik)-i\hat{q}_0(-k)\right]dk\\
-\frac{1}{\pi}\int_{\partial D_1^+} e^{ikx+ik^4t}\left[k^3(1-i)G_0(-ik^4,T)+k^2\left(1-i\right)G_1(-ik^4,T)\right]dk\\
-\frac{1}{2\pi}\int_{\partial D_1^+}e^{ikx+ik^4t}\left[\int_0^Te^{-ik^4s}\left((1-i)\hat{f}(-ik,s)-\hat{f}(-k,s)\right)ds\right]dk\\
-\frac{1}{2\pi}\int_{\partial D_2^+} e^{ikx+ik^4t}\left[(1-i)\hat{q}_0(ik)+i\hat{q}_0(-k)\right]dk\\
-\frac{1}{\pi}\int_{\partial D_2^+} e^{ikx+ik^4t}\left[k^3(1+i)G_0(-ik^4,T)-k^2(1+i)G_1(-ik^4,T)\right]dk\\
-\frac{1}{2\pi}\int_{\partial D_2^+}e^{ikx+ik^4t}\left[\int_0^Te^{-ik^4s}\left(\hat{f}(-k,s)-(1+i)\hat{f}(ik,s)\right)ds\right]dk.
\end{multline}
\subsection{Cauchy problem} In this section, we consider the biharmonic linear Schrödinger equation on the whole real line $\mathbb{R}$:
\begin{align}
&iy_{t}+\partial_x^4y=0, \quad (x,t)\in\mathbb{R}\times (0,T),\label{w1.1}\\
&y(x,0)=y_{0}(x), \quad x\in\mathbb{R}.\label{w1.2}
\end{align}
One has the following regularity properties regarding the Cauchy problem \eqref{w1.1}-\eqref{w1.2}.
\begin{lem}\label{Wrty0}
Let $s\in \mathbb{R}$ and $y_0\in H^s(\mathbb{R})$. Then the solution of \eqref{w1.1}-\eqref{w1.2}, denoted $y(t)=S_{\mathbb{R}}(t)y_0$, satisfies $y\in C([0,T];H^s(\mathbb{R}))$ with the conservation law \begin{equation}\label{conservationlaw}
\|y(\cdot,t)\|_{H^s(\mathbb{R})}=\|y_0\|_{H^s(\mathbb{R})},\,\,\,t\in [0,T].
\end{equation}
If $s\ge -\frac{3}{2}$, then $\displaystyle y\in C(\mathbb{R};H^{\frac{2s+3}{8}}(0,T))$ and there exists a constant $c_s\ge 0$ such that \begin{equation}\label{timeest001}\sup_{x\in\mathbb{R}}\|y(x,\cdot)\|_{H^{\frac{2s+3}{8}}(0,T)}\le c_s(1+T^{\frac{1}{2}})\|y_{0}\|_{H^{s}(\mathbb{R})}.\end{equation} Moreover, if $s\ge -\frac{1}{2}$, then $y$ has the additional regularity $y_x\in C(\mathbb{R};H^{\frac{2s+1}{8}}(0,T))$ with the estimate
\begin{equation}\label{timeest002}\sup_{x\in\mathbb{R}}\|\partial_xy(x,\cdot)\|_{H^{\frac{2s+1}{8}}(0,T)}\le c_s(1+T^{\frac{1}{2}})\|y_{0}\|_{H^{s}(\mathbb{R})}\end{equation} for some constant $c_s\ge 0$.
\end{lem}
\begin{proof} We prove this lemma by using arguments similar to the ones given in the proof of \cite[Theorem 4]{fokas}, which are based on the Fourier representation of the solution. There are a few crucial differences, particularly in the temporal regularity, compared to the classical Schrödinger equation due to the biharmonic evolution operator considered here.
Upon taking the Fourier transform of \eqref{w1.1}-\eqref{w1.2} in the spatial variable, we find
\begin{eqnarray}
\hat{y}(\xi,t)=e^{i\xi^{4}t}\hat{y}_{0}(\xi)\label{2.3}
\end{eqnarray} where $\hat{y}_{0}$ is the Fourier transform of $y_0$. Note that $$|\hat{y}(\xi,t)|=|e^{i\xi^{4}t}\hat{y}_{0}(\xi)|=|\hat{y}_0(\xi)|$$ for $\xi\in\mathbb{R}$, $t\in [0,T]$. Therefore,
\begin{equation}\|y(t)\|_{H^s(\mathbb{R})}=\|y_0\|_{H^s(\mathbb{R})}\end{equation} and $y(t)\in H^s(\mathbb{R})$ for $t\in [0,T]$. Moreover, $t\rightarrow y(\cdot,t)$ is continuous from $[0,T]$ into $H^s(\mathbb{R}).$ In order to see this, let $t,t_n\in [0,T]$ be such that $t_n\rightarrow t$. Then,
\begin{eqnarray*}
\|y(t_{n})-y(t)\|_{H^{s}(\mathbb{R})}^2=\int_{\mathbb{R}}(1+\xi^{2})^{s}|e^{i\xi^{4}t_{n}}-e^{i\xi^{4}t}|^{2}|\hat{y}_{0}(\xi)|^{2}d\xi,
\end{eqnarray*} where $$\displaystyle\lim_{n\rightarrow\infty}(1+\xi^{2})^{s}|e^{i\xi^{4}t_{n}}-e^{i\xi^{4}t}|^{2}|\hat{y}_{0}(\xi)|^{2}\rightarrow 0$$ for $\xi\in\mathbb{R}$ and $$(1+\xi^{2})^{s}|e^{i\xi^{4}t_{n}}-e^{i\xi^{4}t}|^{2}|\hat{y}_{0}(\xi)|^{2}\le 4(1+\xi^{2})^{s}|\hat{y}_{0}(\xi)|^{2}$$ for $\xi\in\mathbb{R}.$ Note that the right hand side of the above inequality is a nonnegative integrable function since $$4\int_{\mathbb{R}}(1+\xi^{2})^{s}|\hat{y}_{0}(\xi)|^{2}d\xi=4\|y_0\|_{H^s(\mathbb{R})}^2<\infty.$$ Hence, by the dominated convergence theorem $\displaystyle\lim_{n\rightarrow \infty}\|y(t_{n})-y(t)\|_{H^{s}(\mathbb{R})}=0$. In other words, $y(t_n)\rightarrow y(t)$ in $H^s(\mathbb{R})$. Hence, we have just proved that $y\in C([0,T];H^s(\mathbb{R}))$.
In order to prove $y\in C(\mathbb{R};H^{\frac{2s+3}{8}}(0,T))$, we first write $y=y_1+y_2$, where $$y_1(x,t)\equiv \frac{1}{2\pi}\int_\mathbb{R}e^{i\xi x+i\xi^{4}t}\theta(\xi)\hat{y}_{0}(\xi)d\xi,$$
$$y_2(x,t)\equiv \frac{1}{2\pi}\int_\mathbb{R}e^{i\xi x+i\xi^{4}t}\left(1-\theta(\xi)\right)\hat{y}_{0}(\xi)d\xi,$$ and $\theta$ is a smooth cut-off function satisfying $\theta\equiv 1$ for $|\xi|\le 1$, $0\le \theta\le 1$ for $1<|\xi|<2$, and $\theta\equiv 0$ for $|\xi|\ge 2$. Taking the $j^{\text{th}}$ order time derivative of $y_1$ with $0\le j\le m$, using the definition of $\theta$, Cauchy-Schwarz inequality, and the definition of the Sobolev norm, one deduces that there exists a non-negative constant $c(s,m)\ge 0$ such that
\begin{eqnarray}
\|y_{1}(x)\|_{H^m(0,T)}\leq c(s,m)T^{\frac{1}{2}}\|y_{0}\|_{H^{s}(\mathbb{R})}\label{2.14},
\end{eqnarray}
at first for all $m\in\mathbb{N}$, and then by interpolation for all $m\ge 0$.
In order to obtain a similar estimate for $y_2$, we split it into two terms and write $y_2=y_{2,1}+y_{2,2}$, where
$$y_{2,1}(x,t)\equiv \frac{1}{2\pi}\int_{-\infty}^{-1}e^{i\xi x+i\xi^{4}t}\left(1-\theta(\xi)\right)\hat{y}_{0}(\xi)d\xi$$ and
$$y_{2,2}(x,t)\equiv \frac{1}{2\pi}\int_1^\infty e^{i\xi x+i\xi^{4}t}\left(1-\theta(\xi)\right)\hat{y}_{0}(\xi)d\xi.$$ Let us first consider the integral given by $y_{2,1}$. We change the variables in this integral by setting $\xi^4=-\tau$ and we define $\displaystyle z^{\frac{1}{4}}$ for $z\in \mathbb{R}_+$ to be the negative real number which is obtained by taking the argument of $z$ as $4\pi$. Then, $y_{2,1}$ can be rewritten as
$$y_{2,1}(x,t)= -\frac{1}{8\pi}\int_{-\infty}^{-1}e^{i(-\tau)^{\frac{1}{4}}x+i\tau t}\left(1-\theta\left((-\tau)^{\frac{1}{4}}\right)\right)\hat{y}_{0}\left((-\tau)^{\frac{1}{4}}\right)\frac{d\tau}{(-\tau)^{\frac{3}{4}}}.$$ The above formula can be thought of as the inverse (temporal) Fourier transform of the function
\[\displaystyle
\hat{y}_{2,1}(x,\tau):=\left\{
\begin{array}{ll}
0, &\tau\in[-1,\infty), \\
-e^{i(-\tau)^{\frac{1}{4}}x}\left(1-\theta\left((-\tau)^{\frac{1}{4}}\right)\right)\hat{y}_{0}\left((-\tau)^{\frac{1}{4}}\right)\frac{1}{4(-\tau)^{\frac{3}{4}}}, &\tau\in(-\infty,-1).
\end{array}
\right.
\] Note then,
\begin{multline}\label{timeest1}
\|y_{2,1}(x,\cdot)\|_{H^{\frac{2s+3}{8}}(0,T)}^2\le \|y_{2,1}(x,\cdot)\|_{H^{\frac{2s+3}{8}}(\mathbb{R})}^2=\int_{\mathbb{R}}(1+\tau^2)^{\frac{2s+3}{8}}|\hat{y}_{2,1}(x,\tau)|^2d\tau \\
= \int_{-\infty}^{-1}(1+\tau^2)^{\frac{2s+3}{8}}\left(1-\theta\left((-\tau)^{\frac{1}{4}}\right)\right)^2\left|\hat{y}_{0}\left((-\tau)^{\frac{1}{4}}\right)\right|^2\frac{1}{16(-\tau)^{\frac{3}{2}}}d\tau\\
=-\int_{-\infty}^{-1}(1+\xi^8)^{\frac{2s+3}{8}}\left(1-\theta\left(\xi\right)\right)^2\left|\hat{y}_{0}\left(\xi\right)\right|^2\frac{1}{4\xi^3}d\xi
\le \int_{-\infty}^{-1}(1+\xi^8)^{\frac{2s+3}{8}}\left|\hat{y}_{0}\left(\xi\right)\right|^2\frac{1}{4\xi^3}d\xi\\
\lesssim \int_{\mathbb{R}}(1+\xi^2)^{s}\left|\hat{y}_{0}\left(\xi\right)\right|^2d\xi=\|y_0\|_{H^s(\mathbb{R})}^2.
\end{multline} The same estimate is also true for $y_{2,2}$ by similar arguments. Now, using these two estimates together with \eqref{2.14}, one obtains
$$\|y(x,\cdot)\|_{H^{\frac{2s+3}{8}}(0,T)}\le c_s(1+T^{\frac{1}{2}})\|y_{0}\|_{H^{s}(\mathbb{R})}$$ for $s\ge -\frac{3}{2}$. Continuity of the map $x\rightarrow y(x,\cdot)$ can be shown by using the dominated convergence theorem again.
Differentiating the problem \eqref{w1.1}-\eqref{w1.2} in $x$ and repeating the above analysis with initial data $y_0'\in H^{s-1}(\mathbb{R})$, we deduce that $y_x\in C(\mathbb{R};H^{\frac{2s+1}{8}}(0,T))$ if $s\ge -\frac{1}{2}$.
\end{proof}
\subsection{Boundary data-to-solution operator}\label{bdrtosol} In this section, we consider the biharmonic Schrödinger equation with zero initial datum and inhomogeneous Dirichlet-Neumann boundary inputs:
\begin{align}
&iz_{t}+\partial_x^4z=0, \quad (x,t)\in\mathbb{R}_+\times (0,T'),\label{3.1}\\
&z(x,0)=0, \quad x\in\mathbb{R}_+,\label{3.2}\\
&z(0,t)=h_{0}(t), \quad t\in(0,T'),\label{3.3} \\
&z_{x}(0,t)=h_{1}(t), \quad t\in(0,T').\label{3.4}
\end{align}
\subsubsection*{Compatibility conditions} In order for solutions to be continuous at the space-time corner point $(x,t)=(0,0)$, we find a set of necessary (compability) conditions based on the analysis of traces. Suppose that $s\ge 0$, $h_0\in H^{\frac{2s+3}{8}}(\mathbb{R})$ and $h_1\in H^{\frac{2s+1}{8}}(\mathbb{R})$. Note that if $s\in \left(\frac{1}{2},\frac{9}{2}\right)$, then $\frac{2s+3}{8}>\frac{1}{2}$, and therefore $h_0(0)$ is well-defined, but $z(0,0)=0$, and hence we must have $h_0(0)=0.$ If $s\in\left(\frac{9}{2},\frac{17}{2}\right)$, then $\frac{2s+3}{8}>\frac{3}{2}$, and therefore $h_0(0), h_0'(0)$ are both well-defined, but $z_x(0,0)=0$, and hence we must have $h_0'(0)=0$ in addition to $h_0(0)=0.$ More generally, if $s\in \left(\frac{1}{2}+4(j-1),\frac{1}{2}+4j\right)$ for some $j\ge 1$, then using also the main equation, we deduce that $\partial_t^kh_0(0)=0$ is a necessary condition for all $0\le k\le j-1$. By similar arguments, we deduce that if $s\in \left(\frac{3}{2}+4(l-1),\frac{3}{2}+4l\right)$ for some $l\ge 1$, then $\partial_t^kh_1(0)=0$ is a necessary condition for all $0\le k\le l-1$.
\subsubsection*{Wellposedness}
We prove the following wellposedness result for the half-line problem \eqref{3.1}-\eqref{3.4}.
\begin{lem}\label{lem239}
Let $s\ge 0$, $s\neq 4j+\frac{1}{2},\,4j+\frac{3}{2}$ for $j\in \mathbb{N}$, $h_0\in H^{\frac{2s+3}{8}}(\mathbb{R})$ and $h_1\in H^{\frac{2s+1}{8}}(\mathbb{R})$, both of which vanish for $t\in \mathbb{R}-(0,T')$ and satisfy the necessary compatibility conditions at $(x,t)=(0,0)$. Then the solution of \eqref{3.1}-\eqref{3.4}, denoted $z(t)=S_{b}([h_0,h_1])(t)$, satisfies $z\in C([0,T'];H^s(\mathbb{R}_+))\cap C(\mathbb{R}_+;H^{\frac{2s+3}{8}}(0,T'))$ with the estimate
\begin{multline}\sup_{t\in [0,T']}\|z(\cdot,t)\|_{H^s(\mathbb{R}_+)}+\sup_{x\in \mathbb{R}_+}\|z(x,\cdot)\|_{H^{\frac{2s+3}{8}}(0,T')}\\
\lesssim \|h_0\|_{H^{\frac{2s+3}{8}}(\mathbb{R})}+\left(1+T'^{\frac{1}{2}}\right)\|h_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R})}.\end{multline}
Moreover, $z$ has the additional regularity $z_x\in C(\mathbb{R}_+;H^{\frac{2s+1}{8}}(0,T'))$.
\end{lem}
\begin{proof}
Using the representation formula \eqref{qDeformednew2}, the solution of \eqref{3.1}-\eqref{3.4} is given by
\begin{multline}\label{zsol}
{z}(x,t)=-\frac{1}{\pi}\int_{\partial D_1^+} e^{ikx+ik^4t}\left[k^3(1-i)H_0(-ik^4,T')+k^2\left(1-i\right)H_1(-ik^4,T')\right]dk\\
-\frac{1}{\pi}\int_{\partial D_2^+} e^{ikx+ik^4t}\left[k^3(1+i)H_0(-ik^4,T')-k^2(1+i)H_1(-ik^4,T')\right]dk,
\end{multline} where $H_0$ and $H_1$ are defined in terms of the formula \eqref{Gj} with data $h_0$ and $h_1$, respectively.
\subsubsection*{Space estimate}
Note that $\partial D_1^+ = \gamma_1\cup \gamma_2$ and $\partial D_2^+ = \gamma_3\cup \gamma_4$ (see Figure \ref{gammas}), where
\begin{eqnarray*}
\gamma_{1}(k):=k+ik, \ \ 0\leq k<\infty, \\
\gamma_{2}(k):=-ik, \ \ -\infty<k\leq 0, \\
\gamma_{3}(k):=-k+ik, \ \ 0\le k< \infty, \\
\gamma_{4}(k):=k, \ \ -\infty<k\leq 0. \\
\end{eqnarray*}
\begin{figure}
\caption{Partitioning the boundary}
\label{gammas}
\end{figure}
Therefore, \eqref{zsol} can be rewritten as
\begin{multline}\label{zsol2}
{z}(x,t)=-\frac{4}{\pi}\int_{0}^\infty e^{-kx+ikx-4ik^4t}\left[(i-1)k^3H_0(4ik^4,T')+ik^2H_1(4ik^4,T')\right]dk\\
+\frac{1-i}{\pi}\int_{0}^\infty e^{-kx+ik^4t}\left[k^3H_0(-ik^4,T')-ik^2H_1(-ik^4,T')\right]dk\\
+\frac{4}{\pi}\int_{0}^\infty e^{-kx-ikx-4ik^4t}\left[(1+i)k^3H_0(4ik^4,T')+ik^2H_1(4ik^4,T')\right]dk\\
+\frac{(1+i)}{\pi}\int_{0}^\infty e^{-ikx+ik^4t}\left[k^3H_0(-ik^4,T')+k^2H_1(-ik^4,T')\right]dk=:\sum_{j=1}^8z_j(x,t).
\end{multline}
We will start by considering the first component of \eqref{zsol2}, which is given by
\begin{equation}\label{z1}z_1(x,t)=-\frac{4}{\pi}\int_{0}^\infty e^{-kx+ikx-4ik^4t}\left[(i-1)k^3H_0(4ik^4,T')\right]dk.\end{equation}
Let us first consider the case $s=0$. In this case, by using the change of variables $\tau=-4k^4$, we get
\begin{multline}\label{z1est0}\|z_1(\cdot,t)\|_{L^2(\mathbb{R}_+)}^2=\frac{16}{\pi^2}\int_{0}^{\infty}\left|\int_{0}^\infty e^{-kx+ikx-4ik^4t}\left[(i-1)k^3H_0(4ik^4,T')\right]dk\right|^{2}dx\\
\le \frac{32}{\pi^2}\int_{0}^{\infty}\left(\int_{0}^\infty e^{-kx}k^3|H_0(4ik^4,T')|dk\right)^{2}dx\le \frac{32}{\pi}\int_{0}^\infty k^6|H_0(4ik^4,T')|^2dk\\
=\frac{32}{\pi}\int_{0}^\infty k^6|\hat{h}_0(-4k^4)|^2dk=\frac{1}{\sqrt{2}\pi}\int_{-\infty}^{0} (-\tau)^\frac{3}{4}|\hat{h}_0(\tau)|^2d\tau\\
\le \frac{1}{\sqrt{2}\pi}\int_{-\infty}^\infty \left(1+{\tau}^2\right)^\frac{3}{8}|\hat{h}_0(\tau)|^2d\tau,\end{multline} where the second inequality above is a property of the Laplace transform (see for example \cite[Lemma 3.2]{fokas}). Taking the square root in \eqref{z1est0}, we obtain \begin{equation}\label{z1est1}
\|z_1(\cdot,t)\|_{L^2(\mathbb{R}_+)}\le \frac{1}{2^{\frac{1}{4}}\sqrt{\pi}}\|h_0\|_{H^{\frac{3}{8}}(\mathbb{R})}.
\end{equation}
Similarly,
\begin{multline}\label{z2est0}\|z_2(\cdot,t)\|_{L^2(\mathbb{R}_+)}^2=\frac{16}{\pi^2}\int_{0}^{\infty}\left|\int_{0}^\infty e^{-kx+ikx-4ik^4t}\left[ik^2H_1(4ik^4,T')\right]dk\right|^{2}dx\\
\le \frac{16}{\pi^2}\int_{0}^{\infty}\left(\int_{0}^\infty e^{-kx}k^2|H_1(4ik^4,T')|dk\right)^{2}dx\le \frac{16}{\pi}\int_{0}^\infty k^4|H_1(4ik^4,T')|^2dk\\
=\frac{16}{\pi}\int_{0}^\infty k^4|\hat{h}_1(-4k^4)|^2dk=\frac{1}{\sqrt{2}\pi}\int_{-\infty}^{0} (-\tau)^\frac{1}{4}|\hat{h}_1(\tau)|^2d\tau\\
\le \frac{1}{\sqrt{2}\pi}\int_{-\infty}^\infty \left(1+{\tau}^2\right)^\frac{1}{8}|\hat{h}_1(\tau)|^2d\tau.\end{multline} Taking the square root in \eqref{z2est0}, we obtain \begin{equation}\label{z2est1}
\|z_2(\cdot,t)\|_{L^2(\mathbb{R}_+)}\le \frac{1}{2^{\frac{1}{4}}\sqrt{\pi}}\|h_1\|_{H^{\frac{1}{8}}(\mathbb{R})}.
\end{equation}
Now, let $s=m\in \mathbb{Z}_+$ and $1\le j\le m$. Then, \begin{multline}\label{z1est2}\|\partial_x^jz_1(\cdot,t)\|_{L^2(\mathbb{R}_+)}^2=\frac{16}{\pi^2}\int_{0}^{\infty}\left|\int_{0}^\infty e^{-kx+ikx-4ik^4t}\left[(i-1)^{j+1}k^{3+j}H_0(4ik^4,T')\right]dk\right|^{2}dx\\
\le \frac{2^{j+5}}{\pi^2}\int_{0}^{\infty}\left(\int_{0}^\infty e^{-kx}k^{3+j}|H_0(4ik^4,T')|dk\right)^{2}dx\le \frac{2^{j+5}}{\pi}\int_{0}^\infty k^{2j+6}|H_0(4ik^4,T')|^2dk\\
=\frac{2^{j+5}}{\pi}\int_{0}^\infty k^{2j+6}|\hat{h}_0(-4k^4)|^2dk=\frac{1}{\sqrt{2}\pi}\int_{-\infty}^{0} (-\tau)^\frac{2j+3}{4}|\hat{h}_0(\tau)|^2d\tau\\
\le \frac{1}{\sqrt{2}\pi}\int_{-\infty}^\infty \left(1+{\tau}^2\right)^\frac{2j+3}{8}|\hat{h}_0(\tau)|^2d\tau,\end{multline} which gives
\begin{equation}\label{z1est3}
\|z_1(\cdot,t)\|_{H^m(\mathbb{R}_+)}\le \frac{1}{2^{\frac{1}{4}}\sqrt{\pi}}\|h_0\|_{H^{\frac{2m+3}{8}}(\mathbb{R})}.
\end{equation}
Similarly,
\begin{multline}\label{z2est2}\|\partial_x^jz_2(\cdot,t)\|_{L^2(\mathbb{R}_+)}^2=\frac{16}{\pi^2}\int_{0}^{\infty}\left|\int_{0}^\infty e^{-kx+ikx-4ik^4t}\left[(i-1)^{j}ik^{2+j}H_1(4ik^4,T')\right]dk\right|^{2}dx\\
\le \frac{2^{j+4}}{\pi^2}\int_{0}^{\infty}\left(\int_{0}^\infty e^{-kx}k^{2+j}|H_1(4ik^4,T')|dk\right)^{2}dx\le \frac{2^{j+4}}{\pi}\int_{0}^\infty k^{2j+4}|H_1(4ik^4,T')|^2dk\\
=\frac{2^{j+4}}{\pi}\int_{0}^\infty k^{2j+4}|\hat{h}_1(-4k^4)|^2dk=\frac{1}{\sqrt{2}\pi}\int_{-\infty}^{0} (-\tau)^\frac{2j+1}{4}|\hat{h}_1(\tau)|^2d\tau\\
\le \frac{1}{\sqrt{2}\pi}\int_{-\infty}^\infty \left(1+{\tau}^2\right)^\frac{2j+1}{8}|\hat{h}_1(\tau)|^2d\tau,\end{multline} which gives
\begin{equation}\label{z2est3}
\|z_2(\cdot,t)\|_{H^m(\mathbb{R}_+)}\le \frac{1}{2^{\frac{1}{4}}\sqrt{\pi}}\|h_1\|_{H^{\frac{2m+1}{8}}(\mathbb{R})}.
\end{equation}
Now, by interpolation between \eqref{z1est1} and \eqref{z1est3}, we obtain
\begin{equation}\label{z1est4}
\|z_1(\cdot,t)\|_{H^s(\mathbb{R}_+)}\lesssim\|h_0\|_{H^{\frac{2s+3}{8}}(\mathbb{R})}
\end{equation} for all $s\ge 0$, including the non-integer values of $s$. Similarly, by interpolating between \eqref{z2est1} and \eqref{z2est3}, we get \begin{equation}\label{z2est4}
\|z_2(\cdot,t)\|_{H^s(\mathbb{R}_+)}\lesssim\|h_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R})}
\end{equation} for $s\ge 0$. Applying the same arguments also to $z_j$ for $3\le j\le 8$, we get
\begin{equation}\label{z34est4}
\|z_3(\cdot,t)\|_{H^s(\mathbb{R}_+)}\lesssim\|h_0\|_{H^{\frac{2s+3}{8}}(\mathbb{R})}\text{ and } \|z_4(\cdot,t)\|_{H^s(\mathbb{R}_+)}\lesssim\|h_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R})},
\end{equation}
\begin{equation}\label{z56est4}
\|z_5(\cdot,t)\|_{H^s(\mathbb{R}_+)}\lesssim\|h_0\|_{H^{\frac{2s+3}{8}}(\mathbb{R})}\text{ and } \|z_6(\cdot,t)\|_{H^s(\mathbb{R}_+)}\lesssim\|h_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R})},
\end{equation}
\begin{equation}\label{z78est4}
\|z_7(\cdot,t)\|_{H^s(\mathbb{R}_+)}\lesssim\|h_0\|_{H^{\frac{2s+3}{8}}(\mathbb{R})}\text{ and } \|z_8(\cdot,t)\|_{H^s(\mathbb{R}_+)}\lesssim\|h_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R})}.
\end{equation}
Combining \eqref{z1est4}-\eqref{z78est4}, we deduce that
\begin{equation}\label{zCnt1}
\|z(\cdot,t)\|_{H^s(\mathbb{R}_+)}\lesssim \|h_0\|_{H^{\frac{2s+3}{8}}(\mathbb{R})}+\|h_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R})}.
\end{equation} Continuity in $t$ can easily be justified by means of the dominated convergence theorem. Therefore we just proved that $z\in C([0,T'];H^s(\mathbb{R}))$ under the given assumptions on $s$, $h_0$ and $h_1$.
\subsubsection*{Time estimate}
We can rewrite \eqref{zsol} as
\begin{multline}\label{zsol3}
{z}(x,t)=-\frac{1}{\pi}\int_{0}^{(1+i)\infty} e^{ikx+ik^4t}\left[k^3(1-i)\hat{h}_0(k^4)+k^2\left(1-i\right)\hat{h}_1(k^4)\right]dk\\
-\frac{1}{\pi}\int_{i\infty}^{0} e^{ikx+ik^4t}\left[k^3(1-i)\hat{h}_0(k^4)+k^2\left(1-i\right)\hat{h}_1(k^4)\right]dk\\
-\frac{1}{\pi}\int_{0}^{(-1+i)\infty} e^{ikx+ik^4t}\left[k^3(1+i)\hat{h}_0(k^4)-k^2(1+i)\hat{h}_1(k^4)\right]dk\\
-\frac{1}{\pi}\int_{-\infty}^{0} e^{ikx+ik^4t}\left[k^3(1+i)\hat{h}_0(k^4)-k^2(1+i)\hat{h}_1(k^4)\right]dk=:\sum_{j=1}^8z_j(x,t).
\end{multline} We will start with considering the first component of \eqref{zsol3}, which is given by
$$z_1(x,t)=-\frac{1}{\pi}\int_{0}^{(1+i)\infty} e^{ikx+ik^4t}\left[k^3(1-i)\hat{h}_0(k^4)\right]dk.$$ By using the change of variables $\tau=k^4$, we get
$$z_1(x,t)=\frac{1-i}{4\pi}\int_{-\infty}^{0} e^{i\tau^\frac{1}{4}x+it\tau}\hat{h}_0(\tau)d\tau,$$ where $z^{\frac{1}{4}}$ for $z\in\mathbb{R}_{-}$ is defined by using the argument equal to $\pi$ so that $\tau^{\frac{1}{4}}\in \gamma_1$ for $\tau\in (-\infty,0)$. Note that we can regard $z_1(x,\cdot)$ as the inverse Fourier transform of the function \[\displaystyle
\hat{z}_{1}(x,\tau):=\left\{
\begin{array}{ll}
0, &\tau\in[0,\infty), \\
\frac{1}{2}e^{i\tau^\frac{1}{4}x}\left[(1-i)\hat{h}_0(\tau)\right], &\tau\in(-\infty,0),
\end{array}
\right.
\] from which it follows that
\begin{multline}\label{z1257}\|z_1(x,\cdot)\|_{H^{\frac{2s+3}{8}}(0,T')}^2\le \|z_1(x,\cdot)\|_{H^{\frac{2s+3}{8}}(\mathbb{R})}^2\\
\le \frac{1}{2}\int_{\mathbb{R}}(1+\tau^2)^{\frac{2s+3}{8}}|\hat{h}_0(\tau)|^2d\tau=\frac{1}{2}\|h_0\|_{H^{\frac{2s+3}{8}}(\mathbb{R})}^2\end{multline} taking into account that $\text{Im}\left(\tau^{\frac{1}{4}}\right)\ge 0$ so that $|e^{i\tau^\frac{1}{4}x}|\le 1$.
Now, we will estimate the term
$$z_2(x,t)=-\frac{1}{\pi}\int_{0}^{(1+i)\infty} e^{ikx+ik^4t}\left[k^2\left(1-i\right)\hat{h}_1(k^4)\right]dk.$$
In order to do this, we write $z_2=z_{2,1}+z_{2,2}$ where
$$z_{2,1}(x,t)=-\frac{1}{\pi}\int_{0}^{(1+i)\infty}e^{ikx+ik^4t} \theta(k)\left[k^2\left(1-i\right)\hat{h}_1(k^4)\right]dk,$$
$$z_{2,2}(x,t)=-\frac{1}{\pi}\int_{0}^{(1+i)\infty}e^{ikx+ik^4t}\left(1-\theta(k)\right)\left[k^2\left(1-i\right)\hat{h}_1(k^4)\right]dk,$$ and $\theta$ is a smooth cut-off function satisfying $\theta\equiv 1$ for $|k|\le \sqrt{2}$, $0\le \theta\le 1$ for $\sqrt{2}<|k|<2\sqrt{2}$, and $\theta\equiv 0$ for $|k|\ge 2\sqrt{2}$ for $k\in \gamma_1$. By using the properties of $\theta$, we can rewrite $z_{2,1}$ as
$$z_{2,1}(x,t)=-\frac{1}{\pi}\int_{0}^{2(1+i)}e^{ikx+ik^4t}\theta(k)\left[k^2\left(1-i\right)\hat{h}_1(k^4)\right]dk.$$ Using the same change of variables $\tau=k^4$ as before, we get
$$z_{2,1}(x,t)=\frac{\left(1-i\right)}{4\pi}\int_{-2^6}^{0}e^{i\tau^{\frac{1}{4}}x+it\tau}\theta\left(\tau^{\frac{1}{4}}\right)\hat{h}_1(\tau)\frac{d\tau}{\tau^{\frac{1}{4}}}.$$
The $j^{\text{th}}$ order time derivative of $z_{2,1}(x,\cdot)$ is then written
$$\partial_t^jz_{2,1}(x,t)=\frac{\left(1-i\right)}{4\pi}\int_{-2^6}^{0}e^{i\tau^{\frac{1}{4}}x+it\tau}\theta\left(\tau^{\frac{1}{4}}\right)(i\tau)^j\hat{h}_1(\tau)\frac{d\tau}{\tau^{\frac{1}{4}}}.$$
We will estimate the above identity in two cases: (i) $s<\frac{3}{2}$, (ii) $s\ge \frac{3}{2}$. If $s<\frac{3}{2}$, then $\frac{2s+1}{8}<\frac{1}{2}$ and $2j-\frac{1}{2}-\frac{2s+1}{8}>-1$ for all $j\in \mathbb{N}$. Therefore, by using the Cauchy-Schwarz inequality:
\begin{multline}
|\partial_t^jz_{2,1}(x,t)|\le \frac{\sqrt{2}}{4\pi}\left(\int_{-2^6}^{0}(1+\tau^2)^{-\frac{2s+1}{8}}\tau^{2j-\frac{1}{2}}d\tau\right)^{\frac{1}{2}}
\left(\int_{-2^6}^{0}(1+\tau^2)^{\frac{2s+1}{8}}|\hat{h}_1(\tau)|^2d\tau\right)^{\frac{1}{2}}\\
\le \frac{\sqrt{2}}{4\pi}\left(\int_{-2^6}^{0}\tau^{2j-\frac{1}{2}-\frac{2s+1}{8}}d\tau\right)^{\frac{1}{2}}\|h_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R})}=c_{j,s}\|h_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R})}.
\end{multline}
Note that $c_{j,s}<\infty$ in the above estimate since $2j-\frac{1}{2}-\frac{2s+1}{8}>-1$. Now, consider the case $s\ge \frac{3}{2}$. In this case, pick some $s'<3/2.$ As before, we have
$$|\partial_t^jz_{2,1}(x,t)|\le c_{j,s'}\|h_1\|_{H^{\frac{2s'+1}{8}}(\mathbb{R})}.$$ But since in particular $s'<s$, we have $\|h_1\|_{H^{\frac{2s'+1}{8}}(\mathbb{R})}\le \|h_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R})}$. Therefore, we deduce that $$|\partial_t^jz_{2,1}(x,t)|\le c_{j,s'}\|h_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R})}.$$
From the definition of the Sobolev norm, one obtains
\begin{eqnarray}
\|z_{2,1}(x)\|_{H^m(0,T')}\leq c_{m,s'}T'^{\frac{1}{2}}\|h_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R})}
\end{eqnarray}
at first for all $m\in\mathbb{N}$, and then by interpolation for all $m\ge 0$. In particular, by choosing $m=\frac{2s+3}{8}$ one has
\begin{eqnarray}\label{z22est1}
\|z_{2,1}(x)\|_{H^{\frac{2s+3}{8}}(0,T')}\leq c_sT'^{\frac{1}{2}}\|h_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R})}.
\end{eqnarray} for $s\ge -\frac{3}{2}$.
Now, let us estimate $z_{2,2}(x,\cdot)$. By using the definition of $\theta$, we can rewrite this function as
$$z_{2,2}(x,t)=-\frac{1}{\pi}\int_{1+i}^{(1+i)\infty}e^{ikx+ik^4t}\left(1-\theta(k)\right)\left[k^2\left(1-i\right)\hat{h}_1(k^4)\right]dk.$$ Using the same change of variables $\tau=k^4$ with argument of $\tau$ equal to $\pi$, we have
$$z_{2,2}(x,t)=\frac{\left(1-i\right)}{4\pi}\int_{-\infty}^{-4}e^{i\tau^{\frac{1}{4}}x+it\tau}\left(1-\theta\left(\tau^{\frac{1}{4}}\right)\right)\hat{h}_1(\tau)\frac{d\tau}{\tau^{\frac{1}{4}}}.$$
Note that, we can regard $z_{2,2}(x,\cdot)$ as the inverse Fourier transform of the function \[\displaystyle
\hat{z}_{2,2}(x,\tau):=\left\{
\begin{array}{ll}
0, &\tau\in[-4,\infty), \\
\frac{1}{2}e^{i\tau^\frac{1}{4}x}\left(1-\theta\left(\tau^{\frac{1}{4}}\right)\right)(1-i)\hat{h}_1(\tau)\frac{1}{\tau^{\frac{1}{4}}}, &\tau\in(-\infty,-4),
\end{array}
\right.
\] from which it follows that
\begin{multline}\label{z22est2}\|z_{2,2}(x,\cdot)\|_{H^{\frac{2s+3}{8}}(0,T')}^2\le \|z_{2,2}(x,\cdot)\|_{H^{\frac{2s+3}{8}}(\mathbb{R})}^2=\int_{\mathbb{R}}(1+\tau^2)^{\frac{2s+3}{8}}|\hat{z}_{2,2}(\tau)|^2d\tau\\
\le \frac{1}{2}\int_{-\infty}^{-4}(1+\tau^2)^{\frac{2s+3}{8}}|\tau|^{-\frac{1}{2}}|\hat{h}_1(\tau)|^2d\tau\lesssim\int_{\mathbb{R}}(1+\tau^2)^{\frac{2s+1}{8}}|\hat{h}_1(\tau)|^2d\tau=\|h_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R})}^2.\end{multline}
Combining \eqref{z22est1} and \eqref{z22est2}, we conclude that
\begin{eqnarray}\label{z22est3}
\|z_{2}(x)\|_{H^{\frac{2s+3}{8}}(0,T')}\lesssim \left(1+T'^{\frac{1}{2}}\right)\|h_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R})}.
\end{eqnarray}
Applying similar arguments that we used for $z_1$ and $z_2$ also to $z_j$ for $3\le j\le 8$ and combining the relevant estimates, we deduce that
\begin{eqnarray}\label{ztestson}
\|z(x)\|_{H^{\frac{2s+3}{8}}(0,T')}\lesssim \|h_0\|_{H^{\frac{2s+3}{8}}(\mathbb{R})}+\left(1+T'^{\frac{1}{2}}\right)\|h_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R})}.
\end{eqnarray}
Continuity in $x$ can easily be justified by means of the dominated convergence theorem. Therefore we just proved that $z\in C(\mathbb{R}_+;H^{\frac{2s+3}{8}}(0,T'))$ under the given assumptions on $h_0$ and $h_1$. The time estimate for $z_x$ can be proven using arguments as in \eqref{z1257} after differentiating in $x$. Note that this will bring an extra factor of $|\tau|^{1/4}$ into the integrals in the estimates. Therefore one will obtain $z_x\in C(\mathbb{R}_+;H^{\frac{2s+1}{8}}(0,T'))$.
\end{proof}
\subsection{Representation formula - revisited}Let $s\in \left[0,\frac{9}{2}\right)\setminus\left\{\frac{1}{2},\frac{3}{2}\right\}$.
Let $(\cdot)^*$ denote a bounded extension operator from $H^s(\mathbb{R}_+)$ into $H^s(\mathbb{R})$. Existence of such an extension operator is guaranteed by the definition of the Sobolev space and the associated Sobolev norm on the half-line:
$$H^s(\mathbb{R}_+)=\{\varphi:\mathbb{R}_+ \rightarrow \mathbb{C}\,|\,\exists \psi\in H^s(\mathbb{R}) \text{ s.t. } \psi|_{\mathbb{R}_+} = \varphi\},$$ where
$$\|\varphi\|_{H^s(\mathbb{R}_+)}=\inf_{\substack{{\psi\in H^s(\mathbb{R})} \\ {\psi|_{\mathbb{R}_+}} = \varphi}}\|\psi\|_{H^s(\mathbb{R})}.$$ Now, given $q_0\in H^s(\mathbb{R}_+)$, we let $q_0^*\in H^s(\mathbb{R})$ be the extension of $q$ with respect to the fixed extension operator $(\cdot)^*$ just defined. Note that by the boundedness of this operator we have $$\|q_0^*\|_{H^s(\mathbb{R})}\lesssim \|q_0\|_{H^s(\mathbb{R}^+)}.$$
Therefore, $y(t)=S_\mathbb{R}(t)q_0^*$ solves the problem
\begin{equation}\label{yext}
iy_t + \partial_x^4y=0,y(0,t)=q_0^*(x), x\in \mathbb{R}, t\in (0,T),
\end{equation} where $S_\mathbb{R}(t)$ is the evolution operator for the free linear biharmonic Schrödinger equation given in Lemma \ref{Wrty0}. Similarly, given $f\in L^1(0,T;H^s(\mathbb{R}_+))$, let $f^*\in L^1(0,T;H^s(\mathbb{R}))$ be the extension of $f$ in the spatial variable with respect to the extension operator $(\cdot)^*$. Then the solution of the non-homogeneous Cauchy problem
\begin{equation}\label{nonhomprob}
iw_t+\partial_x^4 w=f^*, w(x,0)=0, x\in \mathbb{R}, t\in (0,T)\end{equation} can be written as $$w(t)=-i\int_0^tS_\mathbb{R}(t-\tau)f^*(\tau)d\tau.$$
For $j=0,1$, we set $$a_j(t):=\left.\partial_x^jS_\mathbb{R}(t)q^*_0\right|_{x=0} \text{ and } b_j(t):=\left.-i\partial_x^j\int_0^tS_\mathbb{R}(t-\tau)f^*(\tau)d\tau\right|_{x=0}.$$ Note that these traces exist by Lemma \ref{Wrty0} as elements of $H^{\frac{2s+3}{8}}(0,T)$ for $j=0$ and $H^{\frac{2s+1}{8}}(0,T)$ for $j=1$. See also the analog argument for the classical Schrödinger equation given in \cite[Prop. 3.7 (iii)]{bona}.
\subsubsection*{Compatibility conditions} In order for solutions to be continuous at the space-time corner point $(x,t)=(0,0)$, we can find the necessary (compability) conditions based on the analysis of traces again. Suppose that $s\in \left[0,\frac{9}{2}\right)\setminus\left\{\frac{1}{2},\frac{3}{2}\right\}$, $g_0\in H^{\frac{2s+3}{8}}(\mathbb{R})$ and $g_1\in H^{\frac{2s+1}{8}}(\mathbb{R})$. Note that if $s\in \left(\frac{1}{2},\frac{9}{2}\right)$, then $\frac{2s+3}{8}>\frac{1}{2}$ and therefore $g_0(0)$ is well-defined, but $q(0,0)=q_0(0)$, and hence we must have $g_0(0)=q_0(0) (=a_0(0)).$ By a similar argument, we deduce that if $s\in \left(\frac{3}{2},\frac{9}{2}\right)$, then $g_1(0)=q_0'(0)(=a_1(0))$ is a necessary condition.
On the other hand, if $[p_0,p_1]\in H^{\frac{2s+3}{8}}(0,T)\times H^{\frac{2s+1}{8}}(0,T)$ such that $p_0(0)=p_1(0)=0$, then we set $[p_0,p_1]_e=[[p_0]_e,[p_1]_e]$ to be an extension of $[p_0,p_1]$ to $\mathbb{R}$, which satisfies the compact support condition $[p_0]_e|_{(0,2T)^c}=[p_1]_e|_{(0,2T)^c}=0$, regularity $[p_0,p_1]_e\in H^{\frac{2s+3}{8}}(\mathbb{R})\times H^{\frac{2s+1}{8}}(\mathbb{R})$, and the estimates
$$\|[p_0]_e\|_{H^{\frac{2s+3}{8}}(\mathbb{R})}\lesssim \|p_0\|_{H^{\frac{2s+3}{8}}(0,T)}\text{ and }\|[p_1]_e\|_{H^{\frac{2s+1}{8}}(\mathbb{R})}\lesssim \|p_1\|_{H^{\frac{2s+3}{8}}(0,T)}.$$ For the existence of such extension, see for instance \cite[Lemma 2.1]{BatalOzsari16}.
Therefore, if we define \begin{multline}\label{RepForm}q_e(t):=\left.S_\mathbb{R}(t)q_0^*\right|_{\mathbb{R}_+}-\left.i\int_0^tS_\mathbb{R}(t-\tau)f^*(\tau)d\tau\right|_{\mathbb{R}_+}\\
+S_{b}([g_0-a_0-b_0,g_1-a_1-b_1]_e)(t),\end{multline} then $q={q_e}|_{[0,T)}$ solves \eqref{1.1}-\eqref{1.4}. Note that both $g_0-a_0-b_0$ and $g_1-a_1-b_1$ satisfy the necessary comptability conditions given in Section \ref{bdrtosol} since $g_0(0)=a_0(0)$, $b_0(0)=b_1(0)=0$, and $g_1(0)=a_1(0)$.
Now, we are ready to state the following lemma which follows by combining the space-time estimates proved for the solution generators $S_\mathbb{R}$ and $S_b$ in the previous sections.
\begin{lem} Let $s\in \left[0,\frac{9}{2}\right)\setminus\left\{\frac{1}{2},\frac{3}{2}\right\}$, $q_0\in H^s(\mathbb{R}_+)$, $g_0\in H^{\frac{2s+3}{8}}(0,T)$, $g_1\in H^{\frac{2s+1}{8}}(0,T)$, and $f\in L^1(0,T;H^s(\mathbb{R}_+))$. Suppose also that the initial-boundary data satisfies the necessary compatibility conditions. Then the following estimate holds true for the solution of \eqref{1.1}-\eqref{1.4}.
\begin{multline}\|q\|_{C([0,T];H^s(\mathbb{R}_+))}\\
\lesssim \|q_0\|_{H^s(\mathbb{R}_+)}+\|g_0\|_{H^{\frac{2s+3}{8}}(\mathbb{R}_+)}+\left(1+T^{\frac{1}{2}}\right)\left[\|g_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R}_+)}+\|f\|_{L^1(0,T;H^s(\mathbb{R}_+))}\right].\end{multline}
\end{lem}
\section{Nonlinear model}
\subsection{Local wellposedness for $s>\frac{1}{2}$}
In this section, we assume $s>\frac{1}{2}$ and consider the nonlinear model \eqref{4th.1}-\eqref{4th.4}.
\subsubsection*{Local existence}
Note that a solution of \eqref{4th.1}-\eqref{4th.4} is a fixed point of the operator $\Psi$ which is given by
\begin{multline}\label{Psi}[\Psi(q)](t):=\left.S_\mathbb{R}(t)q^*_0\right|_{\mathbb{R}_+}-\left.i\int_0^tS_\mathbb{R}(t-\tau)f(q^*(\tau))d\tau\right|_{\mathbb{R}_+}\\
+S_{b}([g_0-a_0-b_0(q^*),g_1-a_1-b_1(q^*)]_e)(t),\end{multline} where
for $j=0,1$, we set $$a_j(t):=\left.\partial_x^jS_\mathbb{R}(t)q^*_0\right|_{x=0} \text{ and } b_j(q^*)(t):=\left.-i\partial_x^j\int_0^tS_\mathbb{R}(t-\tau)f(q^*(\tau))d\tau\right|_{x=0}.$$ We consider the operator $\Psi$ on the Banach space $X_{T_0}:= C([0,T_0];H^s(\mathbb{R}_+))$. In order to prove the local existence of solutions we will use the Banach fixed point theorem on a closed ball $\overline{B}_R(0)$ of the function space $X_{T_0}$ for appropriately chosen $R>0$ and $T_0\in (0,T]$. Let us first show that $\Psi$ maps $\overline{B}_R(0)$ onto itself for appropriate $R$ and sufficiently small $T_0.$ First of all, we have via Lemma \ref{Wrty0} and the boundedness of the extension operator the estimate
\begin{equation}\label{firstest}
\|S_\mathbb{R}(t)q^*_0\|_{H^s(\mathbb{R}_+)}\le \|S_\mathbb{R}(t)q^*_0\|_{H^s(\mathbb{R})}=\|q_0^*\|_{H^s(\mathbb{R})}\lesssim \|q_0\|_{H^s(\mathbb{R}_+)},
\end{equation} which gives $$\|S_\mathbb{R}(\cdot)q^*_0\|_{X_{T_0}}\lesssim \|q_0\|_{H^s(\mathbb{R}_+)}.$$
Let us recall the following lemma, which holds true under the given assumptions (a1) and (a2). The proof can be found for instance in \cite[Lemma 3.1]{BatalOzsari16}:
\begin{lem}\label{fEst} Let $f(y)=|y|^py$ and $s>\frac{1}{2}$. Then \begin{equation}\label{fEstLem1}\|f(y)\|_{H^s(\mathbb{R})}\lesssim\|y\|_{H^s(\mathbb{R})}^{p+1},\end{equation}
\begin{equation}\|f(y)-f(z)\|_{H^s(\mathbb{R})}\lesssim (\|y\|_{H^s(\mathbb{R})}^{p}+\|z\|_{H^s(\mathbb{R})}^{p})\|y-z\|_{H^s(\mathbb{R})}\end{equation} for $y,z\in H^s(\mathbb{R})$.
\end{lem}
Using Lemma \ref{fEst}, we have
\begin{multline}\left\|-i\int_0^tS_\mathbb{R}(t-\tau)f(q^*(\tau))d\tau\right\|_{X_{T_0}^s}\le \int_0^{T_0}\|f(q^*(\tau))\|_{H^s(\mathbb{R})}d\tau\\
\lesssim \int_0^{T_0}\|q^*(\tau)\|_{H^s(\mathbb{R})}^{p+1}d\tau\lesssim \int_0^{T_0}\|q(\tau)\|_{H^s(\mathbb{R}_+)}^{p+1}d\tau\le T_0\|q\|_{X_{T_0}}^{p+1}.\end{multline}
Similarly,
\begin{multline}\left\|-i\int_0^tS_\mathbb{R}(t-\tau)[f(y^*(\tau))-f(z^*(\tau))]d\tau\right\|_{X_{T_0}}\le \int_0^{T_0}\|f(y^*(\tau))-f(z^*(\tau))\|_{H^s(\mathbb{R})}d\tau\\
\lesssim \int_0^{T_0}(\|y^*(\tau)\|_{H^s(\mathbb{R})}^p+\|z^*(\tau)\|_{H^s(\mathbb{R})}^p)\|y^*(\tau)-z^*(\tau)\|_{H^s(\mathbb{R})}d\tau\\
\lesssim \int_0^{T_0}(\|y(\tau)\|_{H^s(\mathbb{R}_+)}^p+\|z(\tau)\|_{H^s(\mathbb{R}_+)}^p)\|y(\tau)-z(\tau)\|_{H^s(\mathbb{R}_+)}d\tau\\
\lesssim {T_0}(\|y\|_{X_{T_0}}^p+\|z\|_{X_{T_0}}^p)\|y-z\|_{X_{T_0}}.\end{multline}
The last term in \eqref{Psi} is estimated as follows
\begin{multline}\label{Wbhgp}
\|S_{b}([g_0-a_0-b_0(q^*),g_1-a_1-b_1(q^*)]_e)(t)\|_{X_{T_0}}\\
\lesssim\|[g_0-a_0-b_0(q^*)]_e\|_{H^{\frac{2s+3}{8}}(\mathbb{R})}+\left(1+T_0^\frac{1}{2}\right)\|[g_1-a_1-b_1(q^*)]_e\|_{H^{\frac{2s+1}{8}}(\mathbb{R})}\\
\lesssim \|g_0\|_{H^{\frac{2s+3}{8}}(0,T_0)}+\|a_0\|_{H^{\frac{2s+3}{8}}(0,T_0)}+\|b_0(q^*)\|_{H^{\frac{2s+3}{8}}(0,T_0)}\\
+\left(1+T_0^\frac{1}{2}\right)\left[\|g_1\|_{H^{\frac{2s+1}{8}}(0,T_0)}+\|a_1\|_{H^{\frac{2s+1}{8}}(0,T_0)}+\|b_1(q^*)\|_{H^{\frac{2s+1}{8}}(0,T_0)}\right].
\end{multline}
Note that \begin{multline}\label{gEst}\|a_1\|_{H^{\frac{2s+1}{8}}(0,T_0)}=\|\partial_x S_\mathbb{R}(t)q^*_0|_{x=0}\|_{H^{\frac{2s+1}{8}}(0,T_0)}\le \sup_{x\in\mathbb{R}_+}\|\partial_x S_\mathbb{R}(t)q^*_0\|_{H^{\frac{2s+1}{8}}(0,T_0)}\\
\le \left\|\frac{d}{dx}q_0^*\right\|_{H^{s-1}(\mathbb{R})}\le \|q_0^*\|_{H^{s}(\mathbb{R})}\lesssim \|q_0\|_{H^s(\mathbb{R}_+)}.\end{multline}
In \eqref{gEst}, the second inequality follows from the fact that $\partial_x S_\mathbb{R}(t)q^*_0$ is a solution of the linear biharmonic Schrödinger equation on $\mathbb{R}$ with initial condition $\displaystyle\frac{d}{dx}q_0^*$.
Similarly,
\begin{multline}\label{pEst}\|b_1(q^*)\|_{H^{\frac{2s+1}{8}}(0,T_0)}=\left\|-i\partial_x\int_0^tS_\mathbb{R}(t-\tau)f(q^*(\tau))d\tau|_{x=0}\right\|_{H^{\frac{2s+1}{8}}(0,T_0)}\\
\le \sup_{x\in\mathbb{R}_+}\left\|-i\partial_x\int_0^tS_\mathbb{R}(t-\tau)f(q^*(\tau))d\tau\right\|_{H^{\frac{2s+1}{8}}(0,T_0)}\\
\lesssim \|\partial_xf(q^*)\|_{L^1(0,T_0;H^{s-1}(\mathbb{R}))}\le \|f(q^*)\|_{L^1(0,T_0;H^{s}(\mathbb{R}))}\lesssim T_0\|q\|_{X_{T_0}}^{p+1}\end{multline}
and \begin{equation}\|b_1(y^*)-b_1(z^*)\|_{H^{\frac{2s+1}{8}}(0,T_0)}\\
\lesssim T_0(\|y\|_{X_{T_0}}^p+\|z\|_{X_{T_0}}^p)\|y-z\|_{X_{T_0}}. \end{equation}
Combining above estimates, we obtain $$\|\Psi(q)\|_{X_{T_0}}\le C\left(\|q_0\|_{H^s(\mathbb{R}_+)}+\|g_0\|_{H^{\frac{2s+3}{8}}(0,T_0)}+\|g_1\|_{H^{\frac{2s+1}{8}}(0,T_0)}+T_0\|q\|_{X_{T_0}}^{p+1}\right).$$
Regarding the differences, again by above estimates, we have $$\|\Psi(y)-\Psi(z)\|_{X_{T_0}}\le C T_0(\|y\|_{X_{T_0}}^p+\|z\|_{X_{T_0}}^p)\|y-z\|_{X_{T_0}}.$$
Now let $A:=C\left(\|q_0\|_{H^s(\mathbb{R}_+)}+\|g_0\|_{H^{\frac{2s+3}{8}}(0,T_0)}+\|g_1\|_{H^{\frac{2s+1}{8}}(0,T_0)}\right)$, $R=2A$ and $T_0$ be small enough that $A+CT_0R^{p+1}<2A$. Now, if necessary we can choose $T_0$ even smaller so that $\Psi$ becomes a contraction on $\overline{B}_R(0)\subset X_{T_0}$, which is a complete space. Hence, $\Psi$ must have a unique fixed point in $\overline{B}_R(0)$ when we look for a solution whose lifespan is sufficiently small.
\subsubsection*{Uniqueness}\label{Uniq}Let $q_1,q_2\in X_{T_0}$ be two local solutions. Then, \begin{multline}q_1(t)-q_2(t)=-i\int_0^tS_\mathbb{R}(t-s)[f(q_1^*(s))-f(q_2^*(s))]ds\\
+S_b(t)([b_0(q_2^*)-b_0(q_2^*),b_1(q_2^*)-b_1(q_2^*)]_e)\end{multline} for a.a. $t\in [0,T_0]$.
Since $s>1/2$, \begin{multline}\|q_1(t)-q_2(t)\|_{H^s(\mathbb{R}_+)}\le C\int_0^{T_0}\|f(q_1^*(s))-f(q_2^*(s))\|_{H^s(\mathbb{R})}ds\\
+C\|b_0(q_2^*)-b_0(q_1^*)\|_{H^{\frac{2s+3}{8}}(0,T_0)}+C(1+T_0^\frac{1}{2})\|b_1(q_2^*)-b_1(q_1^*)\|_{H^{\frac{2s+1}{8}}(0,T_0)}\\
\le C(1+T_0^\frac{1}{2})\int_0^{T_0}\|q_1(s)-q_2(s)\|_{H^s(\mathbb{R}_+)}\left(\|q_1(s)\|_{H^s(\mathbb{R}_+)}^p+\|q_2(s)\|_{H^s(\mathbb{R}_+)}^p\right)ds\\
\le C(1+T_0^\frac{1}{2})\left(\|q_1(s)\|_{X_{T_0}^s}^p+\|q_2(s)\|_{X_{T_0}^s}^p\right)\int_0^{T_0}\|q_1(s)-q_2(s)\|_{H^s(\mathbb{R}_+)}ds.\end{multline} Now, unleashing the Gronwall's inequality we get $\|q_1(t)-q_2(t)\|_{H^s(\mathbb{R}_+)}=0$, which implies $q_1\equiv q_2$ on $[0,T_0]$.
\subsubsection*{Continuous dependence}\label{ContDep}
Regarding continuous dependence on data, let $B$ be a bounded subset of $H^s(\mathbb{R}_+)\times H^{\frac{2s+3}{8}}(0,T_0)\times H^{\frac{2s+1}{8}}(0,T_0)$.
Let $(y_0, g_0,g_1)\in B$ and $(z_0,h_0,h_1)\in B$. Let $y, z$ be two solutions on a common time interval $(0,T_0)$ corresponding to $(y_0, g_0,g_1)$ and$(z_0,h_0,h_1)$, respectively. Then $w=y-z$ satisfies
\begin{equation}\label{StabilityModel} \left\{ \begin{array}{ll}
i\partial_t w + \partial_x^4w = F(x,t)\equiv f(y)-f(z), & \mbox{$x\in \mathbb{R}_+$, $t\in (0,T_0)$},\\
w(x,0)=w_0(x)\equiv (y_0-z_0)(x),\\
w(0,t)=g(t)\equiv(g_0-h_0)(t),\\
\partial_xw(0,t)=h(t)\equiv(g_1-h_1)(t).\end{array} \right.
\end{equation}
Now, using the linear theory together with the nonlinear $H^s$ estimates on the differences, we have
$$\|w\|_{X_{T_0}}\le C\left(\|w_0\|_{H^s(\mathbb{R}_+)}+\|g\|_{H^{\frac{2s+3}{8}}(0,T_0)}+(1+T_0^{\frac{1}{2}})\|h\|_{H^{\frac{2s+1}{8}}(0,T_0)}+\|F\|_{L^1(0,T_0;H^s(\mathbb{R}_+))}\right),$$
where $$\|F\|_{L^1(0,T_0;H^s(\mathbb{R}_+))}\le CT_0\left(\|y\|_{X_{T_0}}^p+\|z\|_{X_{T_0}}^p\right)\|y-z\|_{X_{T_0}}.$$
Choosing $R$, as in the proof of the local existence, and $T_0$ accordingly small enough, we obtain
\begin{equation}\label{ContDep01}\|y-z\|_{X_{T_0}}\le C\left(\|y_0-z_0\|_{H^s(\mathbb{R}_+)}+\|g_0-h_0\|_{H^{\frac{2s+3}{8}}(0,T_0)}+\|g_1-h_1\|_{H^{\frac{2s+1}{8}}(0,T_0)}\right).\end{equation}
\subsubsection*{Blow-up alternative}\label{BlowSec} In this section, we want to obtain a condition which guarantees that a given local solution on $[0,T_0]$ can be extended globally. Let's consider the set $S$ of all $T_0\in (0,T]$ such that there exists a unique local solution in $X_{T_0}^s$. We claim that if $\displaystyle T_{max}:=\sup_{T_0\in S}T_0<T$, then $\displaystyle\lim_{t\uparrow T_{max}}\|q(t)\|_{H^s(\mathbb{R}_+)}=\infty.$ In order to prove the claim, assume to the contrary that $\displaystyle\lim_{t\uparrow T_{max}}\|q(t)\|_{H^s(\mathbb{R}_+)}\neq\infty.$ Then $\exists M$ and $t_n\in S$ such that $t_n\rightarrow T_{max}$ and $\|q(t_n)\|_{H^s(\mathbb{R}_+)}\le M.$ For a fixed $n$, we know that there is a unique local solution $q_1$ on $[0,t_n]$. Now, we consider the following model.
\begin{equation}\label{MainProb3} \left\{ \begin{array}{ll}
i\partial_t q + \partial_x^4q=f(q), & \mbox{$x\in \mathbb{R}_+$, $t\in (t_n,T)$},\\
q(x,t_n)=q_1(x,t_n),\\
q(0,t)=g_0(t),\\
\partial_xq(0,t)=g_1(t).\end{array} \right.
\end{equation}
We know from the local existence theory that the above model has a unique local solution $q_2$ on some interval $[t_n,t_n+\delta]$ for some $$\delta=\delta\left(M,\|g_0\|_{H^\frac{2s+3}{8}(0,T)},\|g_1\|_{H^\frac{2s+1}{8}(0,T)}\right)\in (0,T-t_n].$$ Now, choose $n$ sufficiently large that $t_n+\delta>T_{max}$. If we set \begin{equation} q:=\left\{ \begin{array}{ll}
q_1, & \mbox{$t\in [0,t_n)$},\\
q_2, & \mbox{$t\in [t_n,t_n+\delta]$},\end{array} \right.\end{equation} then $q$ is a solution on $[0,t_n+\delta]$ where $t_n+\delta>T_{max}$, which is a contradiction.
Hence, the proof of the local wellposedness is complete.
\subsection{Strichartz estimates and low regularity}
In this section, we consider the nonlinear problem and we assume $0\le s< \frac{1}{2}$ and $(\lambda,r)$ is biharmonic admissible throughout. We say that the pair $(\lambda,r)$ is \emph{biharmonic admissible} if $$\lambda,r\in [2,\infty] \text{ and }\frac{1}{8}=\frac{1}{4r}+\frac{1}{\lambda}.$$ We also assume that $$p\le \frac{8}{1-2s} \text{ (both \emph{subcritical} '<' and \emph{critical} '=' cases are considered)}.$$
In order to prove wellposedness in this low regularity setting, it is crucial to prove a Strichartz estimate (an estimate of the $L^\lambda_tW^{s,r}_x$ norm) on the solutions of the linear boundary value problem \eqref{3.1}-\eqref{3.4}. These estimates are generally proven on the whole space or special domains such as manifolds without boundary. It is in general more difficult to prove these estimates on domains with boundary. However, the representation formula on the half line obtained by the Fokas method provides a suitable kernel structure which will help us to prove the Strichartz estimate.
Note that in this case, we do not need any compatibility condition as the traces of the initial and boundary data are not defined for $s<\frac{1}{2}$. The solution of \eqref{3.1}-\eqref{3.4} is given by the formula \eqref{zsol2}. Let us consider the first term in this formula, which is given by \eqref{z1}. We can rewrite \eqref{z1} as
\begin{multline}\label{z1r1}z_1(x,t)=-\frac{4}{\pi}\int_{0}^\infty e^{-kx+ikx-4ik^4t}\left[(i-1)k^3H_0(4ik^4,T')\right]dk\\
=-\frac{4}{\pi}\int_{0}^\infty e^{-kx+ikx-4ik^4t}\left[(i-1)k^3\int_0^{T'}e^{4ik^4s}h_0(s)ds\right]dk\\
=-\frac{4}{\pi}\int_{0}^\infty e^{-kx+ikx-4ik^4t}\left[(i-1)k^3\int_{-\infty}^{\infty}e^{4ik^4s}h_0(s)ds\right]dk\\
=\frac{4(1-i)}{\pi}\int_{0}^\infty e^{-kx+ikx-4ik^4t}k^3\hat{h}_0(-4k^4)dk\\
=\frac{4(1-i)}{\pi}\int_{-\infty}^\infty \psi(y)\left[\int_{0}^{\infty}e^{-kx+ikx-4ik^4t-iky}dk\right]dy,\end{multline} where $\hat{\psi}(k):=k^3\hat{h}_0(-4k^4)$ if $k\ge 0$ and $\hat{\psi}(k):=0$ otherwise. We set $$c(x,y,t):=\frac{4(1-i)}{\pi}\int_{0}^{\infty}e^{-kx+ikx-4ik^4t-iky}dk=\frac{4(1-i)}{\pi t^{\frac{1}{4}}}\int_{0}^{\infty}e^{-kxt^{-\frac{1}{4}}+ikxt^{-\frac{1}{4}}-4ik^4-ikyt^{-\frac{1}{4}}}dk.$$ Then $z_1$ is rewritten as $$z_1(x,t)=\int_{-\infty}^\infty c(x,y,t)\psi(y)dy.$$ Let $$\phi(k;x,y,t):=kxt^{-\frac{1}{4}}-4k^4-kyt^{-\frac{1}{4}} \text{ and } p(k;x,t):=e^{-kxt^{-\frac{1}{4}}}.$$ Then, $$c(x,y,t):=\frac{4(1-i)}{\pi t^{\frac{1}{4}}}\int_{0}^{\infty}e^{i\phi(k;x,y,t)}p(k;x,t)dk.$$ Note that $\left|\frac{d^4}{dk^4}\phi(k;x,y,t)\right|=96\ge 1$, which allows us to apply the theory of oscillatory integrals:
\begin{lem}\cite[Section 1.4]{Lin15}
Let $|\phi^{(4)}(k)|\ge 1$ for $k\ge 0$. Then $$\left|\int_0^\infty e^{i\phi(k)}p(k)dk\right|\le c\left(\|p\|_{\infty}+\|p\|_1\right).$$
\end{lem}
Applying the above lemma to our kernel, we obtain the pointwise (uniform with respect to $x$ and $y$) estimate $$|c(x,y,t)|\lesssim t^{-\frac{1}{4}}\left(\|p(k;x,t)\|_{L_k^\infty(0,\infty)}+\left\|\frac{d}{dk}p(k;x,t)\right\|_{L_k^1(0,\infty)}\right)\lesssim t^{-\frac{1}{4}}.$$
It follows that \begin{equation}\label{z1int01}\|z_1(\cdot,t)\|_{L^\infty(\mathbb{R}_+)}\lesssim t^{-\frac{1}{4}}\|\psi\|_{L^1(\mathbb{R})}.\end{equation} Moreover, from \eqref{z1est0}, we already have
\begin{equation}\label{z1int02}\|z_1(\cdot,t)\|_{L^2(\mathbb{R}_+)}\lesssim \|\psi\|_{L^2(\mathbb{R})}.\end{equation} Interpolating between \eqref{z1int01} and \eqref{z1int02}, we get $$\|z_1(\cdot,t)\|_{L^r(\mathbb{R}_+)}\lesssim t^{-(\frac{1}{4}-\frac{1}{2r})}\|\psi\|_{L^{r'}(\mathbb{R})}.$$
Now, we will estimate $\|z_1\|_{L^\lambda(0,T';L^r(\mathbb{R}_+))}$. To this end, let $\eta\in C_c([0,T'];\mathcal{D}(\mathbb{R}_+))$ be an arbitrary test function. Then, utilizing the admissibility of $(\lambda,r)$, we have
$$\left|(z_1,\eta)_{L^2(0,T';L^2(\mathbb{R}_+))}\right|\lesssim \|\psi\|_{L^2(\mathbb{R})}\|\eta\|_{L^{\lambda'}(0,T';L^{r'}(\mathbb{R}_+))},$$ from which it follows that \begin{equation}\label{z1int1}\|z_1\|_{L^\lambda(0,T';L^r(\mathbb{R}_+))}\lesssim \|\psi\|_{L^2(\mathbb{R})}\lesssim \|h_0\|_{H^{\frac{3}{8}}(\mathbb{R})}.\end{equation} The first estimate above uses the fact that $$\left\|\int_0^{T'}\int_{0}^\infty \overline{c(x,\cdot,t)}\eta(x,t)dxdt\right\|_{L^2(\mathbb{R})}\lesssim \|\eta\|_{L^{\lambda'}(0,{T'};L^{r'}(\mathbb{R}_+))},$$ which follows from same the arguments in \cite[pp. 25-26]{bona} with the exception that $(\lambda,r)$ satisfies now biharmonic admissibility condition compared to the case of classical Schrödinger equation. Differentiating in the spatial variable as in \eqref{z1est2} and reapplying the above arguments to the derivative, one obtains \begin{equation}\label{z1int2}\|\partial_x z_1\|_{L^\lambda(0,{T'};L^r(\mathbb{R}_+))}\lesssim \|h_0\|_{H^{\frac{5}{8}}(\mathbb{R})}.\end{equation} Interpolating between \eqref{z1int1} and \eqref{z1int2}, we get
$$\|z_1\|_{L^\lambda(0,{T'};W^{s,r}(\mathbb{R}_+))}\lesssim \|h_0\|_{H^{\frac{2s+3}{8}}(\mathbb{R})}.$$ Similar estimates can also be found for $z_i$, $i=\overline{2,8}$ in terms of either $\|h_0\|_{H^{\frac{2s+3}{8}}(\mathbb{R})}$ or $\|h_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R})}$, and one obtains the desired Strichartz estimate for the boundary value problem
\begin{equation}\label{zStr1}
\|z\|_{L^\lambda(0,{T'};W^{s,r}(\mathbb{R}_+))}\lesssim \|h_0\|_{H^{\frac{2s+3}{8}}(\mathbb{R})}+\|h_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R})}.
\end{equation}
Let $y$ be a solution of \eqref{yext} and $w$ be a solution of \eqref{nonhomprob}. It is well known that the following Strichartz estimates hold true on the whole space \cite{dinh16}:
\begin{equation}\label{yestStr}
\|y\|_{L^\lambda(0,T;{W}^{s,r}(\mathbb{R}))}\lesssim \|q_0^*\|_{{H}^{s}(\mathbb{R})},
\end{equation}
\begin{equation}\label{westStr}
\|w\|_{C([0,T];H^s(\mathbb{R}))}+\|w\|_{L^\lambda(0,T;{W}^{s,r}(\mathbb{R}))}
\lesssim \|f^*\|_{L^{\lambda'}(0,T;{W}^{s,r'}(\mathbb{R}))}.
\end{equation} Moreover, the arguments in the proof of \cite[Proposition 3.8]{audiard} together with the above Strichartz estimates give the following time trace estimates for the solution of the nonhomogeneous Cauchy problem:
\begin{equation}\label{westStr2}
\sup_{x\in\mathbb{R}}\|w(x,\cdot)\|_{H^{\frac{2s+3}{8}}(0,T)} + \sup_{x\in\mathbb{R}}\|\partial_x w(x,\cdot)\|_{H^{\frac{2s+1}{8}}(0,T)}\\
\lesssim \|f^*\|_{L^{\lambda'}(0,T;{W}^{s,r'}(\mathbb{R}))}.
\end{equation}
We set \begin{equation}X_{T_0}:=\left\{q\in L^\lambda(0,{T_0};{W}^{s,r}(\mathbb{R}_+))\right.\\
\left.\,|\,\|q\|_{L^\lambda(0,{T_0};{{W}}^{s,r}(\mathbb{R}_+))}\le M\right\}\end{equation} and equip it with the metric
$$d(q_1,q_2):=\|q_1-q_2\|_{L^\lambda(0,{T_0};{L}^{r}(\mathbb{R}_+))}.$$
\begin{rem}It is well known that $X_{T_0}$ is a complete metric space. Clearly, $X_{T_0}$ is not a linear space. We will still write $$|q|_{X_{T_0}}:=\|q\|_{L^\lambda(0,{T_0};{{W}}^{s,r}(\mathbb{R}_+))}$$ to shorten the text in the remaining part of this section.
\end{rem}
Now consider again the operator $\Psi$ given in \eqref{Psi} on the metric space $(X_{T_0},d)$: \begin{equation}\label{Psiywz}
\Psi(q)=y|_{\mathbb{R}_+}+w|_{\mathbb{R}_+}+z.
\end{equation}
We see from \eqref{yestStr} that \begin{equation}\label{ypsiest}
|y|_{\mathbb{R}_+}|_{X_{T_0}}\lesssim \|q_0\|_{H^s(\mathbb{R}_+)}.
\end{equation} Replacing $f^*$ by the nonlinear term $f(q^*)$ in \eqref{westStr}, with $(\cdot)^*$ denoting a fixed bounded extension operator from $H^s(\mathbb{R}_+)\cap {W}^{s,r}(\mathbb{R}_+))$ to $H^s \cap {W}^{s,r}$, we get
\begin{equation}\label{wpsiest}|w|_{\mathbb{R}_+}|_{X_{T_0}}\lesssim \|f(q^*)\|_{L^{\lambda'}(0,T;{W}^{s,r'}(\mathbb{R}))}.\end{equation}
Moreover, we see from \eqref{zStr1}, \eqref{timeest001}, \eqref{timeest002}, and \eqref{westStr2} that
\begin{multline}\label{zXt0}
|z|_{X_{T_0}}\lesssim \|h_0\|_{H^{\frac{2s+3}{8}}(\mathbb{R})}+\|h_1\|_{H^{\frac{2s+1}{8}}(\mathbb{R})}\\
\lesssim \|g_0\|_{H^{\frac{2s+3}{8}}(0,T_0)}+\|g_1\|_{H^{\frac{2s+1}{8}}(0,T_0)}+c_s(1+T_0^\frac{1}{2})\|q_0\|_{H^s(\mathbb{R}_+)}+\|f(q^*)\|_{L^{\lambda'}(0,T_0;{W}^{s,r'}(\mathbb{R}))}.
\end{multline}
The nonlinear terms at the right hand sides of \eqref{wpsiest} and \eqref{zXt0} can be done by using the particular admissible pair given by $$\lambda:=\frac{8(p+2)}{p(1-2s)},r:=\frac{p+2}{1+sp}.$$ Indeed, one has the following estimates via the fractional Leibniz and chain rules (see the proof of \cite[Theorem 1.1]{Dinh18} for details):
\begin{equation}\label{fqst1}
\|f(q^*)\|_{{L^{\lambda'}(0,T_0;{\dot{W}}^{s,r'}(\mathbb{R}))}}\lesssim T_0^\theta\|q^*\|_{{L^{\lambda}(0,T_0;{\dot{W}}^{s,r}(\mathbb{R}))}}^{p+1}\le T_0^\theta|q|_{X_{T_0}}^{p+1},
\end{equation}
\begin{equation}\label{fqst2}
\|f(q^*)\|_{{L^{\lambda'}(0,T_0;{{L}}^{r'}(\mathbb{R}))}}\lesssim T_0^\theta\|q^*\|_{{L^{\lambda}(0,T_0;{\dot{W}}^{s,r}(\mathbb{R}))}}^{p}\|q^*\|_{{L^{\lambda}(0,T_0;{{L}}^{r}(\mathbb{R}))}}\le T_0^{\theta}|q|_{X_{T_0}}^{p+1},
\end{equation}
\begin{multline}\label{fqst3}
\|f(q_1^*)-f(q_2^*)\|_{{L^{\lambda'}(0,T_0;{{L}}^{r'}(\mathbb{R}))}}\\
\lesssim T_0^\theta\left(\|q_1^*\|_{{L^{\lambda}(0,T_0;{\dot{W}}^{s,r}(\mathbb{R}))}}^{p}+\|q_2^*\|_{{L^{\lambda}(0,T_0;{\dot{W}}^{s,r}(\mathbb{R}))}}^{p}\right)\|q_1^*-q_2^*\|_{{L^{\lambda}(0,T_0;{{L}}^{r}(\mathbb{R}))}}\\
\le T_0^\theta\left(|q_1|_{X_{T_0}}^{p}+|q_2|_{X_{T_0}}^{p}\right)d(q_1,q_2),
\end{multline} where $\theta:=1-\frac{p(1-2s)}{8}.$ Hence, assuming $T_0$ is small, say $T_0<1$, there exists some $c>0$ (independent of initial-boundary data and $T_0$) such that
for any $q,q_1,q_2\in X_{T_0}$, one has
\begin{multline}\label{Psi1}
|\Psi(q)|_{X_{T_0}}\le c(\|g_0\|_{H^{\frac{2s+3}{8}}(0,T_0)}+\|g_1\|_{H^{\frac{2s+1}{8}}(0,T_0)}+\|q_0\|_{H^s(\mathbb{R}_+)})
+2T_0^\theta|q|_{X_{T_0}}^{p+1}\\
\le c(\|g_0\|_{H^{\frac{2s+3}{8}}(0,T_0)}+\|g_1\|_{H^{\frac{2s+1}{8}}(0,T_0)})+\frac{M}{2}+2T_0^\theta M^{p+1},
\end{multline} where $M:=2c\|q_0\|_{H^s(\mathbb{R}_+)}$. Similarly,
\begin{equation}\label{Psi2}
d(\Psi(q_1),\Psi(q_2))\le cT_0^\theta\left(|q_1|_{X_{T_0}}^{p}+|q_2|_{X_{T_0}}^{p}\right)d(q_1,q_2) \le cT_0^\theta M^{p}d(q_1,q_2).
\end{equation} In \eqref{Psi1}, the terms $c(\|g_0\|_{H^{\frac{2s+3}{8}}(0,T_0)}+\|g_1\|_{H^{\frac{2s+1}{8}}(0,T_0)})+2T_0^\theta M^{p+1}$ can be made smaller than $\frac{M}{2}$ by choosing $T_0$ small. Similarly, in \eqref{Psi2}, we can guarantee that the product $cT_0^\theta M^{p}<1$ if $T_0$ is chosen small enough. Now, it is clear that $\Psi$ is a strict contraction on $(X_{T_0},d)$ for small $T_0$ in the subcritical case $\theta>0$ (equivalently if $p<\frac{8}{1-2s}$).
Regarding the critical case, where $\theta = 0$ (equivalently $p=\frac{8}{1-2s}$), we observe that the quantity $|\Psi(q)|_{X_{T_0}}$ is still finite and can be made as small as we wish by choosing $T_0$ small since $|\cdot|_{X_{T_0}}$ is monotone decreasing as $T_0$ decreases. Moreover, we can guarantee that $\Psi$ becomes a contraction on $(X_{T_0},d)$ since the sum $|q_1|_{X_{T_0}}^{p}+|q_2|_{X_{T_0}}^{p}$ can also be made small for small $T_0$.
It follows from \eqref{Psi2} that the uniqueness is guaranteed for small $T_0$, too.
The solution $q\in X_{T_0}$ obtained above in particular belongs to $C([0,T_0];H^s(\mathbb{R}_+))$. Indeed, since $q\in X_{T_0}$ is a fixed point of $\Psi$, we have in view of \eqref{Psiywz}
\begin{equation}\label{qywz}
q=y|_{\mathbb{R}_+}+w|_{\mathbb{R}_+}+z.
\end{equation} The first term at the right hind side of \eqref{qywz} satisfies
$$\|y|_{\mathbb{R}_+}\|_{C([0,T_0];H^s(\mathbb{R}_+))}\lesssim \|q_0\|_{H^s(\mathbb{R}_+)}$$ as in \eqref{firstest}. The second term satisfies
$$\|w|_{\mathbb{R}_+}\|_{C([0,T_0];H^s(\mathbb{R}_+))}\le \|w\|_{C([0,T_0];H^s(\mathbb{R}))}\lesssim \|f(q^*)\|_{{L^{\lambda'}(0,T_0;{{W}}^{s,r'}(\mathbb{R}))}}\lesssim T_0^{\theta}|q|_{X_{T_0}}^{p+1}.$$ Moreover, from Lemma \ref{lem239} and the second inequality in \eqref{zXt0} , we have
\begin{multline}
\|z\|_{C([0,T_0];H^s(\mathbb{R}_+))}\\
\lesssim \|g_0\|_{H^{\frac{2s+3}{8}}(0,T_0)}+\|g_1\|_{H^{\frac{2s+1}{8}}(0,T_0)}+c_s(1+T_0^\frac{1}{2})\|q_0\|_{H^s(\mathbb{R}_+)}+\|f(q^*)\|_{L^{\lambda'}(0,T_0;{W}^{s,r'}(\mathbb{R}))}\\
\lesssim \|g_0\|_{H^{\frac{2s+3}{8}}(0,T_0)}+\|g_1\|_{H^{\frac{2s+1}{8}}(0,T_0)}+c_s(1+T_0^\frac{1}{2})\|q_0\|_{H^s(\mathbb{R}_+)}+T_0^{\theta}|q|_{X_{T_0}}^{p+1}.
\end{multline}
This completes the proof of Theorem \ref{thmlowreg}.
\subsection{Global wellposedness} In this section, we assume $\kappa\in \mathbb{R}_{-}$.
First we multiply the main equation by $\overline{q}$, use integration by parts on $\mathbb{R}_+$, and take the imaginary parts. Then, we have
\begin{equation}\label{iden00}\frac{1}{2}\frac{d}{dt}\int_0^\infty|q|^2dx=\text{Im}\left[q_{xxx}(0,t)\bar{g}_0(t)\right]-\text{Im}\left[q_{xx}(0,t)\bar{g}_1(t)\right].\end{equation}
As we see from the above identity, the conservation of $L^2$-energy is lost in the presence of inhomogenenous boundary inputs. Therefore, the global wellposedness is quite a nontrivial problem. In order to prove the global wellposedness, one needs to gather some information on the second and third order traces. This enforces us to use other multipliers of higher order. To this end, we multiply the main equation by $\overline{q}_x$ and use integration by parts on $\mathbb{R}_+$. Therefore, we first write
\begin{equation}\label{iden01}\int_{0}^\infty q_t\bar{q}_xdx-i\int_{0}^\infty \left(\partial_x^4q\right)\bar{q}_xdx=-i\kappa\int_{0}^\infty|q|^pq\bar{q}_xdx.\end{equation} We can rewrite the first term as
\begin{multline}\int_{0}^\infty q_t\bar{q}_xdx = \frac{d}{dt}\int_{0}^\infty q\bar{q}_{x}dx-\int_{0}^\infty q\bar{q}_{xt}dx\\
= \frac{d}{dt}\int_{0}^\infty q\bar{q}_{x}dx-\left(\int_{0}^\infty \partial_x\left(q\bar{q}_{t}\right)dx-\int_{0}^\infty q_x\bar{q}_{t}dx\right),\end{multline} from which it follows that
\begin{equation}\text{Im}\int_{0}^\infty q_t\bar{q}_xdx =\frac{1}{2}\frac{d}{dt}\text{Im}\int_{0}^\infty {q}\bar{q}_xdx+\frac{1}{2}\text{Im}\left[g_0(t)\bar{g}_0'(t)\right].\end{equation}
Considering the second term in \eqref{iden01}, we have
\begin{multline}\text{Im}\left[i\int_{0}^\infty \left(\partial_x^4q\right)\bar{q}_xdx\right]=\text{Re}\left[\int_{0}^\infty \left(\partial_x^4q\right)\bar{q}_xdx\right]\\
=-\text{Re}\left[q_{xxx}(0,t)\bar{q}_x(0,t)\right]-\frac{1}{2}\int_{0}^\infty \partial_x\left|{q}_{xx}\right|^2dx\\
=-\text{Re}\left[q_{xxx}(0,t)\bar{g}_1(t)\right]+\frac{1}{2}|q_{xx}(0,t)|^2.\end{multline}
Regarding the term at the right hand side of \eqref{iden01}, we have
\begin{equation}\text{Im}\left[i\kappa\int_{0}^\infty|q|^pq\bar{q}_xdx\right]
=\frac{\kappa}{p+2}\text{Re}\left[\int_{0}^\infty\partial_x\left(|q|^{p+2}\right)dx\right]
=-\frac{\kappa}{p+2}|g_0(t)|^{p+2}.\end{equation}
Combining the above, we have
\begin{multline}\label{combinediden01}
\frac{1}{2}|q_{xx}(0,t)|^2=\frac{1}{2}\frac{d}{dt}\text{Im}\int_{0}^\infty {q}\bar{q}_xdx+\frac{1}{2}\text{Im}\left[g_0(t)\bar{g}_0'(t)\right]\\
+\text{Re}\left[q_{xxx}(0,t)\bar{g}_1(t)\right]-\frac{\kappa}{p+2}|g_0(t)|^{p+2}.
\end{multline}
Now, we multiply the main equation by $\overline{q}_{t}$ and use integration by parts on $\mathbb{R}_+$. Therefore, we have
\begin{multline}\label{iden00h2}\frac{d}{dt}\left[\frac{1}{2}\int_0^\infty|q_{xx}|^2dx-\frac{\kappa}{p+2}\int_0^\infty|q|^{p+2}dx\right]\\
=\text{Re}\left[q_{xxx}(0,t)\bar{g}'_0(t)\right]-\text{Re}\left[q_{xx}(0,t)\bar{g}_1'(t)\right].\end{multline}
Next, we multiply the main equation by $\overline{q}_{xxx}$ and use integration by parts on $\mathbb{R}_+$. Therefore, we first write
\begin{equation}\label{3iden01}\int_{0}^\infty q_t\bar{q}_{xxx}dx-i\int_{0}^\infty \left(\partial_x^4q\right)\bar{q}_{xxx}dx=-i\kappa\int_{0}^\infty|q|^pq\bar{q}_{xxx}dx.\end{equation}
Taking the imarginary part of the first term at the left hand side of \eqref{3iden01} and using integration by parts, we obtain
\begin{multline}\label{3iden2a}
\text{Im}\int_{0}^\infty q_t\bar{q}_{xxx}dx = -\text{Im}\left[g_0'(t)\bar{q}_{xx}(0,t)\right ]-\text{Im}\int_{0}^\infty q_{tx}\bar{q}_{xx}dx\\
= -\text{Im}\left[g_0'(t)\bar{q}_{xx}(0,t)\right ]+\text{Im}\left[g_1'(t)\bar{g}_1(t)\right ]+\text{Im}\int_{0}^\infty q_{txx}\bar{q}_{x}dx\\
= -\text{Im}\left[g_0'(t)\bar{q}_{xx}(0,t)\right ]+\text{Im}\left[g_1'(t)\bar{g}_1(t)\right ]+\text{Im}\left[q_{xx}(0,t)\bar{g}'_0(t)\right ]\\
+\text{Im}\int_{0}^\infty q_{xxx}\bar{q}_{t}dx+\frac{d}{dt}\text{Im}\int_{0}^\infty q_{xx}\bar{q}_xdx.
\end{multline} But recall that $2\text{Im}z=\text{Im}z-\text{Im}\bar{z}$ for any $z\in \mathbb{C}$, and hence we get
\begin{multline}\label{3iden2a}
\text{Im}\int_{0}^\infty q_t\bar{q}_{xxx}dx =
= -\frac{1}{2}\text{Im}\left[g_0'(t)\bar{q}_{xx}(0,t)\right ]+\frac{1}{2}\text{Im}\left[g_1'(t)\bar{g}_1(t)\right ]\\+\frac{1}{2}\text{Im}\left[q_{xx}(0,t)\bar{g}'_0(t)\right ]+\frac{1}{2}\frac{d}{dt}\text{Im}\int_{0}^\infty q_{xx}\bar{q}_xdx.
\end{multline}
Considering the second term at the left hand side of \eqref{3iden01}, we have
\begin{multline}\label{3iden02b}
\text{Im}\left[i\int_{0}^\infty \left(\partial_x^4q\right)\bar{q}_{xxx}dx\right] = \text{Re}\left[\int_{0}^\infty \left(\partial_x^4q\right)\bar{q}_{xxx}dx\right]\\
=\frac{1}{2}\left[\int_{0}^\infty \partial_x\left(\left|{q}_{xxx}\right|^2\right)dxdt\right]=-\frac{1}{2}\left|{q}_{xxx}(0,t)\right|^2.
\end{multline}
Regarding the term at the right hand side of \eqref{3iden01}, we have
\begin{multline}\text{Im}\left[i\kappa\int_{0}^\infty|q|^pq\bar{q}_{xxx}dx\right]
=\kappa\text{Re}\left[\int_{0}^\infty|q|^pq\bar{q}_{xxx}dx\right]\\
=-\kappa\text{Re}\left[|g_0(t)|^pg_0(t)\bar{q}_{xx}(0,t)\right]\\
-\frac{\kappa(p+2)}{2}\text{Re}\left[\int_{0}^\infty|q|^pq_x\bar{q}_{xx} dx\right]-\frac{\kappa p}{2}\text{Re}\left[\int_{0}^\infty|q|^{p-2}q^2\bar{q}_x\bar{q}_{xx} dx\right].\end{multline}
Combining the three identities above,
\begin{multline}\label{combinediden02}
\frac{1}{2}|q_{xxx}(0,t)|^2=\frac{1}{2}\text{Im}\left[g_0'(t)\bar{q}_{xx}(0,t)\right ]-\frac{1}{2}\text{Im}\left[g_1'(t)\bar{g}_1(t)\right ]-\frac{1}{2}\text{Im}\left[q_{xx}(0,t)\bar{g}'_0(t)\right ]\\
-\frac{1}{2}\frac{d}{dt}\text{Im}\int_{0}^\infty q_{xx}\bar{q}_xdx+\kappa\text{Re}\left[|g_0(t)|^pg_0(t)\bar{q}_{xx}(0,t)\right]\\
+\frac{\kappa(p+2)}{2}\text{Re}\left[\int_{0}^\infty|q|^pq_x\bar{q}_{xx} dx\right]+\frac{\kappa p}{2}\text{Re}\left[\int_{0}^\infty|q|^{p-2}q^2\bar{q}_x\bar{q}_{xx} dx\right].
\end{multline}
\subsubsection*{Estimates} Using the $L^2$-identity \eqref{iden00} and the $H^2$-identity \eqref{iden00h2}, we have
\begin{equation}\label{Eest01}
0\le E(t)\le E(0) + A(t)\|g_0\|_{H^1(0,t)} + B(t)\|g_1\|_{H^1(0,t)}\,,
\end{equation} where
$$E(t):=\frac{1}{2}\int_0^\infty|q|^2dx+\frac{1}{2}\int_0^\infty|q_{xx}|^2dx-\frac{\kappa}{p+2}\int_0^\infty|q|^{p+2}dx\,,$$
$$A(t):=\sqrt{\int_0^t|q_{xxx}(0,\tau)|^2d\tau}\text{ and } B(t):=\sqrt{\int_0^t|q_{xx}(0,\tau)|^2d\tau}.$$
Note that by integration by parts
\begin{equation}\label{H1est01}
\int_0^\infty|q_x|^2dx = -q(0,t)\bar{q}_x(0,t)-\int_0^\infty q\bar{q}_{xx}dx,
\end{equation}from which it follows that
\begin{multline}\label{H1est02}
\|q_x\|_{L^2(\mathbb{R}_+)}\le |g_0(t)|^\frac{1}{2}\cdot|g_1(t)|^\frac{1}{2}+\|q\|_{L^2(\mathbb{R}_+)}^\frac{1}{2}\|q_{xx}\|_{L^2(\mathbb{R}_+)}^\frac{1}{2}\\
\le \|g_0\|_{L^\infty(0,t)}^\frac{1}{2}\|g_1\|_{L^\infty(0,t)}^\frac{1}{2}+\sqrt{E(t)}.
\end{multline}
Using this in \eqref{combinediden01}, we obtain
\begin{multline}\label{combinediden01est}
B^2(t)\le\|q_0\|_{L^2(\mathbb{R}_+)}\|q_0'\|_{L^2(\mathbb{R}_+)}+\|q\|_{L^2(\mathbb{R}_+)}\|q_x\|_{L^2(\mathbb{R}_+)}+\|g_0\|_{H^1(0,t)}^2\\
+2A(t)\|g_1\|_{L^2(0,t)}-\frac{2\kappa}{p+2}\|g_0\|_{L^{p+2}(0,t)}^{p+2}\\
\le \|q_0\|_{H^1(\mathbb{R}_+)}^2+\|g_0\|_{H^1(0,t)}^2-\frac{2\kappa}{p+2}\|g_0\|_{L^{p+2}(0,t)}^{p+2}+\|g_0\|_{L^\infty(0,t)}\|g_1\|_{L^\infty(0,t)}\\
+{3}{E(t)}+2A(t)\|g_1\|_{L^2(0,t)}\le c+3E(t)+c A(t),
\end{multline} where $c$ is a nonnegative (generic) constant which might depend on $\kappa,p,$ and various norms of $q_0,g_0,$ and $g_1$.
On the other hand, using \eqref{combinediden02}, we obtain
\begin{multline}\label{combinediden02est}
A^2(t)\le 2\|g_0'\|_{L^2(0,t)}B(t)+\|g_1\|_{H^1(0,t)}^2+\|q_0''\|_{L^2(\mathbb{R}_+)}\|q_0'\|_{L^2(\mathbb{R}_+)}\\
+\|q_{xx}\|_{L^2(\mathbb{R}_+)}\|q_x\|_{L^2(\mathbb{R}_+)}+2|\kappa|\cdot\|g_0\|_{L^{2(p+1)}(0,t)}^{2(p+1)}\cdot B(t)\\
+{2(p+1)\int_0^t|\kappa|}\cdot\|q\|_{L^\infty(\mathbb{R}_+)}^p\|q_x\|_{L^2(\mathbb{R}_+)}\|q_{xx}\|_{L^2(\mathbb{R}_+)}ds\\
\le cB(t)+c+2E(t)+c\int_0^t\|q\|_{H^2(\mathbb{R}_+)}^{p+2}ds.
\end{multline}
We deduce from \eqref{H1est01} and the definition of $E(t)$ that $$\|q\|_{H^2(\mathbb{R}_+)}\le c+3\sqrt{2}\sqrt{E(t)}.$$ Using this in \eqref{combinediden02est}, we obtain
\begin{equation}\label{A2}A^2(t)\le cB(t)+c+2E(t)+c\int_0^tE(s)^{\frac{p+2}{2}}ds.\end{equation}
\eqref{Eest01} gives
\begin{equation}\label{son1}
E^2(t)\le c+cA^2(t)+cB^2(t).
\end{equation}
From \eqref{combinediden01est}, we have
\begin{equation}\label{son2}B^2(t)\le {c}+cE(t)+cA(t).\end{equation}
Combining \eqref{son1} and \eqref{son2}, we obtain
\begin{equation}\label{son3}
E^2(t)\le c+cA^2(t)+cE(t)+cA(t).
\end{equation}
Applying $\epsilon-$Young's inequality to the above estimate, we get
\begin{equation}\label{son4}
E^2(t)\le c_\epsilon+\frac{c+\epsilon}{1-\epsilon} A^2(t).
\end{equation}
Using \eqref{son2} in \eqref{A2}, the assumption $p\le 2$, and $\epsilon$-Young's inequality, we get
\begin{equation}\label{son5}A^2(t)\le c_{\epsilon,p,T}+\frac{\epsilon}{1-\epsilon} E^2(t)+\frac{\epsilon c_{p}}{1-\epsilon} \int_0^tE^2(s)ds.\end{equation}
Inserting \eqref{son5} into \eqref{son4}, we can finally close the estimates as
$$\left(1-\frac{\epsilon(c+\epsilon)}{(1-\epsilon)^2}\right)E^2(t)\le c_{\epsilon,p,T}+\frac{\epsilon(c+\epsilon) c_{p}}{(1-\epsilon)^2} \int_0^tE^2(s)ds.$$
Fixing $\epsilon$ to be sufficiently small (so that the coefficients become positive) and unleashing the Gronwall's inequality, we obtain the uniform estimate $E(t)\le c_{\epsilon,p,T}$ for $t\in (0,T]$, which implies
that the solution is global at the $H^2-$level.
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\begin{enumerate}}{\equivnd{enumerate}}
\title{ One--parameter subgroups of topological abelian groups }
\date{}
\author{M. J. Chasco\\{<mall\equivm Dept. de F\'{\i}sica y Matem\'{a}tica Aplicada},
{<mall\equivm University of Navarra}\\
{<mall\equivm e-mail: mjchasco@unav.es}}
\maketitle
\begin{abstract}
It is proved that, for a wide class of topological abelian groups (locally quasi--convex groups for which the canonical evaluation from the group into its Pontryagin bidual group is onto) the arc component of the group is exactly the union of the one--parameter subgroups. We also prove that for metrizable separable locally arc--connected reflexive groups, the exponential map from the Lie algebra into the group is open.
\equivnd{abstract}
Some properties of one--parameter subgroups of locally compact groups have been known for a long time. In a paper published 1967 Rickert proved that in a compact arc--connected group every point lies on a one--parameter subgroup. Previously Gleason had shown in 1950 that every finite dimensional, locally compact group contains a one-parameter subgroup. There are also examples of topological groups without non trivial one parameter subgroups; this is the case for instance of the subgroup of integer--valued functions of the Hilbert space $L^2[0,1]$. For topological abelian groups which are $k-spaces$, the arc component of the dual group is the union of its one--parameter subgroups. This result published by Nickolas in 1977, remained for some years the only available piece of information outside the class of locally compact groups. Recently it was proved in \cite{AChD} that the same is true for a much wider class of topological groups.
In this paper we go deeper into the study of one parameter subgroups of topological abelian groups. We take as a reference the papers \cite{AChD}, \cite{ATChD}, \cite{Nickolas}) and the book by Hofmann and Morris ``The structure of compact groups" (2000) which presents a wide range of properties of the one--parameter subgroups of locally compact abelian groups.
In order to do that we use two ingredients: on one hand Pontryagin duality techniques and on the other the relation between ${\rm CHom}(\mathbb R,G)$ and the group $G$ which is given by the evaluation mapping $${\rm CHom}(\mathbb R,G)\longrightarrow G,\
\varphi\longmapsto \varphi(1)$$
The vector space ${\rm CHom}(\mathbb
R,G)$ endowed with the compact open topology is called the Lie algebra of the topological group $G$ and denoted by ${\cal L}(G)$ in analogy with the classical theory of Lie groups. In that case the evaluation mapping is continuous and it is called the exponential function (${\rm exp}_G$).
The elements of ${\rm
im}\, {\rm exp}_G$ are those lying on one-parameter subgroups, and $G$ is the union of its one-parameter subgroups if and only if ${\rm exp}_G$ is onto.
For a topological group $G$, we denote by $G_a$ its arc--component.
The main results of the paper are:
{\bf Theorem 6.}
Let $G$ be a Hausdorff locally quasi--convex topological abelian group for which the evaluation mapping from the group into its bidual group is onto. Then, ${\rm im}\,{\rm exp}_{G}=G_a$
{\bf Theorem 11.}
If $G$ is a metrizable separable reflexive topological abelian group which has an arc--connected zero neighborhood, then the exponential mapping $\equivxp_{G}:{\cal L}(G)\longrightarrow G_a$ is a quotient map.
Observe that if $G$ is the additive topological group of a topological vector space $E$, then $\equivxp_{G}:{\rm CHom}(\mathbb R,G)\longrightarrow G,\
\varphi\longmapsto \varphi(1)$ is a topological isomorphism of topological vector spaces and in that case all the topological and algebraic information about $G$ is in ${\rm CHom}(\mathbb R,G)$.
We will use Pontryagin duality theory, hence our topological groups need to be abelian.
Pontryagin-van Kampen duality theorem for locally compact abelian groups (LCA groups) is a very deep result. It is the base of Harmonic Analysis and it allows to know the structure of LCA groups. This explains why a consistent Pontryagin-van Kampen duality theory has been developed for general topological abelian groups and why the abelian topological groups satisfying the Pontryagin-van Kampen duality, the so called {\equivm reflexive groups}, have received considerable attention from the late 40's of the last century (see \cite{survey} for a survey on the subject).
For an abelian topological group $G$, the {\equivm character group} or
{\equivm dual group}
$G^\wedge$ of $G$ is defined by
$$G^\wedge:=\{\varphi:G\rightarrow\mathbb T:\ \chi\ \mbox{ is a continuous
homomorphism}\}$$ where $\mathbb T$ denotes the compact group of complex numbers
of modulus $1$. The elements of $G^\wedge$
are called {\equivm characters}.
We say that the group $G$ {\equivm has enough characters} or that $G$ is a {\equivm MAP} group if for any $0\neq x\in
G$, there is some character $\varphi\in G^\wedge$ such that
$\varphi(x)\neq 1$.
Endowed with the compact--open topology $G^\wedge$ is an abelian
Hausdorff group. The {\equivm bidual} group $G^{\wedge\wedge}$ is $(G^\wedge)^\wedge$ and the canonical mapping in the bidual group is defined by:
$$\alpha_G:G\rightarrow G^{\wedge\wedge},\ x\mapsto
\alpha_G(x):\varphi\mapsto \varphi(x) .$$
The group $G$ is called {\equivm Pontryagin reflexive} if $\alpha_G$ is
a topological isomorphism. For the sake of simplicity, we will use the term
{\equivm reflexive} only. The famous Pontryagin--van Kampen theorem
states that every locally compact abelian (LCA) group is reflexive.
The {\equivm annihilator of} a subgroup $H <ubset G$ is defined as the subgroup
$H^\bot := \{ \varphi \in G^\wedge \colon \varphi
(H)=\{1\} \}$. If $L$ is a subgroup of $G^\wedge$, the {\it inverse annihilator} is defined by $L^\bot:= \{ g \in G \colon \varphi (g) = 1, \forall \varphi \in L \}$.
Annihilators are the particularizations for subgroups of the more general notion of polars of subsets. Namely, for $A <ubset G $ and $B<ubset G^\wedge$, the polar of $A$ is $A^\triangleright: = \{ \varphi \in G^\wedge: \varphi(A) <ubset \Bbb T_+ \}$ and the inverse polar of $B$ is $B^{\triangleleft}:= \{ g \in G \colon \varphi (g) \in \Bbb T_+, \forall \varphi \in B \}$, where $\mathbb T_+:=\{z\in\mathbb T:\ {\rm Re}\,z>e
0\}$ .
For a topological
Abelian group $G$, it is not difficult to prove that a set $M<ubset G^\wedge$ is
\equivmph{equicontinuous\/} if there exists a neighborhood $U$ of the
neutral element in $G$ such that $M<ubset U^\triangleright$. Other standard facts in duality theory are: If $U$ is a neighborhood zero in $G$, its polar is compact and the dual group $G^\wedge$ is the union of all polars of zero neighborhoods in $G$. The family $\{K^\triangleright; K \;\mbox{is a compact subset of}\;G \}$ is a neighborhood basis of the neutral element in $G^\wedge$.
A subgroup $H$ of a topological group $G$ is said to be {\equivm dually closed} if, for every element $x$ of $G<etminus H$, there
is a continuous character $\varphi$ in $G^\wedge$ such that
$\varphi (H)=\{1\}$
and $\varphi (x)\not= 1$.
Reflexive groups lie in a wider class of groups, the so called {\it locally quasi-convex groups}. Vilenkin had the seminal idea to define a sort of convexity for abelian topological groups inspired on the Hahn-Banach theorem for locally convex spaces.
A subset $A$ of a topological group $G$ is called {\equivm quasi-convex} if for every $g\in G<etminus A$, there is some $\varphi \in A^\triangleright $ such that $\mbox{Re}\varphi(g)<0$.
It is easy to prove that for any subset $A$ of a topological group $G$, $A^{\triangleright \triangleleft} $ is a quasi-convex set. It will be called the {\it quasi-convex hull} of $A$ since it is the smallest quasi-convex set that contains $A$. Obviously, $A$ is quasi-convex iff $A^{\triangleright \triangleleft} = A$.
If $A$ is a subgroup of $G$, $A$ is quasi-convex if and only if $A$ is dually closed. The abelian topological group $G$ is said to be locally quasi-convex if it admits a neighborhood basis of zero formed by quasi-convex subsets. Dual groups are examples of locally quasi-convex groups. For locally quasi--convex groups the evaluation map $\alpha_G$ is injective and open onto its image. A topological vector space $E$ is locally convex if and only if in its additive structure is a locally quasi-convex topological group (see \cite[2.4]{Ban}).
By a {\equivm real character} (as opposed to a {\equivm character}) on an
abelian topological group $G$ it is commonly understood a continuous
homomorphism from $G $ into the reals $\mathbb{R }$. The real
characters on $G$ constitute the vector space ${\rm CHom}(G,\mathbb
R)$. It is said that the group $G$ has {\equivm enough real characters}
if ${\rm CHom}(G,\mathbb R)$ separates the points of $G$. We denote by ${\rm CHom}_{co}(G,\mathbb R)$
the group of real characters endowed with the compact open topology. We say that a
character $\varphi: G\rightarrow \mathbb{T}$ can be \equivmph{lifted to
a real character}, if there exist a real character $f$ such that $e^{2\pi if}=\varphi$.
We denote
by $ G^{\wedge}_{\rm lift}$ the subgroup of $G^\wedge$ formed by the characters that can be lifted to a continuous real
character.
Given a topological abelian group $G$, we denote by
$\omega(G,G^\wedge)$ the weak topology on $G$ that is, the
topology on $G$ induced by the elements of $G^\wedge$. This topology
coincides with the Bohr topology.
On the other hand
$\omega(G^\wedge, G)$ denotes the topology on $G^\wedge$ of
pointwise convergence.
In a topological group $G$, the arc components are homeomorphic and
it makes sense to refer to the arc component of the neutral element
$G_a$ as the arc component.
There are well known results about the
arc component of locally compact groups.
Let $G$ be an LCA group, $G_a$ its arc component and $G_0$ its
connected component.
\begin{enumerate}
\item
$G_a= {\rm im}\,{\rm exp}_{G}$ (see \cite[8.30]{HM}).
\item
The arc component $G_a$ is dense in the connected component $G_0$ (see \cite[7.71]{HM}).
\item
The group $G$ has enough real characters iff the dual group $G^\wedge$ is connected (see \cite[24.35]{HR}).
\equivnd{enumerate}
What can be said about the arc component in more general classes of groups?
It is known that for topological abelian groups which are $k-spaces$,
the arc component of the dual group is exactly the union of its one--parameter subgroups (see \cite{Nickolas}).
It was proved recently that the same is true for a much wider class of topological groups: groups satisfying the EAP condition (A topological group $G$ satisfies the EAP if every arc in $G^\wedge$ is equicontinuous). This is introduced and studied for different groups in \cite{AChD}.
Next, we find new classes of groups for which $G_a= {\rm im}\,{\rm exp}_{G}$.
We start with a Lemma that can be proved in a straightforward way.
\begin{Lemma}\label{dense}
Let $G$ be a Hausdorff topological abelian group, $H$ a
subgroup and $L$ a subgroup of $G^\wedge$ then,
\par
$H$ is dense in the weak topology $\omega(G,G^\wedge)$ iff
$H^\bot=\{e\}$
\par
$L$ is dense in the pointwise topology $\omega(G^\wedge, G)$ if $L^\bot=\{e\}$.
If $G^\wedge$ is a MAP group, the
reverse implication is also true.
\equivnd{Lemma}
\begin{Lemma}\label{arco}
Let $G$ be a Hausdorff topological abelian group. If $K$ is a compact
subset of $G$, then $\alpha_G(K)$ is an equicontinuous subset of $G^{\wedge\wedge}$.
\equivnd{Lemma}
\begin{proof}
The polar set $K^\triangleright$ is a neighborhood of $0$ for the compact open topology of $G^\wedge$, then $K^{\triangleright\triangleright}$ is an equicontinuous subset of $G^{\wedge\wedge}$. Moreover, $\alpha_G(K^{\triangleright\triangleleft})=K^{\triangleright\triangleright}\bigcap \alpha_G(G)$. Therefore
$\alpha_G(K)<ubseteq \alpha_G(K^{\triangleright\triangleleft})=K^{\triangleright\triangleright}\bigcap \alpha_G(G)<ubseteq K^{\triangleright\triangleright}$. Since $\alpha_G(K)$ is contained in an equicontinuous subset, it is itself equicontinuous.
\equivnd{proof}
\begin{Proposition}\label{continuous}
Let $G$ be a Hausdorff topological abelian group and $>amma: \mathbb I\rightarrow G$ a continuous arc, then the mapping $\mathbb Phi_>amma: G^\wedge \times \mathbb I \rightarrow \mathbb T$ defined by $\mathbb Phi_>amma(\chi, t )=\chi(>amma(t))$ is continuous.
\equivnd{Proposition}
\begin{proof}
Take $\chi_0 \in G^\wedge$ and $t_0\in \mathbb I$. Let us see that $\mathbb Phi_>amma$ is continuous at $(\chi_0,t_0)$.
For every $n\in \mathbb N$, let us denote by $\mathbb T_n$ the neighborhood of $1$ in $\mathbb T$, $ \{e^{2\pi i t}:\ | t|\le \frac{1}{4n}\}.$
The mapping $\chi_0\circ >amma:\mathbb I \rightarrow \mathbb T$ is continuous so, for every $n\in \mathbb N$ there exists $V_{t_0}$ neighborhood of $t_0$ in $\mathbb I$ such that $\chi_0(>amma(t))\overline{\chi_0(>amma(t_0))} \in \mathbb T_{{2n}}$, for every $t\in V_{t_0}$.\\
On the other hand since $>amma(\mathbb I)$ is a compact subset of $G$, by \ref{arco}, $\alpha_G(>amma(\mathbb I))$ is equicontinuous at $\chi_0$ hence, for every $n\in \mathbb N$, there exists a neighborhood $U_{\chi_0}$ of $\chi_0$ in $G^\wedge$ such that $\chi(>amma(t))\overline{\chi_0(>amma(t))} \in \mathbb T_{{2n}}$ for every $t\in \mathbb I$ and $\chi \in U_{\chi_0}$.\\
Therefore $\chi(>amma(t))\overline{\chi_0(>amma(t))}\chi_0(>amma(t))\overline{\chi_0(>amma(t_0))} \in \mathbb T_{{2n}}\mathbb T_{{2n}}<ubset \mathbb T_{{n}}$ for every $t\in V_{t_0}$ and $\chi \in U_{\chi_0}$. This proves $\mathbb Phi_>amma$ is continuous at $(\chi_0, t_0)$.
\equivnd{proof}
\begin{Proposition}\label{contained}
Let $G$ be a Hausdorff topological abelian group and $G_a$ its arc component, then $G_a\leq \alpha_G^{-1}(G^{^{\wedge\wedge}}_{\rm lift})$
\equivnd{Proposition}
\begin{proof}
Let $x$ be an element in $G_a$, and let $>amma:\mathbb I\to G$ be a continuous mapping joining $0$ and $x$. Then, $\mathbb Phi_>amma: G^\wedge \times \mathbb I \rightarrow \mathbb T$ defined by $\mathbb Phi_>amma(\chi,t )=\chi(>amma(t))$ is continuous as we have seen in \ref{continuous}. Denote by $\psi: G^\wedge \times\{0\} \to \mathbb R$ the null real character. By the homotopy lifting property we can find an homotopy $F:G^\wedge\times \mathbb I\to \mathbb R$ such that $p\circ F = \mathbb Phi_>amma$ and $F\mid_{G^\wedge\times\{0\}}= \psi$. Now the unique path lifting property of $p:\mathbb R \to \mathbb T$ allows us to show that $\varphi: G^\wedge \to \mathbb R$ defined as the restriction of $F$ to $G^\wedge\times \{1\}$ is a homomorphism and hence a continuos real character lifting $\alpha_G(x)$: $p\circ \varphi(l)=p\circ F(l,1)=\mathbb Phi_>amma(l,1)=l(>amma(1))=l(x)=\alpha_G(x)(l)$, for all $l\in G^\wedge$.
\equivnd{proof}
\begin{Proposition}\label{alfasobre}
Let $G$ be a Hausdorff locally quasi--convex topological abelian group for which $\alpha_G$ is onto. Then, $\alpha_G^{-1}(G^{^{\wedge\wedge}}_{\rm lift})\leq {\rm im}\,{\rm exp}_{G}\leq G_a$.
\equivnd{Proposition}
\begin{proof}
Observe that ${\rm im}\,{\rm exp}_{G}$ is arc-connected and so it is contained in $G_a$. Let $x\in G$ such that $\alpha_G(x)\in G^{^{\wedge\wedge}}_{\rm lift}$ and a continuous homomorphism $\widetilde{\alpha_G(x)}: G^\wedge \rightarrow \mathbb R$ such that $p\circ \widetilde{\alpha_G(x)}=\alpha_G(x)$.
Denote by $S$ the topological isomorphisms $S:\mathbb R\to \mathbb R^\wedge,\ s\to \chi_s:\chi_s(t)=e^{2\pi i s t}$.
Since $G$ is locally quasi--convex, the evaluation mapping $\alpha_G$ is injective and open, so we can consider the homomorphism $f\in{\rm CHom}(G, \mathbb R)$ given by $f=\alpha_G^{-1}\circ (\widetilde{\alpha_G(x)})^\wedge\circ S$. Let us see that $f(1)=x$.
Observe first that $(\widetilde{\alpha_G(x)})^\wedge(S(1))=\alpha_G(x)$ [for $\psi\in G^\wedge$: $(\widetilde{\alpha_G(x)})^\wedge(S(1))(\psi)=(\chi_1\circ \widetilde{\alpha_G(x)})(\psi)=e^{2\pi i \widetilde{\alpha_G(x)}(\psi)}=(p\circ \widetilde{\alpha_G(x)})(\psi)=\alpha_G(x)(\psi)$]. Therefore, $f(1)=\alpha_G^{-1}\circ (\widetilde{\alpha_G(x)})^\wedge (S(1))=\alpha_G^{-1}(\alpha_G(x))=x$.
\equivnd{proof}
The following Theorem, obtained from the previous propositions, shows in particular that in an arc--connected locally quasi--convex topological abelian group for which $\alpha_G$ is onto, every point lies in one one--parameter subgroup. The same was proved by Rickert in 1967 for compact arc--connected groups (see \cite{Ric}).
\begin{Theorem}\label{main}
Let $G$ be a Hausdorff locally quasi--convex topological abelian group for which $\alpha_G$ is onto. Then, ${\rm im}\,{\rm exp}_{G}=G_a=\alpha_G^{-1}(G^{^{\wedge\wedge}}_{\rm lift})$
\equivnd{Theorem}
Some classes of Hausdorff locally quasi--convex topological abelian group for which $\alpha_G$ is onto are the following:
reflexive groups, duals of pseudocompact groups, $P$-groups (see \cite{survey}), groups of continuous functions $C(X,\mathbb T)$ where $X$ is a completely regular $k$-space (see \cite[14.8]{Diss}) and the wide class of nuclear complete topological abelian groups \cite[21.5]{Diss}.
The class of nuclear groups was formally introduced by Banaszczyk in \cite{Ban}. It is a class of topological groups embracing nuclear spaces and locally compact abelian groups (as natural generalizations of finite-dimensional vector spaces). The definition of nuclear groups
is very technical, as could be expected from its virtue of joining together objects of such different classes. A nice survey on nuclear groups is also provided by L. Au$<s$enhofer in \cite{Lydia2}. The following are important facts concerning the class of nuclear groups:\\
1. Nuclear groups are locally quasi-convex, $\cite[8.5]{Ban}$.
2. Products, subgroups and quotients of nuclear groups are again nuclear, $\cite[7.5 ]{Ban}$.
3. Every locally compact abelian group is nuclear, $\cite[7.10]{Ban}$ .
4. Every closed subgroup in a nuclear topological group is dually closed $\cite[8.6 ]{Ban}$.
5. A nuclear locally convex space is a nuclear group, $\cite[7.4]{Ban}$. Furthermore, if a topological vector space $E$ is a nuclear group, then it is a locally convex nuclear space, $\cite[8.9]{Ban}$.
\begin{Proposition}\label{LieLydia}\cite[1.4]{LydiaLie}
Let $G$ be a Hausdorff locally quasi--convex topological abelian group for which $\alpha_G$ is continuous, then the mapping $\mathbb Phi_0:{\cal L}(G)\to {\rm CHom}_{co}(G^\wedge,\mathbb{R^\wedge})$ given by $\varphi\mapsto \varphi^{\wedge}$: $\varphi^{\wedge}(\chi)=\chi\circ \varphi$ is an embedding.
\equivnd{Proposition}
\begin{Proposition}\label{LieMJ}
Let $G$ be a reflexive topological abelian group, then the mapping $\mathbb Phi_0:{\cal L}(G)\to {\rm CHom}_{co}(G^\wedge,\mathbb{R^\wedge})$ given by $\varphi\mapsto \varphi^{\wedge}$ is a topological isomorphism.
\equivnd{Proposition}
\begin{proof}
Using Proposition \ref{LieLydia} we only need to check that $\mathbb Phi_0$ is onto:
Let $\psi: G^\wedge \to \mathbb{R^\wedge}$ be a continuous homomorphism and $\varphi=\alpha_G^{-1}\circ \psi^\wedge\circ \alpha_{\mathbb R}$. Let us see that $\varphi^{\wedge}=\psi$ or which is the same, that
for every $\chi\in G^\wedge$, $\varphi^{\wedge}(\chi)=\chi\circ \varphi=\chi\circ \alpha_G^{-1}\circ \psi^\wedge\circ \alpha_{\mathbb R}=\psi(\chi)$.
So, take $t\in \mathbb R$, if $x=(\alpha_G^{-1}\circ \psi^\wedge\circ \alpha_{\mathbb R})(t)=\alpha_G^{-1}( \psi)$, then $\alpha_G(x)=\alpha_{\mathbb R}(t)\circ \psi$
$\chi(x)=\alpha_G(x)(\chi)=(\alpha_{\mathbb R}(t)\circ \psi)(\chi)=\alpha_{\mathbb R}(t)(\psi(\chi))=\psi(\chi)(t)$,
Therefore $\varphi^{\wedge}(\chi)(t)=\psi(\chi)(t)$, for all $\chi \in G^\wedge$ and $t\in \mathbb R$, that is
$\varphi^{\wedge}=\psi$.
\equivnd{proof}
\begin{Corollary}\label{LieR}
Let $G$ be a Hausdorff locally quasi--convex topological abelian group for which $\alpha_G$ is continuous, then ${\cal L}(G)$ is topologically isomorphic to a subgroup of ${\rm CHom}_{co}(G^\wedge,\mathbb{R})$. Moreover, if $G$ is
reflexive then ${\cal L}(G) \cong {\rm CHom}_{co}(G^\wedge,\mathbb{R})$
\equivnd{Corollary}
\begin{proof}
Take into account that $S: \mathbb R \to \mathbb R^\wedge$ given by $S(t)(s)=e^{2\pi ist}$ is a topological isomorphism.
\equivnd{proof}
We recall now that some properties of the group preserved by ${\cal L}(G)$ are
1. ${\cal L}(G)$ is Hausdorff, if the topological group $G$ is Hausdorff.
2. ${\cal L}(G)$ is complete, if the topological group $G$ is complete.
3. ${\cal L}(G)$ is a locally convex space,
if the topological group $G$ is a Hausdorff locally quasi-convex group (see \cite[1.2]{LydiaLie}).
4. ${\cal L}(G)$ is nuclear if the topological group $G$ is nuclear (see \cite[2.6]{LydiaLie}).
\begin{Proposition}
If $G$ is a locally quasi--convex metrizable and separable topological abelian group, ${\rm CHom}_{co}(G^\wedge,\mathbb{R})$ is metrizable complete and separable. The Lie algebra ${\cal L}(G)$ is also metrizable and separable, and it is complete if the topological group $G$ is complete.
\equivnd{Proposition}
\begin{proof}
Since $G$ is metrizable, $G^\wedge$ is hemicompact and a $k$-space \cite{MJ}. By the hemicompactness of $G^\wedge$, ${\rm CHom}_{co}(G^\wedge,\mathbb{R})$ is metrizable and because $G^\wedge$ is a $k$-space, ${\rm CHom}_{co}(G^\wedge,\mathbb{R})$ is complete.
On the other hand since $G$ is metrizable and separable it holds (see \cite[1.7]{ChMPT}) that
compact subsets of $G^\wedge$ are metrizable. But for a hemicompact $k$-space $X$ whose compact subsets are metrizable, $C_{co}(X)$ is metrizable and separable (see \cite{KLMPT} and \cite{Warner}), hence ${\rm CHom}_{co}(G^\wedge,\mathbb{R})$ is metrizable separable and complete. Observe now that ${\cal L}(G)$ is complete because $G$ is complete and it is metrizable and separable because it is topologically isomorphic to a subgroup of ${\rm CHom}_{co}(G^\wedge,\mathbb{R})$.
\equivnd{proof}
{\bf Remark.} For torus groups
$\mathbb T^X$ were $X$ is an arbitrary set,we may identify the exponential function with
the canonical quotient map $\mathbb R^X \to \mathbb T^X$, therefore, it is open. There are also compact
connected groups which are not torus groups but for which the exponential function
is open onto its image (see \cite{HMP}). We next find other classes of abelian groups for which the
corestriction $\equivxp_{G}:{\cal L}(G)\longrightarrow G_a$ is open.
\begin{Theorem}\label{exp.open}
If $G$ is a metrizable separable reflexive topological abelian group which has an arc--connected zero neighborhood, then the exponential mapping $\equivxp_{G}:{\cal L}(G)\longrightarrow G_a$ is a quotient map.
\equivnd{Theorem}
\begin{proof}
By the above Proposition, $ {\cal L}(G)$ is metrizable complete and separable. On the other hand, since $G$ is a locally quasi-convex topological group for which $\alpha_G$ is onto ${\rm im}\,{\rm exp}_{G}=G_a$. By hypothesis, $G_a$ is an open subgroup of the group $G$, hence it is closed. Therefore $G_a$ is metrizable complete. Since the exponential mapping is a continuous and onto homomorphism, by the open mapping theorem, it is open.
\equivnd{proof}
\begin{Corollary}
Let $G$ be a metrizable separable reflexive topological group:
1. If $G$ is locally arc-connected, then the exponential map $\equivxp_{G}:{\cal L}(G)\longrightarrow G_a$ is a quotient map.
2. If $G$ is arc-connected, then the exponential map $\equivxp_{G}:{\cal L}(G)\longrightarrow G$ is a quotient map.
3. If the exponential map $\equivxp_{G}:{\cal L}(G)\longrightarrow G_a$ is a quotient map and $G$ has an arc--connected zero neighborhood, then $G$ is locally arc--connected.
\equivnd{Corollary}
\begin{proof}
For 1) and 2) it is enough to know that metrizable reflexive groups are complete, so all the hypothesis of the above Corollary are fulfilled.
3)
Since ${\cal L}(G)$ is a locally convex topological
vector space it has arbitrarily small arc--connected neighborhoods of
zero which are mapped onto open identity neighborhoods of $G_a$ by $\equivxp_{G}$.
\equivnd{proof}
The following Lemma shows that for a topological abelian group, to have enough real characters and to have enough characters that can be lifted, are equivalent properties.
\begin{Lemma}\label{propliftingofchars}\cite[2.1]{ATChD}
Let $G$ be a topological abelian group and $e_G\neq x\in
G$. The following assertions are equivalent:
1. There exists a real character $f$ such that $f(x)\neq 0$.
2. There exists a character
$\varphi\in G^\wedge_{\rm lift}$ such that $\varphi(x)\neq 1$.
\equivnd{Lemma}
\begin{Corollary}
A Hausdorff topological abelian group $G$ has enough real characters if and only if $(G^{\wedge}_{\rm lift})^\bot=\{0\}$
\equivnd{Corollary}
\begin{Theorem} \label{AChD}\cite{AChD}
Let $(G,\tau)$ be an abelian Hausdorff satisfying EAP.
Then $$G^{\wedge}_{\rm lift} = {\rm im}\,{\rm exp}_{G^{\wedge}}= (G^{\wedge})_a$$
\equivnd{Theorem}
\begin{Theorem}\label{section:maintheorem}
Let $(G,\tau)$ be an abelian Hausdorff group such that every arc in the character group is equicontinuous. Consider the
following conditions:
\begin{itemize}
\item[a)] $G$ has enough real characters.
\item[b)] $(G^{\wedge})_a$ is dense in $G^\wedge$, with the pointwise convergence topology.
\equivnd{itemize}
Then a)$\mathbb Rightarrow$ b). If $G$ is a MAP group, b)$\mathbb Rightarrow$ a).
\equivnd{Theorem}
\begin{proof}
By Lemma \ref{dense} and Theorem \ref{AChD}, $((G^{\wedge})_a)^\bot=(G^{\wedge}_{\rm lift})^\bot$. If the group $G$ has enough real characters $(G^{\wedge}_{\rm lift})^\bot$ is trivial and then $(G^{\wedge})_a$ is dense in $G^\wedge$, with the pointwise convergence topology. For the reverse implication take into account that for a MAP group $G$, $(G^{\wedge})_a$ dense in $G^\wedge$ with the pointwise convergence topology, implies $((G^{\wedge})_a)^\bot$ is trivial.
\equivnd{proof}
\begin{Corollary}\label{LieMJ}
Let $G$ be a locally quasi-convex topological abelian group such that $\alpha_G$ is onto then,
\begin{enumerate}
\item
$G_a$ is $\omega(G,G^\wedge)$--dense iff $G^\wedge$ has enough real characters.
\item
$G$ is arc--connected iff every character of $G^{\wedge}$ can be lifted.
\equivnd{enumerate}
\equivnd{Corollary}
\begin{proof}
(1) By Theorem \ref{main} and Lemma \ref{dense}, $G_a$ is $\omega(G,G^\wedge)$--dense iff $(G_a)^\bot=(\alpha_G^{-1}(G^{^{\wedge\wedge}}_{\rm lift}))^\bot=(G^{^{\wedge\wedge}}_{\rm lift})^\bot=\{0\}$, iff $G^\wedge$ has enough real characters.
(2) Again by Theorem \ref{main}, $G_a=G$ iff $G^{^{\wedge\wedge}}_{\rm lift}
=G^{^{\wedge\wedge}}$.
\equivnd{proof}
\begin{Corollary}
Let $G$ be a locally quasi--convex topological abelian group with $\alpha_G$ onto and such that closed subgroups are dually closed.
If $G^\wedge$ has enough real characters, then $G$ is connected.
\equivnd{Corollary}
\begin{proof}
Having $G^\wedge$
enough real characters the arc--component
$G_a$ of $G$ is $\omega(G,G^\wedge)$--dense. Let $G_0$ be the
connected component of $G$. It is clear that $G_a<ubset G_0$
therefore $G_0$ is $\omega(G,G^\wedge)$--dense. Since $G_0$ is a closed
subgroup of the group $G$, $G_0$ is dually closed
and therefore it is $\omega(G,G^\wedge)$--closed. Then $G_0=G$, that
is: $G$ is connected.
\equivnd{proof}
{\bf Examples}: Nuclear complete topological abelian groups are locally quasi--convex topological abelian groups with $\alpha_G$ onto and such that closed subgroups are dually closed. Therefore every nuclear complete topological group, such that its dual group, $G^\wedge$ has enough real characters, is connected.
The previous results allow us to give an alternative proof for the following well known fact.
\begin{Corollary}
Let $G$ be an LCA group then $G$ is connected iff $G^\wedge$ has enough real characters.
\equivnd{Corollary}
\begin{proof}
Observe that LCA groups are nuclear complete, then the if part is true. Let us prove the reverse implication. Since $G$ is an LCA group, $G_a$ is dense in $G_0$. But $G_0=G$ because $G$ is connected, so $G_a$ is dense and therefore $\omega(G,G^\wedge)$ dense in $G$. Therefore, by Corollary \ref{LieMJ}, $G^\wedge$ has enough real characters.
\equivnd{proof}
Acknowledgement. The author is indebted to Professors E. Martin Peinador and
X. Dominguez and S. Ardanza-Trevijano for very helpful suggestions.
\begin{thebibliography}{10}
\bibitem{AChD} L. Au<s enhofer, M.J. Chasco and X. Dom\'{\i}nguez. Arcs in the Pontryagin dual of a topological abelian group. {\equivm Preprint}.
\bibitem{ATChD}S. Ardanza-Trevijano, M. J. Chasco\ and\ X. Dom\'\i nguez, The role of real characters in the Pontryagin duality of topological abelian groups, J. Lie Theory {\bf 18} (2008), no.~1, 193--203.
\bibitem{Diss}
L. Aussenhofer, Contributions to the duality theory of abelian topological groups and to the theory of nuclear groups, Dissertationes Math. (Rozprawy Mat.) {\bf 384} (1999), 113
\bibitem{Lydia2}
L. Au<s enhofer, A survey on nuclear groups, in {\it Nuclear groups and Lie groups (Madrid, 1999)}, 1--30, Res. Exp. Math., 24, Heldermann, Lemgo.
\bibitem{LydiaLie}
L. Au<s enhofer, The Lie algebra of a nuclear group, J. Lie Theory {\bf 13} (2003), no.~1, 263--270.
\bibitem{Ban}W. Banaszczyk, {\it Additive subgroups of topological vector spaces}, Lecture Notes in Mathematics, 1466, Springer, Berlin, 1991.
\bibitem{MJ}
M. J. Chasco, Pontryagin duality for metrizable groups, Arch. Math. (Basel) {\bf 70} (1998), no.~1, 22--28.
\bibitem{survey} M. J. Chasco, D. Dikranjan\ and\ E. Mart\'\i n-Peinador, A survey on reflexivity of abelian topological groups, Topology Appl. {\bf 159} (2012), no.~9, 2290--2309.
\bibitem{ChMPT} M. J. Chasco, E. Mart\'\i n-Peinador\ and\ V. Tarieladze, A class of angelic sequential non-Fr\'echet-Urysohn topological groups, Topology Appl. {\bf 154} (2007), no.~3, 741--748.
\bibitem{Gl} A. M. Gleason, Arcs in locally compact groups, Proc. Nat. Acad. Sci. U. S. A. {\bf 36} (1950), 663--667.
\bibitem{HR} E. Hewitt\ and\ K. A. Ross, {\it Abstract harmonic analysis. Vol. I}, second edition, Grundlehren der Mathematischen Wissenschaften, 115, Springer, Berlin, 1979.
\bibitem{HM} K. H. Hofmann\ and\ S. A. Morris, {\it The structure of compact groups}, third edition, revised and augmented., De Gruyter Studies in Mathematics, 25, de Gruyter, Berlin, 2013.
\bibitem{HMP}
K.H. Hofmann, S.A. Morris and D. Poguntke.
The exponential function of locally connected compact abelian
groups, Forum Math. {\bf 16}, no.1 (2004) 1-16.
\bibitem{KLMPT}
J. K\c akol\ et al., Lindel\"of spaces $C(X)$ over topological groups, Forum Math. {\bf 20} (2008), no.~2, 201--212.
\bibitem{Nickolas}
P. Nickolas, Reflexivity of topological groups, Proc. Amer. Math. Soc. {\bf 65} (1977), no.~1, 137--141.
\bibitem{Ric}
N. W. Rickert, Arcs in locally compact groups, Math. Ann. {\bf 172} (1967), 222--228.
\bibitem{Warner}
S. Warner, The topology of compact convergence on continuous function spaces, Duke Math. J. {\bf 25} (1958), 265--282.
\equivnd{thebibliography}
\equivnd{document} |
\begin{document}
\begin{center}
\textbf{Explicit formulas of the Bergman kernel for some Reinhardt domains}
\end{center}
\vskip1em
\begin{center}
Tomasz Beberok
\end{center}
\vskip3em
In this paper we obtain the closed forms of some hypergeometric functions. As an application, we obtain the explicit forms of the Bergman kernel functions for Reinhardt domains $\{|z_3|^{\lambda} < |z_1|^{2p} + |z_2|^2, \quad |z_1|^{2p} + |z_2|^2 < |z_1|^{p} \}$ and \\ $\{|z_4|^{\lambda} < (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2, \quad (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2 < (|z_1|^2 + |z_2|^2 )^{p/2} \}$.
\vskip1em
\textbf{Keyword:} Bergman kernel, hypergeometric functions
\vskip1em
\textbf{AMS Subject Classifications:} 32A25; 33D70
\section{\bf Introduction}
In 1921, S. Bergman introduced a kernel function, which is now known as the Bergman kernel function. It is well known that there exists a unique Bergman kernel function for each bounded domain in $\mathbb{C}^n$. Computation of the Bergman kernel function by explicit formulas is an important research direction in several
complex variables. For which domains can the Bergman kernel function be computed by explicit formulas? Many mathematicians (\cite{DA1}, \cite{FH}, \cite{F2}, \cite{F3}, \cite{Park1}) have made efforts to find the explicit formulas of the Bergman kernel for nonhomogeneous domains. Consider the complex ellipsoids or egg domains $\Omega_{p} := \{z \in \mathbb{C}^n : \sum\limits_{j=1}^{n} |z_j |^{2p_j} <1\}$, where $p = (p_1, . . . , p_n)$ for $p_j > 0$. The precise growth estimate of the Bergman kernel near a boundary point on the complex ellipsoid was studied in \cite{ZY}. However, it is not easy to get the closed forms of the Bergman kernel for $D_p$.
In the case when $p_1, . . . , p_n$ are reciprocals of positive integers, Zinov'ev \cite{Z} computed the Bergman kernel for $D_p$ explicitly. What happens if each $p_j$ is a positive integer? The known case is when $p = (1, . . . , 1, p_n), p_n > 0$, for which J. P. D’Angelo \cite{DA1, DA2} obtained the Bergman kernel. J.-D. Park computed the Bergman kernel for $p =(2,2)$ and $p=(2,2,2)$ in \cite{Park1} and \cite{Park2} respectively. The goal of this paper is to give explicit formula for the domains $\{|z_3|^{\lambda} < |z_1|^{2p} + |z_2|^2, \ |z_1|^{2p} + |z_2|^2 < |z_1|^{p} \}$ and $\{|z_4|^{\lambda} < (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2, \ (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2 < (|z_1|^2 + |z_2|^2)^{p/2} \}$.
\begin{theorem}
For any positive real numbers $\lambda, p$ the Bergman kernel for the domain $$D_1=\{z \in \mathbb{C}^4 \colon |z_4|^{\lambda} < (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2, \ (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2 < (|z_1|^2 + |z_2|^2)^{\frac{p}{2}}\}$$ is given by
\begin{align*}
K_{D_1}((z_1,z_2,z_3,z_4),(\zeta_1,\zeta_2,\zeta_3, \zeta_4))= D_{p,\lambda, \alpha} \left\{ C \frac { ( \frac{2}{\lambda} -1 )(\sqrt{1-4\nu_3}-1) + \frac{8\nu_3}{\lambda} }{(1-\mu_4)^2 (1- \mu_1 - \mu_2)^2 } \right. \\ \left. + C \frac{ 4 \sqrt{1-4\nu_3} + 16\nu_3 - 4} {p (1-\mu_4)^2 (1- \mu_1 - \mu_2)^3} + C \frac{4\mu_4 \left(\sqrt{1-4\nu_3} + 4\nu_3 - 1\right) }{\lambda (1-\mu_4)^3 (1- \mu_1 - \mu_2)^2} \right\} ,
\end{align*}
where
$$ \nu_1 = z_1 \overline{\zeta_1}, \quad \nu_2 = z_2 \overline{\zeta_2}, \quad \nu_3 = z_3 \overline{\zeta_3}, \quad \nu_4 = z_4 \overline{\zeta_4}, $$
$$D_{p,\lambda, \alpha}f = \frac{p}{ \pi^4} \left(\frac{2}{p} \frac{\partial}{\partial \nu_1} \nu_1 + \frac{2}{p} \frac{\partial}{\partial \nu_2} \nu_2 + \frac{\partial}{\partial \nu_3} \nu_3 + \frac{2}{\lambda} \frac{\partial}{\partial \nu_4} \nu_4 \right)f ,$$
$$\mu_1= \frac{2^{2/p} \nu_1} {\left(1+ \sqrt{1- 4\nu_3}\right)^{2/p} }, \quad \mu_2= \frac{2^{2/p} \nu_2} {\left(1+ \sqrt{1- 4\nu_3}\right)^{2/p} } , \quad \mu_4= \frac{2^{2/\lambda} \nu_4} {\left(1+ \sqrt{1- 4\nu_3}\right)^{2/\lambda} }$$
and
$$ C=\frac{ 2^{4/p + 2/\lambda} }{4\nu_3 (1-4\nu_3)^{3/2} (1+ \sqrt{1- 4\nu_3})^{4/p + 2/\lambda -1}}.$$
\end{theorem}
\begin{theorem}
The Bergman kernel for $$D_2=\{z \in \mathbb{C}^3 \colon |z_3|^{2} < |z_1|^{4} + |z_2|^2, \quad |z_1|^{4} + |z_2|^2 < |z_1|^{2} \}$$ is given by
\begin{align*}
K_{D_2}((z_1,z_2,z_3),(\zeta_1,\zeta_2,\zeta_3))= \frac{2\nu_1^4 -( \nu_1^2 \nu_3 + \nu_1^3)(\nu_1^2 + \nu_2)}{ \pi^3 (\nu_1 - \nu_3)^3 (\nu_1 - \nu_1^2 - \nu_2)^3} ,
\end{align*}
where $ \nu_1 = z_1 \overline{\zeta_1}, \quad \nu_2 = z_2 \overline{\zeta_2}, \quad \nu_3 = z_3 \overline{\zeta_3}$.
\end{theorem}
In 1995 Francsics and Hanges \cite{FH} expressed the Bergman kernel for complex ellipsoids $\Omega_{p_1,...,p_n}$ in terms of Appell’s multivariable hypergeometric functions which are still infinite series. Recall that an Appell’s hypergeometric function \cite{AK} is defined by $$F^{(n)}_A (\alpha; \beta; \gamma; \zeta)=\sum_{m_1=0}^{\infty} \cdots \sum_{m_n=0}^{\infty} \frac{(\alpha)_{m_1+\ldots+m_n} (\beta_1)_{m_1} \cdots (\beta_n)_{m_n} }{m_1! \cdots m_n! (\gamma_1)_{m_1} \cdots (\gamma_n)_{m_n} } \zeta_1^{m_1} \cdots \zeta_n^{m_n}, $$
\noindent where $(a)_m = \Gamma(a+m)/\Gamma (a)$. In particular we write $F_2 = F^{(2)}_A$ and $F = F^{(1)}_A$. In fact, the Bergman kernel $K(z,w)$ for $\Omega_{p_1,...,p_n}$ is given in \cite{FH} by
\begin{eqnarray*}
K(z,w)=\frac{\prod\limits_{j=1}^n p_j}{\pi^n} \sum\limits_{k_1=0}^{p_1 -1} \cdots \sum\limits_{k_n=0}^{p_n - 1} \frac{\Gamma\left(1+ \sum\limits_{j=1}^n \frac{k_j +1}{p_j}\right)}{\prod\limits_{j=1}^n \Gamma\left(\frac{k_j +1}{p_j}\right)} (z\overline{w})^k \\ \times F^{(n)}_A \left(1+ \sum\limits_{j=1}^n \frac{k_j +1}{p_j};\mathbf{1};\frac{\mathbf{k+1}}{\mathbf{p}};(z\overline{w})^p\right),
\end{eqnarray*}
\noindent where $(z\overline{w})^k = (z_1\overline{w_1})^{k_1}\cdots(z_n \overline{w_n})^{k_n} $. Here we following by Park used the notation $\mathbf{1} = \underbrace{(1,\ldots,1)}_n$ and $\frac{\mathbf{k+1}}{\mathbf{p}}=\underbrace{\left(\frac{k_1 +1}{p_1},\ldots, \frac{k_n +1}{p_n} \right)}_n.$
\section{\bf Explicit formulas of hypergeometric functions}
In this section we will express the sum of the series $\sum\limits_{m=0}^{\infty} \frac{(a)_{2m_1+\ldots + 2m_n} x_1^{m_1} \cdots x_n^{m_n} }{(c)_{m_1+\ldots + m_n} m_1! \cdots m_n! }$ in terms of Gauss hypergeometric function.
\begin{lemma}\label{lem3}
\noindent For $|x_1| + \ldots + |x_r| <1/4$, we have
\begin{align*}
F \left(\frac{a}{2},\frac{a+1}{2} ; c ; 4(x_1+ \ldots + x_r ) \right)= \sum_{m=0}^{\infty} \frac{(a)_{2m_1+\ldots + 2m_n} x_1^{m_1} \cdots x_n^{m_n} }{(c)_{m_1+\ldots + m_n} m_1! \cdots m_n! }
\end{align*}
\end{lemma}
\begin{proof}
Using well known rules for Pochhammer symbol $(2z)_{2k}=4^k (z)_k (z+1/2)_k$ we have
$$\sum_{m=0}^{\infty} \frac{(a)_{2m_1+\ldots + 2m_n} x_1^{m_1} \cdots x_n^{m_n} }{(c)_{m_1+\ldots + m_n} m_1! \cdots m_n! }= \sum_{m=0}^{\infty} \frac{4^{|m|} (a/2)_{|m|} (a/2+1/2)_{|m|}}{(c)_{|m|} m_1! \cdots m_n!} x_1^{m_1} \cdots x_n^{m_n} .$$
Now using $(z)_{n+k}=(z)_n (z+n)_k$ and sum out of $x_1$ variable we have
\begin{eqnarray*}
\sum_{m=0}^{\infty} \frac{4^{|m|} (a/2)_{|m|} (a/2+1/2)_{|m|}}{(c)_{|m|} m_1! \cdots m_n!} x_1^{m_1} \cdots x_n^{m_n} =\\ \sum_{m_2,\ldots,m_n=0}^{\infty} \frac{4^{m_2 + \ldots + m_n} (a/2)_{m_2 + \ldots + m_n} (a/2+1/2)_{m_2 + \ldots + m_n}}{(c)_{m_2 + \ldots + m_n} m_2! \cdots m_n!} x_2^{m_2} \cdots x_n^{m_n} \\ \times F\left(\frac{a}{2}+ m_2 + \ldots + m_n , \frac{a+1}{2} + m_2 + \ldots + m_n; c+ m_2 + \ldots + m_n; 4x_1 \right).
\end{eqnarray*}
In the other hand by decomposition formulas for the Appell function $F^{(r)}_A$ in $r$ ($r>1$) variables we have (see for more details \cite{HS})
\begin{align*}
&F^{(r)}_A (a,b_1,\ldots,b_r ; c_1,\ldots, c_r; y_1,\ldots,y_r )= \\ &\sum_{m_2,\ldots,m_r=0}^{\infty} \frac{ (a)_{m_2 + \ldots + m_r} (b_1)_{m_2 + \ldots + m_r} (b_2)_{m_2} \cdots (b_r)_{m_r} }{m_2! \cdots m_r! (c_1)_{m_2 + \ldots + m_r} (c_2)_{m_2} \cdots (c_r)_{m_r} } y_1^{m_2 + \ldots + m_r} y_2^{m_2} \cdots y_r^{m_r} \\ & \cdot F\left(a + m_2 + \ldots + m_r , b_1 + m_2 + \ldots + m_r; c_1+ m_2 + \ldots + m_r; y_1 \right)\\ & \cdot F^{(r-1)}_A( a + m_2 + \ldots + m_r, b_2+m_2, \ldots , b_r+ m_r; c_2 + m_2, \ldots, c_r + m_r; y_2,\ldots,y_r)
\end{align*}
Next we set $b_i=c_i$ for $i=2,\ldots,r$. After doing so, and using well know formula $F^{(s)}_A (a,b_1,\ldots,b_s; b_1,\ldots,b_s;z_1,\ldots,z_s) = \frac{1}{(1-z_1-\ldots - z_s)^a} $ we obtain
\begin{align*}
&F^{(r)}_A (a,b_1,b_2,\ldots,b_r ; c_1,b_2\ldots, b_r; y_1,\ldots,y_r )= \\ &\sum_{m_2,\ldots,m_r=0}^{\infty} \frac{ (a)_{m_2 + \ldots + m_r} (b_1)_{m_2 + \ldots + m_r} }{m_2! \cdots m_r! (c_1)_{m_2 + \ldots + m_r} } y_1^{m_2 + \ldots + m_r} y_2^{m_2} \cdots y_r^{m_r} \\ & \cdot F\left(a + m_2 + \ldots + m_r , b_1 + m_2 + \ldots + m_r; c_1+ m_2 + \ldots + m_r; y_1 \right)\\& \cdot \frac{1}{(1-y_2-\ldots -y_r)^{a+m_2 + \ldots + m_r}}
\end{align*}
Finally putting $y_1=4 x_1$ and $y_i = \frac{ x_i}{|x|}$ for $i=2, \ldots, r$, where $|x|=x_1+\ldots + x_r$ and using well know formula $F^{(s)}_A(a,b_1,\ldots,b_i,\ldots,b_s; c_1,\ldots,b_i,\ldots,c_s;z_1,\ldots,z_s)= (1-z_i)^{-a} F^{(s-1)}_A(a,b_1,\ldots,b_{i-1},b_{i+1},\ldots,b_s; c_1,\ldots,c_{i-1},c_{i+1},\ldots,c_s;\frac{z_1}{1-z_i},\ldots,\frac{z_s}{1-z_i}) $ we obtain the desired result.
\end{proof}
The following lemma will be useful to explicit computation of Bergman kernel function for the domain $$\Omega= \left\{ (z_1,z_2,z_3) \in \mathbb{C}^{n+m+k} \colon \|z_1\|^{\lambda} < \|z_2\|^{2p} + \|z_3\|^2, \ \|z_2\|^{2p} + \|z_3\|^2 < \|z_2\|^{p} \right\}$$ for $k=2$ and $k=3$.
\begin{lemma}\label{lem2f1}
\begin{align*}
&F\left( \frac{3+a}{2}, \frac{4+a}{2} ; a ; z \right)= \frac{ \left(-a-1 + \left(a - \frac{1}{2}\right )z \right) \left(\left(a - \frac{5}{2} \right)z - a\right) - \left(\frac{1-a}{2}\right) \left(\frac{2-a}{2}\right)z }{ a (a+1) (a+2) z (z-1) (1-z)^{3/2} ( \sqrt{1-z} +1 )^a } \\ & \cdot 2^{a+1} (a (\sqrt{1-z} -1 ) + (a+1)z) - \frac{2^a \left(\left(a - \frac{5}{2} \right)z - a\right) \left(a+a^2\right)z }{a (a+1) (a+2) (z-1)^2 \sqrt{1-z} (\sqrt{1-z}+1)^{a+2} }
\end{align*}
\begin{align*}
F\left( \frac{2+a}{2}, \frac{3+a}{2} ; a ; z \right)=& \frac{ 2^a (a-a^2)z }{2 a (a+1)(z-1) \sqrt{1-z} (\sqrt{1-z} +1)^{a+1} } \\ &+ \frac{ (-3/2 z -a) 2^a ((a-1) (\sqrt{1-z} - 1 )+ az) }{ a (a+1) z (z-1) (1-z)^{3/2} (\sqrt{1-z} +1)^{a-1} }
\end{align*}
\end{lemma}
\begin{proof}
In order to prove the above lemma, we need the following well-known formulas:
$$ F\left( \frac{a+3}{2}, \frac{a+4}{2}; a+2; z \right) = \frac{2^{a+1}(a (\sqrt{1-z}-1) + (a+1)z) }{(a+2)z (1-z)^{3/2} ( \sqrt{1-z} +1)^a } \quad \text{and}$$
$$ F\left( \frac{a+3}{2}, \frac{a+4}{2}; a+3; z \right) = \frac{2^{a+2}}{ \sqrt{1-z} ( \sqrt{1-z} +1 )^{a+2} }$$
Now lemma \ref{lem2f1} follows from recurrence identity for Gauss hypergeometric function
\begin{align*}
& F\left( \frac{a+3}{2}, \frac{a+4}{2}; a; z \right)= C_2 F\left( \frac{a+3}{2}, \frac{a+4}{2}; a+2; z \right) \\& - \frac{\left(a+a^2 \right) z }{ 4(a+1)(a+2)(z-1) } \cdot C_1 F\left( \frac{a+3}{2}, \frac{a+4}{2}; a+3; z \right),
\end{align*}
where $C_2= \frac{ \left(-a-1 + \left(a - \frac{1}{2}\right )z \right) \left(\left(a - \frac{5}{2} \right)z - a\right) - \left(\frac{1-a}{2}\right) \left(\frac{2-a}{2}\right)z }{ a (a+1) (z-1)}$ and $C_1= \frac{\left(a - \frac{5}{2} \right)z - a}{a(z-1)}$.
\end{proof}
\section{\bf Computation of the kernel.}
Let $\Omega$ be a bounded domain in $\mathbb{C}^N$. The Bergman projection operator is the orthogonal projection $P$ from $L^2(\Omega)$ to the closed subspace of holomorphic square integrable functions. The Bergman kernel function is the integral kernel associated with the Bergman projection $P$. The operator $P$ and the function $K$ are therefore related by $$Pf(\zeta)=\int_{\Omega} K(\zeta,\xi) f (\xi) dV(\xi). $$
It is well known that $K$ can be expressed by summation of an orthonormal series. More precisely, suppose that $\{ \Phi_{\alpha}\}$ from a complete orthonormal set for the Hilbert space of holomorphic functions in $L^2(\Omega)$. Then we have $$K(\zeta,\xi)= \sum_{\alpha} \Phi_{\alpha}(\zeta) \overline{\Phi_{\alpha}(\xi)} .$$ Let $\zeta=(z_1,z_2,z_3,z_4) \in \mathbb{C}^4.$ Put $\Phi_{\alpha}(\zeta)= z_1^{\alpha_1} z_2^{\alpha_2} z_3^{\alpha_3} z_4^{\alpha_4}$. It is well known, that function $f$ holomorphic in a Reinhardt domain $D \subset \mathbb{C}^n$ has a “global” expansion into a Laurent series $f(z)=\sum_{\alpha \in \mathbb{Z}^n} a_{\alpha} z^{\alpha}$, $z \in D$ (see Proposition 1.7.15 (c) in \cite{JP}). Moreover if $D \cap (\mathbb{C}^{j-1} \times \{0\} \times \mathbb{C}^{n-j} ) \neq \emptyset $, $j=1,\ldots, n$ then $a_{\alpha}=0$ for $\alpha \in \mathbb{Z}^n \setminus \mathbb{Z}^n_{+}$ (see Proposition 1.6.5 (c) in \cite{JP}). Therefore $\{\Phi_{\alpha} \}$ such that each $\alpha_i \geq 0$ is a complete orthogonal set for $L^2(D_1)$. \newline If $D$ is a Reinhardt domain, $f \in L^2_a(D) := \mathcal{O}(D) \cap L^2(D)$, $f(z)=\sum_{\alpha \in \mathbb{Z}^n} a_{\alpha} z^{\alpha}$, then $\{ z^{\alpha} \colon \alpha \in \sum(f) \} \subset L^2_a(D), $ where $\sum(f):=\{ \alpha \in \mathbb{Z}^n \colon a_{\alpha} \neq 0 \}$ (for proof see \cite{JP} p. 67). Thus it is easy to check, that the set $\{z_1^{\alpha_1} z_2^{\alpha_2} z_3^{\alpha_3}\colon \alpha_2 \geq 0, \alpha_3 \geq 0, \alpha_1 \geq -2 - \alpha_2 - \alpha_3\}$ is a complete orthogonal set for $L^2_a(D_2)$.
\begin{pr}
The squared $L^2(D_2)$-norms satisfy
\begin{equation}
\|z^{\alpha} \|^2_{L^2} = \frac{ \pi^3 \Gamma\left( \alpha_2 +1 \right) \Gamma\left( \alpha_1 + \alpha_2 + \alpha_3 + 3 \right)}{ ( \alpha_3 +1) \left( \alpha_1 + 2\alpha_2 + 2\alpha_3 +5 \right) \Gamma\left( \alpha_1 + 2\alpha_2 + \alpha_3 + 4 \right) },
\end{equation}
where $\alpha_2 \geq 0, \alpha_3 \geq 0, \alpha_1 \geq -2 - \alpha_2 - \alpha_3$.
\end{pr}
\begin{proof}
$$\|z^{\alpha} \|^2_{L^2} = \int\limits_{D_2} |z|^{2\alpha} dV(z)$$
we introduce polar coordinate in each variable by putting $z_j=r_j e^{i\varphi_j}$, for $j=1,2,3$. After doing so, and integrating out the angular variables we have
$$(2 \pi)^3 \int_{Re(D_2)} r^{2 \alpha +1} r_2^{2 \alpha_2+1} r_3^{2 \alpha_3 +1}dV(r),$$
where $Re(D_2)=\{ r \in \mathbb{R}_{+}^3 : r_3^{2} < r_1^{4} + r_2^2, \ r_1^{4} + r_2^2 < r_1^{2} \}$. Next we set $r_1^2=t$ and change variables again. We obtain
$$ \frac{(2 \pi)^3}{2} \int_{Re(D_2')} t^{ \alpha_1} r_2^{ 2\alpha_2 +1} r_3^{ 2\alpha_3 +1} dt dr_2 dr_3,$$
where $Re(D_2')=\{ (t,r_2,r_3) \in \mathbb{R}_{+}^3 : r_3^{2} < t^{2} + r_2^2, \ t^{2} + r_2^2 < t \}$. Next we use spherical coordinate in the $t,r_2$ variables to obtain
$$ 4 \pi^3 \int_{0}^{\pi/2} \int_0^{\sin \theta } \int_0^{\rho} \rho^{ \alpha_1 + 2\alpha_2 + 2 } (\cos \theta )^{2\alpha_2 + 1} (\sin \theta)^{ \alpha_1} r_3^{2 \alpha_3 +1} dr_3 d\rho d \theta$$
After integrating out $r_3, \rho$ and $\theta$ we obtain the desired result.
\end{proof}
\begin{pr}
The squared $L^2(D_1)$-norms satisfy
\begin{equation}
\|z^{\alpha} \|^2_{L^2} = \frac{ \pi^4 \Gamma\left( \alpha_1 +1 \right) \Gamma\left( \alpha_2 +1 \right) \Gamma\left( \alpha_3 + 1 \right) \Gamma\left( \frac{2\alpha_1 + 2\alpha_2 + 4}{p} + \alpha_3 + \frac{2\alpha_4 +2}{ \lambda} + 1 \right)}{ p \Gamma\left( \alpha_1 + \alpha_2 + 2 \right) ( \alpha_4 +1) \left( s + \frac{\alpha_4 + 1}{ \lambda} \right) \Gamma\left(2 s \right) },
\end{equation}
where $s= \frac{\alpha_1 + \alpha_2 + 2}{p} + \alpha_3 + \frac{\alpha_4 + 1}{ \lambda} + 1 $.
\end{pr}
\begin{proof}
$$\|z^{\alpha} \|^2_{L^2} = \int\limits_{D_1} |z|^{2\alpha} dV(z)$$
we introduce polar coordinate in each variable by putting $z_j=r_j e^{i\varphi_j}$, for $j=1,2,3,4$. After doing so, and integrating out the angular variables we have
$$(2 \pi)^4 \int_{Re(D_1)} r^{2 \alpha +1}\, dV(r),$$
where $Re(D_1)=\{ r \in \mathbb{R}_{+}^4 : r_4^{\lambda} < (r_1^2 + r_2^2)^{p} + r_3^2, \ (r_1^2 + r_2^2)^p + r_3^2 < r_1^{p/2} \}$. Next we set $r_1=\rho \cos \omega$, $r_2=\rho \sin \omega$ and change variables again. We obtain
$$(2 \pi)^4 \int\limits_{\substack {r_4^{\lambda} < \rho^{2p} + r_3^2 \\ \rho^{2p} + r_3^2 < \rho^{p}}} \int_0^{\pi/2} \rho^{2\alpha_1 + 2\alpha_2 + 3 } (\cos \omega )^{2\alpha_1 + 1} (\sin \omega )^{2\alpha_2+1} r_3^{ 2\alpha_3 +1} r_4^{ 2\alpha_4 +1}\, d\omega d\rho dr_3 dr_4, $$
integrating out of $\omega$ variable we have
$$\frac{(2 \pi)^4 \Gamma(\alpha_1 +1 ) \Gamma(\alpha_2 +1 ) }{2 \Gamma(\alpha_1 + \alpha_2 + 2 )} \int\limits_{\substack {r_4^{\lambda} < \rho^{2p} + r_3^2 \\ \rho^{2p} + r_3^2 < \rho^{p}}} \rho^{2\alpha_1 + 2\alpha_2 + 3 } r_3^{ 2\alpha_3 +1} r_4^{ 2\alpha_4 +1}\, d\omega d\rho dr_3 dr_4, $$
After little calculation we obtain
$$ C \int_{0}^{\pi/2} \int_0^{\cos \theta } \int_0^{R^{2/ \lambda}} R^{\frac{2\alpha_1 + 2\alpha_2 + 4}{p} + 2\alpha_3 + 1 } (\sin \theta )^{ 2\alpha_3 + 1} (\cos \theta)^{\frac{2\alpha_1 + 2\alpha_2 + 4}{p} -1} r_4^{2 \alpha_4 +1} dr_4 dR d \theta,$$
where $C= \frac{8 \pi^4 \Gamma(\alpha_1 +1 ) \Gamma(\alpha_2 +1 )}{p \Gamma( \alpha_1 + \alpha_2 +2) }$. After integrating out $r_4, R$ and $\theta$ we obtain the desired result.
\end{proof}
Now we will prove main theorem. We set $\nu_j=z_j \overline{w_j}$, for $j=1,2,3,4$. By the series representation Bergman kernel for $D_1$ is given by
\begin{eqnarray*}
K(z,w)= \frac{p}{ \pi^4} \sum_{\alpha=0}^{\infty} \frac{ \Gamma\left( \alpha_1 + \alpha_2 + 2 \right) ( \alpha_4 +1) \left( s + \frac{\alpha_4 + 1}{ \lambda} \right) \Gamma\left( 2 s \right) }{ \Gamma\left( \frac{2\alpha_1 + 2\alpha_2 + 4}{p} + \alpha_3 + \frac{2\alpha_4 +2}{ \lambda} + 1 \right) \alpha_1! \alpha_2! \alpha_3! } \nu^{\alpha}
\end{eqnarray*}
If we define
\begin{eqnarray*}
G= \sum_{\alpha=0}^{\infty} \frac{ \Gamma\left( \alpha_1 + \alpha_2 + 2 \right) ( \alpha_4 +1) \Gamma\left( 2 s \right) }{ \Gamma\left( \frac{2\alpha_1 + 2\alpha_2 + 4}{p} + \alpha_3 + \frac{2\alpha_4 +2}{ \lambda} + 1 \right) \alpha_1! \alpha_2! \alpha_3! } \nu^{\alpha}
\end{eqnarray*}
then we can write
\begin{eqnarray*}
K(z,w)= D_{p,\lambda, \alpha} G,
\end{eqnarray*}
where $ D_{p,\lambda, \alpha}$ is a differential operator defined by
$$ D_{p,\lambda, \alpha}f= \frac{p}{ \pi^4} \left(\frac{2}{p} \frac{\partial}{\partial \nu_1} \nu_1 + \frac{2}{p} \frac{\partial}{\partial \nu_2} \nu_2 + \frac{\partial}{\partial \nu_3} \nu_3 + \frac{2}{\lambda} \frac{\partial}{\partial \nu_4} \nu_4 \right)f. $$
Now we sum out the $\nu_3$ variable using lemma \ref{lem3}, we obtain
\begin{align*}
G= \sum_{\alpha_1,\alpha_2,\alpha_4=0}^{\infty} \frac{ \Gamma\left( \alpha_1 + \alpha_2 + 2 \right) (\alpha_4 +1) \Gamma(a+1) F\left( \frac{1+a}{2}, \frac{2+a}{2} ; a ; 4 \nu_3 \right) }{\Gamma(a) \alpha_1! \alpha_2! } \nu_1^{\alpha_1} \nu_2^{\alpha_2} \nu_4^{\alpha_4} ,
\end{align*}
where $a=\frac{2\alpha_1 + 2\alpha_2 + 4}{p} + \frac{2\alpha_4 +2}{ \lambda} + 1 $.
After some calculation using $$ F\left( \frac{a+1}{2}, \frac{a+2}{2}; a; z \right) = \frac{2^{a-1}((a-2) (\sqrt{1-z}-1) + (a-1)z) }{a z (1-z)^{3/2} ( \sqrt{1-z} +1)^{a-2} } $$
and $z\Gamma(z)=\Gamma(z+1)$, we have
$$G=C \sum_{\alpha_1,\alpha_2,\alpha_4=0}^{\infty} \frac{ \Gamma\left( \alpha_1 + \alpha_2 + 2 \right) (\alpha_4 +1) }{ {[(a-2) (\sqrt{1-4\nu_3}-1) + (a-1)4\nu_3]}^{-1} \alpha_1! \alpha_2! } \mu_1^{\alpha_1} \mu_2^{\alpha_2} \mu_4^{\alpha_4}, $$
where $C=\frac{ 2^{4/p + 2/\lambda} }{4\nu_3 (1-4\nu_3)^{3/2} (1+ \sqrt{1- 4\nu_3})^{4/p + 2/\lambda -1}}$, $\mu_i= \frac{2^{2/p} \nu_i} {(1+ \sqrt{1- 4\nu_3})^{2/p} }$ for $i=1,2$ and $\mu_4= \frac{2^{2/\lambda} \nu_4} {(1+ \sqrt{1- 4\nu_3})^{2/\lambda} } .$
\begin{align*}
&G= C \sum_{\alpha_1,\alpha_2,\alpha_4=0}^{\infty} \frac{ \Gamma\left( \alpha_1 + \alpha_2 + 2 \right) (\alpha_4 +1) } { \alpha_1! \alpha_2! } \\ & \cdot \left[( \frac{2}{\lambda} -1 )(\sqrt{1-4\nu_3}-1) + \frac{8\nu_3}{\lambda}\right] \mu_1^{\alpha_1} \mu_2^{\alpha_2} \mu_4^{\alpha_4} \\& + C \sum_{\alpha_1,\alpha_2,\alpha_4=0}^{\infty} \frac{2 \Gamma\left( \alpha_1 + \alpha_2 + 3 \right) (\alpha_4 +1)} { p \alpha_1! \alpha_2! } \left[\sqrt{1-4\nu_3} + 4\nu_3 - 1\right] \mu_1^{\alpha_1} \mu_2^{\alpha_2} \mu_4^{\alpha_4} \\& + C \sum_{\alpha_1,\alpha_2,\alpha_4=0}^{\infty} \frac{2 \Gamma\left( \alpha_1 + \alpha_2 + 2 \right) (\alpha_4 +1) \alpha_4 } { \lambda \alpha_1! \alpha_2! } \left[\sqrt{1-4\nu_3} + 4\nu_3 - 1\right] \mu_1^{\alpha_1} \mu_2^{\alpha_2} \mu_4^{\alpha_4}
\end{align*}
Finally using well knows formulas $\sum\limits_{k=0}^{\infty } k (k+1) x^k= \frac{2 x}{(1-x)^3}$, $\sum\limits_{k=0}^{\infty } (k+1) x^k=\frac{1}{(1-x)^2}$ and $F_2(r,1,1;1,1,x,y)= \frac{1}{(1-x-y)^r}$, we obtain
\begin{align*}
&G= C \frac { ( \frac{2}{\lambda} -1 )(\sqrt{1-4\nu_3}-1) + \frac{8\nu_3}{\lambda} }{(1-\mu_4)^2 (1- \mu_1 - \mu_2)^2 } + C \frac{ 4 \sqrt{1-4\nu_3} + 16\nu_3 - 4} {p (1-\mu_4)^2 (1- \mu_1 - \mu_2)^3} \\& + C \left[\sqrt{1-4\nu_3} + 4\nu_3 - 1\right] \frac{4\mu_4}{\lambda (1-\mu_4)^3 (1- \mu_1 - \mu_2)^2}
\end{align*}
Similarly we can compute Bergman kernel for $$\Omega= \left\{ (z_1,z_2,z_3) \in \mathbb{C}^{n+m+k} \colon \|z_1\|^{\lambda} < \|z_2\|^{2p} + \|z_3\|^2, \ \|z_2\|^{2p} + \|z_3\|^2 < \|z_2\|^{p} \right\}.$$
Now we will prove Theorem 1.2. Similarly as above Bergman kernel for $D_2$ is given by
\begin{eqnarray*} \sum_{\alpha_1, \alpha_2=0}^{\infty} \sum_{\alpha_1=-2-\alpha_2 - \alpha_3}^{\infty} \frac{ ( \alpha_3 +1) \left( \alpha_1 + 2\alpha_2 + 2\alpha_3 +5 \right) \Gamma\left( \alpha_1 + 2\alpha_2 + \alpha_3 + 4 \right) }{ \pi^3 \Gamma\left( \alpha_2 +1 \right) \Gamma\left( \alpha_1 + \alpha_2 + \alpha_3 + 3 \right) } \nu^{\alpha}
\end{eqnarray*}
Changing the summation index $\alpha_1=k -2 -\alpha_2 - \alpha_3$, we can write
\begin{eqnarray*} \sum_{\alpha_1, \alpha_2=0,k=0}^{\infty} \frac{1}{\nu_1^2} \frac{ ( \alpha_3 +1) \left( k + \alpha_2 + \alpha_3 + 3 \right) \Gamma\left( k + \alpha_2 + 2 \right) }{ \pi^3 \Gamma\left( \alpha_2 +1 \right) \Gamma\left( k + 1 \right) } \nu_1^{k} \left( \frac{\nu_2}{\nu_1} \right)^{\alpha_2} \left(\frac{\nu_3}{\nu_1}\right)^{\alpha_3}
\end{eqnarray*}
After little calculation using $z\Gamma(z)=\Gamma(z+1)$, we have
\begin{eqnarray*} \sum_{\alpha_1, \alpha_2=0,k=0}^{\infty} \frac{1}{\nu_1^2} \frac{ ( \alpha_3 +1)^2 \Gamma\left( k + \alpha_2 + 2 \right) }{ \pi^3 \Gamma\left( \alpha_2 +1 \right) \Gamma\left( k + 1 \right) } \nu_1^{k} \left( \frac{\nu_2}{\nu_1} \right)^{\alpha_2} \left(\frac{\nu_3}{\nu_1}\right)^{\alpha_3} \\ + \sum_{\alpha_1, \alpha_2=0,k=0}^{\infty} \frac{1}{\nu_1^2} \frac{ ( \alpha_3 +1) \Gamma\left( k + \alpha_2 + 3 \right) }{ \pi^3 \Gamma\left( \alpha_2 +1 \right) \Gamma\left( k + 1 \right) } \nu_1^{k} \left( \frac{\nu_2}{\nu_1} \right)^{\alpha_2} \left(\frac{\nu_3}{\nu_1}\right)^{\alpha_3}
\end{eqnarray*}
Finally using well knows formulas $\sum\limits_{k=0}^{\infty } (k+1)^2 x^k= \frac{x+1}{(1-x)^3}$, $\sum\limits_{k=0}^{\infty } (k+1) x^k=\frac{1}{(1-x)^2}$ and $F_2(r,1,1;1,1,x,y)= \frac{1}{(1-x-y)^r}$, we obtain the desired result.
Similarly we can compute Bergman kernel for $$\Omega= \left\{ (z_1,z_2,z_3) \in \mathbb{C}^{1+n+m} \colon \|z_3\|^{\lambda} < |z_1|^{2p} + \|z_2\|^2, \ |z_1|^{2p} + \|z_2\|^2 < |z_1|^{p} \right\},$$ for $p, \lambda > 0$.
\noindent Tomasz Beberok\\
Department of Applied Mathematics\\
University of Agriculture in Krakow\\
ul. Balicka 253c, 30-198 Krakow, Poland\\
email: tbeberok@ar.krakow.pl
\end{document} |
\begin{document}
\title{Practical continuous-variable quantum key distribution with composable security
}
\author{Nitin Jain}
\email{nitinj@iitbombay.org}
\affiliation{\mbox{Center for Macroscopic Quantum States (bigQ), Department of Physics,} Technical University of Denmark, 2800 Kongens Lyngby, Denmark}
\author{Hou-Man Chin}
\affiliation{\mbox{Department of Photonics, Technical University of Denmark}, 2800 Kongens Lyngby, Denmark}
\affiliation{\mbox{Center for Macroscopic Quantum States (bigQ), Department of Physics,} Technical University of Denmark, 2800 Kongens Lyngby, Denmark}
\author{Hossein Mani}
\affiliation{\mbox{Center for Macroscopic Quantum States (bigQ), Department of Physics,} Technical University of Denmark, 2800 Kongens Lyngby, Denmark}
\author{\mbox{Cosmo Lupo}}
\affiliation{\mbox{Department of Physics and Astronomy,} University of Sheffield, S3 7RH Sheffield, UK}
\affiliation{\mbox{Department of Computer Science,} University of York, York YO10 5GH, UK}
\author{\mbox{Dino Solar Nikolic}}
\affiliation{\mbox{Center for Macroscopic Quantum States (bigQ), Department of Physics,} Technical University of Denmark, 2800 Kongens Lyngby, Denmark}
\author{Arne Kordts}
\affiliation{\mbox{Center for Macroscopic Quantum States (bigQ), Department of Physics,} Technical University of Denmark, 2800 Kongens Lyngby, Denmark}
\author{Stefano Pirandola}
\affiliation{\mbox{Department of Computer Science,} University of York, York YO10 5GH, UK}
\author{Thomas Brochmann Pedersen}
\affiliation{Cryptomathic A/S, Aaboulevarden 22, 8000 Aarhus, Denmark}
\author{\mbox{Matthias Kolb}}
\author{\mbox{Bernhard {\"O}mer}}
\author{Christoph Pacher}
\affiliation{Center for Digital Safety \& Security, AIT Austrian Institute of Technology GmbH, 1210 Vienna, Austria.}
\author{Tobias Gehring}
\email{tobias.gehring@fysik.dtu.dk}
\author{Ulrik L. Andersen}
\email{ulrik.andersen@fysik.dtu.dk}
\affiliation{\mbox{Center for Macroscopic Quantum States (bigQ), Department of Physics,} Technical University of Denmark, 2800 Kongens Lyngby, Denmark}
\date{\today}
\begin{abstract}
A quantum key distribution (QKD) system must fulfill the requirement of universal composability to ensure that any cryptographic application (using the QKD system) is also secure. Furthermore, the theoretical proof responsible for security analysis and key generation should cater to the number $N$ of the distributed quantum states being finite in practice. Continuous-variable (CV) QKD based on coherent states, despite being a suitable candidate for integration in the telecom infrastructure, has so far been unable to demonstrate composability as existing proofs require a rather large $N$ for successful key generation. Here we report the first Gaussian-modulated coherent state CVQKD system that is able to overcome these challenges and can generate composable keys secure against collective attacks with $N \lesssim 3.5\times10^8$ coherent states. With this advance, possible due to novel improvements to the security proof and a fast, yet low-noise and highly stable system operation, CVQKD implementations take a significant step towards their discrete-variable counterparts in practicality, performance, and security.
\begin{description}
\item[PACS numbers]
May be entered using the \verb+\pacs{#1}+ command.
\end{description}
\end{abstract}
\pacs{Valid PACS appear here}
\maketitle
\section{Introduction}
Quantum key distribution (QKD) is the only known cryptographic solution for distributing secret keys to users across a public communication channel while being able to detect the presence of an eavesdropper~\cite{Scarani2009, Pirandola2019}. Legitimate QKD users (Alice and Bob) encrypt their messages with the secret keys and exchange them with the assurance that the eavesdropper (Eve) cannot break the confidentiality of the encrypted messages. In particular, if the obtained secret key is (at least) as long as the length of the message, information theoretic security guarantees that Eve cannot break the security even if equipped with unlimited computing resources.
Alice and Bob perform a sequence of steps, shown in Fig.~\ref{fig:scheme}, to obtain a key of a certain length. Such a `QKD protocol' begins with preparation, transmission (on a quantum channel), measurement of quantum states, and concludes with classical data processing and security analysis, performed in accordance with a mathematical proof.
\begin{figure*}
\caption{Composability in continuous-variable quantum key distribution (CVQKD). Alice and Bob obtain bitstreams $s_A$ and $s_B$, respectively, after going through the different steps of the QKD protocol that involve both the quantum and authenticated channels, assumed to be in full control of Eve. Various dashed lines (with arrows) indicate local operations and classical communication. An application using a CVQKD system (transmitter and receiver) to provide composable security must satisfy certain criteria associated with correctness, robustness, and secrecy of the protocol~\cite{Muller-Quade2009, Leverrier2015}
\label{fig:scheme}
\end{figure*}
Amongst the many physical considerations included in the security proof, one is that the number of quantum states available to Alice and Bob are not infinite. Such \textit{finite-size corrections} adversely affect the key length but are essential for the security assurance.
Another related property of a cryptographic key is \textit{composability}~\cite{Canetti2001}, which allows specifying the security requirements for combining different cryptographic applications in a unified and systematic way. In the context of practical QKD, composability is of utmost importance because the secret keys obtained from a QKD protocol are almost always used in other applications, e.g.\ data encryption~\cite{Muller-Quade2009}. A QKD implementation that outputs a key not proven to be composable is thus practically useless.
In one of the most well-known flavours of QKD, the quantum information is coded in continuous variables, such as the amplitude and phase quadratures, of the optical field~\cite{Ralph99, Diamanti2015, Laudenbach2018, Pirandola2019}. Typical continuous-variable (CV)QKD protocols have been Gaussian-modulated coherent state (GMCS) implementations~\cite{Jouguet2013, Huang2016, Wang2018, Wang2020}, and finite-size effects were also considered, though the proof~\cite{Leverrier2010} was non-composable. Composable security in CVQKD was first proven and experimentally demonstrated using two-mode squeezed states, however, since the employed entropic uncertainty relation is not tight, the achievable communication distance was rather limited~\cite{Furrer2012, Gehring2015}.
Composable security proofs for CVQKD systems using coherent states and dual quadrature detection, first proposed in 2015~\cite{Leverrier2015}, have been progressively improved~\cite{Lupo2018, Papanastasiou2021, Pirandola2021}. These proofs promise keys at distances much longer than in Ref.~\cite{Furrer2012} apart from the advantage of dealing with coherent states, which are much easier to generate than squeezed states. Nonetheless, an experimental demonstration of composability has remained elusive, due to a combination of the strict security bounds (because of a complex parameter estimation routine), the large number of required quantum state transmissions (to keep the finite-size terms sufficiently low), and the stringent requirements on the tolerable excess noise.
In this article, we demonstrate a practical GMCS-CVQKD system that is capable of generating composable keys secure against collective attacks. We achieve this by deriving a new method for establishing confidence intervals that is compatible with collective attacks, which allows us to work on smaller (and thus more practical) block sizes than originally required~\cite{Leverrier2015}. On the experimental front, we are able to keep the excess noise below the null key length threshold by performing a careful analysis (followed by eradication or avoidance) of the various spurious noise components, and by implementing a machine learning framework for phase compensation~\cite{Chin2020}.
After taking finite-size effects as well as confidence intervals from various system calibrations into account, we achieve a positive composable key length with merely $N \lesssim 3.5\times10^8$ coherent states (also referred to as `quantum symbols'). With $N = 10^9$, we obtain $>53.4$ Mbits worth of composably secure key material in the worst case.
\section{Composably secure key}\label{sec:Theory}
In the security analysis, we assume collective attacks and take into account the finite number of coherent states transmitted by Alice and measured by Bob. A digital signal processing (DSP) routine yields the digital quantum symbols $\bar{Y}$ discretized with $d$ bits per quadrature and this stream is divided into $M$ frames for information reconciliation (IR), after which we perform parameter estimation (PE) and privacy amplification (PA); as visualized in Fig.~\ref{fig:scheme}. We derive the secret key bound for reverse reconciliation, i.e., Alice correcting her data according to Bob's quantum symbols.
The (composable) secret key length $s_n$ for $n$ coherent state transmissions is calculated using tools from Refs.~\cite{Leverrier2015, Pirandola2021} as well as new results presented in the following. The key length is bounded per the leftover hash lemma in terms of the smooth min-entropy $H_{\min}$ of the alphabet of $\bar{Y}$, conditioned on the quantum state of the eavesdropper $E$~\cite{Tomamichel2012}. From this we subtract the information reconciliation leakage $\mathrm{leak_{IR}}(n,\epsilon_{\mathrm{IR}})$ and obtain,
\begin{equation}
s_{n}^{\epsilon_h+\epsilon_s+\epsilon_{\mathrm{IR}}}
\geq
H_{\min}^{\epsilon_s}(\bar Y | E )_{ \rho^{n} } -\mathrm{leak_{IR}}(n,\epsilon_{\mathrm{IR}})
+ 2 \log_2{(\sqrt{2}\epsilon_h)} \, .
\label{eq:skfSimple}
\end{equation}
The security parameter $\epsilon_h$ characterizes the hashing function, $\epsilon_s$ is the smoothing parameter entering the smooth conditional min-entropy, and $\epsilon_{\mathrm{IR}}$ describes the failure probability of the correctness test after IR.
The probability $p^\prime$ that IR succeeds in a frame is related to the frame error rate (FER) by $p^\prime = 1 - $FER. All frames in which IR failed are discarded from the raw key stream, and this step thereby projects the original tensor product state $\rho^{n} \equiv \rho^{\otimes n}$ into a non i.i.d. state $\tau^{n}$. To take this into account, one replaces the smooth min-entropy term in Eq.~\eqref{eq:skfSimple} with the expression~\cite{ Pirandola2021}:
\begin{align}\label{new-min}
H_{\min}^{\epsilon_s}(\bar Y | E)_{\tau^{n'}}
\geq H_{\min}^{\frac{p^\prime}{3} \epsilon_s^2}(\bar Y |E)_{ \rho^{\otimes n'} }
+ \log_2{\left( p^\prime - \frac{p^\prime}{3}\epsilon_s^2\right) } \, ,
\end{align}
where $n' = n p^\prime$ is the number of quantum symbols remaining after error correction.
The asymptotic equipartition property (AEP) bounds the conditional min-entropy by the von-Neumann conditional entropy,
\begin{equation*}
H_{\min}^{\delta}(\bar Y | E)_{\rho^{\otimes n'}}
\geq n' H(\bar Y |E)_{\rho}-\sqrt{n'}\,\Delta_{\mathrm{AEP}}(\delta,d)\ ,
\end{equation*}
where
\begin{equation}
\Delta_{\mathrm{AEP}}(\delta,d)\leq 4(d+1)\sqrt{\log_2{(2/\delta^{2})}}\ ,
\end{equation}
is an improved penalty in comparison to Ref.~\cite{Leverrier2015,Pirandola2021}, and is proven in the Supplement.
The conditional von-Neumann entropy is given by
\begin{equation}
H(\bar Y | E )_\rho = H(\bar Y)_\rho - I(\bar Y ; E)_\rho \, .
\end{equation}
We estimate the first term directly from the data (up to a probability not larger than $\epsilon_\mathrm{ent}$; further details regarding the confidence intervals are in the Supplement). The second term is bound by the Holevo information,
\begin{equation*}
I(\bar Y ; E)_\rho \leq I( Y ; E )_\rho \leq I( Y ; E )_{\rho_G} \ ,
\end{equation*}
where $Y$ is the continuous version of $\bar Y$ and $I( Y ; E )_{\rho_G}$ is the Holevo information obtained after using the extremality property of Gaussian attacks.
The Holevo information is estimated by evaluating the covariance matrix using worst-case estimates for its entries based on confidence intervals. We improved the confidence intervals of Ref.~\cite{Leverrier2015} by exploiting the properties of the beta distribution. Let $\hat{x}$, $\hat{y}$, $\hat{z}$ be the estimators for the variance of the transmitted ensemble of coherent states, the received variance and the co-variance, respectively. The true values $y$ and $z$ are bound by
\begin{align}
y &\le \left(1 + \delta_\text{Var}(n, \epsilon_\text{PE}/2)\right)\hat{y}\ , \label{eq:dvar} \\
z &\ge \left(1 - 2\delta_\text{Cov}(n, \epsilon_\text{PE}/2)\frac{\sqrt{\hat x \hat y}}{\hat z}\right)\hat{z}\ , \label{eq:dcov}
\end{align}
with $\epsilon_\text{PE}$ denoting the failure probability of parameter estimation, and
\begin{align*}
\delta_\mathrm{Var} (n,\epsilon) & = a'\left(\epsilon/6\right) \left( 1 + \frac{120}{\epsilon} e^{-\frac{n}{16}} \right) - 1 \ , \\
\delta_\mathrm{Cov}(n,\epsilon) & = \frac{1}{2}\left[\frac{a'\left(\epsilon/6\right)-b'\left(\epsilon/6\right)}{2}
+ a'\left(\frac{\epsilon^2}{324}\right)
- b'\left(\frac{\epsilon^2}{324} \right)
\right]
\end{align*}
being the new confidence intervals (derived in the Supplement). In the above equations,
\begin{align*}
a'\left(\epsilon\right) & = 2 \left[ 1 - \mathrm{invcdf}_{\mathrm{Beta}(n/2,n/2)}\left(\epsilon\right) \right] \, , \\
b'\left(\epsilon\right) & = 2 \, \mathrm{invcdf}_{\mathrm{Beta}(n/2,n/2)}\left(\epsilon\right) \, .
\end{align*}
As detailed in section~\ref{sec:resNdis}, the (length of the) secret key we eventually obtain in our experiment requires an order of magnitude lower $N$ due to these confidence intervals.
Finally, we remark here on a technical limitation arising due to the digitization of Alice's and Bob's data. In practice, it is impossible to implement a \emph{true} GMCS protocol because the Gaussian distribution is both unbounded and continuous, while the devices used in typical CVQKD systems have a finite range and bit resolution~\cite{Jouguet2012}. In our work, we consider a range of 7 standard deviations and use $d=6$ bits (leading to a constellation with $2^{2d} = 4096$ coherent states), which per recent results~\cite{Lupo2020, Denys2021}, should suffice to minimise the impact of digitization on the security of the protocol.
\section{Experiment}
\begin{figure*}
\caption{\textbf{Schematic of the experiment.}
\label{fig:setup}
\end{figure*}
Figure~\ref{fig:setup} shows the schematic of our setup, with the caption detailing the components and their role briefly. Below we summarize the setup's operation, calibration measurements, and our protocol implementation. In the Supplement, we describe the different functional blocks of Fig.~\ref{fig:setup} in further detail.
\subsection{Transmitter (Tx)}\label{Exp:Tx}
We performed optical single sideband modulation with carrier suppression (OSSB-CS) using an off-the-shelf IQ modulator and automatic bias controller (ABC). An arbitrary waveform generator (AWG) was connected to the RF ports to modulate the sidebands. The coherent states were produced in a $B=100\,$MHz wide frequency sideband, shifted away from the optical carrier~\cite{Lance2005, Jain2021}. The random numbers that formed the complex amplitudes of these coherent states were drawn from a Gaussian distribution, obtained by transforming the uniform distribution of a vacuum-fluctuation based quantum random number generator (QRNG), with a security parameter $\epsilon_\text{qrng} = 2 \times 10^{-6}$~\cite{Gehring2021}.
To this wideband `quantum data' signal, centered at $f_u=200\,$MHz, we multiplexed in frequency a `pilot tone' at $f_p=25\,$MHz for sharing a phase reference with the receiver~\cite{Qi2015, Soh2015, Huang2015, Kleis2017}. The left inset of Fig.~\ref{fig:setup} shows the complex spectra of the RF modulation signal.
\subsection{Receiver (Rx)} \label{Exp:Rx}
After propagating through the quantum channel---a 20 km long standard single mode fiber spool---the signal field's polarization was manually tuned to match the polarization of the real local oscillator (RLO) for heterodyning~\cite{Qi2015,Soh2015,Huang2015}. The Rx laser that supplied the RLO was free-running with respect to the Tx laser and detuned in frequency by $\sim320\,$ MHz, giving rise to a beat signal, as labelled in the solid-red spectral trace in the right inset of Fig.~\ref{fig:setup}. The quantum data band and pilot tone generated by the AWG are also labelled. Due to finite OSSB~\cite{Jain2021}, a suppressed pilot tone is also visible; the corresponding suppressed quantum band was however outside the receiver bandwidth (we used a low pass filter with a cutoff frequency around 360 MHz at the output of the heterodyne detector).
In separate measurements, we also measured the vacuum noise (Tx laser off, Rx laser on) and the electronic noise of the detector (both Tx and Rx lasers off), depicted by the dotted-blue and dashed-green traces, respectively, in the right inset of Fig.~\ref{fig:setup}. The clearance of the vacuum noise over the electronic noise is $>15\,$dB over the entire quantum data band.
\subsection{Noise analysis \& Calibration}\label{Exp:NoiseCal}
\begin{table*}[!t]
\centering
\caption{Experimental parameters. PNU: photon number units. The security parameter $\epsilon_\text{qrng}$ is limited by the digitization error of the ADC used in the QRNG, but could be improved using longer measurement periods~\cite{Gehring2021}.}
\begin{tabular}{l|l}
\textbf{Transmitter} & \\
\hline
Rate of coherent states, $B$ & 100 MSymbols/s \\
Modulation strength (channel input), $\mu$ & 1.45 PNU \\
\hline
\textbf{Receiver calibration} & \\
\hline
Trusted efficiency (incl. optical loss), $\tau$ & 0.69 \\
Trusted electronic noise, $t$ & 25.71$\,\times 10^{-3}$ PNU \\
\hline
\textbf{Channel parameter estimation} & \\
\hline
Untrusted efficiency, $\eta$ & 0.35 \\
Untrusted excess noise, $u$ & 6.30$\,\times 10^{-3}$ PNU \\
\hline
\textbf{Information reconciliation} & \\
\hline
Signal-to-noise ratio & 0.32 \\
Frame error rate, FER & 0.36\% \\
Reconciliation efficiency, $\beta$ & 91.6\% \\
Leaked bits & $1.60 \times 10^9$ \\
\hline
\textbf{Secret key calculation} & \\
\hline
Raw key length (symbols), $N_\text{PA}$ & $9.84 \times 10^8$ \\
Security parameters & $\epsilon_h = \epsilon_\text{cal} = \epsilon_s = \epsilon_\text{PE} = 10^{-10}$, $ \epsilon_\text{qrng} = 2 \times 10^{-6}$, $\epsilon_\text{IR} = 10^{-12}$ \\
Final secret key length (bits) & 53452436 \\
\end{tabular}
\label{tab:exp_params}
\end{table*}
A careful choice of the parameters defining the pilot tone and the quantum data band, and their locations with respect to the beat signal is crucial in minimizing the excess noise. A strong pilot tone enables more accurate phase reference but at the expense of higher leakage in the quantum band and an increased number of spurious tones. The latter may arise as a result of frequency mixing of the (desired) pilot tone with e.g., the beat signal or the suppressed pilot tone. As can be observed in the right inset of Fig.~\ref{fig:setup}, we avoided spurious noise peaks resulting from sum- or difference-frequency generation of the various discrete components (in the solid-red trace) from landing inside the wide quantum data band.
As is well known in CVQKD implementations, Alice needs to optimize the modulation strength of the coherent state alphabet at the input of the quantum channel to maximize the secret key length. For this, we connected the transmitter and receiver directly, i.e., without the quantum channel, and performed heterodyne measurements to calibrate the \emph{mean photon number} $\mu$ of the resulting thermal state from the ensemble of generated coherent states, as explained in section~\ref{Exp:Tx}. The modulation strength can be controlled in a fine-grained manner using the electronic gain of the AWG and the optical attenuation from the VATT. The DSP that aided in this calibration is explained in detail in the supplement.
Since we conducted our experiment in the non-paranoid scenario~\cite{Scarani2009, Jouguet2012}, i.e., we trusted some parts of the overall loss and excess noise by assuming them to be beyond Eve's control, some extra measurements and calibrations for the estimation of trusted parameters become necessary. More specifically, we decomposed the total transmittance and excess noise into respective trusted and untrusted components. In the Supplement, we present the details of how we evaluated the trusted transmittance $\tau$ and trusted noise $t$ for our setup.
Table~\ref{tab:exp_params} presents the values of $\mu$, $\tau$ and $t$ pertinent to the experimental measurement described in section~\ref{Exp:Protocol}. Let us remark here that in our work, we express the noise and other variance-like quantities, e.g., the modulation strength, in photon number units (PNU) as opposed to the traditional shot noise units (SNU) because the former is independent of quadratures, and in case of $\mu$, facilitates a comparison with discrete-variable (DV) QKD systems\footnote{Assuming symmetry between the quadratures, the modulation variance $V_{\text{mod}} = 2 \mu$ in SNU.}. Finally, note that we recorded a total of $10^{10}$ ADC samples for each of the calibration measurements, and all the acquired data was stored on a hard drive for offline processing.
\subsection{Protocol operation}\label{Exp:Protocol}
We connected the transmitter and receiver using the 20 km channel, optimized the signal polarization, and then collected heterodyne data using the same Gaussian distributed random numbers as mentioned in section~\ref{Exp:NoiseCal}. Offline DSP~\cite{Chin2020} was performed at the receiver workstation to obtain the symbols that formed the raw key. The preparation and measurement was performed with a total of $10^9$ complex symbols, modulated and acquired in 25 blocks, each block containing $4 \times 10^7$ symbols. After discarding some symbols due to a synchronization delay, Alice and Bob had a total of $N_\text{IR} = 9.88 \times 10^8$ correlated symbols at the beginning of the classical phase of the protocol; see Fig.~\ref{fig:scheme}.
Below we provide details of the actual protocol we implemented, where we assumed that the classical channel connecting Alice and Bob was already authenticated.
\begin{enumerate}
\item IR was based on a multi-dimensional scheme~\cite{MD-Recon-PRA.77.042325} using multi-edge-type low-density-parity-check error correcting codes~\cite{Mani2018}. Table~\ref{tab:exp_params} lists some parameters related to the operating regime and the performance of these codes; more information is available in the Supplement. As shown in Fig.~\ref{fig:scheme}, Bob sent the mapping and the syndromes, together with the hashes computed using a randomly chosen Toeplitz function, to Alice, who performed correctness confirmation and communicated it to Bob.
\item During PE, Alice estimated the entropy of the corrected symbols, and together with the symbols from the erroneous frames, i.e., frames that could not be reconciled successfully (and were publicly announced by Bob), Alice evaluated the covariance matrix. This was followed by evaluating the channel parameters as well as performing the `parameter estimation test' (refer Theorem 2 in Ref.~\cite{Leverrier2015}) and getting a bound on Eve's Holevo information. Using the expression for the secret key length with the security parameters from Table~\ref{tab:exp_params}, Alice then calculated the number of bits expected in the output secret key in the worst-case scenario. This length was communicated together with a seed to Bob.
\item For PA, the shared seed from the previous step was used to select a random Toeplitz hash function by Alice and Bob, who then employed the high-speed and large-scale PA scheme~\cite{Tang2019} to generate the final secret key.
\end{enumerate}
\section{Results \& Discussion}\label{sec:resNdis}
\begin{figure*}
\caption{Composable SKF results. (a) Pseudo-temporal evolution of the composable SKF with the time parameter calculated as the ratio of the cumulative number $N$ of complex symbols available for the classical steps of the protocol and the rate $B$ at which these symbols are modulated. (b) Variation of untrusted noise $u$ measured in the experiment (lower point) and its worst-case estimator (upper point), and the noise threshold to beat in order to get a positive composable SKF. The reason for the deviation of the traces in (a) from the experimental data between 1 and 4 seconds is due to the slight increase in $u$. (c) and (d) Comparison of confidence intervals derived in this manuscript (Beta; solid-red trace and Gaussian; dashed-green trace) with those derived in the original composable security proof (Ref.~\cite{Leverrier2015}
\label{fig:results}
\end{figure*}
Table~\ref{tab:exp_params} summarizes the relevant parameters in our experiment. Alice prepared an ensemble of $10^9$ coherent states, characterized by a modulation strength of 1.45 PNU, transmitted them over a 20 km channel to Bob, who measured them with a total excess noise $u+t = 6.3 + 25.7 = 32.0\,$mPNU and a total transmittance $\eta \cdot \tau = 0.35 \cdot 0.69 = 0.24$ averaged over the amplitude and phase (I and Q) quadratures. With a total of $N_\text{IR} = 9.88 \times 10^8$ correlated symbols, Bob and Alice performed reverse reconciliation with an efficiency $\beta = $ 91.6\% as explained in section~\ref{Exp:Protocol}. Notably, due to the low frame error rate (FER = 0.0036) during IR, Alice and Bob were left with $N_\text{PA} = 9.84 \times 10^8$ symbols for performing the last classical step of the protocol.
Using the equations presented in section~\ref{sec:Theory}, we can calculate the composably secure key length (in bits) for a certain number $N$ of the quantum symbols. We partitioned $N_\text{IR}$ in 25 blocks, estimated the key length considering the total number $N_k$ of symbols accumulated from the first $k$ blocks, for $k \in \{1, 2, \ldots, 25\}$. Dividing this length by $N_k$ yields the composable secret key fraction (SKF) in bits/symbols. If we neglect the time taken by data acquisition, DSP, and the classical steps of the protocol, i.e., only consider the time taken to modulate $N=N_k$ coherent states at the transmitter (at a rate $B = 100\,$MSymbols/s), we can construct a hypothetical time axis to show the evolution of the CVQKD system.
Figure~\ref{fig:results}(a) depicts such a time evolution of the SKF after proper consideration to the finite-size corrections due to the average and worst-case (red and blue data points, respectively) values of the underlying parameters. Similarly, Fig.~\ref{fig:results}(b) shows the experimentally measured untrusted noise $u$ (lower squares) together with the worst-case estimator (upper dashes) calculated using $N_k$ in the security analysis. To obtain a positive key length, the worst-case estimator must be below the maximum tolerable noise---null key fraction threshold---shown by the solid line, and this occurs at $N/B \lesssim 3.5$ seconds.
Note that in reality, the DSP and classical data processing consume a significantly long time: In fact, we store the data from the state preparation and measurement stages on disks and perform these steps offline. The plots in Fig.~\ref{fig:results} therefore may be understood to be depicting the time evolution of the SKF and the untrusted noise \emph{if} the entire protocol operation was in real time.
Joining data from both I and Q quadratures bestowed $2 N_\text{PA} = 2 \times 9.84 \times 10^8$ \emph{real} symbols, from which we then obtain a secret key with length $l = 53452436$ bits, implying a worst-case SKF = $0.027\,$bits/symbol. Referring to Fig.~\ref{fig:results}(a), the solid-blue and dashed-red traces simulate the SKF in the worst-case and average scenarios, respectively, while the dotted-black trace shows the asymptotic SKF value obtainable with the given channel parameters; refer Table~\ref{tab:exp_params}. Per projections based on the simulation, the worst-case composable SKF should be within 1\% of the asymptotic value for $N \approx 10^{12}$ complex symbols.
From a theoretical perspective, the reason for being able to generate a positive composable key length with a relatively small number of coherent states ($N \lesssim 3.5\times10^8$) can mainly be attributed to the improvement in confidence intervals during PE; refer equations~\ref{eq:dvar} and \ref{eq:dcov}. Figures~\ref{fig:results}(c) and (d) quantitatively compare the scaling factor in the RHS of these equations, respectively, as a function of $N$ for three different distributions. The estimators $\hat{x}$, $\hat{y}$, $\hat{z}$ for this purpose are the actual values obtained in our experiment and we used an $\epsilon_\text{PE} = 10^{-10}$. The difference between the confidence intervals used in Ref.~\cite{Leverrier2015} (suitably modified here for a fair comparison) with those derived here, based on the Beta distribution, is quite evident at lower values of $N$, as visualized by comparing the dashed-blue trace with the solid-red one.
Since the untrusted noise has a quadratic dependence on the covariance in contrast to variance where the dependence is linear, a method that tightens the confidence intervals for the covariance can be expected to have a large impact on the final composable SKF. In fact, according to simulation, our implementation would have required almost an order of magnitude higher $N_\text{PA} $ ($\gtrsim 7.5 \times 10^9$) using the confidence intervals of Ref.~\cite{Leverrier2015} to achieve the peak SKF depicted by the rightmost blue data point in Fig.~\ref{fig:results}(a).
The dashed-green trace shows the confidence intervals also based on the Beta distribution, and a further assumption of the underlying data, i.e., the I and Q quadrature symbols, following a Gaussian distribution (more details provided in the Supplement). This however may restrict the security analysis to Gaussian collective attacks, therefore, we do not make this assumption in our calculations. The advantage of this method would however be even tighter confidence intervals, and thus, even lower requirements on $N$ for obtaining a composable key with positive length.
On the practical front, a reasonably large transmission rate $B = 100\,$MSymbols/s of the coherent states together with the careful analysis and removal of excess noise (refer section~\ref{Exp:NoiseCal} for more details) enables an overall fast, yet low-noise and highly stable system operation, critical in quickly distributing raw correlations of high quality and keeping the finite-size corrections minimal.
\section{Conclusion \& Outlook}
Due to its similarity to coherent telecommunication systems, continuous-variable quantum key distribution (CVQKD) based on coherent states is perhaps the most cost-effective solution for widespread deployment of quantum cryptography at access network scales ($10-50\,$km long quantum channels). However, CVQKD protocols have lagged behind their discrete-variable counterparts in terms of security, particularly, in demonstrating composability and robustness against finite-size effects. In this work, we have implemented a prepare-and-measure Gaussian-modulated coherent state CVQKD protocol that operates over a 20 km long quantum channel connecting Alice and Bob, who, at the end of the protocol obtain a composable secret key that takes finite-size effects into account and is protected against collective attacks. Our achievement was enabled by means of several novel advances in the theoretical security analysis and technical improvements on the experimental front. Furthermore, by using a real local oscillator at the receiver, we enhance the practicality as well as the security of the QKD system against hacking.
In conclusion, we believe this is a significant advance that demonstrates practicality, performance, and security of CVQKD implementations operating in the low-to-moderate channel loss regime. With an order of magnitude larger $N$ and half the current value of $u$, we expect to obtain a non-zero length of the composable key while tolerating channel losses around 8 dB, i.e.,\ distances up to $\sim 40\,$km (assuming an attenuation factor of 0.2 dB/km). This should be easily achievable with some improvements in the hardware as well as the digital signal processing. We therefore expect that in the future, users across a point-to-point link could use the composable keys from our CVQKD implementation to enable real applications such as secure data encryption, thus ushering in a new era for CVQKD. \\
\end{document} |
\begin{document}
\title{Non-Markovian dynamics revealed at a bound state in continuum}
\author{Savannah Garmon}
\email{sgarmon@p.s.osakafu-u.ac.jp}
\author{Kenichi Noba}
\affiliation{Department of Physical Science,
Osaka Prefecture University,
Gakuen-cho 1-1, Sakai 599-8531, Japan}
\author{Gonzalo Ordonez}
\affiliation{Department of Physics and Astronomy, Butler University,
Gallahue Hall, 4600 Sunset Ave.,
Indianapolis, Indiana 46208, USA}
\author{Dvira Segal}
\affiliation{Chemical Physics Theory Group, Department of Chemistry,
University of Toronto, 80 Saint George St., Toronto,
Ontario, M5S 3H6, Canada}
\begin{abstract}
We propose a methodical approach to controlling and enhancing deviations from exponential decay in quantum and optical systems by exploiting recent progress surrounding another subtle effect: the bound states in continuum, which have been observed in optical waveguide array experiments within this past decade. Specifically, we show that by populating an initial state orthogonal to that of the bound state in continuum, it is possible to engineer system parameters for which the usual exponential decay process is suppressed in favor of inverse power law dynamics and coherent effects that typically would be extremely difficult to detect in experiment. We demonstrate our method using a model based on an optical waveguide array experiment, and further show that the method is robust even in the face of significant detuning from the precise location of the bound state in continuum.
\end{abstract}
\date{\today}
\maketitle
A bound state in continuum (BIC) represents a localized eigenmode with energy eigenvalue that, counter-intuitively, resides directly within the scattering continuum of a given physical system. Although the existence of such modes were first predicted in 1929 \cite{BIC_1929}, the phenomenon is so delicate that they were not observed until much more recently \cite{BIC_review}; for example, in optical waveguide array experiments \cite{BIC_opt_expt0,BIC_opt_expt1,BIC_opt_exptA,BIC_opt_expt2}. Lasing action has also recently been reported for a cavity supporting BICs \cite{BIC_lasing}. In this work, we propose to apply these recent technical advances in optical control of the BIC to the study of another often elusive phenomenon: long-time non-exponential decay.
In many familiar circumstances, such as atomic relaxation, we tend to think of quantum decay as essentially an exponential process. More precisely, exponential decay tends to manifest when an unstable eigenmode (such as an excited atomic level) is resonant with an energy continuum (environmental reservoir, such as the electromagnetic vacuum) to which it is coupled.
However, it can be shown that in fact all quantum systems follow non-exponential dynamics on very short and extremely long timescales. These deviations occur as a direct result of the existence of at least one threshold on the energy continuum in such systems \cite{Khalfin,Winter61,Fonda,MDCG06,Muga_review2,GPSS13,OH17,CS18}. While these effects are ubiquitous in quantum systems, they are unfortunately quite difficult to detect under ordinary circumstances and hence have been measured only in a small handful of
experiments \cite{short_time_expt,zeno_expt,zeno_expt_latt,long_time_expt,KurizkiZeno,ZenoSC}.
The short-time deviation, which can give rise to both decelerated \cite{SudarshanZeno} and accelerated \cite{antiZeno} decay under frequent system observations \cite{SudarshanZeno,zeno_expt,KurizkiZeno,ZenoSC} or modulation of the environmental coupling \cite{zeno_expt_latt}, requires ultra-precision to detect that is often difficult to achieve in the lab.
Ref. \cite{XCLYG14} uses the properties of a BIC to study these short-time effects.
The long-time deviation, meanwhile, has proven even more challenging \cite{long_time_expt}. The difficulty originates in that the effect usually does not appear until many lifetimes of the exponential decay have passed, by which time the survival probability is so depleted that it is rendered undetectable.
A handful of theory papers have suggested special circumstances to enhance the long-time effect; these mostly require an initially prepared state near the threshold, usually combined with other conditions \cite{KKS94,JQ94,LNNB00,Jittoh05,GCV06,Longhi06,DBP08,GPSS13,GO17}. See also the recent experiment \cite{Krinner18}.
In this paper we take advantage of the simple geometric shape of the BIC to present a qualitatively different and more easily generalized scheme by which the long-time deviation can be enhanced.
While it is clear from the outset that the usual exponential decay associated with the resonance is suppressed when the BIC condition is satisfied, if one were to directly populate the BIC itself then one would observe a simple stable evolution, as the BIC is of course an eigenstate of the Hamiltonian.
However, we show that by populating a state that is orthogonal to the BIC we can take advantage of the suppression of the exponential effect while avoiding the stability associated with the BIC itself. The non-exponential dynamics can then drive the evolution on all timescales. What's more, we demonstrate in our example below that the exponential effect can be dramatically suppressed even with significant detuning from the BIC, although the choice of BIC-orthogonal initial state is still essential.
We illustrate our method relying on a simple tight-binding model that can be viewed analogously
to one of the previously mentioned optical waveguide array experiments.
Our Hamiltonian is written
\begin{equation}a
H = \epsilon_\textrm{d} | d \rangle \langle d |
& - & J \sum_{n=1}^\infty \left( | n \rangle \langle n+1 | + | n+1 \rangle \langle n | \right)
\nonumber \\
& &
- g \left( | d \rangle \langle 2 | + | 2 \rangle \langle d | \right)
,
\label{ham.n2.n}
\end{equation}a
in which the second term represents the semi-infinite array with nearest-neighbor hopping parameter $-J$ and the chain is side-coupled at site $| 2 \rangle$ to an ``impurity'' element $| d \rangle$.
After we set the energy units according to $J=1$, the adjustable parameters in the system are the chain-impurity coupling $-g$ and the impurity energy level $\epsilon_\textrm{d}$.
This model captures the essential features of the waveguide array experiment in Ref. \cite{BIC_opt_expt2} (see Ref. \cite{LonghiEPJ07} as well as \cite{Fukuta}), when we view time evolution in the present context analogously to longitudinal propagation within the waveguides.
This model can be partially diagonalized by introducing a half-range Fourier series on the chain according to $| n \rangle = \sqrt{\frac{2}{\pi}} \int_0^\pi dk \; \sin nk | k \rangle$, after which we have
\begin{equation}a
H = \epsilon_\textrm{d} | d \rangle \langle d | & + & \int_0^\pi dk \; E_k | k \rangle \langle k |
\nonumber \\
& & + g \int_0^\pi dk \; V_k \left( | d \rangle \langle k | + | k \rangle \langle d | \right)
\label{ham.n2.k}
\end{equation}a
where $V_k = - \sqrt{\frac{2}{\pi}} \sin 2 k$ and the continuum is given by $E_k = - 2 J \cos k$ over
$k \in [0,\pi]$. Note from here we will measure the energy in units of $J=1$.
The discrete spectrum for this model can be obtained, for example, from the resolvent operator
\begin{equation}
\langle d | \frac{1}{z - H} | d \rangle \; = \;
\frac{1}{z - \epsilon_\textrm{d} - \Sigma (z)}
\label{n2.res.dd}
\end{equation}
in which the self-energy function $\Sigma (z) = g^2 \int_0^\pi dk \frac{| V_k |^2}{z - E_k}$
is evaluated as
\begin{equation}
\Sigma (z)
= \frac{z g^2}{2} \left[ z^2 - 2 - z \sqrt{z^2 - 4} \right]
\label{n2.sigma.z}
\end{equation}
in the first Riemann sheet [see Ref. \cite{Supp} for discussion of the analytic properties of $\Sigma(z)$].
Notice that a pole occurs in Eq. (\mathop{\textrm{Re}}f{n2.res.dd}) at $z=0$ after choosing $\epsilon_\textrm{d} = 0$; this is the BIC solution for this model, which resides directly at the center of the continuum $z \in [-2, 2]$ (defined by
the range of $E_k$) and which takes the form
\begin{equation}
| \psi_\textrm{BIC} \rangle
= \frac{1}{\sqrt{1 + g^2}} \left( | d \rangle - g | 1 \rangle \right)
.
\label{n2.bic}
\end{equation}
We here emphasize that the BIC state can be understood as a resonance with vanishing decay width \cite{Fukuta,BIC_review,SH75,FW85,Robnik,ONK06,SBR06,Rotter_review,QBIC,BS08,Moiseyev_BIC,Reichl09,GurvitzBIC,ZBK12,MA14,Boretz14,GTC17}. In this picture, the ordinary resonance represents a generalized eigenstate with complex energy eigenvalue, for which the imaginary part of the eigenvalue gives the exponential decay half-width. When the BIC condition $\epsilon_\textrm{d} = 0$ is fulfilled the imaginary part of this eigenvalue vanishes, yielding a bound state residing directly in the scattering continuum. When $\epsilon_\textrm{d} \neq 0$ the complex eigenvalue is restored and the exponential decay would generally be expected to reappear.
It is easy to show that there exist two further solutions for the $\epsilon_\textrm{d} = 0$ case with eigenvalues given by $z_\pm = \pm z_g$, in which
\begin{equation}
z_g = g + \frac{1}{g}
.
\label{n2.zg}
\end{equation}
For $g>1$ these two solutions constitute localized bound states residing on the first Riemann sheet of the complex energy plane, while for $g < 1$ they transition to so-called virtual bound states (or anti-bound states), which are delocalized pseudo-states with real eigenvalue resting in the second sheet \cite{DBP08,GPSS13,Nussenzveig59,Hogreve,Moiseyev_NHQM,HO14}, see Fig \mathop{\textrm{Re}}f{fig:0.spec}.
While the virtual bound states do not appear in the diagonalized Hamiltonian, they nevertheless have a similar influence on the long-time power law decay as do the bound states \cite{GPSS13}. Specifically, we will show that the timescale characterizing the non-exponential decay is proportional to
$\Delta_g^{-1}$, where
\begin{equation}
\Delta_g
\equiv z_g - 2
.
\label{n2.Deltag}
\end{equation}
is defined as the gap between either of the (virtual) bound state energies and the nearest band edge.
Note we will particularly focus on the $g \le 1$ portion of the parameter space as the absence of bound states here means that nothing inhibits the non-exponential decay. (For comparison, we will also briefly discuss the $g > 1$ evolution.)
\begin{figure}
\caption{(color online) Discrete spectrum of our model as a function of $g$ in the case $\epsilon_\textrm{d}
\label{fig:0.spec}
\end{figure}
As previously discussed, if we were to consider the evolution of the BIC state itself, the initial state would simply remain occupied for all time as $| \psi_\textrm{BIC} \rangle$ is an eigenstate of $H$ with energy eigenvalue $z=0$. However, by instead choosing the (simplest) BIC-orthogonal state
\begin{equation}
| \psi_\perp \rangle
= \frac{1}{\sqrt{1 + g^2}} \left( g | d \rangle + | 1 \rangle \right)
\label{n2.perp}
\end{equation}
as our initial state, we obtain complete non-exponential decay for any value $g \le 1$, as shown below \footnote{One might pause at the inclusion of the site $|1\rangle$ that is technically part of the reservoir in this initial state. However, $|1\rangle$ could equivalently be viewed as a second impurity element \cite{Supp}. }.
To analyze the evolution of $| \psi_\perp \rangle$, we evaluate the survival probability $P_\perp (t) = |A_{\perp} (t)|^2$, in which the survival amplitude is given by
\begin{equation}
A_{\perp} (t)
= \langle \psi_\perp | e^{-iHt} | \psi_\perp \rangle
= \frac{1}{2 \pi i} \int_{\mathcal{C}_E} e^{-izt} \langle \psi_\perp | \frac{1}{z - H} | \psi_\perp \rangle \; dz
.
\label{A.perp.defn}
\end{equation}
Here $\mathcal{C}_E$ is a counter-clockwise integration contour surrounding the real axis in the first Riemann sheet of the complex energy plane, which includes the branch cut along $z \in [-2,2]$ as well as any bound states.
We can apply various methods to evaluate this integral, for example by directly computing the relevant matrix elements of the resolvent operator and integrating over these or by applying an expansion in terms of the eigenstates of the generalized discrete spectrum of the model as in Ref. \cite{HO14}. By either method we obtain the following results.
For the case $g > 1$ there are two bound states included in the contour for Eq. (\mathop{\textrm{Re}}f{A.perp.defn}). The survival amplitude in this case evaluates as
\begin{equation}
A_\perp (t)
= \frac{g^2 - 1}{g^2} \cos z_g t + A_\textrm{br} (t)
,
\label{A.perp.poles}
\end{equation}
in which the first term represents the combined contributions from the two bound states
while
\begin{equation}
A_\textrm{br} (t)
= \frac{1+g^2}{4 \pi i g^2} \int_{\mathcal{C}_\textrm{br}} dz e^{-izt} \frac{\sqrt{z^2 - 4}}{z^2 - z_g^2}
.
\label{A.br.defn}
\end{equation}
is an integration along the contour $\mathcal{C}_\textrm{br}$ surrounding the branch cut in a counter-clockwise manner in the complex energy plane.
The decay in this case is non-exponential but incomplete due to the presence of the bound states \cite{KKS94,JQ94,LNNB00}.
This can be seen for the case $g=1.1$ in Fig. \mathop{\textrm{Re}}f{fig:n2.surv.prob}(a).
\begin{figure*}
\caption{(color online) Numerical simulations for the survival probability of $|\psi_\perp \rangle$ at time $t$ for $\epsilon_\textrm{d}
\label{fig:n2.surv.prob}
\end{figure*}
Meanwhile for the case $g \le 1$, the bound states have become virtual bound states and the evolution is now determined entirely by the non-Markovian branch cut contribution $A_\perp (t) = A_\textrm{br} (t)$.
We find that this expression yields two distinct time regions, in which the integral is most easily estimated by somewhat different methods.
First there is a short/intermediate time region, in which we first apply a fraction decomposition to the denominator of Eq. (\mathop{\textrm{Re}}f{A.br.defn}); this yields two simpler integrals, one associated with the upper virtual bound state and the other associated with the lower.
As outlined in \cite{Supp}, these two integrals can be evaluated in terms of Bessel functions by methods similar to those used in Ref. \cite{GO17}, which yields
\begin{equation}
A_\textrm{br} (t)
\approx \frac{1}{g} J_0 (2 t) - \frac{1-g}{g} \cos 2 t
;
\label{n2.A.early}
\end{equation}
this expression holds for all $t \ll T_\Delta$ where $T_\Delta$ is written as
$T_\Delta = 1/\Delta_g = g/(1-g)^2$ in terms of the energy gap between the virtual bound states and their respective nearby band edges. On the earliest timescale $t \ll T_Z$ with $T_Z = 1$, this expression yields the usual short time parabolic dynamics $P_Z (t) \approx 1 - C t^2$, in which
$C = (g + g^2 + g^3 - 1)/g^2$.
Then, in the intermediate time region $T_Z \ll t \ll T_\Delta$, we can approximate the Bessel function in the first term of Eq. (\mathop{\textrm{Re}}f{n2.A.early}) to write
\begin{equation}
A_\textrm{NZ} (t)
\approx \frac{\cos (2t - \pi / 4)}{g \sqrt{\pi t}} - \frac{1-g}{g} \cos 2 t
.
\label{n2.A.NZ}
\end{equation}
We refer to this time region including characteristic $1/t$ decay as the {\it non-exponential near zone} (NZ) \cite{GPSS13}, which we can roughly think of as having replaced the usual exponential decay regime.
For values $g \lesssim 1$ fairly close to the $g=1$ localization transition, the first term in Eq. (\mathop{\textrm{Re}}f{n2.A.NZ}) tends to dominate the evolution early in the near zone, while the second term provides only a small correction.
Estimating the evolution in this case
yields
\begin{equation}
P_\textrm{NZ,early} (t)
\approx \frac{ \cos^2 \left( 2 t - \pi/4 \right)}{\pi g^2 t}
,
\label{n2.P.NZ}
\end{equation}
which can be seen for the case $g=0.98$ in Fig. \mathop{\textrm{Re}}f{fig:n2.surv.prob}(c,d).
As we move later into the near zone, the first term decays sufficiently so that the second term becomes non-negligible; we can estimate this as when the second term is about 10\% of the first, which gives $t = T_\textrm{VR} \sim 1 / \left[100 \pi (1-g)^2 \right] = 1/ \left[100 \pi g \right] * T_\Delta$. This implies we should be fairly close to the transition point $g=1$ to observe the pure $1/t$ dynamics. For example, in the case $g=0.98$ shown in Fig. \mathop{\textrm{Re}}f{fig:n2.surv.prob}(d), we can already see a small influence from the second term of Eq. (\mathop{\textrm{Re}}f{n2.A.NZ}) around
$t \gtrsim T_\textrm{VR} \approx 8.0$ as the last three visible oscillation cycles show a slight deviation from the Eq. (\mathop{\textrm{Re}}f{n2.P.NZ}) prediction, which the second term of Eq. (\mathop{\textrm{Re}}f{n2.A.NZ}) [not shown] captures very well.
We will return to the physical interpretation of this second term momentarily.
Next appears the asymptotic time region $T_\Delta \ll t$ during which the dynamics are instead described by a $1/t^3$ power law decay. To show this, we return to the (exact) integral expression for the survival amplitude appearing in Eq. (\mathop{\textrm{Re}}f{A.br.defn}) and instead proceed by deforming the contour $\mathcal{C}_\textrm{br}$ surrounding the branch cut by dragging it out to infinity in the lower half of the complex energy plane, as described in \cite{Supp}.
Following this procedure, we obtain
\begin{equation}
P_\textrm{FZ} (t)
\approx \frac{\left(1 + g^2 \right)^2 \cos^2 \left( 2 t - 3 \pi/4 \right)}
{\pi g^4 \Delta_g^2 \left( 2 + z_g \right)^2 t^3}
,
\label{n2.P.FZ}
\end{equation}
with the characteristic $1/t^3$ decay that is typical of odd dimensional systems on long timescales \cite{Sudarshan,Muga95,GPSS13,GCMM95,Cavalcanti98,Zueco17}.
We refer to this as the {\it non-exponential far zone} (FZ).
The far zone dynamics can be seen for the $g=0.98$ case in Fig. \mathop{\textrm{Re}}f{fig:n2.surv.prob}(c,e).
We emphasize three further points about these results as follows.
First, we draw attention more carefully to the occurrence of oscillations in both time zones, which are due to interference between the contributions from the two band edges. These contributions are equally weighted because the BIC occurs at the center of the continuum band in the present case. Notice further that a $\pi/2$ phase shift occurs between the early near zone result Eq. (\mathop{\textrm{Re}}f{n2.P.NZ}) and the far zone Eq. (\mathop{\textrm{Re}}f{n2.P.FZ}).
These oscillations and the resulting phase shift are highlighted in Fig. \mathop{\textrm{Re}}f{fig:n2.surv.prob}(d,e). While similar oscillations have been previously predicted in the far zone \cite{Longhi06,GO17,Zueco17}, we believe the near zone oscillations as well as the resulting phase shift are new --- indeed, outside of our choice for the initial state, these would almost certainly be obscured by the exponential decay.
Second, we return our attention to the second term of Eq. (\mathop{\textrm{Re}}f{n2.A.NZ}), which becomes relatively more pronounced later in the near zone; however, counterintuitively perhaps, it vanishes in the far zone\footnote{The reason for this is discussed in pp. 21-22 of Ref. \cite{HO14}.}.
Notice this term takes the form of a Rabi-like oscillation between the band edges at $z=\pm 2$. We refer to this effect as a {\it virtual Rabi oscillation}, which is intended to reflect its transient nature. A further interesting point is that the virtual Rabi oscillation plays a role in facilitating the phase shift from the early near zone into the far zone \cite{Supp}.
Third, notice that when we are directly at the localization transition at $g=1$, the second term in Eq. (\mathop{\textrm{Re}}f{n2.A.NZ}) vanishes. Further, since the key timescale $T_\Delta$ is inversely proportional to $\Delta_g$, as we approach $g=1$ from below the energy gap $\Delta_g$ closes and $T_\Delta$ diverges. Hence, in this case, Eq. (\mathop{\textrm{Re}}f{n2.P.NZ}) describes the dynamics accurately for all $T_\textrm{Z} \ll t$, which is shown in Fig. \mathop{\textrm{Re}}f{fig:n2.surv.prob}(b) (see also Ref. \cite{GPSS13} for discussion relevant to this point as well as the influence of a virtual bound state on the power law decay).
We can quantify the divergence of the timescale $T_\Delta$ in terms of the distance $\delta$ from the transition point $g=1$ after reparameterizing according to $g \equiv 1 - \delta$; then the timescale diverges like $T_\Delta \sim 1/\delta^2$ as $\delta \rightarrow 0$.
\begin{figure}
\caption{Numerical simulations for the survival probability and the non-escape probability for detuning from the BIC for $g=0.9$ and (a) $\epsilon_\textrm{d}
\label{fig:4}
\end{figure}
While the preceding analysis gives a clear picture of the types of evolution we can expect for the state
$| \psi_\perp \rangle$, it is still a bit idealized in comparison to experiment in two ways that we will account for below. First, in a real experiment it would be difficult to tune exactly to the BIC at $\epsilon_\textrm{d} = 0$; since the BIC is just the special case of a resonance with zero decay width, as we introduce detuning $\epsilon_\textrm{d} \neq 0$ the resonance must reappear, which we could expect might perturb the non-exponential evolution of $P_\perp (t)$. The complex eigenvalue of the resonance state can be expanded in the vicinity of the BIC up to second order in $\epsilon_\textrm{d}$ as $z_\textrm{res} \approx \epsilon_\textrm{d} / (1+g^2) - i \Gamma/2$ with $\Gamma = 2 g^2 \epsilon_\textrm{d}^2 / (1+g^2)^3$, which of course reduces to $z_\textrm{BIC} = 0$ in the limit $\epsilon_\textrm{d} = 0$. However, when we examine $P_\perp (t)$ (red curve in Fig. \mathop{\textrm{Re}}f{fig:4} for $g=0.9$, as an example), we find that the resonance has virtually no influence on the survival probability, even for moderately large detuning values $\epsilon_\textrm{d} \neq 0$. We can obtain an understanding for this by calculating the resonance pole contribution to
$P_\perp (t)$. Performing first a simple calculation for the pole contribution to the amplitude
$\langle \psi_\perp | e^{-iHt} | \psi_\perp \rangle$ reveals that, due to the geometric shape of the BIC-orthogonal state, both the lowest order and next-lowest order contributions in $\epsilon_\textrm{d}$ cancel out, which yields
\begin{equation}
P_{\perp,\textrm{res}} (t)
\approx \frac{g^4 \epsilon_\textrm{d}^4}{\left(1+g^2\right)^8} e^{- \Gamma t}
.
\label{P.perp.res}
\end{equation}
The pre-factor in this expression, which is fourth order in $\epsilon_\textrm{d}$, assures that the exponential effect will be quite small for almost any $\epsilon_\textrm{d} \simeq 0$ regardless of the value of $g$. For example, even for modest detuning $\epsilon_\textrm{d} = 0.2$ and $g=0.9$ in Fig. \mathop{\textrm{Re}}f{fig:4}(b) [red curve], we have $g^4 \epsilon_\textrm{d}^4 / \left(1+g^2\right)^8 \sim 10^{-5}$.
Second, while preparation of the initial state $| \psi_\perp \rangle$ seems feasible, measuring the precise output state $\langle \psi_\perp |$ might prove more challenging.
Instead, it may be more realistic to consider the quantity
\begin{equation}
P_{1\textrm{d}} (t)
\equiv |\langle 1 | e^{-iHt} | \psi_\perp \rangle|^2 + |\langle \textrm{d} | e^{-iHt} | \psi_\perp \rangle|^2
,
\label{P.1d}
\end{equation}
which is equivalent to the {\it non-escape probability} that has appeared in the literature previously \cite{Muga_review2,Muga95,GCMM95,Cavalcanti98,GCMV07}.
It can easily be shown that $P_{1\textrm{d}} (t) = P_\perp (t)$ for the case $\epsilon_\textrm{d} = 0$, and hence all of our preceding detailed analytical results still apply directly at the BIC. As shown in Fig. \mathop{\textrm{Re}}f{fig:4}, the difference between $P_{1\textrm{d}} (t)$ [blue curve] and $P_\perp (t)$ [red curve] appears first well into the long time region for small $\epsilon_\textrm{d} \neq 0$ and moves gradually to earlier times as we increase the detuning. The origin of the difference between the two quantities is easy to understand
as it seems to be entirely attributable to the fact that only the lowest-order contribution in $\epsilon_\textrm{d}$ cancels out when we calculate the resonance pole contribution to the amplitude for the non-escape probability $P_{1\textrm{d}} (t)$. In particular, we find
\begin{equation}
P_{1\textrm{d},\textrm{res}} (t)
\approx \frac{g^2 \epsilon_\textrm{d}^2}{\left(1+g^2\right)^4} e^{- \Gamma t}
,
\label{P.1d.res}
\end{equation}
which is still small, but has some noticeable influence on the spectrum in some cases. For example, in Fig. \mathop{\textrm{Re}}f{fig:4} (b) for $\epsilon_\textrm{d} = 0.2$ we see the resonance pole with magnitude $g^2 \epsilon_\textrm{d}^2 / \left(1+g^2\right)^4 \sim 0.003$ introduces exponential dynamics into $P_{1\textrm{d}} (t)$ around $t \gtrsim 10$, although this only lasts for a few lifetimes $\tau = 2 / \Gamma \sim 360$, which leaves the non-escape probability relatively intact when this quantity rejoins with $P_\perp (t)$ as the $1/t^3$ far zone dynamics kick in. We note that $P_{1\textrm{d}} (t)$ also exhibits the interesting feature of {\it pre}-exponential decay that extends beyond the usual parabolic dynamics in the region $1 \lesssim t \lesssim 10$. As we further increase $\epsilon_\textrm{d}$ as in Fig. \mathop{\textrm{Re}}f{fig:4} (c), we find the exponential decay region lasts even fewer lifetimes as the difference between $P_{1\textrm{d}} (t)$ and $P_\perp (t)$ again becomes diminished.
In this work we have shown that by populating a state that lies orthogonal to a bound state in continuum one can observe non-exponential dynamics that are usually overwhelmingly suppressed when the resonance condition is satisfied.
Note that for the present model we could consider the evolution of more general BIC orthogonal states such as
$g | d \rangle + | 1 \rangle + \sum_{n=2}^{\infty} w_n | n \rangle$ that include elements of the chain beyond the BIC sector. We briefly comment on a representative example of this more general configuration in Ref. \cite{Supp}, where we show that including a single site from the chain can suppress oscillations in the survival probability.
We briefly note we have focused here on bound states in continuum that appear purely due to interference effects as originally proposed by von Neumann and Wigner in 1929 \cite{BIC_1929}. We have not directly addressed ``accidental'' BICs \cite{Hsu13} that exhibit interesting topological properties \cite{BIC_review,BIC_lasing,BM17,ZHLSS14}, although the study of BIC-orthogonal states in this context might prove fruitful as well.
\begin{thebibliography}{99}
\bibitem{BIC_1929} J. von Neumann and E. Wigner,
Physikalische Zeitschrift {\bf 30}, 465--470 (1929).
\bibitem{BIC_review} C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Solja\v{c}i\'c,
Nature Rev. Mater. {\bf 1}, 16048 (2016).
\bibitem{BIC_opt_expt0}
F. Dreisow, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. T\"unnermann, and S. Longhi,
Opt. Lett. {\bf 34}, 2405 (2009).
\bibitem{BIC_opt_expt1} Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev,
Phys. Rev. Lett. {\bf 107}, 183901 (2011).
\bibitem{BIC_opt_exptA} G. Corrielli, G. Della Valle, A. Crespi, R. Osellame, and S. Longhi,
Phys. Rev. Lett. {\bf 111}, 220403 (2013).
\bibitem{BIC_opt_expt2} S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, A. T\"unnermann, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar,
Phys. Rev. Lett. {\bf 111}, 240403 (2013).
\bibitem{BIC_lasing} A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kant\'e,
Nature {\bf 541}, 196 (2017).
\bibitem{Khalfin} L. A. Khalfin,
Sov. Phys. JETP {\bf 6}, 1053 (1958).
\bibitem{Winter61} R. G. Winter,
Phys. Rev. Lett. {\bf 123}, 1503 (1961).
\bibitem{Fonda} L. Fonda, G. C. Ghirardi, and A. Rimini,
Rep. Prog. Phys. {\bf 41}, 587 (1978).
\bibitem{MDCG06} J. G. Muga, F. Delgado, A. del Campo, and G. Garc\'ia-Calder\'on,
Phys. Rev. A {\bf 73}, 052112 (2006).
\bibitem{Muga_review2}
E. Torrontegui, J. G. Muga, J. Martorell, and D. W. L. Spring,
Adv. Quant. Chem. {\bf 60}, 485 (2010).
\bibitem{GPSS13} S. Garmon, T. Petrosky, L. Simine, and D. Segal,
Fortschr. Phys. {\bf 61}, 261 (2013).
\bibitem{OH17} G. Ordonez and N. Hatano,
J. Phys. A: Math. Theor. {\bf 50}, 405304 (2017).
\bibitem{CS18} A. Chakraborty and R. Sensarma,
Phys. Rev. B {\bf 97}, 104306 (2018).
\bibitem{short_time_expt}
S. R. Wilkinson, C. F. Bharucha, M. C. Fischer, K. W. Madison, P. R. Morrow, Q. Niu, B. Sundaram, and M. G. Raizen,
Nature (London) {\bf 387}, 575 (1997).
\bibitem{zeno_expt}
M. C. Fischer, B. Guti\'errez-Medina, and M. G. Raizen,
Phys. Rev. Lett. {\bf 87}, 040402 (2001).
\bibitem{zeno_expt_latt}
A. G. Kofman and G. Kurizki,
Phys. Rev. Lett. {\bf 87}, 270405 (2001);
S. Longhi, Opt. Lett. {\bf 32}, 557 (2007);
F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. T\"unnermann, and S. Longhi,
Phys. Rev. Lett. {\bf 101}, 143602 (2008).
\bibitem{long_time_expt}
C. Rothe, S. I. Hintschich, and A. P. Monkman,
Phys. Rev. Lett. {\bf 96}, 163601 (2006).
\bibitem{KurizkiZeno}
G. A. \'Alvarez, D. D. B. Rao, L. Frydman, and G. Kurizki,
Phys. Rev. Lett. {\bf 105}, 160401 (2010).
\bibitem{ZenoSC} K. Kakuyanagi, T. Baba, Y. Matsuzaki, H. Nakano, S. Saito, and K. Semba,
New J. Phys. {\bf 17}, 063035 (2015).
\bibitem{SudarshanZeno} B. Misra and E. C. G. Sudarshan,
J. Math. Phys. {\bf 18}, 756 (1977).
\bibitem{antiZeno} A. G. Kofman, and G. Kurizki,
Phys. Rev. A {\bf 54}, R3750 (1996);
M. Lewenstein and K. Rz\c{a}\.zewski,
Phys. Rev. A {\bf 61}, 022105 (2000);
A. G. Kofman, and G. Kurizki,
Nature {\bf 405}, 546 (2000).
\bibitem{XCLYG14} L. Xu, Y. Cao, X.-Q. Li, Y. J. Yan, and S. Gurvitz,
Phys. Rev. A {\bf 90}, 022108 (2014).
\bibitem{KKS94} A. G. Kofman, G. Kurizki, and B. Sherman,
J. Mod. Opt. {\bf 41}, 353 (1994).
\bibitem{JQ94} S. John and T. Quang,
Phys. Rev. A {\bf 50}, 1764 (1994).
\bibitem{LNNB00} P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielson, and S. Bay,
Rep. Prog. Phys. {\bf 63}, 455 (2000).
\bibitem{Jittoh05} T. Jittoh, S. Matsumoto, J. Sato, Y. Sato, and K. Takeda,
Phys. Rev. A {\bf 71}, 012109 (2005).
\bibitem{GCV06} G. Garc\'ia-Calder\'on, and J. Villavicencio,
Phys. Rev. A {\bf 73}, 062115 (2006).
\bibitem{Longhi06} S. Longhi,
Phys. Rev. Lett. {\bf 97}, 110402 (2006).
\bibitem{DBP08}
A. D. Dente, R. A. Bustos-Mar\`{u}n, and H. M. Pastawski,
Phys. Rev. A {\bf 78}, 062116 (2008).
\bibitem{GO17} S. Garmon and G. Ordonez,
J. Math. Phys. {\bf 58}, 062101 (2017).
\bibitem{Krinner18} L. Krinner, M. Stewart, A. Pazmi\~no, J. Kwon, and D. Schneble,
Nature {\bf 559}, 589 (2018).
\bibitem{LonghiEPJ07} S. Longhi,
Eur. Phys. J. B {\bf 57}, 45--51 (2007).
\bibitem{Fukuta} T. Fukuta, S. Garmon, K. Kanki, K.-I. Noba, and S. Tanaka,
Phys. Rev. A {\bf 96}, 052511 (2017).
\bibitem{SH75} F. H. Stillinger and D. R. Herrick,
Phys. Rev. A {\bf 11}, 446 (1975).
\bibitem{FW85} H. Friedrich and D. Wintgen,
Phys. Rev. A {\bf 31}, 3964 (1985).
\bibitem{Robnik} M. Robnik,
J. Phys. A: Math. Gen. {\bf 19}, 3845 (1986).
\bibitem{ONK06} G. Ordonez, K. Na, and S. Kim,
Phys. Rev. A {\bf 73}, 022113 (2006).
\bibitem{SBR06} A. F. Sadreev, E. N. Bulgakov, and I. Rotter,
Phys. Rev. B {\bf 73}, 235342 (2006).
\bibitem{QBIC} H. Nakamura, N. Hatano, S. Garmon, and T. Petrosky,
Phys. Rev. Lett. {\bf 99}, 210404 (2007);
S. Garmon, H. Nakamura, N. Hatano, and T. Petrosky,
Phys. Rev. B {\bf 80}, 115318 (2009).
\bibitem{BS08} E. N. Bulgakov and A. F. Sadreev,
Phys. Rev. B {\bf 78}, 075105 (2008).
\bibitem{Rotter_review} I. Rotter,
J. Phys. A: Math. Theor. {\bf 42}, 153001 (2009).
\bibitem{Moiseyev_BIC} N. Moiseyev,
Phys. Rev. Lett. {\bf 102}, 167404 (2009).
\bibitem{Reichl09} H. Lee and L. E. Reichl,
Phys. Rev. B {\bf 79}, 193305 (2009).
\bibitem{GurvitzBIC} J. Ping, X.-Q. Li, and S. Gurvitz,
Phys. Rev. A {\bf 83}, 042112 (2011).
\bibitem{ZBK12} J. M. Zhang, D. Braak, and M. Kollar,
Phys. Rev. Lett. {\bf 109}, 116405 (2012).
\bibitem{MA14} F. Monticone and A. Al\`u,
Phys. Rev. Lett. {\bf 112}, 213903 (2014).
\bibitem{Boretz14} Y. Boretz, G. Ordonez, S. Tanaka, and T. Petrosky,
Phys. Rev. A {\bf 90}, 023853 (2014).
\bibitem{GTC17} A. Gonz\`alez-Tudela and J. I. Cirac,
Phys. Rev. Lett. {\bf 119}, 143602 (2017).
\bibitem{Nussenzveig59} H. M. Nussenzveig,
Nucl. Phys. {\bf 11}, 499 (1959).
\bibitem{Hogreve} H. Hogreve,
Phys. Lett. A {\bf 201}, 111 (1995).
\bibitem{Moiseyev_NHQM} N. Moiseyev,
{\it Non-Hermitian Quantum Mechanics},
Cambridge University Press (2011).
\bibitem{HO14} N. Hatano and G. Ordonez,
J. Math. Phys. {\bf 55}, 122106 (2014).
\bibitem{Supp} See Supplemental Material attached to this document.
\bibitem{Sudarshan}
C. B. Chiu, B. Misra, and E. C. G. Sudarshan,
Phys. Rev. D {\bf 16}, 520 (1977).
\bibitem{GCMM95} G. Garc\'ia-Calder\'on, J. L. Mateos, and M. Moshinsky,
Phys. Rev. Lett. {\bf 74}, 337 (1995);
\bibitem{Cavalcanti98}
R. M. Cavalcanti,
Phys. Rev. Lett. {\bf 80}, 4353 (1998).
\bibitem{Muga95} J. G. Muga, V. Delgado, and R. F. Snider,
Phys. Rev. B {\bf 52}, 16381 (1995).
\bibitem{Zueco17} E. S\'anchez-Burillo, D. Zueco, L. Mart\'in-Moreno, J. J. Garc\'ia-Ripoll,
Phys. Rev. A {\bf 96}, 023831 (2017).
\pagebreak
\bibitem{GCMV07} G. Garc\'ia-Calder\'on, I. Maldonado, and J. Villavicencio,
Phys. Rev. A {\bf 76}, 012103 (2007).
\bibitem{Hsu13} C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Solja\v{c}i\'c,
Nature {\bf 499}, 188 (2013).
\bibitem{ZHLSS14} B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Solja\v{c}i\'c,
Phys. Rev. Lett. {\bf 113}, 257401 (2014).
\bibitem{BM17} E. N. Bulgakov, and D. N. Maksimov,
Phys. Rev. Lett. {\bf 118}, 267401 (2017).
\end{thebibliography}
\pagebreak
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\section{Supplementary Material: Derivation of resolvent operator and self-energy}
\label{sec:resolvent}
To obtain the explicit expression for the resolvent operator, we first rewrite the Hamiltonian from the main text as $H = H_0 + V$, in which
\begin{equation}
H_0 = \epsilon_\textrm{d} | d \rangle \langle d | + \int_0^\pi dk \; E_k | k \rangle \langle k |
\end{equation}
and
\begin{equation}
V = g \int_0^\pi dk \; V_k \left( | d \rangle \langle k | + | k \rangle \langle d | \right)
.
\end{equation}
Then after applying a simple operator expansion
\begin{equation}a
\langle d | \frac{1}{z - H} | d \rangle
& = & \langle d | \left( \frac{1}{z - H_0} + \frac{1}{z - H_0}V \frac{1}{z - H_0} \right.
\nonumber \\
& & + \left. \frac{1}{z - H_0}V \frac{1}{z - H_0}V \frac{1}{z - H} \right) | d \rangle
,
\end{equation}a
we can easily solve for the explicit form of the resolvent operator
$\langle d | (z - H)^{-1} | d \rangle = (z - \epsilon_\textrm{d} - \Sigma (z))^{-1}$ given in the main text (note the second term of the above expansion vanishes).
Our next task is to perform the necessary integration to obtain the explicit form of the self-energy function
$\Sigma(z)$. This can be achieved through a variety of methods; for example, by applying the integration transformation $w=e^{ik}$ we can write
\begin{equation}a
\Sigma (z)
& = & g^2 \int_0^\pi \frac{| V_k |^2}{z - E_k} dk
= \frac{g^2}{\pi} \int_{- \pi}^{\pi} dk \frac{\sin^2 2k}{z + 2 \cos k}
\nonumber \\
& = & - \frac{g^2}{2 \pi i} \oint_\Lambda dw \frac{w^4 - 1}{(w - w_1)(w - w_2)}
\label{SM.sigma.z.int}
\end{equation}a
in which $\Lambda$ is the counter-clockwise contour just inside the unit circle in the complex $w$-plane and $w_{1,2} = (-z \pm \sqrt{z^2 - 4} )/2$.
Since $w_1 w_2 = 1$, if $w_1$ satisfies $|w_1| < 1$ then we must have $|w_2| > 1$ (or vice versa) and hence exactly one solution always falls inside the unit circle (we treat the situation $|w_1| = |w_2| = 1$ as a limit of the other two cases).
Evaluating the integral as a residue then yields the expression for the self-energy reported in the main text, where the sign in front of the square root is a $-$ ($+$) whenever the solution $w_1$ ($w_2$) appears inside the unit circle.
For the case of a bound state satisfying $z>2$, one can show the $|w_1| < 1$ case holds.
Taking the residue in Eq. (\mathop{\textrm{Re}}f{SM.sigma.z.int}) then yields the expression for the self-energy reported in
the main text.
As an explicit example, consider the upper bound state $z_+$ appearing in the case $\epsilon_\textrm{d} = 0$ and $g > 1$ from the main text. With $z_+ = g + 1/g$ we find $w_1 (z_+) = - 1/g$, so that $-1 < w_1 < 0$ inside the unit circle in the complex $w$-plane, as expected.
Note that we can also determine the wave vector $k_+$ that appears in the associated wave function $\langle x | \psi_+ \rangle \sim e^{i k_+ x}$ from the dispersion relation $z_+ = - 2 \cos k_+$.
Taking $e^{i k_+} = - 1/g$ gives $k_+ = \pi + i \log g$ with $\mathop{\textrm{Im}} k_+ > 0$ for $g>1$, which indeed yields a localized wave function. This verifies that $z_+$ is indeed a bound state eigenvalue in this case, residing in the first Riemann sheet of the complex $z$ plane.
For the case $g < 1$, we are forced to analytically continue this solution into the second Riemann sheet as $w_1 < -1$ passes outside the unit circle.
Instead, we now have $-1 < w_2 < 0$ so that $w_2 = -g$ is the pole appearing inside the contour integration of Eq. (\mathop{\textrm{Re}}f{SM.sigma.z.int}), resulting in a sign change for the non-analytic part of the self-energy for the solution $z = z_+$.
The wave function associated with this eigenvalue now becomes divergent as the wave vector is most naturally written in this case as $k_+ = \pi - i \log \bar{g}$ with $\bar{g} = g^{-1} > 1$ and $\mathop{\textrm{Im}} k_+ < 0$.
One can perform a similar analysis for the lower (virtual) bound state satisfying $z_- < -2$, except that $w_1$ and $w_2$ switch roles compared to the above explanation; this implies the sign in front of the non-analytic part of the self-energy is reversed compared to scenario for the upper bound state $z_+$. Note this requires that the sign designation for the lower bound state is opposite that of the upper bound state, without performing analytic continuation into the second Riemann sheet. (This point can also be shown independently from the present analysis by working directly with the expression reported for the self-energy in the main text.) Finally, the wave vector in this case is given by $k_- = i \log g$.
We note this expression for the self-energy from the main text has appeared in the literature previously \cite{LonghiEPJ07,Fukuta}.
\section{Details of approximate dynamics}
\label{sec:derivation}
\begin{figure*}
\caption{Survival probability of $|\psi_\perp \rangle$ at time $t$ for $\epsilon_\textrm{d}
\label{fig:NZ.FZ}
\end{figure*}
We start with the exact expression for the dynamics associated with the branch cut taken from the main text
\begin{equation}
A_\textrm{br} (t)
= \frac{1+g^2}{4 \pi i g^2} \int_{\mathcal{C}_\textrm{br}} dz e^{-izt} \frac{\sqrt{z^2 - 4}}{z^2 - z_g^2}
.
\label{SM.A.br.defn}
\end{equation}
In Sec. \mathop{\textrm{Re}}f{sec:derivation.NZ} we outline the approximations for the short/intermediate time region and make a brief comment on the influence of the virtual Rabi oscillation on the phase in the near zone, while in Sec. \mathop{\textrm{Re}}f{sec:derivation.FZ} we describe corresponding approximations for the far zone. In Sec. \mathop{\textrm{Re}}f{sec:derivation.plots} we present some plots for additional $g$ values compared to the main text and briefly discuss these. Note that $\epsilon_\textrm{d} = 0$ for all simulations in this document.
\subsection{Short/intermediate time region}
\label{sec:derivation.NZ}
We begin the derivation by performing a fraction decomposition on the integrand of Eq. (\mathop{\textrm{Re}}f{SM.A.br.defn}) in order to rewrite this as
\begin{equation}
A_\textrm{br} (t)
= - \frac{1}{2g} \left( I(z_+) - I(z_-) \right)
\label{SM.A.NZ.1}
\end{equation}
in which
\begin{equation}
I (z_n)
\equiv - \frac{1}{2 \pi} \int_{\mathcal{C}_\textrm{br}} dz e^{-izt} \frac{\sqrt{1 - z^2/4}}{z - z_n}
.
\label{SM.A.NZ.I}
\end{equation}
From this point, we can apply methods similar to those appearing in Apps. C and D of Ref. \cite{GO17} to evaluate this integral; we eventually obtain
\begin{equation}
I (\pm z_g)
= e^{\mp i z_g t} \left[ \mp g - i \int_0^t d\tau e^{i z_g \tau} \frac{J_1 (2 \tau)}{\tau} \right]
,
\label{SM.A.NZ.2}
\end{equation}
in which the first term is a pole contribution associated with the virtual bound states.
In the short/intermediate time region delimited by $\Delta_g t \ll 1$ ($t \ll T_\Delta$), we can approximate the integral as
\begin{equation}
I (\pm z_g)
\approx \pm e^{\mp i z_g t} \left[ 1 - g - e^{\pm 2it} \left( J_0 (2t) \mp i J_1 (2t) \right) \right]
,
\label{SM.A.NZ.3}
\end{equation}
Plugging this result into Eq. (\mathop{\textrm{Re}}f{SM.A.NZ.1}) gives
\begin{equation}a
A_\textrm{br} (t)
& \approx & \frac{1}{g}
\left[ \left( g-1 \right) \cos z_g t + \cos (\Delta_g t) J_0 (2t) \right.
\nonumber \\
& & \left. - \sin (\Delta_g t) J_1 (2t) \right]
.
\label{SM.A.NZ.f}
\end{equation}a
After applying the approximation $\Delta_g t \ll 1$ again we obtain the result in the main text Eq. (12).
As mentioned in the main text, the virtual Rabi oscillation plays a role in facilitating the phase shift from the early near zone (with phase $\pi/4$)
to the far zone (with phase $3 \pi/4$).
For example, at time $t = g/4\pi * T_\Delta \sim T_\Delta/10$ in the near zone evolution the coefficient of the second term in Eq. (13) from the main text has exactly half the magnitude of that of the first term; in this moment, the effective phase of the two combined terms can be shown to be
$\phi_{1/2} = \arctan (\sqrt{2}/(\sqrt{2} -1)) \approx 0.4093 \pi$, which indeed satisfies $\pi/4 < \phi_{1/2} < 3\pi/4$.
\subsection{Asymptotic time region (far zone)}
\label{sec:derivation.FZ}
In the case of the asymptotic time zone $t \gg T_\Delta$, we find it most convenient to evaluate the integral Eq. (\mathop{\textrm{Re}}f{SM.A.br.defn}) using methods similar to those used in Ref. \cite{GPSS13}.
We begin by dragging the contour $\mathcal{C}_\textrm{br}$ surrounding the branch cut out to infinity in the lower half of the complex energy plane. After this, the only non-vanishing portions of the integration are the contours $A_\mp (t)$ running from the two branch points out to infinity in the lower half plane. These portions are written as
\begin{equation}
A_\mp (t)
= \frac{1+g^2}{2 \pi i g^2} \int_{\mp 2}^{\mp 2 - i \infty} dz e^{-izt}
\frac{\sqrt{z^2 - 4}}{\left(z - z_- \right) \left(z - z_+ \right)}
.
\label{A.br.mp}
\end{equation}
Applying an integration variable transform $s \equiv it (z \pm 2)$ yields
\begin{equation}
A_\mp (t)
= \frac{i \left(1+g^2\right) e^{\pm 2it }}{2 \pi g^2 t^2}
\int_{0}^{\infty} ds e^{-s} \frac{\sqrt{s^2 \mp 4ist}}
{\mp \Delta_g \left( 2 + z_g \right) + 4i\frac{s}{t} \mp \frac{s^2}{t^2}}
.
\label{A.br.mp.2}
\end{equation}
For very large $t$ the first term in the denominator is much larger than the other two terms, which can be safely neglected. Performing the remaining simplified integration and combining $A_\pm$ we obtain the result reported for the far zone in Eq. (15) of the main text.
\subsection{Near zone/far zone transition: plots for additional cases}
\label{sec:derivation.plots}
In Fig. \mathop{\textrm{Re}}f{fig:NZ.FZ}(a-c) we plot the survival probability $P_\perp (t)$ for $g=0.9$, similar to the case $g=0.98$ that was presented in Fig. 2(c-e) of the main text; only here we are a bit further away from the localization transition at $g=1$.
We see in Fig. \mathop{\textrm{Re}}f{fig:NZ.FZ}(b) for this case that the early near zone $1/t$ prediction gives only a rough description in terms of the amplitude of $P_\perp (t)$; however, the phase prediction $\cos^2 (2t - \pi/4)$ is still accurate. We can improve our approximation for the amplitude by including the second term from Eq. (13) in the main text, which is shown explicitly as the dashed-dotted curve in Fig. \mathop{\textrm{Re}}f{fig:NZ.close}.
We can estimate the point $T_\textrm{br}$ at which this approximation, too, begins to breakdown as about 10\% of $T_\Delta$. For $g=0.9$ we find this occurs around $t \approx 9$, in rough agreement with Fig. \mathop{\textrm{Re}}f{fig:NZ.close}.
We plot the same in Fig. \mathop{\textrm{Re}}f{fig:NZ.FZ}(d-f) for $g=0.7$, significantly further from the $g=1$ localization transition. In this case, our analytic near zone approximation breaks down, as 10\% of $T_\Delta$ occurs at about $t \approx 0.7$, before the near zone dynamics even emerge. However, we still achieve our primary objective of complete non-exponential decay.
If we keep decreasing the value of $g$, eventually around $g \approx 0.38$ we obtain $T_\Delta \sim 1$. For this and any smaller values of $g$ the near zone is entirely squeezed out and the system will instead transition from the early time parabolic (Zeno) dynamics directly into the $1/t^3$ far zone decay. But again, the evolution is still entirely non-exponential.
We comment that all numerical results in this work were obtained by evolving a chain in the site representation (with up to 16000 elements) according to the Schr\"odinger equation using a variable-order variable-step Adams method.
\begin{figure}
\caption{Closer view of near zone plot from Fig. \mathop{\textrm{Re}
\label{fig:NZ.close}
\end{figure}
\section{Chain-induced effective decoherence}
\label{sec:gen}
Here we briefly consider the evolution of a slightly more general BIC orthogonal state, written as
\begin{equation}
| \psi_w \rangle
= N_w \left( g | d \rangle + | 1 \rangle + w | 2 \rangle \right)
\label{psi.w}
\end{equation}
in which $N_w^2 = (1 + g^2 + w^2)^{-1}$. Here we have included a single site $| 2 \rangle$ from the chain outside of the subspace spanned by the BIC itself, with amplitude $w$.
In Fig. \mathop{\textrm{Re}}f{fig:w} we show how the inclusion of this site modifies the evolution for $g=0.9$. In Fig. \mathop{\textrm{Re}}f{fig:w} (a-c), we see that increasing the value of $w$ in the range $w \le 1$ results in the oscillations we observed in the main text becoming gradually damped out, with near total damping occurring for $w=1$. By contrast, the oscillations return for $w$ values much larger than 1 as shown in Fig. \mathop{\textrm{Re}}f{fig:w} (d).
\begin{figure}
\caption{Numerical simulations for the survival probability for the $| \psi_w \rangle$ state with $g = 0.9$, $\epsilon_\textrm{d}
\label{fig:w}
\end{figure}
Extending from this observation, in Fig. \mathop{\textrm{Re}}f{fig:w_1} we plot numerical simulations for the survival probability of $| \psi_w \rangle$ for $w=1$ over a variety of $g$ values. We observe that the near-total suppression of the oscillations occurs for a wide range of $g$ values in the vicinity of $g=1$.
\begin{figure}
\caption{Numerical simulations for the survival probability for the $| \psi_w \rangle$ state for $w=1$ and $\epsilon_\textrm{d}
\label{fig:w_1}
\end{figure}
We can gain insight into the mechanism of this suppression through the following analytic approximations.
We begin by writing the survival probability for this state
$P_w (t) = |A_w (t)|^2$, in which
\begin{equation}a
A_w (t)
& = & \langle \psi_w | e^{-iHt} | \psi_w \rangle
= \frac{1}{2 \pi i} \int_{\mathcal{C}_E} dz \; e^{-izt} \langle \psi_w | \frac{1}{z - H} | \psi_w \rangle
\nonumber \\
& = & \frac{N_w^2}{2 \pi i}
\int_{\mathcal{C}_E} dz \; e^{-izt} \left( \sigma_1 (z) + Q(z) G_\textrm{dd}(z) \right)
.
\label{A.w.defn}
\end{equation}a
Here we have
\begin{equation}
\sigma_1 (z)
= \frac{z - \sqrt{z^2 - 4}}{2}
,
\label{sigma.1}
\end{equation}
\begin{equation}a
Q (z)
& = & g^2 + \left[ \sigma_1 (z) \right]^2 \left( 2g^2 - 2g^2 wz \right.
\nonumber \\
& & \left. + g^2 \left[ \sigma_1 (z) \right]^2 -2wz +w^2 z^2 \right)
,
\label{Q.z}
\end{equation}a
and $G_\textrm{dd}(z) = \left( z - \epsilon_\textrm{d} - \Sigma (z) \right)^{-1}$ is the resolvent operator at the impurity site, Eq. (3) from the main text.
Focusing on the case $w = 1$ as a quick example, it can be shown that $Q(z)$ simplifies a bit as
\begin{equation}
Q(z)
= \left( 1 + g^2 \right) z \left( z - 2 \right) \left[ \sigma_1 (z) \right]^2
.
\label{Q.z.1}
\end{equation}
Note the useful relation $\sigma_1 (z) + 1/\sigma_1 (z) = z$ has been applied here. Eq. (\mathop{\textrm{Re}}f{Q.z.1}) can in turn be used to simplify the integrand of Eq. (\mathop{\textrm{Re}}f{A.w.defn}) such that we obtain
\begin{equation}
A_w (t)
= \frac{N_w^2}{4 \pi i}
\int_{\mathcal{C}_E} dz \; e^{-izt} \frac{\left( z - g z_g \right)^2 \sqrt{z^2 - 4} }
{\left( z - z_g \right) \left( z + z_g \right)}
.
\label{A.w.1}
\end{equation}
Note the presence of the $(z - g z_g)^2$ factor in the numerator of the integrand. Since the dominant contribution to the integration comes from around the branch points $z= \pm 2$ and since $g z_g = 1 + g^2 \approx 2$ in the vicinity of $g \sim 1$, this factor is very small for the contribution coming from the upper branch cut (if we had chosen $w = -1$, it would instead be the lower branch cut contribution that would be very small). Since there is one overwhelmingly dominant contribution, the oscillations between the two band edges are greatly diminished, which explains the effective decoherence observed in Figs. \mathop{\textrm{Re}}f{fig:w} and \mathop{\textrm{Re}}f{fig:w_1}.
To see this more explicitly, we carry out the integration for the far zone in the $g \neq 1$ case by dragging the integration contour out to infinity in the lower half plane similar to Sec. \mathop{\textrm{Re}}f{sec:derivation.FZ}. Doing so we find there are again the two contributions $A_w (t) = A_{+,w} (t) + A_{-,w} (t)$ in which
\begin{equation}
A_{\mp,w} (t)
= \mp \frac{N_w^2 \left( \mp 2 - g z_g \right)^2 e^{\mp i (2t - \pi/4)} }
{2 \sqrt{\pi} g^2 \left( 2 + z_g \right) \Delta_g t^{3/2} }
.
\label{A.w.2}
\end{equation}
Hence we immediately see the $A_+ (t)$ contribution will indeed be quite small for $g \sim 1$.
The far zone evolution is then approximately described by
\begin{equation}
P_{\textrm{FZ},w} (t)
\approx \frac{\left( 2 + g z_g \right)^4 }
{4 \pi g^4 \left( 2 + g^2 \right)^2 \left( 2 + z_g \right)^2 \Delta_g^2 t^{3} }
,
\label{P.w.FZ}
\end{equation}
which predicts the slope in Fig. \mathop{\textrm{Re}}f{fig:w_1} quite well (red dotted lines), even in the case $g=0.7$ not so near the localization transition at $g=1$. [Compare this result with Eq. (15) in the main text.]
In the special case $g=1$ notice that $ z - gz_g = z - 2$ exactly, which results in the upper band edge contribution vanishing entirely. The timescale
$T_\Delta$ also diverges as the gap $\Delta_g$ closes (just as in the main text), which leads to an asymptotic near zone ($1/t$) description
\begin{equation}
P_w (t) \approx 16/ 9 \pi t
.
\label{P.w.NZ.1}
\end{equation}
This again agrees very well (green dashed line) with the numerical simulation in Fig. \mathop{\textrm{Re}}f{fig:w_1}. [Compare this result with Eq. (14) in the main text.]
\section{Brief comment on model geometry}
\label{sec:geo}
\begin{figure}
\caption{
(a) Original geometry from the main text with a single side-coupled impurity $| d \rangle$; (b) alternative geometry with two impurities, $| d \rangle$ and $| 1 \rangle$. The second model reduces to the first for $g_1 = J$ and $\epsilon_1 = 0$.
}
\label{fig:geo}
\end{figure}
In this work, we have relied on the model depicted in Fig. \mathop{\textrm{Re}}f{fig:geo}(a) to illustrate our idea about populating a BIC-orthogonal state to suppress the exponential decay process. This model consists of a semi-infinite tight-binding chain (with sites $n = 1, 2, ... \infty$) coupled to an `impurity' (discrete) state $|d\rangle$. One might pause when considering that the initial state studied in this work (the BIC-orthogonal state) is a combination of the impurity $|d \rangle$ and an element taken from the chain $| 1 \rangle$, the latter of which is included in the environment portion of the Hamiltonian as it is written in Eq. (1) in the main text.
However, we emphasize here that viewing the $|1\rangle$ state as part of the environment is rather arbitrary, as one could easily imagine a slightly more general model, consisting of a double impurity sector as illustrated in Fig. \mathop{\textrm{Re}}f{fig:geo}(b). Here we have two impurity sites $|1\rangle$ and $|d\rangle$ where $|1\rangle$ now has the generalized energy $\epsilon_1$ and is coupled to the chain with strength $-g_1$. The semi-infinite chain now consists of sites $n = 2, 3 ... \infty$. Clearly the double-impurity model (b) reduces to the original model (a) when $g_1 = J$ and $\epsilon_1 = 0$. This illustrates that, at least from a theoretical perspective, whether site $|1\rangle$ is viewed as part of the impurity sector or as part of the reservoir is arbitrary.
Throughout this work we chose to view the model as depicted in Fig. \mathop{\textrm{Re}}f{fig:geo}(a), in part for convenience and in part because that is how this model has previously been presented in the literature, as a specific case of the models in Refs. \cite{LonghiEPJ07,Fukuta}. However, when performing the experiment proposed in the main text, it might be more natural to adopt the perspective from Fig. \mathop{\textrm{Re}}f{fig:geo}(b). For example, we might view the states $\{ | d \rangle, | 1 \rangle \}$ as two modes of a single waveguide in a potential waveguide array experiment; the experimentalist then achieves the initial state $| \psi_\perp \rangle$ by preparing a coherent superposition of the two modes.
\end{document} |
\begin{document}
\title[Quenched LDP and Parisi formula for GREM Perceptron]{A quenched large deviation principle and a Parisi formula for a perceptron
version of the GREM.}
\author[E. Bolthausen]{Erwin Bolthausen}
\address{E. Bolthausen\\ Institut f\"ur Mathematik \\
Universit\"at Z\"urich \\
Winterthurerstrasse 190 \\
CH-8057 Z\"urich}
\email{eb@math.uzh.ch}
\author[N. Kistler]{Nicola Kistler}
\address{N. Kistler\\Institut f\"ur Angewandte Mathematik\\Rheinische
Friedrich-Wilhelms-Universit\"at Bonn \\Endenicher Allee 60\\ 53115
Bonn,
Germany}
\email{nkistler@uni-bonn.de}
\subjclass[2000]{60J80, 60G70, 82B44} \keywords{Disordered systems, Spin Glasses, Quenched Large Deviation Principles}
\thanks{E. Bolthausen is supported in part by the grant No $200020_125247/1$ of the
Swiss Science Foundation. N. Kistler is partially supported by the German Research Council in the SFB 611 and the Hausdorff Center for Mathematics.}
\begin{abstract}
We introduce a perceptron version of the Generalized Random Energy Model, and
prove a quenched Sanov type large deviation principle for the empirical
distribution of the random energies. The dual of the rate function has a
representation through a variational formula which is closely related to the
Parisi variational formula for the SK-model.
\end{abstract}
\date{\today}
\maketitle
\emph{Dedicated to J\"{u}rgen G\"{a}rtner on the occasion of his 60th birthday.}
\section{Introduction \label{introduction}}
There has been important progress in the mathematical study of mean field spin
glasses over the last $10$ years. By results of Guerra \cite{guerra} and
Talagrand \cite{TalagrandParisi}, the free energy of the
Sherrington-Kirkpatrick model is known to be given by the formula predicted by
Parisi \cite{parisi}. Furthermore, the description of the \textit{high}
temperature is remarkably accurate, see \cite{talagrand} and references
therein. On the other hand, results for the Gibbs measure at \textit{low}
temperature are more scarce and are restricted to models with a simpler
structure, like Derrida's generalized random energy model, the GREM,
\cite{BovierKurkova} and \cite{DerridaGREM}, the nonhierarchical GREMs
\cite{bokis_two} and the $p$-spin model with large $p$ \cite{talagrand}. To
put on rigorous ground the full Parisi picture remains a major challenge, and
even more so in view of its alleged universality, at least for mean-field models.
We introduce here a model which hopefully sheds some new light on the issue.
In this paper we derive the free energy, which can be analyzed by large
deviation techniques. The limiting free energy turns out to be given by a
Gibbs variational formula which can be linked to a Parisi-type formula
by a duality principle, so that it becomes evident why an infimum appears in
the latter. This duality also gives an interesting interpretation of the
Parisi order parameter in terms of the sequence of inverse of temperatures
associated to the extremal measures from the Gibbs variational principle.
In a forthcoming paper, we will give a full description of the Gibbs measure
in the thermodynamic limit in terms of the Ruelle cascades.
\section{A Perceptron version of the GREM\label{Sect_perceptron}}
Let $\left\{ X_{\alpha,i}\right\} _{\alpha\in\Sigma_{N},1\leq i\leq N},$ be
random variables which take values in a Polish space $S$ equipped with the
Borel $\sigma$-field $\mathcal{S},$ and defined on a probability space
$\left( \Omega,\mathcal{F},\mathbb{P}\right) .$ We write $\mathcal{M}
_{1}^{+}\left( S\right) $ for the set of probability measures on $\left(
S,\mathcal{S}\right) ,$ which itself is a Polish space. $\Sigma_{N}$ is
exponential in size, typically $\left\vert \Sigma_{N}\right\vert =2^{N}.$ It
is assumed that all $X_{\alpha,i}$ have the same distribution $\mu$, and that
for any fixed $\alpha\in\Sigma_{N},$ the collection $\left\{ X_{\alpha
,i}\right\} _{1\leq i\leq N}$ is independent. It is however not assumed that
they are independent for different $\alpha.$ The perceptron Hamiltonian is
defined by
\begin{equation}
-H_{N,\omega}\left( \alpha\right) \overset{\mathrm{def}}{=}\sum_{i=1}
^{N}\phi\left( X_{\alpha,i}\left( \omega\right) \right)
,\label{general_perceptron}
\end{equation}
where $\phi:S\rightarrow\mathbb{R}$ is a measurable function. One may allow
that the index set for $i$ is rather $\left\{ 1,\ldots,\left[ aN\right]
\right\} $ with $a$ some positive real number, but for convenience, we always
stick to $a=1$ here. The case which is best investigated (see \cite{talagrand}
) takes for $\alpha$ spin sequences: $\alpha=\left( \sigma_{1},\ldots
,\sigma_{N}\right) \in\left\{ -1,1\right\} ^{N},$ $S=\mathbb{R},$ and the
$X_{\alpha,i}$ are centered Gaussians with
\begin{equation}
\mathbb{E}\left( X_{\alpha,i}X_{\alpha^{\prime},i^{\prime}}\right)
=\delta_{i,i^{\prime}}\frac{1}{N}\sum_{j=1}^{N}\sigma_{j}\sigma_{j}^{\prime
}.\label{SK_perceptron}
\end{equation}
This is closely related to the SK-model, and is actually considerably more
difficult. The model has been investigated by Talagrand \cite{talagrand}, but
a full Parisi formula for the free energy is lacking.
The Hamiltonian (\ref{general_perceptron}) can be written in terms of the
empirical measure
\begin{equation}
L_{N,\alpha}\overset{\mathrm{def}}{=}\frac{1}{N}\sum_{i=1}^{N}\delta
_{X_{\mathbf{\sigma},i}}\label{empirical_Def}
\end{equation}
i.e.
\[
-H_{N,\omega}\left( \alpha\right) =N\int\phi\left( x\right) L_{N,\alpha
}\left( dx\right) .
\]
The quenched free energy is the almost sure limit of
\[
\frac{1}{N}\log\sum_{\alpha}\exp\left[ -H_{N,\omega}\left( \alpha\right)
\right] ,
\]
and it appears natural to ask if this free energy can be obtained by a
quenched Sanov type large deviation principle for $L_{N,\alpha}$ in the
following form:
\begin{definition}
We say that $\left\{ L_{N}\right\} $ satisfies a \textbf{quenched large
deviation principle} (in short QLDP) with good rate function $J:\mathcal{M}
_{1}^{+}\left( S\right) \rightarrow\left[ -\infty,\infty\right) ,$
provided the level sets of $J$ are compact, and for any weakly continuous
bounded map $\Phi:\mathcal{M}_{1}^{+}\left( S\right) \rightarrow\mathbb{R},
$ one has
\[
\lim_{N\rightarrow\infty}\frac{1}{N}\log\sum_{\alpha\in\Sigma_{N}}\exp\left[
N\Phi\left( L_{N,\alpha}\right) \right] =\log2+\sup_{\nu\in\mathcal{M}
_{1}^{+}\left( S\right) }\left[ \Phi\left( \nu\right) -J\left(
\mu\right) \right] ,,\ \mathbb{P}\mathrm{-a.s.}
\]
\end{definition}
The annealed version of such a QLDP is just Sanov's theorem:
\begin{align*}
\lim_{N\rightarrow\infty}\frac{1}{N}\log\sum_{\alpha}\mathbb{E}\exp\left[
N\Phi\left( L_{N,\alpha}\right) \right] & =\log2+\lim_{N\rightarrow\infty
}\frac{1}{N}\log\mathbb{E}\exp\left[ N\Phi\left( L_{N,\alpha}\right)
\right] \\
& =\log2+\sup_{\nu}\left( \Phi\left( \nu\right) -H\left( \nu
\vert
\mu\right) \right)
\end{align*}
where $H\left( \nu
\vert
\mu\right) $ is the usual relative entropy of $\nu$ with respect to $\mu,$
the latter being the distribution of the $X_{\alpha,i}:$
\[
H\left( \nu
\vert
\mu\right) \overset{\mathrm{def}}{=}\left\{
\begin{array}
[c]{cc}
\int\log\frac{d\nu}{d\mu}\ d\nu & \mathrm{if\ }\nu\ll\mu\\
\infty & \mathrm{otherwise}
\end{array}
\right. .
\]
There is no reason to believe that $H\left( \nu
\vert
\mu\right) =J\left( \nu\right) .$
\begin{conjecture}
The empirical measures $\left\{ L_{N,\alpha}\right\} $ with
(\ref{SK_perceptron}) satisfy a QLDP.
\end{conjecture}
We don't know how this conjecture could be proved, nor do we have a clear
picture what $J$ should be in this case. The only support we have for the
conjecture is that it is true in a perceptron version of the GREM, a model we
are now going to describe.
For $n\in{\mathbb{N}}$, $\alpha=(\alpha_{1},\dots,\alpha_{n})$ with
$1\leq\alpha_{k}\leq2^{\gamma_{i}N}$, $\sum_{k}\gamma_{k}=1$, and $1\leq i\leq
N$, let
\[
X_{\alpha,i}=\left( X_{\alpha_{1},i}^{1},X_{\alpha_{1},\alpha_{2},i}
^{2},\dots,X_{\alpha_{1},\alpha_{2},\dots,\alpha_{n},i}^{n}\right)
\]
where the $X^{j}$ are independent, taking values in some Polish Space
$(S,{\mathcal{S}})$ with distribution $\mu_{j}$. For notational convenience,
we assume that the $\gamma_{i}N$ are all integers. Put
\[
\Gamma_{j}\overset{\mathrm{def}}{=}\sum_{k=1}^{j}\gamma_{j}.
\]
We assume that all the variables in the bracket are independent. The
$X_{\alpha,i}$ take values in $S^{n}.$ The distribution is
\[
\mu\overset{\mathrm{def}}{=}\mu_{1}\otimes\cdots\otimes\mu_{n}
\]
The empirical measure $L_{N,\alpha}$ is defined by (\ref{empirical_Def}) which
is a random element in ${\mathcal{M}}_{1}^{+}(S^{n})$. $n$ is fixed in all we
are doing.
Given a measure $\nu\in{\mathcal{M}}_{1}^{+}(S^{n})$, and $1\leq j\leq n,$ we
write $\nu^{(j)}$ for its marginal on the first $j$ coordinates. We define
subsets $\mathcal{R}_{j}$ of ${\mathcal{M}}_{1}^{+}(S^{n})$, $1\leq j\leq n$
by
\[
\mathcal{R}_{j}\overset{\mathrm{def}}{=}\left\{ \nu\in{\mathcal{M}}_{1}
^{+}(S^{n}):H\left( \nu^{(j)}\mid\mu^{(j)}\right) \leq\Gamma_{j}
\log2\right\} .
\]
We will also consider the sets
\[
\mathcal{R}_{j}^{=}\overset{\mathrm{def}}{=}\left\{ \nu\in{\mathcal{M}}
_{1}^{+}(S^{n}):H\left( \nu^{(j)}\mid\mu^{(j)}\right) =\Gamma_{j}
\log2\right\} .
\]
For $\nu\in{\mathcal{M}}_{1}^{+}(S^{n})$ let
\[
J\left( \nu\right) =\left\{
\begin{array}
[c]{cc}
H(\nu\mid\mu) & \mathrm{if\ }\nu\in\bigcap\nolimits_{j=1}^{n}\mathcal{R}_{j}\\
\infty & \mathrm{otherwise}
\end{array}
\right. .
\]
It is evident that $J$ is convex and has compact level sets.
Our first main result is:
\begin{theorem}
\label{Th_GREM_perceptron} $\left\{ L_{N,\alpha}\right\} $ satisfies a QLDP
with rate function $J.$
\end{theorem}
For the rest of this section, we will focus on linear functionals, $\Phi
(\nu)=\int\phi(x)\nu({d}x)$, for a bounded continuous function $\phi
:S^{n}\rightarrow\mathbb{R}.$ For a probability measure $\nu$ on $S^{n}$, we
set
\[
\operatorname{Gibbs}(\phi,\nu)\overset{\mathrm{def}}{=}\int\phi(x)\nu
(dx)-H(\nu\mid\mu),
\]
and define the Legendre transform of $J$ by
\[
J^{\ast}\left( \phi\right) \overset{\mathrm{def}}{=}\sup_{\nu}\left[
\int\phi(x)\nu(dx)-J\left( \nu\right) \right] =\sup\left\{
\operatorname{Gibbs}(\phi,\nu):\nu\in\bigcap\nolimits_{j=1}^{n}\mathcal{R}
_{j}\right\} .
\]
whenever the a.s.-limit exists. As a corollary of Theorem
\ref{Th_GREM_perceptron} we have
\begin{corollary}
\label{Cor_GREMperceptron_linear}Assume that $\phi:S\rightarrow{\mathbb{R}} $
is bounded and continuous.
\[
\lim_{N\rightarrow\infty}\frac{1}{N}\log\sum_{\alpha}\exp\left[
\sum\nolimits_{i=1}^{N}\phi\left( X_{\alpha,i}\right) \right] =J^{\ast
}\left( \phi\right) +\log2,\ \mathrm{a.s.}
\]
\end{corollary}
We next discuss a dual representation of $J^{\ast}\left( \phi\right) $.
Essentially, this comes up by investigating which measures solve the
variational problem. Remark that without the restrictions $\nu\in
\bigcap\nolimits_{j=1}^{n}\mathcal{R}_{j},$ we would simply get
\[
d\nu=\frac{\mathrm{e}^{\phi}d\mu}{\int\mathrm{e}^{\phi}d\mu}
\]
as the maximizer.
Let $\Delta$ be the set of sequences $\mathbf{m}=\left( m_{1},\ldots
,m_{n}\right) $ with $0<m_{1}\leq m_{2}\leq\cdots\leq m_{n}\leq1.$ For
$\mathbf{m}\in\Delta,$ and $\phi:S^{n}\rightarrow\mathbb{R}$ bounded, we
define recursively functions $\phi_{j},~0\leq j\leq n,\ \phi_{j}
:S^{j}\rightarrow\mathbb{R},$ by
\begin{equation}
\phi_{n}\overset{\mathrm{def}}{=}\phi,\label{Def_phin}
\end{equation}
\begin{equation}
\phi_{j-1}\left( x_{1},\ldots,x_{j-1}\right) \overset{\mathrm{def}}{=}
\frac{1}{m_{j}}\log\int\operatorname{exp}\left[ {m}_{j}\phi_{j}\left(
x_{1},\dots,x_{j-1},x_{j}\right) \right] \mu_{j}\left( dx_{j}\right)
.\label{Def_phij}
\end{equation}
$\phi_{0}$ is just a real number, which we denote by $\phi_{0}\left(
\mathbf{m}\right) .$
Remark that if some of the $m_{i}$ agree, say $m_{k}=m_{k+1}=\cdots=m_{l},$
$k<l,$ then $\phi_{k-1}$ is obtained from $\phi_{l}$ by
\[
\phi_{k-1}\left( x_{1},\ldots,x_{k-1}\right) =\frac{1}{m_{k}}\log
\int\operatorname{exp}\left[ {m}_{k}\phi_{l}\left( x_{1},\dots,x_{k-1}
,x_{k},\ldots,x_{l}\right) \right] \prod\limits_{j=k}^{l}\mu_{j}\left(
dx_{j}\right) .
\]
In particular, if all the $m_{i}$ are $1,$ then
\[
\phi_{0}=\log\int\exp\left[ \phi\right] d\mu.
\]
This latter case corresponds to the \textquotedblleft replica
symmetric\textquotedblright\ situation. Put
\begin{equation}
\operatorname{Parisi}\left( \mathbf{m},\phi\right) \overset{\mathrm{def}}
{=}\sum\nolimits_{i=1}^{n}{\frac{\gamma_{i}\log2}{m_{i}}}+\phi_{0}\left(
\mathbf{m}\right) -\log2\label{Parisi}
\end{equation}
\begin{theorem}
\label{Th_Parisi_formula}Assume that $\phi:S\rightarrow{\mathbb{R}}$ is
bounded and continuous. Then
\begin{equation}
J^{\ast}\left( \phi\right) =\inf_{\mathbf{m}\in\Delta}\operatorname{Parisi}
\left( \mathbf{m},\phi\right) .\label{Variationformula3}
\end{equation}
\end{theorem}
The expression for $J^{\ast}\left( \phi\right) $ in this theorem is very
similar to the Parisi formula for the SK-model. Essentially the only
difference is the first summand which in the SK-case is a quadratic
expression. In our case (in contrast to the still open situation in the
SK-model), we can prove that the infimum is uniquely attained, as we will
discuss below.
The derivation of the theorem from Corollary \ref{Cor_GREMperceptron_linear}
is done by identifying first the possible maximizers in the variational
formula for $J^{\ast}\left( \phi\right) $. They belong to a family of
distributions, parametrized by $\mathbf{m}.$ The maximizer inside this family
is then obtained by minimizing $\mathbf{m}$ according to
(\ref{Variationformula3}), and one then identifies the two expressions. The
procedure is quite standard in large deviation situations.
Two conventions: $C$ stands for a generic positive constant, not necessarily
the same at different occurences. If there are inequalities stated between
expressions containing $N,$ it is tacitely assumed that they are valid maybe
only for large enough $N.$
\section{Proofs}
\subsection{The Gibbs variational principle: Proof of Theorem
\ref{Th_GREM_perceptron}}
If $A\in{\mathcal{S}}$, we put $H(A\mid\mu)\overset{\mathrm{def}}
{=}\operatorname{inf}_{\nu\in A}H(\nu\mid\mu)$. If $S$ is a Polish Space, and
${\mathcal{S}}$ its Borel $\sigma$-field, then it is well known that
$\nu\rightarrow H(\nu\mid\mu)$ is lower semicontinuous in the weak topology.
This follows from the representation
\begin{equation}
H(\nu\mid\mu)=\sup_{u\in{\mathcal{U}}}\left[ \int u\,d\nu-\log\int
\mathrm{e}^{u}d\mu\right] ,\label{sup_representation}
\end{equation}
where ${\mathcal{U}}$ is the set of bounded continuous functions
$S\rightarrow{\mathbb{R}}$.
For $(S,{\mathcal{S}}),(S^{\prime},{\mathcal{S}}^{\prime})$ two Polish Spaces,
and $\nu\in{\mathcal{M}}_{1}^{+}(S\times S^{\prime})$. If $\mu\in{\mathcal{M}
}_{1}^{+}(S)$, $\mu^{\prime}\in{\mathcal{M}}_{1}^{+}(S^{\prime})$ we have,
\begin{equation}
H\left( \nu\mid\mu\otimes\mu^{\prime}\right) =H\left( \nu^{(1)}\mid
\mu\right) +H\left( \nu\mid\nu^{(1)}\otimes\mu^{\prime}\right)
,\label{Entropy_Add}
\end{equation}
where $\nu^{(1)}$ is the first marginal of $\nu$ on $S$.
\begin{lemma}
\label{lower_semicontinuity} $H(\nu\mid\nu^{(1)}\otimes\mu^{\prime})$ is a
lower semicontinuous function of $\nu$ in the weak topology.
\end{lemma}
\begin{proof}
Applying (\ref{sup_representation}) to
\[
H(\nu\mid\nu^{(1)}\otimes\mu^{\prime})=\sup_{u\in{\mathcal{U}}}\left[ \int
ud\nu-\log\int\mathrm{e}^{u}d\left( \nu^{(1)}\otimes\mu^{\prime}\right)
\right] ,
\]
where ${\mathcal{U}}$ denotes the set of bounded continuous functions $S\times
S^{\prime}\rightarrow{\mathbb{R}}$. For any fixed $u\in{\mathcal{U}}$, both
functions $\nu\rightarrow\int u\,d\nu$ and $\nu\rightarrow\log\int
\mathrm{e}^{u}d\left( \nu^{(1)}\otimes\mu^{\prime}\right) $ are continuous,
and from this the desired semicontinuity property follows.
\end{proof}
We will need the following \textquotedblleft relative\textquotedblright
\ version of Sanov's theorem. Consider three independent sequences of i.i.d.
random variables $(X_{i}),(Y_{i}),(Z_{i})$, taking values in three Polish
spaces $S,S^{\prime},S^{\prime\prime},$ and with laws $\mu,\mu^{\prime}
,\mu^{\prime\prime}$. We consider the empirical processes
\[
L_{N}\overset{\mathrm{def}}{=}{\frac{1}{N}}\sum_{i=1}^{N}\delta_{(X_{i}
,Y_{i})},\ R_{N}\overset{\mathrm{def}}{=}{\frac{1}{N}}\sum_{i=1}^{N}
\delta_{\left( X_{i},Z_{i}\right) }.
\]
The pair $(L_{N},R_{N})$ takes values in ${\mathcal{M}}_{1}^{+}(S\times
S^{\prime})\times{{\mathcal{M}}}_{1}^{+}(S\times S^{\prime\prime}).$
\begin{lemma}
\label{ldp_empirical_measure_couple} The sequence $(L_{N},R_{N})$ satisfies a
LDP with rate function
\[
J(\nu,\theta)=
\begin{cases}
H\left( \nu^{(1)}\mid\mu\right) +H\left( \nu\mid\nu^{(1)}\otimes\mu
^{\prime}\right) +H\left( \theta\mid\theta^{(1)}\otimes\mu^{\prime\prime
}\right) , & \mathrm{if}\;\nu^{(1)}=\theta^{(1)}\\
\infty & \mathrm{otherwise}.
\end{cases}
\]
\end{lemma}
\begin{proof}
We apply the Sanov theorem to the empirical measure
\[
M_{N}={\frac{1}{N}}\sum_{i=1}^{N}\delta_{(X_{i},Y_{i},Z_{i})}\in{{\mathcal{M}
}}_{1}^{+}(S\times S^{\prime}\times S^{\prime\prime}).
\]
We use the two natural projections $p:S\times S^{\prime}\times S^{\prime
\prime}\rightarrow S\times S^{\prime}$ and $q:S\times S^{\prime}\times
S^{\prime\prime}\rightarrow S\times S^{\prime\prime}$. Then $(L_{N}
,R_{N})=M_{N}(p,q)^{-1}$, and by continuous projection, we get that
$(L_{N},R_{N})$ satisfies a good LDP with rate function
\[
J^{\prime}(\nu,\theta)=\operatorname{inf}\left\{ H(\rho\mid\mu\otimes
\mu^{\prime}\otimes\mu^{\prime\prime}):\rho p^{-1}=\nu,\rho q^{-1}
=\theta\right\} .
\]
It only remains to identify this rate function with the function $J$ given above.
Clearly $J^{\prime}(\nu,\theta)=\infty$ if $\nu^{(1)}\neq\theta^{(1)}$.
Therefore, assume $\nu^{(1)}=\theta^{(1)}$. If we define $\hat{\rho}\left(
\nu,\theta\right) \in\mathcal{M}_{1}^{+}\left( S\times S^{\prime}\times
S^{\prime\prime}\right) $ to have marginal $\nu^{(1)}=\theta^{(1)}$ on $S$,
and the conditional distribution on $S^{\prime}\times S^{\prime\prime}$ given
the first projection is the product of the conditional distributions of $\nu$
and $\theta$, then applying twice (\ref{Entropy_Add}), we get
\[
H(\hat{\rho}\mid\mu\otimes\mu^{\prime}\otimes\mu^{\prime\prime})=H\left(
\nu^{(1)}\mid\mu\right) +H\left( \nu\mid\nu^{(1)}\otimes\mu^{\prime}\right)
+H\left( \theta\mid\theta^{(1)}\otimes\mu^{\prime\prime}\right) ,
\]
and therefore $J\geq J^{\prime}$.
To prove the other inquality, consider any $\rho$ satisfying $\rho p^{-1}
=\nu,\rho q^{-1}=\theta$. We want to show that $J(\nu,\theta)\leq H\left(
\rho\mid\mu\otimes\mu^{\prime}\otimes\mu^{\prime\prime}\right) $. For that,
we can assume that the right hand side is finite. Then
\[
H\left( \rho\mid\mu\otimes\mu^{\prime}\otimes\mu^{\prime\prime}\right)
=H\left( \rho\mid\hat{\rho}\left( \nu,\theta\right) \right) +\int
d\rho\log\frac{d\hat{\rho}\left( \nu,\theta\right) }{d\left( \mu\otimes
\mu^{\prime}\otimes\mu^{\prime\prime}\right) }.
\]
The first summand is $\geq0,$ and the second equals
\[
\int d\hat{\rho}\left( \nu,\theta\right) \log\frac{d\hat{\rho}\left(
\nu,\theta\right) }{d\left( \mu\otimes\mu^{\prime}\otimes\mu^{\prime\prime
}\right) }=J(\nu,\theta).
\]
So, we have proved that
\[
J(\nu,\theta)\leq H\left( \rho\mid\mu\otimes\mu^{\prime}\otimes\mu
^{\prime\prime}\right) ,
\]
for any $\rho$ satisfying $\rho p^{-1}=\nu,\rho q^{-1}=\theta.$
\end{proof}
We now step back to the setting of Theorem \ref{Th_GREM_perceptron}: For
$j=1,\dots,n,$ we have sequences $\left\{ X_{\alpha_{1},\dots,\alpha_{j}
,i}^{j}\right\} $ of independent random variables with distribution $\mu_{j}
$. We emphasize that henceforth $\mu=\mu_{1}\otimes\cdots\otimes\mu_{n}$ and
$\mu^{(j)}$ will denote the marginal on the first $k$ components. Moreover,
for $\alpha=(\alpha_{1},\dots,\alpha_{n})$, we write $\alpha^{(j)}=(\alpha
_{1},\dots,\alpha_{j})$ and set
\[
L_{N,\alpha^{(j)}}^{(j)}={\frac{1}{N}}\sum_{i=1}^{N}\delta_{\left(
X_{\alpha_{1},i}^{1},X_{\alpha_{1},\alpha_{2},i}^{2},\dots,X_{\alpha_{1}
,\dots,\alpha_{j},i}^{j}\right) },
\]
for $j\leq n$, which is the marginal of $L_{N,\alpha}$ on $S^{j}$. With the
notation
\begin{align*}
& X_{\alpha,i}^{(j)}\overset{\mathrm{def}}{=}\left( X_{\alpha_{1},i}
^{1},\dots,X_{\alpha_{1},\dots,\alpha_{j},i}^{j}\right) ,\\
& \hat{X}_{\alpha,i}^{(j)}\overset{\mathrm{def}}{=}\left( X_{\alpha_{1}
,\dots,\alpha_{j+1},i}^{j+1},\dots,X_{\alpha_{1},\dots,\alpha_{n},i}
^{n}\right) ,
\end{align*}
we can write
\begin{equation}
L_{N,\alpha}\overset{\mathrm{def}}{=}{\frac{1}{N}}\sum_{i=1}^{N}
\delta_{\left( X_{\alpha,i}^{(j)},\hat{X}_{\alpha,i}^{(j)}\right)
}.\label{splitting_empirical}
\end{equation}
For $A\subset{{\mathcal{M}}}_{1}^{+}(S^{n})$ we put $M_{N}(A)\overset
{\mathrm{def}}{=}\#\left\{ \alpha:L_{N,\alpha}\in A\right\} $.
\begin{lemma}
\label{very_many} Assume $\nu\in{\mathcal{M}}_{1}^{+}(S^{n})$ satisfies
$H(\nu\mid\mu)<\infty$, and let $V$ be an open neighborhood of $\nu$, and
$\varepsilon>0$. Then there exists an open neighborhood $U$ of $\nu$,
$U\subset V$, and $\delta>0$ such that
\[
{\mathbb{P}}\Big [M_{N}(U)\geq\operatorname{exp}\left[ N\left( \log
2-H(\nu\mid\mu)+\varepsilon\right) \right] \Big ]\leq\mathrm{e}^{-\delta N}.
\]
\end{lemma}
\begin{proof}
If $B_{r}(\nu)$ denotes the open $r$-ball around $\nu$ in one of the standard
metrics, e.g. the Prohorov metric, then by the semicontinuity property of the
relative entropy, on has
\[
H(B_{r}(\nu)\mid\mu)\uparrow H(\nu\mid\mu)
\]
as $r\downarrow0.$ We can choose a sequence $r_{k}>0,r_{k}\downarrow0$ with
$H(B_{r_{k}}(\nu)\mid\mu)=H(\operatorname{cl}\left( B_{r_{k}}(\nu)\right)
\mid\mu)\uparrow H(\nu\mid\mu)$. Given $\varepsilon>0,$ and $V,$ we can find
$k$ such that
\[
H(B_{r_{k}}(\nu)\mid\mu)=H(\operatorname{cl}\left( B_{r_{k}}(\nu)\right)
\mid\mu)\geq H(\nu\mid\mu)-\varepsilon/4
\]
and $B_{r_{k}}(\nu)\subset V.$ By Sanov's theorem we therefore get
\[
{\mathbb{P}}\Big [L_{N,\alpha}\in B_{r_{k}}(\nu)\Big ]\leq\operatorname{exp}
\left[ N(-H(\nu\mid\mu)+\varepsilon/2)\right] ,
\]
and therefore
\[
{{\mathbb{E}}}\Big [M_{N}\left( B_{r_{k}}(\nu)\right) \Big ]\leq
\operatorname{exp}\left[ N(\log2-H(\nu\mid\mu)+\varepsilon/2)\right] .
\]
By the Markov inequality, the claim follows by taking $\delta=\varepsilon/3.$
\end{proof}
\begin{lemma}
\label{still_existent} Assume $\nu\in{\mathcal{M}}_{1}^{+}(S^{n})$ satisfies
$H\left( \nu^{(j)}\mid\mu^{(j)}\right) >\Gamma_{j}\log2$ for some $j\leq n$,
and let $V$ be an open neighborhood of $\nu$. Then there is an open
neighborhood $U$ of $\nu$, $U\subset V$ and $\delta>0$ such that
\[
{\mathbb{P}}\big [M_{N}(U)\neq0\big ]\leq\mathrm{e}^{-\delta N}
\]
for large enough $N$.
\end{lemma}
\begin{proof}
As in the previous lemma, we choose a neighborhood $U^{\prime}$ of $\nu^{(j)}$
in $S^{j}$ such that $H(\operatorname{cl}\left( U^{\prime}\right) \mid
\mu^{(j)})=H(U^{\prime}\mid\mu^{(j)})>\Gamma_{j}\log2+\eta,$ for some
$\eta>0.$ Then we put
\[
U\overset{\mathrm{def}}{=}\left\{ \nu\in{\mathcal{M}}_{1}^{+}(S^{n}):\nu\in
V,\nu^{(j)}\in U^{\prime}\right\} .
\]
If $L_{N,\alpha}\in U$ then $L_{N,\alpha}^{(j)}\in U^{\prime}$,
\begin{align*}
{\mathbb{P}}\left[ \exists\alpha:L_{N,\alpha}\in U\right] & \leq
{\mathbb{P}}\left[ \exists\alpha:L_{N,\alpha}^{(j)}\in U^{\prime}\right] \\
& \leq2^{\Gamma_{j}N}{\mathbb{P}}\left[ L_{N,\alpha}^{(j)}\in U^{\prime
}\right] \\
& \leq2^{\Gamma_{j}N}\operatorname{exp}\left[ -NH\left( \operatorname{cl}
\left( U^{\prime}\right) \mid\mu^{(j)}\right) +N\eta/2\right] \\
& \leq2^{\Gamma_{j}N}\operatorname{exp}\left[ -N\Gamma_{j}\log2-N\eta
/2\right] =\mathrm{e}^{-N\eta/2}.
\end{align*}
This proves the claim.
\end{proof}
\begin{lemma}
\label{second_mom} Assume that $\nu\in{\mathcal{M}}_{1}^{+}(S^{n})$ satisfies
$H\left( \nu^{(j)}\mid\mu^{(j)}\right) <\Gamma_{j}\log2$ for all $j$, and
let $V$ be an open neighborhood of $\nu$, and $\varepsilon>0$. Then there
exists an open neighborhood $U$ of $\nu$, $U\subset V$, and a $\delta>0$ such
that
\[
{\mathbb{P}}\Big [M_{N}(U)\leq\operatorname{exp}\left[ N\left( \log
2-H(\nu\mid\mu)-\varepsilon\right) \right] \Big ]\leq\mathrm{e}^{-\delta N}.
\]
\end{lemma}
\begin{proof}
We claim that we can find $U$ as required, and some $\delta>0,$ such that
\begin{equation}
{\operatorname{var}}\left[ M_{N}(U)\right] \leq\mathrm{e}^{-2N\delta
}\left\{ {{\mathbb{E}}}\left[ M_{N}(U)\right] \right\} ^{2}
\label{less_evident_sanov}
\end{equation}
From this estimate, we easily get the claim: From Sanov's theorem, we have for
any $\chi>0$
\begin{equation}
\mathbb{E}M_{N}(U)=2^{N}\mathbb{P}\left( L_{N,\alpha}\in U\right) \geq
\exp\left[ N\left( \log2-H(\nu\mid\mu)-\chi\right) \right]
.\label{Est_Exp_below}
\end{equation}
Using this, we get by taking $\chi=\varepsilon/2$
\begin{align*}
& {\mathbb{P}}\left( M_{N}(U)\leq\mathrm{e}^{N\left( \log2-H(\nu\mid
\mu)-\varepsilon\right) }\right) \\
& ={\mathbb{P}}\left( M_{N}(U)-\mathbb{E}M_{N}(U)\leq\mathrm{e}
^{-N\varepsilon/2}\mathrm{e}^{N\left( \log2-H(\nu\mid\mu)-\varepsilon
/2\right) }-\mathbb{E}M_{N}(U)\right) \\
& \leq{\mathbb{P}}\left( M_{N}(U)-\mathbb{E}M_{N}(U)\leq\left(
\mathrm{e}^{-N\varepsilon/2}-1\right) \mathbb{E}M_{N}(U)\right) \\
& \leq{\mathbb{P}}\left( M_{N}(U)-\mathbb{E}M_{N}(U)\leq-\frac{1}
{2}\mathbb{E}M_{N}(U)\right) \\
& \leq{\mathbb{P}}\left( \left\vert M_{N}(U)-\mathbb{E}M_{N}(U)\right\vert
\geq\frac{1}{2}\mathbb{E}M_{N}(U)\right) \\
& \leq4\frac{{\operatorname{var}}\left[ M_{N}(U)\right] }{\left\{
\mathbb{E}M_{N}(U)\right\} ^{2}}\leq4\mathrm{e}^{-2N\delta}\leq
\mathrm{e}^{-\delta N}.
\end{align*}
So it remains to prove (\ref{less_evident_sanov}). We first claim that for any
$j$
\begin{align}
& \lim_{r\rightarrow0}\operatorname{inf}_{\rho,\theta\in{\operatorname{cl}
}B_{r}(\nu):\rho^{(j)}=\theta^{(j)}}\left\{ H(\rho\mid\mu)+H\left(
\theta\mid\theta^{(j)}\otimes\hat{\mu}^{(j)}\right) \right\}
\label{lim_balls}\\
& =H(\nu\mid\mu)+H\left( \nu\mid\nu^{(j)}\otimes\hat{\mu}^{(j)}\right)
,\nonumber
\end{align}
where $\hat{\mu}^{(j)}\overset{\mathrm{def}}{=}\mu_{j+1}\otimes\cdots
\otimes\mu_{n}$. The inequality $\leq$ is evident by taking $\rho=\theta=\nu$,
and the opposite follows from the semicontinuity properties: One gets that for
a sequence $(\rho_{n},\theta_{n})$ with $\rho_{n}^{(j)}=\theta_{n}^{(j)}$ and
$\rho_{n},\theta_{n}\rightarrow\nu$, we have
\begin{align*}
\liminf_{n\rightarrow\infty}H\left( \rho_{n}\mid\mu\right) & \geq H(\nu
\mid\mu),\\
\liminf_{n\rightarrow\infty}H\left( \theta_{n}\mid\theta_{n}^{(j)}\otimes
\hat{\mu}^{(j)}\right) & \geq H\left( \nu\mid\nu^{(j)}\otimes\hat{\mu
}^{(j)}\right) ,
\end{align*}
the first inequality by the standard semi-continuity, and the second by Lemma
\ref{lower_semicontinuity}. This proves (\ref{lim_balls}).
Choose $\eta>0$ such that $H\left( \nu^{(j)}\mid\mu^{(j)}\right) <\Gamma
_{j}\log2-\eta$, for all $1\leq j\leq n$. By (\ref{lim_balls}) we may choose
$r$ small enough such that ${\operatorname{cl}}B_{r}(\nu)\subset V,$ and for
all $1\leq j\leq n$,
\begin{align*}
& \operatorname{inf}_{\rho,\theta\in{\operatorname{cl}}B_{r}(\nu):\rho
^{(j)}=\theta^{(j)}}\left\{ H(\rho\mid\mu)+H\left( \theta\mid\theta
^{(j)}\otimes\hat{\mu}^{(j)}\right) \right\} \\
& \geq H(\nu\mid\mu)+H\left( \nu\mid\nu^{(j)}\otimes\hat{\mu}^{(j)}\right)
-\eta/2\\
& =2H(\nu\mid\mu)-H\left( \nu^{(j)}\mid\mu^{(j)}\right) -\eta/2\\
& \geq2H(\nu\mid\mu)-\Gamma_{j}\log2+{\eta/2}.
\end{align*}
For two indices $\alpha,\alpha^{\prime}$ we write $q(\alpha,\alpha^{\prime
})\overset{\mathrm{def}}{=}\max\left\{ j:\alpha^{(j)}=\alpha^{\prime
(j)}\right\} $ with $\max\emptyset\overset{\mathrm{def}}{=}0$. Then
\begin{align*}
{{\mathbb{E}}}{M_{N}^{2}(U)} & =\sum_{j=0}^{n}\sum_{\alpha,\alpha^{\prime
}:q(\alpha,\alpha^{\prime})=j}{\mathbb{P}}\left[ L_{N,\alpha}\in
U,L_{N,\alpha^{\prime}}\in U\right] \\
& =\sum_{\alpha,\alpha^{\prime}:q(\alpha,\alpha^{\prime})=0}{\mathbb{P}
}\left[ L_{N,\alpha}\in U\right] {\mathbb{P}}\left[ L_{N,\alpha^{\prime}
}\in U\right] \\
& +\sum_{j=1}^{n}\sum_{\alpha,\alpha^{\prime}:q(\alpha,\alpha^{\prime}
)=j}{\mathbb{P}}\left[ L_{N,\alpha}\in U,L_{N,\alpha^{\prime}}\in U\right] \\
& \leq{{\mathbb{E}}}[M_{N}({\operatorname{cl}}U)]^{2}+\\
& +\sum_{j=1}^{n}\sum_{\alpha,\alpha^{\prime}:q(\alpha,\alpha^{\prime}
)=j}{\mathbb{P}}\left[ L_{N,\alpha}\in{\operatorname{cl}}U,L_{N,\alpha
^{\prime}}\in{\operatorname{cl}}U\right] .
\end{align*}
We write the empirical measure in the form (\ref{splitting_empirical}), and
use Lemma \ref{ldp_empirical_measure_couple}. For any $1\leq j\leq n$ we have
\begin{align*}
& \sum_{\alpha,\alpha^{\prime}:q(\alpha,\alpha^{\prime})=j}{\mathbb{P}}\left[
L_{N,\alpha}\in\operatorname{cl}U,L_{N,\alpha^{\prime}}\in\operatorname{cl}
U\right] \\
& =2^{\Gamma_{j}N}2^{(1-\Gamma_{j})N}\left( 2^{(1-\Gamma_{j})N}-1\right)
{\mathbb{P}}\left[ L_{N,\alpha}\in\operatorname{cl}U,L_{N,\alpha^{\prime}}
\in\operatorname{cl}U\right] ,
\end{align*}
where on the right hand side $\alpha,\alpha^{\prime}$ is an arbitrary pair
with $q(\alpha,\alpha^{\prime})=j$. Using Lemma
\ref{ldp_empirical_measure_couple} we have
\begin{align*}
& {\mathbb{P}}\left[ L_{N,\alpha}\in\operatorname{cl}U,\;L_{N,\alpha}
\in\operatorname{cl}U\right] \\
& \leq\operatorname{exp}\Bigg [-N\operatorname{inf}_{\rho,\theta
\in\operatorname{cl}U,\rho^{(j)}=\theta^{(j)}}\Big \{H\left( \rho^{(j)}
\mid\mu^{(j)}\right) +\\
& +H\left( \rho\mid\rho^{(j)}\otimes\hat{\mu}^{(j)}\right) +H\left(
\theta\mid\theta^{(j)}\otimes\hat{\mu}^{(j)}\right) \Big \}+{\frac{N\eta}{4}
}\Bigg ]\\
& =\operatorname{exp}\left[ -N\operatorname{inf}_{\rho,\theta\in
\operatorname{cl}U,\rho^{(j)}=\theta^{(j)}}\left\{ H(\rho\mid\mu)+H\left(
\theta\mid\theta^{(j)}\otimes\hat{\mu}^{(j)}\right) \right\} +{\frac{N\eta
}{4}}\right] \\
& \leq2^{\Gamma_{j}N}\operatorname{exp}\left[ -2NH(\nu\mid\mu)-{\frac{N\eta
}{4}}\right] ,
\end{align*}
and thus
\[
\sum_{\alpha,\alpha^{\prime}:q(\alpha,\alpha^{\prime})=j}{\mathbb{P}}\left[
L_{N,\alpha}\in\operatorname{cl}U,\;L_{N,\alpha}\in\operatorname{cl}U\right]
\leq2^{2N}\operatorname{exp}\left[ -2NH(\nu\mid\mu)-{\frac{N\eta}{4}}\right]
.
\]
Combining, we obtain by taking $\chi=\eta/16$ in (\ref{Est_Exp_below})
\[
\operatorname{var}\left[ M_{N}(U)\right] \leq2^{2N}\operatorname{exp}\left[
-2NH(\nu\mid\mu)-{\frac{N\eta}{4}}\right] \leq\mathrm{e}^{-N\eta
/8}{{\mathbb{E}}}[M_{N}(U)]^{2},
\]
which proves our claim.
\end{proof}
\begin{proof}
[Proof of Theorem \ref{Th_GREM_perceptron}]We set
\[
{\mathcal{G}}\overset{\mathrm{def}}{=}\left\{ \nu\in{\mathcal{M}}_{1}
^{+}(S^{n}):H\left( \nu^{(j)}\mid\mu^{(j)}\right) \leq\Gamma_{j}
\log2,\ j=1,\dots,n\right\} ,
\]
which is a compact set.
\textit{Step 1.} We first prove the lower bound. By compactness of
${\mathcal{G}}$ and the semicontinuity of $H$ there exists $\nu_{0}
\in{\mathcal{G}}$ such that
\[
\sup_{\nu\in{\mathcal{G}}}\left\{ \Phi(\nu)-H(\nu\mid\mu)\right\} =\Phi
(\nu_{0})-H(\nu_{0}\mid\mu).
\]
We set $\nu_{\lambda}\overset{\mathrm{def}}{=}(1-\lambda)\nu_{0}+\lambda\mu$
for $0<\lambda<1$. By convexity of $H(\nu\mid\mu)$ in $\nu$ we see that
$H\left( \nu_{\lambda}^{(j)}\mid\mu^{(j)}\right) <\Gamma_{j}\log2$ for all
$1\leq j\leq n$. Furthermore $\nu_{\lambda}\rightarrow\nu_{0}$ weakly as
$\lambda\rightarrow0$, and $\Phi(\nu_{\lambda})\rightarrow\Phi(\nu_{0})$,
$H(\nu_{\lambda}\mid\mu)\rightarrow H(\nu_{0}\mid\mu)$.
Given $\varepsilon>0$ we choose $\lambda>0$ such that
\[
\Phi(\nu_{\lambda})-H(\nu_{\lambda}\mid\mu)\geq\Phi(\nu_{0})-H(\nu_{0}\mid
\mu)-\varepsilon.
\]
By the continuity of $\Phi$ and Lemma \ref{second_mom} we find a neighborhood
$U$ of $\nu_{\lambda}$, and $\delta>0$ such that
\[
\Phi(\theta)-\Phi(\nu_{\lambda})\leq\varepsilon,\ \theta\in U,
\]
and
\[
{\mathbb{P}}\left[ M_{N}(U)\leq2^{N}\operatorname{exp}\left[ -NH(\nu
_{\lambda}\mid\mu)-N\varepsilon\right] \right] \leq\mathrm{e}^{-\delta N},
\]
Then, with probability greater than $1-\mathrm{e}^{-\delta N}$,
\begin{align*}
Z_{N} & =2^{-N}\sum_{\alpha}\operatorname{exp}\left[ N\Phi(L_{N,\alpha
})\right] \\
& \geq2^{-N}\sum_{\alpha:L_{N,\alpha}\in U}\operatorname{exp}\left[
N\Phi(L_{N,\alpha})\right] \\
& \geq\operatorname{exp}\left[ N\Phi(\nu_{\lambda})-N\varepsilon\right]
\operatorname{exp}\left[ -NH(\nu_{\lambda}\mid\mu)-N\varepsilon\right] \\
& \geq\operatorname{exp}\left[ N\sup_{\nu\in{\mathcal{G}}}\left\{ \Phi
(\nu)-H(\nu\mid\mu)\right\} -3N\varepsilon\right] .
\end{align*}
By Borel-Cantelli, we therefore get, as $\varepsilon$ is arbitrary,
\[
\liminf_{N\rightarrow\infty}{\frac{1}{N}}\log Z_{N}\geq\sup_{\nu
\in{\mathcal{G}}}\left\{ \Phi(\nu)-H(\nu\mid\mu)\right\}
\]
almost surely.
\textit{Step 2.} We prove the upper bound. Let again $\varepsilon>0$ and set
\[
\overline{{\mathcal{G}}}\overset{\mathrm{def}}{=}\{\nu:H(\nu\mid\mu)\leq
\log2\}.
\]
If $\nu\in{\mathcal{G}}$ we choose $r_{\nu}>0$ such that $\left\vert
\Phi(\theta)-\Phi(\nu)\right\vert \leq\varepsilon$, $\theta\in B_{r_{\nu}}
(\nu)$ and
\[
{\mathbb{P}}\left[ M_{N}(B_{r_{\nu}}(\nu))\geq2^{N}\operatorname{exp}\left[
-NH(\nu\mid\mu)+N\varepsilon\right] \right] \leq\mathrm{e}^{-N\delta_{\nu}},
\]
for some $\delta_{\nu}>0$ and large enough $N$ (using Lemma \ref{very_many}).
If $\nu\in\overline{{\mathcal{G}}}\setminus{\mathcal{G}}$ we choose $r_{\nu}$
such that $\left\vert \Phi(\theta)-\Phi(\nu)\right\vert \leq\varepsilon$,
$\theta\in B_{r_{\nu}}(\nu)$, and
\begin{equation}
{\mathbb{P}}\left[ M_{N}(B_{r_{\nu}}(\nu))\neq0\right] \leq\mathrm{e}
^{-N\delta_{\nu}},\label{still_existent_two}
\end{equation}
again for large enough $N$ (and by Lemma \ref{still_existent}). As
$\overline{{\mathcal{G}}}$ is compact, we can cover it by a finite union of
such balls, i.e.
\[
\overline{{\mathcal{G}}}\subset U\overset{\mathrm{def}}{=}\bigcup_{j=1}
^{m}B_{r_{j}}(\nu_{j}),
\]
where $r_{j}\overset{\mathrm{def}}{=}r_{\nu_{j}}$. We also set $\delta
\overset{\mathrm{def}}{=}\min_{j}\delta_{\nu_{j}}$. We then estimate
\begin{equation}
Z_{N}\leq2^{-N}\sum_{l=1}^{m}\sum_{\alpha:L_{N,\alpha}\in B_{r_{l}}(\nu_{l}
)}\operatorname{exp}\left[ N\Phi(L_{N,\alpha})\right] +2^{-N}\sum
_{\alpha:L_{N,\alpha}\notin U}\operatorname{exp}\left[ N\Phi(L_{N,\alpha
})\right] .\label{upper_bound_abstract}
\end{equation}
we first claim that almost surely the second summand vanishes provided $N$ is
large enough, i.e. that there is no $\alpha$ with $L_{N,\alpha}\notin U$. By
Sanov's theorem, we have
\[
\limsup_{N\rightarrow\infty}{\frac{1}{N}}\log{\mathbb{P}}\left[ L_{N,\alpha
}\notin U\right] \leq-\operatorname{inf}_{\nu\notin U}H(\nu\mid\mu)<-\log2.
\]
Therefore, almost surely, there is no $\alpha$ with $L_{N,\alpha}\notin U$,
and therefore the second summand in (\ref{upper_bound_abstract}) vanishes for
large enough $N$, almost surely. The same applies to those summands in the
first part for which $\nu_{l}\notin{\mathcal{G}}$, using
(\ref{still_existent_two}). We therefore have, almost surely, for large enough
$N$,
\begin{align*}
Z_{N} & \leq2^{-N}\sum_{l:\nu_{l}\in{\mathcal{G}}}\sum_{\alpha:L_{N,\alpha
}\in B_{r_{l}}(\nu_{l})}\operatorname{exp}\left[ N\Phi(L_{N,\alpha})\right]
\\
& \leq\mathrm{e}^{N\varepsilon}\sum_{l:\nu_{l}\in{\mathcal{G}}}
\operatorname{exp}\left[ N\Phi(\nu_{l})\right] M_{N}(B_{r_{l}}(\nu_{l}))\\
& \leq\mathrm{e}^{2N\varepsilon}\sum_{l:\nu_{l}\in{\mathcal{G}}}
\operatorname{exp}\left[ N\Phi(\nu_{l})\right] \operatorname{exp}\left[
-NH(\nu_{l}\mid\mu)\right] \\
& \leq\mathrm{e}^{2N\varepsilon}m\operatorname{exp}\left[ N\sup_{\nu
\in{\mathcal{G}}}\left\{ \Phi(\nu)-H(\nu\mid\mu)\right\} \right] .
\end{align*}
As $\varepsilon$ is arbitrary, we get
\[
\limsup_{N\rightarrow\infty}{\frac{1}{N}}\log Z_{N}\leq\sup_{\nu
\in{\mathcal{G}}}\left[ \Phi(\nu)-H(\nu\mid\mu)\right] .
\]
This finishes the proof of Theorem \ref{Th_GREM_perceptron}.
\end{proof}
\subsection{The dual representation. Proof of the Theorem
\ref{Th_Parisi_formula}}
We define a family $\mathcal{G}\left( \phi\right) =\left\{ G_{\phi
,\mathbf{m}}\right\} $ of probability distributions on $S^{n}$ which depend
on the parameter $\mathbf{m}=\left( m_{1},\ldots,m_{n}\right) \in\Delta.$
The probability measure $G=G_{\phi,\mathbf{m}}$ is described by a
\textquotedblleft starting\textquotedblright\ measure $\gamma$ on $S,$ and for
$2\leq j\leq n$ Markov kernels $K_{j}$ from $S^{j-1}$ to $S,$ so that $G$ is
the semi-direct product
\[
G=\gamma\otimes K_{2}\otimes\cdots\otimes K_{n}.
\]
\[
\gamma\left( dx\right) \overset{\mathrm{def}}{=}\frac{\exp\left[ m_{1}
\phi_{1}\left( x\right) \right] \mu_{1}\left( dx\right) }{\exp\left[
m_{1}\phi_{0}\right] },
\]
\[
K_{j}\left( \mathbf{x}^{\left( j-1\right) },dx_{j}\right) \overset
{\mathrm{def}}{=}\frac{\exp\left[ m_{j}\phi_{j}\left( \mathbf{x}^{\left(
j\right) }\right) \right] \mu_{j}\left( dx_{j}\right) }{\exp\left[
m_{j}\phi_{j-1}\left( \mathbf{x}^{\left( j-1\right) }\right) \right] },
\]
where we write $\mathbf{x}^{\left( j\right) }\overset{\mathrm{def}}
{=}\left( x_{j},\ldots,x_{j}\right) .$ Remember the definition of the
function $\phi_{j}:S^{j}\rightarrow\mathbb{R}$ in (\ref{Def_phin}),
(\ref{Def_phij}). It should be remarked that these objects are defined for all
$\mathbf{m}\in\mathbb{R}^{n}$, and not just for $\mathbf{m}\in\Delta.$ We also
write
\[
G^{\left( j\right) }\overset{\mathrm{def}}{=}\gamma\otimes K_{2}
\otimes\cdots\otimes K_{j}
\]
which is the marginal of $G$ on $S^{j}.$ In order to emphasize the dependence
on $\mathbf{m},$ we occasionally will write $\phi_{j,\mathbf{m}}
,\ \gamma_{\mathbf{m}},\ K_{j,\mathbf{m}}$ etc.
We remark that by a simple computation
\begin{gather}
\int H\left( K_{j}\left( \mathbf{x}^{\left( j-1\right) },\cdot\right)
\mid\mu_{j}\right) G^{\left( j-1\right) }\left( d\mathbf{x}^{\left(
j-1\right) }\right) \label{PAR1}\\
=m_{j}\left[ \int\phi_{j}dG^{\left( j\right) }-\int\phi_{j-1}dG^{\left(
j-1\right) }\right] .\nonumber
\end{gather}
$\phi_{j},\ldots,\phi_{n}$ do not depend on $m_{j},$ but $\phi_{0},\ldots
,\phi_{j-1}$ do. Differentiating the equation
\[
\mathrm{e}^{m_{r+1}\phi_{r}}=\int\mathrm{e}^{m_{r+1}\phi_{r+1}}d\mu_{r+1}
\]
with respect to $m_{j},$ we get for $0\leq r\leq j-2$
\begin{equation}
\frac{\partial\phi_{r}\left( \mathbf{x}^{\left( r\right) }\right)
}{\partial m_{j}}=\int\frac{\partial\phi_{r+1}\left( \mathbf{x}^{\left(
r\right) },x_{r+1}\right) }{\partial m_{j}}K_{r+1}\left( d\mathbf{x}
^{\left( r\right) },x_{r+1}\right) ,\label{PAR2}
\end{equation}
and for $r=j-1$
\[
\phi_{j-1}\mathrm{e}^{m_{j}\phi_{j}}+m_{j}\frac{\partial\phi_{j-1}}{\partial
m_{j}}\mathrm{e}^{m_{j}\phi_{j}}=\int\phi_{j}\mathrm{e}^{m_{j}\phi_{j}}
d\mu_{j},
\]
i.e.
\[
\frac{\partial\phi_{j-1}}{\partial m_{j}}\left( \mathbf{x}^{\left( j\right)
}\right) =\frac{1}{m_{j}}\left[ \int\phi_{j}\left( \mathbf{x}^{\left(
j-1\right) },x_{j}\right) K_{j}\left( \mathbf{x}^{\left( j-1\right)
},dx_{j}\right) -\phi_{j-1}\left( \mathbf{x}^{\left( j-1\right) }\right)
\right] .
\]
Combining that with (\ref{PAR1}), (\ref{PAR2}) we get
\begin{align}
\frac{\partial\phi_{0,\mathbf{m}}}{\partial m_{j}} & =\frac{1}{m_{j}}\left[
\int\phi_{j}dG^{\left( j\right) }-\int\phi_{j-1}dG^{\left( j-1\right)
}\right] \label{PAR3}\\
& =\frac{1}{m_{j}^{2}}\int H\left( K_{j}\left( \mathbf{x}^{\left(
j-1\right) },\cdot\right) \mid\mu_{j}\right) G^{\left( j-1\right)
}\left( d\mathbf{x}^{\left( j-1\right) }\right) .\nonumber
\end{align}
Theorem \ref{Th_Parisi_formula} is immediate from the following result:
\begin{proposition}
\label{Th_Gibbs_maximizer}Assume that $\phi:S^{n}\rightarrow{\mathbb{R}}$ is
bounded and continuous. Then there is a unique measure $\nu$ maximizing
$\operatorname{Gibbs}\left( \nu,\phi\right) $ under the constraint $\nu
\in\bigcap\nolimits_{j=1}^{n}\mathcal{R}_{j}.$ This measure is of the form
$\nu=G_{\phi,\mathbf{m}}$ where $\mathbf{m}$ is the unique element in $\Delta$
minimizing (\ref{Variationformula3}). For this $\mathbf{m},$ we have
\begin{equation}
\operatorname{Gibbs}\left( G,\phi\right) =\operatorname{Parisi}\left(
\phi,\mathbf{m}\right) .\label{Gibbs=Parisi}
\end{equation}
\end{proposition}
\begin{proof}
From strict convexity of the relative entropy, and the fact that
$\bigcap\nolimits_{j=1}^{n}\mathcal{R}_{j}$ is compact and convex, it follows
that there is a unique maximizer $\nu$ of $\operatorname{Gibbs}\left(
\nu,\phi\right) $ under this constraint.
Also, a straightforward application of H\"{o}lder's inequality shows that
$\operatorname{Parisi}\left( \phi,\mathbf{m}\right) $ is a strictly convex
function in the variables $1/m_{j}.$ Therefore, it follows that there is a
uniquely attained minimum of $\operatorname{Parisi}\left( \phi,\mathbf{m}
\right) $ as a function of $\mathbf{m}\in\Delta.$ This minimizing
$\mathbf{m}=\left( m_{1},\ldots,m_{n}\right) $, we can be split into
subblocks of equal values: There is a number $K,\ 0\leq K\leq n,$ and indices
$0<j_{1}<j_{2}<\cdots<j_{K}\leq n$ such that
\begin{align*}
0 & <m_{1}=\cdots=m_{j_{1}}<m_{j_{1}+1}=\cdots=m_{j_{2}}\\
& <m_{j_{2}+1}\cdots<m_{j_{K-1}+1}=\cdots=m_{j_{K}}\\
& <m_{j_{K}+1}=\cdots m_{n}=1.
\end{align*}
$K=0$ just means that all $m_{i}=1.$ If $j_{K}=n,$ then all $m_{i}$ are $<1.$
We write $G=G_{\phi,\mathbf{m}}.$
>From (\ref{PAR3}), we immediately have
\begin{equation}
\frac{\partial\operatorname{Parisi}\left( \phi,\mathbf{m}\right) }{\partial
m_{j}}=\frac{1}{m_{j}^{2}}\left[ \int H\left( K_{j}\left( \mathbf{x}
^{\left( j-1\right) },\cdot\right) \mid\mu_{j}\right) G^{\left(
j-1\right) }\left( d\mathbf{x}^{\left( j-1\right) }\right) -\gamma
_{j}\log2\right] .\label{PAR_Derivative}
\end{equation}
Set $d_{j}\overset{\mathrm{def}}{=}\int H\left( K_{j}\left( \mathbf{x}
^{\left( j-1\right) },\cdot\right) \mid\mu_{j}\right) G_{\mathbf{m}
}^{\left( j-1\right) }\left( d\mathbf{x}^{\left( j-1\right) }\right)
.$We use (\ref{PAR_Derivative}) and the minimality of $\operatorname{Parisi}
\left( \phi,\mathbf{\cdot}\right) $ at $\mathbf{m.}$ We can perturb
$\mathbf{m}$ by moving a whole block $m_{j_{r}+1}=\cdots=m_{j_{r+1}}$ up and
down locally, without leaving $\Delta,$ provided it is not the possibly
present block of values $1.$ This leads to
\[
\sum_{i=j_{r}+1}^{j_{r+1}}d_{i}=\log2\sum_{i=j_{r}+1}^{j_{r+1}}\gamma_{i}.
\]
Furthermore, we can always move first parts of blocks, say $m_{j_{r}+1}
=\cdots=m_{k},\ k\leq j_{r+1}$ locally down, without leaving $\Delta,$ so that
we get
\[
\sum_{i=j_{r}+1}^{j_{k}}d_{i}\leq\log2\sum_{i=j_{r}+1}^{j_{k}}\gamma_{i}.
\]
These two observations imply
\begin{equation}
G\in\bigcap_{j=1}^{n}\mathcal{R}_{j}\cap\bigcap_{r=1}^{K}\mathcal{R}_{j_{r}
}^{=}.\label{PAR4}
\end{equation}
We next prove
\begin{equation}
\operatorname{Gibbs}\left( \nu,\phi\right) \leq\operatorname{Gibbs}\left(
G,\phi\right) \label{PAR5}
\end{equation}
for any $\nu\in\bigcap_{j=1}^{n}\mathcal{R}_{j}.$
We first prove the case $n=1.$ If $m<1,$ then
\[
H\left( G\mid\mu\right) =\log2\geq H\left( \nu\mid\mu\right)
\]
by (\ref{PAR4}) and the assumption $\nu\in\mathcal{R}_{1}.$ Therefore, in any
case
\begin{align*}
\operatorname{Gibbs}\left( G,\phi\right) -\operatorname{Gibbs}\left(
\nu,\phi\right) & \geq\int\phi dG-\frac{1}{m}H\left( G\mid\mu\right) \\
& -\left[ \int\phi d\nu-\frac{1}{m}H\left( \nu\mid\mu\right) \right] \\
& =\frac{1}{m}H\left( \nu\mid G\right) \geq0
\end{align*}
The general case follows by a slight extension of the above argument. Put
\[
D_{k}\overset{\mathrm{def}}{=}\int\phi_{k}dG^{\left( k\right) }-\frac
{1}{m_{k+1}}H\left( G^{\left( k\right) }\mid\mu^{\left( k\right)
}\right) -\int\phi_{k}d\nu^{\left( k\right) }+\frac{1}{m_{k+1}}H\left(
\nu^{\left( k\right) }\mid\mu^{\left( k\right) }\right) ,
\]
$D_{0}\overset{\mathrm{def}}{=}0,D_{n}=\operatorname{Gibbs}\left(
G,\phi\right) -\operatorname{Gibbs}\left( \nu,\phi\right) .$ We prove
$D_{k-1}\leq D_{k}$ for all $k,$ so that the claim follows. Remark that as
above in the $n=1$ case, if $m_{k}<m_{k+1},$ then $H\left( G^{\left(
k+1\right) }\mid\mu^{\left( k+1\right) }\right) =\Gamma_{k}\log2,$ and
therefore, in any case
\begin{align*}
D_{k} & \geq\int\phi_{k}dG^{\left( k\right) }-\frac{1}{m_{k}}H\left(
G^{\left( k\right) }\mid\mu^{\left( k\right) }\right) -\int\phi_{k}
d\nu^{\left( k\right) }+\frac{1}{m_{k}}H\left( \nu^{\left( k\right) }
\mid\mu^{\left( k\right) }\right) \\
& =\int\phi_{k-1}dG^{\left( k-1\right) }-\frac{1}{m_{k}}H\left( G^{\left(
k-1\right) }\mid\mu^{\left( k-1\right) }\right) -\int\phi_{k}d\nu^{\left(
k\right) }+\frac{1}{m_{k}}H\left( \nu^{\left( k\right) }\mid\mu^{\left(
k\right) }\right) .
\end{align*}
As
\begin{align*}
& H\left( \nu^{\left( k\right) }\mid\mu^{\left( k\right) }\right)
-m_{k}\int\phi_{k}d\nu^{\left( k\right) }+m_{k}\int\phi_{k-1}d\nu^{\left(
k-1\right) }\\
& =H\left( \nu^{\left( k-1\right) }\mid\mu^{\left( k-1\right) }\right)
+\int\log\frac{\nu^{\left( k\right) }\left( dx_{k}\mid\mathbf{x}^{\left(
k-1\right) }\right) \mathrm{e}^{m_{k}\phi_{k-1}\left( \mathbf{x}^{\left(
k-1\right) }\right) }}{\mu_{k}\left( dx_{k}\right) \mathrm{e}^{m_{k}
\phi_{k}\left( \mathbf{x}^{\left( k\right) }\right) }}\nu^{\left(
k\right) }\left( d\mathbf{x}^{\left( k\right) }\right) \\
& \geq H\left( \nu^{\left( k-1\right) }\mid\mu^{\left( k-1\right)
}\right) ,
\end{align*}
(\ref{PAR5}) is proved.
(\ref{PAR4}) and (\ref{PAR5}) identify $G=G_{\phi,\mathbf{m}}$ as the unique
maximizer of $G\left( \cdot,\phi\right) $ under the constraint
$\bigcap\nolimits_{j=1}^{n}\mathcal{R}_{j}.$
The identification (\ref{Gibbs=Parisi}) comes by a straightforward computation.
\end{proof}
\end{document} |
\begin{document}
\sloppy
{\rm m}aketitle
\begin{abstract}
Among the efficient numerical methods based on atomistic models, the quasicontinuum (QC) method has attracted growing interest in recent years.
The QC method was first developed for crystalline materials with Bravais lattice and was later extended to multilattices (Tadmor et al, 1999).
Another existing numerical approach to modeling multilattices is homogenization.
In the present paper we review the existing numerical methods for multilattices and propose another concurrent macro-to-micro method in the numerical homogenization framework.
We give a unified mathematical formulation of the new and the existing methods and show their equivalence.
We then consider extensions of the proposed method to time-dependent problems and to random materials.
\end{abstract}
\begin{keywords}
atomistic model,
quasicontinuum method,
multilattice,
homogenization,
multiscale method,
\end{keywords}
\begin{AMS}
65N30, 70C20, 74G15, 74G65\end{AMS}
\pagestyle{myheadings}
\thispagestyle{plain}
\section{Introduction}
In some applications of solid mechanics, such as modeling cracks, structural defects, or nanoelectromechanical systems, the classical continuum description is not suitable, and one is required to utilize an atomistic description of materials.
However, full atomistic simulations are prohibitively expensive, hence one needs to coarse-grain the problem.
The quasicontinuum (QC) method {\rm c}ite{TadmorPhillipsOrtiz1996} is one of the most efficient methods of coarse-graining the atomistic statics.
The idea behind QC is to introduce piecewise affine constraints for the atoms in regions with smooth deformation and use the Cauchy--Born rule to define the energy of the corresponding groups of constrained atoms.
To formulate the QC method for multilattice crystals one must account for relative shifts of Bravais lattices of which the multilattice is comprised {\rm c}ite{TadmorSmithBernsteinEtAl1999}.
The QC method is a multiscale method capable of coupling atomistic and continuum description of materials.
It is intended to model an atomistic material in a continuum manner in the regions where the deformation is smooth and use the fully atomistic model only in the small neighborhood of defects, thus effectively reducing the degrees of freedom of the system.
Originally, the QC method was developed for crystalline materials with a (single) Bravais lattice {\rm c}ite{TadmorPhillipsOrtiz1996}, and the convergence of a few variants of the method has been analyzed under some practical assumptions (see, e.g., {\rm c}ite{DobsonLuskinOrtner2010b, VanKotenLiLuskinEtAl2012, Lin2003, Lin2007, LuMing, MingYang2009, OrtnerShapeev2011, OrtnerSuli2008}).
The QC method is based on the so-called Cauchy--Born rule (see, e.g., {\rm c}ite{BlancLeBrisLions2007a, EMing2007, Ericksen2008, FrieseckeTheil2002}) which states that the energy of a certain volume of a material can be approximated through the deformation energy density, which is computed for a representative atom, assuming that the neighboring atoms follow a uniform deformation.
Later, QC was extended to multilattices {\rm c}ite{TadmorSmithBernsteinEtAl1999} (a multilattice is a union of a number of Bravais lattices) based on the improved Cauchy--Born rule {\rm c}ite{Stakgold1950} which accounts for relative shifts between the Bravais lattices.
Examples of such materials include diamond cubic {\rm m}box{Si}, HCP metals (stacking two simple hexagonal lattices with a shift vector) like {\rm m}box{Zr}, ferroelectric materials, salts like sodium chloride, and intermetallics like {\rm m}box{NiAl}.
More recent developments of QC for multilattices also include adaptive choice of representative cell of multilattices {\rm c}ite{DobsonElliottLuskinEtAl2007}.
It appears that no rigorous analysis is available so far for the multilattice QC except for the authors' preprint {\rm c}ite{AbdulleLinShapeev2010}.
In the present work we propose a treatment of multilattices within the framework of numerical homogenization.
Homogenization techniques for partial differential equations (PDEs) with multiscale coefficients
are known to be successful for obtaining effective equations with coefficients properly averaged out
{\rm c}ite{BensoussanLionsPapanicolaou1978}.
Finite element methods based on homogenization theory have been pioneered by
Bab\u{u}ska {\rm c}ite{Babuska1976} and have attracted growing attention in recent years
(see {\rm c}ite{Abdulle2009, Abd11b, EEL2007, EfendievHou2009, GeersKouznetsovaBrekelmans2010} for textbooks or review papers).
Following the ideas of {\rm c}ite{BensoussanLionsPapanicolaou1978}, we use formal homogenization techniques to describe
the coarse-graining of multilattices and based on that, propose a macro-to-micro numerical algorithm which we call the {\it homogenized QC (HQC)} method.
Here the term {\it macro-to-micro} refers to coupling macroscopic and microscopic scales for the
same physical model, but not coupling models, like in the nonlocal QC.
The macro-to-micro method developed in this paper follows the framework of the finite element
heterogenenous multiscale method (FE-HMM) {\rm c}ite{Abdulle2009, Abd11b, EEL2007}, a numerical
method coupling a macroscopic finite element method (FEM) defined on a macroscopic mesh
with effective data recovered on the fly by microscopic FEM on patches centered at suitable quadrature points
within the macroscopic mesh. This method belongs to the family of numerical homogenization methods
as it provides a homogenized numerical solution, but unlike classical methods, the effective data are not precomputed
but supplemented by micro computations when and where needed during the macro computation. The HMM
provides an efficient way of coupling micro and macro solvers and a suitable framework for
a priori and a posteriori analysis taking into account numerical approximation at different scales {\rm c}ite{Abdulle2009, Abd11b}.
We give a unified mathematical description and establish equivalence between the homogenized QC, the multilattice QC (MQC) of {\rm c}ite{TadmorSmithBernsteinEtAl1999}, and the finite element method applied to the continuously homogenized equations (see {\rm c}ite{AbdulleLinShapeev2010, FishChenLi2007} and references therein for homogenization of atomistic media).
Despite the formal equivalence, we find value in formulating MQC within the homogenization framework and, more generally, in connecting the existing developments in upscaling atomistic models and classical numerical homogenization.
First, this framework allows us to apply the numerical analysis techniques developed for continuum numerical homogenization such as the finite element heterogeneous multiscale method {\rm c}ite{Abdulle2009, EEL2007} to the multilattice QC method (see our preprint {\rm c}ite{AbdulleLinShapeev2010} for an example of such application).
Second, numerical homogenization techniques can be used to upscale the atomistic model in both, time and space, which makes it promising for modeling and especially analyzing motion of atomistic materials at macro- and micro-scale {\rm c}ite{EEL2007, FishChenLi2007, MillerTadmor2002}.
In this work we demonstrate such an application of HQC to a slow (i.e., with no thermal fluctuation) dynamics of an atomistic crystal (Section \ref{sec:unsteady}).
Also, numerical techniques based on the homogenization framework are well suited for materials described on stochastic lattices
at the atomistic level
such as polymers {\rm c}ite{BaumanOdenPrudhomme2009} and glasses (see, e.g., {\rm c}ite{AlicandroCicaleseGloria2011, BlancLeLions2007_stochastic_lattices}),
or for materials with properties (such as, e.g., conductivity, stiffness, etc.${\rm m}athstrut$)
described by random parameters at the continuum level {\rm c}ite{Tor005}.
We give an example of application of numerical homogenization to a stochastic material in Section \ref{sec:stochastic}.
We note that the idea of applying numerical homogenization methods to atomistic media has appeared in the literature before {\rm c}ite{BaumanOdenPrudhomme2009, ChenFish2006, Chung2004, ChungNamburu2003, FishChenLi2007}.
\\
The paper is organized as follows.
We present the atomistic model in Section \ref{sec:problem_formulation}, and in particular we give a simplified illustrative model in Section \ref{sec:problem_formulation:simplified}.
The simplified model will be useful to better illustrate application of the coarse-graining methods and to draw analogies between the concepts discussed in this paper and their counterparts in the classical continuum homogenization.
We then present the quasicontinuum method in Section \ref{sec:QC}.
In Section \ref{sec:homogenization} we present a formal homogenization technique applied to the atomistic equations.
In Section \ref{sec:HQC} we present the HQC method---a concurrent macro-to-micro algorithm based on the discrete homogenization.
Section \ref{sec:equivalence} is devoted to showing the equivalence of the following three methods applied to multilattices: the HQC method, the MQC method, and the finite element method applied to continuously homogenized equations.
In Section \ref{sec:multilattice} we illustrate an application of HQC to a multilattice.
We emphasize that the HQC is formulated in such a way that it allows for a straightforward extension to non-crystalline materials if the microstructure is known; an example of such extension is given in Section \ref{sec:stochastic}.
In Section \ref{sec:unsteady} we apply the proposed macro-to-micro method to a long-wave unsteady evolution of a 1D multilattice crystal.
Concluding remarks are given in Section \ref{sec:conclusion}.
The commonly used notations are collected in the appendix.
\section{Problem Formulation}\label{sec:problem_formulation}
The focus of the present study is on correct treatment of atomistic materials with spatially oscillating or inhomogeneous local properties.
\subsection{Equations of Equilibrium}\label{sec:problem_formulation:full}
We describe the formulation of the problem of finding an equilibrium of an atomistic material in the periodic setting.
We consider the periodic boundary conditions for simplicity, in order to avoid difficulties arising from presence of the boundary of the atomistic material.
Nevertheless, it should be noted that the numerical method and the algorithm proposed in the present work can be applied to Dirichlet, Neumann, or other boundary conditions.
\subsubsection{Deformation}
Consider an atomistic material occupying a region $\Omega=[0,1)^d$ in its reference (i.e., undeformed) configuration and extended periodically outside of $\Omega$.
The set of positions of atoms in the reference configuration is
\[
{{\rm m}athcal M} = \Omega{\rm c}ap\bigcup_{\alpha=0}^{m-1} \big({\epsilon}{{\rm m}athbb Z}^d+{\epsilon} p_\alpha\big),
\]
where $p_\alpha\in[0,1)^d$ is a shift vector of $\alpha$-th species of atoms in the reference configuration; in total we have $m$ species of atoms.
We assume that $p_\alpha\ne p_\beta$ for $\alpha\ne\beta$ and, for convenience, $p_0=0$.
We collect these shift vectors into the set ${{\rm m}athcal P} := \{p_\alpha \,:\ \alpha=0,\ldots,m-1\}$.
Thus, if we denote a Bravais lattice in $\Omega$ by
\[
{{\rm m}athcal L} = \Omega{\rm c}ap{\epsilon}{{\rm m}athbb Z}^d,
\]
then we can write ${{\rm m}athcal M} = {{\rm m}athcal L} + {\epsilon}{{\rm m}athcal P}$.
This identity means that ${{\rm m}athcal M}$ consists of ${\epsilon}{{\rm m}athcal P}$ repeated periodically with the period ${\epsilon}$.
We will call ${{\rm m}athcal M}$ a multilattice.
The sets ${{\rm m}athcal L}$, ${{\rm m}athcal P}$, and ${{\rm m}athcal M}$ are illustrated in Figure \ref{fig:LPM}.
When the material experiences a deformation, the atom positions become $x + u(x)$, where $u(x)$ is the displacement.
We assume that $u(x)$ is periodic; i.e., $u(x+a)=u(x)$ for all $a\in{{\rm m}athbb Z}^d$.
The space of all periodic displacements is denoted by ${{\rm m}athcal U}_{\rm per}({{\rm m}athcal M})$.
Since we consider only the systems invariant with respect to translation in space, we will also need the space of displacements with zero average, ${{\rm m}athcal U}_\#({{\rm m}athcal M})$ (see Appendix \ref{sec:notations:spaces} for the precise definitions).
\begin{figure}\label{fig:LPM}
\end{figure}
\subsubsection{Interaction}
We assume a general (multibody) finite-range interaction between atoms.
For each atom $x\in{{\rm m}athcal M}$ we introduce its ``interaction neighborhood''---a set of vectors ${{\rm m}athcal R}_{\epsilon}(x)$ such that $\{x+{\epsilon} r\,:\,r\in{{\rm m}athcal R}_{\epsilon}(x)\}$ are the atoms that $x$ interacts with.
The energy of an atom $x\in{{\rm m}athcal M}$ is denoted by $V_{\epsilon}(D_{{{\rm m}athcal R}_{\epsilon}(x)} u(x); x)$, where $D_{{{\rm m}athcal R}_{\epsilon}} u = (D_r u)_{r\in{{\rm m}athcal R}_{\epsilon}}$ (see Appendix \ref{sec:notations:vector_indexed}) is a collection of discrete directional derivatives of $u$ corresponding to the set of neighbors ${{\rm m}athcal R}_{\epsilon}$ (these notations were first introduced in {\rm c}ite{HudsonOrtner2012}).
The discrete derivative in direction $r$ of $u$ evaluated at $x\in{{\rm m}athcal M}$ is defined as $D_r u(x) := \frac{u(x+{\epsilon} r)-u(x)}{\epsilon}$.
The needed properties and definitions of discrete directional derivatives can be found in Appendix \ref{sec:notations:operators}, and more details on discrete directional derivatives in Appendix \ref{sec:notations:vector_indexed}.
Thus, the interaction energy of the displacement $u$ is given by the interaction potential $V_{\epsilon}$ as
\[
E(u)
= \frac1{\#({{\rm m}athcal M})}\sum_{x\in{{\rm m}athcal M}} V_{\epsilon}(D_{{{\rm m}athcal R}_{\epsilon}(x)} u(x); x)
= \big\langle V_{\epsilon}(D_{{{\rm m}athcal R}_{\epsilon}} u) \big\rangle_{{{\rm m}athcal M}}
,
\]
where $\langleg\rangle_S$ denotes the average value of a function $g$ defined on a discrete set $S$.
The subscript ${\epsilon}$ in $V_{\epsilon}$ and ${{\rm m}athcal R}_{\epsilon}$ indicates that these objects depend nonsmoothly on $x$: indeed, the interaction energy and the interaction neighborhood may depend on the species of atoms $\alpha$ for $x\in{{\rm m}athcal L}+{\epsilon} p_\alpha$.
For instance, we can consider a Lennard--Jones potential with atom-dependent parameters:
\begin{equation}
\label{eq:problem_formulation:LJ}
V_{\epsilon}(D_{{{\rm m}athcal R}_{\epsilon}} u; x) = \sum_{r\in{{\rm m}athcal R}_{\epsilon}} s_{x,x+{\epsilon} r}\Big(-2\, \big(\smfrac{|r+D_r u|}{\ell_{x,x+{\epsilon} r}}\big)^{-6}+\big(\smfrac{|r+D_r u|}{\ell_{x,x+{\epsilon} r}}\big)^{-12}\Big),
\end{equation}
where $s_{x,x+{\epsilon} r}$ and $\ell_{x,x+{\epsilon} r}$ are, respectively, the strength and the equilibrium distance of interaction of atoms $x$ and $x+{\epsilon} r$.
We assume that the interaction neighborhood ${{\rm m}athcal R}_{\epsilon}(x+{\epsilon} p_\alpha)$ and the interaction potential $V_{\epsilon}(\bullet,x+{\epsilon} p_\alpha)$ for $x\in{{\rm m}athcal L}$ depend only on $\alpha$, the particular species of atoms, but do not depend on $x$; we therefore write ${{\rm m}athcal R}_{\epsilon}(x+{\epsilon} p_\alpha)=:{{\rm m}athcal R}_{{\epsilon},\alpha}$ and $V_{\epsilon}(\bullet,x+{\epsilon} p_\alpha) =: V_{{\epsilon},\alpha}$.
This assumption states, effectively, an ${\epsilon}$-periodicity of ${{\rm m}athcal R}_{\epsilon}$ and $V_{\epsilon}(\bullet,x)$.
Then, we can use the following form of the energy:
\begin{align} \notag
E(u)
=~&
\bigg\langle
\frac1m \sum_{\alpha=0}^{m-1}
V_{\epsilon}(D_{{{\rm m}athcal R}_{\epsilon}(x+{\epsilon} p_\alpha)} u(x+{\epsilon} p_\alpha); x+{\epsilon} p_\alpha)
\bigg\rangle_{x\in{{\rm m}athcal L}}
\\ =~& \label{eq:E_alt}
\bigg\langle
\frac1m \sum_{\alpha=0}^{m-1}
V_{{\epsilon},\alpha}(D_{{{\rm m}athcal R}_{{\epsilon},\alpha}} u(x+{\epsilon} p_\alpha))
\bigg\rangle_{x\in{{\rm m}athcal L}},
\end{align}
where we used a more verbose notation for averaging of a function $g$ defined on a discrete set $S$, $\langleg\rangle_S=:\langleg(x)\rangle_{x\in S}$.
This expression for the energy will be used to write down the energy of the MQC method in a familiar way (see \eqref{eq:Emqc-simplified}).
\begin{remark}
One can exercise the freedom in choosing ${{\rm m}athcal P}$ by assuming that ${{\rm m}athcal P} = \big\{0,\smfrac1m e_1, \ldots,\smfrac{m-1}m e_1\big\}$, where $e_1\in{{\rm m}athbb R}^d$ is the respective unit vector.
In this case ${{\rm m}athcal M}$, up to a dilatation, is a simple lattice (although with several species of atoms).
This allows one to choose ${{\rm m}athcal R}_{\epsilon}(x)$ independent of $x$ (and also ${{\rm m}athcal R}_{{\epsilon},\alpha}$ independent of $\alpha$), and leave only interaction potential $V_{\epsilon}$ to depend on $x$.
We will not pursue this in the present work; however, such notations would significantly simplify presentation of the MQC (Section \ref{sec:MQC}) and would allow one to conveniently write the equilibrium equation in a strong form (in particular, in Section \ref{sec:homogenization:fast-and-slow}).
Our motivation for not pursuing this is to show that the homogenization and the numerical method can, in principle, be generalized to the case when ${{\rm m}athcal R}_{\epsilon}$ depends on $x$.
This is important when modeling non-crystalline materials with no underlying periodic structure.
\end{remark}
\subsubsection{External Force}
The potential energy of the external force $f=f(x)$ is
\[
-F(u) = - \langlef, u \rangle_{{\rm m}athcal M},
\]
where by $\langle w, v \rangle_{{\rm m}athcal M} := \langle w {\rm c}dot v \rangle_{{\rm m}athcal M}$ we denote a scalar product of $w,v\in{{\rm m}athcal U}_{\rm per}({{\rm m}athcal M})$.
(To be precise, it is an inner product on ${{\rm m}athcal U}_\#({{\rm m}athcal M})$ and a semi-inner product on ${{\rm m}athcal U}_{\rm per}({{\rm m}athcal M})$.)
The forces $f=f(x)$ are applied as ``dead loads''; i.e., they are independent of actual atom positions $x+u$.
For the problem to be well-posed, the sum of all forces per period is assumed to be zero; i.e., $\langlef\rangle_{{\rm m}athcal M} = 0$.
\subsubsection{Equation of Equilibrium}
We denote the total potential energy of the atomistic system by
\[
\Pi(u) = E(u) -F(u).
\]
A displacement $u\in{{\rm m}athcal U}_\#({{\rm m}athcal M})$ is a stable equilibrium if it is a local minimizer of $\Pi$, which implies that $u$ is a critical point of $\Pi$:
\begin{equation}
\label{eq:original_equation}
\langle{\delta\hspace{-1pt}ta\hspace{-0.5pt}\Pi}(u), v\rangle_{{\rm m}athcal M} := \frac{{\rm d}}{{\rm d} t} \Pi(u+t v)\big|_{t=0} = 0
\quad \forall v\in{{\rm m}athcal U}_\#({{\rm m}athcal M}).
\end{equation}
We assume that the function $\Pi(u)$ is smooth enough, and hence $\langle{\delta\hspace{-1pt}ta\hspace{-0.5pt}\Pi}(u), v\rangle_{{\rm m}athcal M}$ is a linear functional with respect to $v\in{{\rm m}athcal U}_\#({{\rm m}athcal M})$, which justifies identification of ${\delta\hspace{-1pt}ta\hspace{-0.5pt}\Pi}(u)$ with an element of ${{\rm m}athcal U}_\#$.
Alternatively, the problem of finding the equilibrium configuration of atoms can formally be written as
\[
\frac{\partial \Pi}{\partial u(x)} = 0 \quad \forall x\in{{\rm m}athcal M},
\]
if we consider $\Pi$ as a function of finite number of variables $u(x)$, $x\in{{\rm m}athcal M}$.
A physical potential energy $\Pi(u)$ has to be invariant with respect to a uniform translation of atoms.
Hence, we pose the following additional condition,
\begin{equation}
\langleu\rangle_{{\rm m}athcal M} = 0,
\label{eq:u-averages-to-zero}
\end{equation}
which is necessary (but may not be sufficient) for the equations \eqref{eq:original_equation} to have a {\rm c}hange{locally unique} solution.
The equilibrium equations \eqref{eq:original_equation} together with the additional condition \eqref{eq:u-averages-to-zero} can be written in variational form: find $u \in {{\rm m}athcal U}_{{\rm per}}({{\rm m}athcal M})$ such that
\begin{subeqnarray} \label{eq:variational_equation}
\langle{\delta\hspace{-1pt} E}(u), v\rangle_{{\rm m}athcal M} & = & F(v)
\quad \forall v\in {{\rm m}athcal U}_{{\rm per}}({{\rm m}athcal M})
\\
\langleu\rangle_{{\rm m}athcal M} & = & 0,
\label{eq:variational_problem_generic}
\end{subeqnarray}
where the functional derivative ${\delta\hspace{-1pt} E}: {{\rm m}athcal U}_{{\rm per}}({{\rm m}athcal M}) \to {{\rm m}athcal U}_{{\rm per}}({{\rm m}athcal M})$ is computed as
\begin{align}
\label{eq:Psi}
\langle{\delta\hspace{-1pt} E}(u), v\rangle_{{\rm m}athcal M}
=~&
\Big\langle\sum_{r\in{{\rm m}athcal R}_{\epsilon}} V'_{{\epsilon},r}(D_{{{\rm m}athcal R}_{\epsilon}} u), D_r v\Big\rangle_{{\rm m}athcal M}
,
\end{align}
and $V'_{{\epsilon},r}(D_{{{\rm m}athcal R}_{\epsilon}} u)$ denotes, effectively, the gradient of a scalar function $V_{\epsilon}$ with respect to its vector-valued variable $D_r u$ (note the difference with $V_{{\epsilon}, \beta}$ introduced in \eqref{eq:E_alt}).
Here and in what follows, with a slight abuse of notations, we keep the sign of summation over $r\in{{\rm m}athcal R}_{\epsilon}$ inside the triangular brackets of the scalar product.
\subsection{A Simple Illustrative Example}\label{sec:problem_formulation:simplified}
The following simplified model will be useful in illustrating the concepts presented in this paper (namely, we will give a simplified version of the quasicontinuum method, in Section \ref{sec:QC-heterogeneous:failure}, and illustrate an application of the homogenization, in Section \ref{sec:homogenization:simplfied}).
The reader can find more examples involving a simplified model in our preprint {\rm c}ite{AbdulleLinShapeev2010}.
Assume one space dimension, $d=1$; the domain $\Omega=[0,1)$, the shift vectors in the reference configuration
\begin{equation} \label{eq:simplified_model:P}
{{\rm m}athcal P} = \big\{0,\smfrac1m,\ldots,\smfrac{(m-1)}m\big\},
\end{equation}
the multilattice
\[
{{\rm m}athcal M}
=\bigcup_{\alpha=0}^{m-1}({\epsilon} {{\rm m}athbb Z} + {\epsilon} \smfrac{\alpha}{m}) {\rm c}ap \Omega
=\smfrac{\epsilon} m{{\rm m}athbb Z}{\rm c}ap\Omega
,
\]
and the basic lattice ${{\rm m}athcal L}={\epsilon}{{\rm m}athbb Z}{\rm c}ap\Omega$.
We further assume ${{\rm m}athcal R}=\{\smfrac1m\}$ (nearest neighbor interaction only) and consider the ``linear spring model'' with the atomistic potential
\begin{equation} \label{eq:simplified_model}
V_{\epsilon}(D_r u; x) = \psi_{\epsilon}(x)\, \frac{(D_r u)^2}2
,
\end{equation}
with $r=\smfrac1m$.
Such a system can be interpreted as a system of masses located at positions $x + u$ and connected with ideal springs with spring constants $k_\alpha = \psi_{\epsilon}(x)/{\epsilon}$ (where $\alpha$ and $x$ are related here through $x\in {\epsilon} \smfrac{1+\alpha}{m} + {\epsilon}{{\rm m}athbb Z}$), as illustrated in Figure \ref{fig:springs_heterogeneous}.
\begin{figure}\label{fig:springs_heterogeneous}
\end{figure}
The equilibrium equation then becomes
\begin{equation}
\langle\psi_{\epsilon} D_r u, D_r v\rangle_{{\rm m}athcal M}
= \langlef, v\rangle_{{\rm m}athcal M}
.
\label{eq:problem_1d_linear}
\end{equation}
If we want to find an equilibrium of a very large atomistic system, we need to coarse-grain these equations.
In Section \ref{sec:QC} we present the quasicontinuum method, one of the methods of numerical coarse-graining of such a system.
We can notice that the equation \eqref{eq:problem_1d_linear} closely resembles the continuum equation
\begin{equation}\label{eq:intro:continuum_energy}
\int_\Omega A\big(\smfrac x{\epsilon}\big) \frac{{\rm d} u}{{\rm d} x} \frac{{\rm d} v}{{\rm d} x} {\rm d} x
=
\int_\Omega f v {\rm d} x
,
\end{equation}
for which the homogenization theory is well-developed.
Here $A\big(\smfrac x{\epsilon}\big)$ is an oscillating coefficient defining the local energy density.
The crystal is, by definition, a periodic arrangement of atoms, which translates into periodicity of $A\big(\smfrac x{\epsilon}\big)$.
Non-crystalline solid materials, in contrast, correspond to random arrangements of atoms, which is analogous to random (non-periodic) $A\big(\smfrac x{\epsilon}\big)$.
The spring constants varying on the scale of ${\epsilon}$ are analogous to $A\big(\smfrac x{\epsilon}\big)$ varying on the scale of ${\epsilon}$.
It is well known from homogenization theory ({\rm c}ite{BensoussanLionsPapanicolaou1978}) that the solution $u$ of \eqref{eq:intro:continuum_energy} converges weakly
in the $H^1$ norm to a homogenized solution $\bar u,$ solution to an equation similar to \eqref{eq:intro:continuum_energy} but
with an effective (homogenized) tensor $\bar A(x).$ We note that, in general, strong convergence holds only for the $L^2$ norm.
Based on this similarity between the continuum and the discrete energy, we apply the formal homogenization techniques to the discrete atomistic equations in Section \ref{sec:homogenization} and based on that formulate the HQC method---a concurrent macro-to-micro algorithm (similar to FE-HMM) based on the discrete homogenization.
The method is formulated in such a way that it allows for a straightforward extension to non-crystalline materials if the microstructure is known; an example of such extension is given in Section \ref{sec:stochastic}.
In Section \ref{sec:unsteady} we apply the proposed macro-to-micro method to a long-wave unsteady evolution of a 1D multilattice crystal.
The long-wave unsteady evolution is analogous to a continuum motion corresponding to a Hamiltonian
\[
\smfrac12 \int_\Omega \bigg[m\big(\smfrac x{\epsilon}\big) \Big(\frac{{\rm d} u}{{\rm d} t}\Big)^2
+ A\big(\smfrac x{\epsilon}\big) \Big(\frac{{\rm d} u}{{\rm d} x}\Big)^2
\bigg]{\rm d} x,
\]
where $u=u(t,x)$ is assumed to have no fast (i.e., on the time scale of $\smfrac1{\epsilon}$) oscillations.
\section{Quasicontinuum (QC) Method}\label{sec:QC}
Traditionally, numerical methods such as the finite element method (FEM) are applied to continuum equations which can then be solved on a computer.
The characteristic feature of the atomistic models we are discussing in the paper is their discreteness, with a number of degrees of freedom often too large to keep track of each individual atom.
Therefore, similarly to FEM, the ideas of reducing the number of degrees of freedom are used for atomistic models as well. The difference is that now the reduction is done from a large but finite number of degrees of freedom to a smaller number of degrees of freedom.
The QC method is a representative of such methods.
We first present its simple-lattice version.
The QC method consists of reducing the number of degrees of freedom of the atomistic system by choosing a coarse mesh of nodal atoms and assuming that the positions of the other atoms can be reconstructed by a linear interpolation.
It should be noted that we discuss here only the {\it local} version of QC which is equivalent to applying FEM to the Cauchy--Born continuum model of elasticity.
We are not considering coupling the continuum and discrete models in this paper.
\subsection{Notation}\label{sec:QC:notations}
Assume a partition ${{\rm m}athcal T}h$ of the domain $\Omega$ into simplicial elements $T$, which we will conveniently refer to as the {\it mesh}.
Normally, $\#({{\rm m}athcal T}h)\ll\#({{\rm m}athcal L})$ (recall that by $\#(\bullet)$ we denote the number of elements in a set).
By $|T|$ we denote the Lebesgue measure of $T$.
The QC solution will be denoted by $u^h$.
The space of piecewise linear discrete vector-functions is denoted by
\begin{equation}
{{\rm m}athcal U}^h_{\rm per} = \big\{
u^h\in \big(W^{1,\infty}_{\rm per}(\Omega)\big)^d\,:\ u^h|_T\in P_1(T) ~\forall T\in{{\rm m}athcal T}h\big\}
,
\label{eq:UH-space}
\end{equation}
and the space of piecewise constant vector-functions as
\begin{displaymath}
{{\rm m}athcal Q}^h_{\rm per} = \big\{
q^h\in \big(L^{\infty}_{\rm per}(\Omega)\big)^d\,:\ q^h|_T\in P_0(T) ~\forall T\in{{\rm m}athcal T}h\big\}.
\end{displaymath}
\subsection{QC for simple lattice}\label{sec:QC:homogeneous}
In this (and only this) subsection we make the simple lattice assumption.
That is, we assume that $m=1$ and hence ${{\rm m}athcal M}={{\rm m}athcal L}$.
In particular, in this subsection we write $V_{\epsilon}(D_{{\rm m}athcal R} u; x)=V(D_{{\rm m}athcal R} u)$ and ${{\rm m}athcal R}_{\epsilon}(x)={{\rm m}athcal R}$ as they no longer depend on $x$.
The QC method {\rm c}ite{TadmorPhillipsOrtiz1996} aims at finding a minimizer of
\[
\Pi(u^h)
=
\big\langleV\big(D_{{\rm m}athcal R} u^h\big)\big\rangle_{{\rm m}athcal L}
- F(u^h)
\]
in ${{\rm m}athcal U}^h_{\rm per}$.
Minimizing $\Pi(u^h)$ in ${{\rm m}athcal U}^h_{\rm per}$ indeed reduces the number of degrees of freedom of the system from $O(\#({{\rm m}athcal M}))$ to $O(\#({{\rm m}athcal T}h))$ (recall that $\#({{\rm m}athcal T}h)\ll \#({{\rm m}athcal M})$).
However, one must still spend $O(\#({{\rm m}athcal M}))$ operations to compute the effective forces on the reduced degrees of freedom.
In order to have an efficient numerical method (i.e., a method with $O(\#({{\rm m}athcal T}_h))$ operations) one introduces an approximation to $\Pi(u^h)$ which is called the {\it local QC method} {\rm c}ite{TadmorPhillipsOrtiz1996} (hereinafter referred to as the QC method).
The local QC method first approximates $D_r u^h$ with $\nabla_r u^h$ within each $T$ (hence the name of the method: the nonlocal finite difference $D_r u^h$ is approximated with the ``local'' directional derivative $\nabla_r u^h$).
Then for each $x\in T$ one has
\begin{align*}
V(D_{{\rm m}athcal R} u^h)
\approx~&
V(\nabla_{{\rm m}athcal R} u^h)
= W\big(\nabla u^h|_T\big),
\end{align*}
where $W({\sf F}) := V({\sf F}{{\rm m}athcal R})$ is the Cauchy--Born energy density associated with a displacement gradient ${\sf F}$ (see \eqref{eq:vector_indexed} to obtain the precise definition of ${\sf F}{{\rm m}athcal R}$).
Second, the local QC method changes the sum over $x\in{{\rm m}athcal L}$ effectively to integration over $\Omega$; i.e.,
\[
E^{\rm qc}(u^h) :=
\int_\Omega W\big(\nabla u^h\big) \dd x
=
\sum_{T\in{{\rm m}athcal T}h} |T|\, W\big(\nabla u^h|T\big)
.
\]
The variational formulation of the QC method is thus
\begin{equation}
\label{eq:qc}
\int_\Omega
\sum_{r\in{{\rm m}athcal R}} \delta\hspace{-1pt} W\big(\nabla_r u^h\big) \!:\! \nabla_r v^h
\dd x
= F^h(v^h)
\quad \forall v^h\in{{\rm m}athcal U}^h_{\rm per}
,
\end{equation}
where $\delta\hspace{-1pt} W$ denotes the derivative of $W$, the semicolon denotes the inner product of matrices.
and $F^h(v^h)$ is some approximation to $\langlef,v^h\rangle_{{\rm m}athcal M}$.
Error analysis of the local QC yields a first-order convergence of the deformation gradient (i.e., roughly speaking, of a quantity $\|u^h-u\|_{W^{1,p}(\Omega)}$) with respect to sizes of triangles $T\in{{\rm m}athcal T}_h$ (see, e.g., {\rm c}ite{Lin2003, Lin2007, OrtnerShapeev2011}).
A more refined analysis shows that the local QC can be second-order accurate {\rm c}ite{DobsonLuskinOrtner2010b, EMing2007, MakridakisSuli}.
\subsection{Multilattice QC}\label{sec:MQC}
Approximating the exact minimizer of $\Pi(u)$ with a piecewise linear $u^h \in {{\rm m}athcal U}^h_{\rm per}$ may be accurate enough for the case when the interatomic interaction $V_{\epsilon}(\bullet, x)$ varies smoothly with $x$ (more precisely, if the mesh ${{\rm m}athcal T}h$ resolves the variations in $V_{\epsilon}(\bullet, x)$) well.
However, for many materials with multilattice structure (examples of such materials were given in the introduction) the piecewise linear approximation of the displacement $u$ is not accurate.
In this subsection we present the Multilattice QC (MQC) method first introduced in {\rm c}ite{TadmorSmithBernsteinEtAl1999} which is designed to handle the multilattice microstructure.
Define the space of QC displacements of the multilattice ${{\rm m}athcal M}$:
\begin{equation}
{{\rm m}athcal U}^{h,q} = \bigg\{
u^h + \sum_{\alpha=1}^{m-1} q^h_\alpha w_\alpha
\,:~
u^h\in {{\rm m}athcal U}^h_{\rm per}
,~q^h_\alpha\in {{\rm m}athcal Q}^h_{\rm per}
,~\alpha=1,\ldots,m-1
\bigg\},
\label{eq:QC_general_complex-lattice_space}
\end{equation}
where $q^h_\alpha$ are the deformed shift vectors (recall that $p_\alpha$ are the undeformed shift vectors) and $w_\alpha : {{\rm m}athcal M} \to {{\rm m}athbb R}$ are the associated basis functions defined as
\begin{equation}\label{eq:MQC_basis_funcs_def}
w_\alpha|_{{{\rm m}athcal L}+{\epsilon} p_\beta} = \delta\hspace{-1pt}ta_{\alpha\beta}
\quad (\alpha,\beta=0,\ldots,m-1)
,
\end{equation}
with $\delta\hspace{-1pt}ta_{\alpha\beta}$ denoting the Kronecker delta.
It should be noted that the domain of definition of functions in ${{\rm m}athcal U}^{h,q}$ is ${{\rm m}athcal M}$, whereas the functions in ${{\rm m}athcal U}^h$ are defined on the entire ${{\rm m}athbb R}^d$.
For a more detailed introduction of the space of QC deformations, refer to {\rm c}ite{AbdulleLinShapeev2010}.
In each element $T\in{{\rm m}athcal T}h$ we thus have $m-1$ nonzero shift vectors $q^h_\alpha$, and we set $q^h_0:=0$.
We denote
\[
{{\rm m}athbf q}^h := (q^h_1,\ldots,q^h_{m-1}) \in ({{\rm m}athcal Q}^h_{\rm per})^{m-1}.
\]
Next, form the interaction energy $E(u)$ with $u = u^h + \sum_{\alpha=1}^{m-1} q^h_\alpha w_\alpha \in{{\rm m}athcal U}^{h,q}$:
\begin{align*}
E(u)
=~&
E\Big(u^h + \sum_{\alpha=1}^{m-1} q^h_\alpha w_\alpha\Big)
\\ = ~&
\Big\langle
V_{\epsilon}\Big(
D_{{{\rm m}athcal R}_{\epsilon}(x)} \Big({ u^h(x) + \sum_{\alpha=1}^{m-1} q^h_\alpha(x) w_\alpha(x)}\Big)
; x
\Big)
\Big>_{x\in{{\rm m}athcal M}}
\\ = ~&
\Big\langle
\frac1m \sum_{\beta=0}^{m-1}
V_{\epsilon}\Big(
D_{{{\rm m}athcal R}_{\epsilon}(x+{\epsilon} p_\beta)} \Big({ u^h(x+{\epsilon} p_\beta) + \sum_{\alpha=1}^{m-1} q^h_\alpha(x+{\epsilon} p_\beta) w_\alpha(x+{\epsilon} p_\beta)}\Big)
; x+{\epsilon} p_\beta
\Big)
\Big>_{x\in{{\rm m}athcal L}}
\\ = ~&
\Big\langle
\frac1m \sum_{\beta=0}^{m-1}
V_{{\epsilon},\beta}\Big(
D_{{{\rm m}athcal R}_{{\epsilon},\beta}} { u^h(x+{\epsilon} p_\beta) + \sum_{\alpha=1}^{m-1} D_{{{\rm m}athcal R}_{{\epsilon},\beta}} q^h_\alpha(x+{\epsilon} p_\beta) w_\alpha({\epsilon} p_\beta)}
\Big)
\Big>_{x\in{{\rm m}athcal L}}
,
\end{align*}
where we used periodicity of $V_{\epsilon}$ (see \eqref{eq:E_alt}) and $w_\alpha$ (which follows directly from the definitions of $w_\alpha$ and ${{\rm m}athcal M}$).
Similarly to the simple-lattice QC, we perform a local quasicontinuum approximation which consists of:
(i) changing the summation over $x\in{{\rm m}athcal L}$ to the integration over $\Omega$,
(ii) approximating $D_r u^h$ with $\nabla_r u^h$, and
(iii) approximating $q^h_\alpha(x+{\epsilon} p_\beta)$ with $q^h_\alpha(x)$:
\begin{align*}
E(u)
\approx~&
\int_\Omega
\frac1m \sum_{\beta=0}^{m-1}
V_{{\epsilon},\beta}\Big(
\nabla_{{{\rm m}athcal R}_{{\epsilon},\beta}} u^h + \sum_{\alpha=1}^{m-1} q^h_\alpha D_{{{\rm m}athcal R}_{{\epsilon},\beta}} w_\alpha({\epsilon} p_\beta)
\Big)
\dd x
\\ =~&
\sum_{T\in{{\rm m}athcal T}h} |T|\,
\frac1m \sum_{\beta=0}^{m-1}
V_{{\epsilon},\beta}\Big(
\big(\nabla u^h|_T\big){{\rm m}athcal R}_{{\epsilon},\beta} + \sum_{\alpha=1}^{m-1} \big(q^h_\alpha|_T\big) D_{{{\rm m}athcal R}_{{\epsilon},\beta}} w_\alpha({\epsilon} p_\beta)
\Big)
\\=:~& \tilde E^{\rm mqc}(u^h, {{\rm m}athbf q}^h)
,
\end{align*}
where we used the identity $\nabla_{{{\rm m}athcal R}_{{\epsilon},\beta}} u^h|_T = \big(\nabla u^h|_T\big){{\rm m}athcal R}_{{\epsilon},\beta}$; cf.\ \eqref{eq:vector_indexed}.
\begin{remark}
The expression for $\tilde E^{\rm mqc}(u^h, \{q^h_\alpha\})$ can be further simplified by denoting the species of atoms ${\epsilon}\beta + {{\rm m}athcal R}_{\epsilon}$ as ${{\rm m}athcal A}_{{\epsilon},\beta}$ (formally ${{\rm m}athcal A}_{{\epsilon},\beta} := (a_{\beta, r})_{r\in{{\rm m}athcal R}_{{\epsilon},\beta}}$ where $a_{\beta, r}\in\{0,\ldots,m-1\}$ is defined so that $p_{a_{\beta, r}} \in p_\beta + r + {{\rm m}athbb Z}^d$).
Then the sum in $\tilde E^{\rm mqc}$ can be simplified as the difference between the shift vectors of interacting atoms:
\begin{align*}
\sum_{\alpha=1}^{m-1} \big(q^h_\alpha|_T\big) D_{{{\rm m}athcal R}_{{\epsilon},\beta}} w_\alpha({\epsilon} p_\beta)
=~&
\bigg(
\sum_{\alpha=1}^{m-1} \big(q^h_\alpha|_T\big) D_r w_\alpha({\epsilon} p_\beta)
\bigg)_{r\in{{\rm m}athcal R}_{{\epsilon},\beta}}
\\=~&
\Big(
\big(q^h_{a_{\beta, r}}|_T\big) \big(w_\alpha({\epsilon} p_{a_{\beta, r}})-w_\alpha({\epsilon} p_\beta)\big)
\Big)_{r\in{{\rm m}athcal R}_{{\epsilon},\beta}}
\\=~&
\Big(
\big(q^h_{a_{\beta, r}}|_T\big) - \big(q^h_{\beta}|_T\big)
\Big)_{r\in{{\rm m}athcal R}_{{\epsilon},\beta}}
\end{align*}
This yields
\begin{equation}\label{eq:Emqc-simplified}
\tilde E^{\rm mqc}(u^h, {{\rm m}athbf q}^h)
=
\sum_{T\in{{\rm m}athcal T}h} |T|\,
\frac1m \sum_{\beta=0}^{m-1}
V_{{\epsilon},\beta}\Big(
\big(\nabla_{{{\rm m}athcal R}_{{\epsilon},\beta}} u^h + q^h_{{{\rm m}athcal A}_{{\epsilon},\beta}} - q^h_\beta\big)\big|_T
\Big)
.
\end{equation}
\end{remark}
In the next step, the shift vectors $q_\alpha$ are eliminated from \eqref{eq:QC_general_complex-lattice_space} by requiring that the variation of $\tilde E^{\rm mqc}(u^h, \{q_\alpha\})$ with respect to $q_\gamma$ in each triangle be zero:
\begin{equation}
\label{eq:QC_general_shift-vectors-equation}
\begin{split}
\frac1m \sum_{\beta=0}^{m-1}
\sum_{r\in{{\rm m}athcal R}_\beta}
V'_{{\epsilon},\beta,r}\Big(
\big(\nabla u^h|_T\big){{\rm m}athcal R}_{{\epsilon},\beta} + \sum_{\alpha=1}^{m-1} \big(q^h_\alpha|_T\big) D_{{{\rm m}athcal R}_{{\epsilon},\beta}} w_\alpha({\epsilon} p_\beta)
\Big)
D_r w_\gamma({\epsilon} p_\beta) = 0
& \\
\quad (\gamma=1,2,\ldots,m-1)
& .
\end{split}
\end{equation}
The equations \eqref{eq:QC_general_shift-vectors-equation} form a system of $m-1$ equations for $m-1$ unknowns $(q_\alpha)_{\alpha=1}^{m-1}$ in each $T$.
A solution of this system gives us the shift vectors $q_\alpha$ depending (as a rule, nonlinearly) only on the displacement gradient:
\[
{{\rm m}athbf q}^h|_T = {{\rm m}athbf q}\big(\nabla u^h|_T\big).
\]
Note that the function ${{\rm m}athbf q}({\sf F})$ does not depend on $T$, unless different periodic materials are considered in different elements $T$.
\begin{remark}\label{rem:no_discussion_of_existence}
The function ${{\rm m}athbf q}({\sf F})$ determines the lattice microstructure of a material under the macroscopic displacement gradient ${\sf F}$.
Often there is more than one lattice microstructure corresponding to a particular ${\sf F}$.
Well-posedness of equations \eqref{eq:QC_general_shift-vectors-equation} is studied in {\rm c}ite{EMing2007} under the assumption that the entire atomistic system is $H^1$-stable, and in {\rm c}ite{AbdulleLinShapeev2011_analysis} under the assumption of dominance of nearest-neighbor interaction in 1D.
In different applications there may be different {\rm c}hange{additional} conditions for choosing the unique ${{\rm m}athbf q}({\sf F})$ (this can be the condition of a global minimum of the microenergy, or proximity to a given microfunction).
{\rm c}hange{In this paper we will not focus on such additional conditions, and}
will {\rm c}hange{therefore} not discuss in detail the existence and uniqueness of solutions of the respective microscopic and macroscopic equations.
{\rm c}hange{Thus}, at this point, by ${{\rm m}athbf q}({\sf F})$ we formally denote one of the solutions of \eqref{eq:QC_general_shift-vectors-equation}, or leave ${{\rm m}athbf q}({\sf F})$ undefined if \eqref{eq:QC_general_shift-vectors-equation} admits no solutions.
In Section \ref{sec:equivalence} we will take a slightly more formal account of existence and uniqueness.
\end{remark}
We now form a QC energy with $q_\alpha$ eliminated:
\begin{equation} \label{eq:Emqc}
E^{\rm mqc}(u^h)
:=
\tilde E^{\rm mqc}\big(u^h, {{\rm m}athbf q}\big(\nabla u^h\big)\big)
.
\end{equation}
The QC equation of equilibrium now reads: find $u^h\in{{\rm m}athcal U}^h_{\rm per}$ such that
\[
\langle{\delta\hspace{-1pt} E}^{\rm mqc}(u^h), v^h\rangle_\Omega
=
F^h(v^h)
\quad\forall v^h\in{{\rm m}athcal U}^h_{\rm per},
\]
where $F^h(v^h)$ is some approximation to $\langlef,v^h\rangle_{{\rm m}athcal M}$.
The function $u^h$ gives a macroscopic displacement of the material, and one needs to compute $u^h + \sum_{\alpha=1}^{m-1} q^h_\alpha w_\alpha$ for the microstructure.
We note that since $q_\alpha$ were found by letting the variation of $\tilde E^{\rm mqc}(u^h, q_\alpha)$ with respect to $q_\alpha$ be zero, we have
\begin{equation}
\label{eq:Eintqc_simplified}
{\delta\hspace{-1pt} E}^{\rm mqc}(u^h) = {\delta\hspace{-1pt}_{u^h}\!\tilde E}^{\rm mqc}\big(u^h, {{\rm m}athbf q}\big(\nabla u^h\big)\big).
\end{equation}
\begin{remark}\label{rem:concurrent_coupling}
Instead of eliminating ${{\rm m}athbf q}^h={{\rm m}athbf q}(\nabla u^h)$, one could also look for a critical point (or a minimizer) of the energy $\tilde E^{\rm mqc}(u^h, {{\rm m}athbf q}^h)$ with respect to both $u^h$ and ${{\rm m}athbf q}^h$ {\rm c}hange{(see, e.g., {\rm c}ite{SorkinElliottTadmor})}.
\end{remark}
\subsection{Application of QC to the Simplified Model} \label{sec:QC-heterogeneous:failure}
We illustrate an application of QC to the simplified 1D model \eqref{eq:simplified_model} for two species of atoms (i.e., $m=2$), $\psi_{\epsilon}(0)=\psi_1$, $\psi_{\epsilon}\big(\smfrac{\epsilon}2\big)=\psi_2$.
If we approximate the exact solution with a piecewise affine displacement $u^h\in{{\rm m}athcal U}^h_{\rm per}$ (i.e., without introducing shift vectors, as done in the simple-lattice QC) then we will find the approximate energy
\[
\sum_{T\in{{\rm m}athcal T}h} |T|\,
\frac12 \bigg[
\psi_1\, \frac{(\nabla_r u^h)^2}2
+
\psi_2\, \frac{(\nabla_r u^h)^2}2
\bigg]
=
\sum_{T\in{{\rm m}athcal T}h} |T|\,
\frac{\psi_1+\psi_2}{2}\, \frac{(\nabla_r u^h)^2}2.
\]
Here $\tilde\psi^0 = \frac{\psi_1 + \psi_2}{2}$ is the wrong effective spring constant, since if the two springs in series are replaced with two identical springs with the effective spring constant $\psi_0$ then $\psi_0 = \frac{2\,\psi_1\psi_2}{\psi_1+\psi_2}$ (see, e.g., {\rm c}ite{ChenFish2006}).
If instead we allow for nonzero shift vector $q_1$ then the corresponding MQC energy \eqref{eq:Emqc-simplified} is
\[
\tilde E^{\rm mqc}(u^h, q^h_1)
=
\sum_{T\in{{\rm m}athcal T}h} |T|\,
\frac12 \bigg[
\psi_1\, \frac{(\nabla_r u^h + q^h_1)^2}2
+
\psi_2\, \frac{(\nabla_r u^h - q^h_1)^2}2
\bigg]
\]
with $r=\smfrac12$.
The strong form of \eqref{eq:QC_general_shift-vectors-equation} in this case can be obtained by differentiating the above expression with respect to $q^h_1$ in each $T$:
\[
\psi_1 \big((\nabla_r u^h + q^h_1)|_T\big) -
\psi_2 \big((\nabla_r u^h - q^h_1)|_T\big)
=
0,
\]
from where we find
\[
q^h_1|_T = \frac{\psi_2-\psi_1}{\psi_1+\psi_2} \big(\nabla_r u^h|_T\big)
.
\]
Substituting this back into the MQC energy (cf.\ \eqref{eq:Emqc}) yields
\begin{align*}
E^{\rm mqc}(u^h)
=~&
\sum_{T\in{{\rm m}athcal T}h} |T|\,
\frac12 \bigg[
\psi_1\, \frac12 \Big(\frac{2\,\psi_2}{\psi_1+\psi_2}\,\big(\nabla_r u^h|_T\big)\Big)^2
+
\psi_2\, \frac12 \Big(\frac{2\,\psi_1}{\psi_1+\psi_2}\,\big(\nabla_r u^h|_T\big)\Big)^2
\bigg]
\\ =~&
\sum_{T\in{{\rm m}athcal T}h} |T|\, \frac{2\,\psi_1\psi_2}{\psi_1+\psi_2}\,\frac{(\nabla_r u^h|_T)^2}2
,
\end{align*}
where the effective spring constant $\psi_0 = \frac{2\,\psi_1\psi_2}{\psi_1+\psi_2}$ is now computed correctly.
\section{Homogenization of Atomistic Media}\label{sec:homogenization}
\begin{figure}\label{fig:2d-springs}
\end{figure}
We now present another coarse graining strategy based on homogenization.
We derive below the homogenized model of the atomistic material which will be the basis for formulating and analyzing a quasicontinuum method for multilattices.
We note that there are existing works applying formal homogenization techniques to upscaling atomistic equations, see {\rm c}ite{ChenFish2006, ChenFish2006a, FishChenLi2007} and references therein.
In the present section we derive the upscaled equations for a general model of interaction in many dimensions as opposed to the pairwise interaction in 1D assumed in the upscaled equations {\rm c}ite{ChenFish2006, ChenFish2006a, FishChenLi2007}.
The upscaled equations are derived using a formal asymptotic expansion.
Rigorous error bounds for the homogenized equations can be found in the preprint {\rm c}ite{AbdulleLinShapeev2010} for the case of the linear 1D nearest-neighbor interaction and in {\rm c}ite{AbdulleLinShapeev2011_analysis} for the case of a 1D finite-range nonlinear interaction.
\subsection{Asymptotic expansion}\label{sec:homogenization:fast-and-slow}
In order to take into account the local variation of the atomistic interaction we think of the displacement as depending on a fast and a slow scale $u(x)\sim u(x,x/{\epsilon})$.
We define $x\in{{\rm m}athbb R}^d$, the macro (``slow'') variable, and $y \in {{\rm m}athbb Z}^d+{{\rm m}athcal P}$, the micro (``fast'') variable related to $x$ as $y=x/{\epsilon}$, and consider a series of functions
$u^n:{{\rm m}athbb R}^d\times ({{\rm m}athbb Z}^d+{{\rm m}athcal P})\rightarrow {{\rm m}athbb R}^d$ indexed by $n=0,1,2\ldots$
As we consider the local structure and interaction to be periodic, we assume that the functions $u^n$ are ${{\rm m}athcal P}$-periodic in the fast variable; i.e., they satisfy for all $(x,y)\in \Omega\times{{\rm m}athcal P}$
\[
u^n(x,y+j) = u^n(x,y),
\quad\forall j\in{{\rm m}athbb Z}^d \]
while the behavior with respect to $x$ is similar as considered in the previous sections
\[
u^n(x+i,y) = u^n(x,y),
\quad\forall i\in{{\rm m}athbb Z}^d .
\]
We then consider the asymptotic expansion
\begin{equation}
\label{equ:asympt_exp}
u (x) \sim \big(u^0(x) + {\epsilon} u^1(x, y) + {\epsilon}^2 u^2(x, y) + \ldots\big)\big|_{y=x/{\epsilon}}
\quad\forall x\in{{\rm m}athcal M}
.
\end{equation}
Notice that we directly assume that the homogenized solution, $u^0$, does not depend on $y$.
We now proceed as in the ``classical homogenization" {\rm c}ite{Bakhvalov1974, BensoussanLionsPapanicolaou1978, S'anchez-Palencia1980}
and insert the ansatz \eqref{equ:asympt_exp} into \eqref{eq:variational_equation}:
\begin{align*}
\Big\langle
\Big(\sum_{r\in{{\rm m}athcal R}_{\epsilon}} V'_{{\epsilon},r}\big(
D_{x,{{\rm m}athcal R}_{\epsilon}} u^0+{\epsilon} D_{x,{{\rm m}athcal R}_{\epsilon}} T_{y,{{\rm m}athcal R}_{\epsilon}} u^1+ D_{y,{{\rm m}athcal R}_{\epsilon}}u^1
+\ldots
\big)&
\\,
D_{x,r} T_{y,r} v + {\epsilon}^{-1} D_{y,r} v\Big)&\Big|_{y=x/{\epsilon}} \Big\rangle_{{\rm m}athcal M}
= \langlef, v\rangle_{{\rm m}athcal M}
,
\end{align*}
where the test functions $v=v(x,y)$ are continuous and smooth in $x\in\Omega$ and discrete in $y\in{{\rm m}athcal P}$.
Here we used the relation \eqref{eq:full_and_partial} to expand the full derivative $D_r$ through partial derivatives $D_{x,r}$, $D_{y,r}$, and the translation operator $T_{y,r}$, and used the collection-of-derivatives notation $D_{{\rm m}athcal R}$ (see Appendix \ref{sec:notations:vector_indexed} for more details).
We then extend the equation on the entire ${{\rm m}athcal M}\times{{\rm m}athcal P}$:
\begin{equation}\label{eq:homogenization:extended_discrete}
\begin{split}
\Big\langle
\sum_{r\in{{\rm m}athcal R}_{\epsilon}} V'_{{\epsilon},r}\big(
D_{x,{{\rm m}athcal R}_{\epsilon}} u^0+{\epsilon} D_{x,{{\rm m}athcal R}_{\epsilon}} T_{y,{{\rm m}athcal R}_{\epsilon}} u^1+ D_{y,{{\rm m}athcal R}_{\epsilon}}u^1
+\ldots
\big)&
\\
,
D_{x,r} T_{y,r} v + {\epsilon}^{-1} D_{y,r} v & \Big\rangle_{{{\rm m}athcal M}\times{{\rm m}athcal P}}
= \langlef, v\rangle_{{{\rm m}athcal M}\times{{\rm m}athcal P}}
.
\end{split}
\end{equation}
We now expand this equation in powers of ${\epsilon}$.
For that, we use the approximation $D_{x,r}\approx \nabla_{x,r}$ (i.e., we essentially use Taylor series to expand $D_{x,r}$), and the notations $V_{\epsilon}(\bullet; x) = V(\bullet; y)$ and ${{\rm m}athcal R}_{\epsilon}(x) = {{\rm m}athcal R}(y)$, and change a sum over ${{\rm m}athcal M}$ to an integral:
\begin{equation} \label{eq:hom:scales_separated}
\begin{split}
\Big\langle
\sum_{r\in{{\rm m}athcal R}} V'_r\big(
\nabla_{x,{{\rm m}athcal R}} u^0+{\epsilon} \nabla_{x,{{\rm m}athcal R}} T_{y,{{\rm m}athcal R}} u^1+ D_{y,{{\rm m}athcal R}}u^1+\ldots
\big)&
\\
, \nabla_{x,r} T_{y,r} v + {\epsilon}^{-1} D_{y,r} v & \Big\rangle_{\Omega\times{{\rm m}athcal P}}
= \langlef, v\rangle_{\Omega\times{{\rm m}athcal P}}
,
\end{split}
\end{equation}
where $\langle\bullet\rangle_{\Omega\times {{\rm m}athcal P}}$ is a short-hand for $\int_\Omega \langle\bullet\rangle_{{\rm m}athcal P} \dd x$.
We first collect the $O({\epsilon}^{-1})$ terms in \eqref{eq:hom:scales_separated}:
\begin{align*}
\Big\langle
\sum_{r\in{{\rm m}athcal R}} V'_r\big(\nabla_{x,{{\rm m}athcal R}} u^0+ D_{y,{{\rm m}athcal R}}u^1\big)
, D_{y,r} v \Big\rangle_{\Omega\times{{\rm m}athcal P}}
= 0
.
\end{align*}
As usual in homogenization we write the solution of this equation (of course, equipped with the zero-average boundary conditions) as $u^1(x,y) = {\rm c}hi(\nabla_x u^0(x); y)+\bar u^1(x)$, where ${\rm c}hi={\rm c}hi({\sf F}; y)\,:\,{{\rm m}athbb R}^{d\times d}\times{{\rm m}athcal P}\to{{\rm m}athbb R}^d$ solves
\begin{equation} \label{eq:chi_def}
\text{find ${\rm c}hi({\sf F}, \bullet)\in{{\rm m}athcal U}_\#({{\rm m}athcal P})$ s.t.}\quad
\Big\langle
\sum_{r\in{{\rm m}athcal R}} V'_r\big({\sf F} {{\rm m}athcal R} + D_{y,{{\rm m}athcal R}}{\rm c}hi({\sf F})\big)
, D_{y,r} \sigma \Big\rangle_{{{\rm m}athcal P}}
= 0
\quad \forall \sigma\in{{\rm m}athcal U}_\#({{\rm m}athcal P}).
\end{equation}
As earlier, ${\rm c}hi({\sf F}, y)$ can be formally understood as some solution to \eqref{eq:chi_def}, similarly to the shift vector function ${{\rm m}athbf q}^h({\sf F})$ discussed in Remark \ref{rem:no_discussion_of_existence}.
We will establish the formal equivalence of ${\rm c}hi({\sf F}, y)$ and ${{\rm m}athbf q}^h({\sf F})$ in Theorem \ref{thm:methods-are-equivalent}, hence the results in the cited references {\rm c}ite{AbdulleLinShapeev2011_analysis, EMing2007} are applicable to well-posedness of \eqref{eq:chi_def} also.
To obtain the equation for the homogenized solution $u^0(x)$, we collect the $O({\epsilon}^{0})$ terms in \eqref{eq:hom:scales_separated} and use the test function $\bar{v}$ of $x$ only:
\begin{align*}
\Big\langle
\sum_{r\in{{\rm m}athcal R}} V'_r\big(
\nabla_{x,{{\rm m}athcal R}} u^0 + D_{y,{{\rm m}athcal R}}u^1
\big)
, \nabla_{x,r} \bar{v} \Big\rangle_{\Omega\times{{\rm m}athcal P}}
= \langlef, \bar{v}\rangle_{\Omega\times{{\rm m}athcal P}}
.
\end{align*}
This leads to the homogenized equation
\begin{equation} \label{eq:homogenized}
\langle{\delta\hspace{-1pt}\Phi}^0(\nabla_x u^0), \nabla_x \bar{v}\rangle_\Omega = \langlef,\bar{v}\rangle_\Omega,
\end{equation}
or equivalently, in the strong form $- \nabla_x{\rm c}dot {\delta\hspace{-1pt}\Phi}^0(\nabla_x u^0) = f(x)$, where ${\delta\hspace{-1pt}\Phi}^0 \,:\,{{\rm m}athbb R}^{d\times d}\to{{\rm m}athbb R}^{d\times d}$ satisfies
\begin{equation}
\label{eq:delPhi0-def}
{\delta\hspace{-1pt}\Phi}^0({\sf F}) =
\Big\langle
\sum_{r\in{{\rm m}athcal R}(y)} V'_r\big({\sf F} {{\rm m}athcal R} + D_{y,{{\rm m}athcal R}}{\rm c}hi({\sf F})\big) \, r^{\!\top}
\Big\rangle_{y\in{{\rm m}athcal P}}.
\end{equation}
Thus, we obtained the equation for the homogenized displacement $u^0$ with the homogenized tensor ${\delta\hspace{-1pt}\Phi}^0$.
Equation \eqref{eq:homogenized} needs to be supplemented with boundary conditions, for instance by requiring that $u^0$ is periodic and has zero average.
As an illustrative example, in the case of a pair interaction potential we can write
$V(D_{{{\rm m}athcal R}(y)}u;y)=\sum_{r\in {{\rm m}athcal R}(y)}\Phi_r(D_r u;y)$ (cf. the Lennard--Jones potential in (\ref{eq:problem_formulation:LJ})),
consequently,
\begin{displaymath} {\delta\hspace{-1pt}\Phi}^0({\sf F}) =
\Big\langle
\sum_{r\in{{\rm m}athcal R}(y)} \Phi'_r\big({\sf F} r + D_{y,r}{\rm c}hi({\sf F})\big) \, r^{\!\top}
\Big\rangle_{y\in{{\rm m}athcal P}}.
\end{displaymath}
\begin{remark}\label{rem:homogenization:discrete}
In the above formal arguments we assumed, for simplicity, that the external force $f=f(x)$ is non-oscillating (i.e., effectively does not depend on $y$) and is defined on all of $\Omega$.
We emphasize that oscillatory external forces, (of the form $f(x,y)|_{y=x/{\epsilon}}$) could also be considered.
The homogenized equation would then depend on a proper average of the external forces.
The assumption that $f$ is defined on the entire $\Omega$ can later be relaxed once the homogenized equations are discretized on a finite element mesh.
\end{remark}
\begin{remark}
Instead of upscaling the original discrete problem \eqref{eq:variational_equation} to a continuous problem of nonlinear elasticity \eqref{eq:homogenized}, one can consider an alternative approach where the upscaled model is discrete.
For instance, one can approximate \eqref{eq:homogenization:extended_discrete} by taking discrete $x\in{{\rm m}athcal L}$ and approximating $D_{x,r}\bullet \approx (D_x\bullet) r$, where $D_x u(x) \in{{\rm m}athbb R}^{d\times d}$ is the discrete gradient of $u\in{{\rm m}athcal U}_{\rm per}({{\rm m}athcal L})$ at the point $x\in{{\rm m}athcal L}$ defined as $(D_x u(x))e_k = D_{x,e_k} u(x)$, $k=1,\ldots,d$, $e_k$ is the $k$-th standard basis vector of ${{\rm m}athbb R}^d$.
Following the described above procedure of asymptotic expansion one can derive the following upscaled equation
\begin{equation} \label{eq:discr_homogenized}
\langle{\delta\hspace{-1pt}\Phi}^0(D_x u^0), D_x v\rangle_{{\rm m}athcal L} = \langlef, v\rangle_{{\rm m}athcal L}
\quad\forall v\in{{\rm m}athcal U}_\#({{\rm m}athcal L})
,
\end{equation}
where ${\delta\hspace{-1pt}\Phi}^0$ is defined by \eqref{eq:delPhi0-def}, the same equation as for the continuum homogenization.
The equation \eqref{eq:discr_homogenized} is upscaled in the sense that ${\delta\hspace{-1pt}\Phi}^0$ no longer depends on the fast variable $y$, and we can apply the standard QC to it.
The reader can refer to our preprint {\rm c}ite{AbdulleLinShapeev2010} for a similar approach.
An advantage of the discrete homogenization is that it is not required to assume a continuous force $f$ to derive \eqref{eq:discr_homogenized}.
\end{remark}
\subsubsection*{Underlying Homogenized Energy}
We claim that, formally, the function ${\delta\hspace{-1pt}\Phi}^0({\sf F})$ defined by \eqref{eq:delPhi0-def} is the derivative of the following function
\begin{equation}\label{eq:Phi0-def}
\Phi^0({\sf F}) :=
\big\langle
V\big({\sf F} {{\rm m}athcal R} + D_{y,{{\rm m}athcal R}}{\rm c}hi({\sf F})\big)
\big\rangle_{y\in{{\rm m}athcal P}}
,
\end{equation}
where ${\rm c}hi={\rm c}hi({\sf F})$ is some solution to \eqref{eq:chi_def}.
Indeed, assuming enough regularity of $V$ and ${\rm c}hi$, we can compute the variation of \eqref{eq:Phi0-def} with respect to ${\sf F}$:
\begin{equation}\label{eq:delPhi0-variation}
{\delta\hspace{-1pt}\Phi}^0({\sf F}_0) \!:\! {\sf G}
=
\Big\langle
\sum_{r\in{{\rm m}athcal R}} V'_r\big({\sf F} {{\rm m}athcal R} + D_{y,{{\rm m}athcal R}}{\rm c}hi({\sf F}_0)\big)
{\rm c}dot ({\sf G} r + D_{y,{{\rm m}athcal R}} \delta\hspace{-1pt}{\rm c}hi({\sf F}) \!:\! {\sf G})
\Big\rangle_{y\in{{\rm m}athcal P}}
.
\end{equation}
Since $\delta\hspace{-1pt}{\rm c}hi({\sf F}) \!:\! {\sf G} \in {{\rm m}athcal U}_\#({{\rm m}athcal P})$, the second term in \eqref{eq:delPhi0-variation} drops due to \eqref{eq:chi_def} and we have
\[
{\delta\hspace{-1pt}\Phi}^0({\sf F}) \!:\! {\sf G}
=
\Big\langle
\sum_{r\in{{\rm m}athcal R}} V'_r\big({\sf F} {{\rm m}athcal R} + D_{y,{{\rm m}athcal R}}{\rm c}hi({\sf F})\big)
{\rm c}dot {\sf G} r
\Big\rangle_{y\in{{\rm m}athcal P}}
=
\Big\langle
\sum_{r\in{{\rm m}athcal R}} V'_r\big({\sf F} {{\rm m}athcal R} + D_{y,{{\rm m}athcal R}}{\rm c}hi({\sf F})\big) r^{\!\top} \!:\! {\sf G}
\Big\rangle_{y\in{{\rm m}athcal P}}
,
\]
which is consistent with \eqref{eq:delPhi0-def}.
Hence, the equations \eqref{eq:homogenized} can be written as
\[
\langle{\delta\hspace{-1pt} E}^0(u^0), v\rangle_\Omega = \langlef, v\rangle_\Omega,
\]
where
\begin{equation}
\label{eq:homogenization:generalizations:E}
E^0(u^0)
:=
\int_\Omega \Phi^0(\nabla u^0)\dd x
.
\end{equation}
The fact that the homogenized equations have an underlying energy may be important in some applications where, for instance, one chooses to use nonlinear conjugate gradient algorithms or needs to check for stability of numerical solutions.
\subsection{Application of Homogenization to the Simplified Model}\label{sec:homogenization:simplfied}
In order to make the steps of the above formal homogenization technique more transparent, we apply it to the simplified model \eqref{eq:problem_1d_linear}, written in a strong form as
\[
D_{-r} (\psi_{\epsilon}(x) D_r u(x)) = f(x)
\quad \forall x\in{{\rm m}athcal M}
,
\]
where $\psi(y) := \psi_{\epsilon}({\epsilon} y)$ and $r:=\smfrac1m$ is fixed throughout this subsection.
We calculate the application of the full derivative $D_r$ (see Appendix \ref{sec:notations:operators} for the precise definition) to \eqref{equ:asympt_exp}:
\begin{align*}
D_r u(x,y)
=~&
(D_{x,r} T_{y,r} + \smfrac1{\epsilon} D_{y,r}) (u^0(x) + {\epsilon} u^1(x,y) + \ldots)
\\ =~&
D_{x,r} u^0(x) + D_{y,r} u^1(x,y) + {\epsilon} D_{x,r} T_{y,r} u^1(x,y) + \ldots
\end{align*}
and hence insert \eqref{equ:asympt_exp} into \eqref{eq:variational_equation}:
\[
(D_{x,-r} T_{y,-r} + \smfrac1{\epsilon} D_{y,-r}) \big(
\psi(y) (D_{x,r} u^0(x) + D_{y,r} u^1(x,y) + \ldots)
\big) = f(x)
\quad \forall x\in{{\rm m}athcal M},~\forall y\in{{\rm m}athcal P}
,
\]
and change the discrete derivative with respect to $x\in{{\rm m}athcal M}$ to the continuum derivative with respect to $x\in{{\rm m}athbb R}$:
\[
(\nabla_{x,-r} T_{y,-r} + \smfrac1{\epsilon} D_{y,-r}) \big(
\psi(y) (\nabla_{x,r} u^0(x) + D_{y,r} u^1(x,y) + \ldots)
\big) = f(x)
\quad \forall x\in{{\rm m}athbb R},~\forall y\in{{\rm m}athcal P}
.
\]
Collecting the $O({\epsilon}^{-1})$ terms in this equation yields
\[
D_{y,-r} \big(
\psi(y) (\nabla_{x,r} u^0(x) + D_{y,r} u^1(x,y))
\big) = f(x)
\quad \forall x\in{{\rm m}athbb R},~\forall y\in{{\rm m}athcal P}
\]
and we can formally write the solution to this equation as
\[
u^1(x,y) = {\rm c}hi(\nabla_x u^0(x)),
\]
where
the cell problem \eqref{eq:chi_def} in the strong form reads
\[
D_{y,-r} \big(\psi\, (Fr + D_{y,r} {\rm c}hi(F))\big) = 0.
\]
For our simplified model, the cell problem admits the exact solution
\begin{equation}
D_{y,r} {\rm c}hi(F) = \frac{C}\psi - Fr.
\label{eq:linear-nn_exact_DYchi}
\end{equation}
with $C = Fr\,\langle1/\psi\rangle_{{\rm m}athcal P}$, and ${{\rm m}athcal P}$ given by \eqref{eq:simplified_model:P}.
The homogenized energy density is therefore
\begin{equation}
\Phi^0(F) = \big\langle\psi \smfrac12 (Fr + D_{y,r} {\rm c}hi)^2\big\rangle_{{\rm m}athcal P}
=
\Big\langle\psi \smfrac12 \big(Fr + \smfrac{C}\psi - Fr\big)^2\Big\rangle_{{\rm m}athcal P}
= \smfrac12\,\frac{C^2}{\langle1/\psi\rangle_{{\rm m}athcal P}}
=
\langle1/\psi\rangle_{{\rm m}athcal P}^{-1}\,\frac{(Fr)^2}2,
\label{eq:linear-nn_exact:homogenized-tensor}
\end{equation}
which yields the homogenized energy
\[
E^0(u^0) = \int_0^1 \langle1/\psi\rangle_{{\rm m}athcal P}^{-1} \frac{(\nabla_r u^0)^2}2\, \dd x
,
\]
with the correct form of the effective spring constant $\psi^0 = \langle1/\psi\rangle_{{\rm m}athcal P}^{-1}$, given by the harmonic average of $\psi$.
We note that this homogenization procedure, and the result that the effective energy density coefficient is in the form of a harmonic mean of the original coefficient, are well known for PDEs {\rm c}ite[Chap.\ 1]{BensoussanLionsPapanicolaou1978}.
The expression for $\psi^0$ is also in agreement with the one found in Section \ref{sec:QC-heterogeneous:failure} for $m=2$.
\section{Homogenized QC}\label{sec:HQC}
We formulate a numerical macro-to-micro method for treating multilattices, which we call the homogenized quasicontinuum method (HQC).
We introduce HQC in the framework of numerical homogenization. For the case of materials with known periodic structure (i.e., crystalline materials)
the HQC method will be shown to be equivalent to applying finite elements to the homogenized equations (see Theorem \ref{thm:methods-are-equivalent}).
We emphasize that HQC can be generalized to non-crystalline materials and to time-dependent zero-temperature and, possibly, finite-temperature problems.
Indeed, in Section \ref{sec:stochastic} we give an application of HQC to a stochastic material and in Section \ref{sec:unsteady} we present an application of HQC to a 1D time-dependent zero-temperature evolution.
In addition, the HQC serves a convenient framework for the error analysis {\rm c}ite{AbdulleLinShapeev2011_analysis, AbdulleLinShapeev2010}.
We present the HQC algorithm assuming that the microstructure is a function of the macroscopic displacement.
A reformulation analogous to the concurrent coupling of {\rm c}ite{SorkinElliottTadmor} is also possible (cf.\ also Remark \ref{rem:concurrent_coupling}).
\subsection{HQC Method}\label{sec:HQC:method}
The method will be presented using macro-to-micro framework as used in some numerical homogenization procedures
{\rm c}ite{Abdulle2009, EEL2007, GeersKouznetsovaBrekelmans2010, MieheBayreuther2007, TeradaKikuchi2001}.
We present the method for the case when the external force $f=f_{\epsilon}$ may be microstructure-dependent.
\subsubsection{Macroscopic affine displacement}
We again assume a partition ${{\rm m}athcal T}h$ of the domain $\Omega$ into simplicial elements $T$, recall the definition of the space ${{\rm m}athcal U}^h_{\rm per}$, \eqref{eq:UH-space}, and introduce its subspace of zero-mean functions ${{\rm m}athcal U}^h_\#\subset{{\rm m}athcal U}^h_{\rm per}$.
\subsubsection{Sampling Domains}\label{sec:HQC:method:sampling_domains}
We choose a representative position $x_T^{\rm rep}\in{{\rm m}athcal L}$ and a sampling domain $S_T^{\rm rep} := x_T^{\rm rep} + {\epsilon}{{\rm m}athcal P}$ associated with each $T\in{{\rm m}athcal T}h$.
The sampling domain is normally chosen inside $T$ (the mesh can be highly refined in certain regions and therefore some sampling domains $S_T^{\rm rep}$ may be bigger than $T$).
The sampling domains have the associated operator of averaging over the sampling domain, $\langle\bullet\rangle_{x\in S_T^{\rm rep}}$ and the functional space ${{\rm m}athcal U}_\#(S_T^{\rm rep}) = {{\rm m}athcal U}_\#({\epsilon}{{\rm m}athcal P})$ (see \eqref{eq:Uper} for the precise definition).
\subsubsection{Energy and Macro Nonlinear Form}
Define the atomistic interaction energy of the HQC method
\begin{equation}
\label{eq:Ehqc_def}
E^{\rm hqc}(u^h)
:=
\sum_{T\in{{\rm m}athcal T}h} |T|
\big\langleV_{\epsilon}(D_{{{\rm m}athcal R}_{\epsilon}} R_T(u^h))\big\rangle_{x\in S_T^{\rm rep}},
\end{equation}
where
$R_T(u^h)$, defined by \eqref{eq:HQC:microproblem}, is the microfunction constrained by $u^h$ in the sampling domain $S_T^{\rm rep}$.
The functional derivative of the above energy reads
\begin{equation}
\langle{\delta\hspace{-1pt} E}^{\rm hqc}(u^h), v^h\rangle_\Omega
=
\sum_{T\in{{\rm m}athcal T}h} |T|
\Big\langle\sum_{r\in{{\rm m}athcal R}_{\epsilon}}V'_{{\epsilon},r}(D_{{{\rm m}athcal R}_{\epsilon}} R_T(u^h)), D_r\, {\delta\hspace{-1pt} R}_T(u^h)\,v^h\Big\rangle_{x\in S_T^{\rm rep}},
\label{eq:HQC:bilinear-form}
\end{equation}
where ${\delta\hspace{-1pt} R}_T(u^h)$ is the functional derivative of the reconstruction $R_T(u^h)$ defined below.
\subsubsection{Microproblem}\label{HQC:nonlinear:cell}
Given a function $u^h\in {{\rm m}athcal U}_{\rm per}^h,$ $R_T(u^h)$
is a function such that $R_T(u^h)-u^h_{\rm lin}\in {{\rm m}athcal U}_\#(S_T^{\rm rep})$
and
\begin{equation}\label{eq:HQC:microproblem}
\Big\langle\sum_{r\in{{\rm m}athcal R}_{\epsilon}}V'_{{\epsilon},r}(D_{{{\rm m}athcal R}_{\epsilon}} R_T(u^h)),~ D_rs\Big\rangle_{x\in S_T^{\rm rep}}
=0
\quad \forall s\in {{\rm m}athcal U}_\#(S_T^{\rm rep})
,
\end{equation}
where $u^h_{\rm lin}$ is an affine extrapolation of $u^h|_T$ over the entire ${{\rm m}athbb R}^d$.
If $S_T^{\rm rep} \subset T$ then $u^h_{\rm lin}$ can be substituted with $u^h$.
\begin{remark}\label{rem:nonlinear_reconstruction_stable_equilibrium}
When modeling essentially nonlinear phenomena (e.g., martensite-austenite phase transformation), one should require that the microstructure corresponds to a stable equilibrium.
That is, one should require, in addition to \eqref{eq:HQC:microproblem}, that $w=R_T(u^h)-u^h_{\rm lin}\in {{\rm m}athcal U}_\#(S_T^{\rm rep})$ is a local minimum of $\langleV_{\epsilon}(D_{{{\rm m}athcal R}_{\epsilon}} (u^h_{\rm lin}+w))\rangle_{x\in S_T^{\rm rep}}$ {\rm c}ite[p.\ 238]{TadmorSmithBernsteinEtAl1999}.
\end{remark}
\begin{remark}\label{rem:linear_reconstruction}
In the case of linear interaction, the reconstruction $R_T$ is a linear function and hence ${\delta\hspace{-1pt} R}_T(u^h)v^h = R_T(v^h)$, which makes the derivative of the HQC energy \eqref{eq:HQC:bilinear-form} take the form
\[
\langle{\delta\hspace{-1pt} E}^{\rm hqc}(u^h), v^h\rangle_\Omega
=
\sum_{T\in{{\rm m}athcal T}h} |T|
\Big\langle\sum_{r\in{{\rm m}athcal R}_{\epsilon}}V'_{{\epsilon},r}(D_{{{\rm m}athcal R}_{\epsilon}} R_T(u^h)), D_r R_T(v^h)\Big\rangle_{x\in S_T^{\rm rep}}.
\]
\end{remark}
\begin{remark}\label{rem:simplify-the-bilinear-form}
The functional derivative of the HQC energy \eqref{eq:HQC:bilinear-form} can equivalently be written as
\begin{equation}
\langle{\delta\hspace{-1pt} E}^{\rm hqc}(u^h), v^h\rangle_\Omega
=
\sum_{T\in{{\rm m}athcal T}h} |T|
\Big\langle\sum_{r\in{{\rm m}athcal R}_{\epsilon}}V'_{{\epsilon},r}(D_{{{\rm m}athcal R}_{\epsilon}} R_T(u^h)), (\nabla_r v^h|_T)\Big\rangle_{x\in S_T^{\rm rep}},
\label{eq:HQC:bilinear-form_2}
\end{equation}
by noting that $D_r {\delta\hspace{-1pt} R}_T(u^h)v^h = D_r v^h_{\rm lin}+\big(D_r {\delta\hspace{-1pt} R}_T(u^h)v^h)-D_r v^h_{\rm lin}\big)$,
that
\[
\sum_{r\in{{\rm m}athcal R}_{\epsilon}} \langleV'_{{\epsilon},r}(D_{{{\rm m}athcal R}_{\epsilon}} R_T(u^h)),~ (D_r {\delta\hspace{-1pt} R}_T(u^h)v^h-D_r v^h_{\rm lin})\rangle_{x\in S_T^{\rm rep}}
=0,
\]
in view of \eqref{eq:HQC:microproblem},
and that $D_r v^h_{\rm lin} = \nabla_r v^h$ on each $T$.
Here we used the fact that ${\delta\hspace{-1pt} R}_T(u^h)v^h-v^h_{\rm lin} \in {{\rm m}athcal U}_\#(S_T^{\rm rep})$ which follows from taking the functional derivative of $R_T(u^h)-u^h_{\rm lin} \in {{\rm m}athcal U}_\#(S_T^{\rm rep})$.
\end{remark}
\subsubsection{Reconstruction}\label{HQC:nonlinear:reconstruction}
The functions $R_T(u^h)$ describe the microstructure of the solution inside each $S_T^{\rm rep}$.
One can reconstruct the solution describing the microstructure, $u^{h,{\rm c}}$, from the homogenized solution $u^h$ by combining $R_T(u^h)$ into a single function defined on the entire atomistic lattice ${{\rm m}athcal M}$:
\begin{equation}
\label{eq:HQC:periodic-extension}
u^{h,{\rm c}}(x) = R_T(u^h)(x)
\quad (x\in T{\rm c}ap{{\rm m}athcal M})
.
\end{equation}
That is, we effectively extend $R_T(u^h)$ periodically on each $T$.
It should be noted that \eqref{eq:HQC:periodic-extension} does not uniquely determine $u^{h,{\rm c}}(x)$ if $x\in\partial T$ for some $T\in{{\rm m}athcal T}h$.
\subsubsection{Variational Problem}
We define the homogenized quasicontinuum approximation as the solution $u^h\in {{\rm m}athcal U}_{\#}^h$ of
\begin{equation}
\langle{\delta\hspace{-1pt} E}^{\rm hqc}(u^h), v^h\rangle_\Omega
=
F^{\rm hqc}(v^h)
\quad\forall v^h\in {{\rm m}athcal U}_{\#}^h
\label{eq:HQC:problem}
\end{equation}
where
\begin{equation}
F^{\rm hqc}(v^h) =
\sum_{T\in{{\rm m}athcal T}h} |T| \langlef_{\epsilon}, v^h\rangle_{x\in S_T^{\rm rep}}
.
\label{eq:HQC:RHS}
\end{equation}
If the external force is smooth, it could instead be evaluated for a single representative atom.
In the case of linear nearest-neighbor 1D interaction it can be shown that (5.7) is well-posed and that the homogenized quasicontinuum solution $u^h$ approximates the solution $u$ of the original equations only in the $L^2$-norm. To get a good approximation in the $H^1$-norm, the reconstructed solution $u^{h,{\rm c}}$ should instead be considered. (This is analogous to the case of continuum homogenization, see discussion in Section \ref{sec:problem_formulation:simplified}.) We will report the analysis for the nonlinear case in a separate paper (see the preprint {\rm c}ite[Theorems 4 and 5]{AbdulleLinShapeev2010} for the analysis of a linear model).
\subsection{HQC Algorithm}\label{sec:HQC:algorithm}
The problem \eqref{eq:HQC:problem} is nonlinear, and its practical implementation is usually done by Newton's method.
We briefly sketch below an algorithm for solving \eqref{eq:HQC:problem}.
For Newton's method we need to compute the second derivative of the energy \eqref{eq:Ehqc_def}:
\begin{equation} \label{eq:HQC:second-variation}
\langle{{\rm d}el E}^{\rm hqc}(u^h) w^h, v^h\rangle_\Omega
=
\sum_{T\in{{\rm m}athcal T}h} |T|
\bigg\langle\sum_{r,\rho\in{{\rm m}athcal R}_{\epsilon}} V''_{{\epsilon},r,\rho}(D_{{{\rm m}athcal R}_{\epsilon}} R_T(u^h)) D_\rho {\delta\hspace{-1pt} R}_T(u^h) w^h
,~ D_r {\delta\hspace{-1pt} R}_T(u^h) v^h
\bigg\rangle_{x\in S_T^{\rm rep}}
.
\end{equation}
\subsubsection{Newton's Iterations for the Macroproblem}
The algorithm based on Newton's method consists of choosing an initial guess $u^{h,(0)}\in {{\rm m}athcal U}^h_{\#}$ and performing iterations
\begin{equation}
\big\langle{{\rm d}el E}^{\rm hqc}\big(u^{h,(n)}\big) \big(u^{h,(n+1)} - u^{h,(n)}\big), v^h\big\rangle_\Omega
=
\big\langle{\delta\hspace{-1pt} E}^{\rm hqc}\big(u^{h,(n)}\big), v^h\big\rangle_\Omega
+
F^{\rm hqc}(v^h)
\quad \forall v^h\in {{\rm m}athcal U}_{\#}^h,
\label{eq:HQC:Newton-macro-iterations}
\end{equation}
with $n=0,1,\ldots$, until $u^{h,(n+1)}$ becomes close to $u^{h,(n)}$ in a chosen norm.
To solve the linear system \eqref{eq:HQC:Newton-macro-iterations} for $u^{h,(n+1)} - u^{h,(n)}\in {{\rm m}athcal U}_{\#}^h$, we choose a nodal basis $w_k^h$ ($1\le k\le K$) of ${{\rm m}athcal U}_{\rm per}^h$.
One way to satisfy the condition $\langleu^h\rangle_\Omega=0$ would be to perform all the computations with one basis function eliminated (e.g., to consider $w_k^h$ for $2\le k\le K$), and post-process the final solution as $u^h - \langleu^h\rangle_\Omega$.
The stiffness matrix of the system \eqref{eq:HQC:Newton-macro-iterations} will thus be
\[
A_{lm}
=
\big\langle{{\rm d}el E}^{\rm hqc}\big(u^{h,(n)}\big) w_l^h, w_m^h\big\rangle_\Omega
\]
and the load vector will be
\[
b_m
=
\big\langle{\delta\hspace{-1pt} E}^{\rm hqc}\big(u^{h,(n)}\big), w_m^h\big\rangle_\Omega
+
F^{\rm hqc}(w_m^h).
\]
As given by the formula \eqref{eq:HQC:second-variation} we need to compute the solution of microproblem $R_T\big(u^{h,(n)}\big)$ on each sampling domain $S_T^{\rm rep}$ as well as its derivative ${\delta\hspace{-1pt} R}_T\big(u^{h,(n)}\big)w^h_l$.
\subsubsection{Solution of the Microproblem}
The microproblem \eqref{eq:HQC:microproblem} can also be solved with Newton's method.
For that, in each $T$ one needs to choose an initial guess $R_T^{(0)}$ to $R_T(u^{h,(n)})$, for instance $R_T^{(0)}(x) := u^{h,(n)}(x)$, and solve
\begin{align*}
\Big\langle
\sum_{r\in{{\rm m}athcal R}_{\epsilon}}V'_{{\epsilon},r}\big(D_{{{\rm m}athcal R}_{\epsilon}} R_T^{(\nu)}\big)
+
\sum_{r,\rho\in{{\rm m}athcal R}_{\epsilon}}V''_{{\epsilon},r,\rho}\big(D_{{{\rm m}athcal R}_{\epsilon}} R_T^{(\nu)}\big)
D_\rho \big(R_T^{(\nu+1)}-R_T^{(\nu)}\big)
,~ D_rs\Big\rangle_{x\in S_T^{\rm rep}}
=0
\\
\forall s\in {{\rm m}athcal U}_\#(S_T^{\rm rep})
,
\end{align*}
with respect to $R_T^{(\nu+1)}$ ($\nu=0,1,\ldots$) constrained by $R_T^{(\nu+1)} - u^{h,(n)}_{\rm lin}\in {{\rm m}athcal U}_\#(S_T^{\rm rep})$, until the difference between $R_T^{(\nu+1)}$ and $R_T^{(\nu)}$ is small in a chosen norm.
After that, we can compute ${\delta\hspace{-1pt} R}_T w^h_l = {\delta\hspace{-1pt} R}_T\big(u^{h,(n)}\big)w^h_l$ by solving
\begin{equation}
\Big\langle
\sum_{r,\rho\in{{\rm m}athcal R}_{\epsilon}}V''_{{\epsilon},r,\rho}\big(D_{{{\rm m}athcal R}_{\epsilon}} R_T^{(\nu)}\big)
D_\rho ({\delta\hspace{-1pt} R}_T w^h_l)
,~ D_rs\Big\rangle_{x\in S_T^{\rm rep}}
=0
\quad \forall s\in {{\rm m}athcal U}_\#(S_T^{\rm rep})
\label{eq:HQC:equation-for-reconstruction-variation}
\end{equation}
constrained by ${\delta\hspace{-1pt} R}_T w^h_l - (w^h_l)_{\rm lin}\in {{\rm m}athcal U}_\#(S_T^{\rm rep})$.
Notice that the gradients of all but $d+1$ basis functions $D_r (w^h_l)_{\rm lin}$ inside $T$ are zero, which implies that we essentially need to solve the problem \eqref{eq:HQC:equation-for-reconstruction-variation} $d+1$ times.
Also observe that when computing ${\delta\hspace{-1pt} R}_T\big(u^{h,(n)}\big)w^h_l$, we need to invert the same linear operator as in the final Newton iteration, which allows for some additional optimization.
\subsubsection{Possible Modifications of the Algorithm}\label{sec:HQC:algorithm:modifications}
First, notice that when solving for $u^{h,(n+1)}$ we could linearize the problem on the previous iteration $u^{h,(n)}$.
In that case we would have linear cell problems and thus we would need only outer Newton iteration, but it would be required to keep the values of the micro-solution $R_T(u^{h,(n)})$ from the previous iteration.
We notice, however, that for a practical implementation of the above algorithm it may also be required to keep the values of the micro-solution: one needs these values to initialize the inner Newton iterations; depending on the initial guess for the microproblem the iterations may converge to a wrong microstructure.
Another modification could be to compute the contribution of the external force $f_{\epsilon}$ in \eqref{eq:HQC:RHS} for a single atom in the case of no oscillations in $f_{\epsilon}$.
In the case of linear interaction, the algorithm becomes simpler: one does not need to do Newton iterations.
Nevertheless, the algorithm in Section \ref{sec:HQC:algorithm} is applicable to the linear problem where it converges in just one iteration.
\section{Equivalence of Numerical Methods for Multilattices} \label{sec:equivalence}
In this section we show the equivalence of three different methods for computing equilibrium of multilattice crystals, namely (1) the proposed HQC method, (2) finite element discretization of continuum homogenization, and (3) MQC. We only compare the interaction energy of the method, since the external forces for these methods can always be chosen same.
Below we specify the three methods that we compare.
It should be noted that given the macroscopic displacement $u^h$ we cannot guarantee uniqueness of the energy as there may be several solutions to the micro-problems corresponding to different phases of a multilattice crystal.
To rigorously address such non-uniqueness, we allow for all possible combinations of microfunctions in each element $T\in{{\rm m}athcal T}_h$, and compare the set of the resulting energies on a fixed $u^h\in{{\rm m}athcal U}^h_{\rm per}$ for the three methods.
In the following definitions we adopt the convention that for two sets, $A$ and $B$, and a number, $\gamma$, $A+B := \{a+b : a\in A, b\in B\}$ and $\gamma A := \{\gamma a:a\in A\}$.
\begin{description}
\item[Method 1. (HQC) ] For $u^h\in{{\rm m}athcal U}^h_{\rm per}$ we define the energy of the HQC method as a set $E^{\rm hqc}(u^h)\subset{{\rm m}athbb R}$,
\begin{equation}\label{eq:equivalence:Ehqc}
E^{\rm hqc}(u^h) := \sum_{T\in{{\rm m}athcal T}h} |T| \, e^{\rm hqc}_T(u^h),
\end{equation}
where $e^{\rm hqc}_T(u^h)\subset{{\rm m}athbb R}$ is defined as
\[
e^{\rm hqc}_T(u^h) := \big\{
\big\langleV_{\epsilon}(D_{{{\rm m}athcal R}_{\epsilon}} R_T(u^h))\big\rangle_{x\in S_T^{\rm rep}}
: R_T(u^h) \text{ is a solution to \eqref{eq:HQC:microproblem}}
\big\}
.
\]
\item[Method 2. (FEM for homogenized equations) ]
The energy of FEM discretization of the homogenized energy is $E^0(u^h)$, defined by
\begin{equation}\label{eq:equivalence:Ezero}
E^0(u^h) := \sum_{T\in{{\rm m}athcal T}h} |T| \, \Phi^0(\nabla u^h|_T),
\end{equation}
where $\Phi^0$ is defined as a set
\[
\Phi^0({\sf F}) :=
\big\{
\big\langle
V\big({\sf F} {{\rm m}athcal R} + D_{y,{{\rm m}athcal R}}{\rm c}hi\big)
\big\rangle_{y\in{{\rm m}athcal P}}
: {\rm c}hi \text{ is a solution to \eqref{eq:chi_def}}
\big\}
.
\]
\item[Method 3. (Multilattice QC)]
We define
\begin{equation}\label{eq:equivalence:Emqc}
E^{\rm mqc}(u^h) := \sum_{T\in{{\rm m}athcal T}h} |T| \, e^{\rm mqc}_T(u^h),
\end{equation}
where
\begin{align*}
e^{\rm mqc}_T(u^h) := \Big\{
&
\frac1m \sum_{\beta=0}^{m-1}
V_{{\epsilon},\beta}\Big(
\big(\nabla u^h|_T\big){{\rm m}athcal R}_{{\epsilon},\beta} + \sum_{\alpha=1}^{m-1} \big(q^h_\alpha|_T\big) D_{{{\rm m}athcal R}_{{\epsilon},\beta}} w_\alpha({\epsilon} p_\beta)
\Big)
\\&
: {{\rm m}athbf q}^h = {{\rm m}athbf q}(\nabla u^h) \text{ is a solution to \eqref{eq:QC_general_shift-vectors-equation}}
\Big\}.
\end{align*}
\end{description}
\begin{theorem}\label{thm:methods-are-equivalent}
Let ${{\rm m}athcal T}_h$ be a triangulation of $\Omega$ and ${{\rm m}athcal U}^h_{\rm per}$ be the associate function space defined by \eqref{eq:UH-space}.
Then for any $u^h\in{{\rm m}athcal U}^h_{\rm per}$, there holds
\[
E^{\rm hqc}(u^h) = E^0(u^h) = E^{\rm mqc}(u^h),
\]
where $E^{\rm hqc}(u^h)$, $E^0(u^h)$, $E^{\rm mqc}(u^h)$ are defined by, respectively, \eqref{eq:equivalence:Ehqc}, \eqref{eq:equivalence:Ezero}, \eqref{eq:equivalence:Emqc}.
\end{theorem}
\begin{proof}
{\it Part 1, $E^{\rm hqc}(u^h) = E^0(u^h)$.}
First, we show that the micro-functions of Methods 1 and 2, $R_T$ and ${\rm c}hi$, are related through
\begin{equation}\label{eq:methods-are-equivalent:one_two_microfunctions}
\big(R_T(u^h)\big)(x) = u^h_{\rm lin}(x) + {\epsilon} {\rm c}hi\big(\nabla u^h|_T;\smfrac x{\epsilon}\big).
\end{equation}
Indeed, denote ${\sf F} = \nabla u^h|_T$ and compute $D_r R_T(u^h)$:
\begin{equation}
\label{eq:equiv_thm_DrRk}
D_r R_T(u^h)
= D_r u^h_{\rm lin} + {\epsilon} D_r {\rm c}hi\big({\sf F};\smfrac x{\epsilon}\big)
= {\sf F} r + D_{y,r} {\rm c}hi\big({\sf F};\smfrac x{\epsilon}\big)
.
\end{equation}
The following calculation shows that the left-hand sides of \eqref{eq:HQC:microproblem} and \eqref{eq:chi_def} coincide up to a factor ${\epsilon}^{-1}$:
\begin{align*}
\Big\langle\sum_{r\in{{\rm m}athcal R}} V'_{{\epsilon},r}(D_{{{\rm m}athcal R}_{\epsilon}(x)} R_T(u^h); x), D_r s(x)\Big\rangle_{x\in S_T^{\rm rep}}
=~&
\Big\langle\sum_{r\in{{\rm m}athcal R}} V'_r(D_{{{\rm m}athcal R}(y)} R_T(u^h); y), {\epsilon}^{-1} D_{y,r} s({\epsilon} y)\Big\rangle_{y\in {{\rm m}athcal P}}
\\ =~&
{\epsilon}^{-1} \Big\langle\sum_{r\in{{\rm m}athcal R}} V'_r({\sf F} {{\rm m}athcal R} + D_{y,{{\rm m}athcal R}}\, {\rm c}hi({\sf F};y); y), D_{y,r} \sigma(y)\Big\rangle_{y\in {{\rm m}athcal P}}
,
\end{align*}
where we do the change of the independent variable $y=\smfrac x{\epsilon}$, and of the test function $\sigma(y) = s({\epsilon} y)$.
Hence \eqref{eq:methods-are-equivalent:one_two_microfunctions} indeed relates the set of solutions of \eqref{eq:HQC:microproblem} and \eqref{eq:chi_def} with ${\sf F} = \nabla u^h|_T$.
The following straightforward calculation concludes the proof of $E^{\rm hqc}(u^h) = E^0(u^h)$:
\begin{align*}
e^{\rm hqc}_T(u^h)
=~&
\big\langleV_{\epsilon} \big( D_{{{\rm m}athcal R}_{\epsilon}} R_T(u^h)\big)\big\rangle_{x\in S_T^{\rm rep}}
\\ =~&
\big\langleV_{\epsilon} \big( (\nabla u^h|_T){{\rm m}athcal R}_{\epsilon} + D_{y,{{\rm m}athcal R}_{\epsilon}} {\rm c}hi\big(\nabla u^h|_T;\smfrac x{\epsilon}\big)\big)\big\rangle_{x\in S_T^{\rm rep}}
\\ =~&
\big\langleV \big( (\nabla u^h|_T){{\rm m}athcal R} + D_{y,{{\rm m}athcal R}} {\rm c}hi(\nabla u^h|_T; y)\big)\big\rangle_{y\in {{\rm m}athcal P}}
=
\Phi^0(\nabla u^h|_T)
\end{align*}
where we used \eqref{eq:equiv_thm_DrRk} in the first step of this calculation.
{\it Part 2, $E^{\rm hqc}(u^h) = E^{\rm mqc}(u^h)$.}
The main component of the proof consists of fixing $T\in{{\rm m}athcal T}h$ and showing that $q^h_\alpha|_T$ and $R_T(u^h)$ are related through
\begin{equation}\label{eq:methods-are-equivalent:one_three_microfunctions}
q^h_\alpha|_T = U({\epsilon} p_{\alpha}) - U(0)
,
\qquad \alpha=0,\ldots,m-1
,
\end{equation}
where $U := R_T(u^h)-u^h_{\rm lin} \in {{\rm m}athcal U}_\#(S_T^{\rm rep})$.
First, assume that $R_T(u^h)$ is a solution to \eqref{eq:HQC:microproblem}.
Notice that due to ${\epsilon}{{\rm m}athcal P}$-periodicity of $U$, we can write
\[
U(x) = \sum_{\alpha=0}^{m-1} U({\epsilon} p_\alpha)\, w_\alpha(x),
\]
subtracting the constant $U(0)$ and applying $D_r$ yields
\begin{align*}
D_r U(x)
=~&
D_r \Big(-U(0)+\sum_{\alpha=0}^{m-1} U({\epsilon} p_\alpha)\, w_\alpha(x)\Big)
\\ =~&
D_r \Big(\sum_{\alpha=1}^{m-1} U({\epsilon} p_\alpha)\, w_\alpha(x)\Big)
\\ =~&
D_r\,\sum_{\alpha=1}^{m-1} (\tilde{q}^h_\alpha|_T) w_\alpha(x)
,
\end{align*}
where we used the identity $\sum_{\alpha=0}^{m-1} w_\alpha(x)=1$ for all $x\in{{\rm m}athcal M}$.
We then substitute $q^h_\alpha|_T = U({\epsilon} p_{\alpha}) - U(0)$ into \eqref{eq:QC_general_shift-vectors-equation}.
The argument of $V_{\epsilon}$ in \eqref{eq:QC_general_shift-vectors-equation} can be written as
\begin{equation}
\label{eq:Dq_eq_DR}
\begin{split}
\big(\nabla u^h|_T\big){{\rm m}athcal R}_{{\epsilon},\beta} + \sum_{\alpha=1}^{m-1} \big(q^h_\alpha|_T\big) D_{{{\rm m}athcal R}_{{\epsilon},\beta}} w_\alpha({\epsilon} p_\beta)
=~&
D_{{{\rm m}athcal R}_{{\epsilon},\beta}} \Big(u^h_{\rm lin} + \sum_{\alpha=1}^{m-1} (q^h_\alpha|_T) w_\alpha({\epsilon} p_\beta)\Big)
\\=~&
D_{{{\rm m}athcal R}_{{\epsilon},\beta}} \Big(u^h_{\rm lin} + U({\epsilon} p_\beta)\Big)
=
D_{{{\rm m}athcal R}_{\epsilon}(x)} R_T(u^h)(x)\big|_{x={\epsilon} p_\beta}
\end{split}
\end{equation}
and therefore, upon noticing that summations over $x={\epsilon} p_\beta$ and over $x\in S_T^{\rm rep}$ coincide for the ${\epsilon}{{\rm m}athcal P}$-periodic functions, we conclude that the left-hand sides of \eqref{eq:QC_general_shift-vectors-equation} and \eqref{eq:HQC:microproblem} coincide when $s(x)$ is chosen as $s(x) = w_\gamma(x)-\langlew_\gamma(x)\rangle_{x\in{\epsilon}{{\rm m}athcal P}}$, $\gamma=1\ldots,m-1$ (then $D_r s = D_r w_\gamma$).
This proves that $q^h_\alpha|_T = U({\epsilon} p_{\alpha}) - U(0)$ satisfies \eqref{eq:QC_general_shift-vectors-equation}.
To show the converse, assume that ${{\rm m}athbf q}^h$ is a solution to \eqref{eq:QC_general_shift-vectors-equation} and let $R_T$ be defined through \eqref{eq:methods-are-equivalent:one_three_microfunctions}.
We then notice that, due to calculation \eqref{eq:Dq_eq_DR}, \eqref{eq:HQC:microproblem} holds with the function $s(x) = w_\gamma(x)$, $\gamma=1\ldots,m-1$ and, obviously, with the function $s(x)=1$. These functions form a basis of ${{\rm m}athcal U}_{\rm per}({{\rm m}athcal P})={{\rm m}athcal U}_{\rm per}(S_T^{\rm rep})$, therefore \eqref{eq:HQC:microproblem} holds with any $s\in{{\rm m}athcal U}_\#(S_T^{\rm rep})\subset{{\rm m}athcal U}_{\rm per}(S_T^{\rm rep})$, that is, $R_T$ is a solution to \eqref{eq:HQC:microproblem}.
This concludes the proof that the set of solutions of \eqref{eq:QC_general_shift-vectors-equation} and \eqref{eq:HQC:microproblem} are related through \eqref{eq:methods-are-equivalent:one_three_microfunctions}.
The stated identity $E^{\rm mqc}(u^h)=E^{\rm hqc}(u^h)$ follows from $e^{\rm mqc}_T(u^h)=e^{\rm hqc}_T(u^h)$ which follows directly from \eqref{eq:Dq_eq_DR}.
\end{proof}
\begin{remark}\label{rem:QC_for_discr_hom}
One can consider yet another approach to coarse-graining multilattices, namely consider the discretely homogenized equation \eqref{eq:discr_homogenized} and apply the standard QC method (see Section \ref{sec:QC:homogeneous}) to it.
As a result one will obtain energy of $\langle\Phi^0(\nabla u^h)\rangle_\Omega$ which obviously coincides with the energy of FEM applied to the continuously homogenized equations.
\end{remark}
\begin{figure}\label{fig:methods-are-equivalent}
\end{figure}
As a corollary of Theorem \ref{thm:methods-are-equivalent} and Remark \ref{rem:QC_for_discr_hom}, the solutions corresponding to the different methods considered, being critical points of the energy, also coincide (of course, provided that the external force is treated in the same way for these methods).
Theorem \ref{thm:methods-are-equivalent} and Remark \ref{rem:QC_for_discr_hom} are graphically summarized in Figure \ref{fig:methods-are-equivalent}.
\section{Application of HQC to a Multilattice}\label{sec:multilattice}
In this section we briefly report the results of application of HQC to the multilattice {\rm c}ite[Section 8]{AbdulleLinShapeev2010}.
Note that due to Theorem \ref{thm:methods-are-equivalent}, application of MQC to the multilattice gives the same results.
We apply HQC to the 1D linear model problem, same as the one in Section \ref{sec:problem_formulation:simplified} but with a larger interaction range ${{\rm m}athcal R}$.
We compute the HQC solution, $u^h$ and the reconstructed (corrected) solution $u^{h,{\rm c}}$, and compare it to the exact solution $u$.
\begin{figure}\label{fig:results_multilattice_linear}
\end{figure}
We prove (for nearest-neighbor interaction) and observe in numerical experiments that $\|u^{h,{\rm c}}-u\|_{H^1({{\rm m}athcal M})}$ converges with the first order in $h$, where $h={\rm m}ax_{T\in{{\rm m}athcal T}} {\rm diam(T)}$ and $\|\bullet\|_{H^1({{\rm m}athcal M})}$ denotes the discrete $H^1$-norm on the lattice ${{\rm m}athcal M}$.
Furthermore, we show that $\|u^h-u\|_{L^2({{\rm m}athcal M})} \leq C_1 h^2 + C_2 {\epsilon}$, that is, the $L^2$-error converges with the second order up to some point where it stagnates at the level of $C_2{\epsilon}$ as $h$ is further refined.
The $H^1$-error of $u^h-u$, on the other hand, stays essentially constant as $h$ is refined.
The results of our numerical experiments are shown in Fig. \ref{fig:results_multilattice_linear}.
The results of application of HQC to a nonlinear interaction are qualitatively same as the presented results for the linear interaction.
\section{Application of HQC to Stochastic Materials}\label{sec:stochastic}
The HQC method can readily be generalized for non-crystalline materials such as glasses or complex metallic alloys.
For that, lacking the period of the microstructure ${{\rm m}athcal P}$, one needs only to take $S_T^{\rm rep}$ large enough to accurately represent the material's microstructure.
In this section we present an example of such computation.
In addition to taking $S_T^{\rm rep}$ large enough, one could also average over an ensemble of samples of different microstructures for a given macroscopic displacement gradient $\nabla u^h|_T$ in each element $T$; however, we do not pursue this in the present work.
We refer to {\rm c}ite{BlancLeLions2007_stochastic_lattices, GloriaOtto2011_variance_estimate} and references therein for theoretical studies of stochastic homogenization of lattice energies.
We take an atomistic system of $2048\times 2048$ atoms.
That is, we choose ${\epsilon}=\smfrac1{2048}$ and ${{\rm m}athcal M} = {\epsilon} {{\rm m}athbb Z}^2 {\rm c}ap [0,1)^2$.
The atomistic bonds are chosen to have quadratic interaction energy,
\[
E(u) = \Big\langle \sum_{r\in{{\rm m}athcal R}} \smfrac12 \psi_{{\epsilon},r}(x) |D_r u|^2 \Big\rangle_{x\in{{\rm m}athcal M}}
\]
with ${{\rm m}athcal R}=\{(1,0),(0,1),(1,1),(-1,1)\}$, as illustrated in Fig.\ \ref{fig:2d_springs_short}.
The bonds' strengths $\psi_{{\epsilon},r}$ are randomly generated with a uniform distribution between $0.5$ and $10$ for $r=(1,0)$ and $r=(0,1)$ (i.e., vertical and horizontal bonds) and between $0.1$ and $5$ for $r=(1,1)$ and $r=(-1,1)$ (i.e., diagonal bonds).
Such choice of $\psi_{{\epsilon},r}$ leads to interaction energy $E(u)$ being a convex function of $u$.
Only a single realization of $\psi_{{\epsilon},r}$ is used for this test.
The external force is chosen as
\[
f(x_1,x_2) = 10 e^{-{\rm c}os(\pi x_1)^2-{\rm c}os(\pi x_2)^2} \left(\begin{array}{c} \sin(2\pi x_1) \\ \sin(2\pi x_2) \end{array}\right)
-\bar{f},
\]
where $\bar{f}$ is determined so that the average of $f$ is zero.
The equilibrium configuration for a system with $32\times32$ atoms is illustrated in Fig.\ \ref{fig:2d-solution-rand-at-large}.
We stress that we no longer have the period of the microstructure ${{\rm m}athcal P}$, and the associated representation of the energy \eqref{eq:E_alt} which was needed in formulation of the MQC method or applying the formal homogenization techniques.
\begin{figure}\label{fig:2d_springs_short}
\label{fig:2d-solution-rand-at-large}
\label{fig:2d_springs_illustration}
\end{figure}
We apply the HQC algorithm to the described system.
We choose the sampling domain $S_T^{\rm rep}$ as a subsystem of $N_{\rm rep}\times N_{\rm rep}$ atoms.
We then compute the HQC solution and compare it to the exact solution of the problem.
A structured triangular uniform mesh with right-angled triangular elements with the leg size $h=\smfrac14,\smfrac18,\ldots$ is used.
For comparison, we also produce the results of calculation with an affine displacements for computing the effective elasticity tensor in each element $T$; i.e., when atoms are not allowed to relax to equilibrium when an external displacement gradient ${\sf F}$ is applied.
The relative errors of the interaction energy of HQC and affine-displacement solutions ($E^{\rm hqc}$ and $E^{\rm ad}$, respectively) as compared to the energy of the exact solution $E$, are plotted in Fig.\ \ref{fig:2d-testcase2-error} for different mesh size $h$ and different sampling domain size $N_{\rm rep}$.
A second-order convergence of HQC and absence of convergence of the solution computed according to the affine deformation can be observed.
One can also see that with $N_{\rm rep}=128$ (and even with $N_{\rm rep}=16$) one can get a rather accurate numerical solution.
\begin{figure}\label{fig:2d-testcase2-error}
\end{figure}
\section{Application of HQC to Time-dependent Problems}\label{sec:unsteady}
We apply the proposed HQC method to the 1D evolution of a multilattice, assumed to be slow (i.e., with no thermal oscillations) described by the following equations
\begin{subeqnarray}
\langle M^{\epsilon} {\rm d}ot u, v\rangle_{{\rm m}athcal M} &=& \langle{\delta\hspace{-1pt} E}(u), v\rangle_{{\rm m}athcal M}
\quad \forall v\in {{\rm m}athcal U}_{{\rm per}}({{\rm m}athcal M})
\\
u|_{t=0} &=& u^0
\\
\dot{u}|_{t=0} &=& 0.
\label{eq:unsteady:variational_problem_generic}
\end{subeqnarray}
Here $u=u(t,x) \in C^2([0,T]; {{\rm m}athcal U}_{{\rm per}}({{\rm m}athcal M}))$ is the time-dependent displacement of atom $x$, $u^0 = u^0(x) \in {{\rm m}athcal U}_{{\rm per}}({{\rm m}athcal M})$ is the initial displacement, $M^{\epsilon}(x) = M\big(\smfrac x{\epsilon}\big)$ is the mass of atom $x$, $\dot u = \frac{{\rm d}}{{\rm d} t} u$, ${\rm d}ot u = \frac{{\rm d}^2}{{\rm d} t^2} u$.
The energy $E(u)$ of a deformation of the multilattice ${{\rm m}athcal M}$ is as defined in Section \ref{sec:problem_formulation:full}.
The masses $M=M(y)$, as well as the interaction, is a ${{\rm m}athcal P}$-periodic function.
We assume no external forces.
One can, assuming no fast oscillations in time of the microstructure, perform the two-scale expansion procedure for the time-depend case (which closely follows the continuum case {\rm c}ite{BensoussanLionsPapanicolaou1978})
\begin{equation}\label{eq:unsteady:homogenized}
\langle M^0 {\rm d}ot u, v\rangle_{{\rm m}athcal M} =\langle{\delta\hspace{-1pt} E}^0(u), v\rangle_{{\rm m}athcal M}
\quad \forall v\in {{\rm m}athcal U}_{{\rm per}}({{\rm m}athcal M}),
\end{equation}
where $E^0(u)$ is given by \eqref{eq:homogenization:generalizations:E} and $M^0 = \langleM\rangle_{{\rm m}athcal P}$, and likewise formulate the macro-to-micro discretization {\rm c}ite{AbdulleGrote2011, EngquistHolstRunborg2011}
\[
\langle M^0 {\rm d}ot u^h, v^h\rangle_{{\rm m}athcal M} =\langle{\delta\hspace{-1pt} E}^{\rm hqc}(u^h), v^h\rangle_{{\rm m}athcal M}
\quad \forall v\in {{\rm m}athcal U}_{{\rm per}}^h.
\]
For the numerical test we take the same lattices as for the simplified model with $m=2$ (see Section \ref{sec:problem_formulation:simplified}).
The atoms interact with the Lennard--Jones potential \eqref{eq:problem_formulation:LJ} with
\[
s_{x,x+{\epsilon} r} = \begin{cases}
0.4 & \frac x{\epsilon} \text{ is half-integer} \\
1.6 & \frac x{\epsilon} \text{ is integer},
\end{cases}
\qquad
\ell_{x,x+{\epsilon} r} = \begin{cases}
1.01 & \frac x{\epsilon} \text{ is half-integer} \\
0.99 & \frac x{\epsilon} \text{ is integer},
\end{cases}
\]
and the cut-off distance $R=3$.
The masses of atoms are
\[
M^{\epsilon}(x) = \begin{cases}
1 & \frac x{\epsilon} \text{ is half-integer} \\
2 & \frac x{\epsilon} \text{ is integer}.
\end{cases}
\]
The atomistic system contains $\#({{\rm m}athcal M})=2^{14}$ atoms.
The initial displacement has to conform with the assumption of absence of fast vibrations of the microstructure.
It is chosen in the following way:
First, we compute an equilibrium displacement $u$; i.e., such that $\langle{\delta\hspace{-1pt} E}(u), v\rangle_{{\rm m}athcal M}=0$ for all $v\in{{\rm m}athcal U}_{{\rm per}}({{\rm m}athcal M})$.
Second, we compute an eigenvector of ${{\rm d}el E}(u)$, $u_1$, corresponding to the mode oscillating most slowly.
Then, the initial displacement is taken to be $u^0 = u + 0.01\,\frac{u_1}{\|D u_1\|_{L^{\infty}}}$.
With such an initial displacement, the solution remains smooth (i.e., most of energy of the solution is contained in long wavelength modes) for times comparable to the largest oscillation period, and one can compare a QC approximation of the solution with the exact solution.
Beyond this critical time, the shock waves appear, which cause fast vibrations of the microstructure past them and hence make the approximation \eqref{eq:unsteady:homogenized} invalid.
We compare the reconstructed solution obtained by the HQC discretization in space with the reference solution obtained in the full atomistic computation.
The reconstruction of the HQC solution is performed similarly as described in Section \ref{HQC:nonlinear:reconstruction}.
The sampling domains $S_T^{\rm rep}$ were chosen to be ${\epsilon}{{\rm m}athcal P}$ up to a shift in ${\epsilon}{{\rm m}athbb Z}$.
The HQC discretization is performed on a sequence of meshes with $h=\smfrac14,\smfrac18,\ldots$.
For the time integration, we use the Verlet method with the timestep $\tau = \frac{1}{20} h$ for the HQC solution and $\tau=\frac{1}{20} {\epsilon}$ for the reference atomistic solution.
We run the computation until $T=\frac{1}{20}$, which corresponds to about a quarter of a period of oscillation of the solution.
\begin{figure}\label{fig:dynamic_error}
\end{figure}
The errors in (the discrete analogues of) $L^\infty([0,T]; L^2(\Omega))$-norm and $L^2([0,T]; H^1(\Omega))$-norm are presented in Fig.~\ref{fig:dynamic_error}.
One can clearly observe for relatively large $h$ a second order convergence in the $L^2(\Omega)$-norm and a first order convergence in the $H^1(\Omega)$-norm, and the convergence seems to stagnate as $h$ is further reduced.
\section{Summary and Concluding Remarks}\label{sec:conclusion}
We have considered the problem of equilibrium of multilattice crystalline materials and discussed the application of the (local) QC method {\rm c}ite{TadmorSmithBernsteinEtAl1999} for such materials.
We then have proposed a homogenization framework and, based on it, proposed a numerical macro-to-micro method which we called HQC.
We have shown that the three methods, namely the HQC method, the QC method applied to the discretely homogenized equations, and the multilattice QC, are equivalent.
Despite equivalence of the methods for statics of multilattice, we argue that the homogenization framework developed in this paper has several advantages.
First, it contributes to a better understanding of the multilattice QC method and provides a link to the existing theory of homogenization of PDEs.
In particular, we have generalized and applied the HQC method to the case of random materials and to the unsteady case, numerically demonstrating convergence of the proposed numerical method.
Second, the developed homogenization framework allows for application of analytical techniques available in the homogenization theory and thus seems most promising for convergence analysis of numerical methods for multilattices.
We refer to our preprint {\rm c}ite{AbdulleLinShapeev2010} and an ongoing work {\rm c}ite{AbdulleLinShapeev2011_analysis} for an example of such analysis.
We also note that the extension of the homogenization technique proposed in this paper to atomistic materials at finite temperature is of high interest.
\section*{Acknowledgments}
We thank the three anonymous referees for many comments that led to significant improvement of this paper.
\appendix
\section{Notations}\label{sec:notations}
In this appendix we gather the frequently used notations.
\subsection{Function spaces}\label{sec:notations:spaces}
For any finite set $S\subset {{\rm m}athbb R}^d$, we define the discrete averaging (integration) operator $\langle\bullet\rangle_S$ by
\[
\langleu\rangle_S := \frac{1}{\#(S)}\sum_{x\in S} u(x),
\]
and sometimes, more verbosely, as $\langleu(x)\rangle_{x\in S}$.
Here $\#(S)$ is the number of elements in the set $S$.
We consider discrete periodic functions (e.g., displacements or external forces) with the periodic cell $\Omega = [0,1)^d$ ($d\in{{\rm m}athbb N}$), and the lattice (being, actually, the discrete periodic cell) $S \subset \Omega$ ($S={{\rm m}athcal L}, {{\rm m}athcal M}$) containing a finite number of points: $\#(S)<\infty$.
The periodic extension of the lattice is denoted by $S_{\rm per} = S+{{\rm m}athbb Z}^d$.
Such space of periodic functions is denoted by
\begin{equation}\label{eq:Uper}
{{\rm m}athcal U}_{\rm per}(S)= \big\{u: S_{\rm per}\to{{\rm m}athbb R}: \ u(x+a) = u(x)~\forall x\in S,~\forall a\in{{\rm m}athbb Z}^d \big\},
\end{equation}
and the space of periodic functions with zero average by
\[
{{\rm m}athcal U}_{\#}(S)= \big\{u\in {{\rm m}athcal U}_{\rm per}(S):~\langleu\rangle_S=0\big\}.
\]
We do not have separate notations for scalar and vector-valued functions and explicitly state whether the function is scalar or vector-valued when it may cause ambiguity.
Similarly to the discrete averaging, we also use continuum averaging notation $\langleu\rangle_\Omega := \int_\Omega u(x) \dd x$, and for functions of two variables we write $\langlev\rangle_{S_1\times S_2} := \big\langle\langlev\rangle_{S_2}\big\rangle_{S_1}$, where each $S_i$ ($i=1,2$) can be either continuous or discrete.
For vector-valued $u=u(x)$ and $v=v(x)$ we denote the pointwise scalar product by $u{\rm c}dot v$ (i.e., $(u{\rm c}dot v)(x) = u(x){\rm c}dot v(x)$)
and the semi-inner product in ${{\rm m}athcal U}_{\rm per}({{\rm m}athcal L})$ by
\[
\langleu,v\rangle_{{\rm m}athcal L} = \langleu{\rm c}dot v\rangle_{{\rm m}athcal L} = \frac{1}{\#({{\rm m}athcal L})}\sum_{x\in{{\rm m}athcal L}} u(x){\rm c}dot v(x).
\]
(It is a proper inner product only in ${{\rm m}athcal U}_\#({{\rm m}athcal L})$.)
We similarly define the pointwise scalar product and the (semi-)inner product for functions of continuum variable and for functions of several continuum or discrete variables.
\subsection{Operators}\label{sec:notations:operators}
For $u\,:\,S\to {{\rm m}athbb R}^d$ ($S={{\rm m}athcal L}, {{\rm m}athcal M}$) we introduce the finite difference $D_{x,r} u$
\[
D_{x,r} u(x) := \frac{u(x + {\epsilon} r)-u(x)}{{\epsilon}}
\qquad(\text{for }x\in S,~r\in{{\rm m}athbb R}^d \text{~~such that }x+{\epsilon} r\in S)
.
\]
In addition to differentiation operators, we define for $u\in {{\rm m}athcal U}_{\rm per}({{\rm m}athcal L}_1)$, the translation operator
$T_x u\in {{\rm m}athcal U}_{\rm per}({{\rm m}athcal L}_1)$
\[
T_{x,r} u(x) := u(x+{\epsilon} r)
\qquad(\text{for }x\in S,~r\in{{\rm m}athbb R}^d \text{~~such that }x+{\epsilon} r\in S)
.
\]
The definitions of the discrete derivative and translation generalize to functions of two variables by considering the partial discrete derivative and translation operators; i.e.,
$D_{x,r},T_{x,r}$ applied to $u(\bullet,y)$ and
$D_{y,r},T_{y,r}$ applied to $u(x,\bullet)$.
In homogenization we consider ``traces on diagonal'' of functions of two variables, $v=v(x,\smfrac x{\epsilon})$.
For such functions we introduce full translation and full derivative operators $T_r := T_{x,r} T_{y,r}$, $D_r := \smfrac1{\epsilon} (T_r-I)$ so that
\begin{equation}
\label{eq:full_derivative_relation}
(T_r u)|_{y=\smfrac x{\epsilon}} = T_{x,r} \Big(u|_{y=\smfrac x{\epsilon}}\Big),
\quad
\text{and}
\qquad
(D_r u)|_{y=\smfrac x{\epsilon}} = D_{x,r} \Big(u|_{y=\smfrac x{\epsilon}}\Big)
.
\end{equation}
The following relates the partial and the full derivatives:
\begin{align} \notag
D_r
=~& \smfrac1{\epsilon} (T_{x,r} T_{y,r}-I)
\\ =~& \notag
\smfrac1{\epsilon} (T_{x,r} T_{y,r}-T_{y,r}) + \smfrac1{\epsilon} (T_{y,r} - I)
\\ =~& \label{eq:full_and_partial}
D_{x,r} T_{y,r} + \smfrac 1{\epsilon} D_{y,r}
.
\end{align}
Notice that the variables $x$ and $y$ are not symmetric in the definition of full derivative.
If a function does not depend on $y$ then the full derivative coincides with the derivative in $x$ (likewise for the translation).
Hence, for functions of $x$ only, we sometimes omit the subscript $x$ in the operators $D_{x,r}$ and $T_{x,r}$.
For continuous functions we denote $\nabla u$ a gradient of $u$ and $\nabla_r u=(\nabla u){\rm c}dot r$ a directional derivative.
For a vector-valued function $u$, the directional derivative, $\nabla_r u$ is defined componentwise and the gradient $\nabla u$ is a matrix such that $\nabla_r u = (\nabla u)r$.
\subsection{Functions of Vector-indexed Variables}\label{sec:notations:vector_indexed}
We consider a general form of interaction, where the energy of each atom depends arbitrarily on relative displacements of all the nearby atoms.
Namely, for the ``interaction neighborhood'' ${{\rm m}athcal R}=\{r_1,\ldots,r_k\}$ we consider functions
\[
V(D_{r_1} u, D_{r_2} u, \ldots, D_{r_k} u).
\]
Since the interaction neighborhood may be different for different atoms (recall that we consider multilattices) and contain different number of neighbors $k$, we index derivatives directly with $r\in{{\rm m}athcal R}$.
That is, we use the following notation for tuples $\alpha$ indexed with $r\in{{\rm m}athcal R}$:
\[
(\alpha_r)_{r\in{{\rm m}athcal R}} := (\alpha_{r_1},\ldots,\alpha_{r_k})
\quad \text{for } {{\rm m}athcal R}=\{r_1,\ldots,r_k\}
\]
and define
\[
D_{{\rm m}athcal R} u := (D_r u)_{r\in{{\rm m}athcal R}},
\qquad
\nabla_{{\rm m}athcal R} u := (\nabla_r u)_{r\in{{\rm m}athcal R}}.
\]
Thus, for the functions of ${{\rm m}athcal R}$-indexed tuples we write
\[
V(D_{{\rm m}athcal R} u) := V(D_{r_1} u, D_{r_2} u, \ldots, D_{r_k} u).
\]
The common algebraic operations on ${{\rm m}athcal R}$-indexed tuples are taken componentwise, e.g.:
\begin{equation}
\label{eq:vector_indexed}
D_{{\rm m}athcal R} u + D_{{\rm m}athcal R} v = (D_r u+D_r v)_{r\in{{\rm m}athcal R}},
\qquad
{\sf F} {{\rm m}athcal R} = ({\sf F} r)_{r\in{{\rm m}athcal R}}
\quad\text{ etc.},
\end{equation}
which is fully analogous to the algebraic operations on $k$-dimensional vectors.
A partial derivative of $V(D_{{\rm m}athcal R} u)$ with respect to $D_r u$ ($r\in{{\rm m}athcal R}$) is denoted by $V'_r(D_{{\rm m}athcal R} u)$.
\end{document} |
\betaegin{document}
\betaegin{abstract}
Using the rudiments of pde jets theory in a nonstandard setting, we first deepen and extend previous nonstandard existence results for generalized solutions of linear differential equations and second extend the previous results for linear differential equations to a much broader class of nonlinear differential equations.
\epsilonnd{abstract}
\title{Generalized Solutions of Nonlinear Differential Equations \ A Nonstandard Jets Approach
}
\tableofcontents
\betaegin{abstract} We extend a recently proved result of Todorov
asserting the existence of generalized solutions of very general
linear partial differential operators. To linear operators whose
symbols vanish only to finite order, we prove existence of solutions
to infinite order on $^\sigma\betabr^m$. We prove existence of generalized
solutions for nonlinear operators satisfying $^\sigma\! PCP$, a
condition that implies Todorov's condition in the linear case. In
the conclusion, we prove that generalized solutions on $^\sigma\betabr^m$
are remarkably abundant .
\epsilonnd{abstract}
\sigmaection{Introduction} In this paper, we extend the results
of Todorov \cite{Todorov96} (on the existence of generalized solutions for a general set of differential operators) in two directions.
If $P$ is a linear partial differential operator of order r, written as $P\in LPDO(r)$, with
$C^{\infty}$ coefficients, and ${\lambda}_P$ is its (total) symbol, then Todorov
proves the existence of generalized solutions $f$ for the equation $\rz P(f)(\rz x)=\rz g(\rz x)$ for all $x\in\betabr^m$ outside $\mathcal Z_{{\lambda}_P}$ and for quite
general $g$, where ${\mathcal Z}_h$ denotes the
$\{x\in\betabr^n:h(x)=0\}$. From a slightly different perspective, Todorov's result says that for a general set of standard $g$,
there exists internal $f\in\rzC^\infty(\betabr^m,\betabr)$, such that $(\rz\! P(f)-\rz\! g)|_{^\sigma\betabr^m}=0$.
($^\sigma\betabr^m$ denotes the standard vectors in the internal vector space $\rz\betabr^m$).
That is, $\rz P(f)-g$ vanishes pointwise, ie., has $0^{th}$ order contact with $\rz\betabr^n$ at each point of $^\sigma\betabr^m$. The standard geometry and jet definitions with respect to PDEs (partial differential equations) are recalled in the next section. The first extension of Todorov in this paper is to give a straightforward construction that there exists
internal smooth maps $f$ such that $\rz\! P(f)-\rz\! g$ has
infinite order contact with $\rz\betabr^n$ at all points of $^\sigma\betabr^m$. (See Theorem \ref{thm:lin eqn,infinite contact
soln}.) Another way of stating this is that Todorov solves the equations at each standard $x$; here we solve the equations along with the infinite family of integrability differential equations associated to the given differential equation at each such $x$. But the two corollaries carry the critical import of this
theorem. Corollary \ref{cor: infinite order Todorov} is the
infinite order direct descendant of Todorov's result. It depends
on the standard jet space work done in Section \ref{section: standard jet work,prolong and
rank}. The needed standard statement following from this work lies in Corollary \ref{lem:symb prolong is
surj}. The second corollary becomes possible only within the
perspective of this paper. We can consider those partial differential operators
whose (total) symbols vanish to some finite order, ie., have any finite order contact with $\rz\betabr^m$ along $^\sigma\betabr^m$; see Section \ref{section:prolong and
vanishing}\;. Note here that Todorov only consider the $0^{th}$ order vanishing case. Our Corollary \ref{cor: infinite order solns for finite
contact} says that as long as the vanishing order of $\rz g$ at
standard points is controlled by that of the symbol of $\rz P$, we
can find internal smooth $f$ solving $\rz P(f)-\rz g =0$ to
infinite order on $^\sigma\betabr^m$. Such a theorem is unstatable
within the venue of Todorov's setting. The work in standard
geometry allowing the proof of this result occurs in section \ref{section:prolong and
vanishing}; see Corollary \ref{cor:removing finite zeroes}. Parenthetically, it's conceivable that the PDE jet results of Sections \ref{section:prolong and vanishing} and \ref{section: standard jet work,prolong and rank} exist in the literature, but the author could not find them.
The overwhelming bulk of the work in this paper concerns the linear theory. But, the nonlinear PDE jet framework and NSA fit quite well together, and so the second direction of extension of the result of Todorov is into nonlinear partial differential equations, NLPDEs. There is
a well developed theory of nonlinear partial differential equations within the jet bundle framework,
exemplified in the texts of Pommaret,\cite{Pommaret1978}
Olver, \cite{Olver1993} and Vinogradov, \cite{VinogradovGeomJetSpNLDE1986}. Nonstandard analysis is as comfortable in this framework as in the linear.
So, in Section \ref{section: nonlinear work}, we introduce simple conditions, $PCP$, and
${} ^\sigma PCP$, on the symbols of general NLPDEs of finite order, and give an easy proof of
existence of generalized solutions in the sense of Todorov for
those NPDO's satisfying these criteria. We show that Todorov's
nonvanishing condition on the symbol implies that his $LPDO$'s
satisfy ${}^\sigma PCP$. But, our theorem asserts the existence of generalized solutions in the
far broader nonlinear arena.
The standard import of these results is yet to be
worked out. See the conclusion for a curious result on this. We prove a result
that might appear startling: almost all internal smooth functions are solutions on $^\sigma\betabr^m$ of any standard differential
operator that has the zero function as a solution.
It seems that the work of Baty, etal., \cite{BatyOneDimlGas2007}, might be a useful framing for this. That is, their analysis needs a lot of elbow room on the infinitesimal level to allow adjusting eg., the infinitesimal widths of Heaviside jumps, etc. It seems that the results here might be interpreted as saying that the formal (nonstandard) jet theory of symbols allows such roominess for such empirically motivated adjustments.
The author relies on a jet bundle framework when some might consider it too
big a machine for the job. Yet, from the point of view of
nonstandard analysis, the jet bundle framework is natural and eg., allows an easy
generalization of Todorov's result to the nonlinear case. The total and principal symbols of a differential operator have a natural
geometric setting which when extended to the nonstandard world
allows a geometric consideration of generalized solutions and, in fact generalized differential operators vis *smooth symbols.
Todorov defines his differential equation and constructs his
solutions within spaces of generalized functions defined on $\rz\Omega$, where $\Omega$ an arbitrary
open subset of $\betabr^m$, and gets his localizable differential
algebra of generalized functions by `quotienting' out by the
parts of the $\rz C^{\infty}$ functions defined on
nonnearstandard points of $\rz \Omega$. (Note also his NSA jazzed up version of the constructions of Colombeau, Oberguggenberger, and company in eg., \cite{TodorVern2008}, of which we will say more later).
On the other hand, my paper focuses almost exclusively
on extending Todorov's existence result to more general classes
of differential operators and very little on a broader analysis
of his differential algebra of generalized functions. (In a follow up to this paper, we will refine the results appearing here within the aforementioned nonstandard version of Colombeau's algebra of generalized functions constructed by Todovov and Oberguggenberger, \cite{OT98}).
Accordingly, our constructions occur on all of $\rz\betabr^m_{nes}$.
If we restrict to differential operators whose
finite vanishing order sets don't have infinitely many components
so that they have no
nontrivial *limiting behavior at nonnearstandard points,
the results here should hold without change for Todorov's
localizable differential algebras.
The
geometric theory of differential equations and their symmetries,
as exemplified in eg., Olver, \cite{Olver1993} and Pommaret, \cite{Pommaret1978}.
is a natural framework within which to integrate the
generalizing notions of NSA. This is the first of a series of
papers in which the author intends to attempt a theory of generalized solutions (existence and regularity) and symmetries of
differential equations within the context of the extensive jet theory.
Note that although this approach seems to be new,
there are a growing number of research programs moving beyond classical approaches; see eg., Colombeau, \cite{Colombeau1985}, Oberguggenberger, \cite{Oberguggengerger1992}
and Rossinger, \cite{Rosinger1987}. For a good overview of the new theories of generalized functions, see eg., Hoskins and Pinto, \cite{HoskinsPintoGeneralizFcns2005}, and for specific surveys of the obstacles to the construction of
a nonlinear generalization of distributions and a comparison of
the characteristics of the these new theories, see
Oberguggenberger, \cite{Oberguggenberger1995a} and more recently Colombeau, \cite{ColombeauSurvGeneralizFcnsInfinites2006}.
Note , in particular the flury of work extending the arena of Colombeau algebras into mathematical physics involving diferential geometry and topology that Kunzinger, \cite{Kunzinger2007}, summarizes.
Further note that Oberguggenberger and
Todorov, see eg., \cite{OT98}, have shown how much of the
theoretical foundations of Schwarz type Colombeau algebras can be simplified
and strengthened within the venue of nonstandard constructions and recent work of Todorov and coworkers, see eg., \cite{TodorVern2008}, have extended the results with this model. See also Todorov's lecture notes, \cite{TodorovNotes}.
None of these approaches consider the symbol of the
differential operator, and the nonstandard extension of its geometric milleu as the primary
object of study. This is the perspective of the current work. Finally, it seems that nonstandard methods are much more encompassing than the impressive work of the Colombeau school of generalized functions; eg., consider the work of the mathematical physicists working around Baty, eg., see \cite{BatyOneDimlGas2007} and \cite{BatyShockwaveStar2009}. Note, in particular the perspective of Baty, etal on p37 of \cite{BatyOneDimlGas2007} (with respect to the benefits of nonstandard methods) where they note that the generalized functions of the Columbeau school
\betaegin{quote}
...are not smooth functions and do not support all of the operations of ordinary algebra and calculus, the multiplication of singular generalized functions is accomplished via a weak equality called association. In contrast to such calculations, the objects manipulated in equations (4.4) to (4.13) (and indeed in the following section of this report) are smooth nonstandard functions.
\epsilonnd{quote}
In reading their papers, it's clear that their need for ``ordinary algebra'', etc., is critical to their analysis. The author also believes that, here also, having at hand the full capacity of mathematics via transfer straightforwardly allows many of the constructions of this paper.
\sigmaection{Some jet PDE basics and nonstandard variations}
\sigmaubsection{Nonstandard analysis}
\sigmaubsubsection{Resources}
Good introductions to nonstandard analysis abound. One might start with the pedestrian tour of its basics in the introduction of the authors dissertation, \cite{McGaffeyPhD}; then get deeper with the introduction of Lindstr{\o}m, \cite{Lindstrom1988} and follow this with the constructive introduction of Henson, \cite{Henson1997}. There is also Nelson's (axiomatic) internal set theory, a theory with similar goals and achievements to Robinson's (currently superstructure/ultrapower) nonstandard analysis; a good text being that of Lutz and Goze, \cite{LutzGoze1981}. One could also check out strategic outgrowths from these nonstandard schools, eg., the work of Di Nasso and Forti, \cite{DiNassoFortiTopExtens2006}. One might also check out their (and Benci's) constructive survey, \cite{BenciFortiNassoEightfoldPath2006}, of a variety of means to a nonstandard mathematics (which also includes a good introduction). There are yet other approaches to a nonstandard mathematics, notably the Russian school; for a good example, see \cite{InfinitesAnalyGordonKusraevKutateladzeBk2002}.
\sigmaubsubsection{Impressionistic introduction with terminology}
Let's give an impressionistic introduction to nonstandard mathematics via the (extended) ultrapower constructions.
There are many motivations for the need of a nonstandard mathematics. To have our real numbers, $\betabr$, embedded in a much more robust object, $\rz\betabr$, with the properties of the real numbers, but also containing infinite and infinitesimal quantities is a boon to a direct formalization of intuitive strategies. Model theoretically, these have been around for more than 60 years (some would argue much longer) via eg., ultrapowers or the compactness theorem.
The ultrapower is the construction generally least involved with theoretical matters of the foundations of math (but see \cite{DiNassoFortiTopExtens2006}). Thinking of eg., infinite numbers as limiting properties of sequences of real numbers, one might attempt to construct nonstandard real numbers as equivalence classes of such sequences, ie., $\rz\betabr=\betabr^\betabn/\sigmaim$ where $/\!\sigmaim$ denotes the forming of such equivalence classes. Clearly, one can extend the operations and relations on $\betabr$ to $\betabr^\betabn$ coordinate wise, getting a partially ordered ring; but almost all of the nice properties of $\betabr$ are lost.
But, if $\mathcal P(\betabr)=2^\betabn$, the power set of $\betabn$, it turns out that there are objects $\mathcal U\sigmaubset\mathcal P(\betabr)$, the ultrafilters, such that defining our equivalence relation in terms of elements of $\mathcal U$ preserves all ``well stated'' properties of $\betabr$. More specifically, given $(r_i),(s_i)\in\betabr^\betabn$, define $(r_i)\sigmaim (s_i)$ if $\{i:r_i=s_i\}\in\mathcal U$ and $(r_i)<(s_i)$ if, again $\{i:r_i<s_i\}\in\mathcal U$, we find that the extended ring operations and partial order are well defined on the quotient $\betabr^\betabn/\mathcal U$ and, in fact, it's not hard to prove that we get a totally ordered field containing $\betabr$ (the set of equivalence classes of constant sequences) as a subfield.
For $r\in\betabr$, let $\rz r=(r)/\sigmaim$, the equivalence class containing the corresponding constant sequence and $\betasm{{}^\sigma A}=\{\rz r:r\in A\}\sigmaubset\rz\betabr$ denote the image of $A\sigmaubset\betabr$ in our nonstandard model of $\betabr$; eg., ${}^\sigma\betabr$ is the image of $\betabr$. In general, we will let $\betasm{\betak{r_i}}$ denote the equivalence class of a sequence $(r_i)\in\betabr^\betabn$.
Given this, existence of infinite elements is clear: if $(r_i)\in\betabr^\betabn$ with $r_i\rightarrow\infty$ as $i\rightarrow\infty$, then for each $s\in\betabr$, the set $\{i:r_i>s\}$ is in $\mathcal F(\sigmaubset\mathcal U)$ and so, by our definition, $\betak{r_i}>\rz s$. Note that to verify the field properties and total ordering of $\rz\betabr$, we need the full strength of ultrafilters, eg., the maximality property: if $A\sigmaubset\betabn$, then precisely one of $A$ or $\betabn\sigmasm A$ is in $\mathcal U$.
In particular, if $\omega=\betak{m_i}\in\rz\betabn$ where $m_i\uparrow\infty$ as $i\rightarrow\infty$, eg., $\omega$ is infinite
Since we can form the $\mathcal U$ equivalence class of arbitrary sequences of real numbers and get a much larger field with all of the `same' well formed properties as $\betabr$, why can't we do this for $\betabn$, $\betabq$, $\betabq(\sigmaqrt{5})$, etc. and get `enlarged' versions of these? We can, but try to do this with the algebra $F(\betabr)$ of real valued smooth maps on $\betabr$; ie., consider $F(\betabr)^\betabn/\sigmaim$ as before. Clearly, this is a ring, but do these `functions' (on $\rz\betabr$) have, in some good sense, all of the properties of functions in $F(\betabr)$? Ignoring subtleties, the simple answer is yes, simply because these elements are internal and therefore fall under the aegis of the all encompassing {\it principle of transfer}; but let's see, to some extent, how this works in this case.
Let's consider, for example, the nonstandard support (*support) of an equivalence class $\betak{f_i}\in\rz F(\betabr)$. (Recall that if $f\in F(\betabr)$, then the support of $f$, $supp(f)$, is the closure of the set of $t\in\betabr$ where $f(t)\nablaot=0$.) But, then as we seem to be following a recipe of extending everything component wise and then taking the quotient, if $A_i=supp(f_i)$, then $\rz supp(\betak{f_i})$ must be the equivalence class $\betak{A_i}$.
Yet, how is this a subset of $\rz\betabr$?
This is a special case of the next problem of nonstandard analysis: extending `is an element of' to our ultrapower constructions. Miraculously, the properties of ultrafilters (eg., our $\mathcal U$) allow one to (simplemindedly!) define $\betak{r_i}\in\betak{A_i}$ if $\{i:r_i\in A_i\}\in\mathcal U$. (This really should be written $\betak{r_i}\;\rz\!\!\in\betak{A_i}$, but starring all extended operations, relations, etc. can rapidly get confusing.) Note that these subsets of $\mathcal P(\rz\betabr)$ of the form $\betak{A_i}$ are called {\it internal sets} and are {\it precisely those subsets that extend the properties of $\mathcal P(\betabr)$ (and therefore shall be denoted $\rz\mathcal P(\betabr)$) via the principle of transfer}.
For example, the typical bounded subset $\mathcal C$ of $\rz\betabr$ does not have a nonstandard supremum, ie., $\rz\sigmaup \mathcal C$ does not exist; in particular, the transfer principle applied to the theorem that bounded subsets of $\betabr$ have suprema does not transfer to all *bounded elements of $\mathcal P(\rz\betabr)$. (For example, the set of {\it infinitesimals}, denoted $\mu(0)$ here, certainly does not have a supremum.) Nonetheless, the transfer principle certainly does apply to the internal $\betak{A_i}$, and the proper definition is (surprise!) $\rz\sigmaup\betak{A_i}=\betak{\sigmaup A_i}$. Note here that other notable examples of external subsets of $\rz\betabr$ are ${}^\sigma\betabr$ (and in fact ${}^\sigma A$ for any infinite subset $A\sigmaubset\betabr$),
\betaegin{align}
\rz\betabr_{nes}=\{\mathfrak t\in\rz\betabr:|\mathfrak t-\rz s|\in\mu(0)\;\text{for some}\;s\in\betabr\},
\epsilonnd{align}
the {\it nearstandard real numbers}, $\rz\betabr_\infty$, the infinite real numbers and the infinite natural numbers, $\rz\betabn_\infty$ which are said to be {\it *finite}. If $A\sigmaubset\betabr$ is finite, let $|A|\in\betabn$ denote its cardinality, let $\omega=\betak{m_i}\in\rz\betabn$ be an infinite *finite integer and $A_i\sigmaubset\betabr$ be such that $\{i:|A_i|=m_i\}\in\mathcal U$. Then we say that $\betak{A_i}\sigmaubset\rz\betabr$ is a {\it *finite subset of *cardinality} $\omega$. Although (for $\omega$ infinte) these sets are infinite (in fact uncountable!), transfer implies that *finite subsets of $\betabr$ have the `same' properly stated properties that finite subsets have. Nonetheless, for a much stronger {\it sufficiently saturated} ultrapowers, there exists *finite subsets of $\rz\betabr$ containing ${}^\sigma\betabr$. These will play a role in this paper.
We still haven't considered how the elements of $\rz F(\betabr)=F(\betabr)^\betabn/\sigmaim$ can be considered as functions on $\rz\betabr$, but by now the reader can see that we must define $\betak{f_i}(\betak{x_i})=\betak{f_i(x_i)}$ and hope that the properties of $\mathcal U$ ensure that this is well defined (ie., independent of choice of representatives) and is a function. This can indeed be verified and these functions are the {\it internal functions} in $F(\rz\betabr)$; the function $\mathfrak f:\rz\betabr\rightarrow\rz\betabr$ defined by $\mathfrak f(x)=x$ if $x\sigmaim 0$, ie., if $x$ is infinitesimal, and $\mathfrak f(x)=0$ if $x\nablaot\sigmaim 0$ is an {\it external function}, eg., does not satisfy the internality criteria allowing the use of transfer. For example, it is *bounded (bounded in $\rz\betabr$), but $\rz\sigmaup\mathfrak f$ does not exist. Yet again, it's straightforward that for *bounded $\betak{f_i}$, $\rz\sigmaup\betak{f_i}$ is well defined precisely by our recipe: $\betak{\sigmaup f_i}\in\rz\betabr$ (this *supremum may be an infinite element of $\rz\betabr$). Internal subsets of $\rz\betabr$ of the form $\betak{A}$ (ie., the equivalence class containing the constant sequence $(A_i)$ for some $A\sigmaubset\betabr$) are called the {\it standard sets}. Following our recipe for denoting the equivalence class of a constant sequence by starring, $\betak{A}$ is usually denoted $\rz A$.
For perspective, note that the copy of $[0,1]$ lying in $\rz[0,1]$, ie., ${}^\sigma[0,1]\sigmaubset\rz[0,1]$ is very sparse. For example, given an infinitesimal, $0<\mathfrak t=\betak{t_i}\in\rz\betabr$ (eg., suppose $t_i\deltaownarrow 0$ as $i\rightarrow\infty$) and $r\in(0,1)$, then $\rz r+[-\mathfrak t,\mathfrak t]\sigmaubset\rz[0,1]$, but intersects ${}^\sigma[0,1]$ only at $\rz r$.
Let's consider the {\it standard function} $\rz \sigmain(x)$. First of all, $\rz\sigmain$ is defined essentially as we defined standard sets, $\rz A$, ie., the $f_i$ above are all the function $\sigmain$. So if we define the {\it *domain} of $\betak{f_i}$ as we have all else: $\rz dom(\betak{f_i})\deltaot=\betak{dom(f_i)}$, we see that $\rz dom(\rz\sigmain)$ is all of $\rz\betabr$. (Or, as the domain of $\sigmain$ is $\betabr$, transfer says that the *domain of $\rz\sigmain$ is $\rz\betabr$.) A consequence of our constructive approach is the fact that $\rz dom(\rz\sigmain)=\betak{dom(f_i)}$ is internal.
It's not hard to check that $\rz\sigmain$ is really an extension of $\sigmain:\betabr\rightarrow[-1,1]$: first, restricting the graph of $\rz\sigmain$ to ${}^\sigma\betabr$ just the image of the graph of $\sigmain$ in $\rz\betabr^2$; second, all of the symmetry and character properties hold and third, it has all of the (transferred) analytic properties that $\sigmain$ has.
Before we conclude this tour, let's look at the standard part map. We defined the external (subring) $\rz\betabr_{nes}\sigmaubset\rz\betabr$ above. It should not be surprising that this is precisely those $\betak{r_i}\in\rz\betabr$ satisfying $|\betak{r_i}|<\rz t$ for some $t\in\betabr$ (here $|\;|:\rz\betabr\rightarrow\rz[0,\infty)$ is defined as all else). But by it's definition, any $\betak{r_i}\in\rz\betabr_{nes}$ satisfies $\betak{r_i}\sigmaim \rz r$ for some (clearly unique) $r\in\betabr$, eg., there is a well defined map (homomorphism onto!) $\fracst:\rz\betabr_{nes}\rightarrow\betabr$, the {\it standard part map}. Sometimes we will write ${}^o\betak{r_i}$ for $\fracst\betak{r_i}$.
Note then that if $\betak{f_i}:\rz\betabr\rightarrow\rz\betabr$ has image in $\rz\betabr_{nes}$, then we can define $\fracst\betak{f_i}:\betabr\rightarrow\betabr$ to be the map $r\in\betabr\mapsto\fracst(\betak{f_i}(\rz r))$
Given this, if $\omega=\betak{m_i}\in\rz 2\betabn$ with $m_i\uparrow\infty$ as $i\rightarrow\infty$ (eg., $\omega$ is infinite), consider $f_i$ given by $x\mapsto\sigmain(m_ix)$, so that writing $\timesi=\betak{x_i}\in\rz\betabr$, we have $\rz\sigmain(\omega \timesi)=\betak{sin(m_ix_i)}$. By transfer, $\timesi\mapsto\rz\sigmain(\omega\timesi)$ has all of the symmetry and analytic properties of $x\mapsto\sigmain(2mx)$ for some $m\in\betabn$, eg., solves the nonstandard *differential equation $\mathfrak f''-\omega^2\mathfrak f=0$; yet it's standard part is not even Lebesgue measurable!
\sigmaubsubsection{Formal tools}
The four pillars of nonstandard analysis are the internal definition principle, transfer, saturation
and (several versions of) ``overflow''. In order to discern the internal sets among all external sets, one can use the internal definition principle. It is basically an algorithmic way of determining if some object is of the form $\betak{S_i}$ and depends on the fact that all internal sets ar elements of some standard set $\rz T$. It asserts that if $\mathcal B=\betak{B_i}$ is internal and $P(H)$ is a statement about an variable quantity $H$ in an internal set $\mathcal X$ (of *functions, *measures, etc.,), then $\{H\in\mathcal X:P(H)\;\text{is true}\}$ is internal.
As described above transfer allows us to, eg., translate to the
nonstandard world careful statements about regular mathematics. Here we will need it to, eg., transfer to the nonstandard world the existence of maps of a certain type
that have specified values on finite sets.
Next, saturation has a variety of guises, one of which will be important here. Besides the need for the monads associated with neighborhood filters for a given topology, the specific form of saturation (see Stroyan and Luxemburg, \cite{StrLux76}, chapter 8) that will
be needed here, in section 4, ensures that *finite set are
sufficiently large. Specifically, let $X\in\mathcal U$ be an infinite
set of cardinality not bigger than $\mathcal P(C^\infty(\betabr^m,\betabr))$. Then, there
exists a *finite $\mathcal A\in\mathcal U$ with ${}^\sigma X\sigmaubset\mathcal A\sigmaubset\rz
X$. This can be situated so that $\mathcal A$ carries the same, well formed finitely stated, characteristics as $X$ (transfer). We will use this in the situation where $X$ is a particular
collection of smooth maps or smooth section of a bundle. We will
also use an overflow type result that depends on our nonstandard model being
polysaturated. See \cite{StrLux76} chapter 7; below we
paraphrase their theorem 7.6.2 for our use.
\betaegin{theorem}\lambdabel{thm:extend external map}
Suppose that $A\sigmaubset\rz\betabr^m$, not necessarily internal with cardinality less than that of $\rz\betabr^n$.
Suppose that $F:A\rightarrow\rz C^\infty(\betabr^m,\betabr^n)$ is any map. Then there is an
internal subset $\betaar{A}\sigmaubset\rz\betabr^m$ and an internal map
$\betaar{F}:\betaar{A}\rightarrow\rz C^\infty(\betabr^m,\betabr^n)$ such that $A\sigmaubset \betaar{A}$ and
$\betaar{F}|_{A}=F$.
\epsilonnd{theorem}
\sigmaubsection{Jet bundle constructions}
In this section we cover enough of the basics of jets and the jet
bundle formulation of (linear) differential operators, sufficient to
formulate and prove our results.
\sigmaubsubsection{Jet bundle setup}
We will briefly summarize that part of jet theory that we need.
Although the following formulation is straightforwardly generalized
to smooth manifolds, for brevity's sake we will restrict to the
Euclidean case.
Let $\betasm{P_k(m,n)}$ denote the polynomial maps of order $k$ from $\betabr^m$ to $\betabr^n$.
If $f\inC^\infty(\betabr^m,\betabr)$, $x\in\betabr^m$ and $k\in\betabn$, let $\betasm{T^k_xf}\in P_k(m,n)$
denote the $k$th order Taylor polynomial of $f$ at $x$. By *
transfer, if $\rz P_k(m,n)$ denotes the $\rz\betabr$ vector space of internal polynomials
from $\rz\betabr^m$ to $\rz\betabr^n$, $f\in\rz C^\infty(\betabr^m,\betabr^n)$ and
$x\in\rz\betabr^m$, we have $\rz T^k_xf$, the internal $k^{th}$
order Taylor polynomial of $f$ at $x$. Note that transfer implies
that this has all of the properties of the Taylor polynomial,
suitably interpreted. Note that although $f=\rz g$ so that $\timesi\mapsto\rz T^k_\timesi f$ is simply the transfer of the standard map $x\mapsto T^k_xf$, for
$\timesi\in\rz\betabr^m_{\infty}$ or $k\in\rz\betabn_{\infty}$, $\rz T^k_\timesi f$ can be
very pathological.
We define an
equivalence relation on $C^\infty(\betabr^m,\betabr^n)$\; as follows. We say that $f\inC^\infty(\betabr^m,\betabr^n)$
\textbf{vanishes to $k$th order at $x$} if $T^k_xf=0$ and for $f,g\inC^\infty(\betabr^m,\betabr^n)$,
we say that f equals g to $k$th order, written
$\betasm{f\overlineerset{x_k}\sigmaim g}$ if $T^k_x(f-g)=0$. This defines an equivalence relation on $C^\infty(\betabr^m,\betabr^n)$.
Let $\betasm{j^k_xf}$ denote the equivalence class containing $f$. We denote
the set of equivalence classes by $\betasm{\mathcal J^k_{m,n,x}}$. There are a
variety of definitions of $\mathcal J^k_{m,n,x}$ and its not hard to
show that one can identify $\mathcal J^k_{m,n,x}$ with the set of Taylor polynomials of order $k$ at $x$ of
smooth maps $(\betabr^m,x)\rightarrow\betabr^n$ and we can identify $j^k_xf$ with
$T^k_xf$. (An equivalence class consists of all maps with a given $k^{th}$ order Taylor polynomial at $x$.) Let $\betasm{\mathcal J^k_{m,n}}=\cup_{x\in\betabr^m}\mathcal J^k_{m,n,x}$.
$\mathcal J^k_{m,n}$ is a smooth fiber bundle, in fact, as our maps have range $\betabr^m$, a vector bundle, over
$\betabr^m$ with fiber over $x\in\betabr^n$ given by $\mathcal J^k_{m,n,x}$. Let
$\betasm{\partiali^k_0}:\mathcal J^k_{m,n}\rightarrow\betabr^m$ denote the bundle projection. Note
also that if $l,k\in\betabn$ with $l>k$, then $\mathcal J^l_{m,n}$ is a bundle
over $\mathcal J^k_{m,n}$; let $\betasm{\partiali^l_k}$ denote the bundle projection. We
let $\betasm{C^{\infty}(\mathcal J^k_{m,n})}$ denote the $\betabr$ vector space of $C^{\infty}$
sections of $\mathcal J^k_{m,n}$. If $f\inC^\infty(\betabr^m,\betabr^n)$, there is a canonical
section of $\partiali^k_0$ given by $\betasm{j^kf}:x\mapsto j^k_xf$. There is a
canonical map, the operation of taking the $k$ jet:
\betaegin{align}
\betasm{j^k}:C^\infty(\betabr^m,\betabr^n) \rightarrow C^{\infty}(\mathcal J^k_{m,n})\quad f\mapsto j^kf
\epsilonnd{align}
For later purposes we also need to define the infinite jet,
$\betasm{j^{\infty}_xf}$, for $f\inC^\infty(\betabr^m,\betabr^n)$. Doing a simplified rendering of projective
limits, we will define the vector space of infinite jets at $x\in\betabr^m$, $\betasm{\mathcal J^{\infty}_{m,n,x}}$, to be the set of sequences
$(f^0,f^1,f^2,\ldots)$ such that $f^k\in\mathcal J^k_{m,n,x}$ for all $k$ and for all nonnegative integers $j<k$, $\partiali^k_j(f^k)=f^j$. Then for a given $f\inC^\infty(\betabr^m,\betabr^n)$, the infinite jet of $f$ at $x$, $j^\infty_xf$, is clearly the well defined element of $\mathcal J^\infty_{m,n,x}$ given by $(j^0_xf,j^1_xf,j^2_xf,\ldots)$. It is easy to see that $\mathcal J^{\infty}_{m,n,x}$, is an infinite dimensional
vector space over $\betabr^m$, operations given componentwise, and
that, for each $x\in\betabr^m$, the map $\betasm{j^{\infty}_x}:f\mapsto j^{\infty}_xf:C^\infty(\betabr^m,\betabr^n)\rightarrow\mathcal J^{\infty}_{m,n,x}$ is a
vector space surjection with kernel the subspace of $g\in C^\infty(\betabr^m,\betabr^n)$
such that $j^k_xg=0$ for all integers $k$, ie., $g$ vanishes to
infinite order at $x$. We will also need the forgetful fiber
projection $\betasm{\partiali_{k,x}}:\mathcal J^{\infty}_{m,n,x}\rightarrow\mathcal J^k_{m,n,x}$. As the base range space is
linear, $\partiali_{k,x}$ is a surjective linear morphism and clearly has kernel
the (ideal) of formal power series that vanish to $k^{th}$ at $x$, see
above .
From the canonical (global) coordinate framing $\betasm{x_i}$,
$1\leq i\leq m$ on $\betabr^m$, and $\betasm{y^j}$, $1\leq j\leq n$ on $\betabr^n$,
we get induced coordinates, $\betasm{x_i, y^j_{\alpha}}$ for $|\alpha|\leq k$ and
$1\leq j\leq n$, on $\mathcal J^k_{m,n}$ defined as follows. The $x_i$ are
just the pullback of the coordinates on the base $\betabr^m$. If
$\partialhi\in\mathcal J^k_{m,n}$, then $\partialhi\in\mathcal J^k_{m,n,x_0}$ for some
$x_0\in\betabr^m$ and so $\partialhi$ can be written as $j^k_{x_0}f$ for some
$f\inC^\infty(\betabr^m,\betabr^n)$. Then
\betaegin{align}
x_i(\partialhi)=x_{0,i},\quad y^j_{\alpha}(\partialhi)\deltaoteq\partial^{\alpha}(f^j)(x_0)=\partialhi^j_{\alpha}.
\epsilonnd{align}
where $\betasm{\partial^{\alpha}}$ denotes $\frac{\partial^{|\alpha|}}{{({\partial}x_1)}^{\alpha_1}\cdotots{({\partial}x_m)}^{\alpha_m}}$
where $\alpha=(\alpha_1,\ldots,\alpha_m)$. If $\lambda\in
C^{\infty}(\mathcal J^k_{m,n},\betabr^n)$, then with respect to the induced
coordinates, we write this as $\betasm{\lambda(x_i,y^j_{\alpha})}$. therefore, for
later use, we can Taylor expansion $\lambda$ \textbf{with respect to the $x_i$ coordinates}, around a given $p_0$, as follows. Let
$p_0=(x_{0,i},y^k_{0,\alpha})$ be a coordinate representation as above.
Let $\betasm{\lambda^l}$ denote the $l^{th}$ coordinate of $\lambda$ with respect to
the canonical coordinates on $\betabr^n$. Then the Taylor expansion to
order s with in the base coordinates is
\betaegin{align}\lambdabel{taylor exp 1}
\lambda^l(x_i,y^k_{\alpha})=\sigmaum_{|\alpha|\leq
s}K_{\alpha}(x-x_0)^{\alpha}\partial^{\alpha}(\lambda^l)(p_0)+ \tilde{\lambda}^l(x_i,y^k_{\alpha})
\epsilonnd{align}
where for $|\alpha|=s+1$, $\tilde{\lambda}^l\in
C^{\infty}(\mathcal J^k_{m,n},\betabr^n)$ vanishes to order $s+1$ in the base coordinates at $p_0$ , $K_{\alpha}$ are the usual factorial
constants, $(x-x_0)^{\alpha}=(x_1-x_{0,1})^{\alpha_1}\cdotots (x_m-x_{0,m})^{\alpha_m}$, and
$\betasm{\partial^{\alpha}}$ is the ${\alpha}^{th}$ partial derivative with respect to the
base coordinates. Therefore, for our purposes the vanishing order, normally defined
in terms of the power of the maximal ideal at the given point in terms of the $x$
coordinates, will be defined in terms of the degree of vanishing
derivatives (in $x$ coordinates) at the given point. We need to emphasize that we are considering vanishing order of the smooth maps on the jet bundle only with respect to dependence on the base coordinates.
Given the usual framing $\partial_i=\frac{\partial}{\partial x_i}$ , $1\leq i\leq m$ for
$T\betabr^m$, $\partial_{i,x}$ being the frame for $T_x\betabr^m$; we have an
induced framing of $T\mathcal J^k_{m,n}$ given by adjoining to these tangent horizontal vectors the vectors $\partial_{y^j_{\alpha}}= \frac{\partial}{\partial y^j_\alpha}$ that are tangent to the fibers of the bundle projection $\partiali^k_0$ at $x$, for $j=1,\ldots,n$ and $|\alpha|\leq k$.
The notion of contact is useful in understanding the sharpening of the results here vis a vie the results of Todorov. Given $x\in\betabr^n$ and a nonnegative integer $s$, we say that $f\inC^\infty(\betabr^m,\betabr^n)$ has \textbf{contact }$\betasm{s}$ with $\betabr^m$ at $x$ if $f(x)=0$ and the graph of $f$, $\Gamma_f\sigmaubset\betabr^m\times\betabr^n$ is flat to $s^{th}$ order at $(x,0)$, that is, if $T^k_xf=0$, ie., if $j^s_xf$ is the equivalence containing the $0$ $s$ jet at $x$.
We say that \textbf{$\betasm{f,g\inC^\infty(\betabr^m,\betabr^n)}$ have $\betasm{s^{th}}$ order contact at $x$} if $f-g$ has $s^{th}$ order contact with $\betabr^m$ at $x$, the graph of the $0$ function, at $x$. It should be obvious that this is an equivalence relation and that the set of all $g\inC^\infty(\betabr^m,\betabr^n)$ that belong to the $s^{th}$ order contact class of $f$ is precisely the affine subset with the same $s^{th}$ order jet as $f$.
\sigmaubsubsection{Prolonging jet maps and total derivatives}
Let $\betasm{\betabr^p_m}$ denote the product bundle with fiber $\betabr^p$ and
base $\betabr^m$; if $p=1$, we will denote this bundle by $\betasm{\betabr_m}$. If $x\in\betabr^m$, let
$\betasm{\betabr^p_{m,x}}$ denote the vector space fiber of $\betabr^p_m$ over $x$. In the following we will be using \textbf{vector bundle}
\textbf{maps}, ie., smooth maps of bundles over the same base that preserve fibers and cover the identity map on the base. The symbols of linear differential operators,
$\lambda:\mathcal J^r_{m,1}\rightarrow\betabr^p_m$ are maps of this type. The set of such maps is a $C^{\infty}(\betabr^m,\betabr)$
module and will be denoted by
$\betasm{C^{\infty}(\mathcal J^r_{m,1},\betabr^p_m)}$.
If $\lambda: \mathcal J^k_{m,n}\rightarrow \betabr^p_m$ is such a smooth bundle map, and $l\in\betabn$,
then there exists a smooth bundle map $\betasm{\lambda^{(l)}}:\mathcal J^{k+l}_{m,n}\rightarrow
\mathcal J^l_{m,p}$\;, called the $\betasm{l^{th}}$\textbf{-prolongation of} $\betasm{\lambda}$ such that
the following diagram is commutative
\betaegin{align}\lambdabel{diag1}
\betaegin{CD}
\mathcal J^{k+l}_{m,n} @>\lambda^{(l)} >> \mathcal J^l_{m,p} \\
@V\partiali^{k+l}_k VV @V\partiali^l_0VV\\
\mathcal J^k_{m,n} @>\lambda >>
\betabr^p_m
\epsilonnd{CD}
\epsilonnd{align}
That is, as $j^s_x(\lambda\circ j^rf)$ depends only on derivatives up to order $s$ of $y\mapsto j_y^rf$ at $x$,
and so only on the $r+s$ jet of $f$ at $x$, then the following
definition is well defined. If $f\inC^\infty(\betabr^m,\betabr^n)$, then
\betaegin{align}\lambdabel{eqn:coord free prolng formula}
\betasm{\lambda^{(s)}}(j^{r+s}_x(f))=j^s_x(\lambda\circ j^rf).
\epsilonnd{align}
The prolongation of vector fields on $\betabr^m$ to vector fields on
$\mathcal J^k_{m,n}$ are given by fairly complicated recursion formulas.
For treatments of prolongations of vector fields in somewhat
different contexts, see Olver \cite{Olver1993}, p110 or Pommaret, \cite{Pommaret1978}, p253. Pommaret
gives a remarkably easy derivation of these expressions. We only
need the prolongation of coordinate vector fields, ie., total
derivatives, and these have far simpler expressions. These give
explicit local expressions of prolongations and so
allow us to computationally investigate the
effect of successive prolongations on maps of jets. For each
coordinate tangent field $\partial_i$ on $T\betabr^m$, for $1\leq i\leq m$,
we have an explicit expression for the corresponding lifted local
section of $T\mathcal J^k_{m,n}$, the \textbf{total derivative} $\betasm{\partial^\#_i}$ defined as
follows.
\betaegin{align}\lambdabel{total der sum}
\partial_i^\#=
\partial_i+ \sigmaum_{\sigmaubstack{|\alpha|\leq k\\1\leq j\leq n}}
y^j_{\alpha^i}\partial_{y^j_{\alpha}}
\epsilonnd{align}
where $\alpha^i=(\alpha_1,\ldots,\alpha_i+1,\ldots,\alpha_n)$. Note that $\partial_i^\#$
depends on coordinates of order $k+1$, ie., for $\lambda\in
C^{\infty}(\mathcal J^k_{m,n},\betabr^n_m)$, we have $\partial_i^\#(\lambda)$ are the
coordinates for a map $\lambda^{(1)}:\mathcal J^{k+1}_{m,n}\rightarrow\mathcal J^1_{m,n}$ with respect to the induced jet coordinates. In fact, we
have the following.
\betaegin{lemma}\lambdabel{lem:loc coord of 1-prolong}
Suppose that $\lambda:\mathcal J^k_{m,n}\rightarrow\betabr^n$ is a smooth map. Let $\lambda^j$
for $j=1,\ldots,n$ denote the coordinates of $\lambda$ with respect to
the standard coordinate basis for $\betabr^n$. Then the components of
$\lambda^{(1)}$ with respect to the given coordinates are
$\partial_i^\#(\lambda^j_{\alpha}$)
\epsilonnd{lemma}
\betaegin{proof}
One can verify this lemma and the expression (\ref{total der sum})
using the local version of the definition for prolonging jet maps
(\ref{eqn:coord free prolng formula}) when $s=1$, eg., see \cite{Olver1993}, p109; ie.,
\betaegin{align}\lambdabel{1 prolong coord eqn}
\partial_i^\#(\lambda)(j_x^{r+1}(f)=\partial_i(\lambda\circ j^r f)(x)
\epsilonnd{align}
and then applying the chain rule to the right side of (\ref{1
prolong coord eqn}).
\epsilonnd{proof}
\sigmaubsection{Differential operators and their prolongations}
In order to align with Todorov's setup we will now restrict the
dimension of the range space to be $1$. The superscript $j$ enumerating range space
components will no longer appear.
Let $\betasm{LPDO_r}$ denote the vector space of linear partial differential operators of degree less than or equal to $r$
with coefficients in $C^\infty(\betabr^m,\betabr)$. Suppose that $P\in LPDO_r$. Then there exists
a smooth bundle map $\betasm{\lambda_P}: \mathcal J^r_{m,1}\rightarrow\betabr_m$ called the \textbf{total symbol of $\betasm{P}$}
such that if $f\inC^\infty(\betabr^m,\betabr)$, then $P(f)=\lambda_P\circ j^rf$ as elements
of $C^\infty(\betabr^m,\betabr)$. If $\lambda:\mathcal J^r_{m,1}\rightarrow\betabr_m$ is the symbol of an $r^{th}$
order differential operator, $P$, then the \textbf{$\betasm{s^{th}}$ prolongation}
$\betasm{\lambda^{(s)}}:\mathcal J^{r+s}_{m,1}\rightarrow\mathcal J^s_{m,1}$ is defined
on $r+s$ jets above.
As prolongations of differential operators mapping $C^\infty(\betabr^m,\betabr)$ to $C^\infty(\betabr^m,\betabr)$ are systems, we will use the notation $\betasm{LPDO_{r+s}^s}$ for $r+s$ order linear differential operators on $C^\infty(\betabr^m,\betabr)$ to smooth sections of $\mathcal J^s_{m,1}$. In particular, if $P\in LPDO_r$, and $s\in\betabn$, then there is
$\betasm{P^{(s)}}\in LPDO_{r+s}^s$\;, called the
$\betasm{s^{th}}$ \textbf{prolongation of} $\betasm{P}$, defined as the differential operator
whose symbol is the $s^{th}$ prolongation of $\lambda_P$. We will have more to say about its nature later.
On the nonstandard level, note that if $g\inC^\infty(\betabr^m,\betabr)$, then, at every standard $x$, ie.,
$\rz x\in\;^\sigma\betabr^m$, and for each $k\in\;^\sigma\betabn_0=\betabn\cup\{0\}$,
$\rz\!j_{*x}^k(\rz g)=\rz\!(j^k_xg)$. That is, the internal operator
$\betasm{\rz\!j^k_{*x}}|_{^\sigma\! C^\infty(\betabr^m,\betabr)}$ operating on standard functions is just the transfer of $j^k_x$. (At nonstandard points this is not true.) It
therefore follows that $\rz\lambda\circ\rz j^k$, restricted to a standard
map, $\rz g$ at a standard point $\rz x$, is just the *transfer of $\lambda\circ
j^k_xg\in\betabr_{m,x}$. We shall give a sufficient account of the remainder of the nonstandard material needed after we recall a bit more standard geometry.
\sigmaection{Standard geometry: Prolongation and Vanishing}\lambdabel{section:prolong and vanishing}
This section proves the standard results that allow the proof of
Corollary \ref{cor: infinite order Todorov}.
The idea here is desingularize our total symbol by ``lifting'' it to a sufficiently high jet level where we can then invoke
a version of Todorov's result. In order to do this, we need some sort
of correspondence between solutions of $P$ and those of $P^{(s)}$.
We also need this procedure to decrease the vanishing order the coefficients
of the $P^{(k)}$ as $k$ increases. Here is the idea. On the one hand, we can think of a symbol, $\lambda=\lambda_P$ of an LPDE as a smooth family of linear maps $x\mapsto\lambda_x$, and therefore one can think of vanishing order of $\lambda$ at $x_0$ as the ``flatness'' of the graph of this map at the point $x_0\in\betabr^m$. This relates directly to the Taylor polynomials of the smooth coefficients of $\lambda$ at $x_0$. On the other hand and more abstractly, there is a classic ``desingularization'' machinery for jet bundle map, eg., symbols of differential operators, that carries solutions to solutions, ie., prolongation. In this section we will relate the intuitive vanishing order to this prolongation method in order to get crude controls between jets in the domain and range of the prolongation of $\lambda$, in terms of these singularities. This will be done in this section.
We first need the proper notion of vanishing order of a linear bundle map
$\lambda:\mathcal J^k_{m,1}\rightarrow\betabr$.
\betaegin{definition}
Let $x_0\in\betabr^m$, and $c\in\betabn$. Then we say
that $\betasm{\lambda}$\textbf{ vanishes to order (exactly)} $\betasm{c}$ \textbf{(in $\betasm{x}$) along $\betasm{(\partiali^k_0)^{-1}(x_0)}$}, written $\betasm{x_0\in\mathcal Z^c(\lambda)}$, if
$\partial^{\alpha}(\lambda)(p)=0$ for all $\alpha$ with $|\alpha|\leq c$ and for all $p\in (\partiali^k_0)^{-1}(x_0)$.
and, secondly, there exists some
and $\betaeta$ with $|\betaeta|=c+1$ and $p_0\in (\partiali^k_0)^{-1}(x_0)$ such that
$\partial^{\betaeta}(\lambda)(p_0)\nablaot =0$. Note, as always, that $\partial^{\alpha}$
denotes the $\alpha^{th}$ derivative with respect to the $x_i$
coordinates.
\epsilonnd{definition}
First note that this is more transparently stated as follows. Writing $\lambda=\sigmaum_{\alpha}f_\alpha y_\alpha$ for some smooth $f_\alpha$'s, this condition is equivalent to stating that for each coefficient $f_\alpha$, $\partial^\betaeta f_\alpha (x_0)=0$ for all multiindices $\betaeta$ satisfying $|\betaeta|\leq k$;
with the second condition being that there exists a coefficient $f_\alpha$ and a multiindex $\betaeta$ with $|\betaeta|=k+1$ such that $\partial^\betaeta f_\alpha(x_0)\nablaot=0$.
Note also that although the notion of contact is closely related to vanishing order, we will not pursue this connection here. In the next lemma, we don't need to restrict to jet mappings that
are the symbols of elements of $LPDO_k$. When we consider how total
derivatives change the vanishing order of jet maps, it will be
essential to consider the particular form of jet maps that are
symbols of elements of $LPDO_k$. Below we will be using the following notation. If $\betaeta=(\betaeta_1,\cdotots,b_m)$ is a multiindex of order $k$; ie., $|\betaeta|=\betaeta_1+\cdotots+\betaeta_m=k$ and $1\leq i_0\leq m$, then $\betasm{\betaeta_{i_0}}=(\betaeta_1,\ldots,\betaeta_{i_0}+1,\betaeta_{i_0+1},\ldots,\betaeta_m)$; so eg., $|\betaeta_{i_0}|=|\betaeta|+1$.
\betaegin{lemma}\lambdabel{lem1:decreasing order of 0}
Let $\lambda\in C^{\infty}(\mathcal J^k_{m,1},\betabr_m)$ and suppose that
$p_0\in\mathcal J^k_{m,1}$ is in $\mathcal Z^c(\lambda)$. Then for some $i\in\{1,\ldots,m\}$, we have
$p_0\in\mathcal Z^{c-1}(\partial_i(\lambda))$.
\epsilonnd{lemma}
\betaegin{proof}
By hypothesis, $\partial^{\alpha}(\lambda)(p_0)=0$ for $|\alpha|\leq c$ and there is a multiindex $\betaeta$ with $|\betaeta|=c+1$ and an
such that $\partial^{\betaeta}(\lambda)(p_0)\nablaot =0$. Write $\betaeta=\alpha_i$
for some multiindex $\alpha$ with $|\alpha|=c$ and $i\in\{1,\ldots,m\}$.
That is, $\partial^{\alpha}(\partial_i\lambda)(p_0)\nablaot =0$. But
if $\alpha$ is a multiindex such that $|\alpha|\leq c-1$, then $|\alpha_i|\leq
c$ and so
$\partial^{\alpha}(\partial_i\lambda)(p_0)=\partial^{\alpha_i}(\lambda)(p_0)=0$ by hypothesis. But
then by definition $p_0\in\mathcal Z^{c-1}(\partial_i\lambda)$, as we wanted to
show .
\epsilonnd{proof}
\sigmaubsection{Lifting solutions }
At this point, it is important to note the explicit form of the jet
maps that are symbols of elements of $LPDO_k$.
\betaegin{lemma}\lambdabel{lem: LPDO symbol}
Suppose that $P\in LPDO_k$. Then $\lambda=\lambda_P:\mathcal J^k_{m,1}\rightarrow\betabr_m$ can
be written in local coordinates $(x_i,y_{\alpha})$ in the following
form $\lambda= \sigmaum_{|\alpha|\leq k
}f_{\alpha}y_{\alpha}$ where $f_{\alpha}\inC^\infty(\betabr^m,\betabr)$.
\epsilonnd{lemma}
\betaegin{proof}
This is clear.
\epsilonnd{proof}
A remark is in order. Since for each $x\in\betabr^m$, $\lambda_P=\lambda_{P,x}:\mathcal J^r_{m,1,x}\betabr_{m,x}$ is linear, then we will consider look of the vanishing order of $x\mapsto\lambda_x$
\betaegin{lemma}
Suppose that $P\in LPDO_r$ , $s\in \betabn$ and $f\inC^\infty(\betabr^m,\betabr)$ solves
$P^{(s)}(f)=0$. Then $f$ solves $P(f)=0$.
\epsilonnd{lemma}
\betaegin{proof}
This is an easy unfolding of the definitions. Operationally,
$P^{(s)}$ is defined on $f\inC^\infty(\betabr^m,\betabr)$ as follows.
$P^{(s)}(f)=j^s\circ P(f)$. But then $P^{(s)}(f)=0$ implies that
$j^s\circ P(f)=0$. But for $g\inC^\infty(\betabr^m,\betabr)$, $j^s(g)=0$ as a section of $\mathcal J^s_{m,1}$ if
and only if $g$ is identically $0$, eg., letting $g=P(f)$, we get
$P(f)=0$.
\epsilonnd{proof}
\sigmaubsection{Lifting zero sets}
\sigmaubsubsection{ Prolongation of linear symbols}
\betaegin{remark}
If $\lambda:\mathcal J^r_{m,1}\rightarrow\betabr_m$ is the symbol of $P\in LPDO_r$, then
w\textbf{e will instead write $\mathcal Z^c(\lambda)$ for $\partiali^r(\mathcal Z^c(\lambda))$. In particular,
$\mathcal Z^c(\lambda)$ will now be considered as a subset of the base space,}
$\betabr^m$. Note that since $\lambda$ is linear on the fibers, with particular form as noted in
Lemma \ref{lem: LPDO symbol}, then
to say that $\betasm{x\mapsto\lambda_x}$ \textbf{vanishes to order $c$ at $x_0$ is the same as our fiber condition as this says that the coefficient $f_\alpha$ of our generic jet derivative $y_\alpha$ vanishes to order $c$ at $x_0$.}
So, in the linear case, we have the following definition.
\epsilonnd{remark}
\betaegin{definition}
Suppose that $\lambda$ is the symbol of $P\in LPDO_r$. Let $\betasm{\mathcal Z^c(\lambda)}$ denote the $x\in\betabr^m$ where all of the coefficients of $\lambda$ vanish to order $c$ and at least one does not vanish to order $c+1$. If $\lambda$ is such a linear jet bundle map, let $\betasm{\mathcal Z_\lambda}$ denote those $x\in\betabr^m$ where $\lambda_x$ is the zero linear map. Given this, it should be obvious that the
conclusion of Lemma \ref{lem1:decreasing order of 0} holds with
$\partiali^r(\mathcal Z^c(\lambda))$ in place of $\mathcal Z^c(\lambda)$.
\epsilonnd{definition}
With this development, we have the following initiating lemma.
\betaegin{lemma}\lambdabel{lem2:decreasing order of 0}
Suppose that $P\in LPDO_r$ with $\lambda\in
C^{\infty}(\mathcal J^r_{m,1},\betabr_m)$
the symbol of $P$.
Let $c\in \betabn$
and $x_0\in\mathcal Z^c(\lambda)$. Then $x_0\in\mathcal Z^{c-1}(\lambda^{(1)})$. In particular, if
$x_0\in\mathcal Z^1(\lambda)$, then $\lambda^{(1)}_{x_0}\nablaot=0$.
\epsilonnd{lemma}
\betaegin{proof}
So we have that $x_0$ satisfies $\partial^{\alpha}\lambda(x_0)=0$ if $|\alpha|\leq
c$ , and there exists multiindex $\betaeta$, with
$|\betaeta|=c+1$ such that $\partial^{\betaeta}\lambda(x_0)\nablaot=0$.
By Lemma (\ref{lem:loc coord of 1-prolong}) we only need to verify
that for all $i,\alpha$ with $|\alpha|\leq c-1$,
$\partial^{\alpha}(\partial_i^\#\lambda)(x_0)=0$ and there exists $i_0,\tilde{\alpha}$ with
$|\tilde{\alpha}|=c$ such that
$\partial^{\tilde{\alpha}}(\partial_{i_0}^\#\lambda)(x_0)\nablaot=0$.
To this end, suppose that the following statement is valid.
Let $d\in\betabn$ and suppose that for all $\alpha$ with $|\alpha|\leq d$, we have that
that $\partial^{\alpha}\lambda(p_0)=0$. Then for arbitrary given $i_0$
and $\tilde{\alpha}$ with $|\tilde{\alpha}|=d$,
$\partial^{\tilde{\alpha}}(\partial^\#_{i_0}\lambda)(p_0)=0$ if and only if
$\partial^{\tilde{\alpha}}(\partial_{i_0}\lambda)(p_0)=0$. Then one can see that from this statement and Lemma
\ref{lem1:decreasing order of 0} the proof of our lemma follows immediately; so it suffices to prove the above statement.
First of all, we need only to
verify this statement for $\lambda=\lambda_P$, for $P=\sigmaum_{|\alpha|\leq
r}f_{\alpha}\partial^{\alpha}$ where $f_{\alpha}\inC^\infty(\betabr^m,\betabr)$ for all $\alpha$; that is, if
$\lambda=\sigmaum_{|\alpha|\leq r}f_{\alpha}y_{\alpha}$. In the following calculations we will leave out evaluation at $x_0$; it is implicit. So
\betaegin{align}
\partial^\#_i(\lambda)=(\partial_i+ \sigmaum_{|\alpha|\leq r}y_{\alpha_i}\partial_{y_{\alpha}})(\sigmaum_{|\betaeta|\leq r}f_{\betaeta}y_{\betaeta})\nablaotag \\
=\sigmaum_{|\betaeta|\leq r}\partial_i(f_{\betaeta})y_{\betaeta}+\sigmaum_{|\alpha|\leq
r}f_{\alpha}y_{\alpha_i}.
\epsilonnd{align}
That is
\betaegin{align}
\partial^\#_i(\lambda)=\sigmaum_{|\alpha|\leq
r}((\partial_if_{\alpha})y_{\alpha}+f_{\alpha}y_{\alpha_i}).
\epsilonnd{align}
so that taking $\partial^{\tilde{\alpha}}$, for $|\tilde{\alpha}|\leq d$, of both sides and interchanging
derivatives, we get
\betaegin{align}\lambdabel{total der expression}
\partial^\#_i(\partial^{\tilde{\alpha}}\lambda)=\sigmaum_{|\betaeta|\leq
r}\partial_i(\partial^{\tilde{\alpha}}f_{\betaeta})y_{\betaeta}+\partial^{\tilde{\alpha}}(f_{\betaeta})y_{\betaeta_i}.
\epsilonnd{align}
But by hypothesis, the second term, on the right side of (\ref{total der expression})
is $0$ and the truth of the statement follows.
\epsilonnd{proof}
\sigmaubsubsection{Higher prolongations; coordinate calculations}
We want to prove a higher order prolongation version of the previous lemma. We need some preliminaries before we state the lemma.
Suppose that $\alpha$ is a multiindex, let $\partial_\alpha^\#$ denote the $\alpha^{th}$ iteration of the coordinate total derivatives; ie., if $\alpha=(\alpha_1,\ldots,\alpha_m)$ is a multiindex of order $k=\alpha_1+\cdotots+\alpha_m$, then $\partial^\#_\alpha=(\partial^\#_1)^{\alpha_1}\circ\cdotots\circ (\partial^\#_m)^{\alpha_m}$, it being understood that if $\alpha_j=0$, then the corresponding $j^{th}$ factor is missing. As defined, $\partial^\#_\alpha$ sends functions on $\mathcal J^r_{m,1}$ to functions on $\mathcal J^{r+k}_{m,1}$.
Next, note that if $\lambda=\sigmaum_\alpha f_\alpha y_\alpha$ is the symbol of $P\in LPDO_r$, then $\lambda^{(k)}$ is the symbol of the $r+k^{th}$ order operator $P^{(s)}$. As such, using the induced coordinates $y_\alpha$, $|\alpha|\leq k$ on the range, $\mathcal J^k_{m,1}$, we can write $\lambda^{(k)}_x$ as $((\partial_\alpha^\#\lambda)_x)_{|\alpha|\leq k}$. That is, $\lambda^{(k)}$ is given by the family of linear maps
\betaegin{align}
x\mapsto ((\partial^\#_\alpha\lambda)_x)_\alpha\;:\mathcal J^{r+k}_{m,1,x}\rightarrow\mathcal J^k_{m,1,x}
\epsilonnd{align}
Note then that this family of maps vanishes to order $c$ at $x_0$ precisely when for each $\alpha$, $|\alpha|\leq r+k$ the component $x\mapsto\partial^\#_\alpha(\lambda)_x$ vanishes to order $c$.
Given this, we have the following extension of the previous lemma to general prolongations.
\betaegin{lemma}
Suppose that the bundle map $\lambda:\mathcal J^r_{m,1}\rightarrow\betabr_m$ is the symbol of $P\in LPDO_r$ and suppose that $x_0\in \mathcal Z^c(\lambda)$. Then $\lambda^{(c+1)}_{x_0}$ is a nonzero linear map.
\epsilonnd{lemma}
\betaegin{proof}
By the remarks before the lemma, it suffices to prove that $\lambda^{(c+1)}_{x_0}$ has a nonzero component at $x_0$; that is, for some $\gamma$ with $|\gamma|\leq c+1$, $\partial^\#_\gamma(\lambda)_{x_0}$ is a nonzero linear map. Note also that if $\alpha,\betaeta$ are distinct multiindices, then $\partial^\#_\alpha$
First note that if $g\inC^\infty(\betabr^m,\betabr)$ and $\alpha,\gamma$ are appropriate multiindices, then
\betaegin{align}
\partial^\#_\gamma(gy_\alpha)=\sigmaum_{\epsilon+\rho=\gamma}\partial_\epsilon g\cdotot y_{\alpha+\rho}
\epsilonnd{align}
Suppose that $g$ vanishes to order (exactly) at $x_0$; so that there is an index $i_0$ and multiindex $\betaar{\betaeta}$ with $|\betaar{\betaeta}|=c$, such that $\partial_\alpha g(x_0)=0$ for $|\alpha|\leq c$ and $\partial_{\betaar{\betaeta}_{i_0}}g(x_0)\nablaot= 0$. Then $\partial^\#_{\betaar{\betaeta}_{i_0}}(gy_\alpha)=\partial_{\betaar{\betaeta}_{i_0}}g\cdotot y_\alpha$.
This follows upon inspection: $\betaeta_{i_0}$ is the only multiindex of length $c+1$ occurring in the sum. All other multiindices occurring are of length $\leq c$ and so these terms are zero by the hypotheses on $g$.
So now consider a general symbol $\lambda=\sigmaum_{|\alpha|\leq r}f_\alpha y_\alpha$ and suppose, by hypothesis that $x_0\in \mathcal Z^c(\lambda)$. So there exists a multiindex $\alpha_0$ with $|\alpha_0|\leq r$, a multiindex $\betaar{\betaeta}$ of order $c$ and an index $i_0\in\{1,\ldots,m\}$ such that $\partial_{\betaar{\betaeta}_{i_0}}f_{\alpha_0}(x_0)\nablaot=0$. We will show that the $\betaar{\betaeta}_{i_0}$ component of $\lambda^{(c+1)}_{x_0}$ is nonzero; ie., that $\partial^\#_{\betaar{\betaeta}_{i_0}}(\lambda)_{x_0}$ is a nonzero linear map. Now
\betaegin{align}
\partial^\#_{\betaar{\betaeta}_{i_0}}(\lambda)_{x_0}=\sigmaum_{|\alpha|\leq r}\partial^\#_{\betaar{\betaeta}_{i_0}}(f_\alpha y_\alpha)_{x_0}\qquad\quad\qquad\qquad \nablaotag \\
\qquad\qquad =\sigmaum_{|\alpha|\leq r}\sigmaum_{\gamma\leq\betaar{\betaeta}_{i_0}}\partial_\gamma(f_\alpha)(x_0)(y_{\alpha+(\betaar{\betaeta}_{i_0}-\gamma)})_{x_0} \nablaotag \\
\qquad =\sigmaum_{|\alpha|\leq r}\sigmaum_{\sigmaubstack{\gamma\leq\betaar{\betaeta}_{i_0}\\|\gamma|=c+1}}\partial_\gamma(f_\alpha)(x_0)(y_{\alpha+(\betaar{\betaeta}_{i_0}-\gamma)})_{x_0}\nablaotag \\
=\sigmaum_{|\alpha|\leq r}\partial_{\betaar{\betaeta}_{i_0}}f_\alpha(x_0)y_\alpha|_{x_0}. \quad\quad \qquad\qquad
\epsilonnd{align}
But the linear forms $y_\alpha|_{x_0}$ are linearly independent and the coefficient of $y_{\alpha_0}|_{x_0}$ is nonzero by hypothesis, hence the conclusion follows.
\epsilonnd{proof}
\betaegin{corollary}\lambdabel{cor:removing finite zeroes}
Suppose that $P\in LPDO_r$ with $\lambda\in
C^{\infty}(\mathcal J^r_{m,1},\betabr_m)$
the symbol of $P$ . Suppose that $c_0=\sigmaup\{c\in\betabn:\mathcal Z^c_{\lambda}\;\text{is nonempty}\}$
is finite.
Then $\mathcal Z(\lambda^{(c_0+1)})$ is empty. That is, for each $x\in\betabr^m$, $rank(\lambda^{(c_0)}_x)\gammaeq 1$.
\epsilonnd{corollary}
\betaegin{proof}
This is an immediate consequence of the above lemma.
\epsilonnd{proof}
\sigmaection{ Standard Geometry: Prolongation and rank}\lambdabel{section: standard jet work,prolong and rank}
In section 3, we were concerned with the vanishing order of the (total) symbol of an element of $LPDO_r$. Our proofs involved calculations with the induced local coordinate formulation of prolongations of jet bundle maps. In this section, we will prove the standard results needed in the proof of Corollary \ref{cor: infinite order Todorov}. Here, we are a bit more tradionally concerned with the prolongation effects of regularity hypotheses on the principal symbol of an element of $LPDO_r$.
Our constructions will instead be in the tradition of diagram chasing through commutative diagrams of jet bundles.
Here we will show that if the principal symbol
$\underline{\lambda}:\mathcal J^r_{m,1,x}\rightarrow\betabr_m$ is nonvanishing, then for each
$k\in\betabn$, $\lambda^{(k)}:\mathcal J^{r+k}_{m,1,x}\rightarrow\mathcal J^k_{m,1,x}$ is
maximal rank. We will use this fact in constructing solutions for
$P(f)=g$. We will first look at a coordinate argument that seems
to indicate this. We will follow this with a simple version with a proof of this fact
using a typical jet bundle argument.
Suppose that $\lambda:\mathcal J^r_{m,1}\rightarrow\betabr_m$ is the symbol of $P\in
LPDO_r$. Let's begin by looking at first order prolongations. Suppose that $x_0\in \betabr^m$ and that $\lambda_{x_0}:\mathcal J^r_{m,1,x_0}\rightarrow\betabr_{x_0}$
is nonzero in the sense that some coefficient $a_{\betaar{\alpha}}$, for $|\betaar{\alpha}|=r$ is nonzero at $x_0$. . Then we will verify the
first prolongation of $\lambda$,
$\lambda^{(1)}_{x_0}:\mathcal J^{r+1}_{m,1,x_0}\rightarrow\mathcal J^1_{m,1,x_0}$ has ``top order part'' of
maximal rank.
So we need to show that the rank of the ``top order part'' of
$\lambda^{(1)}_{x_0}:\mathcal J^{r+1}_{m,1,x_0}\rightarrow\mathcal J^1_{m,1,x_0}$ is $m$ as this
is the dimension of the fiber of $\mathcal J^1_{m,1}$. Write
$\lambda_{x}=\sigmaum_{|\alpha|\leq r}a_{\alpha}(x)y_{\alpha}$ and, in coordinates,
\betaegin{align}
\lambda^{(1)}_{x_0}=(\lambda_{x_0},\partial^{\#}_1\lambda_{x_0},\ldots,\partial^{\#}_m\lambda_{x_0})
\epsilonnd{align}
where, as before for each $i$,
\betaegin{align}\lambdabel{eqn:sum formula;tot der}
\partial^{\#}_i\lambda_{x_0}=\sigmaum_{|\alpha|\leq r}(\partial_i
a_{\alpha}(x_0)y_{\alpha}+a_{\alpha}(x_0)y_{\alpha_i}).
\epsilonnd{align}
where by assumption $a_{\tilde{\alpha}}(x_0)\nablaot=0$ for
some $\tilde{\alpha}$ with $|\tilde{\alpha}|=r$. But then the linear forms
$a_{\tilde{\alpha}}(x_0)y_{\tilde{\alpha}_i}$ for $i=1,\ldots,m$ are linearly
independent. Therefore, the linear forms $\partial^{\#}_i\lambda_{x_0}$, by
their above expressions (\ref{eqn:sum formula;tot der}), are also
linear independent. Hence, their image spans the fiber of
$\mathcal J^1_{m,1}\rightarrow\betabr^m$ over $x_0$.
Given that $\lambda$ is the (total) symbol of $P\in LPDO_r$; the ``top order part'' of $\lambda$, $\underline{\lambda}$ is the principal symbol of $P$. We need a little more jet stuff to properly define the principal symbol and proceed with the general statement for arbitrary prolongations.
The set of $j^{k+1}_xf$ of $f\inC^\infty(\betabr^m,\betabr)$ that vanish to $k^{th}$ order at $x$ have a
canonical $\betabr$ vector space identification with $\betasm{\mathcal S_{k+1} T^*_x}$, the ${k+1}^{st}$ symmetric power of $T^*_x$,
the cotangent space to $\betabr^m$ at $x$. (See Pommaret, \cite{Pommaret1978}, p47, 48.) In fact, for every $x\in\betabr^m$ and $k\in\betabn$ we have a canonical injection of vector spaces,
$\mathcal S_{k}T^*_x\sigmatackrel{i_{k}}{\rightarrow}\mathcal J^{k}_{m,1,x}$ which is a canonical isomorphism
when $k=1$, ie., $T^*_x\cong \mathcal J^1_{m,1,x}$. This injection embeds in an exact sequence:
\betaegin{align}
0\rightarrow\mathcal S_{k+1}T^*_x\sigmatackrel{i_{k+1}}{\rightarrow}\mathcal J^{k+1}_{m,1,x}\sigmatackrel{\partiali^{k+1}_k}{\rightarrow}\mathcal J^k_{m,1,x}\rightarrow 0.
\epsilonnd{align}
In fact this and much of the following holds in far greater generality; eg., giving exact sequences of
jets of bundles over any paracompact smooth manifold. Next, when $k=r$ the order of our operator,
note that expression (\ref{eqn:sum formula;tot der}) when
evaluated on $j^{r+1}_xf\in\mathcal S_{k+1}T^*_x$ gives
\betaegin{align}\lambdabel{eqn: principal symbol sum}
\partial^\#_i\lambda_{x}(j^{k+1}_xf)=\sigmaum_{|\alpha|\leq
r}a_\alpha(x)y_{\alpha_i}(j^{k+1}_xf),
\epsilonnd{align}
the other terms being zero as $j^k_xf=0$.
Now for each $k=0,1,2,\ldots$, the \textbf{principal symbol of} $\betasm{P^{(k)}}$, denoted
$\betasm{\underline{\lambda}^{(k)}}:\mathcal S_{r+k}T^*_x\rightarrow\mathcal S_{k}T^*_x$ is the $\betabr$
linear map induced by the restriction to $\mathcal S_kT^*_x$ of
$\lambda^{(k)}$. As the principal symbol
$\underline{\lambda}:\mathcal S_{r}T^*_x\rightarrow\betabr_{m,x}$ can be represented by $\sigmaum_{|\alpha|=r}a_\alpha y_\alpha$,
then $\underline{\lambda}^{(1)}:\mathcal S_{r+1}\rightarrow T^*_x$ can be written $\sigmaum_i\sigmaum_{|\alpha|=r}a_{\alpha}
y_{\alpha_i}\otimes dx_i$. See the discussion below.
Also note that the linear maps $y_\alpha|_{\mathcal S_{k}T^*_x}$ decomposes as the
$\alpha$ symmetric product of the coordinate cotangent vectors and we
have the canonical $\betabr$ vector space identification $\mathcal S_{r+1}T^*_x\cong
\mathcal S_r T^*_x\otimes T^*_x$. With these preliminaries we can prove
the following.
\betaegin{lemma}
Suppose that for $x\in\betabr^m$,
$\underline{\lambda}_x:\mathcal J^r_{m,1,x}\rightarrow\betabr_{m,x}$ is nonzero, ie. a surjection. Then
$\lambda^{(k)}$ is a surjection for all $k\in\betabn$.
\epsilonnd{lemma}
\betaegin{proof}
The remark above allows us to decompose the expression for
$\underline{\lambda}^{(1)}$ as $1_{T^*_x}\otimes\underline{\lambda}$. Actually this
holds at all levels.(SEE POMMARRET, \cite{Pommaret1978} p193) That is,
\betaegin{align}
\underline{\lambda}_x^{(k)}=\underline{\lambda}_x\otimes 1|_{\mathcal S_kT^*_x}.
\epsilonnd{align}
But note that in the category of finite dimensional $\betabr$ vector
spaces, the tensor product of surjections is a surjection, hence
by hypothesis $\underline{\lambda}^{(k)}$ is a surjection for each
$k\in\betabn$.
We will prove, by induction on $k$, the order of prolongation,
that $\lambda^{(k)}$ is a surjection. The result holds for $k=0$.
Suppose that it holds for some $k\gammaeq 0$. We will prove that it
holds for $k+1$.
By the inductive hypothesis, the remark on $\underline{\lambda}^{(k)}$ directly above and general facts on jets, we have a commutative
diagram of exact sequences of linear maps over $x$\\
\betaegin{align}\lambdabel{diag2}
\betaegin{CD}
0 @. 0\\
@VVV @VVV\\
\mathcal S_{r+k+1}T^*_x @> \underline{\lambda}^{(k+1)} >>\mathcal S_{k+1}T^*_x @>>> 0\\
@VV\i_{r+k+1}V @VV\i_{k+1}V\\
\mathcal J^{r+k+1}_{m,1,x} @>\lambda^{(k+1)} >> \mathcal J^{k+1}_{m,1,x} \\
@VV\partiali^{r+k+1}_{r+k} V @VV\partiali^{k+1}_{k} V\\
\mathcal J^{r+k}_{m,1,x} @>\lambda^{(k)} >>
\mathcal J^k_{m,1,x} @>>> 0\\
@VVV @VVV\\
0 @. 0\\
\epsilonnd{CD}
\epsilonnd{align}
and we wish to prove that the middle row is a surjection. Suppose
that $\epsilonta_{k+1}\in\mathcal J^{k+1}_{m,1,x}$. We will find
$\zeta\in\mathcal J^{r+k+1}_{m,1,x}$ such that $\lambda^{(k+1)}(\zeta)=\epsilonta_{k+1}$.
The proof will be a typical 'diagram chase'. Let
$\epsilonta_k=\partiali^{k+1}_k(\epsilonta_{k+1})$. Then, by hypothesis, there
exist $\epsilonta_{r+k}\in\mathcal J^k_{m,1,x}$ such that
$\lambda^{(k)}(\epsilonta_{r+k})=\epsilonta_k$. Let $\epsilonta_{r+k+1}\in
(\partiali^{r+k+1}_{r+k})^{-1}(\epsilonta_{r+k})$. So commutativity of
the lower square implies that
\betaegin{align}
\partiali^{k+1}_k\circ\lambda^{(k+1)}(\epsilonta_{r+k+1})=\lambda^{(k)}\circ\partiali^{r+k+1}_{r+k}(\epsilonta_{r+k+1})=\epsilonta_{k}= \partiali^{k+1}_{k}(\epsilonta_{k+1}).
\epsilonnd{align}
That is, 1) $\partiali^{k+1}_k(\lambda^{(k+1)}(\epsilonta_{r+k+1})-\epsilonta_{k+1})=0$, and so by exactness of the right sequence, there is
$\sigma_{k+1}\in\mathcal S_{k+1}T^*_x$ such that
\betaegin{align}\lambdabel{eqn:comm diagram,1}
i_{k+1}(\sigma_{k+1})=\lambda^{(k+1)}(\epsilonta_{r+k+1})-\epsilonta_{k+1}
\epsilonnd{align}
But $\underline{\lambda}^{(k+1)}$ is
surjective, ie., there exists $\sigma_{r+k+1}\in\mathcal S_{r+k+1}T*_x$
such that \\ $\underline{\lambda}^{(k+1)}(\sigma_{r+k+1})=\sigma_{k+1}$.
So by this and commutativity of the top square, we have
\betaegin{align}\lambdabel{eqn:comm diagram,2}
\lambda^{(k+1)}\circ i_{r+k+1}(\sigma_{r+k+1})=i_{k+1}\circ
\underline{\lambda}^{(k+1)}(\sigma_{r+k+1})=i_{k+1}(\sigma_{k+1}).
\epsilonnd{align}
Combining the equivalences in (\ref{eqn:comm diagram,1}) and (\ref{eqn:comm
diagram,2}), we get
\betaegin{align}
\lambda^{(k+1)}(\epsilonta_{r+k+1}-i_{r+k+1}(\sigma_{r+k+1}))=\epsilonta_{k+1}.
\epsilonnd{align}
That is, $\zeta\deltaoteq\epsilonta_{r+k+1}-i_{r+k+1}(\sigma_{r+k+1})$ is the
element of $\mathcal J^{r+k+1}_{m,1,x}$ we are looking for.
\epsilonnd{proof}
Note that we did not use the full strength of our setting; we did not use
the exactness of the left vertical sequence.
\betaegin{corollary}\lambdabel{lem:symb prolong is surj}
Suppose that $\lambda:\mathcal J^r_{m,1}\rightarrow\betabr_m$ is such that $\underline{\lambda}_{x_0}:\mathcal J^r_{m,1,x}\rightarrow\betabr_{x_0}$ is nonzero as in the previous
lemma. Then if $g\inC^\infty(\betabr^m,\betabr)$, $j^k_{x_0}g\in Im(\lambda^{(k)}_{x_0})$, for
every $k\in\betabn_0$.
\epsilonnd{corollary}
\betaegin{proof}
This is an immediate consequence of the previous lemma.
\epsilonnd{proof}
This corollary will allow us to extend Todorov's pointwise equality to an infinite jetwise equality in the next section.
\sigmaection{The Main Linear Theorem}
Before we begin the transfer of our results to the nonstandard world, we need in place a bit more of the framework for the infinite jet results in this section. First, let $N_s$ denote the finite set of
multiindices of length $m$ and weight less than or equal to $s$\;;
ie., indexing the fiber jet coordinates for $\mathcal J^r_{m,1}$. Let
$\overline{N_s}\sigmaubset N_s$ denote the subset of multiindices of length
equal to $\alpha$.
Given the notational material, we now examine how the lifting works.
Todorov proved a result crudely stated as follows: given $g$, there exists nonstandard $f$ such that $P(f)(x)=g(x)$ at each standard $x$. Our intention is to prove that such $f$ exists such that $j^s_x(P(f))=j^s_xg$ for all standard $x$ and all $s\in\;^\sigma\betabn$. This will be a consequence of the material in the previous section, the transfer of the Borel lemma and a bit more standard preliminaries.
The mapping $\lambda^{(s)}$ can be seen as the intermediary of $j^s(P(f))$ as follows. If $s\in\betabn$, and $|\alpha|\leq s$, we have that
\betaegin{align}
j^s_{x_0}(P(f))=P^{(s)}(f)(x_0)=\lambda^{(s)}_{x_0}(j^{r+s}_{x_0}(f))=j^s_{x_0}(\lambda\circ j^rf)=j^s_{x_0}g.
\epsilonnd{align}
We can therefore get a good estimate on the size of the range the successive prolongations of the range of $P$ at $x_0$ by watching the mapping properties of $\lambda^{(s)}$.
We will denote by $\betasm{\lambda^{(\infty)}}$ the infinite prolongation of
$\lambda$ given by
\betaegin{align}
j^{r,\infty}_xf\mapsto j^{\infty}_x(\lambda\circ j^rf): \mathcal J^{(r,\infty)}_{m,1,x}\rightarrow\mathcal J^{(\infty)}_{m,1,x}
\epsilonnd{align}
where $j^{r,\infty}_xf=(j^r_xf,j^{r+1}_xf,j^{r+2}_xf,\ldots)$,
ie., $\lambda^{(\infty)}$ being the map whose components are
already defined.
In this section and the next the transfer of the Borel Lemma will be used. Here is a statement of the version we will use.
\betaegin{lemma}[Borel Lemma]\lambdabel{lem: borel}
Let $x\in\betabr^m$ and suppose that $\partialhi\in\mathcal J^\infty_{m,1,x}$. Then there exists $f\inC^\infty(\betabr^m,\betabr)$ such that $\partialhi = j^\infty_xf$.
\epsilonnd{lemma}
Note, implicit in this result is the fact that this determination depends only on the germ of $f$ at $x$.
\sigmaubsection{Transfer of jet preliminaries}
To prove the main theorem we need to transfer the above jet formulation to
the internal arena inserting the homogeneous version of Todorov's
result into a jet level high enough so that the symbol has the
correct form.
If $\betasm{{}^\sigma LPDO_r}$ denotes those elements $P$ of $\rz
LPDO_r$ whose coefficients are standard elements of $C^\infty(\betabr^m,\betabr)$ , then
these correspond to symbols $\lambda_P\in{}^\sigma C^{\infty}(\mathcal J^r_{m,1},\betabr)$. Therefore, a special case of the
*transfer of Corollary \ref{cor:removing finite zeroes} is the following statement.
\betaegin{corollary}
Let $r\in\ {}^\sigma\betabn$, $D_a=\{x\in\betabr^m:|x|\leq a\}$ and $\rz P\in {}^\sigma LPDO_r$ with $\lambda$ the symbol of $P$.
Suppose that $\max\{c\in {}^\sigma\betabn:\rz\mathcal Z^c({\lambda})\cap \rz
D_a\;\text{is nonempty}\}$ is bounded in ${}^\sigma\betabn$
independent of $a\in\betabn$. Then there exists $s\in {}^\sigma \betabn$ such that if
$\lambda'$ is the symbol of $P^{(s)}$, then $\mathcal Z_{\lambda'}\cap\rz\betabr^m_{nes}$ is empty.
\epsilonnd{corollary}
\betaegin{proof}
In *transferring Corollary \ref{cor:removing finite zeroes}, we
need only note the following things for this corollary to follow.
First of all, we *transfer this corollary, for the situation
where $\mathcal Z(\lambda_P)\sigmaubset D_a$ for a given $0<a\in\betabn$ noting that
$\cup_{a>0}\rz D_a=\rz\betabr^m_{nes}$ and that the hypothesis
implies that there exists $a_0\in\betabn$ such that $m(a)\deltaoteq\max\{c\in {}^\sigma\betabn:\rz\mathcal Z^c({\lambda})\cap \rz
D_a\;\text{is nonempty}\}$ satisfies $m(a)\leq m(a_0)$ for all
$a\in\betabn$.
\epsilonnd{proof}
\betaegin{remark}
Suppose that
$\lambda_P\in C^{\infty}(\mathcal J^r_{m,1},\betabr)$ is such that for every bounded
$B\sigmaubset\betabr^m$,
$\mathcal Z(\lambda_P)\cap B$ has no accumulation points. Then $\rz\lambda_P$ can't have the
property that $\lambda_P$ vanishes to infinite, but hyperfinite order
at some point in $\rz\betabr^m_{nes}$. It therefore follows that
we can't use *transfer to generalize this result, in the given context, to points where
$ \lambda_P$ vanishes to infinite, hyperfinite order.
In order to proceed we need a particular type of nonstandard partition of unity
construction. For $0<c\in\rz\betabr^m$ and $y\in\rz\betabr^m$, let
$D_c(y)$ denote the disk centered at $y$ with radius $c$.
\epsilonnd{remark}
\betaegin{lemma}[*Weak partition of unity]\lambdabel{lem:POU}
Suppose that for every $ x\in
\betabr^m$, we have $f^x\in \rz C^\infty(\betabr^m,\betabr^n)$. Then there exists $f\in\rz C^\infty(\betabr^m,\betabr^n)$ and $0<\deltaelta\sigmaim 0$
such that for each $x\in\betabr^m$, $f|_{D_\deltaelta(x)}=f^x|_{D_\deltaelta(x)}$.
\epsilonnd{lemma}
\betaegin{proof}
First of all, sufficient saturation implies that the (external) map $^\sigma\betabr^m\rightarrow\rz C^\infty(\betabr^m,\betabr^n):x\mapsto f^x$ extends
to an internal map $\mathcal I:\rz\betabr^m\rightarrow\rz C^\infty(\betabr^m,\betabr):l\in\mathcal I\mapsto f^l$; see Theorem \ref{thm:extend external map}. Let
$\mathcal L\sigmaubset\rz\betabr^m$ be a *finite subset such that
$^\sigma\betabr^m\sigmaubset\mathcal L$. Choose $0<\deltaelta\in\rz\betabr$ such that $\deltaelta<\frac{1}{10}\rz min\{|l-l'|:l,l'\in\mathcal L,l\nablaot=l'\}$. By the *transfer of a variation on a weak form of the
partition of unity construction, there exists $\partialsi_l\in\rz C^\infty(\betabr^m,\betabr)$
for each $l\in\mathcal L$ such that $\sigmaum_{l\in\mathcal L}\partialsi_l(x)=x$ for each
$x\in\rz\betabr^m$ and for each $l\in\mathcal L$, $\partialsi_l|_{D_\deltaelta(l)}\epsilonquiv
1$. (As the *cardinality of $\mathcal L$ is *finite, we don't have to worry about *local finiteness of the sum of the $\partialsi_l$'s.)
Then the function $f\deltaoteq\rz\sigmaum_{l\in\mathcal L}\partialsi_lf^l$ has the
properties we need.
\epsilonnd{proof}
\betaegin{remark}
In a follow up paper, a numerically controlled version of this lemma (and the corresponding one in the nonlinear section) will allow proof of most of the existence results in this paper within the category of Colombeau-Todorov algebras.
\epsilonnd{remark}
Let $\lambda:\mathcal J^r_{m,1}\rightarrow\mathcal J^0_{m,1}$ denote the symbol of a $P\in
LPDO_r$.
\betaegin{definition}
Let $\betasm{finsupp(P)}$ or $\betasm{finsupp(\lambda)}$ denote the subset of $\betabr^m$ given by $\cup\{\mathcal Z^c(\lambda):c=0,1,2,\ldots\}$
For each $x\in\betabr^m$ and $k\in\betabn_0$, let $\betasm{\mathcal J^k_{\lambda,x}}$
denote the subspace of $\mathcal J^k_{m,1}$ given by $\lambda^{(k)}(\mathcal J^{r+k}_{m,1,x})$.
We write $g\nablaot=0(\lambda,x)$, if $j^k_xg\nablaot=0$ for some $k\in\betabn$. Let
$\betasm{\mathcal V^m_x}<C^\infty(\betabr^m,\betabr)$ denote the ideal of $f\inC^\infty(\betabr^m,\betabr)$ such that $j^{\infty}_xf=0$.
\epsilonnd{definition}
\betaegin{lemma}\lambdabel{lem: infin diml soln space at x}
If $x\in finsupp(\lambda)$, then there exists $g\inC^\infty(\betabr^m,\betabr)$ such that
$j^{k_0}_xg\nablaot=0$ for some $k_0\in\betabn$ and
$j^k_xg\in\mathcal J^k_{\lambda,x}$ for every integer $k\gammaeq 0$.
\epsilonnd{lemma}
\betaegin{proof}
Given Corollary \ref{cor:removing finite zeroes} the assertion amounts to specifying that the derivatives at each level must lie in a given set
and hence is an easy consequence of the Borel lemma.
\epsilonnd{proof}
\betaegin{definition}
Let $\betasm{\fracrak I_{\lambda,x}}=\{g\inC^\infty(\betabr^m,\betabr):j^k_xg\in\mathcal J^k_{\lambda,x}\;\text{for
all}\;k\in\betabn_0\}$.
\epsilonnd{definition}
Note, of course, that $\mathcal V^m_x<\fracrak I_{\lambda,x}$.
So by the above lemma, $\fracrak I_{\lambda,x}$ is infinite dimensional.
Therefore, $\rz\fracrak I_{*\lambda,*x}$ is a *infinite dimensional $\rz\betabr$ subspace of $\rz C^\infty(\betabr^m,\betabr)$.
In the nonstandard world, we have the following analogous definition.
\betaegin{definition}
Let
\betaegin{align}
\partialmb{^\sigma\fracrak I_{\lambda,x}}= \{g\in\rz C^\infty(\betabr^m,\betabr): \rz j^k_{*x}g\in\rz\mathcal J^k_{*\lambda,*x}
\;\text{for all}\;k\in\; ^\sigma\betabn_0\}.
\epsilonnd{align}
\epsilonnd{definition}
Note that $^\sigma\fracrak I_{\lambda,x}$ is an external $\rz\betabr$ vector space and $\rz\fracrak I_{\lambda,x}\sigmaubset\;^\sigma\fracrak
I_{\lambda,x}\sigmaubset\rz C^\infty(\betabr^m,\betabr)$. In particular, $^\sigma\fracrak I_{\lambda,x}$ is infinite dimensional.
Note that its *dimensionality is not well defined. We have one more definition.
\betaegin{definition}
Let $\betasm{^\sigma\fracrak I_\lambda}$ denote the set of $g\in\rz C^\infty(\betabr^m,\betabr)$ such that for
all $\rz x\in\;^\sigma\betabr^m$, $g\in\;^\sigma\fracrak I_{\lambda,x}$.
\epsilonnd{definition}
\betaegin{lemma}\lambdabel{lem: finite vanish implies section}
Suppose that $^\sigma\fracrak I_{*\lambda,*x}\nablaot=0$ for some $x\in\betabr^m$.
Then $^\sigma\fracrak I_\lambda\nablaot=0$.
\epsilonnd{lemma}
\betaegin{proof}
For each $x\in\betabr^m$, choose $f^x\in\rz C^\infty(\betabr^m,\betabr)$ with $f^x\in\; ^\sigma\fracrak
I_{*\lambda,*x}$, such that for some $x$, $f^x\nablaot=0(\lambda,x)$.
By Lemma \ref{lem:POU}, there exists $f\in\rz C^\infty(\betabr^m,\betabr)$ and
$0<\deltaelta\sigmaim0$ such that $f|_{D_\deltaelta(x)}=f^x|_{D_\deltaelta(x)}$ for each
$x\in\betabr^m$. But then, for each $x\in\betabr^m$ and each
$k\in\betabn_0$, $\rz\!j^k_{*x}f=\rz\!j^k_{*x}f^x\in\; ^\sigma\mathcal J^k_{*\lambda,*x}$. That
is $f\in ^\sigma\fracrak I_\lambda$, and $f\nablaot=0(\lambda,x)$ for some $x$.
\epsilonnd{proof}
\betaegin{lemma}\lambdabel{lem:infty soln jet at x}
Let $x\in\betabr^m$, and $g\in\fracrak I_{\lambda,x}$. Then
there exists $f\inC^\infty(\betabr^m,\betabr)$ such that
$\lambda^{(\infty)}_x(j^{r,\infty}_xf)=j^{\infty}_xg$.
\epsilonnd{lemma}
\betaegin{proof}
First of all, for every $k\in\betabn_0$, there exists
$\gamma_k\in\mathcal J^{r+k}_{m,1}$ such that $\lambda^{(k)}(\gamma_k)=j^k_xg$. This
just follows from the definition of $\fracrak I_{\lambda,x}$. Since this
holds for all $k$, then there exists $\gamma\in\mathcal J^{r,\infty}_{m,1}$
with $\lambda^{(\infty)}_x(\gamma)=j^{\infty}_xg$. Just let $\gamma$ be such
that $\partiali^{\infty}_k(\gamma)=\gamma_k$ for each $k$. But note that for
$\gamma\in\mathcal J^{r,\infty}_{m,1,x}$, the Borel Lemma, Lemma \ref{lem: borel}, implies that there exists $f\inC^\infty(\betabr^m,\betabr)$ such that
$j^{r,\infty}_xf=\gamma$.
\epsilonnd{proof}
\betaegin{notation}
If $f\in\rz C^\infty(\betabr^m,\betabr)$, we will denote
\betaegin{align}
\betasm{\rz\!j^\sigma_x(f)}=(\rz(j^k_{*x})(f))_{k\in\betabn_0},\;\text{an external sequence}.
\epsilonnd{align}
Similarly, if $\lambda$ is an internal jet map and we are considering, for each $k\in\betabn$, not $\rz\betabn$, $\lambda^{(k)}_{\;* x}$, the \textbf{internal} prolongation of $\lambda$ at the standard point $\rz x$, ie.,$(\lambda^{(k)}_{*x})_{k\in^\sigma\betabn}$, then we will also write this as $\betasm{\lambda^{(\sigma)}_x}$; eg., if $\lambda$ or $f$ are standard and we are considering only this family of internal prolongations of $\rz\lambda$ or $\rz f$, then we will write $\betasm{\rz\!j^\sigma_x(\rz\!f)}$ or $\betasm{\rz\!\lambda^{(\sigma)}_x}$.
\epsilonnd{notation}
In the situation when $f\inC^\infty(\betabr^m,\betabr)$, $\rz\!j^\sigma_x(\rz f)$ is just the external sequence
of standard numbers, $(\rz(j^k_xf))_{k\in\betabn_0}$. This notation
can be unwieldy; some of the parentheses, or *'s may be left out
if the meaning is still clear.
Note that if
\betaegin{align}
\mathcal V^m=\betaigcap_{x\in\betabr^m}\mathcal V^m_x=\{f\in\rz C^\infty(\betabr^m,\betabr):\rz\!j^\sigma_x(f)=0,\;\text{for
all}\;x\in\betabr^m\},
\epsilonnd{align}
then $\mathcal V^m<\;^\sigma\fracrak I_\lambda$. Although $\mathcal V$ is a $\rz \betabr$ vector space, it is nonetheless external. To get a sense of the size of
$\mathcal V^m$ in $\rz C^\infty(M,\betabr^n)$, note that $\mathcal L<\mathcal V$ where $\mathcal L$ is the
*finite codimensional subspace $\rz C^\infty(M,\betabr^n)$\; defined in the
concluding section of the paper. Therefore, we have the following consequence of Lemma \ref{lem: finite vanish implies section}.
\betaegin{corollary}\lambdabel{cor: section at pt implies infnte diml sections}
Suppose that $^\sigma\fracrak I_{*\lambda,*x}\nablaot=0$ for some $x\in\betabr^m$.
Then $^\sigma\fracrak I_\lambda$ is *infinite dimensional.
\epsilonnd{corollary}
\betaegin{remark}
Suppose that $f\inC^\infty(\betabr^m,\betabr),\;\overline{f}\in\rz C^\infty(\betabr^m,\betabr)$ such that for some standard $x$,
and $0<\deltaelta\sigmaim 0$, $\overline{f}|_{D_\deltaelta(*x)}=\rz\!f|_{D_\deltaelta(*x)}$. Then the
internal jet sequence
$\rz j^{\infty}_{*x}\overline{f}\deltaoteq(\rz j^k_{*x}\overline{f})_{k\in*\betabn}$ is just the *transfer of the standard sequence $(j^k_xf)_{k\in\betabn_0}$,
eg., when the set of jet indices is restricted to to the external set
$^\sigma\betabn_0$. That is, in the above notation, $\rz\!j^\sigma_x(\overline{f})=
(\rz\!j^{k}_xf)_{k\in\betabn_0}$.
\epsilonnd{remark}
\sigmaubsection{Many generalized solutions with high contact}
The following result is the main linear result of the paper, although it's import is not apparent without the following corollaries.
\betaegin{theorem}\lambdabel{thm:lin eqn,infinite contact soln}
Suppose that $P\in LPDO_r$. Then, for every $g\in\; ^\sigma\fracrak
I_\lambda$, there exists $f\in\rz C^\infty(\betabr^m,\betabr)$ such that
\betaegin{align}
\rz\!j^{\sigma}_x(\rz P(f))=\rz\!j^{\sigma}_xg\;\text{for
every}\;x\in\betabr^m.
\epsilonnd{align}
That is, $\rz P(f)$ has $^\sigma$infinite order *contact with $g$ at all points of $\rz\betabr^m_{nes}$.
\epsilonnd{theorem}
\betaegin{proof}
Suppose that $^\sigma\fracrak I_{\lambda}$. By Lemma \ref{lem:infty soln jet at x} if $x\in\betabr^m$, there is
$f^x\inC^\infty(\betabr^m,\betabr) $ such that
\betaegin{align}\lambdabel{eqn:infinite jet soln at point}
\lambda^{(\infty)}_{P,x}(j^{\infty}_xf^x)=j^{\infty}_xg.
\epsilonnd{align}
By Lemma
\ref{lem:POU} there exist $\overline{f}\in\rz C^\infty(\betabr^m,\betabr)$ such that for every
$x\in\betabr^m$ $\overline{f}|_{D_\deltaelta(*x)}=\rz f^x|_{D_\deltaelta(*x)}$. By the remark
above, for each such standard $x$,
$\rz\!j^{\sigma}_x\overline{f}=\rz\!j^{\sigma}_{*x}(\rz\!f^x)$. But this
implies that, at each standard $x$,
\betaegin{align}
\rz\!\lambda^{(\sigma)}_{*x}(\rz\!j^{\sigma}_{*x}\overline{f})=\rz\!\lambda^{(\sigma)}_{*x}(\rz\!j^{\sigma}_{*x}\rz\!f^x)
\epsilonnd{align}
Coupling this with the transfer of expression (\ref{eqn:infinite jet soln at
point}) restricted to standard indices, we now have $\overline{f}\in\rz C^\infty(\betabr^m,\betabr)$ such that
\betaegin{align}\lambdabel{eqn:infinite prolong,global soln in symb}
\rz\lambda^{(\sigma)}_{*x}(\rz j^{\sigma}_{*x}\overline{f})=\rz j^{\sigma}_xg
\epsilonnd{align}
for each $x\in\betabr^m$. But, by definition of prolongation,
*transferred
\betaegin{align}\lambdabel{eqn:infin prolong P is infin prolong lam}
\rz j^{\infty}_{*x}(\rz P(\overline{f}))=\rz\lambda^{(\infty)}_{*x}(\rz j^{r,\infty}_{*x}\overline{f}).
\epsilonnd{align}
Stringing together expressions (\ref{eqn:infinite prolong,global soln in
symb}) and (\ref{eqn:infin prolong P is infin prolong lam}), restricted to standard indices, gets
our result, as this holds for every standard $x$.
\epsilonnd{proof}
\betaegin{corollary}\lambdabel{cor: infinite order Todorov}
Suppose that $P\in LPDO_r$ with symbol $\lambda$, and principal symbol $\underline{\lambda}$. Suppose that for each $x\in\betabr^m,\underline{\lambda}_x\nablaot=0$. Then for every
$g\in\rz C^\infty(\betabr^m,\betabr)$, there exists $f\in\rz C^\infty(\betabr^m,\betabr)$ with
\betaegin{align}
*j^{\infty}_{*x}(*P(f))=*j^{\infty}_{*x}g\;\text{for
every}\;x\in\betabr^m.
\epsilonnd{align}
\epsilonnd{corollary}
\betaegin{proof}
By Lemma \ref{lem:symb prolong is surj}, if $g\inC^\infty(\betabr^m,\betabr)$, then $g\in\; ^\sigma\fracrak
I_\lambda$. But then the result is a direct consequence the previous
theorem.
\epsilonnd{proof}
\betaegin{remark}
To put this result in perspective, note that Todorov, \cite{Todorov96},
proves the $0^{th}$ order jet case in his paper, with a slightly
weaker hypothesis.
\epsilonnd{remark}
For those $x\in\betabr^m$ where $\lambda_x=0$, a trivial case for the
$0$-jet, as Todorov notes, becomes a nontrivial thickened result
when the consideration becomes the infinite jet at standard points
where some finite prolongation $\lambda^{(k)}_x$ is nonzero. For
this situation we have the following result.
\betaegin{corollary}\lambdabel{cor: infinite order solns for finite contact}
Suppose that $finsupp(P)=\betabr^m$. Then there exists an *infinite dimensional subspace $^\sigma\fracrak I_P<\rz C^\infty(\betabr^m,\betabr)$
such that if $g\in\; ^\sigma\fracrak I_P$,
then there exists $f\in\rz C^\infty(\betabr^m,\betabr)$ such that
\betaegin{align}
\rz j^{\sigma}_{*x}(\rz P(f))=\rz j^{\sigma}_{*x}g\;\text{for
every}\;x\in\betabr^m.
\epsilonnd{align}
\epsilonnd{corollary}
\betaegin{proof}
As $finsupp(P)=\betabr^m$, if $x\in\betabr^m$, Lemma \ref{lem: infin diml soln space at
x} implies that $\mathcal S^P_x\deltaoteq\{j^{\infty}_xg:g\in \fracrak I_{\lambda,x}\}$ is nonzero. Therefore, the result follows from Corollary \ref{cor: section at pt implies infnte diml sections} and the above
theorem.
\epsilonnd{proof}
That is, even if the symbol vanishes at points of $\betabr^m$, as long as this vanishing order is finite at each such point, then there exists many $g\in\rz C^\infty(\betabr^m,\betabr)$, satisfying the above compatibility conditions, such that $\rz P(f)=g$ is solved to infinite order along $^\sigma\betabr^m$ by $f\in\rz C^\infty(\betabr^m,\betabr)$.
\sigmaubsection{Solutions for singular Lewy operator}
Before we move to the next section, let's look at the Lewy operator, see \cite{Todorov96}, p.679, $\mathcal L=\partial_1+i\partial_2-2i(x_1+ix_2)\partial_3$ acting on smooth complex valued functions on $\betabr^3$. First of all, note that the results just proved hold just as well with complex valued functions; the proofs are identical. Second, note that the principal symbol, $\underline{\lambda}_{\;\mathcal L}$ of $\mathcal L$ is the same as the total symbol $\lambda_\mathcal L=y_1+iy_2-2i(x_1+ix_2)y_3$. Inspection shows that these maps are nonvanishing, hence $\mathcal L$ satisfies the hypotheses in Corollary \ref{cor: infinite order Todorov}, ie., for any $g\in \rz C^\infty(\betabr^3,\betabc)$, there exists (many) $f\in\rz C^\infty (\betabr^3,\betabc)$ such that $\rz\mathcal L(f)(\rz x)=\rz g(\rz x)$ to infinite order at each $x\in\betabr^3$.
But we can say more. Suppose that $h=(h_1,h_2,h_3)$ is such that $h_i\in C^\infty(\betabr^3,\betabr)$ for each $i$ and $h$ vanishes to finite order at each $x\in\betabr^3$. {\it Let $\widehat{\mathcal L}=h_1(x)\partial_1+ih_2(x)\partial_2-2ih_3(x)(x_1+x_2)\partial_3$, a kind of singular Lewy operator with finite singularities at each $x\in\betabr^3$.
Then Corollary \ref{cor: infinite order solns for finite contact} implies that for any $g\in C^\infty(\betabr^3,\betabc)$ that vanishes where $\lambda_{\widehat{\mathcal L}}$ vanishes to order at least that of $h$, there exists $f\in \rz C^\infty(\betabr^3,\betabc)$ such that $\widehat{\mathcal L}(f)(\rz x)=g(\rz x)$ holds to infinite order at all $x\in\betabr^3$}.
\sigmaection{Nonlinear PDE's and the pointwise lifting property}\lambdabel{section: nonlinear work}
In this section $P$ can now be an arbitrary smooth nonlinear PDO of finite order.
Only the rudiments of a nonlinear development parallel to the linear considerations
in the previous sections will be attempted in this paper. The point here is
that the
framework is not an impediment to a consistent consideration of generalized objects.
First, as it is natural within our framework, we straightforwardly extend the notion of solution of a differential equation, as defined in Todorov's paper, to include nonlinear as well as linear differential equations.
In analogy with $LPDO_r$, a (possibly nonlinear) order $r$ partial differential operator, $P:C^\infty(\betabr^m,\betabr)\rightarrow C^\infty(\betabr^m,\betabr)$, is a mapping given by $P(f)(x)=\lambda(j^r_xf)$ where now the total symbol of $P$, $\lambda:\mathcal J^r_{m,1}\rightarrow\betabr_m$ is a possibly nonlinear smooth bundle map. Let $\betasm{NLDO_r}$ denote this set of operators.
\betaegin{definition}
Given $g\in\rz C^\infty(\betabr^m,\betabr)$, we say that $f\in\rz C^\infty(\betabr^m,\betabr)$ is a solution of $\rz P(f)=g$ if $\rz P(f)(\rz x)=g(\rz x)$ for every $x\in\betabr^m$.
\epsilonnd{definition}
We will consider a simple set theoretic condition on pairs $(P,g)$ (or $(\lambda_P,g)$) the \textbf{pointwise covering property}, $\betasm{PCP}$.
An easy (saturation) proof will get that if $(P,g)$ satisfies this
property, written $(P,g)\in PCP$, or $(\lambda_P,g)\in PCP$, then $P(f)=g$ has generalized solutions in a sense of
Todorov. We will then show that the main theorem is a corollary
of this result by verifying that our linear differential equation
satisfies $PCP$.
\betaegin{definition}
Let $\lambda\in C^{\infty}(\mathcal J^k_{m,1},\betabr)$ and $g\inC^\infty(\betabr^m,\betabr)$. We
say that \textbf{the pair $\betasm{(\lambda,g)}$ satisfies $\betasm{PCP}$}, if for each
$x\in\betabr^m$, there exists $p\in (\partiali^k)^{-1}(x)$, such that
$\lambda(p)=g(x)$. If $\lambda\in \rz C^{\infty}(\mathcal J^k_{m,1},\betabr)$ and
$g\in\rz C^\infty(\betabr^m,\betabr)$, then we say that \textbf{the pair $\betasm{(\rz\lambda,g)}$ satisfies $\betasm{{}^\sigma PCP}$}
if for all $x\in {}^\sigma\betabr^m$, there exists $p\in\rz\partiali^{-1}(x)$,
such that $\lambda(p)=g(\rz x)$.
\epsilonnd{definition}
\betaegin{remark}
Note that finding $p\in(\partiali^r)^{-1}(x)$ such that $\lambda(p)=g(x)$
is identical to finding $h\inC^\infty(\betabr^m,\betabr)$ such that $h$ solves $P(h)=g$
at the single point $x$. Also, note the relationship between $PCP$ and ${}^\sigma PCP$.
If $\lambda\in C^{\infty}(\mathcal J^k_{m,1},\betabr)$ and $g\inC^\infty(\betabr^m,\betabr)$ are
such that $(\lambda,g)\in PCP$, then $(\rz\lambda,\rz g)\in {}^\sigma PCP$. On the other hand, if
$\lambda\in SC^{\infty}(\mathcal J^k_{m,1},\betabr)$ and $g\inSC^\infty(\betabr^m,\betabr)$ are
such that $(\lambda,g)\in {}^\sigma PCP$, then $(^o\lambda,^og)\in PCP$. (Recall that if $X,Y$ are Hausdorff topological spaces and $f:\rz X\rightarrow\rz Y$ is such that $f$ maps nearstandard points of $\rz X$ to those of $\rz Y$, then the standard part of $f$, $^of:X\rightarrow Y$ is a welldefined map.)
\epsilonnd{remark}
The following lemma verifies that the the PCP condition restricted to linear differential operators has Todorov's criterion as a special case. We are working with the symbol of the operator.
\betaegin{lemma}
Let $P\in LPDO_r$ and write
$$
\lambda_P=\sigmaum_{|\alpha|\leq r}f_{\alpha}y_{\alpha}.
$$
Suppose that $\sigmaum_{|\alpha|\leq r}|f^{\alpha}(x)|\nablaot=0$ for all
$x\in\betabr^m$
Then for all $g\in\rz C^\infty(\betabr^m,\betabr)$, $(\rz\lambda_P,g)\in {}^\sigma PCP$.
\epsilonnd{lemma}
\betaegin{proof}
Let $x_0\in\betabr^m$. We will not write $\rz x_0$ when we transfer. The condition guarantees that there exists a
multiindex $\alpha$ such that $f^{\alpha}(x_0)\nablaot=0$. Let $\Gamma=\{\alpha:c_{\alpha}\deltaoteq f^{\alpha}(x_0)\nablaot=0\}$.
If $\Gamma$ has only one element, $\alpha_0$, let $h\in\rz C^\infty(\betabr^m,\betabr)$ be such
that
\betaegin{align}
\rz\partial^{\alpha_0}(h)(x_0)=\frac{g(x_0)}{\rz f^{\alpha_0}(x_0)}
\epsilonnd{align}
Then if $\kappaappa\in\rz\mathcal J^k_{m,1}$ is given by $\rz j^r_{x_0}h$, we get that
\betaegin{align}\lambdabel{eqn:PCP}
\rz\lambda_P(\kappaappa)=\rz\lambda_P(\rz j^r_{x_0}(h))= \\ \rz \sigmaum_{|\alpha|\leq \nablaotag
r}f^{\alpha}(x_0)y_{\alpha}(j^r_{x_0}h)= &
f^{\alpha_0}(x_0)y_{\alpha_0}(j^r_{x_0}h)\nablaotag = \\
f^{\alpha_0}(x_0)\frac{g(x_0)}{f^{\alpha_0}(x_0)}= g(x_0)\nablaotag
\epsilonnd{align}
as we wanted. So suppose that $\Gamma$ has at least two elements.
Let $\alpha_0\in\Gamma$ and let $\Lambda=\Gamma-\{\alpha_0\}$. Choose $h\in\rz C^\infty(\betabr^m,\betabr)$ so
that if $\alpha\in\Lambda$, then $\rz\partial^{\alpha}h(x_0)=0$ and (as in the first
case)
such that $\rz\partial^{\alpha_0}(h)(x_0)=\frac{g(x_0)}{f^{\alpha_0}(x_0)}$. Then as
in expressions (\ref{eqn:PCP}), we get $P(h)(x_0)=g(x_0)$.
\epsilonnd{proof}
Given the above lemma we shall see that Todorov's result is a corollary of this lemma and the next theorem proving the existence of solutions of PCP operators.
Before we proceed to the theorem, we need some NSA preliminaries.
First we give a simple example
of the construction we will need. Let $F(\betabr)$ be all maps from $\betabr$ to
$\betabr$ and let $F^{\infty}(\betabr)=\{f\in F(\betabr):f\;\text{is smooth}\}$. Let $f\in F(\betabr)$, then
there exists an (internal) element $\tilde{f}\in\rz F^{\infty}(\betabr)$ such that
$\tilde{f}|_{{}^\sigma\betabr}=f$, as the following argument shows. Let
$\mathcal Y_1\sigmaubset \rz\betabr$ be *finite such that ${}^\sigma\betabr\sigmaubset\mathcal Y_1$ and let
$\mathcal Y_2=\rz f(\mathcal Y_1)$. Then $\mathcal Y_2$ is obviously a *finite subset of $\rz\betabr$.
Now consider the following elementary standard statement. If $S_1,S_2$ are finite subsets of $\betabr$, of the same cardinality,
and $h:S_1\rightarrow S_2$ is a bijection, there exists $\tilde{h}\in F^{\infty}(\betabr)$ such that $\tilde{h}|_{S_1}=h$. This follows from a simple partition of unity argument.
Now *transfer this to get existence of $\tilde{f}\in\rz F^{\infty}(\betabr)$ such that
$\tilde{f}|_{\mathcal Y_1}=\rz f|_{\mathcal Y_1}$. In particular, $\tilde{f}|_{{}^\sigma\betabr}=\rz
f|_{{}^\sigma\betabr}=f|_{\betabr}$, as we wanted. Now we want to do the
same construction in the venue of bundles and their sections.
Let $\betasm{\Gamma(\mathcal J^r_{ m,1})}=\{s:\betabr^m\rightarrow\mathcal J^r_{m,1}|\;\partiali^r\circ
s=\betabi_{\betabr^m}\}$, ie., set theoretic sections of $\partiali^r$. Let
$\betasm{\Gamma^{\infty}(\mathcal J^r_{m,1})}=\{s\in\Gamma(\mathcal J^r_{ m,1}):s\;\text{is a
smooth map}\}$. We have the following lemma.
\betaegin{lemma}\lambdabel{lem:first standard jet approx}
Suppose that $s\in\Gamma(\mathcal J^r_{m,1})$. Then there exists
$\tilde{s}\in\rz\Gamma^{\infty}(\mathcal J^r_{m,1})$, such that
$\tilde{s}|_{{}^\sigma\betabr^m}=s|_{\betabr^m}$.
\epsilonnd{lemma}
\betaegin{proof}
As with the above example, let $\mathcal X\sigmaubset\rz\betabr^m$ be *finite
such that $\betabr^m\sigmaubset\mathcal X$. We have the following elementary
fact. If $B=\{b_1,\ldots,b_l\}$ is a finite subset of the
base and $P=\{p_1,\ldots,p_l\}\sigmaubset\mathcal J^r_{m,1}$ is a finite
subset such that $p_j\in(\partiali^r)^{-1}(b_j)$ for each $j$, then
there exists $s\in\Gamma(\mathcal J^r_{m,1})$ such that $s(x_j)=p_j$ for
all $j$. Now *transnfer this statement, applying the *transferred
statement to the *finite subset $\mathcal X$ in the base and the
*finite subset $\rz s(\mathcal X)$ of points in the *bundle over $\mathcal X$.
That is, we can infer the existence of an internal section
$\tilde{s}\in\rz\Gamma^{\infty}(\mathcal J^r_{m,1})$ such that for all $x\in\mathcal X$,
$\tilde{s}(x)=\rz s(x)$, in particular $\tilde{s}|_{{}^\sigma\betabr}=s$, as
we wanted.
\epsilonnd{proof}
In the context of this lemma, we have that
$(\rz\lambda,g)\in{}^\sigma PCP$ is equivalent to the existence of a set
theoretic section $s\in\rz\Gamma(\mathcal J^r_{m,1})$ such that the pointwise
condition $\rz\lambda\circ \rz s=g$ holds on ${}^\sigma\betabr^m$. It's
important to note that, generally speaking, such sections are
far from integrable; that is, equal to $j^rf$ for some smooth
$f\inC^\infty(\betabr^m,\betabr)$. But again, by a transfer argument, we can find such a
section.
\betaegin{lemma}\lambdabel{lem:second standard jet approx}
Suppose that $s\in\Gamma^{\infty}(\mathcal J^r_{m,1})$ and $\mathcal X\sigmaubset\rz\betabr^m$.
Then there exists $f\in\rz C^\infty(\betabr^m,\betabr)$ such that $\rz
j^rf|_{\mathcal X}=s|_{\mathcal X}$.
\epsilonnd{lemma}
\betaegin{proof}
This just follows from the *transfer of the following obvious
standard statement about jets. If
$\{p_1,p_2,\ldots,p_l\}\sigmaubset\mathcal J^r_{m,1}$ such that
$x_j=\partiali^r(p_j)$ are all distinct. Then there exists $f\inC^\infty(\betabr^m,\betabr)$
such that $j^r_{x_i}f=p_i$ for all i.
\epsilonnd{proof}
With these preliminaries, the proof of the following result is
immediate.
\betaegin{theorem}
Let $\mathcal D\in NLDO_r$ and let
$\lambda_{\mathcal D}=\lambda\in C^{\infty}(\mathcal J^r_{m,1},\betabr)$ and $g\in\rz C^\infty(\betabr^m,\betabr)$.
Suppose that $(\rz\lambda,g)\in {}^\sigma PCP$. Then $\mathcal D(f)=g$ has a generalized
solution, $f$, in the sense of Todorov.
\epsilonnd{theorem}
\betaegin{proof}
By the remark above, $(\rz\lambda,g)\in {}^\sigma PCP$ is equivalent to
the existence of an $s\in\rz\Gamma(\mathcal J^r_{m,1},\betabr)$ such that for
every $x\in\betabr^m$, $\lambda_{\rz\mathcal D}(s(\rz x))=g(\rz x)$. But by
Lemma \ref{lem:first standard jet approx}, there exists $\tilde{s}\in\rz\Gamma^{\infty}(\mathcal J^r_{m,1})$
such that $\tilde{s}(\rz x)=s(x)$ for all $x\in \betabr^m$. And by
Lemma \ref{lem:second standard jet approx}, there exists
$f\in\rz C^\infty(\betabr^m,\betabr)$, such that for all $x\in\betabr^m$, $\rz
j^r_{* x}(f)=\tilde{s}(\rz x)$.
\epsilonnd{proof}
Todorov's existence result (being for linear operators only) is a special consequence of the
previous development.
\betaegin{corollary}
Suppose that $g\in\rz C^\infty(\betabr^m,\betabr)$ and $P\in LPDO_r$ is such that $\lambda_P$ is nonvanishing
on $\betabr^m$. Then there exists $f\in\rz C^\infty(\betabr^m,\betabr)$, such that for all
$x\in\betabr^m$, $P(f)(\rz x)=g(\rz x)$, ie., $f$ is a solution of
$P(f)=g$ in the manner of Todorov.
\epsilonnd{corollary}
\betaegin{proof}
This is clear.
\epsilonnd{proof}
Given the nonlinear setting of this section, proving results analogous to those in the linear sections appear to need much more involved preliminaries and so will be pursued at a later date.
Nonetheless, it seems clear that we can consider some general criteria
revolving around when $(P,g)\in PCP$. In particular, it appears that we can
prove {\it a universal existence theorem asserting that any possible space of
generalized functions that has the $PCP$ property is already
contained in our nonstandard space}. This, too, will appear as time allows.
\sigmaection{Conclusion}
\sigmaubsection{Too many solutions?}
In this paper I have used some of the machinery of the geometry of partial differential equations
to explore the possibilities of the approach of Todorov. (We have yet to work through the nonlinear analogs of
the linear results presented here; this will entail a much more extensive use of the the jet theory
of nonlinear partial differential operators. Note even more starkly than in this paper; no counterpart in standard mathematics exists.)
The implications of the results of this paper are still not clear. Yet one
thing should be obvious, the class of internally smooth maps are
remarkably `flabby', as compared to the standard world.
As an indication of this, we have the following construction.
let $\betasm{\mathcal L}<\rz C^\infty(\betabr^m,\betabr^n)$ be the $\rz\betabr$ linear subspace of $\rz C^\infty(\betabr^m,\betabr^n)$ defined as follows.
Let $\mathcal Y\sigmaubset\rz\betabr^m$ be a *finite subset such that
$^\sigma\betabr^m\sigmaubset\mathcal Y$. Let $\omegaega\in\rz\betabn_{\infty}$. Then,
the set $\betasm{\mathcal L}=\{f\in\rz C^\infty(\betabr^m,\betabr^n):\rz\! j^{\omega}_x f=0\;\text{for all}\;x\in\mathcal Y\}$ is a
*cofinite dimensional subspace of $\rz C^\infty(\betabr^m,\betabr^n)$, as this set of conditions
on elements $f$ in $\rz C^\infty(\betabr^m,\betabr^n)$\; given by specifying the value of $j^{\omega}_{x}f$, a *finite number of *Taylor coefficients at a *finite set of points in $\rz\betabr^m$, ie., the points of $\mathcal Y$ is *finite.
Now, by construction,
$\mathcal L\cap\;^\sigma C^\infty(\betabr^m,\betabr^n)=\{0\}$, and
we have the following diagram
\betaegin{align}
\betaegin{CD}
\mathcal L\\
@VjVV \\
\rz C^\infty(\betabr^m,\betabr^n) @>\text{ *$j^\omega$}>> \rz \mathcal J^{\omega}_{m,n} @>\rho>> \rz \mathcal J^{\omega}_{m,n}|_{^\sigma\!\betabr^m}\\
@AiAA \\
\rz\betabr\otimes\;^\sigma C^\infty(\betabr^m,\betabr^n)
\epsilonnd{CD}
\epsilonnd{align}
where the maps $i$ and $j$ are $\rz\betabr$ subspace injections and $\rho$ is
the highly external restriction to the fibers over $^\sigma\betabr^m$.
Let $\Phi =\rho\circ\rz j^{\omega}$. Then the following holds.
\betaegin{lemma}
$\Phi|_{Im(j)}$ has image $\{0\}$ and $\Phi|_{Im(i)}$ is an injection.
\epsilonnd{lemma}
\betaegin{proof}
By construction, we have $\Phi(f)=0$ for every $f\in\mathcal L$. On the other hand, if for any element
$f\in\; ^\sigma \!C^\infty(\betabr^m,\betabr^n)$ we have $\rz\!j^{\omega}_x(\rz f)=0$ for each
$x\in\mathcal Y$, we have in particular that $j^{\infty}_x f=0$, for
each $x\in\betabr^m$, that is $f=0$. This therefore holds for all
$f\in\rz\betabr^m\underlinederset{\betabr^m}{\otimes}^\sigma \!C^\infty(\betabr^m,\betabr^n)$.
\epsilonnd{proof}
So we have that the subspace of elements of\; $\rz C^\infty(\betabr^m,\betabr^n)$ \; whose $^\sigma$\! infinite *jet vanishes everywhere
on $^\sigma\betabr^m$ is all of \;$\rz C^\infty(\betabr^m,\betabr^n)$\; up to a *finite dimensional subspace containing all
standard smooth maps. It should therefore be clear that we have the immediate corollary that exemplifies the ability to
bend almost all of $\rz C^\infty(\betabr^m,\betabr^n)$ \; away from contact with the world
of standard differential equations, at least at standard points.
\betaegin{corollary}
If $P\in NPDO_r$ for any $r\in\betabn$ such that $P(\text{zero map})= \text{zero map}$, then $\rz P(f)(\rz x)=0$ for all $f\in\mathcal L$ and all $x\in\betabr^m$.
\epsilonnd{corollary}
\betaegin{proof}
All classical differential operators $P$ of order $r$, factor as $\rz P=\rz\lambda_P\circ \rz j^r=\rz\lambda_P\circ\rz\partiali^{\omega}_r\circ
\rz j^{\omega}$ and by above $j^{\omega}(\mathcal L)|\;^\sigma\betabr^m =\{0\}$.
\epsilonnd{proof}
{\it That is, all classical partial differential operators sending the zero map to the zero map operate as zero maps on ``almost all'' of \;$\rz C^\infty(\betabr^m,\betabr^n)$\;.}
One perspective on the results here should not be a surprise: that *smooth functions (and with some thought *analytic functions) are far too flabby on a full infinitesimal scale. From a positive viewpoint, one could see how this might allow an investigator to have wide latitude in `Tayloring' generalized functions (on the monadic level) to get appropriate rigidities-growth or to test various empirical results by infinitesimal adjustings of singular parameters. The remark (in the introduction) with respect to the work of Baty, etal, see eg., \cite{BatyShockWave2008} seems relevant to the second perspective. The algebras of Oberguggenberger and Todorov, \cite{OT98} and the further developments in eg., Todorov and Vernaeve, \cite{TodorVern2008} seems to be good examples of the Tayloring capacities.
\sigmaubsection{Prospects and goals}
Only the rudiments of jets on the one hand, and nonstandard
analysis, on the other have been deployed in this paper.
In follow up articles we intend to use (*transferred) tools from
smooth function theory along with a more extensive use of jet theory to extend
both the linear and nonlinear existence results. Further, deploying more
nuanced version of the jet material of section \ref{section: standard jet work,prolong and
rank} over certain types of infinite points in the jet fibers, we
intend to prove results on regularity of solutions of partial
differential operators, linear or nonlinear, whose symbols satisfy certain properness
conditions. Our first paper along this line, \cite{McGafRegSolnNPDE}, gives a regularity theorem for a broad class of nonlinear differential operators.
We also intend to extend the results here to
include the results of Akiyama, \cite{AkiyamaNSSolvabilityOpsOnVBs} into the framework
established here in the manner we have included the results of Todorov.
The method is by an extension from internal mapping with *finite support
to internal smooth modules of bundle sections with *finite support.
Furthermore, as noted in the introduction, we will refine the arguments in this paper to Todorov's nonstandard Colombeau algebras.
Given that all of the usual constructions on the symmetries of
differential equations (as in eg., Olver, \cite{Olver1993}) are
straightforwardly lifted to the nonstandard universe, we are also
looking into developing a theoretic framework on generalized
symmetries (eg., shock symmetries) of differential equations, continuing within the jet
theoretic framework begun here.
\betaibliographystyle{amsplain}
\betaibliography{nsabooks}
\epsilonnd{document} |
\begin{document}
\title{A Successive Constraint Approach to Solving Parameter-Dependent Linear Matrix Inequalities}
\thispagestyle{fancy}
\subsection*{Abstract:}
\textbf{We present a successive constraint approach that makes it possible to cheaply solve large-scale linear matrix inequalities for a large number of parameter values. The efficiency of our method is made possible by an offline/online decomposition of the workload. Expensive computations are performed beforehand, in the offline stage, so that the problem can be solved very cheaply in the online stage. We also extend the method to approximate solutions to semidefinite programming problems.}
\section{Introduction}
\label{}
Linear matrix inequalities (LMIs) are a general type of convex constraint that include linear as well as quadratic constraints \cite{VB2000,VB1996} and lead to very natural formulations of a large number of problems in control and systems theory \cite{BBF+1993,FAG1996}. They can be solved using a wide range of existing methods~\cite{AHO1998,FKM+2001,HR2000,Toh2003}, but that can be expensive for large-scale problems. In particular, problems resulting from the discretization of partial differential equations can be extremely expensive due to their high dimensionality. The computations are even more expensive if parameter-dependent problems are considered and solutions are needed for a large number of parameter values. In that case traditional solution methods are extremely inefficient. We propose the construction of reduced-order models that take advantage of the parametric nature of the problem and allow us to very cheaply produce solutions for a large number of parameter values.
Let us introduce a finite-dimensional, bounded parameter domain $\mathcal D\in\mathbb R^p$ and the parameter-dependent LMI
\begin{equation}\tag{L}\label{LMI}
F(x;\mu):=\sum_{q=1}^{Q_F}\left[\theta_q^0(\mu)+\theta_q^L(\mu)x\right]F_q\succeq 0.
\end{equation}
Here $F(x;\mu)\in\mathbb R^{\mathcal N\times \mathcal N}$ is a symmetric matrix that depends on both the parameter $\mu\in \mathcal D$ and the decision variable $x\in\mathbb R^n$ and is composed of $Q_F$ parameter-independent matrices $F_q\in\mathbb R^{\mathcal N\times \mathcal N}$. The parameter dependencies of $F(x;\mu)$ are given by the functions $\theta^0(\cdot):\mathcal D\rightarrow \mathbb R^{Q_F}$ and $\theta^L(\cdot):\mathcal D\rightarrow \mathbb R^{Q_F\times n}$. We use subscripts to indicate components of a vector, such that $\theta_q^0(\mu)$ is the $q$\textsuperscript{th} element of $\theta^0(\mu)$. Similarly, we write $\theta_q^L(\mu)$ to indicate the $q$\textsuperscript{th} row of $\theta^L(\mu)$. The symbol $\succeq$ will be used in the sense that $P\succeq0$ indicates that the symmetric matrix $P$ is positive semi-definite.
The goal of this paper is to efficiently solve the following problems for a large number of parameter values $\mu\in\mathcal D$:
\begin{enumerate}
\item The strict feasibility problem: Find an $x\in\mathbb R^n$ such that $F(x;\mu)\succ0$.
\item The semidefinite program (SDP):
\begin{equation}\tag{S}\label{SDP}
\underset{x\in\mathbb R^n}{\text{minimize}}\quad
c(\mu)^Tx \quad
\text{subject to}\quad
F(x;\mu)\succeq 0.
\end{equation}
\end{enumerate}
Rather than directly solving these problems for each new parameter value, we will solve them for only a small set of intelligently chosen parameter values. We will then use the resulting solutions to build a reduced-order model that can approximate the solution anywhere in $\mathcal D$.
The method that we propose can be viewed as a generalization of the successive constraint method (SCM) \cite{CHM+2008,HRS+2007}, which is often used in the field of reduced basis methods to evaluate stability constants \cite{RHP2008}. This method will allow us to very cheaply determine feasible solutions for any $\mu\in\mathcal D$. That is made possible by decomposing the computational workload into offline and online stages. All expensive computations will be performed in advance, during the offline stage. During the online stage the cost to solve the problem for a new parameter value will be independent of the size of the original constraint, $\mathcal N$. In that way the computational cost of each new solution will remain cheap even if the original constraint has very large dimensions.
Our methods are applicable to a wide range of LMIs and can also be used to extend the applicability of SCM. In the context of reduced basis methods, applications could involve bounding stability constants with respect to parameter-dependent norms or the selection of Lyapunov functions for the computation of error bounds \cite{OConnor2016}. In Section \ref{sec:example} we present an example in which we optimize a system while ensuring that it remains stable.
\section{Reduced-order modeling for strict feasibility}\label{sec:feas}
SCM was originally designed to approximate coercivity constants. We will apply a modified version of SCM to the coercivity constant
\begin{equation}
\alpha(x;\mu):=\inf_{v\in \mathbb R^\mathcal N}\frac{v^TF(x;\mu)v}{v^TF_Sv},
\end{equation}
where $F_S\in\mathbb R^{\mathcal N\times \mathcal N}$ is a fixed symmetric positive-definite matrix. From the definition it is clear that $\alpha(x;\mu)\geq0$ is equivalent to $F(x;\mu)\succeq0$ for all symmetric positive definite matrices $F_S$. Nevertheless, an appropriate choice of $F_S$ could be beneficial from a numerical point of view. If we are dealing with PDE discretizations, it can be advantageous to choose a matrix associated with an energy norm.
The first step in applying SCM is reformulating the coercivity constant as follows:
\begin{equation}\label{eq:coer_opt}
\alpha(x;\mu)=\inf_{y\in \mathcal Y}\left[\theta^0(\mu)+\theta^L(\mu)x\right]^Ty,\quad\text{where}\quad
\mathcal Y:=\left\{y\in\mathbb R^{Q_F}\Big|y_q=\frac{v^TF_qv}{v^TF_Sv},v\in\mathbb R^\mathcal N\right\}.
\end{equation}
This formulation has the advantage that the complexity of the problem has been shifted to the definition of the set $\mathcal Y$. That allows us to compute lower and upper bounds for $\alpha(x;\mu)$ by approximating $\mathcal Y$.
A lower bound for $\alpha(x;\mu)$ can be derived by approximating $\mathcal Y$ from the outside. A bounded but primitive approximation for $\mathcal Y$ is given by
\begin{equation}
\mathcal B_Q:=\prod_{i=1}^{Q_F} \left[\inf_{y\in\mathcal Y}y_q,\sup_{y\in\mathcal Y}y_q\right]\subset \mathbb R^{Q_F}.
\end{equation}
$\mathcal B_Q$ will generally be much larger than $\mathcal Y$ so we will add constraints to restrict it and better approximate $\mathcal Y$. Let us assume that we have computed lower bounds $\bar\alpha(\bar x;\bar\mu)\leq\alpha(\bar x;\bar \mu)$ for each pair $(\bar x,\bar \mu)$ in a predetermined set $\mathcal R\subset\mathbb R^n\times\mathcal D$. The set $\mathcal Y$ is then contained in the set
\begin{equation}
\mathcal Y_{out}:=\left\{y\in\mathcal B_Q\Big|\left[\theta^0(\bar\mu)+\theta^L(\bar\mu)\bar x\right]^Ty\geq\bar\alpha(\bar x;\bar\mu),\, \forall\ (\bar x,\bar\mu)\in \mathcal R\right\}.
\end{equation}
Using $\mathcal Y_{out}$ we can define
\begin{equation}\label{eq:coer_lb}
\alpha_{out}(x;\mu):=\inf_{y\in \mathcal Y_{out}}\left[\theta^0(\mu)+\theta^L(\mu)x\right]^Ty,
\end{equation}
such that $\alpha_{out}(x;\mu)\leq\alpha(x;\mu)$ for any $x\in\mathbb R^n$ and any $\mu\in\mathcal D$.
The lower bound $\alpha_{out}(x;\mu)$ can be formulated as the solution to a linear programming problem. To see that we note that $\mathcal Y_{out}$ is a linearly constrained subset of $\mathbb R^{Q_F}$. As a result we can find a matrix $A_{out}\in\mathbb R^{\ell\times Q_F}$ and a vector $b_{out}\in\mathbb R^\ell$ such that $\mathcal Y_{out}=\{y\in\mathbb R^{Q_F}|A_{out}y\geq b_{out}\}$. Here the symbol $\geq$ indicates a componentwise comparison of vectors and $\ell\ll\mathcal N$ is the number of constraints. The lower bound $\alpha_{out}(x;\mu)$ can be calculated, for any $x\in\mathbb R^n$ and any $\mu\in\mathcal D$, as the solution to the linear program (\ref{P}) or its dual (\ref{D}):
\noindent
\fbox{\begin{minipage}{0.482\textwidth}
\begin{equation}\tag{P}\label{P}
\begin{aligned}
& \underset{y\in\mathbb R^{Q_F}}{\text{minimize}}
& & \left[\theta^0(\mu)+\theta^L(\mu)x\right]^Ty \\
& \text{subject to}
& & A_{out}y\geq b_{out}
\end{aligned}
\end{equation}
\end{minipage}}
\fbox{\begin{minipage}{0.482\textwidth}
\begin{equation}\tag{D}\label{D}
\begin{aligned}
& \underset{p\in\mathbb R^\ell}{\text{maximize}}
& & b_{out}^Tp \\
& \text{subject to}
& & A_{out}^Tp=\theta^0(\mu)+\theta^L(\mu)x\\
& & & p\geq0
\end{aligned}
\end{equation}
\end{minipage}}
To find an optimal value of $x$ we will maximize (\ref{D}) over all $x\in\mathbb R^n$ using the problem
\begin{equation}\tag{RF}\label{RF}
\underset{p\in\mathbb R^\ell,x\in\mathbb R^n}{\text{maximize}}\quad b_{out}^Tp\quad \text{subject to}
\quad A_{out}^Tp-\theta^L(\mu)x=\theta^0(\mu) \quad \text{and} \quad p\geq0.
\end{equation}
If $b_{out}^Tp$ is strictly greater than $0$, then the associated $x$ is guaranteed to strictly satisfy (\ref{LMI}). Even if that is not the case, (\ref{RF}) is always feasible. To show that we note that the boundedness of $\mathcal Y_{out}\subset\mathcal B_Q$ implies the boundedness of (\ref{P}). By duality (\ref{D}) is then feasible for any $x\in\mathbb R^n$ and any $\mu\in\mathcal D$ which implies that (\ref{RF}) is feasible for any $\mu\in\mathcal D$. (\ref{RF}) is also bounded if $\max_{x\in\mathbb R^n} \alpha(x;\mu)$ is bounded; otherwise, we will be content with any $x$ associated with a large, positive value of $b_{out}^Tp$. The main reason to use (\ref{RF}) is that it is cheap to solve: It is a small linear program with $n+\ell$ variables, $Q_F$ equality constraints, and $\ell$ inequality constraints.
Since it is not possible to work directly with the infinite set $\mathcal D$, we introduce a large set $\Xi\subset\mathcal D$ of discrete points that are representative of $\mathcal D$ and a much smaller set $\mathcal C_k\subset\Xi$ with cardinality $k$. The set $\mathcal R$ will be made up of parameter values $\bar\mu\in\Xi$ and associated approximate solutions $\bar x\in\mathbb R^n$. For parameter values $\bar\mu$ that are in $\mathcal C_k$ we will use solutions $\bar x$ to the original problem (\ref{LMI}) and set $\bar\alpha(\bar x;\bar\mu)=\alpha(\bar x;\bar\mu)$. For parameter values in the much larger set $\Xi\setminus\mathcal C_k$ we will use solutions $\bar x$ to (\ref{RF}), and set $\bar\alpha(\bar x;\bar\mu)=\alpha_{out}(\bar x;\bar\mu)$. The pairs $(\bar x,\bar\mu)$ that will be included in $\mathcal R$ will depend on the parameter $\mu$ for which we would like a feasible solution. For two natural numbers $M_C$ and $M_\Xi$ we define $\mathcal R$ to be a set of $M_C$ pairs $(\bar x,\bar\mu)$ with $\bar\mu\in\mathcal C_k$ and $M_\Xi$ pairs $(\bar x,\bar\mu)$ with $\bar\mu\in\Xi\setminus\mathcal C_k$. In both cases we will choose the pairs $(\bar x,\bar\mu)$ with $\bar\mu$ closest to $\mu$ in a predetermined metric. If $M_C>k$, we simply include all of $\mathcal C_k$ in $\mathcal R$.
To build our model we will make use of the greedy algorithm that was introduced in the context of reduced basis methods~\cite{VPR+2003} and plays a vital role in SCM. The algorithm is initiated by choosing a small initial set $\mathcal C_k$. For each $\bar\mu\in\mathcal C_k$ the problem (\ref{LMI}) is solved once to determine the associated values of $\bar x$ and $\bar\alpha(\bar x;\bar\mu)$. For $\bar\mu\in\Xi\setminus \mathcal C_k$ we initially set $\bar x=0$ and $\bar \alpha(\bar x;\bar\mu)=-\infty$. The model is then improved iteratively. In each iteration we will solve (\ref{RF}) for each $\bar\mu\in\Xi\setminus\mathcal C_k$ and update the stored values of $\bar x$ and $\bar \alpha(\bar x;\bar \mu)$. The $\bar\mu$ associated with the smallest value of $\bar\alpha(\bar x;\bar\mu)$ is then added to $\mathcal C_k$, and (\ref{LMI}) is solved to update the stored values of $\bar x$ and $\bar\alpha(\bar x;\bar\mu)$. The process terminates when the smallest value of $\bar\alpha(\bar x;\bar\mu)$ is larger than some positive tolerance.
Solving problems with our method involves two stages. During the offline stage the greedy algorithm builds the model. The expensive operations in the offline stage are $k$ solutions of (\ref{LMI}) and nearly $k|\Xi|$ solutions of (\ref{RF}). Here $k$ is the cardinality of the final $\mathcal C_k$ and $|\Xi|$ is the cardinality of $\Xi$. During the online stage (\ref{RF}) can be constructed and solved very cheaply for any new parameter value. The online computational cost is independent of $\mathcal N$ to ensure that it remains cheap even if $\mathcal N$ is very large.
During the greedy algorithm it is necessary to solve (\ref{LMI}) for various parameter values. That can be done using a variety of methods including algorithms that solve semidefinite programming problems \cite{AHO1998,FKM+2001,HR2000,Toh2003}. In this section we propose a method that is based on SCM and allows us to reuse information that we have already computed.
For a predetermined, finite subset $\mathcal Y_{in}$ of $\mathcal Y$ we define
\begin{equation}\label{eq:coer_ub}
\alpha_{in}(x;\mu):=\inf_{\bar y\in \mathcal Y_{in}}\left[\theta^0(\mu)+\theta^L(\mu)x\right]^T\bar y,
\end{equation}
such that $\alpha_{in}(x;\mu)\geq\alpha(x;\mu)$ for all $x\in\mathbb R^n$ and $\mu\in\mathcal D$. Here we will make use of a variation of the greedy algorithm. In each iteration we solve $\max_{x\in\mathbb R^n}\alpha_{in}(x;\mu)$ to find the optimal $x$. For that value of $x$ we then compute $\alpha(x;\mu)$ and an element $y$ of $\mathcal Y$ for which the infimum in (\ref{eq:coer_opt}) is reached. That $y$ is then added to $\mathcal Y_{in}$ to improve the upper bound $\alpha_{in}(\cdot;\mu)$. The algorithm terminates when $\alpha(x;\mu)$ is sufficiently large or sufficiently close to $\alpha_{in}(x;\mu)$.
In each iteration of the algorithm both the small linear program associated with $\max_{x\in\mathbb R^n}\alpha_{in}(x;\mu)$ and the $\mathcal N$-dimensional eigenvalue problem associated with $\alpha(x;\mu)$ need to be solved once. The dominant cost is that of the eigenvalue problems. To reduce the number of iterations and eigenvalue solves we will store the last value of $\bar y$ that is calculated each time we solve (\ref{LMI}) for a new $\mu\in\mathcal C_k$. The set $\mathcal Y_{in}$ can then be initialized using those values of $\bar y$.
\section{The semidefinite programming problem}\label{sec:SDP}
In this section we will show how our methods can be used to approximate solutions to the semidefinite program (\ref{SDP}). The idea is to minimize $c(\mu)^Tx$ over all $x\in\mathbb R^n$ that satisfy $\alpha_{out}(x;\mu)\geq0$. That is done using the following problem:
\begin{equation}\tag{RS}\label{RS}
\underset{x\in\mathbb R^n,p\in\mathbb R^\ell}{\text{minimize}}\quad c(\mu)^Tx \quad\text{subject to}\quad
\begin{bmatrix}A_{out}^T& -\theta^L(\mu)\end{bmatrix}
\begin{bmatrix}p\\x\end{bmatrix}=\theta^0(\mu)\quad \text{and}\quad \begin{bmatrix}I\\b_{out}^T\end{bmatrix}p\geq
\begin{bmatrix}0 \\ 0\end{bmatrix}.
\end{equation}
Here $A_{out}$, $b_{out}^T$ and $\alpha_{out}(x;\mu)$ are all the same as in Section \ref{sec:feas} except that the stored values of $\bar x$ will be calculated using (\ref{SDP}) and (\ref{RS}) rather than (\ref{LMI}) and (\ref{RF}). The new constraint $b_{out}^Tp\geq0$ guarantees that $\alpha_{out}(x;\mu)\geq0$.
Let us write $\mathcal J_{out}$ and $\mathcal J$ to denote the optimal values of (\ref{RS}) and (\ref{SDP}), respectively. Based on the fact that the optimal $x\in\mathbb R^n$ for (\ref{RS}) will be feasible for (\ref{SDP}) we know that $\mathcal J_{out}\geq \mathcal J$. Ideally $\mathcal J_{out}$ will also be a good approximation to $\mathcal J$ despite being much cheaper to compute: It requires the solution of a linear program with $n+\ell$ variables, $Q_F$ equality constraints, and $\ell+1$ inequality constraints.
For any $\mu\in\mathcal D$, the boundedness of (\ref{SDP}) implies the boundedness of (\ref{RS}), but the question of feasibility is more complicated. To ensure feasibility we first build a model (\ref{RF}) to find strictly feasible solutions. We then convert that model to the form of (\ref{RS}). In doing so we consider the stored values of $\bar x\in\mathbb R^n$, which should be feasible, to be approximate solutions to (\ref{SDP}) and we reinitialize $\mathcal C_k$ as an empty set. We can then improve the model using another greedy algorithm.
The goal of this greedy algorithm is to reduce the error $\mathcal J_{out}-\mathcal J$. Since it is too expensive to compute $\mathcal J$ online, we will compute a lower bound for it by considering the problem
\begin{equation}\tag{ER}\label{ER}
\underset{x\in\mathbb R^n}{\text{minimize}}\quad
c(\mu)^Tx \quad
\text{subject to}\quad
\alpha_{in}(x;\mu)\geq0.
\end{equation}
Here $\alpha_{in}(x;\mu)$ is defined like in (\ref{eq:coer_ub}) with $\mathcal Y_{in}$ initially being the set of $\bar y\in\mathcal Y$ associated with parameter values in $\mathcal C_k$. The optimal value of (\ref{ER}), which we will denote $\mathcal J_{in}$, is by definition a lower bound for $\mathcal J$. That allows us to bound the error $\mathcal J_{out}-\mathcal J\geq0$ from above using $\mathcal J_{out}-\mathcal J_{in}$. Like the previous approximations, this bound can be computed online with a number of computations that is independent of $\mathcal N$. Although (\ref{ER}) is always feasible, it could be unbounded. That can be fixed by increasing $\mathcal C_k$ and consequently $\mathcal Y_{in}$.
During each iteration of the greedy algorithm we will solve both (\ref{RS}) and (\ref{ER}), compute the difference $\mathcal J_{out}-\mathcal J_{in}$, and update both the estimates $\bar x\in\mathbb R^n$ and the associated values $\bar\alpha(\bar x;\bar\mu)$ for each $\bar\mu\in\Xi\setminus\mathcal C_k$. The $\bar\mu$ that produced the maximum value of $\mathcal J_{out}-\mathcal J_{in}$ is then added to $\mathcal C_k$ and a more accurate solution to (\ref{SDP}) is computed. That allows us to update both (\ref{RS}) and (\ref{ER}). This process terminates when the maximum value of $\mathcal J_{out}-\mathcal J_{in}$ is below a desired tolerance.
The major computational burdens in the offline stage of our method are the construction of the model (\ref{RF}), $k$ solutions of (\ref{SDP}), and nearly $k|\Xi|$ solutions of both (\ref{RS}) and (\ref{ER}). During the online stage an approximate solution can be determined for any given $\mu\in\mathcal D$ by simply constructing and solving (\ref{RS}). The error can also be bounded online by solving (\ref{ER}).
The more accurate solutions that we need can be computed using either general solvers for semidefinite programming problems \cite{AHO1998,FKM+2001,HR2000,Toh2003} or a method that is similar to the one presented in Section \ref{sec:feas}. For the latter we will consider the following problem with a fixed value of $\mu\in\mathcal D$:
\begin{equation}\tag{T}\label{T}
\underset{x\in\mathbb R^n}{\text{minimize}}\quad
c(\mu)^Tx \quad
\text{subject to}\quad
\alpha_{in}(x;\mu)>\alpha_{min},
\end{equation}
where $\alpha_{min}>0$ is some small constant that is used as a tolerance. We again use a sort of greedy algorithm. In each iteration we solve (\ref{T}) to get the approximate solution $x$ and update $\mathcal Y_{in}$ by adding the new value of $y$. The process is repeated until the stopping condition $\alpha(x;\mu)>0$ is satisfied. If $\alpha_{min}$ is sufficiently small, the resulting $x$ should be a good approximation to the true solution.
\section{Numerical example: System stabilization}\label{sec:example}
To test our method we will consider a reaction-diffusion equation on the unit square, $\Omega$, depicted in Figure \ref{desc:dom}. To facilitate the explanation we consider a problem with just one parameter $\mu\in\mathcal D:=[0,3]$ and one decision variable $x\in\mathbb R$. The problem is related to the following system:
\begin{equation}
\dot y=\Delta y+\mu y \mathbf 1_{\Omega_1}+\mathbf 1_{\Omega_2}u,\quad \forall z\in\Omega;\qquad\frac{\partial y}{\partial \eta}=0, \quad \forall z\in\partial \Omega;\qquad u=-x\zeta;\qquad \zeta=\int_{\Omega_2}y.
\end{equation}
\begin{figure}
\caption{Setup of the numerical example}
\label{desc:dom}
\label{desc:fb}
\label{desc}
\end{figure}
\noindent Here $\mathbf 1_{S}$ indicates the characteristic function of a set $S$ and $\eta$ is the outward-pointing unit vector on the boundary $\partial \Omega$. The decision variable $x$ determines the feedback gain from the system's output $\zeta(t)$ to its input $u(t)$ as depicted in Figure \ref{desc:fb}. We use linear finite elements to discretize the problem in space. That gives us the following $2601$-dimensional closed-loop semi-discrete system
\begin{equation}\label{eq:semi_disc}
M\dot y+A(x;\mu)y=0, \quad\text{where}\quad A(x;\mu)=A_0-\mu A_1+xA_2,\quad \text{with}\quad M,A_0,A_1,A_2\in\mathbb R^{2601\times 2601}.
\end{equation}
Here $M$ is symmetric positive definite and $A(x;\mu)$ is symmetric. Under these assumptions the system (\ref{eq:semi_disc}) is strictly stable for a given $\mu\in\mathcal D$ and $x\in\mathbb R$ iff $A(x;\mu)\succ0$. Our goal will be to minimize the cost of the control, by keeping $x$ small, while ensuring strict stability. Noting that $A_0$, which is associated with the operator $-\Delta$, is positive semidefinite and that $A_0+A_1$ is positive definite, we define $F(x;\mu):=(1-\rho)A_0+(-\mu-\rho)A_1+xA_2$ and $F_S:=A_0+A_1\succ0$ for some small $\rho>0$. It then holds that
\begin{equation}
F(x;\mu)\succeq0\qquad\Longleftrightarrow \qquad A(x;\mu)- \rho F_S\succeq0,
\end{equation}
and hence the system in (\ref{eq:semi_disc}) is strictly stable if $F(x;\mu)\succeq0$. We will search for a minimal stabilizing gain $x\in\mathbb R$ as the solution to (\ref{SDP}).
For our model we will use values of $M_C=4$ and $M_\Xi=3$. $\mathcal C_k$ is initialized using the smallest and the largest values in $\mathcal D$. For this particular problem that guarantees that (\ref{RS}) is feasible for all $\mu\in\mathcal D$. As a result we can start directly with (\ref{RS}) and do not need to build the model given in (\ref{RF}). Another modification of our method is that we will work with the relative error given by $(\mathcal J_{out}-\mathcal J_{in})/\mathcal J_{in}$ rather than the error $\mathcal J_{out}-\mathcal J_{in}$. That is possible because $\mathcal J_{in}$ will always be strictly positive.
Figure \ref{res:fixed_mu} shows the convergence of the solver described in Section \ref{sec:SDP} for the computation of more accurate solutions to (\ref{SDP}). Here the error is measured as $\alpha_{in}(\bar x;\bar\mu)-\alpha(\bar x;\bar\mu).$ Two examples are shown as well as the worst case over $30$ random parameter values. We recall that the most expensive part of this method is the eigenvalue solves and that one such solve is needed for each iteration. Figure \ref{res:greedy} shows the convergence of the greedy algorithm for two different parameter domains $\mathcal D$. The plotted values are the worst relative errors over the respective sets $\Xi$. For $\mathcal D=[0,3]$ we set $\Xi$ to be $300$ uniformly distributed points and for $\mathcal D=[0,1.5]$ we use $150$ points. Figure \ref{res:snaps} compares the results when $\mathcal C_k$ is constructed using the greedy algorithm or simply as uniformly distributed points. The plot shows the worst relative errors over $100$ random values of $\mu\in\mathcal D$. The greedy algorithm performs slightly better than uniform distributions when $\mathcal C_k$ is large and also has the advantage that it is iterative.
\begin{figure}
\caption{Results of the numerical experiments}
\label{res:fixed_mu}
\label{res:greedy}
\label{res:snaps}
\label{res}
\end{figure}
{\small
\setlength{\parskip}{0pt}
\setlength{\itemsep}{0pt plus 0.1ex}
}
\end{document} |
\begin{document}
\title{
On invariants of link maps in dimension four}
\author{Ash Lightfoot}
\partialate{}
\maketitle
\begin{abstract}
We affirmatively address the question of whether the proposed link homotopy invariant $\omega$ of Li is well-defined. It is also shown that if one wishes to adapt the homotopy invariant $\tau$ of Schneiderman-Teichner to a link homotopy invariant of link maps, the result coincides with $\omega$.
\end{abstract}
\section{Introduction}
A link map $S^2\cup S^2\to S^4$ is a map from a union of 2-spheres with pairwise disjoint images, and a link homotopy is a homotopy through link maps. To a link map $f$, Kirk (\cite{Ki1}, \cite{Ki2}) assigned a pair of integer polynomials $\sigma(f)=(\sigma_+(f), \sigma_-(f))$ which is invariant under link homotopy and vanishes if $f$ is link homotopic to a link map that embeds either component. He posed the still-open problem of whether $\sigma(f)=(0,0)$ is sufficient to link nulhomotope $f$. In \cite{Li97}, Li sought to define a link homotopy invariant $\omega(f)=(\omega_+(f), \omega_-(f))$ to detect link maps in the kernel of $\sigma$. When $\sigma_\pm(f)=0$, the mod $2$ integer $\omega_\pm(f)$ obstructs embedding by counting (after a link homotopy) weighted intersections between $f(S^2_\pm)$ and its Whitney disks in the complement of $f(S^2_\mp)$. While the examples with $\sigma(f)=(0,0)$ but $\omega(f)\neq (0,0)$ in that paper were found to be in error by Pilz (\cite{Pilz}), the latter did not address another issue. Namely, the proof that $\omega$ is invariant under link homotopy relies implicitly on the assumption that a pair of link homotopic abelian link maps are link homotopic through \emph{abelian} link maps. The first purpose of this note is to prove this assumption correct and that $\omega$ is a link homotopy invariant.
\begin{theorem}\label{prop:liwelldefined}
If $f$ and $g$ are link homotopic link maps such that $\sigma(f)=(0,0)=\sigma(g)$, then $\omega(f)=\omega(g)$.
\end{theorem}
The invariants $\sigma$ and $\omega$ may then be viewed, respectively, as primary and secondary obstructions to link homotoping to an embedding. For the problem of homotoping a map $S^2\to Y^4$ to an embedding, the homotopy invariants $\mu$ of Wall (\cite{W}) and $\tau$ of Schneiderman and Teichner (\cite{ST}) form an analogous pair of obstructions. Our second purpose is to show that if one adapts $\tau$ to the setting of link homotopy in the natural way, one obtains $\omega$.
For a link map $f:S^2_+\cup S^2_-\to S^4$, where the signs are used to distinguish each component, write $X_\pm = S^4\rsetminus f(S^2_\mp)$ and let $f_{\pm}$ denote the restricted map $f|S^2_{\pm}:S^2_\pm\to X_{\mp}$.
\begin{theorem}\label{thm:omega-equals-li}
Let $f$ be a link map with $\sigma_+(f)=0$. Then $f$ is link homotopic to a link map $g$ such that $\tau(g_+)$ is defined, and one has that $\tau(g_+)=0$ if and only if $\omega_+(f)=0$.
\end{theorem}
Assume all manifolds are equipped with basepoints and orientations arbitrarily unless otherwise specified.
\section{Proof of Theorem \ref{prop:liwelldefined}}
\mycomment{
\section[\texorpdfstring{Link homotopy invariance of $\omega$}
{Link homotopy invariance of omega}]
{Link homotopy invariance of $\omega$}
}
A link map $f$ is said to be \textit{abelian} if $\pi_1(X_{+})\cong {\mathbb Z}$ and $\pi_1(X_{-})\cong {\mathbb Z}$, and an abelian link homotopy is a link homotopy through abelian link maps. We will say $f$ is \textit{good} if it is abelian and each restricted map $f_\pm$ is a self-transverse immersion with vanishing signed self-intersection number. Theorem \ref{prop:liwelldefined} will be shown using the following lemma.
\begin{lemma}\label{lem:reg-htpic}
If $f$ and $g$ are regularly homotopic good link maps such that $\sigma(f)=(0,0)=\sigma(g)$, then $\omega(f)=\omega(g)$.
\end{lemma}
\begin{proof}[Proof of Theorem \ref{prop:liwelldefined}]
A link map $f$ may be first be perturbed so that it restricts to a self-transverse immersion on each component 2-sphere; local cusp homotopies may then be performed so that these immersions each have vanishing signed self-intersection number. Finger moves of $f(S^2_+)$ in the complement of $f(S^2_-)$, followed by finger moves of $f(S^2_-)$ in the complement of $f(S^2_+)$, may then serve to abelianize $\pi_1(X_+)$ and $\pi_1(X_-)$ (see
\cite[p. 205]{C}; also \cite{Ki1}, \cite{Li97}). \mycomment{By performing local cusp homotopies and finger moves, a link map $f$ is link homotopic to a good link map $f'$ (cf. \cite{Ki2}, \cite{Li97}).}
Denote the resulting good link map by $f'$. In \cite{Li97}, the author gives an algorithm for computing the pair of mod 2 integers $\omega(f')=(\omega_+(f'), \omega_-(f'))$ and defines $\omega(f)=\omega(f')$. Suppose $f''$ is another good link homotopy representative of $f$. Then $f'$ and $f''$ are regularly homotopic by \cite[Theorem 2.4]{Ki2}, so $\omega(f'')=\omega(f')$ by Lemma \ref{lem:reg-htpic}. Thus $\omega$ does not depend on the choice of good representative. Now, if $g$ is link homotopic to $f$, and $g'$ is a good link homotopy representative of $g$, the same argument shows that $\omega(g')=\omega(f')$, so (as defined) $\omega(f)=\omega(g)$.
\end{proof}
\mycomment{ REMARK ABOUT DEFINITION OF TAU USING WHITNEY IMMERSION THEOREM:
Define $\tau(f')=\tau(f')$, where $f'$ is self-transverse, zero euler class, immersion approximation. If $g$ is homotopic to $f$, and $g'$ an approx of $g'$, then $f'$ and $g'$ are homotopic and hence regularly homotopic by Whitney (two immersions of $A^k$ in $X^n$, $n\geq 2k+2$ are homotopic iff reg htpic, this is said on first page (pg 281) of Smale "A Classification of Immersions of the Two-sphere.")
}
It remains to prove Lemma \ref{lem:reg-htpic}. We make use of ideas from \cite{K}, to which the reader is referred for more details on finger and Whitney moves along chords. Throughout the rest of this paper, $Y$ will denote a 4-manifold. Let $k: S^2\to Y$ be an immersion. A \emph{chord} $\gamma$ attached to $k(S^2)$ is a continuous arc in $Y$ whose endpoints are distinct points of $k(S^2)$ (minus its double points) and whose interior is disjoint from $k(S^2)$. A chord is \emph{simple} if this arc is an embedding. If two simple chords $\gamma$ and $\gamma'$ for $k(S^2)$ are ambient isotopic in $Y$\mycomment{meaning isotopy $Y\times I\to Y$ such that $H_0=\operatorname{id}$, H_1\circ g=h$} by an isotopy fixing $k(S^2)$, then a finger move along either chord yields an ambient isotopic immersion. \mycomment{Note that $H_t$ 1-1 means $F_t(\alpha)\cap F_t(k(S^2)) = F_t(\alpha\cap k(S^2))$, so $\alpha$ is carried to $\beta$ through simple chords automatically.}
The following result is a ready consequence of transversality and the isotopy extension theorem.
\begin{lemma}\label{lem:ambient}
Let $C$ be a compact subset of a 4-manifold $Y$, and let $k: S^2\to Y$ be an immersion. Suppose $\alpha$ and $\beta$ are simple chords on $k(S^2)$ with common endpoints $p,q$. If $\alpha$ and $\beta$ are path homotopic in $Y\rsetminus C$ through chords on $k(S^2)$, then $\alpha$ and $\beta$ are ambient isotopic in $Y$ by an isotopy that carries $\alpha$ to $\beta$ and fixes $k(S^2)$ and $C$. \mycomment{Note that $H_t$ 1-1 means $F_t(\alpha)\cap F_t(k(S^2)) = F_t(\alpha\cap k(S^2))$, so $\alpha$ is carried to $\beta$ through simple chords automatically.}
\end{lemma}
We first show that, roughly speaking, finger moves and Whitney moves of a single component of a link map in the complement of the other component commute.
\begin{lemma}\label{lem:kamada-2-modify}
Let $f$ be a link map such that the restricted maps $f_\pm$ are self-transverse immersions. Suppose that an immersion $g_+$ is obtained from $f_+$ by performing a Whitney move, followed by a finger move, in $S^4\rsetminus f(S^2_-)$. Then, up to ambient isotopy in $S^4$ fixing $f(S^2_-)$, $g_+$ may be obtained from $f_+$ by performing a finger move, followed by a Whitney move, in $S^4\rsetminus f(S^2_-)$.
\end{lemma}
\begin{proof}
By the hypotheses, there is an intermediate link map $f'$ such that $f'_-=f_-$ and a Whitney move performed in a 4-ball $B\subset S^4\rsetminus f(S^2_-)$ changes $f_+$ to $f_+'$, and a finger move of $f_+'$ along a chord $\gamma\subset S^4\rsetminus f(S^2_-)$ changes $f_+'$ to $g_+$.
If $\gamma$ is disjoint from $B$, then the lemma is immediate. Otherwise, we may assume that $\gamma$ intersects $B\rsetminus f'(S^2_+)$ along the interior of $\gamma$ in a collection of $n\geq 0$ properly embedded arcs. One may then homotop $\gamma$ in a collar of $\partial B\rsetminus f'(S^2_+)$ to a union $\widehat\gamma \cup_{i=1}^n \alpha_i$, of a simple chord $\widehat\gamma\subset S^4\rsetminus f(S^2_-)$ on $f'(S^2_+)$ that intersects $B$ at precisely one point $p\in \partial B$, and $n$ simple loops $\{\alpha_i\}_{i=1}^n$ in $B\rsetminus f'(S^2_+)$ based at $p$. But as inclusion induces a surjection $\pi_1(\partial B\rsetminus f'(S^2_+),p)\to \pi_1(B\rsetminus f'(S^2_+),p)$, we may further deduce that $\gamma$ is path homotopic in $S^4\rsetminus f(S^2_+)$ through chords on $f'(S^2_-)$ to a simple chord $\gamma'$ that misses $B$. Thus, by Lemma \ref{lem:ambient} there is an ambient isotopy in $S^4$ from $\gamma$ to $\gamma'$ that fixes $f'(S^2_+)$ and $f(S^2_-)$.\mycomment{So have $\gamma$ is disjoint from $B$, without moving anything.} \qedhere
\mycomment{ EXPLANATION for surjectivity: let $\alpha$ be a loop in $B\rsetminus f'(S^2_+)$. It bounds a disk in $B$; make the disk transverse to $f'(S^2_+)$. Then you can shrink $D$ into a loop union a bunch of meridinal disks to $f'$, the latter of which can be pushed into $\partial B\rsetminus f'$ along $f'$.}
\end{proof}
We further require that, roughly speaking, a Whitney move of one component of a link map commutes with a finger move of the other. The proof is similar to that of Lemma \ref{lem:kamada-2-modify} but we include it for completeness.
\begin{lemma}\label{lem:kamada-similar}
Suppose that $f$ and $g$ are good link maps such that $g_+$ is obtained from $f_+$ by performing a Whitney move in $S^4\rsetminus f(S^2_-)$ and $g_-$ is obtained from $f_-$ by performing a finger move in $S^4\rsetminus g(S^2_+)$. Then \emph{(}up to ambient isotopy in $S^4$ fixing $f(S^2_+)$\emph{)} $g_-$ may be obtained from $f_-$ by performing a finger move in $S^4\rsetminus f(S^2_+)$ and \emph{(}up to ambient isotopy in $S^4$ fixing $g(S^2_-)$\emph{)} $g_+$ may be obtained by performing a Whitney move of $f_+$ in $S^4\rsetminus g(S^2_-)$.
\end{lemma}
\begin{proof}
Let $B$ be a 4-ball in $S^4\rsetminus f(S^2_-)$ such that a Whitney move performed in $B$ changes $f_+$ to $g_+$, and let $\gamma$ be a simple chord in $S^4\rsetminus g(S^2_+)$ on $f(S^2_-)$ such that a finger move of $f_-$ along $\gamma$ changes $f_-$ to $g_-$. If $\gamma$ is disjoint from $B$ then the lemma holds without the need for an additional isotopy (note that $f(S^2_+)$ and $g(S^2_+)$ coincide outside $B$)\mycomment{Note that if $\gamma$ is disjoint from $B$, then it is also disjoint from $f(S^2_+) = (f(S^2_+)\cap B) \cup (g(S^2_+)\rsetminus B)$}. Otherwise, we may assume that $\gamma$ intersects $B\rsetminus g(S^2_+)$ along the interior of $\gamma$ in a finite collection of properly embedded arcs.
\mycomment{Otherwise, after an isotopy rel endpoints we may assume that $\gamma$ is disjoint from $f(S^2_+)$ and intersects $B$ in a finite collection of simple arcs.}
Since inclusion induces a surjection $\pi_1(\partial B\rsetminus g(S^2_+))\to \pi_1(B\rsetminus g(S^2_+))$, as in the proof of Lemma \ref{lem:kamada-2-modify} one may path homotop $\gamma$ through chords on $g(S^2_+)$ to a simple chord that misses $B$. \mycomment{
\begin{lemma}\label{lem:ambient}
Let $C$ be a compact subset of a 4-manifold $Y$. Suppose $\alpha$ and $\beta$ are simple chords on $k(S^2)$ with common endpoints $p,q$. If $\alpha$ and $\beta$ are path homotopic in $Y\rsetminus C$ through chords on $k(S^2)$, then $\alpha$ and $\beta$ are ambient isotopic in $Y$ by an isotopy that carries $\alpha$ to $\beta$ through chords and fixes $k(S^2)$ and $C$. \mycomment{Note that $H_t$ 1-1 means $F_t(\alpha)\cap F_t(k(S^2)) = F_t(\alpha\cap k(S^2))$, so $\alpha$ is carried to $\beta$ through simple chords automatically.}
\end{lemma}}
Now apply Lemma \ref{lem:ambient}.
\end{proof}
We can now prove that if two good link maps are regularly link homotopic then they are connected by an \emph{abelian} link homotopy.
\begin{lemma}\label{prop:abelian-reg-htpy}
If $f$ and $g$ are regularly homotopic good link maps, then there is a regular homotopy from $f$ to $g$ consisting of a sequence of abelian link homotopies that alternately fix one component.
\end{lemma}
\begin{proof}
As in the proof of \cite[Theorem 2.4]{Ki2}, there is a regular homotopy taking $f$ to $g$ consisting of a sequence of regular homotopies that alternately fix one component. By Lemmas \ref{lem:kamada-2-modify} and \ref{lem:kamada-similar} this sequence can be chosen to first consist of finger moves (and ambient isotopies) alternately fixing one component, carrying $f$ to a link map $f'$, then a sequence of Whitney moves (and ambient isotopies) alternately fixing one component, carrying $f'$ to $g$.
Now, finger moves and ambient isotopy preserve abelianess, so $f$ and $f'$ are connected by a sequence of abelian, regular link homotopies alternately fixing one component. On the other hand, $f'$ is obtained from $g$ by a sequence of finger moves (and ambient isotopies) alternatively fixing one component, so these link maps are also connected by abelian, regular link homotopies alternately fixing one component.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:reg-htpic}]
By Lemma \ref{prop:abelian-reg-htpy}, the link map $f$ is carried to $g$ by a sequence of abelian, regular link homotopies that alternately fix each component. Lemma \ref{lem:reg-htpic} then follows from the following proposition, which can be deduced from the proof of \cite[Proposition 4.2]{Li97} (and which is expounded upon in \cite[Satz 4.14]{Pilz}).
\end{proof}
\begin{proposition}\label{prop:Li}
Let $h$ be a good link map with $\sigma_+(h)=0$.
\begin{itemize}\itemsep -0.1cm
\item[(i)] If a good link map $h'$ is obtained from $h$ by performing a regular homotopy of $h_+$ in $S^4\rsetminus h(S^2_-)$, then $\omega_+(h')=\omega_+(h).$\hspace{\stretch1}\ensuremath\qedsymbol
\item[(ii)] \mycomment{$\omega_+$ is invariant under a regular homotopy of $f_-$ in $S^4\rsetminus f(S^2_+)$ to a map $g_-$ \emph{if $\pi_1(S^4\rsetminus g(S^2_-))$ is abelian;}}
If a good link map $h'$ is obtained from $h$ by performing a regular homotopy of $h_-$ in $S^4\rsetminus h(S^2_+)$, then $\omega_+(h')=\omega_+(h)$.
\end{itemize}
\end{proposition}
\begin{remark}
Part (i) of this proposition is essentially a special case of the proof in $\cite{ST}$ that the $\tau$-invariant is well-defined, while part (ii) is unique in that the ambient manifold $X_-$, into which $h_+$ maps, is allowed to change.
\end{remark}
\section{Proof of Theorem \ref{thm:omega-equals-li}}
We begin with some preliminary definitions concerning algebraic intersections of immersed surfaces in 4-manifolds. The reader is referred to \cite{FQ} for more details on the subject.
\subsection{Intersection numbers in 4-manifolds}
Suppose $A$ and $B$ are properly immersed, self-transverse 2-spheres or 2-disks in a 4-manifold $Y$. Suppose further that $A$ and $B$ are transverse and that each is equipped with a path (a \emph{whisker}) connecting it to the basepoint of $Y$ .
For an intersection point $x\in A\cap B$, let $\lambda(A,B)_x \in \pi_1(Y)$ denote the homotopy class of a loop that runs from the basepoint of $Y$ to $A$ along its whisker, then along $A$ to $x$, and back to the basepoint along $B$ and its whisker. Define $\operatorname{sign}_{A,B}(x)$ to be $1$ or $-1$ depending on whether or not, respectively, the orientations of $A$ and $B$ induce the orientation of $Y$ at $x$. The (algebraic) intersection ``number'' $\lambda(A,B)$ between $A$ and $B$ is then defined as the sum in the group ring ${\mathbb Z}[\pi_1(Y)]$ of $\operatorname{sign}(x)\lambda(A,B)_x$ over all such intersection points. The value of $\lambda(A,B)$ is invariant under homotopy rel boundary of $A$ or $B$ (\cite{FQ}), but depends on the choice of basepoint of $Y$ and the choices of whiskers and orientations.
The following two observations will be useful. If $x, y\in A\cap B$, then the product of $\pi_1(Y)$-elements $\lambda(A,B)_x\hskip0.02cm(\lambda(A,B)_y)^{-1}$ is represented by a loop that runs from the basepoint to $A$ along its whisker, along $A$ to $x$, then along $B$ to $y$, and back to the basepoint along $A$ and its whisker.
Secondly, if $D_A\subset A$ is a 2-disk that is equipped with the same whisker and oriented consistently with $A$, then $\lambda(A,B)_x = \lambda(D_A,B)_x$ and $\operatorname{sign}_{A,B}(x)=\operatorname{sign}_{D_A,B}(x)$ for each $x\in D_A\cap B$.
\subsubsection{Surgering tori to 2-spheres}\label{surger}
Suppose $T$ is an embedded torus {(}or punctured torus, resp.{)} in $Y\setminus \operatorname{int} B$ and suppose there is a circle $\partialelta_1\subset T$ that is nulhomotopic in $Y$. Choose an immersed 2-disk $D$ in $Y$ that is bound by $\partialelta_1$ and transverse to $T$ and $B$, and choose a normal vector field $\phi$ to $\partialelta_1$ on $T$. Let $\partialelta_1'\subset T$ denote a nearby push-off of $\partialelta_1$ along $\phi$. Extend $\phi$ over $D$ and let $D'$ denote a pushoff of $D$ along $\phi$, bound by $\partialelta_1'$ and which we may assume is also transverse to $B$\mycomment{bend $\phi$ near the intersects if necessary}. If $D$ is oriented and $D'$ has the orientation induced as the pushoff, then intersections between $B$ and $D\cup D'$ occur as finitely many nearby pairs of points $\{x_i, x_i'\}_{i=1}^n$, where $x_i\in \operatorname{int} D$ and $x_i'\in \operatorname{int} D'$ are of opposite sign. Thus, removing from $T$ the interior of the annulus bound by $\partialelta_1 \cup \partialelta_1'$ and attaching $D\cup D'$ yields an immersed 2-sphere (or 2-disk with boundary $\partial T$, resp.) $S$ in $Y$ such that the intersections between $B$ and $S$ are transverse and occur precisely at the pairs of points $\{x_i, x_i'\}_{i=1}^n$. Furthermore, the algebraic intersections between $S$ and $B$ may be calculated using the following lemma. Let $[\alpha]$ denote the class in $\pi_1(Y)$ of a based loop $\alpha$ in $Y$, let $\overline{\gamma}$ denote the reverse of a path $\gamma$, and let $\ast$ denote composition of paths.
\begin{lemma}\label{lem:lambda-surgery}
Let $\partialelta_2$ be an oriented, simple circle on $T$ that intersects each of $\partialelta_1$ and $\partialelta_1'$ exactly once, at points $z$ and $z'$ (respectively), and is tangent to $\phi$ at $z$. Let $\iota$ be a path in $Y$ from its basepoint to $z$. If $S$ and $D$ are oriented consistently and both equipped with the whisker $\iota$, then
\[
\lambda(S, B) = (1 - [\iota\ast\partialelta_2\ast\overline{\iota}])\lambda(D, B).
\]
\end{lemma}
\begin{proof}
For each $i$, let $\gamma_i$ be a path on $D$ connecting $z$ to $x_i$ (that does not pass through any double points) and let $\gamma_i'$ be its pushoff along $\phi$, connecting $z'$ to $x_i'$. Let $\beta_i$ be a path on $B$ from $x_i$ to $x_i'$ (that does not pass through any double points), and let $\widehat \partialelta_2$ be the arc $\partialelta_2\cap S$, oriented to run from $z'$ to $z$. Then the product $\lambda(S,B)_{x_i}(\lambda(S,B)_{x_i'})^{-1}$ is represented by the loop
\begin{align}\label{eqn:loop}
\iota\ast \gamma_i \ast \beta_i\ast \overline{\gamma_i'}\ast \widehat\partialelta_2\ast \overline{\iota}.
\end{align}
Homotoping $S$ (rel boundary) by collapsing $D'$ onto $D$ except near its intersections with $B$\mycomment{because want transverse to $A$}, one sees that the loop \eqref{eqn:loop} is homotopic in $Y$ to the loop $\iota\ast\partialelta_2\ast\overline{\iota}$. Thus, equipping $D$ with the whisker $\iota$ and the same orientation as $S$, we have
\[
\lambda(S,B)_{x_i'} = [\iota \ast\overline{\partialelta_2}\ast\overline{\iota}] \lambda(S,B)_{x_i} = [\iota \ast \overline{\partialelta_2}\ast\overline{\iota}] \lambda(D,B)_{x_i}
\]
and $\operatorname{sign}_{S,B}(x_i') = \operatorname{sign}_{D',B}(x_i') = -\operatorname{sign}_{D,B}(x_i)$. Summing over all such pairs of intersections yields
\begin{align*}
\lambda(S,B) &= \mysum{i}{} \operatorname{sign}_{S,B}(x_i)\lambda(S,B)_{x_i} + \operatorname{sign}_{S,B}(x_i')\lambda(S,B)_{x_i'}\\
&= \mysum{i}{} (1 - [\iota \ast\overline{\partialelta_2}\ast\overline{\iota}])\operatorname{sign}_{D,B}(x_i)\lambda(D,B)_{x_i}\\
&= (1 - [\iota \ast\overline{\partialelta_2}\ast\overline{\iota}])\lambda(D,B).\qedhere
\end{align*}
\mycomment{ explaining change of basepoint:
If basepoint is $z$. Then $\lambda(S,B)_x\overline{\lambda(S,B)}_x$ is $z$ to $x$ along $D$, to $x'$ along $B$, along $D'$ to $x$, along $\partialelta_2$ to $z$. Homotopic to $z$ to $x$ along $D$, to $x$ along $B$, along $D$ to $z$, $\partialelta_2$ to $z$, which is homotopic to $\partialelta_2$.
If basepoint is $z'$. Then conjugate by path $z'$ to $z\in \partialelta_2$.
}
\end{proof}
\mycomment{
\begin{remark}
Referring to the notation above, the construction of $S$ required the data $T$, $\partialelta_1$, $D$ and $\phi$
\end{remark}
{\color{red} Referring to the notation of Lemma \ref{lem:lambda-surgery}, we shall say that $T'$ is the result of surgery on $T$ along $\partialelta_1$.}
}
\mycomment{
\begin{lemma}\label{lem:lambda-surgery}
Let $B$ be a 2-disk (rel $\partial$) or 2-sphere in a 4-manifold $Y$, and let $T$ be an embedded torus (punctured torus) in the complement of $B$.
Suppose $\partialelta_1\subset T$ is a curve that is nulhomotopic in $Y$ and let $\partialelta_2\subset T$ be a curve that is dual to $\partialelta_1$.
Then surgery on $T$ along $\partialelta_1$ yields a 2-sphere (or 2-disk) in $Y$ such that
\[
\lambda(A, B) = (1 - [\partialelta_2])\lambda(D, B),
\]
where $D$ is a 2-disk in $X$ bound by $\partialelta_1$ oriented appropriately, and $[\cdot]$ denotes the class in $\pi_1(Y,\ast)$.
\end{lemma}
}
\subsection{Unknotted immersions and link maps}
Two immersions $k_0, k_1:S^2\to {\mathbb R}^4$ are said to be \emph{equivalent} if there are orientation-preserving self-diffeomorphisms $h$ of $S^2$ and $H$ of ${\mathbb R}^4$, respectively, such that $k_1\circ h = H\circ k_0$. Denote the standard embedding $S^2\subset {\mathbb R}^4$ by $u_0^0$. By applying local cusp homotopies, $d$ of positive sign and $e$ of negative sign, to $u_0^0$, one obtains an \emph{unknotted} immersion, denoted $u_d^e:S^2\to {\mathbb R}^4$. Note that $u_d^e$ is unique up to equivalence; we say that an immersion $k:S^2\to {\mathbb R}^4$ (or its image) is unknotted if $k$ is equivalent to $u_d^e$ for some $d, e\geq 0$. See \cite{K} for more details.
Identify $S^4 = {\mathbb R}^4 \cup \{\infty\}$.
\begin{lemma}\label{lem:good}
A link map $f$ is link homotopic to a good link map $g$ such that $g(S^2_-)$ is unknotted in ${\mathbb R}^4\subset S^4$.
\end{lemma}
\begin{proof}
As in the proof of Theorem \ref{lem:reg-htpic}, we may assume after a link homotopy that $f$ is a good link map.
By \cite[Lemma 3]{K}, there is a family of disjoint chords attached to $f(S^2_-)$ such that finger moves along them change $f(S^2_-)$ into an unknotted immersion in ${\mathbb R}^4 = S^4 \rsetminus \{\infty\}$. As these chords may be assumed to miss $f(S^2_+)$, we have the required result.
\end{proof}
For an immersed 2-sphere $A$ in a 4-manifold $Y$, let $\omega_2(A)\in {\mathbb Z}_2$ denote the second Stiefel Whitney number of the normal bundle of $A$ in $Y$. The results in \cite{me1} readily generalize to give the following.
\begin{lemma}\label{lem:pi2}
Suppose $k:S^2\to {\mathbb R}^4$ is an unknotted, self-transverse immersion with $d$ double points, and let $Y$ denote the complement in $S^4$ of $k(S^2)$. Then $\pi_2(X_-)$ is a free ${\mathbb Z}[{\mathbb Z}]$-module on $d$ generators, the Hurewisc map $\pi_2(Y)\to H_2(Y)$ surjects and $\omega_2(A)=0$ for any immersed 2-sphere $A$ in $Y$.
\end{lemma}
\begin{remark}
Indeed, the complement of an open tubular neighborhood of $k(S^2)$ in $S^4$ has a handlebody decomposition consisting of one 0-handle, one 1-handle, and $d$ zero-framed 2-handles attached along unknotted circles in $S^3$ which are nulhomotopic in the boundary of the union of the $0$- and $1$-handle.
\end{remark}
Let $k:S^2\to Y$ be a self-transverse immersion and suppose $p$ is a double point of $k(S^2)$
An \emph{accessory circle} for $p$ is an (oriented) simple circle on $k(S^2)$ that passes through exactly one double point, $p$, and changes sheets there.
\begin{lemma}\label{lem:2-sphere-generators}
Let $f$ be a good link map such that $f(S^2_-)$ is unknotted in ${\mathbb R}^4\subset S^4$. Equip $f(S^2_+)$ with a whisker in $X_-$ and fix an identification of $\pi_1(X_-)$ with ${\mathbb Z}\langle s\rangle$ so as to write ${\mathbb Z}[\pi_1(X_-)] = {\mathbb Z}[s,s^{-1}]$. Label the double points of $f(S^2_-)$ by $\{p_i\}_{i=1}^d$ and choose an accessory circle $\alpha_i$ for $p_i$ for each $1\leq i\leq d$. Then $\pi_2(X_-)\cong (\hskip -0.03cm\underset{i=1}{\overset{d}{\oplus}}{\mathbb Z})[s,s^{-1}]\mycomment{({\mathbb Z}[{\mathbb Z}])^d}$ and there is a ${\mathbb Z}[s,s^{-1}]$-basis represented by self-transverse, immersed, whiskered 2-spheres $\{A_i\}_{i=1}^d$ in $X_-$ with the following properties. For each $1\leq i\leq d$, there is an integer Laurent polynomial $q_i\in {\mathbb Z}[s,s^{-1}]$ such that
\[
\lambda(f(S^2_+), A_i) = (1-s)^2q_i(s)
\]
and $q_i(1) = \operatorname{lk}(f(S^2_+),\alpha_i)$. Moreover, if for any $1\leq j\leq d$ the loop $\alpha_j$ bounds a 2-disk in $S^4$ that intersects $f(S^2_+)$ exactly once, then we may choose $A_j$ so that
\[
\lambda(f(S^2_+), A_j) = (1-s)^2.
\]
\end{lemma}
\begin{proof}
For $t_0, t_0'\in {\mathbb R}$, $t_0'>t_0$, let ${\mathbb R}^3[t_0]$ denote the hyperplane of ${\mathbb R}^4$ whose fourth coordinate $t$ is $t_0$, and let ${\mathbb R}^3[t_0,t_0'] = \{(x,t)\in {\mathbb R}^4: x\in {\mathbb R}^3, t_0\leq t\leq t_0'\}$. Figure \ref{fig:model} gives a ``moving picture'' description of an immersed 2-disk $U$ (appearing as an arc in each slice ${\mathbb R}^3[t_0]$, $t_0\in [-1,1]$) in a 4-ball $N\subset {\mathbb R}^3[-1,1]$, with a single self-transverse double point $p\subset {\mathbb R}^3[0]$. In this figure we have labeled a loop $\alpha\subset{\mathbb R}^3[0]$ on $U$ that changes sheets at $p$ and bounds a 2-disk $D$. For each $1\leq i\leq d$, let $\widehat U_i$ be a 2-disk on $S^2_-$ that contains the two preimages of the double point $p_i$, and no other double point preimages. There is a diffeomorphism $\Gamma_i$ of $N$ onto a 4-ball neighborhood of $p_i$ in $S^4$ that takes $U$ to $f(\widehat U_i)$, $\alpha$ to $\alpha_i$ and $p$ to $p_i$. Choose a 4-ball neighborhood $N^+\subset N$ of $p$ so that (the smaller 4-ball) $\Gamma_i(N^+)$ is disjoint from $f(S^2_+)$.
\newlength{\myheight}
\setlength{\myheight}{0.25\textwidth}
\begin{figure}\label{fig:model}
\end{figure}
There is a torus $T$ in $N^+\rsetminus U$ that intersects $D$ exactly once; see Figure \ref{fig:model-with-torus}. The torus appears as a cylinder in each of ${\mathbb R}^3[-1]$ and ${\mathbb R}^3[1]$, and appears as a pair of circles in ${\mathbb R}^3[t_0]$ for $t_0\in (-1,1)$.
\begin{figure}\label{fig:model-with-torus}
\end{figure}
By Alexander duality the linking pairing
\[
H_2(X_-)\times H_1(f(S^2_-))\to {\mathbb Z}
\]
defined by $(R,\upsilon)\mapsto R\cdot\Upsilon$, where $\upsilon=\partial \Upsilon\subset S^4$, is nondegenerate. Thus, as the loops $\{\alpha_i\}_i$ represent a basis for $H_1(f(S^2_-))\cong {\mathbb Z}^{d}$, we have that $H_2(X_-) \cong {\mathbb Z}^{d}$ and (after orienting) the so-called \textit{linking tori} $\{T_i\}_i$, defined by $T_i=\Gamma_i(T)$, represent a basis.
We proceed to apply the construction of \eqref{surger} (twice, successively) to turn these tori into 2-spheres.
In Figure \ref{fig:model-with-torus} we have illustrated an oriented circle $\partialelta$ on $T$ which intersects ${\mathbb R}^3[-1]$ and ${\mathbb R}^3[1]$ each in an arc, and appears as a pair of points in ${\mathbb R}^3[t_0]$ for $t_0\in (-1,1)$.
Notice that $\partialelta$ is isotopic in $N^+\rsetminus U$ to the circle $\widehat \partialelta\subset {\mathbb R}^3[1]$ that is also illustrated in Figure \ref{fig:model-with-torus}. By attaching the trace of such an isotopy to the 2-disk $\widehat \Delta\subset {\mathbb R}^3[1]$ illustrated, bound by $\widehat \partialelta$, one may obtain an embedded 2-disk $\Delta\subset N^+$ that is bound by $\partialelta$ and intersects $U$ precisely where $\widehat \Delta$ does.
These intersection points are the endpoints of an arc $\gamma\subset {\mathbb R}^3[1]$ on $U$, shown in Figure \ref{fig:model-with-torus}. Let $\gamma_i = \Gamma_i(\gamma)\subset f(S^2_-)$. In Figure \ref{fig:tubular-nbd} we have illustrated in ${\mathbb R}^3[1]$ the restriction of a tubular neighborhood of $U$ to $\gamma$. Identifying this tubular neighborhood with $\gamma\times D^2$, we may assume the embedding $\Gamma_i$ carries $\gamma\times D^2$ onto the restriction over $\gamma_i$ of a tubular neighborhood of $f(S^2_-)$ that is disjoint from $f(S^2_+)$.
\begin{figure}\label{fig:tubular-nbd}
\label{fig:tubular-nbd-with-beta}
\label{}
\end{figure}
Let $\Theta$ be the punctured torus in $N\rsetminus U$ given by
\[
\Theta = \Delta \rsetminus (\partial \gamma\times \operatorname{int} D^2) \mycup{\partial\gamma\times S^1}{} (\gamma\times S^1),
\]
which has boundary $\partialelta$. Note that $\Gamma_i(\Theta)$ is disjoint from $f(S^2_+)$. Form a loop $\beta$ on ${\mathbb R}^3[1]\cap \Theta$ by connecting the endpoints of $\gamma\times \{1\}$ by an arc on $\widehat \Delta \rsetminus (\partial \gamma\times \operatorname{int} D^2)$. Since $\pi_1(X_-)\cong {\mathbb Z}$ and a loop of the form $\Gamma_i(\{b\}\times S^1)$ ($b\in \operatorname{int} \gamma$) is meridinal to $f(S^2_-)$, by replacing $\gamma\times \{1\}\subset \beta$ by its band sum with oriented copies of $\{b\} \times S^1$ if necessary (see Figure \ref{fig:tubular-nbd-with-beta}) we may assume that $\beta_i=\Gamma_i(\beta)$ is a simple circle bounding an immersed, self-transverse 2-disk $D_i$ in $X_-$ that is transverse to $f(S^2_+)$.
Now, as $f(S^2_+)$ misses $\Gamma_i(N^+)$, the loop $\beta_i$ is freely homotopic in $S^4\rsetminus f(S^2_+)$ to $\alpha_i$. Consequently,
\begin{equation}\label{eq:f-D-i}
|f(S^2_+)\cdot D_i| = |\operatorname{lk}(f(S^2_+), \alpha_i)|
\end{equation}
as non-negative integers. Let $\widehat A_i$ be the immersed, self-transverse 2-disk in $X_-$ obtained by performing the construction of \eqref{surger} with the (embedded) punctured torus $\Gamma_i(\Theta)\subset X_-\rsetminus f(S^2_+)$, $\beta_i\subset \Gamma_i(\Theta)$, $D_i$ and some choice of normal vector field to $\beta_i$ on $\Gamma_i(\Theta)$.
Then, since a loop of the form $\Gamma_i(\{b\}\times S^1)$ ($b\in \operatorname{int} \gamma$) is dual to $\beta_i$ on $\Gamma_i(\Theta)$ and hence represents a generator ($s$ or $s^{-1}$) of $ \pi_1(X_-)$, by Equation \eqref{eq:f-D-i} and Lemma \ref{lem:lambda-surgery} we have (after orienting $\widehat A_i$ and connecting it to the basepoint of $X_-$)
\begin{equation}\label{eq:f-hat-A-i}
\lambda(f(S^2_+), \widehat A_i) =(1-s)\widehat q_i(s)
\end{equation}
for some integer Laurent polynomial $\widehat q_i\in {\mathbb Z}[s,s^{-1}]$ such that $\widehat q_i(1)=\operatorname{lk}(f(S^2_+), \alpha_i)$. Moreover, if $| f(S^2_+) \cap D_j|=1$ for some $j$ then (since we are free to choose the orientation and whisker of $\widehat A_j$) we may take $\widehat q_j=1$.
Now, for each $i$, $\widehat A_i$ is bound by the circle $\Gamma_i(\partialelta)$ on the (embedded) linking torus $T_i\subset X_-\rsetminus f(S^2_+)$. Perform the construction of \eqref{surger} with $T_i$, $\Gamma_i(\partialelta)$, $\widehat A_i$ and some choice of normal vector field to $\Gamma_i(\partialelta)$ on $T_i$. Then, since $\partialelta$ has a dual curve on $T$ that is meridinal to $U$, by Equation \eqref{eq:f-hat-A-i} and Lemma \ref{lem:lambda-surgery} we have (after orienting $A_i$ and connecting it to the basepoint of $X_-$)
\begin{equation*}
\lambda(f(S^2_+), A_i) =(1-s)^2 q_i(s)
\end{equation*}
for some $q_i\in {\mathbb Z}[s,s^{-1}]$ such that $q_i(1)=\operatorname{lk}(f(S^2_+), \alpha_i)$. As above, if $| f(S^2_+) \cap D_j|=1$ for some $j$ then we may take $q_j=1$.
By construction, $A_i$ is homologous to $T_i$ for each $i$, so by Lemma \ref{lem:pi2} the immersed 2-spheres $\{A_i\}_{i=1}^d$ represent a ${\mathbb Z}[s,s^{-1}]$-basis for $\pi_2(X_-)$.\qedhere
\mycomment{, It is a well known result that the complement of a self-transverse, immersed 2-sphere in $S^4$ (or, equivalently, a self-transverse, properly immersed 2-disk in $D^4$) has a cellular decomposition consisting of one $0$-cell, $1$-cell and $d$ $2$-cells. From this the hypothesis that $\pi_1(X_-)\cong {\mathbb Z}$ implies that $\pi_2(X_-)=({\mathbb Z}[{\mathbb Z}])^{d}$. See, for example, \cite{me1}. To see that the collection $\{A_i\}_i$ are generators, one notes the exact sequence
\[
\pi_2(X_-)\xrightarrow{\text{Hurewicz}} H_2(X_-)\to H_2(\pi_1(X_-))=0.\qedhere
\]}
\subsection[\texorpdfstring{The invariant $\tau$ applied to link maps}
{The invariant tau applied to link maps}]{The invariant $\tau$ applied to link maps}
In \cite{ST}, the authors define a homotopy invariant $\tau$ which takes as input a map $k:S^2\to Y^4$ with vanishing Wall self-intersection $\mu(k)$ and gives output in a quotient ${\mathbb P}i(Y, k)$ of the group ring ${\mathbb Z}[\pi_1(Y)\times \pi_1(Y)]$ modulo certain relations. The relations are additively generated by the equations
\begin{align}
(a,b)&=-(b,a)\tag{${\mathcal R}_1$}\\
(a,b)&=-(a^{-1}, ba^{-1})\tag{${\mathcal R}_2$}\\
(a,1)&=(a,a)\tag{${\mathcal R}_3$}\\
(a,\lambda(k(S^2),A))&=(a,\omega_2(A)\cdot 1)\label{eq:omega-2-relation}\tag{${\mathcal R}_4$}
\end{align}
where $a,b\in \pi_1(Y)$, $A$ represents an immersed $S^2$ or ${\mathbb R} {\mathbb P}^2$ in $Y$ (in the latter case, the group element $a$ is the image of the nontrivial element in $\pi_1({\mathbb R}{\mathbb P}^2)$).
Let $f$ be a good link map with $\sigma_+(f)=0$ (from which it follows that $\mu(f_+)=0$). For an integer $k$, let $\overline{k}$ denote its image in ${\mathbb Z}_2$. Letting $\rho$ denote the mod $2$ Hurewicz map $\pi_1(X_-)\to H_1(X_-;{\mathbb Z}_2)={\mathbb Z}_2$, define a ring homomorphism $\varphi_f: {\mathbb P}i(X_-, f_+) \to {\mathbb Z}_2\langle t: t^2=1\rangle$ by
\[
(a,b)\mapsto t^{\overline{\rho(a)+\rho(a)\rho(b)+\rho(b)}},
\]
and extending linearly mod $2$. We now prove a stronger form of Theorem \ref{thm:omega-equals-li}.
\begin{lemma}
Let $f$ be a link map with $\sigma_+(f)=0$. After a certain link homotopy of $f$ we have that $\varphi_f$ is an isomorphism and takes $\tau(f_+)$ to $(1+t)\hskip0.03cm\omega_+(f)$.
\end{lemma}
\mycomment{
\begin{lemma}[{{\cite[Proposition 4.3]{me1}}}]\label{lem:newreln}
If $Y$ is the complement in $S^4$ of an unknotted, self-transverse 2-sphere, then the relations \eqref{eq:omega-2-relation} may be replaced by the relations additively generated by the equations
\begin{align}
(a,\lambda(k,A)) = (0,0)\tag{${\mathcal R}_4'$}
\end{align}
for $a\in\pi_1(Y)$ and $A\in \pi_2(Y)$.\qedhere
\end{lemma}
}
\begin{proof}
By Lemma \ref{lem:good} we may assume $f$ is a good link map (and so $\mu(f_+)=\sigma_+(f)=0$) such that $f(S^2_-)$ is unknotted. We may perform a finger move of $f(S^2_-)$ along an chord attached in the complement of and meridinal to $f(S^2_-)$ so that a slice in ${\mathbb R}^3[t_0]$ (for some $t_0$) of the result is illustrated in Figure \ref{fig:finger-moved-dbl-pt}. This produces a pair of oppositely-signed double points $\{p^+, p^-\}$ on $f(S^2_-)$ such that (in particular) $p^+$ has an accessory circle bounding an obvious embedded 2-disk in ${\mathbb R}^3[t_0]$ that intersects $f(S^2_+)$ exactly once. Note that $f(S^2_-)$ is still unknotted (by \cite[Lemma 1]{K}) and, in particular, its complement in $S^4$ still has abelian fundamental group.
Now, fixing $f_-$ and $X_-$, by Lemma \ref{lem:2-sphere-generators} we may thus identify $\pi_1(X_-)$ with ${\mathbb Z}\langle s\rangle$ and $\pi_2(X_-)$ with $(\hskip -0.03cm\underset{i=1}{\overset{d}{\oplus}}{\mathbb Z})[s,s^{-1}]$, for some $d\geq 0$, such that there is an immersed, whiskered 2-sphere $A_0$ in $X_-$ with the property that $\lambda(f(S^2_+), A_0)=(1-s)^2$. Moreover, for any whiskered, immersed 2-sphere $A$ in $X_-$ we have $\lambda(f(S^2_+), A)=(1-s)^2q_A(s)$ for some integer Laurent polynomial $q_A\in {\mathbb Z}[s,s^{-1}]$.
\begin{figure}\label{fig:finger-moved-dbl-pt}
\end{figure}
\mycomment{
Now, from the proof of Lemma \ref{lem:2-sphere-generators} we see that $H_2(X_-)$ is generated by linking tori which have vanishing homological self-intersection. It follows from the Wu formula that the second Stiefel Whitney class of the tangent bundle $TX_-$ vanishes. Thus, as the tangent bundle of $S^2$ has vanishing Stiefel Whitney numbers, the Cartan formula implies that for any immersion $h:S^2\to X_-$ one has $\omega_2(A) = h^\ast\omega_2(TX_-) = 0$. \mycomment{ Or cite (c.f. \cite{me1} Proposition 4.3) }
}
Therefore, if we identify ${\mathbb Z}[\pi_1(X_-)\times \pi_1(X_-)]$ with ${\mathbb Z}[s^{\pm 1},t^{\pm 1}]$ via $(s^n,s^m)=s^nt^m$ (for $n, m\in {\mathbb Z}$), by Lemma \ref{lem:pi2} the ring ${\mathbb P}i(X_-,f_+)$ is the quotient of the group ring ${\mathbb Z}[s^{\pm 1},t^{\pm 1}]$ modulo the relations generated additively by the equations:
\begin{align}
s^{n}t^{n} - s^{n} &= 0\tag{${\mathcal T}_1$}\label{reln1}\\
s^{n}t^{m}+s^{-n}t^{m-n} &= 0\tag{${\mathcal T}_2$}\label{reln2}\\
s^{n}t^{m}+s^{m}t^{n}&=0\tag{${\mathcal T}_3$}\label{reln3}\\
s^{n}t^m(1-t)^2&=0\tag{${\mathcal T}_4$}\label{reln4}
\end{align}
where $n, m\in {\mathbb Z}$. Note that in reformulating Relation \eqref{eq:omega-2-relation} to obtain Relation \eqref{reln4} we have used the action of $\pi_1(X_-)={\mathbb Z}\langle s\rangle$ on $\pi_2(X_-)$.
Let $\equiv$ denote equivalence in ${\mathbb P}i(X_-,f_+)$. Clearly $\varphi_f$ is surjective; to show injectivity we first show that for any integers $n, m$, one has
\begin{align}\label{reln-big}
s^nt^m \equiv t^{\overline{n+nm+m}}.
\end{align}
By Relations \eqref{reln2} and \eqref{reln3} we have $2t^n\equiv 0$ and hence $2s^n\equiv -2t^n\equiv 0$ for each $n\in {\mathbb Z}$. Then Relation \eqref{reln4} implies that $t^{m+2}\equiv t^m$ for any integer $m$, and it follows by an induction argument that
\begin{align}\label{tm}
t^m\equiv t^{\overline{m}}.
\end{align}
Now, $s\equiv -t\equiv t$ and $st\equiv t$ by Relations \eqref{reln1}-\eqref{reln3}. Combining these equivalences with the consequence of Relation \eqref{reln4} that $st^{m+2}\equiv 2st^{m+1}-st^m$, an induction gives
\begin{align}\label{stm}
st^m\equiv t
\end{align}
for any integer $m$.
Finally, fix $n_0\in {\mathbb Z}$. By Relation \eqref{reln3} and Equivalences \eqref{tm} and \eqref{stm}, we have $s^{n_0}\equiv -t^{n_0} \equiv t^{\overline{n_0}}$ and $s^{n_0}t \equiv -st^{n_0} \equiv -t \equiv t$.
Suppose now that for some $k\geq 1$ Equivalence \eqref{reln-big} holds for $n=n_0$ and any $m\in\{0,1,\ldots, k\}$. Then Relation \eqref{reln4} implies that
\begin{align*}
s^{n_0}t^{k+1}&\equiv 2s^{n_0}t^{k}-s^{n_0}t^{k-1}\\
&\equiv 2t^{\overline{{n_0}+{n_0}k+k}}-t^{\overline{{n_0}+{n_0}(k-1)+(k-1)}}\\
&\equiv t^{\overline{{n_0}+{n_0}(k+1)+(k+1)}}.
\end{align*}
On the other hand, suppose that for some $k\leq 0$ Equivalence \eqref{reln-big} holds for $n=n_0$ and any $m\in\{k, k+1, \ldots, 0, 1\}$; then
\begin{align*}
s^{n_0}t^{k-1}&\equiv 2s^{n_0}t^{k}-s^{n_0}t^{k+1}\\
&\equiv 2t^{\overline{{n_0}+{n_0}k+k}}-t^{\overline{{n_0}+{n_0}(k+1)+(k+1)}}\\
&\equiv t^{\overline{{n_0}+{n_0}(k-1)+(k-1)}}.
\end{align*}
Thus, by induction Equivalence \eqref{reln-big} holds for $n=n_0$ and any integer $m$. But $n_0\in {\mathbb Z}$ was arbitrary, so the equivalence holds for all integers $n, m$. As $2\equiv 0\equiv 2t$, we deduce that ${\mathbb P}i(X_-, f_+)$ is the group ring ${\mathbb Z}_2\langle t: t^2=1\rangle$ and $\varphi_f$ is injective.
Turning to the second part of the lemma, we refer the reader to \cite{Li97} and \cite{ST} for detailed descriptions of the $\omega$ and $\tau$ invariants, respectively, and to \cite{FQ} for background on framed Whitney disks. We make only a few summarizing remarks.
Since $\sigma_+(f)=0$, $\pi_1(X_-)={\mathbb Z}\langle s\rangle$, and $f_+$ is self-transverse with vanishing signed sum of its double points, the double points of $f(S_+^2)$ may be decomposed into \emph{canceling} pairs $\{p_i^+,p_i^-\}_{i=1}^k$ in the following sense. For each $1\leq i\leq k$, one has $\operatorname{sign}(p_i^+)=-\operatorname{sign}(p_i^-)$ and the preimages of $p_i^\pm$ in $S^2_+$ may be labeled $\{x_i^\pm, y_i^\pm\}$ so that if $\gamma_i$ is an arc on $S^2_+$ connecting $x_i^+$ to $x_i^-$ (and missing all other double point preimages) and $\gamma_i'$ is an arc on $S^2_+$ connecting $y_i^+$ to $y_i^-$ (and missing $\gamma_i$ and all other double point preimages), then the loop $f(\gamma_i)\cup f(\gamma_i')\subset f(S^2_+)$ is nulhomotopic in $X_-$. The arcs $\{\gamma_i,\gamma_i'\}_{i=1}^k$ may be chosen so that the resulting \emph{Whitney circles} $\{f(\gamma_i\cup \gamma_i')\}_{i=1}^k$ are mutually disjoint, simple circles in $X_-$ such that each bounds an immersed, framed Whitney disk $W_i$ in $X_-$ whose interior is transverse to $f(S^2_+)$. Let $\alpha_i^\pm$ be an arc on $S^2_+$ connecting $x_i^\pm$ to $y_i^\pm$, and let $n_i^\pm$ denote the integer $\operatorname{lk}(f(S^2_-), f(\alpha_i^\pm))$; then $n_i^+=-n_i^-$. (In \cite{Li97} the non-negative integer $|n_i^+|$ is called the \emph{$n$-multiplicity} for the pair $\{p_i^+, p_i^-\}$, and in \cite{ST} the $\pi_1(X_-)$-element $s^{n_i^+}$ is called the \emph{primary group element} for $W_i$.) Note that $\rho(s^{n_i^+})$ is the mod $2$ image of $n_i^+$.
\mycomment{We will refer to a neighborhood in $f(S^2_+)$ of $f(\gamma_i)$ (resp. $\gamma_i''$) as the \emph{positive} (resp. \emph{negative}) \emph{sheet} of $g(S^2_+)$ near $W_i$.}
\mycomment{ In my translation, since $n_i^+$ goes from neg to pos sheet in \cite{ST}, so $\gamma_i$ is neg arc and $\gamma_i'$ is pos arc}
Let $i\in \{1,2,\ldots, k\}$ and suppose $x\in f(S^2_+) \cap \operatorname{int} W_i$. A loop that first goes along $f(S^2_+)$ from its basepoint to $x$, then along $W_i$ to $f(\gamma_i')\subset \partial W_i$, then back along $f(S^2_+)$ to the basepoint of $f(S^2_+)$, determines a $\pi_1(X_-)$-element $s^{m_x}$ (called the \emph{secondary group element} associated to $x$ in \cite{ST}; the non-negative integer $|m_x|$ is called the $m$-\emph{multiplicity} of $x$ in \cite{Li97}).
Associate to $x$ a sign by orienting $W_i$ using the following convention: orient $\partial W_i$ from $p_i^-$ to $p_i^+$ along the $f(\gamma_i')$, then back to $p_i^-$ along $f(\gamma_i)$; the positive tangent to $\partial W_i$ together with an outward-pointing second vector then orient $W_i$.
\mycomment{
Since the positive and negative sheets meet transversely at $p_i^\pm$, there are a pair of smooth vector fields $v_1, v_2$ on $\partial W_i$ such that $v_1$ is tangent to $f(S^2_+)$ along $\alpha_i$ and normal to $f(S^2_+)$ along $\beta_i$, while $v_2$ is normal to $f(S^2_+)$ along $\alpha_i$ and tangent to $f(S^2_+)$ along $\beta_i$. Such a pair defines a normal framing of $W_i$ on the boundary. We say that $\{v_1,v_2\}$ is a \emph{correct framing} of $W_i$, and that $W_i$ is \emph{framed}, if the pair extends to a normal framing of $W_i$.
}
Let
\[
J^i_x= \overline{n_i^+ + n_i^+m_x + m_x} \in {\mathbb Z}_2
\]
and
\[
I^i_x = \operatorname{sign}(x) s^{n_i^+}t^{m_x} \in {\mathbb Z}[s^{\pm 1},t^{\pm 1}].
\]
Then Li's ${\mathbb Z}_2$-valued $\omega_+$-invariant applied to $f$ is defined by
\[
\omega_+(f) = \mysum{i=1}{k}\; \mysum{x\, \in\, f(S^2_+)\,\cap \,\operatorname{int} W_i}{} J^i_x \mod 2;
\]
while, in this special case, the Schneiderman-Teichner invariant $\tau$ applied to $f_+$ is given by the ${\mathbb Z}[s^{\pm 1},t^{\pm 1}]$-sum
\[
\tau(f_+) =\mysum{i=1}{k}\; \mysum{x\, \in\, f(S^2_+)\,\cap \,\operatorname{int} W_i}{} I^i_x
\]
evaluated in the quotient ${\mathbb P}i(X_-, f_+)$.
Now, $\varphi_f(I^i_x)= t^{J^i_x}$, and consequently
\begin{align*}
\varphi_f(\tau(f_+)) &= \mysum{i=1}{k}\; \mysum{x\, \in\, f(S^2_+)\,\cap \,\operatorname{int} W_i}{} {t^{J^i_x}} \;.\mycomment{12-22-15:\\
&= \overline{\scalebox{1.3}{\#}\{(i,x)\hspace*{-0.1cm}: J^i_x=0\}}\; +\; \overline{\scalebox{1.3}{\#}\{(i,x)\hspace*{-0.1cm}: J^i_x=1\}}\cdot t,}
\end{align*}
\mycomment{where $1\leq i\leq k$ and $x\in \operatorname{int} W_i\cap f(S^2_+)$.} But $\varphi(\tau(f_+))\in\{0,1,t,1+t\}$ must map forward to $0$ under the homomorphism ${\mathbb P}i(X_-,f_+)\to {\mathbb P}i(S^4,f_+)={\mathbb Z}_2$ induced by the inclusion $X_-\subset S^4$ and given by sending $s$, $t \mapsto 1$. Thus \mycomment{12-22-15:
\[
\scalebox{1.3}{\#}\{(i,x)\hspace*{-0.1cm}: J^i_x=0\} = \scalebox{1.3}{\#}\{(i,x)\hspace*{-0.1cm}: J^i_x=1\} \mod 2
\]
and so}
\begin{align*}
\varphi_f(\tau(f_+)) &\equiv \mycomment{12-22-15: \overline{\scalebox{1.3}{\#}\{(i,x)\hspace*{-0.1cm}: J^i_x=1\}}\cdot(1+t)\\
&= }\mysum{i=1}{k}\; \mysum{x\, \in\, f(S^2_+)\,\cap \,\operatorname{int} W_i}{} J_i^x\cdot (1+t) \mod 2\\
&= (1+t)\hskip0.03cm\omega_+(f).\qedhere
\end{align*}
\end{proof}
\mycomment{ Pre 12-29:
Now, the Equivalence \eqref{reln-big} in ${\mathbb P}i(X_-, f_+)$ implies that $I^i_x \equiv t^{J^i_x}$, and consequently
\begin{align*}
\tau(f_+) &\equiv \mysum{i=1}{k}\; \mysum{x\, \in\, f(S^2_+)\,\cap \,\operatorname{int} W_i}{} {t^{J^i_x}} \;.\mycomment{12-22-15:\\
&= \overline{\scalebox{1.3}{\#}\{(i,x)\hspace*{-0.1cm}: J^i_x=0\}}\; +\; \overline{\scalebox{1.3}{\#}\{(i,x)\hspace*{-0.1cm}: J^i_x=1\}}\cdot t,}
\end{align*}
\mycomment{where $1\leq i\leq k$ and $x\in \operatorname{int} W_i\cap f(S^2_+)$.} But $\tau(f_+)$ must map forward to $0$ under the homomorphism ${\mathbb P}i(X_-,f_+)\to {\mathbb P}i(S^4,f_+)={\mathbb Z}_2$ induced by the inclusion $X_-\subset S^4$ and given by sending $s$, $t \mapsto 1$. Thus \mycomment{12-22-15:
\[
\scalebox{1.3}{\#}\{(i,x)\hspace*{-0.1cm}: J^i_x=0\} = \scalebox{1.3}{\#}\{(i,x)\hspace*{-0.1cm}: J^i_x=1\} \mod 2
\]
and so}
\begin{align*}
\tau(f_+) &\equiv \mycomment{12-22-15: \overline{\scalebox{1.3}{\#}\{(i,x)\hspace*{-0.1cm}: J^i_x=1\}}\cdot(1+t)\\
&= }\mysum{i=1}{k}\; \mysum{x\, \in\, f(S^2_+)\,\cap \,\operatorname{int} W_i}{} J_i^x\cdot (1+t) \mod 2\\
&= \omega_+(f)(1+t).\qedhere
\end{align*}
}
\end{proof}
\end{document} |
\begin{document}
\thispagestyle{empty}
\begin{center}
\Large
\textsc{Variations of Hodge Structures\\ of Rank Three $k$-Higgs Bundles\\ and Moduli Spaces of Holomorphic Triples}\\
\normalsize
August 28, 2020\\
\emph{Ronald A. Z\'u\~niga-Rojas}\footnote{ \scriptsize
Supported by Universidad de Costa Rica through Escuela de Matem\'atica, specifically through CIMM
(Centro de Investigaciones Matem\'aticas y Metamatem\'aticas), Project {\tt 820-B8-224}. This work is partly based on the Ph.D. Project~\cite{z-r}
called {\em ``Homotopy Groups of the Moduli Space of Higgs Bundles''}, supported by FEDER through Programa Operacional Factores de
Competitividade-COMPETE, and also supported by FCT (Funda\c{c}\~ao para a Ci\^encia e a Tecnologia) through the projects {\tt PTDC/MAT-GEO/0675/2012}
and {\tt PEst-C/MAT/UI0144/2013} with grant reference {\tt SFRH/BD/51174/2010}.}\\[6pt]
\small Centro de Investigaciones Matem\'aticas y Metamatem\'aticas CIMM\\
\small Escuela de Matem\'atica, Universidad de Costa Rica,\\
\small San Jos\'e 11501, Costa Rica\\
\small e-mail: \texttt{ronald.zunigarojas@ucr.ac.cr}\\
\small ORCID: \href{https://orcid.org/0000-0003-3402-2526}{\texttt{0000-0003-3402-2526}}
\end{center}
\begin{abstract}
\noindent There is an isomorphism between the moduli spaces of $\sigma$-stable holomorphic triples and some of the critical submanifolds of the moduli space of
$k$-Higgs bundles of rank three, whose elements $(E,\varphi^k)$ correspond to variations of Hodge structure, VHS.
There are special embeddings on the moduli spaces of $k$-Higgs bundles of rank three. The main objective here is to study the cohomology of the
critical submanifolds of such moduli spaces, extending those embeddings to moduli spaces of holomorphic triples.
\end{abstract}
\begin{flushleft}
\small
\emph{Keywords}:
Higgs bundles, holomorphic triples, moduli spaces, variations of Hodge structure.
\emph{MSC classes}: Primary \texttt{14F45}; Secondaries \texttt{14D07}, \texttt{14H60}.
\end{flushleft}
\section*{Introduction}
\addcontentsline{toc}{section}{Introduction}
\lambdabel{sec:0}
Consider a compact connected Riemann surface $X$ of genus $g \geqslant 2$. Algebraically, $X$ is a complete irreducible non-singular curve over $\mathbb{C}$.
Let $\mathcal{N} = \mathcal{N}(r,d)$ be the moduli space of polystable vector bundles of rank $r$ and degree $d$ over $X$. In this paper, we consider the co-prime
condition $\GCD(r,d) = 1$, which ensures that polystable implies stable. This space has been widely worked by Atiyah~\&~Bott~\cite{atbo},
Desale~\&~Ramanan~\cite{dera}, Earl~\&~Kirwan~\cite{eaki}, among other authors. Here, we consider it as the corresponding minimal critical submanifold
of $\mathcal{M}(r,d)$, the moduli space of polystable Higgs bundles. A Higgs bundle over $X$ is a pair $(E,\varphi)$ where $E\to X$ is a holomorphic vector
bundle and $\varphi\colon E \to E\otimes K$ is an endomorphism twisted by the cotangent bundle $K = T^{*}X$. Fixing rank $r$ and degree $d$ of the
underlying vector bundle $E$, the isomorphism classes of polystable Higgs bundles are parametrized by a quasiprojective variety: the moduli space of
polystable Higgs bundles $\mathcal{M}^{ps}(r,d)$. Again, since $\GCD(r,d) = 1$, polystability implies stability and then, the space
$\mathcal{M}^{ps}(r,d) = \mathcal{M}^{ss}(r,d) = \mathcal{M}^{s}(r,d)$ becomes a smooth projective variety. These spaces were first worked by Hitchin~\cite{hit2} and
Simpson~\cite{sim2}. Since then, they have been around for more than thirty years, and have been studied extensively by a number of authors: \emph{e.g.}
\cite{decataldo-mark-hausel-migliorini:2012,
hausel:2013,
hausel-letellier-rodriguez-villegas:2011,
hausel-rodriguez-villegas:2008,
hath1,hath2,
z-r0}.
\noindent Higgs bundles are an interesting topic of research because they have links with many other areas of mahtematics such as integrable systems,
mirror symmetry, Langlands programme, Hodge theory, among others. We are interested on their link to Hodge theory. The work of
Simpson~\cite{sim1, sim2}, Hausel~\cite{hau}, and Hausel \& Thaddeus~\cite{hath1, hath2} shall be particularly useful for our purposes.
\noindent There is a Morse function $f\colon \mathcal{M}^k(3,d) \to \mathbb{R}$ defined by
\[
f(E,\varphi) = \frac{1}{2\pi}\lVert \varphi \rVert^2_{L_2}
= \frac{i}{2\pi} \int_X \tr(\varphi \varphi^*)
\]
applied to the moduli spaces of stable $k$-Higgs bundles $\mathcal{M}^k(r,d)$. We study the stabilization of the cohomology groups of
the critical submanifolds from this Morse function $f$, for the case of rank $r = 3$. The co-prime condition $(3,d) = 1$ implies
that the moduli space $\mathcal{M}^k(3,d)$ is smooth. A $k$-Higgs bundle or Higgs bundle with poles of order $k$, $(E,\varphi^k)$, is
a Higgs bundle where the morphism $\varphi^k$ is twisted by $L_p$ $k$-times, where $p\in X$ is an arbitrary fixed point and
$L_p = \mathcal{O}_{X}(p)$ is its associated line bundle (local structure sheaf):
\[
\varphi^k\colon E \to E\otimes K\otimes L_p^{\otimes k} = E \otimes K(k\cdot p).
\]
According to Simpson~\cite{sim1} the critical points of $f$, are
{\em variations of the Hodge structure} (VHS), a decomposition of the form:
\begin{equation}
E = \bigoplus_{j=1}^n E_j \word{such that}
\varphi \colon E_j \to E_{j+1} \otimes K \textmd{ for } 1 \leqslant j \leqslant n - 1.
\end{equation}
for general rank $r$. In our case, for $\mathcal{M}^{k}(3,d)$, there are three kind of variations of Hodge structure:
\begin{enumerate}[i.]
\item
$(1,2)$-VHS:
\[
\Big(
E_1\oplus E_2,
\left(
\begin{array}{c c}
0 & 0\\
\phi & 0
\end{array}
\right)
\Big)\in
F_{d_1}^{k}\subseteq
\mathcal{M}^{k}(3,d).
\]
\item
$(2,1)$-VHS:
\[
\Big(
E_2\oplus E_1,
\left(
\begin{array}{c c}
0 & 0\\
\phi & 0
\end{array}
\right)
\Big)\in
F_{d_2}^{k}\subseteq
\mathcal{M}^{k}(3,d).
\]
\item
$(1,1,1)$-VHS:
\[
\Big(
L_1\oplus L_2\oplus L_3,
\left(
\begin{array}{c c c}
0 & 0 & 0\\
\phi_{21} & 0 & 0\\
0 & \phi_{32} & 0
\end{array}
\right)
\Big)\in
F_{m_1 m_2}^{k}\subseteq
\mathcal{M}^{k}(3,d),
\]
\end{enumerate}
Here, $F_{d_1}^{k},\ F_{d_2}^{k}$ and $F_{m_1 m_2}^{k}$ denote the respective critical submanifolds of the moduli space $\mathcal{M}^{k}(3,d)$. The first
two, $F_{d_1}^{k}$ and $F_{d_2}^{k}$, are close related to the space $\mathcal{N}_{\sigma}(r_1,r_2,d_1,d_2)$, the moduli space of $\sigma$-stable holomorphic
triples of type $(\mathbf{r},\mathbf{d}) = (r_1,r_2,d_1,d_2)$.
\noindent A holomorphic triple $T = (E_1,E_2,\phi)$ on $X$ consists of a pair of holomorphic vector bundles $E_1\to X$ and $E_2\to X$,
of ranks $r_1, r_2$ and degrees $d_1,d_2$ respectively, and a holomorphic map $\phi\colon E_2\to E_1$. The stability for triples depends on a
parameter $\sigma \in \mathbb{R}$, which gives a collection of moduli spaces $\mathcal{N}_{\sigma}(r_1,r_2,d_1,d_2)$ widely worked by several authors: \emph{e.g.}
\cite{
brgp,
bgg1,
bgg2,
ggm,
mos,
mov}.
The range of $\sigma$ is an interval $[\sigma_{m}, \sigma_{M}] \subseteq \mathbb{R}$ split by a finite number of critical values $\sigma_c$. The reader may see
Bradlow, Garc\'ia-Prada, Gothen~\cite{bgg1}, Mu\~noz, Oliveira, S\'anchez~\cite{mos}, or Mu\~noz, Ortega, V\'asquez-Gallo~\cite{mov} for the
interval details.
\noindent This paper works with a very particular framework. We study holomorphic triples on $X$ of the form $T = (\tilde{E}_1,\tilde{E}_2,\phi)$ with type
$(2,1,\tilde{d}_1,\tilde{d}_2)$, where ranks $r_1 = 2,\ r_2 = 1$ and degrees $\tilde{d}_1, \tilde{d}_2$ are in terms of $(1,2)$-VHS described before:
\[
\Big(
E_1\oplus E_2,
\left(
\begin{array}{c c}
0 & 0\\
\phi & 0
\end{array}
\right)
\Big)\in
F_{d_1}^{k}\subseteq
\mathcal{M}^{k}(3,d),
\]
where $\tilde{E}_1 = E_2 \otimes K(kp)$, $\tilde{E}_2 = E_1$, $\phi\colon E_1 \to E_2 \otimes K(kp)$, and so the degrees become
$\tilde{d}_1 = \deg(\tilde{E}_1) = d_2 + 2(2g - 2 + k)$, $\tilde{d}_2 = d_1$.
\noindent We study as well the holomorphic triples $T = (\tilde{E}_1,\tilde{E}_2,\phi)$ with type $(1,2,\tilde{d}_1,\tilde{d}_2)$, related to $(2,1)$-VHS of the form
\[
\Big(
E_2\oplus E_1,
\left(
\begin{array}{c c}
0 & 0\\
\phi & 0
\end{array}
\right)
\Big)\in
F_{d_2}^{k}\subseteq
\mathcal{M}^{k}(3,d),
\]
where, in this case $\tilde{E}_1 = E_1 \otimes K(kp)$, $\tilde{E}_2 = E_2$, $\phi\colon E_2 \to E_1 \otimes K(kp)$, and the degrees become
$\tilde{d}_1 = \deg(\tilde{E}_1) = d_1 + 2g - 2 + k$, $\tilde{d}_2 = d_2$.
\noindent Finally, we also study $(1,1,1)$-VHS
\[
\Big(
L_1\oplus L_2\oplus L_3,
\left(
\begin{array}{c c c}
0 & 0 & 0\\
\phi_{21} & 0 & 0\\
0 & \phi_{32} & 0
\end{array}
\right)
\Big)\in
F_{m_1 m_2}^{k}\subseteq
\mathcal{M}^{k}(3,d),
\]
and those are related to symmetric products of the form
\[
\Sym^{m_1}(X)\times \Sym^{m_2}(X)\times \mathcal{J}^{d_3}(X)
\]
where $J^{d_3}(X)$ is the Jacobian of $X$, the moduli space of stable line bundles of degree $d_3$, and $m_1, m_2$ will be described below as the corresponding degrees of auxiliar bundles.
\noindent Our estimates are based on embeddings $\mathcal{M}^k(3,d) \hookrightarrow \mathcal{M}^{k+1}(3,d)$ defined by
\[
i_k\colon
\big[(E,\varphi^k)\big]
\longmapsto
\big[(E,\varphi^k \otimes s_p)\big]
\]
where $0 \neq s_p \in H^0(X, L_p)$ is a nonzero fixed section of~$L_p = \mathcal{O}_X(p)$.
\noindent The paper is organized as follows:
in section \ref{sec:1} we recall
some basic facts about holomorphic
triples, Higgs bundles and $k$-Higgs bundles;
in section \ref{sec:2}, we present
the effect of the embeddings on
$\sigma$-stable triples;
in subsection \ref{ssec:2.1}, we show that the embeddings preserve $\sigma$-stability,
in subsection \ref{ssec:2.2}, we discuss the effect of the embeddings
considering the flip loci, and present an original result,
the so-called \emph{``Roof Theorem''}:
\begin{Th}[Theorem \ref{RoofTheorem}]
There exists an embedding
\[
\tilde{i_k}: \tilde{\mathcal{N}}_{\sigma_c(k)}
\hookrightarrow
\tilde{\mathcal{N}}_{\sigma_c(k+1)}
\]
such that the following diagram commutes:
\begin{align*}
\begin{xy}
(0,-20)*+{\mathcal{N}_{\sigma_c^{-}(k+1)}}="a";
(20,0)*+{\tilde{\mathcal{N}}_{\sigma_c(k+1)}}="b";
(10,-50)*+{\tilde{\mathcal{N}}_{\sigma_c(k)}}="c";
(40,-20)*+{\mathcal{N}_{\sigma_c^{+}(k+1)}}="d";
(-10,-70)*+{\mathcal{N}_{\sigma_c^{-}(k)}}="e";
(30,-70)*+{\mathcal{N}_{\sigma_c^{+}(k)}}="f";
{\ar@{-->}^(.45){\exists \tilde{i_k}} "c";"b" **\dir{--}};
{\ar^(.45){} "b";"a" **\dir{-}};
{\ar^(.45){i_k} "e";"a" **\dir{-}};
{\ar^(.45){} "c";"e" **\dir{-}};
{\ar_(.45){i_k} "f";"d" **\dir{-}};
{\ar^(.45){} "c";"f" **\dir{-}};
{\ar^(.45){} "b";"d" **\dir{-}};
\end{xy}
\end{align*}
where $\tilde{\mathcal{N}}_{\sigma_c(k)}$ is the blow-up of
$
\mathcal{N}_{\sigma_c^{-}(k)} =
\mathcal{N}_{\sigma_c^{-}(k)}(2,1,\tilde{d}_1,\tilde{d}_2)
$
along the flip locus $S_{\sigma_c^{-}(k)}$ and, at the same time, represents the blow-up of
$
\mathcal{N}_{\sigma_c^{+}(k)} =
\mathcal{N}_{\sigma_c^{+}(k)}(2,1,\tilde{d}_1,\tilde{d}_2)
$
along the flip locus $S_{\sigma_c^{+}(k)}$.
\end{Th}
\noindent Here, $\sigma_c(k) \in ]\sigma_m,\sigma_M[$ is a $\sigma$-critical value depending on the parameter $k$, that lies in the interval mentioned above, where
$\sigma_m = \mu_1 - \mu_2 = \tilde{d}_1/2 - \tilde{d}_2$ and $\sigma_M = 4\sigma_m$.
\noindent In section \ref{sec:3} we present the cohomology main results for triples:
in subsection \ref{ssec:3.1} appear some useful results about the cohomology
of the symmetric product $\Sym^{k}(X)$.
In subsection \ref{ssec:3.2} we present
the stabilization of the cohomology (Theorem \ref{CohomologyBlowUp}) for certain indices:
\begin{Th}[Theorem~\ref{CohomologyBlowUp}]
There is an isomorphism
\[
\tilde{i_k^*}\colon
H^{j}(\tilde{\mathcal{N}}_{\sigma_c(k+1)},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(\tilde{\mathcal{N}}_{\sigma_c(k)},\mathbb{Z})\quad
\forall j \leqslantslant n(k)
\]
at the blow-up level, where
$
n(k) = \min \big\{ \tilde{d}_1 - d_M - \tilde{d}_2 - 1,\quad 2\big(\tilde{d}_1 - 2\tilde{d}_2 - (2g - 2)\big) + 1\big\}
$.
\end{Th}
\noindent And hence, the cohomology stabilization of the moduli spaces of triples:
\begin{Cor}[Corollary~\ref{CohomologyCriticalTriples}]
There is an isomorphism
\[
i_k^*\colon
H^{j}(\mathcal{N}_{\sigma_c(k+1)},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(\mathcal{N}_{\sigma_c(k)},\mathbb{Z})\quad
\forall j \leqslantslant n(k)
\]
where
$
n(k)
$ as above.
\end{Cor}
\noindent In subsection \ref{ssec:3.3} we show the stabilization
of the $(1,2)$-VHS cohomology using the isomorphisms between them
and the moduli spaces of triples:
\begin{Cor}[Corollary~\ref{(1,2)-VHS--Cohomology}]
There is an isomorphism
\[
H^{j}(F_{d_1}^{k+1},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(F_{d_1}^{k},\mathbb{Z})
\]
for all
$
j \leqslantslant \sigma_H(k) - 2(\mu_1 - \mu) - 1
$.
\end{Cor}
\noindent Here, $\sigma_{H}(k) \in ]\sigma_m,\sigma_M[$ is a particular $\sigma$-critical value depending on the parameter $k$:
\[
\sigma_{H}(k) = \deg\big(K(k\cdot p)\big) = 2g - 2 + k.
\]
\noindent In subsection \ref{ssec:3.4} we show the analogous dual result for $(2,1)$-VHS:
\begin{Cor}[Corollary~\ref{(2,1)-VHS--Cohomology}]
For $k$ large enough, there is an isomorphism
\[
H^{j}(F_{d_2}^{k+1},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(F_{d_2}^{k},\mathbb{Z})
\]
for all
$
j \leqslantslant \sigma_H(k) - 4(\mu_2 - \mu) - 1.
$
\end{Cor}
\noindent Finally, in subsection \ref{ssec:3.5} we described the cohomology for $(1,1,1)$-VHS and its relationship with the spaces
$\Sym^{m_1}(X)\times \Sym^{m_2}(X)\times \mathcal{J}^{d_3}(X)$:
\begin{Cor}[Corollary~\ref{(1,1,1)-cohomology-iso}]
There is an isomorphism
\[
H^{j}(F_{m_1 m_2}^{\infty},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(F_{m_1 m_2}^{k},\mathbb{Z})
\]
for all $j \leqslant \min \big(\bar{m}_1 + k, \bar{m}_2 + k\big) - 1$.
\end{Cor}
\section{Preliminary definitions}
\lambdabel{sec:1}
Let $X$ be a compact connected Riemann surface of genus $g \geqslant 2$ and let $K = T^*X$ be the canonical line bundle of
$X$. Note that, algebraically, $X$ is also a nonsingular complex projective algebraic curve.
\noindent The $k$-th symmetric product $\Sym^{k}(X)$ is a smooth projective variety of dimension $k\in \mathbb{N}$, that could be interpretated as
the moduli space of degree $k$ effective divisors. In other words, $\Sym^{k}(X) = X^{k}/S_{k}$, the symmetric product with quotient topology,
is the quotient of $X^{k}$ the $k$-times cartesian product by the action of $S_{k}$ the $k$-symmetric group. Obviously $\Sym^{1}(X) = X$.
\begin{Def}
For a (smooth or holomorphic) vector bundle $E \to X$, we denote the \emph{rank} of~$E$ by
$\rk(E) = r$ and the \emph{degree} of $E$ by $\deg(E) = d$. Its
\emph{slope} is defined to be
\begin{equation}
\mu(E) = \frac{\deg(E)}{\rk(E)} = \frac{d}{r}.
\lambdabel{slope}
\end{equation}
A vector bundle $E \to X$ is called \emph{semistable} if $\mu(F) \leqslant \mu(E)$ for any nonzero $F \subseteq E$. Similarly, a vector bundle $E \to X$ is called \emph{stable} if $\mu(F) < \mu(E)$ for any nonzero $F \subsetneq E$. Finally, $E$ is called \emph{polystable} if it is the direct sum of stable subbundles, all of the same slope.
\end{Def}
\subsection{Holomorphic Triples}
\lambdabel{ssec:1.1}
\begin{Def}
A \emph{holomorphic triple} on $X$ is a triple
$T = (E_1, E_2, \phi)$ consisting of two holomorphic vector bundles
$E_1 \to X$ and $E_2 \to X$ and a homomorphism $\phi \colon E_2 \to E_1$,
{\em i.e.} an element
$\phi \in H^0\big(\Hom(E_2,E_1)\big)$.
\end{Def}
\begin{Def}
A \emph{homomorphism} from a triple $T' = (E'_1,E'_2,\phi')$ to
another triple $T = (E_1,E_2,\phi)$ is a commutative diagram of the
form:
\[
\begin{xy}
(0,0)*+{E'_2}="a";
(20,0)*+{E'_1}="b";
(0,-20)*+{E_2}="c";
(20,-20)*+{E_1}="d";
{\ar@{->}^{\phi'} "a";"b"};
{\ar@{->} "b";"d"};
{\ar@{->} "a";"c"};
{\ar@{->}^{\phi} "c";"d"};
\end{xy}
\]
where the vertical arrows represent holomorphic maps.
\end{Def}
\begin{Def}
A triple $T' = (E_1',E_2',\phi')$ is a {\em subtriple} of $T = (E_1,E_2,\phi)$ if
\begin{enumerate}[i.]
\item
$E_j' \subseteq E_j$ is a coherent subsheaf for $j=1,2$
\item
$\phi' = \phi|_{{}_{E_2'}}$, {\em i.e.} $\phi'$ is the restriction of $\phi$.
\end{enumerate}
In other words, we get the commutative diagram
\[
\begin{xy}
(0,0)*+{E'_2}="a";
(20,0)*+{E'_1}="b";
(0,-20)*+{E_2}="c";
(20,-20)*+{E_1}="d";
{\ar@{->}^{\phi'} "a";"b"};
{\ar@{->} "b";"d"};
{\ar@{->} "a";"c"};
{\ar@{->}^{\phi} "c";"d"};
\end{xy}
\]
where the vertical arrows are injective inclusions this time. In such a case, we denote $T' \subseteq T$.
If $E_1' = E_2' = 0$ we call the subtriple $T' \subseteq T$ as {\em the trivial subtriple}.
$T'$ is a {\em non-trivial proper subtriple} if $0 \neq T' \subsetneq T$.
\end{Def}
\begin{Rmk}
For stability criteria, it will be enough to consider saturated subsheaves. In our case, since $X$ is a Riemann surface,
saturated subsheaves are precisely subbundles.
\end{Rmk}
\begin{Def}
A triple $T = (E_1,E_2,\phi)$ is {\em reducible} if there are direct sum decompositions $\displaystyle E_1 = \bigoplus_{i=1}^{n}E_{1i}$,
$\displaystyle E_2 = \bigoplus_{i=1}^{n}E_{2i}$, and $\displaystyle \phi = \bigoplus_{i=1}^{n}\phi_{i}$ such that
$\phi_{i} \in \Hom(E_{2i},E_{1i})$. In such a case, $T$ has a {\em direct sum decomposition}
\[
T = \bigoplus_{i=1}^{n} T_{i} \word{of subtriples} T_{i} = (E_{1i},E_{2i},\phi_{i}).
\]
If $T = (E_1,E_2,\phi)$ is not reducible, we say that $T$ is {\em irreducible}.
\end{Def}
\begin{Rmk}
We adopt Bradlow~and~Garc\'ia-Prada~\cite{brgp} convention that if $E_{2i} = 0$ or $E_{1i} = 0$ for some $i$, then $\phi_{i}$ is the zero map.
\end{Rmk}
\begin{Def}
$\sigma$-Stability, $\sigma$-Semistability and $\sigma$-Polystability:
\begin{enumerate}[i.]
\item
For any $\sigma \in \mathbb{R}$, the $\sigma$-degree and the $\sigma$-slope of
$T = (E_1,E_2,\phi)$ are defined as:
\[
\deg_\sigma(T) = \deg(E_1) + \deg(E_2) + \sigma \cdot \rk(E_2),
\]
and
\[
\mu_\sigma(T) = \frac{\deg_\sigma(T)}{\rk(E_1) + \rk(E_2)}
= \mu(E_1 \oplus E_2)
+ \sigma\, \frac{\rk(E_2)}{\rk(E_1) + \rk(E_2)} \,
\]
respectively.
\item
$T$ is then called $\sigma$-stable [respectively, $\sigma$-semistable] if
$\mu_\sigma(T') < \mu_\sigma(T)$ [respectively, $\mu_\sigma(T') \leqslant \mu_\sigma(T)$]
for any proper subtriple $0 \neq T' \subsetneq T$.
\item
A triple is called $\sigma$-polystable if it is the direct sum of
$\sigma$-stable triples of the same $\sigma$-slope.
\end{enumerate}
\end{Def}
Now we may use the following notation for moduli spaces of triples:
\begin{enumerate}[i.]
\item
Denote $\mathbf{r} = (r_1,r_2)$ and $\mathbf{d} = (d_1,d_2)$, and then regard
\[
\mathcal{N}_\sigma = \mathcal{N}_\sigma(\mathbf{r},\mathbf{d}) = \mathcal{N}_\sigma(r_1,r_2,d_1,d_2)
\]
as the moduli space of $\sigma$-polystable triples $T = (E_1,E_2,\phi)$
such that $\rk(E_j) = r_j$ and $\deg(E_j) = d_j$.
\item
Denote by $\mathcal{N}^s_\sigma = \mathcal{N}^s_\sigma(\mathbf{r},\mathbf{d})$ the open subspace of
$\sigma$-stable triples.
\item
Call $(\mathbf{r},\mathbf{d}) = (r_1,r_2,d_1,d_2)$ the type of the triple
$T = (E_1,E_2,\phi)$.
\end{enumerate}
\noindent The moduli space of $\sigma$-stable triples $\mathcal{N}^s_\sigma = \mathcal{N}^s_\sigma(\mathbf{r},\mathbf{d}) = \mathcal{N}^s_\sigma(r_1,r_2,d_1,d_2)$ is formally constructed by
Bradlow and Garc\'ia-Prada~\cite{brgp} using dimensional reduction. There is also a direct construction by Schmitt~\cite{schmitt}
using Geometric Invariance Theory (GIT). The reader also may consult the work of Bradlow, Garc\'ia-Prada and
Gothen~\cite{bgg1}; Mu\~noz, Oliveira and S\'anchez~\cite{mos}; or Mu\~noz, Ortega and V\'azquez-Gallo~\cite{mov} for the following
details.
\noindent There are certain necessary conditions on $\sigma$ for $\sigma$-stable triples to exist. For triples of type $(\mathbf{r},\mathbf{d}) = (r_1,r_2,d_1,d_2)$,
consider the slopes $\mu_j = \frac{d_j}{r_j}$ for $j = 1,2$ and define
\begin{equation}\lambdabel{sg-m}
\sigma_m = \mu_1 - \mu_2,
\end{equation}
and
\begin{equation}\lambdabel{sg-M}
\sigma_M =
\left(
1 + \frac{r_1 + r_2}{|r_1 - r_2|}
\right)
(\mu_1 - \mu_2), \word{for} r_1 \neq r_2,
\end{equation}
\begin{Th}[{\cite[Th.~6.1.]{brgp}}]
The moduli space of $\sigma$-stable triples $\mathcal{N}_{\sigma}^{s}(r_1,r_2,d_1,d_2)$ is a complex analytic variety, which is projective when $\sigma \in \mathbb{Q}$.
A necessary condition for $\mathcal{N}_{\sigma}^{s}(r_1,r_2,d_1,d_2)$ to be non-empty is
$
0 \leqslant \sigma_m \leqslant \sigma \leqslant \sigma_M, \word{if} r_1 \neq r_2,
$
$
0 \leqslant \sigma_m \leqslant \sigma, \word{if} r_1 = r_2.
$
\end{Th}
\begin{Rmk}
If $\mu_1 = \mu_2$ and $r_1 \neq r_2$ then $\sigma_m = \sigma_M = 0$ and so, $\mathcal{N}^{s}_{\sigma}(r_1,r_2,d_1,d_2)$ is empty unless $\sigma = 0$.
\end{Rmk}
\begin{Prop}[{\cite[Prop.~2.4.]{bgg1}}]\lambdabel{dualtriples}
The $\sigma$-stability of $T = (E_1,E_2,\phi)$ is equivalent to the $\sigma$-stability of the dual triple $T^{*} = (E_2^*,E_1^*,\phi^*)$, where
$\phi^*$ represents the conjugate transpose of $\phi$. The map $T\mapsto T^*$ defines an isomorphism
\[
\mathcal{N}_{\sigma}^{s}(r_1,r_2,d_1,d_2)
\cong
\mathcal{N}_{\sigma}^{s}(r_2,r_1,-d_2,-d_1).
\]
\end{Prop}
\noindent The last result is frequently used to restrict the study of triples to $r_1 \geqslant r_2$ and appeal to duality when $r_1 < r_2$. We shall use
this duality result later to study and compare the cohomology of $(1,2)$-VHS and $(2,1)$-VHS.
\begin{Def}
For triples of type $(r_1,r_2,d_1,d_2)$, the number $\sigma \in [\sigma_m, \infty[$ is a {\em critical value} if there exist integers
$r_1', r_2', d_1'$ and $d_2'$ such that
\[
\sigma =
\frac{(r_1 + r_2)(d_1' + d_2')-(r_1' + r_2')(d_1 + d_2)}{r_1' r_2 - r_1 r_2'}
\]
or equivalently
\[
\frac{d_1 + d_2}{r_1 + r_2} + \frac{\sigma \cdot r_2}{r_1 + r_2}
=
\frac{d_1' + d_2'}{r_1' + r_2'} + \frac{\sigma \cdot r_2'}{r_1' + r_2'}
\]
with
$0 \leqslant r_j' \leqslant r_j$,
$
(r_1',r_2',d_1',d_2')
\neq
(r_1,r_2,d_1,d_2)
$,
$(r_1,r_2) \neq (0,0)$
and
$r_1' r_2 \neq r_1 r_2'$.
We denote $\sigma = \sigma_c$ if it is critical.
The number $\sigma \in [\sigma_m, \infty[$ is called {\em generic} if it is not critical.
\end{Def}
\begin{Prop}[{\cite[Prop.~2.6.]{bgg1}}]
Fix the type $(r_1,r_2,d_1,d_2)$.
\begin{enumerate}[i.]
\item
The critical values $\sigma_c$ form a discrete subset of the interval $[\sigma_m, \infty[$.
\item
If $r_1 \neq r_2$ the critical values $\sigma_c$ are finite and lie in the interval $[\sigma_m,\sigma_M]$.
\item
The stability criteria for two values of $\sigma$ lying between two consecutive critical values are equivalent; thus, the moduli spaces
are isomorphic.
\item
If $\sigma$ is generic and $\GCD(r_2,r_1 + r_2, d_1 + d_2) = 1$, then $\sigma$-semistability is equivalent to $\sigma$-stability.
\end{enumerate}
\end{Prop}
\subsection{Higgs Bundles}
\lambdabel{ssec:1.2}
\begin{Def}\lambdabel{hbdef}
A \emph{Higgs bundle} over $X$ is a pair $(E, \varphi)$ where
$E \to X$ is a holomorphic vector bundle and $\varphi\colon E \to E \otimes K$
is an endomorphism of $E$ twisted by~$K$, which is called a
\emph{Higgs field}. Note that $\varphi \in H^0(X;\End(E) \otimes K)$.
\end{Def}
\begin{Def}\lambdabel{stablehb}
A subbundle $F \subseteq E$ is said to be \emph{$\varphi$-invariant} if $\varphi(F) \subseteq F \otimes K$. A Higgs bundle is said to be
\emph{semistable} [respectively, \emph{stable}] if $\mu(F) \leqslant \mu(E)$ [respectively, $\mu(F) < \mu(E)$] for any nonzero
$\varphi$-invariant subbundle $F \subseteq E$
[respectively, $F \subsetneq E$]. Finally, $(E,\varphi)$ is called \emph{polystable} if it is the direct sum of stable
$\varphi$-invariant subbundles, all of the same slope.
\end{Def}
\noindent Fixing the rank $\rk(E) = r$ and the degree $\deg(E) = d$ of a Higgs bundle $(E,\varphi)$, the isomorphism classes of polystable bundles are
parametrized by a quasi-projective variety: the moduli space $\mathcal{M}(r,d)$. Constructions of this space can be found in the work of
Hitchin~\cite{hit2}, using gauge theory, or in the work of Nitsure~\cite{nit}, using algebraic geometry methods.
{
\noindent Hitchin~\cite{hit2} works with the \emph{Yang-Mills self-duality equations} (SDE)
\begin{equation}
\left\{
\begin{array}{c c c}
F_A + [\varphi, \varphi^{*}] & = & 0 \\
& & \\
\bar{\partial}_A \varphi & = & 0,
\end{array}
\right.
\lambdabel{eq:YM}
\end{equation}
where $\varphi \in \varOmegaega^{1,0}\big(X, \End(\mathcal{E})\big)$ is a complex auxiliary
field and $F_A$ is the curvature of a connection $d_A$ which is compatible with $\bar{\partial}_A$, the holomorphic structure of the bundle $E = (\mathcal{E}, \bar{\partial}_A)$, where $\mathcal{E}$ is a smooth complex bundle of rank $\rk(\mathcal{E})=2$ and degree $\deg(\mathcal{E})=1$. Hitchin calls $\varphi$ {\em Higgs~field}, because it shares a lot of the physical and gauge properties of those of the Higgs boson. Here, $\varphi^{*}$ denotes the adjoint of $\varphi$ with respect to the hermitian metric on $E$,\footnote{By Hitchin~\cite{hit2}, there is a hermitian metric on E.} and $[\cdot,\cdot]$ denotes the extension of the Lie bracket to Lie algebra-valued forms.
\noindent The set of solutions
$$
\beta(\mathcal{E}) =
\{
(\bar{\partial}_A, \varphi)| \word{solution of}(\ref{eq:YM})
\}
\subseteq
\mathcal{A}^{0,1}(\mathcal{E}) \times \varOmegaega^{1,0}\big(X,\End(\mathcal{E})\big)
$$
where $\mathcal{A}^{0,1}(\mathcal{E})$ denotes the space of holomorphic structures on $\mathcal{E}$,
$\varOmegaega^{1,0}\big(X,\End(\mathcal{E})\big)$ denotes $(1,0)$-forms of $X$ with values on $\End(\mathcal{E})$,
and the collection
$$
\beta_{ps}(2,1) =
\{
\beta(\mathcal{E})| \mathcal{E} \word{polystable,} \rk(\mathcal{E}) = 2, \deg(\mathcal{E}) = 1
\},
$$
allow Hitchin to construct the Moduli space of solutions to SDE~(\ref{eq:YM})
$$
\mathcal{M}^{YM}(2,1) = \beta_{ps}(2,1) / \mathcal{G}^{\mathbb{C}},
$$
and
$$
\mathcal{M}^{YM}_{s}(2,1) = \beta_{s}(2,1) / \mathcal{G}^{\mathbb{C}}
\subseteq
\mathcal{M}^{YM}(2,1),
$$
the moduli space of stable solutions to SDE~(\ref{eq:YM}), where $\mathcal{G}^{\mathbb{C}}$ represents the complex gauge group, which acts by conjugation on $\beta_{ps}(2,1)$ and $\beta_{s}(2,1)$.
\begin{Rmk}
Since $\GCD(2,1) = 1$, then $\beta_{ps}(2,1) = \beta_{s}^(2,1)$ and so
$$
\mathcal{M}^{YM}(r,d) = \mathcal{M}^{YM}_{s}(r,d).
$$
\end{Rmk}
\noindent Using definition~\ref{hbdef}, and the notion of stability~\ref{stablehb}, Hitchin~\cite{hit2} presents an alternative algebro-geometric construction of the moduli space of polystable Higgs bundles:
$$
\mathcal{M}^{H}(2,1) = \{(E,\varphi)|\ E \word{polystable} \} / \mathcal{G}^{\mathbb{C}}
$$
and the subspace
$$
\mathcal{M}^{H}_{s}(2,1) =
\{(E,\varphi)|\ E \word{stable} \} / \mathcal{G}^{\mathbb{C}}
\subseteq
\mathcal{M}^{H}(2,1),
$$
of stable Higgs bundles.
\begin{Rmk}
Again, $\GCD(2,1) = 1$ implies $\mathcal{M}^{H}(2,1) = \mathcal{M}^{H}_{s}(2,1)$.
\end{Rmk}
\noindent Finally, Hitchin~\cite{hit2} concludes:
\begin{Th}\textnormal{\cite{hit2}}
There is a homeomorphism of topological spaces
$$
\mathcal{M}^{H}(2,1)
\cong
\mathcal{M}^{YM}(2,1).
\QEDA
$$
\end{Th}
\noindent Because of the last homeomorphism, from now on it will be enough to denote
$
\mathcal{M}(2,1)
=
\mathcal{M}^{H}(2,1)
\cong
\mathcal{M}^{YM}(2,1)
$
for brief. Hitchin~\cite{hit2} computes the real dimension of the moduli space of stable rank two pairs $(E,\varphi)$:
\begin{Th}[{\cite[Th.~5.8.]{hit2}}]\lambdabel{hit5.8.}
Let $X=\Sigma_g$ be a compact Riemann surface of genus $g > 1$. The moduli space $\mathcal{M}(2,1)$ of all stable pairs $(E,\varphi)$, where $E\to X$
is a rank two holomorphic vector bundle of degree one, and $\varphi$ is a trace free holomorphic section of $\End(E)\otimes K$, is a smooth real
manifold of dimension
\[
\dim_{\mathbb{R}}\big(\mathcal{M}(2,1)\big) = 12(g - 1).
\]
\end{Th}
\begin{Cor}
The space $\mathcal{M}(2,1)$ is a quasi--projective variety of complex dimension
$$
\dim_{\mathbb{C}}\big(\mathcal{M}(2,1)\big) = 3(2g - 2).
$$
\end{Cor}
\noindent Nitsure~\cite{nit} constructs the moduli space of Higgs bundles of general rank $r$ and degree $d$ using Geometric Invariant Theory (GIT), and computes its dimension:
\begin{Th}[{\cite{nit}}]
The space $\mathcal{M}(r,d)$ is a quasi--projective variety of complex dimension
$$
\dim_{\mathbb{C}}\big(\mathcal{M}(r,d)\big) = (r^2-1)(2g - 2).
$$
\end{Th}
\begin{Rmk}
Note that the result of Nitsure~\cite{nit} coincides with the result of Hitchin~\cite{hit2} for rank two Higgs bundles.
\end{Rmk}
\noindent Simpson~\cite{sim2} calls the pair $(E,\varphi)$ as {\em Higgs bundle}. His work contributes generalizing Higgs bundles to higher dimensions and proving an analogous proposition to Theorem~\ref{hit5.8.} for general rank, with the same notion of stability in mind, considering the moduli space of Higgs bundles as the quotient
$$
\mathcal{M}^{H}(r,d) = \{(E,\varphi)|\ E \word{polystable} \} / \mathcal{G}^{\mathbb{C}}
$$
and the subspace
$$
\mathcal{M}^{H}_{s}(r,d) =
\{(E,\varphi)|\ E \word{stable} \} / \mathcal{G}^{\mathbb{C}}
\subseteq
\mathcal{M}^{H}(r,d),
$$
of stable Higgs bundles.
\begin{Rmk}
Once again, $\GCD(r,d) = 1$ implies $\mathcal{M}^{H}(r,d) = \mathcal{M}^{H}_{s}(r,d)$. See
Simpson~\cite{sim2} for details.
\end{Rmk}
\begin{Th}[{\cite[Prop.~1.5]{sim2}}]
There is a homeomorphism of topological spaces
$$
\mathcal{M}^{H}(r,d)
\cong
\mathcal{M}^{YM}(r,d).
\QEDA
$$
\end{Th}
}
\noindent Also for general rank, we denote
$
\mathcal{M}(r,d)
=
\mathcal{M}^{H}(r,d)
\cong
\mathcal{M}^{YM}(r,d)
$
for brief, or even $\mathcal{M} = \mathcal{M}(r,d)$ when the rank $r$ and the degree $d$ are clear. Recall that we are considering the co-prime case $\GCD(r,d) = 1$,
in order for $\mathcal{M} = \mathcal{M}(r,d)$ to be a smooth variety. An important feature of $\mathcal{M}(r,d)$ is that it carries an action
of~$\mathbb{C}^*$: $z \cdot (E, \varphi) = (E, z \cdot \varphi)$. According to Hitchin~\cite{hit2}, $(\mathcal{M},I,\varOmega)$ is a K\"ahler manifold, where $I$ is
its complex structure and $\varOmega$ its corresponding
K\"ahler form. Furthermore, $\mathbb{C}^*$ acts on $\mathcal{M}$ biholomorphically with respect to the complex structure $I$ by the aforementioned action,
where the K\"ahler form $\varOmega$ is invariant under the induced action $e^{i\theta} \cdot (E, \varphi) = (E, e^{i\theta} \cdot \varphi)$ of the
circle $\mathbb{S}^1 \subseteq \mathbb{C}^*$. Besides, this circle action is Hamiltonian, with proper moment map $f \colon \mathcal{M} \to \mathbb{R}$ defined by:
\begin{equation}
f(E, \varphi) = \frac{1}{2\pi} \|\varphi\|_{L^2}^2
= \frac{i}{2\pi} \int_X \tr(\varphi \varphi^*)
\lambdabel{momentum_map_intro}
\end{equation}
where $\varphi^*$ is the adjoint of $\varphi$ with respect to the hermitian metric on~$E$, and $f$ has finitely many critical values.
\noindent There is another important fact mentioned by Hitchin~\cite{hit2} (see the original version in the work of Frankel~\cite{fra}, and its application to Higgs bundles
in~\cite{hit2}): the critical points of $f$ are exactly the fixed points of the circle action on~$\mathcal{M}$.
\noindent If $(E, \varphi) = (E, e^{i\theta}\varphi)$ and $\varphi = 0$, then the critical value is $c_0 = 0$. The corresponding critical submanifold is
$F_0 = f^{-1}(c_{0}) = f^{-1}(0) = \mathcal{N}$, the moduli space of stable bundles (see Hitchin~\cite{hit2}, Simpson~\cite{sim2}, or Bradlow,
Garc\'ia-Prada, Gothen~\cite{bgg2} for details). On the other hand, when $\varphi \neq 0$, there is a type of
algebraic structure for Higgs bundles introduced by Simpson~\cite{sim1, sim2}: a \emph{variation of Hodge structure}, or simply a \emph{VHS},
for a Higgs bundle $(E, \varphi)$ is a decomposition:
\begin{equation}
E = \bigoplus_{j=1}^n E_j \word{such that}
\varphi \colon E_j \to E_{j+1} \otimes K \textmd{ for } 1 \leqslant j \leqslant n - 1.
\lambdabel{VHS}
\end{equation}
\noindent It has been proved by Simpson~\cite{sim1} that the fixed points of the circle action on $\mathcal{M}(r,d)$, and so, the critical points of $f$, are
these variations of the Hodge structure VHS, where the critical values $c_\lambda = f(E,\varphi)$ will depend on the degrees $d_j$ of the
components $E_j \subseteq E$, and $\lambdambda$ denotes the index of the critical point for the Morse-Bott function $f$. By Morse theory, we can stratify $\mathcal{M}$ in such a way that there is a non-minimal critical submanifold
$F_\lambda = f^{-1}(c_\lambda)$ for each nonzero critical value $0 \neq c_\lambda = f(E,\varphi)$ where $(E,\varphi)$ represents a fixed point
of the circle action, or equivalently, a VHS. We then say that $(E,\varphi)$ is a $\big(\rk(E_1),\dots,\rk(E_n)\big)$-VHS.
\noindent The Morse indexes of the critical submanifolds of the moduli space of stable Higgs bundles $\mathcal{M}(r,d)$ for general rank $r$ were
calculated by Bradlow~et~al.~\cite{bgg2}:
\begin{Prop}[{\cite[Prop.~3.10.]{bgg2}}]
Let $(E,\varphi)$ be a stable Higgs bundle which corresponds to a critical point of $f$. Then the Morse index of the corresponding critical submanifold $(E,\varphi)\in F_{\lambdambda}$ is
\[
{index}(F_{\lambda}) =
2 \sum_{k > 0} \dim \Big(\mathbb{H}^1\big(C_{k}^{\bullet}(E,\varphi) \big) \Big)
\]
where
\[
\dim \Big(\mathbb{H}^1\big(C_{k}^{\bullet}(E,\varphi) \big) \Big)
=
-\chi\big(C_{k}^{\bullet}(E,\varphi) \big)
\]
and $C_{k}^{\bullet}(E,\varphi)$ is the deformation complex of the pair $(E,\varphi)$. \QEDA
\end{Prop}
\begin{Prop}[{\cite[Prop.~3.12.(2)]{bgg2}}]
For $\mathcal{M}(r,d)$
\[
{index}(F_{\lambda}) \geqslant
(r - 1)(2g - 2)
\]
for every non-minimal critical submanifold $F_{\lambdambda} \subseteq \mathcal{M}(r,d)$.\QEDA
\end{Prop}
\begin{Prop}[{\cite[Prop.~3.14.]{bgg2}}]
Let $F_{0} \subseteq \mathcal{M}(r,d)$ be the set of local minima. Then
\[
F_{0} =
\left\{
(E,\varphi) \in \mathcal{M}(r,d) |\ \varphi = 0
\right\}.
\]
Hence, $F_{0}$ coincides with $\mathcal{N}(r,d)$, the moduli space of semistable bundles of rank $r$ and degree $d$, which equals the subvariety $\mathcal{N}^{s}(r,d) \subseteq \mathcal{N}(r,d)$ corresponding to stable bundles if $\GCD(r,d) = 1$. \QEDA
\end{Prop}
\subsection{k-Higgs Bundles}
\lambdabel{ssec:1.3}
\begin{Def}
Fix a point $p \in X$, and let $L_p = \mathcal{O}_X(p)$ be the associated line bundle to the divisor $p \in \Sym^1(X) = X$. A
\emph{$k$-Higgs bundle} (or \emph{Higgs bundle with poles of order~$k$}) is a pair $(E,\varphi^k)$ where:
\[
E \xrightarrow{\ \varphi^k\ } E \otimes K \otimes L_p^{ \otimes k} = E \otimes K(k\cdot p)
\]
and where the morphism $\varphi^k \in H^0\big(X, \End(E) \otimes K(k\cdot p)\big)$
is what we call a \emph{Higgs field with poles of order~$k$}. The moduli space of $k$-Higgs bundles of rank~$r$ and degree~$d$
is denoted by $\mathcal{M}^k(r,d)$. For simplicity, we will suppose that $\GCD(r,d) = 1$, and so, $\mathcal{M}^k(r,d)$ will be smooth.
\end{Def}
\begin{Rmk}
So far, everything we have said for Higgs bundles and the moduli space $\mathcal{M}(r,d)$ also hold for $k$-Higgs bundles and the moduli spaces $\mathcal{M}^k(r,d)$.
\end{Rmk}
\noindent There is a new tool for $k$-Higgs bundles: an embedding of the form
\begin{equation}
i_k \colon \mathcal{M}^k(r,d) \to \mathcal{M}^{k+1}(r,d)
\colon [(E,\varphi^k)] \longmapsto [(E,\varphi^k \otimes s_p)]
\lambdabel{eq:twist-embed}
\end{equation}
where $0 \neq s_p \in H^0(X, L_p)$ is a nonzero fixed section of~$L_p$.
\noindent When the rank is $r = 2$ or $r = 3$, the map $i_k$ induces embeddings of the form
\[
F^k_{\lambdambda}
\xrightarrow{\quad i_k \quad}
F^{k+1}_{\lambdambda}\quad \forall \lambdambda,
\]
for non-minimal\footnote{For stable pairs in $F^k_ 0 = \mathcal{N}_k$, the embeddings are trivial. Cf.~Hausel~\cite[Ch.~3.~Sec.~3.4.]{hau}.} critical
submanifolds $F^k_{\lambda}$, where $\lambda$ is the Morse index of the submanifold.
\noindent For $\mathcal{M}^k(2,1)$, when $r = 2$, the Morse index is $\lambdambda = 2(g + 2d_1 - 2) + k$, which depends just on the parameter
$d_1 \in ]\frac{1}{2},g - \frac{1}{2} + \frac{k}{2}[\cap \mathbb{Z}$ since $d = \deg(E) = 1$ (co-prime case $\GCD(r,d) = 1$), $g \geqslant 2$ is fixed,
and $k$ is the order of the pole.
\noindent Hence, we may index the $(1,1)$-critical submanifolds as $F^{k}_{d_1}$, and the embeddings are well defined:
\[
\begin{xy}
(0,0)*+{F^{k}_{d_1}\cong \Sym^{\bar{d_1}+k}(X)}="a";
(65,0)*+{\Sym^{\bar{d_1}+k+1}(X)\cong F^{k+1}_{d_1}}="b";
{\ar@{->} "a";"b"};
(5,-10)*+{i_k\colon D}="c";
(60,-10)*+{D + p}="d";
{\ar@{|->} "c";"d"};
\end{xy}
\]
where $D\in \Sym^{\bar{d_1}+k}(X)$ is a divisor and $\bar{d_1} = 2g - 2d_1 - 1$ for simplicity. The reader may see
Bento~\cite{ben}, Hausel~\cite{hau}, Hausel~and~Thaddeus~\cite{hath1, hath2} or Hitchin~\cite{hit2} for details.
\noindent For $\mathcal{M}^k(3,d)$, when $r = 3$, we have three types of critical submanifolds. For $(1,2)$-critical submanifolds $F^k_{\lambdambda}$,
the Morse index is given by $\lambdambda = 2(3d_1 - d + 2g - 2 + k)$
where once again $d_1 = \deg(E_1)$ is the degree of the maximal destabilizing line bundle $E_1\subseteq E$, and so, we are in a very similar situation
than before. Without lost of generality, we may pick the index $d_1$ for the $(1,2)$-critical submanifolds, and the embeddings become
\[
\begin{xy}
(0,0)*+{i_k\colon F^{k}_{d_1}}="a";
(65,0)*+{F^{k+1}_{d_1}}="b";
{\ar@{->} "a";"b"};
(0,-10)*+{(E,\varphi^k) =
\Big(
E_1\oplus E_2,
\left(
\begin{array}{c c}
0 & 0\\
\phi^k & 0
\end{array}
\right)
\Big)
}="c";
(80,-10)*+{(E,\varphi^k\otimes s_p) =
\Big(
E_1\oplus E_2,
\left(
\begin{array}{c c}
0 & 0\\
\phi^k \otimes s_p & 0
\end{array}
\right)
\Big)
}="d";
{\ar@{|->} "c";"d"};
\end{xy}
\]
where $\phi^k\colon E_1 \to E_2\otimes K(kp)$ and $\frac{d}{3} < d_1 < \frac{d}{3} + g - 1 + \frac{k}{2}$ as we shall see below
(see Bento~\cite{ben}, Gothen~\cite{got} or Z-R~\cite{z-r} for interval details). Moreover
\[
(\phi^k \otimes s_p)(E_1) \subseteq
\phi^k(E_1) \otimes L_p \subseteq
E_2 \otimes K \otimes L_p^{\otimes k} \otimes L_p =
E_2 \otimes K \otimes L_p^{\otimes k+1}
\]
and therefore
\[
i_k(F_{d_1}^{k}) \subseteq F_{d_1}^{k+1}.
\]
\begin{Lem}[{\cite[Lema~2.3.1.]{ben}}]\lambdabel{bento2.3.1.}
Let $(E,\varphi^k)\in F_{d_1}^{k}$ be a $k$-Higgs bundle of the form
\[
(E,\varphi^k) =
\Big(
E_1\oplus E_2,
\left(
\begin{array}{c c}
0 & 0\\
\phi^k & 0
\end{array}
\right)
\Big).
\]
Hence, $(E,\varphi^k)$ is stable if and only if the holomorphic triple $T = (E_2\otimes K(k\cdot p),E_1,\phi^k)$ is $\sigma_{H}$-stable where
$\sigma_H = \sigma_H(k) = \deg\big(K(k\cdot p)\big) = 2g - 2 + k$.
\end{Lem}
\begin{proof}
The pair $(E,\varphi^k)$ is stable if and only if the holomorphic chain
\[
\mathcal{C}\colon \mathcal{E}_1 = E_1 \to \mathcal{E}_2 = E_2\otimes K(k\cdot p)
\]
is $\alphapha = \big(\sigma_H(k),0\big)$ stable, which means that any proper subchain $\mathcal{C}'\colon \mathcal{E}_1'\to \mathcal{E}_2'$ has $\alphapha$-slope
$\mu_{\alphapha}(\mathcal{C}') < \mu_{\alphapha}(\mathcal{C})$; considering a subbundle $\mathcal{E}_1'\subseteq \mathcal{E}_1 = E_1$ with degree $\deg(\mathcal{E}_1') = d'_1$
and a subbundle $\mathcal{E}_2' \subseteq \mathcal{E}_2$ with degree $\deg(\mathcal{E}_2') = d'_2$, then $\mathcal{E}_2'\otimes \big(K(k\cdot p) \big)^{*}\subseteq E_2$ is a
subbundle with degree $\deg\big(\mathcal{E}_2'\otimes K(k\cdot p)^{*}\big) = d_2' - r_2'(2g - 2 + k)$, and we have
\[
(E,\varphi^k) \word{stable}
\Longleftrightarrow
\frac{d_1' + d_2' - r_2'(2g - 2 + k)}{r_1' + r_2'} < \frac{d_1 + d_2}{3}
\]
where $r_j' = \rk(\mathcal{E}_j')$.
\noindent On the other hand, suppose
$(E,\varphi^k) =
\Big(
E_1\oplus E_2,
\left(
\begin{array}{c c}
0 & 0\\
\phi^k & 0
\end{array}
\right)
\Big)
$
is stable. The holomorphic triple $T = \big(E_2\otimes K(k\cdot p), E_1, \phi^k\big)$ is $\sigma$-stable if and only if any proper subtriple
$T' = \big(E'_2\otimes K(k\cdot p), E'_1, (\phi^k)'\big)$ satisfies
$
\mu_{\sigma}(T') < \mu_{\sigma}(T)
\Leftrightarrow
$
\[
\frac{\deg(E_1') + \deg\big(E_2'\otimes K(k\cdot p)\big) + \rk(E_1') \cdot \sigma}{\rk(E_1') + \rk(E_2')} <
\]
\[
\frac{\deg(E_1) + \deg\big(E_2\otimes K(k\cdot p)\big) + \rk(E_1)\cdot \sigma}{\rk(E_1) + \rk(E_2)}
\Leftrightarrow
\]
\[
\frac{d_1' + d_2' + r_2'(2g - 2 + k) + r_1' \sigma}{r_1' + r_2'} < \frac{d_1 + d_2 + 2(2g - 2 + k) + \sigma}{1 + 2}
\]
\[
\Leftrightarrow
\frac{d_1' + d_2' + r_2'\sigma_H(k) + r_1' \sigma}{r_1' + r_2'} < \frac{d_1 + d_2 + 2\sigma_H(k) + \sigma}{3}.
\]
Since $(E,\varphi^k)$ is stable, it is enough to take
\[
\frac{r_2'\cdot \sigma_H(k) + r_1'\cdot \sigma}{r_1' + r_2'} = \frac{2\cdot \sigma_H(k) + \sigma}{3}
\Leftrightarrow
r_2' \sigma_H(k) - 2 r_1' \sigma_H(k) = r_2' \sigma - 2 r_1' \sigma
\]
\[
\Leftrightarrow
(r_2' - 2r_1') \sigma_H(k) = (r_2' - 2r_1') \sigma
\Leftrightarrow
\sigma_H(k) = \sigma
\]
and so, the triple $T = (E_2\otimes K(k\cdot p),E_1,\phi^k)$ is $\sigma_H(k)$-stable.
\end{proof}
\begin{Rmk}
Note that, at the last part of the proof above, the equality $r_2' = 2r_1'$ does not hold, because of the stability of $(E,\varphi^k)$.
\end{Rmk}
\begin{Prop}[{\cite[Proposi\c{c}\~ao~2.3.2.]{ben}}]\lambdabel{bento2.3.2.}
For each $d_1 \in \big]\frac{d}{3},\frac{d}{3}+\frac{\sigma_{H}(k)}{2}\big[\cap \mathbb{Z}$ there is a $(1,2)$ critical submanifold of $\mathcal{M}^k(3,d)$ of the form
\[
F_{d_1}^{k} =
\big\{
(E,\varphi^k) =
\Big(
E_1\oplus E_2,
\left(
\begin{array}{c c}
0 & 0\\
\phi^k & 0
\end{array}
\right)
\Big)\colon
d_1 = \deg(E_1),\
\rk(E_1) = 1,\
\rk(E_2) = 2
\big\}.
\]
Furthermore, there is an isomorphism
\[
F_{d_1}^{k} \cong \mathcal{N}_{\sigma_H(k)}(2,1,d - d_1 + 2\sigma_H(k),d_1)
\]
with the moduli space of $\sigma_{H}(k)$-stable triples of this type.
\end{Prop}
\begin{proof}
The isomorphism is given by:
\[
\begin{xy}
(0,0)*+{F^{k}_{d_1}}="a";
(65,0)*+{\mathcal{N}_{\sigma_H(k)}(2,1,d - d_1 + 2\sigma_H(k),d_1)}="b";
{\ar@{->} "a";"b"};
(0,-10)*+{(E,\varphi^k) =
(
E_1\oplus E_2,
\left(
\begin{array}{c c}
0 & 0\\
\phi^k & 0
\end{array}
\right)
)
}="c";
(80,-10)*+{(E_2 \otimes K(k\cdot p), E_1,\phi^k)}="d";
{\ar@{|->} "c";"d"};
\end{xy}
\]
where $\sigma_{H}(k) = 2g - 2 + k$ as above.
\noindent In general, for the critical values $\sigma_{c}$, we know that the interval is $\sigma_{m} \leqslant \sigma_{c} \leqslant \sigma_{M}$
where
\[
\sigma_{m} = \mu_2 - \mu_1 =
\frac{\deg\big(E_2\otimes K(k\cdot p)\big)}{r_2} - \frac{\deg(E_1)}{r_1} =
\frac{d - d_1 + 2\sigma_H(k)}{2} - d_1
\]
and
\[
\sigma_{M} = \Big(1 + \frac{r_2 + r_1}{|r_2 - r_1|} \Big)(\mu_2 - \mu_1) =
4 \sigma_{m} = 2\big(d - 3d_1 +2\sigma_H(k)\big)
\]
(see \cite{bgg1}). So, in particular we have
\[
\sigma_{H}(k) = 2g - 2 + k >
\sigma_{m} = \frac{d - d_1 + 2\sigma_{H}(k)}{2} - d_1
\Longleftrightarrow
d_1 > \frac{d}{3}.
\]
On the other hand, we have
\[
\sigma_{H}(k) < \sigma_{M} =
2\big(d - 3d_1 +2\sigma_{H}(k)\big)
\Longleftrightarrow
d_1 < \frac{d}{3} + \frac{\sigma_H(k)}{2}.
\]
Therefore,
\[
d_1\in
\Big]
\frac{d}{3},
\frac{d}{3} + \frac{\sigma_H(k)}{2}
\Big[\cap \mathbb{Z}.
\]
\end{proof}
\begin{Rmk}
In general, for the critical values $\sigma_{c}$, the interval $[\sigma_{m}, \sigma_{M}]$ is closed. Nevertheless, for our particular case of
interest $\sigma_m < \sigma_H(k) < \sigma_M$, so the interval will be open.
\end{Rmk}
\noindent For $(2,1)$-critical submanifolds $F^k_{\lambda} = F^k_{d_2}$, the Morse index is given by $\lambda = 2(3d_2 - 2d + 2g - 2 + k)$;
here $d_2 = \deg(E_2)$ is the degree of the maximal destabilizing bundle $E_2\subseteq E$ of rank two this time, and so, we are in
a very similar situation than before:
\[
\begin{xy}
(0,0)*+{i_k\colon F^{k}_{d_2}}="a";
(65,0)*+{F^{k+1}_{d_2}}="b";
{\ar@{->} "a";"b"};
(0,-10)*+{(E,\varphi^k) =
\Big(
E_2\oplus E_1,
\left(
\begin{array}{c c}
0 & 0\\
\phi^k & 0
\end{array}
\right)
\Big)
}="c";
(80,-10)*+{(E,\varphi^k\otimes s_p) =
\Big(
E_2\oplus E_1,
\left(
\begin{array}{c c}
0 & 0\\
\phi^k \otimes s_p & 0
\end{array}
\right)
\Big)
}="d";
{\ar@{|->} "c";"d"};
\end{xy}
\]
with $\phi^k\colon E_2 \to E_1\otimes K(kp)$ and $\frac{2d}{3} < d_2 < \frac{2d}{3} + g - 1 + \frac{k}{2}$ instead (see Bento~\cite{ben},
Gothen~\cite{got} or Z-R~\cite{z-r}). Furthermore,
\[
(\phi^k \otimes s_p)(E_2) \subseteq
\phi^k(E_2) \otimes L_p \subseteq
E_1 \otimes K \otimes L_p^{\otimes k} \otimes L_p =
E_1 \otimes K \otimes L_p^{\otimes k+1}
\]
and hence
\[
i_k(F_{d_2}^{k}) \subseteq F_{d_2}^{k+1}.
\]
\begin{Lem}[{\cite[Lema~2.3.5.]{ben}}]\lambdabel{bento2.3.5.}
Let $(E,\varphi^k)\in F_{d_2}^{k}$ be a $k$-Higgs bundle of the form
\[
(E,\varphi^k) =
\Big(
E_2\oplus E_1,
\left(
\begin{array}{c c}
0 & 0\\
\phi^k & 0
\end{array}
\right)
\Big).
\]
Hence, $(E,\varphi^k)$ is stable if and only if the holomorphic triple $T = (E_1\otimes K(k\cdot p),E_2,\phi^k)$ is $\sigma_{H}$-stable where
$\sigma_H = \sigma_H(k) = \deg\big(K(k\cdot p)\big) = 2g - 2 + k$.
\end{Lem}
\begin{proof}
The proof is very similar to that presented for the $(1,2)$-case in Lemma~\ref{bento2.3.1.}.
\end{proof}
\begin{Prop}[{\cite[Proposi\c{c}\~ao~2.3.6.]{ben}}]\lambdabel{bento2.3.6.}
For each $d_2 \in \big]\frac{2d}{3},\frac{2d}{3}+\frac{\sigma_H(k)}{2}\big[\cap \mathbb{Z}$ there is a $(2,1)$ critical submanifold of $\mathcal{M}^k(3,d)$
of the form
\[
F_{d_2}^{k} =
\big\{
(E,\varphi^k) =
\Big(
E_2\oplus E_1,
\left(
\begin{array}{c c}
0 & 0\\
\phi^k & 0
\end{array}
\right)
\Big)\colon
d_2 = \deg(E_2),\
\rk(E_2) = 2,\
\rk(E_1) = 1
\big\}.
\]
Furthermore, there is an isomorphism
\[
F_{d_2}^{k} \cong \mathcal{N}_{\sigma_H(k)}(1,2,d - d_2 + \sigma_H(k),d_2)
\]
with the moduli space of $\sigma_{H}(k)$-stable triples.
\end{Prop}
\begin{proof}
In this case, the isomorphism is given by:
\[
\begin{xy}
(0,0)*+{F^{k}_{d_2}}="a";
(65,0)*+{\mathcal{N}_{\sigma_H(k)}(1,2,d - d_2 + \sigma_H(k),d_2)}="b";
{\ar@{->} "a";"b"};
(0,-10)*+{(E,\varphi^k) =
(
E_2\oplus E_1,
\left(
\begin{array}{c c}
0 & 0\\
\phi^k & 0
\end{array}
\right)
)
}="c";
(80,-10)*+{(E_1 \otimes K(k\cdot p), E_2,\phi^k)}="d";
{\ar@{|->} "c";"d"}.
\end{xy}
\]
The rest of the proof is very similar to the $(1,2)$-case presented in Proposition~\ref{bento2.3.2.}.
\end{proof}
\noindent Finally, we consider the $(1,1,1)$-critical submanifolds of the form
\[
F^k_{\lambdambda} =
F_{d_1 d_2 d_3}^{k} =
\Bigg\{
\Big(
L_1\oplus L_2\oplus L_3,
\left(
\begin{array}{c c c}
0 & 0 & 0\\
\phi^k_{21} & 0 & 0\\
0 & \phi^k_{32} & 0
\end{array}
\right)
\Big)
\Bigg\}\subseteq
\mathcal{M}^{k}(3,d),
\]
where $L_j \subseteq E$ is a line bundle for $j = \{1,2,3\}$, we denote $d_j = \deg(L_j)$ and so, the degree of $E\to X$ could be write as
$\deg(E) = d = d_1 + d_2 + d_3$. Using the fact that $d_3 = d - d_1 - d_2$ and considering auxiliar bundles
$M_{j} = L_j^{*}\otimes L_{j+1} \otimes K(k\cdot p) \to X$ of degree $m_j = \deg(M_j) = d_{j+1} - d_{j} + \sigma_H(k)$, we may write, for simplicity
$\varphi_{j} \in H^{0}(M_j)$ where $\varphi_{1}^k=\phi_{21}^k$ and $\varphi_{2}^k=\phi_{32}^k$, and hence
$M_j = \mathcal{O}(D_j)$ where $D_j = \divv(\varphi_j)$.
\noindent Note that $\varphi_j \neq 0 \Rightarrow m_j \geqslant 0$. Furthermore
\[
d_3 = d - d_1 - d_2 \Longleftrightarrow d_3 = \frac{d + 2m_2 + m_1 - 3\sigma_H(k)}{3}
\]
and hence $d + m_1 + 2m_2 = 0\mod3$.
\noindent Using all the above notation, we can re-write the $(1,1,1)$-critical submanifolds as
\[
F^k_{\lambdambda} =
F_{m_1 m_2}^{k} =
\Bigg\{
\Big(
L_1\oplus L_2\oplus L_3,
\left(
\begin{array}{c c c}
0 & 0 & 0\\
\varphi^k_{1} & 0 & 0\\
0 & \varphi^k_{2} & 0
\end{array}
\right)
\Big)
\Bigg\}\subseteq
\mathcal{M}^{k}(3,d),
\]
and conclude that
\begin{Prop}[{\cite[Proposi\c{c}\~ao~2.3.9.]{ben}}]\lambdabel{bento2.3.9.}
For each pair $(m_1,m_2)\in \varOmegaega$, there is a $(1,1,1)$-critical submanifold $F^{k}_{m_1 m_2}\subseteq \mathcal{M}^k(3,d)$, where
\[
\varOmegaega =
\Bigg\{
(x,y)\in \mathbb{N}^{*}\times \mathbb{N}^{*}\colon
\begin{array}{c}
d + x + 2y = 0\mod3\\
2x + y < 3\sigma_H(k)\\
x + 2y < 3\sigma_H(k)
\end{array}
\Bigg\}.
\]
\end{Prop}
\begin{proof}
The stability conditions in this case are
\[
\mu(L_2 \oplus L_3) < \mu(E)
\Longleftrightarrow
\frac{d_2 + d_3}{2} < \frac{d}{3}
\word{and}
\mu(L_3) < \mu(E)
\Longleftrightarrow
d_3 < \frac{d}{3}.
\]
In terms of $m_j \geqslant 0$ we get
\[
d_3 < \frac{d}{3} \Longleftrightarrow
2m_2 + m_1 < 3\sigma_H(k)
\word{and}
\frac{d_2 + d_3}{2} < \frac{d}{3}
\Longleftrightarrow
2m_1 + m_2 < 3\sigma_H(k)
\]
\end{proof}
\begin{Rmk}
For the particular case of $(1,1,1)$-critical submanifolds, note that
\[
(\varphi_j^k \otimes s_p)(L_j) \subseteq
(\varphi_j^k)(L_j) \otimes L_p =
L_{j+1} \otimes K \otimes L_p^{\otimes k} \otimes L_p =
L_{j+1} \otimes K \otimes L_p^{\otimes k+1}
\]
and hence
\[
i_k(F_{m_1 m_2}^{k}) \subseteq F_{m_1 m_2}^{k+1}.
\]
\end{Rmk}
\begin{Th}\lambdabel{(1,1,1)-VHS--SymP-Iso}
There is an isomorphism
\[
F_{m_1 m_2}^{k} \cong
\Sym^{\bar{m}_1+k}(X) \times \Sym^{\bar{m}_2+k}(X) \times \mathcal{J}^{d_3}(X)
\]
for each pair $(m_1,m_2)\in \varOmegaega$, where
$J^{d_3}(X)$ is the Jacobian of $X$, the moduli space of stable line bundles of degree $d_3$,
and $\bar{m}_j = m_j - k = d_{j+1} - d_j + 2g - 2$.
\end{Th}
\begin{proof}
It is enough to take
\[
\begin{xy}
(0,0)*+{F^{k}_{m_1 m_2}}="a";
(65,0)*+{\Sym^{\bar{m}_1+k}(X) \times \Sym^{\bar{m}_2+k}(X) \times \mathcal{J}^{d_3}(X)}="b";
{\ar@{->} "a";"b"};
(0,-15)*+{(E,\varphi^k) =
(
L_1\oplus L_2\oplus L_3,
\left(
\begin{array}{c c c}
0 & 0 & 0\\
\varphi^k_{1} & 0 & 0\\
0 & \varphi^k_{2} & 0
\end{array}
\right)
)
}="c";
(80,-15)*+{(\divv(\varphi^k_1),\divv(\varphi^k_2),L_3)}="d";
{\ar@{|->} "c";"d"}.
\end{xy}
\]
\end{proof}
\noindent In this case, the Morse index for $(1,1,1)$-critical submanifolds $F^k_{\lambdambda} = F^k_{m_1m_2}$ is given by
$\lambdambda = 2\big(4(2g - 2)- m _1 - m_2 + 3k\big)$. The reader may consult Gothen~\cite{got} or Bento~\cite{ben}
for details.
\begin{Rmk}
Lemma~\ref{bento2.3.1.}, Proposition~\ref{bento2.3.2.}, Lemma~\ref{bento2.3.5.}, Proposition~\ref{bento2.3.6.},
and Proposition~\ref{bento2.3.9.} are presented by Bento~\cite{ben} for the general case of rank three Hitchin
pairs. Here, we presented them for the particular case of rank three $k$-Higgs bundles.
\end{Rmk}
\noindent From the embeddings
\[
F^k_{\lambda}
\xrightarrow{\quad i_k \quad}
F^{k+1}_{\lambda}\quad \forall \lambda,
\]
above mentioned, we get induced isomorphisms in cohomology:
\[
H^{j}(F_{\lambda}^{k+1},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(F_{\lambda}^{k},\mathbb{Z})
\]
for all $\lambda$, for certain values of $j$ in terms of $k$. Our goal is to find the range of~$j$ for which these isomorphisms hold.
\noindent The embeddings restricted to $(1,1)$-critical submanifolds in the rank two case, were studied by Hausel~\cite{hau} and presented by
Hausel and Thaddeus~\cite{hath2}. Here, we focus on rank three.
\noindent If we restrict the embeddings to critical manifolds of type $(1,2)$:
\begin{equation}
\begin{array}{r c l}
F^k_{d_1} & \xrightarrow{\quad i_k \quad} & F^{k+1}_{d_1}\\
& & \\
\biggl(E_1 \oplus E_2,
\left(
\begin{array}{c c}
0 & 0\\
\varphi^{k}_{21} & 0
\end{array}
\right)
\biggl) &
\longmapsto &
\biggl(E_1 \oplus E_2,
\left(
\begin{array}{c c}
0 & 0\\
\varphi^{k}_{21} \otimes s_p & 0
\end{array}
\right)
\biggl)
\end{array}\lambdabel{eq:(1,2)-VHS-embedding}
\end{equation}
then, the isomorphisms
\[
\begin{array}{r c l}
F^k_{d_1} & \xrightarrow{\quad \cong \quad} & \mathcal{N}_{\sigma_H(k)}(2,1,\tilde{d}_1,\tilde{d}_2)\\
& & \\
\biggl(E_1 \oplus E_2,
\left(
\begin{array}{c c}
0 & 0\\
\varphi^{k}_{21} & 0
\end{array}
\right)
\biggl) &
\longmapsto &
(V_1,V_2,\varphi)
\end{array}
\]
between $(1,2)$ critical submanifolds and moduli spaces of triples, where we denote
by $V_1 = E_2 \otimes K(kp)$,
by $V_2 = E_1$,
by $\varphi = \varphi^{k}_{21}$
and $\sigma_H(k) = \deg(K(kp)) = 2g - 2 + k$,
induce another embeddings:
\[
i_k:
\mathcal{N}_{\sigma_H(k)}(2,1,\tilde{d}_1,\tilde{d}_2)
\to
\mathcal{N}_{\sigma_H(k+1)}(2,1,\tilde{d}_1+2,\tilde{d}_2)
\]
\[
(V_1,V_2,\varphi)
\mapsto
(V_1 \otimes L_p,V_2,\varphi \otimes s_p)
\]
where $\tilde{d}_1 = \deg(V_1) = d_2 + 2\sigma_H(k)$
and $\tilde{d}_2 = \deg(V_2) = d_1$, and so,
induce embeddings on the flips:
\[
i_k: \mathcal{N}_{\sigma_H^{-}(k)}(2,1,\tilde{d}_1,\tilde{d}_2)
\hookrightarrow
\mathcal{N}_{\sigma_H^{-}(k+1)}(2,1,\tilde{d}_1+2,\tilde{d}_2)
\]
and
\[
i_k: \mathcal{N}_{\sigma_H^{+}(k)}(2,1,\tilde{d}_1,\tilde{d}_2)
\hookrightarrow
\mathcal{N}_{\sigma_H^{+}(k+1)}(2,1,\tilde{d}_1+2,\tilde{d}_2).
\]
\noindent The situation with critical manifolds of type $(2,1)$
\begin{equation}
\begin{array}{r c l}
F^k_{d_2} & \xrightarrow{\quad i_k \quad} & F^{k+1}_{d_2}\\
& & \\
\biggl(E_2 \oplus E_1,
\left(
\begin{array}{c c}
0 & 0\\
\varphi^{k}_{21} & 0
\end{array}
\right)
\biggl) &
\longmapsto &
\biggl(E_2 \oplus E_1,
\left(
\begin{array}{c c}
0 & 0\\
\varphi^{k}_{21} \otimes s_p & 0
\end{array}
\right)
\biggl)
\end{array}\lambdabel{eq:(2,1)-VHS-embedding}
\end{equation}
is very similar to the $(1,2)$ situation, using now isomorphisms
\[
\begin{array}{r c l}
F^k_{d_2} & \xrightarrow{\quad \cong \quad} & \mathcal{N}_{\sigma_H(k)}(1,2,\tilde{d}_1,\tilde{d}_2)\\
& & \\
\biggl(E_2 \oplus E_1,
\left(
\begin{array}{c c}
0 & 0\\
\varphi^{k}_{21} & 0
\end{array}
\right)
\biggl) &
\longmapsto &
(V_1,V_2,\varphi)
\end{array}
\]
where
by $V_1 = E_1 \otimes K(k\cdot p)$,
by $V_2 = E_2$,
by $\varphi = \varphi^{k}_{21}$
and $\sigma_H(k) = \deg\big(K(k\cdot p)\big) = 2g - 2 + k$,
and the induced embeddings become:
\[
i_k:
\mathcal{N}_{\sigma_H(k)}(1,2,\tilde{d}_1,\tilde{d}_2)
\to
\mathcal{N}_{\sigma_H(k+1)}(1,2,\tilde{d}_1 + 1,\tilde{d}_2)
\]
\[
(V_1,V_2,\varphi)
\mapsto
(V_1 \otimes L_p,V_2,\varphi \otimes s_p)
\]
where $\tilde{d}_1 = \deg(V_1) = d_1 + \sigma_H(k)$
and $\tilde{d}_2 = \deg(V_2) = d_2$. Hence, for the flips on $\sigma_H(k)$, the induced embeddings become:
\[
i_k\colon \mathcal{N}_{\sigma_H^{-}(k)}(1,2,\tilde{d}_1,\tilde{d}_2)
\hookrightarrow
\mathcal{N}_{\sigma_H^{-}(k+1)}(1,2,\tilde{d}_1 + 1,\tilde{d}_2)
\]
and
\[
i_k: \mathcal{N}_{\sigma_H^{+}(k)}(1,2,\tilde{d}_1,\tilde{d}_2)
\hookrightarrow
\mathcal{N}_{\sigma_H^{+}(k+1)}(1,2,\tilde{d}_1 + 1,\tilde{d}_2).
\]
\noindent Critical submanifolds of type $(1,1,1)$ are different from the other two. The embeddings
\begin{equation}
\begin{array}{r c l}
F^k_{m_1 m_2} & \xrightarrow{\quad i_k \quad} & F^{k+1}_{m_1 m_2}\\
& & \\
(
L_1\oplus L_2\oplus L_3,
\left(
\begin{array}{c c c}
0 & 0 & 0\\
\varphi_{1} & 0 & 0\\
0 & \varphi_{2} & 0
\end{array}
\right)
) &
\longmapsto &
(
L_1\oplus L_2\oplus L_3,
\left(
\begin{array}{c c c}
0 & 0 & 0\\
\varphi_{1}\otimes s_p & 0 & 0\\
0 & \varphi_{2}\otimes s_p & 0
\end{array}
\right)
)
\end{array}\lambdabel{eq:(1,1,1)-VHS-embedding}
\end{equation}
together with the isomorphisms
\[
F_{m_1 m_2}^{k} \cong
\Sym^{\bar{m}_1+k}(X) \times \Sym^{\bar{m}_2+k}(X) \times \mathcal{J}^{d_3}(X)
\]
induce embeddings of the form:
{\footnotesize
\[
\begin{array}{r c l}
\Sym^{\bar{m}_1 + k}(X)\times \Sym^{\bar{m}_2 + k}(X)\times \mathcal{J}^{d_3}(X) & \to & \Sym^{\bar{m}_1 + k + 1}(X)\times \Sym^{\bar{m}_2 + k + 1}(X)\times \mathcal{J}^{d_3}(X)\\
\big(\divv(\varphi^k_1),\divv(\varphi^k_2),L_3\big) & \mapsto & \big(\divv(\varphi^k_1 + p),\divv(\varphi^k_2 + p),L_3\big).
\end{array}
\]
}
\section[Triples and Roof Theorem]{Stable Holomorphic Triples and Roof Theorem}
\lambdabel{sec:2}
\subsection[sigma-Stability]{$\sigma$-Stability}
\lambdabel{ssec:2.1}
For $(2,1,\tilde{d}_2,\tilde{d}_1)$-triples and $(1,2,\tilde{d}_1,\tilde{d}_2)$-triples, the embeddings $i_k$ preserve $\sigma$-stability:
\begin{Lem}\lambdabel{[GoZR3]}
A triple $T$ of type $(2,1,\tilde{d}_2,\tilde{d}_1)$ or type $(1,2,\tilde{d}_1,\tilde{d}_2)$ is $\sigma$-stable $\Leftrightarrow i_k(T)$ is $(\sigma + 1)$-stable.
\end{Lem}
\begin{proof}
We will show the result holds for $(2,1,\tilde{d}_2,\tilde{d}_1)$-triples, the proof of $(1,2,\tilde{d}_1,\tilde{d}_2)$-triples is analogous.
\noindent Recall that $T = (V_1,V_2,\varphi)$ is $\sigma$-stable if and only if $\mu_{\sigma}(T') < \mu_{\sigma}(T)$ for any $T'$ proper subtriple of $T$.
\noindent Denote by $S = i_k(T) = (V_1 \otimes L_p,V_2,\varphi \otimes s_p)$. Is easy to check that $\mu_{\sigma + 1}(S) = \mu_{\sigma}(T) + 1$:
\[
\mu_{\sigma+1}(S) =
\frac{\deg_{\sigma + 1}(S)}{\rk(V_1\otimes L_p) \oplus \rk(V_2)} =
\]
\[
\frac{\deg(V_1\otimes L_p) + \deg(V_2) + (\sigma + 1)\rk(V_2)}{1 + 2} =
\]
\[
\frac{\deg(V_1) + \deg(L_p) + \deg(V_2) + \sigma \rk(V_2) + \rk(V_2)}{3} =
\]
\[
\frac{\deg(V_1) + \deg(V_2) + \sigma \rk(V_2)}{3} +
\frac{\deg(L_p) + \rk(V_2)}{3} =
\mu_{\sigma}(T) + 1
\]
since $\deg(L_p) = 1$ and $\rk(V_2) = 2$.
\noindent Any $S'$ proper subtriple of $S$ is of the form $S' = i_k(T')$ for some $T'$ subtriple of $T$, or equivalently:
\[
S' = (V'_1 \otimes L_p,V'_2,\varphi \otimes s_p)
\]
and there are injective sheaf homomorphisms $V'_1 \rightarrow V_1$ and $V'_2 \rightarrow V_2$. This
statement is justified since the following diagram commutes:
\[
\begin{xy}
(0,0)*+{S}="a";
(20,0)*+{S'}="b";
(40,0)*+{T'}="c";
(60,0)*+{T}="d";
(0,-10)*+{V_2}="e";
(20,-10)*+{B}="f";
(40,-10)*+{B}="g";
(60,-10)*+{V_2}="h";
(-10,-30)*+{V_1 \otimes L_p}="i";
(20,-30)*+{A}="j";
(50,-30)*+{A \otimes L_{p}^{*}}="k";
(80,-30)*+{V_1}="l";
{\ar@{->>}^{\supsetneq} "a";"b"};
{\ar@{^{(}->}^{\subsetneq} "c";"d"};
{\ar@{->} "b";"c"};
{\ar@{|->}^{\supsetneq} "e";"f"};
{\ar@{|->}^{\subsetneq} "g";"h"};
{\ar@{|->}^{=} "f";"g"};
{\ar@{|->}^{\supsetneq} "i";"j"};
{\ar@{|->}^{\subsetneq} "k";"l"};
{\ar@{|->} "j";"k"};
{\ar@{->}_{\varphi \otimes s_p} "e";"i"};
{\ar@{->}_{\varphi \otimes s_p} "f";"j"};
{\ar@{->}^{(\varphi \otimes s_p) \otimes s_p^{-1}} "g";"k"};
{\ar@{->}^{\varphi} "h";"l"};
\end{xy}
\]
where the first floor of the diagram contains the first entries of the triples, second floor contains the second entries,
the diagonal arrows are the coresponding morphisms, and we consider the subbundles $A = V'_1 \otimes L_p,\ B = V'_2$ and
$T' = (V'_1,V'_2,\varphi) \subseteq (V_1,V_2,\varphi) = T$. So, there is a one--to--one correspondence between the proper
subtriples $S' \subseteq S$ and the proper subtriples $T' \subseteq T$. We can easily see that $\mu_{\sigma+1}(S') = \mu_{\sigma}(T')+1$ and hence:
\[
\mu_{\sigma+1}(S') < \mu_{\sigma+1}(S)
\Leftrightarrow
\mu_{\sigma}(T')+1 < \mu_{\sigma}(T)+1
\Leftrightarrow
\mu_{\sigma}(T') < \mu_{\sigma}(T).
\]
Therefore, $T$ is $\sigma$-stable $\Leftrightarrow S = i_k(T)$ is $(\sigma + 1)$-stable.
\end{proof}
\begin{Cor}\lambdabel{(2,1)-embedding}
The embedding
$$
i_k: \mathcal{N}_{\sigma(k)}(2,1,\tilde{d}_1,\tilde{d}_2)
\to
\mathcal{N}_{\sigma(k+1)}(2,1,\tilde{d}_1+2,\tilde{d}_2)
$$
is well defined for any $\sigma(k)$ such that $
\sigma_m < \sigma(k) < \sigma_M$. In particular,
the embedding $i_k$ restricted to $F_{d_1}^{k}$
(see (\ref{eq:(1,2)-VHS-embedding})) is well defined and
we have a commutative diagram of the form:
\begin{align*}
\begin{xy}
(0,0)*+{(\tilde{E}_1,\tilde{E}_2,\varphi_{21}^{k})}="a0";
(0,10)*+{\mathcal{N}_{\sigma_H(k)}}="a1";
(0,40)*+{F_{d_1}^{k}}="a2";
(0,50)*+{(E_1 \oplus E_2, \varphi^k)}="a3";
(50,0)*+{(\tilde{E}_1,\tilde{E}_2,\varphi_{21}^{k}\otimes s_p),}="b0";
(50,10)*+{\mathcal{N}_{\sigma_H(k+1)}}="b1";
(50,40)*+{F_{d_1}^{k+1}}="b2";
(50,50)*+{(E_1 \oplus E_2, \varphi^k \otimes s_p)}="b3";
{\ar@{<->}^{\cong} "a1";"a2" **\dir{--}};
{\ar@{<->}_{\cong} "b1";"b2" **\dir{--}};
{\ar@{->}_{i_k} "a1";"b1" **\dir{--}};
{\ar@{->}^{i_k} "a2";"b2" **\dir{--}};
{\ar@{|->} "a0";"b0" **\dir{--}};
{\ar@{|->} "a3";"b3" **\dir{--}};
\end{xy}
\end{align*}
where $\tilde{E}_1 = E_2 \otimes K(k\cdot p)$, $\tilde{E}_2 = E_1$, and
$\varphi_{21}^{k}: E_1 \to E_2 \otimes K(k\cdot p)$. \QEDA
\end{Cor}
\begin{Cor}\lambdabel{(1,2)-embedding}
The embedding
\[
i_k: \mathcal{N}_{\sigma(k)}(1,2,\tilde{d}_2,\tilde{d}_1)
\to
\mathcal{N}_{\sigma(k+1)}(1,2,\tilde{d}_2 + 1,\tilde{d}_1)
\]
is well defined for any $\sigma(k)$ such that $
\sigma_m < \sigma(k) < \sigma_M$. In particular,
the embedding $i_k$ restricted to $F_{d_2}^{k}$
(see (\ref{eq:(2,1)-VHS-embedding})) is well defined and
we have a commutative diagram of the form:
\begin{align*}
\begin{xy}
(0,0)*+{(\tilde{E}_1,\tilde{E}_2,\varphi_{21}^{k})}="a0";
(0,10)*+{\mathcal{N}_{\sigma_H(k)}}="a1";
(0,40)*+{F_{d_2}^{k}}="a2";
(0,50)*+{(E_2 \oplus E_1, \varphi^k)}="a3";
(50,0)*+{(\tilde{E}_1,\tilde{E}_2,\varphi_{21}^{k}\otimes s_p),}="b0";
(50,10)*+{\mathcal{N}_{\sigma_H(k+1)}}="b1";
(50,40)*+{F_{d_2}^{k+1}}="b2";
(50,50)*+{(E_2 \oplus E_1, \varphi^k \otimes s_p)}="b3";
{\ar@{<->}^{\cong} "a1";"a2" **\dir{--}};
{\ar@{<->}_{\cong} "b1";"b2" **\dir{--}};
{\ar@{->}_{i_k} "a1";"b1" **\dir{--}};
{\ar@{->}^{i_k} "a2";"b2" **\dir{--}};
{\ar@{|->} "a0";"b0" **\dir{--}};
{\ar@{|->} "a3";"b3" **\dir{--}};
\end{xy}
\end{align*}
where $\tilde{E}_1 = E_1 \otimes K(k\cdot p)$, $\tilde{E}_2 = E_2$, and
$\varphi_{21}^{k}: E_2 \to E_1 \otimes K(k\cdot p)$. \QEDA
\end{Cor}
\noindent These results allow us to conclude that there is an interesting and important correspondence between the $\sigma$-stability
values of moduli spaces of holomorphic triples:
\begin{align*}
\begin{xy}
(0,35)*+{\sigma_m(k)}="a1";
(40,35)*+{\sigma_H(k)}="b1";
(80,35)*+{\sigma_M(k)}="e1";
(0,30)*+{|}="a2";
(40,30)*+{*}="b2";
(65,30)*+{|}="c2";
(70,30)*+{\cdot}="d2";
(75,30)*+{\cdot}="e2";
(80,30)*+{|}="f2";
(100,30)*+{}="g2";
(0,5)*+{\cdot}="a3";
(5,5)*+{|}="b3";
(40,5)*+{\cdot}="c3";
(45,5)*+{*}="d3";
(70,5)*+{|}="e3";
(75,5)*+{\cdot}="f3";
(80,5)*+{\cdot}="g3";
(85,5)*+{|}="h3";
(90,5)*+{\cdot}="i3";
(95,5)*+{\cdot}="j3";
(100,5)*+{|}="k3";
(120,5)*+{}="l3";
(5,0)*+{\sigma_m(k+1)}="a4";
(45,0)*+{\sigma_H(k+1)}="b4";
(85,0)*+{\sigma'}="c4";
(100,0)*+{\sigma_M(k+1)}="e4";
{\ar@{-} "a2";"b2"};
{\ar@{-} "b2";"c2"};
{\ar@{-} "c2";"d2"};
{\ar@{-} "d2";"e2"};
{\ar@{-} "e2";"f2"};
{\ar@{->} "f2";"g2"};
{\ar@{->}^{i_k} "a2";"b3"};
{\ar@{->}^{i_k} "b2";"d3"};
{\ar@{->}^{i_k} "f2";"h3"};
{\ar@{->}^{i_k} "c2";"e3"};
{\ar@{-->} "f2";"k3"};
{\ar@{-} "a3";"b3"};
{\ar@{-} "b3";"c3"};
{\ar@{-} "c3";"d3"};
{\ar@{-} "d3";"e3"};
{\ar@{-} "e3";"f3"};
{\ar@{-} "f3";"g3"};
{\ar@{-} "g3";"h3"};
{\ar@{-} "h3";"i3"};
{\ar@{-} "i3";"j3"};
{\ar@{-} "j3";"k3"};
{\ar@{->} "k3";"l3"};
\end{xy}
\end{align*}
where $\sigma_m(k) = \tilde{\mu}_1 - \tilde{\mu}_2$,
$\sigma_M(k) = 4(\tilde{\mu}_1 - \tilde{\mu}_2)$,
$\sigma_H(k) = \deg(K(kp)) = 2g - 2 + k$,
and the correspondence gives us
$\sigma_m(k+1) = \sigma_m(k) + 1$,
$\sigma' = \sigma_M(k) + 1$,
$\sigma_M(k+1) = \sigma_M(k)+3$, and
$\sigma_H(k+1) = \sigma_H(k) + 1$. First and second floor are representations of the real line, where the second floor of the diagram corresponds to the interval $[\sigma_m,\sigma_M]$ for poles of order $k$, and the first floor for poles of order $(k + 1)$ after the embedding $i_k$.
\begin{Rmk}
An interesting fact from the correspondence represented by last diagram is that
\[
i_k\colon
\sigma_M(k) \mapsto \sigma' < \sigma_M(k+1).
\]
\end{Rmk}
\subsection[Blow-up and The Roof Theorem]{Blow-up and The Roof Theorem}
\lambdabel{ssec:2.2}
At this point, a brief description of the flip loci of the moduli spaces of holomorphic triples will be useful to understand the coming results and notation. The reader may see Mu\~noz~et~al.~\cite{mov} for details.
\noindent Fixing the type $(r_1,r_2,d_1,d_2)$ for the moduli spaces of holomorphic triples, we shall describe the differences between $\mathcal{N}_{\sigma_1}(r_1,r_2,d_1,d_2)$ and $\mathcal{N}_{\sigma_2}(r_1,r_2,d_1,d_2)$ where $\sigma_1$ and $\sigma_2$ are separated by a critical value $\sigma_c \in [\sigma_m,\sigma_M]$. Here, we suppose $r_1 \neq r_2$, since for our purposes, the case $r_1 = 2$ and $r_2 = 1$ will be particularly useful.
\noindent Let
\[
\sigma_{c}^{+} = \sigma_c + \varepsilon
\word{and}
\sigma_{c}^{-} = \sigma_c - \varepsilon
\]
where $\varepsilon > 0$ is small enough so that $\sigma_c \in \ ]\sigma_{c}^{-},\sigma_{c}^{+}[$ is the only critical value in that subinterval.
\begin{Def}\lambdabel{fliploci}
Define the {\em flip loci} as the sets
\[
S_{\sigma_{c}^{+}} =
\left\{
T\in \mathcal{N}_{\sigma_{c}^{+}} |\ T \word{is} \sigma_{c}^{-}-\textmd{unstable}
\right\}
\subseteq \mathcal{N}_{\sigma_{c}^{+}}(r_1,r_2,d_1,d_2)
\]
and
\[
S_{\sigma_{c}^{-}} =
\left\{
T\in \mathcal{N}_{\sigma_{c}^{-}} |\ T \word{is} \sigma_{c}^{+}-\textmd{unstable}
\right\}
\subseteq \mathcal{N}_{\sigma_{c}^{-}}(r_1,r_2,d_1,d_2),
\]
and denote
$
S_{\sigma_{c}^{\pm}}^{s} =
S_{\sigma_{c}^{\pm}} \cap
\mathcal{N}_{\sigma_{c}^{\pm}}^{s}(r_1,r_2,d_1,d_2)
$
as the {\em stable part of the flip loci}, where $\sigma_{c}^{\pm}$ means any of both $\sigma_{c}^{+}$ or $\sigma_{c}^{-}$.
\end{Def}
\noindent Denote $\tilde{\mathcal{N}}_{\sigma_c^{-}(k)}$ as the blow-up of $\mathcal{N}_{\sigma_c^{-}(k)} = \mathcal{N}_{\sigma_c^{-}(k)}(2,1,\tilde{d}_1,\tilde{d}_2)$
along the flip locus $S_{\sigma_c^{-}(k)}$, which is isomorphic to $\tilde{\mathcal{N}}_{\sigma_c^{+}(k)}$, the blow-up of
$\mathcal{N}_{\sigma_c^{+}(k)} = \mathcal{N}_{\sigma_c^{+}(k)}(2,1,\tilde{d}_1,\tilde{d}_2)$ along the flip locus $S_{\sigma_c^{+}(k)}$. From now on, we
will denote just $\tilde{\mathcal{N}}_{\sigma_c(k)}$ whenever no confusion is likely to arise.
\begin{Th}\lambdabel{RoofTheorem}
For each $k$, there exists an embedding at the blow-up level
\[
\tilde{i_k}: \tilde{\mathcal{N}}_{\sigma_c(k)} \hookrightarrow \tilde{\mathcal{N}}_{\sigma_c(k+1)}
\]
such that the following diagram commutes:
\begin{align*}
\begin{xy}
(0,-20)*+{\mathcal{N}_{\sigma_c^{-}(k+1)}}="a";
(20,0)*+{\tilde{\mathcal{N}}_{\sigma_c(k+1)}}="b";
(10,-50)*+{\tilde{\mathcal{N}}_{\sigma_c(k)}}="c";
(40,-20)*+{\mathcal{N}_{\sigma_c^{+}(k+1)}}="d";
(-10,-70)*+{\mathcal{N}_{\sigma_c^{-}(k)}}="e";
(30,-70)*+{\mathcal{N}_{\sigma_c^{+}(k)}}="f";
{\ar@{-->}^(.45){\exists \tilde{i_k}} "c";"b" **\dir{--}};
{\ar^(.45){} "b";"a" **\dir{-}};
{\ar^(.45){i_k} "e";"a" **\dir{-}};
{\ar^(.45){} "c";"e" **\dir{-}};
{\ar_(.45){i_k} "f";"d" **\dir{-}};
{\ar^(.45){} "c";"f" **\dir{-}};
{\ar^(.45){} "b";"d" **\dir{-}};
\end{xy}
\end{align*}
where $\tilde{\mathcal{N}}_{\sigma_c(k)}$ is the blow-up of
$\mathcal{N}_{\sigma_c^{-}(k)} = \mathcal{N}_{\sigma_c^{-}(k)}(2,1,\tilde{d}_1,\tilde{d}_2)$ along the flip locus $S_{\sigma_c^{-}(k)}$
and, at the same time, represents the blow-up of
$\mathcal{N}_{\sigma_c^{+}(k)} = \mathcal{N}_{\sigma_c^{+}(k)}(2,1,\tilde{d}_1,\tilde{d}_2)$ along the flip locus $S_{\sigma_c^{+}(k)}$.
\end{Th}
\begin{proof}
Recall that $T$ is $\sigma$-stable if and only if $i_k(T)$ is $(\sigma+1)$-stable. Furthermore, by~\cite{mov}, note that any triple
\[
T = (V_1,V_2,\varphi)\in S_{\sigma_c^{+}(k)} \subseteq \mathcal{N}_{\sigma_c^{+}(k)}(2,1,\tilde{d}_1,\tilde{d}_2)
\]
is a non-trivial extension of a subtriple $T' \subseteq T$ of the form $T' = (V'_1,V'_2,\varphi') = (M,0,\varphi')$ by a quotient triple of
the form $T'' = (V''_1,V''_2,\varphi'') = (L,V_2,\varphi'')$, where $M$ is a line bundle of degree $\deg(M) = d_M$
and $L$ is a line bundle of degree $\deg(L) = d_L = \tilde{d}_1 - d_M$. Besides, also by ~\cite{mov}, the non-trivial critical
values $\sigma_c \neq \sigma_m$ for $\sigma_m < \sigma_c < \sigma_M$ are of the form $\sigma_c = 3d_M - \tilde{d}_1 - \tilde{d}_2$. Then,
we can visualize the embedding $i_k: T \hookrightarrow i_k(T)$ as follows:
\begin{align*}
\begin{xy}
(0,0)*+{0}="a1";
(20,0)*+{T'}="b1";
(40,0)*+{T}="c1";
(60,0)*+{T''}="d1";
(80,0)*+{0}="e1";
(0,-10)*+{0}="a2";
(20,-10)*+{0}="b2";
(40,-10)*+{V_2}="c2";
(60,-10)*+{V_2}="d2";
(80,-10)*+{0}="e2";
(-20,-30)*+{0}="a3";
(0,-30)*+{M}="b3";
(40,-30)*+{V_1}="c3";
(80,-30)*+{L}="d3";
(100,-30)*+{0}="e3";
(0,-50)*+{0}="a4";
(20,-50)*+{0}="b4";
(40,-50)*+{V_2}="c4";
(60,-50)*+{V_2}="d4";
(80,-50)*+{0}="e4";
(-20,-70)*+{0}="a5";
(0,-70)*+{M\otimes L_p}="b5";
(40,-70)*+{V_1\otimes L_p}="c5";
(80,-70)*+{L\otimes L_p}="d5";
(100,-70)*+{0}="e5";
{\ar@{->} "a1";"b1"};
{\ar@{->} "b1";"c1"};
{\ar@{->} "c1";"d1"};
{\ar@{->} "d1";"e1"};
{\ar@{->} "a2";"b2"};
{\ar@{->} "b2";"c2"};
{\ar@{->}^{=} "c2";"d2"};
{\ar@{->} "d2";"e2"};
{\ar@{->}_{\varphi'} "b2";"b3"};
{\ar@{->}_{\varphi} "c2";"c3"};
{\ar@{->}_{\varphi''} "d2";"d3"};
{\ar@{->} "a3";"b3"};
{\ar@{->} "b3";"c3"};
{\ar@{->} "c3";"d3"};
{\ar@{->} "d3";"e3"};
{\ar@{|->}^{i_k} "c3";"c4"};
{\ar@{->} "a4";"b4"};
{\ar@{->} "b4";"c4"};
{\ar@{->}^{=} "c4";"d4"};
{\ar@{->} "d4";"e4"};
{\ar@{->}_{\varphi' \otimes s_p} "b4";"b5"};
{\ar@{->}_{\varphi \otimes s_p} "c4";"c5"};
{\ar@{->}_{\varphi'' \otimes s_p} "d4";"d5"};
{\ar@{->} "a5";"b5"};
{\ar@{->} "b5";"c5"};
{\ar@{->} "c5";"d5"};
{\ar@{->} "d5";"e5"};
\end{xy}
\end{align*}
where $\deg(V_1 \otimes L_p) = \tilde{d}_1 + 2$ and $\deg(M \otimes L_p) = d_M + 1$, and so $L \otimes L_p$
verifies that $\deg(L \otimes L_p) = \deg(V_1 \otimes L_p) - \deg(M \otimes L_p)$:
$$
\deg(L \otimes L_p) =
d_L + 1 =
\tilde{d}_1 - d_M + 1 =
$$
$$
(\tilde{d}_1 + 2) - (d_M + 1) =
\deg(V_1 \otimes L_p) - \deg(M \otimes L_p).
$$
Hence, $\sigma_c(k+1)$ verifies that $\sigma_c(k+1) = \sigma_c(k) + 1$:
$$\sigma_c(k+1) = 3\deg(M \otimes L_p) - \deg(V_1 \otimes L_p) - \deg(V_2) = $$
$$3d_M + 3 - \tilde{d}_1 - 2 - \tilde{d}_2 = (3d_M - \tilde{d}_1 - \tilde{d}_2) + 1 = \sigma_c(k) + 1$$
and where $i_k(T') = (M \otimes L_p,0,\varphi' \otimes s_p)$ is the maximal $\sigma_c^{+}(k+1)$-destabilizing subtriple of $i_k(T)$,
verifying exactness at the image level of the embedding.
\noindent Similarly, also by~\cite{mov}, any triple $T \in S_{\sigma_c^{-}(k)} \subseteq \mathcal{N}_{\sigma_c^{-}(k)}(2,1,\tilde{d}_1,\tilde{d}_2)$
is a non-trivial extension of a subtriple $T' \subseteq T$ of the form $T' = (V'_1,V'_2,\varphi') = (L,V_2,\varphi')$
by a quotient triple of the form $T'' = (V''_1,V''_2,\varphi'') = (M,0,\varphi'')$, where $M$ is a line bundle
of degree $\deg(M) = d_M$ and $L$ is a line bundle of degree $\deg(L) = d_L = \tilde{d}_1 - d_M$.
Then, the embedding
$$
i_k: T \hookrightarrow i_k(T)
$$
looks like:
\begin{align*}
\begin{xy}
(0,0)*+{0}="a1";
(20,0)*+{T'}="b1";
(40,0)*+{T}="c1";
(60,0)*+{T''}="d1";
(80,0)*+{0}="e1";
(0,-10)*+{0}="a2";
(20,-10)*+{V_2}="b2";
(40,-10)*+{V_2}="c2";
(60,-10)*+{0}="d2";
(80,-10)*+{0}="e2";
(-20,-30)*+{0}="a3";
(0,-30)*+{L}="b3";
(40,-30)*+{V_1}="c3";
(80,-30)*+{M}="d3";
(100,-30)*+{0}="e3";
(0,-50)*+{0}="a4";
(20,-50)*+{V_2}="b4";
(40,-50)*+{V_2}="c4";
(60,-50)*+{0}="d4";
(80,-50)*+{0}="e4";
(-20,-70)*+{0}="a5";
(0,-70)*+{L\otimes L_p}="b5";
(40,-70)*+{V_1\otimes L_p}="c5";
(80,-70)*+{M\otimes L_p}="d5";
(100,-70)*+{0}="e5";
{\ar@{->} "a1";"b1"};
{\ar@{->} "b1";"c1"};
{\ar@{->} "c1";"d1"};
{\ar@{->} "d1";"e1"};
{\ar@{->} "a2";"b2"};
{\ar@{->}^{=} "b2";"c2"};
{\ar@{->} "c2";"d2"};
{\ar@{->} "d2";"e2"};
{\ar@{->}_{\varphi'} "b2";"b3"};
{\ar@{->}_{\varphi} "c2";"c3"};
{\ar@{->}_{\varphi''} "d2";"d3"};
{\ar@{->} "a3";"b3"};
{\ar@{->} "b3";"c3"};
{\ar@{->} "c3";"d3"};
{\ar@{->} "d3";"e3"};
{\ar@{|->}^{i_k} "c3";"c4"};
{\ar@{->} "a4";"b4"};
{\ar@{->}^{=} "b4";"c4"};
{\ar@{->} "c4";"d4"};
{\ar@{->} "d4";"e4"};
{\ar@{->}_{\varphi' \otimes s_p} "b4";"b5"};
{\ar@{->}_{\varphi \otimes s_p} "c4";"c5"};
{\ar@{->}_{\varphi'' \otimes s_p} "d4";"d5"};
{\ar@{->} "a5";"b5"};
{\ar@{->} "b5";"c5"};
{\ar@{->} "c5";"d5"};
{\ar@{->} "d5";"e5"};
\end{xy}
\end{align*}
where $i_k(T') = (L,V_2,\varphi')$ is the maximal $\sigma_c^{+}(k+1)$-destabilizing subtriple of $i_k(T)$.
\noindent Hence, $i_k$ restricts to the flip loci $S_{\sigma_c^{+}(k)}$ and $S_{\sigma_c^{-}(k)}$. Recall that, by definition, the blow-up of
$\mathcal{N}_{\sigma_c^{+}(k)}$ along the flip locus $S_{\sigma_c^{+}(k)}$, is the space $\tilde{\mathcal{N}}_{\sigma_c(k)}$ together
with the projection
\[
\pi:\ \tilde{\mathcal{N}}_{\sigma_c(k)} \rightarrow \mathcal{N}_{\sigma_c^{+}(k)}
\]
where $\pi$ restricted to $\mathcal{N}_{\sigma_c^{+}(k)} - S_{\sigma_c^{+}(k)}$ is an isomorphism and the \emph{exceptional divisor}
$\mathcal{E}^{+} = \pi^{-1}(S_{\sigma_c^{+}(k)}) \subseteq \tilde{\mathcal{N}}_{\sigma_c(k)}$ is a fiber bundle over $S_{\sigma_c^{+}(k)}$
with fiber $\mathbb{P}^{n-k-1}$, where $n = \mathrm{dim}(\mathcal{N}_{\sigma_c^{+}(k)})$ and $k = \mathrm{dim}(S_{\sigma_c^{+}(k)})$. So,
the embedding can be extended to $\mathcal{E}^{+}$ in a natural way. Same argument remains valid when we consider
$\tilde{\mathcal{N}}_{\sigma_c(k)}$ as the blow-up of $\mathcal{N}_{\sigma_c^{-}(k)}$ along the flip locus $S_{\sigma_c^{-}(k)}$ with
exceptional divisor $\mathcal{E}^{-} = \pi^{-1}(S_{\sigma_c^{-}(k)}) \subseteq \tilde{\mathcal{N}}_{\sigma_c(k)}$. Therefore, the embedding
can be extended to the whole $\tilde{\mathcal{N}}_{\sigma_c(k)}$.
\end{proof}
\noindent Recall that there is an isomorphism
\[
\mathcal{N}_{\sigma}(1,2,d_1,d_2)
\cong
\mathcal{N}_{\sigma}(2,1,-d_2,-d_1)
\]
for all $\sigma$ by Proposition~\ref{dualtriples}. Hence, the following corollary represents the analogous dual Roof-Theorem for the $(1,2)$-case, and also holds:
\begin{Cor}\lambdabel{(1,2)-RoofTheorem}
For each $k$, there exists an embedding at the blow-up level
\[
\tilde{i_k}: \tilde{\mathcal{N}}_{\sigma_c(k)} \hookrightarrow \tilde{\mathcal{N}}_{\sigma_c(k+1)}
\]
such that the following diagram commutes:
\begin{align*}
\begin{xy}
(0,-20)*+{\mathcal{N}_{\sigma_c^{-}(k+1)}}="a";
(20,0)*+{\tilde{\mathcal{N}}_{\sigma_c(k+1)}}="b";
(10,-50)*+{\tilde{\mathcal{N}}_{\sigma_c(k)}}="c";
(40,-20)*+{\mathcal{N}_{\sigma_c^{+}(k+1)}}="d";
(-10,-70)*+{\mathcal{N}_{\sigma_c^{-}(k)}}="e";
(30,-70)*+{\mathcal{N}_{\sigma_c^{+}(k)}}="f";
{\ar@{-->}^(.45){\exists \tilde{i_k}} "c";"b" **\dir{--}};
{\ar^(.45){} "b";"a" **\dir{-}};
{\ar^(.45){i_k} "e";"a" **\dir{-}};
{\ar^(.45){} "c";"e" **\dir{-}};
{\ar_(.45){i_k} "f";"d" **\dir{-}};
{\ar^(.45){} "c";"f" **\dir{-}};
{\ar^(.45){} "b";"d" **\dir{-}};
\end{xy}
\end{align*}
where $\tilde{\mathcal{N}}_{\sigma_c(k)}$ is the blow-up of
$\mathcal{N}_{\sigma_c^{-}(k)} = \mathcal{N}_{\sigma_c^{-}(k)}(1,2,\tilde{d}_2,\tilde{d}_1)$ along the flip locus $S_{\sigma_c^{-}(k)}$
and, at the same time, represents the blow-up of
$\mathcal{N}_{\sigma_c^{+}(k)} = \mathcal{N}_{\sigma_c^{+}(k)}(1,2,\tilde{d}_2,\tilde{d}_1)$ along the flip locus $S_{\sigma_c^{+}(k)}$.
\end{Cor}
\begin{proof}
Follows from Theorem~\ref{RoofTheorem} and Proposition~\ref{dualtriples}.
\end{proof}
\begin{Rmk}
The construction of the blow-up may be found in the book of Griffiths and Harris~\cite{grha}.
\end{Rmk}
\section[Cohomology]{Cohomology}
\lambdabel{sec:3}
We want to show that the embeddings
$
i_k\colon F_{\lambdambda}^{k} \hookrightarrow F_{\lambdambda}^{k+1}
$
induce covariant isomorphisms in cohomology:
\[
H^{j}(F_{\lambdambda}^{k+1},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(F_{\lambdambda}^{k},\mathbb{Z})
\]
for all $\lambdambda$ and certain $j$. To do that, we need to study $F_{d_1}^k$, $F_{d_2}^k$ and $F_{m_1 m_2}^k$ separately. Because of
Proposition \ref{dualtriples}, the cohomology of $F_{d_1}^k$ and $F_{d_2}^k$ are similar, so it will be enough to analyze $F_{d_1}^k$. The
cohomology of $F_{m_1 m_2}^k$ will be completely different.
\noindent We shall start by describing the cohomology of $\Sym^{k}(X) = X^{k}/S_{k}$, the $k$-th symmetric product in subsection \ref{ssec:3.1}, which is related
to the cohomology of the rank three VHS.
\noindent For $(1,2)$-VHS, we will prove that the embeddings
$
i_k\colon F_{d_1}^{k} \hookrightarrow F_{d_1}^{k+1}
$
induce isomorphisms in cohomology:
\[
H^{j}(F_{d_1}^{k+1},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(F_{d_1}^{k},\mathbb{Z})
\]
for certain $j$, or equivalently:
\[
H^{j}(\mathcal{N}_{\sigma_H}^{k+1},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(\mathcal{N}_{\sigma_H}^{k},\mathbb{Z}),
\]
where we denote
$
\mathcal{N}_{\sigma_H}^{k} =
\mathcal{N}_{\sigma_H(k)}(2,1,\tilde{d}_1,\tilde{d}_2)
$. We do that in two steps. First, in subsection \ref{ssec:3.2}, we get that
\[
H^{j}(\mathcal{N}_{\sigma_c}^{k+1},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(\mathcal{N}_{\sigma_c}^{k},\mathbb{Z})
\]
for all critical $\sigma_c = \sigma_c(k)$ such that $\sigma_m(k) < \sigma_c(k) < \sigma_M(k)$, and for
all $j \leqslantslant n(k)$, where the bound $n(k)$ is known. We first analize the
embedding restricted to the flip loci,
$
i_k: S_{\sigma_c^-(k)}
\hookrightarrow
S_{\sigma_c^-(k+1)}
$
and
$
i_k: S_{\sigma_c^+(k)}
\hookrightarrow
S_{\sigma_c^+(k+1)}
$.
For simplicity, we will denote from now on
$
S_{-}^{k} =
S_{\sigma_c^-(k)}
$
and
$
S_{+}^{k} =
S_{\sigma_c^+(k)}
$
whenever no confusion is likely to arise about the critical value.
\noindent In subsection \ref{ssec:3.3}, we stabilize the cohomology of the $(1,2)$-VHS, using useful results from the work of {Bradlow,~Garc\'ia-Prada,~Gothen~\cite{bgg1}}. In subsection \ref{ssec:3.4}, we present the dual results for $(2,1)$-VHS.
\noindent Finally, in subsection \ref{ssec:3.5}, we study the case of the $(1,1,1)$-VHS.
\subsection[Cohomology of Symmetric Products]{Cohomology of Symmetric Products}
\lambdabel{ssec:3.1}
\noindent We begin by recalling some cohomology features of $\Sym^{k}(X) = X^{k}/S_{k}$, the symmetric product with quotient topology, where $X^{k}$ is the
$k$-times cartesian product and $S_{k}$ is the order $k$ symmetric group. Obviously $\Sym^{1}(X) = X$.
\noindent As mentioned before, the $k$-th symmetric product $\Sym^{k}(X)$ is a smooth projective variety of dimension $k\in \mathbb{N}$, that could be
interpretated as the moduli space of degree $k$ effective divisors.
\noindent It is well known that
\[
H^{0}(X,\mathbb{Z}) = \mathbb{Z},\quad
H^{1}(X,\mathbb{Z}) = \mathbb{Z}^{2g},\quad
H^{2}(X,\mathbb{Z}) = \mathbb{Z}.
\]
There is a generator $\beta\in H^{2}(X,\mathbb{Z})$ induced by the orientation of $X$. Moreover, there are $2g$ generators
$\alpha_1,\ \alpha_2,\ \dots,\ \alpha_{2g}\in H^{1}(X,\mathbb{Z})$ such that
\[
\alpha_{i} \cup \alpha_{j} =
-\alpha_{j} \cup \alpha_{i} =
0 \word{if} i - j \neq \pm g \word{for} i,j\in \{1,\ \dots, 2g\}
\]
and
\[
\alpha_{i} \cup \alpha_{i + g} =
-\alpha_{i+g} \cup \alpha_{i} =
\beta \word{for} i\in \{1,\ \dots, g\}
\]
with the usual cup product $\cup$. Hence
\[
\alpha_{i} \cup \beta =
\beta \cup \alpha_{i} = 0
\word{and}
\beta^2 = \beta \cup \beta = 0.
\]
\noindent For the usual cartesian product $X^{k}$, we get that the ring $H^{*}(X^{k},\mathbb{Z}) \cong H^{*}(X,\mathbb{Z})^{\otimes k}$ is generated by
$\{\alpha_{ir}\}_{i=1}^{2g}$ and $\beta_{r}$ with $1\leqslant r \leqslant k$, which are elements of the form
\[
\alpha_{ir} =
1\otimes \dots \otimes 1 \otimes \alpha_{i} \otimes 1 \otimes \dots \otimes 1 \in H^{1}(X^{k},\mathbb{Z})
\]
and
\[
\beta_{r} =
1\otimes \dots \otimes 1 \otimes \beta \otimes 1 \otimes \dots \otimes 1 \in H^{2}(X^{k},\mathbb{Z})
\]
where $\alpha_{i}$ and $\beta$ fill the $r$-th entry of $\alpha_{ir}$ and $\beta_{r}$ respectively, and they are subject to the relations
\[
\alpha_{ir} \cup \alpha_{jr} =
-\alpha_{jr} \cup \alpha_{ir} =
0 \word{if} i - j \neq \pm g \word{for} i,j\in \{1,\ \dots, 2g\}
\]
and
\[
\alpha_{ir} \cup \alpha_{i + g\ r} =
-\alpha_{i + g\ r} \cup \alpha_{ir} =
\beta_r \word{for} i\in \{1,\ \dots, g\}.
\]
Hence
\[
\alpha_{ir} \cup \beta_{r} = \beta_{r} \cup \alpha_{ir} = 0
\word{and}
\beta_{r}^2 = \beta_{r} \cup \beta_{r} = 0.
\]
Besides, each $\beta_{r}$ commutes with every element of $H^{*}(X^{k},\mathbb{Z})$.
\noindent Finally, the symmetric product $\Sym^{k}(X)$ has a cohomology ring $H^{*}(\Sym^{k}(X),\mathbb{Z})$ generated by elements of the form
\[
\zeta_{i} =
\alpha_{i1} + \dots + \alpha_{ik}
= \sum_{r=1}^{k}\alpha_{ir}
\in H^{1}(\Sym^{k}(X),\mathbb{Z})
\word{for} 1\leqslant i \leqslant 2g
\]
and
\[
\eta =
\beta_{1} + \dots + \beta_{k}
= \sum_{r=1}^{k}\beta_{r}
\in H^{2}(\Sym^{k}(X),\mathbb{Z})
\]
where
\[
\zeta_{i} \cup \zeta_{j}
=
-\zeta_{j} \cup \zeta_{i}
\word{and}
\zeta_{i} \cup \eta
=
\eta \cup \zeta_{i}
\]
for any $i$ and $j$. The reader may consult Macdonald~\cite{mac} or Arbarello-Cornalba-Griffiths-Harris~\cite{acgh} for details.
\noindent According to Arbarello~et al.~\cite{acgh}, there is $\bigtriangleup_{k}\in \Sym^{k+1}(X)$
a universal divisor such that
\[
\bigtriangleup_{k}\Big|_{\{D\}\times X} = D
\word{for every divisor}
D\in \Sym^{k}(X).
\]
Therefore, the first Chern class $c_1(\bigtriangleup_{k})\in H^{2}(\Sym^{k+1}(X),\mathbb{Z})$ of this universal divisor is given by
\begin{equation}\lambdabel{c1udiv}
c_1(\bigtriangleup_{k})
= \gamma \otimes k +
\sum_{i=1}^{g}(\zeta_{i} \otimes \alpha_{i+g} - \zeta_{i + g} \otimes \alpha_{i})
+ \eta \otimes 1
\in
H^{2}(\Sym^{k+1}(X),\mathbb{Z})
\end{equation}
where
\[
H^{2}(\Sym^{k+1}(X),\mathbb{Z}) =
\sum_{j=0}^{2}H^{j}(\Sym^{k}(X),\mathbb{Z})\otimes H^{2-j}(X,\mathbb{Z})
\]
and
\[
\gamma =
\sum_{i = 1}^{g}\zeta_{i}\cup \zeta_{i+g}
\in H^{2}(\Sym^{k}(X),\mathbb{Z}).
\]
Macdonald~\cite{mac} compute the Poincar\'e polynomial of $H^{*}(\Sym^{k}(X),\mathbb{Z})$:
\begin{equation}\lambdabel{poincaresymk}
P_{t}\big(\Sym^{k}(X)\big) =
\begin{array}{r}
\word{Coeff}\\
x^k
\end{array}
\left(
\frac{(1 + xt)^{2g}}{(1 - x)(1 - xt^2)}
\right).
\end{equation}
\noindent For $k > 2g - 2$ there is the Abel--Jacobi map $\Sym^k(X) \to \mathcal{J}^{k}$, which is a locally trivial fibration with fibre $\mathbb{P}^{k - g}$,
and gives the Poincar\'e polynomial:
\begin{equation}\lambdabel{poincaresymk2}
P_{t}\big(\Sym^{k}(X)\big) =
\left(
\frac{(1 + t)^{2g}(1 + t^{2(k-g+1)})}{(1 - t^2)}
\right).
\end{equation}
\noindent The reader may see Macdonald~\cite{mac}, Arbarello~et al.~\cite{acgh}, or Hausel~\cite{hau} for details.
\noindent Our embedding $i_k\colon F_{\lambda}^{k}\to F_{\lambda}^{k+1}$ is in fact related to the embedding
\[
\Sym^{k}(X)\to \Sym^{k+1}(X)
\]
\[
D\mapsto D+p
\]
for a fixed point $p\in X$. We will abuse notation and call this last embedding also $i_k$. We get a sequence
\[
X =
\Sym^{1}(X) \subseteq
\Sym^{2}(X) \subseteq
\dots \subseteq
\Sym^{k}(X) \subseteq
\dots
\]
and so, we may consider its direct limit
\[
\Sym^{\infty}(X) =
\lim_{k\to \infty}\Sym^{k}(X),
\]
which is a $\mathbb{P}^{\infty}$-bundle over the Jacobian $\mathcal{J}$, and hence its Poincar\'e polynomial is:
\begin{equation}\lambdabel{poincaresyminfty}
P_{t}\big(\Sym^{\infty}(X)\big) =
\left(
\frac{(1 + t)^{2g}}{(1 - t^2)}
\right).
\end{equation}
The reader may consult Hausel~\cite{hau} for all the details.
\begin{Th}\lambdabel{symmetric-pullback}
The pull-back
\[
i_{k}^{*}\colon
H^{*}(\Sym^{k+1}(X),\mathbb{Z})
\to
H^{*}(\Sym^{k}(X),\mathbb{Z})
\]
induced by the embedding $i_{k}\colon \Sym^{k}(X)\to \Sym^{k+1}(X)$, is surjective.
\end{Th}
\begin{proof}
It is enough to see that the cohomology ring $H^{*}(\Sym^{k}(X),\mathbb{Z})$ is generated by the universal classes $\{\zeta_i\}_{i=1}^{g}$ and $\eta$
mentioned above, and that the universal divisor $\bigtriangleup_{k}$ has first Chern class of the form~\ref{c1udiv}. See Hausel~\cite{hau} for details.
\end{proof}
\begin{Cor}\lambdabel{symmetric-directlimit}
The cohomology ring of the direct limit $\Sym^{\infty}(X)$ is the covariant limit
\[
H^{*}(\Sym^{\infty}(X),\mathbb{Z})
=
\lim_{\infty \leftarrow k}
H^{*}(\Sym^{k}(X),\mathbb{Z})
\]
which is a graded commutative free algebra generated by the classes $\{\zeta_i\}_{i=1}^{g}$ and $\eta$.
\end{Cor}
\begin{proof}
This is a consequence of Theorem~\ref{symmetric-pullback} and the Poincar\'e polynomial~(\ref{poincaresyminfty}) found by Hausel~\cite{hau}.
\end{proof}
\begin{Th}[{\cite[(12.2)]{mac}}]\lambdabel{macdonald-12.2}
There is a cohomology isomorphism
\[
H^{j}(\Sym^{k+1}(X),\mathbb{Z})
\to
H^{j}(\Sym^{k}(X),\mathbb{Z})
\]
for all $j \leqslant k-1$.\QEDA
\end{Th}
\begin{Cor}\lambdabel{symmetric-cohomology-iso}
There is an isomorphism
\[
H^{j}(\Sym^{\infty}(X),\mathbb{Z})
\to
H^{j}(\Sym^{k}(X),\mathbb{Z})
\]
for all $j \leqslant k-1$.
\end{Cor}
\begin{proof}
It follows directly from Theorem~\ref{symmetric-pullback}, Corollary~\ref{symmetric-directlimit} and Theorem~\ref{macdonald-12.2}.
\end{proof}
\subsection[Cohomology of Triples]{Cohomology of Triples}
\lambdabel{ssec:3.2}
A few words about notation. Recall that we are using $\tilde{d}_j = \deg(V_j)$
because of the correspondence
\[
V_1 = E_2 \otimes K(k\cdot p)
\word{and}
V_2 = E_1
\]
through the isomorphism
$
F_{d_1}^k \cong \mathcal{N}_{\sigma_H(k)}(2,1,\tilde{d}_1,\tilde{d}_2)
$
where
\[
\tilde{d}_1 = \deg(V_1) = \deg\big(E_2 \otimes K(k\cdot p)\big) = d_2 + 2\sigma_H(k)
\word{and}
\tilde{d}_2 = \deg(V_2) = \deg(E_1) = d_1.
\]
Similarly, the notation becomes
\[
V_1 = E_1 \otimes K(k\cdot p)
\word{and}
V_2 = E_2
\]
through the isomorphism
$
F_{d_2}^k \cong \mathcal{N}_{\sigma_H(k)}(1,2,\tilde{d}_1,\tilde{d}_2)
$
for the dual cases, and so
\[
\tilde{d}_1 = \deg(V_1) = \deg(E_1 \otimes K(k\cdot p)) = d_1 + \sigma_H(k)
\word{and}
\tilde{d}_2 = \deg(V_2) = \deg(E_2) = d_2.
\]
\begin{Th}\lambdabel{CohomologyNegativeFlipLocus}
There is an isomorphism
\[
i_k^*: H^{j}(S_{-}^{k+1},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(S_{-}^{k},\mathbb{Z})
\]
for all
$
j \leqslantslant \tilde{d}_1 - d_M - \tilde{d}_2 - 1
= d_2 - d_1 + 2\sigma_H(k) - d_M
$,
where $d_j = \deg(E_j)
$,
$\tilde{d}_j = \deg(V_j)$,
$M\to X$ is a line bundle of degree
$d_M=\deg(M)$, and
$\sigma_H(k)= \deg(K(kp)) = 2g-2+k$.
\end{Th}
\begin{proof}
Recall that, according to~\cite[Theorem~4.8.]{mov},
$S_{-}^{k} = \mathbb{P}(\mathcal{V})$ is the
projectivization of a bundle
$\mathcal{V} \to \mathcal{N}'_{\sigma_c} \times \mathcal{N}''_{\sigma_c}$
of rank $\mathrm{rk}(\mathcal{V}) = -\chi(T'',T')$, where
\[
\mathcal{N}'_{\sigma_c} = \mathcal{N}_{\sigma_c}(1,1,\tilde{d}_1-d_M,\tilde{d}_2)
\cong
\mathcal{J}^{\tilde{d}_2} \times \mathrm{Sym}^{\tilde{d}_1-d_M-\tilde{d}_2}(X)
\]
and
\[
\mathcal{N}''_{\sigma_c} =
\mathcal{N}_{\sigma_c}(1,0,d_M,0)
\cong \mathcal{J}^{d_M}(X)
\]
where any triple
$
T = (V_1,V_2,\varphi) \in S_{-}^{k} \subseteq
\mathcal{N}_{\sigma_c^{-}(k)}(2,1,\tilde{d}_1,\tilde{d}_2)
$
is a non-trivial extension of a subtriple
$ T' \subseteq T$ of the form
$
T' = (V'_1,V'_2,\varphi')
= (L,V_2,\varphi')
$
by a quotient triple of the form
$
T'' = (V''_1,V''_2,\varphi'') =
(M,0,\varphi'')$, where $M
$
is a line bundle of degree $\deg(M) = d_M$ and
$L$ is a line bundle of degree $\deg(L) = d_L = \tilde{d}_1 - d_M$.
\noindent Then, the embedding $i_k: T \rightarrow i_k(T)$ restricts to:
\begin{align*}
\begin{xy}
(0,0)*+{\big([V_2'],\mathrm{div}(\varphi')\big)}="a0";
(0,10)*+{\mathcal{J}^{\tilde{d}_2} \times \mathrm{Sym}^{\tilde{d}_1-d_M-\tilde{d}_2}(X)}="a1";
(0,40)*+{\mathcal{N}'_{\sigma_c}}="a2";
(0,50)*+{(V_1',V_2',\varphi')}="a3";
(70,0)*+{\big([V_2'],\mathrm{div}(\varphi'\otimes s_p)\big)}="b0";
(70,10)*+{\mathcal{J}^{\tilde{d}_2} \times \mathrm{Sym}^{\tilde{d}_1-d_M-\tilde{d}_2+1}(X)}="b1";
(70,40)*+{\mathcal{N}'_{\sigma_c+1}}="b2";
(70,50)*+{(V_1'\otimes L_p,V_2',\varphi'\otimes s_p)}="b3";
{\ar@{<->}^{\cong} "a1";"a2" **\dir{--}};
{\ar@{<->}_{\cong} "b1";"b2" **\dir{--}};
{\ar@{->}_{i_k} "a1";"b1" **\dir{--}};
{\ar@{->}^{i_k} "a2";"b2" **\dir{--}};
{\ar@{|->} "a0";"b0" **\dir{--}};
{\ar@{|->} "a3";"b3" **\dir{--}};
\end{xy}
\end{align*}
because $\sigma_c(k+1) = \sigma_c(k) + 1$, and $d_M(k+1) = d_M(k) + 1$,
and because, by the proof of the Roof Theorem \ref{RoofTheorem},
$i_k$ restricts to the flip locus $S_{-}^{k}$.
\noindent Recall that in our case $\sigma_c = \sigma_c(k) > \sigma_m$. Then, for subtriples of the form $T' = (V_1',V_2',\varphi')$ we get that
$\varphi' \neq 0$ and so, they are entirely parametrized by $\big([V_2'], \mathrm{div}(\varphi)\big)$. That is why the map from $\mathcal{N}'_{\sigma_c}$
to $\mathcal{J}^{\tilde{d}_2} \times \mathrm{Sym}^{\tilde{d}_1-d_M-\tilde{d}_2}(X)$ is an isomorphism at the Jacobian.
\noindent Similarly, $i_k$ restricts to:
\begin{align*}
\begin{xy}
(0,0)*+{[V_1'']}="a0";
(0,10)*+{\mathcal{J}^{d_M}}="a1";
(0,40)*+{\mathcal{N}''_{\sigma_c}}="a2";
(0,50)*+{(V_1'',0,0)}="a3";
(50,0)*+{[V_1''\otimes L_p]}="b0";
(50,10)*+{\mathcal{J}^{d_M}}="b1";
(50,40)*+{\mathcal{N}''_{\sigma_c+1}}="b2";
(50,50)*+{(V_1''\otimes L_p,0,0)}="b3";
{\ar@{<->}^{\cong} "a1";"a2" **\dir{--}};
{\ar@{<->}_{\cong} "b1";"b2" **\dir{--}};
{\ar@{->}_{i_k} "a1";"b1" **\dir{--}};
{\ar@{->}^{i_k} "a2";"b2" **\dir{--}};
{\ar@{|->} "a0";"b0" **\dir{--}};
{\ar@{|->} "a3";"b3" **\dir{--}};
\end{xy}
\end{align*}
Here, the quotient triples of the form $T'' = (V_1'',0,0)$ are trivially parametrized by $[V_1'']$ and so, the map from $\mathcal{N}''_{\sigma_c}$ to
$\mathcal{J}^{d_M}$ is also an isomorphism at the Jacobian.
\noindent Hence, by Corollary~\ref{macdonald-12.2},
\[
i_k^*: H^{j}(\mathcal{N}'_{\sigma_c +1},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(\mathcal{N}'_{\sigma_c},\mathbb{Z})\quad
\forall j \leqslantslant \tilde{d}_1 - d_M - \tilde{d}_2 - 1,
\]
and hence
\[
i_k^*: H^{j}(S_{-}^{k+1},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(S_{-}^{k},\mathbb{Z})\quad
\forall j \leqslantslant \tilde{d}_1 - d_M - \tilde{d}_2 - 1.
\qed
\]
\hideqed
\end{proof}
\noindent Similarly, for the flip locus $S_{+}^{k} = S_{\sigma_c^+(k)}$ we have:
\begin{Th}\lambdabel{CohomologyPositiveFlipLocus}
There is an isomorphism
\[
i_{k}^{*}: H^{j}(S_{+}^{k+1},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(S_{+}^{k},\mathbb{Z})
\]
for all
$
j \leqslantslant \tilde{d}_1 - d_M - \tilde{d}_2 - 1
= d_2 - d_1 + 2\sigma_H(k) - d_M
$,
where $d_j = \deg(E_j)
$,
$\tilde{d}_j = \deg(\tilde{E}_j)$,
$M\to X$ is a line bundle of degree
$d_M=\deg(M)$, and
$\sigma_H(k)= \deg(K(kp)) = 2g-2+k$.
\end{Th}
\begin{proof}
Quite similar argument to the one presented above, except
for the detail that this time is the other way around:
according also to~\cite[Theorem~4.8.]{mov},
$
S_{+}^{k} = \mathbb{P}(\mathcal{V})
$
is the projectivization of a bundle
$
\mathcal{V} \to \mathcal{N}'_c \times \mathcal{N}''_c
$
of rank $\mathrm{rk}(\mathcal{V}) = -\chi(T'',T')$,
but this time
$
\mathcal{N}'_c = \mathcal{N}_c(1,0,d_M,0)
\cong \mathcal{J}^{d_M}(X)$,
and
$
\mathcal{N}''_c = \mathcal{N}_c(1,1,\tilde{d}_1-d_M,\tilde{d}_2)
\cong \mathcal{J}^{\tilde{d}_2} \times \mathrm{Sym}^{\tilde{d}_1-d_M-\tilde{d}_2}(X)
$
where any triple
$
T = (V_1,V_2,\varphi) \in S_{+}^{k}
\subseteq
\mathcal{N}_{\sigma_c^{+}(k)}(2,1,\tilde{d}_1,\tilde{d}_2)
$
is a non-trivial extension of a subtriple
$T' \subseteq T$ of the form $T' = (V'_1,V'_2,\varphi') = (M,0,\varphi')$
by a quotient triple of the form
$T'' = (V''_1,V''_2,\varphi'') = (L,V_2,\varphi'')$,
where $M$ is a line bundle of degree $\deg(M) = d_M$ and
$L$ is a line bundle of degree $\deg(L) = d_L = \tilde{d}_1 - d_M$.
\end{proof}
\begin{Th}\lambdabel{CohomologyNegative}
There is an isomorphism
\[
i_k^*: H^{j}(\mathcal{N}_{\sigma^{-}_c(k+1)},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(\mathcal{N}_{\sigma^{-}_c(k)},\mathbb{Z})\quad
\forall j \leqslantslant 2\big(\tilde{d}_1 - 2\tilde{d}_2 - (2g - 2)\big) + 1.
\]
\end{Th}
\noindent Since the behavior of $\mathcal{N}_{\sigma^{-}_c}$,
where $\sigma^{-}_c = \sigma_c - \varepsilon$, is
the same that the one of
$\mathcal{N}_{\sigma^{+}_m}$, where
$\sigma^{+}_m = \sigma_m + \varepsilon$, is enough
to prove the following lemma:
\begin{Lem}
The relative cohomology groups
\[
H^{j}(\mathcal{N}_{\sigma^{+}_m(k+1)},\mathcal{N}_{\sigma^{+}_m(k)};\mathbb{Z}) = 0
\]
are trivial for all
$
j \leqslantslant 2\big(\tilde{d}_1 - 2\tilde{d}_2 - (2g - 2)\big).
$
\end{Lem}
\begin{proof}
Note that $\mathcal{N}_{\sigma^{-}_m(k)} = \emptyset$,
hence $\mathcal{N}_{\sigma^{+}_m(k)} = S_{+}^{k}$, and
according to~\cite[Theorem~4.10.]{mov}, any triple
$
T = (V_1,V_2,\varphi) \in S_{+}^{k} =
\mathcal{N}_{\sigma_m^{+}(k)}(2,1,\tilde{d}_1,\tilde{d}_2)
$
is a non-trivial extension of a subtriple $T' \subseteq T$
of the form $T' = (V'_1,V'_2,\varphi') = (V_1,0,0)$
by a quotient triple of the form
$T'' = (V''_1,V''_2,\varphi'') = (0,V_2,0)$. Hence,
there is a map
\[
\pi: \mathcal{N}_{\sigma^{+}_m} \to
\mathcal{N}(2,\tilde{d}_1) \times \mathcal{J}^{\tilde{d}_2}(X)
\]
\[
(V_1,V_2,\varphi) \mapsto ([V_1],[V_2])
\]
where the inverse image
$
\pi^{-1}\big( \mathcal{N}(2,\tilde{d}_1) \times
\mathcal{J}^{\tilde{d}_2}(X) \big)= \mathbb{P}^{N}
$
has rank
$
N = -\chi(T'',T') = \tilde{d}_1 - 2\tilde{d}_2 - (2g-2)
$, and the proof follows.
\end{proof}
\begin{Th}\lambdabel{CohomologyBlowUp}
There is an isomorphism
\[
\tilde{i_k^*}\colon
H^{j}(\tilde{\mathcal{N}}_{\sigma_c(k+1)},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(\tilde{\mathcal{N}}_{\sigma_c(k)},\mathbb{Z})\quad
\forall j \leqslantslant n(k)
\]
at the blow-up level, where
$
n(k) =
\min(\tilde{d}_1 - d_M - \tilde{d}_2 - 1,\quad 2\big(\tilde{d}_1 - 2\tilde{d}_2 - (2g - 2)\big) + 1)
$.
\end{Th}
\begin{proof}
By the Roof Theorem~\ref{RoofTheorem}, $i_k$
lifts to the blow-up level. We will denote
$
\mathcal{N}^{k}_{-} =
\mathcal{N}_{\sigma^{-}_c(k)}(2,1,\tilde{d}_1,\tilde{d}_2)
$ and
$\tilde{\mathcal{N}}^{k} = \tilde{\mathcal{N}}_{\sigma_c(k)}$ its blow-up along
the flip locus $S_{-}^{k} = S_{\sigma_c^-(k)}$. Recall that, from the construction of the
blow-up, there is a map $\pi_{-}: \tilde{\mathcal{N}}^{k} \to \mathcal{N}^{k}_{-}$
such that
\[
0\to
\pi_{-}^{*}\big( H^{j}(\mathcal{N}^{k}_{-}) \big)\to
H^{j}(\tilde{\mathcal{N}}^{k})\to H^{j}(\mathcal{E}^{k}) / \pi_{-}^{*}\big( H^{j}(\mathcal{S}^{k}_{-}) \big)
\to 0
\]
splits where $\mathcal{E}^{k} = \pi_{-}^{-1}(S_{-}^{k})$ is the so-called exceptional divisor. Hence, the following diagram
\begin{align}
\begin{xy}
(-20,0)*+{0}="a1";
(10,0)*+{\pi_{-}^{*}\big( H^{j}(\mathcal{N}^{k}_{-}) \big)}="b1";
(50,0)*+{H^{j}(\tilde{\mathcal{N}}^{k})}="c1";
(90,0)*+{H^{j}(\mathcal{E}^{k}) / \pi_{-}^{*}\big( H^{j}(\mathcal{S}^{k}_{-}) \big)}="d1";
(120,0)*+{0}="e1";
(-20,-20)*+{0}="a2";
(10,-20)*+{\pi_{-}^{*}\big( H^{j}(\mathcal{N}^{k+1}_{-}) \big)}="b2";
(50,-20)*+{H^{j}(\tilde{\mathcal{N}}^{k+1})}="c2";
(90,-20)*+{H^{j}(\mathcal{E}^{k+1}) / \pi_{-}^{*}\big( H^{j}(\mathcal{S}^{k+1}_{-}) \big)}="d2";
(120,-20)*+{0}="e2";
{\ar@{->} "a1";"b1"};
{\ar@{->} "b1";"c1"};
{\ar@{->} "c1";"d1"};
{\ar@{->} "d1";"e1"};
{\ar@{->} "a2";"b2"};
{\ar@{->} "b2";"c2"};
{\ar@{->} "c2";"d2"};
{\ar@{->} "d2";"e2"};
{\ar@{->}^{\cong} "b2";"b1"};
{\ar@{-->}^{\tilde{{i_k^*}}} "c2";"c1"};
{\ar@{->}_{\cong} "d2";"d1"};
\end{xy}
\end{align}
commutes for all $j \leqslantslant n(k)$, and the theorem follows.
\end{proof}
\begin{Cor}\lambdabel{CohomologyPositive}
There is an isomorphism
\[
i_k^*: H^{j}(\mathcal{N}_{\sigma^{+}_c(k+1)},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(\mathcal{N}_{\sigma^{+}_c(k)},\mathbb{Z})\quad
\forall j \leqslantslant n(k)
\]
where
$
n(k) =
\min(\tilde{d}_1 - d_M - \tilde{d}_2 - 1,\quad 2\big(\tilde{d}_1 - 2\tilde{d}_2 - (2g - 2)\big) + 1)
$ as before.
\end{Cor}
\begin{proof}
Recall that $\tilde{\mathcal{N}}^{k} = \tilde{\mathcal{N}}_{\sigma_c(k)}$ is also the blow-up
of $\mathcal{N}^{k}_{+} = \mathcal{N}_{\sigma^{+}_c(k)}(2,1,\tilde{d}_1,\tilde{d}_2)$ along
the flip locus $S_{+}^{k} = S_{\sigma_c^+(k)}$, so there is a map $\pi_{+}: \tilde{\mathcal{N}}^{k} \to \mathcal{N}^{k}_{+}$
such that
\[
0\to \pi_{+}^{*}\big( H^{j}(\mathcal{N}^{k}_{+}) \big)\to
H^{j}(\tilde{\mathcal{N}}^{k})\to
H^{j}(\mathcal{E}^{k}) / \pi_{+}^{*}\big( H^{j}(\mathcal{S}^{k}_{+}) \big)\to 0
\]
splits:
\[
H^{j}(\tilde{\mathcal{N}}^{k}) =
\pi_{+}^{*}\big( H^{j}(\mathcal{N}^{k}_{+}) \big)
\oplus
H^{j}(\mathcal{E}^{k}) / \pi_{+}^{*}\big( H^{j}(\mathcal{S}^{k}_{+}) \big),
\]
and by Theorem~\ref{CohomologyPositiveFlipLocus} and
Theorem~\ref{CohomologyBlowUp}, the result follows.
\end{proof}
\begin{Cor}\lambdabel{CohomologyCriticalTriples}
There is an isomorphism
\[
i_k^*\colon
H^{j}(\mathcal{N}_{\sigma_c(k+1)},\mathbb{Z}) \xrightarrow{\quad \cong \quad}
H^{j}(\mathcal{N}_{\sigma_c(k)},\mathbb{Z})\quad \forall j \leqslantslant n(k).
\eqno \QEDA
\]
\end{Cor}
\subsection[Cohomology of (1,2)-VHS]{Cohomology of the $(1,2)$-VHS}
\lambdabel{ssec:3.3}
So far, we stabilize the cohomology of $\mathcal{N}_{\sigma_c(k)}$ for any critical $\sigma_c(k)$.
Here and after, $\sigma_L$ respresents the largest critical value in the open interval
$]\sigma_m,\ \sigma_M[$, and $\mathcal{N}_{\sigma_L^+}$ (respectively $\mathcal{N}_{\sigma_L^+}^s$) denotes the moduli space
of $\sigma_L$-polystable (respectively $\sigma_L$-stable) triples for values
$\sigma_L < \sigma < \sigma_M$. The space $\mathcal{N}_{\sigma_L^+}$ is so-called the `large $\sigma$' moduli space
(see \cite{bgg1}).
The following results will
allow us to generalize the stabilization for all
$
\displaystyle
\sigma \in \left]\sigma_m(k),\ \sigma_M(k)\right[
$:
\begin{Th}[{\cite[Th.~7.7.]{bgg1}}]\lambdabel{bgg1_7.7}
Assume that $r_1 > r_2$ and $\displaystyle \frac{d_1}{r_1} > \frac{d_2}{r_2}$.
Then the moduli space $\mathcal{N}_{\sigma_L^+}^s = \mathcal{N}_{\sigma_L^+}^s(r_1,r_2,d_1,d_2)$ is smooth of dimension
\[
(g - 1)(r_1^2 + r_2^2 - r_1 r_2)- r_1 d_2 + r_2 d_1 +1,
\]
and is birationally equivalent to a $\mathbb{P}^{\tilde{n}}$-fibration over
$
\mathcal{N}^s(r_1 - r_2, d_1 - d_2)\times \mathcal{N}^s(r_2, d_2),
$
where $\mathcal{N}^s(r,d)$ is the moduli space of stable bundles of degree $r$ and degree
$d$, and
\[
\tilde{n} = r_2 d_1 - r_1 d_2 +r_1(r_1 - r_2)(g - 1) - 1.
\]
In particular, $\mathcal{N}_{\sigma_L^+}^s(r_1,r_2,d_1,d_2)$ is non-empty and irreducible.
\noindent If $\GCD(r_1 - r_2, d_1 - d_2) = 1$ and $\GCD(r_2, d_2) = 1$, the birational
equivalence is an isomorphism.
\noindent Moreover, in all cases, $\mathcal{N}_{\sigma_L^+} = \mathcal{N}_{\sigma_L^+}(r_1,r_2,d_1,d_2)$ is irreducible and hence,
birationally equivalent to $\mathcal{N}_{\sigma_L^+}^s$.
\QEDA
\end{Th}
\begin{Th}[{\cite[Th.~7.9.]{bgg1}}]\lambdabel{bgg1_7.9}
Let $\sigma$ be any value in the range $\sigma_m < 2g - 2 \leqslant \sigma < \sigma_M$, then
$\mathcal{N}^s_{\sigma}$ is birationally equivalent to $\mathcal{N}_{\sigma_L^+}^s$. In particular it is non-empty
and irreducible.
\QEDA
\end{Th}
\begin{Cor}[{\cite[Cor.~7.10.]{bgg1}}]\lambdabel{bgg1_7.10}
Let $(\mathbf{r},\mathbf{d}) = (r_1,r_2,d_1,d_2)$ be such that
$$
\GCD(r_2, r_1 + r_2, d_1 + d_2) = 1.
$$
If $\sigma$ is a generic value satisfying $\sigma_m < 2g - 2 \leqslant \sigma < \sigma_M$,
then $\mathcal{N}_{\sigma}$ is birationally equivalent to $\mathcal{N}_{\sigma_L^+}$, and in particular it is
irreducible.
\end{Cor}
\begin{proof}
$\mathcal{N}_{\sigma} = \mathcal{N}^s_{\sigma}$ if
$
\GCD(r_2, r_1 + r_2, d_1 + d_2) = 1
$
and $\sigma$ is generic. In particular, we have
$\mathcal{N}_{\sigma_L^+} = \mathcal{N}_{\sigma_L^+}^s$, and the result follows from the last theorem.
The reader may see the full details in \cite{bgg1}.
\end{proof}
\begin{Th}\lambdabel{z-r_01}
There is an isomorphism
\[
i_k^*\colon
H^{j}(\mathcal{N}_{\sigma_H}^{k+1},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(\mathcal{N}_{\sigma_H}^{k},\mathbb{Z})\quad
\forall j \leqslantslant \sigma_H(k) - 2(\mu_1 - \mu) - 1
\]
where
$
\mathcal{N}_{\sigma_H}^{k} =
\mathcal{N}_{\sigma_{H}(k)}(2,1,\tilde{d}_1,\tilde{d}_2)
$,\quad
$
\sigma_{H} =
\sigma_{H}(k)
= 2g - 2 + k
$,
and $\mu_1 = \mu(E_1) > \mu(E) = \mu$.
\end{Th}
\begin{proof}
In this case $\GCD(1,3,\tilde{d}_1 + \tilde{d}_2) = 1$ trivially, and
$\sigma_H = \sigma_H(k)$ is a $\sigma$-critical value that satisfies
\[
\sigma_m < 2g - 2 \leqslant \sigma_H(k) < \sigma_M.
\]
Therefore, by the description of Mu\~noz~et al.~\cite{mov} of the critical values (\cite{mov} Lemma~5.2. and Lemma~5.3.),
the line bundle $M\to X$ satisfies in this case, the following:
\[
\sigma_m <
\sigma_H(k) =
3d_M - \tilde{d}_1 - \tilde{d}_2
\]
equivalently
\[
d_M = \sigma_H(k) + \mu
\]
and hence
\[
\tilde{d}_1 - d_M - \tilde{d}_2 - 1 =
\sigma_H(k) - 2(\mu_1 - \mu) - 1.
\]
In such a case
\[
\tilde{d}_1 - 2\tilde{d}_2 - (2g-2) =
\sigma_H(k) - 2(\mu_1 - \mu) + k
\geqslant
\sigma_H(k) - 2(\mu_1 - \mu) - 1
\]
and then
\[
2(\tilde{d}_1 - 2\tilde{d}_2 - (2g-2))
\geqslant
\tilde{d}_1 - d_M - \tilde{d}_2 - 1.
\]
Therefore, in this case
\[
n(k) =
\tilde{d}_1 - d_M - \tilde{d}_2 - 1
=
\sigma_H(k) - 2(\mu_1 - \mu) - 1.
\]
\noindent Finally, by Theorem~\ref{bgg1_7.9} and by Corollary~\ref{bgg1_7.10},
the space
$
\mathcal{N}_{\sigma_H}^{k} =
\mathcal{N}_{\sigma_{H}(k)}(2,1,\tilde{d}_1,\tilde{d}_2)
$
is birationally equivalent to
$
\mathcal{N}_{\sigma_L^{+}(k)} =
\mathcal{N}_{\sigma_L^{+}(k)}(2,1,\tilde{d}_1,\tilde{d}_2)
$,
which is equal to the moduli space
$
\mathcal{N}_{\sigma_L^{+}(k)}^s =
\mathcal{N}_{\sigma_L^{+}(k)}^s(2,1,\tilde{d}_1,\tilde{d}_2)
$
of holomorphic stable triples
also by Theorem~\ref{bgg1_7.9}, where $\sigma_L^{+}(k)$ is the maximal critical
value, depending on $k$ in this case.
The isomorphism then follows by Corollary~\ref{CohomologyCriticalTriples}.
\end{proof}
\begin{Cor}\lambdabel{(1,2)-VHS--Cohomology}
There is an isomorphism
\[
H^{j}(F_{d_1}^{k+1},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(F_{d_1}^{k},\mathbb{Z})
\]
for all
$
j \leqslantslant \sigma_H(k) - 2(\mu_1 - \mu) - 1
$
induced by the embedding~\ref{eq:(1,2)-VHS-embedding}.
\QEDA
\end{Cor}
\subsection[Cohomology of (2,1)-VHS]{Cohomology of the $(2,1)$-VHS}
\lambdabel{ssec:3.4}
Because of the duality
\[
\mathcal{N}_{\sigma}(1,2,\tilde{d}_1,\tilde{d}_2)
\cong
\mathcal{N}_{\sigma}(2,1,-\tilde{d}_2,-\tilde{d}_1)
\]
from Proposition~\ref{dualtriples},
we get
\begin{Th}
There is an isomorphism
\[
i_k^*\colon
H^{j}(\mathcal{N}_{\sigma_c(k+1)},\mathbb{Z}) \xrightarrow{\quad \cong \quad}
H^{j}(\mathcal{N}_{\sigma_c(k)},\mathbb{Z})\quad \forall j \leqslantslant m(k)
\]
where
$
m(k) =
\min(-\tilde{d}_1 - d_M + \tilde{d}_2 - 1,\quad 2\big(-\tilde{d}_1 + 2\tilde{d}_2 - (2g - 2)\big) + 1)
$.
\end{Th}
\begin{proof}
The result follows as the analogous to Corollary~\ref{CohomologyCriticalTriples} varying $\tilde{d}_1$ and $\tilde{d}_2$
according to the duality from Proposition~\ref{dualtriples}.
\end{proof}
\begin{Th}\lambdabel{z-r_02}
For $k$ large enough, there is an isomorphism
\[
i_k^*\colon
H^{j}(\mathcal{N}_{\sigma_H}^{k+1},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(\mathcal{N}_{\sigma_H}^{k},\mathbb{Z})\quad
\forall j \leqslantslant \sigma_H(k) - 4(\mu_2 - \mu) - 1
\]
where
$
\mathcal{N}_{\sigma_H}^{k} =
\mathcal{N}_{\sigma_{H}(k)}(1,2,\tilde{d}_1,\tilde{d}_2)
$,\quad
$
\sigma_{H} =
\sigma_{H}(k)
= 2g - 2 + k
$,
and $\mu_2 = \mu(E_2) > \mu(E) = \mu$.
\end{Th}
\begin{proof}
In this case, by the duality from Proposition~\ref{dualtriples}, and by the description of the $\sigma_c$ critical values
(Mu\~noz~et al.~\cite{mov} Lemma~5.2. and Lemma~5.3.), the line bundle $M\to X$ satisfies in this case, the following:
\[
\sigma_m <
\sigma_H(k) =
3d_M + \tilde{d}_1 + \tilde{d}_2
\]
equivalently
\[
d_M = -\mu > -\mu_2
\]
and hence
\[
-\tilde{d}_1 - d_M + \tilde{d}_2 - 1 =
\sigma_H(k) - 4(\mu_2 - \mu) - 1,
\]
where, once again, $\sigma_H = \sigma_H(k)$ is a $\sigma$-critical value satisfying
\[
\sigma_m < 2g - 2 \leqslant \sigma_H(k) < \sigma_M.
\]
\noindent In such a case
\[
-\tilde{d}_1 + 2\tilde{d}_2 - (2g-2) =
\sigma_H(k) - 6(\mu_2 - \mu) + k \geqslant
\]
\[
\sigma_H(k) - 4(\mu_2 - \mu)
\geqslant
\sigma_H(k) - 4(\mu_2 - \mu) - 1
\]
if $k > 2(\mu_2 - \mu) > 0$ is large enough. Then
\[
2(-\tilde{d}_1 + 2\tilde{d}_2 - (2g-2))
\geqslant
-\tilde{d}_1 - d_M + \tilde{d}_2 - 1.
\]
Therefore, in this case
\[
m(k) =
-\tilde{d}_1 - d_M + \tilde{d}_2 - 1
=
\sigma_H(k) - 4(\mu_2 - \mu) - 1.
\]
\noindent Hence, the result follows as the dual analogous to Theorem~\ref{z-r_01}.
\end{proof}
\begin{Cor}\lambdabel{(2,1)-VHS--Cohomology}
For $k$ large enough, there is an isomorphism
\[
H^{j}(F_{d_2}^{k+1},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(F_{d_2}^{k},\mathbb{Z})
\]
for all
$
j \leqslantslant \sigma_H(k) - 4(\mu_2 - \mu) - 1
$
induced by the embedding~\ref{eq:(2,1)-VHS-embedding}.
\QEDA
\end{Cor}
\subsection[Cohomology of (1,1,1)-VHS]{Cohomology of the $(1,1,1)$-VHS}
\lambdabel{ssec:3.5}
\begin{Th}\lambdabel{(1,1,1)-VHS-pullback}
The pull-back
\[
i_{k}^{*}\colon
H^{*}(F_{m_1 m_2}^{k+1},\mathbb{Z})
\to
H^{*}(F_{m_1 m_2}^{k},\mathbb{Z})
\]
induced by the embedding $i_{k}\colon F_{m_1 m_2}^{k}\to F_{m_1 m_2}^{k+1}$, is surjective.
\end{Th}
\begin{proof}
This is a direct consequence of Theorem~\ref{(1,1,1)-VHS--SymP-Iso}, Theorem~\ref{symmetric-pullback} and Corollary~\ref{macdonald-12.2}.
\end{proof}
\begin{Cor}\lambdabel{(1,1,1)-cohomology-iso}
There is an isomorphism
\[
H^{j}(F_{m_1 m_2}^{\infty},\mathbb{Z})
\xrightarrow{\quad \cong \quad}
H^{j}(F_{m_1 m_2}^{k},\mathbb{Z})
\]
for all $j \leqslant \min \big(\bar{m}_1 + k, \bar{m}_2 + k\big) - 1$.
\end{Cor}
\begin{proof}
It follows directly from Theorem~\ref{(1,1,1)-VHS--SymP-Iso}, Corollary~\ref{macdonald-12.2}
and Corollary~\ref{symmetric-cohomology-iso}.
\end{proof}
\section*{References}
\addcontentsline{toc}{section}{References}
\end{document} |
\begin{document}
\setlength{\oddsidemargin}{0cm} \setlength{\evensidemargin}{0cm}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conj}[theorem]{Conjecture}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{exam}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\title[]{Non-Riemannian Einstein-Randers metrics on $E_6/A_4$ and $E_6/A_1$}
\author{Xiaosheng Li}
\address{School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, P.R. China} \email{xiaosheng0526@126.com}
\author{Chao Chen}
\address{School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, P.R. China} \email{}
\author{Zhiqi Chen}
\address{School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P.R. China} \email{chenzhiqi@nankai.edu.cn}
\author{Yuwang Hu}
\address{School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, P.R. China} \email{hywzrn@163.com}
\subjclass[2010]{Primary 53C25, 53C30; Secondary 17B20, 22E46.}
\keywords{Einstein metric, Einstein-Randers metric, homogeneous manifold.}
\begin{abstract}
In this paper, we first prove that homogeneous spaces $E_6/A_4$ and $E_6/A_1$ admit Einstein metrics which are $Ad(T\times A_1\times A_4)$-invariant, and then show that they admit Non-Riemannian Einstein-Randers metrics.
\end{abstract}
\maketitle
\setcounter{section}{0}
\section{Introduction}
Randers metrics were introduced by Randers in the context of general
relativity, and named after him by Ingarden. It shows the importance
of Randers metrics in physics. Moreover, Randers metrics are useful
in other fields; see Ingarden's account on \cite{AIM,In1} for their
application in the study on the Lagrangian of relativistic electrons.
Just as in the Riemannian case, it is a fundamental problem to
classify homogeneous Einstein-Randers spaces. In particular, it is
very important to know if a homogeneous manifold admits invariant Einstein-Randers metrics. There are a lot of studies on Einstein-Randers metrics on homogeneous manifolds, see \cite{CDL1,CDL2,KC1,LD1,NT1,WD1,WD2,WD3,WHD1}.
In this paper, we will discuss Einstein metrics and Einstein-Randers metrics on homogeneous spaces $E_6/A_4$ and $E_6/A_1$. In Section 2, we prove that there are only four Einstein metrics on $E_6/A_4$ and two Einstein metrics on $E_6/A_1$ which are $Ad(T\times A_1\times A_4)$-invariant. Furthermore, in Section 3, we prove that there are at least four and two families of $E_6$-invariant non-Riemannian Einstein-Randers metrics on $E_6/A_4$ and $E_6/A_1$ respectively.
\section{Einstein metrics}
Consider the symmetric spaces $E_6/A_1\times A_5$ and $A_5/T\times A_4$. Let ${E_6}={A_1}\oplus{A_5}\oplus{\mathfrak m}_1$ and $A_5={\mathfrak h_0}\oplus A_4\oplus {\mathfrak m}_2$ be the corresponding decompositions of the Lie algebras. Here $\mathfrak h_0$ is the Lie algebra of $T$. Let $H$ be the Lie group $T\times A_4\times A_1$ with the Lie algebra ${\mathfrak h}_0\oplus A_4\oplus A_1$. Then we have the following decomposition of the Lie algebra $E_6$:
\begin{equation}\label{dec}
{E_6}={\mathfrak h}_0\oplus{A_4}\oplus{A_1}\oplus{\mathfrak m}_1\oplus{\mathfrak m}_2.
\end{equation}
Here the $Ad(H)$-modules ${\mathfrak m}_i,i=1,2$ are irreducible and
mutually non-equivalent and $\dim {\mathfrak
h}_0=1$. For the structure of such decomposition, see \cite{AMS1,DK1}. Clearly $\dim {A_4}=24$, $\dim {A_1}=3$, $\dim {\mathfrak m}_1=40$ and $\dim {\mathfrak m}_2=10$. The left-invariant metric on $E_6$ which
is $Ad(H)$-invariant must be of the form
\begin{equation}\label{metricII}
\langle\cdot,\cdot\rangle=u_0\cdot B|_{\mathfrak h_0}+u_1\cdot
B|_{A_4}+u_2\cdot B|_{A_1}+x_1\cdot
B|_{\mathfrak m_1}+x_2\cdot B|_{\mathfrak m_2}, \end{equation} where
$u_0,u_1,u_2,x_1,x_2\in {\mathbb R}^+$, and the space of left-invariant symmetric covariant 2-tensors on $E_6$ which are
$Ad(H)$-invariant is given by
\begin{equation}\label{non-naturalII}
v_0\cdot B|_{\mathfrak h_0}+ v_1\cdot B|_{A_4}+v_2\cdot
B|_{A_1}+v_3\cdot B|_{\mathfrak m_{1}}+v_4\cdot
B|_{\mathfrak m_{2}},
\end{equation}
where $v_0, v_1, v_2, v_3, v_4\in {\mathbb R}$. In particular, the
Ricci tensor $r$ of a left-invariant Riemannian metric $\langle\cdot,\cdot\rangle$ on $G$ is a left invariant symmetric covariant 2-tensor on
$G$ which is $Ad(H)$-invariant. Thus $r$ is of the form
(\ref{non-naturalII}). The authors give in \cite{AMS1} the formulae of the Ricci tensor corresponding to the metric~(\ref{metricII}) on $E_6$. In fact, the case for $E_6$ is just one case of what the authors study in \cite{AMS1}.
For this decomposition of $E_6$, the left-invariant metric on $E_6/A_4\times A_1$ which is $Ad(H)$-invariant must be of the form
\begin{equation}\label{metric2II}
\langle\cdot,\cdot\rangle=u_0\cdot B|_{\mathfrak h_0}+x_1\cdot
B|_{\mathfrak m_1}+x_2\cdot B|_{\mathfrak m_2},
\end{equation} where $u_0,x_1,x_2\in {\mathbb R}^+$. Based on the formulae given in \cite{AMS1}, the authors classify in \cite{CDL2} Einstein metrics on $E_6/A_4\times A_1$ which are $Ad(H)$-invariant.
The following is to discuss the left-invariant metrics on $E_6/A_4$ and $E_6/A_1$ which are $Ad(H)$-invariant corresponding to the decomposition~(\ref{dec}) of $E_6$.
\subsection{The case of $E_6/A_4$}
The left-invariant metric on $E_6/A_4$ which is $Ad(H)$-invariant must be of the form
\begin{equation}\label{met1}
\langle\cdot,\cdot\rangle=u_0\cdot B|_{\mathfrak h_0}+u_2\cdot B|_{A_1}+x_1\cdot
B|_{\mathfrak m_1}+x_2\cdot B|_{\mathfrak m_2}, \end{equation} where
$u_0,u_2,x_1,x_2\in {\mathbb R}^+$. Based on the formulae given in \cite{AMS1}, we have the components of the Ricci tensor $\widetilde{r}$ of the metric (\ref{met1}) on $E_6/A_4$:
\[ \left\{ \begin{aligned}
&
\widetilde{r}_{{\mathfrak h}_0}=\frac{u_0}{8x_1^2}+\frac{u_0}{8x_2^2},\\
&
\widetilde{r}_{A_1}=\frac{1}{24u_2}+ \frac{5u_2}{24x_1^2}£¬ \\
&
\widetilde{r}_{{\mathfrak
m}_1}=\frac{1}{2x_1}-\frac{x_2}{16x_1^2}-\frac{u_0}{160x_1^2}-\frac{u_2}{32x_1^2},\\
&
\widetilde{r}_{{\mathfrak
m}_2}=\frac{1}{4x_2}+\frac{x_2}{8x_1^2}- \frac{u_0}{40x_2^2}.
\end{aligned} \right. \]
Furthermore, the metric is Einstein if and only if there
exists a positive solution $\{u_0, u_2, x_1, x_2\}$ of the system of
equations
\begin{equation}\label{1}
\widetilde{r}_{{\mathfrak h}_0}=\widetilde{r}_{A_1}=\widetilde{r}_{{\mathfrak
m}_1}=\widetilde{r}_{{\mathfrak m}_2}.
\end{equation}
The following is to solve the equations by the theory of Gr$\rm{\ddot{o}}$bner basis. Putting $u_0=1$ and by $\widetilde{r}_{{\mathfrak h}_0}=\widetilde{r}_{A_1}, \widetilde{r}_{{\mathfrak h}_0}=\widetilde{r}_{{\mathfrak m}_1}, \widetilde{r}_{{\mathfrak h}_0}=\widetilde{r}_{{\mathfrak m}_2}$, we have
\[ \left\{ \begin{aligned}
& f_1=-x_1^2x_2^2-5x_2^2u_2^2+3x_1^2u_2+3x_2^2u_2=0, \\
& f_2=-80x_1x_2^2+10x_2^3+20x_1^2+21x_2^2+5x_2^2u_2=0, \\
& f_3=-10x_1^2x_2-5x_2^3+6x_1^2+5x_2^2=0.
\end{aligned} \right. \]
Consider the polynomial ring $R={\mathbb Q}[z, x_1, x_2, u_2]$ and an ideal $I$ generated by $\{f_1, f_2, f_3,
zx_1x_2u_2-1\}$ to find non-zero solutions of $(\ref{1})$. Take a lexicographic order $>$ with
$z >u_2> x_1> x_2$ for a monomial ordering on $R$. By the help of computer, we have the polynomial of $x_2$ containing in the Gr$\rm{\ddot{o}}$bner basis of the ideal $I$:
\begin{eqnarray*}
f(x_2)&=& 27263765625x_2^8-82709987500x_2^7+94104102500x_2^6-48116787500x_2^5 \\
&& +9948352750x_2^4-491681700x_2^3+74376100x_2^2-1183780x_2+142129.
\end{eqnarray*}
In the Gr$\rm{\ddot{o}}$bner basis of the ideal $I$, $x_1$ and $u_2$ can be written into polynomials of $x_2$. The equation $f(x_2)=0$ has four solutions:
$$x_2\approx 0.6513015810, \quad x_2\approx0.6770950751, \quad x_2\approx0.8288266917, \quad x_2\approx0.8641265950.$$
In fact, we have the following solutions of $(\ref{1})$:
\begin{eqnarray*}
\{u_2\approx0.1141930615, & x_1\approx1.200678505, & x_2\approx0.6513015810\}, \\
\{u_2\approx1.746579387, & x_1\approx0.9798479028, & x_2\approx0.6770950751\}, \\
\{u_2\approx0.7564861893, & x_1\approx0.5068895851, & x_2\approx0.8288266917\}, \\
\{u_2\approx0.0549236976, & x_1\approx0.4382514353, & x_2\approx0.8641265950\}.
\end{eqnarray*}
In summary, we have:
\begin{theorem}\label{theorem1}
Let the notations be as above. Then every left invariant metric on
$E_6/A_4$ which is $Ad(H)$-invariant is of the form $(\ref{met1})$. Up to scaling, there are four Einstein metrics on $E_6/A_4$ which are $Ad(H)$-invariant.
\end{theorem}
\subsection{The case of $E_6/A_1$}
The left-invariant metric on $E_6/A_1$ which is $Ad(H)$-invariant must be of the form
\begin{equation}\label{met2}
\langle\cdot,\cdot\rangle=u_0\cdot B|_{\mathfrak h_0}+u_1\cdot B|_{A_4}+x_1\cdot
B|_{\mathfrak m_1}+x_2\cdot B|_{\mathfrak m_2}, \end{equation} where
$u_0,u_1,x_1,x_2\in {\mathbb R}^+$. Based on the formulae given in \cite{AMS1}, we have the components of the Ricci tensor $\widetilde{r}$ of the metric (\ref{met2}) on $E_6/A_1$:
\[ \left\{ \begin{aligned}
&
\widetilde{r}_{{\mathfrak h}_0}=\frac{u_0}{8x_1^2}+\frac{u_0}{8x_2^2},\\
&
\widetilde{r}_{A_4}=\frac{5}{48u_1}+ \frac{u_1}{8x_1^2}+\frac{u_1}{48x_2^2}£¬ \\
&
\widetilde{r}_{{\mathfrak
m}_1}=\frac{1}{2x_1}-\frac{x_2}{16x_1^2}-\frac{u_0}{160x_1^2}-\frac{3u_1}{20x_1^2},\\
&
\widetilde{r}_{{\mathfrak
m}_2}=\frac{1}{4x_2}+\frac{x_2}{8x_1^2}- \frac{u_0}{40x_2^2}-\frac{u_1}{10x_2^2}.
\end{aligned} \right. \]
Furthermore, the metric is Einstein if and only if there
exists a positive solution $\{u_0, u_1, x_1, x_2\}$ of the system of
equations
\begin{equation}\label{2}
\widetilde{r}_{{\mathfrak h}_0}=\widetilde{r}_{A_4}=\widetilde{r}_{{\mathfrak
m}_1}=\widetilde{r}_{{\mathfrak m}_2}.
\end{equation}
Similar to the discussion on $E_6/A_4$. Putting $u_0=1$ and by $\widetilde{r}_{{\mathfrak h}_0}=\widetilde{r}_{A_4}, \widetilde{r}_{{\mathfrak h}_0}=\widetilde{r}_{{\mathfrak m}_1}, \widetilde{r}_{{\mathfrak h}_0}=\widetilde{r}_{{\mathfrak m}_2}$, we have
\[ \left\{ \begin{aligned}
& f_1=-5x_1^2x_2^2-6x_2^2u_1^2-x_1^2u_1^2+6x_1^2u_1+6x_2^2u_1=0, \\
& f_2=-80x_1x_2^2+10x_2^3+20x_1^2+21x_2^2+24x_2^2u_1=0, \\
& f_3=-10x_1^2x_2-5x_2^3+6x_1^2+5x_2^2+4x_1^2u_1=0.
\end{aligned} \right. \]
Consider the polynomial ring $R={\mathbb Q}[z, x_1, x_2, u_1]$ and an ideal $I$ generated by $\{f_1, f_2, f_3,
zx_1x_2u_1-1\}$ to find non-zero solutions of $(\ref{2})$. Take a lexicographic order $>$ with
$z >u_1> x_1> x_2$ for a monomial ordering on $R$. By the help of computer, we have the polynomial of $x_2$ containing in the Gr$\rm{\ddot{o}}$bner basis of the ideal $I$:
\begin{eqnarray*}
f(x_2)&=& 40733269776x_2^8-95717471616x_2^7+80248108328x_2^6-31589680504x_2^5 \\
&&+7669961625x_2^4-1207801950x_2^3+120201725x_2^2-5089500x_2+422500.
\end{eqnarray*}
In the Gr$\rm{\ddot{o}}$bner basis of the ideal $I$, $x_1$ and $u_1$ can be written into polynomials of $x_2$. The equation $f(x_2)=0$ has two solutions:
$$x_2\approx0.8651778712,\quad x_2\approx 0.9203114422.$$
In fact, we have the following solutions of $(\ref{2})$:
\begin{eqnarray*}
\{u_1\approx 0.1945580092,& x_1\approx 0.5189654864, & x_2\approx 0.8651778712\}, \\
\{u_1\approx 0.7881276805,& x_1\approx 2.582407960, & x_2\approx 0.9203114422\}.
\end{eqnarray*}
In summary, we have:
\begin{theorem}\label{theorem2}
Let the notations be as above. Then every left invariant metric on
$E_6/A_1$ which is $Ad(H)$-invariant is of the form $(\ref{met2})$. Up to scaling, there are two Einstein metrics on $E_6/A_1$ which are $Ad(H)$-invariant.
\end{theorem}
\section{Einstein-Randers metrics}
A Randers metric $F$ on $M$ is built from a Riemannian metric and a
1-form, i.e.,
$$F=\alpha+\beta,$$ where $\alpha$ is a Riemannian metric and
$\beta$ is a 1-form whose length with respect to the Riemannian
metric $\alpha$ is less than 1 everywhere. Obviously, a Randers
metric is Riemannian if and only if it is reversible, i.e.,
$F(x,y)=F(x,-y)$ for any $x\in M$ and $y\in T_x(M)$. Sometimes it is
convenient to use the following presentation of a Randers metric in
\cite{BR1}, i.e.,
\begin{equation}
F(x,y)=\frac{\sqrt{[h(W,y)]^2+h(y,y)\lambda}}{\lambda}-\frac{h(W,y)}{\lambda}
\end{equation}
here $\lambda=1-h(W,W)>0$. The pair $(h,W)$ is called the navigation data of the
corresponding Randers metric $F$ .
The Ricci scalar $\mathfrak{Ric}(x,y)$ of a Finsler metric is
defined to be the sum of those $n-1$ flag curvatures $K(x,y,e_v)$,
where $\{e_v: v=1,2,\cdots,n-1\}$ is any collection of $n-1$
orthonormal transverse edges perpendicular to the flagpole, i.e.
\begin{equation}
\mathfrak{Ric}(x,y)=\sum^{n-1}_{v=1}R_{vv}.
\end{equation}
The Ricci tensor is defined by
\begin{equation}
Ric_{ij}=(\frac{1}{2}F^2\mathfrak{Ric})_{y^iy^j}.
\end{equation}
Obviously, the Ricci scalar depends on the position $x$ and the
flagpole $y$, but does not depend on the specific $n-1$ flags
with transverse edges orthogonal to $y$ (see \cite{BR1,BRS}). In the Riemannian case, it is a well known fact that the
Ricci scalar depends only on $x$. Thus it is
quite interesting to study a Finsler manifold whose Ricci scalar does
not depend on the flagpole $y$. Generally, a Finsler metric with
such a property is called an Einstein metric, i.e.,
\begin{equation}\label{k}
\mathfrak{Ric}(x,y)=(n-1)K(x)
\end{equation}
for some function $K(x)$ on $M$. In particular, for a Randers
manifold $(M,F)$ with $\dim M\geq 3$, $F$ is an Einstein metric if
and only if there is a constant $K$ such that (\ref{k}) holds (see \cite{BR1}). The
following lemma is an important result on Einstein-Randers metrics.
\begin{lemma}[\cite{BR1}]\label{lem1}
Suppose $(M,F)$ is a Randers space with the navigation data
$(h,W)$. Then $(M,F)$ is Einstein with Ricci scalar
$\mathfrak{Ric}(x)=(n-1)K(x)$ if and only if there exists a constant
$\sigma$ satisfying the following conditions:
\begin{enumerate}
\item $h$ is Einstein with Ricci scalar
$(n-1)K(x)+\frac{1}{16}\sigma^2$, and
\item $W$ is an infinitesimal homothety of $h$, i.e., $\mathfrak{L}_Wh=-\sigma
h$.
\end{enumerate}
Furthermore, $\sigma$ must be zero whenever $h$ is not Ricci-flat.
\end{lemma}
It is well know that $K(x)$ is a
constant if $(M,F)$ is a homogeneous Einstein Finsler manifold. Here a Finsler manifold $(M,F)$ is called homogeneous if
its full group of isometries acts transitively on $M$. Based on
Lemma~\ref{lem1}, Deng-Hou obtained a characterization of
homogeneous Einstein-Randers metrics.
\begin{lemma}[\cite{DH1}]\label{lem2}
Let $G$ be a connected Lie group and $H$ a closed subgroup of $G$
such that $G/H$ is a reductive homogeneous space with a
decomposition ${\mathfrak g}={\mathfrak h}+{\mathfrak m}$. Suppose
$h$ is a $G$-invariant Riemannian metric on $G/H$ and $W\in
{\mathfrak m}$ is invariant under $H$ with $h(W,W)<1$. Let
$\widetilde{W}$ be the corresponding $G$-invariant vector field on
$G/H$ with $\widetilde{W}|_o=W$. Then the Randers metric $F$ with the
navigation data $(h,\widetilde{W})$ is Einstein with Ricci constant
$K$ if and only if $h$ is Einstein with Ricci constant $K$ and $W$
satisfies
\begin{equation}\label{asso}
\langle [W,X]_{\mathfrak m}, Y\rangle+\langle X, [W,Y]_{\mathfrak
m}\rangle=0, \quad \forall X,Y\in {\mathfrak m},
\end{equation}
where $\langle,\rangle$ is the restriction of $h$ on $T_o(G/H)\simeq
{\mathfrak m}$. In this case, $\widetilde{W}$ is a Killing vector
field with respect to the Riemannian metric $h$.
\end{lemma}
For simple, denote $H_i=A_4,{\mathfrak h}_j=A_1$, or $H_i=A_1,{\mathfrak h}_j=A_4$. By the equivalence of the adjoint representation and the isotropy representation of $H_i$ on
${\mathfrak h}_0\oplus{\mathfrak h}_j\oplus{\mathfrak m}_1\oplus{\mathfrak m}_2$, the
vector field
$$\widetilde{W}|_{gH}=d(\tau(g))|_o(W), \forall g\in {G}, W\in
{\mathfrak h}_0$$ is well-defined, and it is $G$-invariant (see
\cite{DH2}). For every metric given in Theorem~\ref{theorem1} and Theorem~\ref{theorem2}, one can
easily verifies the equation
$$ \langle [W,X]_{{\mathfrak h}_0\oplus{\mathfrak h}_j\oplus{\mathfrak m}_1\oplus{\mathfrak m}_2}, Y\rangle_{G/H_i}+\langle X, [W,Y]_{{\mathfrak h}_0\oplus{\mathfrak h}_j\oplus{\mathfrak m}_1\oplus{\mathfrak m}_2}\rangle_{G/H_j}=0$$
holds for any $W\in {\mathfrak h}_0$ and $X,Y\in {\mathfrak
h}_0\oplus{\mathfrak h}_j\oplus{\mathfrak m}_1\oplus{\mathfrak m}_2$, using the facts
that ${\mathfrak h}_0\subset {\mathfrak h}$ and that the metric is
$Ad(H)$-invariant. Then by Lemma~\ref{lem2}, the homogeneous metric
$$F(x,y)=\frac{\sqrt{[\langle W,y\rangle_{G/H_1}]^2+\langle y,y\rangle_{G/H_1}\lambda}}{\lambda}-\frac{\langle W,y\rangle_{G/H_1}}{\lambda}$$
is a $G$-invariant Einstein-Randers metric on $G/H_1$ when $W$ satisfies $\langle W,W\rangle_{G/H_1}<1$, and $F$ is Riemannian if and only if $W=0$. That is, we have the following theorem.
\begin{theorem}
There are at least four families of $E_6$-invariant non-Riemannian Einstein-Randers metrics on $E_6/A_4$, and two families of $E_6$-invariant non-Riemannian Einstein-Randers metrics on $E_6/A_1$.
\end{theorem}
\section{Acknowledgments}
This work is supported in part by NSFC (nos.11571182 and 11547122) and Nanhu Scholars Program for Young Scholars of XYNU.
\end{document} |
\begin{document}
\title{A noncommutative version of Farber's topological complexity}
\begin{abstract}
Topological complexity for spaces was introduced by M. Farber as a minimal number of continuity domains for motion planning algorithms. It turns out that this notion can be extended to the case of not necessarily commutative $C^*$-algebras. Topological complexity for spaces is closely related to the Lusternik--Schnirelmann category, for which we do not know any noncommutative extension, so there is no hope to generalize the known estimation methods, but we are able to evaluate the topological complexity for some very simple examples of noncommutative $C^*$-algebras.
\end{abstract}
\section*{Introduction}
Gelfand duality between compact Hausdorff spaces and unital commutative $C^*$--algebras allows to translate some topological constructions and invariants into the noncommutative setting. The most successful example is $K$-theory, which became a very useful tool in $C^*$-algebra theory. Homotopies between $*$-homomorphisms of $C^*$-algebras also play an important role, but there is no nice general homotopy theory for $C^*$-algebras due to the fact that the loop functor has no left adjoint \cite {Uuye}, Appendix A. Nevertheless, there are some homotopy invariants that allow noncommutative versions.
The aim of our work is to show that M. Farber's topological complexity \cite{Farber} is one of those. In Section \ref{Section1} we recall the original commutative definition of topological complexity, and in Section \ref{Section2} we use Gelfand duality to reverse arrows in this definition, and show that the resulting noncommutative definition generalizes the commutative one. In the remaining two sections we calculate topological complexity for some simple examples of $C^*$-algebras. In particular, we show that introducing noncommutative coefficients may decrease topological complexity. Although in most our examples topological complexity is either 1 or $\infty$, we provide a noncommutative example
with topological complexity 2.
The author is grateful to A. Korchagin for helpful comments.
\section{Farber's topological complexity}\label{Section1}
The topological approach to the robot motion planning problem was initiated by M. Farber in \cite{Farber}. Let us recall his basic construction. Let $X$ be the configuration space of a mechanical system. A continuous path $\gamma:[0,1]\to X$ represents a motion of the system, with $\gamma(0)$ and $\gamma(1)$ being the initial and the final state of the system. If $X$ is path-connected then the system can be moved to an arbitrary state from a given state. Let $PX$ denote the space of paths in $X$ with the compact-open topology, and let
\begin{equation}\label{pi}
\pi:PX\to X\times X
\end{equation}
be the map given by $\pi(\gamma)=(\gamma(0),\gamma(1))$. A continuous {\it motion planning algorithm} is a continuous section $$
s:X\times X\to PX
$$
of $\pi$. Typically, there may be no continuous motion planning algorithm, so one may take a covering of $X\times X$ by sets $V_1,\ldots,V_n$ (domains of continuity) and require existence of continuous sections
$$
s_i:V_i\to PX|_{V_i}
$$
of maps $\pi_i:PX|_{V_i}\to V_i$, $i=1,\ldots,n$. Here $PX|_{V_i}$ denotes the restiction of $\pi$ onto $V_i$, i.e. the subset of paths $\gamma:[0,1]\to X$ such that $(\gamma(0),\gamma(1))\in V_i$. In this case, the collection of the sections $s_i$, $i=1,\ldots,n$, is called a (discontinuous) motion planning algorithm. There are several versions of the definition, which use various kinds of coverings, e.g. coverings by open or closed sets, or by Euclidean neighborhood retracts, etc., but most of them agree on simplicial polyhedra (cf. \cite{Farber_survey}, Theorem 13.1). The topological complexity $TC(X)$ of $X$ is the minimal number $n$ of domains of continuity, i.e. the minimal number $n$, for which there exists a covering $V_1,\ldots,V_n$ and continuous sections $s_i$ as above. This number measures the complexity of the problem of navigation in $X$.
\section{Noncommutative version of topological complexity}\label{Section2}
For a compact Hausdorff space $X$ we can rewrite the above construction in terms of unital commutative $C^*$-algebras and their unital $*$-homomorphisms using Gelfand duality. Let $C(X)$ denote the commutative $C^*$-algebra of complex-valued continuous functions on $X$. A closed covering $V_1,\ldots,V_n$ of $X\times X$ corresponds to $n$ surjective $*$-homomorphisms
$$
j_i:C(X)\otimes C(X)\to C(V_i),
$$
$i=1,\ldots,n$, with $\cap_{i=1}^n{\mathbb K}er j_i=\{0\}$. As the path space $PX$ is not locally compact, it is not Gelfand dual to any $C^*$-algebra, but we can bypass this, replacing the sections $s_i$ by $*$-homomorphisms
$$
\sigma_i:C(X)\to C(V_i)\otimes C[0,1]
$$
defined by
$$
\sigma_i(f)(x,t)=f(s_i(x)(t)),
$$
where $x\in V_i$, $t\in[0,1]$, $f\in C(X)$.
Let us denote by $\operatorname{ev}_t$ the $*$-homomorphism of evaluation at $t\in[0,1]$, and let us consider the compositions
$$
\operatorname{ev}_0\circ\sigma_i, \ \operatorname{ev}_1\circ\sigma_i:C(X)\to C(V_i).
$$
Let
$$
\pi_0,\pi_1:X\times X\to X
$$
denote the projections onto the first and the second copy of $X$ respectively, and let
$$
p_0,p_1:C(X)\to C(X)\otimes C(X)
$$
be the corresponding $*$-homomorphisms. The condition $\pi\circ s_i=\operatorname{id}_{V_i}$ can be written as $\pi_k\circ\pi\circ s_i=\pi_k: V_i\to X$, $k=0,1$, which allows rewriting, in terms of $C^*$-algebras and $*$-homomorphisms, as $j_i\circ p_0=\operatorname{ev}_0\circ\sigma_i$, $j_i\circ p_1=\operatorname{ev}_1\circ\sigma_i$. Thus we have
\begin{lem}\label{L1}
Continuous sections $s_i:V_i\to PX|_{V_i}$ exist iff there exist $*$-homomorphisms $\sigma_i$ making the diagrams
\begin{equation}\label{diagram-definition}
\begin{xymatrix}{
C(X) \ar[r]^-{p_k}\ar[d]_-{\sigma_i} &C(X)\otimes C(X)\ar[d]^-{j_i}\\
C(V_i)\otimes C[0,1] \ar[r]_-{\operatorname{ev}_k} & C(V_i),}
\end{xymatrix}
\end{equation}
$k=0,1$, commute.
\end{lem}
Thus, we may define the topological complexity $TC(A)$ for a unital $C^*$-algebra $A$ as the minimal number $n$ of quotient $C^*$-algebras $B_1,\ldots,B_n$ of $A\otimes A$ with the quotient maps $q_i:A\otimes A\to B_i$, such that
\begin{enumerate}
\item
$\cap_{i=1}^n{\mathbb K}er q_i=\{0\}$;
\item
there exist $*$-homomorphisms
$$
\sigma_i:A\to B_i\otimes C[0,1],\qquad i=1,\ldots,n,
$$
making the diagrams
\begin{equation}\label{diagram}
\begin{xymatrix}{
A \ar[r]^-{p_k}\ar[d]_-{\sigma_i} &A\otimes A\ar[d]^-{q_i}\\
B_i\otimes C[0,1] \ar[r]_-{\operatorname{ev}_k} & B_i,}
\end{xymatrix}
\end{equation}
$k=0,1$, commute for each $i=1,\ldots,n$, where $p_0(a)=a\otimes 1$, $p_1(a)=1\otimes a$, $a\in A$.
\end{enumerate}
Here and further we always use $\otimes$ to denote the {\it minimal} tensor product of $C^*$-algebras.
If there is no such $n$ then we set $TC(A)=\infty$.
\begin{cor}
For a compact Hausdorff space $X$, one has $TC(C(X))=TC(X)$ if $TC(X)$ is defined using {\sl closed coverings}.
\end{cor}
\begin{proof}
Commutativity of $A=C(X)$, hence of $A\otimes A$, implies commutativity of $B_i$, hence $B_i=C(V_i)$ for some $V_i$. Surjectivity of $q_i$ implies that $V_i$ is a closed subset of $X\times X$. The condition $\cap_{i=1}^n{\mathbb K}er q_i=\{0\}$ means that $\{V_1,\ldots,V_n\}$ is a covering for $X\times X$.
\end{proof}
As we shall see later, topological complexity is not well suited for general $C^*$-algebras, e.g. it is infinite for topologically non-trivial simple $C^*$-algebras, but there are two good classes of $C^*$-algebras, for which this characterization may be interesting --- the class of noncommutative CW complexes introduced in \cite{ELP} and the class of $C(X)$-algebras. Most of our examples are from the first class.
Note that in the commutative case, topological complexity makes sense only for path-connected spaces --- otherwise any two points may be not connected by a path, i.e. the map (\ref{pi}) is not surjective. There is no good $C^*$-algebraic analog for that, but the following holds:
\begin{lem}\label{L6}
Let $A=A_1\oplus A_2$. Then $TC(A)=\infty$.
\end{lem}
\begin{proof}
One has $A\otimes A=\oplus_{k,l=1}^2 A_k\otimes A_l$. Let $q_i:A\otimes A\to B_i$, $i=1,\ldots,n$, and $\sigma:A\to B_i\otimes C[0,1]$ be as in the definition of topological complexity, and let $e_1=q_i(1_{A_1}\otimes 1_{A_1})$, $e_2=q_i(1_{A_1}\otimes 1_{A_2})$, $e_3=q_i(1_{A_2}\otimes 1_{A_1})$, $e_4=q_i(1_{A_2}\otimes 1_{A_2})$. Then $e_1,\ldots,e_4$ are projections in $B_i$, and, as $q_i$ is surjective, any element of $B_i$ has the form $\sum_{k=1}^4 e_kbe_k$. In particular, if $e\in B_i$ is a projection then each $e_kee_k$ is a projection, and if $e(t)$, $t\in [0,1]$, is a homotopy of projections, then we have four homotopies $e_ke(t)e_k$.
Let $a=1_{A_1}\oplus 0_{A_2}\in A$. Then $q_i(p_0(a))=e_1+e_2$ and $q_i(p_1(a))=e_1+e_3$ should be connected by a homotopy. This is possible only if $e_2=e_3=0$. As this argument does not depend on $i$, we conclude that $1_{A_1}\otimes 1_{A_2},1_{A_2}\otimes 1_{A_1}\in \cap_{i=1}^n{\mathbb K}er q_i=\{0\}$ --- a contradiction.
\end{proof}
The topological complexity of a space $X$ can be estimated from above by using covering dimension of $X$, and from below using multiplicative structure in cohomology. Regretfully, these estimates cannot work in the noncommutative case, thus making the problem of evaluating topological complexity even more difficult.
\section{Case $TC(A)=1$}\label{Section3}
The condition $TC(A)=1$ means that the two inclusions of $A$ into $A\otimes A$, $p_0:a\mapsto a\otimes 1$ and $p_1:a\mapsto 1\otimes a$, are homotopic. This property is similar to, but different from that of approximately inner half flip \cite{Toms-Winter}, which means that $p_0$ and $p_1$ are approximately unitarily equivalent, i.e. there exist unitaries $u_n\in A\otimes A$ such that $\lim_{n\to\infty}\|p_1(a)-\operatorname{Ad}_{u_n}p_0(a)\|=0$ for any $a\in A$.
The condition $TC(A)=1$ imposes restrictions on the $K$-theory groups of $A$. Let $K_*(A)$ denote the graded $K$-theory group of $A$, and let $\mathbf 1\in K_0(A)$ be the class of the unit element. Recall that if $A$ is in the bootstrap class \cite{Schochet} then it satisfies the K\"unneth formula, hence $K_*(A)\otimes K_*(A)\subset K_*(A\otimes A)$. The bootstrap class is the smallest class which contains all separable type I $C^*$-algebras and is closed under extensions, strong Morita equivalence, inductive limits, and crossed products by $\mathbb R$ and by $\mathbb Z$.
\begin{lem}\label{K}
Let $A$ satisfy $K_*(A)\otimes K_*(A)\subset K_*(A\otimes A)$. If $K_*(A)\otimes\mathbf 1\neq \mathbf 1\otimes K_*(A)$ then $TC(A)>1$.
\end{lem}
\begin{proof}
This follows from homotopy invariance of $K$-theory groups. If $TC(A)=1$ then the flip on $K_*(A\otimes A)$ must induce the identity map.
\end{proof}
For spaces, it is known that $TC(X)=1$ iff $X$ is contractible. For $C^*$-algebras, it is reasonable to call a unital $C^*$-algebra $A$ {\it contractible to a point} if there exists a $*$-homomorphism $h:A\to A\otimes C[0,1]$ and a $*$-homomorphism $i:A\to\mathbb C$ such that $\operatorname{ev}_1\circ h=\operatorname{id}_A$ and $\operatorname{ev}_0\circ h=j\circ i$, where $j:\mathbb C\to A$ is defined by $j(1)=1_A$. If $B$ is a non-unital contractible $C^*$-algebra then its unitalization $B^+$ is contractible to a point.
\begin{lem}\label{L2}
Let $A$ be contractible to a point. Then $TC(A)=1$.
\end{lem}
\begin{proof}
Let $\alpha:A\otimes A\otimes C[0,1]$ be the flip, $\alpha(a_1\otimes a_2\otimes f)=a_2\otimes a_1\otimes f$, where $a_1,a_2\in A$, $f\in C[0,1]$.
Let $h:A\to A\otimes C[0,1]$ be the homotopy as above. We write $h_t$ for $\operatorname{ev}_t\circ h$.
Define a $*$-homomorphism
$$
\sigma:A\to A\otimes A\otimes C[-1,1]
$$
by setting, for $a\in A$,
$$
\sigma(a)(t)=\left\lbrace\begin{array}{cl}
\alpha(1\otimes h_t(a)),&{\mbox{if\ }}t\in[0,1];\\
1\otimes h_{-t}(a),&{\mbox{if\ }}t\in[-1,0].
\end{array}\right.
$$
Then $\operatorname{ev}_1\circ\sigma(a)=a\otimes 1$, $\operatorname{ev}_{-1}\circ\sigma(a)=1\otimes a$. Continuity of $\sigma$ at $t=0$ follows from the equality $i(a)\otimes 1=1\otimes i(a)$.
\end{proof}
\begin{cor}
If $A_n=\{f\in C([0,1];M_n): f(1) \mbox{\ is\ scalar}\}$ then $TC(A_n)=1$.
\end{cor}
\begin{lem}\label{L3}
Let $TC(A)=1$. If there exists a unital $*$-homomorphism $i:A\to\mathbb C$ then $A$ is contractible to a point.
\end{lem}
\begin{proof}
Let $\sigma:A\to A\otimes A\otimes C[0,1]$ satisfy $\operatorname{ev}_0\circ\sigma(a)=a\otimes 1$ and $\operatorname{ev}_1\circ\sigma(a)=1\otimes a$, $a\in A$. Let $\bar\iota:A\otimes A\otimes C[0,1]\to A\otimes C[0,1]$ be the map defined by $\bar\iota(a_1\otimes a_2\otimes f)=i(a_1)\cdot a_2\otimes f$, where $a_1,a_2\in A$, $f\in C[0,1]$. Set $h=\bar\iota\circ\sigma:A\to A\otimes C[0,1]$. Then
$\operatorname{ev}_0\circ h(a)=i(a)\cdot 1$, $\operatorname{ev}_1\circ h(a)=a$, hence $h$ is the required homotopy.
\end{proof}
Below we list three examples of $C^*$-algebras with topological complexity 1. The proofs are known to specialists, but we could not find exact references.
\begin{prop}\label{L3}
One has $TC(M_n)=1$.
\end{prop}
\begin{proof}
Let $U$ be a unitary in $M_{n^2}\cong M_n\otimes M_n$ such that $\operatorname{Ad}_U$ is an automorphism of $M_{n^2}$ that interchanges $M_n\otimes 1$ with $1\otimes M_n$. If $M_n$ acts on an $n$-dimensional space $H_n$ with the orthonormal basis $\{e_i\}_{i=1}^n$ then $U$ interchanges vectors $e_i\otimes e_j$ and $e_j\otimes e_i$ when $i\neq j$. Let $U_t$, $t\in[0,1]$, be the path connecting $U$ with 1 constructed using the standard rotation formula. Define $\sigma:M_n\to M_n\otimes M_n\otimes C[0,1]$ by $\sigma(a)(t)=\operatorname{Ad}_{U_t}(a\otimes 1)$, $a\in M_n$.
\end{proof}
The above example can be extended to UHF algebras:
\begin{prop}\label{L4}
If $A$ is a UHF algebra then $TC(A)=1$.
\end{prop}
\begin{proof}
Let $n,k$ be integers, $\varphi:M_n\to M_{kn}$ a unital $*$-homomorphism. Let $\sigma':M_n\to M_n\otimes M_n\otimes C[0,1]$ and $\sigma'':M_{kn}\to M_{kn}\otimes M_{kn}\otimes C[0,1]$ be the maps constructed in the proof of Lemma \ref{L3},
$\sigma'(a')(t)=\operatorname{Ad}_{U'_t}(a'\otimes 1)$, $\sigma''(a'')(t)=\operatorname{Ad}_{U''_t}(a''\otimes 1)$, $a'\in M_n$, $a''\in M_{kn}$.
Then the diagram
\begin{equation}\label{333}
\begin{xymatrix}{
M_n \ar[r]^-{\sigma'}\ar[d]_-{\varphi} &M_n\otimes M_n\otimes C[0,1]\ar[d]^-{\varphi\otimes\varphi\otimes\operatorname{id}}\\
M_{kn}\ar[r]_-{\sigma''} & M_{kn}\otimes M_{kn}\otimes C[0,1]}
\end{xymatrix}
\end{equation}
commutes. Let $A$ be the direct limit of matrix algebras $A_n=M_{m_n}$, where $m_n$ divides $m_{n+1}$, $n\in\mathbb N$. Commutativity of the diagram (\ref{333}) shows that the maps $\sigma^{(n)}:A_n\to A_n\otimes A_n\otimes C[0,1]$ agree, so, for any $t\in[0,1]$ one can define the limit map $\sigma_t:A\to A\otimes A$ such that $(\sigma_t)|_{A_n}=\operatorname{ev}_t\circ\sigma^{(n)}$. Since $\|\sigma_t(a)\|\leq\|a\|$ for any $a\in A$ and any $t\in[0,1]$, continuity of $\sigma_t(a)$ with respect to $t$ for $a\in\cup_{n=1}^\infty A_n$ implies continuity of $\sigma_t(a)$ for any $a\in A$. This means that the family $\{\sigma_t\}_{t\in[0,1]}$ defines a $*$-homomorphism $\sigma:A\to A\otimes A\otimes C[0,1]$, which provides the required homotopy.
\end{proof}
Let $\mathcal O_2$ be the Cuntz algebra generated by two isometries $s_1$, $s_2$ satisfying $s_1s_1^*+s_2s_2^*=1$.
\begin{prop}
One has $TC(\mathcal O_2)=1$.
\end{prop}
\begin{proof}
Let $u=s_1^*\otimes s_1+s_2^*\otimes s_2\in\mathcal O_2\otimes\mathcal O_2$. It is unitary, and it suffices to check on generators that $p_0=\operatorname{Ad}_u p_1$ (cf. \cite{Rordam}, Theorem 5.1.2). But $\mathcal O_2\otimes\mathcal O_2\cong\mathcal O_2$ (\cite{Rordam}, Theorem 5.2.1), and the unitary group of $\mathcal O_2$ is contractible, hence, $p_0$ and $p_1$ are homotopic.
\end{proof}
\section{General case}\label{Section4}
Let $\mathbb K^+$ be the unitalized algebra of compact operators. In contrast with Lemma \ref{L3}, its topological complexity is infinite. This often happens for $C^*$-algebras with few ideals.
\begin{lem}
One has $TC(\mathbb K^+)=\infty$.
\end{lem}
\begin{proof}
Let $B_1,\ldots, B_n$ be the quotients of $\mathbb K^+\otimes\mathbb K^+$. If they satisfy the definition of topological complexity then one of them must coincide with $\mathbb K^+\otimes\mathbb K^+$ itself, in which case other quotients are redundant. Therefore, if $TC(\mathbb K^+)\neq\infty$ then $TC(\mathbb K^+)=1$. To show that this is not the case, recall that $K_0(\mathbb K^+)\cong \mathbb Z^2$ and use Lemma \ref{K}.
\end{proof}
\begin{lem}\label{L5}
Let $TC(A)>1$. If $A$ is simple then $TC(A)=\infty$.
\end{lem}
\begin{proof}
It follows from \cite{Takesaki} that $A\otimes A$ is simple, hence any possible quotient $B$ must equal $A\otimes A$.
\end{proof}
It follows that topological complexity distinguishes commutative $C^*$-algebras from their non-commutative deformations.
For example, consider an irrational rotation algebra $A_\theta$, $\theta\in[0,1]\setminus\mathbb Q$, often called a non-commutative torus. It is simple and has the same $K$-theory as the usual torus $\mathbb T^2$ \cite{Davidson}, hence $TC(A_\theta)=\infty$, while for a usual torus $\mathbb T^2$ one has $TC(C(\mathbb T^2))=3$ (cf. \cite{Farber_survey}, Example 16.4).
Nevertheless, tensoring by matrices does not increase topological complexity.
\begin{prop}\label{L9}
For any compact Hausdorff space $X$, one has $TC(C(X)\otimes M_n)\leq TC(C(X))$.
\end{prop}
\begin{proof}
Let $TC(C(X))=k$, and let $q_i:C(X)\otimes C(X)\to B_i$ and $\sigma_i:C(X)\to B_i\otimes C[0,1]$, $i=1,\ldots,k$, be as in the definition of topological complexity. Set $\overline{B}_i=B_i\otimes M_n\otimes M_n$, $\overline{q}_i=q_i\otimes\operatorname{id}:C(X)\otimes C(X)\otimes M_n\otimes M_n\to \overline{B}_i$. Define $\overline{\sigma}_i:C(X)\otimes M_n\to \overline{B}_i\otimes C[0,1]$ by $\overline{\sigma}_i(f\otimes m)(t)=\sigma_i(f)\otimes \operatorname{Ad}_{U_t}(m\otimes 1)\in B_i\otimes C[0,1]\otimes M_n\otimes M_n$, $f\in C(X)$, $m\in M_n$, $t\in[0,1]$, and $U_t$ as in the proof of Lemma \ref{L3}. Then the maps $\overline{q}_i$, $\overline{\sigma}_i$ make the corresponding diagrams commute, hence $TC(C(X)\otimes M_n)\leq TC(C(X))$.
\end{proof}
More generally, one has
\begin{prop}
Let $TC(A)=n$, $TC(C)=m$. Then $TC(A\otimes C)\leq nm$.
\end{prop}
\begin{proof}
Let $q^A_i:A\otimes A\to B_i$, $\sigma^A_i:A\to B_i\otimes C[0,1]$, $i=1,\ldots,n$, and $q^C_j:C\otimes C\to D_j$, $\sigma^C_j:C\to D_j\otimes C[0,1]$, $j=1,\ldots,m$, be as in the definition of topological complexity. Let $\Delta:C([0,1]^2)\to C[0,1]$ be the map induced by the diagonal embedding $[0,1]\to[0,1]^2$ and define the composition
$$
\begin{xymatrix}{
\sigma_{ij}:A\otimes C\ar[rr]^-{\sigma^A_i\otimes\sigma^C_j}&&B_i\otimes D_j\otimes C([0,1]^2)\ar[rr]^-{\operatorname{id}\otimes\Delta}&&B_i\otimes D_j\otimes C[0,1].
}\end{xymatrix}
$$
Then the diagram
$$
\begin{xymatrix}{
A\otimes C \ar[rr]^-{p^A_k\otimes p^C_k}\ar[d]_-{\sigma_{ij}} &&A\otimes C\otimes A\otimes C\ar[d]^-{q^A_i\otimes q^C_j}\\
B_i\otimes D_j\otimes C[0,1] \ar[rr]_-{\operatorname{ev}_k} && B_i\otimes D_j,}
\end{xymatrix}
$$
$k=0,1$, commutes for all $i$, $j$.
\end{proof}
Remark that in the commutative case the tensor product of $C^*$-algebras is Gelfand dual to the product of spaces, and
there is a much better estimate $TC(A\otimes C)\leq n+m-1$ (\cite{Farber}, Theorem 11).
We have no examples with $TC(C(X)\otimes M_n)<TC(C(X))$, but tensoring by a more general $C^*$-algebra may decrease topological complexity. Let $U(A)$ denote the group of unitaries of a $C^*$-algebra $A$. Recall that $U(\mathcal O_2)$ is contractible \cite{Thomsen}.
Let $\mathbb S$ denote the circle. It is known that $TC(C(\mathbb S))=2$.
\begin{thm}
Let $A$ satisfy $TC(A)=1$, $\pi_0(U(A))=\pi_1(U(A))=0$ $($e.g. $A=\mathcal O_2$$)$. Then $TC(C(\mathbb S)\otimes A)=1$.
\end{thm}
\begin{proof}
We have to connect by a homotopy the two $*$-homomorphisms $\sigma_i:C(\mathbb S)\otimes A\to C(\mathbb S)\otimes A\otimes C(\mathbb S)\otimes A$, $i=0,1$, given by $\sigma_0(f\otimes a)=f\otimes a\otimes 1\otimes 1$ and $\sigma_1(f\otimes a)=1\otimes 1\otimes f\otimes a$, $f\in C(\mathbb S)$, $a\in A$. Note that these maps are determined by their values on $u\otimes a$, where $u(x)=e^{2\pi ix}$, $u\in C(\mathbb S)$. By assumption, any unitary in $C(\mathbb S)\otimes A$ has a homotopy that connects it with $1\otimes 1$. Let $u_t$, $t\in[2/3,1]$, be a homotopy, in the unitary group of $C(\mathbb S)\otimes A$, that connects $u\otimes 1$ with $1\otimes 1$. Then the homotopy $\sigma_t$, given by $\sigma_t(u\otimes a)=1\otimes u_t\otimes a$ connects $\sigma_1$ with $\sigma_{2/3}$ given by $\sigma_{2/3}(u\otimes a)=1\otimes 1\otimes 1\otimes a$. Similarly, one can connect $\sigma_0$ with $\sigma_{1/3}$ given by $\sigma_{1/3}(u\otimes a)=1\otimes a\otimes 1\otimes 1$. Finally, as $TC(A)=1$, $\sigma_{1/3}$ and $\sigma_{2/3}$ are homotopic.
\end{proof}
Our next examples show how sensitive topological complexity may be. Let
$$
A_2=\{f\in C([0,1];M_2):f(1)\mbox{\ is\ diagonal}\}.
$$
This algebra is considered as a noncommutative version of the non-Hausdorff $T_1$ space $X_2$ obtained from two intervals $\{(x,y)\in[0,1]^2:y=0\mbox{\ or\ 1}\}$ by identifying the points $(x,0)$ and $(x,1)$ for each $x\in[0,1)$ \cite{Connes}. Although $X_2$ is not Hausdorff, it is contractible, hence $TC(X_2)=1$.
\begin{lem}\label{L7}
One has $TC(A_2)=\infty$.
\end{lem}
\begin{proof}
Suppose that $TC(A_2)=n<\infty$. Let $q_i:A_2\otimes A_2\to B_i$, $i=1,\ldots,n$, be as in the definition of topological complexity. There are two $*$-homomorphisms from $A_2$ to $\mathbb C$, given by $r_0(f)=f_{11}(1)$ and $r_1(f)=f_{22}(1)$, where $f\in A_2$. It is easy to see that each quotient map from $A_2$ factorizes through the restriction map on
a closed subset of $[0,1]^2$. As $\cap_{i=1}^n{\mathbb K}er q_i=\{0\}$, there is at least one $i$ such that $r_0\otimes r_1$ factorizes through $q_i$. Further, we may argue as in Lemma \ref{L6}: the maps $(r_0\otimes r_1)\circ p_0$ and $(r_0\otimes r_1)\circ p_1$ from $A_2$ to $\mathbb C$ shoud be homotopic. Let $a=\left(\begin{smallmatrix}1&0\\0&0\end{smallmatrix}\right)\in A_2$. Then $(r_0\otimes r_1)\circ p_0(a)=1$, $(r_0\otimes r_1)\circ p_1(a)=0$, which makes homotopy between $(r_0\otimes r_1)\circ p_0$ and $(r_0\otimes r_1)\circ p_1$ impossible.
\end{proof}
Let
$$
D_n=\{f\in C([0,1]; M_n): f(0),f(1)\mbox{\ are\ scalars}\}
$$
be a (unital) dimension-drop algebra.
\begin{lem}\label{DDT}
If $n>1$ then $TC(D_n)=\infty$.
\end{lem}
\begin{proof}
We identify $D_n\otimes D_n$ with the subalgebra of functions $f=f(x,y)$ in $C([0,1]^2;M_n\otimes M_n)$ satisfying the obvious boundary conditions.
As above, if there exist $k$ quotients $B_1\ldots,B_k$ of $D_n\otimes D_n$ then at least one of them surjects onto a copy of $\mathbb C$ that identifies with restrictions of functions $f$ onto the point $(1,0)\in[0,1]^2$. Denote this map by $\mu:B_{i_0}\to\mathbb C$. If there is a homotopy $\sigma_{i_0}:D_n\to B\otimes C[0,1]$ then it restricts to a homotopy $D_n\to\mathbb C\otimes C[0,1]$. If the diagram (\ref{diagram}) commutes then $\mu\circ\operatorname{ev}_0\circ\sigma_{i_0}(f)=f(1)$ and $\mu\circ\operatorname{ev}_1\circ\sigma_{i_0}(f)=f(0)$, $f\in D_n$. But these two maps are not homotopic.
\end{proof}
In both examples, $TC$ infinite means that there is no ``path'' connecting 0 and 1 in the noncommutative versions of an interval.
In contrast with these examples is our next one.
Let
$$
S_n=\{f\in C([0,1]; M_n): f(0)=f(1)\mbox{\ is\ scalar}\}.
$$
This is an algebra of matrix-valued functions on a circle, with the dimension drop at one point. If $n=1$ then $S_1$ is exactly the algebra of continuous functions on a circle.
\begin{thm}
For any $n\in\mathbb N$, $TC(S_n)=2$.
\end{thm}
\begin{proof}
We identify $S_n\otimes S_n$ with the algebra of $M_n\otimes M_n$-valued functions on $[0,1]^2$ with obvious boundary conditions. Let
$$
Y_1=\{(x,y)\in[0,1]^2:|x-y|\leq 2/3\},
$$
$$
Y_2=\{(x,y)\in[0,1]^2:x\geq 2/3, y\leq 1/3\}\cup\{(x,y)\in[0,1]^2:x\leq 1/3, y\geq 2/3\}.
$$
Then $Y_1\cup Y_2=[0,1]^2$. Let $B_i$, $i=1,2$, be the algebras of continuous $M_n\otimes M_n$-valued functions with the same boundary conditions as in $S_n\otimes S_n$, and let $q_i:S_n\otimes S_n\to B_i$ be the quotient $*$-homomorphisms induced by restrictions onto $Y_i$.
We have to construct homotopies $\sigma_i:S_n\to B_i\otimes C[0,1]$ such that
\begin{equation}\label{kontsy}
\operatorname{ev}_0\circ\sigma_i(f)(x,y)=f(x)\otimes 1,\quad \operatorname{ev}_1\circ\sigma_i(f)=1\otimes f(y).
\end{equation}
For $i=1$, $\operatorname{ev}_0\circ\sigma_1$ is homotopic to $\sigma'$ defined by
$$
\sigma'(f)(x,y)=\left\lbrace\begin{array}{cl}
f(0)\otimes 1,&\mbox{for\ }x+y\geq 4/3\mbox{\ or\ }x+y\leq 2/3;\\f(\frac{x+y}{2/3}-1)\otimes 1,&\mbox{for\ }2/3\leq x+y\leq 4/3.\end{array}\right.
$$
Similarly, $\operatorname{ev}_1\circ\sigma_1$ is homotopic to $\sigma''=\operatorname{Ad}_U(\sigma')$, where $U$ intertwines $M_n\otimes 1$ and $1\otimes M_n$. Finally, $\sigma'$ is homotopic to $\sigma''$, as $\operatorname{Ad}_{U_t}$ maps scalars into scalars for any $t$, where $U_t$ is a path connecting $U$ with 1, so the boundary conditions on $Y_1$ hold.
For $i=2$, as
$$
\{(x,y)\in[0,1]^2:x\geq 2/3, y\leq 1/3\}\cap\{(x,y)\in[0,1]^2:x\leq 1/3, y\geq 2/3\}=\emptyset,
$$
so after identifying 0 and 1, there is a single common point $(0,1)=(1,0)$. That's why we can construct the required homotopy separately for each of the $C^*$-algebras corresponding to these sets, but with the additional requirement that the two homotopies should agree at this common point. And as these sets are symmetric, it suffices to construct a homotopy for only one of them. Let $B_0$ denote the $C^*$-algebra of $M_{n^2}$-valued functions on
$$
\{(x,y)\in[0,1]^2:x\geq 2/3, y\leq 1/3\}
$$
with the obvious boundary conditions, and let $q_0:S_n\otimes S_n\to B_0$ be the restriction quotient map.
Note that the maps $\operatorname{ev}_0\circ\sigma_0$ and $\operatorname{ev}_1\circ\sigma_0$ (\ref{kontsy}) factorize through $A_0$ and $A_1$ respectively, where
$$
A_0=\{f\in C([2/3,1];M_n):f(1)\mbox{\ is\ scalar}\},
$$
$$
A_1=\{f\in C([0,1/3];M_n):f(0)\mbox{\ is\ scalar}\}
$$
(i.e. with no restrictions at one of the end-points), hence the map $\operatorname{ev}_0\circ\sigma_0$ is homotopic to $\sigma'_0$ given by $$
\sigma'_0(f)(x,y)=f(1)\otimes 1,
$$
and the map $\operatorname{ev}_1\circ\sigma_0$ is homotopic to $\sigma''_0$ given by
$$
\sigma'_0(f)(x,y)=1\otimes f(0).
$$
But, as $f(0)=f(1)$, they are homotopic. Along all these homotopies, their values at the point $(1,0)$ are the same.
Thus, $TC(S_n)\leq 2$.
To show that $TC(S_n)\neq 1$, let us calculate its $K$-theory groups. As $S_n$ is a split extension of $\mathbb C$ by the suspension $SM_n$ over $M_n$, one has $K_0(S_n)\cong K_1(S_n)\cong\mathbb Z$. Then
$$
K_1(S_n\otimes S_n)\cong K_0(S_n)\otimes K_1(S_n)\oplus K_1(S_n)\otimes K_0(S_n).
$$
Let $(p_k)_*:K_1(S_n)\to K_1(S_n\otimes S_n)$ be the maps induced by the $*$-homomorphisms $p_k:S_n\to S_n\otimes S_n$, $k=0,1$, and let $e$ and $u$ be generators for $K_0(S_n)$ and for $K_1(S_n)$ respectively. Then
$$
(p_0)_*(u)=u\otimes e\in K_1(S_n)\otimes K_0(S_n)\subset K_1(S_n\otimes S_n),
$$
$$
(p_1)_*(u)=e\otimes u\in K_0(S_n)\otimes K_1(S_n)\subset K_1(S_n\otimes S_n).
$$
As these elements are different, there is no homotopy that connects $p_0$ with $p_1$.
\end{proof}
\end{document} |
\begin{document}
\title{There are no 76 equiangular lines in $\mathbb{R}^{19}$}
\author{Wei-Hsuan Yu}
\subjclass[2010]{Primary 52C35; Secondary 14N20, 90C22, 90C05}
\keywords{strongly regular graph, equiangular lines}
\mathop{\mathrm{ad}}\nolimitsdress{Department of Mathematics, Michigan State University,
619 Red Cedar Road, East Lansing, MI 48824}
{\bold e}mail{u690604@gmail.com}
\date{}
\maketitle
\begin{abstract}
Maximum size of equiangular lines in $\mathbb{R}^{19}$ has been known in the range between 72 to 76 since 1973. Acoording to the nonexistence of strongly regular graph $(75,32,10,16)$ \cite{aza15}, Larmen-Rogers-Seidel Theorem \cite{lar77} and Lemmen-Seidel bounds on equiangular lines with common angle $\frac 1 3$ \cite{lem73}, we can prove that there are no 76 equiangular lines in $\mathbb{R}^{19}$. As a corollary, there is no strongly regular graph $(76,35,18,14)$. Similar discussion can prove that there are no 96 equiangular lines in $\mathbb{R}^{20}$.
{\bold e}nd{abstract}
\section{Introduction}
A set of lines in $\mathbb{R}^n$ is called equiangular if the angle between each
pair of lines is the same. We are interested in the upper bounds on the number of
equiangular lines in $\mathbb{R}^n$. Denote this quantity by $M(n)$. The purpose of this paper is to prove that there are no 76 equiangular lines in $\mathbb{R}^{19}$. Since 1973, $M(19)$ has been known to be between 72 and 76. The past four decades have seen no improvement on this bound. From Witt design, we can construct 72 equiangular lines in $\mathbb{R}^{19}$\cite[p. 148]{tay72}. The upper bound 76 is the relative bound for equiangular lines in \cite{lem73} and also a semidefinite programming bound in \cite{barg14}. After our results, $M(19)$ will be reduced to the range 72-75 and we conjecture that 72 is the maximum in $\mathbb{R}^{19}$.
The problem of determining $M(n)$ has been studied for almost seven decades, yet we still know very little.
Until recently the maximum number of equiangular lines in $\mathbb{R}
^n$ was known only for 35 values of the dimension $n$. We know $M(n)$ for most values of $n$ if $n \leq 43$. However, the cases for $n=14, 16, 17, 18, 19, 20$ and $42$ are still open. The history of this problem started with Hanntjes \cite{han48} who found M(n) for $n = 2$ and $3$ in $1948$. Van Lint and Seidel \cite{lin66} found the
largest number of equiangular lines for $4 \leq n \leq 7$. In 1973, Lemmens and Seidel
\cite{lem73} determine M(n) for most values for $8 \leq n \leq 23$. Barg and Yu \cite{barg14} determine $M(n)$ for $ 24 \leq n \leq 41$ and $ n=43$. For other works on the bounds for equiangular lines, please see \cite{grea14}, \cite{bur15} and \cite{oy14}.
We sketch below the approach of our main results. First, we will prove that if there exist 76 equiangular lines, the common angle has to be $\frac 1 3$ or $ \frac 1 5$. Then, by Lemmen-Seidel's bounds for equiangular lines with the angle $\frac 1 3 $, we know that $\frac 1 5$ is the only possible angle. Furthermore, if there exist 76 equiangular lines in $\mathbb{R}^{19}$ with the common angle $\frac 1 5$, it gives rise to an equiangular tight frame (ETF), which implies the existence of the strongly regular graph (75,32,10,16) \cite{wal09}. Azarija and Marc \cite{aza15} proved the nonexistence of the strongly regular graph (75,32,10,16). Therefore, there are no 76 equiangular lines in $\mathbb{R}^{19}$.
By the classical treatment of strongly regular graphs (SRGs), the projection of the vertex set onto a
non-trivial eigenspace is a spherical 2-distance set and a 2-design \cite{car01}. The projection of srg(76,35,18,14) and its complement srg(76,40,18,24) will both form a spherical 2-distance set with inner product values $\pm \frac 1 5$ in $\mathbb{R}^{19}$. Namely, it gives rise to a 76 equiangular line set in $\mathbb{R}^{19}$ with the angle $\frac 1 5$ which contradicts our main result. As a corrolary, we can show the nonexistence of these two SRGs. If the srg(76,30,8,14) or its complement srg(76,45,28,24) exists, there exist 76 equiangular lines in $\mathbb{R}^{57}$ with the common angle $\frac 1 {15}$. By the existence of complementary ETFs, we will have 76 equiangular lines in $\mathbb{R}^{19}$ with the common angle $\frac 1 5$. Therefore, these two SRGs do not exist. Similar discussion also can prove that there is no 96 equiangular lines in $\mathbb{R}^{20}$ and the nonexistence of srg $(96,45,24,18)$ and srg$(96,38,10,18)$. In the last section, we discuss the connection between ETFs, SRGs, spherical few-distance sets and spherical designs.
\section{Prelimanaries}
A set of lines in $\mathbb{R}^n$ is called equiangular if the angle between each
pair of lines is the same. If we have $M$ equiangular lines in $\mathbb{R}^n$, then we will have a set of unit vectors $\{x_i\}_{i=1}^M$ such that $|\langle x_i, x_j\rangle | =c$ for all $ 1\leq i \neq j \leq M$ , where $c$ is a positive constant. We call $c$ the {\bold e}mph{common angle} of the equiangular lines. When the number of equiangular lines is large enough, the common angle will be the reciprocal of an odd integer. Neumann proved the following theorem.
\begin{thm}\label{thm:Neu}
(Neumann \cite{lem73}) If we have $M$ equiangular lines in $\mathbb{R}^n$ and $M>2n$, then the common angle will be $\frac 1{2k-1}$, where $k \in \mathbb{N}$.
{\bold e}nd{thm}
Then, Larman, Rogers and Seidel proved a similar result for spherical two-distance sets.
A set of unit vectors $S = \{x_1, x_2, . . . \} \subset \mathbb{R}^n$ is called a spherical two-distance set if $\langle x_i
, x_j \rangle \in
\{a, b\}$ for some $a, b$ and all $i \neq j$. The study of upper bounds of spherical two-distance sets can be found in \cite{barg13}.
\begin{thm}\label{thm:LRS}
(Larman, Rogers, and Seidel \cite{lar77}). Let $S$ be a spherical two-distance
set in $\mathbb{R}^n$. If $|S| > 2n + 3$ and $a > b$, then $b = \frac{ka-1} {k-1}$
for some integer $k$
such that $2 \leq k \leq \frac {1 + \sqrt{2n}}2$.
{\bold e}nd{thm}
The condition $|S| > 2n+ 3$ was improved to $|S| > 2n+ 1$ by Neumaier \cite{neu81}.
If the spherical two-distance set gives rise to equiangular lines, then $a = -b$. So
Theorem \ref{thm:LRS} implies that $a = \frac 1 {2k-1}$, where $k \in \mathbb{N}$ , which is the statement of the Neumann
theorem in \cite{lem73}. The assumption of Theorem \ref{thm:LRS} is more restrictive than that of Neumann's theorem, but in return we obtain an upper bound on k. For instance when $n=19$, the common angle has to be $\frac 1 3$ or $\frac 1 5$ which cannot be deduced from Neumann's Theorem.
A finite collection of vectors $S = \{x_i \}_{i=1}^M \subset \mathbb{R}^n$
is called a finite frame for the Euclidean space
$\mathbb{R}^n$ if there are constants $0 < A \leq B < \infty$ such that for all $x \in \mathbb{R}^n$
$$ A||x||^2 \leq \sum_ { i=1}^M |\langle x, x_i \rangle |^2 \leq B||x||^2.$$
If $A = B$, then $S$ is called a tight frame. Benedetto and Fickus \cite{ben03} introduced a useful parameter of the frame, called the {{\bold e}m frame potential}. For our purposes it
suffices to define it as $ FP(S)=\sum_{i,j=1}^{M}|\langle
x_i,x_j\rangle|^2.$ We can derive the lower bounds of frame potential and the minimizers are tight frames.
\begin{thm}{\cite[Theorem.6.2]{ben03}}\label{thm:BF}
If $S$ is a set of unit vectors $\{x_i\}_{i=1}^M$ in $\mathbb{R}^n$ and $M>n$, then
\begin{equation}\label{eq:fp}
FP(S)\ge \frac {M^2} n
{\bold e}nd{equation}
with equality if and only if $S$ is a tight frame.
{\bold e}nd{thm}
If the set $S$ is a tight frame and equiangular, i.e. $|\langle x_i, x_j \rangle |=c$ for all $ i \neq j$, then $S$ is called an equiangular tight frame (ETF). ETFs have many nice properties. For instnace, they are Grassmanian frames \cite{str03} and attain the classical {\bold e}mph{Welch bound}\cite{wel74}. The {\bold e}mph{Welch bound} is the famous lower bound on the coherence.
\begin{thm}
If we have a set of unit vectors $S=\{x_i\}_{i=1}^M$ in $\mathbb{R}^n$, then
$$ \max_{ i \neq j} |\langle x_i, x_j\rangle | \geq \sqrt{\frac{M-n}{n(M-1)}},$$
where equality holds if and if $S$ is an ETF.
{\bold e}nd{thm}
When $M$ and $n$ are given for an ETF, the common angle is determined as $\sqrt{\frac{M-n}{n(M-1)}}$. We use ETF(n, M, c) to denote $M$ points ETF in $\mathbb{R}^n$ with the common angle $c$.
ETFs are closely related to strongly regular graphs (SRGs) which form the main source of their constructions. A regular graph of degree $k$ on $v$ vertices is called strongly regular if every two adjacent vertices
have $\lambda$ common neighbors and every two non-adjacent vertices have $\mu$ common neighbors.
Below we denote such strongly regular graph by srg$(v, k, \lambda, \mu)$ .
Waldron \cite{wal09} proved that the existence of ETFs is equivalent to the existence of SRGs with certain parameters.
\begin{thm} \cite[Corollary 5.6]{wal09}
There exists an equiangular tight frame of $M>n+1$ vectors for $\mathbb{R}^n$ if and only there exists a strongly regular graph $G$ of the type
$$
\text{srg}(M-1,k,\frac{3k-M}2,\frac k 2 ),\quad k=\frac 1 2 M- 1 + (1-\frac M {2n}) \sqrt{\frac{n(M-1)}{M-n}}.
$$
{\bold e}nd{thm}
Consequently, we have the following lemma.
\begin{lem} \label{lem:srg75}
The srg$(75,32,10,16)$ exsits if and only if ETF$(19,76,1/5)$ exists.
{\bold e}nd{lem}
Furthermore, we can have two other SRGs connected to the existence of ETF(19,76,1/5).
\begin{lem} \label{lem:srg76}
If either srg$(76,30,8,14)$ or srg$(76,35,18,14)$ exsits then ETF$(19,76,1/5)$ exists.
{\bold e}nd{lem}
Lemma \ref{lem:srg76} is a new result and we have two different ways to prove it.
The first approach is based on the fact that the projection of the vertex set of an SRG onto a
non-trivial eigenspace is a spherical 2-distance set and a spherical 2-design \cite{car01}. Every spherical 2-design is a tight frame. Therefore, the projections of SRGs are {\bold e}mph{two-distance tight frames} which have been discussed in \cite{bgoy15}. We define the notion of general spherical $t$-designs as follows.
\begin{defn}\cite{del77b}
Let $\text{Harm}_t(\mathbb{R}^n)$ be the set of homogeneous harmonic polynomials of degree $t$ in $\mathbb{R}^n$. Let $t$ be a natural number. A finite subset $X$ of the unit sphere $S^{n-1}$
is called a spherical t-design if
$$\sum _{x \in X} f(x) = 0, \quad \forall f(x) \in \text{Harm}_j (\mathbb{R}^n), 1 \leq j \leq t.$$
{\bold e}nd{defn}
We are interested
in the minimum cardinality of a spherical design when $t$ and $n$
are given. Delsarte, Goethals and Seidel \cite{del77b} proved
that the cardinality of a spherical $t$-design $X$ is bounded
below,
$$
|X| \geq \binom {n+e-1}{n-1} + \binom {n+e-2}{n-1}, \quad |X| \geq 2 \binom {n+e-1}{n-1}
$$
for $t=2e$ and $t=2e+1$, where $e \in \mathbb{N}$. The
spherical $t$-design is called tight if the above bounds are
attained. If $X$ is a tight spherical $2s$-design, it is
immediately a maximum spherical $s$-distance set attaining
the linear programming bound in \cite{del77b}.
Also, $X$ is a spherical 2-design if and only if X is a tight frame with the center of mass at
the origin \cite{car01} \cite[Chapter 1]{yu14}.
\begin{thm}\cite{del77b}\cite[Theorem 5.3 and Proposition 5.1 ]{car01} \label{thm:2d2s}
Let G be a connected strongly regular graph which is not complete
multipartite, and let X be the projection of the vertex set of G onto a
non-trivial eigenspace, re-scaled to lie on the unit sphere. Then
X is a spherical two-distance set and a spherical 2-design.
{\bold e}nd{thm}
Notice that an SRG has two non-trivial eigenspaces. Therefore, every SRG gives rise to two different spherical two-distance sets which are also spherical 2-designs.
\begin{example}
If srg$(76,35,18,14)$ exists, then we will have two different spherical two-disance sets and 2-designs. The first, $S_1$, has 76 points in $\mathbb{R}^{19}$ with inner product values $\pm \frac 1 5$ and the second, $S_2$, has 76 points in $\mathbb{R}^{56}$ with inner product values $\frac {-3}{35}$ and $\frac 1 {20}$. Both of them are also spherical 2-designs.
{\bold e}nd{example}
$S_1$ gives rise to 76 equiangular lines in $\mathbb{R}^{19}$. Therefore, if srg$(76,35,18,14)$ exists, then ETF$(19,76,1/5)$ exists. By \cite[Proposition 3.1]{bgoy15}, the shifted 2-design of $S_2$ is a two-distance tight frame with inner product values $\pm \frac 1 {15}$ in one higher dimension ($\mathbb{R}^{57}$), i.e. it is an ETF(57,76,1/15). By the results of complementary equiangular tight frame \cite[Corollary 3.2]{cas13}, an ETF with $M$ elements in $\mathbb{R}^n$ exists if and only an ETF with $M$ elements in $\mathbb{R}^{M-n}$ exists. Therefore, ETF(57,76,1/15) exists if and only if ETF(19,76,1/5) exists.
Applying Theorem \ref{thm:2d2s} again, the projection of srg(76,30,8,14) results in an ETF(57,76,1/15) and a spherical two-distance set with 76 points in $\mathbb{R}^{18}$ with inner product values $\frac {-4}{15}$ and $\frac{7}{45}$. The shifted 2-design \cite{bgoy15} of the latter case has inner product values $\pm \frac 1 5$ in $\mathbb{R}^{19}$, i.e. it gives rise to ETF(19,76,1/5). Both of the projections of srg(76,30,18,14) give rise to ETF(19,76,1/5).
To summerize, if either srg(76,30,8,14) or srg (76,35,18,14) exsits, then ETF(19,76,1/5) exists. Lemma \ref{lem:srg76} follows.
The second approach is as follows : if the Gram matrix of an ETF has the regular property (i.e. each row has the same number of $c$), we can use two different SRGs to construct the Gram matrix of an ETF. Conversly, if an ETF has the regular property, based on the tight frame conditon for Gram matrix, it gives rise to two different SRGs. This approach has been discussed in \cite{yu14}, \cite{bgoy15} and \cite{fjg15}. Following this approach, we can use the adjacency matrix of srg(76,30,8,14) or srg (76,35,18,14) to construct the Gram matrix of ETF(19,76,1/5). We work indepedently and notice that \cite[Corollary 4.5]{fjg15} implies the same result for Lemma \ref{lem:srg76}.
\begin{thm} \cite[Corollary 4.5]{fjg15}
If there exists an srg$(v, k, \lambda, \mu)$ with $v = 4k - 2\lambda - 2\mu$ then there exists an
srg$(v - 1, k \frac{v-2k}{v-2k-1}, \frac{3k-v}2 +\frac{3k}{2(v-2k-1)} , \frac k 2 \frac{v-2k} {v-2k-1}).$
{\bold e}nd{thm}
\section{Main results}\label{sec:main}
In general, there are no constraints on the common angle of equiangular lines. However, if there are 76 equiangular lines in $\mathbb{R}^{19}$, the common angle has to be $\frac 1 5$.
\begin{lem} \label{lem1}
If there are 76 equiangular lines in $\mathbb{R}^{19}$, then the common angle of those equiangular lines is $\frac 1 5$.
{\bold e}nd{lem}
\proof
By Theorem \ref{thm:LRS}, since $76 > 2 \cdot 19+3$, then the common angle of 76 equiangular lines in $\mathbb{R}^{19}$ has to be $\frac 1 3$ or $\frac 1 5$. By Theorem 4.5 in \cite{lem73}, we know that if the common angle of equiangular lines is $\frac 1 3$ and $n \geq 15$, then the upper bound for such equiangular lines is $2n-2$. Since $76 > 2 \cdot 19-2$, then the common angle cannot be $\frac 13$. \qed
\begin{lem} \label{lem2}
If there are 76 equiangular lines in $\mathbb{R}^{19}$ with the common angle $\frac 1 5$, then ETF(19,76,1/5) exists.
{\bold e}nd{lem}
\proof
If there are 76 equiangular lines in $\mathbb{R}^{19}$ with the common angle $\frac 1 5$, then there exists a set of unit vectors $S= \{x_i\}_{i=1}^{76} \in \mathbb{R}^{19}$ such that $|\langle x_i,x_j\rangle| = \frac 1 5$ for all $ 1 \leq i \neq j \leq 76$. Then,
$$ FP(S) =\sum_{i,j=1}^{76}|\langle
x_i,x_j\rangle|^2 = 76+ 76 \cdot 75 \cdot (\frac{1}5)^2 = 76 \cdot 4 = \frac {76^2}{19}.$$
By Theorem \ref{thm:BF}, since FP($S$) attains equality {\bold e}qref{eq:fp}, then $S$ is an tight frame. Namely, $S$ is an equiangular tight frame, and hence ETF(19,76,1/5) exists. \qed
\begin{thm}\label{main}
There are no 76 equiangular lines in $\mathbb{R}^{19}$.
{\bold e}nd{thm}
\proof
By Lemma \ref{lem1} and \ref{lem2}, if there are 76 equiangular lines in $\mathbb{R}^{19}$, then ETF(19,76,1/5) exists. Furthermore, by Lemma \ref{lem:srg75}, there exists srg(75,32,10,16). However, this contradicts Azarija and Marc's result that there is no srg(75,32,10,16) \cite{aza15}.
\qed
\begin{cor}
srg$(76,30,8,14)$ and srg$(76,35,18,14)$ do not exist.
{\bold e}nd{cor}
\proof
By Lemma \ref{lem:srg76}, if either srg(76,30,8,14) or srg(76,35,18,14) exists, then ETF(19,76,1/5) exists, i.e. there are 76 equiangular lines in $\mathbb{R}^{19}$. It contradicts Theorem \ref{main}. \qed
A. V. Bondarenko, A. Prymak and D. Radchenko proved the nonexistence of srg(76,30,8,14) \cite{bon14}. Here, we use the connetion between an SRG and a two-distance set to offer an alternative proof. For srg(76,35,18,14), the paper \cite{aza15} indicated that the proof is obtained from personal communication with Haemers. Here, we offer the prove by the notion that the sphere embedding of an SRG will obtain a sphericla two-distance set which is also a spherical 2-design.
\begin{rem}\label{rem}
Recently, nonexistence of the srg $(95, 40, 12, 20)$ is proved in \cite{aza16}. Similar discussion also can prove that there are no 96 equiangular lines in $\mathbb{R}^{20}$ and the nonexistence of srg $(96,45,24,18)$ and srg$(96,38,10,18)$.
{\bold e}nd{rem}
\section{Discussion}
We are interested in determining the maximum number of equiangular lines in $\mathbb{R}^n$. For $n=2,3,5,6,7,21,22,23$ and $43$, the maximum equiangular lines are ETFs. Previously, $n=19$ and 20 are conjectured the existence of ETFs to attain the upper bounds 76 and 96 respectively. However, we prove the case $n=19$ not attained. For n=20, there are three SRGs connected to the existence of ETF(20,96,1/5). Using the same ideas in Lemma \ref{lem:srg75} and \ref{lem:srg76}, we can show that srg(95,40,12,20) exists if and only if ETF(20,96,1/5) exists and if srg(96,45,24,18) or srg(96,38,10,18) exists, then ETF(20,96,1/5) exists. However, by \cite{aza16}, we know none of them exist. (Ref: Remark \ref{rem})
We note that for several of the sets of parameters that correspond to open
cases in Table \ref{table:ETFs}, their cardinality matches the
best known upper bound on the size of equiangular line set in that
dimension (the semidefinite programming, or SDP, bound of
\cite{barg14}). If we know the existence of any SRGs in Table \ref{table:ETFs}, we will obtain new results for maximum equiangular lines in that dimension. Specifically, this applies to $n=42,45,46.$
For instance, in the case of $n=42$ the SDP bound gives $M=288$
and $c=1/7$ (it is not known whether a set of 288 equiangular
lines in $\mathbb{R}^{42}$ exists). Using our approach, we observe that
such a set could be constructed from srg(287,126,45,63), srg(288,140,76,60) and
srg(288,164,100,84). Unfortunately, neither of them is
known to exist (or not). Notice that in table \ref{table:ETFs}, we know the nonexistence of srg(540, 308,190,156). However, this is not sufficient to show the nonexistence of ETF(45, 540, 1/7). Therefore, the existence of 540 equiangular lines in $\mathbb{R}^{45}$ remains an open question. Furthermore, we want to connect the notion of tight spherical 5-designs with ETFs.
\begin{thm}[Gerzon]\cite{lem73}
If there are $M$ equiangular lines in $\mathbb{R}^n$, then
\begin{equation}\label{eq:gerzon}
M \leq \frac{n(n+1)}{2}
{\bold e}nd{equation}
{\bold e}nd{thm}
Currently, Gerzon's bounds are attained only for
$n=2,3,7,$ and $23.$
Note that if there are equiangular lines attaining the Gerzon bound, then the common angle is $\frac1{\sqrt{(n+2)}}$
\cite[Thm.3.5]{lem73}.
\begin{thm}\cite[Theorem 5.12]{del77b}
If $S$ is a tight spherical 5-design in $\mathbb{R}^n$, then $|S|=n(n+1)$ and the inner product values between each pair of points in $S$ are -1 and the zeros of the polynomial $C_2(x)=1+ \frac{ (n+2)(nx^2-1)}2.$
{\bold e}nd{thm}
The zeros of $C_2(x)$ are $\pm \frac{1}{\sqrt{n-2}}$. If $S$ is a spherical tight 5-design, then $S$ is antipodal and inner product values are -1 and $ \pm \frac{1}{\sqrt{n+2}}$. Therefore, tight spherical 5-designs will give arise to $\frac{n(n+1)}2$ equiangular lines in $\mathbb{R}^n$ and vice versa. By Neumann's Theorem, when $n>2$, $\frac 1{\sqrt{n+2}}= \frac 1 {2m+1}$, where $m \in \mathbb{N}$. Therefore, $n$ has to be an odd square minus two, i.e. $n= (2m+1)^2 -2$ for some positive integer $m$. The existence of a tight spherical 5-design in $\mathbb{R}^n$ is equivalent to the existence of an ETF$(n,\frac{n(n+1)}2, \frac{1}{\sqrt{n+2}})$. Futhermore, such ETFs minimize potential energy for each completely monotonic potential function, i.e. they are universal optimal codes \cite{ck07}. For instance, for the cases of $m=1$ and $2$, ETF(7,28,1/3) and ETF(23,276,1/5) form very nice configurations in the corresponding dimension. Based on the results of E. Bannai, A. Munemasa, and B. Venkov \cite{ban04} and Nebe and Venkov \cite{neb12}, there are no tight 5-designs in $\mathbb{R}^n$, where $n=(2m+1)^2-2$ with an infinite sequence of values of $m$ that begins with $m =3, 4, 6, 10, 12, 22, 38, 30, 34, 42, 46.$ For $m=3$ and $4$, we that ETF(47,1128,1/7) and ETF(79,3160,1/9) do not exist. Using the same ideas in Lemma \ref{lem:srg75} and \ref{lem:srg76}, we can show the nonexistence of srg(1127,640,396,320), srg(1128,644,400,324), srg(1128,560,316,240), srg(3159,1408,1064,702), srg(3160,1575,870,700) and srg(3160,1755,1050,880). Note that the first two cases are known to not exist in Brouwer's table \cite{bro15}.
By \cite[Theorem 5.11]{del77b}, tight spherical 4-designs in $\mathbb{R}^n$ are the maximum spherical two-distance sets with inner product values $\frac{-1 \pm\sqrt{n+3}}{n+2}$. Also, $n$ has to be odd square minus three. For instance, if $n=22$, the tight spherical 4-designs in $\mathbb{R}^{22}$ are the maximum spherical two-distance set in $\mathbb{R}^{22}$ with 275 points and inner product values are $\frac 1 6$ and $-\frac 14$. Such a spherical two-distance set can be obtained from the projection of srg(275,112,30,56) and the projection of 276 equiangular lines in $\mathbb{R}^{23}$. This observation may offer another point of view in recognizing that tight spherical 5-designs in $\mathbb{R}^n$ are equivalent to tight spherical 4-designs in $\mathbb{R}^{n-1}$. There are also more connections between tight spherical designs of hamonic index $T$ and spherical few-distance sets in \cite{ban09}, \cite{ban13}, \cite{oy14}, and \cite{zhu15}. In this discussion, we like to relate different notions for mathematicians who are interested in frame theory, SRGs, equiangular lines, spherical few-distance sets, spherical t-designs and some related topics in algebraic combinatorics.
\begin{table}\begin{center}
{\small \begin{tabular}{|c|c|c|c|c|} \hline
$n$ & $N$ & $c$ & comments\\
\hline
42&288&1/7& srg(287,126,45,63)(o)\\ &&& srg(288,140,76,60) (o)\\&&& srg(288,164,100,84) (o)\\
45&540&1/7 & srg(539,234,81,117)(o)\\ &&& srg(540,266,148,114) (o)\\&&& srg(540,308,190,156) (N)\\
46&736&1/7 & srg(735,318,109,159)(o)\\ &&& srg(736,364,204,156) (o)\\ &&& srg(736,420,260,212) (o)\\
\hline
{\bold e}nd{tabular}}
\vspace*{.1in}\caption{Parameter sets of possible maximum ETFs. The
label `o' means that the existence of an SRG with these parameters
is an open problem. `N' means that the srg does not exist.}\label{table:ETFs}
{\bold e}nd{center}
{\bold e}nd{table}
\section*{Acknowledgements.}
The author would like to thank Alexander Barg and Alexey Glazyrin whose suggestions and comments were of inestimable value for this paper. The author also thanks Ye-Kai Wang and Aditya Viswanathan for their valuable comments.
\providecommand{\bysame}{\leavevmode\hbox
to3em{\hrulefill}\thinspace} \providecommand{\href}[2]{#2}
\begin{thebibliography}{A}
\bibitem{aza15}
J. Azarija and T. Marc, {\bold e}mph{There is no $(75,32,10,16)$ strongly regular graph}, preprint,
http://arxiv.org/abs/1509.05933.
\bibitem{aza16}
J. Azarija and T. Marc, {\bold e}mph{There is no $(95, 40, 12, 20)$ strongly regular graph}, preprint,
http://arxiv.org/abs/1603.02032.
\bibitem{ban04}
E. Bannai, A. Munemasa, and B. Venkov, {\bold e}mph{The nonexistence of certain
tight spherical designs,} St. Petersburg Math. J. \textbf{16} (2005), 609-625.
\bibitem{ban09}
E. Bannaia and Et. Bannai, {\bold e}mph{A survey on spherical designs and algebraic combinatorics on spheres}, European J. of Comb, \text{Volume 30, Issue 6}, August (2009), 1392-1425
\bibitem{bgoy15}
A. Barg, A. Glazyrin, K. Okoudjou, and W.-H. Yu, {\bold e}mph{Finite two-distance tight
frames} Linear Algebra and its Applications, \textbf{475} (2015), 163-175.
\bibitem{barg13}
A. Barg and W.-H. Yu, {\bold e}mph{New bounds for spherical two-distance
set,} Experimental Mathematics, \textbf{22} (2013),
\href{http://www.tandfonline.com/doi/abs/10.1080/10586458.2013.767725#.VCJb9_l_txE}{187--194}.
\bibitem{barg14}
A. Barg and W.-H. Yu, {\bold e}mph{New bounds for equiangular lines}, Discrete Geometry and Algebraic Combinatorics, A. Barg and O. Musin, Editors, AMS Series: Contemporary Mathematics, \href{http://www.ams.org/books/conm/625/}{vol. 625}, (2014), 111-121.
\bibitem{ban13}
E. Bannai, T. Okuda, and M. Tagami, {\bold e}mph{Spherical designs of
harmonic index $t$,} J.~Approx.~Theory, \textbf{195} (2015), 1-18.
\bibitem{ben03}
J. J. Benedetto and M. Fickus, {\bold e}mph{Finite normalized tight
frames}, Advances in Computational Math, \textbf{Vol 18},
nos.~2-4, (2003), 357-385.
\bibitem{bon14}
A. V. Bondarenko, A. Prymak and D. Radchenko,
{\bold e}mph{Non-existence of $(76,30,8,14)$ strongly regular graph and some structural tools}, preprint,
http://arxiv.org/abs/1410.6748, Oct. 2014.
\bibitem{bro15}
A. E. Brouwer, {\bold e}mph{Parameters of Strongly Regular Graphs}, http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html, cited Nov 2015.
\bibitem{bur15}
B. Burk, {\bold e}mph{Bounds on equiangular lines and on related spherical codes}, preprint, http://arxiv.org/abs/1508.00136.
\bibitem{car01}
P. J. Cameron, {\bold e}mph{Strongly regular graph}, preprint,
http://designtheory.org/library/preprints/srg.pdf (2001).
\bibitem{cas13}
P. G. Casazza, D. Redmond and J. C. Tremain, {\bold e}mph{Real
equiangular frames,} preprint,
http://www.math.missouri.edu/~pete/pdf/136-Equiang-corrected.pdf
\bibitem{ck07}
H. Cohn and A. Kumar, {\bold e}mph{Universal optimal distribution of
points on spheres,} J. Amer. Math. Soc. \textbf{20} (2007),
99-148.
\bibitem{del77b}
P. Delsarte, J.~M. Goethals, and J.~J. Seidel, {\bold e}mph{Spherical codes and
designs}, Geometriae Dedicata \textbf{6} (1977), 363-388.
\bibitem{fjg15}
M. Fickus, J. Jasper, D. Mixon, J. D. Peterson and C. E. Watson, {\bold e}mph{Equiangular tight frames with centroidal symmetry}, preprint, http://arxiv.org/pdf/1509.04059v1.pdf.
\bibitem{grea14}
G. Greaves, J. H. Koolen, A. Munemasa, and F. Sz{\"o}ll{\H o}si,
{\bold e}mph{Equiangular lines in {E}uclidean spaces}, J. Combin. Theory Ser. A, \textbf{138} (2016) 208-235.
\bibitem{han48}
J. Haantjes, {\bold e}mph{Equilateral point-sets in elliptic two and three-dimensional spaces}, Nieuw Arch. Wisk., \textbf{22} (1948) 355-362.
\bibitem{lar77}
D.~G. Larman, C.~A. Rogers, and J.~J. Seidel, {\bold e}mph{On two-distance sets in
{E}uclidean space}, Bull. London Math. Soc. \textbf{9} (1977), 261-267.
\bibitem{lin66}
J. H. van Lint and J.J. Seidel, {\bold e}mph{Equiangular point sets in
elliptic geometry} Proc. Nedert. Akad. Wetensh. Series \textbf{69}
(1966), 335-348.
\bibitem{lem73}
P.~W.~H. Lemmens and J.~J. Seidel, {\bold e}mph{Equiangular lines},
Journal of Algebra \textbf{24} (1973), \href{http://www.sciencedirect.com/science/article/pii/0021869373901233}{494-512}.
\bibitem{neb12}
G. Nebe and B. Venkov, {\bold e}mph{On tight spherical designs},
St. Petersburg Math. J. 24 (2013) 485-491.
\bibitem{neu81}
A. Neumaier, {\bold e}mph{Distance matrices, dimension, and conference graphs}, Indag. Math., \textbf{43} (1981), no. 4, 385-391.
\bibitem{oy14}
T. Okuda and W.-H. Yu, {\bold e}mph{A new relative bound for equiangular lines and nonexistence of tight spherical designs of harmonic index 4}, to appear in European J. of Comb, preprint http://arxiv.org/abs/1409.6995
\bibitem{str03}
T. Strohmer, and R. W. Heath Jr., {\bold e}mph{Grassmannian frames with applications to coding and communication}, Applied and Computational Harmonic Analysis, \textbf{Volume 14, Issue 3}, (2003), 257-275
\bibitem{tay72}
D. E. Taylor, {\bold e}mph{Some topics in the theory of finite groups}, PhD thesis, University of Oxford
(1972).
\bibitem{wel74}
L. R. Welch, {\bold e}mph{Lower bounds on the maximum cross correlation of signals}, IEEE Trans. Inform. Theory, \textbf{20} (1974) 397-399.
\bibitem{yu14}
W.-H. Yu {\bold e}mph{Spherical two-distance set and related topics in harmonic analysis} Ph.D thesis, University of Maryland (2014).
\bibitem{wal09}
S. Waldron, {\bold e}mph{On the construction of equiangular frames from
graphs}, Linear Alg. and its applications \textbf{431}, no. 11
(2009), 2228-2242.
\bibitem{zhu15}
Y. Zhu, E. Bannai, Et. Bannai, K.-T. Kim, W.-H. Yu, {\bold e}mph{More on spherical designs of harmonic index $T$}, preprint, http://arxiv.org/abs/1507.05373.
{\bold e}nd{thebibliography}
{\bold e}nd{document} |
\begin{document}
\title{Quantum state protection in cavities}
\author{D. Vitali and P. Tombesi}
\address {Dipartimento di Matematica e Fisica, Universit\`a di
Camerino, via Madonna delle Carceri I-62032 Camerino \\
and Istituto Nazionale di Fisica della Materia, Camerino, Italy}
\author{G.J. Milburn}
\address{Physics Department, University of Queensland, St. Lucia, 4072,
Brisbane, Australia}
\date{\today}
\maketitle
\begin{abstract}
We show how an initially prepared quantum state of a radiation mode in a
cavity can be preserved for a long time using a feedback scheme
based on the injection of appropriately prepared atoms. We present a
feedback scheme both for optical cavities, which can be continuously
monitored by a photodetector, and for microwave cavities, which can be
monitored only indirectly via the detection of atoms that have interacted
with the cavity field.
We also discuss the possibility of applying these methods for
decoherence control in quantum information processing.
\end{abstract}
\pacs{42.50.Lc, 03.65.Bz}
\section{Introduction}
Quantum optics is usually concerned with the generation of nonclassical
states of the electromagnetic field and their experimental detection.
However with the recent rapid progress in the theory of quantum information
processing
the {\it protection} of quantum states and their quantum dynamics also is
becoming a very important issue.
In fact what makes quantum information processing much more attractive
than its classical counterpart is its capability of using entangled
states and of processing generic linear
superpositions of input states. The entanglement between a pair of
systems is capable of connecting two observers separated by a space-like
interval, it can neither be copied nor eavesdropped on without
disturbance, nor can it be used by itself to send a classical message
\cite{bennet}.
The possibility of using linear superposition states has given rise to
quantum computation, which is essentially equivalent to have massive
parallel computation \cite{eke}.
However all these applications crucially
rely on the possibility of maintaining quantum coherence, that is, a
defined phase relationship between the different components of linear
superposition states, over long distances and for long times.
This means that one has to minimize as much as possible the effects of
the interaction of the quantum system with its environment and, in
particular, {\it decoherence}, i.e., the rapid destruction of the
phase relation between two quantum states of a system caused by the
entanglement of these two states with two different states of the
environment \cite{zur,leg1}.
Quantum
optics is a natural candidate for the experimental implementation of
quantum information processing systems,
thanks to the recent achievements in the manipulation of
single atoms, ions and single cavity modes. In fact two quantum gates
have been already demonstrated \cite{turchette,wine1} in quantum
optical systems and
it would be very important to develop strategies capable of {\it
controlling the decoherence} in experimental situations such
as those described in Refs.~\cite{turchette,wine1}.
The possibility of an experimental control of decoherence is important
also from a more fundamental point of view. In fact decoherence is the
practical explanation of why linear superposition of
macroscopically distinguishable states, the states involved in the famous
Schr\"odinger cat paradox \cite{cat}, are never observed and how the classical
macroscopic world emerges from the quantum one \cite{zur}.
In the case of macroscopic systems, the
interaction with the environment can never be escaped; since the
decoherence rate is proportional to the ``macroscopic separation''
between the two states \cite{zur,leg2,milwal}, a linear
superposition of macroscopically distinguishable states is
immediately changed into the corresponding statistical mixture, with
no quantum coherence left. Nonetheless, a full comprehension of the
fuzzy boundary between classical and quantum world is not yet
reached \cite{whee,tod}, and therefore the study of
``Schr\"odinger cat'' states in {\it mesoscopic}
systems where one can hope to observe the decoherence is
important. A first achievement has been obtained by Monroe {\it et
al.} \cite{wine}, who have prepared a trapped ${\rm ^{9}Be^{+}}$ ion
in a superposition of spatially separated coherent states and
detected the quantum coherence between the two localized states.
However, in this experiment the decoherence of the
superposition state has not been studied.
The progressive decoherence of a mesoscopic Schr\"odinger cat has been
observed for the first time in the experiment of Brune
{\it et al}. \cite{prlha}, where the linear
superposition of two coherent states of the electromagnetic field
in a cavity with classically distinct phases has been generated and
detected.
In this paper we propose a simple physical way to control decoherence
and protect a given quantum state against the destructive effects of the
interaction with the environment: applying an appropriate feedback.
We shall consider a radiation mode in a cavity as the quantum system to
protect and we shall show that the ``lifetime'' of an initial quantum
state can be significantly increased and its quantum coherence properties
preserved for quite a long time. The feedback scheme considered here
has a quantum nature, since it is based on the injection of an
appropriately prepared atom in the cavity and some preliminary aspects of the
scheme, and its
performance, have been described in Refs.~\cite{prlno,jmo}. The
present paper is a much more detailed description of our approach to
quantum state protection and is organized as follows. In Section II, the
main idea is presented and a continuous feedback scheme for optical
cavities is studied. In Section III, a possible application of this
continuous feedback scheme to quantum
information processing systems as the quantum phase gate of Ref.~
\cite{turchette} is presented. In the remaining sections, the stroboscopic
version of the continuous feedback scheme, more suited for the microwave
cavity of the Brune {\it et al.} experiment \cite{prlha}, and
first introduced in \cite{prlno}, is discussed in detail.
\section{A feedback loop for optical cavities}
Applying a feedback loop to a quantum system means subjecting it to a
series of measurements and then using the result of these
measurements to modify the dynamics of the system.
Very often the system is continuously monitored and
the associated feedback scheme provides a continuous control of the
quantum dynamics. An example is the measurement of an optical
field mode, such as photodetection and homodyne measurements, and for
these cases, Wiseman and Milburn have developed a quantum theory of
continuous feedback \cite{feed}. This theory has been applied in
Refs.~\cite{noi} to show that homodyne-mediated feedback
can be used to slow down the decoherence of a Schr\"odinger
cat state in an optical cavity.
Here we propose a different feedback scheme,
based on direct photodection rather than homodyne
detection. The idea is very simple: whenever the cavity looses a
photon, a feedback loop supplies the cavity mode with another
photon, through the injection of an appropriately prepared atom.
This kind of feedback is naturally suggested by the quantum
trajectory picture of a decaying cavity field \cite{Carmichael}, in which
time evolution is driven by the non-unitary evolution operator
$\exp\{-\gamma t a^{\dagger}a/2\}$ interrupted at random times by an
instantaneous jump describing the loss of a photon.
The proposed feedback almost instantaneously ``cures'' the effect
of a quantum jump and is able therefore to minimize the destructive effects of
dissipation on the quantum state of the cavity mode.
In more general terms, the
application of a feedback loop modifies the master
equation of the system and therefore it is equivalent to an effective
modification of the dissipative environment of the cavity field. For
example, Ref.~\cite{squee} shows that a squeezed bath \cite{qnoise}
can be simulated by the application of a feedback loop based on a
quantum non-demolition (QND) measurement of a quadrature of a cavity
mode. In other words, feedback is the main tool for realizing,
in the optical domain, the so called ``quantum reservoir
engineering'' \cite{poyatos}.
The master equation for continuous feedback has been
derived by Wiseman and Milburn
\cite{feed}, and, in the case of perfect detection via a single
loss source, is given by
\begin{equation}
\dot{\rho }= \gamma \Phi (a \rho a^{\dagger}) -\frac{\gamma
}{2}a^{\dagger}a \rho -\frac{\gamma }{2} \rho a^{\dagger}a \;,
\label{feedeq}
\end{equation}
where $\gamma $ is the cavity decay rate and $\Phi (\rho )$ is a generic
superoperator describing the
effect of the feedback atom on the cavity state $\rho$.
Eq.~(\ref{feedeq}) assumes perfect detection, i.e.,
all the photons leaving the cavity are absorbed by a unit-efficiency
photodetector and trigger the cavity loop. It is practically impossible to
realize such an ideal situation and therefore it is more realistic to
generalize this feedback master equation to the situation where
only a fraction $\eta < 1$ of the photons leaking out of the cavity
is actually detected and switches on the atomic injector. It is
immediate to see that (\ref{feedeq}) generalizes to
\begin{equation}
\dot{\rho }= \eta \gamma \Phi (a \rho a^{\dagger})
+(1-\eta) \gamma a \rho a^{\dagger}-\frac{\gamma
}{2}a^{\dagger}a \rho -\frac{\gamma }{2} \rho a^{\dagger}a \;.
\label{feedeq2}
\end{equation}
Now, we have to determine the action of the feedback atom on the cavity
field $\Phi (\rho )$; this atom has to release exactly one photon
in the cavity, possibly regardless of the field state in the cavity. In the
optical domain this could be realized using
{\it adiabatic transfer of Zeeman coherence} \cite{adia}.
\subsection{Adiabatic passage in a three level $\Lambda $ atom}
A scheme based on the adiabatic passage of an atom with Zeeman
substructure through overlapping cavity and laser fields has been
proposed \cite{adia} for the generation of linear superpositions of Fock
states in optical cavities. This technique allows for coherent
superpositions of atomic ground state Zeeman sublevels to be ``mapped''
directly onto coherent superpositions of cavity-mode number states.
If one applies this scheme in the simplest case of a three-level $\Lambda
$ atom one obtains just the feedback superoperator we are looking for, that is
\begin{equation}
\Phi (\rho ) =
a^{\dagger} (aa^{\dagger})^{-1/2} \rho (aa^{\dagger})^{-1/2} a \;,
\label{adiafi}
\end{equation}
corresponding to the feedback atom releasing exactly one photon into the cavity,
regardless the state of the field.
To see this, let us consider
a three level $\Lambda$ atom with two ground states $|g_{1}\rangle$
and $|g_{2}\rangle$, coupled to the excited state $|e\rangle $
via, respectively, a classical laser field $\Omega (t)$ of frequency
$\omega _{L}$, and a cavity field mode of frequency $\omega $. The
corresponding Hamiltonian is
\begin{eqnarray}
&& H(t)= \hbar \omega a^{\dagger}a +\hbar \omega_{eg} |e\rangle \langle e|
-i\hbar g(t) \left(|e\rangle \langle g_{2}| a-|g_{2}\rangle \langle e|
a^{\dagger}\right) \nonumber \\
&&+ i \hbar \Omega(t) \left(|e\rangle \langle
g_{1}|e^{-i\omega _{L} t}-|g_{1}\rangle \langle e| e^{i \omega _{L} t}\right)
\;.
\label{jcm}
\end{eqnarray}
The time dependence of $\Omega(t)$ and $g(t)$ is provided by the
motion of the atom across the laser and cavity profiles. This Hamiltonian
couples only states within the three-dimensional manifold spanned by
$|g_{1},n\rangle ,\,|e,n\rangle,\,|g_{2},n+1\rangle $, where $n$ denotes
a Fock state of the cavity mode. Of particular interest within this
manifold is the eigenstate corresponding to the adiabatic energy
eigenvalue (in the frame rotating at the frequency $\omega $)
$E_{n}=n\hbar \omega$,
\begin{equation}
|E_{n}(t)\rangle = \frac{g(t) \sqrt{n+1}|g_{1},n\rangle +\Omega(t)
|g_{2},n+1 \rangle }{\sqrt{\Omega ^{2}(t)+(n+1) g^{2}(t)}}
\end{equation}
which does not contain any contribution from the excited state and for
this reason is called the ``dark state''. This eigenstate exhibits the
following asymptotic behavior as a function of time
\begin{equation}
|E_{n}\rangle \rightarrow \left\{\matrix{
|g_{1},n\rangle & {\rm for} & \Omega(t)/g(t) \rightarrow 0 \cr
|g_{2},n+1\rangle & {\rm for} & g(t)/\Omega(t) \rightarrow 0 \cr} \right.
\label{asymp}
\end{equation}
Now, according to the adiabatic theorem \cite{messiah},
when the evolution from time $t_{0}$ to time $t_{1}$ is sufficiently slow, a
system starting from an eigenstate of $H(t_{0})$ will pass into the
corresponding eigenstate of $H(t_{1})$ that derives from it by continuity.
This means that if the atom crossing is such that adiabaticity is satisfied,
when the atom enters the interaction region in the ground state
$|g_{1}\rangle$, the following adiabatic transformation
of the atom-cavity system state takes place
\begin{eqnarray}
\label{reali}
&& |g_{1}\rangle \langle g_{1}| \otimes \sum_{n,m}\rho _{n,m}
|n\rangle \langle m| \\
&& \rightarrow |g_{2}\rangle \langle g_{2}| \otimes
\sum_{n,m}\rho _{n,m} |n+1\rangle \langle m+1| \nonumber \\
&& = |g_{2}\rangle \langle g_{2}| \otimes
a^{\dagger} (aa^{\dagger})^{-1/2} \rho (aa^{\dagger})^{-1/2} a \nonumber \;.
\end{eqnarray}
Roughly speaking, this transformation amounts to a single photon
transfer from the classical laser field to the quantized cavity mode
realized by the crossing atom, provided that a counterintuitive
pulse sequence in which the classical laser field $\Omega (t)$ is
time-delayed with respect to $g(t)$ is applied. Figure \ref{appa1}
shows a simple diagram of the feedback scheme, together with the
appropriate atomic configuration, cavity and laser field profiles
needed for the adiabatic transformation considered.
The quantitative conditions under which adiabaticity is satisfied are
obtained from the requirement that the transition from the dark state
$|E_{n}(t)\rangle $ to the other states be very small. One obtains
\cite{adia,chim}
\begin{equation}
\Omega_{max},g_{max} \gg T_{cross}^{-1} \;,
\end{equation}
where $T_{cross}$ is the cavity crossing time and $\Omega_{max},g_{max}$
are the two peak intensities.
The above arguments completely neglect dissipative effects due to cavity
losses and atomic spontaneous emission. For example, cavity dissipation
couples a given manifold $|g_{1},n\rangle ,\,|e,n\rangle,\,|g_{2},n+1\rangle $
with those with a smaller number of photons. Since ideal adiabatic
transfer occurs when the passage involves a single manifold, optimization
is obtained when the photon leakage through the cavity is
negligible during the atomic crossing, that is
\begin{equation}
T_{cross}^{-1} \gg \bar{n} \gamma \;,
\end{equation}
where $\bar{n}$ is mean number of photons in the cavity. On the contrary,
the technique of adiabatic passage is robust against the effects of
spontaneous emission as, in principle, the excited atomic state
$|e\rangle$ is never populated. Of course, in practice some fraction of
the population does reach the excited state and hence large values of
$g_{max}$ and $\Omega _{max}$ relative to the spontaneous emission rate
$\gamma _{e}$ are desirable.
To summarize, the quantitative conditions for a practical realization of
the adiabatic transformation (\ref{reali}) are
\begin{equation}
\Omega_{max},g_{max} \gg T_{cross}^{-1} \gg \bar{n} \gamma,
\gamma_{e} \;,
\label{condi}
\end{equation}
which, as pointed out in \cite{adia}, could be realized in optical cavity QED
experiments.
We note that when the adiabaticity conditions (\ref{condi}) are
satisfied, then also the Markovian assumptions at the basis of the feedback
master equation (\ref{feedeq2}) are automatically justified.
In fact, the continuous feedback theory of Ref.~\cite{feed}
is a Markovian theory derived assuming that
the delay time associated to the
feedback loop can be neglected with respect to the typical
timescale of the cavity mode dynamics. In the present
scheme the feedback delay time is due to the electronic trasmission time of the
detection signal and, most importantly, by the interaction time $T_{cross} $
of the atoms with the field, while the typical timescale of the cavity field
dynamics is $1/\gamma \bar{n}$. Therefore, the inequality on the right of
Eq.~(\ref{condi}) is essentially the condition for the validity of the Markovian
approximation and this {\it a posteriori} justifies our use of the Markovian
feedback master equation (\ref{feedeq2}) from the beginning.
\subsection{Properties of the adiabatic transfer feedback model}
When we insert the explicit expression (\ref{adiafi}) of the feedback
superoperator into Eq.~(\ref{feedeq2}), the feedback master equation
can be rewritten in the more transparent form
\begin{equation}
\dot{\rho }= \frac{(1-\eta) \gamma}{2}\left(2 a \rho a^{\dagger}-
a^{\dagger}a \rho - \rho a^{\dagger}a \right)
-\frac{\eta \gamma}{2} \left [ \sqrt{\hat{n}},\left[\sqrt{\hat{n}},
\rho \right] \right]
\label{sqroot}
\end{equation}
that is, a standard vacuum bath master equation with
effective damping coefficient $(1-\eta) \gamma $ plus an unconventional
phase diffusion term, in which the photon number operator is replaced by
its square root and which can be called ``square root of phase diffusion''.
In the ideal case $\eta=1$, vacuum damping vanishes
and only the unconventional phase diffusion survives. As shown in
Ref.~\cite{wise}, this is equivalent to say that ideal photodetection
feedback is able to transform standard photodetection into a quantum
non-demolition (QND) measurement of the photon number. In this ideal case,
a generic Fock state $|n\rangle$ is obviously preserved for an
infinite time, since each photon lost by the cavity triggers the feedback
loop which, in a negligible time, is able to give the photon back through
adiabatic transfer. However, the photon injected by feedback
has no phase relationship with the photons already present in the cavity
and, as shown by (\ref{sqroot}), this results in phase diffusion. An
alternative description of this phenomenon is that the photon
injection process is essentially a nonlinear number amplifier which is
necessarily accompanied by diffusion in the conjugate
variable \cite{Wagner93}.
This means that feedback does not guarantee perfect state protection for
a generic {\it superposition of number states}, even in
the ideal condition $\eta =1$. In fact in this case, only the diagonal matrix
elements in the Fock basis of the initial pure state are perfectly conserved,
while the off-diagonal ones always decay to zero, ultimately leading
to a phase-invariant state. However this does not mean that the
proposed feedback scheme is good for preserving number states only,
because the unconventional ``square-root of phase
diffusion'' is much slower than the conventional one (described by a
double commutator with the number operator).
In fact the time evolution of a generic density matrix element in the
case of feedback with ideal photodetection $\eta =1$ is
\begin{equation}
\rho _{n,m}(t)=\exp\left\{-\frac{\gamma
t}{2}\left(\sqrt{n}-\sqrt{m}\right)^{2}\right\} \rho _{n,m}(0) \;,
\label{sqrt2}
\end{equation}
while the corresponding evolution in the presence of
standard phase diffusion is
\begin{equation}
\rho _{n,m}(t)=\exp\left\{-\frac{\gamma
t}{2}\left(n-m\right)^{2}\right\} \rho _{n,m}(0) \;.
\label{phadif}
\end{equation}
Since
\begin{equation}
(n-m)^{2} \geq \left(\sqrt{n}-\sqrt{m}\right)^{2} =
\frac{(n-m)^{2}}{\left(\sqrt{n}+\sqrt{m}\right)^{2}} \;\;\;\;\;
\forall\;n,m \,
\label{ineq}
\end{equation}
each off-diagonal matrix element decays slower in the square root
case and this means that the feedback-induced unconventional phase
diffusion is slower than the conventional one.
A semiclassical estimation
of the diffusion constant can be obtained from the representation
of the master equation in terms of the Wigner function. When a generic
state is expanded in the Fock basis as
\begin{equation}
\rho =\sum_{n,m}\rho _{n,m}|n\rangle \langle m| \;,
\end{equation}
the corresponding Wigner function is given by \cite{luc} (in polar
coordinates $r,\theta $)
\begin{eqnarray}
\label{polar}
&& W(r,\theta )=\sum_{n}\rho _{n,n}\frac{2}{\pi }(-1)^n
e^{-2r^2}L_{n}(4r^2) \\
&& + 2 {\rm Re}\left\{\sum_{n\neq m} \rho _{n,m}\frac{2}{\pi }(-1)^n
\sqrt{\frac{n!}{m!}}e^{i\theta (m-n)} \right. \nonumber \\
&& \times \left. (2r)^{m-n} e^{-2r^2}L^{m-n} _{n}(4r^2)
\right\} \;,\nonumber
\end{eqnarray}
where $L_{n}^{m-n}$ are the generalized Laguerre polynomials
and using this expression it is easy to see that
\begin{equation}
-\left[n,\left[n,\rho \right]\right] \leftrightarrow \frac{\partial
^2}{\partial \theta ^2} W(r,\theta ) \;.
\label{derise}
\end{equation}
In the case of the square root of phase diffusion, one has instead
\begin{eqnarray}
\label{compli}
&&-\left[\sqrt{n},\left[\sqrt{n},\rho \right]\right] \leftrightarrow
2 {\rm
Re}\left\{\sum_{n\neq m} \rho _{n,m}\frac{2}{\pi }(-1)^n
\sqrt{\frac{n!}{m!}} \right. \\
&& \times \left.
\left(\sqrt{n}-\sqrt{m}\right)^{2}
e^{i\theta (m-n)} (2r)^{m-n} e^{-2r^2}L^{m-n} _{n}(4r^2)
\right\} \;;\nonumber
\end{eqnarray}
using (\ref{ineq}) and considering the semiclassical limit $n,m \gg 1$,
$n \sim m \sim \bar{n}\gg 1$, where $\bar{n}$ is the mean photon number,
(\ref{compli}) can be simplified to
\begin{equation}
-\left[\sqrt{n},\left[\sqrt{n},\rho \right]\right]
\leftrightarrow \frac{1}{4 \bar{n}}\frac{\partial
^2}{\partial \theta ^2} W(r,\theta ) \;,
\label{derise2}
\end{equation}
showing that (at least at large photon number)
in the case of the feedback-induced unconventional phase diffusion, the
diffusion constant is scaled by a factor $4 \bar{n} \gg 1$.
A complementary description of the feedback-induced phase diffusion
can be
given by the time evolution of the mean coherent amplitude $\langle
a(t) \rangle $. In fact, phase diffusion causes a decay of
this amplitude as the phase spreads around $2\pi $, even if the
photon number is conserved. In the presence of ordinary phase diffusion the
amplitude decays at the rate $\gamma /2$; in fact
\begin{equation}
\langle a(t) \rangle = {\rm Tr}\left\{a \rho(t)\right\}=
\sum _{n=0}^{\infty}\sqrt{n+1} \rho_{n+1,n}(t) \;,
\label{amedio}
\end{equation}
and using Eq.~(\ref{phadif}) one gets
$$
\langle a(t) \rangle =e^{-\gamma t/2} \langle a(0) \rangle \;.
$$
In the case of the square root of phase diffusion,
Eqs.~(\ref{sqrt2}) and (\ref{amedio}) instead yield
\begin{equation}
\langle a(t) \rangle = {\rm Tr}\left\{a(t) \rho(0)\right\} \;,
\label{amedisq}
\end{equation}
where the Heisenberg-like time evolved amplitude operator $a(t)$ is
given by
\begin{equation}
a(t) = \exp\left\{-\frac{\gamma t}{2}\left(\sqrt{a a
^{\dagger}}-\sqrt{a^{\dagger} a}\right)^{2}\right\} a \;.
\label{amediti}
\end{equation}
In the semiclassical limit it is reasonable to assume
a complete factorization of averages
(\ref{amedisq}), so to get
\begin{equation}
\langle a(t) \rangle = \exp\left\{-\frac{\gamma
t}{2}\left(\sqrt{\bar{n}+1}-\sqrt{\bar{n}}\right)^{2}\right\}
\langle a(0) \rangle \;,
\label{amediti2}
\end{equation}
which, in the limit of large mean photon number $\bar{n}$, yields a
result analogous to that of Eq.~(\ref{derise2})
\begin{equation}
\langle a(t) \rangle = \exp\left\{-\frac{\gamma
t}{8 \bar{n}}\right\} \langle a(0) \rangle \;.
\label{amediti3}
\end{equation}
This slowing down of phase diffusion (similar to that taking place
in a laser well above threshold) means that,
when the feedback efficiency $\eta $ is not too
low, the ``lifetime'' of generic pure quantum states of
the cavity field can be significantly increased with respect to the
standard case with no feedback (see Eq.~\ref{sqroot}).
\subsection{Description of the dynamics in the presence of feedback}
For a quantitative characterization of how the feedback scheme is able
to protect an initial pure state we study the fidelity $F(t)$
\begin{equation}
F(t) = {\rm Tr}\left\{\rho(0) \rho(t)\right\}
\label{fido}
\end{equation}
i.e., the overlap between the final and the initial state $\rho(0)$
after a time $t$. In general $0\leq F(t)\leq 1$. For an initially pure state
$|\psi
(0)\rangle$, $F(t)$ is in fact the probability to find the system in the intial
state at a later
time. A decay to an asymptotic limit is given by the overlap $\langle \psi (0)
|\rho
(\infty)|\psi (0)\rangle$.
A clear demonstration of the protection capabilities of the proposed
feedback scheme is given when considering the preservation of initial
Schr\"odinger cat state,
i.e., the typical example of nonclassical state whose oscillating and
non-positive definite Wigner
function is a clear signature of quantum coherence \cite{zur}.
In fact, if the initial state is an even $(+)$ or odd $(-)$
Schr\"{o}dinger cat state
\begin{equation}
\label{eocat}
|\alpha_{\pm}\rangle= N_{\pm}\left
(|\alpha\rangle \pm |-\alpha\rangle\right )
\end{equation}
where
\begin{equation}
N_{\pm}^{-2}=2\left(1\pm e^{-2 |\alpha |^{2}}\right) \;,
\label{npm}
\end{equation}
the corresponding fidelity $F(t)$ in absence of feedback ($\eta =0$
in (\ref{sqroot})) is given by
\begin{eqnarray}
&&F_{\pm}(t) = \frac{1+e^{-2 |\alpha |^{2}(1-e^{-\gamma t})}}{2}
e^{-|\alpha |^{2}(1-e^{-\gamma t/2})^{2}} \nonumber \\
&& \times \left(\frac{1\pm
e^{-2 |\alpha |^{2}e^{-\gamma t/2}}}{1\pm e^{-2 |\alpha
|^{2}}}\right)^{2} \;.
\label{fideteo}
\end{eqnarray}
The corresponding function $F(t)$ in the presence of feedback can be
easily obtained from the numerical solution of the master equation
(\ref{sqroot}) and using the general expression
\begin{equation}
F(t) = \sum_{n,m}\rho_{n,m}^{*}(0) \rho_{n,m}(t) \;.
\label{fidege}
\end{equation}
The numerical results (Fig.~\ref{fidecat}) show that
$F(t)$ in the presence of feedback is, at any time, significantly
larger than the corresponding function in absence of feedback, even
when the photodetection efficiency $\eta$ is far from the ideal
value $\eta = 1$.
Figure \ref{fidecat} refers to an initial odd cat state with
$\alpha = \sqrt{5}$; the full line refers to the feedback model in the
ideal case $\eta =1$; the dotted line to the feedback case with $\eta
=0.75$, small dashes refer to the case $\eta =0.5$; big dashes
refer to $\eta =0.25$ and the dot-dashed line to the evolution in absence
of feedback ($\eta =0$).
As
expected, the preservation properties of the proposed scheme
worsen as the photedetection efficiency $\eta $ is decreased.
Nonetheless, Figure \ref{fidecat} clearly shows how
this photodetection-mediated feedback increases the ``lifetime'' of a
generic pure state in the cavity, in the sense that the probability
of finding the initial state at any time $t$ is larger
than the corresponding probability in absence of feedback.
A qualitative confirmation of how well an initial odd cat state with
$\alpha = \sqrt{5}$ is protected by feedback is given by Figure
\ref{wigcats}: (a) shows the Wigner function of the initial cat
state, (b) the Wigner function of the same cat state evolved for a
time $t = 0.2/\gamma $ in the presence of feedback ($\eta=1$) and (c) the
Wigner function of the same state again after a time $t=0.2/\gamma$,
but evolved in absence of feedback. This elapsed time is twice the
decoherence time of the Schr\"odinger cat state, $t_{dec}=(2 \gamma
|\alpha |^{2})^{-1}$ \cite{leg2,milwal}, i.e., the lifetime of the
interference terms in the cat state density matrix in the presence of the
usual vacuum damping. As it is shown by (c), this means that after
this short time the cat state has already lost the oscillating part
of the Wigner function associated to quantum interference and has
become a statistical mixture of two coherent states. This is no longer
true in the presence of our feedback scheme: (b) shows that,
after $t \sim 2 t_{dec}$, the state is almost indistinguishable
from the initial one and that the quantum wiggles of the Wigner function
are still well visible. The capability of the feedback scheme of
preserving the quantum coherence of the initial cat state for quite a
long time is shown also by Fig.~\ref{wigcats2}, in which the Wigner
function both in the presence ($\eta =1$) (a) and in absence of feedback
(b) of the
initial odd cat state of Fig.~\ref{wigcats} evolved after one relaxation
time $t=\gamma ^{-1}$ is shown. In the presence of feedback the
oscillating part between the two peaks is still visible, even if the
state begins to be distorted with respect to the initial one because of
the action of the unconventional phase diffusion which makes it more
``rounded''. We obtain, so to speak, a ``mangy'' cat.
Another clear example of how the quantum coherence associated to
nonclassical superposition states of the radiation field inside the
cavity is well preserved by the feedback scheme based on the adiabatic
passage, is given by the study of the evolution of linear
superpositions of two Fock number states
\begin{equation}
|\psi(0)\rangle = \alpha |n\rangle +\beta |m \rangle \;.
\label{foklin}
\end{equation}
These states have not been experimentally generated in optical
cavities yet, but there are now a number of proposals for their generation
\cite{fok,eberly}. In this case $F(t)$ can be easily evaluated analytically
($m>n$)
\begin{eqnarray}
&& F(t) = |\alpha|^{4}e^{-n (1-\eta)\gamma t}+|\beta|^{4}e^{-m (1-\eta)\gamma t}
\nonumber \\
&&+2 |\alpha|^{2} |\beta|^{2} e^{-\gamma t \left[(m+n)/2-\eta \sqrt{n
m}\right]} \\
&& +|\alpha|^{2} |\beta|^{2} e^{-n (1-\eta)\gamma t}
\left(1-e^{-(1-\eta)\gamma t}\right)^{m-n}\frac{m! n!}{(m-n)!} \nonumber
\end{eqnarray}
and when this expression is plotted for different values of $\eta$
and compared with that in absence of feedback ($\eta =0$), we see, as
in Figure \ref{fidecat}, a significant increase of the ``lifetime'' of
the state (\ref{foklin}).
This comparison is shown in Figure \ref{fidefok},
which refers to the initial state $(|2 \rangle +\sqrt{2} |4\rangle
)/\sqrt{3}$ and where the notation is as in Fig.~\ref{fidecat}:
the full line refers to the feedback model in the
ideal case $\eta =1$; the dotted line to the feedback case with $\eta
=0.75$, small dashes refer to the case $\eta =0.5$; big dashes
refer to $\eta =0.25$ and the dot-dashed line to the evolution in absence
of feedback ($\eta =0$).
\section{Optical feedback scheme for the protection of qubits}
Photon states are known to retain their phase coherence over considerable
distances and for long times and for this reason high-Q optical cavities
have been proposed as a promising example for the realization of simple
quantum circuits for quantum information processing. To act as an
information carrying quantum state, the electromagnetic fields must
consist of a superposition of few distinguishable states. The most
straightforward choice is to consider the superposition of the vacuum and
the one photon state $\alpha |0\rangle +\beta |1\rangle $. However it is easy
to understand that this is not convenient because any interaction
coupling $|0\rangle $ and $|1\rangle $ also couples $|1\rangle $ with
states with more photons and this leads to information losses. Moreover
the vacuum state is not easy to observe because it cannot be
distinguished from a failed detection of the one photon state. A more
convenient and natural choice is {\it polarization coding}, i.e., using
two degenerate polarized modes and qubits of the following form
\begin{equation}
|\psi \rangle = \left(\alpha a_{+}^{\dagger}+\beta a_{-}^{\dagger}\right)
|0\rangle = \alpha |0,1\rangle + \beta |1,0\rangle \;,
\label{twoqub}
\end{equation}
in which one photon is shared by the two modes \cite{sten}.
In fact this is a ``natural'' two-state system, in which the two basis
states can be easily distinguished with polarization measurements;
moreover they can be easily transformed into each other using polarizers.
Polarization coding has been already employed in one of the few
experimental realization of a quantum gate, the quantum phase gate
realized at Caltech \cite{turchette}. This experiment has demonstrated
conditional quantum dynamics between two frequency-distinct fields
in a high-finesse optical cavity. The implementation of this gate
employs two single-photon pulses with frequency separation large
compared to the individual bandwidth, and whose internal state is
specified by the circular polarization basis as in (\ref{twoqub}).
The conditional dynamics between the two fields is obtained
through an effective strong
Kerr-type nonlinearity provided by a beam of cesium atoms.
In the preceding section we have shown that the proposed feedback scheme
is able to increase the ``lifetime'' of linear
superpositions of Fock states. Therefore it is quite natural to look
if our scheme can be used to protect qubits like those of the Caltech
experiment, against the destructive effects of cavity damping. To be more
specific, here we shall not be concerned with the protection of the
quantum gate dynamics, but we shall focus on a simpler but still
important problem: protecting an unknown input state for the longest
possible time against decoherence. For this reason we shall not consider
the two interacting fields, but a single frequency mode with a generic
polarization, i.e., a single qubit. We shall consider a class of
initial states more general than those of Eq.~(\ref{twoqub}), i.e.,
\begin{equation}
|\psi \rangle = \alpha |n,m\rangle + \beta |m,n\rangle \;,
\label{twoqubgen}
\end{equation}
where $m+n$ photons are shared by the two polarized modes.
If we want to apply the adiabatic transfer feedback scheme described
above for protecting qubits as those of
Eq.~(\ref{twoqubgen}), one has to consider a feedback loop
as that of Fig.~1 for each polarized mode. This can be done
using polarization-sensitive detectors which electronically control
the polarization of the classical laser field and the initial state
of the injected atoms. In fact one has to release in the cavity
a left or right
circularly polarized photon depending on which detector has fired
and this can be easily achieved when the $|g_{1}\rangle \rightarrow |e\rangle $
and $|g_{2}\rangle \rightarrow |e\rangle $ transitions are characterized
by opposite angular momentum difference
$\Delta m_{J} = \pm 1$. In this case a left polarized photon,
for example, is given back to the cavity with the adiabatic
transition $|g_{1}\rangle \rightarrow |g_{2}\rangle $ of Fig.~1, while
the right polarized one is released into the cavity
through the reversed adiabatic transition
$|g_{2}\rangle \rightarrow |g_{1}\rangle $ and the two possibilities
are controlled by the polarization sensitive detectors.
Since the input state we seek to protect is unknown, the protection capabilities
of the feedback scheme are better characterized by the minimum fidelity,
i.e., the fidelity of Eq.~(\ref{fido}) minimized over all
possible initial states. This minimum fidelity can be easily evaluated
by solving the master equation (\ref{sqroot}) for each polarized
mode and one gets the following expression
\begin{equation}
F_{min}(t)=\frac{1}{2}\left(
e^{-(1-\eta )\gamma t (n+m)}+e^{-\gamma t (n+m-2\eta \sqrt{n m})}\right)
\;.
\label{minfid}
\end{equation}
In the absence of feedback ($\eta =0$), this expression becomes
$F_{min}(t)=\exp\{-\gamma t(n+m)\}$ showing that in this case, the
states most robust against cavity damping are those with the smallest number
of photons, $m+n=1$, i.e., the states of the form of Eq.~(\ref{twoqub}).
Moreover, in a typical quantum information processing situation, one has
to consider small qubit ``storage'' times $t$ with respect to $\gamma
^{-1}$ so to have reasonably small error probabilities in quantum
information storage. Therefore the protection capability of an optical
cavity with no feedback applied is described by
\begin{equation}
F_{min}(t) = 1-\gamma t \;.
\label{nofed}
\end{equation}
If we now consider the situation in the presence of feedback
(Eq.~(\ref{minfid})), the best protected states for a given nonzero
efficiency $\eta $, may be different from the states with only one
photon, $\alpha |0,1\rangle+\beta
|1,0\rangle$, and they depend upon the explicit value of the feedback efficiency
$\eta$. For the determination of the optimal qubit of the form of
(\ref{twoqubgen}) (i.e., the optimal values for $m$ and $n$), one has to
minimize the deviation from the perfect protection condition $F(t)=1$.
For $\gamma t \ll 1$ one gets
\begin{eqnarray}
&&\min_{m \neq n} \left[(2-\eta)(n+m)-2\eta \sqrt{nm}\right]
\nonumber \\
&& =\min_{m \neq n} \left[(\sqrt{m}-\sqrt{n})^{2}+(1-\eta )
(\sqrt{m}+\sqrt{n})^{2}\right] \\
&& =\min_{n \geq 0,\, p \geq 1} \left[\frac{p^{2}}{(\sqrt{n+p}+\sqrt{n})^{2}}
+(1-\eta )
(\sqrt{n+p}+\sqrt{n})^{2}\right] \nonumber \;,
\end{eqnarray}
where $p=m-n$.
From these expression it can be easily seen that one has
to choose $p=1$, and
therefore the optimal qubits are those of the form
\begin{equation}
|\psi \rangle =\alpha |n_{opt},n_{opt}+1\rangle +\beta
|n_{opt}+1,n_{opt}\rangle \;,
\label{larpho}
\end{equation}
where $n_{opt}$ is determined by the minimization condition
\begin{equation}
\min_{n \geq 0} \left[\frac{1}{(\sqrt{n+1}+\sqrt{n})^{2}}
+(1-\eta )
(\sqrt{n+1}+\sqrt{n})^{2}\right] \;.
\label{mincond}
\end{equation}
As long as
\begin{equation}
\eta \leq 2/(1+\sqrt{2}) \simeq 0.83 \;,
\label{p83}
\end{equation}
one has
$n_{opt}=0$ and therefore the situation is similar to that of the
no-feedback case: the states of the form (\ref{twoqub}) are the best
protected states
and the corresponding minimum fidelity is given by
\begin{equation}
F_{min}(t)=\frac{1}{2}\left(
e^{-(1-\eta )\gamma t}+e^{-\gamma t }\right) \simeq 1-\gamma t
\left(1-\frac{\eta }{2}\right) \;.
\label{minfid2}
\end{equation}
In this case, feedback leads to a very poor qubit protection
with respect to the no-feedback case and therefore our scheme proves
to be practically useless for the protection of single photon qubits of
(\ref{twoqub}) employed in the Caltech experiment of
Ref.~\cite{turchette}.
However, when the feedback efficiency $\eta$ becomes larger than 0.83,
the situation can improve considerably. In fact $n_{opt}$ becomes
nonzero and can become very large in the limit $\eta \rightarrow 1$,
and in this case the minimum fidelity decays very slowly. To be more
specific, $n_{opt}$ is approximately given by the condition
\begin{equation}
\left(\sqrt{n_{opt}+1}+\sqrt{n_{opt}}\right)^{2}=\left(1-\eta\right)^{-1/2}
\label{fgfg}
\end{equation}
and the corresponding small time behavior of $F_{min}(t)$ is given by
\begin{equation}
F_{min}(t) \simeq 1-\frac{\gamma t}{\left(\sqrt{n_{opt}+1}+\sqrt{n_{opt}}
\right)^{2}} \simeq 1- \gamma t \sqrt{1-\eta} \;.
\end{equation}
This means that in the limit of a feedback efficiency very close to
one, it becomes convenient to work with a large number of photons per
mode, since in this limit the probability of errors in the storage of quantum
information can be made very small. This can be easily understood
from Eq.~(\ref{sqroot}), because in this limit the square-root of
phase diffusion term prevails in the master equation and its quantum
state protection capabilities improve for increasing photon number
(see Eq.~(\ref{amediti3})).
In the ideal case $\eta =1$, $n_{opt}$ becomes infinite and therefore
the minimum fidelity can remain arbitrarily close to one.
It is convenient to work with the largest possible number of photons,
that is,
\begin{equation}
|\psi \rangle =\alpha |n,n+1\rangle +\beta |n+1,n\rangle \;\;\;\;\; n \gg1
\label{larpho2}
\end{equation}
and the corresponding minimum fidelity is
$$
F_{min}(t)\simeq \frac{1}{2}\left(1+
e^{-\gamma t/4n}\right) \simeq 1-\frac{\gamma t}{8n} \;.
$$
The feedback method proposed here to deal with decoherence in quantum
information processing is different from most of the proposals made in
this research field, which are based
on the so called quantum error correction codes \cite{error}, which
are a way to use {\it software} to preserve linear superposition states.
In our case, feedback allows a physical control of decoherence, through a
continuous monitoring and eventual correction of the dynamics and in this
sense our approach is similar in spirit to the approach of
Ref.~\cite{pell,mabuchi}.
The present feedback scheme
is not very useful in the case of one-photon qubits
(\ref{twoqub}) of the quantum phase gate experiment of
Ref.~\cite{turchette}; however it predicts a very good decoherence
control in the case of high feedback efficiency $\eta > 0.83$ and for
larger photon numbers (see Eq.~(\ref{larpho})). It is very difficult
to achieve these experimental conditions with the present technology,
but our scheme could become very promising in the future.
\section{A feedback scheme for microwave cavities}
In the case of measurements of an optical
field mode, such as photodetection and homodyne measurements, the system
is {\it continuously} measured and in these cases applying a feedback loop
can be quite effective in controlling the decoherence of an
optical Schr\"odinger cat.
It is therefore quite natural to see if a similar control of
decoherence can be achieved in the only (up to now) experimental
generation and detection of Schr\"odinger cat states of a radiation mode,
the experiment of Brune {\it et al} \cite{prlha}. However, in this experiment,
it is not possible
to monitor continuously the state of the radiation in the cavity, since
the involved field is in the microwave range and there are not good
enough detectors in this wavelength region.
The detection of the cat state is obtained through
measurements performed on a second probe atom crossing the cavity after a
delay time $T$ and that provides a sort of impulsive measurements of the
cavity field state.
This suggests that in this microwave case,
continuous measurement can be replaced
at best by a series of {\it repeated} measurements, performed by
off-resonance atoms crossing the high-Q microwave cavity one by one with a
time interval $T$. As a consequence, one could try to apply a sort of
``discrete'' feedback scheme modifying in a ``stroboscopic''
way the cavity field dynamics according to the result of the
atomic detection.
\subsection{A simplified description of the experiment of
Brune et al.}
In Ref.~\cite{prlha}, a Schr\"odinger cat state for the microwave field
in a superconducting cavity $C$ has been generated using circular Rydberg
atoms crossing the cavity in which a coherent state has been
previously injected.
All the atoms have an appropriately selected velocity and the relevant
levels are two adjacent circular Rydberg states with principal
quantum numbers $n=50$ and $n=51$, which
we denote as $|g\rangle$ and $|e\rangle$ respectively. These two
states have a very long lifetime ($30$ ms) and a very strong coupling
to the radiation and
the atoms are initially prepared in the state $|e\rangle$.
The high-Q superconducting cavity is sandwiched between two low-Q
cavities $R_{1}$ and $R_{2}$,
in which classical microwave fields can be applied and which are
resonant with the transition between the state $|e\rangle$ and the
nearby lower circular state $|g\rangle$. The
intensity of the field in the first cavity $R_{1}$
is then chosen so that, for the
selected atom velocity, a $\pi/2$ pulse is applied to the atom as it
crosses $R_{1}$. As a consequence, the atomic state before entering
the cavity $C$ is
\begin{equation}
|\psi_{atom}\rangle = \frac{1}{\sqrt{2}}\left(|e\rangle
+|g\rangle\right) \;.
\label{at1}
\end{equation}
The high-Q cavity $C$ is slightly off-resonance with respect to the $e\,
\rightarrow \, g$ transition, with detuning
\begin{equation}
\delta = \omega - \omega_{eg}\;,
\label{detu}
\end{equation}
where $\omega $ is the cavity mode frequency and
$\omega_{eg}=(E_{e}-E_{g})/\hbar $. The Hamiltonian of the
atom-microwave cavity mode system is the usual Jaynes-Cummings Hamiltonian,
given by
\begin{eqnarray}
&&H_{JC}=E_{e}|e\rangle \langle e | + E_{g}|g\rangle \langle g |+
\hbar \omega a^{\dagger} a \nonumber \\
&&+\hbar \Omega \left(|e \rangle \langle g |a+|g \rangle \langle e |
a^{\dagger}\right) \;,
\label{jc}
\end{eqnarray}
where $\Omega $ is the vacuum Rabi coupling between the atomic dipole
on the $e\, \rightarrow \, g$ transition and the cavity mode.
In the off-resonant case and perturbative limit $\Omega \ll \delta $, the
atom and the field essentially do not exchange energy
but only undergo dispersive
frequency shifts depending on the atomic level \cite{harray,brune}, and
the Hamiltonian (\ref{jc}) becomes equivalent to
\begin{eqnarray}
&&H_{disp}=E_{e}|e\rangle \langle e | + E_{g}|g\rangle \langle g |+
\hbar \omega a^{\dagger} a \nonumber \\
&&+\hbar \frac{\Omega ^{2}}{\delta} \left(|g \rangle \langle g
|a^{\dagger}a
-|e \rangle \langle e | a a^{\dagger}\right) \nonumber \\
&&=\left(E_{e}-\hbar \frac{\Omega ^{2}}{\delta}\right)
|e\rangle \langle e | + E_{g}|g\rangle \langle g |+
\hbar \left(\omega +\frac{\Omega ^{2}}{\delta}\right)
a^{\dagger} a \nonumber \\
&&-2\hbar \frac{\Omega ^{2}}{\delta} |e \rangle \langle e |
a^{\dagger} a \;.
\label{heff}
\end{eqnarray}
This means that in this dispersive limit, besides a negligible shift
of the cavity frequency and of the $e$ level energy, the atom-field
interaction induces a phase shift $\phi = 2 \Omega ^{2}t_{int}/\delta $
when the atom is in the state $e$, while there is no shift when the
atom is in the state $g$ ($t_{int}$ is the interaction time).
Therefore, using (\ref{at1}), the state of the atom-field system when
the atom has just exited the cavity $C$ is the entangled state
\begin{equation}
|\psi_{atom+field}\rangle = \frac{1}{\sqrt{2}}\left(|e,\alpha e^{i\phi}
\rangle
+|g,\alpha \rangle\right) \;,
\label{atf1}
\end{equation}
where $\alpha$ denotes the coherent state initially present within
the cavity. In the experiment of Ref.~\cite{prlha}, different values
of the phase shift $\phi $
have been considered; however we shall restrict from now on
to the case $\phi=\pi$, which corresponds to the generation of a linear
superposition of two coherent states with {\it opposite} phases.
In the state (\ref{atf1}), each atomic state is correlated to a different
field phase; for the generation of a cat state, however, one has to
correlate each atomic state to a {\it superposition} of coherent
states with different phases, and this is achieved by submitting the
atom to a second $\pi/2$ pulse in the second microwave cavity $R_{2}$.
The $\pi/2$ pulse yields the following transformation
\begin{eqnarray}
|e\rangle & \rightarrow & \frac{1}{\sqrt{2}}\left(|e\rangle
+|g\rangle\right) \nonumber \\
|g\rangle & \rightarrow & \frac{1}{\sqrt{2}}\left(-|e\rangle
+|g\rangle\right) \;,
\label{pi2}
\end{eqnarray}
so that the state (\ref{atf1}) becomes
\begin{equation}
|\psi'_{atom+field}\rangle =
\frac{1}{\sqrt{2}}\left(N_{-}^{-1}|e\rangle|\alpha_-\rangle
+ N_{+}^{-1}|g\rangle |\alpha_+\rangle\right) \;.
\label{atf2}
\end{equation}
where $|\alpha _{\pm}\rangle $ are the even $(+)$ or odd $(-)$ Schr\"{o}dinger
cat states defined in (\ref{eocat}) and $N_{\pm}$ are defined in
Eq.~(\ref{npm}).
Eq.~(\ref{atf2}) shows that
an even or an odd coherent state is conditionally generated in the
cavity according to whether or not the atom is detected in the level $|g\rangle$
or $|e\rangle$, respectively.
After generation, the Schr\"odinger cat state undergoes
a vary fast decoherence process \cite{leg2,milwal},
that is, a fast decay of
interference terms, caused by the inevitable presence of
dissipation in the superconducting cavity. In fact the dissipative
time evolution of the generated cat state is described by the
following density matrix
\begin{eqnarray}
&&\rho(t)=\frac{1}{N_{\pm}^{2}}\left[|\alpha e^{-\gamma t/2}\rangle
\langle \alpha e^{-\gamma t/2}|+|-\alpha e^{-\gamma t/2}\rangle
\langle -\alpha e^{-\gamma t/2}| \right. \nonumber \\
&&\left.\pm e^{-2|\alpha|^{2}(1-e^{-\gamma
t})}\left(|-\alpha e^{-\gamma t/2}\rangle
\langle \alpha e^{-\gamma t/2}| \right. \right. \nonumber \\
&& \left. \left. +|\alpha e^{-\gamma t/2}\rangle
\langle -\alpha e^{-\gamma t/2}|\right)\right] \;,
\label{dec}
\end{eqnarray}
where
$\gamma$ is the cavity decay rate and where the plus (minus) sign
correspond to the even (odd) coherent state. Decoherence is governed by
the factor $\exp\left[-2|\alpha|^{2}(1-e^{-\gamma
t})\right]$, which for $\gamma t \ll 1$ becomes
$\exp\left[-2|\alpha|^{2}\gamma t\right]$, implying therefore that the
interference terms decay to zero with a
lifetime $t_{dec}=(2\gamma |\alpha |^{2})^{-1}$.
The relevance of the experiment of Brune {\it et al}. \cite{prlha}
lies in the fact that this progressive decoherence of the cat state
has been observed for the first time and the theoretical prediction
checked with no fitting parameters.
This monitoring of decoherence has been obtained by sending a
second atom through
the same arrangements of cavities. The atom has exactly the same
velocity of the first atom generating the cat and is sent through the
cavities after a time delay $T$, which is much larger
than the time of flight of the atom through the whole system (which is
of the order of $10^{-5} s$ in the experiment).
The state of the system composed by the second atom and the microwave
field undergoes the same transformation described above for the first
Rydberg atom, i.e.,
\begin{eqnarray}
\label{2at2}
&&\rho_{atom+field} \\
&&=U_{\frac{\pi}{2}}e^{i
\pi a^{\dagger}a |e\rangle \langle e|}U_{\frac{\pi}{2}}(\rho(T) \otimes
|e\rangle \langle e|) U_{\frac{\pi}{2}}^{\dagger}
e^{-i\pi a^{\dagger}a |e\rangle \langle
e|}U_{\frac{\pi}{2}}^{\dagger} \nonumber \;,
\end{eqnarray}
where $U_{\frac{\pi}{2}}$ describes the $\pi/2$ pulse and
$\rho(T)$ is the cavity field at a time $T$ after the passage
of the first atom and it is given by Eq.~(\ref{dec}).
Using (\ref{pi2}) one finally gets the state of the probe atom+field
system just before the field ionization detectors for the measurement
of the $e$ or $g$ atomic state, that is,
\begin{eqnarray}
&&\rho_{atom+field}= |e\rangle \langle e| \otimes \rho_{e} +
|g\rangle \langle g| \otimes \rho_{g} \nonumber \\
&& +|e\rangle \langle g| \otimes \rho_{+} +
|g\rangle \langle e| \otimes \rho_{-} \;,
\label{2at3}
\end{eqnarray}
where
\begin{eqnarray}
\rho_{\frac{g}{e}}&=&\frac{1}{4}\left[P\rho P + \rho \pm P \rho \pm \rho
P\right]
\label{offre} \\
\rho_{\pm}&=&\frac{1}{4}\left[P\rho P - \rho \pm P \rho \mp \rho
P\right] \;,
\end{eqnarray}
and
\begin{equation}
P = e^{\pm i \pi a ^{\dagger}a}
\label{pari}
\end{equation}
is the parity operator of the microwave cavity mode .
From these expressions, the probability of detecting the second atom in
the $e$ or $g$ state is readily obtained
\begin{equation}
P_{\frac{g}{e}}= \frac{1}{2}\left(1\pm \langle P \rangle \right) \;,
\label{prob}
\end{equation}
where $\langle P \rangle $ is the mean value of the
parity of the cavity mode state $\rho(T)$.
If one inserts in (\ref{prob}) the explicit expression of $\rho(T)$
given by (\ref{dec}), one gets the four conditional probabilities
$P_{ij}$, ($i,j=e$ or $g$), of detecting the second atom in the state
$j$ after detecting the first atom in the state $i$ and which
give a satisfactory description of the decoherence process of
the cat state in the cavity \cite{dav}.
Let us consider for example the case of
two successive detections of the circular Rydberg state $e$: in this
case the detection of the first atom projects the microwave field in
the superconducting cavity in an odd coherent state and the
corresponding conditional probability is given by
\begin{equation}
P_{ee}(T)=\frac{1}{2}\left[1-\frac{e^{-2|\alpha |^{2}e^{-\gamma T}}-
e^{-2|\alpha |^{2}\left(1-e^{-\gamma T}\right)}}{1-e^{-2|\alpha
|^{2}}}\right] \;.
\label{pee}
\end{equation}
The dependence of this conditional probability upon the time delay
between the two atom crossings gives a clear description of the cat
state decoherence. In fact, if there is no dissipation in the cavity,
i.e., $\gamma T =0$, it is $P_{ee}=1$ and this perfect correlation
between the atomic state and the cavity state is the experimental
signature of the presence of an odd coherent state in the high-Q
cavity. As long as $\gamma \neq 0$, the conditional probability
decreases for increasing delay time $T$. At a first stage one has a
decay to the value $P_{ee}=1/2$ in the decoherence time
$t_{dec}=1/2\gamma |\alpha|^{2}$; this is the decoherence process
itself, that is, the fast transition from the quantum
linear superposition state to the statistical mixture
\begin{equation}
\rho_{mixt}=\frac{1}{2}\left[|\alpha \rangle \langle \alpha |+
|-\alpha \rangle \langle -\alpha |\right]
\label{mixt}
\end{equation}
describing a {\it classical} superposition of fields with opposite
phases. At larger delays $T$, the plateau $P_{ee}=1/2$ turns to a
slow decay to zero because the two coherent states of the mixture both
tend to the vacuum state and start to overlap, due to field energy
dissipation \cite{dav}.
This conditional probability decay can be experimentally
reconstructed by sending a large number of atom pairs for each delay
time $T$, obtaining therefore a clear observation of the
decoherence phenomenon in its time development. Actually,
in Ref.~\cite{prlha}, the experimental demonstration of decoherence
has been given by considering not simply $P_{ee}$ but the difference
between conditional probabilities $\eta = P_{ee}-P_{ge}$.
\section{The stroboscopic feedback model}
We now propose a modification of the experiment of
Brune {\it et al}. \cite{prlha} in which the cat decoherence is not simply
monitored but also controlled in an active way. The idea is to apply
the same feedback scheme described above for optical cavities, which
gives a photon back to the cavity whenever the photodetector clicks.
However in this microwave case one has to find a different way to determine
if the cavity mode has lost a photon or not, because there are no good
photodetector available in this wavelength region.
Ref.~\cite{prlha} suggests using
off-resonant atoms crossing the cavity to measure the cavity field and
therefore
in this case one could replace continuous photodetection with a
stroboscopic measurement performed by a sequence of off-resonant
probe atoms, separated by a time interval $T$. A sort of indirect
microwave photodetection can be obtained by using the fact that, as
suggested by Eq.~(\ref{prob}), the detection of the $e$ or $g$ atomic
level is equivalent to the measurement of the parity of the cavity
mode state. In fact, Eq.~(\ref{offre}) for the conditioned
cavity mode density matrices $\rho_{g/e}$ can be rewritten in the
following way
\begin{eqnarray}
\rho_{e} &=& P_{odd}\rho P_{odd}
\label{offre21} \\
\rho_{g} &=& P_{even}\rho P_{even} \;,
\label{offre22}
\end{eqnarray}
where $P_{odd}$ ($P_{even}$) is the projector onto the subspace
with an odd (even) number of photons and therefore finding the atom
in the state $e$ ($g$) means measuring a parity $P=-1$ ($P=+1$) for
the state of the microwave mode within the cavity $C$.
To fix the ideas, let us consider from now on the case when the cat
state generated by the first off-resonant atom is an odd coherent
state (first atom detected in $e$). When a second probe atom crosses
the cavities arrangement after a time interval $T$ and is detected in
$e$, it means that the cavity mode state has remained in the odd
subspace, or, equivalently, that the cavity has lost an {\it even}
number of photons. If the time interval $T$ is much smaller than the
cavity decay time $\gamma^{-1}$, $\gamma T \ll 1$, then the
probability of loosing two or more photons is negligible and one can
say that finding the state $e$ means that no photon has leaked out
from the high-Q cavity $C$. On the contrary, when the probe atom is
detected in $g$, the cavity mode state is projected into the even
subspace and this is equivalent to say that the cavity has lost an
{\it odd} number of photons. Again, in the limit of enough closely
spaced sequence of probe atoms, $\gamma T \ll 1$, the probability of
loosing three or more photons is negligible and therefore finding the
level $g$ means that one photon has exited the cavity.
Therefore, for achieving a good protection of the initial odd cat
state, the feedback loop
has to supply the superconducting cavity with a photon whenever the
probe atom is detected in $g$, while feedback must not act
when the atom is detected in the $e$ state.
This feedback loop can be realized with a switch connecting the
$g$ state field-ionization detector with another atom injector, sending
an atom in the excited state $e$ into the high-Q cavity.
This feedback atom has to be {\it resonant} with the radiation mode in the
superconducting cavity and this can be obtained with another
switch turning on an electric field in the cavity $C$ when the atom enters it,
so that the level $e$ is Stark-shifted into resonance with the cavity mode.
A schematic representation of the experimental apparatus of
Ref.~\cite{prlha} together with the feedback loop is given by
Fig.~\ref{appaha}.
The time evolution of the microwave field in the high-Q cavity can be
described stroboscopically by the transformation from the state just
before the crossing of $n$-th non-resonant probe atom $\rho(nT)$,
to the state of the radiation
mode before the next non-resonant atom crossing $\rho(nT+T)$.
This transformation
is given by the composition of two successive mappings:
\begin{equation}
\rho(nT+T) =
\Phi(\rho(nT))=\Phi_{diss}\left(\Phi_{fb}(\rho(nT))\right)
\;, \label{mapp}
\end{equation}
where $\Phi_{fb}$ describes the effect of the interaction with the
non-resonant atom followed by the effect of the resonant feedback
atom, which interacts with the cavity field or not according to the
result of the measurement performed on the off-resonant atoms. The
operation $\Phi_{diss}$ describes instead the dissipative evolution
of the field mode during the time interval $T$ between two successive
atom injections and it is characterized by the energy relaxation rate
$\gamma$.
The feedback mechanism acts only on the density matrix $\rho_{g}$,
conditioned to the detection of level $g$ and is described by the
resonant interaction part of the Hamiltonian (\ref{jc})
\begin{equation}
H_{r}=\hbar \Omega \left(|e\rangle \langle g| a+|g\rangle \langle e|
a^{\dagger}\right) \;;
\end{equation}
the effect on the cavity mode density matrix $\rho$ is then given by
(the feedback atoms are not detected after exiting the microwave
cavity $C$)
\begin{equation}
\rho ' = {\rm
Tr_{at}}\left\{\exp\left\{-\frac{i}{\hbar}H_{r}\tau \right\}\left(|e\rangle
\langle e|\otimes
\rho
\right)\exp\left\{\frac{i}{\hbar}H_{r}\tau \right\} \right\} \;,
\end{equation}
where $\tau$ is the interaction time of the feedback atom.
Performing the trace, one gets
\begin{eqnarray}
\label{firo}
&&\rho ' =
\cos(\mu \sqrt{aa^{\dagger}})\rho \cos(\mu \sqrt{aa^{\dagger}}) \\
&&+a^{\dagger} (aa^{\dagger})^{-1/2}\sin(\mu \sqrt{aa^{\dagger}})\rho
\sin(\mu \sqrt{aa^{\dagger}}) (aa^{\dagger})^{-1/2} a \nonumber \;,
\end{eqnarray}
where $\mu =\Omega \tau $.
Then, we have to take into account the effect of the non-unit efficiency
of the atomic detectors $\eta $, which is of the order of $\eta =0.4$ in
the actual experiment. This means that the off-resonant atoms are not
detected with probability $1-\eta $ and when this happens, the feedback
loop does not act. Using both Eqs.~(\ref{offre}) and (\ref{firo}),
we derive the
explicit expression of the feedback operator $\Phi_{fb}$:
\begin{eqnarray}
\label{fb}
&&\Phi_{fb}(\rho) = \eta\rho_e +\eta\cos(\mu \sqrt{aa^{\dagger}})\rho_g
\cos(\mu \sqrt{aa^{\dagger}}) \\
&&+\eta a^{\dagger} \frac{\sin(\mu \sqrt{aa^{\dagger}})}{(aa^{\dagger})^{1/2}}
\rho_g
\frac{\sin(\mu \sqrt{aa^{\dagger}})}{(aa^{\dagger})^{1/2}} a
+(1-\eta)\left[\rho_e+\rho_g\right ] \;. \nonumber
\end{eqnarray}
In writing this expression we have implicitely assumed that not only
the off-resonant atom time of flight, but also the feedback loop delay
time are much smaller than the typical timescales of the system and that
they can be neglected. This assumption is essentially equivalent to
the Markovian assumption made for the continuous photodetection
feedback described above and it simplifies considerably the discussion.
The operator $\Phi_{diss}$ describing the dissipative time evolution
between two successive atom crossings can be obtained from the exact
evolution of a cavity in a standard vacuum bath \cite{herzog} and it
can be written as
\begin{equation}
\Phi_{diss}(\rho)=\sum_{k=0}^{\infty}A_{k}\rho A_{k}^{\dagger}\;,
\label{disso}
\end{equation}
where
\begin{equation}
A_{k}=\sum_{n=0}^{\infty}\sqrt{\frac{(n+k)!}{n! k!} e^{-n \gamma
T}\left(1-e^{-\gamma T}\right)^{k}} |n\rangle \langle n+k | \;.
\label{cnk}
\end{equation}
If we now use the explicit expressions (\ref{fb}) and (\ref{disso}),
we get the general expression of the transformation $\Phi$ of
Eq.~(\ref{mapp}),
which can be written for the density
matrix elements in the following way ($\langle n
|\Phi(\rho)|n+p \rangle = \rho'_{n,n+p}$):
\begin{eqnarray}
&&\rho'_{n,n+p}=\sum_{k=0}^{\infty}\left\{\frac{c_{n,k}c_{n+p,k}}{4}
\left[\eta s_-(n,k)^{2} +4(1-\eta )
\right. \right. \label{mapoele} \\
&&+ \eta s_+(n,k)^{2}
\cos(\mu\sqrt{n+k+1})\nonumber \\
&&\left . \mbox{}\cos(\mu\sqrt{n+p+k+1})\right]
\nonumber \\
&&+\eta \frac{c_{n,k+1}c_{n+p,k+1}}{4} s_{+}(n,k)^{2}
\sin(\mu\sqrt{n+k+1})\nonumber \\
&&\left .\mbox{}\sin(\mu\sqrt{n+p+k+1}) \right\}
\rho_{n+k,n+p+k} \nonumber \\
&&+ \eta \frac{c_{n,0}c_{n+p,0}}{4}\sin(\mu \sqrt{n})\sin(\mu \sqrt{n+p})
s_{-}(n,0)^{2} \rho_{n-1,n+p-1} \nonumber \;,
\end{eqnarray}
where
\begin{eqnarray*}
c_{n,k} & = & \sqrt{\frac{(n+k)!}{n! k!} e^{-n \gamma
T}\left(1-e^{-\gamma T}\right)^{k}}\\
s_\pm(n,k) & = &1\pm(-1)^{n+k} \;.
\end{eqnarray*}
An important aspect of the above equation is that the time evolution of a
given density matrix element depends only upon
the matrix elements with the same ``off-diagonal'' index $p$.
This implies in particular that only even values of $p$ can be
considered in (\ref{mapoele}), because one starts from an odd coherent
state and the matrix elements with $p$ odd, being zero initially,
remain zero at any subsequent time. To state it in other words,
if the initial state has a definite parity, the
dynamical evolution is such that the cavity mode state evolves
within the two subspaces with given parity and the projection into the
space with no definite parity always remains zero. We have already
used this fact in Eq.~(\ref{fb}) where we have written $\rho =
\rho_{e}+\rho_{g}$, since, as showed by (\ref{offre21}) and
(\ref{offre22}), these two matrices are just the odd and even
components of the density matrix.
Generally speaking, the parity of the cavity mode state plays such a
fundamental role that our stroboscopic feedback scheme is able to
protect only even and odd coherent states (we have considered an initial
odd cat state only, but the scheme can be simply adapted to the even case).
In fact one could generalize the scheme described above and consider
the generation of more general cat states. For example,
one can consider generic phase shifts $\phi\neq \pi $
(as it is done in \cite{prlha}) and generic microwave pulses in the two
cavities $R_{1}$ and $R_{2}$
\begin{eqnarray}
|e\rangle & \rightarrow & c_{e}|e\rangle
+c_{g} |g\rangle \nonumber \\
|g\rangle & \rightarrow & -c_{g}^{*}|e\rangle
+c_{e}^{*} |g\rangle \;,
\label{nopi2}
\end{eqnarray}
where $c_{e}$ and $c_{g}$ depend on the intensity and phase of the microwave
pulses in
$R_{1}$ and $R_{2}$ and on the interaction time.
This allows to generate a large class of linear
superpositions of coherent states with different phases, but only in the case
of cat
states with a given parity our stroboscopic scheme can be implemented.
In fact the essential condition for the stroboscopic protection
scheme to be applied is the existence of relations like (\ref{offre21}) and
(\ref{offre22}) in which the cavity mode states conditioned to the
detection of the two atomic levels are expressed as projections into
given, orthogonal subspaces. Only in this case in fact, is it
possible to correlate with no ambiguity one atomic detection with a
state or property of the cavity mode and then consequently apply a
feedback scheme. It is then easy to prove that the two microwave
pulses in $R_{1}$ and $R_{2}$ and the dispersive interaction in $C$
(see Eq.~(\ref{2at2}))
determine two projection operators only for the situation considered here,
($\phi =\pi $ and two $\pi/2$ pulses) and these projectors are just
the projectors into the even and odd subspace.
\section{Dynamics in the presence of stroboscopic feedback}
The experimental study of this stroboscopic feedback scheme can be done
performing a series of atomic detections of the state of the
off-resonant probe atoms separated by a given time interval $T$ and
repeating this series of measurements many times, always starting from
a first detection in the state $e$. This allows one to reconstruct the
time evolution of the probability of finding the state $e$,
$P_{e}(nT)$ (see Eq.~(\ref{prob})) in the presence of feedback.
The time evolution of this probability is plotted in Fig.~\ref{peef} where an
initial odd coherent state with $|\alpha |^{2}=3.3$ (just the value
corresponding to that of the actual experiment) is considered. The full
line refers to the no feedback case ($\mu =0$), that is, the theoretical
prediction of Eq.~(\ref{pee}); the dashed line refers to $\mu =\pi /6$
and $\gamma T=0.02$; the dotted line to $\mu =\pi /2$
and $\gamma T=0.02$; horizontal crosses to $\mu =\pi /2$
and $\gamma T=0.2$ and diagonal crosses to $\mu =\pi /6$
and $\gamma T=0.2$. In a) the ideal case of perfect atomic detection
is considered, while b) refers to the case $\eta =0.4$, which is the
actual efficiency of the detector employed in \cite{prlha}.
These two figures show the dependence on the three feedback
parameters $\gamma T$, $\mu$ and $\eta$ and, as
expected, the most relevant one is the time between two
successive measurements $T$. This time has to be as small as possible,
because decoherence
can be best inhibited if one can ``check'' the cavity state,
and try to restore it, as soon as possible. Moreover we
have seen that the indirect measurement of the cavity with the atoms
becomes optimal only in the continuous limit $\gamma T \ll 1$ and only
in this limit (and for ideal detection efficiency $\eta=1$)
the initial photon number distribution is perfectly preserved.
The coupling parameter $\mu = \Omega \tau$ is instead
connected to the probability of releasing the
photon within the high-Q cavity. We have assumed that the feedback
atoms come from an independent source just to have the possibility
of varying their velocity and therefore the parameter $ \mu$.
This probability of releasing the photon in the cavity
is maximized when the sine term in
(\ref{fb}) is maximum, i.e., when
\begin{equation}
\mu \sqrt{n}=\pi (m+1/2)\;\;\;\; m \;\;{\rm integer}
\label{sqreso}
\end{equation}
This resonance condition depends on the photon number $n$
which however is not determined in general and moreover decreases as time
evolves
(when $\gamma T \neq 0$). In the case of the Schr\"odinger cat state
studied here, (\ref{sqreso}) roughly corresponds to the condition
$\mu |\alpha |=\pi (m+1/2)$ and this explains why at small times the
case $\mu =\pi/6$ gives a good result ($|\alpha |^{2}=3.3$ in the figures).
At longer times the value $\mu
=\pi/2$ gives the better result and this is due to the fact that the
cavity mean photon number has become approximately one. A complete
explanation of the asymptotic behavior of the curves of Fig.~\ref{peef}
is given by the fact that, as long as $T \neq 0$, the
stationary state of the cavity field is a mixture of the vacuum and the
one-photon state, given by
\begin{eqnarray}
&&\rho^{stat}=\frac{e^{\gamma T}-1}{e^{\gamma
T}-1+\eta \sin^{2}\mu }|0\rangle \langle 0| \nonumber \\
&& +\frac{\eta \sin^{2}\mu}{e^{\gamma
T}-1+\eta \sin^{2}\mu}|1\rangle \langle 1| \;.
\label{statio}
\end{eqnarray}
It is immediate to see that this means
\begin{equation}
P_{e}(\infty)=\rho^{stat}_{11}=\frac{\eta \sin^{2}\mu}{e^{\gamma
T}-1+\eta \sin^{2}\mu} \;.
\label{statio2}
\end{equation}
which is verified by the plots shown in Fig.~\ref{peef}.
The form of the stationary state can be obtained from the general
expression of the mapping (\ref{mapoele}). In fact, since the time
evolution of a given matrix element is coupled only to those with the
same off-diagonal index $p$, this mapping can be written in the simpler
form
\begin{equation}
\vec{V}'_{p} = A_{p} \vec{V}_{p} \;,
\label{sint}
\end{equation}
where $\rho _{n,n+p}$ is the $n$-th component of the vector
$\vec{V}_{p}$
and $A_{p}$ is a matrix whose expression can be obtained from
(\ref{mapoele}). The state of the cavity field after $K$ measurements
(and eventual feedback corrections) is therefore obtained applying the
matrix $A_{p}$ $K$ times. Since the evolution of the cavity field is
dissipative, one can easily check that all the eigenvalues $\lambda$
of the family $A_{p}$ are such that $|\lambda | \leq 1$. The
stationary state will correspond to the eigenvectors associated to the
eigenvalue $\lambda =1$. It is possible to see that there is only one
eigenvalue $\lambda =1$, for the matrix determining the evolution of the
diagonal elements $A_{0}$, and that the associated eigenvector is the one
corresponding to the diagonal stationary state of Eq.~(\ref{statio}).
At first sight, the comparison between the curves in the presence of
feedback, with $P_{e}$ remaining close to one, and that in
absence of feedback, seems to suggest that the initial odd cat state
can be preserved almost perfectly. However, this is an incorrect
interpretation because the quantity $P_{e}$ gives only
a partial information on the state of the radiation mode within the
cavity: it is a measurement of its parity (see Eq.~(\ref{prob}))
and Fig.~\ref{peef} only shows that our feedback scheme
is able to preserve almost perfectly the initial parity.
Perfect cat state ``freezing'' can be
realized only in cavities with an infinite $Q$; the proposed feedback
scheme inevitably modifies the initial state, even in the ideal conditions of
perfect detection efficiency $\eta =1$ and continuous feedback $\gamma T
\approx 0$. In fact the stroboscopic feedback model shows the same
behavior of the continuous feedback model discussed above for optical
cavities, which (when restricting to initial states with given parity)
represents its continuous measurement limit $\gamma T \rightarrow 0$.
It is characterized by
phase diffusion, because the photon left in the cavity by the resonant atom
has no phase relationship with those in the cavity. However this
phase diffusion proves to be slower than the usual phase
diffusion, so that also in this stroboscopic case, the protection of
the initial cat state is extremely good.
This is clearly shown by Fig.~\ref{strobo1},
where the Wigner function of the same initial odd coherent state
considered in Fig.~\ref{peef},
is plotted in a) and compared with the Wigner function
of the cavity state after a time
$t=0.44/\gamma$ ($t \sim 3 t_{dec}$) in the presence of feedback (b).
The two states are almost indistinguishable, even if in
Fig.~\ref{strobo1}b the actual experimental value $\eta =0.4$
is considered (the other parameters are $\mu =\pi /6$, $\gamma T=0.02$).
The comparison with Fig.~\ref{strobo1}c, where the Wigner function evolved
for the same time interval in {\it absence} of feedback is plotted,
clearly shows
the effectiveness of our scheme. Since $t\sim 3 t_{dec}$, the state in absence
of feedback has become a mixture of two coherent states
with opposite phases, and the oscillations associated to quantum
coherence have essentially disappeared.
On the contrary, the state evolved in presence of feedback is almost
indistinguishable from the initial one and the interference
oscillations are still very visible. Fig.~\ref{strobo1}b also shows that
the unconventional, feedback-induced phase diffusion is actually very
slow, since its effects are not yet visible after $t\sim 3 t_{dec}$.
The effects of phase diffusion begin to be visible after one relaxation
time $t=\gamma ^{-1}$, as shown by Fig.~\ref{strobo2}, where the Wigner
functions at this time, both in the presence (a) and in absence (b) of
feedback are compared (other parameter values are the same as in
Fig.~\ref{strobo1}). Quantum coherence is quite visible in (a), while it has
completely disappeared in (b); however the state in the presence of
feedback begins to distort with respect to the initial state, as
the two peaks associated with the two coherent state become broader
and more rounded due to phase diffusion.
\section{Concluding remarks}
In this paper we have presented a way for protecting a generic initial
quantum state of a radiation mode in a cavity against decoherence. The
initial quantum state is not perfectly preserved for an infinite time
(this is possible only in a cavity with an infinite Q); nonetheless its
quantum coherence properties can be preserved for a long time and the
``lifetime'' of the state significantly increased.
The model presented here is a ``physical'' way to control decoherence
based on feedback, that is, measuring the system and modify its dynamics
according to the result of the measurement. In this sense it is very
similar in spirit, to the proposals of Refs.~\cite{pell}.
Our approach is complementary to those based on quantum
error correction codes \cite{error}, using software to deal with
decoherence.
The present feedback acts in
a very simple way: one checks if the cavity has lost a photon, and when
this happens, one gives the photon back through the injection of an
appropriately prepared atom. In the case of a continuous monitoring of
the system and in the ideal limit of unit detector efficiency, the model
preserves perfectly the photon number distribution of the initial quantum
state of the cavity. This is obtained at the price of introducing an
unconventional phase diffusion, slower than the usual phase diffusion, (see
Eqs.~(\ref{sqroot}) and (\ref{derise2})), that modifies the state at
sufficiently long times. To be more specific, feedback
protects very well the relative phase of the
coefficients of the components of the initial state,
generating at the same time the diffusion of
the phase of the field.
The above description of the feedback scheme explicitely considers all
the experimental limitations (non-unit efficiency of the detectors,
comparison between the various timescales) except one:
here we have assumed that one has an extemely good control of the atomic
injection and that it is possible to send {\it exactly} one atom at a
time in the cavity. This is not experimentally possible at the moment:
for example, in \cite{prlha} sending an ``atom'' explicitely means
sending an atomic pulse with an average
number $\bar{n}\sim 0.2$, so that the probability of having
two atoms simultaneously in the cavity is negligible.
This fact makes the proposed feedback scheme much less effective; in fact,
this is essentially equivalent to have, in the stroboscopic case,
an effective quantum efficiency $\eta _{eff}=\eta \bar{n}^{2}$, because
one has a probability $\bar{n}^{2} \sim 0.04$ of having one probe atom
and one feedback atom in each feedback loop. As a consequence,
the dynamics in the presence of feedback becomes hardly
distinguishable from the standard dissipative evolution.
In the continuous feedback scheme for optical cavities one has the
feedback atomic beam only and the effective efficiency is $\eta \bar{n}$.
Anyway in the optical case, the problem of having {\it exactly} one feedback
atom at a time with certainty, could be overcome, at least in principle,
replacing the beam of
feedback atoms with a single fixed feedback atom, optically trapped by
the cavity (for a similar configuration, see for example
\cite{pell}). The trapped atom must have the same $\Lambda $
configuration described in Fig.~1 and the adiabatic photon transfer
between the classical laser field and the quantized optical mode could be
obtained with an appropriate shaping of the laser pulse $\Omega (t)$. The
possibility of simulating the adiabatic transfer with an appropriately
designed laser pulse has been recently discussed by
Kimble and Law \cite{pistol} in a proposal for the realization of a
``photon pistol'',
able to release exactly one photon on demand (see also \cite{eberly}).
In this case, the feedback loop would be simply activated by turning on the
appropriately shaped laser pulse focused on the trapped atom. During the time
interval between two photodetections, the atom
has to remain in the ``ready'' state $|g_{1}\rangle $,
which is decoupled from the cavity mode, and this could be obtained with
an appropriate recycling process, driven, for example, by supplementary
laser pulses \cite{pistol}.
\section{Acknowledgments}
This work has been partially supported by the Istituto Nazionale Fisica
della Materia (INFM) through the ``Progetto di Ricerca Avanzata INFM-CAT''.
\begin{figure}
\caption{Schematic diagram of the photodetection-mediated feedback scheme
proposed for optical cavities, together with the appropriate atomic
configuration for the adiabatic transfer.}
\label{appa1}
\end{figure}
\begin{figure}
\caption{Time evolution of the fidelity $F(t)$ for an initial
odd cat state with
$|\alpha |^{2}
\label{fidecat}
\end{figure}
\begin{figure}
\caption{(a) Wigner function of the initial odd cat
state, $|\psi \rangle= N_{-}
\label{wigcats}
\end{figure}
\begin{figure}
\caption{Wigner function of the odd cat state of Fig.~3,
evolved for a time $t = 1/\gamma $ in the presence of ideal
feedback $\eta=1$ (a), and in absence of feedback (b).}
\label{wigcats2}
\end{figure}
\begin{figure}
\caption{Time evolution of the fidelity $F(t)$ for the initial
superposition state of two Fock states $(|2\rangle +2^{1/2}
\label{fidefok}
\end{figure}
\begin{figure}
\caption{Schematic diagram of the stroboscopic feedback scheme
for the experiment of Brune {\it et al}
\label{appaha}
\end{figure}
\begin{figure}
\caption{Time evolution of the probability of detecting the off-resonant
atoms in the state $e$ in the case when $|\alpha |^{2}
\label{peef}
\end{figure}
\begin{figure}
\caption{(a) Wigner function of the initial odd cat
state, $|\psi \rangle = N_{-}
\label{strobo1}
\end{figure}
\begin{figure}
\caption{Wigner function of the odd cat state of Fig.~8,
evolved for a time $t = 1/\gamma $ in the presence of
feedback with the same parameters as Fig.~8b (a),
and in absence of feedback (b).}
\label{strobo2}
\end{figure}
\end{document} |
\betagin{document}
\title[Lagrange and Markov spectra from the dynamical viewpoint]{The Lagrange and Markov spectra from the dynamical point of view}
\author[Carlos Matheus]{Carlos Matheus}
\address{Carlos Matheus: Universit\'e Paris 13, Sorbonne Paris Cit\'e, LAGA, CNRS (UMR 7539), F-93439, Villetaneuse, France.}
\email{matheus@impa.br}
\date{\today}
\betagin{abstract}
This text grew out of my lecture notes for a 4-hours minicourse delivered on October 17 \& 19, 2016 during the research school ``Applications of Ergodic Theory in Number Theory'' -- an activity related to the Jean-Molet Chair project of Mariusz Lema\'nczyk and S\'ebastien Ferenczi -- realized at CIRM, Marseille, France. The subject of this text is the same of my minicourse, namely, the structure of the so-called Lagrange and Markov spectra (with an special emphasis on a recent theorem of C. G. Moreira).
\end{abstract}
\maketitle
\tableofcontents
\section{Diophantine approximations \& Lagrange and Markov spectra}
\subsection{Rational approximations of real numbers}
Given a real number $\alphapha\in\mathbb{R}$, it is natural to compare the quality $|\alphapha-p/q|$ of a rational approximation $p/q\in\mathbb{Q}$ and the size $q$ of its denominator.
Since any real number lies between two consecutive integers, for every $\alphapha\in\mathbb{R}$ and $q\in\mathbb{N}$, there exists $p\in\mathbb{Z}$ such that $|q\alphapha-p|\leq 1/2$, i.e.
\betagin{equation}\label{e.pre-Dirichlet}
\left|\alphapha-\frac{p}{q}\right|\leq \frac{1}{2q}
\end{equation}
In 1842, Dirichlet \cite{Di} used his famous \emph{pigeonhole principle} to improve \eqref{e.pre-Dirichlet}.
\betagin{theorem}[Dirichlet] For any $\alphapha\in\mathbb{R}-\mathbb{Q}$, the inequality
$$\left|\alphapha-\frac{p}{q}\right|\leq\frac{1}{q^2}$$
has infinitely many rational solutions $p/q\in\mathbb{Q}$.
\end{theorem}
\betagin{proof} Given $Q\in\mathbb{N}$, we decompose the interval $[0,1)$ into $Q$ disjoint subintervals as follows:
$$[0,1)=\bigcup\limits_{j=0}^{Q-1} \left[\frac{j}{Q}, \frac{j+1}{Q}\right)$$
Next, we consider the $Q+1$ distinct\footnote{$\alphapha\notin\mathbb{Q}$ is used here} numbers $\{i\alphapha\}$, $i=0, \dots, Q$, where $\{x\}$ denotes the \emph{fractional part}\footnote{$\{x\}:=x-\lfloor x\rfloor$ and $\lfloor x\rfloor:=\max\{n\in\mathbb{Z}: n\leq x\}$ is the integer part of $x$.} of $x$. By the \emph{pigeonhole principle}, some interval $\left[\frac{j}{Q}, \frac{j+1}{Q}\right)$ must contain two such numbers, say $\{n\alphapha\}$ and $\{m\alphapha\}$, $0\leq n < m\leq Q$. It follows that
$$|\{m\alphapha\}-\{n\alphapha\}|<\frac{1}{Q},$$
i.e., $|q\alphapha-p|<1/Q$ where $0<q:=m-n\leq Q$ and $p:=\lfloor m\alphapha\rfloor - \lfloor n\alphapha\rfloor$. Therefore,
$$\left|\alphapha-\frac{p}{q}\right|<\frac{1}{qQ}\leq\frac{1}{q^2}$$
This completes the proof of the theorem.
\end{proof}
In 1891, Hurwitz \cite{Hu} showed that Dirichlet's theorem is essentially optimal:
\betagin{theorem}[Hurwitz] For any $\alphapha\in\mathbb{R}-\mathbb{Q}$, the inequality
$$\left|\alphapha-\frac{p}{q}\right|\leq\frac{1}{\sqrt{5}q^2}$$
has infinitely many rational solutions $p/q\in\mathbb{Q}$.
Moreover, for all $\varepsilon>0$, the inequality
$$\left|\frac{1+\sqrt{5}}{2}-\frac{p}{q}\right|\leq\frac{1}{(\sqrt{5}+\varepsilon)q^2}$$
has only finitely many rational solutions $p/q\in\mathbb{Q}$.
\end{theorem}
The first part of Hurwitz theorem is proved in Appendix \ref{a.Hurwitz}, while the second part of Hurwitz theorem is left as an exercise to the reader:
\betagin{exercise} Show the second part of Hurwitz theorem. (Hint: use the identity $p^2-pq-q^2 = \left(q\frac{1+\sqrt{5}}{2} - p\right) \left(q\frac{1-\sqrt{5}}{2}-p\right)$ relating $\frac{1+\sqrt{5}}{2}$ and its Galois conjugate $\frac{1-\sqrt{5}}{2}$).
Moreover, use your argument to give a bound on
$$\#\left\{\frac{p}{q}\in\mathbb{Q}: \left|\frac{1+\sqrt{5}}{2} - \frac{p}{q} \right|\leq \frac{1}{(\sqrt{5}+\varepsilon)q^2}\right\}$$ in terms of $\varepsilon>0$.
\end{exercise}
Note that Hurwitz theorem does \emph{not} forbid an improvement of ``$\left|\alphapha-\frac{p}{q}\right|\leq \frac{1}{\sqrt{5} q^2}$ has infinitely many rational solutions $p/q\in\mathbb{Q}$'' for \emph{certain} $\alphapha\in\mathbb{R}-\mathbb{Q}$. This motivates the following definition:
\betagin{definition} The constant
$$\ell(\alphapha) := \limsup\limits_{p, q\to\infty} \frac{1}{|q(q\alphapha-p)|}$$
is called the \emph{best constant of Diophantine approximation} of $\alphapha$.
\end{definition}
Intuitively, $\ell(\alphapha)$ is the best constant $\ell$ such that $|\alphapha-\frac{p}{q}|\leq \frac{1}{\ell q^2}$
has infinitely many rational solutions $p/q\in\mathbb{Q}$.
\betagin{remark} By Hurwitz theorem, $\ell(\alphapha)\geq\sqrt{5}$ for all $\alphapha\in\mathbb{R}-\mathbb{Q}$ and $\ell(\frac{1+\sqrt{5}}{2})=\sqrt{5}$.
\end{remark}
The collection of \emph{finite} best constants of Diophantine approximations is the \emph{Lagrange spectrum}:
\betagin{definition} The \emph{Lagrange spectrum} is
$$L:=\{\ell(\alphapha): \alphapha\in\mathbb{R}-\mathbb{Q}, \ell(\alphapha)<\infty\}\subset\mathbb{R}$$
\end{definition}
\betagin{remark} Khinchin proved in 1926 a famous theorem implying that $\ell(\alphapha)=\infty$ for Lebesgue almost every $\alphapha\in\mathbb{R}-\mathbb{Q}$ (see, e.g., Khinchin's book \cite{Kh} for more details).
\end{remark}
\subsection{Integral values of binary quadratic forms}
Let $q(x,y)=ax^2+bxy+cy^2$ be a \emph{binary quadratic form} with real coefficients $a, b, c\in \mathbb{R}$. Suppose that $q$ is \emph{indefinite}\footnote{I.e., $q$ takes both positive and negative values.} with positive \emph{discriminant} $\mc{D}elta(q):=b^2-4ac$. What is the smallest value of $q(x,y)$ at non-trivial integral vectors $(x,y)\in\mathbb{Z}^2-\{(0,0)\}$?
\betagin{definition} The \emph{Markov spectrum} is
$$M:=\left\{\frac{\sqrt{\mc{D}elta(q)}}{\inf\limits_{(x,y)\in\mathbb{Z}^2-\{(0,0)\}}|q(x,y)|}\in\mathbb{R}: q \textrm{ is an indefinite binary quadratic form with } \mc{D}elta(q)>0\right\}$$
\end{definition}
\betagin{remark} A similar Diophantine problem for \emph{ternary} (and $n$-ary, $n\geq 3$) quadratic forms was proposed by Oppenheim in 1929. Oppenheim's conjecture was famously solved in 1987 by Margulis using \emph{dynamics on homogeneous spaces}: the reader is invited to consult Witte Morris book \cite{WM} for more details about this beautiful portion of Mathematics.
\end{remark}
In 1880, Markov \cite{Ma} noticed a relationship between certain binary quadratic forms and rational approximations of certain irrational numbers. This allowed him to prove the following result:
\betagin{theorem}[Markov]\label{t.Markov} $L \cap (-\infty, 3) = M \cap (-\infty, 3) = \{k_1<k_2<k_3<k_4<\dots\}$
where $k_1=\sqrt{5}$, $k_2=\sqrt{8}$, $k_3=\frac{\sqrt{221}}{5}$, $k_4=\frac{\sqrt{1517}}{13}$, $\dots$ is an explicit increasing sequence of quadratic surds\footnote{I.e., $k_n^2\in\mathbb{Q}$ for all $n\in\mathbb{N}$.} accumulating at $3$.
In fact, $k_n=\sqrt{9-\frac{4}{m_n^2}}$ where $m_n\in\mathbb{N}$ is the $n$-th Markov number, and a Markov number is the largest coordinate of a Markov triple $(x,y,z)$, i.e., an integral solution of $x^2+y^2+z^2=3xyz$.
\end{theorem}
\betagin{remark} All Markov triples can be deduced from $(1,1,1)$ by applying the so-called \emph{Vieta involutions} $V_1, V_2, V_3$ given by
$$V_1(x,y,z) = (x',y,z)$$
where $x'=3yz-x$ is the other solution of the second degree equation $X^2-3yzX+(y^2+z^2)=0$, etc. In other terms, all Markov triples appear in \emph{Markov tree}\footnote{Namely, the tree where Markov triples $(x,y,z)$ are displayed after applying permutations to put them in normalized form $x\leq y\leq z$, and two normalized Markov triples are connected if we can obtain one from the other by applying Vieta involutions.}:
\betagin{figure}[htb!]
\includegraphics[scale=0.4]{markoffCIRM.pdf}
\end{figure}
\end{remark}
\betagin{remark} For more informations on Markov numbers, the reader might consult Zagier's paper \cite{Za} on this subject. Among many conjectures and results mentioned in this paper, we have:
\betagin{itemize}
\item Conjecturally, each Markov number $z$ determines \emph{uniquely} Markov triples $(x,y,z)$ with $x\leq y\leq z$;
\item If $M(x):=\#\{m \textrm{ Markov number}: m\leq x\}$, then $M(x)=c(\log x)^2 + O(\log x (\log\log x)^2)$ for an \emph{explicit} constant $c\simeq 0.18071704711507...$; conjecturally, $M(x)=c(\log(3x))^2 + o(\log x)$, i.e., if $m_n$ is the $n$-th Markov number (counted with multiplicity), then $m_n\sim \frac{1}{3} A^{\sqrt{n}}$ with $A=e^{1/\sqrt{c}}\simeq 10.5101504...$
\end{itemize}
\end{remark}
\subsection{Best rational approximations and continued fractions}
The constant $\ell(\alphapha)$ was defined in terms of rational approximations of $\alphapha\in\mathbb{R}-\mathbb{Q}$. In particular,
$$\ell(\alphapha)=\limsup\limits_{n\to\infty}\frac{1}{|s_n(s_n\alphapha-r_n)|}$$
where $(r_n/s_n)_{n\in\mathbb{N}}$ is the sequence of best rational approximations of $\alphapha$. Here, $p/q$ is called a \emph{best rational approximation}\footnote{This nomenclature will be justified later by Propositions \ref{p.finding-best} and \ref{p.best-nomenclature} below.} whenever
$$\left|\alphapha-\frac{p}{q}\right|<\frac{1}{2q^2}$$
The sequence $(r_n/s_n)_{n\in\mathbb{N}}$ of best rational approximations of $\alphapha$ is produced by the so-called \emph{continued fraction algorithm}.
Given $\alphapha=\alphapha_0\notin\mathbb{Q}$, we define recursively $a_n=\lfloor\alphapha_n\rfloor$ and $\alphapha_{n+1} = \frac{1}{\alphapha_n-a_n}$ for all $n\in\mathbb{N}$. We can write $\alphapha$ as a \emph{continued fraction}
$$\alphapha=a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{\ddots}}}=:[a_0; a_1, a_2,\dots]$$
and we denote
$$\mathbb{Q}\ni\frac{p_n}{q_n}:=a_0+\frac{1}{a_1+\frac{1}{\ddots+\frac{1}{a_n}}}:=[a_0; a_1,\dots, a_n]$$
\betagin{remark} L\'evy's theorem \cite{Le} (from 1936) says that $\sqrt[n]{q_n}\to e^{\varphii^2/12\log 2}\simeq 3.27582291872...$ for Lebesgue almost every $\alphapha\in\mathbb{R}$. By elementary properties of continued fractions (recalled below), it follows from L\'evy's theorem that $\sqrt[n]{|\alphapha-\frac{p_n}{q_n}|}\to e^{-\varphii^2/6\log 2}\simeq 0.093187822954...$ for Lebesgue almost every $\alphapha\in\mathbb{R}$.
\end{remark}
\betagin{proposition}\label{p.pnqn} $p_n$ and $q_n$ are recursively given by
$$\left\{\betagin{array}{cc} p_{n+2} = a_{n+2} p_{n+1} + p_n, & p_{-1} = 1, p_{-2} = 0 \\ q_{n+2} = a_{n+2} q_{n+1} + q_n, & q_{-1} = 0, q_{-2} = 1 \end{array}\right.$$
\end{proposition}
\betagin{proof} Exercise\footnote{Hint: Use induction and the fact that $[t_0; t_1,\dots, t_n, t_{n+1}] = [t_0; t_1,\dots, t_n+\frac{1}{t_{n+1}}]$.}.
\end{proof}
In other words, we have
\betagin{equation}\label{e.Mobius-fraction}
[a_0; a_1,\dots, a_{n-1}, z] = \frac{zp_{n-1}+p_{n-2}}{zq_{n-1}+q_{n-2}}
\end{equation}
or, equivalently,
\betagin{equation}\label{e.SL2Z-fraction}
\left(\betagin{array}{cc} p_{n+1} & p_n \\ q_{n+1} & q_n \end{array}\right) \cdot
\left(\betagin{array}{cc} a_{n+2} & 1 \\ 1 & 0 \end{array}\right) =
\left(\betagin{array}{cc} p_{n+2} & p_{n+1} \\ q_{n+2} & q_{n+1} \end{array}\right)
\end{equation}
\betagin{corollary}\label{c.determinant-fraction} $p_{n+1}q_n-p_nq_{n+1} = (-1)^n$ for all $n\geq 0$.
\end{corollary}
\betagin{proof} This follows from \eqref{e.SL2Z-fraction} because the matrix $\left(\betagin{array}{cc}\ast & 1 \\ 1 & 0 \end{array}\right)$ has determinant $-1$.
\end{proof}
\betagin{corollary}\label{c.Mobius-fraction} $\alphapha=\frac{\alphapha_n p_{n-1} + p_{n-2}}{\alphapha_n q_{n-1} + q_{n-2}}$ and $\alphapha_n = \frac{p_{n-2} - q_{n-2}\alphapha}{q_{n-1}\alphapha-p_{n-1}}$.
\end{corollary}
\betagin{proof} This is a consequence of \eqref{e.Mobius-fraction} and the fact that $\alphapha =: [a_0; a_1,\dots, a_{n-1},\alphapha_n]$.
\end{proof}
The relationship between $\frac{p_n}{q_n}$ and the sequence of best rational approximations is explained by the following two propositions:
\betagin{proposition} $\left|\alphapha-\frac{p_n}{q_n}\right|\leq \frac{1}{q_n q_{n+1}} < \frac{1}{a_{n+1} q_n^2}\leq \frac{1}{q_n^2}$ and, moreover, for all $n\in\mathbb{N}$,
$$\textrm{either } \left|\alphapha-\frac{p_n}{q_n}\right| < \frac{1}{2q_n^2} \textrm{ or } \left|\alphapha-\frac{p_{n+1}}{q_{n+1}}\right|< \frac{1}{2q_{n+1}^2}.$$
\end{proposition}
\betagin{proof} Note that $\alphapha$ belongs to the interval with extremities $p_n/q_n$ and $p_{n+1}/q_{n+1}$ (by Corollary \ref{c.Mobius-fraction}). Since this interval has size
$$\left|\frac{p_{n+1}}{q_{n+1}} - \frac{p_n}{q_n}\right| = \left|\frac{p_{n+1}q_n - p_n q_{n+1}}{q_nq_{n+1}} \right| = \left|\frac{(-1)^n}{q_n q_{n+1}}\right| = \frac{1}{q_n q_{n+1}}$$
(by Corollary \ref{c.determinant-fraction}), we conclude that $|\alphapha-\frac{p_n}{q_n}|\leq \frac{1}{q_n q_{n+1}}$.
Furthermore, $\frac{1}{q_n q_{n+1}} = |\frac{p_{n+1}}{q_{n+1}}-\alphapha| + |\alphapha-\frac{p_n}{q_n}|$. Thus, if $$\left|\alphapha-\frac{p_n}{q_n}\right| \geq \frac{1}{2q_n^2} \quad \textrm{ and } \quad \left|\alphapha-\frac{p_{n+1}}{q_{n+1}}\right|\geq \frac{1}{2q_{n+1}^2},$$
then
$$\frac{1}{q_n q_{n+1}}\geq \frac{1}{2q_n^2}+\frac{1}{2q_{n+1}^2},$$
i.e., $2q_n q_{n+1}\geq q_n^2+q_{n+1}^2$, i.e., $q_n=q_{n+1}$, a contradiction.
\end{proof}
In other terms, the sequence $(p_n/q_n)_{n\in\mathbb{N}}$ produced by the continued fraction algorithm contains best rational approximations with frequency at least $1/2$.
Conversely, the continued fraction algorithm detects \emph{all} best rational approximations:
\betagin{proposition}\label{p.finding-best} If $|\alphapha-\frac{p}{q}|<\frac{1}{2q^2}$, then $p/q = p_n/q_n$ for some $n\in\mathbb{N}$.
\end{proposition}
\betagin{proof} Exercise\footnote{Hint: Take $q_{n-1}< q\leq q_n$, suppose that $p/q\neq p_n/q_n$ and derive a contradiction in each case $q=q_n$, $q_n/2\leq q<q_n$ and $q<q_n/2$ by analysing $|\alphapha-\frac{p}{q}|$ and $|\frac{p}{q}-\frac{p_n}{q_n}|$ like in the proof of Proposition \ref{p.best-nomenclature}.}.
\end{proof}
The terminology ``best rational approximation'' is motivated by the previous proposition and the following result:
\betagin{proposition}\label{p.best-nomenclature} For all $q < q_n$, we have $|\alphapha-\frac{p_n}{q_n}| < |\alphapha-\frac{p}{q}|$.
\end{proposition}
\betagin{proof} If $q<q_{n+1}$ and $p/q\neq p_n/q_n$, then
$$\left|\frac{p}{q}-\frac{p_n}{q_n}\right|\geq \frac{1}{q q_n} > \frac{1}{q_n q_{n+1}} = \left|\frac{p_{n+1}}{q_{n+1}}-\frac{p_n}{q_n}\right|$$
Hence, $p/q$ does not belong to the interval with extremities $p_n/q_n$ and $p_{n+1}/q_{n+1}$, and so
$$\left|\alphapha-\frac{p_n}{q_n}\right|< \left|\alphapha-\frac{p}{q}\right|$$
because $\alphapha$ lies between $p_n/q_n$ and $p_{n+1}/q_{n+1}$.
\end{proof}
In fact, the approximations $(p_n/q_n)$ of $\alphapha$ are usually quite impressive:
\betagin{example} $\varphii=[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, \dots]$ so that
$$\frac{p_0}{q_0}=3, \quad \frac{p_1}{q_1}=\frac{22}{7}, \quad \frac{p_2}{q_2}=\frac{333}{106}, \quad \frac{p_3}{q_3}=\frac{355}{113}, \quad \dots$$
The approximations $p_1/q_1$ and $p_3/q_3$ are called Yuel\"u and Mil\"u (after Wikipedia) and they are somewhat spectacular:
$$\left|\varphii-\frac{22}{7}\right|<\frac{1}{700}<\left|\varphii-\frac{314}{100}\right| \quad \textrm{ and } \quad \left|\varphii-\frac{355}{113}\right|<\frac{1}{3,000,000}<\left|\varphii-\frac{3141592}{1,000,000}\right|$$
\end{example}
\subsection{Perron's characterization of Lagrange and Markov spectra}
In 1921, Perron interpreted $\ell(\alphapha)$ in terms of Dynamical Systems as follows.
\betagin{proposition}\label{p.Perron} $\alphapha-\frac{p_n}{q_n} = \frac{(-1)^n}{(\alphapha_{n+1}+\betata_{n+1})q_n^2}$ where $\betata_{n+1}:=\frac{q_{n-1}}{q_n}=[0; a_n, a_{n-1}, \dots, a_1]$.
\end{proposition}
\betagin{proof} Recall that $\alphapha_{n+1} = \frac{p_{n-1}-q_{n-1}\alphapha}{q_n\alphapha-p_n}$ (cf. Corollary \ref{c.Mobius-fraction}). Hence, $\alphapha_{n+1}+\betata_{n+1} = \frac{p_{n-1}q_n-p_nq_{n-1}}{q_n(q_n\alphapha-p_n)} = \frac{(-1)^n}{q_n(q_n\alphapha-p_n)}$ (by Corollary \ref{c.determinant-fraction}). This proves the proposition.
\end{proof}
Therefore, the proposition says that $\ell(\alphapha)=\limsup\limits_{n\to\infty}(\alphapha_n+\betata_n)$. From the dynamical point of view, we consider the \emph{symbolic space} $\Sigma = (\mathbb{N}^*)^{\mathbb{Z}}=: \Sigma^-\times\Sigma^+ = (\mathbb{N}^*)^{\mathbb{Z}^-}\times (\mathbb{N}^*)^{\mathbb{N}}$ equipped with the left \emph{shift dynamics} $\sigma:\Sigma\to\Sigma$, $\sigma((a_n)_{n\in\mathbb{Z}}):=(a_{n+1})_{n\in\mathbb{Z}}$ and the \emph{height function} $f:\Sigma\to\mathbb{R}$, $f((a_n)_{n\in\mathbb{Z}}) = [a_0; a_1, a_2, \dots] + [0; a_{-1}, a_{-2}, \dots]$. Then, the proposition above implies that
$$\ell(\alphapha) = \limsup\limits_{n\to+\infty} f(\sigma^n(\underline{\theta}))$$
where $\alphapha=[a_0; a_1, a_2, \dots]$ and $\underline{\theta} = (\dots, a_{-1}, a_0, a_1, \dots)$. In particular,
\betagin{equation}\label{e.Perron-Lagrange}
L = \{\ell(\underline{\theta}): \underline{\theta}\in\Sigma, \ell(\underline{\theta})<\infty\}
\end{equation}
where $\ell(\underline{\theta}):=\limsup\limits_{n\to+\infty} f(\sigma^n(\underline{\theta}))$.
Also, the Markov spectrum has a \emph{similar description}:
\betagin{equation}\label{e.Perron-Markov}
M=\{m(\underline{\theta}): \underline{\theta}\in\Sigma, m(\underline{\theta})<\infty\}
\end{equation}
where $m(\underline{\theta}):=\sup\limits_{n\in\mathbb{Z}} f(\sigma^n(\underline{\theta}))$.
\betagin{remark}\label{r.Perron-Gauss} A \emph{geometrical interpretation} of $\sigma:\Sigma\to\Sigma$ is provided by the so-called \emph{Gauss map}\footnote{From Number Theory rather than Differential Geometry.}:
\betagin{equation}\label{e.Gauss-map}
G(x)=\left\{\frac{1}{x}\right\}
\end{equation}
for $0<x\leq 1$.
\betagin{figure}[htb!]
\input{gaussCIRM.pdf_tex}
\end{figure}
Indeed, $G([0; a_1, a_2, \dots]) = [0; a_2, \dots]$, so that $\sigma:\Sigma\to\Sigma$ is a symbolic version of the \emph{natural extension} of $G$.
Furthermore, the identification $(\dots, a_{-1}, a_0, a_1, \dots)\simeq ([0; a_{-1}, a_{-2}, \dots], [a_0; a_1, a_2, \dots]) = (y,x)$ allows us to write the height function as $f((a_n)_{n\in\mathbb{Z}}) = x+y$.
\betagin{figure}[htb!]
\input{curso4.pdf_tex}
\end{figure}
\end{remark}
Perron's dynamical interpretation of the Lagrange and Markov spectra is the starting point of many results about $L$ and $M$ which are not so easy to guess from their definitions:
\betagin{exercise} Show that $L\subset M$ are closed subsets of $\mathbb{R}$.
\end{exercise}
\betagin{remark}\label{r.Freiman-M-L} $M-L\neq\emptyset$: for example, Freiman \cite{Fr68} proved in 1968 that
$$s=\overline{221221122}11\overline{221122122}\in(\mathbb{N}^*)^{\mathbb{Z}}$$
has the property that $3.118120178\simeq m(s)\in M-L$. (Here $\overline{\theta_1\dots\theta_n}$ means infinite repetition of the block $\theta_1\dots\theta_n$.)
Also, Freiman \cite{Fr73} showed in 1973 that $m(s_n)\in M-L$ and $m(s_n)\to m(s_{\infty})\simeq 3.293044265 \in M-L$ where
$$s_n = \overline{2221121}\underbrace{22\dots 22}_{n \textrm{ times}}121122212\overline{1122212}$$
for $n\geq 4$, and
$$s_{\infty} = \overline{2}121122212\overline{1122212}$$
\end{remark}
\subsection{Digression: Lagrange spectrum and cusp excursions on the modular surface}
The Lagrange spectrum is related to the values of a certain height function $H$ along the orbits of the geodesic flow $g_t$ on the (unit cotangent bundle to) the modular surface: indeed, we will show that
$$L=\{\limsup\limits_{t\to+\infty} H(g_t(x))<\infty: x \textrm{ is a unit cotangent vector to the modular surface}\}$$
\betagin{remark} This fact is not surprising to experts: the Gauss map appears naturally by quotienting out the weak-stable manifolds of $g_t$ as observed by Artin, Series, Arnoux, ... (see, e.g., \cite{Ar}).
\end{remark}
An \emph{unimodular lattice} in $\mathbb{R}^2$ has the form $g(\mathbb{Z}^2)$, $g\in SL(2,\mathbb{Z})$, and the stabilizer in $SL(2,\mathbb{R})$ of the standard lattice $\mathbb{Z}^2$ is $SL(2,\mathbb{Z})$. In particular, the space of unimodular lattices in $\mathbb{R}^2$ is $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$.
As it turns out, $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$ is the unit cotangent bundle to the \emph{modular surface} $\mathbb{H}/SL(2,\mathbb{Z})$ (where $\mathbb{H}=\{z\in\mathbb{C}: \textrm{Im}(z)>0\}$ is the hyperbolic upper-half plane and $\left(\betagin{array}{cc} a & b \\ c & d \end{array}\right)\in SL(2,\mathbb{R})$ acts on $z\in\mathbb{H}$ via $\left(\betagin{array}{cc} a & b \\ c & d \end{array}\right)\cdot z = \frac{az+b}{cz+d}$).
The \emph{geodesic flow} of the modular surface is the action of $g_t=\left(\betagin{array}{cc} e^t & 0 \\ 0 & e^{-t} \end{array}\right)$ on $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$. The \emph{stable} and \emph{unstable manifolds} of $g_t$ are the orbits of the \emph{stable} and \emph{unstable horocycle flows} $h_s=\left(\betagin{array}{cc} 1 & 0 \\ s & 1 \end{array}\right)$ and $u_s =\left(\betagin{array}{cc} 1 & s \\ 0 & 1 \end{array}\right)$: indeed, this follows from the facts that $g_t h_s = h_{s e^{-2t}} g_t$ and $g_t u_s = u_{s e^t} g_t$.
The set of \emph{holonomy} (or \emph{primitive}) \emph{vectors} of $\mathbb{Z}^2$ is
$$\textrm{Hol}(\mathbb{Z}^2):=\{(p,q)\in\mathbb{Z}^2: \textrm{gcd}(p,q)=1\}$$
In general, the set $\textrm{Hol}(X)$ of holonomy vectors of $X=g(\mathbb{Z}^2)$, $g\in SL(2,\mathbb{Z})$, is
$$\textrm{Hol}(X):=g(\textrm{Hol}(\mathbb{Z}^2))\subset \mathbb{R}^2$$
The \emph{systole} $\textrm{sys}(X)$ of $X=g(\mathbb{Z}^2)$ is
$$\textrm{sys}(X) := \min\{\|v\|_{\mathbb{R}^2}: v\in\textrm{Hol}(X)\}$$
\betagin{remark} By Mahler's compactness criterion \cite{Mahler}, $X\mapsto \frac{1}{\textrm{sys}(X)}$ is a proper function on $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$.
\end{remark}
\betagin{remark} For later reference, we write $\textrm{Area}(v):=|\textrm{Re}(v)|\cdot |\textrm{Im}(v)|$ for the area of the rectangle in $\mathbb{R}^2$ with diagonal $v=(\textrm{Re}(v), \textrm{Im}(v))\in \mathbb{R}^2$.
\end{remark}
\betagin{proposition} The forward geodesic flow orbit of $X\in SL(2,\mathbb{R})/SL(2,\mathbb{Z})$ does not go straight to infinity (i.e., $\textrm{sys}(g_t(X))\to 0$ as $t\to+\infty$) if and only if there is no vertical vector in $\textrm{Hol}(X)$. In this case, there are (unique) parameters $s, t, \alphapha\in\mathbb{R}$ such that
$$X = h_s g_t u_{-\alphapha}(\mathbb{Z}^2)$$
\end{proposition}
\betagin{proof} By unimodularity, any $X = g(\mathbb{Z}^2)$ has a single \emph{short} holonomy vector. Since $g_t$ contracts vertical vectors and expands horizontal vectors for $t>0$, we have that $\textrm{sys}(g_t(X))\to 0$ as $t\to+\infty$ if and only if $\textrm{Hol}(X)$ contains a vertical vector.
By Iwasawa decomposition, there are (unique) parameters $s, t, \theta\in\mathbb{R}$ such that $X=h_s g_t r_{\theta}$, where $r_{\theta} = \left( \betagin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array} \right)$. Since $\cos\theta\neq 0$ when $\textrm{Hol}(X)$ contains no vertical vector and, in this situation,
$$r_{\theta} = h_{\tan\theta} g_{\log\cos\theta} u_{-\tan\theta},$$
we see that $X=h_{s+e^{-2t}\tan\theta} \cdot g_{t+\log\cos\theta} \cdot u_{-\tan\theta}(\mathbb{Z}^2)$
(because $h_s g_t r_{\theta}= h_s g_t h_{\tan\theta} g_{\log\cos\theta} u_{-\tan\theta}=h_{s+e^{-2t}\tan\theta} \cdot g_{t+\log\cos\theta} \cdot u_{-\tan\theta}$). This ends the proof of the proposition.
\end{proof}
\betagin{proposition} Let $X=h_s g_t u_{-\alphapha}(\mathbb{Z}^2)$ be an unimodular lattice without vertical holonomy vectors. Then,
$$\ell(\alphapha) = \limsup\limits_{\substack{|\textrm{Im}(v)|\to\infty \\ v\in\textrm{Hol}(X)}} \frac{1}{\textrm{Area}(v)} = \limsup\limits_{T\to+\infty} \frac{2}{\textrm{sys}(g_T(X))^2}$$
\end{proposition}
\betagin{remark} This proposition says that the dynamical quantity $\limsup\limits_{T\to+\infty} \frac{2}{\textrm{sys}(g_T(X))^2}$ does \emph{not} depend on the ``weak-stable part'' $h_s g_t$ (but only on $\alphapha$) and it can be computed \emph{without} dynamics by simply studying almost vertical holonomy vectors in $X$.
\end{remark}
\betagin{proof} Note that $\textrm{Area}(g_t(v)) = \textrm{Area}(v)$ for all $t\in\mathbb{R}$ and $v\in\mathbb{R}^2$. Since $\textrm{Area}(v) = \frac{\|g_{t(v)}(v)\|^2}{2}$ for $t(v):=\frac{1}{2}\log\frac{|\textrm{Im}(v)|}{|\textrm{Re}(v)|}$, the equality $\limsup\limits_{\substack{|\textrm{Im}(v)|\to\infty \\ v\in\textrm{Hol}(X)}} \frac{1}{\textrm{Area}(v)} = \limsup\limits_{T\to+\infty} \frac{2}{\textrm{sys}(g_T(X))^2}$ follows.
The relation $g_T h_s = h_{s e^{-2T}} g_T$ and the continuity of the systole function imply that $\limsup\limits_{T\to+\infty} \frac{2}{\textrm{sys}(g_T(X))^2}$ depends only on $\alphapha$. Because any $v\in\textrm{Hol}(u_{-\alphapha}(\mathbb{Z}^2))$ has the form $v=(p-q\alphapha, q) = u_{-\alphapha}(p,q)$ with $(p,q)\in\textrm{Hol}(\mathbb{Z}^2)$, the equality $\limsup\limits_{\substack{|\textrm{Im}(v)|\to\infty \\ v\in\textrm{Hol}(X)}} \frac{1}{\textrm{Area}(v)} = \ell(\alphapha)$.
\end{proof}
In summary, the previous proposition says that the Lagrange spectrum $L$ coincides with
$$\{\limsup\limits_{T\to+\infty} H(g_T(x))<\infty: x\in SL(2,\mathbb{R})/SL(2,\mathbb{Z})\}$$
where $H(y) = \frac{2}{\textrm{sys}(y)^2}$ is a (proper) height function and $g_t$ is the geodesic flow on $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$.
\betagin{figure}[htb!]
\input{curso1.pdf_tex}
\end{figure}
\betagin{remark} Several number-theoretical problems translate into dynamical questions on the modular surface: for example, Zagier \cite{Za-RH} showed that the Riemann hypothesis is equivalent to a certain speed of equidistribution of $u_s$-orbits on $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$.
\end{remark}
\subsection{Hall's ray and Freiman's constant}
In 1947, M. Hall \cite{Hall} proved that:
\betagin{theorem}[Hall]\label{t.Hall} The half-line $[6,+\infty)$ is contained in $L$.
\end{theorem}
This result motivates the following nomenclature: the biggest half-line $[c_F,+\infty)\subset L (\subset M)$ is called \emph{Hall's ray}.
In 1975, G. Freiman \cite{Fr75} determined Hall's ray:
\betagin{theorem}[Freiman] $c_F = 4+\frac{253589820+283798\sqrt{462}}{491993569}\simeq 4.527829566...$
\end{theorem}
The constant $c_F$ is called \emph{Freiman's constant}.
Let us sketch the proof of Hall's theorem based on the following lemma:
\betagin{lemma}[Hall]\label{l.Hall} Denote by $C(4):=\{[0; a_1, a_2, \dots]\in\mathbb{R}: a_i\in\{1, 2, 3, 4\} \,\,\forall\, i\in\mathbb{N}\}$. Then,
$$C(4)+C(4):=\{x+y\in\mathbb{R}: x, y\in C(4)\} = [\sqrt{2}-1, 4(\sqrt{2}-1)] = [0.414\dots, 1.656\dots]$$
\end{lemma}
\betagin{remark} The reader can find a proof of this lemma in Cusick-Flahive's book \cite{CF}. Interestingly enough, some of the techniques in the proof of Hall's lemma were rediscovered much later (in 1979) in the context of Dynamical Systems by Newhouse \cite{Newhouse} (in the proof of his \emph{gap lemma}).
\end{remark}
\betagin{remark} $C(4)$ is a \emph{dynamical Cantor set}\footnote{See Subsections \ref{ss.dynamical-Cantor} and \ref{ss.Gauss-Cantor} below.} whose Hausdorff dimension is $>1/2$ (see Remark \ref{r.JP} below). In particular, $C(4)\times C(4)$ is a planar Cantor set of Hausdorff dimension $>1$ and Hall's lemma says that its image $f(C(4)\times C(4)) = C(4)+C(4)$ under the the projection $f(x,y) = x+y$ contains an interval. Hence, Hall's lemma can be thought as a sort of ``particular case'' of \emph{Marstrand's theorem} \cite{Marstrand} (ensuring that typical projections of planar sets with Hausdorff dimension $>1$ has positive Lebesgue measure).
\end{remark}
For our purposes, the specific form $C(4)+C(4)$ is \emph{not} important: the \emph{key point} is that $C(4)+C(4)$ is an interval of length $>1$.
Indeed, given $6\leq\ell<\infty$, Hall's lemma guarantees the existence of $c_0\in\mathbb{N}$, $5\leq c_0\leq\ell$ such that $\ell-c_0\in C(4)+C(4)$. Thus,
$$\ell = c_0 + [0; a_1, a_2,\dots] + [0; b_1, b_2,\dots]$$
with $a_i, b_i\in\{1,2,3,4\}$ for all $i\in\mathbb{N}$.
Define
$$\alphapha:=[0; \underbrace{b_1, c_0, a_1}_{1^{st} \textrm{ block}}, \dots, \underbrace{b_n, \dots, b_1, c_0, a_1, \dots, a_n}_{n^{th} \textrm{ block}}, \dots]$$
Since $c_0\geq 5 > 4\geq a_i, b_i$ for all $i\in\mathbb{N}$, Perron's characterization of $\ell(\alphapha)$ implies that
$$L\ni \ell(\alphapha) = \lim\limits_{n\to\infty} (c_0 + [0; a_1, a_2, \dots, a_n] + [0; b_1, b_2, \dots, b_n]) = \ell$$
This proves Theorem \ref{t.Hall}.
\subsection{Statement of Moreira's theorem}
Our discussion so far can be summarized as follows:
\betagin{itemize}
\item $L\cap (-\infty, 3) = M\cap (-\infty, 3) = \{k_1<k_2<\dots<k_n<\dots\}$ is an \emph{explicit} discrete set;
\item $L\cap [c_F,\infty) = M\cap [c_F,\infty)$ is an \emph{explicit} ray.
\end{itemize}
Moreira's theorem \cite{Moreira} says that the \emph{intermediate parts} $L\cap [3, c_F]$ and $M\cap [3, c_F]$ of the Lagrange and Markov spectra have an intricate structure:
\betagin{theorem}[Moreira]\label{t.Gugu} For each $t\in\mathbb{R}$, the sets $L\cap (-\infty, t)$ and $M\cap (-\infty, t)$ have the same Hausdorff dimension, say $d(t)\in [0,1]$.
Moreover, the function $t\mapsto d(t)$ is continuous, $d(3+\varepsilon)>0$ for all $\varepsilon>0$ and $d(\sqrt{12})=1$ (even though $\sqrt{12}=3.4641... < 4.5278...=c_F$).
\end{theorem}
\betagin{remark} Many results about $L$ and $M$ are \emph{dynamical}\footnote{I.e., they involve Perron's characterization of $L$ and $M$, the study of Gauss map and/or the geodesic flow on the modular surface, etc.}. In particular, it is not surprising that many facts about $L$ and $M$ have counterparts for \emph{dynamical Lagrange and Markov spectra}\footnote{I.e., the collections of ``records'' of height functions along orbits of dynamical systems.}: for example, Hall ray or intervals in dynamical Lagrange spectra were found by Parkkonen-Paulin \cite{PP}, Hubert-Marchese-Ulcigrai \cite{HMU} and Moreira-Roma\~na \cite{MR}, and the continuity result in Moreira's theorem \ref{t.Gugu} was recently extended by Cerqueira, Moreira and the author in \cite{CMM}.
\end{remark}
Before entering into the proof of Moreira's theorem, let us close this section by briefly recalling the notion of Hausdorff dimension.
\subsection{Hausdorff dimension}
The $s$-\emph{Hausdorff measure} $m_s(X)$ of a subset $X\subset\mathbb{R}^n$ is
$$m_s(X):=\lim\limits_{\delta\to 0}\inf\limits_{\substack{\bigcup\limits_{i\in\mathbb{N}} U_i \supset X, \\ \textrm{diam}(U_i)\leq \delta \,\, \forall\, i\in\mathbb{N}}} \sum\limits_{i\in\mathbb{N}} \textrm{diam}(U_i)^s$$
The \emph{Hausdorff dimension} of $X$ is
$$HD(X):=\sup\{s\in\mathbb{R}: m_s(X)=\infty\} = \inf\{s\in\mathbb{R}: m_s(X)=0\}$$
\betagin{remark}\label{r.box-counting} There are many notions of dimension in the literature: for example, the \emph{box-counting dimension} of $X$ is $\lim\limits_{\delta\to 0}\frac{\log N_X(\delta)}{\log(1/\delta)}$ where $N_X(\delta)$ is the smallest number of boxes of side lengths $\leq\delta$ needed to cover $X$. As an exercise, the reader is invited to show that the Hausdorff dimension is always smaller than or equal to the box-counting dimension.
\end{remark}
The following exercise (whose solution can be found in Falconer's book \cite{Falconer}) describes several elementary properties of the Hausdorff dimension:
\betagin{exercise}\label{exercise.HD} Show that:
\betagin{itemize}
\item[(a)] if $X\subset Y$, then $HD(X)\leq HD(Y)$;
\item[(b)] $HD(\bigcup\limits_{i\in\mathbb{N}} X_i) = \sup\limits_{i\in\mathbb{N}} HD(X_i)$; in particular, $HD(X)=0$ whenever $X$ is a countable set (such as $X=\{p\}$ or $X=\mathbb{Q}^n$);
\item[(c)] if $f:X\to Y$ is $\alphapha$-H\"older continuous\footnote{I.e., for some constant $C>0$, one has $|f(x)-f(x')|\leq C |x-x'|^{\alphapha}$ for all $x, x'\in X$.}, then $\alphapha\cdot HD(f(X))\leq HD(X)$;
\item[(d)] $HD(\mathbb{R}^n) = n$ and, more generally, $HD(X)=m$ when $X\subset\mathbb{R}^n$ is a smooth $m$-dimensional submanifold.
\end{itemize}
\end{exercise}
\betagin{example} Cantor's middle-third set $C=\{\sum\limits_{i=1}^{\infty}\frac{a_i}{3^i}: a_i\in\{0,2\}\,\, \forall \, i\in\mathbb{N}\}$ has Hausdorff dimension $\frac{\log 2}{\log 3}\in (0,1)$: see Falconer's book \cite{Falconer} for more details.
\end{example}
Using item (c) of Exercise \ref{exercise.HD} above, we have the following corollary of Moreira's theorem \ref{t.Gugu}:
\betagin{corollary}[Moreira] The function $t\mapsto HD(L\cap(-\infty,t))$ is not $\alphapha$-H\"older continuous for any $\alphapha>0$.
\end{corollary}
\betagin{proof} By Theorem \ref{t.Gugu}, $d$ maps $L\cap[3,3+\varepsilon]$ to the non-trivial interval $[0, d(3+\varepsilon)]$ for any $\varepsilon>0$. By item (c) of Exercise \ref{exercise.HD}, if $t\mapsto d(t)=HD(L\cap(-\infty, t))$ were $\alphapha$-H\"older continuous for some $\alphapha>0$, then it would follow that
$$0<\alphapha=\alphapha\cdot HD([0,d(3+\varepsilon)])\leq HD(L\cap [3,3+\varepsilon]) = d(3+\varepsilon)$$
for all $\varepsilon>0$. On the other hand, Theorem \ref{t.Gugu} (and item (b) of Exercise \ref{exercise.HD}) also says that
$$\lim\limits_{\varepsilon\to 0} d(3+\varepsilon)=d(3)=HD(L\cap(-\infty,3))=0$$
In summary, $0<\alphapha\leq\lim\limits_{\varepsilon\to 0}d(3+\varepsilon) = 0$, a contradiction.
\end{proof}
\section{Proof of Moreira's theorem}
\subsection{Strategy of proof of Moreira's theorem} Roughly speaking, the continuity of $d(t)=HD(L\cap(-\infty, t))$ is proved in four steps:
\betagin{itemize}
\item if $0<d(t)<1$, then for all $\eta>0$ there exists $\delta>0$ such that $L\cap(-\infty, t-\delta)$ can be ``\emph{approximated from inside}'' by $K+K'=f(K\times K')$ where $K$ and $K'$ are \emph{Gauss-Cantor sets} with $HD(K)+HD(K')=HD(K\times K')>(1-\eta)d(t)$ (and $f(x,y)=x+y$);
\item by \emph{Moreira's dimension formula} (derived from profound works of Moreira and Yoccoz on the geometry of Cantor sets), we have that
$$HD(f(K\times K')) = HD(K\times K')$$
\item thus, if $0<d(t)<1$, then for all $\eta>0$ there exists $\delta>0$ such that
$$d(t-\delta)\geq HD(f(K\times K')) = HD(K\times K')\geq (1-\eta)d(t);$$
hence, $d(t)$ is \emph{lower semicontinuous};
\item finally, an elementary compactness argument shows the \emph{upper semicontinuity} of $d(t)$.
\end{itemize}
\betagin{remark} This strategy is \emph{purely dynamical} because the particular forms of the height function $f$ and the Gauss map $G$ are \emph{not} used. Instead, we just need the \emph{transversality} of the gradient of $f$ to the stable and unstable manifolds (vertical and horizontal axis) and the \emph{non-essential affinity} of Gauss-Cantor sets. (See \cite{CMM} for more explanations.)
\end{remark}
In the remainder of this section, we will implement (a version of) this strategy in order to deduce the continuity result in Theorem \ref{t.Gugu}.
\subsection{Dynamical Cantor sets}\label{ss.dynamical-Cantor} A \emph{dynamically defined Cantor set} $K\subset \mathbb{R}$ is
$$K=\bigcap\limits_{n\in\mathbb{N}}\varphisi^{-n}(I_1\cup\dots\cup I_k)$$
where $I_1,\dots, I_k$ are pairwise disjoint compact intervals, and $\varphisi:I_1\cup\dots\cup I_k\to I$ is a $C^r$-map from $I_1\cup\dots\cup I_k$ to its convex hull $I$ such that:
\betagin{itemize}
\item $\varphisi$ is \emph{uniformly expanding}: $|\varphisi'(x)|>1$ for all $x\in I_1\cup\dots\cup I_k$;
\item $\varphisi$ is a (full) \emph{Markov map}: $\varphisi(I_j)=I$ for all $1\leq j\leq k$.
\end{itemize}
\betagin{remark} Dynamical Cantor sets are usually defined with a weaker Markov condition, but we stick to this definition for simplicity.
\end{remark}
\betagin{example} Cantor's middle-third set $C=\{\sum\limits_{i=1}^{\infty} \frac{a_i}{3^i}: a_i\in\{0,2\} \,\, \forall\,i\in\mathbb{N}\}$ is
$$C=\bigcap\limits_{n\in\mathbb{N}} \varphisi^{-n}([0,1/3]\cup[2/3,1])$$
where $\varphisi:[0, 1/3]\cup [2/3, 1]\to [0,1]$ is given by
$$\varphisi(x)=\left\{\betagin{array}{cl} 3x, & \textrm{if } 0\leq x\leq 1/3 \\ 3x-2, & \textrm{if } 2/3\leq x\leq 1\end{array}\right.$$
\betagin{figure}[htb!]
\input{curso5.pdf_tex}
\end{figure}
\end{example}
\betagin{remark} A dynamical Cantor set is called \emph{affine} when $\varphisi|_{I_j}$ is affine for all $j$. In this language, Cantor's middle-third set is an \emph{affine dynamical Cantor set}.
\end{remark}
\betagin{example} Given $A\geq 2$, let $C(A):=\{[0;a_1, a_2,\dots]: 1\leq a_i\leq A\,\,\forall\,i\in\mathbb{N}\}$. This is a dynamical Cantor set associated to Gauss map: for example,
$$C(2) = \bigcap\limits_{n\in\mathbb{N}} G^{-n}(I_1\cup I_2)$$
where $I_1$ and $I_2$ are the intervals depicted below.
\betagin{figure}[htb!]\label{f.C(2)}
\input{curso3.pdf_tex}
\end{figure}
\end{example}
\betagin{remark}\label{r.JP} Hensley \cite{He} showed that
$$HD(C(A)) = 1-\frac{6}{\varphii^2 A} - \frac{72\log A}{\varphii^4 A^2} + O(\frac{1}{A^2}) = 1-\frac{1+o(1)}{\zeta(2)A}$$
and Jenkinson-Pollicott \cite{JePo1}, \cite{JePo2} used thermodynamical formalism methods to obtain that
$$HD(C(2)) = 0.53128050627720514162446864736847178549305910901839\dots,$$
$$HD(C(3)) \simeq 0.705\dots, \quad HD(C(4)) \simeq 0.788\dots$$
\end{remark}
\subsection{Gauss-Cantor sets}\label{ss.Gauss-Cantor}
The set $C(A)$ above is a particular case of \emph{Gauss-Cantor set}:
\betagin{definition} Given $B=\{\betata_1,\dots,\betata_l\}$, $l\geq 2$, a finite, primitive\footnote{I.e., $\betata_i$ doesn't begin by $\betata_j$ for all $i\neq j$.} alphabet of finite words $\betata_j\in(\mathbb{N}^*)^{r_j}$, the Gauss-Cantor set $K(B)\subset [0,1]$ associated to $B$ is
$$K(B):=\{[0;\gamma_1, \gamma_2, \dots]: \gamma_i\in B\,\,\forall\, i\}$$
\end{definition}
\betagin{example} $C(A) = K(\{1,\dots, A\})$.
\end{example}
\betagin{exercise}\label{ex.Gauss-Cantor} Show that any Gauss-Cantor set $K(B)$ is dynamically defined.\footnote{Hint: For each word $\betata_j\in(\mathbb{N}^*)^{r_j}$, let $I(\betata_j)=\{[0;\betata_j, a_1,\dots]:a_i\in\mathbb{N}\,\,\forall\,i\}=I_j$ and $\varphisi|_{I_j}:=G^{r_j}$ where $G(x)=\{1/x\}$ is the Gauss map.}
\end{exercise}
From the \emph{symbolic} point of view, $B=\{\betata_1,\dots,\betata_l\}$ as above induces a subshift $$\Sigma(B)=\{(\gamma_i)_{i\in\mathbb{Z}}:\gamma_i\in B \,\,\forall\,i\}\subset \Sigma=(\mathbb{N}^*)^{\mathbb{Z}} = \Sigma^-\times\Sigma^+:=(\mathbb{N}^*)^{\mathbb{Z}^-}\times (\mathbb{N}^*)^{\mathbb{N}}$$
Also, the corresponding Gauss-Cantor is $K(B)=\{[0;\gamma]:\gamma\in\Sigma^+(B)\}$ where $\Sigma^+(B) = \varphii^+(\Sigma(B))$ and $\varphii^+:\Sigma\to\Sigma^+$ is the natural projection (related to local unstable manifolds of the left shift map on $\Sigma$).
For later use, denote by $B^T=\{\betata^T:\betata\in B\}$ the \emph{transpose} of $B$, where $\betata^T:=(a_n,\dots, a_1)$ for $\betata=(a_1,\dots,a_n)$.
The following proposition (due to Euler) is proved in Appendix \ref{a.Euler}:
\betagin{proposition}[Euler]\label{p.Euler} If $[0;\betata]=\frac{p_n}{q_n}$, then $[0;\betata^T]=\frac{r_n}{q_n}$.
\end{proposition}
A striking consequence of this proposition is:
\betagin{corollary}\label{c.Euler} $HD(K(B)) = HD(K(B^T))$.
\end{corollary}
\betagin{proof}[Sketch of proof] The lengths of the intervals $I(\betata) = \{[0;\betata, a_1,\dots]: a_i\in\mathbb{N} \,\,\forall\,i \}$ in the construction of $K(B)$ depend only on the denominators of the partial quotients of $[0;\betata]$. Therefore, we have from Proposition \ref{p.Euler} that $K(B)$ and $K(B^T)$ are Cantor sets constructed from intervals with same lengths, and, \emph{a fortiori}, they have the Hausdorff dimension.
\end{proof}
\betagin{remark} This corollary is closely related to the existence of \emph{area-preserving} natural extensions of Gauss map (see \cite{Ar}) and the coincidence of stable and unstable dimensions of a horseshoe of an area-preserving surface diffeomorphism (see \cite{McCMa}).
\end{remark}
\subsection{Non-essentially affine Cantor sets} We say that
$$K=\bigcap\limits_{n\in\mathbb{N}} \varphisi^{-n}(I_1\cup\dots\cup I_r)$$
is \emph{non-essentially affine} if there is \emph{no} global conjugation $h\circ\varphisi\circ h^{-1}$ such that \emph{all} branches
$$(h\circ \varphisi\circ h^{-1})|_{h(I_j)}, \,\,\, j=1, \dots, r$$
are affine maps of the real line.
Equivalently, if $p\in K$ is a periodic point of $\varphisi$ of period $k$ and $h:I\to I$ is a diffeomorphism of the convex hull $I$ of $I_1\cup\dots \cup I_r$ such that $h\circ\varphisi^k\circ h^{-1}$ is affine\footnote{Such a diffeomorphism $h$ linearizing \emph{one} branch of $\varphisi$ always exists by Poincar\'e's linearization theorem.} on $h(J)$ where $J$ is the connected component of the domain of $\varphisi^k$ containing $p$, then $K$ is non-essentially affine if and only if $(h\circ\varphisi\circ h^{-1})''(x)\neq 0$ for some $x\in h(K)$.
\betagin{proposition}\label{p.non-ess-aff-Gauss} Gauss-Cantor sets are non-essentially affine.
\end{proposition}
\betagin{proof} The basic idea is to explore the fact that the second derivative of a non-affine M\"obius transformation never vanishes.
More concretely, let $B=\{\betata_1,\dots,\betata_m\}$, $\betata_j\in(\mathbb{N}^*)^{r_j}$, $1\leq j\leq m$. For each $\betata_j$, let
$$x_j:=[0;\betata_j,\betata_j,\dots]\in I_j=I(\betata_j)\subset\{[0;\betata_j,\alphapha]:\alphapha\geq 1\}$$
be the fixed point of the branch $\varphisi|_{I_j}=G^{r_j}$ of the expanding map $\varphisi$ naturally\footnote{Cf. Exercise \ref{ex.Gauss-Cantor}.} defining the Gauss-Cantor set $K(B)$.
By Corollary \ref{c.Mobius-fraction}, $\varphisi|_{I_j}(x)=\frac{q^{(j)}_{r_{j}-1}x - p^{(j)}_{r_{j}-1}}{p^{(j)}_{r_j} - q^{(j)}_{r_j}x}$ where $\frac{p^{(j)}_k}{q^{(j)}_k} = [0; b^{(j)}_1, \dots, b^{(j)}_k]$ and $\betata_j = (b^{(j)}_1, \dots, b^{(j)}_{r_j})$.
Note that the fixed point $x_j$ of $\varphisi|_{I_j}$ is the positive solution of the second degree equation
$$q^{(j)}_{r_j} x^2 + (q^{(j)}_{r_{j}-1} - p^{(j)}_{r_j}) x - p^{(j)}_{r_{j}-1} = 0$$
In particular, $x_j$ is a \emph{quadratic surd}.
For each $1\leq j\leq k$, the M\"obius transformation $\varphisi|_{I_j}$ has a hyperbolic fixed point $x_j$. It follows (from Poincar\'e linearization theorem) that there exists a M\"obius transformation
$$\alphapha_j(x) = \frac{a_j x + b_j}{c_j x + d_j}$$
linearizing $\varphisi|_{I_j}$, i.e., $\alphapha_j(x_j)=x_j$, $\alphapha'(x_j)=1$ and $\alphapha_j\circ (\varphisi|_{I_j})\circ \alphapha_j^{-1}$ is an affine map.
Since non-affine M\"obius transformations have non-vanishing second derivative, the proof of the proposition will be complete once we show that $\alphapha_1\circ(\varphisi|_{I_2})\circ\alphapha_1^{-1}$ is not affine. So, let us suppose by contradiction that $\alphapha_1\circ(\varphisi|_{I_2})\circ\alphapha_1^{-1}$ is affine. In this case, $\infty$ is a common fixed point of the (affine) maps $\alphapha_1\circ(\varphisi|_{I_2})\circ\alphapha_1^{-1}$ and $\alphapha_1\circ(\varphisi|_{I_1})\circ\alphapha_1^{-1}$, and, \emph{a fortiori}, $\alphapha_1^{-1}(\infty) = -d_1/c_1$ is a common fixed point of $\varphisi|_{I_1}$ and $\varphisi|_{I_2}$. Thus, the second degree equations
$$q^{(1)}_{r_1} x^2 + (q^{(1)}_{r_1-1} - p^{(1)}_{r_1}) x - p^{(j)}_{r_1-1} = 0 \quad \textrm{and} \quad q^{(2)}_{r_2} x^2 + (q^{(2)}_{r_2-1} - p^{(2)}_{r_2}) x - p^{(2)}_{r_2-1} = 0$$
would have a common root. This implies that these polynomials coincide (because they are polynomials in $\mathbb{Z}[x]$ which are irreducible\footnote{Thanks to the fact that their roots $x_1, x_2\notin\mathbb{Q}$.}) and, hence, their other roots $x_1$, $x_2$ must coincide, a contradiction.
\end{proof}
\subsection{Moreira's dimension formula} The Hausdorff dimension of projections of products of non-essentially affine Cantor sets is given by the following formula:
\betagin{theorem}[Moreira]\label{t.Moreira-dim} Let $K$ and $K'$ be two $C^2$ dynamical Cantor sets. If $K$ is non-essentially affine, then the projection $f(K\times K')=K+K'$ of $K\times K'$ under $f(x,y)=x+y$ has Hausdorff dimension
$$HD(f(K+K')) = \min\{1, HD(K)+HD(K')\}$$
\end{theorem}
\betagin{remark} This statement is a \emph{particular} case of Moreira's dimension formula (which is sufficient for our current purposes because Gauss-Cantor sets are non-essentially affine).
\end{remark}
The proof of this result is out of the scope of these notes: indeed, it depends on the techniques introduced in two works (from 2001 and 2010) by Moreira and Yoccoz \cite{MY01}, \cite{MY10} such as fine analysis of \emph{limit geometries} and \emph{renormalization operators}, ``recurrence on scales'', ``compact recurrent sets of relative configurations'', and \emph{Marstrand's theorem}. We refer the reader to \cite{Moreira-dim} for more details.
\betagin{remark} Moreira's dimension formula is coherent with Hall's Lemma \ref{l.Hall}: in fact, since $HD(C(4))>1/2$, it is natural that $HD(C(4)+C(4))=1$.
\end{remark}
\subsection{First step towards Moreira's theorem \ref{t.Gugu}: projections of Gauss-Cantor sets}
Let $\Sigma(B)\subset(\mathbb{N}^*)^{\mathbb{Z}}$ be a complete shift of finite type. Denote by $\ell(\Sigma(B))$, resp. $m(\Sigma(B))$, the pieces of the Lagrange, resp. Markov, spectrum generated by $\Sigma(B)$, i.e.,
$$\ell(\Sigma(B)) = \{\ell(\underline{\theta}): \underline{\theta}\in\Sigma(B)\}, \,\, \textrm{resp.} \quad
m(\Sigma(B)) = \{m(\underline{\theta}): \underline{\theta}\in\Sigma(B)\}$$
where $\ell(\underline{\theta})=\limsup\limits_{n\to\infty}f(\sigma^n(\underline{\theta}))$, $m(\underline{\theta})=\sup\limits_{n\in\mathbb{Z}}f(\sigma^n(\underline{\theta}))$, $f((\theta_i)_{i\in\mathbb{Z}}) = [\theta_0;\theta_1,\dots]+[0;\theta_{-1},\dots]$ and $\sigma((\theta_i)_{i\in\mathbb{Z}}) = (\theta_{i+1})_{i\in\mathbb{Z}}$ is the shift map.
The following proposition relates the Hausdorff dimensions of the pieces of the Langrange and Markov spectra associated to $\Sigma(B)$ and the projection $f(K(B)\times K(B^T))$:
\betagin{proposition}\label{p.Lagrange-Cantor} One has $HD(\ell(\Sigma(B))) = HD(m(\Sigma(B))) = \min\{1, 2\cdot HD(K(B))\}$.
\end{proposition}
\betagin{proof}[Sketch of proof] By definition,
$$\ell(\Sigma(B))\subset m(\Sigma(B))\subset \bigcup\limits_{a=1}^R(a+K(B)+K(B^T))$$
where $R\in\mathbb{N}$ is the largest entry among all words of $B$.
Thus, $HD(\ell(\Sigma(B))) \leq HD(m(\Sigma(B))) \leq HD(K(B)) + HD(K(B^T))$. By Corollary \ref{c.Euler}, it follows that
$$HD(\ell(\Sigma(B))) \leq HD(m(\Sigma(B))) \leq \min\{1, 2\cdot HD(K(B))\}$$
By Moreira's dimension formula (cf. Theorem \ref{t.Moreira-dim}), our task is now reduced to show that for all $\varepsilon>0$, there are ``replicas'' $K$ and $K'$ of Gauss-Cantor sets such that
$$HD(K), HD(K') > HD(K(B))-\varepsilon \quad \textrm{and} \quad f(K\times K')=K+K'\subset \ell(\Sigma(B))$$
In this direction, let us order $B$ and $B^T$ by declaring that $\gamma<\gamma'$ if and only if $[0;\gamma] < [0;\gamma']$.
Given $\varepsilon>0$, we can replace if necessary $B$ and/or $B^T$ by $B^n=\{\gamma_1\dots\gamma_n: \gamma_i\in B\,\,\forall\,i\}$ and/or $(B^T)^n$ for some large $n=n(\varepsilon)\in\mathbb{N}$ in such a way that
$$HD(K(B^*)), HD(K((B^T)^*)) > HD(K(B))-\varepsilon$$
where $A^*:=\{\min A, \max A\}$. Indeed, this holds because the Hausdorff dimension of a Gauss-Cantor set $K(A)$ associated to an alphabet $A$ with a large number of words does not decrease too much after removing only two words from $A$.
We \emph{expect} the values of $\ell$ on $((B^T)^*)^{\mathbb{Z}^-}\times (B^*)^{\mathbb{N}}$ to \emph{decrease} because we removed the minimal and maximal elements of $B$ and $B^T$ (and, in general, $[a_0; a_1, a_2, \dots]<[b_0; b_1, b_2, \dots]$ if and only if $(-1)^k(a_k-b_k)<0$ where $k$ is the smallest integer with $a_k\neq b_k$).
In particular, this gives \emph{some} control on the values of $\ell$ on $((B^T)^*)^{\mathbb{Z}^-}\times (B^*)^{\mathbb{N}}$, but this does \emph{not} mean that $K(B^*)+K((B^T)^*)\subset\ell(\Sigma(B))$.
We overcome this problem by studying \emph{replicas} of $K(B^*)$ and $K((B^T)^*)$. More precisely, let $\widetilde{\theta} = (\dots,\widetilde{\gamma}_0,\widetilde{\gamma}_1,\dots)\in\Sigma(B)$, $\widetilde{\gamma}_i\in B$ for all $i\in\mathbb{Z}$, such that
$$m(\widetilde{\theta}) = \max m(\Sigma(B))$$
is attained at a position in the block $\widetilde{\gamma}_0$.
By compactness, there exists $\eta>0$ and $m\in\mathbb{N}$ such that any
$$\theta=(\dots,\gamma_{-m-2},\gamma_{-m-1},\widetilde{\gamma}_{-m},\dots,\widetilde{\gamma}_0,\dots,\widetilde{\gamma}_m,\gamma_{m+1},\gamma_{m+2},\dots)$$
with $\gamma_i\in B^*$ for all $i>m$ and $\gamma_i\in (B^T)^*$ for all $i<-m$ satisfies:
\betagin{itemize}
\item $m(\theta)$ is attained in a position in the \emph{central block} $(\widetilde{\gamma}_{-m}, \dots, \widetilde{\gamma}_0, \dots, \widetilde{\gamma}_m)$;
\item $f(\sigma^n(\theta)) < m(\theta)-\eta$ for any \emph{non-central position} $n$.
\end{itemize}
By exploring these properties, it is possible to enlarge the central block to get a word called $\tau^{\#}=(a_{-N_1},\dots, a_0,\dots, a_{N_2})$ in Moreira's paper \cite{Moreira} such that the replicas
$$K=\{[a_0; a_1,\dots, a_{N_2}, \gamma_1, \gamma_2,\dots]: \gamma_i\in B^*\,\,\forall\,i>0\}$$
and
$$K'=\{[0; a_{-1},\dots, a_{-N_1}, \gamma_{-1}, \gamma_{-2},\dots]: \gamma_i\in (B^T)^*\,\,\forall\,i<0\}$$
of $K(B^*)$ and $K((B^T)^*)$ have the desired properties that
$$K+K' = f(K\times K')\subset \ell(\Sigma(B))$$
and
$$HD(K)=HD(K(B^*))>HD(K)-\varepsilon, \quad HD(K')=HD(K((B^T)^*))>HD(K(B^T))-\varepsilon$$
This completes our sketch of proof of the proposition.
\end{proof}
\subsection{Second step towards Moreira's theorem \ref{t.Gugu}: upper semi-continuity}
Let $\Sigma_t:=\{\theta\in (\mathbb{N}^*)^{\mathbb{Z}}: m(\theta)\leq t\}$ for $3\leq t < 5$.
Our long term goal is to compare $\Sigma_t$ with its projection $K_t^+:=\{[0;\gamma]:\gamma\in\varphii^+(\Sigma_t)\}$ on the unstable part (where $\varphii^+:(\mathbb{N}^*)^{\mathbb{Z}}\to (\mathbb{N}^*)^{\mathbb{N}}$ is the natural projection).
Given $\alphapha=(a_1,\dots,a_n)$, its \emph{unstable scale} $r^+(\alphapha)$ is
$$r^+(\alphapha) = \lfloor\log 1/(\textrm{length of }I^+(\alphapha))\rfloor$$
where $I^+(\alphapha)$ is the interval with extremities $[0;a_1,\dots,a_n]$ and $[0; a_1,\dots,a_n+1]$.
Denote by
$$P_r^+:=\{\alphapha=(a_1,\dots, a_n): r^+(\alphapha)\geq r, r^+(a_1,\dots,a_{n-1})<r\}$$
and
$$C^+(t,r):=\{\alphapha\in P_r^+: I^+(\alphapha)\cap K_t^+\neq\emptyset\}.$$
\betagin{remark} By symmetry (i.e., replacing $\gamma$'s by $\gamma^T$'s), we can define $K^-_t$, $r^-(\alphapha)$, etc.
\end{remark}
For later use, we observe that the unstable scales have the following behaviour under concatenations of words:
\betagin{exercise}\label{ex.subadditive} Show that $r^+(\alphapha\betata k)\geq r^+(\alphapha)+r^+(\betata)$ for all $\alphapha$, $\betata$ finite words and for all $k\in\{1,2,3,4\}$.
\end{exercise}
In particular, since the family of intervals
$$\{I^+(\alphapha\betata k): \alphapha\in C^+(t,r), \betata\in C^+(t,s), 1\leq k\leq 4\}$$
covers $K_t^+$, it follows from Exercise \ref{ex.subadditive} that
$$\# C^+(t,r+s)\leq 4\#C^+(t,r)\#C^+(t,s)$$
for all $r, s\in\mathbb{N}$ and, hence, the sequence $(4\#C^+(t,r))_{r\in\mathbb{N}}$ is \emph{submultiplicative}.
So, the \emph{box-counting dimension} (cf. Remark \ref{r.box-counting}) $\mc{D}elta^+(t)$ of $K_t^+$ is
$$\mc{D}elta^+(t) = \inf\limits_{m\in\mathbb{N}}\frac{1}{m}\log(4\#C^+(t,m)) =
\lim\limits_{m\to\infty} \frac{1}{m}\log\#C^+(t,m)$$
An elementary compactness argument shows that the upper-semicontinuity of $\mc{D}elta^+(t)$:
\betagin{proposition}\label{p.upper-sc} The function $t\mapsto \mc{D}elta^+(t)$ is upper-semicontinuous.
\end{proposition}
\betagin{proof} For the sake of contradiction, assume that there exist $\eta>0$ and $t_0$ such that $\mc{D}elta^+(t)>\mc{D}elta^+(t_0)+\eta$ for all $t>t_0$.
By definition, this means that there exists $r_0\in\mathbb{N}$ such that
$$\frac{1}{r}\log\#C^+(t,r) > \mc{D}elta^+(t_0)+\eta$$
for all $r\geq r_0$ and $t>t_0$.
On the other hand, $C^+(t,r)\subset C^+(s,r)$ for all $t\leq s$ and, by compactness, $C^+(t_0,r)=\bigcap\limits_{t>t_0} C^+(t,r)$. Thus, if $r\to\infty$ and $t\to t_0$, the inequality of the previous paragraph would imply that
$$\mc{D}elta^+(t_0) > \mc{D}elta^+(t_0)+\eta,$$
a contradiction.
\end{proof}
\subsection{Third step towards Moreira's theorem \ref{t.Gugu}: lower semi-continuity}
The main result of this subsection is the following theorem allowing us to ``approximate from inside'' $\Sigma_t$ by Gauss-Cantor sets.
\betagin{theorem}\label{t.lower-sc} Given $\eta>0$ and $3\leq t<5$ with $d(t):=HD(L\cap (-\infty,t))>0$, we can find $\delta>0$ and a Gauss-Cantor set $K(B)$ associated to $\Sigma(B)\subset\{1,2,3,4\}^{\mathbb{Z}}$ such that
$$\Sigma(B)\subset \Sigma_{t-\delta} \quad \textrm{and} \quad HD(K(B))\geq (1-\eta)\mc{D}elta^+(t)$$
\end{theorem}
This theorem allows us to derive the continuity statement in Moreira's theorem \ref{t.Gugu}:
\betagin{corollary}\label{c.Gugu-continuity} $\mc{D}elta^-(t)=\mc{D}elta^+(t)$ is a continuous function of $t$ and $d(t)=\min\{1,2\cdot\mc{D}elta^+(t)\}$.
\end{corollary}
\betagin{proof} By Corollary \ref{c.Euler} and Theorem \ref{t.lower-sc}, we have that
$$\mc{D}elta^-(t-\delta)\geq HD(K(B^T)) = HD(K(B))\geq (1-\eta)\mc{D}elta^+(t).$$
Also, a ``symmetric'' estimate holds after exchanging the roles of $\mc{D}elta^-$ and $\mc{D}elta^+$. Hence, $\mc{D}elta^-(t)=\mc{D}elta^+(t)$. Moreover, the inequality above says that $\mc{D}elta^-(t)=\mc{D}elta^+(t)$ is a lower-semicontinuous function of $t$. Since we already know that $\mc{D}elta^+(t)$ is an upper-semicontinuous function of $t$ thanks to Proposition \ref{p.upper-sc}, we conclude that $t\mapsto \mc{D}elta^-(t)=\mc{D}elta^+(t)$ is continuous. Finally, by Proposition \ref{p.Lagrange-Cantor}, from $\Sigma(B)\subset\Sigma_{t-\delta}$, we also have that
$$d(t-\delta)\geq HD(\ell(\Sigma(B))) = \min\{1,2\cdot HD(K(B))\}\geq (1-\eta)\min\{1,2\mc{D}elta^+(t)\}$$
Since $d(t)\leq \min\{1,\mc{D}elta^+(t)+\mc{D}elta^-(t)\}$ (because $\Sigma_t\subset\varphii^-(\Sigma_t)\times\varphii^+(\Sigma_t)$), the proof is complete.
\end{proof}
Let us now sketch the construction of the Gauss-Cantor sets $K(B)$ approaching $\Sigma_t$ from inside.
\betagin{proof}[Sketch of proof of Theorem \ref{t.lower-sc}] Fix $r_0\in\mathbb{N}$ large enough so that
$$\left|\frac{\log\#C^+(t,r)}{r} - \mc{D}elta^+(t)\right|<\frac{\eta}{80}\mc{D}elta^+(t)$$
for all $r\geq r_0$.
Set $B_0:=C^+(t,r_0)$, $k=8(\# B_0)^2\lceil 80/\eta\rceil$ and
$$\widetilde{B}:=\{\betata=(\betata_1,\dots,\betata_k):\betata_j\in B_0 \textrm{ and } I^+(\betata)\cap K_t^+\neq\emptyset\}\subset B_0^k$$
It is not hard to show that $\widetilde{B}$ has a significant cardinality in the sense that
$$\#\widetilde{B} > 2 (\# B_0)^{(1-\tfrac{\eta}{40})k}$$
In particular, one can use this information to prove that $HD(K(\widetilde{B}))$ is not far from $\mc{D}elta^+(t)$, i.e.
$$HD(K(\widetilde{B}))\geq (1-\frac{\eta}{20})\mc{D}elta^+(t)$$
Unfortunately, since we have no control on the values of $m$ on $\Sigma(\widetilde{B})$, there is no guarantee that $\Sigma(\widetilde{B})\subset\Sigma_{t-\delta}$ for some $\delta>0$.
We can overcome this issue with the aid of the notion of \emph{left-good} and \emph{right-good} positions. More concretely, we say that $1\leq j\leq k$ is a right-good position of $\betata=(\betata_1,\dots,\betata_k)\in\widetilde{B}$ whenever there are two elements $\betata^{(s)}=\betata_1\dots\betata_j\betata_{j+1}^{(s)}\dots\betata_k^{(s)}\in\widetilde{B}$, $s\in\{1,2\}$ such that
$$[0;\betata_j^{(1)}]<[0;\betata_j]<[0;\betata_j^{(2)}]$$
Similarly, $1\leq j\leq k$ is a left-good position $\betata=(\betata_1,\dots,\betata_k)\in\widetilde{B}$ whenever there are two elements $\betata^{(s)}=\betata_1\dots\betata_j\betata_{j+1}^{(s)}\dots\betata_k^{(s)}\in\widetilde{B}$, $s\in\{3,4\}$ such that
$$[0;(\betata_j^{(3)})^T]<[0;\betata_j^T]<[0;(\betata_j^{(2)})^T]$$
Furthermore, we say that $1\leq j\leq k$ is a \emph{good position} of $\betata=(\betata_1,\dots,\betata_k)\in\widetilde{B}$ when it is both a left-good and a right-good position.
Since there are at most two choices of $\betata_j\in B_0$ when $\betata_1,\dots,\betata_{j-1}$ are fixed and $j$ is a right-good position, one has that the subset
$$\mathcal{E}:=\{\betata\in\widetilde{B}:\betata \textrm{ has } 9k/10 \textrm{ good positions (at least)}\}$$ of \emph{excellent} words in $\widetilde{B}$ has cardinality
$$\#\mathcal{E} > \frac{1}{2} \#\widetilde{B} > (\# B_0)^{(1-\tfrac{\eta}{40})k}$$
We \emph{expect} the values of $m$ on $\Sigma(\mathcal{E})$ to \emph{decrease} because excellent words have many good positions. Also, the Hausdorff dimension of $K(\mathcal{E})$ is not far from $\mc{D}elta^+(t)$ thanks to the estimate above on the cardinality of $\mathcal{E}$. However, there is no reason for $\Sigma(\mathcal{E})\subset\Sigma_{t-\delta}$ for some $\delta>0$ because an \emph{arbitrary} concatenation of words in $\mathcal{E}$ might not belong to $\Sigma_t$.
At this point, the idea is to build a complete shift $\Sigma(B)\subset\Sigma_{t-\delta}$ from $\mathcal{E}$ with the following combinatorial argument. Since $\betata=(\betata_1,\dots,\betata_k)\in\mathcal{E}$ has $9k/10$ good positions, we can find good positions $1\leq i_1\leq i_2\leq\dots\leq i_{\lceil 2k/5\rceil}\leq k-1$ such that $i_s+2\leq i_{s+1}$ for all $1\leq s\leq\lceil 2k/5\rceil-1$ and $i_s+1$ are also good positions for all $1\leq s\leq \lceil 2k/5\rceil$. Because $k:=8(\# B_0)^2\lceil 80/\eta\rceil$, the pigeonhole principle reveals that we can choose positions $j_1\leq \dots\leq j_{3(\# B_0)^2}$ and words $\widehat{\betata}_{j_1}, \widehat{\betata}_{j_1+1},\dots, \widehat{\betata}_{j_{3(\# B_0)^2}}, \widehat{\betata}_{j_{3(\# B_0)^2}+1}\in B_0$ such that $j_s+2\lceil 80/\eta\rceil\leq j_{s+1}$ for all $s<3(\# B_0)^2$ and the subset
$$X=\{(\betata_1,\dots,\betata_k)\in\mathcal{E}: j_s, j_s+1 \textrm{ are good positions and } \betata_{j_s}=\widehat{\betata}_{j_s}, \betata_{j_s+1}=\widehat{\betata}_{j_s+1} \,\forall\,\,s\leq 3(\# B_0)^2 \}$$
of excellent words with prescribed subwords $\widehat{\betata}_{j_s}$, $\widehat{\betata}_{j_s+1}$ at the good positions $j_s$, $j_s+1$ has cardinality
$$\#X > (\# B_0)^{(1-\tfrac{\eta}{20})k}$$
Next, we convert $X$ into the alphabet $B$ of an appropriate complete shift with the help of the projections $\varphii_{a,b}:X\to B_0^{j_b-j_a}$, $\varphii_{a,b}(\betata_1,\dots,\betata_k) = (\betata_{j_a+1},\betata_{j_a+2},\dots,\betata_{j_b})$. More precisely, an elementary counting argument shows that we can take $1\leq a<b\leq 3(\# B_0)^2$ such that $\widehat{\betata}_{j_a}=\widehat{\betata}_{j_b}$, $\widehat{\betata}_{j_a+1} = \widehat{\betata}_{j_b+1}$, and the image $\varphii_{a,b}(X)$ of some projection $\varphii_{a,b}$ has a significant cardinality
$$\#\varphii_{a,b}(X) > (\# B_0)^{(1-\tfrac{\eta}{4})(j_b-j_a)}$$
From these properties, we get an alphabet $B=\varphii_{a,b}(X)$ whose words concatenate in an appropriate way (because $\widehat{\betata}_{j_a}=\widehat{\betata}_{j_b}$, $\widehat{\betata}_{j_a+1} = \widehat{\betata}_{j_b+1}$), the Hausdorff dimension of $K(B)$ is $HD(K(B))>(1-\eta)\mc{D}elta^+(t)$ (because $\# B >(\# B_0)^{(1-\tfrac{\eta}{4})(j_b-j_a)}$ and $j_b-j_a>2\lceil\tfrac{80}{\eta}\rceil$), and $\Sigma(B)\subset\Sigma_{t-\delta}$ for some $\delta>0$ (because the features of good positions forces the values of $m$ on $\Sigma(B)$ to decrease). This completes our sketch of proof.
\end{proof}
\subsection{End of proof of Moreira's theorem \ref{t.Gugu}}
By Corollary \ref{c.Gugu-continuity}, the function
$$t\mapsto d(t)=HD(L\cap (-\infty,t))$$
is continuous. Moreover, an inspection of the proof of Corollary \ref{c.Gugu-continuity} shows that we have also proved the equality $HD(M\cap(-\infty,t)) = HD(L\cap(-\infty,t))$.
Therefore, our task is reduced to prove that $d(3+\varepsilon)>0$ for all $\varepsilon>0$ and $d(\sqrt{12})=1$.
The fact that $d(3+\varepsilon)>0$ for any $\varepsilon$ uses explicit sequences $\theta_m\in\{1,2\}^{\mathbb{Z}}$ such that $\lim\limits_{m\to\infty} m(\theta_m) = 3$ in order to exhibit non-trivial Cantor sets in $M\cap (-\infty,3+\varepsilon)$. More precisely, consider\footnote{This choice of $\theta_m$ is motivated by the discussion in Chapter 1 of Cusick-Flahive book \cite{CF}.} the periodic sequences $$\theta_m:=\overline{2\underbrace{1\dots 1}_{2m \textrm{ times}} 2}$$
where $\overline{a_1\dots a_k}:=\dots a_1\dots a_k \,\, a_1\dots a_k\dots$. Since the sequence $\theta_{\infty} = \overline{1}, 2, 2, \overline{1}$ has the property that $m(\theta_{\infty}) = [2; \overline{1}]+[0;2,\overline{1}] =3$, and $|[a_0;a_1,\dots, a_n, b_1,\dots]-[a_0;a_1,\dots,a_n,c_1,\dots]|<\frac{1}{2^{n-1}}$ in general\footnote{See Lemma 2 in Chapter 1 of \cite{CF}.}, we have that the alphabet $B_m$ consisting of the two words $2\underbrace{1\dots 1}_{2m \textrm{ times}} 2$ and $2\underbrace{1\dots 1}_{2m+2 \textrm{ times}} 2$ satisfies
$$\Sigma(B_m)\subset \Sigma_{3+\frac{1}{2^m}}$$
Thus, $d(3+\tfrac{1}{2^m})=HD(M\cap(-\infty, 3+\frac{1}{2^m}))\geq HD(\Sigma(B_m)) = 2\cdot HD(K(B_m))>0$ for all $m\in\mathbb{N}$.
Finally, the fact that $d(\sqrt{12})=1$ follows from Corollary \ref{c.Gugu-continuity} and Remark \ref{r.JP}. Indeed, Perron showed that $m(\theta)\leq\sqrt{12}$ if and only if $\theta\in\{1,2\}^{\mathbb{Z}}$ (see the proof of Lemma 7 in Chapter 1 of Cusick-Flahive book \cite{CF}). Thus, $K_{\sqrt{12}}^+ = C(2)$. By Corollary \ref{c.Gugu-continuity}, it follows that
$$d(\sqrt{12})=\min\{1,2\cdot \mc{D}elta^+(\sqrt{12})\} = \min\{1,2\cdot HD(C(2))\}$$
Since Remark \ref{r.JP} tells us that $HD(C(2))>1/2$, we conclude that $d(\sqrt{12})=1$.
\appendix
\section{Proof of Hurwitz theorem}\label{a.Hurwitz}
Given $\alphapha\notin\mathbb{Q}$, we want to show that the inequality
$$\left|\alphapha-\frac{p}{q}\right|\leq\frac{1}{\sqrt{5}q^2}$$
has infinitely many rational solutions.
In this direction, let $\alphapha=[a_0;a_1,\dots]$ be the continued fraction expansion of $\alphapha$ and denote by $[a_0;a_1,\dots,a_n] = p_n/q_n$. We affirm that, for every $\alphapha\notin\mathbb{Q}$ and every $n\geq 1$, we have
$$\left|\alphapha-\frac{p}{q}\right|<\frac{1}{\sqrt{5}q^2}$$
for some $\frac{p}{q}\in\{\frac{p_{n-1}}{q_{n-1}}, \frac{p_n}{q_n}, \frac{p_{n+1}}{q_{n+1}}\}$.
\betagin{remark} Of course, this last statement provides infinitely many solutions to the inequality $\left|\alphapha-\frac{p}{q}\right|\leq\frac{1}{\sqrt{5}q^2}$. So, our task is reduced to prove the affirmation above.
\end{remark}
The proof of the claim starts by recalling Perron's Proposition \ref{p.Perron}:
$$\alphapha-\frac{p_n}{q_n} = \frac{(-1)^n}{(\alphapha_{n+1}+\betata_{n+1})q_n^2}$$
where $\alphapha_{n+1}:=[a_{n+1};a_{n+2},\dots]$ and $\betata_{n+1} =
\frac{q_{n-1}}{q_n} = [0;a_n,\dots,a_1]$.
For the sake of contradiction, suppose that the claim is false, i.e., there exists $k\geq 1$ such that \betagin{equation}\label{e.A1}
\max\{(\alphapha_k+\betata_k), (\alphapha_{k+1}+\betata_{k+1}), (\alphapha_{k+2}+\betata_{k+2})\}\leq \sqrt{5}
\end{equation}
Since $\sqrt{5}<3$ and $a_m\leq\alphapha_m+\betata_m$ for all $m\geq 1$, it follows from \eqref{e.A1} that
\betagin{equation}\label{e.A2}
\max\{a_k,a_{k+1},a_{k+2}\}\leq 2
\end{equation}
If $a_m=2$ for some $k\leq m\leq k+2$, then \eqref{e.A2} would imply that $\alphapha_m+\betata_m\geq 2+[0;2,1] = 2+\frac{1}{3}>\sqrt{5}$, a contradiction with our assumption \eqref{e.A1}.
So, our hypothesis \eqref{e.A1} forces
\betagin{equation}\label{e.A3}
a_k=a_{k+1}=a_{k+2}=1
\end{equation}
Denoting by $x=\frac{1}{\alphapha_{k+2}}$ and $y=\betata_{k+1} = q_{k-1}/q_k\in\mathbb{Q}$, we have from \eqref{e.A3} that
$$\alphapha_{k+1}=1+x, \quad \alphapha_k = 1+\frac{1}{1+x}, \quad \betata_{k} = \frac{1}{y}-1, \quad \betata_{k+2} = \frac{1}{1+y}$$
By plugging this into \eqref{e.A1}, we obtain
\betagin{equation}\label{e.A4}
\max\left\{\frac{1}{1+x}+\frac{1}{y}, 1+x+y, \frac{1}{x}+\frac{1}{1+y}\right\}\leq \sqrt{5}
\end{equation}
On one hand, \eqref{e.A4} implies that
$$\frac{1}{1+x}+\frac{1}{y}\leq \sqrt{5} \quad \textrm{and} \quad 1+x\leq \sqrt{5}-y.$$
Thus,
$$\frac{\sqrt{5}}{y(\sqrt{5}-y)} = \frac{1}{\sqrt{5}-y}+\frac{1}{y}\leq \frac{1}{1+x}+\frac{1}{y}\leq\sqrt{5},$$
and, \emph{a fortiori}, $y(\sqrt{5}-y)\geq 1$, i.e.,
\betagin{equation}\label{e.A5}
\frac{\sqrt{5}-1}{2}\leq y\leq \frac{\sqrt{5}+1}{2}
\end{equation}
On the other hand, \eqref{e.A4} implies that
$$x\leq \sqrt{5}-1-y \quad \textrm{and} \quad \frac{1}{x}+\frac{1}{1+y}\leq \sqrt{5}.$$
Hence,
$$\frac{\sqrt{5}}{(1+y)(\sqrt{5}-1-y)} = \frac{1}{\sqrt{5}-1-y}+\frac{1}{1+y}\leq \frac{1}{x}+\frac{1}{1+y}\leq\sqrt{5},$$
and, \emph{a fortiori}, $(1+y)(\sqrt{5}-1-y)\geq 1$, i.e.,
\betagin{equation}\label{e.A6}
\frac{\sqrt{5}-1}{2}\leq y\leq \frac{\sqrt{5}+1}{2}
\end{equation}
It follows from \eqref{e.A5} and \eqref{e.A6} that $y=(\sqrt{5}-1)/2$, a contradiction because $y=\betata_{k+1}= q_{k-1}/q_k\in\mathbb{Q}$. This completes the argument.
\section{Proof of Euler's remark}\label{a.Euler}
Denote by $[0; a_1, a_2,\dots, a_n] = \frac{p(a_1,\dots,a_n)}{q(a_1,\dots,a_n)} = \frac{p_n}{q_n}$. It is not hard to see that
$$q(a_1)=a_1, \quad q(a_1,a_2) = a_1a_2+1, \quad q(a_1,\dots,a_n) = a_n q(a_1,\dots,a_{n-1}) + q(a_1,\dots,a_{n-2}) \,\,\,\,\forall\,\,n\geq 3.$$
From this formula, we see that $q(a_1,\dots,a_n)$ is a sum of the following products of elements of $\{a_1,\dots,a_n\}$. First, we take the product $a_1\dots a_n$ of all $a_i$'s. Secondly, we take all products obtained by removing any pair $a_i a_{i+1}$ of adjacent elements. Then, we iterate this procedure until no pairs can be omitted (with the convention that if $n$ is even, then the empty product gives $1$). This rule to describe $q(a_1,\dots,a_n)$ was discovered by Euler.
It follows immediately from Euler's rule that $q(a_1,\dots,a_n) = q(a_n,\dots,a_1)$. This proves Proposition \ref{p.Euler}.
\betagin{thebibliography}{99}
\bibitem{Ar}
{\bf P. Arnoux},
\newblock{\it Le codage du flot g\'eod\'esique sur la surface modulaire},
\newblock Enseign. Math. (2) 40 (1994), no. 1-2, 29--48.
\bibitem{CMM}
{\bf A. Cerqueira, C. Matheus and C. G. Moreira},
\newblock{\it Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra},
\newblock Preprint (2016) available at arXiv:1602.04649.
\bibitem{CF}
{\bf T. Cusick and M. Flahive},
\newblock{\it The Markoff and Lagrange spectra},
\newblock Mathematical Surveys and Monographs, 30. American Mathematical Society, Providence, RI, 1989. x+97 pp.
\bibitem{Di}
{\bf P. G. Dirichlet},
\newblock{\it Verallgemeinerung eines Satzes aus der Lehre von den Kettenbr\"uchen nebst einigen Anwendungen auf die Theorie der Zahlen},
\newblock p.633-638
Bericht \"uber die Verhandlungen der K\"oniglich Preussischen Akademie der Wissenschaften. Jahrg. 1842, S. 93-95
\bibitem{Falconer}
{\bf K. Falconer},
\newblock{\it The geometry of fractal sets},
\newblock Cambridge Tracts in Mathematics, 85. Cambridge University Press, Cambridge, 1986. xiv+162 pp.
\bibitem{Fr68}
{\bf G. Freiman},
\newblock{\it Non-coincidence of the spectra of Markov and of Lagrange},
\newblock Mat. Zametki 3 1968 195--200.
\bibitem{Fr73}
{\bf G. Freiman},
\newblock{\it Non-coincidence of the spectra of Markov and of Lagrange},
\newblock Number-theoretic studies in the Markov spectrum and in the structural theory of set addition (Russian), pp. 10--15, 121--125. Kalinin. Gos. Univ., Moscow, 1973.
\bibitem{Fr75}
{\bf G. Freiman},
\newblock{\it Diophantine approximations and the geometry of numbers (Markov's problem)},
\newblock Kalinin. Gosudarstv. Univ., Kalinin, 1975. 144 pp.
\bibitem{Hall}
{\bf M. Hall},
\newblock{\it On the sum and product of continued fractions},
\newblock Ann. of Math. (2) 48, (1947). 966--993.
\bibitem{He}
{\bf D. Hensley},
\newblock{\it Continued fraction Cantor sets, Hausdorff dimension, and functional analysis},
\newblock J. Number Theory 40 (1992), no. 3, 336--358.
\bibitem{HMU}
{\bf P. Hubert, L. Marchese and C. Ulcigrai},
\newblock{\it Lagrange spectra in Teichm\"uller dynamics via renormalization},
\newblock Geom. Funct. Anal. 25 (2015), no. 1, 180--255.
\bibitem{Hu}
{\bf A. Hurwitz},
\newblock{\it Ueber die angen\"aherte Darstellung der Irrationalzahlen durch rationale Br\"uche},
\newblock Math. Ann. 39 (1891), no. 2, 279--284.
\bibitem{JePo1}
{\bf O. Jenkinson and M. Pollicott},
\newblock{\it Computing the dimension of dynamically defined sets: E2 and bounded continued fractions},
\newblock Ergodic Theory Dynam. Systems 21 (2001), no. 5, 1429--1445.
\bibitem{JePo2}
{\bf O. Jenkinson and M. Pollicott},
\newblock{\it Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: a hundred decimal digits for the dimension of $E_2$},
\newblock Preprint (2016) available at arXiv:1611.09276.
\bibitem{Kh}
{\bf A. Khinchin},
\newblock {\it Continued fractions},
\newblock The University of Chicago Press, Chicago, Ill.-London 1964 xi+95 pp.
\bibitem{Le}
{\bf P. L\'evy},
\newblock {\it Sur le d\'eveloppement en fraction continue d'un nombre choisi au hasard},
\newblock Compositio Math. 3 (1936), 286--303.
\bibitem{Ma}
{\bf A. Markoff},
\newblock{\it Sur les formes quadratiques binaires ind\'efinies},
\newblock Math. Ann. 17 (1880), no. 3, 379--399.
\bibitem{Marstrand}
{\bf J. Marstrand},
\newblock{\it Some fundamental geometrical properties of plane sets of fractional dimensions},
\newblock Proc. London Math. Soc. (3) 4, (1954). 257--302.
\bibitem{Mahler}
{\bf K. Mahler},
\newblock{\it On lattice points in n-dimensional star bodies. I. Existence theorems},
\newblock Proc. Roy. Soc. London. Ser. A. 187, (1946). 151--187.
\bibitem{McCMa}
{\bf H. McCluskey and A. Manning},
\newblock{\it Hausdorff dimension for horseshoes},
\newblock Ergodic Theory Dynam. Systems 3 (1983), no. 2, 251--260.
\bibitem{Moreira}
{\bf C. G. Moreira},
\newblock{\it Geometric properties of the Markov and Lagrange spectra},
\newblock Preprint (2016) available at arXiv:1612.05782.
\bibitem{Moreira-dim}
{\bf C. G. Moreira},
\newblock{\it Geometric properties of images of cartesian products of regular Cantor sets by differentiable real maps},
\newblock Preprint (2016) available at arXiv:1611.00933.
\bibitem{MR}
{\bf C. G. Moreira and S. Roma\~na},
\newblock{\it On the Lagrange and Markov dynamical spectra},
\newblock Ergodic Theory and Dynamical Systems (2016), pp. 1--22. doi: 10.1017/etds.2015.121.
\newblock
\bibitem{MY01}
{\bf C. G. Moreira and J.-C. Yoccoz},
\newblock{\it Stable intersections of regular Cantor sets with large Hausdorff dimensions},
\newblock Ann. of Math. (2) 154 (2001), no. 1, 45--96.
\bibitem{MY10}
{\bf C. G. Moreira and J.-C. Yoccoz},
\newblock{\it Tangences homoclines stables pour des ensembles hyperboliques de grande dimension fractale},
\newblock Ann. Sci. \'Ec. Norm. Sup\'er. (4) 43 (2010), no. 1, 1--68.
\bibitem{Newhouse}
{\bf S. Newhouse},
\newblock{\it The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms},
\newblock Inst. Hautes \'Etudes Sci. Publ. Math. No. 50 (1979), 101--151.
\bibitem{PP}
{\bf J. Parkkonen and F. Paulin},
\newblock{\it Prescribing the behaviour of geodesics in negative curvature},
\newblock Geom. Topol. 14 (2010), no. 1, 277--392.
\bibitem{WM}
{\bf D. Witte Morris},
\newblock {\it Ratner's theorems on unipotent flows},
\newblock Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2005. xii+203 pp.
\bibitem{Za}
{\bf D. Zagier},
\newblock{\it On the number of Markoff numbers below a given bound},
\newblock Math. Comp. 39 (1982), no. 160, 709--723.
\bibitem{Za-RH}
{\bf D. Zagier},
\newblock{\it Eisenstein series and the Riemann zeta function},
\newblock Automorphic forms, representation theory and arithmetic (Bombay, 1979), pp. 275–301,
Tata Inst. Fund. Res. Studies in Math., 10, Tata Inst. Fundamental Res., Bombay, 1981.
\end{thebibliography}
\end{document} |
\begin{document}
\title{Vector Representation of Preferences on $\sigma$-Algebras and Fair Division in Saturated Measure Spaces\thanks{I am grateful to M. Ali Khan, who insisted a more suitable usage on the terminology for axiomatization, and a referee for helpful comments. This research was supported by JSPS KAKENHI Grant Number JP18K01518 from the Ministry of Education, Culture, Sports, Science and Technology, Japan.}
\setcounter{page}{0}
\thispagestyle{empty}
\begin{abstract}
The purpose of this paper is twofold. First, we axiomatize preference relations on a $\sigma$-\hspace{0pt}algebra of a saturated measure space represented by a vector measure and furnish a utility representation in terms of a nonadditive measure satisfying the appropriate requirement of continuity and convexity. Second, we investigate the fair division problems in which each individual has nonadditive preferences on a $\sigma$-\hspace{0pt}algebra invoking our utility representation result. We show the existence of individually rational Pareto optimal partitions, Walrasian equilibria, core partitions, and Pareto optimal envy-free partitions. \\
\noindent
{\bfseries Key words:} preference relation on $\sigma$-algebra, vector measure, nonadditive measure, Lyapunov convexity theorem, saturated measure space, integral transformation, individual rationality, Pareto optimality, envy-freeness, core, Walrasian equilibrium. \\
\noindent
{\bfseries MSC 2000:} Primary: 28B05, 28E10, 46B22,46G10; secondary: 91B16, 91B50.
\end{abstract}
\section{Introduction}
Dividing a fixed amount of resources among members of a society to achieve efficiency and fairness is a central theme of social decision making. The first constructive solution to the problem of dividing a ``cake'' fairly was provided by \citet{st48}. Here, cake is simply a metaphor for a heterogeneously divisible commodity, and subsequently, the problem was formulated elegantly in \citet{ds61} as a partitioning problem in a measure space in which the preferences of each individual for pieces of a cake are represented by a nonatomic probability measure. Although the mathematical problem under investigation has attracted considerable attention in recent years, it has a long history. For the historical background, see \cite[Chapter 1]{ba05} and \cite[Chapter 2]{bt96}. Regarding the existence of various solutions for efficiency and fairness and the construction of the protocol/algorithm to obtain fair solutions, satisfactory results have been obtained under the additivity of preferences of each individual; see \cite{ba05,ds61,we85} for the nonconstructive existence of solutions and \cite{bt96,rw98} for the construction of the protocol/algorithm for fair solutions.
Incidentally, representing preference relations on a $\sigma$-algebra by means of a probability measure means that the utility function inevitably exhibits ``constant marginal utility''. This implies that given an additional element $X$ in a $\sigma$-\hspace{0pt}algebra ${\mathcal{S}}igma$, the utility function $\nu$ on ${\mathcal{S}}igma$ given by a probability measure satisfies $\nu(A\cup X)-\nu(A)=\nu(X)$ for every $A\in {\mathcal{S}}igma$ that is disjoint from $X$, which reveals that the marginal utility is independent of $A$. Obviously, this is a severe restriction on preference relations that is difficult to justify from an economics point of view. Thus, the reasonable conditions under which the preferences of each individual can have nonadditive representations should be addressed for fair division problems. For earlier attempts to axiomatize the preference relations on a finite measure space with their continuous representation by means of a nonadditive measure, see \cite{dm09,sv09}.
The purpose of this paper is twofold. First, we axiomatize preference relations on a $\sigma$-\hspace{0pt}algebra of a finite measure space represented by a vector measure and furnish a utility representation in terms of a nonadditive measure that satisfies the appropriate requirement of continuity and convexity, which presents a different approach from \cite{dm09,sv09} with a numerical representation of preference relations by means of a nonatomic finite measure. The axioms we introduce here guarantee that the utility functions on a $\sigma$-\hspace{0pt}algebra are continuous quasiconcave transformations of a vector measure with values in a Banach space, which lays the axiomatic foundation for nonadditive utility functions exploited in \cite{hu08,hu11,hs13,sa06} for fair division problems. In particular, if the underlying measure space is ``saturated'' as formulated in \cite{fk02,hk84,ks09}, then such a utility representation is always viable in a separable Banach space along with the Lyapunov convexity theorem in infinite dimensions established in \cite{ks13,ks15,ks16}. This utility representation has a great advantage because the preference relations on a $\sigma$-\hspace{0pt}algebra have a continuous convex extension to the set of measurable functions with values in the unit interval. We characterize the saturation of measure spaces in terms of the continuous extensions of preference relations on $\sigma$-algebras.
Second, we investigate the fair division problems in which each individual has nonadditive preferences on a $\sigma$-\hspace{0pt}algebra invoking our utility representation result. If the preferences of each individual are represented by a nonatomic probability measure, then the classical Lyapunov convexity theorem \cite{ly40} guarantees the compactness and convexity of the utility possibility set and thereby makes it possible to establish the existence of solutions with respect to efficiency and fairness; see \cite{ba05,ds61,we85}. In consideration of nonadditive utility functions on $\sigma$-\hspace{0pt}algebras, such favorable properties are no longer valid. For the clarification of the role of the nonatomicity hypothesis in the fair division problems with additive utility functions on $\sigma$-\hspace{0pt}algebras, see the survey article \cite{sa11}.
To overcome the difficulty with nonadditive utility functions on $\sigma$-\hspace{0pt}algebras, as in \cite{ak95,da01,hs13}, we subsume partitions of an economy in which the preferences of each individual are represented by a continuous transformation of a vector measure into allocations of its extended economy with the commodity space of $L^\infty$ in which the preferences of each individual on a $\sigma$-algebra are continuously extended to the subset of functions in $L^\infty$ with values in the unit interval. We clarify the role of saturation of the underlying measure space to formulate the indifference relation for each individual between partitions and allocations, which provides another characterization of saturation. Under the saturation hypothesis, we show the existence of individually rational Pareto optimal partitions without any convexity assumption on the preferences of each individual and the existence of Walrasian equilibria, core partitions, and Pareto optimal envy-free partitions under the convexity assumption.
The paper is organized as follows. After the brief introduction of saturated measure spaces and the Lyapunov convexity theorem in separable Banach spaces in Section 2, the axiomatization for the preference relations on a $\sigma$-\hspace{0pt}algebra is given in Section 3. The fair division problems are framed in Section 4 and the existence results on fair partitions are provided under the saturation hypothesis. Lastly, an open question is stated in Section 5 as a concluding remark.
\section{Preliminaries}
\subsection{Lyapunov Convexity Theorem in Saturated Measure Spaces}
Throughout this paper, we always assume that $(\Omega,{\mathcal{S}}igma,\mu)$ is a finite measure space. A measure space $(\Omega,{\mathcal{S}}igma,\mu)$ is said to be \textit{essentially countably generated} if its $\sigma$-\hspace{0pt}algebra can be generated by a countable number of subsets together with the null sets; $(\Omega,{\mathcal{S}}igma,\mu)$ is said to be \textit{essentially uncountably generated} whenever it is not essentially countably generated. Let ${\mathcal{S}}igma_X=\{ X\cap A\mid A\in {\mathcal{S}}igma \}$ be the $\sigma$-\hspace{0pt}algebra restricted to $X\in {\mathcal{S}}igma$. Denote by $L^1(X,{\mathcal{S}}igma_X,\mu)$ the space of $\mu$-\hspace{0pt}integrable functions on the measurable space $(X,{\mathcal{S}}igma_X)$ whose elements are restrictions of functions in $L^1(\Omega,{\mathcal{S}}igma,\mu)$ to $X$. An equivalence relation $\equiv$ on ${\mathcal{S}}igma$ is given by $A\equiv B \Longleftrightarrow \mu(A\triangle B)=0$, where $A\triangle B$ is the symmetric difference of $A$ and $B$ in ${\mathcal{S}}igma$. The collection of equivalence classes is denoted by ${\mathcal{S}}igma(\mu)={\mathcal{S}}igma/\equiv$ and its generic element $\widehat{A}$ is the equivalence class of $A\in {\mathcal{S}}igma$. The metric $\rho$ on ${\mathcal{S}}igma(\mu)$ is defined by $\rho(\widehat{A},\widehat{B})=\mu(A\triangle B)$. Then $({\mathcal{S}}igma(\mu),\rho)$ is a complete metric space (see \cite[Lemma 13.13]{ab06}) and $({\mathcal{S}}igma(\mu),\rho)$ is separable if and only if $L^1(\Omega,{\mathcal{S}}igma,\mu)$ is separable (see \cite[Lemma 13.14]{ab06}). The \textit{density} of $({\mathcal{S}}igma(\mu),\rho)$ is the smallest cardinal number of the form $|{\mathcal{U}}|$, where ${\mathcal{U}}$ is a dense subset of ${\mathcal{S}}igma(\mu)$.
\begin{dfn}
A finite measure space $(\Omega,{\mathcal{S}}igma,\mu)$ is \textit{saturated} if $L^1(X,{\mathcal{S}}igma_X,\mu)$ is nonseparable for every $X\in {\mathcal{S}}igma$ with $\mu(X)>0$. We say that a finite measure space has the \textit{saturation property} if it is saturated.
\end{dfn}
Saturation implies nonatomicity and several equivalent definitions for saturation are known; see \cite{fk02,fr12,hk84,ks09}. One of the simple characterizations of the saturation property is as follows. A finite measure space $(\Omega,{\mathcal{S}}igma,\mu)$ is saturated if and only if $(X,{\mathcal{S}}igma_X,\mu)$ is essentially uncountably generated for every $X\in {\mathcal{S}}igma$ with $\mu(X)>0$. The saturation of finite measure spaces is also synonymous with the uncountability of the density of ${\mathcal{S}}igma_X(\mu)$ for every $X\in {\mathcal{S}}igma$ with $\mu(X)>0$; see \cite[331Y(e)]{fr12}. An inceptive notion of saturation already appeared in \cite{ka44,ma42}.
Let $E$ be a Banach space. For a vector measure $m:{\mathcal{S}}igma\to E$, a set $N\in {\mathcal{S}}igma$ is said to be \textit{$m$-null} if $m(A\cap N)=0$ for every $A\in {\mathcal{S}}igma$. A vector measure $m:{\mathcal{S}}igma\to E$ is said to be \textit{$\mu$-\hspace{0pt}continuous} (or \textit{absolutely continuous} with respect to $\mu$) if every $\mu$-null set is $m$-null. Let ${\mathcal{S}}:=\{f\in L^\infty(\Omega,{\mathcal{S}}igma,\mu)\mid 0\le f\le 1 \}$, which is a weakly$^*\!$ compact, convex subset of $L^\infty(\Omega,{\mathcal{S}}igma,\mu)$, and define $m({\mathcal{S}}):=\{ \int fdm\in E\mid f\in {\mathcal{S}} \}$. Since the integration operator $T_m:L^\infty(\Omega,{\mathcal{S}}igma,\mu)\to E$ defined by $T_m(f):=\int fdm$ is continuous with respect to the weak$^*\!$ topology of $L^\infty(\Omega,{\mathcal{S}}igma,\mu)$ and the weak topology of $E$ (see \cite[Lemma IX.1.3]{du77}), the set $m({\mathcal{S}})$ is weakly compact and convex in $E$.
The saturation property and the Lyapunov convexity theorem in separable Banach spaces are a major apparatus in this paper.
\begin{prop}[\citet{ks13}]
\label{lyap}
Let $E$ be a separable Banach space. If $(\Omega,{\mathcal{S}}igma,\mu)$ is saturated, then for every $\mu$-continuous vector measure $m:{\mathcal{S}}igma\to E$, its range $m({\mathcal{S}}igma)$ is weakly compact and convex with $m({\mathcal{S}}igma)=m({\mathcal{S}})$. Conversely, if every $\mu$-continuous vector measure $m:{\mathcal{S}}igma\to E$ has the weakly compact convex range, then $(\Omega,{\mathcal{S}}igma,\mu)$ is saturated whenever $E$ is infinite dimensional.
\end{prop}
\begin{rem}
The significance of the saturation property lies in the fact that it is necessary and sufficient for the weak compactness and the convexity of the Bochner integral of a multifunction as well as the Lyapunov convexity theorem; see \cite{ks13,ks14,ks15,ks16,po08,sy08}. For the further generalization of Proposition \ref{lyap} to nonseparable locally convex spaces, see \cite{gp13,ks15,ks16,sa17}. Another intriguing characterization of saturation in terms of the existence of Nash equilibria in large games is found in \cite{ks09}.
\end{rem}
\section{Vector Representation of Preference Relations on $\sigma$-\hspace{0pt}Algebras}
\subsection{Axioms for Preference Relations}
Let $(\Omega,{\mathcal{S}}igma,\mu)$ be a finite measure space. A binary relation on the $\sigma$-algebra ${\mathcal{S}}igma$ is a subset of the product space ${\mathcal{S}}igma\times {\mathcal{S}}igma$. The \textit{preference relation} $\succsim$ on ${\mathcal{S}}igma$ is a complete transitive binary relation on ${\mathcal{S}}igma$. We denote by $A\succsim B$ the relation $(A,B)\in{}\succsim$. The indifference and strict preference relations are defined respectively by $A\sim B\Longleftrightarrow A\succsim B$ and $B\succsim A$ and by $A\succ B \Longleftrightarrow A\succsim B$ and $A\not\sim B$. A real-valued function $\nu:{\mathcal{S}}igma\to {\mathbb{R}}$ is called a \textit{utility function} representing $\succsim$ if for every $A,B\in {\mathcal{S}}igma$: $\nu(A)\ge \nu(B) \Longleftrightarrow A\succsim B$. A set function $\nu:{\mathcal{S}}igma\to {\mathbb{R}}$ is called a \textit{nonadditive measure} if $\nu(\emptyset)=0$.
\begin{ax}[vector representation]
\label{ax1}
There exist a Banach space $E$ and a $\mu$-\hspace{0pt}continuous vector measure $m:{\mathcal{S}}igma\to E$ such that $m(A)=m(B)$ implies $A\sim B$.
\end{ax}
\noindent
The axiom enables one to define the preference relation $\mathscr{R}$ on $m({\mathcal{S}}igma)$ by
\begin{equation}
\label{eq1}
\forall x,y\in m({\mathcal{S}}igma): x\,\mathscr{R}\,y \stackrel{\text{def}}{\Longleftrightarrow} A\succsim B \text{ with $x=m(A)$ and $y=m(B)$}.
\end{equation}
A preference relation on ${\mathcal{S}}igma$ satisfying Axiom \ref{ax1} is said to admit a \textit{vector representation} in $E$. A vector representation for $\succsim$ is not unique because any scalar multiplication of $m$ is consistent with Axiom \ref{ax1}. In addition, the observation that $\succsim$ may admit a vector representation in another Banach space motivates one to introduce the following axiom. (In what follows, we assume that $\succsim$ admits a vector representation in a Banach space $E$ via a $\mu$-continuous vector measure $m:{\mathcal{S}}igma\to E$.)
\begin{ax}[commutativity]
\label{ax2}
If $\succsim$ admits another vector representation in a Banach space $F$ via a $\mu$-continuous vector measure $n:{\mathcal{S}}igma\to F$, then there exists a unique continuous linear operator $T:E\to F$ such that $n=T\circ m$.
\end{ax}
\noindent
By the symmetric treatment of $E$ and $F$, the axiom guarantees that there exists a unique continuous linear operator $U:F\to E$ such that $m=U\circ n$. Then
$(U\circ T)m=m$ and $(T\circ U)n=n$. The situation is illustrated in the diagram below:
$$
\xymatrix{
& \ E \ar[dd]_{T} \ar@{<-}@<1.0ex>[dd]^{U} \\
{\mathcal{S}}igma \ar[ur]^m \ar[dr]_n \\
& \ F
}
$$
A preference relation on ${\mathcal{S}}igma$ satisfying Axiom \ref{ax2} is said to admit \textit{commutativity} in $E$. Commutativity guarantees that the vector representation for $\succsim$ is unique up to the equivalence in the following sense. Let ${\mathcal{V}}({\mathcal{S}}igma,\mu)$ be the set of $\mu$-continuous vector measures on ${\mathcal{S}}igma$ with values in any Banach space. A typical element in ${\mathcal{V}}({\mathcal{S}}igma,\mu)$ is denoted by $(m,E)$ to distinguish explicitly the range spaces of vector measures. Define the equivalence relation on ${\mathcal{V}}({\mathcal{S}}igma,\mu)$ by $(m,E)\simeq(n,F)\stackrel{\text{def}}\Longleftrightarrow$ there exists a unique pair of continuous linear operators $T:E\to F$ and $U:F\to E$ such that $n=T\circ m$ and $m=U\circ n$. As defined in \eqref{eq1}, the commutativity for $\succsim$ yields an induced preference relation $\mathscr{Q}$ on $n({\mathcal{S}}igma)$. Thus, for every $x,y\in m({\mathcal{S}}igma)$: $x\,\mathscr{R}\,y \Longleftrightarrow Tx\,\mathscr{Q}\,Ty$ and for every $v,w\in n({\mathcal{S}}igma)$: $v\,\mathscr{Q}\,w \Longleftrightarrow Uv\,\mathscr{R}\,Uw$.
\begin{ax}[monotonicity]
\label{ax3}
For every $A,B\in {\mathcal{S}}igma$ with $A\supset B$ and $m(A\setminus B)\ne 0$, we have $A\,\succ\,B$.
\end{ax}
\noindent
The axiom is equivalent to the monotonicity of $\mathscr{R}$ on $m({\mathcal{S}}igma)$: for every $x,y\in m({\mathcal{S}}igma)$ with $x+y\in m({\mathcal{S}}igma)$ and $y\ne 0$, we have $x+y\,\mathscr{R}\,x$, but not $x\,\mathscr{R}\,x+y$. A preference relation on ${\mathcal{S}}igma$ satisfying Axiom \ref{ax3} is said to admit a \textit{monotone representation} in $E$.
\begin{ax}[continuity]
\label{ax4}
For every $A,B\in {\mathcal{S}}igma$: $A^k\succsim B$ for each $k\in {\mathbb{N}}$ with $m(A^k)\to m(A)$ as $k\to \infty$ implies $A\succsim B$ and $A\succsim B^k$ for each $k\in {\mathbb{N}}$ with $m(B^k)\to m(B)$ implies $A\succsim B$.
\end{ax}
\noindent
The axiom is equivalent to the continuity of $\mathscr{R}$ on $m({\mathcal{S}}igma)$: for every $x\in m({\mathcal{S}}igma)$, both the upper contour set $\{ y\in m({\mathcal{S}}igma)\mid y\,\mathscr{R}\,x \}$ and the lower contour set $\{ y\in m({\mathcal{S}}igma)\mid x\,\mathscr{R}\,y \}$ are closed in $m({\mathcal{S}}igma)$. A preference relation on ${\mathcal{S}}igma$ satisfying Axiom \ref{ax4} is said to admit a \textit{continuous representation} in $E$.
\begin{ax}[convexity]
\label{ax5}
Suppose that $m({\mathcal{S}}igma)$ is convex. For every $A\in {\mathcal{S}}igma$, the set $m(\{ B\in {\mathcal{S}}igma\mid B\,\succsim\,A \})$ is convex.
\end{ax}
\noindent
The axiom is equivalent to the convexity of $\mathscr{R}$ on $m({\mathcal{S}}igma)$: for every $x\in m({\mathcal{S}}igma)$, the upper contour set $\{ y\in m({\mathcal{S}}igma)\mid y\,\mathscr{R}\,x \}$ is convex. A preference relation on ${\mathcal{S}}igma$ satisfying Axiom \ref{ax5} is said to admit a \textit{convex representation} in $E$. Note that the convexity of $m({\mathcal{S}}igma)$ is not automatic even when $m$ is nonatomic because of the well-known failure of the Lyapunov convexity theorem in infinite dimensions (for counterexamples in which the Lyapunov convexity theorem fails in infinite dimensions, see \cite[Examples IX.1.1 and IX.1.2]{du77}).
\subsection{Utility Functions on ${\mathcal{S}}igma$}
Axioms \ref{ax1}, \ref{ax2}, and \ref{ax4} guarantee the unique representation of $\succsim$ in terms of a nonadditive measure on ${\mathcal{S}}igma$ given by a continuous transformation of a vector measure. If, moreover, Axiom \ref{ax5} is imposed, then the representation is a quasiconcave transformation of a vector measure.
\begin{thm}
\label{thm1}
A preference relation $\succsim$ on ${\mathcal{S}}igma$ admits a continuous (and convex) representation in a separable Banach space $E$ if and only if there exist a $\mu$-continuous vector measure $m:{\mathcal{S}}igma\to E$ (with $m({\mathcal{S}}igma)$ being convex) and a continuous (and quasiconcave) function $\varphi:m({\mathcal{S}}igma)\to {\mathbb{R}}$ with $\varphi(0)=0$ such that:
\begin{equation}
\label{eq2}
\forall A,B\in {\mathcal{S}}igma: A\succsim B \Longleftrightarrow \varphi(m(A))\ge \varphi(m(B)).
\end{equation}
If, furthermore, $\succsim$ admits commutativity, then the continuous (and convex) representation via $(m,E)\in {\mathcal{V}}({\mathcal{S}}igma,\mu)$ is unique up to the equivalence classes in ${\mathcal{V}}({\mathcal{S}}igma,\mu)/\simeq$.
\end{thm}
\begin{proof}
Suppose that $\succsim$ admits a continuous (and convex) representation in a separable Banach space $E$. Then there exists a $\mu$-continuous vector measure $m:{\mathcal{S}}igma\to E$ (with $m({\mathcal{S}}igma)$ being convex) such that \eqref{eq1} holds. Since the induced preference relation $\mathscr{R}$ on $m({\mathcal{S}}igma)$ is continuous (and convex) and $E$ is separable, by the celebrated theorem of \citet{de64}, there exists a continuous (and quasiconcave) function $\varphi:m({\mathcal{S}}igma)\to {\mathbb{R}}$ representing the preference relation $\mathscr{R}$ on $m({\mathcal{S}}igma)$, that is, $x\,\mathscr{R}\,y\Longleftrightarrow \varphi(x)\ge \varphi(y)$. Therefore, $\varphi\circ m$ is a utility function on ${\mathcal{S}}igma$ representing the preference relation $\succsim$ on ${\mathcal{S}}igma$. By replacing $\varphi(x)$ with $\varphi(x)-\varphi(0)$ if necessary, without loss of generality one may assume that $\varphi(0)=0$. The converse implication is obvious without assuming the separability of $E$.
\end{proof}
Denote by $L^1_E(\Omega,{\mathcal{S}}igma,\mu)$ the space of $E$-\hspace{0pt}valued Bochner integrable functions on $\Omega$. A Banach space $E$ has the \textit{Radon--Nikodym property} (\textit{RNP}) for a finite measure space $(\Omega,{\mathcal{S}}igma,\mu)$ if for every $\mu$-\hspace{0pt}continuous vector measure $m:{\mathcal{S}}igma\to E$ of bounded variation, there exists $u\in L^1_E(\Omega,{\mathcal{S}}igma,\mu)$ such that $m(A)=\int_Aud\mu$ for every $A\in {\mathcal{S}}igma$. Note that for every Banach space $E$, given a Bochner integrable function $u\in L^1_E(\Omega,{\mathcal{S}}igma,\mu)$, the vector measure $m^u:{\mathcal{S}}igma\to E$ defined by the indefinite integral $m^u(A):=\int_Aud\mu$ for $A\in {\mathcal{S}}igma$ is of bounded variation; see \cite[Theorem II.2.4]{du77}. This observation yields the following result.
\begin{cor}
A preference relation $\succsim$ on ${\mathcal{S}}igma$ admits a continuous (and convex) representation in a separable Banach space $E$ with the RNP for $(\Omega,{\mathcal{S}}igma,\mu)$ if and only if there exist a Bochner integrable function $u\in L^1_E(\Omega,{\mathcal{S}}igma,\mu)$ (with $m^u({\mathcal{S}}igma)$ being convex) and a continuous (and quasiconcave) function $\varphi:m^u({\mathcal{S}}igma)\to {\mathbb{R}}$ with $\varphi(0)=0$ such that:
$$
\forall A,B\in {\mathcal{S}}igma: A\succsim B \Longleftrightarrow \varphi\left( \int_Au(\omega)d\mu \right)\ge \varphi\left( \int_Bu(\omega)d\mu \right).
$$
\end{cor}
\begin{ex}
Let $(\Omega,{\mathcal{S}}igma,\mu)$ be a nonatomic finite measure space and $f\in L^1(\Omega,{\mathcal{S}}igma,\mu)$ be a nonnegative integrable function. If $\succsim$ is such that for every $A,B\in {\mathcal{S}}igma$: $A\succsim B \stackrel{\text{def}}{\Longleftrightarrow} \int_Afd\mu\ge \int_Bfd\mu$, then $\succsim$ admits a vector representation in the real field ${\mathbb{R}}$ via the $\mu$-continuous nonatomic finite measure $\nu$ defined by $\nu(A):=\int_Afd\mu$ and satisfies Axioms \ref{ax1}, \ref{ax3}, \ref{ax4}, and \ref{ax5}. Conversely, if $\succsim$ admits a vector representation in ${\mathbb{R}}$ via the $\mu$-continuous $\sigma$-finite measure $\nu$ and satisfies Axioms \ref{ax1}, \ref{ax3}, \ref{ax4}, and \ref{ax5}, then by Theorem \ref{thm1}, there exists a continuous quasiconcave function $\varphi:\nu({\mathcal{S}}igma)\to {\mathbb{R}}$ with $\varphi(0)=0$ such that for every $A,B\in {\mathcal{S}}igma$: $A\succsim B \Longleftrightarrow \varphi(\int_Afd\mu)\ge \varphi(\int_Bfd\mu)$, where $f\in L^1(\Omega,{\mathcal{S}}igma,\mu)$ is a Radon--Nikodym derivative of $\nu$. In this specific case, $\succsim$ admits a continuous convex representation with a quasiconcave integral transformation on ${\mathcal{S}}igma$.
\end{ex}
\begin{rem}
Note that the proof of Theorem \ref{thm1} demonstrates that if $(\Omega,{\mathcal{S}}igma,\mu)$ is a nonatomic finite measure space, then $\succsim$ admits a continuous (and convex) representation in ${\mathbb{R}}^l$ if and only if there exist a $\mu$-continuous vector measure $m:{\mathcal{S}}igma\to {\mathbb{R}}^l$ with its range $m({\mathcal{S}}igma)$ being compact and convex and a continuous (and quasiconcave) function $\varphi:m({\mathcal{S}}igma)\to {\mathbb{R}}$ with $\varphi(0)=0$ satisfying \eqref{eq2}. This is an immediate consequence of the classical Lyapunov convexity theorem in view of the nonatomicity of $m$.
\end{rem}
\subsection{Continuous Extensions to ${\mathcal{S}}$}
Denote by $\chi_A$ the characteristic function of $A\in {\mathcal{S}}igma$. Then $\chi_A\in {\mathcal{S}}$ for every $A\in {\mathcal{S}}igma$. A preference relation ${\succsim}^{\diamond}$ on ${\mathcal{S}}$ is a complete transitive binary relation on ${\mathcal{S}}$. A preference relation ${\succsim}^{\diamond}$ is said to be \textit{continuous} if for every $f\in {\mathcal{S}}$ both the upper contour set $\{ g\in {\mathcal{S}}\mid g\,{\succsim}^{\diamond}\,f \}$ and the lower contour set $\{ g\in {\mathcal{S}}\mid f\,{\succsim}^{\diamond}\,g \}$ are weakly$^*\!$ closed in ${\mathcal{S}}$; ${\succsim}^{\diamond}$ is said to be \textit{convex} if for every $f\in {\mathcal{S}}$ the upper contour set $\{ g\in {\mathcal{S}}\mid g\,{\succsim}^{\diamond}\,f \}$ is convex; ${\succsim}^{\diamond}$ is said to be \textit{monotone} if $f\ge g$ with $f\ne g$ imply $f\,\succ\,g$; ${\succsim}^{\diamond}$ is called an \textit{extension} of the preference relation $\succsim$ on ${\mathcal{S}}igma$ if for every $A,B\in {\mathcal{S}}igma$: $\chi_A\,{\succsim}^{\diamond}\,\chi_B \Longleftrightarrow A\succsim B$.
If $\succsim$ is represented by a utility function of the form $\varphi\circ m$ on ${\mathcal{S}}igma$, where $m$ is a $\mu$-\hspace{0pt}continuous vector measure with values in a Banach space $E$ (with $m({\mathcal{S}}igma)$ being convex) and $\varphi$ is a continuous (and quasiconcave) function on $m({\mathcal{S}}igma)$ with a continuous (and quasiconcave) extension $\varphi^\diamond$ to $m({\mathcal{S}})$, then the continuous (and convex) extension $\succsim^\diamond$ of $\succsim$ to ${\mathcal{S}}$ is given by:
\begin{equation}
\label{eq3}
\forall f,g\in {\mathcal{S}}: f\,{\succsim}^{\diamond}g \stackrel{\text{def}}\Longleftrightarrow \varphi^\diamond\left( \int_\Omega f(\omega)dm \right)\ge \varphi^\diamond\left( \int_\Omega g(\omega)dm \right).
\end{equation}
The continuous (and convex) extension $\succsim^\diamond$ defined in this way is called a \textit{$($quasiconcave$)$ integral transformation} on ${\mathcal{S}}$. Then the utility function $\nu^\diamond:{\mathcal{S}}\to {\mathbb{R}}$ defined by $\nu^\diamond(f):=\varphi^\diamond(\int fdm)$ is weakly$^*\!$ continuous (and quasiconcave) on $S$ in view of the fact that the integration operator $T_m$ is continuous with respect to the weak$^*\!$ topology of $L^\infty(\Omega,{\mathcal{S}}igma,\mu)$ and the weak topology of $E$, as noted above.
\begin{thm}
\label{thm2}
Let $(\Omega,{\mathcal{S}}igma,\mu)$ be a saturated finite measure space. Then every preference relation $\succsim$ on ${\mathcal{S}}igma$ with a continuous representation in a separable Banach space has a continuous extension $\succsim^\diamond$ to ${\mathcal{S}}$ with an integral transformation such that for every $f\in {\mathcal{S}}$ there exists $A\in {\mathcal{S}}igma$ satisfying $f\sim^\diamond \chi_A$.
\end{thm}
\begin{proof}
Let $\succsim$ be a preference relation on ${\mathcal{S}}igma$ that admits a continuous representation in a separable Banach space $E$. By Theorem \ref{thm1}, $\succsim$ has a utility function of the form $\varphi\circ m$, where $m$ is a $\mu$-continuous vector measure with values in a separable Banach space $E$ and $\varphi$ is a continuous function on $m({\mathcal{S}}igma)$. In view of Proposition \ref{lyap}, $m({\mathcal{S}}igma)=m({\mathcal{S}})$, and hence, the desired extension ${\succsim}^{\diamond}$ of $\succsim$ to ${\mathcal{S}}$ is given by the formula \eqref{eq3} with $\varphi=\varphi^\diamond$.
\end{proof}
The converse of Theorem \ref{thm2} is true in the following sense, which provides another characterization of saturation.
\begin{thm}
\label{thm3}
A finite measure space $(\Omega,{\mathcal{S}}igma,\mu)$ is saturated if every preference relation $\succsim$ on ${\mathcal{S}}igma$ with a continuous representation in an infinite-\hspace{0pt}dimensional Banach space and its continuous extension $\succsim^\diamond$ to ${\mathcal{S}}$ with an integral transformation are such that for every $f\in {\mathcal{S}}$ there exists $A\in {\mathcal{S}}igma$ satisfying $f\sim^\diamond\chi_A$.
\end{thm}
\begin{proof}
Suppose that $(\Omega,{\mathcal{S}}igma,\mu)$ is not saturated. Then for every infinite-dimensional Banach space $E$, there exists a $\mu$-continuous vector measure $m:{\mathcal{S}}igma\to E$ such that its range $m({\mathcal{S}}igma)$ is not convex; see \cite[Lemma 4]{po08} and \cite[Remark 1]{sy08}. Then there exist $x,y\in m({\mathcal{S}}igma)$ and $t\in (0,1)$ such that $tx+(1-t)y\not\in m({\mathcal{S}}igma)$. Let $C,D\in {\mathcal{S}}igma$ be such that $x=m(C)$ and $y=m(D)$ and define $\hat{f}:=t\chi_C+(1-t)\chi_D\in {\mathcal{S}}$. Let $\varphi^\diamond:m({\mathcal{S}})\to {\mathbb{R}}$ be a continuous function with $\varphi^\diamond(0)=0$ that attains the maximum at the unique point $\hat{x}:=\int \hat{f}dm\in m({\mathcal{S}})$ (such a function can be simply given by, for instance, the strictly concave continuous function $\varphi^\diamond(z)=\| \hat{x}\|-\| \hat{x}-z \|$). Define the preference relation $\succsim^\diamond$ on ${\mathcal{S}}$ via \eqref{eq3} and denote its restriction to ${\mathcal{S}}igma$ by $\succsim$. Then $\succsim$ admits a continuous representation in the Banach space $E$ by Theorem \ref{thm1} (the separability of $E$ is unnecessary here) and $\succsim^\diamond$ is a continuous extension of $\succsim$ with an integral transformation. Hence, there exists $A\in {\mathcal{S}}igma$ such that $\hat{f}\sim^\diamond\chi_A$. This means that $\varphi^\diamond(\hat{x})=\varphi^\diamond(m(A))$, and hence, $\hat{x}=tx+(1-t)y=m(A)\in m({\mathcal{S}}igma)$, a contradiction.
\end{proof}
\begin{rem}
The continuous extension of nonadditive measures on ${\mathcal{S}}igma$ to ${\mathcal{S}}$ with an integral transformation explored in Theorem \ref{thm2} is a further improvement of \cite{hs12} under the saturation hypothesis incorporating infinite dimensional vector representation. An alternative formulation of the convexity of preference relations on $\sigma$-\hspace{0pt}algebras and their representation in terms of a nonadditive measure with the quasiconcavity-\hspace{0pt}like property based on the Lyapunov convexity theorem in finite dimensions were studied in \cite{sv09}. For continuous concave extensions of nonadditive measures to $L^\infty(\Omega,{\mathcal{S}}igma,\mu)$ based on Choquet integrals and the support functionals of the cores of exact games, see \cite{sa13}.
\end{rem}
\section{Fair Division in Saturated Measure Spaces}
\subsection{Extensions of Economies}
The problem of dividing a heterogeneous commodity among a finite number of individuals is formulated as partitioning in a finite measure space $(\Omega,{\mathcal{S}}igma,\mu)$. Here, the set $\Omega$ is a heterogeneously divisible commodity, the $\sigma$-\hspace{0pt}algebra ${\mathcal{S}}igma$ of subsets of $\Omega$ denotes the collection of possible pieces of $\Omega$, and the finite measure $\mu$ describes an objective evaluation for a cardinal attribute of each piece in ${\mathcal{S}}igma$. There are $n$ individuals and the set of individuals is denoted by $I=\{ 1,2,\dots,n \}$, each of whom is indexed by $i\in I$, whose preference relation $\succsim_i$ is defined on a common consumption set ${\mathcal{S}}igma$. Each individual possesses an initial endowment $\Omega_i\in {\mathcal{S}}igma$, for which $\Omega_i\cap \Omega_j=\emptyset$ for each $i\ne j$ and $\bigcup_{i=1}^n\Omega_i=\Omega$. A \textit{partition} of $\Omega$ is an ordered $n$-\hspace{0pt}tuple $(A_1,\dots,A_n)$ of mutually disjoint sets $A_1,\dots,A_n$ in ${\mathcal{S}}igma$ whose union is $\Omega$, where each $A_i$ is a share of the divisible commodity $\Omega$ given to individual $i\in I$. Thus, $(\Omega_1,\dots,\Omega_n)$ is an \textit{initial partition} of $\Omega$. An \textit{economy} ${\mathcal{E}}=\{ (\Omega,{\mathcal{S}}igma,\mu),(\succsim_i,\Omega_i)_{i=1}^n \}$ consists of the commodity space $(\Omega,{\mathcal{S}}igma,\mu)$, the common consumption set ${\mathcal{S}}igma$, on which a preference relation $\succsim_i$ is defined for each $i\in I$, and the initial endowment $\Omega_i$ of each individual.
Following \cite{ak95,da01,hs12}, we subsume economies with commodity space $(\Omega,{\mathcal{S}}igma,\mu)$ into ones with commodity space $L^\infty(\Omega,{\mathcal{S}}igma,\mu)$. If each $\succsim_i^\diamond$ is a preference relation on ${\mathcal{S}}$ that is an extension of $\succsim_i$ given an economy ${\mathcal{E}}$, then ${\mathcal{E}}^\diamond=\{ L^\infty(\Omega,{\mathcal{S}}igma,\mu),{\mathcal{S}},({\succsim_i^\diamond},\chi_{\Omega_i})_{i=1}^n \}$ is called an \textit{extended economy} of ${\mathcal{E}}$, which consists of the commodity space $L^\infty(\Omega,{\mathcal{S}}igma,\mu)$, the common consumption set ${\mathcal{S}}$, on which a preference relation $\succsim_i^\diamond$ is defined for each $i\in I$, and the initial endowment $\chi_{\Omega_i}\in {\mathcal{S}}$ of each individual. If $\succsim_i$ is represented by a utility function of the form $\varphi_i\circ m_i$ on ${\mathcal{S}}igma$, where $m_i$ is a $\mu$-continuous vector measure and $\varphi_i$ is a continuous (and quasiconcave) function on $m({\mathcal{S}}igma)$ (with $m({\mathcal{S}}igma)$ being convex) that has an continuous (and quasiconcave) extension $\varphi_i^\diamond$ to $m_i({\mathcal{S}})$, then a continuous (and convex) extension $\succsim_i^\diamond$ to ${\mathcal{S}}$ with a (quasiconcave) integral transformation of the preference relation $\succsim_i$ on ${\mathcal{S}}igma$ is given by the formula \eqref{eq3}. In this case, ${\mathcal{E}}^\diamond$ is called an extended economy of ${\mathcal{E}}$ with a \textit{(quasiconcave) integral transformation}. An ordered $n$-\hspace{0pt}tuple $(f_1,\dots,f_n)$ of functions in $L^\infty(\Omega,{\mathcal{S}}igma,\mu)$ is called an \textit{allocation} of $\chi_\Omega$ if $\sum_{i=1}^nf_i(\omega)=1$ for every $\omega\in \Omega$ and $f_i\in {\mathcal{S}}$ for each $i\in I$. Every $A\in {\mathcal{S}}igma$ is identified with its characteristic function $\chi_A$ in ${\mathcal{S}}$. Note that $(A_1,\dots,A_n)$ is a partition of $\Omega$ if and only if $(\chi_{A_1},\dots,\chi_{A_n})$ is an allocation of $\chi_\Omega$.
\begin{dfn}
An allocation $(f_1,\dots,f_n)$ of $\chi_\Omega$ is said to be:
\begin{enumerate}[(i)]
\item \textit{individually rational} if $f_i\succsim_i^\diamond \chi_{\Omega_i}$ for each $i\in I$;
\item \textit{Pareto optimal} if there is no allocation $(g_1,\dots,g_n)$ such that $g_i\succsim_i^\diamond f_i$ for each $i\in I$ and $g_j\succ_j^\diamond f_j$ for some $j\in I$;
\item \textit{envy-free} if $f_j\succsim_i^\diamond f_i$ for each $i,j\in I$.
\end{enumerate}
\end{dfn}
A nonempty subset of $I$ is called a \textit{coalition}. The family of coalitions is denoted by ${\mathcal{I}}$ with its typical element denoted by $S\in {\mathcal{I}}$. For a coalition $S\in {\mathcal{I}}$, an allocation $(f_1,\dots,f_n)$ of $\chi_\Omega$ is called an \textit{$S$-\hspace{0pt}allocation} if $\sum_{i\in S}f_i=\sum_{i\in S}\chi_{\Omega_i}$. Then an $I$-allocation is nothing but an allocation of $\chi_\Omega$.
\begin{dfn}
A coalition $S\in {\mathcal{I}}$ is said to \textit{improve upon} an allocation $(f_1,\dots,f_n)$ of $\chi_\Omega$ if there exists an $S$-allocation $(g_1,\dots,g_n)$ such that $g_i\,\succ_i^\diamond\,f_i$ for each $i\in S$. An allocation of $\chi_\Omega$ that cannot be improved upon by any coalition is called a \textit{core allocation}.
\end{dfn}
Let $\mathit{ba}({\mathcal{S}}igma,\mu)$ be the space of finitely additive signed measures vanishing on $\mu$-null sets, which is the price space for the heterogeneously divisible commodity $(\Omega,{\mathcal{S}}igma,\mu)$. A price $p\in \mathit{ba}({\mathcal{S}}igma,\mu)$ is said to be \textit{positive} if $p(A)\ge 0$ for every $A\in {\mathcal{S}}igma$ and $p\ne 0$. The dual space of $L^\infty(\Omega,{\mathcal{S}}igma,\mu)$ is $\mathit{ba}({\mathcal{S}}igma,\mu)$, whose duality relation is denoted by $\langle p,f \rangle=\int fdp$ for $p\in \mathit{ba}({\mathcal{S}}igma,\mu)$ and $f\in L^\infty(\Omega,{\mathcal{S}}igma,\mu)$. Given price $p\in \mathit{ba}({\mathcal{S}}igma,\mu)$, the \textit{budget set} of each individual is given by $B_i^\diamond(p,\chi_{\Omega_i}):=\{ f\in {\mathcal{S}} \mid \langle p,f \rangle\le \langle p,\chi_{\Omega_i} \rangle \}$.
\begin{dfn}
A price-allocation pair $(p,(f_1,\dots,f_n))$ is called a \textit{Walrasian equilibrium} if $f_i$ is a maximal element for $\succsim_i$ on $B_i^\diamond(p,\chi_{\Omega_i})$ for each $i\in I$.
\end{dfn}
Denote by ${\mathcal{P}}$ the set of partitions of $\Omega$, by ${\mathcal{P}}_\mathrm{IR}$ the set of individually rational partitions, by ${\mathcal{P}}_\mathrm{PO}$ the set of Pareto optimal partitions, by ${\mathcal{P}}_\mathrm{EF}$ the set of envy-free partitions, by ${\mathcal{P}}_\mathrm{C}$ the set of core partitions, and by ${\mathcal{P}}_\mathrm{WE}$ the set of Walrasian equilibrium partitions respectively in ${\mathcal{E}}$. Then the inclusion ${\mathcal{P}}_\mathrm{WE}\subset {\mathcal{P}}_\mathrm{C}\subset {\mathcal{P}}_\mathrm{IR}\cap{\mathcal{P}}_\mathrm{PO}$ is trivial by definition. Denote by ${\mathcal{A}}$ the set of partitions of $\chi_\Omega$, by ${\mathcal{A}}_\mathrm{IR}$ the set of individually rational partitions, by ${\mathcal{A}}_\mathrm{PO}$ the set of Pareto optimal partitions, by ${\mathcal{A}}_\mathrm{EF}$ the set of envy-free partitions, by ${\mathcal{A}}_\mathrm{C}$ the set of core allocations, and by ${\mathcal{A}}_\mathrm{WE}$ the set of Walrasian equilibrium allocations respectively. Then the inclusion ${\mathcal{A}}_\mathrm{WE}\subset {\mathcal{A}}_\mathrm{C}\subset {\mathcal{A}}_\mathrm{IR}\cap {\mathcal{A}}_\mathrm{PO}$ is trivial by definition. For every economy ${\mathcal{E}}$ and its extended economy ${\mathcal{E}}^\diamond$, the inclusions ${\mathcal{P}}\subset {\mathcal{A}}$, ${\mathcal{P}}_\mathrm{IR}\subset {\mathcal{A}}_\mathrm{IR}$, and ${\mathcal{P}}_\mathrm{EF}\subset {\mathcal{A}}_\mathrm{EF}$ are automatic.
In what follows, we assume that each $\succsim_i$ admits a continuous representation in a separable Banach space $E_i$. In view of Theorem \ref{thm1}, each $\succsim_i$ is represented by a utility function $\varphi_i\circ m_i$ on ${\mathcal{S}}igma$, where $m_i:{\mathcal{S}}igma\to E_i$ is a $\mu$-continuous vector measure and $\varphi_i:m_i({\mathcal{S}}igma)\to {\mathbb{R}}$ is a continuous function. The utility function of this form means that each individual evaluates any measurable subset $A$ of the heterogeneously divisible commodity $(\Omega,{\mathcal{S}}igma,\mu)$ in terms of the subjective cardinal attribute $m_i(A)$ with a vector representation in $E_i$. When $E_i={\mathbb{R}}^{l_i}$, there are $l_i$ cardinal attributes for the heterogeneously divisible commodity that is significant to individual $i$, which is the specification modelled in \cite{hu08,hu11,hs12,sa06}.
\begin{rem}
Since an extended economy ${\mathcal{E}}^\diamond$ of ${\mathcal{E}}$ is nothing but a standard exchange economy with commodity space $L^\infty(\Omega,{\mathcal{S}}igma,\mu)$, the existence of Walrasian equilibria for ${\mathcal{E}}^\diamond$ simply follows from exactly the same argument as in \cite{be72} if one imposes Mackey continuity and convexity on each $\succsim_i^\diamond$. In the proof of Theorem \ref{thm6} below, we demonstrate the existence of Walrasian equilibria for ${\mathcal{E}}^\diamond$ in a more specific setting.
\end{rem}
\subsection{A Characterization of Saturation}
The power of saturation is exemplified in the indifference relation between partitions in ${\mathcal{E}}$ and allocations in the extended economy ${\mathcal{E}}^\diamond$ of the original economy ${\mathcal{E}}$. The proof presented here is based on the Lyapunov convexity theorem in separable Banach spaces (Proposition \ref{lyap}) and the extreme point argument in the weakly$^*\!$ compact subset of $L^\infty$ inspired by a functional analytic approach originated in \cite{li66} and then explored in \cite{ak95,hs12}.
\begin{thm}
\label{thm4}
Let $(\Omega,{\mathcal{S}}igma,\mu)$ be a saturated finite measure space. Then every economy ${\mathcal{E}}=\{ (\Omega,{\mathcal{S}}igma,\mu),(\succsim_i,\Omega_i)_{i=1}^n \}$ for which each $\succsim_i$ admits a continuous representation in a separable Banach space has its extended economy ${\mathcal{E}}^\diamond=\{ L^\infty(\Omega,{\mathcal{S}}igma,\mu),{\mathcal{S}},(\succsim_i^\diamond,\chi_{\Omega_i})_{i=1}^n \}$ with an integral transformation such that for every allocation $(f_1,\dots,f_n)$ in ${\mathcal{E}}^\diamond$, there exists a partition $(A_1,\dots,A_n)$ in ${\mathcal{E}}$ satisfying $f_i\sim_i^\diamond\chi_{A_i}$ for each $i\in I$.
\end{thm}
\begin{proof}
Let ${\mathcal{E}}=\{ (\Omega,{\mathcal{S}}igma,\mu),(\succsim_i,\Omega_i)_{i=1}^n \}$ be an economy satisfying the hypothesis of the theorem. In view of Theorem \ref{thm1}, each $\succsim_i$ is represented by a utility function of the form $\varphi_i\circ m_i$, where $m_i$ is a $\mu$-continuous vector measure with values in a separable Banach space $E_i$ and $\varphi_i$ is a continuous function on $m_i({\mathcal{S}}igma)$. Since $m_i({\mathcal{S}}igma)=m_i({\mathcal{S}})$ for each $i\in I$ by Proposition \ref{lyap}, ${\mathcal{E}}$ possesses its extended economy ${\mathcal{E}}^\diamond=\{ L^\infty(\Omega,{\mathcal{S}}igma,\mu),{\mathcal{S}},(\succsim_i^\diamond,\chi_{\Omega_i})_{i=1}^n \}$ with an integral transformation determined by the formula \eqref{eq3}. Then as demonstrated in \cite[Lemma 3.2]{hs13}, by the Banach--Alaoglu theorem \cite[Corollary V.4.4]{ds58}, ${\mathcal{A}}$ is weakly$^*\!$ compact in the $n$-fold product space $[L^\infty(\Omega,{\mathcal{S}}igma,\mu)]^n$ of $L^\infty(\Omega,{\mathcal{S}}igma,\mu)$. Take any $(f_1,\dots,f_n)\in {\mathcal{A}}$ and let
$$
{\mathcal{A}}{(f_1,\dots,f_n)}:=\left\{ (g_1,\dots,g_n)\in {\mathcal{A}}\mid g_i\sim_i^\diamond f_i \ \forall i\in I \right\}.
$$
Since each $\succsim_i^\diamond$ is continuous, the set ${\mathcal{A}}{(f_1,\dots,f_n)}$ is nonempty and weakly$^*\!$ compact in $[L^\infty(\Omega,{\mathcal{S}}igma,\mu)]^n$. According to the Krein--\hspace{0pt}Milman theorem \cite[Lemma V.8.2]{ds58}, ${\mathcal{A}}{(f_1,\dots,f_n)}$ has an extreme point $(g_1,\dots,g_n)$. We claim that each of $g_i$ is a characteristic function. Suppose, to the contrary, that $g_j$ is not a characteristic function for some $j$. By virtue of the fact that $\sum_{i=1}^ng_i=1$ and $g_i\ge 0$ for each $i\in I$, we may assume without loss of generality that there exist $\varepsilon>0$ and $A\in {\mathcal{S}}igma$ with $\mu(A)>0$ such that $\varepsilon<g_1,g_2<1-\varepsilon$ on $A$. Define the $\mu$-continuous vector measure $m:{\mathcal{S}}igma\to {\mathbb{R}}\times \prod_{i=1}^nE_i$ by $m:=(\mu,m_1,\dots,m_n)$. It follows from Proposition \ref{lyap} that there exists a measurable subset $B\subset A$ such that $m(B)=m(A)/2$. Set $h=\varepsilon(\chi_A-2\chi_B)$. Then $h\ne 0$, $0\le g_1\pm h,g_2\pm h\le 1$, and $\int hdm=0$. Since $\varphi_i(\int(g_i\pm h)dm_i)=\varphi_i(\int g_idm_i)=\varphi_i(\int f_idm_i)$ for $i=1,2$, we have $(g_1\pm h,g_2\mp h,g_3,\dots,g_n) \in {\mathcal{A}}{(f_1,\dots,f_n)}$. This yields:
$$
(g_1,\dots,g_n)=\frac{1}{2}\left[ (g_1+h,g_2-h,g_3,\dots,g_n)+(g_1-h,g_2+h,g_3,\dots,g_n) \right],
$$
which means that $(g_1,\dots,g_n)$ is a convex combination of two distinct elements $(g_1+h,g_2-h,g_3,\dots,g_n)$ and $(g_1-h,g_2+h,g_3,\dots,g_n)$ in ${\mathcal{A}}{(f_1,\dots,f_n)}$, an obvious contradiction to the fact that $(g_1,\dots,g_n)$ is an extreme point in ${\mathcal{A}}{(f_1,\dots,f_n)}$.
\end{proof}
\begin{cor}
If ${\mathcal{E}}$ has its extended economy ${\mathcal{E}}^\diamond$ with an integral transformation, then the inclusions ${\mathcal{P}}_\mathrm{PO}\subset {\mathcal{A}}_\mathrm{PO}$ and ${\mathcal{P}}_\mathrm{C}\subset {\mathcal{A}}_\mathrm{C}$ hold whenever $(\Omega,{\mathcal{S}}igma,\mu)$ is saturated.
\end{cor}
\begin{proof}
To demonstrate the first inclusion, take any Pareto optimal partition $(A_1,\dots,A_n)$ in ${\mathcal{E}}$. If $(\chi_{A_1},\dots,\chi_{A_n})$ is not a Pareto optimal allocation in ${\mathcal{E}}^\diamond$, then there exists another allocation $(f_1,\dots,f_n)$ such that $f_i\,\succsim_i^\diamond\,\chi_{A_i}$ for each $i\in I$ and $f_j\,\succ_j^\diamond\,\chi_{A_j}$ for some $j\in I$. It follows from Theorem \ref{thm4} that there exists a partition $(B_1,\dots,B_n)$ in ${\mathcal{E}}$ such that $f_i\sim_i^\diamond\chi_{B_i}$ for each $i\in I$, which yields $B_i\,\succsim_i\,A_i$ for each $i\in I$ and $B_j\,\succ_j\,A_j$ for $j$, a contradiction to the Pareto optimality of $(A_1,\dots,A_n)$ in ${\mathcal{E}}$.
To demonstrate the second inclusion, take any core partition $(A_1,\dots,A_n)$ in ${\mathcal{E}}$. If $(\chi_{A_1},\dots,\chi_{A_n})$ is not a core allocation in ${\mathcal{E}}^\diamond$, then some coalition $S\in {\mathcal{I}}$ improves upon $(\chi_{A_1},\dots,\chi_{A_n})$, and hence, there exists an $S$-allocation $(f_1,\dots,f_n)$ such that $f_i\,\succ_i^\diamond\,\chi_{A_i}$ for each $i\in S$. It follows from Theorem \ref{thm4} that there exists a partition $(B_1,\dots,B_n)$ in ${\mathcal{E}}$ such that $f_i\sim_i^\diamond\chi_{B_i}$ for each $i\in I$, which yields $B_i\,\succ_i\,A_i$ for each $i\in S$, a contradiction to the fact that $(A_1,\dots,A_n)$ is a core partition in ${\mathcal{E}}$.
\end{proof}
The following converse of Theorem \ref{thm4} yields an intriguing characterization of saturation in terms of the interplay between the economy ${\mathcal{E}}$ with preference relations with a continuous representation in a Banach space and its extended economy ${\mathcal{E}}^\diamond$ with an integral transformation.
\begin{thm}
A finite measure space $(\Omega,{\mathcal{S}}igma,\mu)$ is saturated if every economy ${\mathcal{E}}=\{ (\Omega,{\mathcal{S}}igma,\mu),(\succsim_i,\Omega_i)_{i=1}^n \}$ for which each $\succsim_i$ admits a continuous representation in an infinite-dimensional Banach space and its extended economy ${\mathcal{E}}^\diamond=\{ L^\infty(\Omega,{\mathcal{S}}igma,\mu),{\mathcal{S}},(\succsim_i^\diamond,\chi_{\Omega_i})_{i=1}^n \}$ with an integral transformation are such that for every allocation $(f_1,\dots,f_n)$ in ${\mathcal{E}}^\diamond$ there exists a partition $(A_1,\dots,A_n)$ in ${\mathcal{E}}$ satisfying $f_i\sim_i^\diamond\chi_{A_i}$ for each $i\in I$.
\end{thm}
\begin{proof}
Suppose that $(\Omega,{\mathcal{S}}igma,\mu)$ is not saturated. As noted in the proof of Theorem \ref{thm3}, for every infinite-\hspace{0pt}dimensional Banach space $E$, there exists a $\mu$-continuous vector measure $m:{\mathcal{S}}igma\to E$ such that its range $m({\mathcal{S}}igma)$ is not convex. Then there exist $x,y\in m({\mathcal{S}}igma)$ and $t\in (0,1)$ such that $tx+(1-t)y\not\in m({\mathcal{S}}igma)$. Let $C,D\in {\mathcal{S}}igma$ be such that $x=m(C)$ and $y=m(D)$ and define $f_1:=t\chi_C+(1-t)\chi_D\in {\mathcal{S}}$ and $f_j:=(n-1)^{-1}(1-f_1)\in {\mathcal{S}}$ for $j=2,3,\dots,n$. By construction, $\sum_{i=1}^nf_i=1$. Let $\varphi_1^\diamond:m({\mathcal{S}})\to {\mathbb{R}}$ be a continuous function with $\varphi_1^\diamond(0)=0$ that attains the maximum at the unique point $x_1:=\int f_1dm\in m({\mathcal{S}})$. For $j=2,3,\dots,n$, take any continuous function $\varphi_j^\diamond:m({\mathcal{S}})\to {\mathbb{R}}$ with $\varphi_j^\diamond(0)=0$, define the preference relation $\succsim_i^\diamond$ on ${\mathcal{S}}$ via the formula \eqref{eq3}, and denote its restriction to ${\mathcal{S}}igma$ by $\succsim_i$ for each $i\in I$. Then ${\mathcal{E}}=\{ (\Omega,{\mathcal{S}}igma,\mu),(\succsim_i,\Omega_i)_{i=1}^n \}$ is an economy for which each $\succsim_i$ admits a continuous representation in the common Banach space $E$ by Theorem \ref{thm1} and ${\mathcal{E}}^\diamond=\{ L^\infty(\Omega,{\mathcal{S}}igma,\mu),{\mathcal{S}},(\succsim_i^\diamond,\chi_{\Omega_i})_{i=1}^n \}$ is an extended economy of ${\mathcal{E}}$ with an integral transformation. Since $(f_1,\dots,f_n)$ is an allocation in ${\mathcal{E}}^\diamond$, there exists a partition $(A_1,\dots,A_n)$ in ${\mathcal{E}}$ such that $f_i\sim_i^\diamond\chi_{A_i}$ for each $i\in I$. This means that $\varphi_1^\diamond(x_1)=\varphi_1^\diamond(m(A_1))$, and hence, $x_1=tx+(1-t)y=m(A_1)\in m({\mathcal{S}}igma)$, a contradiction.
\end{proof}
\subsection{Existence Results}
We present four existence results on partitions in ${\mathcal{E}}$. We first show under the saturation hypothesis the existence of individually rational Pareto optimal partitions without any convexity assumption on the preferences of each individual.
\begin{thm}
\label{thm5}
If $(\Omega,{\mathcal{S}}igma,\mu)$ is saturated, then for every economy ${\mathcal{E}}=\linebreak\{ (\Omega,{\mathcal{S}}igma,\mu),(\succsim_i,\Omega_i)_{i=1}^n \}$ for which each $\succsim_i$ admits a continuous representation in a separable Banach space, there exists an individually rational Pareto optimal partition in ${\mathcal{E}}$.
\end{thm}
\begin{proof}
It follows from Theorem \ref{thm4} that the economy ${\mathcal{E}}$ has its extended economy ${\mathcal{E}}^\diamond=\{ L^\infty(\Omega,{\mathcal{S}}igma,\mu),{\mathcal{S}},(\succsim_i^\diamond,\chi_{\Omega_i})_{i=1}^n \}$ with an integral transformation such that for every allocation $(f_1,\dots,f_n)$ in ${\mathcal{E}}^\diamond$, there exists a partition $(A_1,\dots,A_n)$ in ${\mathcal{E}}$ satisfying $f_i\sim_i^\diamond\chi_{A_i}$ for each $i\in I$. Since each $\succsim_i^\diamond$ is represented by the form \eqref{eq3}, where $m_i:{\mathcal{S}}igma \to E_i$ is a $\mu$-continuous vector measure with values in a separable Banach space $E_i$ and $\varphi_i^\diamond:m_i({\mathcal{S}})\to {\mathbb{R}}$ is a continuous function, and ${\mathcal{A}}$ is weakly$^*\!$ compact in $[L^\infty(\Omega,{\mathcal{S}}igma,\mu)]^n$, so is ${\mathcal{A}}_{\text{IR}}$. Then there exists a solution to the maximization problem:
\begin{equation}
\label{P}
\max\left\{ \sum_{i=1}^n\varphi_i^\diamond\left( \int_\Omega f_i(\omega)dm_i \right) \mid (f_1,\dots,f_n)\in {\mathcal{A}}_{\text{IR}} \right \} \tag{P}.
\end{equation}
The individual rationality of a solution $(f_1,\dots,f_n)$ to \eqref{P} is obvious. If $(f_1,\dots,f_n)$ is not Pareto optimal, then there is another allocation $(g_1,\dots,g_n)$ such that $g_i\succsim_i^\diamond f_i$ for each $i\in I$ and $g_j\succ_j^\diamond f_j$ for some $j\in I$. This means that $\sum_{i=1}^n\varphi_i^\diamond(\int g_idm_i)>\sum_{i=1}^n\varphi_i^\diamond(\int f_idm_i)$, a contradiction to the fact that $(f_1,\dots,f_n)$ is a solution to \eqref{P}. Therefore, any partition $(A_1,\dots,A_n)$ satisfying $f_i\sim_i^\diamond\chi_{A_i}$ for each $i\in I$ is an individually rational Pareto optimal partition in ${\mathcal{E}}$.
\end{proof}
For the existence of core partitions, the convexity of preferences of each individual are imposed additionally.
\begin{thm}
If $(\Omega,{\mathcal{S}}igma,\mu)$ is saturated, then for every economy ${\mathcal{E}}=\linebreak\{ (\Omega,{\mathcal{S}}igma,\mu),(\succsim_i,\Omega_i)_{i=1}^n \}$ for which each $\succsim_i$ admits a continuous convex representation in a separable Banach space, there exists a core partition in ${\mathcal{E}}$.
\end{thm}
\begin{proof}
It follows from Theorem \ref{thm4} that the economy ${\mathcal{E}}$ has its extended economy ${\mathcal{E}}^\diamond=\{ L^\infty(\Omega,{\mathcal{S}}igma,\mu),{\mathcal{S}},(\succsim_i^\diamond,\chi_{\Omega_i})_{i=1}^n \}$ with a quasiconcave integral transformation such that for every allocation $(f_1,\dots,f_n)$ in ${\mathcal{E}}^\diamond$, there exists a partition $(A_1,\dots,A_n)$ in ${\mathcal{E}}$ satisfying $f_i\sim_i^\diamond\chi_{A_i}$ for each $i\in I$. Since each $\succsim_i^\diamond$ is represented by the form \eqref{eq3}, where $m_i:{\mathcal{S}}igma \to E_i$ is a $\mu$-continuous vector measure with values in a separable Banach space $E_i$ and $\varphi_i^\diamond:m_i({\mathcal{S}})\to {\mathbb{R}}$ is a continuous quasiconcave function, the utility function $\nu_i^\diamond:{\mathcal{S}}\to {\mathbb{R}}$ defined by $\nu_i^\diamond(f):=\varphi_i^\diamond(\int fdm_i)$ is weakly$^*\!$ continuous and quasiconcave on ${\mathcal{S}}$.
The market game $V:{\mathcal{I}}\to 2^{{\mathbb{R}}^n}$ with nontransferrable utility (NTU) for the extended economy ${\mathcal{E}}^\diamond$ is a set-valued mapping defined by:
\[
V(S)=\left\{ (x_1,\dots,x_n)\in {\mathbb{R}}^n \left|
\begin{array}{ll}
\exists\,\text{$S$-\hspace{0pt}allocation $(f_1,\dots,f_n)\in {\mathcal{A}}$:} \\
\text{$x_i\le \nu_i^\diamond(f_i)$ $\forall i\in S$}
\end{array}
\right.\right\}.
\]
By construction, $V(S)$ is the subset of the utility possibility set of the individuals in which payoff vectors are attainable via some coalition $S\in {\mathcal{I}}$. The \textit{core} $C(V)$ of the NTU game $V$ is given by:
\[
C(V)=\left\{ (x_1,\dots,x_n)\in V(I) \mid \not\exists (S,y)\in {\mathcal{I}}\times V(S): x_i<y_i\ \forall i\in S \right \}.
\]
By the celebrated theorem of \citet{sc67}, $C(V)$ is nonempty if $V$ is comprehensive below and balanced, $V(S)$ is closed and bounded from above for every $S\in {\mathcal{I}}$, and $x=(x_1,\dots,x_n)\in {\mathbb{R}}^n$, $y=(y_1,\dots,y_n)\in V(S)$ and $x_i=y_i$ for each $i\in S$ imply $x\in V(S)$. We show that $V$ satisfies these conditions.
It is easy to see that each $V(S)$ is comprehensive from below, i.e., $x=(x_1,\dots,x_n)\in {\mathbb{R}}^n$, $y=(y_1,\dots,y_n)\in V(S)$ and $x_i\le y_i$ for each $i\in I$ imply $x\in V(S)$. Since each $\nu_i^\diamond$ is weakly$^*\!$ continuous, and hence, bounded on the weakly$^*\!$ compact set ${\mathcal{S}}$, for each $S\in {\mathcal{I}}$ there exists $M_S\in {\mathbb{R}}$ such that $x_i\le M_S$ for every $x\in V(S)$ and $i\in S$.
We shall show that $V$ is a balanced game. To this end, let ${\mathcal{B}}$ be a balanced family of ${\mathcal{I}}$ with balanced weights $\{\lambda^S\ge 0 \mid S\in {\mathcal{B}} \}$ and let ${\mathcal{B}}_i=\{ S\in {\mathcal{B}} \mid i\in S \}$. We then have $\sum_{S\in {\mathcal{B}}_i}\lambda^S=1$ for each $i\in I$. Define:
\[
\chi_i^S=
\begin{cases}
1 & \text{if $S\in {\mathcal{B}}_i$}, \\
0 & \text{otherwise}
\end{cases}
\quad\text{and} \quad t^S=\frac{1}{n}\sum_{i\in I}\lambda^S\chi_i^S.
\]
Then, we have:
$$
\sum_{S\in {\mathcal{B}}}t^S=\frac{1}{n}\sum_{S\in {\mathcal{B}}}\sum_{i\in I}\lambda^S\chi_i^S=\frac{1}{n}\sum_{i\in I}\sum_{S\in {\mathcal{B}}_i}\lambda^S=1.
$$
Choose any $x=(x_1,\dots,x_n)\in \bigcap_{S\in {\mathcal{B}}}V(S)$. Then, for every $S\in {\mathcal{B}}$, there exists an $S$-\hspace{0pt}allocation $(f_1^S,\dots,f_n^S)$ such that $x_i\le \nu_i^\diamond(f_i^S)$ for each $i\in S$. Let $f_i=\sum_{S\in {\mathcal{B}}}t^Sf_i^S$ for each $i\in I$. Then, $(f_1,\dots,f_n)$ is an allocation because ${\mathcal{A}}$ is convex. Since $x_i\le \nu_i^\diamond(f_i)$ for each $i\in I$ by the quasiconcavity of $\nu_i^\diamond$, we have $x\in V(I)$. Therefore, $\bigcap_{S\in {\mathcal{B}}}V(S)\subset V(I)$, and consequently, $V$ is balanced.
We finally show that $V(S)$ is closed for every $S\in {\mathcal{B}}$. Let $\{ x^k \}_{k\in {\mathbb{N}}}$ be a sequence in $V(S)$ converging to $x\in {\mathbb{R}}^n$. Then, there exists an allocation $(f_1^k,\dots,f_n^k)$ such that $x_i^k\le \nu_i^\diamond(f_i^k)$ for each $i\in S$ and $k\in {\mathbb{N}}$. Since ${\mathcal{A}}$ is weakly$^*\!$ compact, the sequence $\{ (f_1^k,\dots,f_n^k) \}_{k\in {\mathbb{N}}}$ contains a subsequence that is weakly$^*\!$ convergent to $(f_1,\dots,f_n)\in {\mathcal{A}}$. Then we have $x_i\le \nu_i^\diamond(f_i)$ for each $i\in S$ by the weak$^*\!$ continuity of $\nu_i^\diamond$. It is easy to verify that $(f_1,\dots,f_n)$ is an $S$-\hspace{0pt}allocation. Thus, we obtain $x\in V(S)$, and hence, $V(S)$ is closed.
Since $C(V)$ is nonempty, one can choose an element $(x_1,\dots,x_n)$ in $C(V)$. Then there exists an $I$-\hspace{0pt}allocation $(f_1,\dots,f_n)$ such that $x_i\le \nu_i^\diamond(f_i)$ for each $i\in I$. Suppose that $(f_1,\dots,f_n)$ is not a core allocation in ${\mathcal{E}}^\diamond$. Then there exists an $S$-\hspace{0pt}allocation $(g_1,\dots,g_n)$ such that $\nu_i^\diamond(f_i)<\nu_i^\diamond(g_i)$ for each $i\in S$. Then we have $(\nu_1^\diamond(g_1),\dots,\nu_n^\diamond(g_n))\in V(S)$ and $x_i<\nu_i^\diamond(g_i)$ for each $i\in S$, which contradicts the fact that $(x_1,\dots,x_n)$ is in $C(V)$. Let $(x_1,\dots,x_n)$ be in $C(V)$ and define the set ${\mathcal{A}}(x_1,\dots,x_n)$ by:
$$
{\mathcal{A}}(x_1,\dots,x_n)=\left\{ (f_1,\dots,f_n)\in {\mathcal{A}}\mid x_i\le \nu_i^\diamond(f_i) \ \forall i\in I \right\}.
$$
It follows from the above argument that ${\mathcal{A}}(x_1,\dots,x_n)$ is a nonempty subset of ${\mathcal{A}}_\mathrm{C}$. Since ${\mathcal{A}}(x_1,\dots,x_n)$ is nonempty and weakly$^*\!$ compact in $[L^\infty(\Omega,{\mathcal{S}}igma,\mu)]^n$, according to the Krein--\hspace{0pt}Milman theorem, ${\mathcal{A}}(x_1,\dots,x_n)$ has an extreme point $(g_1,\dots,g_n)$. Precisely in the same way with the proof of Theorem \ref{thm4}, we can show that each $g_i$ is a characteristic function. Therefore, there exists a partition $(A_1,\dots,A_n)$ of $\Omega$ such that $(\chi_{A_1},\dots,\chi_{A_n})\in {\mathcal{A}}(x_1,\dots,x_n)$. This means that $(A_1,\dots,A_n)$ is a core partition in ${\mathcal{E}}$.
\end{proof}
For the existence of Walrasian equilibrium partitions with a positive price, the monotonicity of preferences of each individual are imposed further.
\begin{thm}
\label{thm6}
If $(\Omega,{\mathcal{S}}igma,\mu)$ is saturated, then for every economy ${\mathcal{E}}=\linebreak\{ (\Omega,{\mathcal{S}}igma,\mu),(\succsim_i,\Omega_i)_{i=1}^n \}$ for which each $\succsim_i$ admits a continuous convex monotone representation in a separable Banach space, there exists a Walrasian equilibrium partition in ${\mathcal{E}}$ with a positive price.
\end{thm}
\begin{proof}
By Proposition \ref{lyap} and Theorem \ref{thm1}, the economy ${\mathcal{E}}$ has its extended economy ${\mathcal{E}}^\diamond=\{ L^\infty(\Omega,{\mathcal{S}}igma,\mu),{\mathcal{S}},{(\succsim_i^\diamond,\chi_{\Omega_i})}_{i=1}^n \}$ with a quasiconcave integral transformation such that each $\succsim_i^\diamond$ is represented by the form \eqref{eq3}, where $m_i:{\mathcal{S}}igma \to E_i$ is a vector measure with values in a separable Banach space $E_i$ and $\varphi_i^\diamond:m_i({\mathcal{S}})\to {\mathbb{R}}$ is a continuous quasiconcave function. Since ${\mathcal{S}}$ is weakly$^*\!$ closed in $L^\infty(\Omega,{\mathcal{S}}igma,\mu)$ and each utility function defined by $\nu_i^\diamond(f)=\varphi_i^\diamond(\int fdm_i)$ is weakly$^*\!$ continuous and quasiconcave on ${\mathcal{S}}$ satisfying \cite[Monotonicity Assumption]{be72}, and hence, all the assumptions of \cite[Theorem 1]{be72} are met for ${\mathcal{E}}^\diamond$. Then there exists a Walrasian equilibrium $(\hat{p},(\hat{f}_1,\dots,\hat{f}_n))$ with positive price $\hat{p}$ in ${\mathcal{E}}^\diamond$. Define the set of Walrasian allocations associated with the equilibrium price $\hat{p}$ by:
$$
{\mathcal{A}}_\mathit{W}(\hat{p})=\left\{ (f_1,\dots,f_n)\in {\mathcal{A}} \left| \begin{array}{l}\text{$f_i$ is a maximal element for $\succsim_i$} \\ \text{on $B_i^\diamond(\hat{p},\chi_{\Omega_i})$ $\forall i\in I$} \end{array} \right.\right\}.
$$
Then ${\mathcal{A}}_\mathit{W}(\hat{p})$ is nonempty and weakly$^*\!$ compact in $[L^\infty(\Omega,{\mathcal{S}}igma,\mu)]^n$. According to the Krein--\hspace{0pt}Milman theorem, ${\mathcal{A}}_\mathit{W}(\hat{p})$ has an extreme point $(g_1,\dots,g_n)$. Precisely in the same way as the proof of Theorem \ref{thm4}, we can show that each $g_i$ is a characteristic function. Therefore, there exists a partition $(\hat{A}_1,\dots,\hat{A}_n)$ of $\Omega$ such that $(\chi_{\hat{A}_1},\dots,\chi_{\hat{A}_n})\in {\mathcal{A}}_\mathit{W}(\hat{p})$. This means that the price-\hspace{0pt}partition pair $(\hat{p},(\hat{A}_1,\dots,\hat{A}_n))$ is a Walrasian equilibrium with positive price $\hat{p}$ in ${\mathcal{E}}$.
\end{proof}
Under the same assumption as Theorem \ref{thm6} the existence of Pareto optimal envy-free partitions is guaranteed.
\begin{thm}
If $(\Omega,{\mathcal{S}}igma,\mu)$ is saturated, then for every economy ${\mathcal{E}}=\linebreak\{ (\Omega,{\mathcal{S}}igma,\mu),(\succsim_i,\Omega_i)_{i=1}^n \}$ for which each $\succsim_i$ admits a continuous convex monotone representation in a separable Banach space, there exists a Pareto optimal envy-free partition in ${\mathcal{E}}$.
\end{thm}
\begin{proof}
Let ${\mathcal{E}}^\diamond=\{ L^\infty(\Omega,{\mathcal{S}}igma,\mu),{\mathcal{S}},{(\succsim_i^\diamond,\chi_{\Omega_i})}_{i=1}^n \}$ be an extended economy of ${\mathcal{E}}$ with a quasiconcave integral transformation such that for every allocation $(f_1,\dots,f_n)$ in ${\mathcal{E}}^\diamond$, there exists a partition $(A_1,\dots,A_n)$ in ${\mathcal{E}}$ satisfying $f_i\sim_i^\diamond\chi_{A_i}$ for each $i\in I$. Consider the extended economy $\overline{{\mathcal{E}}^\diamond}=\{ L^\infty(\Omega,{\mathcal{S}}igma,\mu),{\mathcal{S}},\linebreak {(\succsim_i^\diamond)}_{i=1}^n,n^{-1}\chi_\Omega \}$ in which each individual possesses the equal initial endowment $n^{-1}\chi_\Omega\in {\mathcal{S}}$. Then a Walrasian equilibrium $(\bar{p},(\bar{f}_1,\dots,\bar{f}_n))$ with positive price $\bar{p}$ in $\overline{{\mathcal{E}}^\diamond}$ exists as proved in Theorem \ref{thm6}. The Walrasian allocation $(\bar{f}_1,\dots,\bar{f}_n)$ is Pareto optimal in ${\mathcal{E}}^\diamond$ (because of the the first welfare theorem) and in view of the equal budget set among all individuals, it is evidently envy free in ${\mathcal{E}}^\diamond$. Take any partition $(\bar{A}_1,\dots,\bar{A}_n)$ in ${\mathcal{E}}$ satisfying $\bar{f}_i\sim_i^\diamond\chi_{\bar{A}_i}$ for each $i\in I$. If $(\bar{A}_1,\dots,\bar{A}_n)$ is not Pareto optimal in ${\mathcal{E}}$, then there exists another partition $(A_1,\dots,A_n)$ such that $A_i\succsim_i \bar{A}_i$ for each $i\in I$ and $A_j\succ_j \bar{A}_j$ for some $j\in I$. This means that $\chi_{\bar{A}_i}\succsim_i^\diamond \bar{f}_i$ or each $i\in I$ and $\chi_{\bar{A}_j}\succsim_j^\diamond \bar{f}_j$ for $j$, a contradiction to the Pareto optimality of $(\bar{f}_1,\dots,\bar{f}_n)$ in ${\mathcal{E}}^\diamond$. Hence, $(\bar{A}_1,\dots,\bar{A}_n)$ is a Pareto optimal partition in ${\mathcal{E}}$. If $(\bar{A}_1,\dots,\bar{A}_n)$ is not envy free in ${\mathcal{E}}$, then $\bar{A}_j\,\succ_i\,\bar{A}_i$ for some $i\ne j$. This implies that $\bar{f}_j\,\succ_i^\diamond\,\bar{f}_i$, a contradiction to the fact that $(\bar{f}_1,\dots,\bar{f}_n)$ is envy free in ${\mathcal{E}}^\diamond$.
\end{proof}
\begin{rem}
To obtain a countably additive equilibrium price in Theorem \ref{thm6}, the same argument as in \cite{be72}, which uses the Yosida--Hewitt decomposition in $\mathit{ba}({\mathcal{S}}igma,\mu)$, is valid. In particular, if the extended economy ${\mathcal{E}}^\diamond$ is such that each $\succsim_i^\diamond$ is monotone and has a Mackey continuous extension to the positive cone of $L^\infty(\Omega,{\mathcal{S}}igma,\mu)$ preserving monotonicity, then \cite[Theorem 2]{be72} is applicable to the proof of Theorem \ref{thm6}, and thereby, $\hat{p}$ can be taken in $L^1(\Omega,{\mathcal{S}}igma,\mu)$.
\end{rem}
\begin{rem}
For an economy ${\mathcal{E}}$ in which each $\succsim_i$ admits a continuous representation in ${\mathbb{R}}^{l_i}$, the existence of Walrasian equilibria (see \cite{hu11}), the nonemptiness of the core (see \cite{hu08,sa06}), the nonemptiness of the fuzzy core and the existence of supporting prices (see \cite{hs12}), and the existence of Pareto optimal $\alpha$-fair partitions (see \cite{sa11}) were established under the convexity hypothesis on $\succsim_i$. Under the alternative continuity hypothesis on $\succsim_i$, the existence of Pareto optimal envy-free partitions was given in \cite{hs13} without any convexity hypothesis on $\succsim_i$. For other solution concepts regarding the fair division problems, see \cite{sa08,sa11,sv10b,sv11}.
\end{rem}
\section{Concluding Remarks}
We conclude this paper by raising an open problem. We leave aside the existence of supporting prices for Pareto optimal partitions under the convexity assumption. The existence of supporting prices in $L^1(\Omega,{\mathcal{S}}igma,\mu)$ was obtained in \cite{hs12} with a finite-dimensional vector representation of preference relations. The problem is a standard application of the separation theorem in $L^\infty(\Omega,{\mathcal{S}}igma,\mu)$. It should be noted that thanks to Theorem \ref{thm4}, supporting prices in an extended economy are automatically the ones in the original economy. As demonstrated in Theorem \ref{thm5}, however, for the existence of Pareto optimal partitions, the convexity assumption is unnecessary. Thus, the challenging problem is instead to demonstrate the second fundamental theorem of welfare economics for economies without any convexity assumption. For nonconvex production economies with an infinite-dimensional commodity space, \cite{kv88} derived a price system in the Clarke normal cone that is consistent with a Pareto optimal allocation, and their result covers the commodity space of $L^\infty(\Omega,{\mathcal{S}}igma,\mu)$. It is well-known that in the presence of nonconvexity Clarke normal cones are strictly larger than \citeauthor{mo00} normal cones (see \cite{mo06a}), so that price systems in the latter are called for to derive sharper necessary conditions for Pareto optimality. It is \cite{kh91} who first introduced \citeauthor{mo00} normal cones to obtain the second welfare theorem into finite-dimensional nonconvex production economies. For the extension of \cite{kh91} to Banach or Asplund spaces of commodities, see \cite{bm10,mm01,mo00,mo05,mo06b}. The second welfare theorem without convexity assumptions specific to optimal partitions based on \citeauthor{mo00} normal cones is still unknown.
\end{document} |
\begin{document}
\title{Entanglement Manipulation and Concentration}
\author{R.~T.~Thew\cite{RTT} and W. J. Munro\cite{WJM}}
\address{Special Research Centre for Quantum Computer Technology,\\
University of Queensland, Brisbane, Australia}
\date{\today}
\maketitle
\begin{abstract}
We introduce a simple, experimentally realisable, entanglement
manipulation protocol for exploring mixed state entanglement. We show
that for both non-maximally entangled pure, and mixed
polarisation-entangled two qubit states, an increase in the degree of
entanglement and purity, which we define as concentration, is achievable.
\end{abstract}
\pacs{03.67.-a, 42.50.-p,03.65.Bz}
\label{INTRO}
The increasing interest in quantum information and computing as well as
other quantum mechanical dependent operations such as teleportation
\cite{Bennett2:96} and cryptography \cite{Deutsch:96} have as their
cornerstone a reliance on entanglement. There has been a great deal of
discussion of measures and manipulation of entanglement in recent years with
respect to purification \cite{Bennett1:96}, concentration
\cite{Bennett3:96}, and distillable entanglement
\cite{Rains1:99,Vedral:97} especially concerning states
subject to environmental noise. It is this noise that takes the initially
pure maximally entangled resource and leaves us with, at best, a non maximally
entangled state, or at worst a mixed state, both less pure and less entangled.
We introduce a simple, experimentally realisable \cite{Kwiat:00}, protocol to
manipulate and explore both pure and mixed-state entanglement. While
the scheme will have limitations, in part due to its simplicity, it
will allow experimental investigation of the large Hilbert space
associated with mixed states.
The motivation for this scheme comes from focusing ideas and
proposals of several groups from the past few years into a
simple realisation of mixed state entanglement
manipulation. It was proposed that quantum correlations on mixed
states could be enhanced by positive operator valued measurements
\cite{Popescu1:95}. A more specific example by Gisin\cite{Gisin1:96}
considered the manipulation of a $ 2\times 2 $ system using local
filters. The scheme we propose here combines these ideas and uses
an arrangement similar to the original Procrustean method \cite{Bennett3:96}
which dealt solely with pure states. The primary motivation here is in
proposing a scheme that can be easily realised experimentally.
With the recent advances in the preparation of nonmaximally entangled
pure\cite{White:99} and mixed\cite{White:00} polarisation-entangled
quantum states we now have a source for which there is a high
degree of control over the degree of entanglement and purity of the
state. This allows us to consider a wide variety of states and
examine what operations can be performed so as to make the
state more useful in the context of an entanglement resource.
For the purposes of describing the possible manipulation of a state
we will define the following three concepts of distillation,
purification and concentration (illustrated schematically in
Figure (\ref{fig:ent-pur})) as follows,
\begin{itemize}
\item \it Distillation\rm: Increasing the entanglement of a state.
\item\it Purification\rm: Increasing the purity of a state
(decreasing its entropy). This is not
purification with respect to some particular state, for example obtaining a
singlet state from a mixed state.
\item \it Concentration\rm: Increasing both the entanglement and the
purity of a mixed state.
\end{itemize}
These concepts have been used almost interchangeable in the literature
but we will follow our primitive definitions to avoid potential confusion.
In this letter it is the concentration of a state that is the main aim
for the maintenance or recovery of an entanglement resource.
\begin{center}
\begin{figure}
\caption{\label{fig:ent-pur}
\label{fig:ent-pur}
\end{figure}
\end{center}
Let us now specify the measures which we will be using
to characterise the degree of entanglement and purity of a state.
The entanglement and purity of a state can be determined using distinct
measures. Here we will restrict our attention to $2 \times 2$ systems
and hence will use analytic expressions for
The Entanglement of Formation and Entropy as our respective
measures. The Entanglement of formation as introduced by Wootters
\cite{Wooters1:98} is found by considering that for a general two
qubit state, $ \rho $, the "spin-flipped state" $\tilde{\rho}$ is given by
\begin{eqnarray}
\tilde{\rho} = (\sigma _y \otimes \sigma _y) \rho^{*} (\sigma _y \otimes
\sigma _y)
\end{eqnarray}
where $\sigma_y$ is the Pauli operator in the computational basis. We
calculate the square root of the eigenvalues $ \tilde{\lambda}_{i} $ of
$\rho \tilde \rho$, in descending order, to determine the
``Concurrence'',
\begin{eqnarray}
C(\rho) = max\{\tilde{\lambda}_1 - \tilde{\lambda}_2 -\tilde{\lambda}_3
-\tilde{\lambda}_4,0 \}
\end{eqnarray}
The Entanglement of Formation (EOF) is then given by
\begin{eqnarray}
E(C(\rho)) = h\left(\frac{1 + \sqrt{1 - C(\rho)^2}}{2} \right)
\end{eqnarray}
where $ h $ is the binary entropy function
\begin{eqnarray}
h(x) = -x\log(x) - (1 - x)\log(1 - x)
\end{eqnarray}
The entropy of the density matrix $ \rho $ (our purity measure)
is given by
\begin{eqnarray}
S = -\sum_{i=1}^{4} \lambda_i \log_{4} \lambda_i
\end{eqnarray}
where $ \lambda_i $ are the eigenvalues of $ \rho $.
We will now describe our entanglement manipulation protocol and emphasise its
simplicity. The experimental arrangement for our
protocol is described by the schematic in figure(\ref{fig:exp2}).
The aim of our protocol is to manipulate mixed states and
enhance their degree of entanglement. Let us consider an initial
state composed of two subsystems, A and B,
each represented by a general $2 \times 2$
matrix. We will describe the joint state of the system, $AB$,
in the polarisation basis, $\{|VV\rangle,|VH\rangle, |HV\rangle,
|HH\rangle\}$, as
\begin{eqnarray}
\hat{\rho}_{ABin} &=& \left( \begin{array}{cccc}
\rho_{11} & \rho_{12} & \rho_{13} & \rho_{14} \\
\rho_{12}^* & \rho_{22} & \rho_{23} & \rho_{24} \\
\rho_{13}^* & \rho_{23}^* & \rho_{33} & \rho_{34} \\
\rho_{14}^* & \rho_{24}^* & \rho_{34}^* & \rho_{44} \\ \end{array} \right)
\end{eqnarray}
with the $\hat{\rho}_{ij}$ satisfying the requirements for a legitimate
density matrix. From our source (see figure(\ref{fig:exp2}))
we have four polarisation modes (two for A and two for
B). These polarisation modes are spatially separated and
input onto beam splitters (BS), with independent and variable
transmission coefficients. The second input port of each of these
beam splitters are assumed to be vacuums. With perfectly efficient
photodetectors it would be possible to monitor the second output mode
of each of these beamsplitter and use the results to conditionally
select the concentrated state we wish to produce. We know that if the detection
of a photon is made in any of the second output ports then the preparation
process is considered to have failed. Non-detection (with perfectly
efficient detectors) at all the second output ports is required to
prepare our state and here is the problem with current single photon detection
efficiencies. Photon detectors have a finite efficiency and it
possible that a photon present at these second output ports will not
be detected. Hence we will not get the conditioned state we desire.
Instead we will examine the transmitted modes of the beamsplitter
and consider the situations where joint
coincidences are registered at the photodetectors
of the two subsystems A and B, or Alice and Bob if you prefer. While
this is a post selective process it has the advantage that poor
detection efficiency only decreases the coincidence count rate. As we
discard any information present at the second output of the
beamsplitters, the protocol we describe is not unitary.
\begin{center}
\begin{figure}
\caption{\label{fig:exp2}
\label{fig:exp2}
\end{figure}
\end{center}
If we consider that having each mode incident on a BS has the
effect of expanding the Hilbert space of the system, then in the
expanded Hilbert space we can manipulate the state and then project
it back onto the polarisation coincidence basis. The BSs transform each mode in
the following way
\begin{eqnarray}
|V,H \rangle |0 \rangle \rightarrow \eta_{v,h}|V,H \rangle |0 \rangle +
\sqrt{(1-\eta_{v,h}^2)}|0\rangle |1\rangle
\end{eqnarray}
and hence we obtain an output density matrix for this reduced system of the form
\begin{eqnarray}
\hat{\rho}_{ABout} &=&{\cal N} \left( \begin{array}{cccc}
\rho_{11}\eta_{va}^2\eta_{vb}^2&\rho_{12}\eta_{va}^2\eta_{vb}\eta_{hb}& \rho_{13}\eta_{va}\eta_{ha}\eta_{vb}^2 & \rho_{14}\eta \\
\rho_{12}^*\eta_{va}^2\eta_{vb}\eta_{hb}&\rho_{22}\eta_{va}^2\eta_{hb}^2&\rho_{23}\eta&\rho_{24}\eta_{va}\eta_{ha}\eta_{hb}^2 \\
\rho_{13}^*\eta_{va}\eta_{ha}&\rho_{23}^*\eta&\rho_{33}\eta_{ha}^2\eta_{vb}^2&\rho_{34}\eta_{ha}^2\eta_{vb}\eta_{hb} \\
\rho_{14}^*\eta&\rho_{24}^*\eta_{va}\eta_{ha}\eta_{hb}^2&\rho_{34}^*\eta_{ha}^2\eta_{vb}\eta_{hb} &\rho_{44}\eta_{ha}^2\eta_{hb}^2 \\
\end{array}
\right)
\end{eqnarray}
where $\eta = \eta_{va}\eta_{ha}\eta_{vb}\eta_{hb}$ and
$\eta_{v,h|a,b} $ are the vertical and horizontal polarisation
transmission coefficients for subsystems A and B. The normalisation is given by
\begin{eqnarray}
{\cal N} = [\rho_{11}\eta_{va}^2\eta_{vb}^2 + \rho_{22}\eta_{va}^2\eta_{hb}^2 + \rho_{33}\eta_{ha}^2\eta_{vb}^2 + \rho_{44}\eta_{ha}^2\eta_{hb}^2]^{-1}
\end{eqnarray}
and the probability of obtaining the desired output state is determined from
the trace of the unnormalized BS-transformed density matrix, ${\cal
N}^{-1}$, and thus is dependent on the transmission coefficients. This is the probability of obtaining the output state
once the BS parameters have been determined.
This scheme is more easily understood by considering the behaviour
of pure states under the protocol. As such we now illustrate the
distillation process with a specific example. We will examine a non-maximally
entangled pure state and show how to recover a maximally entangled
state via our protocol. Consider an initial state produced by our source of the form
\begin{eqnarray}\label{eq:nmeps1}
|\varphi_{in} \rangle_{ab} = {\cal N}_{1}[\epsilon_1|VV\rangle_{ab} + \epsilon_2e^{i\phi}|HH\rangle_{ab}]
\end{eqnarray}
or alternatively
\begin{eqnarray}\label{eq:nmeps2}
|\varphi_{in} \rangle_{ab} = {\cal N}_{1}[\epsilon_1|VH\rangle_{ab} + \epsilon_2e^{i\phi}|HV\rangle_{ab}]
\end{eqnarray}
where
\begin{eqnarray}
{\cal N}_{1}^2=[|\epsilon_1|^2 + |\epsilon_2|^2]^{-1}
\end{eqnarray}
Assuming the polarisation modes are all spatially separated we input
them onto separate BSs (see figure (\ref{fig:exp2})). We can choose
to manipulate the BSs at A and B independently to find the optimal
output for a given state. For convenience we consider a state of the form of
(\ref{eq:nmeps1}) which allows us to simplify the analysis. With
this in mind we can set $\eta_{va} = \eta_{vb} = \eta_v$ and
$\eta_{ha} = \eta_{hb} = \eta_b$.
The state of our system after the BSs (assuming vacuum
inputs to the second BS ports) is
\begin{eqnarray}
|\varphi_{total} \rangle_{AB} &=&{\cal N}_{1} \left[
\epsilon_1\eta_v^2|VV\rangle_{AB}|00\rangle +
\epsilon_2e^{i\phi}\eta_h^2|HH\rangle_{AB} |00\rangle \right. \nonumber \\
&\;&\;\;\;+\epsilon_1\eta_v \sqrt{(1-\eta_{v}^2)}\left\{
|V0\rangle_{AB}|01\rangle+
|0V\rangle_{AB}|10\rangle \right\} \nonumber \\
&\;&\;\;\;+\epsilon_2e^{i\phi}\eta_h \sqrt{(1-\eta_{h}^2)}\left\{
|H0\rangle_{AB}|01\rangle+
|0H\rangle_{AB}|10\rangle \right\} \nonumber \\
&\;&\;\;\; + \epsilon_1 \left(1-\eta_v^2\right)|00\rangle_{AB}|11\rangle
\nonumber \\
&\;&\;\;\;\left. + \epsilon_2e^{i\phi}\left(1-\eta_h^2\right)|00\rangle_{AB}
|11\rangle\right]
\end{eqnarray}
The outcomes we are interested in are in the joint coincidence basis of A,B and
hence the vacuum state components are removed from consideration
leaving an effective output state of the form
\begin{eqnarray}
|\varphi_{out} \rangle_{AB} = {\cal
N}_{2}[\epsilon_1\eta_v^2|VV\rangle_{AB} +
\epsilon_2e^{i\phi}\eta_h^2|HH\rangle_{AB}]
\end{eqnarray}
where the normalisation in this coincidence basis is
\begin{eqnarray}
{\cal N}_{2}^2 = [|\epsilon_1|^2\eta_v^4 + |\epsilon_2|^2\eta_h^4]^{-1}
\end{eqnarray}
For maximal entanglement we have the following simple relationship
\begin{eqnarray}
|\epsilon_1| \eta_v^2 = |\epsilon_2| \eta_h^2
\end{eqnarray}
We observe that the entanglement of the output state is dependent
on the transmission coefficients of the BSs. Further, this protocol
can always take a non-maximally entangled state and obtain a pure
maximally entangled one. This protocol can also incorporate a phase
adjuster at either A or B to tune any relative phase difference for
the state. If we had considered states of the form of
(\ref{eq:nmeps2}) then we would need to consider the tuning
parameters independently such that the requirement for a pure
maximally entangled state is then
\begin{eqnarray}
|\epsilon_1| \eta_{va}\eta_{hb} = |\epsilon_2| \eta_{vb}\eta_{ha}
\end{eqnarray}
This is where the protocol differs from the Procrustean method of
Bennett \it et.al \rm \cite{Bennett3:96}. We have introduced individual
depolarising channels, thus obtaining more degrees of freedom, and
so allowing the protocol to be extended to mixed states. It is
important to mention again that with perfect single photon detection
it is possible to monitor the discarded ports for each of the modes,
thus preparing the desired state by conditioned measurements.
Let us now turn our attention to the concentration of mixed
states. As an extension to the distillation process we take the
density matrix $\hat{\rho}_{ABin}$ to be a mixture of the density
matrices of two of the Bell-type states, (\ref{eq:nmeps1}) and
(\ref{eq:nmeps2}), one of which, say (\ref{eq:nmeps1}), is maximally
entangled, $\epsilon_1 = \epsilon_2 = 1$. The mixing can be controlled
by the parameter $\gamma$, that is,
\begin{eqnarray}\label{eq:rhoineg}
\hat{\rho}_{ABin} &=& \gamma {\cal N}_{1}^{2} \left( \begin{array}{cccc}
|\epsilon_1|^2 & 0 & 0 & \epsilon_1^{*}\epsilon_2 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\epsilon_1\epsilon_2^{*} & 0 & 0 & |\epsilon_2|^2 \\ \end{array} \right)
+ \frac{1-\gamma}{2}\left( \begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 \\ \end{array} \right)
\end{eqnarray}
This state is one of many of the range of mixed states that can be
concentrated and has been chosen to easily show the protocols
extension from pure to mixed states, from distillation to concentration.
Using the BS protocol illustrated in figure(\ref{fig:exp2}) the
output state for (\ref{eq:rhoineg}) in the coincidence basis, $AB$, can
be represented as
\begin{eqnarray}
\hat{\rho}_{ABout}&=&{\cal N}_{3}^{2} \left( \begin{array}{cccc}
\gamma|\epsilon_1|^2\eta_{va}^2\eta_{vb}^2&0&0&\gamma \epsilon_1\epsilon_2^{*}\eta \\
0 &\Gamma \eta_{va}^2\eta_{hb}^2 &\Gamma \eta & 0 \\
0 & \Gamma \eta & \Gamma \eta_{ha}^2\eta_{vb}^2 & 0 \\
\gamma \epsilon_1^{*}\epsilon_2\eta & 0 & 0 &\gamma |\epsilon_2|^2\eta_{ha}^2\eta_{hb}^2 \\ \end{array} \right)
\end{eqnarray}
with $\Gamma = \frac{(1-\gamma)}{2}$
and the normalisation ${\cal N}_{3}$ given by
\begin{eqnarray}
{\cal N}_{3}^{2} =
[\gamma(|\epsilon_1|^2\eta_{va}^2\eta_{vb}^2+|\epsilon_2^2|\eta_{ha}^2\eta_{hb}^2)+\Gamma(\eta_{va}^2\eta_{h_{b}}^2 + \eta_{ha}^2\eta_{vb}^2)]^{-1}
\end{eqnarray}
In figure (\ref{fig:conc}) we display the effect of our protocol for
a range of $\gamma$ values with $\epsilon_1=1$ and $\epsilon_2=0.1$
(the $\gamma$ values are labeled at the peak of each curve). The initial points for the fixed $\gamma$, $\epsilon_1$ and $\epsilon_2$
are displayed as solid dots. These curves represent the behaviour
of the Entropy and EOF of the states as the BSs are tuned to optimise
both. We see how this class of state can be improved is dependent on the
amount mixing. The behaviour of the state is similarly dependent
on the degree of entanglement in the pure state components of the
mixed state of (\ref{eq:rhoineg}), variations in $\epsilon_{1,2}$,
though this is not explicitly shown here.
\begin{center}
\begin{figure}
\caption{\label{fig:conc}
\label{fig:conc}
\end{figure}
\end{center}
The curves in figure (\ref{fig:conc}) represent the range of (S,EOF)
values for the output states from our protocol. We take the
specific case of $\gamma = 0.1$ and observe the variation of (S,EOF) as we tune
$\eta_{va} = \eta_{vb} = \eta_v $. From the initial state marked with a
black circle at (S,EOF) = (0.23,0.84) with $\eta_v=1$ we then adjust
the BSs, moving up the curve, to a state with (S,EOF) =
(0.075,0.94) for $\eta_v= 0.32$. This constitutes a turning point on the plane and if we
continue decreasing $\eta_v$ we follow the curve back to our initial point in the plane after which
the entanglement-entropy properties of the state deteriorate from the
original values.
What does the state look like? We observe that with
($\epsilon_1,\epsilon_2,\gamma) = (1.00,0.10,0.30)$ and allowing all
the light through the horizontal BS (an optimal setting provided
$|\epsilon_1| > |\epsilon_2|$ to maximise the output), and tuning the
vertical beam splitters transmission to $\eta_{v} = 0.32$ we can take
an initial state
\begin{eqnarray}\label{eqn:rhoinsp}
\hat{\rho}_{ABin} = \left( \begin{array}{cccc}
0.297 & 0 & 0 & 0.030 \\
0 & 0.350 & 0.350 & 0 \\
0 & 0.350 & 0.350 & 0 \\
0.030 & 0 & 0 & 0.003 \\ \end{array} \right)
\end{eqnarray}
to an output state
\begin{eqnarray}
\hat{\rho}_{ABout} = \left( \begin{array}{cccc}
0.039 & 0 & 0 & 0.039 \\
0 & 0.461 & 0.461 & 0 \\
0 & 0.461 & 0.461 & 0 \\
0.039 & 0 & 0 & 0.039 \\ \end{array} \right)
\end{eqnarray}
This output state has an increase in the Entanglement of Formation
from EOF = 0.52 to EOF = 0.78, while the entropy of the system has
decreased from S = 0.30 to S = 0.20, this result is achieved with a
finite probability P = 7.6\%.
There exists a critical point with respect to
concentration at $\gamma = 0.5$ which corresponds to the case
where the two pure states of (\ref{eq:rhoineg}) are evenly mixed. For
those states with the mixing parameter $\gamma \le 0.5$ concentration
is possible whilst for those states above this value the entanglement
can be increased but this is at the cost of purity. All of these
states can be concentrated if we choose to tune another BS, thus
highlighting the need for all four BSs. Similarly if we considered a
mixture of the pure states of (\ref{eq:nmeps1}) and (\ref{eq:nmeps2}),
where both had $\epsilon_{1,2} \ne 1$, then we find that concentration
is still achievable.
Now let us consider the incoherent sum of a pure state and a mixed
state and take as an example of this the Werner state, a mixture of
the identity and some fraction of a pure state. If the pure state
fraction of the Werner state is a non-maximally entangled pure state,
then it is possible to increase the entanglement of the state. However
this entanglement increase comes at the cost of purity and is bound by
the amount of entanglement that would be inherent in a Werner state using a
maximally entangled pure state.
\label{CONCLUSION}
In conclusion, we have proposed an entanglement concentration protocol that is
experimentally realisable and can produce a finite concentration of Bell pairs
from some initially mixed states. The key point here is that whilst this
is achievable we are more interested in the entanglement properties
then the final form of the state. Indeed with such a simple protocol the range of possible
tests with respect to quantum information and entanglement are quite
diverse, and whilst this protocol does require some knowledge of the
state in determining the tuning parameters and is a non-unitary
operation, we believe it should provide a most useful tool in the
exploration of mixed state entanglement.\\
The authors would like to thank A.G. White and P.G. Kwiat for useful
discussions with respect to the practicality of the experimental
implementation of this scheme. WJM would like to acknowledge the support of the Australian
Research Council.
\begin{references}
\small
\bibitem[*]{RTT} Electronic address: thew@physics.uq.edu.au
\bibitem[\dagger]{WJM} Electronic address: billm@physics.uq.edu.au
\bibitem{Bennett2:96} C.H. Bennett, G.Brassard, S. Popescu, and B.
Schumacher, J.A. Smolin, and W.K. Wooters, Phys. Rev. Lett. {\bf 76}, 722
(1996).
\bibitem{Deutsch:96} D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello,
S. Popescu, and A. Sanpera, Phys. Rev. Lett. {\bf 77}, 2818 (1996).
\bibitem{Bennett1:96} C.H. Bennett,D.P. Vincenzo, J.A. Smolin and
W.K. Wootters, Phys. Rev. A {\bf 54}, 3824 (1996).
\bibitem{Bennett3:96} C.H. Bennett, H.J. Bernstein, S. Popescu, and
B. Schumacher, Phys. Rev. A {\bf 53}, 2046 (1996).
\bibitem{Rains1:99} E.M. Rains, Phys. Rev. A {\bf 60}, 173 (1999).
\bibitem{Vedral:97} V. Vedral, M.B. Plenio, M.A. Rippen, and
P.L. Knight, Phys.Rev. Lett. {\bf 78}, 2275 (1997).
\bibitem{Kwiat:00} P.G. Kwiat, private communication.
\bibitem{Popescu1:95} S. Popescu, Phys. Rev. Lett. {\bf 74}, 2619 (1995).
\bibitem{Gisin1:96} N. Gisin, Phys. Lett. A {\bf 210}, 151 (1996).
\bibitem{White:99} A.G. White, D.F.V. James, P.H. Eberhard, and
P.G. Kwiat, Phys.Rev. Lett. {\bf 83}, 3103 (1999).
\bibitem{White:00} A.G. White, D.F.V. James, W.J.Munro, and P.G. Kwiat, In preparation.
\bibitem{Wooters1:98} W.K. Wooters, Phys. Rev. Lett. {\bf 80}, 2245 (1998).
\end{references}
\end{document} |
\begin{document}
\title{Evaluating individualized treatment effect predictions: \protect\\a new perspective on discrimination and \protect\\calibration assessment}
\begin{abstract}
Personalized medicine constitutes a growing area of research that benefits from the many new developments in statistical learning. A key domain concerns the prediction of individualized treatment effects, and models for this purpose are increasingly common in the published literature. Aiming to facilitate the validation of prediction models for individualized treatment effects, we extend the classical concepts of discrimination and calibration performance to assess causal (rather than associative) prediction models. Working within the potential outcomes framework, we first evaluate properties of existing statistics (including the c-for-benefit) and subsequently propose novel model-based statistics. The main focus is on randomized trials with binary endpoints. We use simulated data to provide insight into the characteristics of discrimination and calibration statistics, and further illustrate all methods in a trial in acute ischemic stroke treatment. Results demonstrate that the proposed model-based statistics had the best characteristics in terms of bias and variance. While resampling methods to adjust for optimism of performance estimates in the development data were effective on average, they had a high variance across replications that limits their accuracy in any particular applied analysis. Thereto, individualized treatment effect models are best validated in external data rather than in the original development sample.
\end{abstract}
\section{Introduction}
The prediction of individualized treatment effect conditional on patient characteristics has received much interest recently \cite{kent_personalized_2018, kent_predictive_2020, senn_statistical_2018, rekkas_predictive_2020, lin_scoping_2021, hoogland_tutorial_2021}. Such models typically predict a clinically relevant outcome under two different treatment conditions, and the difference between these predictions is attributed to the effect of treatment. This information is of clear interest in the context of clinical decision-making if the underlying model is of sufficient quality. However, the evaluation of individualized treatment effect (ITE) models is still a key methodological challenge and little guidance is currently available on how to quantify their performance \cite{hoogland_tutorial_2021}.
In this paper, we focus on ITE models that contrast the effect of two treatment conditions on the risk of a binary endpoint. More specifically, we focus on assessment of their performance; guidance on their development is available elsewhere (\textit{e.g.}, \cite{kent_predictive_2020, hoogland_tutorial_2021, wager_estimation_2018}). Typical measures of prediction model performance with respect to \textit{outcome risk} predictions include measures of calibration and discrimination \cite{steyerberg_clinical_2019, harrell_regression_2015, van_calster_calibration_2019}. However, our specific interest here is in predictions of \textit{risk difference} attributed to the effect of treatment (\textit{i.e.}, in absolute individualized treatment effect predictions). Although calibration and discrimination performance can also be assessed at the \textit{risk difference} (treatment effect) level, existing measures (\textit{e.g.}, calibration intercept, calibration slope, c-statistic) do not apply without modification because individual treatment effects (in contrast to regular outcomes) are unobservable \cite{hoogland_tutorial_2021}. For this reason, a new c-statistic was recently proposed that applies to absolute treatment effect predictions in settings with a binary endpoint, along with a quantile-based assessment of calibration \cite{van_klaveren_proposed_2018}.
We expand on this previous work by casting the entire prediction and evaluation process in the potential outcomes framework \cite{rubin_estimating_1974, rubin_causal_2005} and by developing model-based measures of discrimination and calibration performance with respect to individualized treatment effect predictions. Herein, the potential outcomes framework provides a way to deepen understanding of what is actually being measured. The model-based measures make more efficient use of the data without relying on matching on arbitrary cut-offs.
Section \ref{sec:ite} sets the scene and describes the challenge of individualized causal prediction in terms of the potential outcomes framework. Subsequently, Section \ref{sec:discrimination} and Section \ref{sec:calibration} describe existing and novel measures of discrimination and calibration with respect to absolute treatment effect respectively. Simulation results are provided for illustrative purposes. An applied example using data from the third International Stroke Trial (IST-3) \cite{the_ist-3_collaborative_group_benefits_2012} is described in Section \ref{sec:ae}. Lastly, Section \ref{sec:discussion} provides a general discussion.
\section{Individualized treatment effect prediction} \label{sec:ite}
Most outcome prediction research focuses on capturing statistical association in absence of interventions. Individualized treatment effect (ITE) prediction is a different type of prediction since it has a causal interpretation: the quantity to be predicted is the effect caused by the treatment (or intervention, in a larger sense) on the outcome. Therefore, before moving to the performance measures of interest, this section shortly outlines causal prediction. Subsequently, issues surrounding the use of binomial outcome data for ITE modeling are shortly discussed (further details are available as online supplementary material \ref{app:binom_challenges}).
\subsection{Causal prediction} \label{sec:causal_prediction}
To emphasize the causal nature of the predictions, it is helpful to write the individualized treatment effect of interest in terms of the potential outcomes framework \cite{rubin_estimating_1974, rubin_causal_2005}. For treatment taking values $a \in \mathcal{A}$, $Y^{A=a}$ denotes the potential outcome under treatment $a$. When comparing two treatments, the ITE for individual $i,\ldots,n$ can be defined as
\begin{equation} \label{eq:deltai_potential}
\delta(\bm{x}_i) = \EX(Y_i^{a=1} =1 | \bm{X} = \bm{x}_i) - \EX(Y_i^{a=0} = 1 | \bm{X}_i = \bm{x}_i)
\end{equation}
where $\bm{x}_i$ is a row vector of individual-level characteristics in matrix $\bm{X}$. The degree of granularity or individualization reflected by $\delta(\bm{x}_i)$ relates to the number of predictors included in $\bm{X}$, to the strength and shape of their association with the potential outcomes, and especially to the degree to which they have a differential effect across potential outcomes (\textit{i.e.}, modify the effect of treatment). Ideally, the set of measured individual-level characteristics includes all relevant characteristics with respect to individualized treatment effect. In practice however, this set of all relevant characteristics is often unknown and the best way forward is to aim for conditioning on the most important characteristics. Correspondingly, equation \eqref{eq:deltai_potential} reflects ITE as a conditional treatment effect for some set of characteristics.
Since in practice only one potential outcome is observed per individual \cite{holland_statistics_1986}, assumptions are required to estimate $\delta(\bm{x}_i)$ based on the observed data. These assumptions are discussed in detail elsewhere \cite{hoogland_tutorial_2021, hernan_causal_2020}. In short, the key assumptions are \textit{exchangeability} (the potential outcomes do not depend on the assigned treatment), \textit{consistency} (the observed outcome under treatment $a \in \mathcal{A}$ corresponds to the potential outcomes $Y^{A=a}$), and positivity (each individual has a non-zero probability of each treatment assignment. An additional assumption that eases inference is \textit{no interference} (the potential outcomes for individual $i$ do not depend on treatment assignment to other individuals). Based on these assumptions, the individualized treatment effect can be identified given the observed data:
\begin{align}
\delta(\bm{x}_i) &= \EX(Y_i^{a=1}=1|\bm{X}=\bm{x_i}) - \EX(Y_i^{a=0}=1|\bm{X}=\bm{x_i}) \nonumber \\
&= \EX(Y_i^{a=1}=1| A=1, \bm{X}=\bm{x_i}) - \EX(Y_i^{a=0}=1| A=0, \bm{X}=\bm{x_i}) \quad \textnormal{(by exchangeability)} \nonumber \\
&= \EX(Y_i=1| A=1, \bm{X}=\bm{x_i}) - \EX(Y_i=1| A=0, \bm{X}=\bm{x_i}) \quad \textnormal{(by consistency)} \label{eq:delta_iobs}
\end{align}
Equation \eqref{eq:delta_iobs} shows that ITE predictions ($\hat{\delta}(\bm{x}_i)$) can be estimated using a prediction model for outcome risk $\EX(Y_i = 1 | A=a_i, \bm{X}=\bm{x}_i)$. Many modeling tools can be used for this endeavor and the details are beyond the scope of this paper and are given elsewhere (\textit{e.g.,} \cite{hoogland_tutorial_2021, lamont_identification_2018}).
\subsection{Binary outcome data}
Focusing on binary outcomes, we observe outcome $Y_i \in \{0,1\}$ and covariate status $\bm{x}_i$ for each individual $i$. In this context, the ITE estimate $\delta(\bm{x}_i)$ is a difference between two risk predictions ($p(Y_i = 1|A = 1, \bm{X}=\bm{x}_i) - p(Y_i = 1|A = 0, \bm{X}=\bm{x}_i)$). The range of $\hat{\delta}(\bm{x}_i)$ includes all values in the $[-1,1]$ interval, while the observed difference between any two outcomes can only be one of $\{-1, 0, 1\}$. Therefore, in addition to the challenge that each individual only has an observed outcome for one treatment condition, the observations come with large and irreducible binomial error and hence provide only limited information. A further consideration with predictions for binary outcome data is that they are commonly non-linear functions of the covariates, and hence the effects of treatment and the covariates are usually not additive on the risk difference scale of interest here. Consequently, the resulting ITE predictions conflate variability from different sources: between-subject variability in $P(Y^{a=0}|X=x)$ and genuine treatment effect heterogeneity on the scale used for modeling. This is the price to pay for the benefit in terms of interpretation of measures on the scale of $\delta(x)$ \cite{murray_patients_2018}.
\subsection{The challenge}
In practice, the fundamentally limited nature of observed data when it comes to causal inference (\textit{i.e.}, with only one potential outcome being observed), the irreducible binomial error affecting the risk difference twice, and the challenge of model specification and estimation are all present at the same time. This evidently poses a challenge at the time of model development, but within the scope of this paper, it certainly also poses challenges during evaluation of models predicting individualized treatment effects. Most notably, and in contrast with regular prediction modeling, a direct comparison between predictions and observed outcomes is not feasible.
In this paper, we evaluate discrimination and calibration at the level of predicted individualized treatment effects. To this end, we first evaluate a recently introduced metric to assess discriminative performance \cite{van_klaveren_proposed_2018} and subsequently propose alternative procedures that aim to alleviate some of the shortcomings. Thereafter, we address calibration of predicted treatment effect. With respect to the detection of overfitting (\textit{i.e.}, overly complex models that fail to generalize), we examine performance in both internal and external validation settings.
\section{Discrimination for individualized treatment effects} \label{sec:discrimination}
Discriminative model performance reflects the degree to which model predictions are correctly rank-ordered and is a common performance measure in regular prediction modeling \cite{harrell_regression_2015, steyerberg_clinical_2019}. In the context of outcome risk prediction, the observed outcomes provide an immediate reference to check rank-ordering at the level of the predictions. However, such a direct reference is not available for ITE models since individual treatment effects cannot be observed directly, which necessitates approximations. One of the possibilities is to use matching and is used in a recent proposal, the c-for-benefit, for a measure of discriminative performance on the ITE level \cite{van_klaveren_proposed_2018}. The section shortly outlines the c-for-benefit and subsequently discusses its properties, limitations, and possible extensions.
\subsection{C-for-benefit definition} \label{sec:c-for-benefit_overview}
In the setting of a two-arm study measuring a binary outcome of interest, the c-for-benefit aims to assess discrimination at the level of ITE predictions (referred to as 'predicted [treatment] benefit' in the original paper).\footnote{The original paper did not focus on the required conditions for \textit{causal} interpretation of the predicted individualized treatment effects; here we assume that these assumptions, as described in Section \ref{sec:causal_prediction}, are met.} The problem of unobserved individual treatment effects is approached from a matching perspective. One-to-one matching is used to match treated individuals to control individuals based on their predicted treatment effects. The subsequent data pairs hence consist of a treated individual and a control individual with similar predicted treatment effect. Observed treatment effect in the pair is defined as the difference in outcomes between these two individuals. Of note, observed (within-pair) treatment effect can only be \{-1,0,1\}. Subsequently, c-for-benefit has been defined as "the proportion of all possible pairs of matched individual pairs with unequal observed benefit in which the individual pair receiving greater treatment benefit was predicted to do so" \cite{van_klaveren_proposed_2018}. The predicted treatment effect within each pair used in this definition is taken to be the (within-pair) average of predicted treatment effects. That is, for a pair comprising control individual $i$ out of $1,\ldots,n_i$ and treated individual $j$ out of $1,\ldots,n_j$, predicted treatment effects are taken to be
\begin{align} \label{eq:deltahat_ij}
\hat{\delta}_{ij}(\bm{x}_i, \bm{x}_j) = \{ &(\hat{P}(Y_i | A_i=1, \bm{X} = \bm{x}_i) - \hat{P}(Y_i | A_i=0, \bm{X} = \bm{x}_i)) + \nonumber \\
&(\hat{P}(Y_j | A_j=1, \bm{X} = \bm{x}_j) - \hat{P}(Y_j | A_j=0, \bm{X} = \bm{x}_j)) \} /2
\end{align}
The 'observed' treatment effect is subsequently taken to be $O_{ij}=Y_i - Y_j$. Therefore, the c-for-benefit is a regular concordance statistic (c-statistic) as commonly applied in survival data \cite{harrell_evaluating_1982, harrell_regression_2015}, but applied to pairs of individuals that underwent different treatments. If the two (binary) outcomes in such a pair are discordant, then there supposedly is some evidence of a treatment effect (\textit{i.e.}, benefit or harm); conversely, there is no such evidence when the outcomes are concordant (\textit{i.e.}, the predicted treatment effect did not manifest as a difference in outcomes).
The implicit assumption is that individual $i$ and $j$ are similar enough to serve as pseudo-observations of the unobserved potential outcomes. In the ideal case where $\bm{x}_i=\bm{x}_j$ this is indeed the case, but such perfect matches are unlikely to be available for multivariable prediction models.\footnote{Note that we here forgo the notion of including \textit{all} relevant covariates, since $\bm{x}_i=\bm{x}_j$ is sufficient for the degree of individualization reflected by the model.} An alternative (unsupervised) matching procedure that was proposed in the same paper is to match on covariates in terms of Mahalanobis distance.\footnote{ \label{footnote:Mahalanobis} where the distance between $\bm{x}_i$ and $\bm{x}_j$ is defined as $d(\bm{x}_i,\bm{x}_j) = \sqrt{(\bm{x}_i-\bm{x}_j)' \bm{S}^{-1} (\bm{x}_i-\bm{x}_j)}$ with $\bm{S}$ the covariance matrix of the covariates in $\bm{X}$} For the remainder of this paper, we will use cben-$\hat{\delta}$ to refer to the original c-for-benefit using 1:1 matching on predicted treatment effect.
\subsection{C-for-benefit challenges} \label{sec:c-for-benefit_procedure}
Although the c-for-benefit has been applied on several occasions (\textit{e.g.}, \cite{bress_patient_2021,olsen_which_2021, duan_clinical_2019}), its properties have not been fully elucidated. Van Klaveren et al. \cite{van_klaveren_proposed_2018} recommended further work on its theoretical basis and simulations studies, which we here present. Evidently, many issues that apply to the regular concordance statistic also apply to the c-for-benefit. However, since the c-for-benefit relates to risk differences and depends on outcomes that cannot be observed directly, additional challenges arise which we outline below.
\subsubsection{Difficult interpretation}
As described, the c-for-benefit uses 1:1 matching \textit{and} averages ITEs within each pair of matched individuals (\textit{i.e.}, $\hat{\delta}_{ij}(\bm{x}_i, \bm{x}_j)$ in equation \eqref{eq:deltahat_ij}). As we will see below (section \ref{sec:alternatives}), this average of two ITEs does not generally correspond to the treatment effect induced by the study design even if the model is correctly specified. Also, the observed outcome difference $O_{ij}$ reflects more than just $\hat{\delta}_{ij}(\bm{x}_i, \bm{x}_j)$ unless both control outcome risk and treated outcome risk are the same for matched individuals. These two issues obfuscate the interpretation of the index.
\subsubsection{Sensitivity to matching procedure}
Two matching procedures were proposed for the c-for-benefit: i) based on $\hat{\bm{\delta}}$ (\textit{i.e.}, minimize the distance between pairs $\hat{\delta}_i$ and $\hat{\delta}_j$), and ii) based on the Mahalanobis distance between covariate vectors\textsuperscript{\ref{footnote:Mahalanobis}} \cite{van_klaveren_proposed_2018}. In theory, matching on covariates $\bm{X}$ leads to appropriate matches on predicted treatment effects since the latter is a function of the covariates. However, the reverse is not true: matching on $\hat{\bm{\delta}}$ does not necessarily lead to appropriate matches on $\bm{X}$. The reason is that multiple configurations of $\bm{X}$ can give rise to the same value of $\hat{{\delta}}$, which does not satisfy equation \eqref{eq:delta_iobs}. Importantly, this is even the case for a correctly specified model. The only setting in which matching on predicted ITEs is guaranteed to generate appropriate matches on $\bm{X}$ is when $\hat{\bm{\delta}}$ is a bijective function of $\bm{X}$ (\textit{i.e.}, $\hat{\bm{\delta}}$ and $\bm{X}$ have a one-to-one correspondence). However, this is highly atypical in prediction modeling (\textit{e.g.}, a model with only one covariate that has a functional form with a strictly positive or negative first derivative).
Also, both matching procedures were proposed for $1:1$ matching, which requires either equal groups size for both study arms or loss of data. A simple remedy that stays close to the original idea is to perform repeated analysis with random sub-samples of the larger arm. Alternatively, the implementation of many-to-one matching (\textit{e.g.}, full matching) or many-to-many matching \cite{rosenbaum_characterization_1991, hansen_full_2004, colannino_efficient_2007} might be implemented, but none of these has been studied in the context of the c-for-benefit.
\subsection{Towards a more principled concordance statistic for benefit} \label{sec:alternatives}
The c-for-benefit compares concordance between differences in i) average predicted treatment effect within matched control-treated pairs $\hat{\delta}_{ij}(\bm{x}_i, \bm{x}_j)$ and ii) observed outcome differences within those same pairs $O_{ij}$. However, in general $\delta_{ij}(\bm{x}_i, \bm{x}_j) \neq \EX(O_{ij}|\bm{x}_i, \bm{x}_j)$ unless $\bm{x}_i = \bm{x}_j$. This section decomposes $\EX(O_{ij}|\bm{x}_i, \bm{x}_j)$ to find conditions under which unbiased comparison to ITE predictions is available. Thereto, for controls $i \in 1,\ldots,n_i$ and treated individuals $j \in 1,\ldots,n_j$ and writing ${g}_0(\bm{x})$ for $P(Y_i=1| A=0, \bm{X}=\bm{x})$ and ${g}_1(\bm{x})$ for $P(Y_i=1| A=1, \bm{X}=\bm{x}$),
\begin{align}
\EX(O_{ij}|\bm{x}_i, \bm{x}_j) =& \EX(Y_j|\bm{x}_j - Y_i|\bm{x}_i) \nonumber \\
=& \EX(Y_j|\bm{x}_j) - \EX(Y_i|\bm{x}_i) \nonumber \\
=& {g}_1(\bm{x}_j) - {g}_0(\bm{x}_i) \label{eq:EXOij_0} \\
=& [g_0(\bm{x}_j) + \delta(\bm{x}_j)] - g_0(\bm{x}_i) \label{eq:EXOij_a} \\
=& g_1(\bm{x}_j) - [g_1(\bm{x}_i) - \delta(\bm{x}_i)] \label{eq:EXOij_b}
\end{align}
Hence, from equation \eqref{eq:EXOij_a} and given $g_0(\cdot)$, the expected observed outcome difference between individual $j$ receiving treatment and individual $i$ receiving control equals the true equals the true individualized treatment effect for individuals sharing the same characteristics $\bm{x}$ as $j$
\begin{align} \label{eq:delta_trt}
\EX(O_{ij} | g_0(\bm{x}_i), g_0(\bm{x}_j)) = \EX(Y_j - g_0(\bm{x}_j)) - \underbrace{\EX(Y_i - g_0(\bm{x}_i))}_\text{0} = \delta(\bm{x}_j),
\end{align}
and analogously, from equation \eqref{eq:EXOij_b} and given $g_1(\cdot)$, the expected observed outcome difference between individual $j$ receiving treatment and individual $i$ receiving control equals the true individualized treatment effect for individuals sharing the same characteristics $\bm{x}$ as $i$
\begin{align} \label{eq:delta_ctrl}
\EX(O_{ij} | g_1(\bm{x}_i), g_1(\bm{x}_j)) = \underbrace{\EX(Y_j - g_1(\bm{x}_j))}_\text{0} - \EX(Y_i - g_1(\bm{x}_i)) = \delta(\bm{x}_i)
\end{align}
Conditioning on $g_0(\cdot)$ in equation \eqref{eq:delta_trt} aims to achieve prognostic balance, which bears resemblance to prognostic score analysis \cite{hansen_prognostic_2008, nguyen_use_2019}. Conditioning on $g_1(\cdot)$ in equation \eqref{eq:delta_ctrl} is just the mirror image for $g_1(\cdot)$. In practice, $g_0(\cdot)$ and/or $g_1(\cdot)$ will of course have to be estimated and the exact equalities will become approximations. For continuous outcomes, equation \eqref{eq:delta_trt} allows evaluation of predictions $\hat{\delta}(\bm{x}_j)$ against residuals $Y_j - \hat{g}_0(\bm{x}_j)$ for $j=1\ldots,n_j$, and equation \eqref{eq:delta_ctrl} allows evaluation of predictions $-\hat{\delta}(\bm{x}_i)$ against residuals $Y_i - \hat{g}_1(\bm{x}_i)$ for $i=1\ldots,n_i$.\footnote{Either way, one arm can be used to estimate the relevant $\hat{g}(\cdot)$ and the other arm to evaluate $\hat{\delta}(\cdot)$. Alternatively, an external model $\hat{g}(\cdot)$ might be used, but, as demonstrated in the simulation study, bias will be introduced if this external model does not fit the data well.} A key benefit of this approach is that matching is not required. However, extension of such a residual-based approach to binary outcome data is not clear. Hence, we implemented a 1:1 matching procedure similar to cben-$\hat{\delta}$, but with two important differences. First, matching was performed based on $\hat{g}_0(\bm{x})$ as opposed to predicted treatment effect. Thereby, whenever $\hat{g}_0(\bm{x}_i)=\hat{g}_0(\bm{x}_j)$, the expected difference between $Y_i$ and $Y_j$ \textit{does} equal $\hat{\delta}(\bm{x}_j)$ if the models for $g_0$ and $g_1$ are correct. Second, and following from the previous, this implementation evaluated concordance between $\hat{\delta}(\bm{x}_j)$ (as opposed to $\hat{\delta}_{ij}$) and the corresponding $O_{ij}$'s. We will further refer to this implementation as cben-$\hat{y}^0$. Note that a mirror image alternative could be performed when matching on $\hat{g}_1(\bm{x})$; the choice between the two might be guided by the expected quality in terms of prediction accuracy of $\hat{g}_0(\cdot)$ and $\hat{g}_1(\cdot)$, and the size of the group in which ITE predictions will be evaluated.
\subsection{Model-based c-statistics for individualized treatment effect} \label{sec:model_based_c_ite}
Extending earlier work on model-based concordance assessment in the context of risk prediction \cite{van_klaveren_new_2016}, we propose model-based concordance assessment on the level of absolute individualized treatment effect prediction. The concordance probability that we aim for is the probability that any randomly selected pair of patients (regardless of treatment assignment) has concordant ITE predictions and outcomes, divided by the probability that their outcomes are different. While the ITE predictions are clearly defined (equation \eqref{eq:delta_iobs}), the outcomes (individualized treatment effects) are never observed directly and can be approximated in multiple ways. The model-based approach is to use the model's predictions of the potential outcomes when deriving the concordance statistic. Therefore, it reflects the concordance statistic that would be expected under the assumption that the model is correct and given a specific set of data. Note that all information required for a model-based estimate is in the model, so there is no problem with respect to unobserved potential outcomes.
Let us first define concordance between ITE predictions and potential outcome patterns in line with the c-for-benefit. As above, take an event Y=1 to be harmful. For a randomly selected individual $k$ with a lower predicted ITE than another individual $l$ ($\hat{\delta}_k < \hat{\delta}_l$, where $k,l \in 1,\ldots,n$ and $k \neq l$), treatment is predicted to be more beneficial (or less harmful) for individual $k$ as compared to individual $l$. The potential outcome patterns that are concordant with $\hat{\delta}_k < \hat{\delta}_l$ are
\begin{enumerate}
\item $Y_k^{a=1}=0, Y_k^{a=0}=1, Y_l^{a=1}=0, Y_l^{a=0}=0$ (benefit for $k$, no benefit for $l$)
\item $Y_k^{a=1}=0, Y_k^{a=0}=1, Y_l^{a=1}=1, Y_l^{a=0}=1$ (benefit for $k$, no benefit for $l$)
\item $Y_k^{a=1}=0, Y_k^{a=0}=1, Y_l^{a=1}=1, Y_l^{a=0}=0$ (benefit for $k$, harm for $l$)
\item $Y_k^{a=1}=0, Y_k^{a=0}=0, Y_l^{a=1}=1, Y_l^{a=0}=0$ (no benefit for $k$, harm for $l$)
\item $Y_k^{a=1}=1, Y_k^{a=0}=1, Y_l^{a=1}=1, Y_l^{a=0}=0$ (no benefit for $k$, harm for $l$).
\end{enumerate}
The corresponding estimated probabilities of these patterns follow easily from the model(s) for both potential outcomes. For instance, for the first pattern: $[1-\hat{P}(Y_k^{a=1}=1)] \cdot \hat{P}(Y_k^{a=0}=1) \cdot [1-\hat{P}(Y_l^{a=1}=1)] \cdot [1-\hat{P}(Y_l^{a=0}=1)]$. The sum of the five patterns is further referred to as $P_{\textnormal{benefit}, k, l}$. Likewise, let $P_{\textnormal{harm}, k, l}$ denote the total probability of observing relative harm for case $k$ with respect to case $l$, which can be obtained in a similar manner. Returning to our definition of concordance probability, the probability of concordant ITE predictions and potential outcomes for two randomly chosen patients $k$ and $l$ for any given model can be written as
\begin{equation}
P(\textnormal{concordant})=\frac{1}{n(n-1)} \sum_k \sum_{l \neq k} \left[
I(\hat{\delta}_k < \hat{\delta}_l) \hat{P}_{\textnormal{benefit}, k, l} +
I(\hat{\delta}_k > \hat{\delta}_l) \hat{P}_{\textnormal{harm}, k, l} \right]
\end{equation}
Subsequently, the model-based probability of a potential outcome pattern reflecting either relative benefit or relative harm for a randomly selected pair of patients is
\begin{equation}
\hat{P}(\textnormal{treatment effect in potential outcomes})=\frac{1}{n(n-1)} \sum_k \sum_{l \neq k} \left[
\hat{P}_{\textnormal{benefit}, k, l} + \hat{P}_{\textnormal{harm}, k, l} \right],
\end{equation}
and hence the concordance probability is
\begin{equation}
\frac{\sum_k \sum_{l \neq k} \left[
I(\hat{\delta}_k < \hat{\delta}_l) \hat{P}_{\textnormal{benefit}, k, l} +
I(\hat{\delta}_k > \hat{\delta}_l) \hat{P}_{\textnormal{harm}, k, l} \right]}
{\sum_k \sum_{l \neq k} \left[ \hat{P}_{\textnormal{benefit}, k, l} + \hat{P}_{\textnormal{harm}, k, l} \right]}
\end{equation}
A formulation that allows for ties $\hat{\delta}_k = \hat{\delta}_l$ and avoids the need to derive $\hat{P}_{\textnormal{harm}, k, l}$ is, in line with the Harrell's c-statistic \cite{harrell_evaluating_1982, harrell_regression_2015},
\begin{equation} \label{eq:mbcb}
\textnormal{mbcb} = \frac{\sum_k \sum_{l \neq k} \left[
I(\hat{\delta}_k < \hat{\delta}_l) \hat{P}_{\textnormal{benefit}, k, l} +
\frac{1}{2} I(\hat{\delta}_k = \hat{\delta}_l) \hat{P}_{\textnormal{benefit}, k, l} \right]}
{\sum_k \sum_{l \neq k} \left[ \hat{P}_{\textnormal{benefit}, k, l} \right]}
\end{equation}
We propose the model-based c-for-benefit (mbcb) as a model-based alternative to the c-for-benefit, hence the name. Estimating both $\hat{\delta}(\bm{x})$ and $\hat{P}_{\textnormal{benefit}, k, l}$ from the same model, the mbcb provides the theoretical concordance probability between ITE predictions and potential outcomes that would be achieved if the model is correct. Important to note, such a model-based statistic only depends on the observed outcomes in the development data through the model, and hence does not provide any insight into model fit. For instance, estimating both $\hat{\delta}(\bm{x})$ and $\hat{P}_{\textnormal{benefit}, k, l}$ from some ITE model $\hat{\delta}_m$ in new data $D$, the mbcb would return the expected concordance probability for $\hat{\delta}_m$ at the ITE-level as based on the distribution of $\bm{X}$ in $D$, and assuming $\hat{\delta}_m$ is correct; it does not depend on the outcomes measured in $D$. In other words, it provides a case-mix adjusted (\textit{i.e.}, adjusted for the sampled $\bm{X}$) expected mbcb for new data \cite{van_klaveren_new_2016}. This is of interest since concordance statistics are known to be sensitive to case-mix. For instance, discriminative performance in terms of a concordance statistic for a new sample that is truncated in terms of $\bm{X}$ (\textit{e.g.}, due to inclusion criteria) will be lower, even if the model is perfectly adequate, just because it is harder to discriminate in the new sample \cite{nieboer_assessing_2016}. Hence, the case-mix adjusted expected mbcb is a better reference than the mbcb in the development data when validating a model. To obtain a validation estimate of ITE-level concordance probability (\textit{i.e.}, without assuming that the ITE model is correct), the estimator for $\hat{P}_{\textnormal{benefit}, k, l}$ should be based on independent validation data that were not used for ITE model development. The main goal is to obtain estimates of $\hat{P}_{\textnormal{benefit}, k, l}$ as accurate as possible for the new data, since these take the role of the 'observed' outcomes for the model-based c-for-benefit. A way to do so it by refitting the original ITE model in the independent data and using the resulting outcome risk predictions to calculate $\hat{P}_{\textnormal{benefit}, k, l}$.
\section{Calibration of individualized treatment effect predictions} \label{sec:calibration}
A calibration measure reflects the degree to which predictions (predicted treatment effect) agree with observations (observed treatment effect). Several routes can be taken when interest is in predicted individualized treatment effect.
\begin{enumerate}[label=(\Alph*)]
\item Classical calibration: compare predicted outcome risk under the assigned treatment conditions versus observed outcomes. \label{classical}
\item Within-arm classical calibration: classical calibration within treatment arms. \label{within-arm}
\item Quantile-group calibration of average individualized versus observed treatment effect. \label{quantile}
\item Model-based calibration of individualized versus observed treatment effect. \label{model-based}
\end{enumerate}
Method \ref{classical} has been described at length in the literature (\textit{e.g.}, \cite{steyerberg_clinical_2019, harrell_regression_2015, van_calster_calibration_2019}) and method \ref{within-arm} is a straightforward extension. Both have the disadvantage that they do not directly assess absolute treatment effect: overall calibration of outcome risk may look good when prognostic factors are well modeled and explain most of the outcome risk, even though a comparatively small (but possibly important) treatment effect is not well represented.
A common way to proceed in the direction of direct predicted treatment effect evaluation is to form quantile groups of the predictions and to compare (average) predicted and observed treatment effect within these groups \cite{van_klaveren_proposed_2018, hoogland_tutorial_2021} (method \eqref{quantile}). However, the cut-off points to form these groups are always arbitrary and smooth model-based calibration plots have become the preferred method of choice in regular (outcome risk) calibration assessment \cite{van_calster_calibration_2019}. This leaves method \ref{model-based} which, in theory, provides the desired direct ITE assessment while avoiding the disadvantages associated with cut-offs. However, to our knowledge, such a method has not been described yet. We propose a model-based approach that isolates the calibration of $\hat{\delta}$ based on equation \eqref{eq:delta_trt}.
According to equation \eqref{eq:delta_trt}, $\EX(Y_j - \hat{g}_0(\bm{x}_j))$ can be directly compared against predictions $\hat{\delta}(\bm{x}_j)$ for treated individuals $j \in 1,\ldots,n_j$. In case of dichotomous $Y_j$, a natural way to do so is to model $Y_j$ with offset $\hat{g}_0(\bm{x}_j)$. For instance, for a logistic model, a calibration model could be formulated as
\begin{align}
\text{logit}(Y_j) = \beta_0 + \beta_1 \hat{\delta}_{lp}(\bm{x}_j) + \hat{g}_{lp,0}(\bm{x}_j) \label{eq:cal_model}
\end{align}
where $\hat{\delta}_{lp}(\bm{x}_j) = \text{logit}(\hat{g}_1(\bm{x}_j)) - \text{logit}(\hat{g}_0(\bm{x}_j))$ and $\hat{g}_{lp,0}(\bm{x}_j) = \text{logit}(\hat{g}_{0}(\bm{x}_j))$. The anticipated intercept $\beta_0$ and slope $\beta_1$ in case of a perfect prediction are 0 and 1 respectively, as for regular prognostic model calibration \cite{steyerberg_clinical_2019, harrell_regression_2015}. Assuming that $\hat{g}_{0}(\bm{x}_j)$ is correct, the estimated slope $\hat{\beta}_1$ directly reflects ITE overfitting (slope below 1) or underfitting (slope above 1), and the estimated intercept $\hat{\beta}_0$ reflects average error in the ITE predictions. When $\hat{g}_{0}(\bm{x}_j)$ is misspecified, $\hat{\beta}_0$ and $\hat{\beta}_0$ amalgamate ITE calibration and errors in $\hat{g}_{lp,0}(\bm{x}_j)$. Given the importance of $\hat{g}_{0}(\bm{x}_j)$, which essentially anchors the ITE predictions, it might be preferable to derive predictions $\hat{g}_{0}$ based on a new model fitted in the external control arm data to reduce bias in the assessment of ITE calibration. A more direct way to assess average error in predicted ITEs is to examine the difference between observed and expected average treatment effect: $[\frac{1}{n_j} \sum_j Y_j - \frac{1}{n_i} \sum_i Y_i] - [\frac{1}{n_j} \sum_j \hat{g}_{1}(\bm{x}_j) - \frac{1}{n_i} \sum_i \hat{g}_{0}(\bm{x}_i)]$.
As a side note, continuous outcomes $Y_j$ allow for direct analysis of residuals $Y_j - \hat{g}_0(\bm{x}_j)$ by means of a linear regression model.
\begin{align}
Y_j - \hat{g}_0(\bm{x}_j) = \beta_0 + \beta_1 \hat{\delta}(\bm{x}_j) + \epsilon_j
\end{align}
for individuals $j \in 1,\ldots,n_j$ and with $\epsilon_j \sim N(0,\sigma^2)$. The anticipated intercept, slope, and procedure to derive calibration in the large are the same as for \eqref{eq:cal_model}. In addition to model-based evaluation, a smooth curve such as a loess (locally estimated scatterplot smoothing) estimate can be drawn through a scatterplot of $Y_j - \hat{g}_0(\bm{x}_j))$ versus $\delta(\bm{x}_j)$ to provide a visual evaluation of ITE calibration for continuous outcomes.
\section{Simulation study} \label{sec:sim}
A simulation study was performed with the aim to compare performance of the different discrimination and calibration measures for ITE predictions discussed across varying sample sizes. The simulation study was performed and reported in line with recommendations by Morris et al. \cite{morris_using_2019} and using \texttt{R} statistical software version 4.2 \cite{r_core_team_r_2022}.
\subsection{Simulation study procedures}
\textbf{Data generating mechanisms:} Synthetic trial data were simulated for a trial comparing two treatments on a binary outcome. Covariates $\bm{x}_1$ and $\bm{x}_2$ were generated from independent standard normal distributions and treatment assignment was 1:1 and independent of $\bm{X}$. Data were simulated for both potential outcomes based on a logistic data generating mechanism (DGM) according to model
\begin{align}
\textnormal{logit}(P(Y^{A=a}_i=1)) = &-1 - 0.75 a_i + x_{i1} + 0.5 a_i x_{i2} \label{eq:dgm1}
\end{align}
for a population of size 100,000, and will be further referred to as DGM-1. DGM-1 includes main effects of treatment and $X_1$ and an interaction between treatment and $X_2$. For each of $n_{sim}=500$ simulation runs, development (D) and validation data (V1) sets of size 500, 750, and 1000 were randomly drawn from the population. Marginal event probabilities were $P(Y^{a=0}) \approx 0.31$ and $P(Y^{a=1}) \approx 0.20$. Additionally, independent validation sets of 1000 cases (V2) were sampled from a population of size 100,000 generated from a second DGM (DGM-2) with changes in the coefficients to reflect a different population
\begin{align}
\textnormal{logit}(P(Y^{A=a}_i=1)) = &-0.5 - 0.5 a_i + 0.75 x_{i1} + 0.25 x_{i2} +
0.25 a_i x_{i1} + 0.25 a_i x_{i2}, \label{eq:dgm2}
\end{align}
Marginal event probabilities for the second DGM were $P(Y^{a=0}) \approx 0.39$ and $P(Y^{a=1}) \approx 0.31$. With differences in both average treatment effect and heterogeneity of treatment effect between DGM-1 and DGM-2, a model developed in a sample from DGM-1 should not perform well in individuals from DGM-2.
\textbf{Estimands:} For discrimination, our estimand $\theta_d$ is the concordance statistic between ITE predictions and the true probabilities to observe benefit. For a fixed ITE model, a given data generating mechanism, and a fixed matrix of observed covariates, this is exactly the definition of the mbcb in equation \eqref{eq:mbcb} when substituting true the value of $P_{\textnormal{benefit}, k, l}$ (know from the DGM) for the estimated $\hat{P}_{\textnormal{benefit}, k, l}$. This provides the expected ITE concordance statistic with the expectation taken over repeated samples of the potential outcomes. Due to its dependence on the matrix of observed covariates in a sample, it is further referred to as the 'sample reference'. For the estimand in the population, the mbcb is a considerable computational burden due to the fast-growing number of observation pairs with increasing sample size. Instead, the 'population reference' was not based on the expectation over potential outcomes given the covariates, but on a single sample of the potential outcomes. Hence it is still unbiased with respect to the true estimand in the population. Further details are provided in online supplementary material \ref{app:mbcb}) and also show the relation between the mbcb and Harrell's c-statistics \cite{harrell_evaluating_1982, harrell_regression_2015} applied to ITE predictions and simulations of both potential outcomes for each individual.
For calibration performance, our estimands were the calibration intercept $\beta_0$ and calibration slope $\beta_1$ as defined in equation \eqref{eq:cal_model}. The true values follow directly from equation \eqref{eq:cal_model} when taking, for $j \in 1,\ldots,n_j$, the known probabilities $P(Y_j^{a=1})$ based on the appropriate DGM, $\hat{\delta}_{lp}(\bm{x}_j)$ as the sample-based ITE predictions under evaluation, and $\hat{g}_{lp,0}(\bm{x}_j)$ based on known probabilities $P(Y_j^{a=0})$. Both a sample reference (the value of the estimand for the distribution of $\bm{X}$ in the given sample) and a population reference (the value of the estimand for the distribution of $\bm{X}$ in the given population) were derived.
\textbf{Methods:} The ITE model fitted to the development data was a logistic regression model estimated by means of maximum likelihood of the form
\begin{align}
\textnormal{logit}(P(Y_i=1)) = &\beta_0 + \beta_1 a_i + \beta_2 x_{i1} +
\beta_3 x_{i2} + \beta_4 a_i x_{i1} + \beta_5 a_i x_{i2} \label{eq:ITEmodel}
\end{align}
Discrimination performance was assessed by means of the c-for-benefit using 1:1 benefit matching (cben-$\hat{\delta}$), the c-for-benefit using 1:1 matching on predicted outcome risk under the control treatment (cben-$\hat{y}^0$), and the mbcb. Calibration performance was assessed according to equation \eqref{eq:cal_model}. Each of the performance measures was evaluated (1) without correction in the same samples as in which the ITE model was developed (sample D, apparent performance\cite{harrell_regression_2015}), (2) in interval validation using bootstrap 0.632+ adjustment (3) in interval validation using bootstrap optimism correction, (4) in external validation samples V1, (5) in external validation data samples V2, (6) in the external population from DGM-1, and (7) in the external population from DGM-2. A more detailed account of the procedures in available in online supplementary material \ref{app:sim_evals}.
\textbf{Performance measures:} Writing $\theta_s$ for the reference value in simulation run $s$, and $\hat{\theta}_s$ for the corresponding estimate, performance measures were averaged across simulations $s \in 1, \ldots, n_{sim}$ in terms of bias $\frac{1}{n_{sim}}\sum_{s=1}^{n_{sim}}(\theta_s - \hat{\theta}_s)$ and root mean squared prediction error $\sqrt{\frac{1}{n_{sim}}\sum_{s=1}^{n_{sim}}(\theta_s - \hat{\theta}_s)^2}$.
\textbf{Additional reference:} For calibration evaluation only, a 'naive' population reference value was derived for each ITE model to demonstrate that ITE calibration heavily relies on the accuracy of control outcome risk predictions ($\hat{g}_0$). This reference does not correspond to an estimand of interest, but instead corresponds to 'naive' adjustment for $\hat{g}_0$ as predicted by the evaluated ITE model (\textit{i.e.}, instead of $\hat{g}_0$ as predicted from a local model in data independent from the development data). Thereby, it serves to illustrate the large-sample error that occurs under misspecification of the model for $\hat{g}_0$.
\subsection{Discrimination results}
Figure \ref{fig:simResults_discr} and Table \ref{tab:simResults_discr} show the main simulation results with respect to the discrimination statistics. Tabulated results corresponding to Figure \ref{fig:simResults_discr} are available as online supplementary Table \ref{app:discr_means}. First, note that the sample reference and population reference show near perfect agreement across all panels. This shows that the estimand in a validation sample generalized well to entire population (\textit{i.e.}, did not greatly depend on the specific sample of covariate values in a given validation sample).
With respect to the estimates, and starting with apparent evaluations (top left), all statistics showed optimism with respect to the reference standards, which decreased with increasing sample size. As expected, direct evaluation in new data from the same DGM (top-middle) removes optimism for cben-$\hat{\delta}$ and cben-$\hat{y}^0$, and \textit{did not} remove optimism in the model-based c-for-benefit. The latter preserves overfitting since it only estimates the c-statistic that would be obtained for the new data if the model were correct. Note that the estimated cben-$\hat{\delta}$ in V1 was actually too low, indicating bias in the estimator.
As shown in the bootstrap panels in Figure \ref{fig:simResults_discr}, both types of bootstrap evaluations adjusted for optimism in apparent evaluations. On average, bias was almost eliminated from cben-$\hat{y}^0$ and the mbcb. For cben-$\hat{\delta}$ the bootstrap adjusted estimates were too low, which is in line the findings in V1. Nonetheless, bootstrap procedures were generally not able to decease the root mean squared prediction error between the estimated statistic and population reference statistic (Table \ref{tab:simResults_discr}). The 0.632+ procedure for the cben-$\hat{y}^0$ forms an exception, decreasing both bias and rmse for all sample sizes.
With respect to evaluation in new data from a different DGM (top-right), the large systematic error for cben-$\hat{y}^0$ and mbcb is apparent. For the cben-$\hat{y}^0$, this is because it relies on predictions $\hat{g}^0$ for 1:1 matching that are not suitable for the data at hand. Consequently, the observed outcome difference within matched pairs cannot be fully attributed to treatment, resulting in biased estimates. For the mbcb, this actually a feature, showing the expected model performance adjusted to the case-mix in V2 assuming the ITE model is correct. Local estimates of $\hat{P}_{\textnormal{benefit}, k, l}$ are required for actual external validation. Lastly, the original cben-$\hat{\delta}$ was a little too low (as in V1), but still quite close to the reference standards, with 1:1 matching on $\hat{\delta}$ apparently reasonable for this DGM.
Sub-figures and rows indicated with a star (in Figure \ref{fig:simResults_discr} and Table \ref{tab:simResults_discr}) show the results after local estimation of control outcome risk (for the cben-$\hat{y}^0$) and $\hat{P}_{\textnormal{benefit}, k, l}$ (for the mbcb). For the cben-$\hat{y}^0$, this is required for accurate matching in new data.\footnote{External data set V1 is an exception, since the DGM was exactly the same as for the development set, but this is never known in practice} For the mbcb, local estimates of $\hat{P}_{\textnormal{benefit}, k, l}$ are always required for validation purposes. For both V1 and V2, this results in essentially unbiased estimates for both the cben-$\hat{y}^0$ and the mbcb. However, rmse of the cben-$\hat{y}^0$ was still large and the mbcb is clearly to be preferred in terms of error when compared against the population reference estimates.
Summarizing, the cben-$\hat{\delta}$ showed some bias in all settings but was very stable throughout. Both the cben-$\hat{y}^0$ and mbcb were essentially unbiased when evaluated in external data, but only the latter was sufficiently precise to also outperform the cben-$\hat{\delta}$ in terms of rmse. Lastly, bootstrap procedures removed optimism across the board, but also increased variability.
\begin{figure}
\caption{Simulation results for the discrimination statistics in terms of mean c-statistic $\pm$ 1 SD. Top row: apparent evaluations in the original data (left), new data from the same DGM (middle), and new data from a different DGM (right). Bottom row, left to right: adjusted evaluations in the original data (bootstrap corrected 0.632+ and optimism correction), adjusted evaluations in new data from the same DGM, and adjusted evaluations in new data from a different DGM.}
\label{fig:simResults_discr}
\end{figure}
\begin{table}[ht]
\centering
\begin{tabular}{lrrr}
Statistic & cben-$\hat{\delta}$ & cben-$\hat{y}^0$ & mbcb \\
\hline
\textbf{Development data} &&& \\
Apparent & 0.036, 0.031, 0.025 & 0.045, 0.035, 0.030 & \textBF{0.035, 0.027, 0.023} \\
0.632+ & 0.041, 0.035, 0.030 & 0.037, 0.030, 0.026 & 0.039, 0.030, 0.027 \\
Opt. corrected & 0.042, 0.036, 0.030 & 0.046, 0.036, 0.031 & 0.037, 0.029, 0.026 \\
&&& \\
\textbf{External} &&& \\
DGM-1 & 0.028, 0.028, 0.028 & 0.032, 0.030, 0.030 & 0.036, 0.027, 0.023 \\
DGM-2 & 0.023, 0.024, 0.024 & 0.042, 0.043, 0.041 & 0.085, 0.079, 0.076 \\
DGM-1* & na & 0.031, 0.030, 0.031 & \textBF{0.024, 0.023, 0.024} \\
DGM-2* & na & 0.028, 0.028, 0.029 & \textBF{0.022, 0.023, 0.022} \\
\hline
\end{tabular}
\caption{Root mean squared error against population reference as averaged over simulation runs for each measure and for each of the sample sizes (500, 750, and 1000; left to right). * after local estimation of control outcome risk (for cben-$\hat{y}^0$) and $\hat{P}_{\textnormal{benefit}, k, l}$ (for mbcb). Bold numbers denote the best performance for each sample size in the following groups: development data, external data from DGM-1, and external data from DGM-2.} \label{tab:simResults_discr}
\end{table}
\subsection{Calibration results}
Simulation results for the calibration estimates are shown in Figure \ref{fig:simResults_cal_slope} (slopes), online supplementary material Figure \ref{fig:simResults_cal_int} (intercepts) and Table \ref{tab:simResults_cal}. Tabulated results corresponding to these figures is in online supplementary Tables \ref{app:cal_int_means} and \ref{app:cal_slope_means}. As expected, the apparent intercept and slope evaluations were uniformly 0 and 1 respectively. While this is a useful check of procedures, it also illustrates a challenge in calibration procedures: the apparent assessment is not just optimistic, but wholly uninformative.
Naive calibration assessment in V1 (\textit{i.e.}, with offset $\hat{g}_{lp,0}(\bm{x}_j)$ based on the ITE model under evaluation) showed optimistic slope estimates. Performing the same assessment in the whole population for DGM-1 ('population naive') gave similarly biased estimates. These findings are exaggerated when assessing calibration in V2, with the naive findings for all sample sizes seemingly very good, yet with large deviation from the true slopes for either the sample reference (\textit{i.e.}, given the distribution of covariates in the validation data) or the population reference.
Both bootstrap procedures removed optimism from apparent estimates, but the 0.632+ estimate was on average 0.062 too low for all sample sizes. Optimism correction performed better and was on average 0.036 below the population reference. Nonetheless, in terms of rmse, bootstrap estimates were all worse than the non-informative apparent evaluation. This implies that there does not seem to be enough information in a single sample to obtain reliable ITE calibration estimates.
The only consistently unbiased estimates were obtained in external data V1 and V2 after locally estimating a model for $\hat{g}_{lp,0}(\bm{x}_j)$ in the control arm. Note that this approach does not use data twice, since the ITE calibration model is fitted in the treated arm only. Regardless of sample size or data generating mechanism, these estimates were almost unbiased and had the best rmse as compared to any of the other estimates. The only exception was for n1000 in V1, which had a similar rmse based on the original model for $\hat{g}_{lp,0}(\bm{x}_j)$. Since D and V1 are both from DGM-1, re-estimation of $\hat{g}_{lp,0}(\bm{x}_j)$ is only beneficial if V1 has a larger control arm, which was the case for n500 and n750 for D. Nonetheless, since rmse is on the scale of the estimates, the absolute size of the errors was still large, casting doubt on the practical utility of ITE calibration.
Summarizing, relying on the ITE model for predictions $\hat{g}_{lp,0}(\bm{x}_j)$ (control outcome risk) can induce large bias and local estimation of $\hat{g}_{lp,0}(\bm{x}_j)$ in independent data is preferable. Bootstrap estimates removed optimism and performed well in terms of bias, but were highly variable for particular data sets, which limits practical applicability. In external validation data, performance of the ITE calibration metrics provided a large improvement over apparent and bootstrap estimated in terms of both bias and root mean squared prediction error.
\begin{figure}
\caption{Simulation results for the ITE calibration slope estimates (mean $\pm$ 1 SD). Top row: apparent evaluations in the original data (left), new data from the same DGM (middle), and new data from a different DGM (right). Bottom row, left to right: adjusted evaluations in the original data (bootstrap corrected 0.632+ and optimism correction), adjusted evaluations in new data from the same DGM, and adjusted evaluations in new data from a different DGM.}
\label{fig:simResults_cal_slope}
\end{figure}
\begin{table}[ht]
\centering
\begin{tabular}{lrr}
Statistic & $\hat{\beta}_0$ & $\hat{\beta}_1$ \\
\hline
\textbf{Development data} && \\
Apparent & \textBF{0.563, 0.378, 0.319} & \textBF{0.724, 0.491, 0.409} \\
0.632+ & na & 0.893, 0.679, 0.573 \\
Opt. corrected & 0.627, 0.450, 0.384 & 0.817, 0.600, 0.506 \\
&& \\
\textbf{External} && \\
DGM-1 & 0.367, 0.336, \textBF{0.315} & 0.473, 0.413, \textBF{0.386} \\
DGM-2 & 0.924, 0.911, 0.898 & 0.730, 0.701, 0.665 \\
DGM-1* & \textBF{0.337, 0.322,} 0.319 & \textBF{0.412, 0.375}, 0.393 \\
DGM-2* & \textBF{0.373, 0.284, 0.269} & \textBF{0.455, 0.316, 0.332} \\
\hline
\end{tabular}
\caption{Root mean squared error against population reference as averaged over simulation runs for calibration intercept and slope estimates for each of the sample sizes (500, 750, and 1000; left to right). * after local estimation of control outcome risk. Bold numbers denote the best performance for each sample size in the following groups: development data, external data from DGM-1, and external data from DGM-2.} \label{tab:simResults_cal}
\end{table}
\section{Applied example: the third International Stroke Trial} \label{sec:ae}
Patients with an ischemic stroke have sudden onset of neurological symptoms due to a blood clot that narrows or blocks an artery that supplies the brain. A key component in the emergency medical treatment of these patients includes clot-busting drug alteplase (intravenous thrombolysis recombinant tissue-type plasminogen activator) \cite{powers_guidelines_2019}.
The third International Stroke Trial (IST-3) was a randomized trial and investigated the benefits and harms of intravenous thrombolysis with alteplase in acute ischemic stroke \cite{the_ist-3_collaborative_group_benefits_2012}. This large trial included 3035 patients receiving either alteplase or placebo in a 1:1 ratio. The primary outcome was proportion of patients that was alive and independent at 6-month follow-up, which we used as outcome of interest here. Primary analyses of the treatment effect were performed by with logistic regression adjusted for linear effects of age, National Institutes of Health stroke scale (NIHSS) score, time from onset of stroke symptoms to randomization, and presence (vs absence) of ischemic change on the pre-randomization brain scan according to expert assessment. This analysis showed weak evidence of an effect (OR 1.13, 95\% CI 0.95-1.35), but subgroup analyses suggested possibly heterogeneous treatment effect by age, NIHSS score, and predicted probability of a poor outcome.
For illustrative purposes, we here compare a main effects logistic regression model similar to the original adjusted analysis (model 1) with a model where all covariate-treatment interactions were included (model 2). The outcome was coded as 0 for those independent and alive after 6 months and 1 otherwise. The included variables were treatment, age, NIHSS, time (from onset of stroke symptoms to randomization), and imaging status (presence vs absence of ischemic change on the pre-randomization brain scan). Continuous variables age, NIHSS, and time, were modeled using smoothing splines. We also included covariate-treatment interactions for these variables. Continuous variables age, NIHSS, and time, were modeled using smoothing splines with shrinkage. We also included covariate-treatment interactions for these variables. Models were fitted using the \texttt{mgcv} package in \texttt{R} with defaults smoothing parameter selection based on generalized cross-validation \cite{wood_generalized_2017}. All in all, this applied example illustrates different ways to assess the quality of individualized treatment effect predictions. The evaluated models were emphatically chosen for this purpose and were not developed in collaboration with clinical experts in the field. Hence, they are not meant to me applied in practice.
The exact parameter estimates for both models are not of key interest, but the apparent performance with respect to outcome risk prediction was good for both: c-statistics were 0.826 and 0.831 for model 1 and 2 respectively, with accompanying Brier scores of 0.160 and 0.158 and Nagelkerke $R^2$ of 0.389 and 0.402. The Spearman correlation between outcome risk predictions for both models, conditional on the assigned treatments, was 0.99.
\begin{table}[ht]
\centering
\begin{tabular}{lrrrrr}
Model & cben-$\hat{\delta}$ & cben-$\hat{y}^0$ & mbcb & $\hat{\beta_0}$ & $\hat{\beta_1}$ \\
\hline
\textbf{Apparent} &&&&& \\
M1 & 0.488 & 0.489 & 0.510 & -0.117 & \\
M2 & 0.562 & 0.570 & 0.567 & 0.011 & 1.071 \\
&&&&& \\
\textbf{bootstrap 0.632+} &&&&& \\
M1 & 0.489 & 0.499 & 0.505 & & \\
M2 & 0.535 & 0.559 & 0.536 & & 0.522 \\
&&&&& \\
\textbf{Optimism corrected} &&&&& \\
M1 & 0.485 & 0.475 & 0.507 & -0.068 & \\
M2 & 0.534 & 0.544 & 0.518 & -0.022 & 0.895 \\
\hline
\end{tabular}
\caption{Applied example discrimination and calibration statistics for predicted individualized treatment effect.} \label{tab:aeResults}
\end{table}
Nonetheless, the range of predicted ITEs (\textit{i.e.}, on the risk difference scale) was very different. Model 1 predicted ITEs with median -0.020 (IQR -0.027, -0.011 and range -0.029, 0.000), while model 2 predicted ITEs with median -0.026 (IQR -0.059, 0.027 and range -0.811, 0.213). That is, the predicted treatment effect was very similar across individuals when predicted by model 1 (assuming a constant treatment effect on the log odds scale), but not when predicted by model 2 (assuming a heterogeneous treatment effect on the log odds scale). Table \ref{tab:aeResults} shows the apparent and bootstrap corrected results for discrimination and calibration assessment at the ITE level for the applied example as averaged over 1000 bootstrap samples.
With respect to ITE discrimination, both apparent and bootstrap-corrected discrimination estimates favored model 2 over model 1, with model 1 estimates around the no discriminative ability value of 0.5. If model 2 were entirely correct, the expected c for benefit for samples with similar characteristics was estimated to be 0.567 (apparent mbcb). While we know that model 2 is just a model and not exactly correct, this value is relevant since it provides the upper bound of ITE concordance for the combination of model and data. Subsequently, the bootstrap procedures uncovered evidence of overfitting of ITE’s and provided downward adjusted estimates.
Calibration slope estimates suggested that model 2 ITE estimates are more heterogeneous than justified by the data, and require shrinkage. The amount of shrinkage suggested varies considerably between the 0.632+ and optimism corrected estimates. Based on the simulation study, optimism correction was already conservative and was to be preferred over the 0.632+ slope which were yet more conservative. Note that the calibration slope for model 1 is not estimable (since the ITEs have no variability on the logit scale) and the intercept estimate for model 1 clearly show that the degree of predicted benefit is underestimated.
In summary, the results indicate that model 1 did not provide useful differentiation in terms of ITEs. While the discriminative ability of model 2 seems modest, clear benchmarks are lacking. After updating based on the optimism corrected calibration estimates, model 2 ITE predictions may still be meaningful, having a median of -0.02 (IQR -0.52, 0.02 and range -0.58, 0.19). Comparing the 1969 patients predicted to have benefit ($\hat{\delta}_{model2} < 0$) with the remaining 1066 patients ($\hat{\delta}_{model2} \geq 0$), the first were older [median(IQR) age 83 (78-87) vs 73 (63-82)], had worse symptoms [median(IQR) nihss 15(10-20) vs 6(4-9)], were treated earlier [median(IQR) time in hours 3.5 (2.5-4.9) vs 4.2 (3.6-4.8)], were more likely to have visual infarction on imaging (43\% vs 36\%), and had a less favorable outcome on average (alive and independent after 6 months in 23\% vs 58\% respectively).
\section{Software}
\texttt{R} package \textbf{iteval} (\url{https://github.com/jeroenhoogland/iteval}) provides a free software implementation on the freely available \texttt{R} software environment for statistical computing \cite{r_core_team_r_2022} for the cben-$\hat{\delta}$, cben-$\hat{y}^0$, mbcb, and calibration measures as defined in this paper.
\section{Discussion} \label{sec:discussion}
Measures of calibration and discrimination have a long history in the context of prediction models for observed outcome data, especially of the binary type. However, the evaluation of individualized treatment effect (ITE) prediction models is more challenging, first and foremost because of the causal nature of the predictions and the ensuing unobservable nature of individualized treatment effects. In this paper, we utilized the potential outcomes framework \cite{rubin_causal_2005} to obtain insight into existing performance measures \cite{van_klaveren_proposed_2018} and to develop novel measures of discrimination and calibration for ITE prediction models. We proposed model-based statistics to address challenges of existing methods. Importantly, these statistics are applicable regardless of the modeling approach used to generate ITE predictions, as long as predictions for each potential outcome are available. This means that our methods are usable for both statistical, as well as machine learning methods. Also, while the primary focus was on dichotomous outcomes, we also provided residual-based approaches for continuous outcome models. As such, our work provides generally applicable tools for the endeavor of ITE prediction model evaluation \cite{hoogland_tutorial_2021}.
With respect to discriminative ability, the model-based c-for-benefit (mbcb) provides both a normal performance measure and an expected (case-mix adjusted) reference level for new data. The latter is relevant since concordance probabilities are known to be sensitive to case-mix \cite{nieboer_assessing_2016}. Also, bootstrap procedures are available to adjust for optimism during model development. In the simulation study, the mbcb estimates were best in terms of both bias and root mean squared error across simulation settings. Both matching-based measures of discriminative performance had specific downsides. The original cben-$\hat{\delta}$ has a difficult interpretation and was downward biased in the simulation study, but was very stable throughout. The adaptations implemented in the cben-$\hat{y}^0$ did remove the bias, but at the cost of much larger variability. We hypothesize that the stability of the mbcb is due to the lack of a need for a matching algorithm. The large variability of cben-$\hat{y}^0$ likely relates to strong reliance of the matching procedure on the accuracy of predicted control outcome risk.
With respect to calibration, the potential outcomes framework provided a model-based method that evaluates ITE prediction in the treated individuals, against an offset of prediction outcome risk under control treatment. Compared to traditional calibration measures at the level of the outcome of interest, calibration of ITE predictions has the additional challenge that ITEs are in fact relative effects. As such, correct calibration of ITEs depends on correct calibration of outcome risk under control treatment. In line, local updating of the model predicting control outcome risk proved paramount for valid assessment of ITE calibration.
A key finding for both ITE discrimination and calibration measures was that bootstrap procedures were able to remove optimism (\textit{i.e.} reduce bias), but that the increase in variance of the estimator generally led to increased root mean squared error. This implies that external data is required to accurately assess ITE predictions. The underlying reason is the need for local estimates (\textit{i.e.}, independent of the ITE model under evaluation) of control and treated outcome risk. While these steps were incorporated in the bootstrap procedures, they are necessarily noisy since they have to rely on only 36,8\% of the data for any particular bootstrap run.
While this paper focused on measures specifically targeting ITE predictions, in practice we recommend assessing prediction performance with respect to the observed outcomes first \cite{harrell_regression_2015, steyerberg_clinical_2019}. For instance, performance with respect to outcome risk can be evaluated in the control arm and in the treated arm separately. If performance is good, one can move on to ITE evaluation. The motivation for this hierarchy is that ITEs reflect differences and that they hence compound errors in both potential outcome predictions.
Limitations of the current work include the limited nature of the simulation study which was mainly performed for illustrative purposes. While both discrimination and calibration are well researched in classical settings, their application to ITE predictions is relatively novel. While we did elucidate several aspects of ITE prediction model evaluation in terms of discrimination and calibration, important questions remain. These include questions with respect to the best strategy for model comparison, the uncertainty of the estimated statistics, the relation between discrimination and calibration on the outcome risk level and the ITE level, and the relation between discrimination and calibration statistics and clinical usefulness of the models. With respect to uncertainty estimates, bootstrap procedures provide a good option, but many of the challenges are still open.
In terms of future research, it would be interesting to evaluate whether some level of grouping is beneficial for the evaluation of model performance. Paradoxically, the aim for precision underlying the development of ITE models may hamper the possibility to evaluate them, since individual level treatment effects are inherently unobservable, and their evaluation hence involves approximations based on the very model under evaluation. Also, especially in the binary event case, even if individual-level treatment effects would be observable, they would still be very noisy. This is the underlying reason that the c-statistics for benefit are so much lower than c-statistics on the outcome level.
Summarizing, we used the potential outcomes framework to obtain insight into measures of discrimination and calibration at the level of individualized treatment effect predictions. This allowed for a principled examination of existing measures and a proposal of new measures that may enrich the toolbox for ITE model evaluation. Further research is necessary to improve understanding of the exact characteristics of these measures under varying conditions with respect to sample size, degree of treatment effect heterogeneity, and explained variation.
\section*{Data Availability Statement}
Data for the International Stroke Trial-3 applied example are publicly available \cite{sandercock_p_third_2016}. \texttt{R} package \textbf{iteval} is available on GitHub (\url{https://github.com/jeroenhoogland/iteval}) and provides functions to derive the cben-$\hat{\delta}$, cben-$\hat{y}^0$, mbcb, and calibration measures as defined in this paper. Github repository \textbf{iteval-sims} (\url{https://github.com/jeroenhoogland/iteval-sims}) provides the required files and instructions for replication of the simulation study.
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\section{Binomial outcome data} \label{app:binom_challenges}
\subsection{Absolute risk, risk difference, and binomial error}
Focusing on binary outcomes, assume we observe outcome $Y_i \in \{0,1\}$ and covariate status $\bm{x}_i$ for each individual $i$. Using data on $n$ individuals, we can model the outcome risk $P(Y_i = 1 | A=a_i, \bm{X}=\bm{x}_i)$. There are two sources of error when using such a model to predict binary outcomes. There is the reducible error in modeling the risk (\textit{i.e.}, how well the modelled probability approximates the actual probability of an event), and there is the irreducible error in the difference between the actual probability of an event and its manifestation as a $\{0,1\}$ outcome (binomial error).
Adding to this, actual interest is in the difference in outcome risk under different treatment assignment $a \in \{0,1\}$. That is, interest is in $p(Y_i = 1|A = 1, \bm{X}=\bm{x}_i) - p(Y_i = 1|A = 0, \bm{X}=\bm{x}_i)$. The range of possible true (and estimated) treatment effects (risk differences) includes all values in the $[-1,1]$ interval, but the observed difference between any two outcomes can only be one of $\{-1, 0, 1\}$. An example may be helpful to appreciate the large influence of irreducible error in this setting. For instance, regardless of any modeling, assume that an active treatment (as compared to a control condition) reduces outcome risk from 25\% to 20\% for a certain individual. Moreover, assume that these probabilities are known exactly and that this individual can be observed under both treatment conditions. A simple probabilistic exercise\footnote{For instance, $P(Y^0=0, Y^1=0)=(1-P(Y^0=0))(1-P(Y^1=0))=(1-0.25)(1-0.2)=0.6$} shows that the different outcome probabilities are $P(Y^0=0, Y^1=0)=0.6, P(Y^0=0, Y^1=1)=0.15, P(Y^0=1, Y^1=0)=0.2$, and $P(Y^0=1, Y^1=1)=0.05$. That is, the probability that the active treatment induces any observed outcome difference is 35\%, and only 20\% is in the expected direction (\textit{i.e.} in the direction of the treatment effect). This is just due to the \textit{irreducible} error, apart from any modeling issues, and ignoring the fact that in practice only one potential outcome is observed of each individual. The insensitivity of binary endpoints is of course well known in the context of trials, where a larger number of replications can provide a solution when the average treatment effect is of interest. In the case of individualized treatment effect estimation however, the required number of replications is more challenging to control due to its complex dependence on all individual-level characteristics of interest.
\subsection{Scale matters}
Models that predict the risk of a binary event commonly make use of a link function in order to map a function of the covariates in $\mathbb{R}$ onto the probability scale \cite{agresti_categorical_2013}. Such link functions, such as the logit or inverse Gaussian, are inherently non-linear and hence do not preserve additivity. Consequently, a treatment effect that is constant (\textit{i.e.}, does not vary with other covariates) before applying the link function \textit{shall} vary with other covariates on the risk scale and vice versa. As an example, we write $h^{-1}$ for an inverse link function and take control risk to be a function $f(\cdot)$ of only one random variable $X$ (\textit{i.e.}, $P(Y^{a=0}|X=x) = h^{-1}(f(X))$). Subsequently, assume a constant (homogeneous) relative treatment effect $d$ such that $P(Y^{a=1}|X=x)=h^{-1}(f(X) + d)$, then the absolute treatment effect necessarily depends on $X$, since
\begin{align} \label{eq:abs-rel}
\delta(x) &= P(Y^{a=1}|X=x) - P(Y^{a=0}|X=x) \\ \nonumber
&= h^{-1}[f(X) + d] - h^{-1}[f(X)] \neq h^{-1}[f(X) + d - f(X)] = h^{-1}(d)
\end{align}
unless $h^{-1}(\cdot)$ is linear. Consequently, between-individual variability (\textit{i.e.}, variability in terms of $X$) directly changes control outcome risk \textit{and} affects the absolute effect of $d$ on the probability scale even if $d$ is constant. For instance, a constant treatment effect on the log-odds scale translates into heterogeneous treatment effect on the risk difference scale. Thereby, relatively simple treatment effect structures may lead to meaningful between-individual treatment effect variability at the risk difference level if there is large variability in $h^{-1}[f(X)]$ \cite{hoogland_tutorial_2021, harrell_viewpoints_2018}. In addition, treatment effect may interact with $X$ in the domain of $h^{-1}(\cdot)$, \textit{i.e.}, we may directly model treatment effect heterogeneity. These two sources of variability in $\delta(x)$ can no longer be discerned when evaluating just the estimates $\hat{\delta}(x)$. Hence, the benefit in terms of interpretation of measures on the scale of $\delta(x)$ \cite{murray_patients_2018}, as of interest in this paper, has a price in that they conflate variability in $\hat{\delta}(x)$ from different sources: between-subject variability in $P(Y^{a=0}|X=x)$ and genuine treatment effect heterogeneity on the scale used for modeling.
\section{Discrimination estimand} \label{app:mbcb}
Harrell's c-statistic \cite{harrell_evaluating_1982, harrell_regression_2015} can be applied to ordered predictions and (possibly censored) ordered outcomes. Applying Harrell's c-statistic with ITE predictions and within-individual differences in potential outcomes $Y^{a=1}_k - Y^{a=0}_k$ as simulated from some data generating mechanism, the equation for the concordance probability can be written as
\begin{equation} \label{eq:c_delta_ben}
c_{\hat{\delta}, ben} = \frac{\sum_k \sum_{l \neq k} \left[
I(\hat{\delta}_k < \hat{\delta}_l) \textnormal{ben}_{kl} +
\frac{1}{2} I(\hat{\delta}_k = \hat{\delta}_l) \textnormal{ben}_{kl} \right]}
{\sum_k \sum_{l \neq k} \left[ \textnormal{ben}_{kl} \right]}
\end{equation}
with
\begin{equation}
\textnormal{ben}_{kl} = I([Y^{a=1}_k - Y^{a=0}_k] < [Y^{a=1}_l - Y^{a=0}_l])
\end{equation}
However, the within-individual differences in sampled potential outcomes $Y^{a=1}_k - Y^{a=0}_k$ are just a single manifestation of treatment effect for covariate matrix $\bm{X}$, and interest is in the expected value over repeated samples of potential outcomes given $\bm{X}$. Taking this expectation $\EX_{\bm{Y}^{a=A}|\bm{X}}$ over equation \eqref{eq:c_delta_ben} does not affect ITE predictions, since these are invariant conditional on a fixed ITE model and fixed $\bm{X}$. For $\textnormal{ben}_{kl}$,
\begin{align}
\EX_{\bm{Y}^{a=A}|\bm{X}}(\textnormal{ben}_{kl}) &= \EX_{\bm{Y}^{a=A}|\bm{X}}(I([Y^{a=1}_k - Y^{a=0}_k] < [Y^{a=1}_l - Y^{a=0}_l])) \nonumber \\
&= P([Y^{a=1}_k - Y^{a=0}_k] < [Y^{a=1}_l - Y^{a=0}_l]) \nonumber \\
&= P_{\textnormal{benefit}, k, l} \quad \textnormal{(as defined in Section } \ref{sec:model_based_c_ite})
\end{align}
In turn, substituting an estimate of $P_{\textnormal{benefit}, k, l}$ for $\textnormal{ben}_{kl}$ in equation \eqref{eq:c_delta_ben} gives the equation for the mbcb (equation \eqref{eq:mbcb}). Instead substituting the true probabilities $P_{\textnormal{benefit}, k, l}$, as known in a simulation context given the data generating mechanism, provides the expected value for the ITE concordance statistic for a given data generating mechanism, a fixed matrix of observed covariate values $\bm{X}$, and a fixed ITE model. Therefore, it was used as the estimand value of the ITE concordance statistic for a given sample (denoted 'sample reference').
When the sample size is very large, such as for the simulated populations of size 100,000, calculation of the mbcb is computationally very intensive and an approximation based on $c_{\hat{\delta}, ben}$ (equation \eqref{eq:c_delta_ben}) is accurate. Note that $c_{\hat{\delta}, ben}$ is still unbiased, but of course more variable.
\section{Performance evaluation details} \label{app:sim_evals}
\subsection{Apparent performance}
Apparent ITE model performance was evaluated, without any adjustment, in the development sample D in which the ITE model was fitted. Consequently, apparent estimates can be expected to be optimistic. For apparent calibration performance of a logistic ITE model based on maximum likelihood estimation, note that the estimates will invariably be $\hat{\beta}_0=0$ and $\hat{\beta}_1=1$ since the calibration model is of the exact same type. All in all, apparent performance was primarily assessed to show the need for internal validation procedures that correct for optimism or, better yet, new data.
\subsection{Internal validation}
\textit{Discrimination} \newline
Internal validation was performed based on a non-parametric bootstrapping procedure based on 100 bootstrap samples. Performance estimates were based on either a 0.632+ method \cite{efron_improvements_1997} adapted for application in the context of c-statistics or on optimism correction \cite{harrell_regression_2015}.
The adapted 0.632+ method provides a weighted average of apparent performance and average out-of-sample performance as based on predictions from bootstrap models for the cases not in the bootstrap sample. Writing $\hat{c}_{app}$ (scalar) for the apparent c-statistic and $\hat{c}_{oos}$ (scalar) for the average out-of-sample c-statistic across bootstrap replications,
\begin{align}
\hat{c}_{oos} &= \left\{ \begin{array}{l}
\textnormal{min}(\gamma, \hat{c}_{oos}), \quad \hat{c}_{app} \geq \gamma \\
\textnormal{max}(\gamma, \hat{c}_{oos}), \quad \hat{c}_{app} < \gamma \\
\end{array} \right. \\
R &= \left\{ \begin{array}{l}
\frac{|\hat{c}_{app} - \hat{c}_{oos}|}{|\hat{c}_{app} - \gamma|}, \quad |\hat{c}_{oos} - \gamma| < |\hat{c}_{app} - \gamma| \\
0, \quad \textnormal{otherwise} \\
\end{array} \right. \\
w &= \frac{0.632}{1 - 0.368 R} \\
\hat{c}_{0.632+} &= \hat{c}_{app}(1-w) + w \hat{c}_{oos}
\end{align}
where $\gamma$ is the value of the statistic for an uninformative model (so $\gamma=0.5$ for c-statistics), and $w$ is a weight that depends on the discrepancy between apparent and out-of-sample performance. To prevent that $R$ falls outside of the $(0,1)$, we avoid the possibility of bootstrap correction towards a point beyond the no information threshold by replacement of $\hat{c}_{oos}$ with $\hat{c}_{oos}'$ throughout, with
\begin{align}
\hat{c}_{oos}' &= \left\{ \begin{array}{l}
\textnormal{min}(\gamma, \hat{c}_{oos}), \quad \hat{c}_{app} \geq \gamma \\ \label{eq:coos_prime}
\textnormal{max}(\gamma, \hat{c}_{oos}), \quad \hat{c}_{app} < \gamma \\
\end{array} \right. \\ \nonumber
\end{align}
Subsequently, $R$ reflect the degree of overfitting and ranges from zero to one, with $w$ depending only on $R$ and ranging from 0.632 and 1. Thereby, $\hat{c}_{0.632+}$ moves towards $\hat{c}_{oos}'$ when the amount of overfitting ($|\hat{c}_{app} - \hat{c}_{oos}'|$) is large with respect to the models gain relative to no information ($|\hat{c}_{app} - \gamma|$). The choice to use $\hat{c}_{oos}'$ instead of $\hat{c}_{oos}'$ in \eqref{eq:coos_prime} was to avoid correction of an apparent estimate beyond the no information threshold.
Alternatively, optimism correction estimates optimism as the average difference between performance of bootstrap models as evaluated in a) the original full data set D and b) within the bootstrap sample. In case of overfitting, the discrepancy between the two will increase. The apparent estimate is subsequently corrected for this bootstrap estimate of optimism.
Obtaining either the 0.632+ or optimism corrected estimates for cben-$\hat{\delta}$ and cben-$\hat{y}^0$ is straightforward. One subtlety is that in case of unequal group sizes (treated vs control), the average over 1000 repeated analyses of subsamples of the larger arm was taken to accommodate for 1:1 matching. For the model-based estimates, a choice with respect to the estimation of $\hat{P}_{\textnormal{benefit}, k, l}$ has to be made with respect to out-of-sample evaluation. To avoid bias, the out-of-sample evaluation of $\hat{P}_{\textnormal{benefit}, k, l}$ for the 0.632+ estimate was based on a model for $\hat{g}_0$ and $\hat{g}_1$ is the out-of-sample cases (with the same specification as the model under evaluation). For the optimism correction, $\hat{P}_{\textnormal{benefit}, k, l}$ for the whole of D was based on the ITE model as developed in the full sample D. That is, the 0.632+ model-based c-statistic estimates were obtained from (1) out-of-sample predictions $\hat{\delta}(\bm{x}_{i \in oos})$ from bootstrap models and (2) $\hat{P}_{\textnormal{benefit}, k, l}$ based on an out-of-sample model. Optimism corrected model-based c-statistic estimates were obtained from (1) predictions $\hat{\delta}(\bm{x}_{i})$ from bootstrap models and $\hat{P}_{\textnormal{benefit}, k, l}$ based on the development model.
\textit{Calibration} \newline
Bootstrap evaluation of the calibration parameters was also performed. A 0.632+ estimate was derived for the slope estimates in analogy to the derivation for c-statistics, but using $\gamma=0$ for the value that slope $\beta_1$ takes for an uninformative model. Out-of-sample estimates of $\hat{g}_0(\cdot)$ were based on a model fitted in just the out-of-sample controls to serve as an offset in the calibration model fitted in the out-of-sample treated arm. A 0.632+ estimate for the calibration intercept parameter is not readily available since a $\gamma$ value for a non-informative intercept cannot be defined. Optimism corrected bootstrap estimates were obtained for both intercepts and slopes. Estimates of $\hat{g}_0(\cdot)$ in the bootstrap sample were based on the bootstrap model and estimates $\hat{g}_0(\cdot)$ in the original data were based on the original ITE model fitted in development data D.
\subsection{External validation}
\textit{Discrimination} \newline
External validation was performed in both V1 (DGM-1) and V2 (DGM-2). ITE predictions can be evaluated directly using cben-$\hat{\delta}$. For cben-$\hat{y}^0$, which matches based on predicted control outcome risk $\hat{g}_0(\cdot)$, a key question is whether $\hat{g}_0(\cdot)$ may best be based on the ITE model \textit{or} on a \textit{new} model fitted in the control arm of the external data. In practice, the accuracy of $\hat{g}_0(\cdot)$ based on the ITE model can be assessed in the control arm of the external data. If not satisfactory, a new model for $\hat{g}_0(\cdot)$ can be derived in the external data for use with the cben-$\hat{y}^0$. The latter option was taken for the simulation study based on a refitting of the relevant parts of model \eqref{eq:ITEmodel} in the control arm (\textit{i.e.}, omitting parameters relating to $a$ which equal 0 for controls) of the external data. Note that fitting a new model will in general remove bias, but may have a high cost in terms of variance if the external data set is small. For the model-based c-for-benefit (mbcb), the accuracy of $\hat{P}_{\textnormal{benefit}, k, l}$, and hence the underlying $\hat{g}_0(\cdot)$ and $\hat{g}_1(\cdot)$, is paramount. In the mbcb, $\hat{P}_{\textnormal{benefit}, k, l}$ is the sole carrier of information from the external data and acts as the reference for ITE predictions under evaluation. In line with the procedure for the cben-$\hat{y}^0$, performance with respect $\hat{g}_0(\cdot)$ and $\hat{g}_1(\cdot)$ can be examined in the external data (control arm and treated arm respectively) and may indicate the need for a new model. Again, the latter option was chosen for the simulation study. Note that D, V1 and V2 were always of equal size, such that there was the benefit of possibly reducing bias while avoiding the possible harm of increased variance. Finally, note that while cben-$\hat{\delta}$, cben-$\hat{y}^0$, and the mbcb focus on $\hat{\delta}(\bm{x}_{i})$, inadequate prediction performance with respect to $\hat{g}_0(\cdot)$ and/or $\hat{g}_1(\cdot)$ is an ominous sign for ITE model performance. Nonetheless, the attention with respect to accurate potential outcome prediction in the external data is required even if only to reliably show bad performance with respect to $\hat{\delta}(\bm{x}_{i})$.
\textit{Calibration} \newline
Direct calibration assessment in external data exactly followed the lines of apparent calibration assessment with all predictions (both $\hat{\delta}_{lp}(\bm{x}_j)$ and $\hat{g}_{lp,0}(\bm{x}_j)$) based on the ITE model as derived in D and applied in V1 and V2. Adjusted estimates were obtained based on local predictions $\hat{g}_{lp,0}(\bm{x}_j)$ based on a refitting of the relevant parts of model \eqref{eq:ITEmodel} in the control arm (\textit{i.e.}, omitting parameters relating to $a$ which equal 0 for controls) of the external data.
\textit{Calibration} \newline
As for discrimination, the estimands as defined in Section \ref{sec:sim} reflect the target reference parameter values for a specific ITE model as evaluated in a specific sample (D, V1 or V2). To remove dependence of the performance measure on a (small) sample, population reference values were derived per data generating mechanism as for the discrimination measures. Derivation was exactly analogous to the description under 'estimands' in Section \ref{sec:sim}. In addition, a naive population reference for calibration assessment that mirrored assessment in D (\textit{i.e.}, naive referring to adjustment based on $\hat{g}_0(\cdot)$ based on the ITE model instead of independent data). Therein, this naive population reference helps to remove the influence of sample size from the assessment of bias due to misspecification of $\hat{g}_0(\cdot)$.
\section{Additional simulation study results}
\subsection{Discrimination} \label{app:mean_tables}
\begin{table}[ht]
\centering
\begin{tabular}{lrrrrr}
\hline
Statistic & cben-$\hat{\delta}$ & cben-$\hat{y}^0$ & mbcb & sample & population \\
& & & & ref. & ref. \\
\hline
\textbf{Development data, n=500} &&&&& \\
Apparent & 0.587 & 0.599 & 0.598 & 0.578 & 0.580 \\
0.632+ & 0.568 & 0.583 & 0.576 & 0.578 & 0.580 \\
Opt. corrected & 0.569 & 0.581 & 0.578 & 0.578 & 0.580 \\
\textbf{External, n=500} &&&&& \\
DGM-1 & 0.569 & 0.581 & 0.598 & 0.578 & 0.580 \\
DGM-2 & 0.518 & 0.551 & 0.598 & 0.520 & 0.520 \\
DGM-1* & & 0.580 & 0.578 & 0.578 & 0.580 \\
DGM-2* & & 0.521 & 0.520 & 0.520 & 0.520 \\
&&&&& \\
\textbf{Development data, n=750} &&&&& \\
Apparent & 0.584 & 0.595 & 0.594 & 0.582 & 0.584 \\
0.632+ & 0.572 & 0.585 & 0.580 & 0.582 & 0.584 \\
Opt. corrected & 0.572 & 0.583 & 0.580 & 0.582 & 0.584 \\
\textbf{External} &&&&& \\
DGM-1 & 0.574 & 0.584 & 0.594 & 0.582 & 0.584 \\
DGM-2 & 0.520 & 0.553 & 0.594 & 0.520 & 0.520 \\
DGM-1* & & 0.581 & 0.582 & 0.582 & 0.584 \\
DGM-2* & & 0.523 & 0.523 & 0.520 & 0.520 \\
&&&&& \\
\textbf{Development data, n=1000} &&&&& \\
Apparent & 0.581 & 0.594 & 0.592 & 0.584 & 0.586 \\
0.632+ & 0.573 & 0.585 & 0.582 & 0.584 & 0.586 \\
Opt. corrected & 0.573 & 0.584 & 0.582 & 0.584 & 0.586 \\
\textbf{External} &&&&& \\
DGM-1 & 0.575 & 0.584 & 0.592 & 0.584 & 0.586 \\
DGM-2 & 0.518 & 0.552 & 0.592 & 0.521 & 0.521 \\
DGM-1* & & 0.584 & 0.585 & 0.584 & 0.586 \\
DGM-2* & & 0.521 & 0.520 & 0.521 & 0.521 \\
\hline
\end{tabular}
\caption{Mean over simulation runs for each discrimination measure and for each of the sample sizes (500, 750, and 1000; left to right). *) After local estimation of control outcome risk (for cben-$\hat{y}^0$) and $\hat{P}_{\textnormal{benefit}, k, l}$ (for mbcb).} \label{app:discr_means}
\end{table}
\subsection{Calibration} \label{app:cal_int}
\begin{figure}
\caption{Simulation results for the ITE calibration intercept estimates (mean $\pm$ 1 SD). Top row: apparent evaluations in the original data (left), new data from the same DGM (middle), and new data from a different DGM (right). Bottom row, left to right: adjusted evaluations in the original data (bootstrap corrected 0.632+ and optimism correction), adjusted evaluations in new data from the same DGM (middle), and adjusted evaluations in new data from a different DGM (right)}
\label{fig:simResults_cal_int}
\end{figure}
\begin{table}[ht]
\centering
\begin{tabular}{lrrrr}
\hline
Statistic & $\hat{\beta}_0$ & population & sample & population \\
& & naive & ref. & ref. \\
\hline
\textbf{Development data, n=500} &&&& \\
Apparent & 0.000 & -0.046 & -0.105 & -0.107 \\
0.632+ & & -0.046 & -0.105 & -0.107 \\
Opt. corrected & -0.108 & -0.046 & -0.105 & -0.107 \\
\textbf{External, n=500} &&&& \\
DGM-1 & -0.052 & -0.046 & -0.108 & -0.107 \\
DGM-2 & 0.687 & 0.694 & -0.140 & -0.140 \\
DGM-1* & -0.109 & -0.103 & -0.108 & -0.107 \\
DGM-2* & -0.162 & -0.154 & -0.140 & -0.140 \\
&&&& \\
\textbf{Development data, n=750} &&&& \\
Apparent & 0.000 & -0.011 & -0.034 & -0.037 \\
0.632+ & & -0.011 & -0.034 & -0.037 \\
Opt. corrected & -0.063 & -0.011 & -0.034 & -0.037 \\
\textbf{External} &&&& \\
DGM-1 & -0.031 & -0.011 & -0.038 & -0.037 \\
DGM-2 & 0.746 & 0.727 & -0.100 & -0.100 \\
DGM-1* & -0.064 & -0.043 & -0.038 & -0.037 \\
DGM-2* & -0.088 & -0.105 & -0.100 & -0.100 \\
&&&& \\
\textbf{Development data, n=1000} &&&& \\
Apparent & 0.000 & 0.007 & -0.011 & -0.010 \\
0.632+ & & 0.007 & -0.011 & -0.010 \\
Opt. corrected & -0.036 & 0.007 & -0.011 & -0.010 \\
\textbf{External} &&&& \\
DGM-1 & -0.006 & 0.007 & -0.009 & -0.010 \\
DGM-2 & 0.748 & 0.745 & -0.102 & -0.101 \\
DGM-1* & -0.003 & 0.009 & -0.009 & -0.010 \\
DGM-2* & -0.094 & -0.096 & -0.102 & -0.101 \\
\hline
\end{tabular}
\caption{Mean over simulation runs for calibration intercept estimates each of the sample sizes (500, 750, and 1000; left to right). *) after local estimation of control outcome risk.} \label{app:cal_int_means}
\end{table}
\begin{table}[ht]
\centering
\begin{tabular}{lrrrrr}
\hline
Statistic & $\hat{\beta}_1$ & population & sample & population \\
& & naive & ref. & ref. \\
\hline
\textbf{Development data, n=500} &&&& \\
Apparent & 1.000 & 0.907 & 0.856 & 0.853 \\
0.632+ & 0.785 & 0.907 & 0.856 & 0.853 \\
Opt. corrected & 0.831 & 0.907 & 0.856 & 0.853 \\
\textbf{External, n=500} &&&& \\
DGM-1 & 0.932 & 0.907 & 0.852 & 0.853 \\
DGM-2 & 0.908 & 0.913 & 0.384 & 0.383 \\
DGM-1* & 0.868 & 0.843 & 0.852 & 0.853 \\
DGM-2* & 0.374 & 0.379 & 0.384 & 0.383 \\
&&&& \\
\textbf{Development data, n=750} &&&& \\
Apparent & 1.000 & 0.942 & 0.939 & 0.935 \\
0.632+ & 0.878 & 0.942 & 0.939 & 0.935 \\
Opt. corrected & 0.901 & 0.942 & 0.939 & 0.935 \\
\textbf{External} &&&& \\
DGM-1 & 0.956 & 0.942 & 0.935 & 0.935 \\
DGM-2 & 0.989 & 0.948 & 0.426 & 0.425 \\
DGM-1* & 0.920 & 0.907 & 0.935 & 0.935 \\
DGM-2* & 0.455 & 0.416 & 0.426 & 0.425 \\
&&&& \\
\textbf{Development data, n=1000} &&&& \\
Apparent & 1.000 & 0.981 & 0.990 & 0.991 \\
0.632+ & 0.930 & 0.981 & 0.990 & 0.991 \\
Opt. corrected & 0.939 & 0.981 & 0.990 & 0.991 \\
\textbf{External} &&&& \\
DGM-1 & 0.992 & 0.981 & 0.992 & 0.991 \\
DGM-2 & 0.993 & 0.985 & 0.439 & 0.439 \\
DGM-1* & 1.004 & 0.991 & 0.992 & 0.991 \\
DGM-2* & 0.443 & 0.435 & 0.439 & 0.439 \\
\hline
\end{tabular}
\caption{Mean over simulation runs for calibration slope estimates each of the sample sizes (500, 750, and 1000; left to right). *) after local estimation of control outcome risk.} \label{app:cal_slope_means}
\end{table}
\end{document} |
\begin{document}
\title{An Obstacle Control problem involving the $p$-Laplacian}
\begin{abstract}
In this paper, we consider the analogous of the obtacle problem in
$H_0^1(\Omega)$, on the space $W^{1,p}_0(\Omega)$.
We prove an existence and uniqueness of the result.
In a second time, we define the optimal control problem associated.
The results, here enclosed, generalize the one obtained by D.R. Adams, S. Lenhard
in \cite{1}, \cite{2} in the case $p=2$.
\end{abstract}
{\small {\bf \sc Key words and phrases:}
Obstacle problem, Approximation, Optimal Pair.}
\section{Introduction}
Let $\Omega$ be a bounded domain in $\mathbb{R}^N, N\geq2$, whose boundary is $\mathcal C^1$
piecewise. For $p >1$ and for $\psi$ given in
$W^{1,p}_0(\Omega)$, define
$$K(\psi)=\{v \in W_0^{1,p}(\Omega), v \geq \psi \ \text{a.e. in} \ \Omega\}.$$
It is clear that $K(\psi)$ is a convex and weakly closed set in $L^{p}(\Omega)$.
Let $p'$ be the conjugate of $p$, and $f \in L^{p'}(\Omega)$.
We consider the following variational inequality called {\it the obstacle problem}:
\begin{equation}\label{P-ob:eq:1.1}
\begin{cases}
u \in K(\psi),\\
{\displaystyle \int_\Omega} \sigma(u) \cdot \nabla(v-u) \ dx \geq {\displaystyle \int_\Omega} f(v-u) \ dx,
\ \forall v \in K(\psi),
\end{cases}
\end{equation}
where $\sigma(u)= |\nabla u|^{p-2} \nabla u$.
We shall say that $\psi$ is {\it the obstacle} and $f$ is {\it the
source term}.
We begin to prove existence and uniqueness of a solution $u$ to (\ref{P-ob:eq:1.1}),
using variational formulation of the obstacle problem on the set $K(\psi)$.
We shall then denote $u$ by: $u=T_f(\psi)$.
Secondly, we characterize $T_f(\psi)$ as the lowest $f-$superharmonic
function greater than $\psi$.
\section{Existence and uniqueness of the solution}
\begin{proposition}\label{P-ob:prop:1.1}
A function $u$ is a solution to the problem {\rm{(\ref{P-ob:eq:1.1})}} if and only if $u$
satisfies the following:
\begin{equation}\label{P-ob:eq:1.2}
\begin{cases}
u \in K(\psi),\\
-\Delta_p u \geq f, \ \text{a.e. \ in} \ \Omega,\\
{\displaystyle \int_\Omega }\sigma(u) \cdot \nabla (\psi-u) \ dx={\displaystyle \int_\Omega} f(\psi-u) \ dx.
\end{cases}
\end{equation}
\end{proposition}
\begin{proof}[Proof of Proposition \ref{P-ob:prop:1.1}]
Suppose that $u$ satisfies (\ref{P-ob:eq:1.1}). Then taking $v=u+\phi\in
K(\psi)$ for $\phi \in \mathcal D(\Omega), \ \phi \geq 0$, one gets that
$-\Delta_p u \geq f$ in $\Omega$.
Moreover, For $v=\psi$ and $v= 2u-\psi$, one gets that
$$\int_\Omega \sigma(u) \cdot \nabla (\psi-u) \ dx=\int_\Omega f(\psi-u) \ dx,$$
hence $u$ satisfies (\ref{P-ob:eq:1.2}).
Conversely, let $u \in K(\psi)$ such that $-\Delta_p u \geq
f$, let $v$ be in $K(\psi)$ and $\phi_n \in
\mathcal D(\Omega), \phi_n \geq 0$ such that
$\phi_n \rightarrow v-\psi \ \text{in} \ W^{1,p}_0(\Omega)$. Then one gets
\begin{align*}
\int_\Omega \sigma(u) \cdot \nabla (v-\psi)
&=\lim_{n\rightarrow \infty} \int_\Omega \sigma(u) \cdot \nabla \phi_n\\
&=\lim_{n\rightarrow \infty} \int_\Omega -\Delta_p u \ \phi_n\\
&\geq \lim_{n\rightarrow \infty} \int_\Omega f\phi_n =
\int_\Omega f (v-\psi), \ \forall \ v\in K(\psi).
\end{align*}
Using the last equality of (\ref{P-ob:eq:1.2}), one gets that
$$\int_\Omega \sigma(u) \cdot \nabla (v-u) \geq \int_\Omega f (v-u),
\ \forall \ v\in K(\psi),$$
hence $u$ satisfies (\ref{P-ob:eq:1.1}).
\end{proof}
Let us prove now the existence and uniqueness of a solution to the obstacle problem (\ref{P-ob:eq:1.1}).
\begin{proposition}\label{P-ob:prop:1.2}
There exists a solution to {\rm{(\ref{P-ob:eq:1.1})}}, which can be obtained as the
minimizer of the following minimization problem
\begin{equation}\label{P-ob:eq:1.3}
\inf_{v \in K(\psi)} I(v),
\end{equation}
where $I$ is the following energy functional
$$I(v)=\frac{1}{p}\int_\Omega |\nabla v|^p -
\int_\Omega f v.$$
\end{proposition}
\begin{proof}[Proof of Proposition \ref{P-ob:prop:1.2}]
Using classical arguments in the calculus of variations, since $K(\psi)$
is a weakly closed convex set in $W^{1,p}_0(\Omega)$,
and the functional $I$ is convex and coercive on $W^{1,p}_0(\Omega)$, then
one obtains that there exists a solution $u$ to (\ref{P-ob:eq:1.3}).
\end{proof}
\begin{proposition}\label{P-ob:prop:1.3}
The inequation {\rm{(\ref{P-ob:eq:1.1})}} possesses a unique solution.
\end{proposition}
\begin{proof}[Proof of Proposition \ref{P-ob:prop:1.3}]
Suppose that $u_1, u_2 \in W^{1,p}_0(\Omega)$ are two solutions of the variational
inequality (\ref{P-ob:eq:1.1})
$$u_i \in K(\psi):
\int_\Omega \sigma(u_i) \cdot \nabla(v-u_i) \ dx \geq \int_\Omega
f(v-u_i) \ dx, \ \forall \ v \in K(\psi), \
i=1,2$$
Taking $v=u_1$ for $i=2$ and $v=u_2$ for $i=1$ and adding, we have
$$\int_\Omega \left[\sigma(u_1)-\sigma(u_2)\right] \cdot \nabla(u_1-u_2) \leq 0.$$
Recall that we have
$$\int_\Omega \left[\sigma(u_1)-\sigma(u_2)\right] \cdot \nabla(u_1-u_2)\geq 0,$$
which implies that
$$\int_\Omega \left[\sigma(u_1)-\sigma(u_2)\right] \cdot \nabla(u_1-u_2)=0,$$
and then, $u_1=u_2$ a.e in $\Omega$.
\end{proof}
Thus, we get the existence and uniqueness of a solution to (\ref{P-ob:eq:1.1}).
\begin{definition}\label{P-ob:def:1.1}
We shall say that $u$ is {\it $f-$superhamonic} in $\Omega$,
if $u \in W^{1,p}_0(\Omega)$ is a weak solution to
$-\Delta_p u \geq f$, in the sense of distributions.
\end{definition}
\begin{proposition}\label{P-ob:prop:1.4}
A function $u$ is a solution of {\rm{(\ref{P-ob:eq:1.1})}}, if and only if $u$ is the lowest
$f-$superharmonic function, greater than $\psi$.
\end{proposition}
\begin{proof}[Proof of Proposition \ref{P-ob:prop:1.4}]
Let $u$ be a solution of (\ref{P-ob:eq:1.1}) and $v$ be an $f-$super\\harmonic
function, greater than $\psi$.
Let $\xi=\max(u,v), \ \xi \in K(\psi)$.
Recalling that $v^-=\sup(0, -v)$, one has then $(\xi-u)=-(v-u)^-$.
From (\ref{P-ob:eq:1.1}), one gets
$$\int_\Omega \sigma(u) \cdot \nabla (\xi-u) \geq \int_\Omega f
(\xi-u).$$
On the other hand, since $\xi-u \leq 0$ and $-\Delta_p v \geq f$, we have
$$\int_\Omega \sigma(v) \cdot \nabla (\xi-u) \leq \int_\Omega f (\xi-u).$$
We obtain, subtracting the above two inequalities:
$$\int_\Omega \left[\sigma(v)-\sigma(u)\right] \cdot \nabla (\xi-u) \leq 0,$$
which implies that
$$-\int_\Omega \left[\sigma(v)-\sigma(u)\right] \cdot \nabla (v-u)^- \leq 0,$$
and then $(v-u)^-=0$, or equivalently $u\leq v$ in $\Omega$.
\end{proof}
Recall that we define by $T_f(\psi)$ the lowest
$f-$superharmonic function, greater than $\psi$.
\begin{lemma}\label{P-ob:lemma:1.1}
The mapping $\psi \mapsto T_f(\psi)$ is increasing.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{P-ob:lemma:1.1}]
Let $u_1=T_f(\psi_1)$ and $u_2=T_f(\psi_2)$,
which are respectively solutions to the following variational inequalities
$$\begin{cases}
-\Delta_p u_i \geq f\\
u_i \geq \psi_i, \ i=1,2
\end{cases}$$
and let $\psi_1 \leq \psi_2$. It is clear that $u_2 \geq \psi_1$.
Hence $u_2$ is $f-$superharmonic and using Proposition \ref{P-ob:prop:1.4},
one obtains $u_1\leq u_2$.
\end{proof}
\begin{proposition}\label{P-ob:prop:1.5}
The mapping $\psi \mapsto T_f(\psi)$ is weak lower semicontinuous,
in the sense that:
\begin{itemize}
\item If $\psi_k \rightharpoonup \psi$ weakly in
$W^{1,p}_0(\Omega)$, then
$T_f(\psi) \leq {\displaystyle \liminf_{k \rightarrow \infty}} T_f(\psi_k)$.
\item ${\displaystyle \int_\Omega} |\nabla(T_f(\psi))|^p \leq {\displaystyle \liminf_{k \rightarrow \infty}} {\displaystyle \int_\Omega} |\nabla(T_f(\psi_k))|^p$.
\end{itemize}
\end{proposition}
\begin{proof}[Proof of Proposition \ref{P-ob:prop:1.5}]
Let $(\psi_k)$ be a sequence in $W^{1,p}_0(\Omega)$ which converges weakly
in $W^{1,p}_0(\Omega)$ to $\psi$,
and let $\phi_k= {\rm{min}}(\psi_{k},\psi)$.
Since $T_f$ is increasing, one gets that $T_f(\phi_k) \leq T_f(\psi_k)$.
We now prove that $T_f(\phi_k)$ converges strongly in $W^{1,p}_0(\Omega)$
towards $T_f(\psi)$. This will imply that
$$T_f(\psi)= \lim _{k \rightarrow \infty}T_f(\phi_k) \leq
\liminf_{k \rightarrow \infty} T_f(\psi_{k}).$$
We denote $u_k$ as $T_f(\phi_k)$. It is clear that $u_k$ is
bounded in $W^{1,p}_0(\Omega)$ since $\phi_k \leq \psi$.
Hence for a subsequence, still denoted
$u_k$, there exists some $u$ in $W^{1,p}_0(\Omega)$ such that
\begin{equation}\label{P-ob:eq:1.4}
\nabla u_k \rightharpoonup \nabla u \ {\rm{weakly \ in}} \
L^p(\Omega) , \ u_k \rightarrow u \ {\rm{strongly \ in}} \ L^p(\Omega).
\end{equation}
On the other hand, using the fact that $\phi_k$
converges weakly to $\psi$ in $W^{1,p}_0(\Omega)$ (see
Lemma \ref{P-ob:lemma:1.2} below),
one gets the following assertion:
$$u_k \geq \phi_k \Longrightarrow u \geq \psi.$$
Let us prove now that $u$ is a solution of the minimizing problem (\ref{P-ob:eq:1.3}).
For that aim, for $v\in K(\psi)$, since $v \geq \psi \geq \phi_k$, we have
\begin{align*}
\frac{1}{p} \int_\Omega |\nabla u|^p -\int_\Omega f u
&\leq \liminf_{k \rightarrow \infty}\frac{1}{p} \int_\Omega
|\nabla u_k|^p -\int_\Omega f u_k\\
&\leq \liminf_{k \rightarrow \infty}
\inf_{w \geq \phi_k}\left\{\frac{1}{p} \int_\Omega
|\nabla w|^p -\int_\Omega f w\right\}\\
&\leq \frac{1}{p} \int_\Omega |\nabla v|^p -\int_\Omega f v.
\end{align*}
Then $u$ realizes the infimum in (\ref{P-ob:eq:1.3}).
At the same time, since $u \in K(\psi)$,
one has the following convergence
$$\frac{1}{p} \int_\Omega |\nabla u_k|^p -\int_\Omega f u_k \longrightarrow
\frac{1}{p} \int_\Omega |\nabla u|^p -\int_\Omega f u,
\ {\rm{when}} \ k \rightarrow \infty,$$
which implies that $u_k$ converges strongly to $u$ in $
W^{1,p}_0(\Omega)$. We can conclude that $T_f(\phi_k)$ converges
strongly to $T_f(\psi)$.
\end{proof}
\begin{lemma}\label{P-ob:lemma:1.2}
Suppose that $\psi_k$ converges weakly to some $\psi$ in
$W^{1,p}_0(\Omega)$. Then, $\phi_k=\min(\psi_k,\psi)$
converges weakly to $\psi$ in $W^{1,p}_0(\Omega)$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{P-ob:lemma:1.2}]
We have
$$\psi_k \longrightarrow \psi \quad \text{in} \ L^p(\Omega).$$
Then
$$\phi_k=\frac{\psi_k+\psi-|\psi_k-\psi|}{2} \longrightarrow \psi \quad \text{in} \ L^p(\Omega).$$
Let us prove now that $|\nabla \phi_k|$ is bounded in
$L^p(\Omega)$. For that aim, we write
\begin{align*}
\int_\Omega |\nabla \phi_k|^p
&=\int_\Omega \left|\nabla
\left(\frac{\psi_k+\psi-|\psi_k-\psi|}{2}\right)\right|^p\\
&\leq C_p \left(\int_\Omega |\nabla \psi_k|^p +\int_\Omega |\nabla \psi|^p\right).
\end{align*}
Therefore the sequence $\phi_k$ is bounded in $W^{1,p}_0(\Omega)$, so it converges weakly, up to a subsequence, to $\psi$ in $W^{1,p}_0(\Omega)$.
\end{proof}
\begin{proposition}\label{P-ob:prop:1.6}
The mapping $T_f$ is an involution, i.e. $T_f^2=T_f$.
\end{proposition}
\begin{proof}[Proof of Proposition \ref{P-ob:prop:1.6}]
Up to replacing $\psi$ by $u$ in the variational inequalities
(\ref{P-ob:eq:1.1}), and using proposition
\ref{P-ob:prop:1.4}, one gets that $u=T_f(u)$.
Then, we conclude that $ T_f^2(\psi)=T_f(\psi)$.
\end{proof}
\section{A method of penalization}
Let $\mathcal M^+(\Omega)$ be the set of all nonnegative Radon
measures on $\Omega$ and $W^{-1,p^\prime}(\Omega)$ be the dual space of
$W^{1,p}(\Omega)$ on $\Omega$ where $p^\prime$ is the conjugate of $p \
(1<p<\infty)$.
Suppose that $u$ solves (\ref{P-ob:eq:1.1}). Using the fact that a nonnegative
distribution on $\Omega$ is a nonnegative measure on $\Omega$ (cf. \cite{DF9}), one gets the existence of
$\mu \geq 0, \ \mu \in \mathcal M^+(\Omega)$, such that
\begin{equation}\label{P-ob:eq:1.5}
\int_\Omega \sigma(u) \cdot \nabla \Phi \ dx - \int_\Omega f \Phi \ dx=
\langle \mu,\Phi \rangle, \quad \forall \ \Phi \in \mathcal D(\Omega),
\end{equation}
that we shall also write
$-\Delta_p u =f+\mu, \quad \mu\geq0 \ \text{in} \ \Omega.$
Let us introduce
\begin{equation}\label{P-ob:eq:1.6}
\beta(x)=
\begin{cases}
0, \quad x >0,\\
x, \quad x\leq 0.
\end{cases}
\end{equation}
Clearly, $\beta$ is $\mathcal C^1$ piecewise, $\beta(x) \leq 0$ and is nondecreasing.
Let us consider, for some $\delta >0$, the following semilinear elliptic equation:
\begin{equation}\label{P-ob:eq:1.7}
\begin{cases}
-\Delta_p u+ \frac{1}{\delta} \beta(u-\psi)=f, \quad
\text{in} \ \Omega\\
u_{|{\partial \Omega}}=0.
\end{cases}
\end{equation}
We have the following existence result:
\begin{theorem}\label{P-ob:thm:2.1}
For any given $\psi \in W_0^{1,p}(\Omega)$ and $\delta>0$, {\rm(\ref{P-ob:eq:1.7})} possesses a unique
solution $u^\delta$. Moreover,
\begin{itemize}
\item[{\rm(1)}]$u^\delta \longrightarrow u \ {\rm{strongly \ in}} \ {W_0^{1,p}(\Omega)}$, as $\delta \longrightarrow 0$, with $u:=T_f(\psi)$.
\item[{\rm(2)}] There exists a unique $\mu \in W^{-1,p^\prime}(\Omega) \cap \mathcal M^+(\Omega)$ such that:
\begin{itemize}
\item[{\rm(i)}]$- \frac{1}{\delta} \beta(u^\delta-\psi) \rightharpoonup \mu \ {\rm{in}} \ W^{-1,p^\prime}(\Omega) \cap \mathcal M^+(\Omega).$
\item[{\rm(ii)}]$\langle \mu, T_f(\psi)-\psi \rangle=0$.
\end{itemize}
\end{itemize}
\end{theorem}
\begin{proof}[Proof of Theorem \ref{P-ob:thm:2.1}]
(1) Let $B$ be defined as $B(r)={\displaystyle \int_0^r} \beta(s) ds, \ \forall \ r\in \mathbb{R}$.
We introduce the following variational problem
\begin{equation}\label{P-ob:eq:1.8}
\inf_{v \in W_0^{1,p}(\Omega)} \left\{\frac{1}{p} \int_\Omega |\nabla v|^p +
\frac{1}{\delta} \int_\Omega B(v-\psi)-\int_\Omega f v \right\}.
\end{equation}
The functional in (\ref{P-ob:eq:1.8}) is coercive, strictly convex and continuous.
As a consequence it possesses a unique solution
$u^\delta \in W_0^{1,p}(\Omega)$.
Since $B(0)=0$, one has
$$\frac{1}{p} \int_\Omega |\nabla u^\delta|^p+ \frac{1}{\delta}
\int_\Omega B(u^\delta-\psi) - \int_\Omega f u^\delta
\leq \frac{1}{p} \int_\Omega |\nabla \psi|^p - \int_\Omega f \psi,$$
since $B\geq 0$, then $u^\delta$ is bounded in $W_0^{1,p}(\Omega)$.
Extracting from $u^\delta$ a subsequence,
there exists $u$ in $W_0^{1,p}(\Omega)$, such that
$$\nabla u^\delta \rightharpoonup \nabla u \ {\rm{weakly \ in}} \
L^p(\Omega), \ u^\delta \rightarrow u \ {\rm{strongly \ in}} \ L^p (\Omega).$$
Using $\frac{1}{\delta} {\displaystyle \int_\Omega} B(u^\delta-\psi) \leq C$
and the continuity of $B$ one has
$$0\leq \int_\Omega B(u-\psi) \leq \liminf_{\delta \rightarrow 0}
\int_\Omega B(u^\delta-\psi)=0,$$
hence $u \in K(\psi)$.
We want to prove now that $u$ solves (\ref{P-ob:eq:1.1}).
Let $v \in K(\psi)$, since $B(r)\geq0, \ \forall \ r \in \mathbb{R}$ one gets:
\begin{align*}
\frac{1}{p} \int_\Omega |\nabla u|^p -\int_\Omega f u
&\leq {\displaystyle \liminf_{\delta \rightarrow 0}}
\left(\frac{1}{p} \int_\Omega |\nabla u^\delta|^p- \int_\Omega f u^\delta\right)\\
&\leq {\displaystyle \liminf_{\delta \rightarrow 0}}
\left(\frac{1}{p} \int_\Omega |\nabla u^\delta|^p+ \frac{1}{\delta}
\int_\Omega B(u^\delta-\psi) - \int_\Omega f u^\delta\right)\\
&\leq {\displaystyle \liminf_{\delta \rightarrow 0}} \inf_{u \geq \psi}
\left\{\frac{1}{p} \int_\Omega |\nabla u|^p+ \frac{1}{\delta}
\int_\Omega B(u-\psi) - \int_\Omega f u \right\}\\
&\leq \frac{1}{p} \int_\Omega |\nabla v|^p -\int_\Omega f v.
\end{align*}
Then, one concludes that
$ \nabla u^\delta \longrightarrow \nabla u \ {\rm{strongly \ in}} \ L^p(\Omega)$
and since $u \in K(\psi)$, then $u$ solves (\ref{P-ob:eq:1.1}).
(2)
(i) let $u^\delta$ be the solution of
(\ref{P-ob:eq:1.7}), since $\nabla u^\delta$
is uniformly bounded in $L^p(\Omega)$ by some constant $C$, we get that
$-\Delta_p u^\delta -f$ is bounded in $W^{-1,p^\prime}(\Omega)$, so it converges weakly, up to a subsequence, in $W^{-1,p^\prime}(\Omega)$.
Hence, $-\frac{1}{\delta} \beta(u^\delta-\psi)$ converges too, up to a subsequence, in $W^{-1,p^\prime}(\Omega)$, and we have
$$-\frac{1}{\delta} \beta(u^\delta-\psi) \rightharpoonup \mu \
{\rm{weakly \ in}} \ W^{-1,p^\prime}(\Omega),$$
where $\mu$ is a positive distribution, hence a positive measure.
Then, by (1), we see that $u$ and $\mu$ are linked by the relation (\ref{P-ob:eq:1.5}).
We now prove
(ii): let $u$ be the solution of (\ref{P-ob:eq:1.1}).
Taking $\phi=(\psi-u) \in W^{1,p}_0(\Omega) $ in the above inequalities,
one gets
$$- \frac{1}{\delta} \int_\Omega \beta(u^\delta-\psi) \ (u-\psi) \ dx
\leq \|\nabla u^\delta\|^{p-1}_p \|\nabla (\psi-u)\|_p
+\|f\|_{p^\prime} \| \psi-u\|_p .$$
Since $u \in K(\psi)$, passing to the limit we obtain:
$$\langle \mu, \psi-u \rangle = \int_\Omega |\nabla u|^{p-2} \nabla u
\cdot\nabla (\psi-u)- \int_\Omega f (\psi-u) = 0,\ \text{by} \ (\ref{P-ob:eq:1.2})$$
Then (ii) follows.
\end{proof}
\section{Optimal Control for a Non-Positive Source Term}
\subsection{Optimal control for a non positive source term}
\begin{proposition}\label{POC:prop:1.1}
Let $f, \psi$ and $T_f(\psi)$ be as in {\rm{(\ref{P-ob:eq:1.1})}}. One has
$$\frac{1}{p}\int_\Omega |\nabla T_f(\psi)|^p \leq \frac{1}{p}\int_\Omega |\nabla \psi|^p \ dx+\int_\Omega f [T_f(\psi)- \psi] \ dx.$$
\end{proposition}
\begin{proof}[Proof of Proposition \ref{POC:prop:1.1}]
From {\rm{(\ref{P-ob:eq:1.1})}} taking $v=\psi$ and using H\"older's inequality, we have
$$\int_\Omega |\nabla T_f(\psi)|^p \leq
\frac{p-1}{p} \|\nabla T_f(\psi)\|^p_p+
\frac{1}{p}\|\nabla \psi\|^p_p +\int_\Omega f [T_f(\psi)- \psi] \ dx.$$
\end{proof}
Note that since $T_f(\psi)\geq \psi$, it follows that if $f\leq 0$, then
\begin{equation}\label{POC:eq:1.1}
\int_\Omega |\nabla T_f(\psi)|^p \ dx \leq \int_\Omega |\nabla \psi|^p \ dx.
\end{equation}
Let us now introduce the following problem, said ``{\it{optimal control problem}}'':
\begin{equation}\label{POC:eq:1.2}
\inf_{\widetilde \psi \in W_0^{1,p}(\Omega)} J_f(\widetilde \psi),
\end{equation}
where
\begin{equation}\label{POC:eq:1.3}
J_f(\widetilde \psi)=\frac{1}{p}\int_\Omega \left\{|T_f(\widetilde \psi)- z|^p+|\nabla \widetilde \psi|^p \right\} \ dx,
\end{equation}
for some given $z \in L^p(\Omega)$. $z$ is said to be {\it the
initial profile}, $\psi$ is {\it the control variable} and $T_f(\psi)$ is {\it the state variable}. The pair $(\psi^*,T_f(\psi^*))$ where $\psi^*$ is a solution
for (\ref{POC:eq:1.2}) is called an optimal pair and $\psi^*$ an optimal control.
In this section, we establish the existence and uniqueness of
the optimal pair in the case where $f\leq 0$.
\begin{theorem}\label{POC:thm:1.1}
If $f \in L^{p'}(\Omega), \ f \leq 0$ on $\Omega$, then there exists
a unique optimal control $\psi^*\in W_0^{1,p}(\Omega)$ for
{\rm{(\ref{POC:eq:1.2})}}.
Moreover, the corresponding state $u^*$ coincides with $\psi^*$,
i.e. $T_f(\psi^*)=\psi^*$.
\end{theorem}
\begin{proof}[Proof of Theorem \ref{POC:thm:1.1}]
In a first time we prove that there exists a pair of solutions of the form
$(u^*, u^*)$, hence $(u^*=T_f(u^*))$.
Let $(\psi_k)_k$ be a minimizing sequence for (\ref{POC:eq:1.3}), then $T_f(\psi_k)$ is bounded
in $W^{1, p}(\Omega)$, therefore $T_f(\psi_k)$ converges for a subsequence towards some $u^* \in W_0^{1,p}(\Omega)$.
Moreover, using the lower semicontinuity of $T_f$ as in proposition \ref{P-ob:prop:1.5}, one gets
$$T_f(u^*) \leq \liminf_{k \rightarrow \infty}T_f(T_f(\psi_k))
\leq \lim_{k \rightarrow \infty}T_f(\psi_k)=u^*,$$
and by the definition of $T_f, \ T_f(u^*)\geq u^*$.
Hence $u^*=T_f(u^*)$.
We prove that $(u^*,u^*)$ is an optimal pair.
Using proposition \ref{P-ob:prop:1.5}, by the lower semicontinuity in $W_0^{1,p}(\Omega)$
of $T_f$:
\begin{align*}
J_f(u^*)
&=\frac{1}{p}\int_\Omega \left\{|u^*- z|^p+|\nabla u^*|^p \right\} \ dx\\
&\leq \liminf_{k \rightarrow \infty}
\frac{1}{p}\int_\Omega \left\{|T_f(\psi_k)- z)|^p+|\nabla \psi_k|^p \right\} \ dx\\
&=\inf_{\psi \in W_0^{1,p}(\Omega)} J_f(\psi).
\end{align*}
Secondly, we prove that every optimal pair is of the form $(u^*, u^*)$.
Observe that if $(\psi^*, T_f(\psi^*))$ is a solution then
$(T_f(\psi^*), T_f(\psi^*))$ is a solution.
Indeed
$$\int_\Omega \left\{|T_f(\psi^*)- z)|^p+|\nabla T_f(\psi^*)|^p \right\} \ dx
\leq \int_\Omega \left\{|T_f(\psi^*)- z)|^p+|\nabla \psi^*|^p \right\} \ dx.$$
So
\begin{equation}\label{POC:eq:1.4}
\int_\Omega |\nabla T_f(\psi^*)|^p \ dx= \int_\Omega |\nabla \psi^*|^p \ dx,
\end{equation}
by inequality (\ref{POC:eq:1.1}),
using the H\"older's inequality, one obtains then
\begin{align*}
0
&\leq \int_\Omega f(\psi^*-T_f(\psi^*)) \ dx \\
&\leq \int_\Omega \sigma(T_f(\psi^*)) \cdot \nabla(\psi^*-T_f(\psi^*)) \ dx\\
&\leq \int_\Omega |\nabla T_f(\psi^*)|^{p-2}\nabla T_f(\psi^*)\cdot \nabla\psi^*-
\int_\Omega |\nabla T_f(\psi^*)|^p\\
&\leq \left(\int_\Omega |\nabla T_f(\psi^*)|^p\right)^{\frac{p-1}{p}}
\left(\int_\Omega |\nabla T_f(\psi^*)|^p\right)^{\frac{1}{p}}-
\int_\Omega |\nabla T_f(\psi^*)|^p=0,
\end{align*}
which implies
$$\int_\Omega |\nabla T_f(\psi^*)|^{p-2}\nabla T_f(\psi^*) \cdot \nabla\psi^*
-\int_\Omega |\nabla T_f(\psi^*)|^p=0.$$
Let us recall that by convexity, one has the following inequality
$$\frac{1}{p}\int_\Omega |\nabla \psi^*|^p+\frac{p-1}{p} \int_\Omega
|\nabla T_f(\psi^*)|^p- \int_\Omega |\nabla T_f(\psi^*)|^{p-2}\nabla
T_f(\psi^*) \cdot \nabla\psi^* \geq 0.$$
Then the equality holds and by the strict convexity, one gets $\nabla (\psi^*)=\nabla
(T_f(\psi^*)) \ \text{a.e.}$, hence $\psi^*=T_f(\psi^*)$.
Finally, we deduce from the two previous steps that the pair is unique.
Suppose that $(u_1, u_1)$ and $(u_2, u_2)$ are two solutions, and consider
$(\frac{u_1+u_2}{2}, T_f(\frac{u_1+u_2}{2}))$. We prove that it is
also a solution.
Indeed:
\begin{align*}
\int_\Omega \left|\frac{u_1+u_2}{2}- z\right|^p
&+\left|\nabla T_f(\frac{u_1+u_2}{2})\right|^p \ dx\\
&\leq \int_\Omega \left|\frac{u_1+u_2}{2}- z\right|^p+\left|\nabla \left(\frac{u_1+u_2}{2}\right)\right|^p \ dx\\
&\leq \frac{1}{2} \left(J_f(u_1)+ J_f(u_2)\right)=
\inf_{\psi \in W_0^{1,p}(\Omega)} J_f(\psi),
\end{align*}
which implies that $u_1=u_2$. Thus, the uniqueness of the optimal pair
for $f\leq0$ holds.
\end{proof}
\subsection{Optimal control for a nonnegative source term}
We are interested here to the case $f\geq 0$ on $\Omega$.
In what follows we will denote by $Gf$ the unique function in $W^{1, p}_0(\Omega)$ which verifies
$$\begin{cases}
-\Delta_p (Gf)=f,\ \text{in} \ \Omega \ \text{a.e.}\\
Gf=0, \ \text{on} \ \partial \Omega,
\end{cases}$$
where $f \in L^{p'}(\Omega)$ and $Gf \in W^{1,p}_0(\Omega)$.
\begin{theorem}\label{POC:thm:1.2}
Suppose that $f\in L^{p^\prime}(\Omega)$ is a nonnegative function. Suppose that $z \in L^p(\Omega)$, satisfying $z\leq Gf$ a.e on $\Omega$. Then the minimizing problem {\rm{(\ref{POC:eq:1.2})}}
has a unique optimal pair $(0, Gf)$.
\end{theorem}
\begin{lemma}\label{POC:lemma:1.1}
Let $T_f(\psi)$ be a solution to {\rm{(\ref{P-ob:eq:1.1})}} and $Gf$ defined as above.
Then $T_f(\psi)$ is greater than $Gf$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{POC:lemma:1.1}]
We have that $-\Delta_p(Gf)=f$, and $T_f(\psi)$ realizes $-\Delta_p(T_f(\psi))\geq f$.
Then, by the Comparison Theorem for $-\Delta_p$ we get that $Gf\leq T_f(\psi).$
\end{proof}
\begin{proof}[Proof of Theorem \ref{POC:thm:1.2}]
In a first time we prove that $(0,Gf)$ is an optimal pair. Indeed, for all $\psi \in W^{1,p}_0(\Omega)$
\begin{align*}
J_f(\psi)
&=\frac{1}{p} \int_\Omega \left\{|Gf-z+T_f(\psi)-Gf|^p+|\nabla \psi|^p\right\}\\
&\geq \frac{1}{p} \int_\Omega \left\{|Gf- z|^p +p|Gf- z|^{p-2}(Gf- z)(T_f(\psi)- Gf)\right\}\\
&\geq \frac{1}{p} \int_\Omega \left\{|Gf- z|^p\right\}\\
&=J_f(0).
\end{align*}
The equality with $(\psi^*, T_f(\psi^*))$ implies that we have equality in each step, so we get $\|\nabla \psi^*\|_p=0$, then $\psi^*=0 \ \text{a.e. \ in} \ \Omega$.
Thus, $(0,Gf)$ is the unique optimal control pair.
\end{proof}
\addcontentsline{toc}{section}{Bibliographie}
\end{document} |
\begin{document}
\title{Relation between discrete and continuous
teleportation using linear elements}
\author{Dirk Witthaut and Michael Fleischhauer}
\affiliation{Fachbereich Physik,
Universit\"{a}t Kaiserslautern, D-67663 Kaiserslautern,
Germany}
{\rm d}ate{\today}
\begin{abstract}
We discuss the relation between discrete and continuous linear teleportation,
i.e. teleportation schemes that use only linear optical elements and
photodetectors. For this the existing
qubit protocols are generalized to qudits with a discrete and
finite spectrum but with an arbitrary number of states or alternatively
to continuous variables. Correspondingly a generalization of
linear optical operations
and detection is made. It is shown that linear teleportation is
only possible in a probabilistic sense.
A general expression for the
success probability is derived which is shown to depend only on the dimensions of
the input and ancilla Hilbert spaces. From this the known results $P=1/2$ and
$P=1$ for the discrete and continuous cases can be recovered.
We also discuss the probabilistic teleportation scheme of Knill, Laflame and
Milburn and argue that it does not make optimum use of resources.
\end{abstract}
\pacs{03.67.Hk, 42.50.Dv, 03.67.Lx, 03.67.-a}
\maketitle
\section{Introduction}
Recently Knill, Laflamme and Milburn (KLM) \cite{Knill-Nature-2001} proposed a scheme
for linear optical quantum computing using photons as qubits.
This scheme works efficient if one can implement deterministic
discrete quantum teleportation.
The original proposal for discrete teleportation by
Bennett {\it et.al.} \cite{Bennett-PRL-1993} involves the projection
on two-particle Bell states which, as shown in
\cite{Lutkenhaus-PRA-1999,Calsamiglia-PRA-2002}, cannot
be implemented with linear elements in a deterministic way, unless
entanglement in additional degrees of freedom is used
\cite{Cerf-PRA-1998,Kwiat-PRA-1998}.
{\it Discrete} teleportation with linear elements
is only possible using post selection
with a success probability of 50 \% e.g. using the scheme suggested by
Weinfurter \cite{Weinfurter-EPL-1994}.
On the other hand in the case of {\it continuous} variables,
perfect teleportation with linear elements can be achieved
as shown by Braunstein and Kimble \cite{Braunstein-PRL-1998}.
This raises the
question about the origin of this manifestly different behavior.
Vaidman and Yoran suggested that the beam-splitter used in the
Braunstein-Kimble experiment would lead to an effective ``quantum-quantum''
interaction \cite{Vaidman-PRA-1999,Vaidman-PRA-1994}. The proposal of KLM hints
in a different direction: The success probability
of the KLM teleportation protocol tends toward 100 \% when an increasing
number of additional ancilla photons is used \cite{Knill-Nature-2001},
which indicates that the difference could be connected to the
resources used.
We here discuss a general relation between discrete and continuous
teleportation using linear elements. We show that the
success probability of a generalized linear protocol
is only limited by the
dimensions of the input and ancilla spaces. It is argued that the
KLM scheme is inefficient from this point of view as it does not
reach this limit. Recently Franson et.al. \cite{Franson-PRL-2002} suggested a
modification of the KLM
scheme with a better scaling of the fidelity with the number
of ancilla photons at the expense that the
input state is never reconstructed exactly. This scheme also does not reach
the limit obtained in the present paper.
Guided by continuous variable teleportation protocols
\cite{Braunstein-PRL-1998,Vaidman-PRA-1994} we
generalize teleportation from qubits to qudits with
a bounded discrete spectrum of states.
Also the notion of linear elements is generalized
reducing however to photodetection and / or homodyne
detection in the two limits.
We will restrict the discussion first to schemes
that faithfully reproduce the input state, i.e. in which Alice and Bob
can decide whether the teleportation was successful.
At the end we will also discuss the connection to teleportation schemes of the
Franson-type with non-unity fidelity but higher success probability.
\section{A generalized linear teleportation protocol}
Let us briefly recall the general properties of a teleportation protocol:
Alice possesses an unknown quantum state $\left| \Psi {\rm i}ght\rangle$
in a Hilbert space of dimension $d$ (qubits for $d=2$
or qudits in the general case),
which she wants to transfer to Bob by only applying local operations including
measurements and exchanging classical information.
In the following we will restrict ourselves to discrete generalizations
of a qubit, note however that also continuous generalizations can be
discussed in a similar way.
For the teleportation Alice and Bob make use
of an ancilla system with state
$\left| \Phi {\rm i}ght\rangle$, which is
an entangled pair of qudits which they share.
Alice then performs an appropriate measurement on her qudit of
the ancilla and the unknown quantum state. This measurement projects Bob's
ancilla qudit to a certain state which is up to local operations identical to
the unknown one. Alice's measurement must be such that the knowledge about its
outcome, transmitted via classical channels, is
sufficient for Bob to reconstruct the initial state $|\Psi\rangle$
by local operations.
If Alice is able to perform all possible
measurements in the joint Hilbert space of her qudit of the ancilla state
and the unknown state,
unconditional teleportation is always possible. Unfortunately
this is in general a very difficult task. For example in order to
measure the four Bell states in the Hilbert space of two qubits, as required
in the teleportation protocol of Bennett {\it et.al.}, quantum gate operations
must be performed. As shown in \cite{Lutkenhaus-PRA-1999,Calsamiglia-PRA-2002}
for the case of photonic qubits, only
two out of the four Bell states can be distinguished without gate
operations by linear optical elements and photodetection.
On the other hand in the case of continuous teleportation linear elements
suffice. Until now the limitations of teleportation with linear elements are
not completely understood. Since linear elements are easy to implement such an
understanding is very important for the practical realization
of quantum information processing.
We thus restrict the discussion
in the following to teleportation protocols that use only
certain generalizations of linear optical
operations and photodetection.
Let us consider an unknown input state decomposed
into a set of orthonormal basis
states $\left| q {\rm i}ght\rangle$, which are (non-degenerate) eigenvectors
of an observable $\hat q$
\be
\hat q \, |q\rangle =q\, |q\rangle.
\ee
The real numbers $q$ are the eigenvalues
which are assumed to lie in the symmetric interval $\{-a,a\}$ with integer
steps. Their total number is $2a+1$.
Thus Alice's state is given by
\be
|\Psi\rangle &=& \sum_{q_1 = -a}^{a} \alpha_{q_1} \left| q_1 {\rm i}ght\rangle_A.
\ee
The ancilla state shared by Alice and Bob has the general form:
\be
|\Phi\rangle &=&
\sum_{q_2,q_3 = -b}^b \beta_{q_2q_3} \left| q_2 {\rm i}ght\rangle_A \left|
q_3 {\rm i}ght\rangle_B,
\ee
where we have assumed for the sake of simplicity that $q_2$ and $q_3$
have the same symmetric interval of allowed values $\{-b,b\}$, again with
integer steps.
In order to teleport an unknown state from a Hilbert space of dimension
$2a+1$, there must be at least the same number of
pairs of bi-orthogonal ancilla
states, i.e.
the Schmidt number of the ancilla state must be larger or equal
to $2a+1$, implying $b \ge a$.
Without loss of generality we here require that the
coefficients $\beta$ are such that
only states that fulfill the relation $q_3=-q_2$ have a non-vanishing
amplitude, and that the Schmidt number is $b$.
This choice is quite general since any state in which
there is a unique relation between
the eigenvalues $q_2$ and $q_3$ can be brought into this form by reordering.
One further needs that
\[
\abs{\beta_{q_2}} = \textrm{const.} \quad \forall q_2.
\]
This condition can be deduced
in a general way for tight teleportation schemes \cite{Werner-JPA-2001},
but will also become clear in the course of the present discussion.
For
simplicity we choose all $\beta_{q_2}$ to be equal, i.e.
\be
|\Phi\rangle &=& \frac{1}{\sqrt{2b+1}} \sum_{q_2= -b}^b
\left| q_2 {\rm i}ght\rangle_A \left| -q_2 {\rm i}ght\rangle_B.
\ee
The above scheme includes in particular the input and ancilla states of the
Bennett protocol
as well as the KLM protocol. In the latter case the
state $\left| q_i {\rm i}ght\rangle$
is a quantum state of $n$ modes occupied by $q_i$ photons.
For the teleportation Alice needs to measure the total state
$|\Gamma_0\rangle= |\Phi\rangle \otimes |\Psi\rangle$
in a way that the input state is restored in Bob's qudit
up to a local operation which is uniquely defined by the
outcome of the measurement. As discussed above, we will
not allow for all
possible measurements but restrict ourselves to generalized
{\it linear} measurements, which will be defined in the following.
In an optical realization the
$\hat q_i$ correspond to either photon numbers
or quadrature amplitudes of an electromagnetic field
which can be measured by direct photon counting
or homodyne detection. A measurement scheme that uses
only linear optical elements like beamsplitters etc. can only
project onto eigenstates of linear combinations of photon number
or quadrature amplitude operators.
We thus call a generalized linear measurement a
projection onto eigenstates of any linear combination
of operators $\hat x_1$ and $\hat x_2$, i.e. of operators
$\hat X$ in ${\cal H}_1\oplus {\cal H}_2$.
Without loss of generality we consider here a
projection onto eigenstates of
\be
\hat Q_+=\hat q_1 +\hat q_2,
\ee
which corresponds to the setup used in continuous teleportation.
Measuring $\hat Q_+$ can lead to only $2a+2b+1$ different
outcomes. The total Hilbert space ${\cal H}_1\otimes {\cal H}_2$
is however of dimension $(2a+1)\times(2b+1)$ and thus the spectrum
of eigenstates of $\hat Q_+$ must be degenerate. In order to
project onto a non-degenerate state a second measurement is required.
In the continuous teleportation protocol of Braunstein and Kimble
\cite{Braunstein-PRL-1998}, where $\hat q_{1,2}$ are equivalent to
position operators, the difference between the two momenta $\hat P_-=\hat p_1
-\hat p_2$ is measured in addition to $\hat Q_+$. \\
We now define an analogue measurement in the discrete case. To this end we
construct another orthonormal basis $| p \rangle$ with the property
\be
\abs{\langle p | q \rangle} = \text{const.} \quad \forall \, p,q.
\ee
We define the states
\begin{eqnarray}
| p \rangle &:=& \frac{1}{\sqrt{2b+1}} \sum_{q = -b}^b \exp\left\{
{\frac{2\pi {\rm i}
qp}{2b+1}}{\rm i}ght\}\, | q \rangle \label{eqn-transformation_pq}
\end{eqnarray}
with $p$ running in integer steps from $-b$ to $b$. Furthermore we assume that
the quantum number $p$ can be detected by measuring the observable
\be
\hat p = \sum_{p = -b}^b p |p\rangle\langle p |.
\ee
Again we allow only measurements of linear combinations of
the operators $\hat p_1$ and $\hat p_2$, i.e.
linear measurements.
In particular Alice could measure
in addition to $\hat Q_+$ the quantity
\be
\hat P_- = \hat p_1 - \hat p_2,
\ee
in close analogy to continuous teleportation.
The measurement operators $\hat Q_+$ and $\hat P_-$ do not commute, except
in a limiting case, but
this does not matter as we will see. For simplicity
of the discussion we
consider a measurement outcome with $Q \ge 0$ and $P \ge 0$. Then the
measurement outcome is described by the projection operators:
\begin{eqnarray}
\hat \Pi_Q &=& \sum_{q = Q-b}^b |Q-q,q \rangle \langle Q-q,q| \\
\hat \Pi_P &=& \sum_{p=-b}^{b-P} |P+p,p \rangle \langle P+p,p|.
\end{eqnarray}
The subsequent measurement of $\hat Q_+$ and $\hat P_-$ thus projects the initial
state $| \Gamma_0 \rangle$ onto
\be
| \Gamma_1 \rangle = \frac{\hat \Pi_P \hat \Pi_Q | \Gamma_0 \rangle}{
\| \hat \Pi_P \hat \Pi_Q | \Gamma_0 \rangle \| }.
\ee
To calculate this expression
it is convenient to express the projection
operators in one common basis, so we express $\hat \Pi_P$ in terms of
the states $| q \rangle$ via eqn. ({\rm e}f{eqn-transformation_pq}):
\be
\hat \Pi_P &=& \frac{1}{2b+1} \sum_{p = -b}^{b-P} \,\,
\sum_{q', \bar q' = -b}^b
\,\,\sum_{q'',\bar q'' = -b}^b |q',q''\rangle \langle \bar q',\bar q''| \nn \\
&& \qquad \times \exp\left\{ \frac{2\pi {\rm i}}{2b+1} \bigl(
q'(P+p) + q''p - \bar q'(P+p) - \bar q'' p\bigr) {\rm i}ght\}.
\ee
The total projection operator is thus given by
\begin{eqnarray}
\hat \Pi_P \hat \Pi_Q &=& \frac{1}{2b+1} \sum_{p = -b}^{b-P}
\,\, \sum_{q', \bar q' = -b}^b\,\,
\sum_{q'',\bar q'' = -b}^b\,\, \sum_{q = Q-b}^b
|q',q''\rangle \langle \bar q',\bar q'' | Q- q, q \rangle \langle Q-q,q | \nn \\
&& \qquad \times \exp\left\{ \frac{2\pi {\rm i}}{2b+1}\left(q'P+ q'p + q''p - \bar q'P
-\bar q' p - \bar q'' p{\rm i}ght) {\rm i}ght\} \nn \\
&=& \frac{1}{2b+1} \sum_{p = -b}^{b-P}\,\, \sum_{q', q'' = -b}^b
\,\, \sum_{q = Q-b}^b |q',q''\rangle \langle Q-q,q | \exp\left\{ \frac{2\pi {\rm i}}{2b+1}
\left(q'P + q'p + q''p - QP +qP -Qp {\rm i}ght) {\rm i}ght\}.
\end{eqnarray}
Applying this operator on the initial state $| \Gamma_0 \rangle$ finally yields
\begin{eqnarray}
\hat \Pi_P \hat \Pi_Q | \Gamma_0 \rangle &=& \frac{1}{(2b+1)^{3/2}}
\sum_{p = -b}^{b-P} \sum_{q', q'' = -b}^b \sum_{q = Q-b}^b
|q',q''\rangle_A \otimes | -q \rangle_B \nn \\
&& \qquad \times \alpha_{Q-q} \exp\left\{ \frac{2 \pi {\rm i}}{2b+1} \left(q'P
+ q'p + q''p - QP +qP -Qp {\rm i}ght) {\rm i}ght\}\\
&=& \Biggl[\frac{1}{(2b+1)^{3/2}} \sum_{p = -b}^{b-P} \sum_{q', q'' = -b}^b
\exp\left\{ \frac{2 \pi {\rm i}}{2b+1} \left(q'P
+ (q'+ q'')p - Q(P-p) {\rm i}ght) {\rm i}ght\} |q',q''\rangle_A\Biggr]
\otimes \bigl\vert\Gamma_2\bigr\rangle_B
\end{eqnarray}
where $\bigl\vert\Gamma_2\bigr\rangle_B$ is Bob`s
(unnormalized) state after the measurement of $\hat Q_+$
and $\hat P_-$
\be
| \Gamma_2 \rangle_B =\sum_{q = Q-b}^b \alpha_{Q-q}\,
\exp\left\{{\frac{2\pi {\rm i} qP}{2b+1}}{\rm i}ght\}\, | -q \rangle_B.
\label{eqn-Gamma_pre}
\ee
This represents the initial unknown state with
shifted amplitudes $\alpha_{Q-q} $ and some phase
factors ${\rm e}^{\frac{2\pi {\rm i} qP}{2b+1}}$. Both, the index shift $Q$
and the phase factor proportional to $P$ are
known to Alice after the joint measurement of $\hat Q_+$ and $\hat P_-$
and the corresponding information can be transmitted to Bob by
classical channels. Bob can then
apply appropriate shift and phase-rotation operations
to his quantum state to recover a replica of the input state.
We do see however that due to the index shift and the
finite dimension of the Hilbert space of Bob`s state, some
of the original state amplitudes $\alpha_q$ may be lost, depending on
the value of $Q$. Consequently the
teleportation has only a finite success probability.
\section{Probability of success}
We now discuss the probability of success of the described
teleportation protocol.
For this we generalize the above
discussion and drop the requirements $Q \ge 0$ and $P \ge 0$. Then eqn.
({\rm e}f{eqn-Gamma_pre}) can be rewritten as
\be
| \Gamma_2 \rangle \sim \sum_{q={\rm max}[-b,-b+Q]}^{{\rm min}[b,b+Q]}
\alpha_{q}\, \exp\left\{{\frac{2 \pi {\rm i}}{2b+1}(q+Q)P}{\rm i}ght\}
\, |q+Q\rangle. \label{Gamma-dis}
\ee
We see that only state amplitudes $\alpha_q$ survive for which
$q\in\{{\rm max}[-b,-b+Q],{\rm min}[b,b+Q]\}$.
Thus a sufficient condition for a successful teleportation of
an arbitrary qudit state is that the measured eigenvalue $Q$
fulfills
\be
|Q| \le b-a.
\label{condition_success}
\ee
If this inequality holds, the knowledge of $Q$ and $P$ is sufficient for
Bob to reproduce the initial state by
local operations.
We now want to calculate the success rate for the given
teleportation protocol, i.e.
the probability that condition
({\rm e}f{condition_success}) is fulfilled.
The probability to measure a specific eigenvalue $Q$ is
\begin{eqnarray}
p(Q) &=& \abs{\left\langle Q {\rm i}ght| \left. \Gamma {\rm i}ght\rangle}^2 \nn \\
&=& \frac{1}{2b+1} \sum_{q_1=-a}^a \sum_{q_2=-b}^b \abs{\alpha_{q_1}}^2
{\rm d}elta_{q_1+q_2,Q}
\end{eqnarray}
The probability of success of the teleportation protocol is therefore
\begin{equation}
P =\sum_{Q = -(b-a)}^{b-a}p(Q)=
\frac{1}{2b+1} \sum_{Q = -(b-a)}^{b-a} \sum_{q_1,q_2}
\abs{\alpha_{q_1}}^2 {\rm d}elta_{q_1+q_2,Q}.
\end{equation}
Independent of the value of $\alpha_{q_1}$ the summation over $q_2$ and $Q$
yields
\[
\sum_{Q=-(b-a)}^{b-a} \sum_{q_2 = -b}^b {\rm d}elta_{q_2,Q-q_1} = 2(b-a)+1.
\]
This is because $a-b \le Q \le
b-a$ and $-a \le q_1 \le a $ always imply $-b \le Q - q_1 \le b$ and
so one has $2(b-a)+1$ non-vanishing contributions regardless of $q_1$.
With this we eventually arrive at the success probability
\begin{eqnarray}
P &=& \frac{ 2(b-a)+1 }{2b+1} \sum_{q_1} \abs{\alpha_{q_1}}^2 \nonumber \\
&=& \frac{ 2(b-a)+1 }{2b+1}= 1 - \frac{2a}{2b+1},
\end{eqnarray}
which does not depend on the input state.
We can rewrite this result noting that the
dimension of the Hilbert space of the input state is
dim$\{{\cal H}_i\}=2a+1$ and the dimension of the
ancilla Hilbert space is dim$\{{\cal H}_a\}=2b+1$:
\be
P = 1 - \frac{{\rm dim}\{{\cal H}_i\}-1}{{\rm dim}\{{\cal H}_a\}}.
\label{eqn-success-dim1}
\ee
Eq.({\rm e}f{eqn-success-dim1}) is the main result of our paper. It shows that a
linear teleportation protocol is in general always probabilistic and that
its success probability only depends on the relative Hilbert dimensions
of the input and ancilla spaces. In the case of qubits, i.e. for
${\rm dim}\{{\cal H}_i\}={\rm dim}\{{\cal H}_a\}=2$ we obtain $P=1/2$
as realized
e.g. in the scheme of Weinfurter \cite{Weinfurter-EPL-1994}. Likewise
in the case of large Hilbert spaces with
${\rm dim}\{{\cal H}_a\}\gg{\rm dim}\{{\cal H}_i \}$ as in the
Braunstein-Kimble proposal \cite{Braunstein-PRL-1998} in the limit of
perfect squeezing, a unit success
probability can be achieved.
It should be noted
that an analogous calculation can be performed in the case of continuous variables
with a constant density of states but bounded spectrum.
In this case the input and the ancilla
states are given by
\be
|\Psi\rangle &=& \int_{-a}^a \!\! {\rm d} q_1 \alpha(q_1) |q_1 \rangle \\
|\Phi\rangle &=& \frac{1}{\sqrt{2b}} \int_{-b}^b \!\! {\rm d} q_2 \int_{-b}^b \!\!
| q_2 \rangle | -q_2 \rangle.
\ee
Then $\hat q$ and $\hat p$ are position and momentum operators or
quadrature amplitudes. One obtains the similar result
\begin{equation}
P_{\text{cont}} = 1 - \frac{a}{b}.
\end{equation}
The only difference in this formula is the missing of the term $+1$
which arises from the different treatment of the end
points.
\section{Efficiency of the KLM teleportation scheme}
At first sight the KLM \cite{Knill-Nature-2001} teleportation scheme seems
to implement the generalized discrete teleportation scheme discussed
in this paper where the quantum number $q$ is the total number of
photons in Alice's modes since its probability of success scales as
\be
P = 1 - \frac{1}{n+1}.
\ee
The entangled ancilla state consist of $n+1$ terms containing
$n$ photons each. This is exactly the result
one would obtain from eqn. ({\rm e}f{eqn-success-dim1})
if we insert dim$\{ {\cal H}_i\} = 2$ (qubits) and
dim$\{ {\cal H}_a \} = n+1$.
But this interpretation is not correct because the dimension of the
Hilbert space ${\cal H}_a$ of the $n$ ancilla photons is much larger.
Although the true Hilbert-space dimension of $n$ photons distributed over
$2n$ modes is much larger that $n+1$, one could
argue that in fact only a small subspace of this large Hilbert space is
actually used. This leads us to the question what is the relevant dimension of
the Hilbert space of the ancilla photons.
One definition that gives a lower bound for the dimension of the used
Hilbert space is the number of distinguishable measurement outcomes.
In the KLM case one applies a discrete $n+1$ point Fourier transformation
on the first $n+1$ modes followed by photodetection of these modes. One
has to measure not only the total photon number but also their distribution
over the modes. In fact every possible
distribution of $k = 0,\ldots,n+1$
photons over $n+1$ modes does occur. As photons in the same mode are
indistinguishable the number of different measurements is smaller
than the physical dimension of the Hilbert space and one has
a number of
\be
N_{\rm KLM} &=& \sum_{k = 0}^{n+1} \left({n+k} \atop {k} {\rm i}ght) \nn \\
&=& \frac{(2n+2)!}{((n+1)!)^2}
\ee
different measurement outcomes. In contrast, the measurement of $\hat Q_+$
and $\hat P_-$ discussed in this paper can only result into $2(a+b)+1$
resp. $4b+1$ different measurement outcomes. I.e. one can distinguish
\be
N = \bigl( 2(a+b)+1 \bigr) \left(4b+1 {\rm i}ght)
\ee
different measurement results. We see that for the discussed teleportation
scheme the number of distinguishable measurement outcomes scales quadratically
in $b$ whereas for the KLM teleportation scheme this number scales much less
favorable with
$(2n+2)!/((n+1)!)^2$.
Since the KLM scheme does not
scale optimal in the sense of the probability of
success calculated above,
it seems feasible that other linear teleportation
schemes may be developed
whose success probability scales better.
This is of particular interest for the practical implementation of linear
optical quantum computing, for example in the protocol described by Yoran
and Reznik \cite{Yoran2003}, where the resources depend critical on the
scaling of the probability of success of the single quantum gates.
\section{Success vs. Fidelity}
Recently Franson, {\it et al.}~\cite{Franson-PRL-2002} suggested a modification
of the KLM scheme which always succeeds but whose fidelity, i.e. the
overlap of the teleported to the input state is not unity. According to
reference \cite{Franson-PRL-2002} we will rather consider the square
of the fidelity
\be
F = \left| \left\langle \Psi_{\rm in} {\rm i}ght. \left| \Psi_{\rm out}
{\rm i}ght\rangle {\rm i}ght|^2.
\ee
In the proposal of Franson, {\it et.al} the square of the fidelity
scales as $F=1-1/n^2$. This approach is different from the
present one and the one of Knill, Laflame, and Milburn, where the teleportation
reproduces exactly the input state if it is successful. Whether or not the
teleportation is successful is hereby uniquely determined by the measurement
result. This has the advantage that
by post-measurement selection of a teleported ensemble a sub-ensemble
of exact replica of the input state can be generated. The obvious advantage
of the Franson scheme is that in any case a state is teleported which
is similar to the input one.
To make a comparison to the Franson scheme we could ask the question
what is the average fidelity of our scheme if we do not discard
those events where $Q$
falls outside of the
interval given by eq.({\rm e}f{condition_success}).
Then the mean
squared fidelity of the discussed teleportation scheme is:
\begin{eqnarray}
\bar F &=& \sum_{Q = -(b+a)}^{b+a} p(Q) \bigl\vert\langle
\Gamma_2^Q | \Psi \rangle\bigr\vert^2 \nn \\
&=& P \cdot 1 + \sum_{\abs{Q} > b-a} p(Q)\bigl\vert
\langle \Gamma_2^Q
| \Psi \rangle\bigr\vert^2
\end{eqnarray}
where $| \Gamma_2^Q \rangle$ is Bob's state after the measurement
with result $Q$. $p(Q)$ denotes the probability
to obtain this outcome and $P$ denotes the probability of
success derived in
previous sections, corresponding to an exact reproduction of the
input state.
To estimate the fidelity we assume in the following that
the basis states $| q_i \rangle $ have an equal a priori
probability to appear in the input state of the teleportation
such that we have the mean value of the coefficients $\alpha_{q_i}$:
\be
\langle \abs{\alpha_{q_i}} \rangle = {\rm const.} = \frac{1}{\sqrt{2a+1}}
\ee
Then given a measurement outcome with $Q > b-a$,
the overlap between the input and the output state is
\begin{eqnarray}
\bigl\vert \langle \Gamma_2^Q | \Psi \rangle\bigr\vert &=&
\frac{1}{2a+1} \sum_{q = Q - b}^{b} \sum_{q' = -a}^a {\rm d}elta_{q,q'} \nn \\
&=& \frac{a+1+b-Q}{2a+1}.
\end{eqnarray}
Considering as an example qubits that can only assume two values (i.e.
$a = 1/2$)
the teleportation does not reproduce the exact input state iff $Q
= b + 1/2$ or $Q = -b - 1/2$. In this case it is $\bigl\vert
\langle \Gamma_2^Q |
\Psi \rangle\bigr\vert^2 = \frac{1}{4}$ and $p(b+1/2) = p (-b-1/2) =
\frac{1}{2(2b+1)}$ and thus we find for the mean squared fidelity
of our teleportation protocol:
\be
\bar F &=& 1 - \frac{1}{2b+1} + \frac{1}{4(2b+1)}.
\ee
The first two terms represent the non unity probability
to exactly reproduce the input state, whereas the last term
describes the finite but nonvanishing overlap of the teleported
state with the initial state in the previously unsuccessful cases.
We see that allowing for a nonperfect reproduction of the state
the fidelity can be enhanced as compared to the result of the last section.
Still $1-F$ scales linear in $1/b$ as compared to the quadratic scaling
in $1/n$ of the Franson scheme.
One has to keep in mind however that the number $n$ in the KLM
or Franson scheme is not the dimension of the ancilla Hilbert space,
as we pointed out in the preceding section.
\section{Conclusion}
In the present paper we discussed the question why continuous
teleportation can be performed with linear elements in a determinstic way
while the discrete counterpart requires nonlinear elements in form
of quantum gates in order to be successful in all cases.
For this we introduced a teleportation protocol
similar to that used in the continuous case applied to qudits with
a discrete and finite set of basis states. We also generalized the
notion of linear elements and detection to the measurements of operators
$\hat Q_+=\hat q_1+\hat q_2$ and $\hat P_- = \hat p_1-\hat p_2$ which are
linear combinations of the basic observables $\hat q$ and $\hat p$ in the
input and ancilla spaces. We have shown that this protocol which uses
only linear elements allows only for a probabilistic teleportation, a result
recently shown to be true for any linear protocol \cite{Luetkenhaus-preprint},
and has a success probability $P$ which is determined only by the Hilbert space
dimensions of the input (${\rm dim}\{{\cal H}_i\}$) and ancilla states
(${\rm dim}\{{\cal H}_a\}$). In the case of qubits we recover
the value $P=1/2$, which is the known limit for discrete teleportation with
linear elements. On the other hand for the continuous teleportation
protocol of Braunstein and Kimble the requirement for a local oscillator
field with infinite squeezing, and thus infinite photon number, implies that
the effective
dimension of the ancilla Hilbert space is much larger than that of the
input space. In this case the success probability of linear teleportation
approaches unity. Thus the difference between discrete and
continuous teleportation appears to result from the difference
in the ancilla resources used rather than hidden nonlinearities
as suggested in \cite{Vaidman-PRA-1999}.
We have also compared the Knill Laflame Milburn
proposal for linear teleportation
\cite{Knill-Nature-2001}
as well as the one by Franson and coworkers
\cite{Franson-PRL-2002}
with our abstract scheme and found that both do not scale optimal.
This suggests that it may be possible to construct specific linear
teleportation schemes for photons which use much less resources than the
KLM and Franson schemes.
\section*{Acknowledgment}
We would like to thank Norbert L\"utkenhaus and Peter van Loock
for stimulating discussions and making results available prior to
publication. The financial support of the Deutsche Forschungsgemeinschaft
within the Schwerpunktprogramm ``Quanteninformation'' is gratefully
acknowledged.
\end{document} |
\begin{document}
\title{Bounded weight modules of the Lie algebra of vector fields on ${\mathbb C}^2$}
\author{Andrew Cavaness and Dimitar Grantcharov$^1$}
\address{Department of Mathematics \\
University of Texas at Arlington \\ Arlington, TX 76021, USA}
\varepsilonilonmail{cavaness@uta.edu}
\address{Department of Mathematics \\
University of Texas at Arlington \\ Arlington, TX 76021, USA}
\varepsilonilonmail{grandim@uta.edu}
\thanks{$^{1}$This work was partially supported by Simons Collaboration Grant 358245}
\maketitle
\begin{abstract}
We study weight modules of the Lie algebra $W_2$ of vector fields on $\mathbb C^2$. A classification of all simple weight modules of $W_2$ with a uniformly bounded set of weight multiplicities is provided. To achieve this classification we introduce a new family of generalized tensor $W_n$-modules. Our classification result is an important step in the classification of all simple weight $W_n$-modules with finite weight multiplicities.\\
\noindent 2000 MSC: 17B66, 17B10 \\
\noindent Keywords and phrases: Lie algebra, Cartan type, weight module, localization.
\varepsilonilonnd{abstract}
\section*{Introduction}
Lie algebras of vector fields have been studied since the appearance of infinite Lie groups in the works of S. Lie in the late 19th century. Based on the fundamental works of E. Cartan in the early 20th century, some of these infinite-dimensional Lie algebras are known as Lie algebras of Cartan type. A classical example of a Cartan type Lie algebra is the Lie algebra $W_n$ consisting of derivations of the polynomial algebra $\mathbb C[x_1,...,x_n]$, or, equivalently, the Lie algebra of polynomial vector fields on $\mathbb C^n$. The first classification results concerning representations of $W_n$ and other Cartan type Lie algebras were obtained by A. Rudakov in 1974-1975, \cite{Rud1}, \cite{Rud2}. These results address the classification of a class of irreducible $W_n$-representations that satisfy some natural topological conditions. The so-called {\it tensor modules}, that is modules $T({\boldsymbol{\nu}}, S)$ whose underlying spaces are tensor products $\boldsymbol{x}^{\boldsymbol{\nu}}\mathbb C[x_1^{\pm 1},..,x_n^{\pm 1}] \otimes S$ of a ``shifted'' Laurent polynomial ring and a finite-dimensional $\mathfrak{gl}_n$-module $S$, play an important role in the works of Rudakov. Tensor $W_1$-modules and extensions of tensor modules were studied extensively in the 1970's and in the 1980's by B. Feigin, D. Fuks, I. Gelfand, and others, see for example, \cite{FF}, \cite{Fuk}.
In this paper we focus on the category of weight representations of $W_n$, namely those that decompose as direct sums of weight spaces relative to the subalgebra $\mathfrak{h}_{W_n}$ of $W_n$ spanned by the derivations $x_1\partial_1$,...,$x_n\partial_n$. Weight representations of Lie algebras of vector fields (in particular, of $W_n)$) are subject of interest by both mathematicians and theoretical physicists in the last 30 years. Another important example of a Lie algebra of vector fields is the Witt algebra $\rm{Witt}_n$ consisting of the derivations of the Laurent polynomial algebra $\mathbb C[x_1^{\pm},...,x_n^{\pm 1}]$, or, equivalently, the Lie algebra of polynomial vector fields on the $n$-dimensional complex torus. In particular, $\rm{Witt}_1$ is the centerless Virasoro algebra. The classification of all simple weight representations with finite weight multiplicities of $W_1$ and ${\rm Witt}_1$ (and hence of the Virasoro algebra) was obtained by O. Mathieu in 1992, \cite{M-Vir}, proving a conjecture of V. Kac, \cite{Kac}. Following a sequence of works of S. Berman, Y. Billig, C. Conley, S. Eswara Rao, X. Guo, C. Martin, O. Mathieu, V. Mazorchuk, V. Kac, G. Liu, R. Lu, A. Piard, Y. Su, K. Zhao, very recently, Y. Billig and V. Futorny managed to extend Mathieu's classification result to ${\rm Witt}_n$ for arbitrary $n\mathfrak{g}eq 1$ (see \cite{BF} and the references therein). The classification theorem in \cite{BF} states roughly that every nontrivial simple weight ${\rm Witt}_n$-module with finite weight multiplicities is either a submodule of a tensor module or a module of highest weight type.
In contrast with ${\rm Witt}_n$, the classification of the simple weight $W_n$-modules $M$ with finite weight multiplicities is still an open problem for $n>1$. The possible supports (sets of weights) of all such $M$ have been described by I. Penkov and V. Serganova in \cite{PS}. In addition, in \cite{PS}, a parabolic induction theorem for such modules $M$ is proven. More precisely, it is shown that $M$ is a quotient of a parabolically induced module from a parabolic subalgebra $\mathfrak p$ of $W_n$. Unfortunately, the parabolic subalgebras $\mathfrak p = \mathfrak l \oplus \mathfrak n^+$ that appear in the parabolic induction theorem are quite complicated and have Levi components $\mathfrak l$ isomorphic to a semi-direct sum of Lie algebras of Cartan type and finite dimensional-reductive Lie algebras. Another obstacle in the study of weight $W_n$-modules is the fact that the $W_n$-modules $T({\boldsymbol{\nu}}, S)$ are highly reducible - they may contain $2^n$ simple subquotients.
The purpose of this paper is to make the first step towards the classification of the simple weight $W_n$-modules with finite weight multiplicities. Namely, we classify the simple bounded modules of $W_2$, that is, all simple weight $W_2$-modules whose sets of weight multiplicities is uniformly bounded. The classification is given in Theorem \ref{th-main} (the tensor modules are introduced in Definition \ref{def-tensor}). The second step in the weight module classification is to classify all simple bounded $\mathfrak l$-modules, where $\mathfrak l$ is a Levi subalgebra of a parabolic subalgebra of $W_n$. The last step is, based on the parabolic induction theorem of Penkov-Serganova, to complete the classification in question. The second and the third steps will be addressed in a subsequent paper. We note that for $n=2$, the classification of simple bounded $\mathfrak l$-modules, where $\mathfrak l$ is a Levi subalgebra of a parabolic subalgebra of $W_2$, is obtained in the present paper and, in fact, is used to classify the simple bounded $W_2$-modules. It turns out that in this case $\mathfrak l \simeq \Der \mathbb C[x] \ltimes \mathbb C[x]$.
In addition to obtaining the classification of simple weight modules with finite weight multiplicities, the results in the present paper will be essential for the study of the category $\mathcal B$ of bounded representations of $W_n$. This category is intimately related to the corresponding category of bounded $\mathfrak{sl}_{n+1}$-modules. It is expected that, like in the case of $\mathfrak{sl}_{n+1}$, the indecomposable injectives of $\mathcal B$ will have a nice geometric realizations in terms of twisted functions and twisted differential forms on algebraic varieties, \cite{GS2}, \cite{GS3}.
An important tool in the present paper is the twisted localization functor, a functor used by O. Mathieu in the proof of another fundamental result: the classification of all simple weight modules with finite weight multiplicities of finite-dimensional reductive Lie algebras, \cite{M}. Also, in order to deal with the reducibility of $T({\boldsymbol{\nu}}, S)$, we introduce a family of (generalized) tensor modules $T({\boldsymbol{\nu}}, S, J)$. The modules $T({\boldsymbol{\nu}}, \boldsymbol{\lambda}, J) = T({\boldsymbol{\nu}}, S, J)$ are defined for a tuple $J = (a_1,...,a_k)$ of signed integers $a_i = b_i^+$ or $a_i = b_i^-$, where the $b_i$'s are in the set of all indices $j$ such that $\lambda_j - \nu_j \in \mathbb Z$, and $\boldsymbol{\lambda}$ is the highest weight of $S$. Our main result is that all nonzero simple bounded $W_2$-modules are isomorphic to $T({\boldsymbol{\nu}}, \boldsymbol{\lambda}, J) $ for some ${\boldsymbol{\nu}}, \boldsymbol{\lambda}, J$. A similar result will hold for the simple bounded weight $W_n$-modules ($n\mathfrak{g}eq 2$) as well, but additional conditions for the tensor modules $T({\boldsymbol{\nu}}, \boldsymbol{\lambda}, J)$ corresponding to fundamental weights $\boldsymbol{\lambda}$ have to be imposed.
The content of the paper is as follows. In Section 2 we collect important results on the twisted localization functor, parabolic subalgebras and tensor modules of $W_n$. In particular we provide an explicit list of the possible parabolic subalgebras $\mathfrak p$ of $W_2$. In Section 3, we classify all simple bounded modules over the Lie algebra $\mathcal A = \Der \mathbb C[x] \ltimes \mathbb C[x]$. In Section 4, based on the results of Section 3, we complete the classification of simple bounded $W_2$-modules.
\noindent{\it Acknowledgements.} We would like to thank M. Gorelik and V. Serganova for the fruitful discussions and helpful suggestions. We also would like to thank the referee for the valuable remarks and the careful reading of the manuscript.
\section{Notation and Conventions}
Throughout the paper the ground field is $\mathbb C$. All vector spaces, algebras, and tensor products are assumed to be over $\mathbb C$ unless otherwise stated.
By $W_n$ we denote the Lie algebra $ \Der \mathbb C [x_1,...,x_n] $ of derivations of ${\mathbb C} [x_1,...,x_n]$. Also, $\mathcal A_n$ will be the semi-direct product $\mathcal A_n= \Der \mathbb C [x_1,...,x_n] \ltimes \mathbb C [x_1,..,x_n]$. For simplicity we set $\mathcal A := \mathcal A_1$ and $\partial_i :=\frac{\partial}{\partial x_i}$. Every element $w$ of $W_n$ can be written uniquely as $w= \sum_{i=1}^n f_i\partial_i$, for some $f_i \in \mathbb C[x_1,...,x_n]$.
By $ {\mathbb Z}_{\mathfrak{g}eq k}$ we denote the set of all integers $n$ such that $n \mathfrak{g}eq k$. We similarly define $ {\mathbb Z}_{\leq k}$, $ {\mathbb Z}_{> k}$, $ {\mathbb R}_{\mathfrak{g}eq k}$, etc. If $M$ is a set of real numbers, and $S$ is
a subset of a real vector space $V$, then by $MS$ we denote the set of all $M$-linear combinations of elements in $S$.
For a Lie algebra $\mathfrak a$ by $U(\mathfrak a)$ we denote the universal enveloping algebra of $\mathfrak a$.
Throughout the paper we use the multi index-notation for monomials: $\boldsymbol{x}^{\boldsymbol{\nu}} = x_1^{\nu_1}...x_n^{\nu_n}$ if
$\boldsymbol{x} = (x_1,...,x_n)$ and $\boldsymbol{\nu} = (\nu_1,...,\nu_n)$. If $n$ is fixed, we set ${\mathbb C}[\boldsymbol{x}] = {\mathbb C}[x_1,...,x_n]$, ${\mathbb C}[\boldsymbol{x}^{\pm 1}] = {\mathbb C}[x_1^{\pm 1},...,x_n^{\pm 1}]$, and $\boldsymbol{x}^{\boldsymbol{\nu}} {\mathbb C}[\boldsymbol{x}^{\pm 1}] = x_1^{\nu_1}...x_n^{\nu_n}{\mathbb C}[x_1^{\pm 1},...,x_n^{\pm 1}]$, where the latter is the span of all (formal) monomials $x_1^{\nu_1+k_1}...x_n^{\nu_n + k_n}$, $k_i \in {\mathbb Z}$.
For an $n$-tuple $\boldsymbol{\nu} = (\nu_1,...,\nu_n)$ in ${\mathbb C}^n$, we set $\mbox{Int} (\boldsymbol{\nu}):=\{ i \; | \; \nu_i \in {\mathbb Z}\}$.
\section{Preliminaries}
\subsection{The Lie algebra $\mathcal A$}
Recall that $\mathcal A = \Der \mathbb C[x] \ltimes \mathbb C[x]$ and $W_1 = \Der \mathbb C[x]$. Note that by definition $[D,f] = Df$ for $D \in \Der \mathbb C[x]$ and $f \in \mathbb C[x]$. In terms of generators and relations, $\mathcal A$ can be defined as follows: $$\mathcal A = \Span \{D_i, I_j \; | \; i \in \mathbb Z_{\mathfrak{g}eq -1}, j \in \mathbb Z_{\mathfrak{g}eq 0} \}$$ with
\begin{eqnarray*}
\left[D_i, D_j\right] &=& (j-i) D_{i+j},\\
\left[D_i, I_j\right] &=& j I_{i+j},\\
\left[I_i, I_j\right] &=& 0.
\varepsilonilonnd{eqnarray*}
Here $D_i$ and $I_j$ correspond to $x^{i+1} \partial$ and $x^j$, respectively. Note that the center of $\mathcal A$ is generated by $I_0$. We say that an ${\mathcal A}$-module $M$ has central charge $c$ if $I_0m = cm$ for every $m \in M$. In particular, every irreducible ${\mathcal A}$-module $M$ has a central charge.
We say that $\mathfrak{h} \subset \mathcal A$ is a \varepsilonilonmph{Cartan subalgebra} of $\mathcal A$ if $\mathfrak{h}$ is both self-normalizing and nilpotent. In what follows, we fix the Cartan subalgebra of $\mathcal A$ to be $\mathfrak{h}_{\mathcal A} = \Span \{D_0, I_0\}$. We also have the triangular decomposition $\mathcal A = \mathcal A^- \oplus \mathcal A^0 \oplus \mathcal A^+$, where $\mathcal A^- = \Span \{ D_{-1}\}$, $\mathcal A^0 = \mathfrak{h}_{\mathcal A}$, and $\mathcal A^+ = \Span \{D_i, I_j \; | \; i,j\mathfrak{g}eq 1\}$. Define $\varepsilonilon, \delta \in \mathfrak{h}_{\mathcal A}^*$ by the identities $$\varepsilonilon(D_0) = 1, \; \; \varepsilonilon(I_0) = 0; \; \; \delta(D_0) = 0, \; \; \delta(I_0)=1.$$
\subsection{Injective and finite actions}
Let $\mathfrak{g}$ be Lie algebra, and $M$ be a $\mathfrak{g}$-module. We say that an element $x$ of $\mathfrak{g}$ acts \varepsilonilonmph{locally nilpotently} (or, \varepsilonilonmph{finitely}) on a vector $m$ in $M$, if there is $N = N(x,m)$ such that $x^N(m) = 0$. If such $N$ does not exists we say that $x$ {\it acts injectively} on $m$. We say that $x$ acts injectively (respectively, finitely) on $M$ if $x$ acts injectively (respectively, finitely) on all $m \in M$.
We will often use the following setting. Let $x$ be an $\ad$-nilpotent element in $\mathfrak{g}$ and let $M$ be a $\mathfrak{g}$-module. Then the set $M^{\langle x \rangle}$ of all $m$ on which $x$ acts finitely is a submodule of $M$. In particular, if $M$ is simple, then every ad-nilpotent element $x$ of $\mathfrak{g}$ acts either finitely or injectively on $M$.
\subsection{Weight modules} \label{subsec-wht} We first introduce weight modules in a general setting.
Let $\mathcal U$ be an associative unital algebra and $\mathcal H\subset\mathcal U$
be a commutative subalgebra. We assume in addition that
$\mathcal H$
is a polynomial algebra identified with the symmetric algebra of a vector space ${\mathfrak h}$, and that we
have a decomposition
$$\mathcal U=\bigoplus_{\mu\in {{\mathfrak h}^*}}\mathcal U^\mu,$$
where
$$\mathcal U^\mu=\{x\in\mathcal U | [h,x]=\mu(h)x, \forall h\in\mathfrak h\}.$$
Let $Q_{\mathcal U} = {\mathbb Z}\Delta_{\mathcal U}$ be the ${\mathbb Z}$-lattice in ${\mathfrak h}^*$
generated by $\Delta_{\mathcal U}= \{ \mu \in {\mathfrak h}^* \; | \; {\mathcal U}^{\mu} \neq 0\}$. We obviously have
${\mathcal U}^\mu {\mathcal U}^\nu\subset {\mathcal U}^{\mu+\nu}$.
We call a ${\mathcal U}$-module $M$ {\it a generalized weight $({\mathcal U}, {\mathcal H})$-module} (or just\varepsilonilonmph{ generalized weight $\mathcal U$-module}) if $M = \bigoplus_{\lambda \in {\mathfrak h}^*} M^{(\lambda)}$, where
$$
M^{(\lambda)} = \{m\in M \; |\; (h- \lambda (h)\mbox{Id})^N m=0\,\text{for some}\, N>0\, \text{and all}\, h \in \mathfrak h\}.
$$
We call $M^{(\lambda)}$ the generalized weight space of $M$ and $\dim
M^{(\lambda)}$ the weight multiplicity of the weight $\lambda$. A vector $v$ in $M^{(\lambda)}$ is called a \varepsilonilonmph{weight vector of weight $\lambda$} and we write $\wht(v) = \lambda$.
Note that
\begin{equation}\label{rootweight}
\mathcal U^\mu M^{(\lambda)}\subset M^{(\mu+\lambda)}.
\varepsilonilonnd{equation}
A generalized weight module $M$ is called a {\it weight $({\mathcal U}, {\mathcal H})$-module} (or just \varepsilonilonmph{weight} \varepsilonilonmph{$\mathcal U$-module}) if $M^{(\lambda)} = M^{\lambda},$ where
$$
M^{\lambda} = \{m\in M \; |\; (h - \lambda(h) \mbox{Id}) m=0\,\text{ for all } h \in {\mathfrak h}\}.
$$
In the case when ${\mathcal U} = U(\mathfrak g)$ is the universal enveloping algebra of a Lie algebra $\mathfrak g$ and $\mathfrak{h}$ is a subalgebra of $\mathfrak g$, a (generalized) weight $({\mathcal U}, {\mathcal H})$-module will be called (generalized) weight $(\mathfrak g, \mathfrak h)$-module.
\begin{definition}
\begin{itemize}
\item[(i)]An $\mathcal A$-module $M$ is a \varepsilonilonmph{weight $\mathcal A$-module} if $M$ is a weight $(\mathcal A , \mathcal{H})$-module for $\mathcal H = \mathbb C [\mathfrak{h}_{\mathcal A}]$. If $M$ is a weight $\mathcal A$-module we call the set of weights $\lambda \in \mathfrak{h}_{\mathcal A}^*$ such that $M^{\lambda} \neq 0$ the \varepsilonilonmph{$\mathcal A$-support} (or simply the \varepsilonilonmph{support}) of $M$ and denote it by $\Supp M$.
\item[(ii)]We say that a weight $\mathcal A$-module $M$ is \varepsilonilonmph{bounded} if there is $N>0$ such that $\dim M^\lambda < N$ for all $\lambda \in \mathfrak{h}_{\mathcal A}^*$. If $M$ is bounded we call $\sup \{ \dim M^{\lambda}\; | \; \lambda \in \mathfrak{h}_{\mathcal A}^* \}$ the $\mathcal A$-\varepsilonilonmph{degree} (or simply the \varepsilonilonmph{degree}) of $M$.
\varepsilonilonnd{itemize}
\varepsilonilonnd{definition}
The adjoint module $\mathcal A$ is a weight module of central charge $0$ such that $\mathcal A^{\lambda} \neq 0$ if and only if $\lambda = n \varepsilonilon$ for $n \in \mathbb Z_{\mathfrak{g}eq -1}$. The set $\Delta_{\mathcal A} = \{ - \varepsilonilon, n\varepsilonilon \; | \; n\in \mathbb Z_{>0}\}$ is the \varepsilonilonmph{root system} of $\mathcal A$, and
$$
\mathcal A^{-\varepsilonilon} = \Span \{ D_{-1}\}; \, \mathcal A^{n\varepsilonilon} = \Span \{ I_n, D_n \}, n\in \mathbb Z_{>0}
$$
are the root spaces of $\mathcal A$.
If $M$ has central charge $c$, then a weight of $M$ is of the form $\lambda \varepsilonilon + c \delta$, for some $\lambda \in \mathbb C$. If $c$ is fixed, with a slight abuse of notation we set $M^{\lambda} = M^{\lambda \varepsilonilon + c \delta}$, for all weight modules $M$ with central charge $c$. In particular, $M = \bigoplus _{\lambda \in \mathbb C} M^\lambda$ and $\Supp M \subset \mathbb C$.
We similarly introduce the notions of weight and bounded $W_n$-modules. More precisely, let $\mathfrak{h}_{W_n}$ (or simply $\mathfrak{h}_{W}$ if $n$ is fixed) be the subalgebra $\Span \{ x_1\partial_1,..., x_n\partial_n \}$. Then $\mathfrak{h}_W$ is a Cartan subalgebra of $W_n$.
\begin{definition}
A $W_n$-module $M$ is a \varepsilonilonmph{weight $W_n$-module} if $M$ is a weight $(W_n, {\mathcal H})$-module with $\mathcal H = \mathbb C [\mathfrak{h}_W]$. We say that a weight $W_n$-module $M$ is \varepsilonilonmph{bounded} if there is $N>0$ such that $\dim M^\lambda < N$ for all $\lambda \in \mathfrak{h}_{W}^*$. If $M$ is bounded we call $\sup \{ \dim M^{\lambda}\; | \; \lambda \in \mathfrak{h}_{W}^* \}$ the $W_n$-\varepsilonilonmph{degree} (or simply the \varepsilonilonmph{degree}) of $M$.
\varepsilonilonnd{definition}
Note that $W_n$ is a weight $W_n$-module whose support is $\Delta_{W_n} \cup \{ 0\}$, where
$\Delta_{W_n}$ is the root system of $W_n$. We identify the root lattice of $W_n$ with ${\mathbb Z}^n$ and will often write every element $\alpha$ of ${\mathbb Z} \Delta_{W_n}$ as an $n$-tuple $(\alpha_1,...,\alpha_n)$ of integers. In particular,
$$
\Delta_{W_n} \cup \{ 0\}= \{ (\alpha_1,...,\alpha_n) \; | \; \alpha_i \mathfrak{g}eq 0\} \sqcup \{ (\alpha_1,...,\alpha_n) \; | \; \varepsilonilonxists i: \alpha_i = -1 \mbox{ and } \alpha_j \mathfrak{g}eq 0 \mbox{ for all } j\neq i \}.
$$
For simplicity we will often write $\Delta_{W}$ for $\Delta_{W_n}$. In what follows we use the (root) basis of $W_n$ consisting of the elements $\boldsymbol{x}^{\boldsymbol{\alpha}} (x_i\partial_i)$, $\boldsymbol{\alpha} \in \Delta_{W_n} \cup \{ 0\}$, $i=1,...,n$.
\subsection{Tensor modules}
We say that $(\lambda_1,...,\lambda_n) \in {\mathbb C}^n$ is a dominant integral $\mathfrak{gl}_n$-weight if $\lambda_i - \lambda_{i+1} \in {\mathbb Z}_{\mathfrak{g}eq 0}$ for all $i = 1,..,n-1$. If $\boldsymbol{\lambda} = (\lambda_1,...,\lambda_n)$ is a dominant integral weight, by $L_{\mathfrak{gl}} (\boldsymbol{\lambda} ) = L_{\mathfrak{gl}} (\lambda_1,...,\lambda_n )$ we denote the simple finite-dimensional module with highest weight $\boldsymbol{\lambda}$.
For a dominant integral $\mathfrak{gl}_n$-weight $\boldsymbol{\lambda} = (\lambda_1,...,\lambda_n)$ and any $\boldsymbol{\nu} = (\nu_1,...,\nu_n)$ in $\mathbb C^n$, we define the $W_n$-modules $T(\boldsymbol{\nu},\boldsymbol{\lambda})$ as follows:
$$
T(\boldsymbol{\nu},\boldsymbol{\lambda}) = \boldsymbol{x}^{\boldsymbol{\nu}} {\mathbb C}[\boldsymbol{x}^{\pm 1}] \otimes L_{\mathfrak{gl}} (\boldsymbol{\lambda} )
$$
with $W_n$-action defined by
\begin{equation} \label{def-tensor-action}
( \boldsymbol{x}^{\boldsymbol{\alpha}} x_i\partial_i)\cdot (\boldsymbol{x}^{\bf s} \otimes v) = s_i \boldsymbol{x}^{\boldsymbol{\alpha + s}} \otimes v+ \sum_{j=1}^n \alpha_j \boldsymbol{x}^{\boldsymbol{\alpha + s}} \otimes E_{ji}v,
\varepsilonilonnd{equation}
where $\boldsymbol{\alpha} \in \Delta_{W_n} \cup\{ 0 \}$, ${\bf s} \in \boldsymbol{\nu} + {\mathbb Z}^n$, $v \in L_{\mathfrak{gl}} (\boldsymbol{\lambda} )$, and $E_{ji}$ is the $(j,i)$th elementary matrix of $\mathfrak{gl}_n$. As indicated in the introduction, these modules play important role in the classification of simple weight modules with finite weight multiplicities over various classes of Lie algebras. We easily extend the $W_n$-action on $T(\boldsymbol{\nu},\boldsymbol{\lambda})$ to an ${\mathcal A}_n$-action. Namely, for $c \in {\mathbb C}$ we define $T(\boldsymbol{\nu},\boldsymbol{\lambda}, c) = T(\boldsymbol{\nu},\boldsymbol{\lambda}) $ as vector space and set
\begin{equation} \label{x-i-action}
\boldsymbol{x}^{\bf j}\cdot (\boldsymbol{x}^{\bf s} \otimes v) := c \boldsymbol{x}^{\bf{j + s}} \otimes v.
\varepsilonilonnd{equation}
The next theorem gives a necessary and sufficient condition when two tensor modules are isomorphic as $W_n$-modules and ${\mathcal A}_n$-modules. The fact is well-known but for reader's convenience a short proof suggested by M. Gorelik is provided.
\begin{proposition}
The following are equivalent.
\begin{itemize}
\item[(i)] $T(\boldsymbol{\nu},\boldsymbol{\lambda}) \simeq T(\boldsymbol{\nu}',\boldsymbol{\lambda}')$ as $W_n$-modules.
\item[(ii)] $T(\boldsymbol{\nu},\boldsymbol{\lambda}) \simeq T(\boldsymbol{\nu}',\boldsymbol{\lambda}')$ as ${\mathcal A}_n$-modules.
\item[(iii)] $\boldsymbol{\nu} - \boldsymbol{\nu}' \in {\mathbb Z}^n$ and $\boldsymbol{\lambda} = \boldsymbol{\lambda}'$.
\varepsilonilonnd{itemize}
\varepsilonilonnd{proposition}
\begin{proof}
The fact that (iii) implies (i) and (ii) is straightforward. Also, obviously (ii) implies (i). It remains to show that (i) implies (iii).
Let $\psi: T(\boldsymbol{\nu},\boldsymbol{\lambda})\to T(\boldsymbol{\nu}' ,\boldsymbol{\lambda}' )$ be an isomorphism. Since the $\boldsymbol{\mu}$-weight space of $T(\boldsymbol{\nu},\boldsymbol{\lambda})$ is $\boldsymbol{x}^{\boldsymbol{\mu}} \otimes L(\boldsymbol{\lambda})$, we have that for every $\bf{s} \in \boldsymbol{\nu} + \mathbb Z^n$ and $u \in L(\boldsymbol{\lambda})$, $\psi (\boldsymbol{x}^{\boldsymbol{s}} \otimes u) = \boldsymbol{x}^{\boldsymbol{s}} \otimes u'$ for some $u' \in L(\boldsymbol{\lambda}')$. Also, $\dim L(\boldsymbol{\lambda}) = \dim L(\boldsymbol{\lambda}')$.
Let $v_{\lambda}$ be a highest weight vector of $L(\boldsymbol{\lambda})$, and let us fix ${\bf s} \in \boldsymbol{\nu} + \mathbb Z^n$ such that $s_i \neq 0$ for every $i$. Also, let $\psi(\boldsymbol{x}^{\bf s}\otimes v_{\lambda})=\boldsymbol{x}^{\bf s} \otimes v$ for some
$v\in L(\boldsymbol{\lambda}')$. Denote by $v_i$ the $\mathfrak{gl}_n$-weight components of $v$, i.e.
$v=\sum_{i=1}^t v_i$, where
$v_i\in L(\boldsymbol{\lambda})^{\boldsymbol{\varepsilonilonta}_i}$ are nonzero vectors and $\boldsymbol{\varepsilonilonta}_1,\ldots,\boldsymbol{\varepsilonilonta}_t$
are distinct weights (of $\mathfrak{gl}_n$).
Assume that $\boldsymbol{\varepsilonilonta}_1$ is a minimal element in $\{\boldsymbol{\varepsilonilonta}_1,..., \boldsymbol{\varepsilonilonta}_n\}$ with respect to the standard
partial order on ${\mathfrak h}_{\mathfrak{gl}_n}^*$. Then for $1\leq j<i\leq n$ we have
\begin{eqnarray*}
(x_ix_j \partial_i)\partial_j(\boldsymbol{x}^{\boldsymbol{s}}\otimes v)&=&\boldsymbol{x}^{\boldsymbol{s}}\otimes (s_i+E_{ji})(s_j-E_{jj}) v; \\
(x_ix_j \partial_i)\partial_j(\boldsymbol{x}^{\boldsymbol{s}}\otimes v_{\lambda})&=&s_i(s_j-\lambda_j)\cdot \boldsymbol{x}^{\boldsymbol{s}}\otimes v_{\lambda},
\varepsilonilonnd{eqnarray*}
where $\boldsymbol{\lambda} = (\lambda_1,...,\lambda_n)$. Thus
$$(s_i+E_{ji})(s_j-E_{jj})v=s_i(s_j-\lambda_j)v.$$
Using the minimality of $\boldsymbol{\varepsilonilonta}_1$, after taking the $\boldsymbol{\varepsilonilonta}_1$-components of the vector above, we obtain $E_{jj}v_1 = \lambda_j v_1$ for all $j<n$. Thus $\boldsymbol{\lambda}-\boldsymbol{\varepsilonilonta}_1 = c \varepsilon_n$ some $c \in \mathbb C$. But since $\boldsymbol{\varepsilonilonta}_1$ is in the support of $L(\boldsymbol{\lambda}')$ and $\boldsymbol{\lambda}'$ is a maximal weight in this support, we have $\boldsymbol{\lambda}'-\boldsymbol{\lambda} \mathfrak{g}eq - c \varepsilon_n$. With similar reasoning we obtain $\boldsymbol{\lambda}-\boldsymbol{\lambda}' \mathfrak{g}eq - c' \varepsilon_n$ for some $c' \in \mathbb C$. Thus $\boldsymbol{\lambda}-\boldsymbol{\lambda}' \in \mathbb C \varepsilon_n$. Now using this and the Weyl dimension formula for $\dim L(\boldsymbol{\lambda}) = \dim L(\boldsymbol{\lambda}')$, we prove that $\boldsymbol{\lambda} = \boldsymbol{\lambda}'$.
\varepsilonilonnd{proof}
In what follows we introduce some important subquotients of the $W_n$-modules $T(\boldsymbol{\nu},\boldsymbol{\lambda}) $ defined above. First, for any ${\bf z} \in {\mathbb C}^n$, we set $\mathcal{PM}({\bf z}) = \{+,-\}^{{\rm Int} (\bf z)}$. Every element $J: {\rm Int} (\bf z) \to \{ +,-\}$ of $\mathcal{PM}({\bf z}) $ will be written as $({i_1}^{J(i_1)},...,{i_k}^{J(i_k)})$, where ${\rm Int} ({\bf z}) = \{ i_1,...,i_k\}$ and $i_1<\cdots <i_k$. For example, if ${\rm Int} ({\bf z}) = \{1, 2\}$, then the elements of $\mathcal{PM}({\bf z})$ are: $(1^+, 2^+)$, $(1^+, 2^-)$, $(1^-, 2^+)$, $(1^-, 2^-)$.
For every element $J$ in $\mathcal{PM}({\bf z})$ we write $J^+$ (respectively, $J^-$) for the subset of $J$ consisting of all $i_j^{J(i_j)}$ with $J(i_j) = ``+''$ (respectively, $J(i_j) = ``-''$).
\begin{definition} \label{def-tensor} Let $\boldsymbol{\lambda}$ be a dominant integral $\mathfrak{gl}_n$-weight, $\boldsymbol{\nu} \in {\mathbb C}^n$, and $J$ be in $\mathcal{PM} (\boldsymbol{\lambda} - \boldsymbol{\nu})$. We define $T(\boldsymbol{\nu}, \boldsymbol{\lambda}, J)$ as follows.
\begin{itemize}
\item[(i)] $T(\boldsymbol{\nu}, \boldsymbol{\lambda}, \varepsilonilonmptyset) := T(\boldsymbol{\nu}, \boldsymbol{\lambda})$.
\item[(ii)] If $J^+ \neq \varepsilonilonmptyset$ and $J^- = \varepsilonilonmptyset$, then
$$
T(\boldsymbol{\nu}, \boldsymbol{\lambda}, J) := \Span \{ x^{\varepsilonilonta} \otimes v_{\mu} \; | \; v_{\mu} \in L_{\mathfrak{gl}} (\lambda)^{\mu}, \varepsilonilonta_i - \mu_i \in {\mathbb Z}_{\mathfrak{g}eq 0} \mbox{ for all } i \in J\}.
$$
\item[(iii)] If $J^- \neq \varepsilonilonmptyset$, then
$$
T(\boldsymbol{\nu}, \boldsymbol{\lambda}, J) := T(\boldsymbol{\nu}, \boldsymbol{\lambda}, J^+)/\left( \sum_{j^- \in J^-}T(\boldsymbol{\nu}, \boldsymbol{\lambda}, J^+ \cup \{j^+\}) \right)
$$
\varepsilonilonnd{itemize}
\varepsilonilonnd{definition}
It is easy to check that if $J^- = \varepsilonilonmptyset$, then $T(\boldsymbol{\nu}, \boldsymbol{\lambda}, J)$ is a submodule of $T(\boldsymbol{\nu}, \boldsymbol{\lambda})$. Therefore all $T(\boldsymbol{\nu}, \boldsymbol{\lambda}, J)$ are bounded weight $W_n$-modules of degree $\dim L_{\mathfrak{gl}} (\boldsymbol{\lambda})$.
Using (\ref{x-i-action}), we easily endow $T(\boldsymbol{\nu}, \boldsymbol{\lambda}, J)$ with an $\mathcal A_n$-module structure and the resulting module $T(\boldsymbol{\nu}, \boldsymbol{\lambda}, J, c)$ has central charge $c$. In what follows, we will call both $T(\boldsymbol{\nu}, \boldsymbol{\lambda}, J)$ and $T(\boldsymbol{\nu}, \boldsymbol{\lambda}, J, c)$ \varepsilonilonmph{generalized tensor modules}, or simply \varepsilonilonmph{tensor modules}.
\subsection{Tensor $\mathcal A_1$- and $W_1$-modules and classification of simple weight $W_1$-modules with finite weight multiplicities}
In the case $n=1$ we will use the notation $T(\nu, \lambda)$ and $T(\nu, \lambda, c)$ for the tensor modules corresponding to $\nu, \lambda \in {\mathbb C}$. More explicitly, $T(\nu, \lambda, c) = \Span \{ x^{\nu + \varepsilonilonll} \otimes v_{\lambda} \; | \; \varepsilonilonll \in {\mathbb Z}\}$, with action of $\mathcal A$ defined by:
\begin{eqnarray*}
D_i (x^{\nu + \varepsilonilonll} \otimes v_{\lambda}) & = & (\nu + \varepsilonilonll + i \lambda) x^{\nu + \varepsilonilonll + i} \otimes v_{\lambda},\\
I_j (x^{\nu + \varepsilonilonll} \otimes v_{\lambda}) & = & c x^{\nu + \varepsilonilonll + j} \otimes v_{\lambda}
\varepsilonilonnd{eqnarray*}
\begin{remark}
Another important class of modules that appears in the literature consists of the \varepsilonilonmph{tensor densities modules}. Namely these are the $\mathcal A$-modules $\mathcal F (\nu , \lambda, c) = x^{\mu} {\mathbb C}[x^{\pm 1}] (dx)^{\lambda}$ of central charge $c$ and with the natural action of the generators $D_i$ and $I_j$. One easily can show that $\mathcal F (\nu, \lambda, c) \simeq T(\nu + \lambda, \lambda, c)$.
\varepsilonilonnd{remark}
We will also write $T(\lambda, \lambda, +) = T(\lambda, \lambda, 1^+)$ and $T(\lambda, \lambda, -) = T(\lambda, \lambda, 1^-)$. In particular, $T(\lambda, \lambda, +) = \Span \{ x^{\lambda + n} \otimes v_{\lambda}\; | \; n \in {\mathbb Z}_{\mathfrak{g}eq 0}\}$ and $T(\lambda, \lambda, -) = T(\lambda, \lambda) / T(\lambda, \lambda, +)$. Furthermore, the corresponding $\mathcal A$-modules to $T(\lambda, \lambda, \pm)$ will be denoted by $T(\lambda, \lambda, c, \pm)$. The Jordan-H\"older decomposition of the modules $T(\nu, \lambda)$ and $T(\nu, \lambda,c)$ is described in the following two propositions. The proof is standard and is omitted.
\begin{proposition} \label{prop-tens-w1-mod} Let $\lambda, \nu \in {\mathbb C}$
\begin{itemize}
\item[(i)] The $W_1$-modules $T(\nu_1, \lambda_1)$ and $T(\nu_2, \lambda_2)$ are isomorphic if and only if:
\begin{itemize}
\item[(a)] $\nu_1-\nu_2 \in {\mathbb Z}$ and $\lambda_1 = \lambda_2$, or
\item[(b)] $\nu_1-\nu_2 \in {\mathbb Z}$, $\nu_1 \notin \mathbb Z$, and $\{\lambda_1, \lambda_2\} = \{ 0,1\} $.
\varepsilonilonnd{itemize}
\item[(ii)] The $W_1$-module $T(\nu, \lambda)$ is simple if and only if $\lambda - \nu \notin {\mathbb Z}$.
\item[(iii)] The $W_1$-module $T(\lambda, \lambda, +)$ is simple if and only if $\lambda \neq 0$. The $W_1$-module $T(0, 0, +)$ has length two with a simple submodule isomorphic to $\mathbb C$ and a simple quotient isomorphic to $T(1,1,+)$.
\item[(iv)] The $W_1$-module $T(\lambda, \lambda, -)$ is simple if and only if $\lambda \neq 1$. The $W_1$-module $T(1, 1, -)$ has length two with a simple submodule isomorphic to $T(0,0,-)$, and a simple quotient isomorphic to $\mathbb C$.
\varepsilonilonnd{itemize}
\varepsilonilonnd{proposition}
In view of the above proposition, for $\lambda \neq 0$ we set $L(\lambda,+):=T(\lambda, \lambda, +)$ and $L(\lambda, -) = T(\lambda+1, \lambda+1, -)$. We also set $L(0) := \mathbb C$. Note that $L(\lambda, +)$ and $L(\lambda, -)$ are highest weight modules with highest weight $\lambda$ with respect to the Borel subalgebras ${\mathfrak b} (+) = \Span \{ D_i \; | \; i \in {\mathbb Z}_{\mathfrak{g}eq 0} \}$ and ${\mathfrak b} (-) = \Span \{ D_i \; | \; i \in \{0,-1\}\}$, respectively.
We now look at the structure of the tensor $\mathcal A$-modules. In the case $c = 0$, we simply restate Proposition \ref{prop-tens-w1-mod} replacing the statements for $T(\nu,\lambda)$, $T(\lambda ,\lambda,\pm)$ by $T(\nu,\lambda,0)$, $T(\lambda ,\lambda,0,\pm)$, respectively. We also write $L(\lambda,0,+)$ and $L(\lambda,0,-)$ for $T(\lambda,\lambda,0,+)$ and $T(\lambda+1,\lambda+1,0,-)$ if $\lambda \neq 0$, and $L(0,0) = \mathbb C$.
For tensor $\mathcal A$-modules with nonzero central charge we have the following.
\begin{proposition} \label{prop-tens-a1-mod} Let $\lambda, \nu \in {\mathbb C}$ and let $c \neq 0$.
\begin{itemize}
\item[(i)] The $\mathcal A$-modules $T(\nu_1, \lambda_1,c)$ and $T(\nu_2, \lambda_2,c)$ are isomorphic if and only if $\nu_1-\nu_2 \in {\mathbb Z}$ and $\lambda_1 = \lambda_2$.
\item[(ii)] The $\mathcal A$-module $T(\nu, \lambda,c)$ is simple if and only if $\lambda - \nu \notin {\mathbb Z}$.
\item[(iii)] The $\mathcal A$-modules $T(\lambda, \lambda, c, +)$ and $T(\lambda, \lambda, c, -)$ are simple for all $\lambda \in \mathbb C$.
\varepsilonilonnd{itemize}
\varepsilonilonnd{proposition}
Naturally, for $c \neq 0$ we set $L(\lambda,c,+) := T(\lambda,\lambda,c,+)$ and $L(\lambda,c,-) := T(\lambda+1,\lambda+1,c,+)$.
We finish this subsection with the classification theorem for all simple weight $W_1$-modules with finite weight multiplicities due to O. Mathieu, see \cite{M-Vir}.
\begin{theorem} \label{th-class-w1}
Every simple weight $W_1$-module with finite weight multiplicities is isomorphic to a module in the following list: $T(\nu, \lambda)$, $\lambda -\nu \notin {\mathbb Z}$, $L(\varepsilonilonta, +)$, $L(\varepsilonilonta, -)$, $\varepsilonilonta \neq 0$, $L(0)$. The only isomorphisms among the modules in the list are: $T(\nu, \lambda) \simeq T(\nu + n, \lambda)$ for $n \in {\mathbb Z}$ and $\lambda - \nu \notin {\mathbb Z}$; $T(\nu,0) \simeq T(\nu,1)$, for $\nu \notin \mathbb Z$.
\varepsilonilonnd{theorem}
\subsection{Twisted localization of weight modules}
We first introduce the twisted localization functor in a general setting. Let $\mathcal U$, $\mathcal H = \mathbb C [\mathfrak{h}]$ be as in \S \ref{subsec-wht}. Let $a$ be an ad-nilpotent element of $\mathcal U$. Then the set $\langle a \rangle = \{ a^n \; | \; n \mathfrak{g}eq 0\}$ is an Ore subset of $\mathcal U$ (see for example \S 4 in \cite{M}) which allows us to define the $\langle a \rangle$-localization $D_{\langle a \rangle} \mathcal U$ of $\mathcal U$. For a $\mathcal U$-module $M$ by $D_{\langle a \rangle} M = D_{\langle a \rangle} {\mathcal U} \otimes_{\mathcal U} M$ we denote the $\langle a \rangle$-localization of $M$. Note that if $a$ is injective on $M$, then $M$ is isomorphic to a submodule of $D_{\langle a \rangle} M$. In the latter case we will identify $M$ with that submodule, and will consider $M$ as a submodule of $D_{\langle a \rangle} M$.
We next recall the definition of the generalized conjugation of $D_{\langle a \rangle} \mathcal U$ relative to $x \in {\mathbb C}$. This is the automorphism $\phi_x : D_{\langle a \rangle} \mathcal U \to D_{\langle a \rangle} \mathcal U$ given by $$\phi_x(u) = \sum_{i\mathfrak{g}eq 0} \binom{x}{i} \ad (a)^i (u) a^{-i}.$$ If $x \in \mathbb Z$, then $\phi_x(u) = a^xua^{-x}$. With the aid of $\phi_x$ we define the twisted module $\mathbb Phi_x(M) = M^{\phi_x}$ of any $D_{\langle a \rangle} \mathcal U$-module $M$. Finally, we set $D_{\langle a \rangle}^x M = \mathbb Phi_x D_{\langle a \rangle} M$ for any $\mathcal U$-module $M$ and call it the \varepsilonilonmph{twisted localization} of $M$ relative to $a$ and $x$. We will use the notation $a^x\cdot m$ (or simply $a^x m$) for the element in $D_{\langle a \rangle}^x M$ corresponding to $m \in D_{\langle a \rangle} M$. In particular, the following formula holds in $D_{\langle a \rangle}^{x} M$:
$$
u (a^{x} m) = a^{x} \left( \sum_{i\mathfrak{g}eq 0} \binom{-x}{i} \ad (a)^i (u) a^{-i}m\right)
$$
for $u \in \mathcal U$, $m \in D_{\langle a \rangle} M$.
We easily check that if $a$ is a weight element of $\mathcal U$ (i.e. $a \in {\mathcal U}^{\mu}$ for some $\mu \in {\mathfrak h}^*$) and $M$ is a (generalized) $({\mathcal U}, {\mathcal H})$-weight module, then $D_{\langle a \rangle}^x M$ is a (generalized) $({\mathcal U}, {\mathcal H})$-weight module. We will apply the twisted localization functor for several pairs $({\mathcal U}, {\mathcal H})$, and in particular in the following two cases: $\mathcal U = U(\mathcal A)$, $\mathcal H = \mathbb C[\mathfrak{h}_{\mathcal A}]$; and $\mathcal U = U(W_2)$, $\mathcal H= \mathbb C[\mathfrak{h}_{W_2}]$.
\begin{lemma} \label{lem-tw-loc-simple}
Let $a \in {\mathcal U}$ be an $\ad$-nilpotent weight element in ${\mathcal U}$, $M$ be a simple $a$-injective weight ${\mathcal U}$-module, and let $z \in \mathbb C$. If $N$ is any simple submodule of $D^z_{\langle a \rangle}M$, then $D_{\langle a \rangle}M \simeq D^{-z}_{\langle a \rangle}N$. In particular, if $a$ acts bijectively on $M$, $M \simeq D^{-z}_{\langle a \rangle}N$.
\varepsilonilonnd{lemma}
\begin{proof}
We use the fact that if $M$ is a simple weight ${\mathcal U}$-module, then $D_{\langle a \rangle}M$ and $D_{\langle a \rangle}^z M$ are simple $D_{\langle a \rangle} {\mathcal U}$-modules. So, if $N$ is any simple submodule of $D^z_{\langle a \rangle}M$, then $D_{\langle a \rangle} N$ is a submodule of $D^z_{\langle a \rangle}M$. This forces $D_{\langle a \rangle} N \simeq D^z_{\langle a \rangle}M$ and $D^{-z}_{\langle a \rangle} N \simeq D_{\langle a \rangle}M$. If $a$ acts bijectively, then $M \simeq D_{\langle a \rangle}M$. \varepsilonilonnd{proof}
The above lemma will be applied both for $\mathcal U = U(\mathcal A)$ and $\mathcal U = U(W_2)$. For reader's convenience, we list some (but not all) ad-locally nilpotent weight elements $a$ in $\mathcal U$ in these two cases:
\begin{itemize}
\item $a = D_{-1}, I_j$, $j \mathfrak{g}eq 0$ for $\mathcal U = U(\mathcal A)$;
\item $a = \partial_1, \partial_2, x_1\partial_2, x_2\partial_1$ for $\mathcal U = U(W_2)$.
\varepsilonilonnd{itemize}
\begin{lemma} \label{lem-iso-loc-a} Let $\nu \notin \mathbb Z$. Then the following $\mathcal A$-module isomorphisms hold.
\begin{itemize}
\item[(i)] $D_{\langle I_1 \rangle}^{\nu} T (\lambda, \lambda, c, +) \simeq T (\lambda + \nu, \lambda, c)$, whenever $c \neq 0$.
\item[(ii)] $D_{\langle D_{-1} \rangle}^{-\nu} T (\lambda, \lambda, c, -) \simeq T (\lambda + \nu, \lambda,c)$.
\item[(iii)] $D_{\langle D_{-1} \rangle}^{\nu-\varepsilonilonta} T (\lambda+ \nu, \lambda,c) \simeq T (\lambda + \varepsilonilonta, \lambda,c)$, whenever $ \varepsilonilonta \notin {\mathbb Z}$.
\varepsilonilonnd{itemize}
\varepsilonilonnd{lemma}
\begin{proof}
(i) Note that if $c \neq 0$, the set $\{ I_1^{\nu + \varepsilonilonll} (x^{\lambda} \otimes v_{\lambda})\; | \; \varepsilonilonll \in {\mathbb Z} \}$ forms a basis of $D_{\langle I_1 \rangle}^{\nu} T (\lambda, \lambda,c, +)$. Then it is easy to check that the map
$$ I_1^{\nu + \varepsilonilonll} (x^{\lambda} \otimes v_{\lambda}) \mapsto c^{\nu + \varepsilonilonll} x^{\lambda + \nu + \varepsilonilonll} \otimes v_{\lambda}$$
provides an isomorphism $D_{\langle I_1 \rangle}^{\nu} T (\lambda, \lambda, c,+) \simeq T (\lambda + \nu, \lambda,c)$.
(ii) Note that the set $\{ D_{-1}^{-\nu - \varepsilonilonll -1} (x^{\lambda-1} \otimes v_{\lambda} + T(\lambda, \lambda, c, +))\; | \; \varepsilonilonll \in {\mathbb Z} \}$ forms a basis of $D_{\langle D_{-1} \rangle}^{-\nu} T (\lambda, \lambda, c, -)$. Then it is easy to check that the map
$$ D_{-1}^{-\nu - \varepsilonilonll -1} (x^{\lambda-1} \otimes v_{\lambda} + T(\lambda, \lambda, c, +)) \mapsto D_{-1}(\nu, \varepsilonilonll)x^{\lambda + \nu + \varepsilonilonll} \otimes v_{\lambda},$$
where $D_{-1}(\nu, \varepsilonilonll) = \nu (\nu - 1)...(\nu - (-\varepsilonilonll -1))$ if $l < 0$ and $D_{-1}(\nu, \varepsilonilonll) = \frac{1}{(\nu+1)...(\nu+\varepsilonilonll)}$ if $\varepsilonilonll \mathfrak{g}eq 0$, provides an isomorphism $D_{\langle D_{-1} \rangle}^{-\nu} T (\lambda, \lambda, c, -) \simeq T (\lambda + \nu, \lambda,c)$.
Part (iii) easily follows from (ii) and the additive property of the twisted localization functors: $D_{\langle D_{-1} \rangle}^{\nu_1 + \nu_2} M \simeq D_{\langle D_{-1} \rangle}^{\nu_1} D_{\langle D_{-1} \rangle}^{\nu_2} M$. \varepsilonilonnd{proof}
\begin{remark}
Note that the isomorphism in Lemma \ref{lem-iso-loc-a}(ii) holds even in the case $(\lambda, c) = (1,0)$, namely when $ T (\lambda, \lambda, c, -)$ and $T (\lambda + \nu, \lambda,c)$ are not simple. Also, the coefficients $D_{-1} (\nu, \varepsilonilonll)$ in the proof of Lemma \ref{lem-iso-loc-a}(ii) can also be defined in terms of Gamma-functions: $D_{-1}(\nu, \varepsilonilonll) = \frac{\Gamma (\nu + 1)}{\Gamma (\nu + \varepsilonilonll + 1)}$.
\varepsilonilonnd{remark}
\subsection{Parabolic subalgebras of $W_2$ and Penkov-Serganova Parabolic Induction Theorem}
To keep the content of the paper short we will avoid the general definition of a parabolic subalgebra of $W_n$. Such a definition in terms of flags of real subspaces can be found in \S1 of \cite{PS}. Alternatively, one can use the general definition of a parabolic set of roots $P $ and then define a parabolic subalgebra $\mathfrak{p}_P$ associated with $P$ following \cite{GY}. The problem is that the root system of $W_n$ is neither symmetric nor finite.
In what follows we fix $\sigma$ to be the automorphism of $\Delta_{W_2}$ interchanging $\varepsilonilon_1$ and $\varepsilonilon_2$. This automorphism naturally defines an automorphism of $\mathfrak{h}_{W_2}^*$ , $\mathfrak{h}_{W_2}$ and $W_2$. With a slight abuse of notation we will denote all resulting automorphisms by $\sigma$.
We have a natural embedding of $\mathfrak{sl}_{n+1}$ in $W_n$ arising from the infinitesimal action of the group $PSL (n+1)$ on ${\mathbb C}P^n$. In explicit terms, the embedding $\mathbb Phi$ is defined by $E_{ij} \mapsto x_i \partial_j$, $1\leq i,j \leq n$, $E_{0k} \mapsto x_i {\mathcal E}$, $E_{k0} \mapsto -\partial_k$, $1\leq k \leq n$, where ${\mathcal E} = \sum_{k=1}^nx_k\partial_k$.
With the embedding of $\mathfrak{sl}(n+1)$ in $W_n$ in mind we fix the Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{sl}(n+1)$ to be the one corresponding to $\mathfrak{h}_W$ under the embedding $\mathbb Phi$. Again with a slight abuse of notation, the root system of $\mathfrak{sl}_{n+1}$ relative to $\mathfrak h$ will be denoted by $\Delta_{\mathfrak{s}} = \{ \varepsilonilon_i - \varepsilonilon_j \; | \; 0 \leq i \neq j \leq n\}$. Since we will deal with parabolic subalgebras of $W_n$ that are induced from parabolic subalgebras of $\mathfrak{sl}_{n+1}$, we will limit out attention to this case only.
We now recall one of the few equivalent definitions of a parabolic subalgebra of $\mathfrak{sl}_{n+1}$.
A \varepsilonilonmph{parabolic subset of roots} of $\mathfrak{s} = \mathfrak{sl}_{n+1}$ is a proper subset $P_\mathfrak{s}$ of $\Delta_{\mathfrak{s}} $ for which
$$
\Delta_{\mathfrak{s}} = P_\mathfrak{s} \cup(-P_\mathfrak{s}) \; \; \; \mbox{and} \; \;
\alpha, \beta \in P_\mathfrak{s} \; {\text { with }} \; \alpha + \beta \in \Delta_\mathfrak{s}
\; \;
{\text { implies }} \; \; \alpha + \beta \in P_\mathfrak{s}.
$$
For a parabolic subset of roots
$P_\mathfrak{s}$ of $\Delta_{\mathfrak{s}} $, we call $L_\mathfrak{s} := P_\mathfrak{s} \cap (-P_\mathfrak{s})$
the {\varepsilonilonm{Levi component}} of $P_\mathfrak{s}$, $N_\mathfrak{s}^+ := P_\mathfrak{s} \backslash (-P_\mathfrak{s})$ the
{\varepsilonilonm{nilradical}} of $P_\mathfrak{s}$, and $P_\mathfrak{s} = L_\mathfrak{s} \sqcup N_\mathfrak{s}^+$ the Levi decomposition
of $P$. We call
$$
\mathfrak{p}_{P_\mathfrak{s}} := \mathfrak{h} \oplus \left( \bigoplus_{\alpha \in P_\mathfrak{s}} \, \mathfrak{s}^\alpha \right).
$$
a parabolic subalgebra of $\mathfrak{s}$ associated with $P_\mathfrak{s}$.
If $P_{\mathfrak{s}}= L_{\mathfrak{s}} \sqcup N_{\mathfrak{s}}^+ $ is a parabolic subset of roots of $\mathfrak s$, then we call $P= L \sqcup N^+ $ \varepsilonilonmph{a parabolic subset of roots of $W_n$ induced from $P_{\mathfrak s}$}, where $L:= {\mathbb Z} L_{\mathfrak{s}} \cap \Delta_W$ and $N^+ := \left( {\mathbb Z}_{\mathfrak{g}eq 0}P_{\mathfrak s} \cap \Delta_W \right) \setminus L$. If $P_{\mathfrak{s}}= L_{\mathfrak{s}} \sqcup N_{\mathfrak{s}}^+ $ is a parabolic subset of roots of $\mathfrak s$, and $P = L \sqcup N^+$ is the parabolic subset of roots of $W_n$ induced from $P_{\mathfrak s}$, we set $N_{\mathfrak s}^{-} = \Delta_{\mathfrak s} \setminus P_{\mathfrak s}$ and $N^{-} = \Delta_W \setminus P$. The reader is referred to Lemma 3 in \cite{PS} for a proof of the fact that every parabolic subset of roots of $W_n$ induced from one of $\mathfrak s$ is indeed a parabolic subset of roots of $W_n$. We call $P^- = L \sqcup N^-$ the \varepsilonilonmph{opposite to $P$} parabolic subset. Analogously to the case of $\mathfrak{sl}_{n+1}$ we define
$$
\mathfrak{p}_{P} := \mathfrak{h} \oplus \left( \bigoplus_{\alpha \in P} \, \mathfrak{s}^\alpha \right).
$$
to be the parabolic subalgebra of $W_n$ associated with $P$ (or with $P_{\mathfrak s}$). By $\mathfrak p^-$ we denote the parabolic subalgebra associated with $P^-$ and call it the opposite to $\mathfrak p$ parabolic subalgebra.
If $P_{\mathfrak s} = L_{\mathfrak{s}} \sqcup N_{\mathfrak{s}}^+ $ is a parabolic subset of roots of $\mathfrak s$ for which $L_{\mathfrak{s}} = \varepsilonilonmptyset$, we will call the corresponding subalgebras $\mathfrak{p}_{P_\mathfrak{s}} $ and $\mathfrak{p}_{P}$ \varepsilonilonmph{Borel subalgebras} of $\mathfrak{sl}_{n+1}$ and $W_n$, respectively.
Below we list all parabolic subsets of roots of $W_2$ induced from parabolic subsets of roots of $\Delta_{\mathfrak{s}}$ of $\mathfrak{sl} (3)$ together with the corresponding parabolic subalgebras.
\begin{example} \label{ex-par-w2} For a subset $J$ of the real vector space ${\mathbb R} \Delta_W$, let $P(J) = \{\alpha \in \Delta_W \; | \; (\alpha, s) \in {\mathbb R}_{\leq 0} \mbox{ for every } s\in J\}$. Then all parabolical subset of roots of $W_2$ induced by parabolic subsets of roots of $\mathfrak{sl} (3)$ are of the form $P(J)$ for a set $J$ consisting of one or two elements. All possible sets $J$ together with the notation that will be used for the corresponding parabolic subset of roots $P(J)$ and the parabolic subalgebra $\mathfrak p (J) = \mathfrak{p}_{P(J)}$ are listed below.
\begin{itemize}
\item[(i)] $J = \{ \varepsilonilon_1 - \varepsilonilon_0\}$, $P(1^+)$, ${\mathfrak p}(1^+)$.
\item[(ii)] $J = \{ \varepsilonilon_0 - \varepsilonilon_1\}$, $P(1^-)$, ${\mathfrak p}(1^-)$.
\item[(iii)] $J = \{ \varepsilonilon_2 - \varepsilonilon_0\}$, $P(2^+)$, ${\mathfrak p}(2^+)$.
\item[(iv)] $J = \{ \varepsilonilon_0 - \varepsilonilon_2\}$, $P(2^-)$, ${\mathfrak p}(2^-)$.
\item[(v)] $J = \{ \varepsilonilon_1 + \varepsilonilon_2\}$, $P(12^+)$, ${\mathfrak p}(12^+)$.
\item[(vi)] $J = \{ -\varepsilonilon_1 - \varepsilonilon_2\}$, $P(12^-)$, ${\mathfrak p}(12^-)$.
\item[(vii)] $J = \{\varepsilonilon_1 - \varepsilonilon_0, \varepsilonilon_0 - \varepsilonilon_2\}$, $P(1^+,2^-)$, ${\mathfrak p}(1^+,2^-)$.
\item[(viii)] $J = \{\varepsilonilon_0 - \varepsilonilon_1, \varepsilonilon_2 - \varepsilonilon_0\}$, $P(1^-,2^+)$, ${\mathfrak p}(1^-,2^+)$.
\item[(ix)] $J = \{\varepsilonilon_0 - \varepsilonilon_2, - \varepsilonilon_1 - \varepsilonilon_2\}$, $P(2^-,12^-)$, ${\mathfrak p}(2^-,12^-)$.
\item[(x)] $J = \{\varepsilonilon_1 - \varepsilonilon_0, \varepsilonilon_1 + \varepsilonilon_2\}$, $P(1^+,12^+)$, ${\mathfrak p}(1^+,12^+)$.
\item[(xi)] $J = \{\varepsilonilon_0 - \varepsilonilon_1, - \varepsilonilon_1 - \varepsilonilon_2\}$, $P(1^-,12^-)$, ${\mathfrak p}(1^-,12^-)$.
\item[(xii)] $J = \{\varepsilonilon_2 - \varepsilonilon_0, \varepsilonilon_1 + \varepsilonilon_2\}$, $P(2^+,12^+)$, ${\mathfrak p}(2^+,12^+)$.
\varepsilonilonnd{itemize}
\varepsilonilonnd{example}
\begin{remark}
Note for example that $P(1^+) = {\mathbb Z}_{\leq 0} \varepsilonilon_1 + {\mathbb Z} \varepsilonilon_2$ which seems counterintuitive. This sign change is imposed in order to match the notation of the parabolic subalgebras with the notation of the corresponding induced modules, see for example Proposition \ref{prop-w2-bnd-nec}.
\varepsilonilonnd{remark}
The following parabolic induction theorem follows from Lemma 11 in \cite{PS}.
\begin{theorem} \label{th-par-ind}
Let $M$ be a simple weight $W_n$-module with finite weight multiplicities. Then there is a parabolic subalgebra $\mathfrak p = \mathfrak{l} \oplus \mathfrak{n}^+$ of $W_n$ induced from a parabolic subalgebra of $\mathfrak s = \mathfrak{sl}_{n+1}$ and a simple $\mathfrak{p}$-module $S$ with a trivial action of $ \mathfrak{n}^+$ such that $M$ is a quotient of the induced module $U(W_n) \otimes_{U(\mathfrak p)} S$. Moreover, there is $\lambda \in \Supp L$ such that $\lambda + \alpha \notin \Supp M$ for all $\alpha$ in $N^+ = \Delta_{\mathfrak{n}^+}$.
\varepsilonilonnd{theorem}
\subsection{Supports of $W_2$-modules}
In what follows we describe all possible supports of simple weight $W_2$-modules with finite weight mutliplicities, see Example 2 in \cite{PS}.
\begin{proposition} \label{prop-w2-par}
All possible supports of simple weight $W_2$-modules with finite weight multiplicities are exactly in one of the following forms:
\begin{itemize}
\item[(i)--(xii)] $\lambda + P(J)$ for $J$ being one in the list {\rm (i)-(xii)} of Example \ref{ex-par-w2},
\item[(xiii)] $\left(\lambda + P(2^-,12^-) \right) \cap \left(\sigma (\lambda) + P(1^-,12^-) \right) $,
\item[(xiv)] $\left(\lambda + P(1^+,12^+) \right) \cap \left(\sigma (\lambda) + P(2^+,12^+) \right) $,
\item[(xv)] $\lambda + {\mathbb Z}^2$,
\item[(xvi)] $\{ 0\}$,
\varepsilonilonnd{itemize}
with the conditions on $\lambda$ as follows: no restrictions on $\lambda$ in cases {\rm (i)-(vi), (xv)}; $\lambda \neq 0$ in cases {\rm (vii)-(viii)}; $\lambda_2 - \lambda_1 \notin {\mathbb Z}_{\mathfrak{g}eq 0}$ in cases {\rm (ix)-(x)}; $\lambda_1 - \lambda_2 \notin {\mathbb Z}_{\mathfrak{g}eq 0}$ in cases {\rm (xi)-(xii)}; $\lambda_1 - \lambda_2 \in {\mathbb Z}_{\mathfrak{g}eq 0}$, $\lambda \neq 0$ in cases {\rm (xiii)-(xiv)}.
\varepsilonilonnd{proposition}
A simple weight module $M$ of type (xv), i.e. such that $\Supp M = \lambda + \mathbb Z^2$, will be called a \varepsilonilonmph{dense module}.
\section{Classification of simple bounded $\mathcal A$-Modules}
We start with a general property of the bounded $\mathcal A_n$-modules.
\begin{proposition} \label{prop-fin-len}
Every bounded $\mathcal A_n$-module (and, hence $W_n$-module) whose support is a subset of $\lambda + {\mathbb Z}^n$ for some $\lambda$ has finite length.
\varepsilonilonnd{proposition}
\begin{proof} This follows from the fact that $\mathcal A_n$ has a subalgebra isomorphic to $\mathfrak{sl}_{n+1}$ and that the statement holds for bounded $\mathfrak{sl}_{n+1}$-modules with the same support property, see Lemma 3.3 in \cite{M}.
\varepsilonilonnd{proof}
In the rest of this section we work with $n=1$, i.e. with $\mathcal A$.
\begin{lemma} \label{lem-loc-fin}
Let $M$ be a simple weight $\mathcal A$-module of central charge $c$. If $D_{-1}$ acts finitely on $M$, then either $M \simeq L (\lambda , c, +)$ for some $\lambda, c$, $(\lambda, c) \neq (0,0)$, or $M \simeq L (0,0)$.
\varepsilonilonnd{lemma}
\begin{proof} Let $v \in M^{\lambda}$ be such that $D_{-1} v = 0$. Then $M = U(\mathcal A)v $ is an $\mathcal A^-$-highest weight module. Thus $U( \mathcal A) v$ is the unique simple quotient of the induced module\\ $U(\mathcal A) \otimes _{U(\mathcal A ^- \oplus \mathcal A^0)} \mathbb C_{(\lambda, c)}$, where $\mathbb C_{(\lambda, c)}$ is the 1-dimensional $\mathcal A^0$-module of weight $(\lambda,c)$ on which $\mathcal A^-$ acts trivially. However, we know that $L(\lambda , c, +)$ and $L (0 , 0)$ are such simple highest weight modules.
\varepsilonilonnd{proof}
\begin{theorem} \label{th-bounded-a}
Let $M$ be a simple bounded $\mathcal A$-module of central charge $c$. If $c = 0$, then $M$ is a simple bounded $W_1$-module, i.e. it is isomorphic to one of the modules listed in Theorem \ref{th-class-w1} with trivial action of $I_k$, $k\mathfrak{g}eq 0$. If $c\neq 0$, then $M$ is isomorphic to one of the following: $T (\nu , \lambda, c)$ ($\lambda - \nu \notin {\mathbb Z}$), $L(\lambda, c, +)$, $L (\lambda, c, -)$. The only isomorphisms of the listed $\mathcal A$-modules for $c\neq 0$ are: $ T (\nu , \lambda, c) \simeq T(\nu +n , \lambda, c)$, $n \in {\mathbb Z}$, for $\nu \notin \mathbb Z$.
\varepsilonilonnd{theorem}
\begin{proof} By Lemma \ref{lem-loc-fin} we know that the result holds if $D_{-1}$ acts finitely on $M$. So, for the rest of the proof, we can assume that $D_{-1}$ acts injectively on $M$. We split the proof in two parts depending on the central charge $c$. In all statements we assume that $M$ is a simple bounded $\mathcal A$-module of central charge $c$.
\noindent{\it Case 1: Nonzero central charge, i.e. $c \neq 0$.}
We split this case into two subclasses depending on the action of $I_1$ on $M$.
\begin{lemma} \label{lem-1} If $c \neq 0$, $D_{-1}$ acts injectively on $M$, and $I_1$ acts finitely on $M$, then $M \simeq L(\lambda , c, -)$ for some $\lambda$.
\varepsilonilonnd{lemma}
\textit{Proof of Lemma \ref{lem-1}.} Let $\mathfrak a = \Span \{ D_{-1}, I_0, I_1\}$. Note that $\mathfrak a$ is a Lie subalgebra of $\mathcal A$ which is isomorphic to the three-dimensional Heisenberg Lie algebra. Furthermore, each weight space $M^{\lambda}$, $\lambda \in \mathfrak{h}_{\mathcal A}^*$, is $I_1D_{-1}$-invariant, so $M$ considered as $\mathfrak a$-module is a generalized weight $(\mathfrak a, \mathfrak{h}_{\mathfrak a})$-module for $ \mathfrak{h}_{\mathfrak a} = \Span \{ I_1D_{-1}\}.$
The classification of simple generalized weight $(\mathfrak a, \mathfrak{h}_{\mathfrak a})$-modules with nonzero central charge $c$ (equivalently, generalized weight modules of the Weyl algebra $U(\mathfrak a)/(I_0 - c)$) on which $I_1$ acts finitely is well-known. All such modules are simple weight $(\mathfrak a, \mathfrak{h}_{\mathfrak a})$-modules and are isomorphic to the module $\mathbb C[D_{-1}]$, such that $D_{-1} (D_{-1}^k) = D_{-1}^{k+1}$, $I_0 (D_{-1}^k)= c D_{-1}^k $, and $I_1 (D_{-1}^k) = - c k D_{-1}^{k-1}$ (see for example \S2 in \cite{GS1}). Note that if $c=0$ then we should add the trivial module ${\mathbb C}$ in that list, but this case is addressed in Case 2 below.
Let $d$ be the degree of $M$. Looking at the $\mathcal A$-support of $M$ we see that $M$ can not have more than $d$ simple ${\mathfrak a} $-subquotients. Indeed, if the converse is true, all such subquotients will be isomorphic as ${\mathfrak a} $-modules to $\mathbb C[D_{-1}]$, and then we can easily find an $\mathcal A$-weight space of $M$ of dimension bigger than $d$. In particular, $M$ has finitely many simple ${\mathfrak a}$-suqbquotients $M_i$ and $M_i = \Span \{ D_{-1}^k m_i \; | \; k \mathfrak{g}eq 0\}$ for some $m_i \in M$. Hence the $\mathcal A$ support of $M$ is bounded from the right, i.e. there is $\lambda \in \Supp M$ such that $\mbox{Supp} M \subset \lambda + {\mathbb Z}_{\leq 0}$. Therefore $M$ is a simple $\mathcal A^+$-highest weight module whose highest weight is $\lambda$, that is $M \simeq L(\lambda , c, -)$.
\begin{lemma} \label{lem-2} Let $c \neq 0$ and let both $D_{-1}$ and $I_1$ act injectively on $M$.
\begin{itemize}
\item[(i)]There are $\nu \in \mathbb C$ and a simple $\mathcal A$-module $N$ on which $D_{-1}$ acts finitely, such that $M \simeq D_{\langle I_1 \rangle}^\nu N$.
\item[(ii)] $M \simeq T (\nu, \lambda , c)$ for some $\lambda \in \mathbb C$.
\varepsilonilonnd{itemize}
\varepsilonilonnd{lemma}
\textit{Proof of Lemma \ref{lem-2}.} First note that since $D_{-1}$ and $I_1$ act injectively on $M$, then $I_1$ acts bijectively on $M$. In particular, $D_{\langle I_1 \rangle} M \simeq M$. Now consider $D_{\langle I_1 \rangle}^{-\nu} M$, for any $\nu \in \mathbb C$. Let $\lambda \in \mbox{Supp} M$. Then $I_1 D_{-1} |_{M^\lambda}$ is an endomorphism on the finite-dimensional vector space $M^\lambda$. Let $\alpha$ be an eigenvalue of this endomorphism and let $I_1 D_{-1}v = \alpha v$ for $v \in M^{\lambda}$. Then
\begin{align*}
D_{-1} (I_1 ^{-\nu} v) = & I_1^{-\nu} \left ( \sum _{i \mathfrak{g}eq 0} \binom{\nu}{i} \left (\ad I_1\right)^i (D_{-1})I_1^{-i} (v) \right ) \\
= & I_1 ^{-\nu } \left ((D_{-1} + \nu I_0 I_{1}^{-1}) (v) \right ) \\
= & I_1 ^{-\nu -1 } \left ((\alpha+ \nu c ) v\right )
\varepsilonilonnd{align*} We first note that since both $D_{-1}$ and $I_1$ act injectively on $M$ and the weight space of $M$ are finite dimensional, then $I_1$ (and $D_{-1}$) act bijectively on $M$, hence $M \simeq D_{\langle I_1 \rangle} M$.
If $\nu = - \frac{\alpha }{c}$, then $D_{-1} (I_1 ^\nu m) = 0 $. The elements of $D_{\langle I_1 \rangle}^{-\nu} M$ on which $D_{-1}$ acts finitely form a submodule $N'$ of $D_{\langle I_1 \rangle}^{-\nu} M$. Then by Proposition \ref{prop-fin-len}, $N'$ has finite length, so we can choose a simple $\mathcal A$-submodule $N$ of $N'$. Then by Lemma \ref{lem-tw-loc-simple}, $M \simeq D_{\langle I_1 \rangle}^\nu N$ which proves part (i).
To prove (ii), we apply Lemma \ref{lem-loc-fin} and obtain that $N \simeq L (\lambda, c, +)$. Therefore $M \simeq D_{\langle I_1 \rangle} M \simeq D_{\langle I_1 \rangle}^{\nu} L (\lambda, c,+) \simeq T (\lambda+ \nu, \lambda, c)$. The last isomorphism follows from Lemma \ref{lem-iso-loc-a}(i).
\noindent{\it Case 2: Zero central charge, i.e. $c = 0$.}
In this case we have the following lemma.
\begin{lemma} \label{lem-3} Suppose that $c =0$ and $D_{-1}$ acts injectively on $M$. Then $I_k =0$ on $M$ for all $k\mathfrak{g}eq 0$. In particular, $M$ is a simple $W_1$-module, and thus is isomorphic to one of the modules $T(\nu, \lambda , 0)$ ($\lambda - \nu \notin \mathbb Z$), $L (\varepsilonilonta , 0,-)$, $\varepsilonilonta \neq 0$.
\varepsilonilonnd{lemma}
\textit{Proof of Lemma \ref{lem-3}.} Let $d$ be the degree of $M$, let $\lambda \in \mbox{Supp}\, M$ and consider the endomorphisms $S = D_{-1}^2 I_2$ and $T = D_{-1} I_1 |_{M^{\lambda}}$ on the vector space $M^{\lambda}$ of dimension at most $d$. Using that $D_{-1}$ and $I_1$ commute, we easily check that $[T,S] = 2T^2$ and $[T^N,S] = 2N T^{N+2}$. Therefore the trace of the endomorphism $T^N = \left[\frac{1}{2N-4}T^{N-2}, S\right]$ is zero for all $N>2$. But then the sum of the $N$-th powers, $N > 2$, of the eigenvalues of $T$ is zero and hence $T$ is nilpotent. Thus $T^d = 0$. But using again that $I_1$ and $D_{-1}$ commute we find that $I_1^d = 0$ on $M$. Fix $N_0>0$ such that $I_1^{N_0} = 0$ and $I_1^{N_0-1} \neq 0$ on $M$. Let $v_0 \in M$ be such that $I_1^{N_0-1}(v_0) \neq 0$. Then for $k\mathfrak{g}eq 1$, we have
$$
0 = D_{k-1}(I_1^{N_0}(v_0)) = I_1^{N_0} (D_{k-1}(v_0)) + N_0I_kI_1^{N_0-1}(v_0).
$$
Therefore $I_k(v) = 0$ for every $k\mathfrak{g}eq 1$ where $v = I_1^{N_0-1}(v_0)$. This implies that the set $M'$ of all $w$ with the property $I_kw =0$ for all $k\mathfrak{g}eq 0$ is nonzero. Since $M'$ is an $\mathcal A$-submodule of $M$, we have $M'=M$, which proves the first assertion of the lemma. The second part of the lemma follows from the classification of the simple weight $W_1$-modules, i.e. from Theorem \ref{th-class-w1}.
\varepsilonilonnd{proof}
\section{Classification of simple bounded $W_2$-modules}
In this section we classify all simple bounded $W_2$-modules, i.e. weight $W_2$-modules with uniformly bounded set of weight multiplicities.
In what follows we set $\mathfrak a := \Span \{ x_i\partial_j \; | \; 1\leq i,j\leq 2\}$. We know that the Cartan subalgebras of $W_2$, $\mathfrak a$ and $\mathfrak s$ coincide with $\mathfrak h_{W_2}$. Using this and the isomorphisms $\mathfrak s \simeq \mathfrak{sl}_3$, $\mathfrak a \simeq \mathfrak{gl}_2$ we will often write the weights of $\mathfrak{sl}_3$ and $\mathfrak{gl}_2$ as pairs $(\lambda_1,\lambda_2)$. In some cases, when using representation theory results for $\mathfrak{sl}_3$ we will write an $\mathfrak{sl}_3$-weight $\lambda$ as $\lambda = \lambda_0 \varepsilonilon_0 + \lambda_1 \varepsilonilon_1 + \lambda_2 \varepsilonilon_2$ with $\lambda_0+\lambda_1+\lambda_2 = 0$ (in particular $\lambda = (\lambda_1,\lambda_2)$ as an element of $\mathfrak{h}_{W_2}^*$).
We also set $W(x_i) := \Der (\mathbb C[x_i])$, $i=1,2$, $ \mathcal A (x_1) := W(x_1) \ltimes \left( \mathbb C[x_1] (x_2\partial_2)\right)$, and $ \mathcal A (x_2) := W(x_2) \ltimes \left( \mathbb C[x_2] (x_1\partial_1)\right)$. In particular, $\mathcal A (x_1) \simeq \mathcal A (x_2) \simeq \mathcal A$.
\begin{definition}
Let $J$ be a set from the list {\rm(i)-(xii)} of Example \ref{ex-par-w2}. We say that a simple weight $W_2$-module $M$ with finite weight multiplicities is \varepsilonilonmph{of type $J$} if $M$ is the simple quotient of the module $U(W_2) \otimes_{U(\mathfrak p (J))} L$ for some simple $\mathfrak p (J)$-module $L$.
\varepsilonilonnd{definition}
We proceed with the classification of simple bounded $W_2$-modules $M$ in three steps depending on the type of $M$.
\subsection{Classification of simple bounded highest weight $W_2$-modules}
In this subsection we classify all bounded highest weight $W_2$-modules, namely all modules from cases (vii)--(xii) in the list of Example \ref{ex-par-w2} and Proposition \ref{prop-w2-par}. For simplicity, in this section, we will not use bold symbols for the vectors and multi-indexes. For example, we write $\lambda$ for $\boldsymbol{\lambda}$, etc.
Recall that for every $\lambda \in \mathbb C^2$, the tensor modules $T(\lambda,\lambda)$ has four subquotients $T(\lambda, \lambda, (1^+,2^+))$, $T(\lambda, \lambda, (1^+,2^-))$, $T(\lambda, \lambda, (1^-,2^+))$, $T(\lambda, \lambda, (1^-,2^-))$. Some important properties of the these four modules are collected in the next proposition. A proof is provided in \cite{Cav} and is based on the description of the highest weight bounded $\mathfrak{sl}_3$-modules.
\begin{proposition} \label{prop-tensor-simple-w2}
Let $\lambda \in \mathbb C^2$.
\begin{itemize}
\item[(i)] Let $J = (1^+,2^+)$. The $W_2$-module $T(\lambda, \lambda, J)$ is simple if and only if $\lambda \neq (0,0)$ and $\lambda \neq (1,0)$. The $W_2$-module $T((0,0),(0,0),J)$ has length $2$ with simple submodule isomorphic to $\mathbb C$. The $W_2$-module $T((1,0),(1,0),J)$ has length $2$ with simple submodule isomorphic to $T((0,0),(0,0),J)/\mathbb C$ and simple quotient isomorphic to $T((1,1),(1,1),J)$.
\item[(ii)] Let $J = (1^+,2^-)$ or $J = (1^-,2^+)$. The $W_2$-module $T(\lambda, \lambda, J)$ is simple if and only if $\lambda \neq (1,0)$. The module $T((1,0), (1,0), J)$ has length $3$ with simple submodule $T((0,0), (0,0), J)$, simple quotient $T((1,1), (1,1), J)$, and simple subquotient $\mathbb C$.
\item[(iii)] Let $J = (1^-,2^-)$. The $W_2$-module $T(\lambda, \lambda, J)$ is simple if and only if $\lambda \neq (1,0)$ and $\lambda \neq (1,1)$. The $W_2$-module $T((1,0),(1,0),J)$ has length $2$ with simple submodule isomorphic to isomorphic to $T((0,0),(0,0),J)$. The $W_2$-module $T((1,1),(1,1),J)$ has length $2$ with simple quotient isomorphic to $\mathbb C$ and simple submodule isomorphic to $T((1,0),(1,0),J)/T((0,0),(0,0),J)$.
\varepsilonilonnd{itemize}
\varepsilonilonnd{proposition} \label{prop-char-for}
The character formulae of all tensor $W_2$-modules $T(\lambda, \lambda, J)$ follow directly from their definition. For a weight module $M$ with finite weight multiplicities, we write $\ch M = \sum_{\lambda \in \Supp M} \dim M^{\lambda} e^{\lambda}$.
\begin{proposition}
Let $\lambda \in \mathbb C^2$. Then the following identities hold.
\begin{itemize}
\item[(i)] $\displaystyle \ch T(\lambda,\lambda, (1^+, 2^+)) = \frac{\ch L_{\mathfrak{gl}}(\lambda)}{(1-e^{\varepsilonilon_1})(1-e^{\varepsilonilon_2})}$.
\item[(ii)] $\displaystyle \ch T(\lambda,\lambda, (1^+, 2^-)) = \frac{e^{-\varepsilonilon_2}\ch L_{\mathfrak{gl}}(\lambda)}{(1-e^{\varepsilonilon_1})(1-e^{-\varepsilonilon_2})}$,\; $\displaystyle \ch T(\lambda,\lambda, (1^-, 2^+)) = \frac{e^{-\varepsilonilon_1}\ch L_{\mathfrak{gl}}(\lambda)}{(1-e^{-\varepsilonilon_1})(1-e^{\varepsilonilon_2})}$.
\item[(iii)] $\displaystyle \ch T(\lambda,\lambda, (1^-, 2^-)) = \frac{e^{-\varepsilonilon_1-\varepsilonilon_2}\ch L_{\mathfrak{gl}}(\lambda)}{(1-e^{-\varepsilonilon_1})(1-e^{-\varepsilonilon_2})}$.
\varepsilonilonnd{itemize}
In particular, the degrees of all four modules equal $\dim L_{\mathfrak{gl}}(\lambda) = \lambda_1 - \lambda_2 + 1$.
\varepsilonilonnd{proposition}
For any Borel subalgebra $\mathfrak b$ of $W_2$ induced by a Borel subalgebra $\mathfrak b_{\mathfrak s}$ of ${\mathfrak s} \simeq \mathfrak{sl}_3$, by $L_{\mathfrak b}(\lambda)$ (respectively, by $L_{\mathfrak b_{\mathfrak s}}^{\mathfrak{sl}}(\lambda)$) we denote the simple highest weight $W_2$-module (respectively, $\mathfrak s$-module) relative to $\mathfrak b$ (respectively, to ${\mathfrak b}_{\mathfrak s}$) with highest weight $\lambda$. In the case when $\mathfrak b_{\mathfrak s}$ is the standard Borel subalgebra $\mathfrak b_{st}$ of ${\mathfrak s} \simeq \mathfrak{sl}_3$, i.e. the one with base $\mathbb Pi_{st} = \{ \varepsilonilon_0 - \varepsilonilon_1, \varepsilonilon_1 - \varepsilonilon_2\}$, we will write $L(\lambda)$ and $L^{\mathfrak{sl}}(\lambda)$ for $L_{\mathfrak b}(\lambda)$ and $L_{\mathfrak b_{\mathfrak s}}^{\mathfrak{sl}}(\lambda)$, respectively. Note that the Borel subalgebra of $W_2$ induced by $\mathfrak b_{st}$ is ${\mathfrak p} (2^+, 12^+)$. For $\mathfrak b_{st}^-$ (the opposite to the standard Borel subalegbra) and its induced Borel subalgebra ${\mathfrak p} (2^-, 12^-)$ of $W_2$, the corresponding modules will be denoted by $\tilde{L}(\lambda)$ and $\tilde{L}^{\mathfrak{sl}}(\lambda)$, respectively.
For a weight $W_2$-module $M = \bigoplus_{\lambda \in \mathfrak{h}^*} M^{\lambda}$ with finite weight multiplicities, by $M^*$ we denote the restricted dual of $M$, namely the module $\bigoplus_{\lambda \in \mathfrak{h}^*} \Hom_{\mathbb C} (M^{\lambda}, {\mathbb C})$ with action defined by $ (u f) (m) = f (-um)$. It is clear that $M^*$ is also a weight module with finite weight multiplicities. Moreover, $\left(L_{\mathfrak b} (\lambda)\right)^* \simeq L_{\mathfrak b^-}(-\lambda)$, where recall that ${\mathfrak b^-}$ is the opposite to ${\mathfrak b}$ Borel subalgebra. Certainly, the same isomorphism holds for the corresponding
highest weight $\mathfrak s$-modules, and Borel subalgebras of $\mathfrak{sl}_3$.
A weight $\lambda$ will be called \varepsilonilonmph{$(W_2, \mathfrak b)$-bounded} (respectively, \varepsilonilonmph{$(\mathfrak{sl}_3, \mathfrak b_{\mathfrak s})$-bounded}) if $L_{\mathfrak b}(\lambda)$ (respectively $L_{\mathfrak b_{\mathfrak s}}^{\mathfrak{sl}}(\lambda)$) is a bounded module. We will use the following classification of the $(\mathfrak{sl}_3, \mathfrak b_{\mathfrak s})$-bounded weights, see Lemma 7.1 in \cite{M}.
\begin{lemma} \label{lem-sl3-bw}
A weight $\lambda$ of $\mathfrak{sl}_3$ is $(\mathfrak{sl}_3, \mathfrak b_{\mathfrak s})$-bounded if and only if
$(\lambda + \rho_{\mathfrak b_{\mathfrak s}}, \alpha) \in \mathbb Z_{\mathfrak{g}eq 0}$ for some root $\alpha$ of $\mathfrak b_{\mathfrak s}$, where $\rho_{\mathfrak b_{\mathfrak s}}$ is the half-sum of the ${\mathfrak b_{\mathfrak s}}$-positive roots of $\Delta_{\mathfrak{sl}_3}$.
\varepsilonilonnd{lemma}
Note that in the lemma above we may have more than one root $\alpha$ that satisfy the stated condition. In particular, if all three roots satisfy the condition, then $L_{\mathfrak b_{\mathfrak s}}^{\mathfrak{sl}}(\lambda)$ is finite dimensional.
\begin{lemma} \label{lem-hw-wbounded}
Let $\lambda \in \mathbb C^2$. Then $L(\lambda)$ is a bounded module if and only if $\lambda_1 - \lambda_2 \in \mathbb Z_{\mathfrak{g}eq 0}$.
\varepsilonilonnd{lemma}
\begin{proof}
If $\lambda_1 - \lambda_2 \in \mathbb Z_{\mathfrak{g}eq 0}$, by Proposition \ref{prop-tensor-simple-w2} we know that $L(\lambda)$ is a subquotient of $T(\lambda, \lambda, (1^+,2^+))$ (in fact, $L(\lambda) \simeq T(\lambda, \lambda, (1^+,2^+))$ if $\lambda \neq (0,0), (1,0)$). Hence, $L(\lambda)$ is bounded.
For the ``only if'' direction, we will prove the following equivalent statement: If $\tilde{L}(\mu)$ is bounded, then $\mu_2 - \mu_1 \in \mathbb Z_{\mathfrak{g}eq 0}$. The two statements are equivalent because $L(\lambda)^* = \tilde{L}(-\lambda)$. Since $\mu$ is an $(\mathfrak{sl}_3, \mathfrak{b}^-_{st})$-bounded weight and $\rho_{\mathfrak b_{st}^-} = \varepsilonilon_2 - \varepsilonilon_0$, by Lemma \ref{lem-sl3-bw}, $\mu$ is one (or more than one) of the following three types:
\noindent {\it Type 1:} $\mu_2 - \mu_1 \in \mathbb Z_{\mathfrak{g}eq 0}$;
\noindent {\it Type 2:} $2 \mu_1 + \mu_2 \in \mathbb Z_{\mathfrak{g}eq 0}$;
\noindent {\it Type 3:} $ \mu_1 + 2\mu_2 + 1\in \mathbb Z_{\mathfrak{g}eq 0}$.
Assume for the sake of contradiction that $\mu_2 - \mu_1 \notin \mathbb Z_{\mathfrak{g}eq 0}$, in particular, $\mu$ is of Type 2 or of Type 3. Then $\tilde{L} (\mu)$ is $\partial_1$-injective module. Indeed, if $\tilde{L} (\mu)$ is $\partial_1$-finite, then the $\mathcal A (x_1)$-module generated by a highest weight vector of $\tilde{L} (\mu)$ must have finite support. But the only possible finite-dimensional $\mathcal A$-modules are the trivial modules, i.e. $\mu_1=\mu_2=0$, contradicting $\mu_2 - \mu_1 \notin \mathbb Z_{\mathfrak{g}eq 0}$.
Since $\tilde{L} (\mu)$ is $\partial_1$-injective, it can be considered as a submodule of $D_{\langle \partial_1\rangle }\tilde{L} (\mu)$. But then the quotient $D_{\langle \partial_1\rangle }\tilde{L} (\mu)/ \tilde{L} (\mu)$ has a primitive vector relative to the Borel subalgebra $\mathfrak p (1^+, 2^-)$. Namely, this is the vector $\partial_1^{-1} v$ where $v$ is a highest weight vector of $ \tilde{L} (\mu)$. As a result $(\mu_1+1,\mu_2)$ is a $(W_2, \mathfrak p (1^+, 2^-))$-bounded weight. This implies that $(-\mu_1 - \mu_2 -1)\varepsilonilon_0 + (\mu_1 + 1)\varepsilonilon_1 + \mu_2\varepsilonilon_2$ is $(\mathfrak{sl}_3, s_{\varepsilonilon_0-\varepsilonilon_1} {\mathfrak b}_{st}^-)$-bounded. Here $s_{\beta}$ denotes the reflection of the Weyl group reflection corresponding to the $\mathfrak{sl}_3$-root $\beta$. We apply Lemma \ref{lem-sl3-bw} again but this time for the weight $(\mu_1 +1,\mu_2)$. Then one of the following conditions hold:\\
(a) $ \mu_1 + 2\mu_2 + 1\in \mathbb Z_{\mathfrak{g}eq 0}$; (b) $ -2 \mu_1 - \mu_2 - 2 \in \mathbb Z_{\mathfrak{g}eq 0}$; (c) $\mu_2 - \mu_1 \in \mathbb Z_{\mathfrak{g}eq 0}$.\\
We already assumed that (c) does not hold. If (a) holds then $\mu$ is of Type 3. If (b) holds then $\mu$ can not be of Type 2. Hence, it remains to consider the case when $\mu$ is of Type 3. Look again at the simple highest weight $W_2$-module $L = L_{\mathfrak p (1^+, 2^-)} (\mu_1+1,\mu_2)$. As mentioned above, this module has a simple $\mathfrak{sl}_3$-subquotient $L_0$ with highest weight $(-\mu_1 - \mu_2 -1)\varepsilonilon_0 + (\mu_1 + 1)\varepsilonilon_1 + \mu_2\varepsilonilon_2$ relative to ${\mathfrak b}_{st}^-$. Since $\mu_2 - (-\mu_1 - \mu_2 -1) \in \mathbb Z_{\mathfrak{g}eq 0}$, $L_0$ is $\partial_2$-finite. Therefore $L$ has a simple $\mathfrak{sl}_3$-subquotient isomorphic to $L_{s_{\varepsilonilon_0-\varepsilonilon_1} {\mathfrak b}_{st}^-}^{\mathfrak{sl}}(\mu_2+1,-\mu_1-\mu_2-2)$. However one easily checks that since $\mu_2 - \mu_1 \notin \mathbb Z_{\mathfrak{g}eq 0}$ and $ \mu_1 + 2\mu_2 + 1\in \mathbb Z_{\mathfrak{g}eq 0}$, the weight $(\mu_2+1,-\mu_1-\mu_2-2)$ is not $(\mathfrak{sl}_3, s_{\varepsilonilon_0-\varepsilonilon_1}{\mathfrak b}_{st}^-)$-bounded. This contradicts with the fact that $L_{s_{\varepsilonilon_0-\varepsilonilon_1} {\mathfrak b}_{st}^-}^{\mathfrak{sl}}(\mu_2+1,-\mu_1-\mu_2-2)$ is a subquotient of the bounded module $L$.
\varepsilonilonnd{proof}
\begin{theorem} \label{th-hw-bounded} Let $\lambda \in \mathbb C^2$. Then the highest weight $W_2$-module $L_{\mathfrak b} (\lambda)$ is bounded if and only if:
\begin{itemize}
\item[(i)] $\lambda_1 - \lambda_2 \in \mathbb Z_{\mathfrak{g}eq 0}$ for $\mathfrak b = {\mathfrak p}(2^+,12^+)$ and $\mathfrak b = {\mathfrak p}(1^-,12^-)$,
\item[(ii)] $\lambda_2 - \lambda_1 \in \mathbb Z_{\mathfrak{g}eq 0}$ for $\mathfrak b = {\mathfrak p}(1^+,12^+)$ and $\mathfrak b = {\mathfrak p}(2^-,12^-)$,
\item[(iii)] $\lambda_1 - \lambda_2 + 1 \in \mathbb Z_{\mathfrak{g}eq 0}$ for $\mathfrak b = {\mathfrak p}(1^-,2^+)$.
\item[(iv)] $\lambda_2 - \lambda_1 + 1 \in \mathbb Z_{\mathfrak{g}eq 0}$ for $\mathfrak b = {\mathfrak p}(1^+,2^-)$,
\varepsilonilonnd{itemize}
\varepsilonilonnd{theorem}
\begin{proof} Using Lemma \ref{lem-hw-wbounded} and applying the duality functor $M\mapsto M^*$ and the twist by the automorphism $\sigma$, we easily prove (i) and (ii).
Again by duality and because ${\mathfrak p}(1^+,2^-)^- = {\mathfrak p}(1^-,2^+)$, we see that it is enough to show (iii). For the ``only if'' direction we use that $L_{{\mathfrak p}(1^-,2^+)}(\lambda_1, \lambda_2)$ is isomorphic to a subquotient of the bounded module $T((\lambda_1+1,\lambda_2), (\lambda_1+1,\lambda_2), (1^-,2^+))$. Assume now that $L = L_{{\mathfrak p}(1^-,2^+)}(\lambda_1, \lambda_2)$ is bounded. We reason as in the proof of Lemma \ref{lem-sl3-bw}. Namely, we first observe that $L$ is $\partial_1$-injective. Then the module $D_{\langle \partial_1 \rangle} L / L$ has a ${\mathfrak p}(2^+, 12^+)$-primitive vector of weight $(\lambda_1 + 1, \lambda_2)$ (namely the vector $\partial_1^{-1}w$ where $w$ is a highest weight vector of $L$). Then we use (i) for $(\lambda_1 + 1, \lambda_2)$ and ${\mathfrak p}(2^+, 12^+)$ and complete the proof.
\varepsilonilonnd{proof}
\subsection{Classification of simple bounded half-plane $W_2$-modules}
In this subsection we classify all simple bounded $W_2$-modules $M$ whose supports are half-planes. Namely we give a necessary and sufficient conditions for the modules listed in (i)--(vi) of Example \ref{ex-par-w2} and Proposition \ref{prop-w2-par} to be bounded. We call modules $M$ in that list ((i)--(vi)) \varepsilonilonmph{simple weight half-plane modules}.
We first provide the decomposition of the half-plane tensor modules. It is not surprising that in this case the result is much more simple than the highest weight case described in Proposition \ref{prop-tensor-simple-w2}. The proof of the following proposition is provided in \cite{Cav}.
\begin{proposition}
Let $\nu,\lambda \in \mathbb C^2$ be such that ${\rm Int} (\lambda - \nu) = \{i\}$, where $i=1$ or $i=2$. Let $J \in \mathcal{PM} (\lambda - \nu) $, i.e. $J \in \{ i^{+}, i^{-}\}$. Then $T(\nu,\lambda,J)$ is a simple $W_2$-module if and only $\lambda \neq (1,0)$.The module $T(\nu,(1,0),J)$ has length $2$ with simple submodule isomorphic to $T(\nu,(0,0),J)$ and simple quotient isomorphic to $T(\nu,(1,1),J)$.
\varepsilonilonnd{proposition}
\begin{remark} \label{rem-char-half}
One easily can write character formulae for all tensor half-plane modules $T(\nu,\lambda,J)$. Naturally, these formulas contain more terms than the ones for highest weight modules, see Proposition \ref{prop-char-for}. For example, the character formula for $T(\nu,\lambda, 2^-)$ is:
$$
\ch T(\nu,\lambda, 2^-) = \frac{\left(\sum_{k \in \mathbb Z} e^{(\nu_1-\lambda_1 + k) \varepsilonilon_1 -\varepsilonilon_2}\right) \ch L_{\mathfrak{gl}} (\lambda)}{1- e^{-\varepsilonilon_2}}.
$$
\varepsilonilonnd{remark}
In this section we will use two parabolic induction functors. For a parabolic subalgebra $\mathfrak p = \mathfrak l \oplus \mathfrak n^+$ of $W_2$ induced from a parabolic subalgebra of $\mathfrak{sl}_3$, and a simple $\mathfrak l$-module $S$ with trivial action of $\mathfrak n^+$, we define $M_{\mathfrak p} (S) = U(W_2) \otimes_{U(\mathfrak p)} S$. Also, by $L_{\mathfrak p} (S)$ we denote the simple quotient of $M_{\mathfrak p} (S)$. Similarly we define the two parabolic induction functors for the algebras $\mathcal A_1$, $\mathfrak{sl}_3$, and $\mathfrak{gl}_2$. We will use numerous times the facts that if $S$ is dense $\mathfrak{sl}_3$- or $\mathfrak{gl}_2$-module, then $S$ is a twisted localization of a bounded highest weight module, and that the twisted localization functors commute with the parabolic induction functors $M_{\mathfrak p}$ and $L_{\mathfrak p}$, see for example Proposition 6.2 and Lemma 13.2 in \cite{M}. The proof that the twisted localization and the parabolic induction functors commute in \cite{M} concerns the case of a finite-dimensional reductive Lie algebra $\mathfrak{g}$, but one naturally extends Mathieu's proof for $W_2$. For further properties and a more detailed exposition of the twisted localization functor, the reader is referred for example to \cite{Gr1}.
We first deal with the last two cases in the list (i)--(vi) of Example \ref{ex-par-w2}.
\begin{lemma} Let $\mathfrak p = \mathfrak p (12^+)$ or $\mathfrak p = \mathfrak p (12^-)$, and let $S$ be a simple $\mathfrak p$-module with a trivial action of $\mathfrak n^+$. Assume that the support of $M = L_{\mathfrak p} (S)$ is a half-plane. Then $M$ is not bounded.
\varepsilonilonnd{lemma}
\begin{proof} Assume that $M$ is bounded. In both cases for $\mathfrak p$, the Levi subalgebra of $\mathfrak p$ is $\mathfrak a \simeq \mathfrak{gl}_2$. Then since the support of $M$ is a half-plane, $S$ is a dense $x_2\partial_1$-injective $\mathfrak a$-module. So, let us consider $\lambda \in \mathfrak{h}^*_{\mathfrak a} \simeq \mathbb C^2$ and $\nu \in \mathbb C^2$ so that $S = D_{\langle x_2\partial_1\rangle}^{\nu} L_{\mathfrak{gl}} (\lambda)$, where recall that $L_{\mathfrak{gl}} (\lambda)$ is the simple highest weight $\mathfrak a$-module relative to the Borel subalgebra $\mathfrak b_{\mathfrak a} = \Span \{x_1\partial_2, x_1\partial_1, x_2\partial_2\}$ of $\mathfrak a$. But then
$$
L_{\mathfrak p} (S) \simeq L_{\mathfrak p} \left( D_{\langle x_2\partial_1\rangle}^{\nu} L_{\mathfrak{gl}} (\lambda) \right) \simeq D_{\langle x_2\partial_1\rangle}^{\nu} (L_{\mathfrak b}(\lambda)),
$$
where $\mathfrak b = \mathfrak b_{\mathfrak a} + \mathfrak{n}^+$. Since $L_{\mathfrak b}(\lambda)$ is bounded and $\mathfrak b = {\mathfrak p}(2^+,12^+)$ or $\mathfrak b = {\mathfrak p}(1^-,12^-)$, by Theorem \ref{th-hw-bounded}, we have that $\lambda_1 - \lambda_2 \in \mathbb Z_{\mathfrak{g}eq 0}$. But this implies that $L_{\mathfrak{gl}} (\lambda)$ is finite dimensional which contradicts to the fact that it is $x_2 \partial_1$-injective.
\varepsilonilonnd{proof}
For the four remaining cases (i)--(iv) of simple bounded half-plane modules $L_{\mathfrak p} (S)$, the parabolic subalgebra $\mathfrak p$ has Levi component isomorphic to $\mathcal A$. More precisely, we have the following straightforward result.
\begin{lemma} \label{lem-levi-a}
The Levi component of $\mathfrak p = \mathfrak p (1^+)$ and $\mathfrak p = \mathfrak p (1^-)$ is $ \mathcal A (x_2)$, while the Levi component of $\mathfrak p = \mathfrak p (2^+)$ and $\mathfrak p = \mathfrak p (2^-)$ is $ \mathcal A (x_1)$.
\varepsilonilonnd{lemma}
Before we state our classification result for the bounded simple half-plane modules, recall that, by Theorem \ref{th-bounded-a}, every simple dense bounded weight $\mathcal A$-module is isomorphic to $T(\nu, \lambda, c)$ for some $\lambda,\nu,c$, such that $\lambda - \nu \notin \mathbb Z$.
\begin{proposition} \label{prop-w2-bnd-nec} Let $\nu, \lambda,c \in \mathbb C$ be such that $\lambda - \nu \notin \mathbb Z$. Then the following isomorphisms hold.
\begin{itemize}
\item[(i)] If $\lambda - c \in \mathbb Z_{\mathfrak{g}eq 0}$ and $(\lambda, c) \neq (1,0)$, then $L_{\mathfrak p (1^+)} T(\nu,\lambda,c) \simeq T((\lambda,\nu),(\lambda,c),1^+)$ and $L_{\mathfrak p (2^+)} T(\nu,\lambda,c) \simeq T((\nu,\lambda),(\lambda,c),2^+)$.
Moreover, for any $\nu \notin \mathbb Z$, \\$L_{\mathfrak p (1^+)} T(\nu,1,0) \simeq T((0,\nu),(0,0),1^+)$ and $L_{\mathfrak p (2^+)} T(\nu,1,0) \simeq T((\nu,0),(0,0),2^+)$.
\item[(ii)] If $c+ 1 - \lambda \in \mathbb Z_{\mathfrak{g}eq 0}$ and $(\lambda,c) \neq (0,0)$, then $L_{\mathfrak p (1^-)} T(\nu,\lambda,c) \simeq T((\lambda,\nu),(c+1,\lambda),1^-)$ and $L_{\mathfrak p (2^-)} T(\nu,\lambda,c) \simeq T((\nu,\lambda),(c+1,\lambda),2^-)$. Moreover, for any $\nu \notin \mathbb Z$, $L_{\mathfrak p (1^-)} T(\nu,0,0) \simeq T((1,\nu),(1,1),1^-)$ and $L_{\mathfrak p (2^-)} T(\nu,0,0) \simeq T((\nu,1),(1,1),2^-)$.
\varepsilonilonnd{itemize}
\varepsilonilonnd{proposition}
\begin{proof}
We prove (i) for the parabolic subalgebra ${\mathfrak p} (1^+)$. The statements for the remaining three parabolic subalgebras are analogous. Let $\lambda - c \in \mathbb Z_{\mathfrak{g}eq 0}$ and $(\lambda, c) \neq (1,0)$. To show that $L_{\mathfrak p (1^+)} T(\nu,\lambda,c) \simeq T((\lambda,\nu),(\lambda,c),1^+)$, observe that the nilradical of $\mathfrak p = \mathfrak p (1^+)$ is $\mathfrak n^+ = \Span \{ x_2^k\partial_1 \; | \; k \mathfrak{g}eq 0\}$. We easily check that if $x^s \otimes v \in T((\lambda,\nu),(\lambda,c),1^+)$ is such that $x_2^k \partial_1 (x^s \otimes v) = 0$, then $s_1 = c$ and the weight of $v$ must be $(c,\lambda)$. Therefore the $\mathfrak n^+$-invariants of $T((\lambda,\nu),(\lambda,c),1^+)$ form an $\mathcal A (x_2)$-module isomorphic to $T(\nu,\lambda,c)$, which proves the desired isomorphism for ${\mathfrak p} (1^+)$. The isomorphism for ${\mathfrak p} (2^+)$ follows with similar reasoning. It remains to consider the case $(\lambda,c) =(1,0)$. In this case we use that $T(\nu,1,0) \simeq T(\nu,0,0)$ as $\mathcal A (x_2)$-modules and apply the isomorphism we just proved for $(\lambda,c) =(0,0)$ (possible because $(\lambda,c) \neq (1,0)$). Part (ii) follows in a similar way.
\varepsilonilonnd{proof}
\begin{theorem} \label{th-half-plane} Let $\nu, \lambda,c \in \mathbb C$ be such that $\lambda - \nu \notin \mathbb Z$. The simple weight half-plane module $M \simeq L_{\mathfrak p} T(\nu,\lambda,c)$ is bounded if and only if the following conditions hold.
\begin{itemize}
\item[(i)] $\lambda - c \in \mathbb Z_{\mathfrak{g}eq 0}$ for $\mathfrak p = \mathfrak p (1^+)$ or $\mathfrak p = \mathfrak p (2^+)$ .
\item[(ii)] $c+ 1 - \lambda \in \mathbb Z_{\mathfrak{g}eq 0}$ for $\mathfrak p = \mathfrak p (1^-)$ or $\mathfrak p = \mathfrak p (2^-)$ .
\varepsilonilonnd{itemize}
\varepsilonilonnd{theorem}
\begin{proof}
The ``if'' directions follow from Proposition \ref{prop-w2-bnd-nec}. For the ``only if'' directions, we prove again the condition only for the parabolic subalgebra $\mathfrak p(1^+)$ and then use similar reasoning for the remaining three parabolic subalgebras. We need to show that if $L_{\mathfrak p (1^+)} T(\nu,\lambda,c) $ is bounded, then $\lambda - c \in \mathbb Z_{\mathfrak{g}eq 0}$. If $(\lambda,c) = (1,0)$ the statement follows from the third isomorphism of Proposition \ref{prop-w2-bnd-nec}(i). Assume now that $(\lambda,c) \neq (1,0)$. To prove the desired condition, we use that $T(\nu,\lambda,c) \simeq D_{\langle \partial_2 \rangle}^{\lambda - \nu} T(\lambda,\lambda,c,-)$, see Lemma \ref{lem-iso-loc-a}(ii). Then
$$
L_{\mathfrak p (1^+)} T(\nu,\lambda,c) \simeq L_{\mathfrak p (1^+)} D_{\langle \partial_2 \rangle}^{\lambda - \nu} T(\lambda,\lambda,c,-) \simeq D_{\langle \partial_2 \rangle}^{\lambda - \nu} L_{\mathfrak p (1^+)} L(\lambda - 1, c,-) \simeq D_{\langle \partial_2 \rangle}^{\lambda - \nu} L_{\mathfrak p (1^+,2^-)} (c, \lambda - 1).
$$
The last isomorphism uses the fact that the Levi subalgebra of $ L_{\mathfrak p (1^+)}$ is $\mathcal A (x_2)$, see Lemma \ref{lem-levi-a}. Hence $L_{\mathfrak p (1^+,2^-)} (c, \lambda - 1)$ is bounded and the condition $\lambda - c \in \mathbb Z_{\mathfrak{g}eq 0}$ follows from Theorem \ref{th-hw-bounded}(iv). \varepsilonilonnd{proof}
\subsection{Classification of simple bounded dense $W_2$-modules}
Recall that $M$ is a dense module if $\Supp M = \lambda + \mathbb Z^2$ for some $\lambda$.
\begin{lemma} \label{lem-12-inj}
Let $M$ be a simple bounded $W_2$-module on which $x_1 \partial_ 2$ or $x_2 \partial_1$ act finitely. Then the support of $M$ is contained in a horizontal or vertical half-plane. In particular, if $M$ is dense, then $x_1 \partial_ 2$ and $x_2 \partial_1$ act injectively (hence bijectively) on $M$.
\varepsilonilonnd{lemma}
\begin{proof}
Assume that $M$ is not isomorphic to $\mathbb C$ and that $x_2 \partial_1$ acts finitely on $M$. To identify the possible types of $M$ we use representation theory of $\mathfrak{gl}_2$. Let $\alpha = \varepsilonilon_1 - \varepsilonilon_2$. Recall that ${\mathfrak a}\simeq \mathfrak{gl}_2$ is the subalgebra of $W_2$ generated by $x_i \partial_i$, $i,j = 1,2$. For a weight $\mu$ in $\Supp M$, consider the $\mathfrak a$-module $M[\mu] = \bigoplus_{k \in \mathbb Z} M^{\mu + k \alpha}$. This is a bounded $\mathfrak a$-module on which $x_2 \partial_1$ acts finitely. Then the number of weights of $(x_2\partial_1)$-primitive vectors in $M[\mu]$ is bounded by $2 \deg M [\mu]$. Hence $M [\mu]$ has an $(x_2\partial_1)$-maximal weight, say $\mu' = \mu + k \alpha$. Namely $\Supp M [\mu]$ is a subset of the $\alpha$-half-line $\mu'+ {\mathbb Z}_{\leq 0}\alpha$. Thus the set $\left( \mu+ {\mathbb Z}_{\leq 0}\alpha\right) \cap \Supp M$ is also on the $\alpha$-half-line $\mu'+ {\mathbb Z}_{\leq 0}\alpha$. By Proposition \ref{prop-w2-par}, the possible supports of $M$ with empty $\alpha$-half-lines, are contained either in a horizontal, or a vertical, or a diagonal (i.e. $M$ is of type $12^+$ or $12^-$) half-plane. Assume now that $M$ is of type $12^+$ (the case of type $12^-$ is analogous). Then $M$ is a quotient of $U(W_2)\otimes_{U(\mathfrak p (12^+))} S$ for some simple $\mathfrak p (12^+)$-module $S$ whose support is a whole $\alpha$-line. But, on the other hand, $M$, and therefore $S$, is $x_2\partial_1$-finite. Thus $S$ can not be simple, which is a contradiction. \varepsilonilonnd{proof}
\begin{lemma} \label{lem-3-cases} Let $M$ be a simple bounded dense $W_2$-module.
Then there is $\nu \notin {\mathbb Z}$ such that $M \simeq D_{\langle x_1\partial_2 \rangle}^{\nu} M_0$, where
\begin{itemize}
\item[(i)] $M_0 = T(\nu',\lambda', 2^-)$ for some $\nu', \lambda'$ with $\lambda'_1- \nu'_1 \notin \mathbb Z$, $\lambda'_2 - \nu'_2 \in \mathbb Z$, or
\item[(ii)] $M_0 = T(\nu',\lambda', 1^+)$ for some $\nu', \lambda'$ with $\lambda'_1- \nu'_1 \in \mathbb Z$, $\lambda'_2 - \nu'_2 \notin \mathbb Z$, or
\item[(iii)] $M_0 = T(\nu',\lambda', (1^+, 2^-))$ for some $\nu', \lambda'$ with $\lambda'_1- \nu'_1 \in \mathbb Z$, $\lambda'_2 - \nu'_2 \in \mathbb Z$.
\varepsilonilonnd{itemize}
\varepsilonilonnd{lemma}
\begin{proof}
By Lemma \ref{lem-12-inj}, $x_1\partial_2$ and $x_2\partial_1$ act injectively on $M$.
Let $\lambda = (\lambda_1, \lambda_2)$ be in $\Supp M$ and consider the module $D_{\langle x_1\partial_2 \rangle}^{-\nu} M$ for any $\nu \in \mathbb C$. For any $m \in M^{\lambda}$ we have
\begin{equation} \label{eq-comp-loc}
x_2\partial_1 \left( (x_1 \partial_2)^{-\nu} m\right) = (x_1 \partial_2)^{-\nu} \left( x_2\partial_1 + \nu(\lambda_1 - \lambda_2 - \nu -1) (x_1\partial_2)^{-1}\right)m.
\varepsilonilonnd{equation}
Consider the endomorphism $(x_1\partial_2)(x_2\partial_1)|_{M^{\lambda}}$ and choose an eigenvector $m$ with eigenvalue $x$. If we choose now $\nu$ to be a root of $x + \nu(\lambda_1 - \lambda_2 - \nu -1) = 0$, we have that $x_2\partial_1 \left( (x_1 \partial_2)^{-\nu} m\right) = 0$. In particular, the submodule $\left( D_{\langle x_1\partial_2 \rangle}^{-\nu} M \right)^{\langle x_2 \partial_1\rangle}$ consisting of all $x_2\partial_1$-finite vectors in $D_{\langle x_1\partial_2 \rangle}^{-\nu} M$ is nonzero. Since this is a bounded module, by Proposition \ref{prop-fin-len} it has a simple submodule $M_0$. Then by Lemma \ref{lem-tw-loc-simple}, $M \simeq D_{\langle x_1\partial_2 \rangle}^{\nu} M_0$. Since, $M_0$ is bounded, $(x_1 \partial_2)$-injective, and $(x_2\partial_1)$-finite, then by Lemma \ref{lem-12-inj}, $\Supp M_0$ is contained in a horizontal or vertical half-plane. But we classified all such modules in the last two subsections. After applying Theorem \ref{th-hw-bounded}, Proposition \ref{prop-w2-bnd-nec}, and Theorem \ref{th-half-plane}, we show that $M_0$ is indeed (exactly) one of the three types listed in (i)--(iii).
\varepsilonilonnd{proof}
To achieve our goal, it remains to show that the modules $ D_{\langle x_1\partial_2 \rangle}^{\nu} M_0$ for all $M_0$ listed in (i)--(iii) of Lemma \ref{lem-3-cases} are tensor $W_2$-modules. The strategy is to identify each $M_0$ as a submodule of a twisted localization $ D_{\langle x_1\partial_2 \rangle}^{-\nu} T(s,\lambda)$ of a dense tensor module $T(s,\lambda)$. For this we will use that $M_0$ can easily be detected by the weights of its $(x_2\partial_1)$-primitive vectors. More precisely, for an associative algebra $\mathcal U$, $u \in \mathcal U$, and a weight $\mathcal U$-module $M$ (recall the definition of a weight $\mathcal U$-module in \S\ref{subsec-wht}), let
$$
\mbox{WP}_{M} (u) = \{\lambda \; | \; \varepsilonilonxists m \in M^{\lambda}: um=0\}
$$
be the set of weights of all $u$-primitive weight vectors in $M$. If $M$ is fixed we will write $\mbox{WP} (u)$ for $\mbox{WP}_M (u)$. The following lemma concerns the case of $\mathcal U = U(\mathfrak{gl}_2)$ and it follows from the representation theory of $\mathfrak{gl}_2$.
\begin{lemma} \label{lem-gl-2}
Let $\mathfrak{g} = \mathfrak{gl}_2$, $\alpha = \varepsilonilon_1 - \varepsilonilon_2$, $e \in \mathfrak{g}^{\alpha}$, and $f \in \mathfrak{g}^{-\alpha}$. If $M$ is a weight $\mathfrak{gl}_2$-module for which $\ch M = \frac{e^{\lambda}}{1 - e^{\alpha}}$, then $\mbox{WP}_{D_{\langle e\rangle} M} (f) = \{ \lambda, s_{\alpha}\cdot \lambda\}$ if $\lambda_1 - \lambda_2 \in \mathbb Z$ and $\mbox{WP}_{D_{\langle e\rangle} M} (f) = \{ \lambda \}$, otherwise. Here $s_{\alpha} \cdot (\lambda_1,\lambda_2) = (\lambda_2-1,\lambda_1+1)$.
\varepsilonilonnd{lemma}
For convenience, in what follows we fix $\alpha = \varepsilonilon_1 - \varepsilonilon_2$ as a root of $\mathfrak a = \Span \{x_i\partial_j\; | \; i,j =1,2 \}$. To describe sets of weights of primitive vectors of localized tensor modules we introduce some subsets of $\mathbb C^2$. For $y,z_1,z_2 \in \mathbb C$ with $z_1 - z_2 \in \mathbb Z_{\mathfrak{g}eq 0}$, set: \\$\mbox{Hor}(y,[z_1,z_2]) = \left(y + \mathbb Z\right) \times \left( [z_1,z_2] \cap (z_1 + \mathbb Z) \right)$ (horizontal strip in $(y,z_1) + \mathbb Z^2$), and $\mbox{Ver}([z_1,z_2], y) = \left( [z_1,z_2] \cap (z_1 + \mathbb Z) \right) \times \left(y + \mathbb Z\right ) $ (vertical strip in $(z_1,y) + \mathbb Z^2$).
\begin{lemma} \label{lem-prim-wht} Let $M = D_{\langle x_1\partial_2 \rangle}M_0$ and $M_0$ be one of the three modules listed in (i)--(iii) in Lemma \ref{lem-3-cases}. Then the following hold.
\begin{itemize}
\item[(i)] If $M_0 = T(\nu, \lambda, 2^-)$, then $\mbox{WP}_{M} (x_2\partial_1) = {\rm Hor} (\nu_1, [\lambda_1-1, \lambda_2-1])$.
\item[(ii)] If $M_0 = T(\nu, \lambda, 1^+)$, then $\mbox{WP}_{M} (x_2\partial_1) = {\rm Ver} ([\lambda_1, \lambda_2], \nu_2)$.
\item[(iii)] If $M_0 = T(\nu, \lambda, (1^+, 2^-))$, then $\mbox{WP}_{M} (x_2\partial_1) \subset {\rm Hor} (\nu_1, [\lambda_1-1, \lambda_2-1]) \cup {\rm Ver} ([\lambda_1, \lambda_2], \nu_2)$.
\varepsilonilonnd{itemize}
\varepsilonilonnd{lemma}
\begin{proof}
For part (i) we look at the character formula for $M_0$, see Remark \ref{rem-char-half}. More precisely, if $s \in \Supp M_0$, then the character of the $\mathfrak{a}$-module $M_0[s] = \oplus_{k \in \mathbb Z} M^{s+ k\alpha}$ equals the character of the $\mathfrak{gl_2}$-module $L_0 = \bigoplus_{i=0}^{\lambda_1-\lambda_2} L_{\mathfrak{gl}}(s_1+s_2 - \lambda_1- i+1, \lambda_1+i-1)$. But since the central characters of the direct summands of $L_0$ are distinct, we have $M_0 [s] \simeq L_0$. Hence $\mbox{WP}_{M_0[s]} (x_2\partial_1) = \mbox{WP}_{L_0} (E_{21})$. We use the identity $M_0 = \bigoplus_{\varepsilonilonll \in \mathbb Z} M_0[s+\varepsilonilonll\varepsilonilon_1]$ of non-integral $\mathfrak{a}$-modules. After applying the functor $D_{\langle x_1\partial_2\rangle}$ on that identity, and Lemma \ref{lem-gl-2} on each $D_{\langle x_1\partial_2\rangle}M_0[s]$, we prove the desired identity in part (i). The proof of part (ii) is similar to the proof of (i). For part (iii) we use the same reasoning, and in particular apply Lemma \ref{lem-gl-2} for the integral case. Unfortunately in this case, some direct summands of $M_0[s]$ may have equal central characters, so we can not claim that $M_0[s]$ is a direct sum of simple highest modules. For that reason, we cannot claim that identity holds for $\mbox{WP}_{M} (x_2\partial_1)$, but we can prove that we have an inclusion. \varepsilonilonnd{proof}
In order to explicitly write the weights of the $x_2\partial_1$-primitive vectors in $ D_{\langle x_1\partial_2 \rangle}^{-\nu} T(s,\lambda)$, we need to introduce additional notation. Recall that the elementary matrices of $\mathfrak{gl}_2$ are denoted by $E_{ij}$, $i,j=1,2$. For the simple finite-dimensional $\mathfrak{gl}_2$-module $L_{\mathfrak{gl}} (\lambda)$ we use the following setting: $L_{\mathfrak{gl}} (\lambda) = \Span \{ v_0,...,v_n \}$, $n=\lambda_1-\lambda_2$, with $\mathfrak{gl}_2$-action defined by
\begin{eqnarray*}
E_{12} v_i & = & iv_{i-1},\\
E_{21} v_i & = & (n-i)v_{i+1},\\
E_{11} v_i & = & (\lambda_1 - i)v_{i},\\
E_{22} v_i & = & (\lambda_2+i)v_{i},
\varepsilonilonnd{eqnarray*}
for $i=0,...,n$.
For any $x,y \in \mathbb C$ we introduce the following $(n+1)\times(n+1)$ matrices:
\begin{equation}
A_n(x) = \begin{bmatrix}
x & 0 & \dots & 0 & 0 \\
1 & x-1& \dots & 0 & 0 \\
\vdots & \vdots & & \vdots & \vdots \\
0 & 0 & \dots & n & x-n
\varepsilonilonnd{bmatrix}, \;
B_n(y) = \begin{bmatrix}
y-n & 0 & \dots & 0 & 0 \\
n & y-n+1& \dots & 0 & 0 \\
\vdots & \vdots & & \vdots & \vdots \\
0 & 0 & \dots & 1 & y
\varepsilonilonnd{bmatrix}.
\varepsilonilonnd{equation}
In particular, $B_n(x)$ is the anti-diagonal transpose of $A_n(x)$.
\begin{lemma} \label{lem-ab-eigen}
Let $s, \lambda$ be in $\mathbb C^2$, and let $T = T(s,\lambda)$.
\begin{itemize}
\item[(i)] The matrix $M(s,\lambda) $ of the endomorphism $(x_1\partial_2)(x_2\partial_1)|_{T^{s}}$ relative to the basis $\{ x^s\otimes v_0,...,x^s\otimes v_n \}$ of $T^{s} = T(s,\lambda)^s$ is
$$
M(s,\lambda) := A_n(s_2-\lambda_1+1)B_n(s_1-\lambda_2).
$$
\item[(ii)] Let $v \in L_{\mathfrak{gl}}(\lambda)$ be such that $x_1 \partial_2$ is injective on $x^s \otimes v$. Then for all $\nu \in \mathbb C$,
$$
(x_2\partial_1) (x_1 \partial_2)^{-\nu} (x^s \otimes v)= (x_1 \partial_2)^{-\nu-1} \left( (x_1\partial_2) (x_2 \partial_1) + \nu(s_1-s_2 - \nu-1) \Id\right) (x^s \otimes v)
$$
in $D_{\langle x_1\partial_2 \rangle }^{-\nu} T (s,\lambda)$.
\item[(iii)] The characteristic polynomial of $A_n(x)B_n(y)$ is:
\begin{equation} \label{eq-char-poly}
\det \left( \mu I_{n+1}- A_n(x)B_n(y) \right) = (\mu - xy)(\mu - (x-1)(y-1))...(\mu - (x-n)(y-n)).
\varepsilonilonnd{equation}
\varepsilonilonnd{itemize}
\varepsilonilonnd{lemma}
\begin{proof}
Part (i) follows by a direct verification using the formulas (\ref{def-tensor-action}) and the explicit $\mathfrak{gl}_2$-action on $L_{\mathfrak{gl}}(\lambda)$. Part (ii) is also a subject of direct verification (see also (\ref{eq-comp-loc})).
We can prove part (iii) with purely technical tools, but there is a more elegant proof using the structure of the modules $D_{\langle x_1\partial_2 \rangle }^{-\nu} T(s,\lambda)$. Let $P(\mu,x,y)$ be the characteristic polynomial of $A_n(x)B_n(y)$. Note that $P(\mu,x,y)$ is polynomial in $\mu,x,y$.
Consider $s,\lambda$ such that $s_1 - \lambda_1 \notin \mathbb Z$ and $s_2 - \lambda_2 \in \mathbb Z$, and let $\nu \in \mathbb Z$. In this case $x_1 \partial_2$ is injective on $x^s \otimes v$ if and only if $x^s \otimes v \notin T(s,\lambda, 2^+)$. If the latter holds, by part (ii) we have that $s-\nu\alpha$ is a weight of an $(x_2\partial_1)$-primitive vector in $D_{\langle x_1\partial_2 \rangle }^{-\nu} T(s,\lambda)$ if and only if $\nu(\nu+1-s_1+s_2)$ is an eigenvalue of $M(s,\lambda)$. On the other hand, since $L \mapsto D_{\langle x_1\partial_2 \rangle }L$ is an exact functor, $\nu \in \mathbb Z$, and $D_{\langle x_1\partial_2 \rangle } T(s,\lambda, 2^+) =0$, we have
$$
D_{\langle x_1\partial_2 \rangle }^{-\nu} T(s,\lambda) \simeq D_{\langle x_1\partial_2 \rangle } T(s,\lambda) \simeq D_{\langle x_1\partial_2 \rangle } T(s,\lambda, 2^-).
$$
By Lemma \ref{lem-prim-wht}(i),
$$
(s_1+s_2 - \lambda_2-i+1, \lambda_2+i-1) = s - (\lambda_2-s_2+i-1)\alpha, \; i=0,...,\lambda_1-\lambda_2,
$$ are the weights of the set of $(x_2\partial_1)$-primitive vectors of $D_{\langle x_1\partial_2 \rangle } T(s,\lambda, 2^-)$ that are on the $\alpha$-line $s+ \mathbb Z\alpha$. Therefore, $xy$, $(x-1)(y-1)$,... $(x-n)(y-n)$ are all eigenvalues (with possible repetitions) of $(x_1\partial_2) (x_2 \partial_1)|_{T^s}$, where $x = s_2 - \lambda_2 + 1$ and $y = s_1 - \lambda_2$. Thus (\ref{eq-char-poly}) holds for all $\mu$, $x \in \mathbb Z$, $y \notin \mathbb Z$. Since $\mathbb Z \times \left( \mathbb C\setminus \mathbb Z\right)$ is a Zariski dense subset of $\mathbb C^2$, we have that (\ref{eq-char-poly}) holds for all $\mu,x,y$. \varepsilonilonnd{proof}
\begin{lemma} \label{lem-reverse-loc}
Let $\lambda, s \in \mathbb C^2$ be such that $\lambda_i - s_i \notin \mathbb Z$, $i=1,2$, and $\lambda \neq (1,0)$.
\begin{itemize}
\item[(i)] If $\lambda_1 + \lambda_2 - s_1 - s_2 \notin \mathbb Z$, then the following isomorphisms hold:
\begin{itemize}
\item[(a)] $D_{\langle x_1\partial_2\rangle}^{\nu_2}T(s- \nu_2\alpha, \lambda, 2^-) \simeq T(s, \lambda)$,
\item[(b)] $D_{\langle x_1\partial_2\rangle}^{\nu_1}T(s- \nu_1\alpha, \lambda, 1^+) \simeq T(s, \lambda)$,
\varepsilonilonnd{itemize}
for $\nu_2 = s_2-\lambda_2 + 1$ and $\nu_1 = s_1- \lambda_1$.
\item[(ii)] If $\lambda_1 + \lambda_2 - s_1 - s_2 \in \mathbb Z$, then
$$
D_{\langle x_1\partial_2\rangle}^{\nu}T(s- \nu \alpha, \lambda, (1^+,2^-)) \simeq T(s, \lambda)
$$
for $\nu_2 = s_2-\lambda_2 + 1$ and for $\nu_1 = s_1-\lambda_1$.
\varepsilonilonnd{itemize}
\begin{proof}
For part (i)(a) consider first the module $D_{\langle x_2\partial_2\rangle}^{-\nu_2}T(s, \lambda)$ (where $\nu_2 = s_2-\lambda_2$), and let $\nu \in \mathbb C$ be such that $\nu - \nu_2 \in \mathbb Z$. By Lemma \ref{lem-ab-eigen}(ii) we know that $s-\nu\alpha$ is a weight of an $(x_2\partial_1)$-primitive vector in $D_{\langle x_1\partial_2 \rangle }^{-\nu_2} T(s,\lambda)$ if and only if $\nu(\nu+1-s_1+s_2)$ is an eigenvalue of $M(s,\lambda)$. But by Lemma \ref{lem-ab-eigen}(ii), all such eigenvalues are $(s_2-\lambda_2-i+1)(s_1-\lambda_2-i)$, $i=0,...,\lambda_1-\lambda_2$. Recall that $\nu - s_2+\lambda_2 \in \mathbb Z$ and $\lambda_1 + \lambda_2 - s_1 - s_2 \notin \mathbb Z$. Hence, $s-\nu\alpha$ is a weight of an $(x_2\partial_1)$-primitive vector in $D_{\langle x_1\partial_2 \rangle }^{-\nu_2} T(s,\lambda)$ if and only if $\nu = s_2-\lambda_2-i+1$ for some $i \in \{0,1,...,\lambda_1-\lambda_2\}$.
On the other hand, $T(s,\lambda)$ is dense and $D_{\langle x_1\partial_2 \rangle }^{-\nu_2} T(s,\lambda)$ has $(x_2\partial_1)$-primitive vectors. Thus by Lemma \ref{lem-3-cases}, we have $T(s,\lambda) = D_{\langle x_1\partial_2 \rangle }^{\nu_2} T(s-\nu_2\alpha,\lambda', 2^-)$ for some $\lambda' \in \mathbb C^2$. Hence, $D_{\langle x_1\partial_2 \rangle }^{-\nu_2} T(s,\lambda) \simeq D_{\langle x_1\partial_2 \rangle } T(s-\nu_2\alpha,\lambda', 2^-)$. Using Lemma \ref{lem-prim-wht}(i) and the description of the weights of $(x_2\partial_1)$-primitive vectors in $D_{\langle x_1\partial_2 \rangle }^{-\nu_2} T(s,\lambda)$, we obtain
$$
{\rm Hor} (s_1 - \nu_2, [\lambda_1-1, \lambda_2-1]) = {\rm Hor} (s_1 - \nu_2, [\lambda'_1-1, \lambda'_2-1]).
$$
Thus $\lambda = \lambda'$, which completes the proof of (i)(a).
For parts (i)(b) and (ii) we use the same reasoning, namely we apply again the corresponding parts of Lemma \ref{lem-ab-eigen}, Lemma \ref{lem-3-cases}, Lemma \ref{lem-prim-wht}. Part (i)(b) will automatically follow, while for part (ii) we will have at the end
$$
{\rm Hor} (s_1 - \nu_2, [\lambda_1-1, \lambda_2-1]) \cup {\rm Ver} ([\lambda_1, \lambda_2], s_2 + \nu_2) \subset {\rm Hor} (s_1 - \nu_2, [\lambda'_1-1, \lambda'_2-1]) \cup {\rm Ver} ([\lambda'_1, \lambda'_2], s_2 + \nu_2).
$$
However, the above condition is sufficient to conclude that $\lambda = \lambda'$. \varepsilonilonnd{proof}
\varepsilonilonnd{lemma}
Using Lemma \ref{lem-3-cases} and Lemma \ref{lem-reverse-loc}, we obtain the classification of simple bounded dense $W_2$-modules.
\begin{theorem} \label{th-dense}
If $M$ is a simple bounded dense $W_2$-module, then $M \simeq T(\nu,\lambda)$ for some $\nu, \lambda$, such that $\lambda_i - \nu_i \notin \mathbb Z$, $i=1,2$, $\lambda \neq (1,0)$.
\varepsilonilonnd{theorem}
\subsection{Main Theorem}
Combining Theorems \ref{th-hw-bounded}, \ref{th-half-plane}, and \ref{th-dense} we obtain our main result in the paper.
\begin{theorem} \label{th-main}
Let $M $ be a simple bounded $W_2$-module. Then either $M \simeq \mathbb C$ or $M \simeq T(\nu,\lambda, J)$ for some $\nu,\lambda \in \mathbb C^2$, and $J \in \mathcal{PM} (\lambda - \nu)$, such that:
$$
\lambda \neq (1,0), (\nu,\lambda, J) \neq ((0,0),(0,0),(1^+,2^+)), (\nu,\lambda, J) \neq ((1,1),(1,1),(1^-,2^-)).
$$
Furthermore, two modules $T(\nu,\lambda, J) $ and $T(\nu',\lambda', J')$ in this list are isomorphic if and only if $\nu - \nu' \in \mathbb Z^2$, $\lambda = \lambda'$, and $J = J'$.
\varepsilonilonnd{theorem}
\begin{thebibliography}{99}
\bibitem{BF} Y. Billig, V. Futorny, Classification of simple $W_n$-modules with finite-dimensional weight spaces, to appear in {\it J. reine angew. Math.}\\
\bibitem{Cav} A. Cavaness, Bounded representations of the Lie algebra of vector fields on $\mathbb C^n$, Ph. D. Thesis, University of Texas at Arlington, in preparation.\\
\bibitem{FF} B. Feigin, D. Fuks, Homology of the Lie algebra of vector fields on the line, {\it Funct. Anal. Appl.} {\bf 14} (1980), 201--212.\\
\bibitem{Fuk} D. Fuks, Cohomology of infinite-dimensional Lie algebras, Contemporary Soviet Mathematics. Consultants Bureau, New York, 1986.\\
\bibitem{Gr1} D. Grantcharov, Twisted localization of weight modules, {\it “Developments and Retrospectives in Lie Theory: Algebraic Methods”}, Volume 38, 2014, 185--206, Springer Verlag.\\
\bibitem{GS1} D. Grantcharov, V. Serganova, Category of $\mathfrak{sp}(2n)$-modules with bounded weight multiplicities, {\it Mosc. Math. J.} {\bf 6} (2006) 119--134.\\
\bibitem{GS2} D. Grantcharov, V. Serganova, Cuspidal representations of $\mathfrak{sl}(n+1)$, {\it Adv. Math.} {\bf 224} (2010) 1517--1547.\\
\bibitem{GS3} D. Grantcharov, V. Serganova, On weight modules of algebras of twisted differential operators on the projective space, {\it Transform. Groups.} {\bf 21} (2016) 87--114.\\
\bibitem{GY} D. Grantcharov, M. Yakimov, Cominuscule parabolics of simple finite dimensional Lie superalgebras, J. Pure Appl. Algebra 217 (2013), 1844--1863.\\
\bibitem{Kac} V. Kac, Some problems of infinite-dimensional Lie algebras and their representations, {\it Lecture Notes in Mathematics}, {\bf 933} (1982), 117-126. Berlin, Heidelberg, New York: Springer.\\
\bibitem{M-Vir} O. Mathieu, Classification of Harish-Chandra modules over the Virasoro algebra, {\it Invent. Math.} {\bf 107} (1992), 225--234.\\
\bibitem{M} O. Mathieu, Classification of irreducible weight modules, {\it
Ann. Inst. Fourier} {\bf 50} (2000), 537--592.\\
\bibitem{PS} I. Penkov, V. Serganova, Weight representations of the polynomial Cartan type Lie algebras $W\sb n$ and $\overline S\sb n$, {\it Math. Res. Lett.} {\bf 6} (1999), 397--416.\\
\bibitem{Rud1} A. Rudakov, Irreducible representations of infinite-dimensional Lie algebras of Cartan type, {\it Izv. Akad.
Nauk SSSR Ser. Mat.} {\bf 38} (1974), 835--866 (Russian); English version: {\it USSR-Izv.} {\bf 8} (1974), 836--866.\\
\bibitem{Rud2} A. N. Rudakov, Irreducible representations of infinite-dimensional Lie algebras of types $S$ and $H$, {\it Izv. Akad. Nauk SSSR Ser. Mat.} {\bf 39} (1975), 496--511 (Russian); English version: {\it USSR-Izv.} {\bf 9} (1975), 465--480.\\
\varepsilonilonnd{thebibliography}
\varepsilonilonnd{document} |
\begin{document}
\title[Analysis of a class of variational multiscale methods]
{An analysis of a class of variational multiscale methods
based on subspace decomposition}
\author[Kornhuber]{Ralf Kornhuber}
\address{Institut f\"{u}r Mathematik,
Freie Universit\"{a}t Berlin,
14195 Berlin, Germany}
\email{kornhuber@math.fu-berlin.de}
\author[Peterseim]{Daniel Peterseim}
\address{Institut f\"{u}r Mathematik,
Universit\"{a}t Augsburg,
86135 Augsburg, Germany}
\email{daniel.peterseim@math.uni-augsburg.de}
\author[Yserentant]{Harry Yserentant}
\address{Institut f\"{u}r Mathematik,
Technische Universit\"{a}t Berlin,
10623 Berlin, Germany}
\email{yserentant@math.tu-berlin.de}
\thanks{This research was supported by Deutsche
Forschungsgemeinschaft through SFB 1114}
\subjclass[2010]{65N12, 65N30, 65N55}
\begin{abstract}
Numerical homogenization tries to approximate the solutions
of elliptic partial differential equations with strongly
oscillating coefficients by functions from modified finite
element spaces. We present a class of such methods that
are closely related to the methods that have recently been
proposed by M\r{a}lqvist and Peterseim
[Math. Comp. 83, 2014, pp. 2583--2603]. Like these methods,
the new methods do not make explicit or implicit use of a
scale separation. Their comparatively simple analysis is
based on the theory of additive Schwarz or subspace
decomposition methods.
\end{abstract}
\maketitle
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\numberwithin{equation}{section}
\def \I {\mathcal{I}}
\def \N {\mathcal{N}}
\def \S {\mathcal{S}}
\def \T {\mathcal{T}}
\def \V {\mathcal{V}}
\def \W {\mathcal{W}}
\def \dx {\,\mathrm{d}x}
\def \uS {\widetilde{u}}
\def \ker {\mathrm{ker}\,}
\def \span {\mathrm{span}\,}
\def \sp {\mspace{1mu}}
\section{Introduction}
\label{sec1}
Numerical homogenization aims at a modification of standard
finite element discretizations that preserve the accuracy
known from smooth coefficients functions to the case of highly
oscillatory coefficient functions. A method of this kind,
which utilizes no separation of scales at all and is founded
on a comprehensive convergence theory, has recently been
proposed by M\r{a}lqvist and one of the authors
\cite{Malqvist-Peterseim}. The central idea of this paper,
which can be considered as a late descendant of the work
of Babu\v{s}ka and Osborn \cite{Babuska-Osborn}, is to
assign to the vertices of the finite elements new basis
functions that span the modified finite element space
and reflect the multiscale structure of the problem under
consideration. In the basic version of the method, the new
basis functions are the standard piecewise linear hat
functions minus their orthogonal projection onto a space
of rapidly oscillating functions, orthogonal with respect
to the symmetric, coercive bilinear form associated with
the boundary value problem. They possess a global support
but decay exponentially from one shell of elements
surrounding the assigned vertex to the next. It is
therefore possible to replace them by local counterparts
without sacrificing the accuracy. The support of these
localized basis functions consists of a fixed number of
shells of elements surrounding the associated node. The
number of these shells increases logarithmically with
increasing accuracy, that is, decreasing gridsize.
Here we present a considerably simplified analysis of a class
of closely related methods that is based on the theory of
iterative methods, more precisely of additive Schwarz or
subspace decomposition methods \cite{Xu}, \cite{Yserentant},
and essentially utilizes a reinterpretation \cite{Peterseim}
of the method proposed in \cite{Malqvist-Peterseim} in terms
of variational multiscale methods. Arguments used in the
present paper are inspired by the work of Demko~\cite{Demko}
on the inverses of band matrices and have been used, for
example, in the proof of the $H^1$-stability of the
$L_2$-orthogonal projection onto finite element spaces
\cite{Bank-Yserentant}. Our proof works on a more abstract
level than that in \cite{Malqvist-Peterseim} and is less
centered around the single finite element basis functions.
Implicitly, it compares their projections onto the mentioned
space of rapidly oscillating functions with their iteratively
calculated approximations and shows in this way that these
decay exponentially with the distance to the assigned nodes.
It rests, as that of all related results, upon a local version
of the Poincar\'{e} inequality and thus depends on the local
contrast ratio. A different class of methods not suffering
from such restrictions and condensing, similar to numerical
homogenization methods, the features of the problem under
consideration in a relatively low dimensional matrix can
possibly be based on hierarchical matrices \cite{Hackbusch};
see the recent work of Bebendorf \cite{Bebendorf} in
conjunction with the work of Hackbusch and Drechsler
\cite{Hackbusch-Drechsler}.
The present paper builds a bridge between numerical
homogenization methods and multigrid-like iterative solvers.
As a matter of fact, many numerical upscaling techniques
bear close and often overlooked resemblance to techniques
used by the multigrid community to develop fast iterative
methods suitable for problems with rough coefficient
functions. A prominent example of this kind, not unsimilar
to our approach, is the technique used by Xu and Zikatanov
\cite{Xu-Zikatanov} to construct coarse-level basis functions
for algebraic multigrid methods, another the reference
\cite{Kornhuber-Yserentant}, in which the direct application
of a two-grid iterative method to numerical homogenization
is studied and compared to methods like ours.
\section{The equation and the basic approximation of its solutions}
\label{sec2}
The model problem considered in this paper is a standard second
order differential equation in weak form with homogeneous
Dirichlet boundary conditions on a polygonal domain $\Omega$
in $d=2$ or $3$ space dimensions. Its solution space is the
Sobolev space $H_0^1(\Omega)$ and the associated bilinear
form reads
\begin{equation} \label{eq2.1}
a(u,v)=\int_\Omega \nabla u\cdot A\nabla v\dx.
\end{equation}
The matrix $A$ is a function of the spatial variable $x$
with measurable entries and is assumed to be symmetric
positive definite. We assume for simplicity that
\begin{equation} \label{eq2.2}
\delta\sp|\eta|^2\leq\,\eta\cdot A(x)\eta\,\leq M|\eta|^2
\end{equation}
holds for all $\eta\in\mathbb{R}^d$ and almost all
$x\in\Omega$, where $|\eta|$ denotes the euclidean norm
of $\eta$ and $\delta$ and $M$ are positive constants.
In the same way as in \cite{Kornhuber-Yserentant} it is
possible to replace this condition by an, at least in the
quantitative sense, weaker local condition on the contrast
ratio. The condition (\ref{eq2.2}) guarantees that the
bilinear form (\ref{eq2.1}) is an inner product on
$H_0^1(\Omega)$. It induces the energy norm $\|\cdot\|$,
which is equivalent to the original norm on this space.
The boundary value problem
\begin{equation} \label{eq2.3}
a(u,v)=f^*(v),\quad v\in H_0^1(\Omega),
\end{equation}
possesses by the Lax-Milgram theorem under the condition
(\ref{eq2.2}) for all bounded linear functionals $f^*$
on $H_0^1(\Omega)$ a unique solution $u\in H_0^1(\Omega)$.
We cover the domain $\Omega$ with a triangulation $\T$,
consisting of triangles in two and of tetrahedrons in
three space dimensions. We assume that the elements in
$\T$ are shape regular but do not require that $\T$ is
quasiuniform. Associated with $\T$ is the conforming,
piecewise linear finite element subspace $\S$ of
$H_0^1(\Omega)$. A key ingredient of the methods
discussed here is a bounded local linear projection
operator
\begin{equation} \label{eq2.4}
\Pi:H_0^1(\Omega)\,\to\,\S:u\,\to\,\Pi u
\end{equation}
like that defined as follows. At first, the given function
$u\in H_0^1(\Omega)$ is locally, on the single elements
$t\in\T$, approximated by its $L_2$-orthogonal projection
onto the space of linear functions, regardless of the
continuity across the boundaries of the elements.
In a second step, the values of these approximants at
a vertex in the interior of the domain are replaced by
a weighted mean, according to the contribution of the
involved elements to the area or the volume of their
union. The values at the vertices on the boundary are
set to zero. Together, these values fix the projection
$\Pi u$ of $u$ onto $\S$. For functions
$u\in H_0^1(\Omega)$ then the estimates
\begin{equation} \label{eq2.5}
|\Pi u|_1\leq c_1|\sp u\sp |_1, \quad
\|h^{-1}(u-\Pi u)\|_0\leq c_2|\sp u\sp |_1
\end{equation}
hold, where $\|\cdot\|_0$ denotes the $L_2$-norm,
$|\cdot|_1$ the $H^1$-seminorm, and $h$ is an
elementwise constant function whose value on the
interior of a given element is its diameter. The
first condition means that the projection operator
(\ref{eq2.4}) is stable with respect to the
$H^1$-norm and therefore also with respect to the
energy norm. The second condition is an approximation
property. The constants $c_1$ and $c_2$ depend, as
with any other reasonable choice of $\Pi$, only on
the shape regularity of the finite elements, but
not on their size. Quasi-interpolation operators
are a common tool in finite element theory. The
use of quasi-interpolation operators that are at
the same time projections onto the finite element
space under consideration can be traced back
to the work of Brenner \cite{Brenner} and Oswald
\cite{Oswald}. The operator described above falls
into this category and is analyzed in the appendix
to this paper. A comprehensive recent presentation
of such constructions can be found in
\cite{Ern-Guermond}.
The kernel $\V=\ker\Pi$ of $\Pi$ is a closed subspace of
$H_0^1(\Omega)$ and therefore itself a Hilbert space.
We can therefore introduce the $a$-orthogonal projection
operator~$C$ from~$H_0^1(\Omega)$ onto the kernel of
$\Pi$ and moreover the finite dimensional subspace
\begin{equation} \label{eq2.6}
\W=\{v-Cv\,|\,v\in\sp \S\}
\end{equation}
of the $a$-orthogonal complement of the kernel of $\Pi$.
The dimension of $\W$ and of the finite element space
$\S$ coincide as $v\in\S$ can be recovered from
$v-Cv$ via
\begin{equation} \label{eq2.7}
v\,=\,\Pi\sp (v-Cv).
\end{equation}
Analogously to the approach in \cite{Malqvist-Peterseim},
we discretize the equation (\ref{eq2.3}) using $\W$
both as trial and test space. The following representation
of the approximate solution is based on observations made
in \cite{Peterseim}, where a computationally more
advantageous nonsymmetric variant with $\S$ as trial
and $\W$ as test space is advocated.
\begin{lemma}[Peterseim \cite{Peterseim}, cf. also
M\r{a}lqvist and Peterseim \cite{Malqvist-Peterseim}]
\label{lm2.1}
The equation
\begin{equation} \label{eq2.8}
a(w,\chi)=f^*(\chi), \quad \chi\in\W,
\end{equation}
possesses a unique solution $w\in\W$, namely
the projection
\begin{equation} \label{eq2.9}
w=\Pi u-C\Pi u
\end{equation}
of the exact solution $u\in H_0^1(\Omega)$
of equation {\rm (\ref{eq2.3})}
onto the space $\W$. The error
\begin{equation} \label{eq2.10}
u-w=Cu
\end{equation}
is the $a$-orthogonal projection $Cu$ of
the solution $u$ onto the kernel of $\Pi$.
\end{lemma}
\begin{proof}
As $\Pi$ is a projection operator, $u-\Pi u$ is
contained in the kernel of $\Pi$. The difference
of the exact solution $u$ and the function
(\ref{eq2.9}) can therefore be written as
\begin{displaymath}
u-w=(u-\Pi u)-C(u-\Pi u)+Cu=Cu.
\end{displaymath}
Because the functions in $\W$ are $a$-orthogonal
to the functions in the range of $C$,
\begin{displaymath}
a(u-w,\chi)=\sp 0, \quad \chi\in\W,
\end{displaymath}
follows. That is, this $w$ is the unique solution
of the equation (\ref{eq2.8}).
\end{proof}
The energy norm of $Cu$ can easily be estimated
for a right-hand side
\begin{equation} \label{eq2.11}
f^*(v)=\int_\Omega fv\dx.
\end{equation}
One obtains in this way the following, rather
surprising error estimate.
\begin{theorem}
[M\r{a}lqvist and Peterseim \cite{Malqvist-Peterseim}]
\label{thm2.2}
For right-hand sides of the form {\rm (\ref{eq2.11})},
the approximate solution {\rm (\ref{eq2.9})}
satisfies the energy norm error estimate
\begin{equation} \label{eq2.12}
\|u-w\|\leq c\,\|hf\|_0,
\end{equation}
where the constant $c$ depends only on the constants
$c_2$ from {\rm (\ref{eq2.5})} and $\delta$ from
{\rm (\ref{eq2.2})}.
\end{theorem}
\begin{proof}
The proof is based on the representation (\ref{eq2.10})
of the error as $a$-orthogonal projection $Cu$ of
the solution $u$ onto the kernel of $\Pi$ and starts
from the identity
\begin{displaymath}
\|Cu\|^2=\,a(u,Cu-\Pi\sp Cu)=(f,Cu-\Pi\sp Cu),
\end{displaymath}
from which one obtains the estimate
\begin{displaymath}
\|Cu\|^2\leq\,\|hf\|_0\|h^{-1}(Cu-\Pi\sp Cu)\|_0.
\end{displaymath}
The second factor on the right-hand side is estimated
with help of the error estimate from (\ref{eq2.5}),
and the $H^1$-seminorm of $Cu$, then with (\ref{eq2.2})
by its energy norm.
\end{proof}
Remarkably, neither the smoothness of the solution $u$
nor the regularity properties of the equation enter
into this error estimate. The size of the error bound
is determined by the local behavior of the right-hand
side $f$.
\section{Localization}
\label{sec3}
Let $x_1,x_2,\ldots,x_n$ be the vertices of the
elements in the triangulation $\T$ and
let~$\varphi_1,\varphi_2,\ldots,\varphi_n$ be the
piecewise linear hat functions assigned to these
nodes. The $\varphi_i$ assigned to the nodes in
the interior of the domain $\Omega$ then form a
basis of the finite element space $\S$, and the
corresponding functions $\varphi_i-C\varphi_i$ a
basis of the trial space (\ref{eq2.6}). It has
been shown in \cite{Malqvist-Peterseim} that
these basis functions decay exponentially with
the distance to the assigned nodes and can
therefore be replaced by localized counterparts.
We deviate here from the arguments there and
utilize the theory of iterative methods to prove
a result of similar kind. Let $\omega_i$ be the
union of the finite elements with vertex $x_i$
and let
\begin{equation} \label{eq3.1}
\V_i=\{v-\Pi v\,|\,v\in H_0^1(\omega_i)\}.
\end{equation}
The functions in $\V_i$ vanish outside a small
neighborhood of the vertex $x_i$, depending on
the choice of $\Pi$. For the exemplary operator
mentioned earlier, this neighborhood consists of
the two shells of elements surrounding $x_i$.
The $\V_i$ are closed subspaces of the kernel $\V$
of $\Pi$. This can be seen as follows. Let the
$v_k\in\V_i$ converge to the function $v\in\V$.
As the $v_k$ are piecewise linear outside
$\omega_i$, the same holds for $v$. Thus there
exists a function $\varphi$ in the finite element
space $\S$ such that $v-\varphi\in H_0^1(\omega_i)$.
Because $\Pi v=0$ and $\Pi\varphi=\varphi$, then
\begin{equation} \label{eq3.2}
v=(v-\varphi)-\Pi(v-\varphi)\in\V_i.
\end{equation}
Let $P_i$ be the $a$-orthogonal projection from
$H_0^1(\Omega)$ to $\V_i$, defined via the
equation
\begin{equation} \label{eq3.3}
a(P_iv,v_i)=a(v,v_i),\quad v_i\in\V_i.
\end{equation}
Introducing the, with respect to the bilinear
form (\ref{eq2.1}), symmetric operator
\begin{equation} \label{eq3.4}
T=P_1+P_2+\cdots+P_n,
\end{equation}
the approximation spaces replacing $\W$ are built
up with the help of the bounded linear operators
$F_\nu$ from $H_0^1(\Omega)$ to $\V$ that are,
starting from $F_0u=0$, defined via
\begin{equation} \label{eq3.5}
F_{\nu+1}u=F_\nu u+T(u-F_\nu u).
\end{equation}
The correction $T(u-F_\nu u)$ is the sum of its
components $d_i=P_i(u-F_\nu u)$ in the subspaces
$\V_i$ of $\V$, the solutions $d_i\in\V_i$ of
the local equations
\begin{equation} \label{eq3.6}
a(d_i,v_i)=a(u,v_i)-a(F_\nu u,v_i), \quad v_i\in\V_i.
\end{equation}
The new trial and test spaces are the spaces
$\W_\ell$ spanned by the functions
\begin{equation} \label{eq3.7}
\varphi_i-F_\nu\varphi_i, \quad \nu=0,1,\ldots,\ell,
\end{equation}
attached to the nodes $x_i$ in the interior
of the domain $\Omega$. In contrast to their
counterparts $\varphi_i-C\varphi_i$ spanning
the original space $\W$ they have a local
support, which expands layer by layer with
the number $\nu$ of iterations.
To study the approximation properties of these
spaces $\W_\ell$, we consider optimally or almost
optimally chosen fixed linear combinations
\begin{equation} \label{eq3.8}
C_\ell=\sum_{\nu=0}^\ell\alpha_{\ell\nu}F_\nu,
\quad
\sum_{\nu=0}^\ell\alpha_{\ell\nu}=1,
\end{equation}
of the operators $F_\nu$ as approximations of the
$a$-orthogonal projection $C$. These operators
$C_\ell$ serve here solely as a tool and do not
need to be explicitly accessible. Our analysis
is based on the theory of additive Schwarz or
subspace decomposition methods, here applied to
an equation in the kernel $\V$. Key is the
following lemma.
\begin{lemma} \label{lm3.1}
For all $v\in\V$, there is a with respect
to the energy norm stable decomposition
$v=v_1+\cdots +v_n$ of $v$ into functions
$v_i$ in the spaces $\V_i$, such that
\begin{equation} \label{eq3.9}
\sum_{i=1}^n\|v_i\|^2\leq K_1\|v\|^2
\end{equation}
holds, where the constant $K_1$ depends only
on the constants $c_1$ and $c_2$ from
{\rm (\ref{eq2.5})}, on the shape regularity
of the finite elements, and on the contrast
ratio $M/\delta$. Moreover, there is a
constant $K_2$ such that
\begin{equation} \label{eq3.10}
\|v\|^2\leq K_2\sum_{i=1}^n\|v_i\|^2
\end{equation}
holds for all such decompositions of $v$
into functions $v_i$ in the subspaces
$\V_i$ of the kernel. This constant
depends only on the shape regularity of
the finite elements.
\end{lemma}
\begin{proof}
The upper estimate (\ref{eq3.10}) is rather trivial
because $K_2$ can be bounded in terms of the maximum
number of the parts $v_i$ that do not vanish on a
given element. We use that the $\varphi_i$ form a
partition of unity and prove that (\ref{eq3.9})
holds for the decomposition of a function $v$
in the kernel $\V$ of $\Pi$ into the parts
\begin{displaymath}
v_i=\varphi_i v-\Pi\sp (\varphi_i v),
\quad i=1,\ldots,n,
\end{displaymath}
in $\V_i$. It suffices to prove that this decomposition
is $H^1$-stable. By the $H^1$-stability of the projection
$\Pi$ and the shape regularity of the finite elements
one obtains
\begin{displaymath}
\sum_{i=1}^n|\sp v_i\sp |_1^2
\,\lesssim\,
\sum_{i=1}^n|\varphi_i v|_1^2
\,\lesssim\,
|\sp v\sp |_1^2\sp +\|h^{-1}v\|_0^2.
\end{displaymath}
Using once more that $\Pi v=0$, the second term
on the right-hand side can, by the approximation
property from (\ref{eq2.5}), be estimated as
\begin{displaymath}
\|h^{-1}v\|_0=\sp \|h^{-1}(v-\Pi v)\|_0
\lesssim\sp |\sp v\sp |_1.
\end{displaymath}
Since the $H^1$-seminorm can, because of the assumption
(\ref{eq2.2}), be estimated by the energy norm and vice
versa, the estimate (\ref{eq3.9}) follows and the proof
is complete.
\end{proof}
With the help of the estimates (\ref{eq3.9}) and
(\ref{eq3.10}) one can show that
\begin{equation} \label{eq3.11}
1/K_1\sp a(v,v)\leq a(Tv,v)\leq K_2\sp a(v,v)
\end{equation}
holds for the functions $v$ in the kernel $\V$
of $\Pi$. The spectrum of $T$, seen as an
operator from $\V$ to itself, is therefore a
compact subset of the interval with the endpoints
$1/K_1$ and $K_2$. Because
$I-F_\nu=(I-T)^\nu$ and $F_\nu C=F_\nu$,
\begin{equation} \label{eq3.12}
C-C_\ell=
\bigg\{\sum_{\nu=0}^\ell\alpha_{\ell\nu}(I-T)^\nu\bigg\}\,C.
\end{equation}
Using the spectral mapping theorem and the fact
that the norm of a bounded, symmetric operator
from a Hilbert space to itself is equal to its
spectral radius, one gets therefore similarly
to the finite dimensional case the following
error estimate.
\begin{lemma} \label{lm3.2}
If the weights $\alpha_{\ell\nu}$ are optimally
chosen, the estimate
\begin{equation} \label{eq3.13}
\|C u-C_\ell u\|\leq
\frac{2\sp q^{\sp \ell}}{1+q^{\sp 2\ell}}
\,\|Cu\|
\end{equation}
holds for all $u\in H_0^1(\Omega)$, where
the convergence rate
\begin{equation} \label{eq3.14}
q=\frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}
\end{equation}
is determined by the condition number
$\kappa\leq K_1K_2$ of the operator
{\rm (\ref{eq3.4})} seen as bounded,
symmetric operator from the subspace
$\V$ of $H_0^1(\Omega)$ to itself.
\end{lemma}
The distance of $C_\ell u$ to $C u$ can thus be
estimated in terms of the two constants $K_1$ and
$K_2$ and the norm of $Cu$ and tends exponentially
in the number $\ell$ of iterations to zero. The
lemma is basically a reformulation of a standard
result from the theory of subspace decomposition
methods \cite{Xu}, \cite{Yserentant}, where $Cu$ is
here the solution of the equation and the $C_\ell u$
are its iteratively generated approximations. We
refer to \cite{Kornhuber-Yserentant} for the
missing details of the here only sketched proof.
\begin{theorem} \label{thm3.3}
Let $w$ and $w_\ell$ be the best approximations
of the solution $u$ of the original equation
{\rm (\ref{eq2.3})} in $\W$ and $\W_\ell$ with
respect to the energy norm. Then
\begin{equation} \label{eq3.15}
\|u-w_\ell\|\,\leq\,
\bigg(1+\frac{2\sp q^{\sp \ell}}{1+q^{\sp 2\ell}}\bigg)\|u-w\|+
\frac{2\sp q^{\sp \ell}}{1+q^{\sp 2\ell}}\,\|u-\Pi u\|.
\end{equation}
\end{theorem}
\begin{proof}
Because $w_\ell$ is the best approximation of $u$
by a function in $\W_\ell$, because the function
$\Pi u-C_\ell\Pi u$ is contained in this space,
and because $w=\Pi u-C\Pi u$, we~have
\begin{displaymath}
\|u-w_\ell\|\leq\|u-(\Pi u-C_\ell\Pi u)\|
=\|(u-w)-(C\Pi u-C_\ell\Pi u)\|.
\end{displaymath}
The distance of $C\Pi u$ and $C_\ell\Pi u$ can
according to Lemma~\ref{lm3.2} be estimated by
the energy norm of $C\Pi u$. As
$C\Pi u=(u-w)-(u-\Pi u)$, this leads to
(\ref{eq3.15}).
\end{proof}
We conclude that logarithmically many iteration steps
$\nu$ or even less, depending on the behavior of the
energy norm of $u-\Pi u$, suffice to reach the same
level of accuracy as with the original space $\W$
based on the exact projection $C$.
The best approximation $w_\ell$ of the solution $u$
in the space $\W_\ell$ can itself again be calculated
iteratively by an additive or multiplicative Schwarz
or subspace decomposition method based on the splitting
of $\W_\ell$ into the finite element space $\S$, that
provides for the global exchange of information, and
local subspaces like
\begin{equation} \label{eq3.16}
\W_{\ell,i}=\,
\span\{\sp\varphi_i-F_\nu\varphi_i\,|\,\nu=0,\ldots,\ell\,\},
\end{equation}
bearing the information on the fine scale structure
of the solution.
The proof of Theorem~\ref{thm3.3} shows that one can
replace the $\ell+1$ functions (\ref{eq3.7}) attached
to the nodes $x_i$ in the interior of $\Omega$ by a
single, fixed linear combination
\begin{equation} \label{eq3.17}
\sum_{\nu=0}^\ell\alpha_{\ell\nu}(\varphi_i-F_\nu\varphi_i),
\quad
\sum_{\nu=0}^\ell\alpha_{\ell\nu}=1,
\end{equation}
without sacrificing the error bound (\ref{eq3.15}).
This considerably reduces the size of the spaces
$\W_\ell$. The problem is that the optimum
coefficients $\alpha_{\ell\nu}$ depend on the
entire spectrum of the operator $T$ from the
kernel $\V$ to itself. In a much simplified
variant, approximations $C_\ell$ of the projection
$C$ are determined via a recursion
\begin{equation} \label{eq3.18}
C_{\ell+1}u=C_\ell u+\omega\sp T(u-C_\ell u),
\end{equation}
where $C_0 u=0$ is set and $\omega\geq 1/K_2$ is
a damping parameter whose optimal value depends
again on the end points of the spectrum. Because
in this case
\begin{equation} \label{eq3.19}
C-C_\ell=(I-\omega\sp T)^\ell C,
\end{equation}
convergence is guaranteed at least for $\omega<2/K_2$,
and for $\omega=1/K_2$ in particular. The functions
$\varphi_i-C_\ell\varphi_i$ spanning the new spaces
$\W_\ell$ can be calculated in the same way as the
functions $\varphi_i-F_\nu\varphi_i$. The convergence
rate degrades, however, with this version from
(\ref{eq3.14}) to a value not better than
\begin{equation} \label{eq3.20}
q=\frac{\kappa-1}{\kappa+1},
\end{equation}
and in the extreme case to $q=1-1/(K_1K_2)$ if
$\omega=1/K_2$ is chosen. This means that
possibly a larger number $\ell$ of iterations
and layers, respectively, is needed.
\section{Discretization}
\label{sec4}
The infinite dimensional subspaces $\V_i$ of the kernel
of the projection $\Pi$ have to be replaced by discrete
counterparts to obtain a computationally feasible method.
We start from a potentially very strong, uniform or
nonuniform refinement $\T'$ of the triangulation $\T$,
bridging the scales and resolving the oscillations of
the coefficient functions, and a finite element space
$\S'\subseteq H_0^1(\Omega)$ that consists of the
continuous functions whose restrictions to the elements
in $\T'$ are linear. The whole theory then literally
transfers to the present situation replacing only
the continuous solution space $H_0^1(\Omega)$ and its
subspaces by their discrete counterparts. The only
modification concerns the construction of the stable
decomposition of the functions~$v$ in the kernel of
$\Pi$ into a sum of functions in the corresponding
local subspaces
\begin{equation} \label{eq4.1}
\V_i=\{v-\Pi v\,|\,v\in \S'\cap H_0^1(\omega_i)\}.
\end{equation}
To construct such a decomposition, we use the interpolation
operator $\I:C(\bar{\Omega})\to\S'$ that interpolates
at the nodes of usual kind and reproduces the functions
in $\S'$. As the operator $\I$ is linear and the
$\varphi_i$ form a partition of unity, we can decompose
the functions $v\in\S'$ in the kernel of $\Pi$ into
the sum of the functions
\begin{equation} \label{eq4.2}
v_i=\I(\varphi_i v)-\Pi\sp (\I(\varphi_i v))
\end{equation}
in the modified subspaces $\V_i$. The stability of
this decomposition in the sense of (\ref{eq3.9})
can be deduced in the same way as the stability of
the decomposition in the proof of Lemma~\ref{lm3.1}
since for the functions $v\in\S'$ in the kernel of
$\Pi$ an estimate
\begin{equation} \label{eq4.3}
|\I(\varphi_i v)|_1\leq c\,|\varphi_i v|_1
\end{equation}
holds, which is shown separately for the single
elements $t\in\T'$ using that the restrictions of
such functions $\varphi_i v$ to these elements are
second order polynomials.
\section*{Appendix. Analysis of a local projection operator}
For the convenience of the reader, we give in this appendix
a comparatively detailed proof of the estimates (\ref{eq2.5})
for the exemplary local linear projection operator~$\Pi$
described in Section~\ref{sec2}. Let $\N$ be the set of the
indices of the vertices $x_i$ of the finite elements in
the interior of $\Omega$. The operator $\Pi$ can then be
written as
\begin{displaymath}
\Pi v=\sum_{i\,\in\N}\alpha_i\varphi_i,
\end{displaymath}
with coefficients $\alpha_i$ that depend linearly on $v$
and are calculated as follows. At first, the given function
$v$ is locally, separately on each single finite element
and regardless of the continuity across the boundaries
of the elements, approximated by its $L_2$-orthogonal
projection onto the space of linear functions. The
coefficient $\alpha_i$ is then a weighted mean of the
values of these linear functions at the node $x_i$ under
consideration, weighted according to the contribution of
the involved elements to the area or the volume of the
patch $\omega_i$, the support of the basis function
$\varphi_i$. The constant functions $x\to\alpha_i$
satisfy the local $L_2$-norm estimate
\begin{displaymath}
\|\alpha_i\|_{0,\omega_i}\lesssim\|v\|_{0,\omega_i}
\end{displaymath}
over these patches, where the constant on the
right-hand side depends solely on the space dimension.
Let the patch $\omega_i$ be the union of the elements
$t_1,\ldots,t_n$ and let $v_1,\ldots,v_n$ be the
corresponding local $L_2$-orthogonal projections of $v$
onto the space of linear functions. Let $|\omega_i|$
and $|t_k|$ be the areas or volumes of $\omega_i$ and
the $t_k$. Then
\begin{displaymath}
\|\alpha_i\|_{0,\omega_i}^2=\,
|\omega_i|\,\bigg(\frac{1}{|\omega_i|}\sum_{k=1}^n|t_k|v_k(x_i)\bigg)^2
\leq\,\sum_{k=1}^n|t_k|v_k(x_i)^2.
\end{displaymath}
Transformation to a reference element yields the estimate
\begin{displaymath}
\sum_{k=1}^n|t_k|v_k(x_i)^2\lesssim\sum_{k=1}^n\|v_k\|_{0,t_k}^2
\end{displaymath}
of the right-hand side, with a constant that depends
only on the space dimension. As the $L_2$-norms of
the linear functions $v_k$ over the $t_k$ are less
than or equal to the $L_2$-norms of $v$ over the $t_k$,
this proves the estimate above for the $\alpha_i$.
To prove the estimates (\ref{eq2.5}), we use that the
functions in $H_0^1(\Omega)$ can be considered as
functions in $H^1(\mathbb{R}^2)$ and $H^1(\mathbb{R}^3)$,
respectively, with value zero outside $\Omega$. Because
the hat functions $\varphi_i$ form a partition of unity
on $\Omega$,
\begin{displaymath}
v-\Pi v=\sum_{i\,\in\N}\varphi_i\sp(v-\alpha_i)
+\sum_{i\,\notin\N}\varphi_i v.
\end{displaymath}
On a given element only the functions $\varphi_i$
assigned to its vertices are different from zero.
The square of the $H^1$-seminorm of the error can
therefore be estimated as
\begin{displaymath}
|v-\Pi v|_1^2\,\lesssim
\sum_{i\,\in\N}|\varphi_i\sp (v-\alpha_i)|_1^2
+\sum_{i\,\notin\N}|\varphi_i v|_1^2.
\end{displaymath}
Let $B_i$ be the ball with center $x_i$ of minimum
diameter that covers the patch $\omega_i$ and let
$h_i$ be its radius. As
$|\nabla\varphi_i|\lesssim h_i^{-1}$ by the shape
regularity of the elements,
\begin{displaymath}
|\varphi_i\sp(v-\alpha_i)|_1\lesssim
h_i^{-1}\|v-\alpha_i\|_{0,\omega_i}+
|\sp v\sp|_{1,\omega_i}.
\end{displaymath}
As the linear functional $v\to\alpha_i$ reproduces
the value of constant functions,
\begin{displaymath}
\|v-\alpha_i\|_{0,\omega_i}\lesssim
\|v-\alpha\|_{0,\omega_i}\leq\|v-\alpha\|_{0,B_i}
\end{displaymath}
holds for all constants $\alpha$, and, in particular,
for the mean value $\alpha$ of the function $v$ over
the ball $B_i$. The Poincar\'{e} inequality for
balls leads therefore to the estimate
\begin{displaymath}
\|v-\alpha_i\|_{0,\omega_i}\lesssim h_i|\sp v\sp|_{1,B_i},
\end{displaymath}
with a constant that is, of course, independent
of the radius $h_i$ of $B_i$. For the terms
associated with the inner vertices, thus finally
\begin{displaymath}
|\varphi_i\sp (v-\alpha_i)|_1\lesssim|\sp v\sp|_{1,B_i}.
\end{displaymath}
The boundary terms can be treated with a local variant
of the Friedrichs inequality. Here we prefer to proceed
in a similar way as with the inner vertices. At first,
\begin{displaymath}
|\varphi_i v|_1\lesssim
h_i^{-1}\|v\|_{0,\omega_i}+|\sp v\sp|_{1,\omega_i}.
\end{displaymath}
As polygonal domain, $\Omega$ satisfies an exterior
cone condition. That is, for the $B_i$ assigned to
the vertices $x_i$ on the boundary of $\Omega$
there is constant $c$ such that
\begin{displaymath}
|B_i\cap\Omega|\leq c\,|B_i\!\setminus\!\Omega\sp|.
\end{displaymath}
This constant is independent of $i$ and depends at
most on an upper bound for the diameters of the balls.
The $L_2$-distance of a function $v$ in $L_2(B_i)$
to its mean value over the part of $B_i$ outside
$\Omega$ can therefore, analogously to the reasoning
above, be estimated by its $L_2$-distance to the mean
value over $B_i$ itself. As the mean value of the
given functions $v$ over the part of $B_i$ outside
of $\Omega$ is zero, this leads, again by means of
the Poincar\'{e} inequality for balls, to the
estimate
\begin{displaymath}
\|v\|_{0,\omega_i}\lesssim h_i|\sp v\sp|_{1,B_i}.
\end{displaymath}
For the terms associated with the vertices on the
boundary, then
\begin{displaymath}
|\varphi_i v|_1\lesssim |\sp v\sp|_{1,B_i}.
\end{displaymath}
Since the balls $B_i$ form, because of the shape regularity
of the finite elements, a locally finite covering of $\Omega$,
one obtains finally the estimate
\begin{displaymath}
|v-\Pi v|_1\lesssim |\sp v\sp|_1,
\end{displaymath}
with a constant that depends only on the shape regularity
of the finite elements and the constant from the exterior
cone condition in the form given above. This implies the
stability of $\Pi$. Using that for all square integrable
functions $w$
\begin{displaymath}
\|h^{-1}\varphi_i w\|_0\lesssim h_i^{-1}\|w\|_{0,\omega_i}
\end{displaymath}
holds, the approximation property follows by the same
arguments.
\end{document} |
\begin{document}
\title{Eguchi-Hanson singularities in U(2)-invariant Ricci flow}
\author{Alexander Appleton}
\address{Department of Mathematics, UC Berkeley, CA 94720, USA}
\email{aja44@berkeley.edu}
\maketitle
\begin{abstract}
We show that a Ricci flow in four dimensions can develop singularities modeled on the Eguchi-Hanson space. In particular, we prove that starting from a class of asymptotically cylindrical $U(2)$-invariant initial metrics on $TS^2$, a Type II singularity modeled on the Eguchi-Hanson space develops in finite time. Furthermore, we show that for these Ricci flows the only possible blow-up limits are (i) the Eguchi-Hanson space, (ii) the flat $\mathbb{R}^4 /\mathbb{Z}_2$ orbifold, (iii) the 4d Bryant soliton quotiented by $\mathbb{Z}_2$, and (iv) the shrinking cylinder $\mathbb{R} \times \mathbb{R} P^3$. As a byproduct of our work, we also prove the existence of a new family of Type II singularities caused by the collapse of a two-sphere of self-intersection $|k| \geq 3$.
\end{abstract}
\setcounter{tocdepth}{2}
\makeatletter
\def\@tocline{2}{0pt}{2.5pc}{5pc}{}{\@tocline{2}{0pt}{2.5pc}{5pc}{}}
\makeatother
\tableofcontents
\section{Introduction}
The main result of this paper is to show
\noindent
\begin{center}
\parbox{0.8\linewidth}{
\noindent \textit{A Ricci flow on a four dimensional non-compact manifold may develop a Type II singularity modeled on the Eguchi-Hanson space in finite time.}
}
\end{center}
The Eguchi-Hanson space is diffeomorphic to the cotangent bundle of the two-sphere and asymptotic to the flat cone $\mathbb{R}^4 / \mathbb{Z}_2$. It is the simplest example of a Ricci flat asymptotically locally euclidean (ALE) manifold and in the physics literature known as a gravitational instanton. The Eguchi-Hanson singularities constructed in this paper are the first examples of orbifold singularities in Ricci flow, and are also the first examples of singularities with Ricci flat blow-up limits. As a byproduct of our work we also show that
\noindent
\begin{center}
\parbox{0.8\linewidth}{
\noindent \textit{A Ricci flow on a four dimensional non-compact manifold may collapse an embedded two-dimensional sphere with self-intersection $k \in \mathbb{Z}$ to a point in finite time and thereby produce a singularity.}
}
\end{center}
The singularities we construct when $|k| \geq 3$ are of Type II and the author conjectures that their blow-up limits are homothetic to the steady Ricci solitons found in \cite{A17}.
\subsection{Background}
A family of time-dependent metrics $g(t)$ on a manifold $M$ is called a Ricci flow if it solves the equation
\begin{equation}
\label{RF}
\partial_t g(t) = -2 Ric_{g(t)}.
\end{equation}
Here $Ric_{g(t)}$ is the Ricci tensor of the metric $g(t)$. In local coordinates the Ricci flow equation can be written as a coupled system of second order non-linear parabolic equations. Heuristically speaking, the Ricci flow smoothens the metric $g(t)$, while simultaneously shrinking positively curved and expanding negatively curved directions at each point of the manifold.
Ricci flow was introduced by Hamilton \cite{Ham82} in 1982 to prove that a closed three dimensional manifold admitting a metric of positive Ricci curvature also admits a metric of constant positive sectional curvature. This success demonstrated the power of Ricci flow and ignited much research in this area, culminating in Perelman's proof of the Poincar\'e and Geometrization Conjectures for three dimensional manifolds.
Even though every complete Riemannian manifold $(M,g)$ of bounded curvature admits a short-time Ricci flow starting from $g$, singularities may develop in finite time. Understanding their geometry is central to the study of Ricci flow and has topological implications. For instance, Perelman proved the Geometrization Conjecture by analyzing the singularity formation in three dimensional Ricci flow and showing that a Ricci flow nearing its singular time exhibits one of the following two behaviors:
\begin{itemize}
\item Extinction: The manifold becomes asymptotically round before shrinking to a point
\item (Degenerate or non-degenerate) Neckpinch: A region of the shape of a small cylinder $\mathbb{R} \times S^2$ develops
\end{itemize}
Based on this knowledge Perelman was able to construct a Ricci flow with surgery, which performs the decomposition of a three manifold into pieces corresponding to the eight Thurston geometries, yielding a proof of the Geometrization Conjecture.
In order to understand the formation of singularities in Ricci flow it is very useful to take blow-up limits. Roughly speaking one zooms into the region in which the singularity is forming by parabolically rescaling space and time. The resulting blow-up limit is an ancient Ricci flow called the \textbf{singularity model}. It encapsulates most of the geometric information of the singularity. Note that a Ricci flow is called \textbf{ancient} if it can be extended infinitely into the past. To date all known singularity models are either shrinking or steady \textbf{Ricci solitons}. These are self-similar solutions to the Ricci flow equation that, up to diffeomorphism, homothetically shrink or remain steady, and can be understood as a generalization of Einstein manifolds of positive or zero scalar curvature, respectively. Hamilton distinguishes between \textbf{Type I and Type II singularities}, depending on the rate at which the curvature blows up to infinity as one approaches the singular time. It has been proven that Type I singularities are modeled on shrinking Ricci solitons \cite{EMT11}, however it is unknown whether all Type II singularity models are steady Ricci solitons. In three dimensions the only Type I singularity models are $S^3$, $\mathbb{R} \times S^2$ and their quotients.
As three dimensional singularity formation is now well understood the next step is to analyze the four dimensional case, where currently very little is known other than that the possibilities are far more numerous. Below we list all Type I singularity models known in four dimensions:
\begin{enumerate}
\item $S^3 \times \mathbb{R} $ and its quotients
\item $S^2 \times \mathbb{R}^2$ and its quotients
\item Einstein manifolds of positive scalar curvature (e.g. $S^4$, $\mathbb{C} P^2$, etc.)
\item Compact gradient shrinking Ricci solitons that are not Einstein
\item The FIK shrinker \cite{FIK03}
\end{enumerate}
Note that (1) and (2) are just products of a three dimensional Type I singularity model with the real line. As Einstein manifolds in four dimensions remain to be classified, list item (3) may contain a very large set of manifolds. As for (4), to date only few examples of compact shrinking Ricci solitons are known and a list of these can be found in Cao's survey \cite{Cao10}. The FIK shrinker is a non-compact $U(2)$-invariant shrinking K\"ahler-Ricci soliton, which is diffeomorphic to the blow-up of $\mathbb{C}^2$ at the origin. It is an open question whether there are other non-flat one-ended shrinking Ricci solitons in four dimensions. Maximo proved that Type I singularities modeled on the FIK shrinker may occur in $U(2)$-invariant K\"ahler-Ricci flow \cite{M14}.
The FIK shrinker models an interesting singularity in four dimensional Ricci flow --- namely the collapse of an embedded two-dimensional sphere with non-trivial normal bundle. Topologically, real rank 2 vector bundles over the two-dimensional sphere are classified by their Euler class, which is an integer multiple of the generator of $H^2(S^2, \mathbb{Z})$. We call this multiple the twisting number and denote it by $k$. Recall that the self-intersection of an embedded two-dimensional sphere in a four-dimensional manifold is equal to the twisting number of its normal bundle. Unlike in K\"ahler geometry, where there is a canonical choice for the generator of $H^2(S^2, \mathbb{Z})$ and the sign of the self-intersection number is crucial, in the Riemannian case only its absolute value affects the geometry and behavior of embedded two-spheres under Ricci flow. Heuristically speaking, the larger $|k|$, the more negative curvature there is in the vicinity of the sphere and the less likely it collapses to a point. In the list above $S^2 \times \mathbb{R}^2$ and the FIK shrinker model the collapse of two dimensional spheres with self-intersection equal to $0$ and $-1$, respectively. The main goal of this paper is to show that embedded spheres of self-intersection number $|k| \geq 2$ may also collapse in finite time. To explain this in greater detail we give an overview of our setup below.
\subsection{Overview of setup}
Let $M_k$, $k \geq 1$, be diffeomorphic to the blow-up of $\mathbb{C}^2/\mathbb{Z}_k$ at the origin, and denote by $S^2_o$ the two-sphere stemming from the blow-up. Alternatively one can also view $M_k$ as a plane bundle over $S^2_o$. Fix an arbitrary point o, for 'origin', on $S^2_o$. Note that $S^2_o$, with respect to the orientation inherited from $\mathbb{C}^2$, has self-intersection $-k$. Then equip $M_k$ with an $U(2)$-invariant metric $g$. It turns out that with help of the Hopf fibration
$$\pi: S^3/\mathbb{Z}_k \rightarrow S^2$$
such $U(2)$-invariant metrics can be conveniently written as a warped product metric of the form
\begin{equation}
\label{metric-intro-s}
g = ds^2 + a^2(s) \omega \otimes \omega + b^2(s) \pi^{\ast} g_{S^2(\frac{1}{2})},
\end{equation}
on the open dense subset $\mathbb{R}_{> 0 } \times S^3/\mathbb{Z}_k \subset M_k$. $\omega$ is the 1-form dual to the vertical directions of the Hopf fibration and $s$ is a parametrisation of the $\mathbb{R}_{>0}$ factor. Note that $g$ pulls back to a Berger sphere metric on the cross-sections $S^3/\mathbb{Z}_k$. One can complete $g$ to a smooth metric on all of $M_k$ by requiring
\begin{align}
\label{boundary-cond-intro-s}
a(0) &= 0 \\ \nonumber
a_s(0) &= k \\ \nonumber
b(0) &> 0
\end{align}
and that $a(s)$ and $b(s)$ can be extended to an odd and even function around $s=0$, respectively. Via the boundary condition $a_s(0) = k$ is how topology enters the analysis of the Ricci flow equation. We would like to mention here that throughout this paper we often take the warping functions $a$ and $b$ to be functions of points $(p,t)$ in spacetime rather than of $(s,t)$. This will always be clear from context.
An upshot of writing the metric $g$ in the form (\ref{metric-intro-s}) is that the Ricci flow equation (\ref{RF}) reduces to a $(1+1)$-dimensional system of parabolic equations for the warping functions $a$ and $b$, which simplifies the analysis greatly. In addition to this, both the FIK shrinker, which is diffeomorphic to $M_1$, and the Eguchi-Hanson space, which is diffeomorphic to $M_2$, are $U(2)$-invariant, and therefore their metrics can be written in the form (\ref{metric-intro-s}). In this paper we only study Riemannian manifolds diffeomorphic to $M_k$, $k \in \mathbb{N}$, equipped with a $U(2)$-invariant metric of the form (\ref{metric-intro-s}).
We will consider numerous \emph{scale-invariant} quantities, the most fundamental and important of which we introduce here:
\begin{align*}
Q &\coloneqq \frac{a}{b} \\
x &\coloneqq a_s + Q^2 - 2 \\
y &\coloneqq b_s - Q
\end{align*}
The quantity $Q$ measures the `roundness' of the cross-sectional $S^3/\mathbb{Z}_k$. That is, when $Q =1$ the metric on the cross-section is round. The quantities $x$ and $y$ are more interesting, as they measure the deviation of the metric $g$ from being K\"ahler. In particular, when
$$y = 0$$
the manifold $(M_k,g)$, $k\geq 1$, is K\"ahler with respect to the standard complex structure induced from $\mathbb{C}^2$. Moreover, when
$$x = y = 0$$
the manifold $(M_k, g)$ is hyperk\"ahler, and as we show in section \ref{kahler-E-H-section}, homothetic to the Eguchi-Hanson space.
\subsection{Overview of results}
The first main result of this paper is to show that
\begin{center}
\parbox{0.8\linewidth}{
\noindent \textit{For a large class of $U(2)$-invariant asymptotically cylindrical initial metrics on $M_k$, $k \geq 2$, the Ricci flow develops a Type II singularity in finite time, as the area of $S^2_o$ decreases to zero. When $k=2$ the blow-up limit of the singularity is homothetic to the Eguchi-Hanson space.}
}
\end{center}
We define the class of metrics for which this result holds in subsection \ref{subsec:preciseresults}. Note that in the $k=2$ case the Eguchi-Hanson space is the first example of a Ricci flat singularity model. Based on numerical simulations the author believes that the Type II singularities in the $k \geq 3$ case are modeled on the steady solitons found in \cite{A17}. A paper on the numerical results is in preparation.
The above result should be contrasted with the behavior of a Ricci flow starting from a K\"ahler metric. It is well known that the K\"ahler condition is preserved by Ricci flow, and that for such a flow the area of a complex submanifold evolves in a fixed manner. In particular, if $(M, g)$ is a K\"ahler manifold with K\"ahler form $\omega$, then under Ricci flow the K\"ahler class evolves by
$$[\omega(t)] = [\omega(0)] - 4 \pi t c_1(M),$$
where $c_1(M)$ is the first Chern class of $M$. If we integrate the above equation over a complex curve $\Sigma$ in $M$ then one sees that
$$ |\Sigma|_t = |\Sigma|_0 - 4 \pi t \langle c_1(M), [\Sigma] \rangle,$$
where $|\Sigma|_t$ denotes the area of $\Sigma$ at time $t$. In the case that $M \cong M_k$, $\Sigma = S^2_o$ and $g$ is K\"ahler, it was shown in \cite[Proof of Lemma 1.2]{FIK03} that
$$\langle c_1(M_k), [S^2_o] \rangle = 2 - k$$
and hence
\begin{equation}
\label{area-evol-kahler}
|S^2_o|_t = |S^2_o|_0 - 4 \pi t \left( 2 - k \right).
\end{equation}
This shows that for a K\"ahler-Ricci flow $(M_k, g(t))$, $k \in \mathbb{N}$ the two sphere $S^2_o$ can only collapse to a point when $k = 1$. In fact, when $k=2$ the area of $S^2_o$ is stationary and for $k\geq 3$ increasing. Maximo in \cite{M14} uses the K\"ahler condition and (\ref{area-evol-kahler}) to show that an embedded sphere of self-intersection $-1$ may collapse to a point in finite time under Ricci flow. Note that in our construction the metrics are not assumed to be K\"ahler and hence we cannot rely on (\ref{area-evol-kahler}).
The second main result of this paper is the classification of all possible blow-up limits in the $k=2$ case, including those at larger distance scales from the tip of $M_2$. In particular, we show that
\begin{center}
\parbox{0.8\linewidth}{
\noindent \textit{For a large class of $U(2)$-invariant asymptotically cylindrical initial metrics on $M_2$ any sequence of blow-ups subsequentially converges to one of the following spaces:
\begin{enumerate}[label=(\roman*)]
\item The Eguchi-Hanson space
\item The flat $\mathbb{R}^4 / \mathbb{Z}_2$ orbifold
\item The 4d Bryant soliton quotiented by $\mathbb{Z}_2$
\item The shrinking cylinder $\mathbb{R} \times \mathbb{R} P^3$
\end{enumerate}
}
}
\end{center}
The blow-up limits (ii) and (iii) show that the Eguchi-Hanson singularity results in the formation of an orbifold point, which to our knowledge the first concrete example of such in four dimensional Ricci flow.
We expect that many of our methods generalize to the analysis of Ricci flow on other cohomogeneity one manifolds. These are manifolds that admit an action by isometries of a compact Lie group $G$ for which the quotient is one dimensional. The author believes that this work could contribute towards a complete picture of Ricci solitons and ancient Ricci flows on cohomogeneity one manifolds in four dimensions.
\subsection{Precise statement of results}
\label{subsec:preciseresults}
Before presenting the main theorems of this paper, we list the definition of a class $\mathcal{I}$ of metrics needed to state our results.
\noindent \textbf{Definition 7.2.}\textit{
For $K>0$ let $\mathcal{I}_K$ be the set of all complete \emph{bounded curvature} metrics of the form (\ref{metric-intro-s}) on $M_k$, $k \geq 1$, with \emph{positive injectivity radius} that satisfy the following scale-invariant inequalities:
\begin{align*}
Q &\leq 1 \\
a_s, b_s &\geq 0 \\
y &\leq 0 \\
\sup a_s &< K \\
\sup |b b_{ss}| &< K
\end{align*}
Denote by $\mathcal{I}$ the set of metrics $g$ such that for sufficiently large $K>0$ we have $g\in \mathcal{I}_K$.
}
For any $k \in \mathbb{N}$ the set $\mathcal{I}$ of metrics on $M_k$ is non-empty, as for example the metric on $M_k$ defined by
\begin{align*}
a(s) &= Q = \tanh(k s) \\
b(s) &= 1
\end{align*}
is contained in $\mathcal{I}$. Moreover, as we prove in Lemma \ref{I-preserved}, the class $\mathcal{I}_K$ of metrics is preserved by the Ricci flow for sufficiently large $K>0$. In our paper we will mainly consider Ricci flows $(M_k, g(t))$, $t \in [0,T)$, starting from an initial metric $g(0) \in \mathcal{I}$.
Now we list the precise statements of the main results of this paper.
\noindent \textbf{Theorem 9.1}\hspace{0.5em}(Type II singularities)\textbf{.}\textit{
Let $(M_k, g(t))$, $k \geq 2$, be a Ricci flow starting from an initial metric $g(0) \in \mathcal{I}$ (see Definition \ref{def:I}) with
\begin{equation*}
\sup_{p \in M_2} b(p,0) < \infty.
\end{equation*}
Then $g(t)$ encounters a Type II curvature singularity in finite time $T_{sing}>0$ and
\begin{equation*}
\sup_{0 \leq t < T_{sing}} \left(T_{sing} - t\right) b^{-2}(o,t) = \infty.
\end{equation*}
Furthermore, there exists a sequence of times $t_i \rightarrow T_{sing}$ such that the following holds:
Consider the rescaled metrics
\begin{equation*}
g_i(t) = \frac{1}{b^2(o,t_i)} g\left( t_i + b^2(o,t_i) t \right), \quad t \in \big[-b(o, t_i)^{-2} t_i, b(o, t_i)^{-2}\left(T_{sing} - t_i\right) \big).
\end{equation*}
Then $(M_k, g_i(t), o)$ subsequentially converges, in the pointed Gromov-Cheeger sense, to an eternal Ricci flow $(M_k, g_\infty(t), o)$, $t \in (-\infty, \infty)$. When $k=2$ the metric $g_\infty(t)$ is stationary and homothetic to the Eguchi-Hanson metric.
}
\begin{remark}
We would like to make the following remarks:
\begin{enumerate}
\item In Theorem 9.1, case $k=2$, we only prove that there exists a blow-up sequence which converges to the Eguchi-Hanson space. In Theorem 12.1 below we extend this result and show that in fact any blow-up around the tip of $M_2$ is homothetic to the Eguchi-Hanson space.
\item The initial metric $g(0) \in \mathcal{I}$ with $\sup_{p \in M} b(p,0)<\infty$ is asymptotic to $\mathbb{R} \times S^3 / \mathbb{Z}_k$, where $S^3/\mathbb{Z}_k$ is equipped with a squashed Berger sphere metric. This is because metrics in $\mathcal{I}$ satisfy $a_s, b_s \geq 0$ and $Q \leq 1$.
\end{enumerate}
\end{remark}
The second main result of our paper is the classification of all possible blow-up limits in the $k=2$ case:
\noindent \textbf{Theorem 12.1}\hspace{0.5em}(Blow-up limits)\textbf{.}\textit{
Let $(M_2, g(t))$, $t \in [0, T_{sing})$, be a Ricci flow starting from an initial metric $g(0) \in \mathcal{I}$ (see Definition \ref{def:I}) with $\sup_{p \in M_2} b(p,0) < \infty$. Let $(p_i,t_i)$ be a sequence of points in spacetime with $b(p_i, t_i) \rightarrow 0$. Passing to a subsequence, we may assume that we are in one of the four cases listed below.
\begin{enumerate}[label=(\roman*)]
\item $\lim_{i \rightarrow \infty} \frac{b(p_i,t_i)}{b(o,t_i)} < \infty$
\item $\lim_{i \rightarrow \infty} \frac{b(p_i,t_i)}{b(o,t_i)} = \infty$ and $\lim_{i \rightarrow \infty} b_s(p_i, t_i) = 1$
\item $\lim_{i \rightarrow \infty} \frac{b(p_i,t_i)}{b(o,t_i)} = \infty$ and $\lim_{i \rightarrow \infty} b_s(p_i, t_i) \in (0,1)$
\item $\lim_{i \rightarrow \infty} \frac{b(p_i,t_i)}{b(o,t_i)} = \infty$ and $\lim_{i \rightarrow \infty} b_s(p_i, t_i) = 0$
\end{enumerate}
Consider the dilated Ricci flows
$$g_i (t) = \frac{1}{b^2(p_i,t_i)} g\left(t_i + b^2(p_i, t_i) t \right), \quad t \in [- b(p_i,t_i)^{-2} t_i , 0].$$
Then $(M_2, g_i(t), p_i)$, $t \in [- b(p_i,t_i)^{-2} t_i , 0]$, subsequentially converges, in the Cheeger-Gromov sense, to an ancient Ricci flow $(M_\infty, g_\infty(t), p_\infty)$, $t \in (-\infty, 0]$. Depending on the limiting property of the sequence $(p_i, t_i)$ we have:
\begin{enumerate}[label=(\roman*)]
\item $M_\infty \cong M_2$ and $g_\infty(t)$ is stationary and homothetic to the Eguchi-Hanson metric
\item $M_\infty \cong \mathbb{R}^4\setminus\{0\} / \mathbb{Z}_2$ and $g_\infty(t)$ can be extended to a smooth orbifold Ricci flow on $\mathbb{R}^4/\mathbb{Z}_2$ that is stationary and isometric to the flat orbifold $\mathbb{R}^4/\mathbb{Z}_2$
\item $M_\infty \cong \mathbb{R}^4\setminus\{0\} / \mathbb{Z}_2$ and $g_\infty(t)$ can be extended to a smooth orbifold Ricci flow on $\mathbb{R}^4/\mathbb{Z}_2$ that is homothetic to the 4d Bryant soliton quotiented by $\mathbb{Z}_2$
\item $M_\infty \cong \mathbb{R} \times \mathbb{R} P^3$ and $g_\infty(t)$ is homothetic to a shrinking cylinder
\end{enumerate}
}
\begin{remark}
Note that in Theorem 12.1 we do not prove that all blow-up limits (i)-(iv) occur. In fact, it may turn out that the Eguchi-Hanson singularity is isolated, in which case we would only see blow-up limits (i) and (ii).
\end{remark}
As a byproduct of our work we also prove the following two theorems, which are of independent interest. Firstly, we exclude shrinking Ricci solitons in a large class of metrics.
\noindent \textbf{Theorem 6.1} \hspace{0.5em}(No shrinker)\textbf{.}\textit{
On $M_k$, $k \geq 2$, there does not exists a complete $U(2)$-invariant shrinking Ricci soliton of bounded curvature satisfying the conditions
\begin{enumerate}
\item $\sup_{p \in M_k} |b_s| < \infty$
\item $T_1 = a_s + 2 Q^2 - 2 > 0$ for $s > 0$
\item $Q = \frac{a}{b} \leq 1$
\end{enumerate}
}
As we show in section 5 the inequalities $T_1 > 0$ for $s >0$ and $Q \leq 1$ are preserved by a Ricci flow $(M_k, g(t))$, $k \geq 2$, with $g(t) \in \mathcal{I}$. For this reason Theorem 6.1 can be used to exclude Type I singularities for such flows.
Secondly, we prove a uniqueness result for ancient Ricci flows on $M_2$.
\noindent \textbf{Theorem 11.1} \hspace{0.5em}(Unique ancient flow)\textbf{.}\textit{
Let $\kappa > 0$ and $\left(M_2,g(t)\right)$, $t \in (-\infty,0]$, be an ancient Ricci flow that is $\kappa$-non-collapsed at all scales and $g(t) \in \mathcal{I}$, $t \in (-\infty,0]$ (see Definition \ref{def:I}). Then $g(t)$ is stationary and homothetic to the Eguchi-Hanson metric.
}
We rely heavily on this result when we investigate all possible blow-up limits of a Ricci flow $(M_2, g(t))$ encountering a singularity at $S^2_o$.
\subsection{Outline of paper and proofs}
\label{subsec:outline-paper}
Our paper is organized by sections. Section 2 is preliminary and its goal is to set up in more detail the manifolds and metrics considered in this paper. Here we also derive the full curvature tensor and Ricci flow equation for $U(2)$-invariant metrics. In section 3 we prove a maximum principle for degenerate parabolic differential equations on $M_k$. Beginning from section 4 we present new results. Below we outline the main results of those sections and their proofs.
\noindent \textbf{Outline of section 4.}
A key ingredient in our paper are the scale-invariant quantities
$$ x = a_s + Q^2 - 2$$
and
$$ y = b_s - Q,$$
that measure the deviation of a $U(2)$-invariant metric from being K\"ahler with respect to two fixed complex structures $J_1$ and $J_2$ on $M_k$, $k \geq 1$ (see section \ref{kahler-E-H-section} for the precise definition of $J_1$ and $J_2$). In particular, a metric is K\"ahler with respect to $J_1$ whenever $y=0$ and with respect to $J_2$ whenever $x=y=0$.
Interestingly, a $U(2)$-invariant metric of the form (\ref{metric-intro-s}) is K\"ahler with respect to $J_2$ if, and only if, the underlying manifold is diffeomorphic to $M_2$ and the metric is homothetic to the Eguchi-Hanson metric, as we show in Lemma \ref{unique-Kahler-lem}. Therefore the quantities $x$ and $y$ can be used to measure how much a metric on $M_2$ deviates from the Eguchi-Hanson metric --- a tool that is indispensable to our analysis. In the later sections we develop methods to control the behavior of $x$ and $y$ under the Ricci flow. This will allow us to prove that certain singularities of Ricci flows $(M_2, g(t))$ are modeled on the Eguchi-Hanson space.
In Lemma \ref{E-H-properties-lem} of this section we also derive various properties of the Eguchi-Hanson metric. These are frequently used throughout the paper.
\noindent \textbf{Outline of section 5.} The goal of this section is to derive various \emph{scale-invariant} inequalities that are conserved by Ricci flow. We say that on a Riemannian manifold $(M,g)$ a geometric quantity $T_g: M \rightarrow \mathbb{R}$ is scale-invariant if for every point $p \in M$, we have $T_g(p) = T_{\lambda g}(p)$ for all $\lambda > 0$. The scale-invariance of the inequalities derived is crucial, as it ensures that they pass to blow-up limits and thus also constrain their geometry.
We construct these inequalities from the scale-invariant quantities $a_s$, $b_s$ and $Q := \frac{a}{b}$, where $a$ and $b$ are the warping functions of the metric $g$ of the form (\ref{metric-intro-s}). Note that subscript $s$ denotes the derivative with respect to $s$. The key observation is that the evolution equation of the \emph{scale-invariant} quantity
$$T_{(\alpha, \beta, \gamma)} = \alpha a_s + \beta Q b_s + \gamma Q^2, \quad \alpha, \beta, \gamma \in \mathbb{R},$$
can be written in the form
$$ \partial_t T_{(\alpha, \beta, \gamma)} = \left[T_{(\alpha, \beta, \gamma)}\right]_{ss} + \left( 2\frac{b_s}{b} -\frac{a_s}{a} \right)\left[T_{(\alpha, \beta, \gamma)}\right]_{s} + \frac{1}{b^2} C_{(\alpha, \beta, \gamma)},$$
where $C_{(\alpha, \beta, \gamma)}$ is a function of $a_s$, $b_s$ and $Q$. For certain choices of $\alpha$, $\beta$, $\gamma$ and $\delta \in \mathbb{R}$ one can determine the sign of $C_{(\alpha, \beta, \gamma)}$ at a local extremum at which $T_{(\alpha, \beta, \gamma)} = \delta$. Depending on the sign, this allows one to prove, via the maximum principle, that either
$$T_{(\alpha, \beta, \gamma)} \geq \delta$$
or
$$T_{(\alpha, \beta, \gamma)} \leq \delta$$
is a conserved inequality. One of the conserved inequalities of this form is
$$ x \leq 0,$$
however we derive many others.
In this section we also find conserved inequalities not of the above form. For instance, we show that each of the inequalities listed below are conserved by the Ricci flow:
\begin{itemize}
\item $Q \leq 1$
\item $y \leq 0$
\item $a_s, b_s \geq 0$
\end{itemize}
The proof is carried out by applying the maximum principle to their evolution equations or, in the case of $a_s , b_s \geq 0$, to their system of evolution equations. The conserved inequalities $Q \leq 1$, $y \leq 0$ and $a_s, b_s \geq 0$ are especially important, as they are part of the definition of the class of metrics $\mathcal{I}$ mentioned above, and constitute the first step in showing that $\mathcal{I}$ is preserved by the Ricci flow.
\noindent \textbf{Outline of section 6.} The main result of section 6 is Theorem \ref{thm:no-shrinker}, which rules out shrinking solitons on $M_k$, $k \geq 2$, within a large class of $U(2)$-invariant metrics. Before we outline the proof, note that from the evolution equation (\ref{b-evol}) of $b$ under Ricci flow it follows by an application of L'H\^opital's rule that at $s=0$
\begin{equation}
\label{outline:b0-evol}
\partial_t b(0,t)^2 = 4 \left( by_s + k-2 \right).
\end{equation}
This formula is a generalization of (\ref{area-evol-kahler}) to the non-K\"ahler case, as the area of $S^2_o$ at time $t$ equals $b(o,t)^2 \pi$. Hence a shrinking soliton must satisfy
$$ \partial_t b(0,t)^2 < 0,$$
which for $k \geq 2$ implies that $y_s < 0$ at $s = 0$.
For the proof of Theorem \ref{thm:no-shrinker} we have to rely on the inequality
$$ T_1 = a_s + 2 Q^2 - 2 \geq 0,$$
which by Lemma \ref{T1-preserved-lem} is conserved by the Ricci flow. In particular, we show that amongst metrics of the form (\ref{metric-intro-s}) on $M_k$, $k \geq 2$, satisfying $Q \leq 1$, $T_1 > 0$ when $s> 0$, and $\sup_{p\in M_k} |b_s| < \infty$ there are no shrinking solitons. We briefly sketch the proof here: First we show in Lemma \ref{dQ-Lemma} that $Q_s \geq 0$ for shrinking solitons. This follows from the Ricci soliton equation, which for metrics of the form (\ref{metric-intro-s}) reduces to a system of ordinary differential equations. Then we consider the evolution equation
\begin{equation}
\label{outline:y-evol}
\partial_t y = y_{ss} + \frac{a_s}{a} y - \frac{y}{a^2} G,
\end{equation}
of $y$, where $G$ is a function of $a_s$, $b_s$ and $Q$. In Lemma \ref{Gpos-lem} we show that whenever $Q_s, T_1 > 0$ we have $G > 0$. This shows that under Ricci flow satisfying these inequalities a negative minimum of $y$ is strictly increasing and a positive maximum is strictly decreasing. However, since $y$ is a scale-invariant quantity, and a shrinking Ricci soliton, up to diffeomorphism, homothetically shrinks under Ricci flow, we see that the maximum or minimum of $y$ must remain constant throughout the flow. We conclude that $y = 0$ everywhere, excluding a shrinking soliton. In the proof of Theorem \ref{thm:no-shrinker}, rather than working with the evolution equation (\ref{outline:y-evol}) of $y$, we use the corresponding ordinary differential equation on a Ricci soliton background.
\noindent \textbf{Outline of section 7.}
The goal of this section is to prove Theorem \ref{curv-bound}, which states that for a Ricci flow $(M_k, g(t))$, $k \geq 1$, starting from an initial metric $g(0) \in \mathcal{I}$ with $\sup_{p \in M_k} b(p,0) < \infty$ there exists a $C_1 > 0$ such that the curvature bound
$$ |Rm_{g(t)}|_{g(t)} \leq \frac{C_1}{b^{2}}$$
holds. The proof is carried out by a contradiction/blow-up argument: Assume there exists a sequence of numbers $D_i \rightarrow \infty$ and points $(p_i, t_i)$ in spacetime such that
$$K_i := |Rm_{g(t_i)}|_{g(t_i)}(p_i) = \frac{D_i}{b(p_i,t_i)^{2}}.$$
Consider the rescaled metrics
$$g_i(t) = K_i g\left( t_i + \frac{t}{K_i}\right), \quad t \in [-K_i t_i, 0],$$
normalized such that $|Rm_{g_i(0)}|_{g_i(0)}(p_i) = 1$. Then Perelman's no-local-collapsing theorem shows that $(M_k, g_i(t), p_i)$ subconverges to an ancient non-collapsed Ricci flow $(M_\infty, g_\infty(t), p_\infty)$, $t \leq 0$. As $D_i \rightarrow \infty$ the warping functions $b_i$ corresponding to the metrics $g_i(t)$ satisfy $b_i(p_i,0) \rightarrow \infty$. Recalling that the warping function $b$ describes the size of the base manifold $S^2$ in the Hopf fibration of the $S^3/\mathbb{Z}_k$ cross-sections, one can see that $(M_\infty, g_\infty(t))$ splits as $M_\infty = \mathbb{R}^2 \times N$, where $\mathbb{R}^2$ is equipped with the flat euclidean metric and the restriction of $g_\infty(t)$ to $N$ is a 2d non-compact $\kappa$-solution. However, the only $\kappa$-solutions in 2d are either the shrinking sphere or its $\mathbb{Z}_2$ quotient, both of which are compact. This is a contradiction and the proof of the curvature bound follows.
In Corollary \ref{cor:curv-bound-ancient} we show that ancient Ricci flows in $\mathcal{I}$, which are $\kappa$-non-collapsed at all scales, also satisfy the curvature bound
$$ |Rm_{g(t)}|_{g(t)} \leq \frac{C_1}{b^{2}}.$$
This curvature bound will be important in section \ref{E-H-unique-ancient-section}.
\noindent \textbf{Outline of section 8.}
In this section we prove various local and global compactness results for $U(2)$-invariant Ricci flows in the class of metrics $\mathcal{I}$. To state the results we need to first introduce the following notation for a $U(2)$-invariant Riemannian manifold $(M,g)$:
\begin{itemize}
\item Let $\Sigma_p \subset M$ denote the orbit of $p$ under the $U(2)$-action.
\item Let $$C_g(p,r) \coloneqq \left\{ q\in M \: \Big | \: d_g(q, \Sigma_p) < r \right\}$$
\end{itemize}
One sees that $C_g(p,r)$ is the tubular neighborhood of `radial width' $r$ of the orbit $\Sigma_p$ of $p$ under the $U(2)$-action. See Definition \ref{def:C} for more details.
The main result of this section is Theorem \ref{thm:local-compactness}, which states under which conditions a sequence of $U(2)$-invariant Ricci flows of the form $(C_{g_i(0)}(p_i, r), g_i(t), p_i)$, $t \in [-\Delta t, 0]$, $\Delta t > 0 , r > 0$, subsequentially converges, in the Cheeger-Gromov sense, to a limiting $U(2)$-invariant Ricci flow $(\mathcal{C}_\infty, g_\infty(t), p_\infty)$, $t \in [-\Delta, 0]$. Amongst other conditions, we require that $g_i(t)$ is
\begin{itemize}
\item $\kappa$-non-collapsed at some scale $\rho > 0$ at the point $(p_i, 0)$ in spacetime
\item In the class $\mathcal{I}$
\item Normalized such that $b(p_i, 0) = 1$
\item Of uniformly bounded curvature in $C_{g_i(0)}(p_i, r) \times [-\Delta t, 0]$
\end{itemize}
We also show that after choosing suitable coordinates the warping functions of the metrics $g_i(t)$ subsequentially converge to the corresponding warping functions of $g_\infty(t)$. The compactness result of Theorem \ref{thm:local-compactness} is used frequently throughout the paper, especially its variation, stated in Proposition \ref{blow-up-prop}.
\noindent \textbf{Outline of section 9.}
The main goal of this section is to constrain the geometry of ancient Ricci flows $(M_k, g(t))$, $k \in \mathbb{N}$, $t \in (-\infty, 0]$ in the class of metrics $\mathcal{I}$ that are $\kappa$-non-collapsed at all scales. This is achieved by proving that various scale-invariant inequalities hold. For instance, in Theorem \ref{T-ancient} we prove that three inequalities of the form $T_{(\alpha, \beta, \gamma)} \geq 0$, as in introduced in the outline of section 5 above, hold on such ancient flows. Furthermore, we prove in Theorem \ref{kahler-ancient-thm} that an ancient Ricci flow on $M_2$ in $\mathcal{I}$ which is K\"ahler with respect to $J_1$, i.e. $y=0$ everywhere, is stationary and homothetic to the Eguchi-Hanson space. This result will be used in section \ref{E-H-sing-section}, where we construct an eternal blow-up limit of a Ricci flow on $M_2$ that is homothetic to the Eguchi-Hanson space.
The proof of these theorems is via a \textbf{contradiction/compactness argument} frequently employed throughout the paper. We briefly sketch the method here: Assume we want to prove that a scale-invariant inequality $T \geq 0$ holds on $M_k \times (-\infty, 0]$. We argue by contradiction and assume that
$$\iota :=\inf_{M_k \times (-\infty, 0]} T < 0.$$
We then take a sequence of points $(p_i, t_i)$ in spacetime such that $T(p_i, t_i) \rightarrow \iota$ as $i \rightarrow \infty$, and consider the dilated metrics
$$g_i(t) = \frac{1}{b(p_i,t_i)^2} g\left( t + t_i b(p_i,t_i)^2\right), \quad t \in [-\Delta t, 0],$$
on the tubular neighborhoods $C_{g_i(0)}(p_i, \frac{1}{2})$ (see Definition \ref{def:C}) for some small $\Delta t > 0$. By the compactness results of section 8, in particular Proposition \ref{blow-up-prop}, the Ricci flows $(C_{g_i(0)}(p_i, \frac{1}{2}), g_i(t), p_i)$, $[-\Delta t, 0]$, subsequentially converges to a Ricci flow $(\mathcal{C}_\infty, g_\infty(t), p_\infty)$, $[-\Delta t, 0]$, where
$$T(p_\infty, 0) = \inf_{\mathcal{C}_\infty \times [-\Delta t, 0]} T = \iota < 0$$
by the scale invariance of $T$. If, however, the evolution equation of $T$ precludes a negative infimum from being attained, we have arrived at a contradiction and proven the desired result.
\noindent \textbf{Outline of section 10.} The goal of this section is to prove Theorem \ref{E-H-sing-thm}, which states that a Ricci flow $(M_k, g(t))$, $k \geq 2$, starting from an initial metric $g(0) \in \mathcal{I}$ with $\sup_{p \in M_k} b(p,0) < \infty$ encounters a Type II singularity in finite time at the tip of $M_k$ as the area of $S^2_o$ decreases to zero. In the $k=2$ case we show that such a singularity possesses a blow-up limit that is stationary and homothetic to the Eguchi-Hanson space. We do not further investigate the $k \geq 3$ case, however the author conjectures that their blow-up limits are homothetic to the steady Ricci solitons found in \cite{A17}.
The proof is carried out in multiple steps. First we show in Lemma \ref{sing-time-finite} that $g(t)$ encounters a singularity in finite time $T_{sing}\in (0, \infty)$ and $b(o,t) \rightarrow 0$ as $t \rightarrow T_{sing}$. This shows that the two-sphere $S^2_o$ at the tip of $M_k$ collapses to a point in finite time and thereby produces a singularity.
In the second step, we rely on the results of section 6 to show that a blow-up limit around $o \in S^2_o$ cannot be a shrinking Ricci soliton when $k \geq 2$.
As all Type I singularities are modeled on shrinking Ricci solitons we deduce that the singularity is of Type II.
In the third step we borrow a trick due to Hamilton to pick a sequence of times $t_i \rightarrow T_{sing}$ such that the following holds: Take the rescaled metrics
$$g_i(t) = \frac{1}{b(o,t_i)^2} g\left(t_i + b^2(o,t_i) t \right), \quad t \in \big[-b(o, t_i)^{-2} t_i, b(o, t_i)^{-2}\left(T_{sing} - t_i\right) \big),$$
where we recall that $o \in S^2_o$. Then $(M_k, g_i(t), o)$ subsequentially converges to an eternal Ricci flow $(M_\infty, g_{\infty}(t), o)$, $t \in (-\infty, \infty)$, where $M_\infty$ is diffeomorphic to $M_k$.
In the final step we analyze the geometry of $M_\infty$ when $k=2$. It turns out that for the choice of times $t_i$ it follows that
$$ \partial_t b(o,0) = 0$$
on $M_\infty$ background. By the evolution equation (\ref{outline:b0-evol}) of $b$ at $o$ this implies
$$ y_s(o,0) = 0.$$
Applying a strong maximum principle we deduce that $y=0$ everywhere. By the results of section 9 it then follows that $g_\infty(t)$ is stationary and homothetic to the Eguchi-Hanson metric.
We mention here that the $k=2$ case of Theorem \ref{E-H-sing-thm} is superseded by Corollary \ref{E-H-blowup} of Theorem \ref{E-H-ancient-thm}. However, since the proof of Theorem \ref{E-H-sing-thm} is simpler we present it here.
\noindent \textbf{Outline of section 11.} The goal of this section is to show that an ancient Ricci flow $(M_2, g(t)), t\in (-\infty,0]$, which is $\kappa$-non-collapsed at all scales and satisfies $g(t) \in \mathcal{I}$, is stationary and homothetic to the Eguchi-Hanson space. The most important consequence of this is that in Theorem \ref{E-H-sing-thm} in fact \emph{any} blow-up of the singularity forming at the tip of $M_2$ is homothetic to the Eguchi-Hanson space, whereas we had previously only proven that there exists \emph{a} blow-up sequence that converges to the Eguchi-Hanson space.
The proof idea, which we call \textbf{successive constraining}, is to find a continuously varying family of preserved inequalities $Z_{\theta} \geq 0$, $\theta \in [0,1]$, for which $Z_0 \geq 0$ on $M_2 \times (-\infty, 0]$ implies that $g(t)$ is homothetic to the Eguchi-Hanson metric. For our choice of conserved inequalities $Z_\theta \geq 0$, $\theta \in [0,1]$, it follows from the work of section 9 that $Z_1 \geq 0$ on $M_2 \times (-\infty,0]$. Then we deform the inequality $Z_1 \geq 0$ along the path $Z_\theta \geq 0$, $\theta \in [0,1]$, to the inequality $Z_0 \geq 0$ with help of the strong maximum principle applied to the evolution equation of $Z_\theta$. This allows us to deduce that $g(t)$ is stationary and homothetic to the Eguchi-Hanson space. In subsection \ref{E-H-ancient-thm-outline} we give a more detailed outline of the proof of Theorem \ref{E-H-ancient-thm}.
\noindent \textbf{Outline of section 12.} The main result of this section is Theorem \ref{blow-up-thm}, which characterizes all the possible blow-up limits of a Ricci flow $(M_2, g(t))$ starting from an initial metric $g(0) \in \mathcal{I}$ with $\sup_{p \in M_2} b(p,0) < \infty$. We show that the only possible blow-up limits are (i) the Eguchi-Hanson space, (ii) the flat orbifold $\mathbb{R}^4 / \mathbb{Z}_2$, (iii) the 4d Bryant soliton quotiented by $\mathbb{Z}_2$ and (iv) the shrinking cylinder $\mathbb{R} \times \mathbb{R} P^3$.
Below we give a brief outline of the proof of Theorem \ref{blow-up-thm}: Assume we are given a sequence of points $(p_i, t_i)$ in spacetime with $b(p_i, t_i) \rightarrow 0$. Consider the rescaled metrics
$$g_i(t) = \frac{1}{b(p_i,t_i)^2} g( t_i + b(p_i, t_i)^2 t), \quad t \in [- b(p_i, t_i)^{-2} t_i, 0].$$
By passing to a subsequence we may assume that either
$$\text{ (I) } \sup_i \frac{b(p_i,t_i)}{b(o,t_i)} < \infty \quad \text{or} \quad \text{ (II) } \lim_{i \rightarrow \infty} \frac{b(p_i,t_i)}{b(o,t_i)} = \infty.$$
By section 11 we already know that in case (I) we converge to the Eguchi-Hanson space. Therefore we only need to investigate the behavior in case (II), i.e. at scales larger than the forming Eguchi-Hanson singularity. For this we need to divide case (II) into three subcases: By passing to a subsequence we may assume that
$$ \text{(II.a) } b_s(p_i, t_i) \rightarrow 1 \: \text{ or } \: \text{ (II.b) } b_s(p_i,t_i) \rightarrow \eta \in (0,1) \: \text{ or } \: \text{ (II.c) } b_s(p_i,t_i) \rightarrow 0.$$
For (II.a) and (II.c) we show in Lemma \ref{flat-orbifold-blow-up-lem} and Lemma \ref{cylinder-blow-up-lem} that $(M_2, g_i(t), p_i)$ subsequentially converges to the flat orbifold $\mathbb{R}^4 / \mathbb{Z}_2$ and the shrinking cylinder $\mathbb{R} \times \mathbb{R} P^3$, respectively. The proof of these lemmas is relatively easy. Proving in Lemma \ref{lem:orbifold-blowup} that the blow-up limit in case (II.b) is homothetic to the 4d Bryant soliton quotiented by $\mathbb{Z}_2$ is trickier. Here we rely on Lemma \ref{QT2bddb-limit-lem}, which characterizes the geometry of the high curvature regions of $g(t)$ at distance scales larger than the Eguchi-Hanson singularity away from the tip of $M_2$. In subsection \ref{blow-up-thm-outline} we give a more detailed outline of the proof of Theorem \ref{blow-up-thm}.
\subsection{Further questions and conjectures}
\label{subsec:conjectures}
In this section we collect some conjectures and further questions that arise from our results. The central open question remaining in this paper is whether or not the Eguchi-Hanson singularity of Theorem \ref{blow-up-thm} is isolated. By isolated we mean that the only blow-up limits are the Eguchi-Hanson space and its asymptotic cone, the flat orbifold $\mathbb{R}^4/\mathbb{Z}_2$. We conjecture that
\begin{conj}
The Eguchi-Hanson singularity of Theorem \ref{blow-up-thm} is not isolated and all four blow-up limits (i)-(iv) occur. In particular, it is accompanied by a Type I singularity modeled on the shrinking cylinder $\mathbb{R} \times S^3/\mathbb{Z}_2$.
\end{conj}
An affirmative answer to this conjecture would provide evidence for a longstanding conjecture in Ricci flow stating that a Type II singularity is always accompanied by a Type I singularity in its vicinity. The author has an argument showing that if the Eguchi-Hanson singularity were isolated, the curvature would blow up at a rate faster than $(T_{sing}-t)^{-\lambda}$, where $\lambda$ is any positive constant.
Although we have not analyzed the blow-up limits of a $U(2)$-invariant Ricci flow $(M_k, g(t))$, $t \in [0,T_{sing})$, in the $k\neq2$ case, we believe that for each $k\in \mathbb{N}$ there exists a unique blow-up limit of the singularity arising from the collapse of the two sphere $S^2_o$ at the tip of $M_k$. In collaboration with Jon Wilkening the author has already conducted numerical simulations confirming this, and a paper is in preparation \cite{AW19}. In particular, we conjecture that
\begin{conj}
\label{conj:sing}
Let $(M_k, g(t))$ be a $U(2)$-invariant Ricci flow encountering a singularity at the tip of $M_k$, as the area of $S^2_o$ decreases to zero. Then the following picture holds:
\begin{center}
\begin{tabular}{| c| c |c| c| c |}
\hline
$k$ & Blow-up limit at $o \in S^2_o$ & Type & Isolated \\
\hline
$1$ & FIK shrinker & Type I & Yes \\
$2$ & Eguchi-Hanson space & Type II & No\\
$\geq 3$ & Steady Ricci solitons of \cite{A17} & Type II & No\\
\hline
\end{tabular}
\end{center}
\end{conj}
By isolated we mean that the singularity is not accompanied by a Type I singularity in its vicinity. For instance, in the case $k\geq 2$ we expect a singularity caused by the collapse of the two-sphere $S^2_o$ at the tip of $M_k$ to always be accompanied by a Type I singularity modeled on the shrinking cylinder $\mathbb{R} \times S^3/\mathbb{Z}_k$ and therefore not to be isolated. If for each $k\geq 3$ the corresponding steady Ricci soliton of \cite{A17} is in fact the unique blow-up limit at the tip of $M_k$, then these singularities are necessarily accompanied by a Type I singularity modeled on $S^3/\mathbb{Z}_k$, because these solitons are asymptotically cylindrical.
Another interesting question is the following:
\begin{question}
Can the Eguchi-Hanson singularity occur on a closed four dimensional manifold?
\end{question}
The author conjectures that the answer is yes, however only non-generically. The simplest model on which to investigate this question is $M = M_2 \#_{\mathbb{R} P^3} M_2 \cong S^2 \times S^2$ equipped with an $U(2)$-invariant metric. One could carry out a construction as follows: Vary between an initial metric that encounters a $\mathbb{R} \times \mathbb{R} P^3$ neckpinch singularity and an initial metric that leads to the collapse of one of the $S^2$ factors of $M$. On the path between these two metrics there should be a metric whose Ricci flow evolution forms an Eguchi-Hanson singularity in finite time.
This paper has not touched upon the behavior of a general non-$U(2)$-invariant metric on $TS^2$. A first question would be:
\begin{question}
Does the picture of Theorem \ref{blow-up-thm} also hold for Ricci flows starting from non-$U(2)$-invariant perturbations of asymptotically cylindrical $U(2)$-invariant metrics on $TS^2$?
\end{question}
And a final big question mark is the following:
\begin{question}
Are there other four dimensional Ricci flat ALE spaces that can occur as blow-up limits in Ricci flow?
\end{question}
So far all known Ricci flat ALE spaces in four dimensions are hyperk\"ahler and it is not known whether non-hyperk\"ahler examples exist. Kronheimer classified all hyperk\"ahler ALE spaces \cite{KronI89}, \cite{KronII89}. These spaces have one end that is asymptotic to the cone $\mathbb{R}^4 / \Gamma$, where $\Gamma \subset U(2)$ is a certain finite group --- a binary dihedral, tetrahedral, octahedral or icosahedral group.
In the case that $\Gamma = \mathbb{Z}_k$ is cyclic, Gibbons and Hawking \cite{GH78}, \cite{GH79} discovered a closed form $(3k - 6)$-parameter family of such metrics. In the physics literature these metrics are known as multi-center Eguchi Hanson spaces. It would be interesting to see whether the results of this paper can be generalized to prove the existence of singularities modeled on these spaces.
\subsection{Acknowledgments}
The author would like to thank his PhD advisors Richard Bamler and Jon Wilkening for their constant support and encouragement. This work was supported by a GSR fellowship, which was funded by NSF grant DMS-1344991.
\section{Preliminaries}
\subsection{Notation}
Here we collect some of the notation used throughout the paper.
\begin{itemize}
\item $M_k$, $k \in \mathbb{N}$: a manifold diffeomorphic to the blow-up of $\mathbb{C}^2/ \mathbb{Z}_k$, $k \geq 1$, at the origin.
\item $S^2_o$: the two-sphere added during the blow-up of $\mathbb{C}^2/ \mathbb{Z}_k$.
\item $\xi$: the radial coordinate on $M_k$ or the parametrization of the $\mathbb{R}_{>0}$ factor in the product $\mathbb{R}_{>0} \times S^3/\mathbb{Z}_k \subset M_k$.
\item $o$: a fixed point on $S^2_o$.
\item $\Sigma_{p}$: denotes the orbit of $p$ under the $U(2)$-action. For instance if $p \in S^2_o \subset M_k$ we have $\Sigma_{p} = S^2_o$ and when $p \in M_k \setminus S^2_o$ we have $\Sigma_{p}\cong S^3 /\mathbb{Z}_k$.
\item $s$: the geodesic distance from $S^2_o$, and often considered as a function of $\xi$ and $t$.
\item origin: refers to the point $o$.
\item $g$: a metric of the form (\ref{metric1}) or (\ref{metric2}) unless otherwise stated
\item $d_g$: the metric distance induced by $g$.
\item $g(t)$: a time dependent family of metrics of the form (\ref{metric1}) or (\ref{metric2}).
\item $u$, $a$, $b$: the warping functions of the metric (\ref{metric1}). Depending on context these will be considered as functions of $(\xi,t)$, $(s,t)$ or $(p,t)$, where $p$ is a point on $M_k$.
\item $Q\coloneqq \frac{a}{b}$.
\item $B_g(p,r)$: the ball centered at $p$ of radius $r$ with respect to the metric $g$.
\item $C_g(p, r)$, $ r>0$: the subset of a cohomogeneity one $U(2)$-invariant Riemannian manifold $(M,g)$ defined by
$$C_g(p, r) = \left\{ q \in M \Big| \: d_g(q, \Sigma_{p}) < r \right\}.$$ The set $C_g(p, r)$ is diffeomorphic to either $M_k$ or $\mathbb{R} \times S^3/\mathbb{Z}_k$.
\item $\overline{C}_g(p, r)$: the closure of $C_g(p, r)$.
\item $T_{sing}$: the singular time of a Ricci flow.
\item $C^\infty_{U(2)}(M_k \times [0,T])$: The space of smooth $U(2)$-invariant functions $u: M_k \times [0,T] \rightarrow \mathbb{R}$.
\item $x, y$: K\"ahler quantities introduced in section \ref{kahler-E-H-section}.
\end{itemize}
\subsection{The manifold and metric}
\label{manifold-metric-subsec}
For $k \in \mathbb{N}$ let $M_k$ be diffeomorphic to the blow-up of $\mathbb{C}^2 /\mathbb{Z}_k$ at the origin. Denote by $S^2_o$ the embedded two-sphere in $M_k$ stemming from the blow-up, and fix some point $o$ for `origin' on $S^2_o$.
We now describe the $U(2)$-invariant metrics on $M_k$, $k \geq 1$, studied in this paper. Let $z_1, z_2$ be the standard coordinates on $\mathbb{C}^2$ and let $U(2)$ act on $\mathbb{C}^2$ by left multiplication. This action descends to $M_k$, $k \in \mathbb{N}$. Note that $M_k$ can be seen as the total space of the complex line bundle $O(-k)$ via
\begin{align*}
\pi: M_k &\longrightarrow S^2_o \\
(z_1, z_2) &\mapsto [z_1, z_2]
\end{align*}
Then $U(2) \cong U(1) \times SU(2)$ acts on $M_k$ in the following way: The action of $U(1)$ rotates the fibres of $\pi$ and $SU(2)$ acts on the base $S^2_o$ via rotations. Now introduce the Hopf coordinates
\begin{align*}
z_1 &= \xi \sin\eta \, e^{i( \psi + \phi)} = x_1 + i y_1 \\
z_2 &= \xi \cos\eta \, e^{i( \psi - \phi)} = x_2 + i y_2
\end{align*}
on $\mathbb{C}^2_\ast$, where $\xi > 0$, $\eta \in [0, \pi/2]$ and $\psi, \phi \in [0, 2 \pi)$. These coordinates descend to $M_k$. In particular, this allows us to endow $M_k$ with the radial coordinate $\xi: M_k \rightarrow \mathbb{R}_{\geq 0}$, by continuously extending $\xi$ to $S^2_o$ by taking $\xi = 0$ on $S^2_o$. Note that the coordinate $\xi$ is only smooth on $M_k \setminus S^2_o$.
A computation shows that the standard euclidean metric
$$g_{eucl} = dx_1^2 + dy_1^2 + dx_2^2 + dy_2^2$$
may be written as
$$g_{eucl} = d\xi^2 + \xi^2 \left( d\eta^2 + \sin^2(2\eta) d\phi^2 + \left[d \psi - \cos(2\eta)d\phi \right]^2 \right)$$
in Hopf coordinates. The 1-form
$$\omega \coloneqq d \psi - \cos(2\eta)d\phi$$
is dual to the Hopf fibre directions, or equivalently dual to the vector field generated by the $U(1)$ action. Furthermore
\begin{equation}
\label{FS-pullback}
d\eta^2 + \sin^2(2\eta) d\phi^2
\end{equation}
is the pull-back of the Fubini-Study metric $g_{FS}$ on $\mathbb{C} P^1$, normalized to have constant sectional curvature equal to $\frac{1}{4}$.
From the above we see that the warped-product metric
\begin{equation}
\label{metric1}
g = u(\xi)^2 d\xi^2 + a(\xi)^2 \omega \otimes \omega + b(\xi)^2 \pi^\ast( g_{FS})
\end{equation}
is the most general $U(2)$-invariant metric on $\mathbb{C}_\ast^2$ and descends to a $U(2)$-invariant metric on the open dense set $\mathbb{C}^2_\ast /\mathbb{Z}_k \subset M_k$. It will be useful to introduce the change of coordinates defined by
$$ ds = u(\xi) d\xi$$
and $s = 0 $ at $\xi = 0$. Then for $p \in M_k$ the quantity
$s(p) = d_{g}(p, S^2_o)$ describes the radial distance of $p$ from $S^2_o$. In these coordinates the metric becomes
\begin{equation}
\label{metric2}
g = ds^2 + a(s)^2 \omega \otimes \omega + b(s)^2 \pi^\ast( g_{FS}),
\end{equation}
where in a slight abuse of notation we consider $a$ and $b$ as functions of $s$. Depending on the context we will consider $a$ and $b$ either as functions of $s$ or $\xi$.
The metric $g$ can be extended to a metric on all of $M_k$ by taking $a(0) = 0$ and $b(0) > 0$. In other words we shrink the Hopf fibre directions to zero as $s \rightarrow 0$ or equivalently as we approach $S^2_o$. Note that for every $p \in S^2_0$
$$ds^2 + a(s)^2 \omega \otimes \omega$$
is the pull-back of $g$ onto the fibre $\pi^{-1}(p)$. As $U(1)$ acts on the fibre $\pi^{-1}(p)$, we see that $\pi^{-1}(p)$ is a union of $S^1$ orbits and $p$. Furthermore, such a $S^1$ orbit in $\pi^{-1}(p) \subset M_k$ is parametrized by $0\leq \psi < \frac{2\pi}{k}$ and, by the form of the metric (\ref{metric2}), such an $S^1$ orbit at radial distance $s$ from $S^2_o$ has a circumference of length $\frac{2\pi}{k}a(s)$. Because
$$\frac{2\pi}{k}a(s) = \frac{2\pi}{k} a_s(0) s + O(s^2) \: \text{ as } \: s \rightarrow 0$$
we must require that $a_s(0) = k$ in order to avoid a conical singularity at $S^2_o$. This is how the topology of the manifold enters the analysis of the Ricci flow equation. Additionally requiring that $a(s)$ and $b(s)$ can be extended to an odd and even function, respectively, around $s=0$ is a sufficient condition for the metric $g$ to be smoothly extendable to all of $M_k$ \cite{VZ18}. In the rest of the paper all metrics considered will be of the form (\ref{metric1}) or equivalently (\ref{metric2}). In Figure \ref{fig:manifold} the manifold $M_k$ and its metric close to the two sphere $S^2_o$ is schematically depicted.
\begin{figure}
\caption{Diagram of the manifold $M_k$ close to the tip}
\label{fig:manifold}
\end{figure}
$(M_k, g)$, $k \in \mathbb{N}$, are cohomogeneity one manifolds, meaning that the generic orbits of the $U(2)$ action are of codimension 1. The generic orbit is also called the principal orbit. The non-generic orbits are called non-principal orbits. In the case of $M_k$ the principal orbits are diffeomorphic to $S^3/\mathbb{Z}_k$ and the single non-principal orbit is $S^2_o$ and of codimension 2. Below we introduce some notation that we frequently employ throughout the paper:
\begin{definition}
\label{def:C}
Assume $(M,g)$ is a $U(2)$-invariant cohomogeneity one manifold with principal orbit $S^3/\mathbb{Z}_k$ for some fixed $k \in \mathbb{N}$ and $g$ is a metric of the form (\ref{metric2}). Let $p \in M$ and $r >0$. Then
\begin{itemize}
\item Let $\Sigma_p \subset M$ denote the orbit of $p$ under the $U(2)$-action.
\item Let $\Sigma^+_p$ be the set of all points $q \in M$ that can be joined via path $\tau$ to $p$ with $g\left(\dot{\tau},\frac{\partial}{\partial s}\right) \geq 0$.
\item Let $$C_g(p,r) \coloneqq \left\{ q\in M \: \Big | \: d_g(q, \Sigma_p) < r \right\}$$
\item Let $$C^+_g(p,r) \coloneqq \left\{ q\in \Sigma^+_p \: \Big | \: d_g(q, \Sigma_p) < r \right\}$$
\end{itemize}
\end{definition}
Note that we have $\Sigma_p \cong S^3/\mathbb{Z}_k$ if $p$ lies on a principal orbit and $\Sigma_p \cong S^2$ if $p$ lies on a non-principal orbit.
\subsection{The connection, Laplacian and curvature tensor}
\label{con-lap-cur-subsec}
We now compute the connection, Laplacian and curvature tensor for metrics of the form (\ref{metric2}). To obtain the corresponding expressions for the metric (\ref{metric1}) use the relation
$$ \frac{\partial}{\partial s} = \frac{1}{u}\frac{\partial}{\partial \xi}.$$
Take the orthonormal basis
$$e^0 = ds \qquad e^1 = a \left[ d\psi - \cos(2\eta) d \phi \right] \qquad e^2 = b d \eta \qquad e^3 = b \sin(2 \eta ) d \phi$$
of $T^\ast M$. Let $e_i$, $i = 1 , 2, 3, 4$, be the corresponding dual basis of $T_\ast M$. Define the connection 1-forms $\theta_i^j$ by $\nabla e_i = \theta_i^j e_j$ and the curvature 2-forms $\Omega_i^j$ by $R( \cdot, \cdot) e_i = \Omega_i^j e_j$. With help of Cartan's structure equations
\begin{align*}
\theta_i^j &= - \theta_j^i \\
de^i &= - \theta_j^i \wedge e^j \\
\Omega_i^j &= d \theta_i^j + \theta^j_k \wedge \theta^k_i
\end{align*}
one can compute the connection 1-forms and curvature 2-forms. First note that
\begin{align*}
de^0 &= 0 \\
de^1 &= \frac{a_s}{a} e^0 \wedge e^1 + \frac{2a}{b^2} e^2 \wedge e^3 \\
de^2 &= \frac{b_s}{b} e^0 \wedge e^2 \\
de^3 &= \frac{b_s}{b} e^0 \wedge e^3 + \frac{2}{b} \cot(2 \eta) e^2 \wedge e^3.
\end{align*}
Hence we obtain the connection 1-forms $\theta^i_j$:
\begin{align*}
\theta^1_0 &= \frac{a_s}{a} e^1 && \theta^1_2 = \frac{a}{b^2} e^3 \\
\theta^2_0 &= \frac{b_s}{b} e^2 && \theta^2_3 = - \frac{a}{b^2} e^1 - \frac{2}{b} \cot(2\eta) e^3 \\
\theta^3_0 &= \frac{b_s}{b} e^3 && \theta^3_1 = \frac{a}{b^2} e^2
\end{align*}
Therefore
$$ \nabla_{e_0} e_0 = 0 \qquad \nabla_{e_1} e_1 = - \frac{a_s}{a} e_0 \qquad \nabla_{e_2} e_2 = -\frac{b_s}{b} e_0 \qquad \nabla_{e_3} e_3 = - \frac{b_s}{b} e_0$$
from which we can derive the expression for the Laplacian of a $U(2)$-symmetric function $f(s)$ on $M_k$.
\begin{equation}
\label{laplacian}
\Delta f = \sum_{i=0}^3 \nabla^2_{e_i,e_i} f = f_{ss} + \left(\frac{a_s}{a} + 2 \frac{b_s}{b} \right) f_s.
\end{equation}
Finally, we may compute the components
\begin{equation*}
R_{ijkl} = g\left(R(e_k,e_l)e_j,e_i\right).
\end{equation*}
of the curvature tensor. Below we list its non-zero components
\begin{align*}
R_{0101} &= -\frac{a_{ss}}{a}= K_1 \\
R_{0202} &= -\frac{b_{ss}}{b} = K_2 \\
R_{0303} &= -\frac{b_{ss}}{b}= K_3 \\
R_{0123} &= -\frac{2}{b^2} \left( a_s - Q b_s\right) = M_1 \\
R_{0231} &= \frac{1}{b^2} \left( a_s - Q b_s\right) = M_2 \\
R_{0312} &= \frac{1}{b^2} \left( a_s - Q b_s\right) = M_3 \\
R_{1212} &= \frac{a^2}{b^4}- \frac{a_sb_s}{ab} = H_{12} \\
R_{2323} &= \frac{4}{b^2} - 3 \frac{a^2}{b^4} - \left(\frac{b_s}{b}\right)^2 = H_{23} \\
R_{3131} &= \frac{a^2}{b^4}- \frac{a_sb_s}{ab} = H_{31}.
\end{align*}
All other components are either determined by the standard symmetries of the curvature tensor or are zero.
\subsection{The Ricci flow equation}
\label{ricci-flow-equations-sec}
With help of the above list of curvature components one can check that the Ricci tensor is diagonal and hence the form of the metric (\ref{metric1}) is preserved by Ricci flow. Allowing the warping functions $a$, $b$ and $p$ to vary in time, the Ricci flow equation (\ref{RF}) in $(\xi, t)$ coordinates can be expressed as a system of coupled parabolic equations in $a$, $b$ and $u$.
\begin{align}
\label{u-evol}\partial_t u &= \frac{1}{a} \partial_{\xi} \left( \frac{a_{\xi}}{u} \right) + \frac{2}{b} \partial_{\xi} \left( \frac{b_{\xi}}{u} \right) \\
\label{a-evol} \partial_t a &= \frac{1}{u} \partial_{\xi} \left( \frac{a_{\xi}}{u} \right) - 2 \frac{a^3}{b^4} + 2 \frac{a_{\xi}b_{\xi}}{bu^2} \\
\label{b-evol} \partial_t b &= \frac{1}{u} \partial_{\xi} \left( \frac{b_{\xi}}{u} \right) - \frac{4}{b} + 2\frac{a^2}{b^3} + \frac{a_{\xi}b_{\xi}}{au^2} + \frac{b^2_{\xi}}{bu^2}
\end{align}
Define the time dependent radial distance function $s = s(\xi, t)$ by
\begin{equation*}
ds = u(\xi, t) d\xi
\end{equation*}
Then
\begin{equation}
\label{s-def}
s(\xi, t) = \int_0^{\xi} u(\xi,t) \, \mathrm{d}\xi
\end{equation}
and
\begin{equation}
\label{s-deriv}
\frac{\partial}{\partial s} = \frac{1}{u} \frac{\partial}{\partial \xi}.
\end{equation}
Furthermore the commutation relation
\begin{equation*}
[\partial_t, \partial_s] = -\frac{\partial_t u}{u}\partial_s
\end{equation*}
holds. In terms of $s$ we can use (\ref{s-deriv}) to rewrite the Ricci flow equation in a slightly simpler form
\begin{align}
\label{u-evol} \frac{\partial_t u}{u} &= \frac{a_{ss}}{a} + 2 \frac{b_{ss}}{b} \\
\label{a-evol} \partial_t a &= a_{ss} - 2 \frac{a^3}{b^4} + 2 \frac{a_sb_s}{b} \\
\label{b-evol} \partial_t b &= b_{ss} - \frac{4}{b} + 2\frac{a^2}{b^3} + \frac{a_sb_s}{a} + \frac{b^2_s}{b}.
\end{align}
Note that the dependence of the right hand side of this system of equations on $\xi$ is hidden in the variable $s = s(\xi, t)$. However we can write the equations in terms of $(s,t)$ by introducing the functions
\begin{align*}
\tilde{a}(s,t) &= a(\xi, t) \\
\tilde{b}(s,t) &= b(\xi, t)
\end{align*}
and noting that
\begin{align*}
\partial_t a \big|_{\xi} &= \partial_t \tilde{a} \big|_{s} + \partial_s \tilde{a} \big|_{t} \frac{\partial s}{\partial t}\big|_{\xi} \\
\partial_t b \big|_{\xi} &= \partial_t \tilde{b} \big|_{s} + \partial_s \tilde{b} \big|_{t} \frac{\partial s}{\partial t}\big|_{\xi}.
\end{align*}
By slight abuse of notation, however, we will drop the tilde and consider the warping functions $a$,$b$ and $u$ as functions of either $(p,t)$, $p \in M_k$ or $(\xi, t)$ or $(s,t)$, depending on context. In $(s,t)$ coordinates the Ricci flow equation reads
\begin{align}
\label{a-evol-s-coord} \partial_t a \big |_{s} &= a_{ss} - 2 \frac{a^3}{b^4} + 2 \frac{a_sb_s}{b} - a_s \frac{\partial s}{\partial t} \\
\label{b-evol-s-coord} \partial_t b \big |_{s} &= b_{ss} - \frac{4}{b} + 2\frac{a^2}{b^3} + \frac{a_sb_s}{a} + \frac{b^2_s}{b} - b_s \frac{\partial s}{\partial t}
\end{align}
where
\begin{align}
\label{dsdt}
\frac{\partial s}{\partial t}\big|_{\xi} = \int_0^s \frac{a_{ss}}{a} + 2 \frac{b_{ss}}{b} \, \mathrm{d}s
\end{align}
Whenever we differentiate a function $f: M_k \times [0,T]$ with respect to time, unless stated otherwise, assume that the point on the manifold $M_k$ is held fixed. If we differentiate with respect to time while holding $s$ fixed we will denote the partial derivative by $\partial_t f |_{s}$ to avoid confusion. Because $s$ is a function of $(\xi,t)$, in general for fixed $s_0 > 0$ the set $\{ s = s_0\} \subset M_k$ is dependent on time. Therefore holding $s$ or $\xi$ fixed during partial differentiation produces very different results.
This following property of the warping functions $a$ and $b$ will be used throughout the paper.
\begin{lem}
\label{parity-lem}
Let $(M_k, g(t))$, $t \in [0,T)$, be a smooth Ricci flow solution. Then for all $t \in [0,T]$ the warping functions $a(s ,t)$ and $b(s ,t)$ can be extended to an odd and even function, respectively, on $\mathbb{R}$.
\end{lem}
\begin{proof}
Note that a necessary condition for a metric $g$ of the form (\ref{metric2}) to be smooth is that its corresponding warping functions $a$ and $b$ are extendable to odd and even functions, respectively, on $\mathbb{R}$. Therefore the desired result follows. Alternatively, notice that if the warping functions of the initial data $a(s, 0)$ and $b(s, 0)$ can be extended to an odd and even function, respectively, on $\mathbb{R}$, we can also extend the equations (\ref{a-evol-s-coord}), (\ref{b-evol-s-coord}) and (\ref{dsdt}) to all of $\mathbb{R}$. An inspection of these equations shows that the parity of $a$ and $b$ is preserved under the flow.
\end{proof}
\subsection{Recap of blow-up limits of singularities}
As mentioned above, every complete Riemannian manifold $(M,g)$ of bounded curvature admits a short-time Ricci flow starting from $g$, however singularities may develop in finite time. Similar to the study of other nonlinear equations, it is very useful to consider blow-up limits of singularities. We briefly sketch the idea here: Assume $(M, g(t))$, $t \in [0,T_{sing})$, is a Ricci flow encountering a curvature singularity as $t \rightarrow T_{sing}$. Let $(p_i, t_i)$ with $p_i \in M$ and $t_i \rightarrow T_{sing}$ be a sequence of points in spacetime such that
$$K_i := |Rm_{g(t_i)}|_{g(t_i)}(p_i) = \sup_{t \leq t_i} |Rm_{g(t)}|_{g(t)}$$
and
$$K_i \rightarrow \infty \: \text{ as } \: i \rightarrow \infty.$$
Take the rescaled metrics
$$ g_i(t) = K_i g\left( t_i + \frac{t}{K_i} \right), \quad t \in[-K_i t_i, 0].$$
Then $(M_k, g_i(t), p_i)$ subsequentially converge, in the Cheeger-Gromov sense, to a pointed ancient Ricci flow solution $(M_\infty, g_\infty(t), p_\infty), t\in(-\infty, 0]$ (see \cite[Theorem 6.68]{ChI} for more details). Note that in general $M_\infty \neq M$. A Ricci flow is called ancient if it can be extended to a time interval of the form $(-\infty, T)$, $T \in \mathbb{R}$. The blow-up limit $(M_\infty, g_\infty(t), p_\infty)$ is called the singularity model and yields important geometrical information on the shape of the singularity. Hamilton \cite{Ham95} distinguishes between Type I and II singularities, depending on the rate of curvature blow-up, i.e. for Type I
$$\sup_{M \times [0,T)} \left(T_{sing}- t\right)|Rm_{g(t)}|_{g(t)} < \infty$$
and for Type II
$$\sup_{M \times [0,T)} \left(T_{sing}- t\right)|Rm_{g(t)}|_{g(t)} = \infty.$$
By the work of Naber \cite{N10}, and Enders, M\"uller and Topping \cite{EMT11} it is known that Type I singularities are modeled on shrinking Ricci solitons. One hopes --- although it has not been proven --- that all Type II singularities are modeled on steady solitons, as to date all known examples are.
\section{The maximum principle}
\label{sec:maximum-principle}
Assume we are given a Ricci flow $(M_k, g(t))$, $t \in [0,T]$, $k \geq 1$. We make the following definition:
\begin{definition}
\label{def:smoothU2}
Let $C^\infty_{U(2)}(M_k \times [0,T])$ be the space of smooth $U(2)$-invariant functions
$$u: M_k \times [0,T] \rightarrow \mathbb{R}.$$
\end{definition}
In this section we prove a maximum principle for operators
$$P: C^\infty_{U(2)} ( M_k \times [0,T]) \rightarrow C^\infty_{U(2)} ( M_k \times [0,T])$$
that in $(\xi,t)$ coordinates and away from the non-principal orbit $S^2_o$ of $M_k$ can be written in the form
\begin{equation}
\label{Pop}
P[u] = \partial_{ss} u + \left( m \frac{a_s}{a} + n \frac{b_s}{b}\right) u_s + c u - \partial_t u, \quad m,n \in \mathbb{R},
\end{equation}
where $c\in C^\infty_{U(2)}(M_k \times [0,T])$. Recall from section \ref{ricci-flow-equations-sec} that we are interpreting the $s$ derivative as
$$\frac{\partial}{\partial s} = \frac{1}{u} \frac{\partial}{\partial \xi}.$$
It is useful to work in $(s,t)$ coordinates, in which case the operator $P[u]$ can be expressed as
\begin{equation}
\label{Pop-st}
P[u] = \partial_{ss} u + \left( m \frac{a_s}{a} + n \frac{b_s}{b} - \frac{\partial s}{\partial t} \right) u_s + c u - \partial_t \Big|_s u,
\end{equation}
where we recall the expression $(\ref{dsdt})$ for $\frac{\partial s}{\partial t}$. Note $P[u]$ is degenerate at the origin $s=0$ as
$$m \frac{a_s}{a} + n \frac{b_s}{b} - \frac{\partial s}{\partial t} \sim \frac{m}{s} \text{ for } 0<|s|\ll 1.$$
However, in $(s,t)$ coordinates the smoothness of $u$ and $c$ is equivalent to saying that $u(s,t), c(s,t)$ can be extended to smooth even functions around $s=0$ by defining $u(s,t) = u(-s,t)$ and $v(s,t) = v(-s,t)$ for $s \leq 0$. Hence we see that $u_s = c_s = 0$ at $s=0$ and via L'H\^opital's Rule we obtain the following representation of $P[u]$ on the principal orbit $S^2_0$:
$$P[u] = (m+1)\partial_{ss} u + c u - \partial_t u.$$
The maximum principle derived for $P$ below depends on the sign of $(m+1)$:
\begin{thm}
\label{maximum-principle}
Let $(M_k, g(t))$, $k \in \mathbb{N}$, $t\in[0,T]$ be a Ricci flow with bounded curvature. Let $P$ be an operator of the form (\ref{Pop}) and $u \in C^\infty_{U(2)}(M_k \times [0,T])$. If
\begin{equation*}
P[u] \leq 0 \: \text{ in } \: M_k \times [0,T]
\end{equation*}
and there exist constants $M, \sigma > 0$ such that the growth conditions
\begin{align*}
u(s,t) & \geq - M\exp(- \sigma s^2) \\
c(s,t) & \leq M\left( |s|^2 +1 \right)
\end{align*}
are satisfied, then the following holds true:
\begin{description}
\item[Case 1]($1+m \leq 0$) If
\begin{align*}
u(s,0) &\geq 0 \: \text{ for } \: s \geq 0 \\
u(0,t) &\geq 0 \: \text{ for } \: t \in [0,T]
\end{align*}
then $u(s,t) \geq 0$ on $[0,\infty) \times [0,T]$. Furthermore, if $u = 0$ somewhere on $(0,\infty)\times(0,T]$ then $u=0$ everywhere.
\item[Case 2]($1+m > 0$)
If
\begin{align*}
u(s,0) &\geq 0 \: \text{ for } \: s \geq 0
\end{align*}
then $u(s,t) \geq 0$ on $[0,\infty) \times [0,T]$. Furthermore, if $u = 0$ somewhere on $[0,\infty) \times(0,T]$ then $u=0$ everywhere.
\end{description}
\end{thm}
Before proving Theorem \ref{maximum-principle} we need to derive some bounds on $\frac{a_s}{a}$, $\frac{b_s}{b}$ and $\frac{1}{b^2}$ for metrics $g$ of bounded curvature. This will allow us to bound the coefficients appearing in the expression (\ref{Pop-st}) of the operator $P[u]$.
\begin{lem}
\label{b-bound}
Let $(M_k ,g)$, $k \in \mathbb{N}$, and $K > 0$ such that $|Rm_g|_g \leq K \: \text{ on } \: M_k$. Then everywhere on $M_k$ we have
\begin{enumerate}
\item $b^2 \geq \frac{1}{K}$
\item $\left(\frac{b_s}{b}\right)^2 \leq 5 K$
\item $ \frac{Q^2}{b^2} \leq \frac{5}{3} K $
\end{enumerate}
\end{lem}
\begin{proof}
From the curvature components derived in subsection \ref{con-lap-cur-subsec} we see that
\begin{align}
\label{exp1} b^2 H_{23} &= 4 - 3Q^2 - b_s^2 \\
\label{exp2} b^2 H_{12} &= Q^2 - \frac{a_s b_s}{Q}
\end{align}
At a local minimum $b_s =0$ we thus have
\begin{equation*}
b^2 = \frac{4}{3H_{12} + H_{23}} \geq \frac{1}{K}
\end{equation*}
Now we argue by contradiction. Assume that there exists a $s^\ast>0$ and $\delta > 0$ such that at $s= s^\ast$ we have $b^2 < \frac{1-\delta}{K}$. From above it follows that $b_s < 0$ for $s \geq s^{\ast}$ and hence, because $b >0$ everywhere, $\lim_{s \rightarrow \infty} b_s = 0$. Equation (\ref{exp1}) then shows that
\begin{align*}
Q^2 &= \frac{1}{3} \left( 4 - b^2 H_{23} - b_s^2 \right) \\
&\geq 1 + \frac{\delta}{3} - \frac{b_s^2}{3} \\
&\geq 1 + \frac{\delta}{4}
\end{align*}
for $s$ sufficiently large. Then (\ref{exp2}) implies that eventually
\begin{equation*}
a_s \frac{b_s}{Q} \geq \frac{5}{4}\delta.
\end{equation*}
Dividing by $\frac{b_s}{Q}$ shows that
\begin{equation*}
a_s \rightarrow -\infty \: \text{ as } \: s\rightarrow \infty
\end{equation*}
contradicting $a \geq 0$. This proves the first bound. To prove the second bound note that
\begin{equation*}
\left(\frac{b_s}{b}\right)^2 = \frac{4-3Q^2}{b^2} - H_{23} \leq \frac{4}{b^2} + K \leq 5K,
\end{equation*}
where the last inequality follows from (1). For the third bound we have
$$ \frac{Q^2}{b^2} = \frac{1}{3} \left( \frac{4}{b^2} - \left(\frac{b_s}{b}\right)^2 - H_{23} \right) \leq \frac{5}{3}K.$$
\end{proof}
\begin{lem}
\label{asa-bound-lem}
Let $(M_k ,g)$, $k \in \mathbb{N}$, and $K > 0$ such that $|Rm_g|_g \leq K \: \text{ on } \: M_k.$
Then everywhere on $M_k$ we have
\begin{equation}
\label{asa-bound}
- 2 \sqrt{K} < \frac{a_s}{a} < \frac{1}{s} + \sqrt{K}
\end{equation}
\end{lem}
\begin{proof}
The quantity $\phi = \frac{a_s}{a}s$ satisfies the ODE
\begin{equation*}
\frac{d \phi}{ds} = s \frac{a_{ss}}{a} + \frac{\phi(1-\phi)}{s}
\end{equation*}
and by L'H\^opital's rule we have $\phi(0) = 1$. Note that the function $\phi$ can be extended to an even function on $\mathbb{R}$. Therefore $\frac{d \phi}{ds} (0) = 0$ and there exists a small $\epsilon > 0$ such that
\begin{equation*}
\phi \leq 1 + \sqrt{K}s \: \text{ for } \: 0 \leq s \leq \epsilon.
\end{equation*}
Actually the inequality holds for all $s\geq 0$, since whenever $\phi(s) = 1 + \sqrt{K}s$ we have
\begin{equation*}
\frac{d \phi}{ds} = s \left( \frac{a_{ss}}{a} - K\right) - \sqrt{K} < 0,
\end{equation*}
since $|\frac{a_{ss}}{a}| = |R_{0101}| \leq K$. this proves the the upper bound of (\ref{asa-bound}).
To prove the lower bound assume that $a_s(s_0) < 0$. For every $s_1 > s_0$ there exists a $s^\ast \in (s_0, s_1)$ such that
\begin{equation*}
a_s(s_1) - a_s(s_0) = (s_1- s_0) a_{ss} (s^{\ast}) \leq (s_1- s_0) K |a(s^{\ast})|
\end{equation*}
by the mean value theorem. It follows that
\begin{equation*}
a_s(s) \leq \frac{1}{2} a_s(s_0) \: \text{ for } \: s_0 \leq s \leq s_0 + \frac{1}{2K} \Big|\frac{a_s(s_0)}{a(s_0)}\Big|.
\end{equation*}
Therefore
\begin{equation*}
0 \leq a\left(s_0 + \frac{1}{2K} \Big|\frac{a_s(s_0)}{a(s_0)}\Big|\right) \leq a(s_0) + \frac{1}{2K} \Big|\frac{a_s(s_0)}{a(s_0)}\Big| \frac{a_s(s_0)}{2}
\end{equation*}
which implies
\begin{equation*}
\frac{a_s(s_0)}{a(s_0)} \geq -2 \sqrt{K}.
\end{equation*}
This concludes the proof.
\end{proof}
Now we may proceed to proving the maximum principle of Theorem \ref{maximum-principle}.
\begin{proof}[Proof of Case 1 of Theorem \ref{maximum-principle}]
Let $K>0$ such that
\begin{equation*}
\sup_{M_k \times [0,T]} |Rm_{g(t)}|_{g(t)} \leq K.
\end{equation*}
Introduce the new variable $r := r(s, t)$ defined by
\begin{equation*}
r(r+2) = s^2
\end{equation*}
and let
\begin{equation*}
\overline{u}(r,t) = u(\sqrt{r(r+2)},t).
\end{equation*}
Note that $ 2r \sim s^2$ for $s \ll 1$ and $r \sim s$ for $s \gg 1$. We make this substitution to remove the apparent singularity at $s=0$ in the $(s,t)$ coordinate representation (\ref{Pop-st}) of the operator $P[u]$. The function $\overline{u}$ is smooth, because $u$ is extendable to an even function around the origin (see \cite{W43}). Rewriting (\ref{Pop-st}) in terms of $(r,t)$ coordinates we see that $\overline{u}$ satisfies the inequality
\begin{align*}
\partial_t \overline{u} \big|_{r} & \geq A(r) \partial_{rr} \overline{u} + B(r,t) \partial_r \overline{u} + C(r,t) \overline{u}
\end{align*}
where $A$, $B$ and $C$ are smooth functions defined by
\begin{align*}
A(r) &= \frac{r(r+2)}{(r+1)^2} \\
B(r,t) &= \left(m\frac{a_s}{a} + n\frac{b_s}{b} - \frac{\partial s}{\partial t} \right) \frac{s}{r+1} + \frac{1}{(r+1)^3} \\
C(r,t) &= c(s,t)
\end{align*}
Note that above we regard $s$ as a function of $r$. Recall that by Lemma \ref{parity-lem} the functions $a$ and $b$ can be extended to an odd and even function, respectively, around the origin. Therefore the quantity
\begin{equation}
\label{first-order-coef}
\left(m\frac{a_s}{a} + n\frac{b_s}{b} - \frac{\partial s}{\partial t} \right)s,
\end{equation}
considered as a function of $s$, can be extended to an even function around the origin by \cite{W43}. Hence this expression depends smoothly on $r$, showing that $B(r,t)$ is smooth. Similarly, we see that $C(r,t)$ is smooth. From the expression (\ref{dsdt}) for $\frac{\partial s}{\partial t}$ and the curvature components listed in subsection \ref{con-lap-cur-subsec} it follows that
\begin{equation}
\label{dsdt-bound}
\Big|\frac{\partial s}{\partial t}\Big| = \Big|\int_0^s -K_1 - 2 K_2 \, \mathrm{d}s\Big| \leq 3 K s
\end{equation}
By Lemma \ref{b-bound} and Lemma \ref{asa-bound-lem} we hence see that
$$|B(r,t)| \leq M (r + 1)$$
for some some positive constant $M$ depending on $K$. Finally, noting that $A(r)$ is bounded and positive for $r > 0$, we can apply \cite[Theorem 9, p.43]{F64} to deduce that the weak maximum principle holds. Note that for any compact $U \subset M_k \times [0,T]$ we may assume that $c < 0$ on $U$ by performing the transformation $\overline{u} \leftarrow \overline{u} e^{-\gamma t}$, for $\gamma= \gamma(U)$ chosen sufficiently large. Therefore the strong maximum principle follows from a slight adaptation of \cite[Theorem 1, p.34]{F64}.
\end{proof}
\begin{proof}[Proof of Case 2 of Theorem \ref{maximum-principle}]
We first prove the weak maximum principle. Taking $u' = u e^{-\gamma t}$ we see that $u'$ satisfies
$$\partial_t u' \geq \partial_{ss} u' + \left( m \frac{a_s}{a} + n \frac{b_s}{b}\right) u'_s + (c - \gamma) u'.$$
As $c$ is a smooth function of $(s,t)$, we can choose $\gamma$ sufficiently large such that in a neighborhood of $\{s=0\}\times[0,T]$ we have $c - \gamma < 0$. Since $m+1 >0$ we see that $u'$ cannot attain a negative minimum on $\{s=0\} \times (0,T]$, as otherwise
\begin{equation*}
0 \leq \partial_t u' = (1 + m) u'_{ss} + c u' > 0,
\end{equation*}
which is a contradiction. The weak maximum principle now follows by the proof of \cite[Theorem 9, p.43]{F64}.
In this paper we only apply the strong maximum principle for $m\in \mathbb{N}$ and therefore only prove this case here. For the general case refer to \cite[Theorem 5.17]{Fee13}. Given a Ricci flow $(M_k, g(t))$, $t\in [0,T]$, define the corresponding family of rotationally symmetric spaces $(\mathbb{R}^{m+1}, h(t))$, $t \in [0,T]$, by
\begin{equation*}
h = ds^2 + a^2(s,t) g_{S^m(\frac{1}{k})},
\end{equation*}
where $g_{S^{m}(\frac{1}{k})}$ is the round metric on $S^m$ of sectional curvature $k^2$. A sufficient condition for $h$ to be smooth at $s=0$ is that $a$ is extendable to an odd function around the origin and
\begin{equation*}
a_s(0) = k.
\end{equation*}
Both these conditions are satisfied and we conclude that $h$ is a smooth metric. The Laplacian of a rotationally symmetric function $f$ on $(\mathbb{R}^{m+1},h)$ is given by
\begin{equation*}
\Delta_{h} f = f_{ss} + m\frac{a_s}{a} f_s
\end{equation*}
and thus the condition $P[u] \leq 0$ may be written as
\begin{equation*}
\partial_t u \geq \Delta_{h(t)} u + n \frac{b_s}{b} u_s + c u
\end{equation*}
Note that for any bounded $U \subset M_k \times [0,T]$ we may assume that $c < 0$ on $U$ by performing the transformation $u \leftarrow u e^{-\gamma t}$, for $\gamma= \gamma(U)$ chosen sufficiently large. Hence the desired result follows from \cite[Theorem 12.40]{ChII}.
\end{proof}
\begin{remark}
It was crucial in our analysis that $u$ is extendable to an even function around the origin, as the following example demonstrates:
Consider the degenerate parabolic equation
\begin{equation}
\label{u-eqn}
u_t = u_{xx} -2 \frac{u_x}{x} + 2 \frac{u}{x^2}
\end{equation}
on $x, t \geq 0$ with initial data satisfying $u(x,0) \leq 0$. If we take
\begin{equation*}
u = x v
\end{equation*}
a computation shows that the above PDE corresponds to
\begin{equation}
\label{v-eqn}
v_t = v_{xx}
\end{equation}
Now considering (\ref{v-eqn}) as the heat equation on all of $\mathbb{R}$ we can set up initial data $v(x,0)$ such that
\begin{equation*}
v(x,0) \leq 0 \: \text{ for } \: x \geq 0,
\end{equation*}
however the solution $v$ to the heat equation becomes positive at some later time $t>0$ and $x >0$. This shows that $u \leq 0$ is not necessarily preserved by (\ref{u-eqn}).
\end{remark}
In this paper we also rely on a maximum principle for a system of parabolic inequalities on $u_1, u_2 \in C^\infty_{U(2)}(M_k \times [0,T])$ of the form
\begin{align}
\label{para-system1}
\partial_t u_1 &\geq (u_1)_{ss} + \left( m \frac{a_s}{a} + n \frac{b_s}{b}\right) (u_1)_s + h_{11} u_1 + h_{12} u_2 \\
\label{para-system2}
\partial_t u_2 &\geq (u_2)_{ss} + \left( m \frac{a_s}{a} + n \frac{b_s}{b}\right) (u_2)_s + h_{21} u_1 + h_{22} u_2,
\end{align}
where $h_{ij}\in C^\infty_{U(2)}(M_k \times [0,T])$, $i,j = 1,2$, are bounded and satisfy
$$ h_{12}, h_{21} \geq 0 \: \text{ on } \: M_k \times [0,T].$$
We prove the following Lemma:
\begin{lem}
\label{maximum-principle-coupled}
Let $(M_k, g(t))$, $t \in [0,T]$, be a Ricci flow with bounded curvature. Assume $u_1, u_2 \in C^\infty_{U(2)}(M_k \times [0,T])$ satisfy the above system (\ref{para-system1})-(\ref{para-system2}) of parabolic inequalities and for some constants $M, \sigma > 0$
$$ u_1(s,t), u_2(s,t) \geq -M \exp(\sigma s^2) \: \text{ for } t \in [0,T].$$
If
\begin{align*}
u_1(s,0), u_2(s,0) &\geq 0 \: \text{ for } s \geq 0 \\
u_1(0,t), u_2(0,t) &\geq 0 \: \text{ for } t \in [0,T]
\end{align*}
then $u_1, u_2 \geq 0$ on $M_k\times[0,T]$.
\end{lem}
\begin{proof}
Writing the equation in terms of $(r,t)$ as in the proof of Theorem \ref{maximum-principle} we obtain
\begin{align*}
\partial_t u_1 \Big |_{r} &\geq A(r) (u_1)_{rr} + B(r,t) (u_1)_r + H_{11} u_1 + H_{12} u_2 \\
\partial_t u_2 \Big |_{r} &\geq A(r) (u_2)_{rr} + B(r,t) (u_2)_r + H_{21} u_1 + H_{22} u_2,
\end{align*}
where $A(r)$, $B(r,t)$ are as in the proof of Theorem \ref{maximum-principle} and
$$H_{ij} (r,t) = h_{ij}(s(r),t), \quad i,j = 1,2.$$
After constructing a barrier function of the form
$$H(s,t) = \exp\left[ \frac{k |s|^2}{1 - \mu t} + \nu t \right], \quad 0 \leq t \leq \frac{1}{2\mu},$$
the result follows combining the arguments of \cite[Theorem 1, p.34]{F64} and \cite[Theorem 13, p. 190]{PW84}.
\end{proof}
\section{K\"ahler quantities and the Eguchi-Hanson space}
\label{kahler-E-H-section}
Recall that a complex structure $J$ on a Riemannian manifold $(M,g)$ satisfying
\begin{enumerate}
\item $g(V_1,V_2) = g(JV_1,JV_2)$ for all $V_1, V_2 \in TM$
\item $\nabla J = 0$
\end{enumerate}
defines a K\"ahler structure. On the manifolds $M_k$, $k \geq 1$, we define two complex structures $J_1$ and $J_2$ by
$$ J_1 e_1 = e_0 \qquad J_1 e_2 = e_3 $$
and
$$ J_2 e_0 = e_2 \qquad J_2 e_1 = e_3.$$
A computation shows that $(M_k, g, J_1)$ is K\"ahler if and only if
$$b_s - Q = 0.$$
Similarly, $(M_k, g, J_2)$ is K\"ahler if and only if
\begin{equation*}
a_s + Q^2 - 2 = 0 \quad \text{and} \quad b_s - Q = 0.
\end{equation*}
Note that being K\"ahler with respect to $J_2$ automatically implies K\"ahlerity with respect to $J_1$. This motivates the definition of the following \emph{scale-invariant} quantities
\begin{align*}
x &:= a_s + Q^2 - 2 \\
y &:= b_s - Q
\end{align*}
to measure the deviation of a metric from being K\"ahler with respect to the complex structures $J_1$ and $J_2$. For example, the FIK shrinker \cite{FIK03} is K\"ahler with respect to the complex structure $J_1$ and in our notation satisfies $y=0$. The Eguchi-Hanson space is the unique K\"ahler manifold with respect to $J_2$ as the following lemma shows.
\begin{lem}
\label{unique-Kahler-lem}
Amongst all Riemannian manifolds $(M_k, g)$,$k \geq 1$, equipped with $U(2)$-invariant metric $g$ of the form (\ref{metric2}), up to scaling the Eguchi-Hanson space is the unique K\"ahler manifold with respect to the complex structure $J_2$. Furthermore being K\"ahler with respect to $J_2$ is equivalent to $x = y = 0$.
\end{lem}
\begin{proof}
By the above discussion being K\"ahler with respect to $J_2$ is equivalent to
\begin{equation}
\label{xynull}
x = y = 0.
\end{equation}
Notice at $s=0$ we have
$$x = a_s -2 = 0$$
forcing the underlying manifold to be diffeomorphic to $M_2$ by the boundary conditions (\ref{boundary-cond-intro-s}). Then in terms of $a$ and $b$ the condition $x = y = 0$ is equivalent to the first order system of equations
\begin{align}
\label{EH-a}
a_s &= 2 - Q^2 \\
\label{EH-b}
b_s &= Q
\end{align}
Let $a^E$ and $b^E$ be a solution to this system of equations satisfying the initial conditions
\begin{align*}
a^E &= 0 \\
b^E &= 1
\end{align*}
at $s=0$. Then by the scale-invariance of condition $(\ref{xynull})$, for every $\lambda >0$ the metric given by the warping functions $\lambda a^{E}(\lambda s)$ and $\lambda b^{E}(\lambda s)$ also satisfies $(\ref{xynull})$. Hence up to rescaling there is a unique K\"ahler manifold with respect to the complex structure $J_2$. From \cite{EH79} or \cite{Cal79} we see that the metric given by $a^E$ and $b^E$ is homothetic to the Eguchi-Hanson metric.
\end{proof}
In the rest of the paper we denote by $g_E$ the Eguchi-Hanson metric with warping functions $a^E$ and $b^E$ normalized such that $b^E = 1$ on $S^2_o$. Note that the normalization condition is equivalent to saying that the area of the exceptional divisor $S^2_o$ is equal to $2 \pi$.
\begin{lem}
\label{E-H-properties-lem}
The warping functions $a^{E}$ and $b^{E}$ of the Eguchi-Hanson metric satisfy the following properties
\begin{align*}
a^{E} ,b^{E} &\sim s \: \text{ as } \: s \rightarrow \infty \\
a^{E}_{ss} &< 0 \: \text{ for } \: s \geq 0 \\
b^{E}_{ss} &> 0 \: \text{ for } \: s \geq 0 \\
\frac{a^{E}}{b^{E}} & < 1 \: \text{ for } \: s \geq 0 \\
\end{align*}
\end{lem}
\begin{proof}
For brevity write $a$ and $b$ for $a^{E}$ and $b^{E}$, respectively. Note that on the Eguchi-Hanson background we have
$$Q_s = \frac{1}{b} \left(a_s - Q b_s \right) = \frac{2}{b} \left( 1- Q^2 \right),$$
where the last equality follows from (\ref{EH-a}) and (\ref{EH-b}). As $Q = 0$ at $s=0$ it follows that
$$Q < 1 \: \text{ for } \: \geq 0$$
and hence
$$Q_s > 0 \: \text{ for } \: s \geq 0.$$
As
\begin{align*}
a_s &= Q_s b + Q b_s = 2 - Q^2
\end{align*}
it follows that
$$a_{ss} = -2 Q Q_s < 0.$$
Similarly
$$b_{ss} = Q_s > 0.$$
Therefore the limits
$$ a_\infty := \lim_{s\rightarrow \infty} a_s $$
and
$$ b_\infty := \lim_{s\rightarrow \infty} b_s $$
both exist. From the system of differential equations (\ref{EH-a}) and (\ref{EH-b}) we then see that
$$a_{\infty} = b_{\infty} = 1.$$ This concludes the proof.
\end{proof}
\section{Some preserved conditions}
\label{section-preserved-conditions}
In this section we derive various \emph{scale-invariant} inequalities that are preserved by a Ricci flow $(M_k, g(t))$, $t \in [0,T]$, $k \in \mathbb{N}$. The scale-invariance is crucial, as it ensures that the inequalities pass to blow-up limits and therefore also constrain their geometry. The preserved inequalities we list in this section will play an important role in all subsequent sections.
\noindent\textbf{Section Outline.}
A central quantity in our analysis is
$$Q = \frac{a}{b}.$$
In geometric terms, $Q$ measures the deviation of the cross-sectional $S^3 / \mathbb{Z}_k$ in $M_k$ from being round. That is, when $Q=1$ the cross-section is round and as $Q \rightarrow 0$ the cross-sectional $S^3 / \mathbb{Z}_k$ collapses along the $S^1$ Hopf fibres to a two-sphere. A computation shows that the evolution equation of $Q$ is
\begin{equation}
\label{Q-evol}
\partial_t Q = Q_{ss} + 3 \frac{b_s}{b} Q_s + \frac{4}{b^2} Q(1-Q^2).
\end{equation}
Therefore one expects that the inequality $Q \leq 1$ is preserved by Ricci flow, which in Lemma \ref{Qleq1} we prove to be the case.
Apart from $Q$, the K\"ahler quantities $x$ and $y$ introduced in section \ref{kahler-E-H-section} are used throughout this paper and are one of the key ingredients in showing that certain Ricci flows on $M_2$ develop singularities modeled on the Eguchi-Hanson space. We show in Lemma \ref{xleq0-lem} and Lemma \ref{yleq0-lem} that the inequalities
$$ x\leq 0$$
and
$$ y \leq 0$$
are both preserved. Furthermore, using the maximum principle for systems of weakly coupled parabolic equations of Lemma \ref{maximum-principle-coupled}, we show in Lemma \ref{ab-mono} that
$$a_s, b_s \geq 0$$
is preserved. In Lemma \ref{da-bounded} we show that on a Ricci flow background satisfying $Q \leq 1$ and $y \leq 0$, for any $C > 2$ the inequality $a_s \leq C$ is preserved. In the following sections we will mainly consider Ricci flows satisfying $a_s, b_s \geq 0$, $y \leq 0$, $Q\leq 1$ and $a_s \leq C$. This gives us enough control on $a$ and $b$ to prove many interesting results.
Finally, we show that whenever a subset of the inequalities $Q \leq 1$, $y\leq 0$ and $a_s, b_s \geq 0$ hold, the details of which are discussed below, the following inequalities
\begin{align*}
T_1 &= a_s + 2Q^2 -2 \geq 0 \\
T_2 &= Qy - x = -a_s + Qb_s + 2\left(1 - Q^2 \right) \geq 0 \\
T_3 &= a_s - Qb_s - Q^2 + 1 \geq 0 \\
\min(T_1, T_4) &\geq 0
\end{align*}
where
$$T_4 = a_s - \frac{1}{2} Q b_s - \left(1 - Q^2\right)$$
are preserved by the Ricci flow. The precise statements and proofs of these preserved inequalities can be found in Lemmas \ref{T1-preserved-lem}, \ref{T2-preserved-lem}, \ref{T3-preserved-lem} and \ref{minT1T4-preserved-lem} below.
The main idea in constructing the above inequalities is to study the evolution equation of the scale-invariant quantities
\begin{equation}
\label{cons-cond-form}
T_{(\alpha, \beta, \gamma)} = \alpha a_s + \beta Q b_s + \gamma Q^2, \quad \alpha, \beta, \gamma \in \mathbb{R}.
\end{equation}
For this we need to compute the evolution equations of $a_s$, $Qb_s$ and $Q^2$. Recall Definition \ref{def:smoothU2} of $C^\infty_{U(2)}(M_k\times [0,T])$. To simplify the formulae slightly, define the linear operator
$$L: C^\infty_{U(2)}(M_k\times [0,T]) \rightarrow C^\infty_{U(2)}(M_k\times [0,T])$$
by
\begin{equation*}
L[u] = u_{ss} + \left( 2\frac{b_s}{b} -\frac{a_s}{a} \right) u_s
\end{equation*}
away from the non-principal orbit $S^2_o$. As in section \ref{sec:maximum-principle} we may use L'H\^optital's rule to find a representation of $L$ on the non-principal orbit $S^2_o$. Then, as we show in the Appendix A, the evolution equations of $a_s$, $Qb_s$ and $Q^2$ can be written as
\begin{align}
\label{as-evol}
\partial_t a_s &= L[a_s] + \frac{1}{b^2}\left( -2 a_s b_s^2- 6 Q^2 a_s + 8 Q^3 b_s \right) \\
\label{Qbs-evol}
\partial_t Q b_s &= L[Qb_s] +\frac{1}{b^2}\left( 4 Q^2 a_s- 10 Q^3 b_s - 2 Q b_s^3 + 8 Q b_s\right) \\
\label{Q2-evol}
\partial_t Q^2 &= L[Q^2] + \frac{1}{b^2}\left(4 Q a_s b_s - 4 Q^2 b_s^2 - 8 Q^4 + 8 Q^2 \right).
\end{align}
Since the operator $L$ is linear, one sees that $T_{(\alpha, \beta,\gamma)}$ satisfies an evolution equation of the form
\begin{equation}
\label{T-schematic-evol}
\partial_t T_{(\alpha, \beta, \gamma)} = L[T_{(\alpha, \beta, \gamma)}] + \frac{1}{b^2} C_{(\alpha, \beta, \gamma)},
\end{equation}
where $C_{(\alpha, \beta, \gamma)}$ is a function of $a_s$, $b_s$ and $Q$. This evolution equation is very useful, as it allows us to systematically search for preserved inequalities. In particular, if we can find $\alpha, \beta, \gamma, \delta \in \mathbb{R}$ for which we can determine the sign of $C_{(\alpha, \beta, \gamma)}$ at a local extrema of $T_{(\alpha, \beta, \gamma)}$ at which $T_{(\alpha, \beta, \gamma)} = \delta$, it follows from the maximum principle of Theorem \ref{maximum-principle} that, depending on the sign, either
$$T_{(\alpha, \beta, \gamma)} \geq \delta$$
or
$$T_{(\alpha, \beta, \gamma)} \leq \delta$$
is a preserved inequality.
We searched for real numbers $\alpha, \beta, \gamma$ and $\delta$ leading to preserved conditions that yield the most useful control of the geometry of the flow. This is how we found the quantities $T_1$, $T_2$, $T_3$ and $T_4$. In later sections we will make heavy use of each of their respective inequalities. For instance, we use the preserved inequalities $T_1 \geq 0$ to exclude shrinking solitons on $M_k$, $k \geq2$, in the next section. Finally, in section \ref{E-H-unique-ancient-section} we generalize the above idea to find a continuously varying family of conserved inequalities.
\noindent\textbf{Statement and proof of results.}
In this subsection we list the precise statements and proofs of the results stated in the section outline. Before we begin, we prove the following technical lemma, which we need for verifying the growth conditions of the maximum principle of Theorem \ref{curv-bound}.
\begin{lem}
\label{lem:growth-cond}
Let $(M_k, g)$, $k \in \mathbb{N}$, satisfy $|Rm_g|_g \leq K$. Then
$$ |a_s|, |Qb_s| , |Q^2| = O(\exp(2\sqrt{K} s)).$$
\end{lem}
\begin{proof}
By the curvature components listed in section \ref{con-lap-cur-subsec} we see that
$$\left|\frac{a_{ss}}{a}\right|, \left|\frac{b_{ss}}{b}\right| \leq K.$$
Integrating
$$b_{ss} \leq b K,$$
shows that
$$b = O(\exp(\sqrt{K} s)).$$
From Lemma \ref{b-bound} we have
$$Q^2 \leq \frac{5}{3} K b^2$$
from which we conclude that
$$Q^2 = O(\exp(2\sqrt{K} s)).$$
Similarly, Lemma \ref{b-bound} shows
$$ |b_s| \leq \sqrt{5K} b$$
from which we deduce that
$$ |Qb_s| \leq \sqrt{\frac{25}{3}} K b^2$$
and hence
$$ |Qb_s| = O(\exp(2\sqrt{K} s)).$$
Finally,
$$ |a_{ss}| \leq a K $$
shows that
$$ |a_s| = O(\exp(\sqrt{K} s)).$$
This concludes the proof.
\end{proof}
Now we begin proving the conserved inequalities listed above.
\begin{lem}
\label{Qleq1}
Let $(M_k, g(t))$, $t\in[0,T]$, $k \geq 1$, be a Ricci flow with bounded curvature. Then the inequality
$$Q \leq 1$$
is preserved by the Ricci flow.
\end{lem}
\begin{proof}
Define the quantity $\tilde{Q} = 1 - Q$. From the evolution equation (\ref{Q-evol}) of $Q$ we see
$$\partial_t \tilde{Q} = \tilde{Q}_{ss} + 3 \frac{b_s}{b} \tilde{Q}_s + \tilde{Q} \left( -\frac{4}{b^2} Q(1+Q) \right).$$
As $Q \geq 0$ everywhere, the coefficient $-\frac{4}{b^2} Q(1+Q)$ is non-positive. Furthermore, by Lemma \ref{lem:growth-cond} we have $|\tilde{Q}| = o(\exp(s^2))$. Therefore we may apply the maximum principle of Theorem \ref{maximum-principle} to deduce that $\tilde{Q} \geq 0$ on $M_k \times [0,T]$. The desired result thus follows.
\end{proof}
\begin{lem}
\label{xleq0-lem}
Let $(M_k, g(t))$, $t\in[0,T]$, $k = 1,2$, be a Ricci flow with bounded curvature. Then the inequality
\begin{equation*}
x \leq 0
\end{equation*}
is preserved by the Ricci flow.
\end{lem}
\begin{proof}
The evolution equation of $x$, as derived in the Appendix A, is
\begin{align}
\label{x-evol}
\partial_t x &= L[x] - \frac{2}{b^2}\left(2 Q^2 + y^2\right) x - \frac{2}{b^2}\left(Q^2 +2 \right) y^2 \\
&\leq L[x] - \frac{2}{b^2}\left(2 Q^2 + y^2\right) x \nonumber
\end{align}
Note that $|x| = o(\exp(s^2))$ by Lemma \ref{lem:growth-cond}. Therefore applying the maximum principle of Theorem \ref{maximum-principle} yields the desired result.
\end{proof}
\begin{remark}
Note that $x = 2 -k$ at $s=0$ by the boundary conditions (\ref{boundary-cond-intro-s}). Therefore the result can only hold for $k = 1,2$.
\end{remark}
\begin{lem}
\label{yleq0-lem}
Let $(M_k, g(t))$, $t\in[0,T]$, $k \geq 1$, be a Ricci flow with bounded curvature. Then the inequality
$$y \leq 0$$
is preserved by the Ricci flow.
\end{lem}
\begin{proof}
Let $K >0$ such that
$$\sup_{M_k \times [0,T]} |Rm_{g(t)}|_{g(t)} < K.$$
Since $y$ is an odd quantity, we consider the quantity $Qy = Qb_s - Q^2$ instead. Its evolution equation is
\begin{equation}
\label{Qy-evol}
\partial_t Qy = L[Qy] - 2 \frac{Qy}{b^2}\left(2(Q^2+x) + Qy + y^2 \right).
\end{equation}
Note that
\begin{align*}
- \frac{2}{b^2}\left(2(Q^2+x) + Qy + y^2 \right) &= - \frac{2}{b^2} \left(4Q^2 -4 + b^2 M_1 + Qb_s + b_s^2 \right) \\
&\leq \frac{8}{b^2} + 2K + 2 \frac{|Q||b_s|}{b^2} \\
&\leq \frac{8}{b^2} + 2K + \frac{Q^2}{b^2}+ \frac{b^2_s}{b^2},
\end{align*}
where $M_1$ is one of the curvature components listed in section \ref{con-lap-cur-subsec}. By Lemma \ref{b-bound} we see that for some $C>0$
$$- \frac{2}{b^2}\left(2(Q^2+x) + Qy + y^2 \right) \leq C K \: \text{ on } \: M_k \times [0,T]$$
Furthermore $|Qy| = o(\exp(s^2))$ by Lemma \ref{lem:growth-cond}. Now the result follows from applying the maximum principle of Theorem \ref{maximum-principle}.
\end{proof}
\begin{lem}
\label{ab-mono}
Let $(M_k, g(t))$, $t\in[0,T]$, $k \geq 1$, be a Ricci flow with bounded curvature. If the initial metric $g(0)$ satisfies $a_s, b_s \geq 0$, then $a_s, b_s \geq 0$ for all times $t\in[0,T]$.
\end{lem}
\begin{proof}
The evolution equations (\ref{as-evol}) and (\ref{Qbs-evol}) of $a_s$ and $Qb_s$ can be written as a system of weakly coupled parabolic equations
\begin{align}
\label{as-evol-2}
\partial_t a_s &= L[a_s] - \left(2 \left(\frac{b_s}{b}\right)^2 + 6\frac{Q^2}{b^2} \right) a_s + 8 \frac{Q^2}{b^2} (Qb_s) \\
\partial_t Qb_s &= L[Qb_s] + 4 \frac{Q^2}{b^2} a_s + \left( \frac{8-10Q^2}{b^2} - 2 \left(\frac{b_s}{b}\right)^2 \right) (Qb_s),
\end{align}
By Lemma \ref{b-bound} and Lemma \ref{asa-bound-lem} the zeroth order coefficients of $a_s$ and $Qb_s$ are bounded. Lemma \ref{lem:growth-cond} shows that $|a_s|, |b_s| = o(\exp(s^2))$. Finally, note that the off-diagonal coefficients $8\frac{Q^3}{b^2}$ and $4 \frac{Q^2}{b^2}$ are non-negative. Thus the desired result follows by the maximum principle for weakly coupled parabolic equations of Lemma \ref{maximum-principle-coupled}.
\end{proof}
\begin{lem}
\label{da-bounded}
Let $(M_k, g(t))$, $t\in[0,T]$, $k \geq 1$, be a Ricci flow with bounded curvature satisfying $y \leq 0$, $Q \leq 1$ and $a_s, b_s \geq 0$. Then for $C\geq 2$ the inequality
$$a_s \leq C$$
is preserved by the Ricci flow.
\end{lem}
\begin{proof}
Define the quantity $A \coloneqq a_s - C$. Then from the evolution equation (\ref{as-evol-2}) of $a_s$ it follows that
\begin{align*}
\partial_t A &= L[A] - \left(2 \left(\frac{b_s}{b}\right)^2 + 6\frac{Q^2}{b^2} \right) A + \frac{1}{b^2}\left(8Q^3 b_s - CQ^2 - 2 Cb_s^2 \right).
\end{align*}
Fix $C \geq 2$. Then
$$8Q^3 b_s - 6CQ^2 - 2 Cb_s^2 \leq 8 Q^4 - C Q^2 \leq (8 - 6C)Q^2 \leq 0,$$
where we used $Q \leq 1$ and $y = b_s - Q \leq 0$. As $|A| = o(\exp(s^2))$ by Lemma \ref{lem:growth-cond} it follows from the maximum principle of Theorem \ref{curv-bound} that the inequality $A \leq 0$ is preserved by the Ricci flow. This proves the desired result.
\end{proof}
\begin{lem}
\label{T1-preserved-lem}
Let $(M_k, g(t))$, $t\in[0,T]$, $k \geq 1$, be a Ricci flow with bounded curvature satisfying $y \leq 0$, $b_s \geq 0$ and $Q \leq 1$. Then the inequality
\begin{equation*}
T_1 = a_s + 2Q^2 - 2 \geq 0
\end{equation*}
is preserved by the Ricci flow.
\end{lem}
\begin{proof}
The evolution equation of $T_1$ is
\begin{align}
\label{T1-evol}
\partial_t T_1 &= L[T_1] + \frac{1}{b^2} \left[- 4\left(1+Q^2\right)y^2 + 8Q\left(1-2Q^2 \right)y + 16Q^2\left(1-Q^2\right)\right] \\ \nonumber
& \qquad \qquad + T_1 \frac{2 y}{b^2}\left( 2Q- y\right),
\end{align}
which can be derived from the evolution equations (\ref{as-evol}), (\ref{Qbs-evol}) and (\ref{Q2-evol}) for $a_s$, $Qb_s$ and $Q^2$ listed above. Inspecting the quadratic expression
\begin{equation}
\label{quadexpr}
- 4\left(1+Q^2\right)y^2 + 8Q\left(1-2Q^2 \right)y + 16Q^2\left(1-Q^2\right)
\end{equation}
we see that when $y=0$ it is equal to
\begin{equation*}
16Q^2\left(1-Q^2\right) \geq 0
\end{equation*}
and when $y = - Q$ it is equal to
\begin{equation*}
4Q^2\left(1- Q^2\right) \geq 0
\end{equation*}
As $y = b_s - Q \in [-Q,0]$ by the assumptions $y \leq 0$, $b_s \geq 0$ and $Q \leq 1$, and furthermore the quadratic expression (\ref{quadexpr}) is concave in $y$, we conclude that
\begin{equation*}
\partial_t T_1 \geq L[T_1] + \frac{2y}{b^2}\left( 2Q- y\right) T_1
\end{equation*}
Note that the zeroth order coefficient of $T_1$ is bounded by Lemma \ref{b-bound}. Furthermore $|T_1| = o(\exp(s^2))$ by Lemma \ref{lem:growth-cond}. Hence the result follows from applying the maximum principle of Theorem \ref{maximum-principle}.
\end{proof}
Below we prove some further preserved conditions. These can be skipped on the first reading of the paper.
\begin{lem}
\label{T2-preserved-lem}
Let $(M_k, g(t))$, $t\in[0,T]$, $k = 1, 2$, be a Ricci flow with bounded curvature satisfying $Q \leq 1$. Then the condition
$$T_2 = Qy - x = - a_s + Qb_s + 2\left( 1 - Q^2 \right) \geq 0$$
is preserved by the Ricci flow.
\end{lem}
\begin{proof}
Note that $T_2 = 2 - k$ when $s = 0$ by the boundary conditions (\ref{boundary-cond-intro-s}). Therefore the result can only hold true for $k = 1,2$. The evolution equations of $T_2$ is
\begin{align}
\label{T2-evol}
\partial_t T_2 &= L[T_2] + \frac{4}{b^2}\left(1- Q^2\right) y^2 -2 \frac{T_2}{b^2} \left( \left(b_s-2Q\right)^2+ Q^2\right).
\end{align}
The coefficients are bounded by Lemma \ref{b-bound}. Furthermore $|T_2| = o(\exp(s^2))$ by Lemma \ref{lem:growth-cond}. Therefore applying the maximum principle of Theorem \ref{maximum-principle} yields the desired result.
\end{proof}
\begin{lem}
\label{T3-preserved-lem}
Let $(M_k, g(t))$, $t\in[0,T]$, $k \geq 1$, be a Ricci flow with bounded curvature satisfying $Q \leq 1$.
Then the inequality
$$T_3 = a_s - Qb_s - Q^2 + 1 \geq 0$$
is preserved by the Ricci flow.
\end{lem}
\begin{proof}
The evolution equations of $T_3$ is
\begin{align}
\label{T3-evol}
\partial_t T_3 &= L[T_3] + \frac{2}{b^2} \left(1 - Q^2\right) y^2 - 2 \frac{T_3}{b^2} \left( (b_s + Q)^2+ 4Q^2\right)
\end{align}
Note that the coefficients are bounded by Lemma \ref{b-bound}. Furthermore $|T_3| = o(\exp(s^2))$ by Lemma \ref{lem:growth-cond}. Applying the maximum principle of Theorem \ref{maximum-principle} yields the desired result.
\end{proof}
\begin{lem}
\label{minT1T4-preserved-lem}
Let $(M_k, g(t))$, $t\in[0,T]$, $k \geq 1$, be a Ricci flow with bounded curvature satisfying $y \leq 0$, $b_s \geq 0$ and $Q \leq 1$.
Then the inequality
\begin{align*}
\min(T_1, T_4) \geq 0
\end{align*}
is preserved by the Ricci flow. Here
\begin{align*}
T_1 &= a_s + 2Q^2 - 2\\
T_4 &= a_s - \frac{1}{2} Q b_s - \left(1 - Q^2\right).
\end{align*}
\end{lem}
\begin{proof}
By Lemma \ref{T1-preserved-lem} we already know that the inequality
$$T_1 = a_s - 2 + 2Q^2 \geq 0$$
is preserved. Thus we only need to show that $T_4\geq0$ is preserved whenever the Ricci flow satisfies $T_1 \geq 0$. The evolution equation of $T_4$ is
\begin{equation*}
\partial_t T_4 = L[T_4] + \frac{1}{b^2}\left(b_s\left(5Q^3-2b_s\right) -2 T_4\left(4Q^2 - 2Q b_s +b_s^2\right) \right).
\end{equation*}
A computation shows
\begin{equation*}
\frac{1}{2}Qb_s = T_1 - T_4 + 1 - Q^2.
\end{equation*}
By the assumption $y \leq 0$ we have
$$ \frac{Q^2}{2} \geq \frac{1}{2}Qb_s$$
and hence it follows that
\begin{equation*}
Q^2 \geq \frac{2}{3}\left(1-T_4\right).
\end{equation*}
Therefore
\begin{equation*}
5Q^3 - 2 b_s \geq 5 Q^3 - 2Q \geq Q\left( \frac{4}{3} - \frac{10}{3} T_4 \right),
\end{equation*}
which implies that
\begin{equation*}
\partial_t T_4 \geq L[T_4] - \frac{2 T_4}{b^2}\left(4 Q^2 - \frac{1}{3}Qb_s + b_s^2 \right)
\end{equation*}
since $b_s \geq 0$. Note that the zeroth order coefficient of $T_4$ is bounded by Lemma \ref{b-bound}. Furthermore $|T_4| = o(\exp(s^2))$ by Lemma \ref{lem:growth-cond}. Applying the maximum principle of Theorem \ref{maximum-principle} yields the desired result.
\end{proof}
\section{Exclusion of shrinking solitons}
\label{non-existence}
In this section we rule out $U(2)$-invariant shrinking solitons on $M_k$, $k\geq 2$, within a large class of metrics. In particular, we show
\begin{restatable}[No shrinker]{thm}{noshrinker}
\label{thm:no-shrinker}
On $M_k$, $k \geq 2$, there does not exists a complete $U(2)$-invariant shrinking Ricci soliton of bounded curvature satisfying the conditions
\begin{enumerate}
\item $\sup_{p \in M_k} |b_s| < \infty$
\item $T_1 = a_s + 2 Q^2 - 2 > 0$ for $s > 0$
\item $Q = \frac{a}{b} \leq 1$
\end{enumerate}
\end{restatable}
This theorem is the key ingredient in section \ref{E-H-sing-section}, where we show that certain Ricci flows on $M_k$, $k \geq 3$, develop Type II singularities in finite time.
\noindent\textbf{Soliton equations.}
Recall that a shrinking Ricci soliton $(M,g(t))$ is a solution to the Ricci flow equation that up to diffeomorphism homothetically shrinks. Such a soliton solution may be written as
$$ g(t) = \sigma^2(t) \Phi^\ast_t g(0),$$
where
$$ \sigma(t) = \sqrt{1 - 2 \rho t}$$
for some $\rho > 0$ and $\Phi_t$ is a family of diffeomorphisms. The reader may consult \cite{Top06} for more details. Hence for a $U(2)$-invariant shrinking Ricci soliton $(M_k, g(t))$, $k \geq 1$, the corresponding warping functions can be written as
\begin{align}
\label{a-sol-ricci-cor}
a(s,t) &= \sigma(t)a\left(\frac{s}{\sigma(t)},0\right) \\
\label{b-sol-ricci-cor}
b(s,t) &= \sigma(t)b\left(\frac{s}{\sigma(t)},0\right).
\end{align}
The above formulae are with respect to the radial coordinate $s$, which is equivalent to fixing a gauge. For this reason the family of diffeomorphisms $\Phi_t$ does not appear explicitly. Differentiating with respect to $t$ at time $0$ yields
\begin{align*}
\partial_t|_{t=0} a(s,t) &= a_s(s,0)\left( \frac{\partial s} {\partial t} + \rho s \right) - \rho a(s,0) \\
&= a_s(s,0) f_s - \rho a(s,0),
\end{align*}
where $f: M_k \rightarrow \mathbb{R}$ is the potential function satisfying
\begin{equation*}
f_{ss} = \rho + \frac{a_{ss}}{a} + 2 \frac{b_{ss}}{b}
\end{equation*}
and we used the expression (\ref{dsdt}) for $\frac{\partial s}{\partial t}$ derived in section \ref{ricci-flow-equations-sec}. Similarly we obtain
$$\partial_t|_{t=0} b(s,t) = b_s(s,0) f_s(s) - \rho b(s,0).$$
Substituting the expressions $\partial_t a$ and $\partial_t b$ from the Ricci flow equations (\ref{a-evol}) and (\ref{b-evol}), respectively, we see that the soliton equations for the warping functions $a$ and $b$ at time $t=0$ read (c.f. \cite{A17})
\begin{align}
\label{p-soliton} f_{ss} &= \frac{a_{ss}}{a}+ 2 \frac{b_{ss}}{b} + \rho \\
\label{a-soliton} a_{ss} &= 2 \frac{a^3}{b^4} - 2 \frac{a_sb_s}{b} + a_s f_s - \rho a\\
\label{b-soliton} b_{ss} &= \frac{4}{b} - 2\frac{a^2}{b^3} - \frac{a_sb_s}{a} - \frac{b^2_s}{b} + b_s f_s - \rho b
\end{align}
In a slight abuse of notation we will denote $a$ and $b$ as functions of $s$ only when we are considering Ricci solitons. In that case $a$ and $b$ should be interpreted as the initial data $a(s,0)$ and $b(s,0)$ at time zero that leads to a Ricci soliton solution, via the correspondence (\ref{a-sol-ricci-cor}) and (\ref{b-sol-ricci-cor}).
\begin{remark}
The above shows that all $U(2)$-invariant Ricci solitons on $M_k$ are automatically gradient Ricci solitons with potential function $f$.
\end{remark}
\noindent\textbf{Evolution of $x$, $y$ and $Q$ on soliton background.}
Since $x$, $y$ and $Q$ are \emph{scale-invariant} quantities, their evolution on a Ricci soliton background can be expressed as follows:
\begin{align*}
x(s, t) &= x\left(\frac{s}{\sigma(t)},0\right) \\
y(s, t) &= y\left(\frac{s}{\sigma(t)},0\right) \\
Q(s, t) &= Q\left(\frac{s}{\sigma(t)},0\right) \\
\end{align*}
Differentiating, we therefore obtain
\begin{align*}
\partial_t|_{t=0} x(s,t) &= x_s(s,0) f_s(s) \\
\partial_t|_{t=0} y(s,t) &= y_s(s,0) f_s(s) \\
\partial_t|_{t=0} Q(s,t) &= Q_s(s,0) f_s(s).
\end{align*}
With help of the evolution equations (\ref{x-evol}), (\ref{y-evol}) and (\ref{Q-evol}) for $x$, $y$ and $Q$, this yields the following ordinary differential equations for $x$, $y$ and $Q$ at time zero on a soliton background
\begin{align}
\label{soliton-x}
0 &= x_{ss} + \left(2\frac{b_s}{b}-\frac{a_s}{a}- f_s\right)x_s - \frac{1}{b^2}\left( 2 Q^2 \left(2x + y^2\right) + 2 y^2\left(2 + x\right)\right)\\
\label{soliton-y}
0 &= y_{ss} + \left(\frac{a_s}{a} - f_s\right)y_s -\frac{y}{a^2} \left( \left(x+2\right)^2 + Q^2 \left(2x + y^2 \right)\right) \\
\label{soliton-Q}
0 &= Q_{ss} + \left(3\frac{b_s}{b} - f_s\right) Q_s + \frac{4}{b^2}Q\left(1-Q^2\right).
\end{align}
Alternatively these equations can be derived from the soliton equations (\ref{p-soliton})-(\ref{b-soliton}). In a slight abuse of notation we will often denote $x$, $y$ and $Q$ as functions of $s$ only when we are considering Ricci solitons.
\noindent\textbf{Exclusion of shrinking solitons.} By \cite{CZ10} we know that the potential function of a non-compact complete shrinking Ricci soliton grows quadratically with the distance to some fixed point. In our setting this translates into the following lemma:
\begin{lem}
\label{f-asymp}
Let $(M_k, g)$, $k \geq 1$, be a complete non-compact shrinking Ricci soliton of bounded curvature. Then
\begin{align*}
f &\sim \frac{\rho}{2} s^2 \\
f_s&\sim \rho s
\end{align*}
as $s\rightarrow \infty$.
\end{lem}
\begin{proof}
See Theorem 1.1, equation (2.3) and equation (2.8) of \cite{CZ10}.
\end{proof}
This allows us to prove the following lemma:
\begin{lem}
\label{dQ-Lemma}
Let $(M_k, g)$, $k \geq 1$, be a complete non-compact shrinking Ricci soliton of bounded curvature with $Q \leq 1$ on $M_k$. Then $Q_s \geq 0$ on $M_k$.
\end{lem}
\begin{proof}
First notice that for a complete shrinking Ricci soliton with $Q \leq 1$, the strong maximum principle applied to the evolution equation (\ref{Q-evol}) of $Q$ forces
\begin{equation*}
Q < 1 \: \text{ for } \: s \geq 0,
\end{equation*}
as otherwise we would have $Q=1$ everywhere, which cannot be the case. Similarly,
\begin{equation*}
Q > 0
\end{equation*}
unless we are at the origin $s=0$. By equation (\ref{soliton-Q}) we have
\begin{equation}
\label{soliton-Q2}
Q_{ss} = \left(f_s - 3 \frac{b_s}{b}\right)Q_s - \frac{4}{b^2}Q\left(1-Q^2\right).
\end{equation}
We now argue by contradiction. Assume there exists an $s_{\ast} > 0$ such that $Q_s(s_{\ast}) < 0$. Then $Q_s(s) < 0$ for all $s> s_{\ast}$, because at any extremum of $Q$ we have $Q_s = 0$ and
\begin{equation*}
Q_{ss} = - \frac{4}{b^2} Q \left(1 - Q^2 \right) < 0.
\end{equation*}
Lemma \ref{b-bound} shows that $\frac{b_s}{b}$ is bounded and from Lemma \ref{f-asymp} it follows that
$$f_s \rightarrow \infty \: \text{ as } \: s \rightarrow \infty.$$
Therefore eventually
$$ f_s - 3 \frac{b_s}{b} > 0$$
from which it follows by equation (\ref{soliton-Q2}) that
$$Q_{ss} <0$$
for sufficiently large $s$. This, however, contradicts that $Q > 0$ unless $s=0$.
\end{proof}
In the lemma below we bound the term
$$G := (x+2)^2 + Q^2(2x+y^2),$$
which appears in the evolution equation (\ref{soliton-y}) of $y$, away from zero.
\begin{lem}
\label{Gpos-lem}
Whenever $Q_s \geq 0$ and $Q, T_1 > 0$ we have $G > 0$.
\end{lem}
\begin{proof}
We have
\begin{equation*}
\frac{Q_s}{Q} = \frac{a_s}{a}-\frac{b_s}{b} = \frac{x}{a} - \frac{y}{b} + \frac{2}{a} - \frac{2a}{b^2}.
\end{equation*}
For $Q_s \geq 0 $ it follows that
\begin{equation*}
x - Q y \geq 2\left(Q^2 - 1\right).
\end{equation*}
Recall the quantity
$$ T_1 = a_s + 2 Q^2 - 2$$
defined in section \ref{section-preserved-conditions}. Then
\begin{align*}
G &\geq \left(x+2\right)^2 + Q^2 \left(2\left(Qy + 2\left(Q^2 - 1\right) \right) + y^2\right) \\
&= x^2 + 4x + 4 + 2 Q^3 y + 4Q^4 - 4Q^2 + Q^2 y^2 \\
&=\left( a_s + Q^2 -2 \right)^2 + 4\left( a_s + Q^2 -2 \right) + 4 + 3 Q^4 - 4 Q^2 + Q^2\left(y + Q\right)^2 \\
&=a_s^2 + 2 Q^2 a_s + 4\left(Q^4 - Q^2\right) + Q^2 \left(y + Q\right)^2 \\
&=a_s^2 + 2Q^2 T_1 + Q^2 \left( y + Q \right)^2 \\
&=a_s^2 + Q^2 b_s^2 + 2 Q^2 T_1 > 0
\end{align*}
\end{proof}
Now we prove the non-existence of shrinking solitons.
\begin{proof}[Proof of Theorem \ref{thm:no-shrinker}]
We argue by contradiction. Assume such a shrinking Ricci soliton exists. Applying L'H\^opital's Rule to the evolution equation (\ref{b-evol}) of $b$ shows that at $s=0$
\begin{align*}
\partial_t b \Big | _{s=0} &= 2 b_{ss} - \frac{4}{b} \\ \nonumber
&= 2 \left( y_s + \frac{k-2}{b}\right).
\end{align*}
Clearly, every shrinking soliton satisfies
$$\partial_t b \Big |_{s=0} < 0.$$
The boundary conditions (\ref{boundary-cond-intro-s}) of $a$ and $b$ at $s=0$ imply that
\begin{equation*}
y(0) = 0.
\end{equation*}
and thus we deduce from the above that
$$y_s(0) < 0,$$
as $k \geq 2$ by assumption. The ordinary differential equation (\ref{soliton-y}) for $y$ can be written as
\begin{equation}
\label{soliton-y-simp}
y_{ss} = \left(f_s - \frac{a_s}{a} \right)y_s + \frac{y}{a^2} G.
\end{equation}
Lemma \ref{dQ-Lemma} and Lemma \ref{Gpos-lem} imply that $$G > 0\:\text{ for }\: s > 0,$$
which in turn shows that $y_s \leq 0$ everywhere, as at a negative local minimum of $y$ we would have
\begin{equation*}
y_{ss} = \frac{y}{a^2} G < 0.
\end{equation*}
The asymptotic properties of $f$ listed in Lemma \ref{f-asymp} and the bounds on $\frac{a_s}{a}$ proven in Lemma \ref{asa-bound-lem} show that eventually
$$f_s - \frac{a_s}{a} > 0$$
and hence from the equation (\ref{soliton-y-simp}) it follows that
$$ y_{ss} < 0$$
for $s$ sufficiently large. From this it follows that $$\lim_{s \rightarrow \infty} y = \lim_{s \rightarrow \infty} \left( b_s - Q \right)= - \infty,$$ which contradicts our assumptions on $b_s$ and $Q$.
\end{proof}
\section{Curvature bound}
\label{sec:curv-bound}
The aim of this section is to prove that a Ricci flow $(M_k, g(t))$, $k \in \mathbb{N}$, $t\in [0,T)$, starting from an initial metric $g(0) \in \mathcal{I}$ --- where $\mathcal{I}$ is a class of metrics to be discussed below --- with $\sup_{p \in M_k} b(p,0) < \infty$ satisfies the curvature bound
$$ |Rm_{g(t)}|_{g(t)} \leq C_1 b^{-2} \: \text{ for } \: t \in (0,T),$$
where $C_1>0$ is some constant. This allows us to control the geometry via the warping function $b$, which will be crucial for constructing blow-up limits in the following parts of the paper. Note that this bound was already derived in the compact case in \cite{IKS17} and we will follow their strategy to prove it in our non-compact setting.
Recall the following definition (see also \cite{ChI}[Definition 8.23]):
\begin{definition}[$\kappa$-non-collapsing]
Let $(M,g(t))$, $t \in [0,T)$, be a Ricci flow and $\kappa > 0$. We say that the Ricci flow is $\kappa$-non-collapsed at a point $(p_0, t_0)$ in spacetime at scale $\rho$ if the following two conditions hold for all $r \leq \rho$:
\begin{itemize}
\item (bounded normalized curvature) We have $|Rm(p,t)|\leq r^{-2}$ for every $(p,t) \in B_{g(t_0)}(p_0,r) \times [ t_0 - r^2, t_0]$. In particular we assume $ [ t_0 - r^2, t_0] \subset [0,T)$.
\item (non collapsed volume) At time $t_0$ the ball $B_{g(t_0)}(p_0, r)$ has volume at least $\kappa r^4$.
\end{itemize}
\end{definition}
We now define the class of metrics $\mathcal{I}$.
\begin{definition}
\label{def:I}
For $K>0$ let $\mathcal{I}_K$ be the set of all complete \emph{bounded curvature} metrics of the form (\ref{metric2}) on $M_k$, $k \geq 1$, with \emph{positive injectivity radius} that satisfy the following scale-invariant inequalities:
\begin{align}
\label{I1} Q &\leq 1 \\
a_s, b_s &\geq 0 \\
y &\leq 0 \\
\label{I4} \sup a_s &< K \\
\label{I5} \sup |b b_{ss}| &< K
\end{align}
Denote by $\mathcal{I}$ the set of metrics $g$ such that for sufficiently large $K>0$ we have $g\in \mathcal{I}_K$.
\end{definition}
Note that for any $k \in \mathbb{N}$ the set $\mathcal{I}$ of metrics on $M_k$ is non-empty, as for example the metric on $M_k$ defined by
\begin{align*}
a(s) &= Q = \tanh(k s), \quad k \in \mathbb{N} \\
b(s) &= 1
\end{align*}
is contained in $\mathcal{I}$. In Lemma \ref{I-preserved} below we show that if $g(0) \in \mathcal{I}_{K_0}$ for some $K_0>0$ then there exists a $K> K_0$ such that $g(t)\in \mathcal{I}_K$ for $t \in [0, T)$. Note that conditions (\ref{I1})- (\ref{I5}) are scale-invariant, and therefore pass to blow-up limits.
An adaptation of \cite[Theorem 8.26]{ChI} to our setting yields the following result:
\begin{thm}[No local collapsing]
\label{thm:no-local-collapsing}
Let $g(t)$, $t \in [0,T)$, $T< \infty$, be a Ricci flow starting from an initial metric $g(0) \in \mathcal{I}$. Then there exists a $\kappa >0$ depending on $T$, $\mathrm{inj}(g(0))$ and $\sup_{M \times [0, T/2]} Ric_{g(t)}$ such that $g(t)$ is $\kappa$-non-collapsed at every $(p,t) \in M\times (\frac{T}{2}, T)$ at every scale $\rho < \sqrt{T/2}$.
\end{thm}
\begin{remark}
Recall that if a Ricci flow $g(t)$ is $\kappa$-non-collapsed at scale $\rho$, then the parabolically dilated Ricci flow $\alpha^2 g( \alpha^{-2} t)$ is $\kappa$-non-collapsed at scale $\alpha \rho$. As the $\kappa$-non-collapsedness property is preserved under Cheeger-Gromov limits, a blow-up limit of a Ricci flow $(M_k, g(t))$, $[0, T_{sing})$ is $\kappa$-non-collapsed at all scales.
\end{remark}
Having set up the necessary terminology, we may now state the main theorem of this section:
\begin{thm}[Curvature bound]
\label{curv-bound}
Let $(M_{k}, g(t))$, $t \in [0, T)$, be a Ricci flow starting from an initial metric $g(0) \in \mathcal{I}$ (see Definition \ref{def:I}) with
\begin{equation*}
\sup _{p \in M_k} b(p,0) < \infty.
\end{equation*}
Then there exists a constant $C_1 > 0$ such that
\begin{equation*}
|Rm_{g(t)}|_{g(t)}(p) \leq C_1 b(p,t)^{-2}
\end{equation*}
for $(p,t) \in M_k \times (0,T)$.
\end{thm}
A useful variant of Theorem \ref{curv-bound} is:
\begin{cor}
\label{cor:curv-bound-ancient}
Let $(M_{k}, g(t))$ with $g(t) \in \mathcal{I}$ (see Definition \ref{def:I}) for $t \in (-\infty, 0]$ be an ancient Ricci flow solution which is $\kappa$-non-collapsed at all scales. Then there exists a constant $C_1 > 0$ such that
\begin{equation*}
|Rm_{g(t)}|_{g(t)}(p) \leq C_1 b(p,t)^{-2}
\end{equation*}
for $(p,t) \in M_k \times (-\infty, 0]$.
\end{cor}
\begin{remark}
Corollary \ref{cor:curv-bound-ancient} follows immediately from Theorem \ref{curv-bound} for ancient $\kappa$-non-collapsed Ricci flows that arise as blow up limits of Ricci flows $(M_k, g(t))$, $t \in [0, T_{sing})$, $g(0) \in \mathcal{I}$, as the curvature bound is scale-invariant. Nevertheless, we give a proof of the general case.
\end{remark}
Let us now prove the assertions made above. We begin with the following lemma:
\begin{lem}
\label{bddb-bound-lem}
Let $K_0 > 0$ and assume that $(M_k,g(t))$, $k \geq 1$, $t\in[0,T)$, is a Ricci flow starting from an initial metric $g(0) \in \mathcal{I}_{K_0}$. Then there exists a constant $K \geq 0$, depending only on the initial metric $g(0)$, such that
\begin{equation}
\label{bddb-bound}
|bb_{ss}| \leq K
\end{equation}
on $M_k \times [0,T)$.
\end{lem}
\begin{proof}
We follow the proof strategy of \cite[Lemma 7]{IKS17}. Consider the quantities
\begin{align*}
H_- &= b b_{ss} + a_s^2 - b_s^2 - C \\
H_+ &= b b_{ss} - a_s^2 - b_s^2 + C,
\end{align*}
where $C > 0$ is a constant to be determined later. The goal is to show that the inequalities $H_+ \geq 0$ and $H_- \leq 0$ are preserved by the Ricci flow for sufficiently large $C> 0$. The quantities $H_\pm$ satisfy the evolution equations
\begin{align*}
\partial_t H_\pm &= [H_\pm]_{ss} + \left(\frac{a_s}{a} -2 \frac{b_s}{b}\right) [H_\pm]_s +H_\pm \left(-\frac{2 a_s^2}{a^2}-\frac{4 a^2}{b^4}-\frac{4b_s^2}{b^2}\right) \\
& \pm C \left(\frac{2 a_s^2}{a^2}+\frac{4 a^2}{b^4}+\frac{4b_s^2}{b^2}\right) \\
&\pm 2 a_{ss}^2 + a_{ss} \left(-\frac{2 b a_s b_s}{a^2} \mp \frac{8 a_s b_s}{b} \pm \frac{4 a_s^2}{a}+\frac{4a}{b^2}\right) \\
&+\frac{2 b a_s^3 b_s}{a^3}-\frac{32 a a_sb_s}{b^3}\mp\frac{16 a^3 a_s b_s}{b^5}+\frac{4 a_s^2}{b^2} \pm \frac{8 a^2a_s^2}{b^4} \\
&\mp \frac{2 a_s^4}{a^2} +\frac{32 a^2 b_s^2}{b^4}-\frac{16 b_s^2}{b^2}.
\end{align*}
In the Appendix A we carry out the derivation of the evolution equation. We now show that $H_- \leq 0$ is preserved. Using Young's inequality to bound the terms involving $a_{ss}$ and then disregarding non-positive terms not involving $C$, we obtain
\begin{align*}
\partial_t H_- &\leq [H_-]_{ss} + \left(\frac{a_s}{a} -2 \frac{b_s}{b}\right) [H_-]_s +H_- \left(-\frac{2 a_s^2}{a^2}-\frac{4 a^2}{b^4}-\frac{4b_s^2}{b^2}\right) \\
&-C \left(\frac{2 a_s^2}{a^2}+\frac{4 a^2}{b^4}+\frac{4b_s^2}{b^2}\right) \\
&+\frac{1}{2}\left(\left(\frac{2 b a_s b_s}{a^2}\right)^2+ \left(\frac{8 a_s b_s}{b}\right)^2 +\left(\frac{4 a_s^2}{a}\right)^2+\left(\frac{4a}{b^2}\right)^2\right) \\
&+\frac{2 b a_s^3 b_s}{a^3}+\frac{16 a^3 a_s b_s}{b^5}+\frac{4 a_s^2}{b^2}+\frac{2 a_s^4}{a^2} +\frac{32 a^2 b_s^2}{b^4}
\end{align*}
Recall that on $M_k \times [0,T)$ we have $y = b_s - Q \leq 0 $, $Q \leq 1$, $a_s,b_s \geq 0$ and $a_s \leq C'$ for some $C'>0$ by Lemma \ref{yleq0-lem}, Lemma \ref{Qleq1}, Lemma \ref{ab-mono} and Lemma \ref{da-bounded}, respectively. Therefore we obtain the following bounds away from the non-principal orbit $S^2_o$:
\begin{align*}
\left( \frac{2b a_s b_s}{a^2}\right)^2 &= \left( \frac{2 a_s b_s}{a Q} \right)^2 \leq \frac{4 a_s^2}{a^2} \\
\left(\frac{8 a_s b_s}{b} \right)^2 &= \left(8\frac{a_s}{a} Qb_s\right)^2 \leq 64 \frac{a_s^2}{a^2} \\
\left(\frac{4 a_s^2}{a}\right)^2 &= 16 C'^2 \frac{a_s^2}{a^2} \\
\frac{2 b a_s^3 b_s}{a^3} &= \frac{ 2a_s^3}{a^2} \frac{b_s}{Q} \leq 2 C' \frac{a_s^2}{a^2} \\
\frac{16 a^3 a_s b_s}{b^5} &= 16 Q^3 \frac{a_sb_s}{b^2} \leq 8Q^3 \left( \frac{a_s^2}{b^2} + \frac{b_s^2}{b^2} \right) \leq 8\left( \frac{a_s^2}{a^2} + \frac{b_s^2}{b^2} \right) \\
\frac{4 a_s^2}{b^2} &\leq 4 \frac{a_s^2}{a^2} \\
\frac{2 a_s^4}{a^2} &\leq 2 C'^2 \frac{a_s^2}{a^2} \\
\frac{32 a^2 b_s^2}{b^4} &= 32 Q^2 \frac{b_s^2}{b^2} \leq 32 \frac{b_s^2}{b^2}
\end{align*}
Hence for a sufficiently large $C>0$ it follows that
\begin{align*}
\partial_t H_- &\leq [H_-]_{ss} + \left(\frac{a_s}{a} -2 \frac{b_s}{b}\right) [H_-]_s + H_- \left(-\frac{2 a_s^2}{a^2}-\frac{4 a^2}{b^4}-\frac{4b_s^2}{b^2}\right)
\end{align*}
away from the non-principal orbit $S^2_o$. Switching to coordinates $(s,t)$ we see that for $s > 0$
\begin{align}
\label{H-evol-ineq}
\partial_t\Big|_{s} H_- &\leq [H_-]_{ss} + \left(\frac{a_s}{a} -2 \frac{b_s}{b}\right) [H_-]_s + H_- \left(-\frac{2 a_s^2}{a^2}-\frac{4 a^2}{b^4}-\frac{4b_s^2}{b^2} - \frac{\partial s}{\partial t} \right).
\end{align}
On the non-principal orbit $S^2_o$, or equivalently when $s=0$, we have
$$H_- = bb_{ss} + k^2 - C \leq bQ_s +k^2 - C \leq k + k^2 -C,$$
where we used that $y = b_s - Q \leq 0$ with equality at $s = 0$. Choosing $C>k^2 + k$ we have $H_- < 0$ on $\{s = 0\} \times [0,T)$. Hence for every $T' \in [0, T)$ there exists a $s_0 > 0$ such that
$$ H_-(s,t) \leq 0 \: \text{ on } \: [0,s_0] \times [0, T'],$$
as $H_-(s,t)$ is a smooth function on $\mathbb{R}_{\geq 0} \times [0, T']$. Furthermore note that
$$ |H_-|\leq |Rm_{g(t)}|_{g(t)} b^2 + C'^2 + 1 + C,$$
where we used the expression for the curvature component $R_{0202}$ derived in section \ref{con-lap-cur-subsec}. This shows that for each time $t < T'$ the function $H_-(s,t)$ grows subexponentially. Note that by Lemma \ref{b-bound} and Lemma \ref{asa-bound-lem} the coefficient
$$\frac{a_s}{a} -2 \frac{b_s}{b}$$
is bounded on $[s_0, \infty) \times [0, T']$. Similarly, we see from the bound (\ref{dsdt-bound}) on $|\frac{\partial s}{\partial t}|$ presented in the proof of the maximum principle of Theorem \ref{maximum-principle}, Case 1, that the coefficient
$$-\frac{2 a_s^2}{a^2}-\frac{4 a^2}{b^4}-\frac{4b_s^2}{b^2} - \frac{\partial s}{\partial t}$$
grows at most linearly on every times slice of $[s_0, \infty) \times [0, T']$. Therefore, applying the weak maximum principle to the evolution equation (\ref{H-evol-ineq}) of $H_-$ on the parabolic neighborhood $[s_0, \infty) \times [0,T']$, we deduce that
$$H_- \leq 0 \: \text{ on } \: M_k \times [0, T'].$$
As $T' \in [0, T)$ was arbitrary it follows that $H_-\leq 0$ is preserved by the Ricci flow.
We repeat the same process to prove that $H_+ \geq 0$ is preserved. Applying Young's inequality to bound terms involving $a_{ss}$ and then disregarding non-negative terms not involving $C$, we see that
\begin{align*}
\partial_t H_+ &\geq [H_+]_{ss} + \left(\frac{a_s}{a} -2 \frac{b_s}{b}\right) [H_+]_s +H_+ \left(-\frac{2 a_s^2}{a^2}-\frac{4 a^2}{b^4}-\frac{4b_s^2}{b^2}\right) \\
& + C \left(\frac{2 a_s^2}{a^2}+\frac{4 a^2}{b^4}+\frac{4b_s^2}{b^2}\right) \\
&-\frac{1}{2}\left(\left(\frac{2 b a_s b_s}{a^2}\right)^2+ \left(\frac{8 a_s b_s}{b}\right)^2 +\left(\frac{4 a_s^2}{a}\right)^2+\left(\frac{4a}{b^2}\right)^2\right) \\
&-\frac{32 a a_sb_s}{b^3}-\frac{16 a^3 a_s b_s}{b^5}- \frac{2 a_s^4}{a^2} -\frac{16 b_s^2}{b^2}.
\end{align*}
Bounding the zeroth order terms via Young's inequality as above, we see that for $C>0$ sufficiently large
\begin{align*}
\partial_t H_+ &\geq [H_+]_{ss} + \left(\frac{a_s}{a} -2 \frac{b_s}{b}\right) [H_+]_s +H_+ \left(-\frac{2 a_s^2}{a^2}-\frac{4 a^2}{b^4}-\frac{4b_s^2}{b^2}\right)
\end{align*}
away from the non-principal orbit $S^2_o$. On the non-principal orbit $S^2_o$ we have
$$H_+ = bb_{ss} - k^2 + C \geq - k^2 + C,$$
where we used that $b_s \geq 0$ with equality at $s =0$ to deduce that $b_{ss} \geq 0$ at $s = 0$. From here the above proof that $H_- \leq 0$ is preserved carries over and we may conclude that $H_+ \geq 0$ is preserved as well. Recalling the bounds on $a_s$ and $b_s$, the desired result now follows.
\end{proof}
Now we can prove that $\mathcal{I}$ (see Definition \ref{def:I}) is preserved by Ricci flow:
\begin{lem}
\label{I-preserved} Let $K_0 > 0$. Then there exists a $K \geq K_0$ such that the following holds: Let $(M_k,g(t))$, $k \geq 1$, $t \in [0,T)$, be a Ricci flow solution starting from an initial metric $g(0) \in \mathcal{I}_{K_0}$. Then $g(t) \in \mathcal{I}_K$ for every $t \in [0,T)$.
\end{lem}
\begin{proof}
By Lemma \ref{Qleq1}, Lemma \ref{ab-mono}, Lemma \ref{yleq0-lem}, Lemma \ref{da-bounded} we see that for $K > 2$ the conditions (\ref{I1}) - (\ref{I4}) are preserved. By Lemma \ref{bddb-bound-lem} we see that there exists a $K \geq K_0$ such that inequality (\ref{I5}) holds for $t \in [0,T)$.
Now we only need to prove that for every time $t \in [0, T)$ the metric $g(t)$ has bounded curvature and positive injectivity radius. As the curvature of $g(0)$ is bounded by the assumption that $g(0) \in \mathcal{I}_K$, it follows by Shi's Theorem \cite{Shi89} that for every time $T' \in [0,T)$ the Ricci flow $g(t)$ has bounded curvature on the time interval $[0,T']$. As $\mathrm{inj}_{g(0)} > 0$ it follows that the metric $g(0)$ is non-collapsed. By standard volume distortion estimates it follows that for each $t \in [0, T/2]$ the metric $g(t)$ is non-collapsed, and hence $\mathrm{inj}_{g(t)} > 0$. By Theorem \ref{thm:no-local-collapsing} there exists a $\kappa >0$ and $\rho>0$ such that for each $t \in [0,T)$ the metric $g(t)$ is $\kappa$-non-collapsed at scale $\rho < \sqrt{T/2}$. This shows that $\mathrm{inj}_{g(t)} > 0$ for all $t \in [T/2, T)$.
\end{proof}
Before proving Theorem \ref{curv-bound}, we need to prove the following two lemmas in preparation:
\begin{lem}
\label{lem:dtbb-bound}
Let $(M_k,g(t))$, $k \geq 1$, $t\in[0,T)$, be a Ricci flow starting from an initial metric $g(0) \in \mathcal{I}$. Then there exists a constant $C_0 \geq 0$, depending only on the initial metric $g(0)$, such that
\begin{equation}
\label{dtbb-bound}
|\partial_t b^2| \leq C_0
\end{equation}
\end{lem}
\begin{proof}
By Lemma \ref{I-preserved} there exists a $K> 0$ such that $g(t) \in \mathcal{I}_K$ for $t \in [0, T)$. From the evolution equation (\ref{b-evol}) of $b$ and Definition \ref{def:I} of $\mathcal{I}_K$ it follows that
\begin{align*}
|\partial_t b^2| &= \left|2 b b_{ss} - 8 + 4 Q^2 + 2 \frac{a_sb_s}{Q} + 2 b_s^2\right| \\ \nonumber
&\leq 2 K + 8 + 4 + 2 K + 2 \\ \nonumber
&= 4K + 14
\end{align*}
This concludes the proof.
\end{proof}
\begin{lem}
\label{bbounded-lem}
Let $(M_k, g(t))$, $k \geq 1$, $t \in [0,T)$, be a Ricci flow starting from an initial metric $g(0) \in \mathcal{I}$. Then
$$\sup_{p \in M_k} b(p,t) \leq \sup_{p \in M_k} b(p,0)$$
for all $t \in [0,T)$.
\end{lem}
\begin{proof}
From the evolution equation (\ref{b-evol}) of $b$ and expression (\ref{laplacian}) for the Laplacian with respect to the background metric $g(t)$ it follows
\begin{align*}
\partial_t b^2 & = \Delta_{g(t)} b^2 - 8 + 4 Q^2 - 4 b_s^2 \\
&\leq \Delta_{g(t)} b^2 - 4.
\end{align*}
Applying the maximum principle \cite[Theorem 12.14]{ChII} yields the desired result.
\end{proof}
We now proceed to proving Theorem \ref{curv-bound}.
\begin{proof}[Proof of Theorem \ref{curv-bound}]
We argue by contradiction. Assume there exists a sequence of points $(p_i, t_i)$ in spacetime and constants $D_i \rightarrow \infty$ as $i \rightarrow \infty$ such that
\begin{equation*}
|Rm_{g(t_i)}|_{g(t_i)}(p_i) = D_i b(p_i,t_i)^{-2} := K_i
\end{equation*}
and
\begin{equation*}
|Rm_{g(t)}|_{g(t)} \leq D_i b^{-2} \: \text{ on } \: M_k \times [0, t_i].
\end{equation*}
By the assumption that $g(0) \in \mathcal{I}$ the initial metric $g(0)$ has bounded curvature. Hence by Shi's theorem \cite{Shi89} we have that for every $T' \in [0,T)$ the metric $g(t)$ has bounded curvature on $M_k \times [0,T']$. As by Lemma \ref{bbounded-lem} the warping function $b$ is uniformly bounded on $M_k \times [0,T)$, we thus see that $D_i \rightarrow \infty$ forces $K_i \rightarrow \infty$ and therefore $t_i \rightarrow T$.
Consider the rescaled Ricci flows
\begin{equation*}
g_i(t) = K_i g \left( t_i + K_i^{-1} t \right), \quad t \in [- K_i \Delta t_i, 0],
\end{equation*}
where $\Delta t_i > 0$ is to be determined below. As $K_i \rightarrow \infty$ we see that $g_i(t)$ are blow-ups rather than blow-downs, which is important for the following reason: By Theorem \ref{thm:no-local-collapsing} there exists a $\kappa>0$ such that $g(t)$ is $\kappa$-non-collapsed at every scale $p \leq \sqrt{T/2}$ at every spacetime point $(p,t) \in M_k \times [T/2, T)$. As $K_i \rightarrow \infty$ we see that $g_i(t)$ are $\kappa$-non-collapsed at scales tending to infinity as $i \rightarrow \infty$.
By Lemma \ref{I-preserved} there exists a $K>0$ such that $g(t) \in \mathcal{I}_K$ for all $t \in [0,T)$. Furthermore, by Lemma \ref{lem:dtbb-bound} there exists a $C_0$ such that $|\partial_t b^2| \leq C_0$ on $M_k \times [0, T)$. Recall the Definition \ref{def:C} of $C_g(p,r)$. Set
$$\Delta t_i = \min\left( \frac{t_i}{2}, \frac{b^2(p_i, t_i)}{8 C_0} \right)$$
and consider the parabolic neighborhoods
$$\Omega_i = C_{g(t_i)}\left(p_i, \frac{b(p_i,t_i)}{2}\right) \times [t_i - \Delta t_i, t_i].$$
As $g(t) \in \mathcal{I}_K$ for $t \in [0,T)$ we have that $y = b_s - Q \leq 0$, $Q \leq 1$ and $b_s \geq 0$ everywhere on $M_k \times [0,T)$. Therefore
$$b(p, t_i) \geq \frac{b(p_i, t_i)}{2} \text{ on } \Omega_i\cap \left\{t = t_i\right\}$$
By Lemma \ref{lem:dtbb-bound}
\begin{equation*}
b^2(p,t_i) - b^2(p,t) \leq C_0 (t_i - t)
\end{equation*}
for all $(p,t) \in \Omega_i$ from which it follows that
$$ \frac{1}{4} b(p_i, t_i)^2 - b(p,t)^2 \leq b^2(p,t_i) - b^2(p,t) \leq C_0 (t_i - t) \leq C_0 \Delta t \leq \frac{1}{8} b(p_i, t_i)^2.$$
Thus we deduce that
\begin{equation}
\label{eqn:b-lower}
b^2(p,t) \geq \frac{1}{8} b^2(p_i,t_i) \text{ on } \Omega_i.
\end{equation}
It follows that for $(p,t) \in \Omega_i$
\begin{align*}
|Rm_{g(t)}|_{g(t)}(p) &\leq D_i b(p,t)^{-2} \\ \nonumber
&\leq 8 D_i b(p_i,t_i)^{-2} \\ \nonumber
&= 8 K_i
\end{align*}
and hence the curvatures of the rescaled metrics $g_i(t)$ satisfy
$$
|Rm_{g_i(t)}|_{g_i(t)} \leq 8
$$
on the parabolic neighborhoods $\Omega'_i$
$$\Omega'_i \coloneqq C_{g_i(0)}\left(p_i, \sqrt{K_i} \frac{b(p_i,t_i)}{2} \right) \times [-K_i \Delta t_i, 0].$$
Note that
$$ K_i \Delta t_i \rightarrow \infty \:\text{ as }\: i \rightarrow \infty$$
and
$$ \sqrt{K_i} \frac{b(p_i,t_i)}{2} \geq \frac{\sqrt{D_i}}{2} \rightarrow \infty \: \text{ as } \: i \rightarrow \infty.$$
Hence $\left(C_{g_i(t)}(p_i, \sqrt{D_i}/2), g_i(t), p_i\right)$, $t \in [-K_i \Delta t_i, 0]$, subsequentially converges, in the Cheeger-Gromov sense, to an ancient pointed Ricci flow $(M_{\infty}, g_{\infty}(t), p_\infty)$, $t \in (-\infty, 0]$.
\begin{claim}
The Ricci flow $(M_\infty, g_\infty(t), p_\infty)$, $t \in (-\infty, 0]$, splits as $(\mathbb{R}^2 \times N, g_{eucl} + g_N(t))$, $t \in (-\infty, 0]$, where $g_{eucl}$ is the flat euclidean metric, and $(N, g_N(t))$ is a non-compact ancient Ricci flow.
\end{claim}
\begin{claimproof}
Denote by $a_i$ and $b_i$ the warping functions of the rescaled metrics $g_i(t)$. Then by (\ref{eqn:b-lower}) we see that
\begin{equation}
\label{ineq:bi}
b_i(p, t) \geq \sqrt{\frac{D_i}{8}} \: \text{ on } \: \Omega'_i
\end{equation}
As $D_i \rightarrow \infty$, the warping functions $b_i$ tend to infinity uniformly. As $b_i$ describes the size of the base $S^2$ in the Hopf fibration,intuitively one can see that this claim is true. Nevertheless, we provide a formal proof below:
As $g(t)\in \mathcal{I}_K$ for $t\in [0,T)$ we have
$$ \left|-\frac{b_{ss}}{b}\right| \leq \frac{K}{b^2} \: \text{ on } \: M_k \times [0, T).$$
Inspecting the curvature components listed in section \ref{con-lap-cur-subsec}, we see that all the curvature components of $g_i(t)$, apart from $R_{0101}$, tend to zero on $\Omega'_i$. Hence the curvature operator of $g_\infty(t)$ is of rank 1. Furthermore, as $g(0)$ has bounded curvature by the assumption that $g(0) \in \mathcal{I}$ we see that the scalar curvature $R_{g(t)}$ is pointwise bounded below by $\inf_{p \in M_k} R_{g(0)}(p) > - \infty$. Hence the blow-up limit $g_\infty(t)$ has non-negative scalar curvature, which in turn implies that the curvature operator is non-negative. By \cite[8.3. Theorem \& p. 178]{Ham86} we conclude that $(M_\infty, g_\infty(t))$ splits as a product $(\mathbb{R}^2 \times N, g_{eucl} + g_N(t))$. Note also that $N$ is diffeomorphic to the leafs of the distribution spanned by $e_0$ and $e_1$, as these are the only planes with non-flat sectional curvature. Recalling that $e_0=\frac{\partial}{\partial s}$ we see that the integral curves of $e_0$ are non-compact and therefore $N$ is non-compact as well.
\end{claimproof}
As $(M_\infty, g_{\infty}(t))$ is $\kappa$-non-collapsed at all scales, the above claim implies that $(N, g_N(t))$ is a 2d $\kappa$-solution. However, by Hamilton's work a two dimensional $\kappa$-solution is either the shrinking round sphere $S^2$ or its $\mathbb{Z}_2$ quotient \cite[\S 1 of Chapter 9]{CLN06}. Since $N$ is non-compact we have arrived at a contradiction. Therefore the desired result follows.
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor:curv-bound-ancient}]
The proof is the same as for Theorem \ref{curv-bound}. Since the Ricci flow is assumed to be $\kappa$-non-collapsed at all scales, we may also take blow-down limits and do not need to assume that $b$ is uniformly bounded. Furthermore, since ancient Ricci flows have non-negative scalar curvature, Claim 1 of the proof of Theorem \ref{curv-bound} also carries over.
\end{proof}
\section{Compactness properties}
In this section we prove some compactness properties of $U(2)$-invariant cohomogeneity one Ricci flows. For general Ricci flows the compactness properties are well-known \cite[Chapter 3]{ChI}. Therefore the main technical difficulty is to show that the $U(2)$-symmetry passes to the limit.
The main theorem of this section is Theorem \ref{thm:local-compactness} which roughly states the following compactness property: Let $(U_i, g_i(t), p_i)$, $[-\Delta t, 0]$, be a sequence of $U(2)$-invariant cohomogeneity one manifolds in the class $\mathcal{I}$ of metrics. Here the $U_i$ are open manifolds and assumed to compactly contain the sets $C_{g_i(0)}(p_i, r)$ (see Definition \ref{def:C}) for some fixed $r> 0$. This condition can be understood as requiring $U_i$ to have `radial diameter' of at least $r$. Furthermore the metrics $g_i(t)$ are normalized such that $b = 1$ at the points $(p_i, 0)$ in spacetime. We show that if the flows $g_i(t)$ are $\kappa$-non-collapsed and of uniformly bounded curvature, then $(U_i, g_i(t), p_i)$, $[-\Delta t, 0]$, subsequentially converges to a limiting $U(2)$-invariant Ricci flow $(\mathcal{C}_\infty, g_\infty(t), p_\infty)$. Moreover, if we correctly pick/normalize the coordinate $\xi$, the warping functions $u_i(\xi, t)$, $a_i(\xi,t)$ and $b_i(\xi,t)$ of the metrics $g_i(t)$ converge on compact parabolic sets in $C^\infty$ to the warping functions $u_\infty(\xi,t)$, $u_\infty(\xi, t)$ and $b_\infty(\xi,t)$ of $g_\infty(t)$. This in essence shows that when taking limits of $U(2)$-invariant Ricci flows, we may work with the warping functions only, without having to concern ourselves with the underlying manifold.
Theorem \ref{thm:local-compactness} has two important applications: Firstly, it implies the corresponding compactness result for complete Ricci flows. In particular, a sequence of uniformly bounded and non-collapsed $U(2)$-invariant cohomogeneity one Ricci flows $(M_k, g_i(t), p_i)$, $t \in [-t_i, 0]$, $t_i\rightarrow \infty$, normalized such that $b(p_i, 0) = 1$, subsequentially converges, in the Cheeger-Gromov sense, to a limiting Ricci flow $(M_\infty, g_\infty(t), p_\infty)$, $t \in [-\infty, 0]$, that is also $U(2)$-invariant and cohomogeneity one. Secondly, we prove a variant of Theorem \ref{thm:local-compactness} in Proposition \ref{blow-up-prop}, where we specialize to the case in which the `radial diameter' of the $U_i$ is equal to $\frac{1}{2}$. This will allow us to alter one assumption of Theorem \ref{thm:local-compactness} and yield a very useful tool for proving certain scale-invariant inequalities via a contradiction/compactness argument, as introduced in the outline of section 9 in section \ref{subsec:outline-paper} of this paper.
Below we state the main results of this section. For this recall Definition \ref{def:C} of $C_g(p,r)$, $C_g^+(p,r)$ and $\Sigma_p$.
\begin{thm} [Local compactness]
\label{thm:local-compactness}
Let $k \in \mathbb{N}$ and $\kappa, \rho, K, r, \Delta t > 0$. Assume that
$$(U_i, g_i(t), p_i), \quad t \in \left[-\Delta t, 0\right],$$
is a sequence of pointed cohomogeneity one $U(2)$-invariant Ricci flows satisfying the following properties:
\begin{enumerate}
\item $U_i$ is an open $U(2)$-invariant manifold with principal orbit $S^3/\mathbb{Z}_k$.
\item For $t \in [-\Delta t, 0]$ we have $g_i(t) \in \mathcal{I}$ (see Definition \ref{def:I}). Denote by $u_i$, $a_i$ and $b_i$ the warping functions of $g_i(t)$.
\item The closed sets $\overline{C_{g_i(0)}\left(p_i, r\right)} \subset U_i$ are compact.
\item $b_i(p_i,0) = 1$.
\item The Ricci flow $(U_i, g_i(t))$ is $\kappa$-non-collapsed at $(p_i, 0)$ at scale $\min(\rho, r, \sqrt{\Delta t})$.
\item $|Rm_{g_i(t)}|_{g_i(t)} \leq K$ in $U_i \times \left[- \Delta t, 0\right]$.
\end{enumerate}
Then $(C_{g_i(0)}\left(p_i, r \right), g_i(t), p_i)$, $t \in [-\Delta t, 0]$, subsequentially converges, in the Cheeger-Gromov sense, to a pointed Ricci flow
$$(\mathcal{C}_\infty, g_\infty(t), p_\infty), \quad t \in \left[- \Delta t, 0\right],$$
satisfying the following properties:
\begin{enumerate}[label=(\alph*)]
\item $\mathcal{C}_\infty$ is a cohomogeneity one $U(2)$-invariant manifold such that either
\begin{enumerate}[label=(\roman*)]
\item All orbits are principal: In this case $\mathcal{C}_\infty$ is diffeomorphic to the cylinder $\mathbb{R} \times S^3/\mathbb{Z}_k$ and we equip $\mathcal{C}_\infty$ with a radial coordinate $\xi: \mathcal{C}_\infty \rightarrow \mathbb{R}$ defined by $\xi(p) = d_{g_\infty(0)}\left(p, \Sigma_{p_\infty} \right)$.
\item There is exactly one non-principal orbit: In this case $\mathcal{C}_\infty$ is diffeomorphic to $M_k$ and we equip $\mathcal{C}_\infty$ with the radial coordinate $\xi: \mathcal{C}_\infty \rightarrow \mathbb{R}$ defined by $\xi(p) = d_{g_\infty(0)}\left(p, S^2_o\right)$.
\end{enumerate}
\item There exist warping functions
$$u_\infty, a_\infty, b_\infty: C_{g_\infty(0)}\left(p_\infty, r \right) \times [-\Delta t, 0] \rightarrow \mathbb{R}_{\geq 0}$$
such that the metric $g_\infty(t)$, $t\in \left[-\Delta t,0 \right]$, is of the form (\ref{metric1}) and in the class $\mathcal{I}$
\item Choosing the coordinate $\xi$ on $(U_i, g_i(t), p_i)$ corresponding to whether we are in case (i) or (ii) above, the warping functions $u_i(\xi,t)$, $a_i(\xi, t)$ and $b_i(\xi,t)$ converge on compact sets to $u_\infty(\xi,t)$, $a_\infty(\xi,t)$ and $b_\infty(\xi, t)$.
\item For every $r' < r$ the closed set $\overline{C_{g_\infty(0)}(p_\infty, r')} \subset \mathcal{C}_\infty$ is compact.
\end{enumerate}
\end{thm}
From Theorem \ref{thm:local-compactness} the following corollary follows immediately:
\begin{cor}[Compactness of complete Ricci flows]
\label{cor:compactness-complete-flows}
Let $k \in \mathbb{N},$ $\kappa, K >0$ and $r_i, t_i, \rho_i \rightarrow \infty$ as $i \rightarrow \infty$. Assume that $(\mathcal{M}_i, g_i(t), p_i)$, $t \in [-t_i, 0]$, is a sequence of pointed $U(2)$-invariant cohomogeneity one Ricci flows satisfying:
\begin{enumerate}
\item For $t \in [-t_i, 0]$ we have $g_i(t) \in \mathcal{I}$. Denote by $u_i$, $a_i$ and $b_i$ the warping functions of $g_i(t)$.
\item $\overline{C_{g_i(0)}(p_i, r_i) } \subset \mathcal{M}_i$ is compact.
\item $b(p_i,0) = 1$
\item $g_i(t)$ is $\kappa$-non-collapsed at scale $\rho_i$
\item $|Rm_{g_i(t)}|_{g_i(t)} \leq K$ on $\mathcal{M}_i \times [-t_i, 0]$
\end{enumerate}
Then $(\mathcal{M}_i, g_i(t), p_i)$, $t\in [-t_i, 0]$, subsequentially converges, in the Cheeger-Gromov sense, to a pointed complete ancient Ricci flow $(\mathcal{M}_\infty, g_\infty(t), p_\infty)$, $ t \in (-\infty, 0]$, with bounded curvature satisfying properties (a) - (d) of Theorem \ref{thm:local-compactness}, when taking $\mathcal{C}_\infty = \mathcal{M}_\infty$ and $r = \infty$.
\end{cor}
\begin{proof}
This follows from Theorem \ref{thm:local-compactness} by a diagonal argument.
\end{proof}
The following proposition is a variant of Theorem \ref{thm:local-compactness} in the case we take $r=\frac{1}{2}$.
\begin{prop}
\label{blow-up-prop}
Let $k \in \mathbb{N}$, $\kappa, \rho, C_1 > 0$, $r = \frac{1}{2}$ and $\Delta t \in (0, \frac{1}{48C_1}]$. Assume
$$(U_i, g_i(t), p_i), \quad t \in \left[-\Delta t, 0\right],$$
is a sequence of pointed $U(2)$-invariant cohomogeneity one Ricci flows satisfying conditions (1)-(5) of Theorem \ref{blow-up-prop}. If, instead of condition (6) of Theorem \ref{thm:local-compactness}, we require
\begin{enumerate}[label=(6')]
\item $|Rm_{g_i(t)}|_{g_i(t)} \leq \frac{C_1}{b^2}$ on $U_i \times [-\Delta t, 0]$
\end{enumerate}
then
$$\left(C_{g_i(0)}\left(p_i, \frac{1}{2}\right), g_i(t), p_i\right), \quad t \in [-\Delta t, 0],$$
subsequentially converges, in the Cheeger-Gromov sense, to a pointed Ricci flow
$$(\mathcal{C}_\infty, g_\infty(t), p_\infty), \quad t \in \left[- \Delta t, 0\right],$$
satisfying the same properties (a)-(d) listed in Theorem \ref{blow-up-prop}.
\end{prop}
\begin{proof}[Proof of Proposition \ref{blow-up-prop}]
For brevity we write $\Omega_ i = C_{g_i(0)}\left( p_i, \frac{1}{2}\right) \times [-\Delta t,0]$. Let $a_i$ and $b_i$ denote the warping functions of $g_i(t)$.
As $g_i(t) \in \mathcal{I}$ we have
$$0 \leq (b_i)_s \leq Q_i \leq 1 \: \text{ in } \Omega_i,$$
where $Q_i = \frac{a_i}{b_i}$ and thus
$$b_i(p, 0) \geq \frac{1}{2} \: \text{ for } p \in C_{g_i(0)}\left( p_i, \frac{1}{2}\right)$$
as $b_i(p_i, 0) = 1$ by assumption. By the Ricci flow equation we have
\begin{align*}
\partial_t b_i^2 &= - 2b^2 \left(R_{0202}+ R_{1212} + R_{2323}\right) \\
& \leq 6 C_1 \: \text{ on } \Omega_i.
\end{align*}
This implies
\begin{equation*}
b_i(p,t) \geq \frac{1}{\sqrt{8}} \: \text{ for } (p,t)\in \Omega_i,
\end{equation*}
as $\Delta t \leq \frac{1}{48C_1}$ by assumption. This yields the uniform curvature bound
\begin{equation*}
|Rm(g_i)|_{g_i} \leq 8C_1 \: \text{ on } \Omega_i.
\end{equation*}
The result now follows from Theorem \ref{thm:local-compactness}.
\end{proof}
The main proof idea of Theorem \ref{thm:local-compactness} is to construct a set of four Killing vector fields $\overline{X}_j$, $j = 1, 2, 3, 4$, generated by the $U(2)$-action on each $(U_i, g_i(t))$, and show that these Killing vector fields pass to the limit $(\mathcal{C}_\infty, g_\infty(t))$. This allows us to reconstruct the $U(2)$-action on $\mathcal{C}_\infty$, proving the desired result. The main difficulty, however, is to show that the orbits corresponding to the flows of the Killing vector fields do not degenerate in the limit and thereby ensure that the full $U(2)$ symmetry group is preserved. For this we will rely on Lemma \ref{lem:non-collapsed-Q} below, where we prove that $\kappa$-non-collapsedness implies a lower positive bound on $Q$ away from a non-principal orbit.
\begin{lem}
\label{lem:non-collapsed-Q}
Let $k \in \mathbb{N}$, $r_0 \in (0, 1]$ and $\kappa, C_1, c > 0$. Assume that $(M,g)$ is a $U(2)$-invariant cohomogeneity one manifold with principal orbit $S^3 /\mathbb{Z}_k$ equipped with a metric $g \in \mathcal{I}$. Take $p \in M$. If
\begin{enumerate}
\item The set $\overline{C^+_g(p,b(p)r_0)}$ (See Definition \ref{def:I}) is compactly contained in $M$
\item $|Rm_{g}|_{g} \leq \frac{C_1}{b^2}$ on $\overline{C^+_g(p,b(p)r_0)}$
\item $g$ is $\kappa$-non-collapsed at scale $cb(p)$: If for $r \leq c b(p)$ the ball $B_g(p,r)$ is compactly contained in $M$ and $|Rm_g|_g < r^{-2}$ on $B_g(p,r)$ then $vol(B_g(p,r)) \geq \kappa r^4$
\end{enumerate}
then there exists an $\epsilon > 0$ depending on $k, \kappa, C_1 , c$ and $r_0$ for which the following holds: If for $q \in C^+_g(p,b(p)r_0)$ the set $C_g(q, b(p) \frac{r_0}{4})$ is compactly contained in $C^+_g(p,b(p)r_0)$ then $Q(q) \geq \epsilon$.
\end{lem}
\begin{proof}
By rescaling we may assume without loss of generality $b(p) =1$ and that the metric $g$ is $\kappa$-non-collapsed at scale $c>0$. The latter follows from the fact that if $g$ is $\kappa$-non-collapsed at scale $\rho$ then $\alpha^2 g$ is $\kappa$-non-collapsed at scale $\alpha \rho$. Fix a $q \in C^+_g(p,b(p)r_0)$ such that the assumptions of the lemma hold. Take $U \coloneqq C_g(q, b(p) \frac{r_0}{4})$. Note that $U$ is a union of orbits of the $U(2)$-action. Recall that non-principal orbits are non-generic and characterized by $a = 0$. As $a_s \geq 0$ we see that all the orbits of $U$ are principal and therefore diffeomorphic to $S^3/\mathbb{Z}_k$. Because $0 \leq b_s \leq Q \leq 1 $ for metrics in $\mathcal{I}$ we see that
$$1 \leq b \leq 2 \text{ in } C^+_g(p,r_0)$$
and hence
$$|Rm_g|_g \leq C_1 \text{ in } C^+_g(p,r_0)$$
by assumption (2). From the expression
$$M_2 = \frac{1}{b^2}\left( a_s - Q b_s \right)$$
for the curvature component $R_{0231}$ derived in section \ref{con-lap-cur-subsec} and the fact that
$$Q_s = \frac{1}{b} \left( a_s - Q b_s\right)$$
we deduce that
$$ |Q_s| \leq C_1 \text{ in } C^+_g(p,r_0).$$
Thus for $r \leq r_1 \coloneqq \min\left(\frac{r_0}{4}, \frac{Q(q)}{C_1}\right)$ we have
$$ Q \leq 2Q(q) \: \text{ on } C_g(q, r_1).$$
\begin{claim}
For $r \leq r_2 \coloneqq \min\left(\frac{1}{100}, r_1\right)$ we have $$ vol(B_g(q,r)) \leq C r^3 Q(q) $$
for some constant $C> 0$ depending on $k$ only.
\end{claim}
\begin{claimproof}
Let $q' \in C_g(q, r)$, $r < r_2$. Then $\Sigma_{q'}$ is isometric to $S^3/\mathbb{Z}_k$ equipped with a squashed Berger metric. In particular, if we denote by $\iota: \Sigma_{q'} \rightarrow M$ the inclusion, then
$$\iota^\ast g = a(q')^2 \omega \otimes \omega + b(q')^2\pi^\ast(g_{FS}),$$
where $g_{FS}$ is the Fubini-Study metric on $S^2$ normalized to have curvature equal to $\frac{1}{4}$ and $\pi: S^3 / \mathbb{Z}_k \rightarrow S^2$ is the Hopf fibration. Note that
$$\Sigma_{q'} \cap B_g(q,r) \subseteq \Sigma_{q'} \cap B_g(q',r) \subseteq \pi^{-1}(B_{g_{FS}}(\pi(q'), r)) \subseteq \Sigma_{q'}.$$
Furthermore, as the Hopf fibers of $\Sigma_{q'} \cong S^3/\mathbb{Z}_k$ have length $\frac{2\pi}{k} a(q')$, we see that
$$vol(\pi^{-1}(B_{g_{FS}}(\pi(q'), r))) = \frac{2\pi}{k} a(q') vol(B_{g_{FS}}(\pi(q'), r)) \leq C a(q') r^2,$$
for some constant $C>0$ depending on $k$ only. Since $Q = \frac{a}{b}$, $Q \leq 2 Q(q)$ and $b \in [1, 2]$ in $C_g(q, r_1)$ it follows that
$$ vol(\Sigma_{q'} \cap B_g(q,r)) \leq 4 C Q(q) r^2.$$
Integrating this inequality proves the claim.
\end{claimproof}
As $|Rm_g|_g \leq C_1$ on $C^+_g(p,r_0)$, the ball $B_g(q,\frac{r_0}{4})$ is compactly contained in $M$, and $g$ is $\kappa$-non-collapsed at scale $c$, we see that for $r \leq r_3 \coloneqq \min\left(\frac{1}{\sqrt{C_2}}, c, \frac{r_0}{4}\right)$
$$vol(B_g(q,r)) \geq \kappa r^4.$$
Setting $r_4 \coloneqq \min\left(r_2, r_3\right)$ we therefore obtain
$$C r_4^3 Q(q) \geq vol(B_g(q,r_4)) \geq \kappa r_4^4.$$
Rearranging this inequality proves the lemma.
\end{proof}
Before proving the compactness theorems listed above, we construct a set of four Killing vector fields on a general $U(2)$-invariant cohomogeneity one manifold $M$ with principal orbit $S^3/\mathbb{Z}_k$. By passing to the universal cover we may assume without loss of generality that $k = 1$. Pick the basis
$$ X_0 = \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix} \qquad X_1 = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} \qquad X_2 = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} \qquad X_3 = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} $$
for the Lie algebra of $U(2)$. Then $X_i$, $i = 1, 2, 3, 4$, satisfy the commutation relations
$$[X_1,X_2] = 2 X_3 \qquad [X_2,X_3] = 2 X_1 \qquad [X_3,X_1] = 2 X_2.$$
and
$$[X_0, X_i] = 0 \: \text{ for } \: i = 1, 2, 3.$$
Extend $X_i$, $i = 1, 2, 3, 4$, to left-invariant vector fields on $U(2)$. Note that the integral curves generated by $X_i$, $i = 1, 2, 3, 4$, have period $2\pi$. The $U(2)$-action generates four corresponding Killing vector fields $\overline{X}_i$, $i = 1, 2, 3, 4$, on $M_k$ by taking
$$\overline{X}_i(p) = \frac{d}{dt} \Big |_{t=0} \exp(t X_i) \cdot p, \quad p \in M_k,\:i = 1,2,3,4.$$
We now prove the following:
\begin{lem}
\label{lem:killing}
For $i = 1, 2, 3, 4$ we have
$$|\overline{X}_i|_{g} \leq \max(a,b).$$
\end{lem}
\begin{proof}
By the form (\ref{metric2}) of the metric we see that $|\overline{X}_0|_g = a$. Hence we only need to prove the result for $i = 1, 2, 3$. First note that the vector fields $\overline{X}_i$, $i = 1, 2, 3$, are orthogonal to $\frac{\partial}{\partial s}$ and therefore parallel to the orbits of the $U(2)$ action on $M_k$. Hence it suffices to study the metric $g$ restricted to these directions. Here we see that
$$a^2 \omega \otimes \omega + b^2 \pi^\ast(g_{FS}) \leq \max(a,b)^2 g_{S^3},$$
where $g_{S^3}$ is the round metric on $S^3$ with sectional curvatures equal to 1. Thus it suffices to show that
$$|\overline{X}_i|_{g_{S^3}} \leq 1.$$
If we identify $S^3$ with $SU(2)$, the vectors $\overline{X}_i$ correspond to right-invariant vector fields on $SU(2)$. Moreover, one can check that these vector fields are orthonormal with respect to the metric $g_{S^3}$. Hence the desired result follows.
\end{proof}
\begin{remark}
In fact one can show that $\min(a,b) \leq |\overline{X}_i|_{g} \leq \max(a,b)$. Recalling that the isometry generated by the Killing vector field $\overline{X}_i$ descends to a rotation of the base $S^2$ in the Hopf fibration $\pi: S^3 \rightarrow S^2$, one can see that the upper bound is attained on $\pi^{-1}(\{\text{Equator of }S^2\})$ and the lower bound is attained on $\pi^{-1}(\{N,S\})$, where $N$, $S$ denote the north and south pole with respect to the rotation induced by $\overline{X}_i$.
\end{remark}
Now we proceed to proving the main theorem of this section:
\begin{proof}[Proof of Theorem \ref{thm:local-compactness}]
As $g_i(t)$ is $\kappa$-non-collapsed at $(p_i,0)$ at scale $\min(\rho, r, \sqrt{\Delta t})$ it follows from \cite[Lemma 6.54]{ChI} that there exists a uniform $\delta >0$ such that
$$\mathrm{inj}_{g_i(0)}(p_i) > \delta.$$
By assumption $g_i(t)$ has bounded curvature on the parabolic neighborhood
$$\Omega_i \coloneqq C_{g_i(0)}\left(p_i, r\right) \times [-\Delta t, 0].$$
By an adaptation of \cite[Theorem 3.16]{ChI} we therefore deduce that after passing to a subsequence
$$\left(C_{g_i(0)}\left( p_i, r\right), g_i(t), p_i\right), \quad t \in [-\Delta t, 0],$$
converges, in the Cheeger-Gromov sense, to a pointed Ricci flow
$$\left(\mathcal{C}_{\infty}, g_{\infty}(t), p_{\infty}\right), \quad t \in [-\Delta t, 0],$$
where $\mathcal{C}_\infty$ is an open manifold.
\begin{claim}
$(\mathcal{C}_{\infty}, g_{\infty}(t))$, $ t \in [-\Delta t, 0]$, is $U(2)$-invariant.
\end{claim}
\begin{claimproof}
Recall the construction of the Killing vector fields $\overline{X}_j$, $j = 1, 2, 3, 4$ for a general $U(2)$-invariant manifold $M$ explained above. Let $\overline{X}_{ij}$, $i \in \mathbb{N}$, $j = 1, 2, 3, 4$, denote the corresponding Killing vector fields on the manifolds $C_{g_i(0)}\left(p_i, r\right)$. Recall that $g_i(t) \in \mathcal{I}$ implies that $0 \leq b_s \leq Q \leq 1$. Therefore $b \leq r+1$ on $C_{g_i(0)}\left(p_i, r\right)$. Note that from the evolution equation (\ref{b-evol}) of $b$ it follows that
$$\left|\frac{\partial_t b}{b}\right| \leq c |Rm_{g_i(t)}|_{g_i(t)},$$
where $c > 0$ is some universal constant. As $|Rm_{g_i(t)}|_{g_i(t)} \leq C_1$ on $\Omega_i$ by assumption, we see that there exists a $C>0$, depending on $r$ and $C_1$ only, such that $b \leq C$ on $\Omega_i$. From Lemma \ref{lem:killing} it hence follows that for $i \in \mathbb{N}$, $j = 1, 2, 3, 4$,
$$ |\overline{X}_{ij}|_{g_i(t)} \leq C \: \text{ on } \Omega_i.$$
Recall that in general a Killing vector field $X^a$ on a manifold satisfies the relation
$$ \nabla_a\nabla_b X^c = -R^c_{\:abd} X^d.$$
Therefore we see that the Killing vector fields $\overline{X}_{ij}$ are uniformly bounded in $C^2(\Omega_i)$, and converge to $C^1$ Killing vector fields $\overline{X}_{\infty, j}$, $j = 1, 2, 3,4$, on $\left(\mathcal{C}_{\infty}, g_{\infty}(t)\right)$ after passing to a subsequence. However, since the group of isometries of a smooth manifold is a smooth Lie group, the vector fields $\overline{X}_{\infty, j}$, $j = 1, 2, 3,4$ are in fact smooth. As the Killing vector fields $\overline{X}_{ij}$, $i \in \mathbb{N}$, $j = 1,2, 3,4$, are complete, so are $\overline{X}_{\infty, j}$, $j = 1, 2, 3,4$. Integrating the Killing vector fields $\overline{X}_{\infty, j}$, $j = 1,2,3,4$, then yields the desired $U(2)$-action on $(\mathcal{C}_\infty, g_\infty(t))$.
\end{claimproof}
It remains to be shown that this action is faithful by proving that the Killing vector fields are non-zero at times $t \in [-\Delta t, 0]$.
\begin{claim}
The $U(2)$-action on $(\mathcal{C}_{\infty}, g_{\infty}(t))$, $ t \in [-\Delta t, 0]$ is faithful.
\end{claim}
\begin{claimproof}
Take $r_1 > 0$ such that $C_{g_\infty(t)}(p_\infty, r_1)$ is compactly contained in $\mathcal{C}_\infty$ for all $t \in [-\Delta t, 0]$. This is possible by standard distance distortion estimates and the fact that $\mathcal{C}_{\infty} \times [-\Delta t, 0]$ has bounded curvature. Furthermore, since $C_{g_\infty(t)}(p_\infty, r_1) \times [-\Delta t, 0]$ is compactly contained in $\mathcal{C}_\infty \times [-\Delta t, 0]$, there exist constants $\rho', \kappa' > 0$ such that for each $t \in [-\Delta t, 0]$ the manifold $(C_{g_\infty(t)}(p_\infty, r_1), g_\infty(t))$ is $\kappa'$-non-collapsed at scale less or equal to $\rho'$. Since $(C_{g_i(t)}(p_i, r), g_i(t), p_i)$ converges in the Cheeger-Gromov sense to $(\mathcal{C}_\infty, g_\infty(t), p_\infty)$ we see that eventually $(C_{g_i(t)}(p_i, r_1), g_i(t))$ is $\kappa'/2$-non-collapsed at scales less or equal to $\rho'/2$.
Fix $t' \in [-\Delta t, 0]$ and choose points $q_i \in \Sigma^+_{p_i}$ (see Definition \ref{def:C}) and $q_\infty \in \mathcal{C}_\infty$ with $d_{g_i(t')}(q_i, \Sigma^+_{p_i}) = \frac{1}{2}r_1$ and $q_i \rightarrow q_\infty$. Checking the conditions of Lemma \ref{lem:non-collapsed-Q}, we see that there exists an $\epsilon > 0$, independent of $i$, such that
$$Q(q_i,t') \geq \epsilon.$$
As $g \in \mathcal{I}$ and therefore $0 \leq b_s \leq Q \leq 1$, we see that
$$1 \leq b(q_i, t') \leq \frac{3}{2}$$
Therefore the geometry of the orbit $\Sigma_{q_i} \cong S^3 /\mathbb{Z}_k$ is controlled --- the curvature and diameter are uniformly bounded from above, and its volume and Hopf fiber lengths are uniformly bounded away from zero. Hence the norms of the Killing vector fields $\overline{X}_{ij}$, $j = 1, 2, 3, 4$, at the points $(q_i, t')$ in spacetime are uniformly bounded away from zero, proving that on the limiting space $(\mathcal{C}_\infty, g_\infty(t'))$ the Killing vector fields $\overline{X}_{\infty, j}$, $j = 1, 2,3,4$, are non-zero. As $t' \in [\Delta t, 0]$ was arbitrary, the desired result follows.
\end{claimproof}
By the slice theorem we we see that either (i) all orbits of $\mathcal{C}_\infty$ are principal and diffeomorphic to $S^3/\mathbb{Z}_k$ or (ii) there exists exactly one non-principal orbit, which is diffeomorphic to $S^2$ and as usual we denote by $S^2_o$. Below it will become clear why $\mathcal{C}_\infty$ cannot possess two non-principal orbits. In case (i) $\mathcal{C}_\infty$ is diffeomorphic to the manifold $\mathbb{R} \times S^3/\mathbb{Z}_k$ and in case (ii) it is diffeomorphic to $M_k$. In both cases there is a dense open set of the form $\mathbb{R} \times S^3/\mathbb{Z}_k \subset \mathcal{C}_\infty$.
We now show that the metrics $g_\infty(t)$ can be expressed in the form (\ref{metric2}). Denote the warping functions of the metrics $g_i(t)$ by $a_i$ and $b_i$. In case (i) we define the radial coordinates
$$\xi_i(p) = \pm d_{g_i(0)}(p, \Sigma_{p_i}),$$
and
$$\xi_\infty(p) = \pm d_{g_\infty(0)}(p, \Sigma_{p_\infty})$$
on $C_{g_i(0)}\left(p_i, r\right)$ and $\mathcal{C}_\infty$, respectively. We choose the sign of $\xi_i(p)$ depending on which side of the hypersurface $\Sigma_{p_i}$ the point $p$ lies, and in such a way that $\partial_{\xi_i} a_i , \partial_{\xi_i} b_i \geq 0$. The sign of $\xi_\infty(p)$ is chosen such that $\xi_i \rightarrow \xi_\infty$ as $i \rightarrow \infty$. In case (ii) we may assume without loss of generality that for all $i \in \mathbb{N}$ the open manifolds $C_{g_i(0)}\left(p_i, r\right)$ contain a point $o_i$ such that the orbit $\Sigma_{o_i}$ is non-principal and $o_i \rightarrow o_\infty \in \mathcal{C}_\infty$ as $i \rightarrow \infty$. Then define radial coordinates
$$\xi_i(p) = d_{g_i(0)}(p, \Sigma_{o_i})$$
and
$$\xi_\infty(p) = d_{g_\infty(0)}(p, \Sigma_{o_\infty})$$
on $C_{g_i(0)}\left(p_i, r\right)$ and $\mathcal{C}_\infty$. Note that the coordinates $\xi_i$ and $\xi_\infty$ are smooth away from a non-principal orbit and furthermore that $\xi_i \rightarrow \xi_\infty$ in $C^\infty$ away from a non-principal orbit. Hence we obtain the one-forms $d\xi_i$ and $d\xi_\infty$ away from a non-principal orbit, which are orthogonal to all orbits of $C_{g_i(0)}\left(p_i, r\right)$ and $\mathcal{C}_\infty$, respectively. For brevity we drop the subscript and write $\xi$ for the coordinates $\xi_i$ or $\xi_\infty$.
Since the metric $g_\infty(t)$ is $U(2)$-invariant, as shown above, there exists warping functions $u_\infty, a_\infty, b_\infty: \mathcal{C}_\infty \times [-\Delta t, 0] \rightarrow \mathbb{R}_{\geq 0}$ such that the metric can be expressed as
$$g_\infty(t) = u^2_\infty(\xi, t) d\xi^2 + a^2_\infty(\xi,t) \omega\otimes\omega + b_\infty^2(\xi, t) \pi^\ast(g_{FS}),$$
where at time $0$ we have
$$u = 1 \: \text{ on } \: \mathcal{C}_\infty.$$
As
$$a_\infty(p,t) = |\overline{X}_{\infty, 0}|_{g_\infty(t)}(p)$$
and $\overline{X}_{i,o} \rightarrow \overline{X}_{\infty,0}$ as $i \rightarrow \infty$ by above, we see that away from a non-principal orbit $a_i \rightarrow a_\infty$ smoothly. Similarly, one can show with help of the remaining Killing vector fields $\overline{X}_{i,j}$, $j = 1, 2, 3$, that away from a non-principal orbit $b_i \rightarrow b_\infty$ smoothly.
Hence away from a non-principal orbit,$a_i(\xi, t), b_i(\xi, t) \rightarrow a_\infty(\xi, t), b_\infty(\xi, t)$ in $C^\infty$ as $i \rightarrow \infty$. Furthermore, from the curvature bounds on $\Omega_i$ and the boundary conditions on $a_i$, $b_i$, $a_\infty$, $b_\infty$ at a non-principal orbit (see section \ref{manifold-metric-subsec} for the smoothness conditions on the warping functions at the non-principal orbit), one can show that in fact $a_i(\xi, t), b_i(\xi, t) \rightarrow a_\infty(\xi, t), b_\infty(\xi, t)$ smoothly everywhere. Hence the metric $g_\infty(t)$, $t \in [-\Delta t,0]$, is in the class $\mathcal{I}$. As $a_s \geq 0$ for metrics in $\mathcal{I}$ we see that $\mathcal{C}_\infty$ can possess at most one non-principal orbit. Finally, we note that by \cite[Theorem 3.16]{ChI} the closed set $\overline{C_{g_{\infty(0)}}(p_\infty, r')} \subset \mathcal{C}_\infty$ is compact for every $r' < r$.
\end{proof}
\section{Ancient Ricci flows Part I}
In this section we prove some properties of ancient Ricci flows $g(t) \in \mathcal{I}$, $-\infty < t \leq 0$, that are \emph{non-collapsed at all scales}. This yields important geometric information on the blow-up limits of singular Ricci flows, which we exploit and refine in later chapters. The main goal is to prove the following theorem:
\begin{thm}
\label{T-ancient}
Let $\kappa >0$ and $(M_k, g(t))$, $k \geq 2$, $t \in (-\infty,0]$, be an ancient Ricci flow, which satisfies the following properties:
\begin{enumerate}[label=(\roman*)]
\item $\kappa$-non-collapsed at all scales
\item $g(t) \in \mathcal{I}$ for $t \in (-\infty,0]$.
\end{enumerate}
Then if $k = 2$ the following inequalities hold:
\begin{align*}
T_1 &= a_s + 2Q^2 - 2 \geq 0 \\
T_2 &= Q y -x \geq 0 \\
T_3 &= a_s - Q b_s - Q^2 + 1 \geq 0
\end{align*}
If $k > 2$ we only have $T_1 \geq 0 $ and $T_3 \geq 0$. For all $k \geq 2$ we have $T_1(p,t) = 0$ if, and only if, $k =2$ and $p \in S_o^2$.
\end{thm}
Furthermore we show
\begin{thm}
\label{kahler-ancient-thm}
Let $\kappa >0$ and $(M_2,g(t))$, $t \in (-\infty,0]$, be an ancient Ricci flow, which satisfies the following properties:
\begin{enumerate}[label=(\roman*)]
\item $\kappa$-non-collapsed at all scales
\item $g(t) \in \mathcal{I}$ for $t \in (-\infty, 0]$
\item K\"ahler with respect to $J_1$, or equivalently $y=0$ everywhere
\end{enumerate}
Then $(M_2,g(t))$ is stationary and homothetic to the Eguchi-Hanson space.
\end{thm}
\noindent\textbf{Proof strategy.}
In both of these theorems we are given an ancient Ricci flow $(M,g(t))$, $t \leq 0$, and want to show that a scale invariant quantity $T$ satisfies
$$T \geq 0 \: \text{ on } \: M \times (-\infty, 0].$$
We prove such statements by a contradiction/compactness argument. First we assume that
$$\iota \coloneqq \inf_{M_k \times (-\infty,0]} T < 0$$
and take a sequence of points $(p_i, t_i)$ in spacetime such that
$$T(p_i, t_i) \rightarrow \iota \: \text{ as } \: i \rightarrow \infty.$$
Then we consider the rescaled Ricci flows
$$g_i(t) = \frac{1}{b^2(p_i,t_i)} g\left( t_i + b^2(p_i, t_i) t \right), \quad t \in [-\Delta t, 0],$$
where $\Delta t > 0$ is chosen such that the conditions of Proposition \ref{blow-up-prop} are met. Then $(C_{g_i(0)}\left(p_i, \frac{1}{2}\right), g_i(t), p_i)$ subsequentially converges to a Ricci flow $(\mathcal{C}_\infty, g_\infty(t), p_\infty)$, $t \in [-\Delta t, 0]$. By construction
$$T(p_\infty, 0) = \inf_{\mathcal{C}_\infty\times [-\Delta t, 0]} T = \iota < 0.$$
However, if the evolution equation of $T$ precludes the possibility of a negative infimum being attained, we have arrived at a contradiction and proven the desired result.
\noindent\textbf{Proof of main theorems of this section.} Before proving Theorem \ref{T-ancient} we need to state a technical lemma in preparation:
\begin{lem}
\label{away-from-origin-lem}
Let $(M_k, g(t))$, $k \in \mathbb{N}$, $t \leq 0$, be an ancient Ricci flow satisfying the conditions of Theorem \ref{T-ancient}. Then for every $\epsilon > 0$ there exists a $\delta > 0$ such that whenever at a point $(p, t)$ in spacetime one of the following inequalities holds
\begin{enumerate}[label=(\roman*)]
\item $T_1(p, t) \leq - \epsilon$ and $k \geq 2$
\item $T_2(p, t) \leq -\epsilon$ and $k \leq 2$
\item $T_3(p, t) \leq - \epsilon$ and $k \in \mathbb{N}$
\item $|x(p, t)| \geq \epsilon$ and $k = 2$
\end{enumerate}
then $s(p, t) \geq \delta b(p,t)$.
\end{lem}
\begin{proof}
Recall that by Corollary \ref{cor:curv-bound-ancient} there exists a $C_1 > 0$ such that $|Rm_{g(t)}|_{g(t)} \leq \frac{C_1}{0}$ on $M_k \times (-\infty, 0]$. We first prove (i). We fix $\epsilon >0$ and argue by contradiction. Assume there exists a sequence of points $(p_i, t_i)$ in spacetime such that
$$T_1(p_i, t_i) \leq - \epsilon$$
and
\begin{equation}
\label{eqn:dist-from-origin}
\frac{s(p_i,t_i)}{b(p_i, t_i)} \rightarrow 0.
\end{equation}
Define the rescaled metrics
$$g_i = \frac{1}{b^2(p_i, t_i)} g\left(t_i + t b^{2}(p_i, t_i) \right), \: t \in [-\Delta t, 0].$$
For sufficiently small $\Delta t > 0$ the conditions of Proposition \ref{blow-up-prop} are satisfied and hence $( C_{g_i(0)}\left(p_i, \frac{1}{2}\right), g_i(t), p_i)$ subsequentially converges to a Ricci flow $(\mathcal{C}_\infty, g_\infty(t), p_\infty)$. By (\ref{eqn:dist-from-origin}) one sees that $p_\infty$ lies on the non-principal orbit $S^2_o$ of $\mathcal{C}_\infty$. By construction we have $T_1(p_\infty, 0) \leq - \epsilon$ as $T_1$ is a scale invariant quantity. This however contradicts the fact that $T_1 = a_s + 2\left(Q^2 -1\right) = k - 2 \geq 0$ on $S^2_o$.
Note that $T_2 = 2 - k$, $T_3 = k + 1$ and $x = k - 2$ on $S^2_o$. Therefore by the same argument applied to $T_2$, $T_3$ and $x$ the desired result holds true.
\end{proof}
Next we prove Theorem \ref{T-ancient}.
\begin{proof}[Proof of Theorem \ref{T-ancient}]
Recall that by Corollary \ref{cor:curv-bound-ancient} there exists a $C_1 > 0$ such that $|Rm_{g(t)}|_{g(t)} \leq \frac{C_1}{0}$ on $M_k \times (-\infty, 0]$.
We first show that $T_1 \geq 0$ in $M_k \times (-\infty, 0]$. We argue by contradiction. Assume that
\begin{equation*}
\iota := \inf_{M_k\times(-\infty, 0]} T_1 < 0.
\end{equation*}
As $g(t) \in \mathcal{I}$ we know that $\iota> - \infty$. Take a sequence of points $(p_i, t_i)$ in spacetime such that
\begin{equation*}
T_1(p_i, t_i) \rightarrow \iota \: \text{ as } \: i \rightarrow \infty.
\end{equation*}
From Lemma \ref{away-from-origin-lem} it follows that for sufficiently large $i$
\begin{equation}
\label{away-from-origin}
s(p_i, t_i) \geq \delta b(0,t_i)
\end{equation}
for some $\delta > 0$. Define the rescaled metrics
$$g_i = \frac{1}{b^2(p_i, t_i)} g\left(t + t_i b^2(p_i, t_i)\right), \quad t \in [-\Delta t, 0].$$
For sufficiently small $\Delta t > 0$ the conditions of Proposition \ref{blow-up-prop} are satisfied and hence $(C_{g_i(0)}\left(p_i, \frac{1}{2}\right), g_i(t), p_i)$ subsequentially converges to a Ricci flow $(\mathcal{C}_\infty, g_\infty(t), p_\infty)$, $t \in [-\Delta t, 0]$, on which by construction
\begin{equation*}
b(p_{\infty},0) = 1.
\end{equation*}
and
\begin{equation}
\label{T1-inf}
T_1(p_{\infty},0) = \inf_{\mathcal{C}_\infty \times [-\Delta t, 0]} T_1 = \iota < 0,
\end{equation}
as $T_1$ is a scale invariant quantity. Since $T_s(p_\infty, 0) = 0$, we see from the evolution equation (\ref{T1-evol}) of $T_1$ that
\begin{align*}
\partial_t T_1 \Big |_{(p_\infty, 0)} &= L[T_1] + \frac{1}{b^2} \left[- 4\left(1+Q^2\right)y^2 + 8Q\left(1-2Q^2 \right)y + 16Q^2\left(1-Q^2\right)\right] \\
& \qquad + \frac{2 y T_1}{b^2} \left( 2Q- y\right) \\ \nonumber
& \geq (T_1)_{ss} + \frac{4Q^2}{b^2}\left(1- Q^2\right) + \frac{2 y T_1}{b^2} \left( 2Q- y\right),
\end{align*}
where we bounded the zeroth order term from below as in the proof of Lemma \ref{T1-preserved-lem}. Hence
\begin{equation*}
\partial_t T_1 \Big|_{(p_{\infty},0)} > 0
\end{equation*}
unless
$$\text{case b)}: \quad Q(p_{\infty},0) = 0 \text{ and } y(p_{\infty},0) = 0$$
or
$$\text{case a)}: \quad Q(p_{\infty},0) = 1 \text{ and } y(p_{\infty},0) = 0$$
However by (\ref{T1-inf}) we have
\begin{equation*}
\partial_t T_1 \Big|_{(p_{\infty},0)} \leq 0.
\end{equation*}
showing that either case a) or case b) must hold. We now show that both of these cases are impossible, thereby arriving at a contradiction. First note that by (\ref{away-from-origin}) we know that $p_\infty$ does not lie on the non-principal orbit $S^2_o$. Therefore the strong maximum principle applied to the evolution equation (\ref{Q-evol}) of $Q$ shows that in case a) $Q= 0$ everywhere on $\mathcal{C}_\infty \times [-\Delta t, 0]$. This, however, contradicts the non-collapsedness of $\mathcal{C}_\infty$ and therefore case a) cannot occur. In case b) the same argument shows that $Q=1$ everywhere on $\mathcal{C}_\infty\times [-\Delta t, 0]$. Then applying the strong maximum principle to the evolution equation (\ref{Qy-evol}) of $Qy$, which simplifies when $Q = 1$, shows that $y =0$ everywhere on $\mathcal{C}_\infty\times [-\Delta t, 0]$. This, however, implies that $T_1 = 1 > 0$ on $\mathcal{C}_\infty \times [-\Delta t, 0]$ contradicting our assumption that $\iota < 0$.
It remains to be shown that $T_1(p,t) = 0$ if, and only if, $k=2$ and $p$ lies on the non-principal orbit $S^2_o$. We argue by contradiction. Assume there exists a point $(p,t)$ in spacetime such that $p \notin S^2_o$ and
$$T_1(p, t) = 0.$$
Then arguing as above, we see that either case a) or case b) must hold true, both of which lead to the same contradiction.
By the same method we may prove that $T_2 \geq 0$ and $T_3 \geq 0$ on $M_k \times (-\infty, 0]$. Note that the evolution equations (\ref{T2-evol}) and (\ref{T3-evol}) show that $T_2$ and $T_3$ cannot attain a negative infimum, leading to the desired contradiction.
\end{proof}
Next we prove Theorem \ref{kahler-ancient-thm}.
\begin{proof}[Proof of Theorem \ref{kahler-ancient-thm}]
Recall that by Corollary \ref{cor:curv-bound-ancient} there exists a $C_1 > 0$ such that $|Rm_{g(t)}|_{g(t)} \leq \frac{C_1}{0}$ on $M_2 \times (-\infty, 0]$. Also recall Lemma \ref{unique-Kahler-lem}, which states that $(M_2, g(t)$, $t \in (-\infty, 0]$, is homothetic to the Eguchi-Hanson space if, and only if,
$$x = y = 0 \: \text{ on } \: M_2 \times (-\infty, 0].$$
Therefore it suffices to show that $x = 0$. We follow the proof strategy of Theorem \ref{T-ancient} and argue by contradiction. Assume
\begin{equation*}
\iota \coloneqq \inf_{M_2\times(\infty, 0]} x < 0
\end{equation*}
and take a sequence of points $(p_i, t_i)$ in spacetime such that
\begin{equation*}
x(p_i, t_i) \rightarrow \iota.
\end{equation*}
Note that $\iota > -\infty$ as $g(t) \in \mathcal{I}$ for $t \in (-\infty, 0]$. From Lemma \ref{away-from-origin-lem} it follows that
\begin{equation}
\label{away-from-origin2}
s(p_i, t_i) \geq \delta b(0,t_i)
\end{equation}
for some $\delta > 0$. Define the rescaled metrics
$$g_i = \frac{1}{b^2(p_i, t_i)} g\left(t + t_i b^2(p_i, t_i)\right), \quad t \in [-\Delta t, 0],$$
where $\Delta t > 0$ is chosen such that the conditions of Proposition \ref{blow-up-prop} are satisfied. Then $(C_{g_i(0)}\left(p_i, \frac{1}{2}\right), g_i(t), p_i)$ subsequentially converges to a Ricci flow $(\mathcal{C}_\infty, g_\infty(t), p_\infty)$, $t \in [-\Delta t, 0]$, on which by construction
\begin{equation*}
x(p_{\infty},0) = \iota < 0,
\end{equation*}
since $x$ is a scale-invariant quantity. Furthermore, we see by (\ref{away-from-origin2}) that $p_\infty$ does not lie on the non-principal orbit $S^2_o$. The evolution equation (\ref{x-evol}) for $x$ in the K\"ahler case $y=0$ simplifies to
\begin{equation*}
\partial_t x = L[x] - \frac{4Q^2}{b^2} x
\end{equation*}
which implies that
\begin{equation*}
\partial_t x \Big|_{(p_{\infty}, 0)} = x_{ss} - \frac{4Q^2}{b^2} x > 0
\end{equation*}
unless $Q(p_\infty,0) = 0$. This, however, cannot happen, as otherwise the strong maximum principle applied to the evolution equation (\ref{Q-evol}) of $Q$ would imply that $Q = 0$ on $\mathcal{C}_\infty \times [-\Delta t, 0]$. Hence we have arrived at a contradiction and conclude
\begin{equation*}
x \geq 0 \: \text{ on } M_2 \times (-\infty, 0].
\end{equation*}
By the same argument one shows that
\begin{equation*}
x \leq 0 \: \text{ on } M_2 \times (-\infty, 0].
\end{equation*}
as well, which concludes the proof.
\end{proof}
\section{Eguchi-Hanson and a family of Type II singularities}
\label{E-H-sing-section}
In this section we show that Ricci flow solutions $(M_k, g(t))$, $k \geq 2$, starting from a large class of initial metrics encounter a Type II singularity in finite time at the origin. In the case $k=2$ we show that the Eguchi-Hanson metric can occur as a blow-up limit. Below we state the precise result:
\begin{thm}[Type II singularities]
\label{E-H-sing-thm}
Let $(M_k, g(t))$, $k \geq 2$, be a Ricci flow starting from an initial metric $g(0) \in \mathcal{I}$ (see Definition \ref{def:I}) with
\begin{equation}
\label{bounded-b-initially}
\sup_{p \in M_2} b(p,0) < \infty.
\end{equation}
Then $g(t)$ encounters a Type II curvature singularity in finite time $T_{sing}>0$ and
\begin{equation*}
\sup_{0 \leq t < T_{sing}} \left(T_{sing} - t\right) b^{-2}(o,t) = \infty.
\end{equation*}
Furthermore, there exists a sequence of times $t_i \rightarrow T_{sing}$ such that the following holds:
Consider the rescaled metrics
\begin{equation*}
g_i(t) = \frac{1}{b^2(o,t_i)} g\left( t_i + b^2(o,t_i) t \right), \quad t \in \big[-b(o, t_i)^{-2} t_i, b(o, t_i)^{-2}\left(T_{sing} - t_i\right) \big).
\end{equation*}
Then $(M_k, g_i(t), o)$ subsequentially converges, in the pointed Gromov-Cheeger sense, to an eternal Ricci flow $(M_k, g_\infty(t), o)$, $t \in (-\infty, \infty)$. When $k=2$ the metric $g_\infty(t)$ is stationary and homothetic to the Eguchi-Hanson metric.
\end{thm}
In this paper we do not study the detailed geometry of the singularity models of the Type II singularities arising in the $k \geq 3$ case, however, as stated in Conjecture \ref{conj:sing} in the introduction, the author believes that these singularities are modeled on the non-collapsed steady Ricci solitons found in \cite{A17}. The author, in collaboration with Jon Wilkening, has carried out numerical simulations supporting this conjecture. A paper summarizing the results is in preparation \cite{AW19}.
\noindent\textbf{Outline of proof.} Here we sketch the proof of Theorem \ref{E-H-sing-thm}. First we show in Lemma \ref{sing-time-finite} that the condition (\ref{bounded-b-initially}) forces a Ricci flow solution $(M_k, g(t))$, $k \geq 2$, to develop a singularity in finite time $T_{sing}>0$ at the origin. Then we take a sequence of times $t'_i \rightarrow T_{sing}$ and define the rescaled metrics
\begin{equation*}
g'_i(t) = \frac{1}{b^2(0,t'_i)} g\left( t'_i + b^2(0,t'_i) t \right).
\end{equation*}
These metrics subsequentially converge to a singularity model $(M_k, g'_\infty(t))$, $-\infty < t \leq 0$ --- an ancient solution of the Ricci flow. Now recall the dichotomy between Type I and Type II singularities and that every Type I singularity is modeled on a shrinking Ricci soliton \cite{EMT11}. Therefore we can prove that the singularity is of Type II by showing that $(M_k, g'_\infty(t))$ is not a shrinking Ricci soliton. For this we apply Theorem \ref{thm:no-shrinker}, which excludes shrinking solitons whenever (i) $\sup |b_s| < \infty$, (ii) $T_1>0$ for $s >0$ and (iii) $Q \leq 1$ hold. By definition, every metric in $\mathcal{I}$ satisfies conditions (i) and (iii). As these conditions are scale-invariant, they pass to the blow-up limit $(M_k, g'_\infty(t))$. From Theorem \ref{T-ancient} it follows that condition (iii) holds true as well, allowing us to conclude that $(M_k, g'_\infty(t))$ is not a shrinking soliton and that the singularity is of Type II. By the work of Hamilton we can then choose a sequence of times $t_i \rightarrow T_{sing}$, possibly different from the sequence $t'_i$, such that the corresponding blow-ups around the origin converge to an eternal Ricci flow $(M_k, g_\infty(t))$, $-\infty< t < \infty$.
In the $k=2$ case we show that $(M_2, g_\infty(t))$ is stationary under Ricci flow and homothetic to the Eguchi-Hanson space. What makes $k=2$ special is that the second term of the right hand side of
\begin{equation*}
\partial_t b(0,t) = 2 \left( y_s + \frac{k-2}{b} \right)
\end{equation*}
is zero and therefore
\begin{equation}
\label{b0-evol-k2}
\partial_t b(0,t) = 2y_s(0,t) \leq 0,
\end{equation}
as $y\leq 0$ with equality at $S^2_0$ for metrics in $\mathcal{I}$. It turns out that for the specific choice of $t_i \rightarrow T_{sing}$ from Hamilton's trick we have that on $(M_2, g_\infty(t))$ at $S^2_o$ at time 0 we have
$$\partial_t b(0,t) = 2y_s(0,t) = 0.$$
An application of L'H\^opital's Rule shows that on $S^2_o$ we have
$$\frac{y}{Q} = \frac{y_s}{k}.$$
Therefore we can apply the strong maximum principle of Theorem \ref{maximum-principle}, Case 2, to the evolution equation (\ref{yQ-evol}) of $\frac{y}{Q}$ to show that $y = 0$ everywhere. From Theorem \ref{kahler-ancient-thm} it then follows that $(M_2, g_\infty(t))$ is homothetic to the Eguchi-Hanson space.
\begin{remark}
A priori it may be possible that other sequences of times give rise to blow-up limits around $o$ that are not homothetic to the Eguchi-Hanson space. However in section \ref{E-H-unique-ancient-section} we show that the Eguchi-Hanson space is in fact the unique blow-up limit.
\end{remark}
\noindent\textbf{Recap of some properties of singular Ricci flow solutions.}
Before proving Theorem \ref{E-H-sing-thm} we summarize some properties of curvature blow-up rates of Ricci flows encountering singularities and their respective singularity models. For this let $(M, g(t))$, $t \in [0,T)$ be a Ricci flow encountering a singularity at time $T$. Let
$$K_{max}(t) := \sup_{M} |Rm_{g(t)}|_{g(t)}.$$
By Shi's result \cite{Shi89} on the short time existence of Ricci flow we have
$$ \limsup_{t \nearrow T} K_{max}(t) = \infty.$$
In fact one can show with help of the evolution equation of $|Rm_{g(t)}|_{g(t)}^2$ that
\begin{equation}
\label{blow-up-rate-lower-bound}
\sup_{M} |Rm_{g(t)}|_{g(t)} \geq \frac{1}{8}\frac{1}{T-t}.
\end{equation}
Hamilton \cite{Ham95} introduced the notion of Type I and Type II Ricci flows, which are defined by the rate at which the curvature blows up as $t \nearrow T$.
In particular, $(M_2, g(t))$ is of Type I if it satisfies if there exists a $C>0$ such that for $t \in [0, T)$
$$K_{max}(t) \leq \frac{C}{T-t},$$
In the case that such a constant $C>0$ does not exists, that is
$$\sup_{t \in [0,T)} (T-t) K_{max}(t) = \infty,$$
we say the singularity is of Type II.
By the work of Naber \cite{N10} and Enders, M\"uller and Topping \cite{EMT11} every Type I singularity model is a non-flat Ricci shrinking soliton. Hamilton showed how for Type II singularities one can extract a blow-up sequence converging to an eternal Ricci flow \cite[Theorem 16.4]{Ham95}. However it remains to be understood whether or not all Type II singularity models are steady solitons. So far all known examples are.
Below we recap the main result of \cite{EMT11}: First note the following definition:
\begin{definition}[see {\cite[Definition 1.2]{EMT11}} ]
\label{singular-point-def}
A spacetime sequence $(p_i, t_i)$ with $p_i \in M$ and $t_i \nearrow T$ in a Ricci flow is
called an \textbf{essential blow-up sequence} if there exists a constant $c>0$ such that
$$|Rm_{g(t_i)}|_{g(t_i)}(p_i) \geq \frac{c}{T-t}.$$
A point $p \in M$ in a Type I Ricci flow is called a (general) \textbf{Type I singular point} if there
exists an essential blow-up sequence with $p_i \rightarrow p$ on $M$.
\end{definition}
Now we state the main result of \cite{EMT11}, asserting that Type I singularities are modeled on shrinking Ricci solitons.
\begin{thm}[see {\cite[Theorem 1.4]{EMT11}} ]
\label{typeI-blow-up}
Let $(M, g(t))$ be a Type I Ricci flow on $[0, T)$ and suppose $p$ is a Type
I singular point as in Definition \ref{singular-point-def}. Then for every sequence $\lambda_j \rightarrow \infty$, the rescaled Ricci
flows $(M, g_j(t), p)$ defined on $[−\lambda_j T, 0)$ by
$$g_j (t) := \lambda_j g\left(T + \frac{t}{\lambda_j} \right) $$
subconverge to a non-flat gradient shrinking soliton.
\end{thm}
We use Theorem \ref{typeI-blow-up} to exclude Type I singularities for Ricci flows satisfying the assumptions of Theorem \ref{E-H-sing-thm}.
\noindent\textbf{Proof of the main theorem.}
First we show that a singularity must occur in finite time:
\begin{lem}
\label{sing-time-finite}
The maximal extension of a Ricci flow $(M_k, g(t))$, $k \geq 1$, starting from an initial metric $g(0) \in \mathcal{I}$ with
\begin{equation*}
\sup_{p \in M_k} b(p,0) < \infty
\end{equation*}
encounters a singularity at the $S^2_o$ in finite time $T_{sing}>0$.
\end{lem}
\begin{proof}
By Shi's short time existence of Ricci flow \cite{Shi89} we have $T_{sing} > 0$. From the evolution equation (\ref{b-evol}) of $b$ under Ricci flow it follows that
\begin{equation*}
\partial_t b^2 \leq \Delta_{g(t)} b^2 - 4,
\end{equation*}
where we used expression (\ref{laplacian}) of the Laplacian. By the maximum principle (see for instance \cite[Theorem 12.14]{ChII}) we see that there exists a $T < \infty$ such that
$$\inf_{p\in M_k} b^2(p,t) \rightarrow 0 \quad \text{ as } t \rightarrow T.$$
As $b_s \geq 0$ we conclude that
\begin{equation*}
\lim_{t\rightarrow T} b(o,t) = 0.
\end{equation*}
From Lemma \ref{b-bound} it follows that the curvature at $S^2_o$ blows up as $t \rightarrow T$. Hence $T = T_{sing}$.
\end{proof}
Below we prove Theorem \ref{E-H-sing-thm}.
\begin{proof}[Proof of Theorem \ref{E-H-sing-thm}]
By Lemma \ref{sing-time-finite} the Ricci flow becomes singular in finite time $T_{sing} > \delta > 0$ and $b(o,t) \rightarrow 0$ as $t \nearrow T_{sing}$. Recall that by Theorem \ref{curv-bound} there exist a $C_1 > 0$ such that
$$|Rm_{g(t)}|_{g(t)} \leq \frac{C_1}{b^2} \: \text{ on } \: M_k \times [0, T_{sing}).$$
Moreover, by Theorem \ref{thm:no-local-collapsing} there exist constants $\kappa, \rho > 0$ such that $g(t)$ is $\kappa$-non-collapsed at scales less or equal to $\rho$.
Now take a sequence of times $t'_i \nearrow T_{sing}$ such that
\begin{enumerate}
\item $b(o, t'_i) \rightarrow 0$ as $i \rightarrow \infty$
\item $b(o, t) \geq b(o,t'_i)$ for $t \leq t'_i$
\end{enumerate}
\begin{claim}
The sequence of points $(o, t'_i)$ in spacetime is an essential blow-up sequence.
\end{claim}
\begin{claimproof}
We argue by contradiction. Assume, after passing to a subsequence, that
$$(T_{sing}-t'_i) |Rm_{g(t'_i)}|_{g(t'_i)} \rightarrow 0 \text{ as } i \rightarrow \infty.$$
Then by Lemma \ref{b-bound} and the fact that $b_s \geq 0$ for metrics in $\mathcal{I}$ we have
$$b^2(p,t) \geq b^2(o,t) = \frac{4}{R_{2323}} \geq \frac{4}{|Rm_{g(t'_i)}|_{g(t'_i)}(o)},$$
where we used the expression for the curvature component $R_{2323}$ derived in section \ref{con-lap-cur-subsec}. This shows that
$$|Rm_{g(t'_i)}|_{g(t'_i)}(p) \leq \frac{C_1}{b^2(p,t'_i)} \leq \frac{C_1}{4} |Rm_{g(t'_i)}|_{g(t'_i)}(o) \: \text{ for } \: p \in M_k.$$
Therefore
$$\lim_{i \rightarrow \infty} \left(T_{sing} - t'_i\right) \sup_{p\in M_2}|Rm_{g(t'_i)}|_{g(t'_i)}(p) = 0,$$
which contradicts (\ref{blow-up-rate-lower-bound}). This proves the claim.
\end{claimproof}
Define the rescaled metrics
\begin{equation*}
g'_i(t) = \frac{1}{b^2(0,t'_i)} g\left( t'_i + b^2(0,t'_i) t \right), \quad t \in [- b^{-2}(o,t'_i)t'_i, 0],
\end{equation*}
By property (2) above and the fact that $b_s\geq 0$ for metrics in $\mathcal{I}$ it follows that
$$|Rm_{g'_i(t)}|_{g'_i(t)} \leq C_1 \: \text{ on } \: M_k \times [- b^{-2}(o,t'_i)t'_i, 0].$$
Note also that the rescaled metrics $g'_i(t)$ are $\kappa$-non-collapsed at scales tending to infinity. Corollary \ref{cor:compactness-complete-flows} then implies that $(M_k, g'_i(t),o)$ subsequentially converges, in the Cheeger-Gromov sense, to an ancient Ricci flow $(M_\infty, g'_{\infty}(t),o)$, $t \in (-\infty,0]$, where $M_\infty \cong M_k$. By Theorem \ref{T-ancient} we have
$$T_1(p,t) > 0 \: \text{ on } \: M_\infty \setminus S^2_o \times (-\infty, 0]$$
on the blow-up limit $g'_\infty(t)$. Theorem \ref{thm:no-shrinker} shows that $g'_\infty(t)$ cannot be a shrinking soliton, which by the contrapositive of Theorem \ref{typeI-blow-up} proves that the singularity is of Type II. Therefore
\begin{equation*}
\sup_{M \times [0,T_{sing})} \left(T_{sing}- t\right) |Rm_{g(t)}|_{g(t)} = \infty
\end{equation*}
from which we see that
\begin{equation*}
\sup_{t \in [0, T_{sing})} \left(T_{sing} - t\right) b^{-2}(0,t) = \infty.
\end{equation*}
Now we mimic the proof of \cite[Theorem 16.4, Type II(a)]{Ham95} to construct an eternal blow-up limit. Pick a sequence of times $T_i < T_{sing}$ satisfying
$$\left(T_{sing} - T_i\right) b^{-2}(o,t) \rightarrow \infty $$ as $i \rightarrow \infty$. Then we can choose $t_i < T_i$ such that
\begin{equation}
\label{bminus2-bound}
\left(T_i - t_i\right) b^{-2}(o,t_i) = \sup_{t \leq T_i} \left(T_i - t\right)b^{-2}(o,t)
\end{equation}
as the latter goes to zero as $t \rightarrow T_i$. Consider the rescaled Ricci flow solutions
$$g_i(t) = \frac{1}{b^2(0,t_i)} g\left( t_i + b^2(0,t_i) t \right),$$
which exist for $-A_i \leq t \leq B_i$ with
\begin{align*}
A_i &= t_i b^{-2}(o,t_i) \rightarrow \infty\\
B_i &= \left(T_i- t_i\right) b^{-2}(o, t_i) \rightarrow \infty.
\end{align*}
If we write $a_i, b_i$ for the warping functions of the rescaled metric $g_i(t)$ we obtain from equation (\ref{bminus2-bound}) the following inequality
\begin{equation*}
\left(B_i - t \right)b_i^{-2} (0,t) \leq B_i.
\end{equation*}
Note that here $t$ is the time variable of the rescaled Ricci flow $g_i(t)$. Therefore for any fixed $t$ we have
\begin{equation}
\label{b-before-limit-everywhere}
b_i^{-2} (o,t) \leq \frac{B_i}{B_i -t} \rightarrow 1 \quad \text{as } i \rightarrow \infty
\end{equation}
and
\begin{equation}
\label{b-before-limit-at-o}
b^{-2}_i(o,0) = 1.
\end{equation}
From this, the fact that $b_s \geq 0$ and the curvature bound of Theorem \ref{curv-bound}, we see that on bounded time intervals the curvatures of $g_i(t)$ eventually become bounded by $2C_1$. In addition to this the metrics $g_i(t)$ are $\kappa$-non-collapsed at larger and larger scales. Therefore Corollary \ref{cor:compactness-complete-flows} implies that $(M_2, g_i(t), o)$ subsequentially converges to an eternal Ricci flow $(M_2, g_\infty(t), o)$. Furthermore (\ref{b-before-limit-everywhere}) and (\ref{b-before-limit-at-o}) show that that
\begin{equation}
\label{b-limit-greater-1}
b_{\infty}(o,t) \geq 1 \: \text{ for } \: t \in (-\infty, \infty)
\end{equation}
and
\begin{equation}
\label{b-limit-at-o}
b_{\infty} (o,0) = 1,
\end{equation}
where we write $a_{\infty}$ and $b_{\infty}$ for the warping functions of the metric $g_{\infty}(t)$. Notice that (\ref{b-limit-greater-1}) and (\ref{b-limit-at-o}) imply that at time $0$ on $S^2_o$ we have
\begin{equation}
\label{origin-behavior}
\partial_t b_{\infty} = 2 \left( (y_{\infty})_s + \frac{k-2}{b_{\infty}}\right) = 0,
\end{equation}
where $y_\infty = (b_\infty)_s - \frac{a_\infty}{b_\infty}$ corresponds to the K\"ahler quantity $y$ on the $g_\infty$ background.
Now it only remains to be shown that in the $k=2$ case $g_\infty(t)$ is stationary and homothetic to the Eguchi-Hanson metric. In the following we drop the $\infty$ subscript and let $a$, $b$, $Q$, $y$ be with respect to the metric $g_\infty(t)$. Note that equation (\ref{origin-behavior}) and an application of L'H\^opital's Rule show that at time $0$ on $S^2_o$ we have
\begin{equation*}
y_s = \frac{y}{Q} = 0
\end{equation*}
The evolution equation for $\frac{y}{Q}$ derived in the Appendix A is
\begin{equation}
\label{yoverQ-evol}
\partial_t \left(\frac{y}{Q}\right) = \left(\frac{y}{Q}\right)_{ss} + \left(3 \frac{a_s}{a} - 2 \frac{b_s}{b} \right) \left(\frac{y}{Q}\right)_s + \frac{2}{b^2}\frac{y}{Q}\left( 2 + \frac{y}{Q}\right) \left( Q b_s - 2 a_s \right).
\end{equation}
Because $g_\infty(t) \in \mathcal{I}$ is of bounded curvature we see that
\begin{equation*}
\frac{1}{b^2} \left( 2 + \frac{y}{Q}\right) \left( Q b_s - 2a_s \right) = \frac{1}{b^2}\left(-\frac{2 a_s b_s}{Q}-2 a_s+Q b_s+b_s^2 \right)
\end{equation*}
is bounded. Note that we applied Lemma \ref{b-bound} to show that $\frac{1}{b^2}$ is bounded. Therefore we may apply the strong maximum principle of Theorem \ref{maximum-principle}, Case 2, to deduce that
\begin{equation*}
\frac{y}{Q} = 0 \: \text{ on } \: M_2 \times (-\infty, 0],
\end{equation*}
yielding that $g_\infty(t)$ is K\"ahler in the the $k=2$ case. By Theorem \ref{kahler-ancient-thm} we then deduce that $g_\infty(t)$ is homothetic to the Eguchi-Hanson metric, which proves the desired result.
\end{proof}
\section{Ancient Ricci flows Part II: $k=2$ case}
\label{E-H-unique-ancient-section}
In this section we prove that every non-collapsed ancient Ricci flow in the class of metrics $\mathcal{I}$ is isometric to the Eguchi-Hanson metric:
\begin{thm}[Unique ancient flow]
\label{E-H-ancient-thm}
Let $\kappa > 0$ and $\left(M_2,g(t)\right)$, $t \in (-\infty,0]$, be an ancient Ricci flow that is $\kappa$-non-collapsed at all scales and $g(t) \in \mathcal{I}$, $t \in (-\infty,0]$ (see Definition \ref{def:I}). Then $g(t)$ is stationary and homothetic to the Eguchi-Hanson metric.
\end{thm}
An immediate consequence of this theorem is that for \emph{every} sequence of times $t_i \rightarrow T_{sing}$ in Theorem \ref{E-H-sing-thm}, the rescaled Ricci flows
$$g_i(t) = \frac{1}{b^2(o,t_i)}g\left(t + t_i b^2(o,t_i) \right), \quad t \in \left[- b(o,t_i)^{-2} t_i, 0\right], $$
subsequentially converge to the Eguchi-Hanson space. In other words, the Eguchi-Hanson space is the \emph{unique} limit of blow-ups around the origin. With a little extra work one can show the slightly more general result, asserting that blow-up limits centered at points close to, but not necessarily on the tip of $M_2$, subsequentially converge to the Eguchi-Hanson space:
\begin{cor}
\label{E-H-blowup}
Let $(M_2, g(t))$, $t \in[0, T_{sing})$, be a Ricci flow starting from an initial metric $g(0) \in \mathcal{I}$ (see Definition \ref{def:I}) that develops a singularity at time $T_{sing}$. Let $(p_i, t_i)$ be a sequence of points in spacetime with $t_i \rightarrow T_{sing}$ satisfying
$$ \sup_i \frac{b(p_i,t_i)}{b(o,t_i)} < \infty$$
and consider the rescaled metrics
\begin{equation*}
g_i(t) = \frac{1}{b^2(p_i,t_i)} g\left(t_i + b^2(p_i,t_i) t\right), \quad t \in \left[- t_i b^{-2}(p_i,t_i) , 0\right].
\end{equation*}
Then $(M_2, g_i(t), p_i)$ subsequentially converges, in the Gromov-Cheeger sense, to a blow-up limit $(M_2, g_\infty(t), p_\infty)$, $t \leq 0$, which is homothetic to the Eguchi-Hanson space.
\end{cor}
We defer the proof of Corollary \ref{E-H-blowup} to the end of subsection \ref{ancient-main-thm-subsec}.
\subsection{Outline of Proof}
\label{E-H-ancient-thm-outline}
Here we outline the proof of Theorem \ref{E-H-ancient-thm}. Below we take $(M_2, g(t))$, $t \in (-\infty,0]$, to be a non-collapsed ancient Ricci flow with $g(t) \in \mathcal{I}$, $t \in (-\infty,0]$. We construct a continuously varying one-parameter family of functions
\begin{equation*}
f_{\theta}: [0,1] \rightarrow [0,1], \quad \theta \in (0,1],
\end{equation*}
satisfying the following five requirements:
\begin{enumerate}
\item For every $\theta \in (0,1]$ the condition
\begin{equation*}
Z_{\theta}(\xi, t) := \frac{x}{Q^2} + f_{\theta}(Q) = \frac{a_s + Q^2 - 2}{Q^2} + f_{\theta} (Q) \geq 0
\end{equation*}
is preserved on the $(M_2, g(t))$ background.
\item For every $0\leq Q < 1$
\begin{equation*}
f_{\theta} (Q) \longrightarrow 0 \: \text{ as } \: \theta \longrightarrow 0
\end{equation*}
\item For every $\theta \in (0,1]$ there exists a $Q_{\theta} \in [0,1)$ such that
$$f(Q) < 1 \: \text{ for } \: Q < Q_{\theta},$$
and
$$ f(Q) = 1 \: \text{ for } \: Q \geq Q_{\theta}.$$
Furthermore $Q_{\theta}$ depends continuously on $\theta$.
\item For $\theta = 1$ $$f_1 = 1 $$ everywhere.
\item For every $\theta \in (0,1]$ the function $f_{\theta}$ is extendable to a smooth even function around $0$.
\end{enumerate}
\begin{remark}
\label{f-constr-rem}
We briefly remark on some of the properties of $f_{\theta}$:
\begin{itemize}
\item In the expression for $Z_{\theta}$ of requirement (1) we take $x$, $Q$ and $a_s$ to be functions of spacetime. For brevity we do not express the dependence explicitly.
\item The term $\frac{x}{Q^2}$ can be extended smoothly to the non-principal orbit $S^2_o$, as $x = x_s = 0$ at $s = 0$. Therefore $Z_{\theta}$ is well-defined on $M_2$.
\item When $\theta = 1$ we already know that
$$Z_1 = \frac{x + Q^2}{Q^2} \geq 0$$
in $M_2 \times (-\infty, 0]$, as from Theorem \ref{T-ancient} it follows that $T_1 = Q^2 Z_1\geq 0$ on $M_2 \times (-\infty, 0]$.
\item At any point $(p, t)$ in spacetime such that $Q(p,t) \geq Q_{\theta}$ we have $$Z_{\theta} (p, t) = Z_1(p, t) \geq 0.$$
\item The family $f_{\theta}(Q)$, $\theta \in (0,1]$, we construct below is smooth everywhere apart from when $Q = Q_{\theta}$. It will become clear later that this does not pose a problem.
\end{itemize}
\end{remark}
In subsection \ref{subsec-evol} we show that at points $(p, t)$ in spacetime at which $Q(p, t) \neq Q_{\theta}$, or equivalently at points where $f$ is smooth, the evolution equation of the corresponding $Z_{\theta}$ can locally be written as
\begin{align*}
\partial_t Z_{\theta} &= [Z_{\theta}]_{ss} + \left(3 \frac{a_s}{a} - 2\frac{b_s}{b}\right) [Z_{\theta}]_s + \frac{1}{b^2}\left( W_{\theta} + Z_{\theta} \tilde{D}_{\theta} \right),
\end{align*}
where $W_{\theta}$ and $\tilde{D}_{\theta}$ are bounded and scale-invariant expressions involving $b_s$, $Q$, $f_{\theta}(Q)$, $f'_{\theta}(Q)$ and $f''_{\theta}(Q)$. Again, all quantities in the evolution equation of $Z_{\theta}$ should be interpreted as functions of spacetime.
In subsection \ref{subsec-constr-f} we construct a family of functions $f_{\theta}$, $\theta \in (0,1]$, by solving an initial value problem for a second order non-linear ordinary differential equation. Subsequently we show that the family satisfies requirements (1)-(5) listed above. In particular, we show in subsection \ref{subsec-mathcalQ-pos} that for the constructed family --- on a non-collapsed ancient Ricci flow background --- the following property holds true: For all points $(p, t)$ in spacetime such that $Q(p, t) < Q_{\theta}$, we have $W_{\theta}(p, t) \geq 0$, $\theta\in (0,1]$. This fact, in conjunction with the fourth bullet point of Remark \ref{f-constr-rem}, essentially shows that for each $\theta\in (0,1]$ the inequality $Z_{\theta} \geq 0$ is preserved on the $g(t)$ background.
Once we have shown that our family $f_{\theta}$, $\theta \in (0,1]$, satisfies requirements (1)-(5), we will use a blow-up argument in conjunction with the strong maximum principle to show that if for some $\theta_0 \in (0,1]$ the inequality $Z_{\theta} \geq 0$ holds for all $\theta \in [\theta_0 , 1]$, then there must exists an $\theta_1 < \theta_0$ such that the inequality also holds for all $\theta \in [\theta_1, 1]$. This shows that the set
$$ \mathcal{E} = \left \{ \theta \in (0,1] \: \big | \: Z_{\theta'} \geq 0 \text{ for all } \theta \leq \theta' \leq 1 \right\} \subseteq (0,1]$$
is open. As $\mathcal{E}$ is defined by a closed condition and therefore closed, it follows that $\mathcal{E} = (0,1]$ and therefore
$$Z_{\theta} \geq 0 \quad \text{ for all } \theta \in (0,1].$$
By property (2) of $f_{\theta}$ we deduce that at all points $(p,t)$ in spacetime such that $Q(p,t) < 1$ we have
$$x(p,t) \geq 0.$$
Note that by the strong maximum principle applied to the evolution equation (\ref{Q-evol}) of $Q$ it follows that $Q < 1$ and hence $x \leq 0$ everywhere. Now recall Theorem \ref{T-ancient} which states that
$$ x \leq Qy \leq 0 \: \text{ on } \: M_2 \times (-\infty, 0].$$
Therefore
$$x = y = 0 \: \text{ on } \: M_2 \times (-\infty, 0]$$
and we conclude that the metric $g(t)$ is homothetic to the Eguchi-Hanson metric by Lemma \ref{unique-Kahler-lem}.
\subsection{Evolution equations}
\label{subsec-evol} The main difficulty in carrying out the proof is to find a family of functions $f_{\theta}$, $\theta \in (0,1]$, for which requirement (1) is satisfied. Our strategy is to first derive the evolution equation of $Z_{\theta}$ for a general $f_{\theta}$ and then reduce the problem to solving a second order ordinary differential equation in $f_\theta$. For this, first note that the evolution equation of $f_{\theta}(Q)$ away from the non-principal orbit $S^2_o$ can be written as
\begin{align*}
\partial_t f_{\theta}(Q) = [f_{\theta}(Q)]_{ss} + \left(3 \frac{a_s}{a} - 2\frac{b_s}{b}\right) [f_{\theta}(Q)]_s + \frac{1}{b^2} C_{f},
\end{align*}
where
\begin{align}
\label{Cf}
C_f &= \left( 8 a_s b_s - 3 \frac{a_s^2}{Q} - 5 Q b_s^2 + 4 Q \left(1 - Q^2\right) \right)f' - \left( a_s - Q b_s \right)^2f''
\end{align}
The computation is carried out in the Appendix A.
\begin{remark} Some remarks on the evolution equation of $f_{\theta}(Q)$:
\begin{itemize}
\item We often omit the dependence of our quantities on spacetime, i.e. by $f_{\theta}(Q)$ we mean $f_{\theta}(Q(p,t))$.
\item For brevity we often omit the dependence of $f$ on $\theta$ and $Q$, as in the expression for $C_f$ above. For instance, we write $f'$ for $f'_{\theta}(Q)$ and $f''$ for $f''_{\theta}(Q)$.
\item Note that by Lemma \ref{parity-lem} the quantity $Q = \frac{a}{b}$ as a function of $s$ can be extended to an odd function around the origin. Therefore as long as $f_{\theta}: [0,1] \rightarrow [0,1]$ is extendable to an even function around the origin, the term $\frac{f'}{Q}$ and hence $C_f$ can be smoothly extended to all of $M_2$.
\end{itemize}
\end{remark}
From equation (\ref{Cf}) and the evolution equations (\ref{as-evol}) and (\ref{Q2-evol}) of $a_s$ and $Q^2$, respectively, we see that the evolution equation of $Z_{\theta}$ can be written as
\begin{equation}
\label{Z-epsilon-evol-1}
\partial_t Z_{\theta} = [Z_{\theta}]_{ss} + \left(3 \frac{a_s}{a} - 2\frac{b_s}{b}\right) [Z_{\theta}]_s + \frac{1}{b^2}\left( C_{Z,0} + C_{Z,1} Z_{\theta} + + C_{Z,2} Z^2_{\theta} \right),
\end{equation}
after having eliminated any occurring $a_s$ by substituting
\begin{equation*}
a_s = Q^2 Z_{\theta}- f Q^2 - Q^2 + 2.
\end{equation*}
A computation carried out in the Appendix A shows that
\begin{align*}
C_{Z,0} &= A_0 +\left[\frac{b_s}{Q}\right] A_1 + \left[\frac{b_s}{Q}\right]^2 A_2,
\end{align*}
where
\begin{align*}
A_0 =& - Q^4 f^2 f''-2 Q^4 f f''-Q^4 f''+4 Q^2 f f''+4 Q^2 f''-4 f''-3 Q^3 f^2 f' \\ \nonumber
&-6 Q^3 f f'-7 Q^3 f'+12 Q f f'+16 Q f'-\frac{12 f'}{Q}-2 Q^2 f+8 f-2 Q^2-4 \\
A_1 =& -2 Q^4 f f''-2 Q^4 f''+4 Q^2 f''-8 Q^3 f f'-8 Q^3 f' \\ \nonumber
& \qquad+16 Q f'-4 Q^2 f^2-8 Q^2 f+8 f+4 Q^2+8 \\
A_2 =& - Q^4f''-5 Q^3 f'-2 Q^2 f-2 Q^2-4 \\
\end{align*}
Similarly, we compute the expressions for $C_{Z,1}$ and $C_{Z,2}$ in the Appendix A, however their exact forms are not important for our analysis. It is only important to note that when $f$ is extendable to an even function around $0$, the quantities $C_{Z,i}$, $A_i$, $i = 0, 1, 2$, are scale-invariant, bounded, and can be extended smoothly to $S^2_o$.
For reasons that will become clear below, we rewrite the equation (\ref{Z-epsilon-evol-1}) in the form
\begin{align}
\label{Z-epsilon-evol-2}
\partial_t Z_{\theta} &= [Z_{\theta}]_{ss} + \left(3 \frac{a_s}{a} - 2\frac{b_s}{b}\right) [Z_{\theta}]_s + \frac{1}{b^2}\left( W_{\theta} + Z_{\theta} D_{\theta} \right),
\end{align}
where
\begin{equation}
\label{mathcalQ}
W_{\theta} = A_0 +\left[\frac{b_s}{Q} - Z_{\theta} \right] A_1 + \left[\frac{b_s}{Q} - Z_{\theta} \right]^2 A_2
\end{equation}
and
\begin{equation*}
D_{\theta} = C_{Z,1} + Z C_{Z,2} + A_1 - A_2\left(Z_{\theta} - 2 \frac{b_s}{Q}\right)
\end{equation*}
Sometimes it will be useful to regard $W_{\theta}$ as a quadratic polynomial. Therefore we define
$$w_{\theta}(z) =A_0 + A_1 z + A_2 z^2$$
Then
$$W_{\theta} = w_{\theta}\left(\frac{b_s}{Q} - Z_{\theta}\right).$$
In the proof of Theorem \ref{E-H-ancient-thm} we also need the evolution equation of
$$Z_1 = \frac{x}{Q^2}+1,$$
which can be written as
\begin{equation}
\label{Z1-evol}
\partial_t Z_1 = [Z_1]_{ss} + \left(3 \frac{a_s}{a} - 2\frac{b_s}{b}\right) [Z_1]_s + \frac{1}{b^2} \left(C_{Z_1,0} + C_{Z_1,1} Z_1 + C_{Z_1,2} Z^2_1 \right)
\end{equation}
where
$$C_{Z_1,0} = \frac{1}{Q^2}\left( - 4\left(1+Q^2\right)y^2 + 8Q\left(1-2Q^2 \right)y + 16Q^2\left(1-Q^2\right) \right)$$
and $C_{Z_1,1}$, $C_{Z_1,2}$ are a bounded scale-invariant functions of $a_s$, $b_s$ and $Q$. The derivation of this evolution equation is carried out in the Appendix A. Note the following lemma:
\begin{lem}
\label{lem:suc-con-start}
Let $(M_2, g(t))$, $t \in (-\infty, 0]$, be an ancient Ricci flow as in Theorem \ref{E-H-ancient-thm}. Then
$$Z_1 \geq 0$$
and
$$C_{Z_1,0} \geq 4\left(1-Q^2\right)$$
everywhere in $M_2 \times (-\infty, 0]$.
\end{lem}
\begin{proof}
By Theorem \ref{T-ancient} we know that $Z_1 = \frac{T_1}{Q^2} \geq 0$ in $M_2 \times (-\infty, 0]$. Moreover, notice the similarity of $C_{Z_1,0}$ to the zeroth order term in the evolution equation (\ref{T1-evol}) of $T_1$. Therefore we see by the proof of Lemma \ref{T1-preserved-lem} that $C_{Z_1,0} \geq 4\left(1-Q^2\right)$ for metrics in $\mathcal{I}$.
\end{proof}
In the proof of Theorem \ref{E-H-ancient-thm} we deform the inequality $Z_1 \geq 0$ along a path of conserved inequalities $Z_{\theta} \geq 0$, $\theta \in (0,1]$. Thus $Z_1 \geq 0$ is the starting point for successively constraining the ancient Ricci flow towards the Eguchi-Hanson space. Below we construct the $f_{\theta}$ leading to the conserved inequalities $Z_{\theta} \geq 0$.
\subsection{Construction of $f_{\theta}$, $\theta \in (0,1]$}
\label{subsec-constr-f}
The goal of the following discussion is to find a family of functions $f_{\theta}: [0,1] \rightarrow [0,1]$, $\theta \in (0,1]$, such that
\begin{equation*}
W_{\theta} \geq 0
\end{equation*}
is non-negative on ancient Ricci flows satisfying $Z_{\theta} \geq 0$. For this we consider solutions to the ordinary differential equation
\begin{align}
\label{C0-ODE}
0 = -4 \left(1-Q^2\right)^2 f'' &- 4 \left(1-Q^2\right) \left(Q^2 f-5 Q^2+3\right) \frac{f'}{Q} \\ \nonumber
& \qquad + 2f\left( f^2 Q^2+3 f Q^2-6 f-6 Q^2+8 \right),
\end{align}
which is equivalent to
\begin{equation}
\label{eqn:alpha-ode-equivalent}
w_{\theta} \left( - f + 1\right) = 0.
\end{equation}
Note that we are now regarding $Q$ as an independent variable and not as a function of spacetime. Before we explain how we arrived at this differential equation, we list some of its properties below. For clarity of exposition we defer their proofs to subsection \ref{technical-lemmas-subsec}.
\begin{lem}
\label{f-existence-lem}
For every $f_0 \in \mathbb{R}$ the ordinary differential equation (\ref{C0-ODE}) possesses an even analytic solution around the origin with initial condition
\begin{equation*}
f(0) = f_0.
\end{equation*}
Furthermore, $f$ varies smoothly with $f_0$.
\end{lem}
\begin{lem}
\label{f-inc-lem}
Let $f: [0,Q_{max}) \rightarrow \mathbb{R}$, $Q_{max} \leq 1$, be the maximal solution to the ordinary differential equation (\ref{C0-ODE}) with initial condition $0 < f(0) < 1$. Then on any interval $(0, Q_\ast)$, $Q_\ast \leq Q_{max}$, on which $0 < f(Q) \leq 1$ we have $f'(Q) > 0$.
\end{lem}
\begin{lem}
\label{f-lem-2}
Let $\theta \in (0,1]$ and $f_{\theta}: 0 \in I \rightarrow \mathbb{R}$ be the maximal solution to the ordinary differential equation (\ref{C0-ODE}) with $f_{\theta}(0) = \theta$. Then there exists a $Q_{\theta} \in [0,1)$ such that
\begin{equation*}
f_{\theta}(Q_{\theta}) = 1
\end{equation*}
and
\begin{equation*}
f_{\theta}(Q) < 1 \: \text{ for } 0 \leq Q < Q_{\theta}.
\end{equation*}
Furthermore,
\begin{enumerate}
\item $Q_{\theta}$ varies continuously with $\theta \in (0,1]$
\item $Q_{\theta} \rightarrow 1 \text{ as } \theta \rightarrow 0$
\item $Q_1 = 0$
\end{enumerate}
\end{lem}
For each $\theta \in (0,1]$ let
$$ \phi_{\theta}: [0, Q_{\theta}] \rightarrow [\theta, 1]$$
be the solution to the differential equation (\ref{C0-ODE}) with initial condition $$\phi_{\theta}(0) = \theta$$ and define $f_{\theta}$, $\theta \in (0,1]$, as follows:
\begin{equation}
\label{f-def}
f_{\theta}(Q) =
\begin{cases*}
\phi_{\theta}(Q) & for $0 \leq Q \leq Q_{\theta}$ \\
1 & for $ Q_{\theta} < Q \leq 1$
\end{cases*}
\end{equation}
Note that $f_{\theta}$ is continuous but in general not smooth at $Q = Q_{\theta}$. This is not a problem, as will become clear later. In summary, we have:
\begin{prop}
\label{prop:f}
There exists a unique continuously varying family of continuous functions $f_{\theta}: [0,1] \rightarrow [0,1]$ and numbers $Q_{\theta} \in [0,1)$ for $\theta \in (0,1]$ satisfying the following properties:
\begin{itemize}
\item $f_{\theta}(Q)$ solves (\ref{C0-ODE}) or equivalently $w_{\theta}(-f_{\theta}(Q) + 1) = 0$ for $0 \leq Q \leq Q_{\theta}$
\item $f_{\theta}(0) = \theta$
\item $f_{\theta}(Q) < 1$ for $Q < Q_{\theta}$ and $f_{\theta}(Q) = 1$ for $Q \geq Q_{\theta}$
\item $f_{\theta}(Q)$ is strictly increasing in $Q$ when $0 < Q < Q_{\theta}$
\item $f_{\theta}(Q)$ is extendable to an even function around the origin
\item $Q_{\theta}$ varies continuously with $\theta$
\item $Q_{\theta} \rightarrow 1 \text{ as } \theta \rightarrow 0$ and $Q_1 = 0$
\item For every $Q \in [0,1)$ we have $f_{\theta}(Q) \rightarrow 0$ as $\theta \rightarrow 0$
\end{itemize}
\end{prop}
\subsection{Non-negativity of $W_{\theta}$}
\label{subsec-mathcalQ-pos}
For the choice of $f_{\theta}$, $\theta \in (0,1]$, defined above the following proposition holds true:
\begin{prop}
\label{Qcal-pos-prop}
Let $\theta \in (0,1]$ and $f_{\theta}$ be as defined in (\ref{f-def}). Assume $(M_2, g(t))$, $t \in (-\infty,0]$, is a non-collapsed ancient Ricci flow with $g(t) \in \mathcal{I}$ for $t\in (-\infty,0]$ and $Z_{\theta} \geq 0$ everywhere. Suppose at the point $(p, t)$ in spacetime $Q(p, t) < Q_{\theta}$. Then
\begin{equation*}
W_{\theta}(p, t) \geq 0
\end{equation*}
with equality if, and only if,
\begin{equation*}
T_2(p,t) = 0.
\end{equation*}
\end{prop}
We prove this proposition in multiple steps. First note
\begin{lem}
\label{A2-negative}
Let $f_{\theta}$, $\theta\in(0,1]$, be the family of functions as defined in Proposition \ref{prop:f}. Then
\begin{equation}
\label{bs2-coeff}
A_2 = - Q^4f''-5 Q^3 f'-2 Q^2 f-2 Q^2-4 < 0
\end{equation}
for $0 \leq Q < Q_{\theta}$. Thus $w_{\theta}(z) = A_2 z^2 + A_1 z + A_0$ is concave in $z$ whenever $0 \leq Q < Q_{\theta}$.
\end{lem}
The proof of this technical lemma can be found in subsection \ref{technical-lemmas-subsec}. Furthermore we have
\begin{lem}
\label{lem:borders}
Let $\theta \in (0,1]$. Let $(M_2, g(t))$, $t \in (-\infty, 0]$, be a non-collapsed ancient Ricci flow with $g(t) \in \mathcal{I}$ for $ t \in (-\infty, 0]$ and $Z_{\theta} \geq 0$ everywhere. Then
\begin{equation*}
- f_{\theta}(Q) + 1 \leq \frac{b_s}{Q} - Z_{\theta} \leq \min\left(1, -f_{\theta}(Q) + \frac{3}{Q^2} - 2 \right).
\end{equation*}
and
\begin{equation}
\label{min}
\min\left(1 , -f_{\theta}(Q) + \frac{3}{Q^2} - 2 \right) =
\begin{cases*}
1 & if $f_{\theta}(Q) \leq 3 \frac{1-Q^2}{Q^2}$ \\
-f + \frac{3}{Q^2} - 2 & otherwise
\end{cases*}
\end{equation}
\end{lem}
\begin{proof}
By Theorem \ref{T-ancient} and since $g(t) \in \mathcal{I}$ we know that
\begin{align*}
y &=b_s - Q \leq 0 \\
T_2 &= Qy - x = -a_s + Q b_s + 2\left(1-Q^2\right) \geq 0 \\
T_3 &=a_s - Qb_s + 1 - Q^2 \geq 0
\end{align*}
on $M_2 \times (-\infty, 0]$. Therefore $y \leq 0$ implies
$$ \frac{b_s}{Q} - Z_{\theta} \leq 1 - Z_{\theta}$$
and $T_2 \geq 0$ implies
$$ \frac{b_s}{Q} - Z_{\theta} = \frac{Qb_s - a_s -Q^2 +2}{Q^2} - f_{\theta}(Q) \geq 1 - f_{\theta}(Q)$$
and finally $T_3 \geq 0$ implies
$$ \frac{b_s}{Q} - Z_{\theta} = \frac{Qb_s - a_s -Q^2 +2}{Q^2} - f_{\theta}(Q) \leq - f_{\theta}(Q) + \frac{3}{Q^2} - 2.$$
Now applying the assumption $Z_{\theta} \geq 0 $ proves the desired result.
\end{proof}
Recalling that by definition $$W_{\theta} = w_{\theta}\left( \frac{b_s}{Q} - Z_{\theta}\right),$$ the above Lemma \ref{lem:borders} and concavity of $w_{\theta}(z)$ show that to prove Proposition \ref{Qcal-pos-prop} it suffices to check that for $\theta \in (0,1]$ and $0 \leq Q < Q_{\theta}$ we have
\begin{align*}
\alpha &:= w_{\theta}\left(- f_{\theta}(Q) + 1 \right) \geq 0, \\
\beta &:= w_{\theta}\left( 1 \right)\geq 0 \: \text{ whenever } \: f_{\theta}(Q) \leq 3 \frac{1-Q^2}{Q^2},\\
\gamma &:= w_{\theta}\left(-f_{\theta}(Q) + \frac{3}{Q^2} - 2\right) \geq 0 \: \text{ whenever } \: f_{\theta}(Q) \geq 3 \frac{1-Q^2}{Q^2},
\end{align*}
where $f_{\theta}$ is as defined in Proposition \ref{prop:f}. Note that $\gamma$ is only defined for $Q^2> 0$. This however does not pose a problem as
$$1 \geq f_{\theta}(Q) \geq 3 \frac{1-Q^2}{Q^2}$$
implies that $Q^2 \geq \frac{3}{4}> 0$. Recall that by the properties of $f_{\theta}(Q)$ summarized in Proposition \ref{prop:f} we have
$$ w_{\theta}\left(- f_{\theta}(Q) + 1 \right) = 0 \: \text{ for } \: 0 \leq Q < Q_{\theta},$$
and therefore only need to investigate the sign of $\beta$ and $\gamma$ in their respective regimes. This explains why we chose to define $f_{\theta}(Q)$ via the ordinary differential equation (\ref{C0-ODE}). In the following technical lemma, the proof of which we defer to subsection \ref{technical-lemmas-subsec}, we show that for our choice of $f_{\theta}$ the functions $\beta$ and $\gamma$ are in fact positive in their respective regimes:
\begin{lem}
\label{quadratic-positive-lem}
Fix $\theta \in (0,1]$ and let $f_{\theta}(Q)$ as defined in Proposition \ref{prop:f}. Then for $0 \leq Q < Q_{\theta}$ we have
\begin{equation*}
\beta > 0 \: \text{ whenever } \: f_{\theta}(Q) \leq 3 \frac{1-Q^2}{Q^2},
\end{equation*}
and
\begin{equation*}
\gamma > 0 \: \text{ whenever } \: f_{\theta}(Q) \geq 3 \frac{1-Q^2}{Q^2}.
\end{equation*}
\end{lem}
Now we can prove Proposition \ref{Qcal-pos-prop}.
\begin{proof}[Proof of Proposition \ref{Qcal-pos-prop}]
By Lemma \ref{A2-negative} and Lemma \ref{quadratic-positive-lem} we know that $W_{\theta} \geq 0$ whenever $0 \leq Q < Q_{\theta}$, with equality if and only if $$\frac{b_s}{Q} - Z_{\theta} = 1 - f_{\theta}(Q),$$ which by the definition of $Z_{\theta}$ is equivalent to $T_2 = 0$.
\end{proof}
\begin{remark}
The proof of Proposition \ref{Qcal-pos-prop} essentially implies that for every $\theta \in (0,1]$ the inequality $Z_{\theta} \geq 0$ is preserved on Ricci flow backgrounds in $\mathcal{I}$ satisfying $T_1 \geq 0$, $T_2 \geq 0$ and $T_3 \geq 0$. We do not prove this here, as our proof of Theorem \ref{E-H-ancient-thm} does not rely on this fact.
\end{remark}
\subsection{Proof of main theorem}
\label{ancient-main-thm-subsec}
Next, we prove that the Eguchi-Hanson space is the unique ancient Ricci flow in the class $\mathcal{I}$.
\begin{proof}[Proof of Theorem \ref{E-H-ancient-thm}]
Recall that by Corollary \ref{cor:curv-bound-ancient} there exists a $C_1 > 0$ such that
$$|Rm_{g(t)}|_{g(t)} \leq \frac{C_1}{b^2}.$$
Moreover, by Theorem \ref{T-ancient}
$$T_1, T_2, T_3\geq 0 \: \text{ on } \: M_2\times (\infty , 0]$$
and by Lemma \ref{lem:suc-con-start}
$$ Z_1 \geq 0 \: \text{ on } \: M_2\times (\infty , 0].$$
Hence we may assume that there exists a $\theta_0 \in (0,1]$ such that for all $\theta \in [\theta_0, 1]$
\begin{equation*}
Z_{\theta} = \frac{x}{Q^2} + f_{\theta}(Q) \geq 0 \: \text{ on } \: M_2\times (\infty , 0].
\end{equation*}
\begin{claim}
For every $0 \leq Q_\ast < 1$ we have
\begin{equation*}
\inf \left\{ Z_{\theta_0}(p,t) \: \Big | \: (p,t) \in M_2 \times (\infty, 0] \text{ such that } Q(p,t) \leq Q_\ast \right\} > 0.
\end{equation*}
\end{claim}
\begin{claimproof}
We argue by contradiction. Assume there exists a sequence of points $(p_i, t_i)$ in spacetime such that
\begin{equation*}
Q(p_i, t_i) \leq Q_\ast
\end{equation*}
and
\begin{equation*}
Z_{\theta_0}(p_i, t_i) \rightarrow 0 \: \text{ as } \: i \rightarrow \infty.
\end{equation*}
Consider the rescaled metrics
\begin{equation*}
g_i(t) = \frac{1}{b^2(p_i,t_i)} g( t_i + b^2(p_i,t_i) t), \quad t\in[-\Delta t, 0].
\end{equation*}
For sufficiently small $\Delta t > 0$ the conditions of Proposition \ref{blow-up-prop} are satisfied and therefore $(C_{g_i(0)}\left(p_i, \frac{1}{2}\right), g_i(t), p_i)$, $t\in[-\Delta t, 0]$ subsequentially converges to a Ricci flow $(\mathcal{C}_\infty, g_\infty(t), p_\infty)$, $t \in [-\Delta t, 0]$. Write
$$\Omega = \mathcal{C}_\infty \times [-\Delta t, 0].$$
By construction
\begin{equation*}
Z_{\theta_0}(p_\infty, 0) = \inf_{\Omega } Z_{\theta_0} = 0.
\end{equation*}
Now we need to distinguish two cases:
\subsubsection*{Case 1: $Q(p_\infty,0) < Q_{\theta_0}$} Then there exists an $r\in (0,\frac{1}{2})$ and $\Delta t' \in (0, \Delta t)$ such that on the parabolic set
$$\Omega' = C_{g_\infty(0)}(p_\infty, r) \times [-\Delta t', 0] \subset \Omega$$
we have $Q < Q_{\theta_0}$. By the strong maximum principle of Theorem \ref{maximum-principle} applied to the evolution equation (\ref{Z-epsilon-evol-2}) of $Z_{\theta_0}$ we have
\begin{equation*}
Z_{\theta_0} = 0 \: \text{ on } \: \Omega'
\end{equation*}
and therefore
$$(Z_{\theta_0})_s = (Z_{\theta_0})_{ss} = 0 \: \text{ on } \: \Omega'.$$
By the evolution equation (\ref{Z-epsilon-evol-2}) of $Z_{\theta_0}$ we see that that
\begin{equation*}
0 = \partial_t Z_{\theta_0} = \mathcal{Q}_{\theta_0} \: \text{ in } \Omega',
\end{equation*}
which by Proposition \ref{Qcal-pos-prop} implies
\begin{equation*}
T_2 = Qy - x = 0 \: \text{ in } \Omega'.
\end{equation*}
However, the evolution equation (\ref{T2-evol}) of $T_2$ then implies
$$y = 0 \: \text{ on } \Omega'.$$
and thus also
$$x = 0 \: \text{ on } \Omega'.$$
That in turn implies
\begin{equation*}
Z_{\theta_0}(p_\infty,0) = f(Q(p_\infty,0)) \geq \theta_0 > 0,
\end{equation*}
which is a contradiction.
\subsubsection*{Case 2: $ Q(p_\infty,0) \geq Q_{\theta_0}$}
Recall that at points $(p,t)$ in spacetime satisfying $Q(p,t) \geq Q_{\theta_0}$ we have $Z_{\theta_0}(p,t) = Z_1(p,t)$. In this case we therefore have
$$Z_1(p_\infty,0) = Z_{\theta_0}(p_\infty, 0) = 0.$$
By the strong maximum principle applied to the evolution equation (\ref{Z1-evol}) of $Z_1$ and Lemma \ref{lem:suc-con-start} we deduce
$$Z_1 = 0 \: \text{ on }\Omega.$$
Furthermore, we see that this is only possible when
$$Q = 1 \: \text{ on } \: \Omega.$$
which contradicts
$$ Q(p_\infty,0) \leq Q_\ast < 1.$$
This concludes the proof of the claim.
\end{claimproof}
Thus for every $Q_\ast \in (0,1)$ there exists a $\delta > 0$ such that for all points $(p,t)$ in spacetime satisfying $0 \leq Q(p,t) \leq Q_\ast$ we have
\begin{equation*}
Z_{\theta_0}(p,t) > \delta.
\end{equation*}
By the continuous dependence of $Z_{\theta}$ and $Q_{\theta}$ on $\theta$, and the fact that $Z_{\theta} = Z_{\theta'}$ at points $(p,t)$ in spacetime at which $Q(p,t) \geq \max(Q_{\theta}, Q_{\theta'})$, there exists an $\theta_1 < \theta_0$ such that for $\theta \in [\theta_1, 1]$
\begin{equation*}
Z_{\theta} \geq 0 \: \text{ on } \: M_2 \times (-\infty, 0].
\end{equation*}
Now consider the set
$$ \mathcal{E} = \left \{ \theta \in (0,1] \: \big | \: Z_{\theta'} \geq 0 \text{ for } \theta \leq \theta' \leq 1 \right\} \subseteq (0,1]$$
The above argument shows that $\mathcal{E}$ is an open subset of $(0,1]$. As the condition $Z_{\theta} \geq 0$ is closed and $f_{\theta}$ depends continuously on $\theta$, it follows that $\mathcal{E}$ is also a closed subset of $(0,1]$. Hence by connectedness of $(0,1]$ it follows that $\mathcal{E} = (0,1]$ and thus for all $\theta \in (0,1]$
\begin{equation*}
Z_{\theta} \geq 0 \: \text{ on } \: M_2 \times (-\infty, 0].
\end{equation*}
Note that by the strong maximum principle applied to the evolution equation (\ref{Q-evol}) of $Q$
$$Q < 1 \: \text{ on } \: M_2 \times (-\infty, 0],$$
as otherwise $Q=1$ everywhere, which is not true. As $Z_{\theta} = \frac{x}{Q^2} + f_{\theta}(Q)$ and by Proposition \ref{prop:f} for every $0 \leq Q < 1$ we have $f_{\theta}(Q) \rightarrow 0$ as $\theta \rightarrow 0$ it follows that
$$x \geq 0 \: \text{ on } \: M_2 \times (-\infty, 0].$$
However, as
$$T_2 = Qy - x \geq 0 \: \text{ on } \: M_2 \times (-\infty, 0]$$
and $y \leq 0$ by the assumption that $g(t) \in \mathcal{I}$ it follows that
$$ x = y = 0 \: \text{ on } \: M_2 \times (-\infty, 0].$$
By Lemma \ref{unique-Kahler-lem} we conclude that $(M_2, g(t)), t\in (-\infty, 0],$ is stationary and homothetic to the Eguchi-Hanson space.
\end{proof}
Now we prove Corollary \ref{E-H-blowup}.
\begin{proof}[Proof of Corollary \ref{E-H-blowup}]
By Theorem \ref{curv-bound} and the fact that $b_s \leq 0$ for metrics in $\mathcal{I}$ there exists a $C_1 > 0$ such that
\begin{equation*}
|Rm_{g(t)}|_{g(t)}(p) \leq \frac{C_1}{b^2(p,t)} \leq \frac{C_1}{b^2(o,t)}.
\end{equation*}
This shows that
$$b(o,t) \rightarrow 0 \text{ as } t \rightarrow T_{sing}.$$
As by assumption
$$ C \coloneqq \sup \frac{b(p_i, t_i)}{b(o,t_i)} < \infty $$
and $y \leq 0$ by the fact that $g(t) \in \mathcal{I}$ it follows that
\begin{equation*}
\partial_t b(o,t) \leq 0, \quad t \in [0,T_{sing}),
\end{equation*}
by (\ref{b0-evol-k2}). We deduce
$$b(p_i,t_i) \rightarrow 0 \text{ as } i \rightarrow \infty.$$
Consider the rescaled metrics
$$g_i(t) = \frac{1}{b^2(p_i,t_i)} g\left(t_i + t b^2(p_i, t_i) \right), \quad t\in[-t_i b^{-2}(p_i,t_i), 0].$$
These satisfy the curvature bound
\begin{equation*}
|Rm_{g_i(t)}|_{g_i(t)} \leq C^2 C_1 \: \text{ on } \: M_2 \times [-t_i b^{-2}(p_i,t_i), 0].
\end{equation*}
By Theorem \ref{thm:no-local-collapsing} the rescaled metrics $g_i(t)$ are $\kappa$-non-collapsed at larger and larger scales. Hence by Corollary \ref{cor:compactness-complete-flows} the Ricci flows $(M_2, g_i(t), p_i)$ subsequentially converge to a pointed ancient Ricci flow $(M_2, g_{\infty}(t), p_\infty)$, $ - \infty< t < 0$, with $g_\infty(t) \in \mathcal{I}$. By Theorem \ref{E-H-ancient-thm} it follows that $g_\infty(t)$ is stationary and homothetic to the Eguchi-Hanson metric.
\end{proof}
\subsection{Proof of technical lemmas}
\label{technical-lemmas-subsec}
In this subsection we collect the proofs of the technical lemmas we relied on above.
\begin{proof}[Proof of Lemma \ref{f-existence-lem}]
We apply \cite[Theorem 9.2]{A17} to prove this lemma. Define
$$r = Q^2.$$
Then
\begin{align*}
f' &= 2 Q f_r \\
f'' &= 2 f_r + 4 r f_{rr},
\end{align*}
where $'$ denotes the derivative with respect to $Q$ and subscript $r$ denotes the derivative with respect to $r$. Rewriting the differential equation (\ref{C0-ODE}) with respect to the independent variable $r$, we obtain
\begin{equation}
\label{f-in-r-eqn}
r f_{rr} = \frac{1}{2(1-r)} \left( 6r - 4 - r f\right)f_r + \frac{f}{8 \left( 1- r\right)^2 }\left(f^2 r + 3 fr - 6f - 6r + 8 \right)
\end{equation}
At $r=0$ the right hand side must equal zero, which can be ensured by requiring
$$ f_r (0) = \frac{1}{2} f_0 - \frac{3}{8} f_0^2 $$
Now define
\begin{align*}
u_1 &= f - f_0 \\
u_2 &= f_r - f_r(0).
\end{align*}
Then (\ref{f-in-r-eqn}) can be written as a system of equations of the form
$$ r (u_i)_r = P_i(\vec{u},r,f_0), \quad i = 1, 2,$$
where
\begin{align*}
P: \mathbb{R}^2 \times (-1,1) \times \mathbb{R} &\longrightarrow \mathbb{R}^2 \\
(\vec{u}, r, f_0) &\longrightarrow P(\vec{u}, r, f_0)
\end{align*}
is an analytic vector-valued function of several variables satisfying
$$P(\vec{0}, 0, f_0) = 0$$
for all $f_0 \in (-1, 1)$. A computation shows that
$$ \frac{\partial P} {\partial u} \Big |_{(\vec{0},0, f_0)} = \begin{pmatrix} 0 & 0 \\ 1- \frac{3}{2} f_0 & -2 \end{pmatrix} $$
This matrix has no positive integer eigenvalues and furthermore
$$
B = \sup_{\substack{m\in \mathbb{N} \\ f_0 \in \mathbb{R}}} \norm{ \left( m I_2 - \frac{\partial P} {\partial u} \Big |_{(\vec{0},0, f_0)}\right)^{-1}} < \infty,
$$
where $I_2$ is the $2\times2$ identity matrix. By \cite[Theorem 9.2]{A17} the desired result follows.
\end{proof}
\begin{proof}[Proof of Lemma \ref{f-inc-lem}]
At $Q=0$, we have by L'H\^opital's Rule that
\begin{equation}
\label{ddf-origin}
f'' = \frac{1}{4} f \left( 4 - 3 f \right) > 0 \: \text{ for } 0 < f(0) < \frac{4}{3}.
\end{equation}
Furthermore, at an extremum of $f$ we have
\begin{equation}
\label{df-extremum}
f'' = \frac{2 f}{4\left(1-Q^2\right)^2}\left( f^2 Q^2+3 f Q^2-6 f-6 Q^2+8 \right).
\end{equation}
Defining the polynomial
\begin{equation*}
p_1(f,Q^2) = f^2 Q^2+3 f Q^2-6 f-6 Q^2+8
\end{equation*}
we see that
\begin{equation*}
\partial_{Q^2} p_1 = f^2 + 3f - 6 < 0\: \text{ for } 0 < f \leq 1.
\end{equation*}
Therefore
\begin{equation*}
p_1(f, Q) > p_1(f, 1) = f^2 - 3f + 2 \geq 0 \: \text{ for } 0 < f \leq 1 \text{ and } 0 \leq Q < 1.
\end{equation*}
From (\ref{df-extremum}) it then follows that $f'>0$ for as long as $0 < f \leq 1$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{f-lem-2}]
We argue by contradiction. Assume there does not exist such a $Q_{\theta} < 1$. Then by Lemma \ref{f-inc-lem} we have $f' >0$ on $Q \in (0,1)$ and hence
\begin{equation*}
\lim_{Q \rightarrow 1^-} f(Q) = l \leq 1
\end{equation*}
exists. By (\ref{C0-ODE}) we have
\begin{align*}
4 \left(1-Q^2\right)^2 f'' = &- 4 \left(1-Q^2\right) \left(Q^2 f-5 Q^2+3\right) \frac{f'}{Q} \\ \nonumber
& \qquad + 2f\left( Q^2\left(f^2+3 f-6\right)+8-6 f \right).
\end{align*}
For $Q^2 > 1 - \frac{\theta}{4}$ and $0 < f < 1$ we have
\begin{equation*}
Q^2 f-5 Q^2+3 < 3 - 4 Q^2 < -1 + \theta
\end{equation*}
and
\begin{equation*}
Q^2\left(f^2+3 f-6\right)+8-6 f > (2-f)(1-f),
\end{equation*}
as for $0<f<1$
\begin{equation*}
f^2+3 f-6 < 0.
\end{equation*}
Hence for $ Q^2 > 1 - \frac{\theta}{4}$ and $0< f< 1$ we obtain the following inequality
\begin{align}
\label{ddf-ODI}
f'' &\geq \alpha \frac{f'}{1-Q} + \beta \frac{1-f}{(1-Q)^2},
\end{align}
where
\begin{align*}
\alpha &= \frac{1-\theta}{Q(1+Q)}\\
\beta &= \frac{f\left(2 - f\right)}{2 \left(1+Q\right)^2}.
\end{align*}
Furthermore we observe that
\begin{align*}
\alpha &\rightarrow \frac{1}{2} \: \text{ as } \: f,Q \rightarrow 1 \\
\beta &\rightarrow \frac{1}{8} \: \text{ as } \: f,Q \rightarrow 1.
\end{align*}
If $l < 1$, then there would exists a $Q_\ast<1$ such that for $Q \geq Q_\ast$ we have
\begin{equation*}
f'' \geq \frac{1}{10} \frac{l(1-l)(2-l)}{(1-Q)^2}.
\end{equation*}
Here $\frac{1}{10}$ can be replaced by any positive number less that $\frac{1}{8}$. However, integrating this differential inequality shows that in this case $f$ would reach $1$ before $Q = 1$, leading to a contradiction of our assumption. Therefore we may assume that $l =1$.
Defining
\begin{equation*}
g(Q) = 1 - f(Q)
\end{equation*}
we obtain the differential inequality
\begin{equation}
\label{ODI-g}
g''(Q) \leq \alpha \frac{g'(Q)}{1-Q} - \beta \frac{g(Q)}{(1-Q)^2}.
\end{equation}
By Lemma \ref{f-inc-lem} we know that $g(Q) > 0$ and $g'(Q) < 0$ on $Q \in (0,1)$.
\begin{claim}
The function $g(Q)$ reaches zero before $Q = 1$.
\end{claim}
\begin{claimproof}
By our assumption that $l=1$ we know that there exists a $Q_\ast < 1$ such that for $Q > Q_\ast$
\begin{equation*}
g(Q) < \theta.
\end{equation*}
Furthermore, by choosing $Q_\ast < 1$ sufficiently close to 1, we may assume that for $Q \geq Q_\ast$
\begin{equation}
\label{ODI-g-2}
g''(Q) \leq \frac{3}{7} \frac{g'(Q)}{1-Q} - \frac{5}{49} \frac{g(Q)}{(1-Q)^2},
\end{equation}
as $ \frac{3}{7} < \frac{1}{2}$ and $\frac{5}{49} < \frac{1}{8}$. Now take the substitution
$$g(Q) = u\left( r \right)$$
for
$$r = -\ln(1- Q).$$
Then the (\ref{ODI-g-2}) becomes
$$\frac{d^2u}{dr^2} + \frac{4}{7} \frac{du}{dr} + \frac{5}{49} u \leq 0.$$
The corresponding ordinary differential equation is of oscillatory type, which motivates the substitution
$$ u(r) = e^{-\frac{2}{7} r} v(r)$$
yielding the inequality
$$ \frac{d^2v}{dr^2} \leq - \frac{1}{49} v.$$
Hence $v$ reaches $0$ in finite $r$, which tracing back the substitutions, shows that $g$ must reach zero before $Q = 1$.
\end{claimproof}
Now it remains to prove the assertion (1), (2) and (3). We prove (1). First fix a $\theta \in (0,1)$. By Lemma \ref{f-inc-lem} we know that $f'_\theta(Q_{\theta}) > 0$. Now extend the solution $f_\theta$ of (\ref{C0-ODE}) to the interval $[0, Q_{\theta} + c]$, $c>0$, such that $f'_{\theta}(Q) > 0$ on $(0, Q_{\theta} + c]$. By the continuous dependence of $f_{\theta}(Q)$ on $\theta$ it follows that for every $\epsilon > 0$ there exists a $\delta > 0$ such that $|\theta - \theta'| \leq \delta$ implies $|Q_\theta - Q_{\theta'}| < \epsilon$. To prove the continuity of $Q_\theta$ at $\theta = 1$, note that $Q_1 = 1$, $f_1(0) = 1$ and $f'_1=0$. Then recall from the proof of Lemma \ref{f-inc-lem} that $f''(0) > 0$ when $0 < f(0) < \frac{4}{3}$. Now applying the same argument as above yields continuity of $Q_\theta$ at $\theta = 1$ and therefore proves (1).
Assertion (2) follows from the fact that for the initial condition $f(0) = 0$ the corresponding solution to the ODE (\ref{C0-ODE}) is $f(Q) = 0$. By the continuous dependence of $f$ on $f(0) = 0$ we deduce that
\begin{equation*}
Q_{\theta} \rightarrow 1 \text{ as } \theta \rightarrow 0.
\end{equation*}
Finally, assertion (3) follows by definition.
\end{proof}
\begin{proof}[Proof of Lemma \ref{A2-negative}]
First note that by Lemma \ref{f-inc-lem} we know that $f,f' \geq 0$ on $[0, Q_{\theta}]$. Solving the ODE (\ref{C0-ODE}) for $f''$, we obtain
\begin{align}
\label{ddf-exp}
f'' &= \frac{1}{2 Q \left(1-Q^2\right)^2} \Big( 2 Q^4 f f'-10 Q^4 f'-2 Q^2 f f'+16 Q^2 f' \\ \nonumber
&\quad -6 f' + Q^3 f^3+3 Q^3 f^2-6 Q^3 f-6 Q f^2+8 Q f \Big)
\end{align}
Substituting expression (\ref{ddf-exp}) into (\ref{bs2-coeff}) yields
\begin{align*}
A_2&= - \frac{1}{2 \left(1-Q^2\right)^2} \Big ( 2 Q^3 \left(1-Q^2\right) \left( 2 - Q^2 f \right) f' \\ \nonumber
& \qquad + f^3 Q^6+3 f^2 Q^6-6 f^2 Q^4-2 f Q^6+4 f Q^2+4 Q^6-12 Q^2+8 \Big )
\end{align*}
Defining
\begin{equation*}
p_2 (f, Q^2) = f^3 Q^6+3 f^2 Q^6-6 f^2 Q^4-2 f Q^6+4 f Q^2+4 Q^6-12 Q^2+8
\end{equation*}
we then only need to check that
\begin{equation*}
p_2 \geq 0 \: \text{ for } \: 0 \leq f, Q \leq 1.
\end{equation*}
Defining
\begin{equation*}
\tilde{p}_2 (F, Q^2) = p_2(f, Q^2)
\end{equation*}
for
\begin{equation*}
F = f Q^2
\end{equation*}
we see that
\begin{equation*}
\partial_{Q^2} \tilde{p}_2 \Big |_F = 3 F^2 - 4FQ^2 + 12 \left( Q^4 -1 \right) \leq 0 \: \text{ for } \: 0\leq F, Q \leq 1
\end{equation*}
with equality only at $F= 0$, $Q = 1$. Therefore the minimum of $p_2$ is attained when $Q = 1$, in which case we have we have
\begin{equation*}
\tilde{p}_2(F, 1) = \left(F-2\right) \left(F-1\right) F \geq 0 \: \text{ for } \: 0\leq F \leq 1.
\end{equation*}
As $ 0 \leq Q \leq Q_{\theta} < 1$, we actually have $p_2(f, Q^2) > 0$ on $(f,Q) \in [0,1] \times [0, Q_\theta]$ and the result follows.
\end{proof}
\begin{proof}[Proof of Lemma \ref{quadratic-positive-lem}]
A computation shows that
\begin{align}
\label{Cupper1}
\beta &= -\left(Q^2f+2 Q^2-2\right)^2 f'' \\ \nonumber
& + \left(-3 Q^4 f^2-14 Q^4 f+12 Q^2 f-20 Q^4+32 Q^2-12\right) \frac{f'}{Q} \\ \nonumber
& -4 Q^2 f^2-12 Q^2 f+16 f
\end{align}
and
\begin{align}
\label{Cupper2}
\gamma &= -\left(1-Q^2\right)^2 f'' + (1- Q^2)(2 Q^2 f+11 Q^2-9)\frac{f'}{Q} \\ \nonumber
& + 2 Q^2 f^3+6 Q^2 f^2-12 f^2-6 Q^2 f+\frac{30 f}{Q^2}-20 f-18 Q^2+\frac{54}{Q^2}-\frac{36}{Q^4},
\end{align}
where we omitted the dependence of $f$ on $\theta$ and $Q$ for brevity. We first show that $\beta > 0$ in the region
\begin{equation*}
R_1 = \left \{(f,Q) \: \Big | \: 0 \leq Q \leq Q_{\theta}, 0 < f \leq \min\left(1, 3 \frac{1-Q^2}{Q^2}\right) \right\}
\end{equation*}
of the $f$-$Q$-plane. Plugging the expression (\ref{ddf-exp}) of $f''$ into the expression (\ref{Cupper1}) for $\beta$, we obtain
\begin{align*}
2 \left(1-Q^2 \right)^2 \beta &= 2 ff' Q(1 - Q^2)\left( Q^4 f^2+2 Q^4 f-4 Q^2 f-2 Q^4-2 Q^2+4 \right) \\ \nonumber
& + f^2 \Big ( -Q^6f^3-7 Q^6 f^2+10 Q^4 f^2-10 Q^6 f+36 Q^4 f \\ \nonumber
& \quad\quad\quad\qquad -28 Q^2 f+4 Q^6+8 Q^4-36 Q^2+24 \Big)
\end{align*}
An application of L'H\^opital's Rule shows that $\beta = 12 f^2 > 0$ at $Q=0$ and therefore we may assume that $Q>0$. Recall that $f,f' > 0$ on $(0,Q_{\theta})$ by Lemma \ref{f-inc-lem}. Hence it suffices to show that the polynomials
\begin{equation*}
p_3(f,Q^2) = Q^4 f^2+2 Q^4 f-4 Q^2 f-2 Q^4-2 Q^2+4
\end{equation*}
and
\begin{align*}
p_4(f,Q^2)&= -Q^6f^3-7 Q^6 f^2+10 Q^4 f^2-10 Q^6 f+36 Q^4 f \\ \nonumber
& \quad\quad-28 Q^2 f+4 Q^6+8 Q^4-36 Q^2+24
\end{align*}
are positive on $R_1 \cap \{ Q > 0 \}$. A computation shows
\begin{equation*}
\partial_{Q^2} p_3 = 2\left(f^2 Q^2 - 1 \right) + 4f \left(Q^2 -1 \right) - 4 Q^2 \leq 0
\end{equation*}
and hence for every $(f,Q) \in R_1$ we have
\begin{equation*}
p_3(f, Q^2) \geq p_3\left(f, \frac{3}{3 +f}\right) = \left(\frac{f}{3+f}\right)^2 > 0.
\end{equation*}
To show that $p_4 > 0$ on $R_1$ is more complicated. For this we introduce the variable
\begin{equation*}
F= fQ^2
\end{equation*}
and polynomial
\begin{equation*}
\tilde{p}_4 (F, Q^2) = p_4( f, Q^2)
\end{equation*}
Then
\begin{align*}
\tilde{p}_4(F, Q^2) &= 24 -28 F+10 F^2 -F^3+\left(-7 F^2+36 F-36\right) Q^2 \\ \nonumber
& \quad +(8-10 F) Q^4+4 Q^6
\end{align*}
which gives
$$\partial_{Q^2} \tilde{p}_4 = -7 F^2 - 4 F \left(-9 + 5 Q^2\right) + 4 \left(-9 + 4 Q^2 + 3 Q^4\right).$$
As this expression is concave in $F$ one can easily check that in the region
$$ 0 < F \leq \min\left( Q^2, 3\left(1-Q^2\right) \right), 0 \leq Q \leq 1$$
of the $Q^2$-$F$-plane we have
$$\partial_{Q^2} \tilde{p}_4 \leq 0$$
and thus
\begin{equation*}
\tilde{p}_4(F, Q^2) \geq \tilde{p}_4\left( F, \frac{3-F}{3}\right) = \frac{1}{27} F \left(2 F^2-3 F+18\right) > 0.
\end{equation*}
From this we conclude that $p_4 > 0$ on $R_1\cap\{Q > 0 \}$ and hence $\beta > 0$ on $R_1$.
We adopt the same procedure to show that $\gamma > 0$ in the region
\begin{equation*}
R_2 = \left \{(f,Q) \: \Big | \: 0 \leq Q \leq Q_{\theta}, 3 \frac{1-Q^2}{Q^2} \leq f \leq 1 \right\}.
\end{equation*}
Substituting the expression (\ref{ddf-exp}) for $f''$ into the expression (\ref{Cupper2}) for $\gamma$ we obtain
\begin{align*}
\gamma &= 3 \left(1-Q^2\right) \left(Q^2 f+2 Q^2-2\right) \frac{f'}{Q} +\frac{1}{Q^4} \Big( \frac{3 f^3 Q^6}{2}+\frac{9 f^2 Q^6}{2}-9 f^2 Q^4 \\ \nonumber
& \quad -3 f Q^6-24 f Q^4+30 f Q^2-18 Q^6+54 Q^2-36\Big)
\end{align*}
First notice that for any point $(f,Q) \in R_2$
$$ 3\frac{1- Q^2}{Q^2} \leq 1$$
and hence
$$ \sqrt{\frac{3}{4}} \leq Q \leq Q_{\theta} < 1.$$
Then from
\begin{equation*}
f \geq 3 \frac{1-Q^2}{Q^2}
\end{equation*}
it follows that
\begin{equation*}
Q^2 f + 2 Q^2 - 2 \geq 1- Q^2 > 0 \: \text{ on } \: R_2.
\end{equation*}
Therefore the first term in the expression for $\gamma$ is positive and we only need to prove non-negativity of the second term. For this define the polynomials
\begin{align*}
p_5(f, Q^2) &= \frac{3 f^3 Q^6}{2}+\frac{9 f^2 Q^6}{2}-9 f^2 Q^4-3 f Q^6-24 f Q^4 \\ \nonumber
& \quad +30 f Q^2-18 Q^6+54 Q^2-36
\end{align*}
and
\begin{align*}
\tilde{p_5}(F,Q^2) &= \frac{3 F^3}{2}+\frac{9 F^2 Q^2}{2}-9 F^2-3 F Q^4-24 F Q^2 \\ \nonumber
& \quad +30 F-18 Q^6+54 Q^2-36,
\end{align*}
where we again took $F = f Q^2$. Computing the partial derivatives
\begin{align*}
\partial_{Q^2} \tilde{p}_5 &= \frac{9 F^2}{2}-6 F Q^2-24 F-54 Q^4+54 \\
\partial_{F} \tilde{p}_5 &= \frac{9 F^2}{2}+9 F Q^2-18 F-3 Q^4-24 Q^2+30
\end{align*}
We deduce that at an local extrema $\partial_{Q^2} \tilde{p}_3 = \partial_{F} \tilde{p}_4=0$
\begin{equation*}
F = \frac{-17 Q^4+8 Q^2+8}{5 Q^2+2}
\end{equation*}
and
\begin{equation*}
80 + 144 Q^2 - 188 Q^4 - 200 Q^6 + 307 Q^8 = 0.
\end{equation*}
In Lemma \ref{lem:Q-pol-zeros} below we show that the equation for $Q^2$ has no zeros in the interval $Q^2 \in [\frac{3}{4}, 1]$. Therefore $p_5(F,Q^2)$ has no local extrema in the region $R_3$ of the $(F,Q^2)$-plane enclosed by the curves
\begin{align*}
L_1&: Q^2 = 1, 0\leq F\leq1 \\
L_2&: \frac{2}{3} \leq Q^2 \leq 1, 0\leq F\leq1 \\
L_3&: \frac{2}{3} \leq Q^2 \leq 1, F = 3 \left(1-Q^2\right)
\end{align*}
As the set of the $(F,Q^2)$-plane corresponding to $R_2$ is a subset of $R_3$ it suffices to check non-negativity of $\tilde{p}_5$ on the boundary of the region $R_3$. There we have
\begin{equation*}
\tilde{p}_5(F, 1) = \frac{3}{2} F\left(1 - F\right) \left(2-F\right) \geq 0 \quad \text{ on $L_1$}
\end{equation*}
and
\begin{equation*}
\tilde{p}_5(1, Q^2) = \frac{3}{2} (1-Q^2) \left(12 Q^4+14 Q^2-9\right) \geq 0 \quad \text{ on $L_2$}
\end{equation*}
and
\begin{equation*}
\tilde{p}_5\left(3(1-Q^2), Q^2\right) = \frac{9}{2} \left(1-Q^2\right) \left(2 Q^4-3 Q^2+3\right) \geq 0 \quad \text{ on $L_3$}
\end{equation*}
This concludes the proof.
\end{proof}
\begin{lem}
\label{lem:Q-pol-zeros}
The equation $$80 + 144 r - 188 r^2 - 200 r^3 + 307 r^4 = 0$$ has no roots in the interval $[0,1]$.
\end{lem}
\begin{proof}
For $r \in [0,1]$ we have
\begin{align*}
80 + 144 r - 188 r^2 - 200 r^3 + 307 r^4 &\geq (80 - 6r) + 150r - 200r^2 - 200 r^3 + 300 r^4 \\
&\geq 24 + 50\left(1 + 3r - 4r^2 - 4r^3 + 6r^4 \right).
\end{align*}
Then we see that
\begin{align*}
1 + 3r - 4r^2 - 4r^3 + 6r^4 &= \left(1 - 2r^2\right)^2 +r \left( 2r^3 - 4r^2 + 3 \right) \\
&\geq \left(1 - 2r^2\right)^2 +r \left( 2r^4 - 4r^2 + 3 \right) \\
&\geq \left(1 - 2r^2\right)^2 +r \left( 2(r^2 -1)^2 + 1\right) \\
&\geq 0
\end{align*}
This concludes the proof.
\end{proof}
\section{Discussion of blow-up limits in $k = 2$ case}
\label{sec:blow-up}
In this section we investigate the possible blow-up limits of a Ricci flow $(M_2, g(t))$, $t\in[0, T_{sing})$, starting from an initial metric $g(0) \in \mathcal{I}$ with $\sup_{p \in M_2} b(p,0) < \infty$. By Lemma \ref{sing-time-finite} and Corollary \ref{E-H-blowup} we know that such flows develop a Type II singularity modeled on the Eguchi-Hanson space as the area of the non-principal orbit $S^2_o$ shrinks to zero. One expects, however, that at larger distance scales from $S^2_o$ one could also see other blow-up limits. The goal of this section is to show that these are in fact limited to the following four possibilities: (i) the Eguchi-Hanson space, (ii) the flat $\mathbb{R}^4 / \mathbb{Z}_2$ orbifold, (iii) the 4d Bryant soliton quotiented by $\mathbb{Z}_2$ and (iv) the shrinking cylinder $\mathbb{R} \times \mathbb{R} P^3$.
Next, we state the main theorem of this section:
\begin{thm}[Blow-up limits]
\label{blow-up-thm}
Let $(M_2, g(t))$, $t \in [0, T_{sing})$, be a Ricci flow starting from an initial metric $g(0) \in \mathcal{I}$ (see Definition \ref{def:I}) with $\sup_{p \in M_2} b(p,0) < \infty$. Let $(p_i,t_i)$ be a sequence of points in spacetime with $b(p_i, t_i) \rightarrow 0$. Passing to a subsequence, we may assume that we are in one of the four cases listed below.
\begin{enumerate}[label=(\roman*)]
\item $\lim_{i \rightarrow \infty} \frac{b(p_i,t_i)}{b(o,t_i)} < \infty$
\item $\lim_{i \rightarrow \infty} \frac{b(p_i,t_i)}{b(o,t_i)} = \infty$ and $\lim_{i \rightarrow \infty} b_s(p_i, t_i) = 1$
\item $\lim_{i \rightarrow \infty} \frac{b(p_i,t_i)}{b(o,t_i)} = \infty$ and $\lim_{i \rightarrow \infty} b_s(p_i, t_i) \in (0,1)$
\item $\lim_{i \rightarrow \infty} \frac{b(p_i,t_i)}{b(o,t_i)} = \infty$ and $\lim_{i \rightarrow \infty} b_s(p_i, t_i) = 0$
\end{enumerate}
Consider the dilated Ricci flows
$$g_i (t) = \frac{1}{b^2(p_i,t_i)} g\left(t_i + b^2(p_i, t_i) t \right), \quad t \in [- b(p_i,t_i)^{-2} t_i , 0].$$
Then $(M_2, g_i(t), p_i)$, $t \in [- b(p_i,t_i)^{-2} t_i , 0]$, subsequentially converges, in the Cheeger-Gromov sense, to an ancient Ricci flow $(M_\infty, g_\infty(t), p_\infty)$, $t \in (-\infty, 0]$. Depending on the limiting property of the sequence $(p_i, t_i)$ we have:
\begin{enumerate}[label=(\roman*)]
\item $M_\infty \cong M_2$ and $g_\infty(t)$ is stationary and homothetic to the Eguchi-Hanson metric
\item $M_\infty \cong \mathbb{R}^4\setminus\{0\} / \mathbb{Z}_2$ and $g_\infty(t)$ can be extended to a smooth orbifold Ricci flow on $\mathbb{R}^4/\mathbb{Z}_2$ that is stationary and isometric to the flat orbifold $\mathbb{R}^4/\mathbb{Z}_2$
\item $M_\infty \cong \mathbb{R}^4\setminus\{0\} / \mathbb{Z}_2$ and $g_\infty(t)$ can be extended to a smooth orbifold Ricci flow on $\mathbb{R}^4/\mathbb{Z}_2$ that is homothetic to the 4d Bryant soliton quotiented by $\mathbb{Z}_2$
\item $M_\infty \cong \mathbb{R} \times \mathbb{R} P^3$ and $g_\infty(t)$ is homothetic to a shrinking cylinder
\end{enumerate}
\end{thm}
\begin{remark}
Notice that in Theorem \ref{blow-up-thm} we do not claim that all blow-up limits (i)-(iv) actually occur. If the Eguchi-Hanson singularity is isolated one would only see (i) and (ii).
\end{remark}
\subsection{Outline of proof}
\label{blow-up-thm-outline}
Assume we are given a sequence of points $(p_i, t_i)$ in spacetime with $b(p_i, t_i) \rightarrow 0$. Consider the rescaled metrics
$$g_i(t) = \frac{1}{b(p_i,t_i)^2} g( t_i + b(p_i, t_i)^2 t), \quad t \in [- b(p_i, t_i)^{-2} t_i, 0],$$
normalized such that $b(p_i, 0) = 1$. By passing to a subsequence we may assume that either
$$\text{ (I) } \sup_i \frac{b(p_i,t_i)}{b(o,t_i)} < \infty \quad \text{or} \quad \text{ (II) } \lim_{i \rightarrow \infty} \frac{b(p_i,t_i)}{b(o,t_i)} = \infty.$$
In case (I) we know by Corollary \ref{E-H-blowup} that $(M_2, g_i(t), p_i)$ subsequentially converges to the Eguchi-Hanson space, which is the blow-up limit (i) from above. Therefore we only need to investigate the behavior in case (II), i.e. at scales larger than the forming Eguchi-Hanson singularity. At these scales Lemma \ref{QT2bddb-limit-lem} yields very important geometric information. In particular, we show that for every $\epsilon > 0$ there exist constants $C,\delta > 0$ such that the following holds: For all points $(p,t)$ in spacetime at which $C b(o,t) \leq b(p,t) \leq \delta$ we have
\begin{itemize}
\item $Q \geq 1 - \epsilon$
\item $T_{F_1} \coloneqq b b_{ss} + 1- b_s^2 \geq -\epsilon$
\item $T_{F_2} \coloneqq b b_{ss} + 1- b_s^2 - \left(1-b_s^2\right)^2 \leq \epsilon$
\item $\partial_t b^2 \leq \epsilon$
\end{itemize}
Hence a blow-up limit $(M_\infty, g_\infty(t), p_\infty)$ in case (II) satisfies $Q = 1$, $T_{F_1} \geq 0$ and $T_{F_2} \leq 0$. Therefore $M_\infty$ is rotationally symmetric and satisfies
$$\frac{1 - b^2_s}{b^2} - \frac{(1-b_s^2)^2}{b^2} \leq -\frac{b_{ss}}{b} \leq \frac{1 - b_s^2}{b^2}.$$
As $-\frac{b_{ss}}{b}$ and $\frac{1-b_s^2}{b^2}$ are the only non-zero components of the curvature tensor of a rotationally symmetric metric, we see that blow-up limits of case (II) satisfy the curvature bound
$$|Rm_{g_\infty(t)}|_{g_\infty(t)} \leq c \frac{1-b_s^2}{b^2}$$
for some universal constant $c>0$.
We now briefly explain some of the geometric intuition behind the quantities $T_{F_1}$ and $T_{F_2}$ for rotationally symmetric metrics. When $T_{F_1} = 0$ the underlying space is of constant curvature and therefore isometric to a sphere, the flat plane or hyperbolic space, depending on the sign of the scalar curvature. On the other hand solving the ODE $T_{F_2} = 0$ one can show that $b_s \rightarrow 0$ as $s \rightarrow \infty$ and the underlying space is asymptotically cylindrical. Thus blow-up limits in case (II) are rotationally symmetric spaces that are `sandwiched' between a sphere and an asymptotically cylindrical space.
We need to divide case (II) into three subcases in order to investigate the possible blow-up limits: By passing to a subsequence we may assume that
$$ \text{(II.a) } b_s(p_i, t_i) \rightarrow 1 \: \text{ or } \: \text{ (II.b) } b_s(p_i,t_i) \rightarrow \eta \in (0,1) \: \text{ or } \: \text{ (II.c) } b_s(p_i,t_i) \rightarrow 0.$$
For (II.a) and (II.c) we show in Lemma \ref{flat-orbifold-blow-up-lem} and Lemma \ref{cylinder-blow-up-lem} that $(M_2, g_i(t), p_i)$ subsequentially converges to the flat orbifold $\mathbb{R}^4 / \mathbb{Z}_2$ and the shrinking cylinder $\mathbb{R} \times \mathbb{R} P^3$, respectively. The main idea is that by the strong maximum principle applied to the evolution equation (\ref{db-evol}) of $b_s$ when $Q=1$ a minimum $b_s = 0$ or a maximum $b_s = 1$ can only be attained if $b_s$ is constant everywhere.
Proving that the blow-up limit in case (II.b) is an ancient orbifold Ricci flow, which is homothetic to the 4d Bryant soliton quotiented by $\mathbb{Z}_2$, is trickier. The construction is carried out in Lemma \ref{lem:orbifold-blowup}, the proof of which we sketch here: Fix a $T > 0$ and define
$$E_{p,t,n} \coloneqq \left\{p' \in M_2 \: \Big | \: b(p',t) > \frac{b(p,t)}{n} \right\} \subseteq M_2.$$
Then consider the rescaled metrics $g_i(t)$, defined as above, on the parabolic neighborhoods
$$\Omega_{i,n} \coloneqq E_{p_i, t_i, n} \times [-T - 1, 0]$$
in spacetime. By Lemma \ref{QT2bddb-limit-lem} we know that $\partial_t b^2 \rightarrow 0$ uniformly as $b \rightarrow 0$. Hence from the curvature bound
\begin{equation}
\label{curv-bound-intro}
|Rm_{g(t)}|_{g(t)} \leq \frac{C_1}{b^{2}}
\end{equation}
of Theorem \ref{curv-bound}, we see that the curvature of $g_i(t)$ is bounded by $Cn^2$ on $\Omega_{i,n}$ for $i$ sufficiently large and $C>0$ some constant. The difficulty in constructing the limiting orbifold flow arises from the fact that the curvature bound (\ref{curv-bound-intro}) degenerates as $n \rightarrow \infty$. We get around this by exploiting the inequalities on $T_{F_1}$ and $T_{F_2}$ derived in Lemma \ref{QT2bddb-limit-lem}, to find a uniform curvature bound independent of $n$. From here it is then easy to construct the orbifold Ricci flow $g_\infty(t)$, $t \in [-T, 0]$, on $\mathbb{R}^4 \setminus \{0\} /\mathbb{Z}_2$ by taking the limit $n\rightarrow \infty$. Via Lemma \ref{lem:C11-regularity}, and Theorem \ref{RF-removable-singularity} in the Appendix B, we show that $g_\infty(t)$ can be extended to a smooth orbifold Ricci flow on $\mathbb{R}^4/\mathbb{Z}_2$. Apriori the curvature bound of $g_\infty(t)$ on $\mathbb{R}^4 / \mathbb{Z}_2 \times [0,T]$ may deteriorate as $T \rightarrow \infty$. Nevertheless we can use a diagonal argument to construct an ancient orbifold Ricci flow on $\mathbb{R}^4 /\mathbb{Z}_2$. Hamilton's trace Harnack inequality then implies that $g_\infty(t)$ has bounded curvature on $\mathbb{R}^4/\mathbb{Z}^2 \times (-\infty,0]$. Finally, we apply the result \cite{LZ18} by Xiaolong Li and Yongjia Zhang to deduce that $g_\infty(t)$ is homothetic to the 4d Bryant soliton quotiented by $\mathbb{Z}_2$.
\subsection{Proof of main theorem}
We begin by proving the central lemma of this section, which yields important geometric information on the high curvature regions of a Ricci flow $(M_2, g(t))$ as in the Theorem \ref{blow-up-thm}.
\begin{lem}
\label{QT2bddb-limit-lem}
Let $(M_2, g(t))$, $t \in [0, T_{sing})$, be a Ricci flow starting from an initial metric $g(0) \in \mathcal{I}$ with $\sup_{p \in M_2} b(p,0) < \infty$. Then for every $\epsilon \in (0,1)$ there exist constants $C, \delta > 0$ such that at all points $(p, t)$ in spacetime with $C b(o,t) \leq b(p,t) \leq \delta$ the following inequalities hold:
\begin{enumerate}[label=(\roman*)]
\item $Q \geq 1 - \epsilon$
\item $bb_{ss} \leq \epsilon$
\item $T_{F_1} \coloneqq b b_{ss} + 1- b_s^2 \geq -\epsilon$
\item $T_{F_2} \coloneqq b b_{ss} + 1- b_s^2 - \left(1-b_s^2\right)^2 \leq \epsilon$
\item $\partial_t b^2 \leq \epsilon$
\end{enumerate}
\end{lem}
\begin{remark}
Inequality (ii) is implied by (iv) for metrics in $\mathcal{I}$. However, we need (ii) as an intermediate result before proving (iv).
\end{remark}
\begin{proof}
Fix $\epsilon \in (0,1)$. Recall the following facts of the Eguchi-Hanson space $(M_2, g^E)$ derived in section \ref{kahler-E-H-section}:
\begin{enumerate}[label=(\alph*)]
\item $x = y = 0$ on $M_2$
\item $Q \rightarrow 1$ as $s \rightarrow \infty$
\item $g^E$ is normalized such that its warping function $b^E$ satisfies $b^E(0)=1$.
\end{enumerate}
Using (a) a computation shows that
\begin{align*}
bb_{ss} &= 2\left(1-Q^2\right) \\
T_{F_1} &= 3\left(1 - Q^2\right) \\
T_{F_2} &= 3\left(1 - Q^2\right) - \left(1-Q^2\right)^2 \\
\partial_t b^2 &= 0
\end{align*}
on $(M_2, g^E)$. Pick $C > 10$ such that on $(M_2, g^E)$ we have $Q > 1 - \epsilon$, $bb_{ss} < \epsilon$,
$T_{F_1} > -\epsilon$ and $T_{F_2} < \epsilon$ whenever $s > C$. This is possible by property (b).
Take a path $\gamma: [0,T_{sing}) \rightarrow M_2$ such that
$$s(\gamma(t), t) = C b(o,t),$$
where we recall that $s(p, t)$ is the distance of a point $p \in M_2$ from the non-principal orbit $S^2_o$ at the tip of $M_2$. By Corollary \ref{E-H-blowup} we know that at distance scales comparable to $b(o,t)$ away from $S^2_o$ we converge to the Eguchi-Hanson space as $t \rightarrow T_{sing}$. From the scale-invariance of $Q$, $bb_{ss}$, $T_{F_1}$, $T_{F_2}$ and $\partial_t b^2$ it follows that at spacetime points $(\gamma(t),t)$ inequalities (i)-(v) eventually hold as $t \rightarrow T_{sing}$.
Let $A$ be the set of all sequences of points $\{(p_i, t_i)\}_{i \in \mathbb{N}}$ in spacetime satisfying the following two properties:
\begin{enumerate}
\item $b(p_i, t_i) \geq C b(o, t_i)$
\item $b(p_i, t_i) \rightarrow 0$ as $i \rightarrow \infty$
\end{enumerate}
Note that property (2) implies that for such sequences $t_i \rightarrow T_{sing}$ as $i \rightarrow \infty$.
We first prove inequality (i), arguing by contradiction. Assume that
$$\iota := \inf\left\{ \liminf_{i \rightarrow \infty} Q(p_i, t_i) \: \big | \: \{(p_i, t_i)\}_{i \in \mathbb{N}} \in A \right\} < 1 - \epsilon.$$
Then there exists a sequence $\{ A_n \}_{n \in \mathbb{N}}$ of sequences $A_n = \{(p_{n,i}, t_{n,i}) \}_{i \in \mathbb{N}}$ of points in spacetime satisfying properties (1) and (2) above, and
$$\lim_{n \rightarrow \infty} \liminf_{i \rightarrow \infty} Q(p_{n,i}, t_{n,i}) = \iota.$$
For each $n \in \mathbb{N}$ take $N(n) \in \mathbb{N}$ such that for $m \geq N(n)$ we have
$$\left| \liminf_{i \rightarrow \infty} Q(p_{m,i}, t_{m,i}) - \iota \right| \leq \frac{1}{n}.$$
For each $n \in \mathbb{N}$ take $I(n) \in \mathbb{N}$ such that
$$\left|Q(p_{n,I(n)}, t_{n,I(n)}) - \liminf_{i \rightarrow \infty} Q(p_{n,i}, t_{n,i})\right| \leq \frac{1}{n}$$
and
$$b(p_{n,I(n)}, t_{n, I(n)}) \leq \frac{1}{n}.$$
Let $(p_n, t_n) = (p_{N(n), I(N(n))}, t_{N(n), I(N(n))})$ for $n \in \mathbb{N}$. Then we see that
$$Q(p_i, t_i) \rightarrow \iota \text{ as } i \rightarrow \infty$$
and both properties (1) and (2) from above hold.
Before we carry on recall that by Theorem \ref{curv-bound} there exists a $C_1>0$ such that
$$|Rm_{g(t)}|_{g(t)} \leq \frac{C_1}{b^2} \: \text{ on } \: M_2 \times [0, T_{sing}).$$
Recall also Theorem \ref{thm:no-local-collapsing}, from which it follows that there exist constants $\kappa, \rho>0$ such that $g(t)$ is $\kappa$-non-collapsed at scale $\rho$. Next, consider the rescaled metrics
$$g_i(t) = \frac{1}{b^2(p_i, t_i)} g( t_i + t b^2(p_i, t_i)), \quad [-\Delta t, 0],$$
where $\Delta t > 0$ is chosen such that Proposition \ref{blow-up-prop} holds. Then $(C_{g_i(0)}\left(p_i, \frac{1}{2}\right), g_i(t), p_i)$ subsequentially converges to a pointed Ricci flow $(\mathcal{C}_\infty, g_\infty(t), p_\infty)$, $t \in [-\Delta t, 0]$. By construction
$$b(p_\infty, 0) = 1$$
and
$$ Q(p_\infty, 0) = \iota < 1 - \epsilon.$$
\begin{claim}
$\lim_{i \rightarrow \infty} \frac{b(p_i, t_i)}{b(o, t_i)} = \infty$
\end{claim}
\begin{claimproof}
We argue by contradiction. Assume there exists a $C'>0$ such that after passing to a subsequence of $(p_i, t_i)$
$$\frac{b(p_i, t_i)}{b(o, t_i)} < C'.$$
Consider the rescaled metrics
$$g_i(t) = \frac{1}{b^2(p_i, t_i)} g( t_i + t b^2(p_i, t_i)), \quad [-b(p_i, t_i)^{-2} t_i, 0].$$
By Corollary \ref{E-H-blowup} we see that $(M_2, g_i(t), p_i)$ subsequentially converges to $(M_2, g_{\infty}(t), p_\infty)$, where $g_\infty(t)$ is stationary and homothetic to the Eguchi-Hanson metric. By construction
$$1 = b(p_\infty, 0) \geq C b(o, 0)$$
and
$$Q(p_\infty, 0) = \iota < 1- \epsilon.$$
Furthermore,
$$s(p_\infty, 0) \geq b(p_\infty, 0) \geq C b(o, 0),$$
where the first inequality follows from the fact that $1 \geq Q \geq b_s \geq 0$ for metrics in $\mathcal{I}$ and the second inequality follows from the definition of $C$. Thus
$$Q(p_\infty, 0) > 1 - \epsilon,$$
which is a contradiction and hence proves the claim.
\end{claimproof}
\begin{claim}
For every $N \in \mathbb{N}$ eventually $\frac{b(p,t)}{b(o,t)} > N$ everywhere in $(C_{g_i(0)}\left(p_i, \frac{1}{2}\right), g_i(t), p_i) \times [-\Delta t, 0]$
\end{claim}
\begin{claimproof}
Fix $N \in \mathbb{N}$. We argue by contradiction. After passing to a subsequence of $(p_i, t_i)$, we may assume that there exists a sequence of spacetime points $(p'_i, t'_i) \in (C_{g_i(0)}\left(p_i, \frac{1}{2}\right), g_i(t), p_i) \times [-\Delta t, 0]$ for which $\frac{b(p'_i,t'_i)}{b(o, t'_i)} \leq N$. Consider the rescaled metrics
$$g'_i(t) = \frac{1}{b^2(p'_i, t'_i)} g(t'_i + t b^2(p'_i, t'_i)), \quad t \in [-b(p'_i, t'_i)^{-2} t'_i, 0].$$
By Corollary \ref{E-H-blowup}, $(M_2, g'_i(t), p'_i)$, $t \in [-b(p'_i, t'_i)^{-2} t'_i, 0]$, converges to an ancient Ricci flow $(M_\infty, g'_\infty(t), p'_\infty)$, $t \in (-\infty, 0]$, which is stationary and homothetic to the Eguchi-Hanson space. Note that on the non-principal orbit $S^2_o$
$$0 \geq \partial_t b^2(o,t) = 4by_s \geq - 4bQ_s = -4k,$$
as $0 \geq y = b_s - Q \geq -Q$ and $y=Q=0$ at $o$ for metrics in $\mathcal{I}$. Hence for $\tau \in (0, \frac{1}{4k})$ the warping function $b_i$ of the metric $g_i(t)$ satisfies
$$b_i(o,t) \geq 1 - 4k \tau > 0 \: \text{ for } \: t \in [-b(p'_i, t'_i)^{-2} t'_i, \tau].$$
We deduce by Theorem \ref{curv-bound} that $g_i(t)$ has bounded curvature on $M_2 \times [-b(p'_i, t'_i)^{-2} t'_i, \tau]$. Hence $(M_2, g'_i(t), p'_i)$, $t \in [-b(p'_i, t'_i)^{-2} t'_i, \tau]$, also converges to the stationary Eguchi-Hanson space. In fact, inductively we can then show that for any $\tau > 0$ we converge to the Eguchi-Hanson space. As $(p'_i, t'_i)$ converges to a point $(p'_\infty, t'_\infty)$ in $\mathcal{C}_\infty \times [-\Delta t, 0]$, this implies that $\mathcal{C}_\infty \times [-\Delta t, 0]$ is a subset of a spacetime corresponding to the Eguchi-Hanson space, and therefore $\lim_{i \rightarrow \infty} \frac{b(p_i,t_i)}{b(o,t_i)} < \infty$. This, however, contradicts Claim 1.
\end{claimproof}
\begin{claim}
$ Q(p_\infty, 0) = \inf_{\mathcal{C}_\infty \times [-\Delta t, 0]} Q$
\end{claim}
\begin{claimproof}
We argue by contradiction. If $Q(p',t') < \iota$ at a point $(p', t') \in \mathcal{C}_\infty \times [-\Delta t, 0]$, one could pick a sequence of points $(p'_i, t'_i) \in C_{g_i(0)}\left(p_i, \frac{1}{2}\right) \times [-\Delta t, 0]$ with $(p'_i, t'_i) \rightarrow (p',t')$ as $i \rightarrow \infty$. Then shifting back to the time of the Ricci flow $(M_2, g(t))$ via $T'_i = t_i + t'_ib(p_i,t_i)^2$ we see that the sequence $(p'_i, T'_i) \in M_2 \times [0, T_{sing})$ satisfies properties (1) and (2). The former property holds because of Claim 2. This, however, would contradict the definition of $\iota$.
\end{claimproof}
By property (1) of the sequence $(p_i, t_i)$ we see that $p_\infty$ does not lie on a non-principal orbit of $\mathcal{C}_\infty$ and therefore $Q(p_\infty, 0) = a(p_\infty, 0) > 0$. However, the evolution equation (\ref{Q-evol}) of $Q$ shows that the only attainable minima are $0$ and $1$, yielding a contradiction. This concludes the proof of (i).
We prove (ii)-(v) by the same strategy. Below we first prove inequality (ii) by contradiction. Assume that
$$\iota := \sup\left\{ \limsup_{i \rightarrow \infty} bb_{ss}\big|_{(p_i, t_i)} \: \Big | \: \{(p_i, t_i)\}_{i \in \mathbb{N}} \in A \right\} > \epsilon.$$
As before we can construct a sequence of points $(p_i, t_i)$ in spacetime satisfying properties (1) and (2), and such that
$$\lim_{i \rightarrow \infty} bb_{ss}\big|_{(p_i, t_i)} = \iota.$$
Consider the rescaled metrics
$$g_i(t) = \frac{1}{b^2(p_i, t_i)} g( t_i + t b^2(p_i, t_i)), \quad [-\Delta t, 0],$$
where $\Delta t > 0$ is chosen such that Proposition \ref{blow-up-prop} holds. Then $(C_{g_i(0)}\left(p_i, \frac{1}{2}\right), g_i(t), p_i)$ subsequentially converges to a pointed Ricci flow $(\mathcal{C}_\infty, g_\infty(t), p_\infty)$. By construction
$$ bb_{ss}\big|_{(p_\infty, 0)} = \iota > \epsilon.$$
Furthermore, by the same arguments as in Claim 1 \& 2 \& 3, we have
$$ bb_{ss}\big|_{(p_\infty, 0)} = \sup_{\mathcal{C}_\infty \times [-\Delta t, 0]} bb_{ss}$$
and hence
$$\partial_t bb_{ss} \Big|_{(p_\infty, 0)} \geq 0.$$
By statement (i) of this lemma we know that $Q = 1$ on $\mathcal{C}_\infty \times [-\Delta t, 0]$. For $Q=1$ the evolution equation for $bb_{ss}$ is
$$\partial_t (bb_{ss}) = (bb_{ss})_{ss} - \frac{b_s}{b} (bb_{ss})_s -4 \frac{b_s^2}{b^2} \left( 1- b_s^2\right) -2 \frac{b_{ss}}{b} \left(b b_{ss}+2 b_s^2\right).$$
The derivation is carried out in the Appendix A, in the subsection on the evolution equations when $Q=1$. From this we see that at the point $(p_\infty,0)$ in spacetime we have
$$\partial_t (bb_{ss}) \Big |_{(p_\infty,0)} < 0,$$
which is a contradiction. This proves (ii).
We prove inequality (iii) similarly. Assume that
$$\iota := \inf\left\{ \liminf_{i \rightarrow \infty} T_{F_1}(p_i, t_i) \: \big | \: \{(p_i, t_i)\}_{i \in \mathbb{N}} \in A \right\} < - \epsilon.$$
Pick $\{(p_i,t_i)\}_{i \in \mathbb{N}} \in A$ such that
$$\lim_{i \rightarrow \infty} T_{F_1}(p_i, t_i) = \iota.$$
As before, $(C_{g_i(0)}\left(p_i, \frac{1}{2}\right), g_i(t), p_i)$ subsequentially converges to a pointed Ricci flow $(\mathcal{C}_\infty, g_\infty(t), p_\infty)$. By construction
$$ T_{F_1} (p_\infty, 0) = \inf_{\mathcal{C}_\infty \times [-\Delta t, 0]} T_{F_1} = \iota < - \epsilon$$
and hence
$$\partial_t T_{F_1} \big|_{(p_\infty, 0)} \leq 0.$$
By inequality (i) of this lemma $Q = 1$ on $\mathcal{C}_\infty \times [-\Delta t, 0]$. For $Q=1$ the evolution equation of $T_{F_1}$ can be written as
$$\partial_t T_{F_1} = (T_{F_1})_{ss} - \frac{b_s}{b} (T_{F_1})_s - 8\frac{b^2_s}{b^2} T_{F_1}.$$
The derivation is carried out in the Appendix A. From this we see that at the point $(p_\infty,0)$ in spacetime we have
$$\partial_t T_{F_1} \Big |_{(p_\infty,0)} \geq 0,$$
with equality if, and only if, $b_s \big|_{(p_\infty,0)} = 0$. Therefore we conclude that $b_s = 0$ at $(p_\infty, 0)$. Applying the strong maximum principle to the evolution equation (\ref{db-evol}) of $b_s$ when $Q=1$, it then follows that $b_s = 0$, and hence $b_{ss}=0$, everywhere in $\mathcal{C}_\infty \times [-\Delta t, 0]$. This, however, implies $T_{F_1} = 1$ at $(p_\infty, 0)$, which is a contradiction and thus proves (ii).
We proceed to prove (iv) in the same fashion. Assume that
$$\iota := \sup\left\{ \limsup_{i \rightarrow \infty} T_{F_2}(p_i, t_i) \: \big | \: \{(p_i, t_i)\}_{i \in \mathbb{N}} \in A \right\} > \epsilon.$$
Pick $\{(p_i,t_i)\}_{i \in \mathbb{N}} \in A$ such that
$$\lim_{i \rightarrow \infty} T_{F_2}(p_i, t_i) = \iota.$$
As before, $(C_{g_i(0)}\left(p_i, \frac{1}{2}\right), g_i(t), p_i)$ subsequentially converges to a pointed Ricci flow $(\mathcal{C}_\infty, g_\infty(t), p_\infty)$. By construction
$$ T_{F_2}(p_\infty, 0) = \sup_{\mathcal{C}_\infty \times [-\Delta t, 0]} T_{F_2} = \iota > \epsilon.$$
Therefore
$$\partial_t T_{F_2} \Big|_{(p_\infty, 0)} \geq 0.$$
By statement (i) and (ii) of this lemma we have $Q=1$ and $bb_{ss} \leq 0$ on $\mathcal{C}_\infty \times [-\Delta t, 0]$. When $Q=1$ the evolution equation of $T_{F_2}$ can be written as
$$\partial_t T_{F_2} = (T_{F_2})_{ss} - \frac{b_s}{b} (T_{F_2})_s + \frac{1}{b^2} C_{F_2},$$
where $C_{F_2}$ is a polynomial expression in $bb_{ss}$ and $1 - b_s^2$. The derivation is carried out in the Appendix A. By Lemma \ref{lem:CF2-neg} in the Appendix A, $C_{F_2} < 0$ whenever $T_{F_2} > 0$ and $bb_{ss} \leq 0$. This, however, implies
$$\partial_t T_{F_2}\Big|_{(p_\infty, 0)} < 0,$$
which is a contradiction and thus proves (iv).
Finally we prove (v), also by contradiction. Assume that
$$\iota := \sup\left\{ \limsup_{i \rightarrow \infty} \partial_t b^2 (p_i, t_i) \: \big | \: \{(p_i, t_i)\}_{i \in \mathbb{N}} \in A \right\} > \epsilon.$$
Pick $\{(p_i,t_i)\}_{i \in \mathbb{N}} \in A$ such that
$$\lim_{i \rightarrow \infty} \partial_t b^2 (p_i, t_i) = \iota.$$
As before, $(C_{g_i(0)}\left(p_i, \frac{1}{2}\right), g_i(t), p_i)$ subsequentially converges to a pointed Ricci flow $(\mathcal{C}_\infty, g_\infty(t), p_\infty)$. By construction
$$\partial_t b^2\Big|_{(p_\infty, 0)} = \iota > \epsilon,$$
as $\partial_t b^2$ is a scale-invariant quantity.
By (i) we have $Q = 1$ on $\mathcal{C}_\infty\times[-\Delta t, 0]$ and the evolution equation (\ref{b-evol}) of $b$ simplifies to
$$\partial_t b^2 = 2bb_{ss} + 4 \left( b_s^2 -1\right).$$
By inequality (iii) of this lemma we have
$$bb_{ss} \leq b_s^2 -1 +\left(1 - b_s^2\right) \leq 0 \: \text{ on } \: \mathcal{C}_\infty \times [-\Delta t, 0],$$
as $b_s \in [0,1]$ for metrics in $\mathcal{I}$. This, however, implies that
$$\partial_t b^2 \leq 0,$$
which is a contradiction and thus proves (v).
\end{proof}
\begin{lem}
\label{bbdot-bounded-lem}
Let $(M_2, g(t))$, $t \in [0, T_{sing})$, be a Ricci flow starting from an initial metric $g(0) \in \mathcal{I}$ with $\sup_{p \in M_2} b(p,0) < \infty$. Then for every $\epsilon \in (0,1)$ there exists a $\delta > 0$ such that at all points $(p,t)$ in spacetime at which $b(p,t) \leq \delta$ we have
$$ \partial_t b^2 \leq \epsilon.$$
\end{lem}
\begin{proof}
Fix $\epsilon > 0$. By Lemma \ref{QT2bddb-limit-lem} we only need to prove that there exists a $\delta > 0$ such that the result holds when $b(p,t)\leq C b(o,t) \leq \delta$, where $C>0$ is as in Lemma \ref{QT2bddb-limit-lem}. Note that
$$\partial_t b^2 = 0$$
on the Eguchi-Hanson space background. By Corollary \ref{E-H-blowup} we know that on the scale $b(p,t) \leq C b(o,t)$ we converge to the Eguchi-Hanson space as $t \rightarrow T_{sing}$. As $b(o,t) \rightarrow 0$ as $t \rightarrow T_{sing}$ we see that there exists a $\delta >0$ such that
$$ \partial_t b^2 \leq \epsilon$$
at all points $(p,t)$ in spacetime at which $b(p,t) \leq C b(o,t) \leq \delta$. This completes the proof.
\end{proof}
Below we prove the simplest case of Theorem \ref{blow-up-thm}.
\begin{lem}
\label{cylinder-blow-up-lem}
Let $(M_2, g(t))$, $t \in [0, T_{sing})$, be a Ricci flow starting from an initial metric $g(0) \in \mathcal{I}$ with $\sup_{p \in M_2} b(p,0) < \infty$. Let $(p_i,t_i)$ be a sequence of points in spacetime satisfying
\begin{enumerate}
\item $b(p_i, t_i) \rightarrow 0$
\item $b_s(p_i,t_i) \rightarrow 0$
\item $b(p_i,t_i) > 2 b(o, t_i)$
\end{enumerate}
Consider the rescaled metrics
\begin{equation*}
g_i(t) = \frac{1}{b^2(p_i,t_i)} g(t_i + b^2(p_i,t_i) t), \quad t \in [-b(p_i, t_i)^{-2}t_i, 0].
\end{equation*}
Then $(M_2, g_i(t), p_i)$ , $t \in [-b(p_i, t_i)^{-2}t_i, 0]$, subsequentially converges, in the Cheeger-Gromov sense, to the shrinking cylinder $\mathbb{R} \times \mathbb{R} P^3$.
\end{lem}
\begin{proof}
Fix $T>0$. By Lemma \ref{bbdot-bounded-lem}, the curvature bound of Theorem \ref{curv-bound} and the fact that $b_s \in [0,1]$ for metrics in $\mathcal{I}$, we see that the curvatures of $g_i(t)$ on the parabolic neighborhoods $C_{g_i(0)}(p_i, \frac{1}{2}) \times [-T, 0]$ are eventually uniformly bounded. By Theorem \ref{thm:no-local-collapsing} the Ricci flows $g_i(t)$ are $\kappa$-non-collapsed. Hence $(C_{g_i(0)}(p_i, \frac{1}{2}), g_i(t), p_i)$, $t \in [-T,0]$, subsequentially converges to a Ricci flow $(\mathcal{C}_\infty, g_\infty(t), p_\infty)$, $t \in [-T,0]$, where by construction $b_s = 0$ at the point $(p_\infty, 0)$ in spacetime. Lemma \ref{QT2bddb-limit-lem} implies $Q = 1$ on $\mathcal{C}_\infty \times [-T, 0]$. Applying the strong maximum principle to the evolution equation (\ref{db-evol}) of $b_s$ when $Q=1$ shows that
$$b_s = 0 \: \text{ on } \: \mathcal{C}_{\infty} \times [-T, 0].$$
That is, the metric $g_\infty(t)$ is cylindrical. From here one can inductively show that for every $r > 0$ the Ricci flows $(C_{g_i(0)}(p_i, r), g_i(t), p_i)$, $t \in [-T,0]$ subsequentially converge to a limiting cylindrical Ricci flow. Hence $(M_2, g_i(t), p_i)$, $t \in [-T, 0]$, subsequentially converges to the shrinking cylinder $(\mathbb{R} \times \mathbb{R} P^3, g_\infty(t), p_\infty)$, $t \in [-T,0]$. As $T>0$ is arbitrary the desired result follows by a diagonal argument.
\end{proof}
Before we carry on constructing the blow-up limit (iii) of Theorem \ref{blow-up-thm}, we need to state two technical lemmas. Their proofs can be skipped on the first reading.
\begin{lem}
\label{lem:ODE-technical}
Let $\hat{\eta} > 0$. There exists a $K= K(\hat{\eta})>1$ such that the following holds: Let $s_0 > 0$ and $b: [s_0, \infty) \rightarrow \mathbb{R}$ satisfy the ordinary differential inequalities
\begin{equation}
\label{bddb-ODI}
b b_{ss} \leq b_s^2 -1 + \left( 1- b_s^2\right)^2
\end{equation}
and
$$b_s > 0.$$
If at $s_0$ we have
\begin{equation}
\label{init-cond-1}
\frac{1- b_s^2}{b^2}\Big|_{s_0} = K
\end{equation}
and
\begin{equation}
\label{init-cond-2}
b_s\big|_{s_0} \in [\hat{\eta}, 1),
\end{equation}
then $b_s < \hat{\eta}$ when $b \geq1$.
\end{lem}
\begin{proof}
First note that $b(s_0) \leq \frac{1}{\sqrt{K}} < 1 $ by (\ref{init-cond-1}) and (\ref{init-cond-2}). Furthermore, as $b_s \in [\hat{\eta}, 1)$ at $s_0$ we see from (\ref{bddb-ODI}) that $b_s < 1$ on $[s_0, \infty)$.
Write $B = b_s$. Then the ODI becomes
$$b B_s \leq B^2 -1 + \left( 1- B^2\right)^2 = - B^2\left( 1- B^2\right).$$
Since $b_s > 0$ we may treat $b$ as the independent variable, yielding the following ODI
$$ \frac{\mathrm{d}B}{\mathrm{d}b}\leq - \frac{B^2 \left(1- B^2\right)}{bB}.$$
Note that as $B=b_s\in(0,1)$ we may rearrange the inequality and integrate to obtain
$$ \int_{B_0}^B \frac{B \mathrm{d}B}{B^2\left(1 - B^2\right)} \leq -\int_{b_0}^b \frac{\mathrm{d}b}{b},$$
where we denote by $b_0$ and $B_0$ the values of $b$ and $B$ at $s_0$, respectively. Evaluating the integrals and rearranging we deduce
$$\frac{B^2}{1 - B^2} \leq \frac{B_0^2 b_0^2}{(1-B_0^2) b^2}.$$
By the initial conditions (\ref{init-cond-1}) and (\ref{init-cond-2}) we have
$$ \frac{b_0^2 B_0^2}{1 - B_0^2} = \frac{B_0^2}{K} = \frac{B_0^4}{K B_0^2}\leq \frac{1}{K \hat{\eta}^2}$$
and therefore
$$ \frac{B^2}{1 - B^2} \leq \frac{1}{K\hat{\eta}^2b^2}.$$
Hence when $b \geq 1$ we have
$$\frac{B^2}{1 - B^2} \leq \frac{1}{K \hat{\eta}^2},$$
which can be rearranged to
$$B^2 \leq \frac{1}{K \hat{\eta}^2 + 1}.$$
Choosing $K$ sufficiently large the desired result follows.
\end{proof}
Now we may construct the orbifold Ricci flow blow-up:
\begin{lem}
\label{lem:orbifold-blowup}
Let $\eta \in (0,1)$ and $(M_2, g(t))$, $t \in [0, T_{sing})$, be a Ricci flow starting from an initial metric $g(0) \in \mathcal{I}$ with $\sup_{p \in M_2} b(p,0) < \infty$. Assume that $(p_i,t_i)$ is a sequence of points in spacetime satisfying
\begin{enumerate}
\item $b(p_i, t_i) \rightarrow 0$ as $i \rightarrow \infty$
\item $\frac{b(p_i,t_i)}{b(o,t_i)} \rightarrow \infty $ as $i \rightarrow \infty$
\item $b_s(p_i,t_i) \rightarrow \eta $ as $i \rightarrow \infty$
\end{enumerate}
Consider the rescaled Ricci flows
\begin{equation*}
g_i(t) = \frac{1}{b^2(p_i,t_i)} g(t_i + b^2(p_i,t_i) t), \quad t\in [-b(p_i, t_i)^{-2} t_i, 0].
\end{equation*}
Then $(M_2, g_i(t), p_i)$, $t\in [-b(p_i, t_i)^{-2} t_i, 0]$, subsequentially converges, in the Cheeger-Gromov sense, to an ancient Ricci flow $(M_\infty, g_\infty(t), p_\infty)$, $t \in (-\infty,0]$, where $M_\infty \cong \mathbb{R}^4\setminus\{0\}/\mathbb{Z}_2$. Moreover, $g_\infty(t)$ can be extended to a smooth orbifold Ricci flow on $\mathbb{R}^4 /\mathbb{Z}_2$ that is homothetic to the 4d Bryant soliton quotiented by $\mathbb{Z}_2$.
\end{lem}
\begin{proof}
Fix $T > 0$. By Lemma \ref{bbdot-bounded-lem} we have that for every $\epsilon > 0$ there exists a $\delta > 0$ such that at points $(p,t)$ in spacetime at which $b(p,t) < \delta$ we have $\partial_t b^2 \leq \epsilon$.
This shows that for every $N > 0$ there exists a $\delta'= \delta'(N) > 0$ such that whenever $b(p,t) < \delta'$ then
\begin{equation}
\label{b-lower-bound}
b(p,t') > \frac{b(p,t)}{2} \: \text{ for } \: t' \in [t- N b^2(p,t), t].
\end{equation}
Consider the rescaled metrics
$$g_i(t) = \frac{1}{b^2(p_i,t_i)} g\left(t_i + b^2(p_i,t_i) t\right).$$
For $n \in \mathbb{N}_{\geq 2}$ and $(p,t) \in M_2 \times [0, T_{sing})$ define the open set
$$E_{p,t,n} := \left\{p' \in M_2 \: \Big | \: b(p',t) > \frac{b(p,t)}{n} \right\} \subseteq M_2$$
Furthermore, define the parabolic neighborhoods
$$\Omega_{i,n} = E_{p_i,t_i,n} \times [-T-1 , 0].$$
Recall that by Theorem \ref{curv-bound} there exists a $C_1 > 0$ such that
$$|Rm_{g(t)}|_{g(t)} \leq \frac{C_1}{b^2} \: \text{ on } \: M_2 \times [0, T_{sing}).$$
Hence for fixed $n$ and sufficiently large $i$ the curvatures of $g_i(t)$ on $\Omega_{i,n}$ are uniformly bounded:
$$|Rm_{g_i(t)}|_{g_i(t)} \leq 4n^2C_1 \: \text{ on } \Omega_{i,n}.$$
This follows from (\ref{b-lower-bound}), $b(p_i, t_i) \rightarrow 0$ and the fact that $b_s \geq 0$ for metrics in $\mathcal{I}$. By Theorem \ref{thm:no-local-collapsing} the Ricci flows $g_i(t)$ are $\kappa$-non-collapsed at larger and larger scales. By a slight adaptation of the local compactness Theorem \ref{thm:local-compactness} we see that for each $n \in \mathbb{N}$ the Ricci flows $(E_{p_i, t_i, n}, g_i(t), p_i)$, $t \in [-T-1,0]$, subsequentially converge to a Ricci flow $(\mathcal{E}_{\infty, n}, g_{\infty,n}(t), p_{\infty,n})$, $t \in [-T-1, 0]$. The manifolds $\mathcal{E}_{\infty, n}$ are diffeomorphic to $\mathbb{R}^4\setminus\{0\}/\mathbb{Z}_2$ and therefore incomplete. By a diagonal argument we may assume that $\mathcal{E}_{\infty, n} \subset \mathcal{E}_{\infty, n+1}$ and $g_{\infty,n}(t) = g_{\infty,n+1}(t)$ on $\mathcal{E}_{\infty, n}$. This allows us to drop the dependence on $n$ and write $g_\infty(t)$ and $p_\infty$ for brevity. By Lemma \ref{QT2bddb-limit-lem} we have $Q = 1$, $bb_{ss} \leq 0$, $T_{F_1} \leq 0$, $T_{F_2} \geq 0$ and $\partial_t b^2 \leq 0$ on $\mathcal{E}_{\infty, n}$.
\begin{claim}
There exists an $\hat{\eta} > 0$, independent of $n$, such that on the word line $(p_{\infty}, t)$, $t \in [-T, 0]$, in $\mathcal{E}_{\infty, n} \times [-T, 0]$ we have $b_s > \hat{\eta}$ uniformly.
\end{claim}
\begin{claimproof}
We argue by contradiction. Assume that $t' \in [-T,0]$ is such that for $t \in [t', 0]$ we have $b_s(p_\infty, t) \geq 0$ with equality if, and only if, $t = t'$. Applying the strong maximum principle to the evolution equation (\ref{db-evol}) of $b_s$ when $Q=1$, we obtain that $b_s = 0$ on $\mathcal{E}_{\infty, n} \times [-T-1, t']$. That is, the metric $g_\infty(t)$ is cylindrical for times $t \in [-T-1, t']$. We now show that this leads to a contradiction. Take times $t'_i = t_i - t' b^2(p_i,t_i)$. Then the spacetime points $(p_i,t'_i) \in M_2 \times [0, T_{sing})$ converge to the spacetime point $(p_\infty, t') \in \mathcal{E}_{\infty, n} \times [-T-1, t']$. Consider the rescaled metrics
$$g'_i(t) = \frac{1}{b(p_i,t'_i)^2} g(t'_i + t b(p_i, t'_i)^2), \quad t\in[-t'_i b(p_i, t'_i)^{-2}, 0].$$
Because $b_s(p_i, t'_i) \rightarrow 0$ as $i \rightarrow \infty$, Lemma \ref{cylinder-blow-up-lem} implies that after passing to a subsequence $(M_2, g'_i(t), p_i)$ converges to the shrinking cylinder $\mathbb{R} \times \mathbb{R} P^3$. For every $n\in \mathbb{N}$ take $N_n \in \mathbb{N}$ such that for $i \geq N_n$ the region $C_{g'_i(0)}(p_i, n) \subset M_2$ is close, in the Cheeger-Gromov sense, to a cylinder $\mathbb{R} \times \mathbb{R} P^3$ of length $2n$ and radius $1$. By Perelman's pseudolocality theorem there exists a $K > 0$ and $\tau > 0$ such that $g'_i(t)$ has bounded curvature on $C_{g'_i(0)}(p_i, n-1) \times [0, \tau]$. Hence $(M_2, g'_i(t), p_i)$, $t \in[-t'_i b(p_i, t'_i)^{-2}, \tau]$, subsequentially converges to a limiting Ricci flow $(M_\infty, g'_\infty(t),p_\infty)$, $ t \in (-\infty, \tau]$, where $M_\infty \cong \mathbb{R} \times \mathbb{R} P^3$ and $g'_\infty(0)$ is cylindrical. By the uniqueness of Ricci flow solutions \cite{CZ06}, we see that $g'_\infty(t)$ remains cylindrical for $t \in [0, \tau]$ and therefore $b_s= 0$ on $M_\infty \times (-\infty, \tau)$. Now we have arrived at a contradiction, as this implies that $t'$ is not the earliest time at which $b_s = 0$ on the wordline through the point $(p_\infty, 0)$ in the spacetime $\mathcal{E}_{\infty, n} \times [-T-1, 0]$. This proves the claim.
\end{claimproof}
As $T_{F_1} \geq 0$ we have
\begin{equation*}
-\frac{b_{ss}}{b} \leq \frac{1-b_s^2}{b^2} \: \text{ on } \: \mathcal{E}_{\infty, n} \times [-T, 0]
\end{equation*}
and hence
\begin{equation}
\label{curv-bound-rotational}
|Rm_{g_{\infty}(t)}|_{g_{\infty}(t)} \leq c \frac{1-b_s^2}{b^2}
\end{equation}
for some universal constant $c >0$, as $\frac{1-b_s^2}{b^2}$ and $-\frac{b_{ss}}{b}$ are the only non-zero curvature components of a rotationally symmetric metric.
\begin{claim}
There exists a $K= K(\hat{\eta}) > 1$, independent of $n$, such that
$$|Rm_{g_{\infty}(t)}|_{g_{\infty}(t)} < cK$$
uniformly on $\mathcal{E}_{\infty, n} \times [-T,0]$.
\end{claim}
\begin{claimproof}
Fix $n \geq 2$. As $bb_{ss} \leq 0$ and $b_s \geq 0$ it follows from Claim 1 that $b_s \geq \hat{\eta}>0$ in the region
$$R = \left\{ (p,t) \in \mathcal{E}_{\infty, n} \times [-T,0] \: \Big| \: b(p,t) \leq b(p_\infty, t) \right\}.$$
As $\partial_t b^2 \leq 0$ we have $b(p_\infty, t) \geq b(p_\infty, 0) = 1$ for $t \in [-T, 0]$.
Now choose a $K = K(\hat{\eta}) > 1$ such that Lemma \ref{lem:ODE-technical} holds true. If at some point $(p',t') \in \mathcal{E}_{\infty, n} \times [-T,0]$ we had
$$ \frac{1- b_s^2}{b^2} \geq K $$
then on the time slice $\{t= t'\} \subset \mathcal{E}_{\infty,n}$ the result of Lemma \ref{lem:ODE-technical} would imply that $b_s < \hat{\eta}$ when $b \geq 1$. This cannot be true, as by Lemma \ref{bbdot-bounded-lem} we have that $\partial_t b^2 \leq 0$ on $\mathcal{E}_{\infty,n} \times [-T, 0]$ and therefore $b(p_\infty, t) \geq 1$ for $t \in [-T,0]$. Hence we deduce by (\ref{curv-bound-rotational}) that the curvature is bounded by $cK$ on the region $R$. As on $\mathcal{E}_{\infty, n} \times [-T,0] \setminus R$ we have $b > 1$, it follows by (\ref{curv-bound-rotational}) and the fact that $b_s \in [0,1]$ for metrics in $\mathcal{I}$ that the curvature is uniformly bounded by $c$ there.
\end{claimproof}
Claim 2 shows that as $n \rightarrow \infty$ we may extract a limiting Ricci flow $(M_\infty, g_\infty(t), p_\infty)$, $t \in [-T, 0]$, with curvature bounded by $cK$. By construction $M_\infty$ is diffeomorphic to $(R^4\setminus\{0\})/ \mathbb{Z}_2$. Define the radial coordinate $\xi: M_{\infty} \rightarrow \mathbb{R}$ by
$$\xi(p) = d_{g_\infty(0)}(p, \Sigma_{p_\infty}) + \xi_0,$$
where $\xi_0 \in \mathbb{R}$ is chosen such that $\xi\rightarrow 0$ as $b \rightarrow 0$.
Note that by the Ricci flow equation (\ref{b-evol}) for $b$ we have
$$|\partial_t b^2| \leq 3 b^2 |Rm_{g(t)}|_{g(t)} \leq 3 b^2 K \: \text{ on } \: M_\infty \times [-T,0].$$
Working in $(\xi, t)$ coordinates we see that
$$b^2(\xi, t) \leq b^2(\xi, 0) e^{3 K T}, \quad t \in [-T,0].$$
Hence for all $t \in [-T,0]$ we have $b(\xi, t) \rightarrow 0$ as $\xi \rightarrow 0$. As $M_\infty$ has bounded curvature, we see that $\frac{1-b_s^2}{b^2}$ is bounded as well and hence $b_s(\xi, t) \rightarrow 1$ as $\xi \rightarrow 0$. From Theorem \ref{RF-removable-singularity} in Appendix B it then follows that $g_\infty(t)$, $t \in (-T,0]$, can be extended to a smooth orbifold Ricci flow on $\mathbb{R}^4\times \mathbb{Z}_2$. Since $T$ was arbitrary, a diagonal argument produces an ancient orbifold Ricci flow $(\mathbb{R}^4\setminus/\mathbb{Z}_2, g_\infty(t), p_\infty)$, $t \in (-\infty, 0]$. Note that apriori $g_\infty(t)$ might have unbounded curvature as $t \rightarrow -\infty$.
As $Q = 1$, $b_s \in [0,1]$ and $b_{ss} \leq 0$ we see that $g_\infty(t)$ is rotationally symmetric and has positive sectional curvature. Furthermore, for each $t \in (-\infty, 0]$ the metric $g_\infty(t)$ is asymptotically cylindrical, as the following argument shows: Either $b$ is bounded, in which case $bb_{ss} \leq 0$ and $b_s \geq 0$ show that $\lim_{s\rightarrow \infty} b_s = 0$, or $b$ is unbounded, in which case the inequality $T_{F_2} \leq 0$ and the proof of Lemma \ref{lem:ODE-technical} show that on each time slice $b_s \rightarrow 0$ as $b \rightarrow \infty$.
By the Hamilton's trace Harnack inequality (see for instance \cite[Theorem D.26]{ChII}) and the fact that for any $T > 0$ the metric $g_\infty(t)$ has bounded curvature on $\mathbb{R}^4/\mathbb{Z}_2 \times [-T,0]$, it follows that
$$\partial_t R_{g_\infty(t)} \geq 0 \: \text{ on } \: \mathbb{R}^4/\mathbb{Z}_2 \times (-\infty,0].$$
Therefore $g_\infty(t)$ has bounded curvature on $\mathbb{R}^4/\mathbb{Z}_2 \times (\infty, 0]$. By the result of Li and Zhang \cite{LZ18} we conclude that $g_\infty(t)$ is homothetic to the four dimensional Bryant soliton quotiented by $\mathbb{Z}_2$.
\end{proof}
\begin{lem}
\label{flat-orbifold-blow-up-lem}
Let $(M_2, g(t))$, $t \in [0, T_{sing})$, be a Ricci flow starting from an initial metric $g(0) \in \mathcal{I}$ with $\sup_{p \in M_2} b(p,0) < \infty$. Let $(p_i,t_i)$ be a sequence of points in spacetime satisfying
\begin{enumerate}
\item $b(p_i,t_i) \rightarrow 0$
\item $b_s(p_i,t_i) \rightarrow 1$
\end{enumerate}
Consider the sequence of rescaled metrics
\begin{equation*}
g_i(t) = \frac{1}{b^2(p_i,t_i)} g(t_i + b^2(p_i,t_i) t), \quad t \in [-b(p_i,t_i)^2 t_i, 0].
\end{equation*}
Then $(M_2, g_i(t), p_i)$, $t \in [-b(p_i,t_i)^2 t_i, 0]$, subsequentially converges, in the Cheeger-Gromov sense, to an ancient Ricci flow $(M_\infty, g_\infty(t), p_\infty)$, $t\in (-\infty, 0]$, where $M_\infty \cong \mathbb{R}^4 \setminus\{0\} /\mathbb{Z}_2$ and $g_\infty(t)$ can be extended to a smooth orbifold Ricci flow on $\mathbb{R}^4/\mathbb{Z}_2$ that is stationary and isometric to the flat orbifold $\mathbb{R}^4/\mathbb{Z}_2$.
\end{lem}
\begin{proof}
First note that
\begin{claim}
$$\frac{b(p_i, t_i)}{b(o,t_i)} \rightarrow \infty \: \text{ as } \: i \rightarrow \infty$$
\end{claim}
\begin{claimproof}
We argue by contradiction. Assume there exists a $C>0$ such that after passing to a subsequence of $(p_i, t_i)$ we have
$$\frac{b(p_i, t_i)}{b(o,t_i)} \leq C.$$
Consider the rescaled metrics
$$g_i(t) = \frac{1}{b(p_i, t_i)^2} g \left(t_i + t b(p_i, t_i)^2 \right), \quad t \in [- b(p_i, t_i)^{-2}t_i, 0].$$
Then by by Corollary \ref{E-H-blowup} the sequence $(M_2, g_i(t), p_i)$ subsequentially converges to a blow-up limit $(M_2, g_\infty(t), p_\infty)$, which is homothetic to the Eguchi-Hanson space. By construction
$$b(p_\infty, 0) = 1$$
and
$$b_s(p_\infty, 0) = 1.$$
The latter follows from the assumption that $b_s(p_i,t_i) \rightarrow 1$ as $i \rightarrow \infty$. However, by Lemma \ref{E-H-properties-lem} we have $b_s < 1$ everywhere on the Eguchi-Hanson space. This is a contradiction and the claim follows.
\end{claimproof}
Fix $T>0$ and consider the rescaled metrics $g_i(t)$ on the parabolic sets $E_{(p_i,t_i,n)} \times [-T, 0]$ as in the proof of Lemma \ref{lem:orbifold-blowup}. By the same reasoning, we see that for all $n \in \mathbb{N}_{\geq 2}$ the flows $(E_{p_i,t_i,n}, g_i(t), p_i)$ subsequentially converges to a Ricci flow $(\mathcal{E}_{\infty, n}, g_{\infty, n}(t), p_{\infty, n})$, $t \in [-T,0]$. As in the proof of Lemma \ref{lem:orbifold-blowup}, we may assume that $\mathcal{E}_{\infty, n} \subset \mathcal{E}_{\infty, n+1}$ and $g_{\infty, n} = g_{\infty, n+1}$ on $\mathcal{E}_{\infty, n}$. Therefore we drop the dependence on $n$ and write $p_\infty$ and $g_{\infty}(t)$. By construction we have
$$b(p_\infty, 0) = 1$$
and
$$b_s(p_\infty, 0) = 1,$$
where the latter follows from the assumption that $b_s(p_i, t_i) \rightarrow 1$ as $i \rightarrow \infty$. Furthermore, by Lemma \ref{QT2bddb-limit-lem} and Claim 1 we have $Q = 1$ on $\mathcal{E}_{\infty, n} \times [-T, 0]$. Applying the strong maximum principle to the evolution equation (\ref{db-evol}) of $b_s$ when $Q=1$ we deduce that $b_s = 1$ everywhere in $\mathcal{E}_{\infty, n} \times [-T, 0]$. Hence $g_\infty(t)$ is flat and $(\mathcal{E}_{\infty, n}, g_\infty(t), p_\infty)$, $t \in [-T, 0]$, converges to the flat orbifold $\mathbb{R}^4 / \mathbb{Z}_2$ as $n \rightarrow \infty$. As $T> 0$ was arbitrary the desired result follows by a diagonal argument.
\end{proof}
\section*{Appendix A}
Here we carry out some of the computations we rely on throughout the paper. Recall
$$ \frac{\partial}{\partial s} = \frac{1}{u(\xi, t)} \frac{\partial}{\partial \xi}$$
and the commutation relation
$$ [\partial_t, \partial_s] = - \frac{a_{ss}}{a} - 2 \frac{b_{ss}}{b}$$
from subsection \ref{ricci-flow-equations-sec}. For the computations it will also be helpful to keep in mind that
$$ bQ_s = a_s - Q b_s$$
which follows from differentiating the expression $Q = \frac{a}{b}$. Finally recall the definition of the K\"ahler quantities $$x = a_s + Q^2 -2$$ and $$y = b_s - Q$$ from section \ref{kahler-E-H-section}.
First we compute the evolution equation of $Q$:
\begin{align*}
\partial_t Q &= \frac{\partial_t a}{b} - \frac{a \partial_t b}{b^2}
\end{align*}
Inserting the expressions for $\partial_t a$ and $\partial_t b$ from the evolution equations (\ref{a-evol}) and (\ref{b-evol}) for $a$ and $b$ we obtain
\begin{align*}
\partial_t Q = Q_{ss} + 3 \frac{b_s}{b} Q_s + \frac{4}{b^2} Q(1-Q^2).
\end{align*}
\subsection*{Evolution equations of $a_s$, $b_s$, $Qb_s$, $x$, $y$ and $\frac{y}{Q}$}
By the commutation relations above we have
\begin{align*}
\partial_t a_s &= \partial_s \partial_t a - \left(\frac{a_{ss}}{a} + 2 \frac{b_{ss}}{b}\right) a_s \\
\partial_t b_s &= \partial_s \partial_t b - \left(\frac{a_{ss}}{a} + 2 \frac{b_{ss}}{b}\right) b_s
\end{align*}
Hence plugging in the expressions for $\partial_t a$ and $\partial_t b$ from the evolution equations (\ref{a-evol}) and (\ref{b-evol}) for $a$ and $b$ we obtain
\begin{align*}
\partial_t a_s &= (a_s)_{ss} + \left(2 \frac{b_s}{b} - \frac{a_s}{a}\right) (a_s)_s + \frac{1}{b^2}\left( -2 a_s b_s^2- 6 Q^2 a_s + 8 Q^3 b_s \right) \\
\partial_t b_s &= (b_s)_{ss} + \frac{a_s}{a} (b_s)_s + \frac{1}{b^2}\left( -\frac{a_s^2 b_s}{Q^2}+4 Q a_s-6 Q^2 b_s-b_s^3+4 b_s \right)
\end{align*}
From here we can compute the evolution equation of $Q b_s$:
\begin{align*}
\partial_t Qb_s &= (\partial_t Q )b_s + Q \partial_t b_s \\
&= (Qb_s)_{ss} + \left(2 \frac{b_s}{b} - \frac{a_s}{a}\right) (Qb_s)_s +\frac{1}{b^2}\left( 4 Q^2 a_s- 10 Q^3 b_s - 2 Q b_s^3 + 8 Q b_s\right)
\end{align*}
Now we may compute the evolution equations of the K\"ahler quantities $x$ and $y$:
\begin{align*}
\partial_t x &= \partial_t a_s + 2 Q \partial_t Q \\
&= x_{ss} + \left(2 \frac{b_s}{b} - \frac{a_s}{a}\right) x_s - \frac{2}{b^2}\left(2 Q^2 + y^2\right) x - \frac{2}{b^2}\left(Q^2 +2 \right) y^2,
\end{align*}
where in the last step we made the substitutions $a_s = x - Q^2 + 2$, $b_s = y + Q$ and $a = Q b$. Similarly,
\begin{align}
\label{y-evol}
\partial_t y &= \partial_t b_s - \partial_t Q \\ \nonumber
&= y_{ss} + \frac{a_s}{a} y - \frac{y}{a^2} \left( \left(x+2\right)^2 + Q^2 \left(2x + y^2\right) \right)
\end{align}
Then we can compute
\begin{align}
\label{yQ-evol}
\partial_t \left(\frac{y}{Q}\right) &=\frac{\partial_t y}{Q} - \frac{y \partial_t Q}{Q^2} \\ \nonumber
&= \left(\frac{y}{Q}\right)_{ss} + \left(3 \frac{a_s}{a} - 2 \frac{b_s}{b} \right) \left(\frac{y}{Q}\right)_s + \frac{2}{b^2}\frac{y}{Q}\left( 2 + \frac{y}{Q}\right) \left( Q b_s - 2 a_s \right)
\end{align}
where we substituted $a_s = x - Q^2 + 2$, $b_s = y + Q$ and $a = Q b$ in the last step.
\subsection*{Evolution equation of $H_{\pm}$}
In section \ref{sec:curv-bound} we define the quantities
$$H_{\pm} \coloneqq bb_{ss} \mp a_s^2 - b_s^2 \pm C$$
for some constant $C> 0$. Below we derive its evolution equation.
First note that we have
$$\partial_t b_{ss} = \partial_s \partial_t b_{s} - \left(\frac{a_{ss}}{a} + 2 \frac{b_{ss}}{b}\right) b_s.$$
Substituting the evolution equation for $b_s$ derived above we obtain
\begin{align*}
\partial_t b_{ss} &= (b_{ss})_{ss} +\frac{a_s}{a}(b_{ss})_s + \frac{4 a a_{ss}}{b^3}-\frac{2 a_s^2 b_{ss}}{a^2}+\frac{2 a_s^3 b_s}{a^3} \\
& \qquad -\frac{24 a a_s b_s}{b^4}+\frac{4 a_s^2}{b^3}-\frac{2 a_s a_{ss} b_s}{a^2}-\frac{6 a^2 b_{ss}}{b^4}+\frac{24 a^2 b_s^2}{b^5}\\
& \qquad -\frac{2 b_{ss}^2}{b}+\frac{4 b_{ss}}{b^2}+\frac{2 b_s^4}{b^3}-\frac{8 b_s^2}{b^3}-\frac{3 b_s^2 b_{ss}}{b^2}
\end{align*}
Hence we can compute the evolution equation of $H$ via
$$\partial_t H = (\partial_t b)b_{ss} + b \partial_t b_{ss} \mp 2 a_s \partial_t a_s - 2 b_s \partial_t b_s$$
and substituting the expressions for $\partial_t b$, $\partial_t b_{ss}$, $\partial_t a_s$ and $\partial_t b_s$ derived above. Noting that
$$H_s = \mp 2 a_s a_{ss}+b (b_{ss})_s-b_{s} b_{ss}$$
and
$$H_{ss} = \mp 2 a_{ss}^2 \mp 2 (a_{ss})_{s} a_s+b (b_{ss})_{ss}-b_{ss}^2$$
a longer computation shows that
\begin{align*}
\partial_t H_\pm &= [H_\pm]_{ss} + \left(\frac{a_s}{a} -2 \frac{b_s}{b}\right) [H_\pm]_s +H_\pm \left(-\frac{2 a_s^2}{a^2}-\frac{4 a^2}{b^4}-\frac{4b_s^2}{b^2}\right) \\
& \pm C \left(\frac{2 a_s^2}{a^2}+\frac{4 a^2}{b^4}+\frac{4b_s^2}{b^2}\right) \\
&\pm 2 a_{ss}^2 + a_{ss} \left(-\frac{2 b a_s b_s}{a^2} \mp \frac{8 a_s b_s}{b} \pm \frac{4 a_s^2}{a}+\frac{4a}{b^2}\right) \\
&+\frac{2 b a_s^3 b_s}{a^3}-\frac{32 a a_sb_s}{b^3}\mp\frac{16 a^3 a_s b_s}{b^5}+\frac{4 a_s^2}{b^2} \pm \frac{8 a^2a_s^2}{b^4} \\
&\mp \frac{2 a_s^4}{a^2} +\frac{32 a^2 b_s^2}{b^4}-\frac{16 b_s^2}{b^2}.
\end{align*}
\subsection*{Evolution equation of $f_{\theta}(Q)$}
\begin{align*}
\partial_t f_{\theta}(Q) &= f' \partial_t Q \\
&= f' \left( Q_{ss} + 3 \frac{b_s}{b} Q_s + \frac{4}{b^2}Q \left(1 - Q^2\right) \right)
\end{align*}
by the evolution equation (\ref{Q-evol}) of $Q$. Note that we omitted the dependence the quantities on spacetime $(\xi, t)$ and and the dependence of $f$ on $Q$ and $\theta$. For example we wrote $f'$ for $f'_{\theta}(Q(\xi,t))$. Noting that
$$\left[f(Q)\right]_{s} = f'(Q)Q_s $$
and
$$ \left[f(Q)\right]_{ss} = f''(Q)Q^2_s + f'(Q) Q_{ss}$$
we obtain
\begin{align*}
\partial_t f(Q) &= [f(Q)]_{ss} - f'' Q_s^2 + 3 \frac{b_s}{b} [f(Q)]_s + \frac{4}{b^2} f' Q \left(1 - Q^2 \right) \\
&= [f(Q)]_{ss} + \left(3 \frac{a_s}{a} - 2 \frac{b_s}{b} \right)[f(Q)]_s \\
& \qquad \qquad + \left(5 \frac{b_s}{b} - 3 \frac{a_s}{a} \right) [f(Q)]_s + \frac{4}{b^2} f' Q \left(1 - Q^2 \right) - f'' Q_s^2\\
&= [f(Q)]_{ss} + \left(3 \frac{a_s}{a} - 2 \frac{b_s}{b} \right)[f(Q)]_s + \frac{1}{b^2} C_f
\end{align*}
where
\begin{align*}
C_f &= \left(5 b_s - 3 \frac{a_s}{Q} \right) b[f(Q)]_s + 4 f' Q \left(1 - Q^2 \right) - f'' b^2 Q_s^2 \\
&= \left(5 b_s - 3 \frac{a_s}{Q} \right)f' \left(a_s - Qb_s\right) + 4 f' Q \left(1 - Q^2 \right) - \left(a_s - Q b_s\right)^2f'' \\
&= \left( 8 a_s b_s - 3 \frac{a_s^2}{Q} - 5 Q b_s^2 + 4 Q \left(1 - Q^2\right) \right)f' - \left( a_s - Q b_s \right)^2f''
\end{align*}
\subsection*{Evolution equation of $Z_{\theta}$}
We have
\begin{align*}
\partial_t Z_{\theta} = \partial_t \left(\frac{x}{Q^2}\right) + \partial_t f_{\theta}(Q)
\end{align*}
We computed the evolution equation for $f_{\theta}(Q)$ above. Therefore it remains to compute $\partial_t \frac{x}{Q^2}$. For this recall the evolution equation (\ref{x-evol}) of $x$
$$ \partial_t x = x_{ss} + \left(2 \frac{b_s}{b} - \frac{a_s}{a} \right) x_{s} + \frac{1}{b^2}C_x $$
where
$$ C_x = - 2 \left(2 Q^2 + y^2\right) x - 2 \left(Q^2 +2 \right) y^2.$$
Differentiation shows that
$$ \partial_s \left(\frac{x}{Q^2}\right) = \frac{x_s}{Q^2}- 2 \frac{xQ_s}{Q^3}$$
and
$$ \partial_{ss} \left(\frac{x}{Q^2}\right) = \frac{x_{ss}}{Q^2}- 4 \frac{x_sQ_s}{Q^3} - 2x \frac{Q_{ss}}{Q^3} + 6x \frac{Q_s^2}{Q^4} .$$
Therefore we get
\begin{align*}
\partial_t \frac{x}{Q^2} &= \frac{1}{Q^2}\partial_t x - 2\frac{x}{Q^3} \partial_t Q \\
&= \left(\frac{x}{Q^2}\right)_{ss}+ \left(3\frac{a_s}{a} - 2 \frac{b_s}{b} \right) \left(\frac{x}{Q^2}\right)_s + \frac{1}{b^2} C_{\frac{x}{Q^2}}
\end{align*}
where
\begin{align*}
C_{\frac{x}{Q^2}} &= 6 \frac{x a_s}{Q^4} (bQ_s) - 10 \frac{x b_s}{Q^3} (bQ_s) - 6\frac{x}{Q^4} (bQ_s)^2 + \frac{C_x}{Q^2} - \frac{8x}{Q^2} (1- Q^2) \\
&= -\frac{4 a_{s}^2 b_{s}}{Q^3}+\frac{2 a_{s} b_{s}^2}{Q^2}+\frac{8 a_{s}
b_{s}}{Q^3}-\frac{8 a_{s}}{Q^2}+2 a_{s}-\frac{8 b_{s}^2}{Q^2}+8 Q
b_{s}+\frac{16}{Q^2}-16
\end{align*}
In the last step, we used the expressions for $x$, $y$ and $Q_s$ in terms of $a_s$, $b_s$ and $Q$ to eliminate $x$, $y$ and $Q_s$ from the expression for $C_{\frac{x}{Q^2}}$. Hence we have
$$\partial_t Z_{\theta} = [Z_{\theta}]_{ss} + \left( 3\frac{a_s}{a} - 2 \frac{b_s}{b}\right) [Z_{\theta}]_s + \frac{1}{b^2} C_Z$$
where
$$C_Z = C_{\frac{x}{Q^2}} + C_f.$$
As
$$ Z_{\theta} = \frac{x}{Q^2} + f_{\theta}(Q) = \frac{a_s + Q^2 - 2}{Q^2} + f_{\theta}(Q)$$
by definition, we can solve for $a_s$ to obtain
$$a_s = Q^2 Z_{\theta} - Q^2 f_{\theta} + 2 - Q^2.$$
Using this substitution to eliminate all occurring $a_s$ from the expression $C_Z$ we obtain
\begin{align*}
C_Z = C_{Z,0} + C_{Z,1} Z_{\theta} + C_{Z,2} Z_{\theta}^2
\end{align*}
where
\begin{align*}
C_{Z,0} &= A_0 + A_1 \left[\frac{b_s}{Q}\right] + A_2 \left[\frac{b_s}{Q}\right]^2 \\
C_{Z,1} &= 2 Q^3 b_s f''+8 Q^2 b_s f'+8 f Q b_s+8 Qb_s-\frac{8 b_s}{Q} +2 b_s^2+2 f Q^4 f'' \\
&+2 Q^4 f''-4Q^2 f''+6 f Q^3 f'+6 Q^3 f'-12 Q f'+2 Q^2-8\\
C_{Z,2} &= -4 Q b_s -Q^4f''-3 Q^3 f'
\end{align*}
and
\begin{align*}
A_0 =& - Q^4 f^2 f''-2 Q^4 f f''-Q^4 f''+4 Q^2 f f''+4 Q^2 f''-4 f''-3 Q^3 f^2 f' \\ \nonumber
&-6 Q^3 f f'-7 Q^3 f'+12 Q f f'+16 Q f'-\frac{12 f'}{Q}-2 Q^2 f+8 f-2 Q^2-4 \\
A_1 =& -2 Q^4 f f''-2 Q^4 f''+4 Q^2 f''-8 Q^3 f f'-8 Q^3 f' \\ \nonumber
& \qquad+16 Q f'-4 Q^2 f^2-8 Q^2 f+8 f+4 Q^2+8 \\
A_2 =& - Q^4f''-5 Q^3 f'-2 Q^2 f-2 Q^2-4 \\
\end{align*}
\subsection*{Evolution equation of $Z_1$}
The evolution equation for $Z_1 = \frac{x}{Q^2} + 1$ follows quickly from the evolution equations for $Z_{\theta}$ by setting $f = 1$. One obtains
\begin{equation}
\partial_t Z_1 = [Z_1]_{ss} + \left( 3\frac{a_s}{a} - 2 \frac{b_s}{b}\right) [Z_1]_s + \frac{1}{b^2}\left( C_{Z_1, 0} + C_{Z_1, 1} Z_1 + C_{Z_1, 2} Z_1^2\right)
\end{equation}
where
\begin{align*}
C_{Z_1,0} &= \frac{1}{Q^2}\left( - 4\left(1+Q^2\right)y^2 + 8Q\left(1-2Q^2 \right)y + 16Q^2\left(1-Q^2\right) \right) \\
C_{Z_1,1} &= 16 Q b_s-\frac{8 b_s}{Q}+2 b_s^2+2 Q^2-8 \\
C_{Z_1,2} &= -4 Q b_s.
\end{align*}
Note that we wrote $C_{Z_1,0}$ in terms of $y = b_s - Q$ instead of $b_s$ in order to see the similarity with the zeroth order term of the evolution equation of $T_1$ presented in the proof of Lemma \ref{T1-preserved-lem}.
\subsection*{Evolution equations when $Q=1$}
When $Q = 1$ we have $a = b$ and the Ricci flow equations simplify. In particular, we obtain
\begin{align*}
\frac{\partial_t u}{u} &= 3\frac{b_{ss}}{b} \\
\partial_t b &= b_{ss} + \frac{2}{b}\left(b_s^2 -1 \right)
\end{align*}
Using the commutation relation of $\partial_t$ and $\partial_s$ we can also compute the evolution equation of $b_s$ and $b_{ss}$:
\begin{align}
\label{db-evol}
\partial_t b_s &= \partial_s \partial_t b - 3 b_s \frac{b_{ss}}{b} \\ \nonumber
&= (b_s)_{ss} + \frac{b_s}{b}(b_s)_s + 2 \frac{b_s}{b^2}\left( 1- b_s^2\right)
\end{align}
Similarly,
\begin{align}
\label{ddb-evol}
\partial_t b_{ss} &= \partial_s \partial_t b_s - 3 b_{ss} \frac{b_{ss}}{b} \\ \nonumber
&= (b_{ss})_{ss} +\frac{b_{s}}{b} (b_{ss})_s -\frac{2 b_{ss}^2}{b}+\frac{4 \left(b_{s}^2-1\right) b_{s}^2}{b^3} \\ \nonumber
&\qquad -\frac{5 b_{s}^2 b_{ss}}{b^2}-\frac{2 \left(b_{s}^2-1\right) b_{ss}}{b^2}
\end{align}
Let us introduce the notation
\begin{align*}
X &\coloneqq 1 - b_s^2 \\
Y &\coloneqq -b b_{ss}.
\end{align*}
We need the evolution equations of \emph{scale-invariant} quantities of the form
$$T_F = -Y + F(X),$$
where $F: [0, 1] \rightarrow \mathbb{R}$ is a smooth function.
In particular, we see that
$$\partial_t T_F = (\partial_t b) b_{ss} + b \partial_t b_{ss} -2 b_s F'\left(1-b_s^2\right) \partial_t b_s$$
Expanding this expression, we obtain
$$\partial_t T_F = (T_F)_{ss} - \frac{b_s}{b}(T_F)_s + \frac{1}{b^2} C_F$$
where
\begin{align*}
C_F &= 4 X^2-4 X Y-4 X-2 Y^2+4 Y \\
& +2\left( 2 X^2-2 X Y-2 X+Y^2+2 Y \right)F'(X) \\
& +4 (X-1) Y^2 F''(X)
\end{align*}
In this paper we make use of three different choices of $F$:
\begin{align*}
F_0(X) &= 0 \\
F_1(X) &= X \\
F_2(X) &= X - X^2
\end{align*}
Plugging these into the expression for $C_F$ above we compute
\begin{align*}
C_{F_0} &= -4 b_s^2 \left( 1- b_s^2\right) -2 b b_{ss} \left(b b_{ss}+2 b_s^2\right) \\
C_{F_1} &= -8 b_s^2 T_{F_1} \\
C_{F_2} &= 4 \left( (2-3 X) Y^2 +\left(2 X^2-4 X+2\right) Y -2 (X-1)^2 X \right)
\end{align*}
We also prove the following lemma here:
\begin{lem}
\label{lem:CF2-neg}
Let $X\in [0,1]$. Then whenever $0 \leq Y < X - X^2$ we have
$$P(X,Y) \coloneqq (2-3 X) Y^2 +\left(2 X^2-4 X+2\right) Y -2 (X-1)^2 X < 0.$$
In other words, $C_{F_2} < 0$ whenever $T_{F_2} > 0$ and $bb_{ss} \leq 0$.
\end{lem}
\begin{proof}
Let $R$ be the region in the $X$-$Y$-plane satisfying the inequalities $X\in [0,1]$ and $0 \leq Y < X - X^2$. Note that $Y < X - X^2$ and $Y \geq 0$ implies that $X \in (0,1)$.
A computation shows
$$P(X, X- X^2) = -3 (X-1)^2 X^3 < 0 \: \text{ for } \: X \in (0,1)$$
Notice that for a fixed $X \in [0, \frac{2}{3}]$ the quadratic polynomial $P(X,Y)$ in $Y$ is convex. As
$$P(X, 0) = -2 (X-1)^2 X < 0 \: \text{ for } \: X \in (0,1)$$
we deduce that $P(X,Y) > 0$ on $R \cap \{ X \leq \frac{2}{3} \}$. To prove that $P(X, Y) > 0$ in $R \cap \{ X \geq \frac{2}{3} \}$ is trickier. For this we prove the following claim:
\begin{claim}
$\partial_X P(X, Y) > 0$ on $R \cap \{ X \geq \frac{2}{3} \}$.
\end{claim}
\begin{claimproof}
A computation shows
$$\partial_X P(X, Y) = -6 X^2+8 X -2 +(4 X-4) Y-3 Y^2.$$
Hence for fixed $X$ is concave in $Y$. Then note that
$$\partial_X P(X, 0) = -6 X^2+8 X -2 > 0 \: \text{ for } \: X \in [\frac{2}{3},1)$$
and
$$\partial_X P(X, X-X^2) = (1-X) \left(3 X^3+X^2+2 X-2\right) > 0 \: \text{ for } \: X \in [\frac{2}{3},1).$$
The last inequality follows by the fact that the polynomial $3 X^3+X^2+2 X-2$ is increasing on $[0,1]$ and evaluates to $\frac{2}{3}$ at $X = \frac{2}{3}$. Hence the claim follows by concavity of $\partial_X P(X, Y)$ in $Y$.
\end{claimproof}
By above we know that $P(X,X^2-X) < 0$ for $X\in (0,1)$. Hence using the result of the claim, we see that $P(X,Y) < 0$ on $R \cap \{ X \geq \frac{2}{3} \}$.
\end{proof}
\section*{Appendix B: Removable singularity}
We prove the following theorem:
\begin{thm}[Removable singularity]
\label{RF-removable-singularity}
Let $(\mathbb{R}^4 \setminus \{0\}, g(t))$, $t \in [0,T]$, be a rotationally symmetric Ricci flow of bounded curvature, i.e. there exists a $K>0$ such that
$$|Rm_{g(t)}|_{g(t)} < K \: \text{ on } \: \mathbb{R}^4 \times [0,T].$$
Taking $\xi \in (0,\infty)$ to be a radial coordinate on $\mathbb{R}^4 \setminus \{0\}$ the metric $g(t)$ may be written as
$$g(t) = u^2(\xi, t) d\xi^2 + b^2(\xi,t) g_{S^3},$$
where $u,b : (0,\infty) \rightarrow \mathbb{R}$ are smooth warping functions, and $g_{S^3}$ is the round metric on $S^3$ with sectional curvatures equal to one. If for all $t \in [0,T]$ the warping function $b(\xi,t) \rightarrow 0$ as $\xi \rightarrow 0$, then $g(t)$ can be extended to a smooth Ricci flow on $\mathbb{R}^4 \times (0,T]$.
\end{thm}
Below we assume $(\mathbb{R}^4 \setminus \{0\}, g(t))$, $t \in [0,T]$, is a Ricci flow as in Theorem \ref{RF-removable-singularity}. The proof strategy will be as follows: First we prove in Lemma \ref{lem:C11-regularity} that for every $t_0 \in [0,T]$ there exist coordinates $x^i, i = 1,2,3,4$, of $\mathbb{R}^4$ for which the metric $g(t_0)$ can be extended to a $C^{1,1}$ metric on $\mathbb{R}^4$. Note, however, without control on the derivative of the curvature tensor the metric $g(t)$ at times $t \neq t_0$ may not to be $C^{1,1}$ with respect to the coordinates $x^i$. To get around this issue we show in Lemma \ref{lem:curv-deriv-bounded} and Lemma \ref{lem:higher-curv-deriv-bounded} that in fact all derivatives $\nabla^m Rm$, $m \in \mathbb{N}$, of the curvature tensor are bounded on $\mathbb{R}^4 \setminus \{0\}\times(\delta,T]$ for any $\delta > 0$. The proof utilizes Shi's interior derivative estimates and is based on a De Giorgi-Nash-Moser iteration argument. With these results in place, we use harmonic coordinates to prove Theorem \ref{RF-removable-singularity}. Let us begin by proving
\begin{lem}
\label{lem:C11-regularity}
Let $g = ds^2 + b(s)^2 g_{S^3}$ be a smooth, rotationally symmetric metric with bounded curvature on $\mathbb{R}^4\setminus\{0\}$. Here $g_{S^3}$ is the round metric of curvature one on $S^3$ and $b: (0,\infty) \rightarrow \mathbb{R}$ is a smooth positive function. If
$$ b \rightarrow 0 \: \text{ as } \: s \rightarrow 0$$
then $g$ can be extended to a $C^{1,1}$ metric on $\mathbb{R}^4$. Furthermore, if we take standard Euclidean coordinates $x_i$, $i = 1, 2, 3,4$, on $\mathbb{R}^4$ we have $g_{ij} = \delta_{ij}$ and $\partial_k g_{ij} = 0$ at the origin, and $\partial_k \partial_l g_{ij}$ locally bounded on $\mathbb{R}^4\setminus \{0\}$.
\end{lem}
\begin{proof}
As $g$ has bounded curvature there exists a $K>0$ such that
$$\Big|\frac{b_{ss}}{b}\Big|, \Big|\frac{1-b_s^2}{b^2}\Big| \leq K,$$
because these are the only non-zero curvature components of a rotationally symmetric metric. In particular, this shows that
$$b_s \rightarrow 1 \: \text{ as } \: s \rightarrow 0^+$$
and
$$b_{ss} \rightarrow 0 \: \text{ as } \: s \rightarrow 0^+.$$
Let $x_i$, $i =1, 2,3,4$, be Euclidean coordinates of $\mathbb{R}^4$, normalized such that $\sum_i (x^i)^2 = s^2$. In these coordinates
$$g = \left[\delta_{ij} + \left(s^2 \delta_{ij} - x_i x_j \right) \Psi(s) \right] \textrm{d}x^i \textrm{d}x^j,$$
where
$$\Psi(s) = \frac{\left(\frac{b}{s}\right)^2 -1}{s^2}.$$
Note that we used the Einstein summation convention.
\begin{claim}
$\Psi$, $s \partial_s \Psi$, $s^2 \partial_{ss} \Psi = O(K)$ as $s\rightarrow 0$.
\end{claim}
\begin{claimproof}
Fix $s>0$. By Taylor's theorem there exist numbers $s_0, s_1 \in(0,s)$ such that
\begin{align*}
b(s) &= s + \frac{1}{2} b_{ss}(s_0) s^2 \\
b_s(s) &= 1 + b_{ss}(s_1)s.
\end{align*}
Hence
\begin{align*}
\Psi(s) &= \frac{b_{ss}(s_0)}{s} + \left(\frac{b_{ss}(s_0)}{2}\right)^2.
\end{align*}
As $\big|\frac{b_{ss}}{b}\big| \leq K$, $b \rightarrow 0$ and $b_s \rightarrow 1$ as $s \rightarrow 0$, we see $\Psi(s) = O(K)$ as $s\rightarrow 0$. By similar reasoning one shows that
\begin{align*}
s \partial_s \Psi(s) &= \frac{ 2 - 4\left(\frac{b}{s}\right)^2 + 2 \left(\frac{b}{s}\right) b_s}{s^2} \\
&= -\frac{3 b_{ss}(s_0)}{s}+\frac{2 b_{ss}(s_1)}{s}-b_{ss}(s_0)^2+b_{ss}(s_1) b_{ss}(s_0)
\end{align*}
and
\begin{align*}
s^2 \partial_{ss} \Psi(s) &= \frac{2 b b_{ss}-16 \left(\frac{b}{s}\right) b_s + 2 b_s^2 +20 \left(\frac{b}{s}\right)^2 -6}{s^2} \\
&= b_{ss}(s_0) b_{ss}(s)+\frac{2 b_{ss}(s)}{s}+\frac{12 b_{ss}(s_0)}{s}-\frac{12 b_{ss}(s_1)}{s} \\
& \quad\qquad +5 b_{ss}(s_0)^2-8 b_{ss}(s_1) b_{ss}(s_0)+2 b_{ss}(s_1)^2
\end{align*}
are of order $O(K)$ as $s \rightarrow 0$.
\end{claimproof}
Next, extend $g$ to the origin by setting $g = \delta_{ij}$ there. As
$$ \left(s^2 \delta_{ij} - x_i x_j \right) = O(s^2)$$
it follows by Claim 1 that this defines a continuous extension of $g$ to the origin.
A computation shows
$$\partial_k g_{ij} = \left(2 x_k \delta_{ij} - \delta_{ik} x_j - x_i \delta_{jk} \right) \Psi(s) +\left( s^2 \delta_{ij} - x_i x_j \right) \frac{x_k}{s}\partial_s \Psi(s).$$
As
$$\left(2 x_k \delta_{ij} - \delta_{ik} x_j - x_i \delta_{jk} \right) = O(s) $$
and
$$\left( s^2 \delta_{ij} - x_i x_j \right) \frac{x_k}{s} = O(s^2),$$
it follows that we may continuously extend $\partial_k g_{ij}$ to the origin by setting $\partial_k g_{ij} = 0$ there. Finally, note that
$$\partial_k \partial_l g_{ij} = O(1) \Psi + O(s)\partial_s \Psi + O(s^2) \partial_{ss} \Psi = O(K).$$
Hence $\partial_k \partial_l g_{ij}$ is bounded in a neighborhood around, but excluding the origin. This shows that $\partial_k g_{ij}$ is Lipshitz.
\end{proof}
Next, we prove the boundedness of the gradient of the curvature tensor. For this we recall some interior curvature estimates. Note the following differential inequalities for the evolution of the curvature tensor and its derivatives under Ricci flow (See for instance \cite[Chapter 7]{BC04}):
\begin{align}
\label{curv-evol} \big(\partial_t - \Delta \big) |Rm|^2 &\leq - 2|\nabla Rm|^2 + 16 |Rm|^3 \\
\label{dcurv-evol} \big(\partial_t - \Delta \big) |\nabla^m Rm|^2 &\leq- 2 |\nabla^{m+1} Rm|^2 \\ \nonumber
& \qquad+ \sum_{j=0}^m c_{mj} |\nabla^j Rm| \cdot |\nabla^{m-j} Rm| \cdot |\nabla^m Rm|
\end{align}
Here $c_{mj}$ are positive constants depending on $j$, $m$ and the dimension of the manifold only. Note also that the laplacian is with respect to the evolving metric $g(t)$. Using these inequalities one can show the following interior derivative estimate (See for instance \cite[Theorem 1.4.2]{CZ06}).
\begin{thm}[Shi's interior estimates]
\label{shi}
There exist positive constants $\theta, C_m, m \in \mathbb{N}$, depending on the dimension $n$ only, such that the following holds: Let $M$ be a manifold of dimension $n$ and $0 < T \leq \frac{\theta}{K}$. Assume that $g(t)$, $t \in [0, T]$, is a solution to the Ricci flow on an open neighborhood $U$ of $M$ and
$$|Rm| < K \: \text{ on } \: B_{g(0)}\left(p, r\right) \times [0,T].$$
If for $p \in U$ and $r>0$ the closed set $\overline{B_{g(0)}(p, r)}$ is contained in $U$ then
$$|\nabla^m Rm|^2<C_m K^2 \left(\frac{1}{r^{2m}} + \frac{1}{t^m} + K^m\right) \: \text{ on } \: B_{g(0)}\left(p, \frac{r}{2}\right) \times (0,T]$$
\end{thm}
Next, we prove that for all $\tau > 0$ the gradient $|\nabla Rm|$ is bounded on $\mathbb{R}^4 \setminus \{0\} \times [\tau, T]$. First note that due to Shi's estimates of Theorem \ref{shi}
$$|\nabla Rm_{g(t)}|_{g(t)}(p) = O\left(\frac{1}{d_{g(t)}(p, 0)}\right) \: \text{ on } \: \mathbb{R}^4 \setminus \{0\}\times [\tau, T].$$
Furthermore, from (\ref{dcurv-evol}) and and an application of Kato's inequality to show that
$$|\nabla |\nabla Rm|| \leq |\nabla^2 Rm|$$
it follows that
$$\big( \partial_t - \Delta \big) |\nabla Rm| \leq C|Rm||\nabla Rm|.$$
Hence when curvature is bounded by $K$, the function $u \coloneqq e^{-CKt} |\nabla Rm|$ is a subsolution to the heat equation, i.e.
$$\big(\partial_t - \Delta \big) u \leq 0.$$
With help of a De Giorgi-Nash-Moser iteration argument, this is enough to prove that $u$ is bounded for $t > \tau$. We carry this out in the lemma below:
\begin{lem}
\label{lem:curv-deriv-bounded}
Let $(\mathbb{R}^4 \setminus \{0\},g(t))$, $ t\in [0,T]$, be a Ricci flow as in Theorem \ref{RF-removable-singularity}. Then for any $\tau > 0$ there exists a $C = C(K, \tau) > 0$ such that
$$|\nabla Rm| < C$$
on $\mathbb{R}^4 \times \setminus \{0\} \times [\tau, T]$.
\end{lem}
\begin{proof}
As shown above, the function $u = e^{-CKt} |\nabla Rm|$ is a subsolution to the heat equation, i.e.
$$\left(\partial_t - \Delta\right) u \leq 0.$$
By Lemma (\ref{lem:C11-regularity}) we may choose Euclidean coordinates $x^i$, $i = 1, 2, 3, 4$, on $\mathbb{R}^4$ for which $g(0)$ is $C^{1,1}$. Take $s^2 = \sum_i (x^i)^2$ and write $B_R(x)$ for the ball centered at $x$ with radius $R$ with respect to $g(0)$.
Since the curvature of $g(t)$ is bounded on $\mathbb{R}^4 \setminus \{0\} \times [0,T]$, there exists a $\lambda > 0$ such that
$$\frac{1}{\lambda} g(0) \leq g(t) \leq \lambda g(0) \: \text{ on } \: \mathbb{R}^4 \setminus \{0\} \times [0,T].$$
Therefore, Shi's interior estimates imply
$$ u( \cdot, t) = O\left(\frac{1}{s}\right) \: \text{ for } \: t \in [\tau, T].$$
Hence it suffices to show that for some $R>0$ the function $u$ is bounded on $B_R(0) \times [\tau, T]$. We achieve this via a De Giorgi-Nash-Moser iteration argument. In the Claim below we derive the crucial estimate.
\begin{claim}
Let $\delta > 0$, $p \geq 2$, $R_0 \in [1,10]$ and $t_0 \in [\delta, T)$. Then there exists a constant $C = C(K, \delta, T) >0$ such that the following holds: If $u \in L^p(B_{R_0} \times [t_0, T])$, then for $R_1 \in [\frac{1}{2}, R_0)$ and $t_1 \in (t_0, T]$
$$ \norm{u}_{L^{2p}(B_{R_1}(0) \times[t_1,T])} \leq \left[C\left(\frac{p^2}{(R_0 - R_1)^2} + \frac{1}{t_1-t_0} \right)\right]^{\frac{1}{p}} \norm{u}_{L^{p}(B_{R_0}(0) \times[t_0,T])}.$$
\end{claim}
\begin{claimproof}
In the following a constant $C$ is assumed to only depend on $K$, $\delta$ and $T$ and might vary from line to line. Fix a number $A > 1$ that we later take to $\infty$. Then choose a $C^2$ function $F: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}$ with the following properties:
\begin{enumerate}
\item $F(s) = s^p$ for $ s \leq A$
\item $F$ is linear for $s \geq A+1$ with slope $pA^{p-1} + 1$
\item On $[A,A+1]$ take $F(s)$ to be defined such that $F''\geq 0$
\end{enumerate}
We see that these properties imply that $F'(s) \leq ps^{p-1}$. Next, define the cut-off functions $\eta_\epsilon$, $\epsilon>0$, and $\phi: \mathbb{R}^4 \rightarrow \mathbb{R}$. For this take a smooth function $h: \mathbb{R} \rightarrow \mathbb{R}$ with $h = 0$ on $(-\infty,\frac{1}{2}]$ and $h = 1$ on $[1,\infty)$. Then define
$$\eta_\epsilon = h\left(\frac{s}{\epsilon}\right)$$
and
$$\phi = h\left(\frac{R_0-s}{R_0-R_1}\right).$$
That is, $\phi = 1$ on $B_{R_1}(0)$ and $\phi = 0$ on $\mathbb{R}^4 \setminus B_{R_0}(0)$. Clearly, $|\nabla \phi|_{g(t)} \leq \frac{c}{R_0 - R_1}$ and $|\nabla \eta_\epsilon|_{g(t)} \leq \frac{c}{\epsilon}$ for some universal constant $c$ depending on $h$ and $\lambda$ only.
Since $u \in L^p(B_{R_0} \times [t_0, T])$ is a positive function there exists a $t' \in [t_0, t_1]$ such that
\begin{equation}
\label{slice-p-bound}
\int_{B_{R_0}(0)} u^p(\cdot, t') \; \mathrm{d}x \leq \frac{1}{t_0-t_1} \int_{t_0}^T \int_{B_{R_0}(0)} u^p \; \mathrm{d}x \; \mathrm{d}t.
\end{equation}
Next, we compute via integration by parts
\begin{align*}
\frac{d}{dt} \int_{\mathbb{R}^4} \eta_\epsilon F(u) \phi^2 \; \mathrm{d} x &= \int_{\mathbb{R}^4} \eta_\epsilon F'(u) \Delta u \phi^2 \; \mathrm{d} x \\
&= -\int_{\mathbb{R}^4} \nabla \eta_\epsilon F'(u) \nabla u \phi^2 \; \mathrm{d} x
-\int_{\mathbb{R}^4} \eta_\epsilon F''(u) |\nabla u|^2 \phi^2 \; \mathrm{d} x \\
& \qquad\qquad -\int_{\mathbb{R}^4} \eta_\epsilon F'(u) \nabla u \nabla \phi^2 \; \mathrm{d} x.
\end{align*}
Integrating with respect to time from $t'$ to $T$ and noting that
$$\int_{\mathbb{R}^4} \eta_\epsilon F(u(\cdot, T)) \phi^2 \; \mathrm{d} x \geq 0,$$
we obtain
\begin{align}
\label{crucial-ineq}
\int_{t'}^T \int_{\mathbb{R}^4} \eta_\epsilon F''(u) |\nabla u|^2 \phi^2 \; \mathrm{d}x \; \mathrm{d}t
&\leq \int_{\mathbb{R}^4} \eta_\epsilon F(u(\cdot, t')) \phi^2 \; \mathrm{d}x
-\int_{t'}^T\int_{\mathbb{R}^4} \nabla \eta_\epsilon F'(u) \nabla u \phi^2 \; \mathrm{d}x \; \mathrm{d}t \\ \nonumber
&\qquad \qquad -\int_{t'}^T\int_{\mathbb{R}^4} \eta_\epsilon F'(u) \nabla u \nabla \phi^2 \; \mathrm{d}x \; \mathrm{d}t \\ \nonumber
&\coloneqq I_1 - I_2 - I_3.
\end{align}
Next we estimate each of these integrals $I_1$, $I_2$ and $I_3$ separately, in order to analyze their behaviors as $\epsilon \rightarrow 0$. For the first integral we have
$$I_1 \leq \int_{B_{R_0}} u^p(\cdot, t') \; \mathrm{d}x \leq \frac{1}{t_1-t_0} \norm{u}^p_{L^p(B_{R_0}(0)\times [t_0,T])}. $$
For the second integral $I_2$, note that Shi's estimates and Kato's inequality yield
$$|\nabla u| = |\nabla \left( e^{-CKt}|\nabla Rm| \right)| \leq |\nabla^2 Rm| = O\left(\frac{1}{s^2}\right) \: \text{ as } \: s \rightarrow 0.$$
As $|\nabla \eta_\epsilon| \leq \frac{c}{\epsilon}$, $|F'| \leq pA^{p-1} +1$ and $\phi^2 =1$ in a neighborhood of $0$, we see that
$$|\nabla \eta_\epsilon| \cdot |F'(u)|\cdot |\nabla u|\cdot |\phi^2| \leq C \epsilon^{-3} (pA^{p-1}+1) \: \text{ on } \: B_{\epsilon}(0).$$
As $\textrm{vol}_{g(t)}(B_\epsilon(0)) \leq C \epsilon^4$ we obtain
$$|I_2| \leq C \epsilon (pA^{p-1}+1).$$
For the final integral $I_3$, recall that by definition $0 \leq F'(u) \leq pu^{p-1}$. Furthermore, $|\nabla \phi^2|$ has support in $B_{R_0}(0)\setminus B_{R_1}(0)$ and is bounded by $\frac{2c}{R_0-R_1}$. As $R_1 \geq \frac{1}{2}$ by assumption, Shi's estimates imply that on this set $|\nabla u|$ and $u$ are bounded by some constant $C$. Thus
\begin{align*}
|I_3| &\leq \frac{2c pC^{p}}{R_0-R_1} \textrm{vol}\left(B_{R_0}(0)\setminus B_{R_1}(0)\right) \\
&\leq pC^{p+1},
\end{align*}
where we used that
$$\textrm{vol}(B_{R_0}(0)\setminus B_{R_1}(0)) \leq C\left( R^4_0 -R^4_1 \right) \leq C \left(R_0 - R_1\right),$$
as $\frac{1}{2} \leq R_0 \leq R_1 \leq 10$ by assumption. This shows that $I_3$ is convergent. Now split the integral $I_3$ as
$$I_3 = \int_{t'}^T \int_{\{u \leq A\}} \eta_\epsilon F'(u) \nabla u \nabla \phi^2 \; \mathrm{d}x \; \mathrm{d}t + I_4,$$
where
$$I_4 = \int_{t'}^T \int_{\{u \geq A\}} \eta_\epsilon F'(u) \nabla u \nabla \phi^2 \; \mathrm{d}x \; \mathrm{d}t.$$
As $I_3$ is convergent, we see that $I_4 \rightarrow 0$ as $A \rightarrow \infty$. Observe that by Young's inequality
$$p u^{p-1} |\nabla u| |\nabla \phi^2| = \phi u^{\frac{p-2}{2}} |\nabla u| \cdot 2p u^{\frac{p}{2}} |\nabla \phi| \leq \frac{1}{2} u^{p-2} |\nabla u|^2 \phi^2 + 2p^2 |\nabla \phi|^2 u^p.$$
Moreover,
$$|\nabla \phi|^2 \leq \left(\frac{c}{R_0-R_1}\right)^2.$$
Hence we obtain
\begin{align*}
|I_3| &\leq \int_{t'}^T\int_{\{u \leq A\}} \frac{1}{2} \eta_\epsilon u^{p-2} |\nabla u|^2 \phi^2 + 2 p^2 \eta_\epsilon |\nabla \phi|^2 u^p \; \mathrm{d} x \; \mathrm{d} t + I_4 \\
&\leq \int_{t'}^T\int_{\{u \leq A\}} \frac{1}{2} \eta_\epsilon u^{p-2} |\nabla u|^2 \phi^2 \; \mathrm{d} x \; \mathrm{d} t + \frac{Cp^2}{(R_0-R_1)^2} \norm{u}^p_{L^p(B_{R_0}(0)\times [t_0,T])} + I_4.
\end{align*}
Next, note that $F''(u) = p(p-1)u^{p-2}$ for $u \leq A$ and $F'' \geq 0$ everywhere. Therefore
\begin{align*}
\int_{t'}^T \int_{\mathbb{R}^4} \eta_\epsilon F''(u) |\nabla u|^2 \phi^2 \; \mathrm{d} x \; \mathrm{d} t &\geq \int_{t'}^T \int_{\{u \leq A\}} \eta_\epsilon p(p-1)u^{p-2} |\nabla u|^2 \phi^2 \; \mathrm{d} x \; \mathrm{d} t.
\end{align*}
Substituting this inequality and the inequalities for $|I_1|$, $|I_2|$ and $|I_3|$ derived above into (\ref{crucial-ineq}), we deduce
\begin{align*}
\left(p(p-1) - \frac{1}{2} \right)& \int_{t'}^T \int_{\{u \leq A\}} \eta_\epsilon u^{p-2} |\nabla u|^2 \phi^2 \; \mathrm{d} x \; \mathrm{d} t \\
& \leq C\left(\frac{p^2}{(R_0-R_1)^2} + \frac{1}{t_1-t_0} \right)\norm{u}^p_{L^p(B_{R_0}(0)\times [t_0,T])} + C \epsilon (pA^{p-1}+1) + I_4
\end{align*}
Taking $\epsilon \rightarrow 0$ and then $A \rightarrow \infty$ yields
\begin{align*}
\left(p(p-1) - \frac{1}{2} \right) \int_{t'}^T \int_{\mathbb{R}^4}u^{p-2} &|\nabla u|^2 \phi^2 \; \mathrm{d} x \; \mathrm{d} t \\
& \leq C\left( \frac{p^2}{(R_0-R_1)^2} + \frac{1}{t_1-t_0} \right) \norm{u}^p_{L^p(B_{R_0}(0)\times [t_0,T])}
\end{align*}
by the monotone convergence theorem. Then note that
$$ u^{p-2} |\nabla u|^2 \phi^2 = \frac{4}{p^2} |\nabla u^{\frac{p}{2}}|^2 \phi^2$$
and
\begin{align*}
|\nabla u^{\frac{p}{2}}|^2 \phi^2 &= \left| \nabla(\phi u^{\frac{p}{2}}) - u^{\frac{p}{2}} \nabla \phi \right|^2 \\
&\geq \left|\nabla(\phi u^{\frac{p}{2}})\right|^2 + \left|u^{\frac{p}{2}} \nabla \phi\right|^2 - 2\left|\nabla(\phi u^{\frac{p}{2}})\right| \cdot \left|u^{\frac{p}{2}} \nabla \phi\right| \\
&\geq \frac{1}{2}|\nabla(\phi u^{\frac{p}{2}})|^2 - u^p |\nabla\phi|^2,
\end{align*}
where in the last line we applied Young's inequality to bound the cross-term. Therefore
\begin{align*}
\int_{t'}^T \int_{\mathbb{R}^4} |\nabla (\phi u^{\frac{p}{2}})|^2 \; \mathrm{d} x \; \mathrm{d} t \leq C\left( \frac{p^2}{(R_0-R_1)^2} + \frac{1}{t_1-t_0} \right) \norm{u}^p_{L^p(B_{R_0}(0)\times [t_0,T])}
\end{align*}
and applying the Sobolev inequality proves Claim 1.
\end{claimproof}
Now we may iterate the estimate of Claim 1 to prove the desired result. First note that due to Shi's estimates, for any $R_0> 0$ and $t_0>0$ we have that $u \in L^2(B_{R_0}(0)\times [t_0, T])$. We take $t_0 = \frac{\tau}{2}$, $R_0 = 2 + \sqrt{\frac{\tau}{2}}$, $\Delta t_i = (\Delta R_i)^2 = \frac{\tau}{2^{i+1}}$, $p_i = 2^{i+1}$ and
\begin{align*}
R_{i+1} &= R_{i} - \Delta R_i \\
t_{i+1} &= t_{i} + \Delta t_i.
\end{align*}
Then inductively applying the estimate of Claim 1 and taking the limit as $i \rightarrow \infty$, we obtain
$$\norm{u}_{L^{\infty}(B_{2}(0)\times [\tau, T])} \leq C_\infty \norm{u}_{L^{2}(B_{R_0}(0)\times [\frac{\tau}{2}, T])} < \infty,$$
where $C_\infty > 0$ is a positive constant. This proves the desired result.
\end{proof}
Next, we prove that the higher derivatives of the curvature tensor are also bounded at positive times. For this we need a generalization of Shi's estimates for the situation in which some of the derivatives of the curvature tensor are known to be bounded. In particular, we have
\begin{thm}[Shi's interior estimates with derivative bounds]
\label{shi-modified}
Let $n \geq 2$ and $m \geq 1$. Then for every choice of constant $K>0$ there exists constants $\theta > 0$ and $C>0$ such that the following holds: Let $M$ be an open manifold $M$ of dimension $n$ and $0 < T \leq \frac{\theta}{K}$. Assume that $g(t)$, $t\in [0,T]$, is a solution to the Ricci flow on an open subset $U$ of $M$ and
\begin{align*}
|\nabla^l Rm| &\leq K \: \text{ on } \: U \times [0, T] \: \text{ and for } \: l \in \{0, 1, 2, \cdots, m\}
\end{align*}
If for $p \in U$ and $r>0$ the closed set $\overline{B_{g(0)}(p, r)}$ is contained in $U$ then
$$|\nabla^{m+1} Rm|^2 \leq C \left( \frac{1}{r^2} + \frac{1}{t} + 1 \right) \: \text{ on } \: B_{g(0)}\left(p, \frac{r}{2}\right) \times (0,T]$$
\end{thm}
\begin{proof}
We follow the proofs of \cite[Theorem 1.4.2]{CZ06} and \cite[Theorem 14.16]{ChII}. In the following the constant $C$ depends on $m$ and $n$ only and may vary from line to line. Consider the quantity
$$ S= \left( BK^2 + |\nabla^m Rm|^2 \right) |\nabla^{m+1} Rm|^2,$$
where $B>0$ is to be fixed later. With help of the differential inequality (\ref{dcurv-evol}) we obtain
\begin{align*}
\partial_t S &\leq \Delta S - 2 \nabla |\nabla^m Rm|^2 \nabla |\nabla^{m+1} Rm|^2 - 2 |\nabla^{m+1} Rm|^4 \\
& + \sum_j c_{mj} \cdot |\nabla^j Rm| \cdot |\nabla^{m-j} Rm| \cdot |\nabla^m Rm| \cdot |\nabla^{m+1} Rm|^2 \\
&-2 \left(BK^2 + |\nabla^m Rm|^2 \right) |\nabla^{m+2} Rm|^2\\
& + \left( BK^2 + |\nabla^{m}Rm|^2 \right) \sum_j c_{m+1 j} \cdot |\nabla^j Rm| \cdot |\nabla^{m+1-j} Rm| \cdot |\nabla^{m+1} Rm|
\end{align*}
Using Cauchy's inequality and the assumption that $|\nabla^l Rm| \leq K$ for $l= 0, 1, 2, \cdots m$, we deduce
\begin{align*}
\partial_t S &\leq \Delta S + 8K |\nabla^{m+1} Rm|^2 |\nabla^{m+2} Rm| - 2 |\nabla^{m+1} Rm|^4 -2 BK^2 |\nabla^{m+2} Rm|^2 \\
& + CK^3 |\nabla^{m+1}Rm|^2 + CK^3(B+1) \left(|\nabla^{m+1} Rm|^2 + K |\nabla^{m+1} Rm| \right)
\end{align*}
Noting that for all $x \in \mathbb{R}$ we have $x^2 + K x \leq 2 x^2 + \frac{1}{4} K^2$ we obtain with help of Young's inequality that
\begin{align*}
\partial_t S &\leq \Delta S - |\nabla^{m+1}Rm|^4 + 2 \left( 32 - B\right)K^2 |\nabla^{m+2} Rm|^2\\
& \qquad \qquad + CK^6 + CK^5(B+1) + CK^6(B+1)^2.
\end{align*}
Taking $B = 32$ and assuming without loss of generality that $K>1$, we obtain
\begin{align*}
\partial_t S \leq \Delta S - \frac{S^2}{C K^4} + C K^6
\end{align*}
From here we may follow the proof of \cite[Theorem 1.4.2]{CZ06} to deduce the desired result.
\end{proof}
With help of Theorem \ref{shi-modified} we inductively prove that the higher derivatives of the curvature tensor are bounded.
\begin{lem}
\label{lem:higher-curv-deriv-bounded}
Let $(\mathbb{R}^4 \setminus \{0\},g(t))$, $ t\in [0,T]$, be a Ricci flow as in Theorem \ref{RF-removable-singularity}. Then for any $\tau > 0$ there exist constants $C_m = C_m(K, \tau) > 0$, $m \in \mathbb{N}$, such that
$$|\nabla^m Rm| < C_m$$
on $\mathbb{R}^4 \times \setminus \{0\} \times \left[\tau, T\right]$.
\end{lem}
\begin{proof}
We prove this lemma by induction. By Lemma \ref{lem:curv-deriv-bounded} the result is true for $m=1$. Assume that the result is true for $m \leq N$. Then there exist constants $C_l>0$, $l = 1, 2, 3, \cdots, N$ such that
$$|\nabla^l Rm| \leq C_l \: \text{ on } \: \mathbb{R}^4 \setminus \{0\} \times \left[\frac{\tau}{4}, T\right] \: \text{ and for } \: l = 1, 2, 3, \cdots, N.$$
As in the proof of Lemma \ref{lem:curv-deriv-bounded}, choose coordinates $x^i$, $i = 1, 2, 3,4$, such that $g(0)$ can be extended to a $C^{1,1}$ metric on $\mathbb{R}^4$, and write $s^2 = \sum_i (x^i)^2$. As the curvature of $g(t)$, $t \in [0,T]$, is bounded there exists a $\lambda > 0$ such that
$$\frac{1}{\lambda} g(0) \leq g(t) \leq \lambda g(0) \: \text{ on } \: \mathbb{R}^4 \setminus \{0\} \times [0, T].$$
By the modified Shi's estimates of Theorem \ref{shi-modified} we see that
$$|\nabla^{N+1} Rm| \leq C\left(\frac{1}{s} + 1\right) \: \text{ on } \: \mathbb{R}^4 \setminus \{0\} \times \left[\frac{\tau}{2}, T\right],$$
for some $C$ that depends on $\tau, K$, and $C_l$, $l = 1, 2, \cdots, N$, only. In particular, this implies that for all $R>0$ the function $u \in L^2(B_R(0))\times [0,T]$. By the differential inequality (\ref{dcurv-evol}) for the evolution of the curvature derivatives we see that
$$\left(\partial_t - \Delta \right) |\nabla^{N+1}Rm|^2 \leq - 2 |\nabla^{N+2} Rm|^2 + CK^2|\nabla^{N+1} Rm| + CK |\nabla^{N+1} Rm|^2$$
and hence
$$\left(\partial_t - \Delta \right) |\nabla^{N+1}Rm| \leq CK \left( K + |\nabla^{N+1} Rm|\right).$$
Thus defining
$$u = e^{-CKt} \left(|\nabla^{N+1} Rm| + K\right)$$
we deduce that
$$\left(\partial_t - \Delta \right) u \leq 0.$$
Now we are in the same setup as in the proof of Lemma \ref{lem:curv-deriv-bounded}. Therefore we may use the same De Giorgi-Nash-Moser iteration argument to show that $u$ and hence $|\nabla^{N+1} Rm|$ are bounded in $\mathbb{R}^4 \times [\tau, T]$. This proves the desired result.
\end{proof}
Now we may prove the main Theorem \ref{RF-removable-singularity}:
\begin{proof}[Proof of Theorem \ref{RF-removable-singularity}]
By Lemma \ref{lem:C11-regularity} we can choose coordinates $x^i$, $i = 1, 2, 3, 4$, for $\mathbb{R}^4$ such that $g(T)$ can be extended to a $C^{1,1}$ metric on all of $\mathbb{R}^4$. Below we write $g = g(T)$ for brevity. By \cite[Lemma 1.2]{DK81} there exist $C^{2,\alpha}$ harmonic coordinates $y^i: U \rightarrow \mathbb{R}$, $i =1 , 2, 3, 4$, in an open neighborhood $U$ of $\mathbb{R}^4$ containing the origin and satisfying
\begin{enumerate}
\item $y^i = 0$
\item $\frac{\partial y^i}{\partial x^j} = \delta^i_j$
\end{enumerate}
at the origin. Furthermore, as $g$ is smooth on $U \setminus \{0\}$, it follows from interior elliptic regularity that $y^i$ are smooth on $U \setminus \{0\}$. Write
$$g_{ij} = g\left(\frac{\partial}{\partial y^i}, \frac{\partial}{\partial y^j}\right) \: \text{ and } \: Ric_{ij} = Ric_g\left(\frac{\partial}{\partial y^i}, \frac{\partial}{\partial y^j}\right).$$
We have that $g_{ij} \in C^{1, \alpha}(U)$ with respect to the $y^i$ coordinates. Furthermore, $g_{ij}$ is smooth on $U \setminus \{0\}$.
By \cite[Chapter 10, Lemma 49]{PP}) we have
\begin{align}
\label{elliptic-harmonic}
\frac{1}{2} \Delta g_{ij} + Q(g, \partial g) = - Ric_{ij} \: \text{ on } \: U \setminus \{0\},
\end{align}
where $Q(g, \partial g)$ is some universal analytic expression that is polynomial
in the matrix $g$, quadratic in $\frac{\partial g}{\partial y^i}$, and has a denominator term depending on $\sqrt{det\,g_{ij}}$. The equation (\ref{elliptic-harmonic}) makes sense on all of $U$ if we interprete it in the weak sense.
\begin{claim}
If $g_{ij}(y) \in C^{k}(U)$ for $k \in \mathbb{N}$ then $R_{ij}(y) \in C^{k-1, 1}(U)$.
\end{claim}
\begin{claimproof}
Write
$$Y_i = \frac{\partial}{\partial y^i} \: \text{ for } \: i = 1,2,3,4.$$
on $U \setminus \{0\}$ we have
\begin{align*}
\frac{\partial^k}{\partial y^{i_1} \partial y^{i_2} \cdots \partial y^{i_k}} Ric_{ij} &= Y_{i_1} Y_{i_2} \cdots Y_{i_k} Ric( Y_i , Y_j) \\
&= \nabla_{Y_{i_1}} \nabla_{Y_{i_2}} \cdots \nabla_{Y_{i_k}} Ric(Y_i, Y_j).
\end{align*}
Since covariant differentiation commutes with contractions, we can use the product rule to express the above derivative as a sum of terms, which only involve $\nabla^m Ric$, $m = 1, 2, \cdots, k$, and $\nabla^m Y_{i_l}$, $m, l = 1, 2, 3, \cdots, k$, contracted with $Y_i$, $Y_j$ and $Y_{i_l}$, $l = 1, 2, \cdots, k$. As by Lemma \ref{lem:higher-curv-deriv-bounded} all the derivatives of the curvature tensor are bounded and $g_{ij}(y) \in C^{k}(U)$ we see that
$$ \frac{\partial^k}{\partial y^{i_1} \partial y^{i_2} \cdots \partial y^{i_k}} Ric_{ij}$$
is bounded as well. Hence the $k$-th spatial derivatives $\partial^k Ric_{ij}$ are bounded, which implies that $\partial^{k-1} Ric_{ij}$ is a Lipshitz function and can be continuously extended to a all of $U$. Similarly, the lower order derivatives $\partial^m Ric_{ij}$, $m = 0, 1, 2, \cdots, k-2$, can be continuously extended to the origin.
\end{claimproof}
First note that $g$ is a $C^{1,1}(U)$ weak solution of the elliptic equation (\ref{elliptic-harmonic}). Furthermore $Q(g, \partial g) \in C^{0,1}(U)$ and by Claim 1 we have $Ric_{ij} \in C^{0,1}(U)$ as well. Since such weak solutions are unique, and there exists a $C^{2,\alpha}(U)$ solution that agrees on the boundary $\partial U$, we see that $g$ is in fact $C^{2,\alpha}(U)$. Bootstrapping standard Schauder estimates and the result of Claim 1, we conclude that $g_{ij}$ is smooth with respect to the harmonic coordinates $y^i$, $i = 1, 2, 3, 4$.
It remains to be shown that $g(t)$ can be extended to a smooth Ricci flow on $\mathbb{R}^4 \times (0,T]$. Recall that by Lemma \ref{lem:higher-curv-deriv-bounded}, for all $\tau >0$ the derivatives of the curvature tensor are bounded on $U \times [\tau, T]$. Moreover $g(T)$ is bi-lipshitz to the euclidean metric $\delta_{ij}$ on $U$ and by the previous paragraph the covariant derivatives of $g(T)$ with respect to $\delta_{ij}$ are all bounded. Therefore we may follow the proof of \cite[Lemma 3.11]{ChI} with $t_0 = T$ to deduce that
$$\frac{\partial^m}{\partial t^m} \frac{\partial^n}{\partial y^{i_1} \cdots \partial y^{i_n}} \left(g(t)\right)_{ij} \leq K_{m,n} \: \text{ on } \: U\setminus \{0\}\times [\tau,T],$$
for some constants $K_{m,n}>0$. This shows that $g(t)$ can be smoothly extended to $U\times [\tau, T]$. As $\tau > 0$ was arbitrary the desired result follows.
\end{proof}
\end{document} |
\begin{document}
\begin{center}\textbf{\textcolor{black}{\Large Generation of higher
order nonclassical states via interaction of intense
electromagnetic
field with third order nonlinear medium}}\textcolor{black}{\Large }\\
\textcolor{black}{\Large }\end{center}{\Large \par}
\begin{center}\textcolor{black}{Anirban Pathak}
\footnote{\textcolor{black}{email: anirban.pathak@jiit.ac.in}
\textcolor{black}{~~~~~~~~~~~anirbanpathak@yahoo.co.in}
}\end{center}
\begin{center}\textcolor{black}{Department of Physics, JIIT, A-10,
Sectror-62, Noida, UP-201307, India.}\\
\end{center}
\begin{abstract}
\textcolor{black}{\normalsize Interaction of intense laser beam
with an inversion symmetric third order nonlinear medium is
modeled as a quartic anharmonic oscillator. A first order operator
solution of the model Hamiltonian is used to study the
possibilities of generation of higher order nonclassical states.
It is found that the higher order squeezed and higher order
antibunched states can be produced by this interaction. It is also
shown that the higher order nonclassical states may appear
separately, i.e. a higher order antibunched state is not
essentially higher order squeezed state and vice versa.
}{\normalsize \par}
\end{abstract}
\section{\textcolor{black}{\normalsize Introduction}}
\textcolor{black}{A nonclassical state of the electromagnetic
field is one for which the Glauber-Sudarshan P-function
{[}\ref{elements of quantum optics}{]} is not as well defined as
the probability density is {[}\ref{the:Dodonov-V-V}{]}. To be
precise, if P-function becomes either negative or more singular
than delta function, then we obtain a nonclassical state. The
nonclassical states do not have any classical analogue. Commonly,
standard deviation of an observable is considered to be the most
natural measure of quantum fluctuation {[}\ref{the:orlowski}{]}
associated with that observable and the reduction of quantum
fluctuation below the coherent state level corresponds to a
nonclassical state. For example, an electromagnetic field is said
to be electrically squeezed field if uncertainties in the
quadrature phase observable $X$ reduces below the coherent state
level (i.e. $\left(\Delta X\right)^{2}<\frac{1}{2}$) and
antibunching is defined as a phenomenon in which the fluctuations
in photon number reduces below the Poisson level (i.e.
$\left(\Delta N\right)^{2}<\langle N\rangle$)
{[}\ref{nonclassical}, \ref{hbt}{]}. Standard deviations can also
be combined to form some complex measures of nonclassicality,
which may increase with the increasing nonclassicality. As an
example, we can note that the total noise} of a quantum state
\textcolor{black}{which,} is a measure of the total fluctuations
of the amplitude\textcolor{black}{, increases with the increasing
nonclassicality in the system {[}\ref{the:hilery2}{]}. }
\textcolor{black}{Probably, antibunching and squeezing are the
most popular examples of nonclassical states and people have shown
serious interest on these states since 1960s. But higher order
extension of these nonclassical states are only introduced in late
1980s {[}\ref{hong1}-\ref{lee2}{]}. Among these higher order
nonclassical effects higher order squeezing is studied in detail
{[}\ref{hong1}, \ref{hong2}, \ref{hillery}, \ref{giri}{]} but the
higher order antibunching {[}\ref{lee1}{]} is not yet studied
rigorously. As a result, still we do not have answers to certain
fundamental questions like: Whether higher order antibunching and
higher order squeezing appears simultaneously or not? The present
work aims to provide answers to this question. In order to do so,
in section 2, we have modeled the interaction of an intense laser
beam with an inversion symmetric third order nonlinear medium as a
quartic anharmonic oscillator. A first order operator solution of
the model Hamiltonian is also used to provide time evolution of
some useful operators. In section 3 and 4 we have theoretically
studied the possibilities of generation of the higher order
squeezed and higher order antibunched states respectively. We have
shown that the generation of higher order squeezed and higher
order antibunched states is possible but they may or may not
appear simultaneously. We finish with some comments and concluding
remarks in section 5. }
\section{\textcolor{black}{\normalsize The model: an intense laser beam interacts
with a 3rd order nonlinear medium}}
\textcolor{black}{An intense electromagnetic field interacting
with a dielectric medium induces a macroscopic polarization
($\overrightarrow{P}$) having a general form \begin{equation}
\overrightarrow{P}=\chi_{1}\overrightarrow{E}+\chi_{2}\overrightarrow{E}\overrightarrow{E}+\chi_{3}\overrightarrow{E}\overrightarrow{E}\overrightarrow{E}+....\label{new1}\end{equation}
where $\overrightarrow{E}$ is the electric field and $\chi_{i}$
is the $i-th$ order susceptibility. Corresponding electromagnetic
energy density is given by \begin{equation}
H_{em}=\frac{1}{8\pi}\left[(\overrightarrow{E}+4\pi\overrightarrow{P}).\overrightarrow{E}+\overrightarrow{B}.\overrightarrow{B}\right]\label{new2}\end{equation}
where $\overrightarrow{B}$ is the magnetic field. Now, if we consider
an inversion symmetric medium then even order susceptibilities
($\chi_{2}$$,\chi_{4}$ etc.) would vanish. Hence the leading
contribution to the nonlinear polarization in an inversion
symmetric medium comes through the third order susceptibility
($\chi_{3}$). If we neglect the macroscopic magnetization (if any)
then the interaction energy will be proportional to the $4$-th
power of the electric field. Normal mode expansion of the field
ensures that the electric field operator $E_{x}$ for $x$-th mode
is proportional to $(a_{x}+a_{x}^{\dagger})$ and the free field
Hamiltonian is \begin{equation}
H_{0}=\sum_{x}\omega_{x}(a_{x}^{\dagger}a_{x}+\frac{1}{2})\label{n1}\end{equation}
where we have chosen $\hbar=1$. }
\textcolor{black}{Thus the total Hamiltonian of a physical system
in which a single mode of intense electromagnetic field having
unit frequency interacts with a 3rd order nonlinear non-absorbing
medium is\begin{equation}
\begin{array}{lcl}
H & = & (a^{\dagger}a+\frac{1}{2})+\frac{\lambda}{16}(a^{\dagger}+a)^{4}\\
& = & \frac{X^{2}}{2}+\frac{\dot{X}^{2}}{2}+\frac{\lambda}{4}X^{4}\end{array}\label{ten.1}\end{equation}
with\begin{equation}
X=\frac{1}{\sqrt{2}}(a^{\dagger}+a)\label{11}\end{equation}
and \begin{equation}
\dot{X}=-\frac{i}{\sqrt{2}}(a^{\dagger}-a).\label{eq:11.1}\end{equation}
The parameter $\lambda$ is the coupling constant}
\textcolor{black}{\emph{}}\textcolor{black}{and is a function of
$\chi_{3}$. Here we can note that the silica crystals which are
used to construct optical fibers are example of third order
nonlinear medium. So third order nonlinear medium described by the
Hamiltonian (\ref{ten.1}) is also important from the point of view
of applicability.}
\textcolor{black}{The above Hamiltonian (\ref{ten.1}) represents a
quartic anharmonic oscillator of unit mass and unit frequency. The
equation of motion corresponding to (\ref{ten.1}) is
\begin{equation} \ddot{X}+X+\lambda
X^{3}=0\label{eqm}\end{equation}
which can not be solved exactly. But in the interaction picture the
potential $V_{I}$ and the time evolution operator $U_{I}$
corresponding to (\ref{ten.1}) are respectively \begin{equation}
V_{I}(t)=\exp(ia^{\dagger}at)\lambda(a+a^{\dagger})^{4}\exp(-ia^{\dagger}at)=\lambda(a\exp(-it)+a^{\dagger}\exp(it))^{4}\label{eq:VI(t)}\end{equation}
and \begin{equation}
U_{I}(t)=1-i\int_{0}^{t}V_{I}(t_{1})dt_{1}+(-i)^{2}\int_{0}^{t}V_{I}(t_{1})dt_{1}\int_{0}^{t_{1}}V_{I}(t_{2})dt_{2}+.....\,\,\,\,.\label{eq:Ui(t)}\end{equation}
Now if we assume \begin{equation}
\int_{0}^{t}V_{I}(t_{1})dt_{1}\int_{0}^{t_{1}}V_{I}(t_{2})dt_{2}\ll1\label{eq:condition}\end{equation}
then we can neglect higher order terms and ( \ref{eq:Ui(t)})
reduces to \begin{equation}
U_{I}(t)=1-i\int_{0}^{t}V_{I}(t_{1})dt_{1}=1-i\int_{0}^{t}\lambda(a\exp(-it_{1})+a^{\dagger}\exp(it_{1}))^{4}dt_{1}\label{eq:uit1}\end{equation}
Thus the first order expression for time evolution of annihilation
operator is }
\textcolor{black}{\begin{equation}
\begin{array}{lcl}
a_{I}(t)=U_{I}^{\dagger}(t)a(0)U_{I}^{\dagger}(t) & = & a-\frac{i\lambda}{8}\left[6ta+6ta^{\dagger}a^{2}+6\exp(it)\sin ta^{\dagger2}a-\exp(2it)\sin(2t)a^{\dagger3}\right.\\
& + & \left.6\exp(it)\sin ta^{\dagger}+2\exp(-it)\sin ta^{3}\right].\end{array}\label{ev6}\end{equation}
The derivation of this first order expression (for a more generalized
Hamiltonian) is shown in detail in {[}\ref{fernandez},
\ref{pathak1}{]}. Here we would like to note that the annihilation
operator in the Heisenberg picture $a_{H}$ and that in the
interaction picture $a_{I}$ are related by
$a_{H}=\exp(-it)a_{I}(t)$ and our solution (\ref{ev6}) is valid
only when (\ref{eq:condition}) is satisfied. Physically this
condition implies that the anharmonic term present in
(\ref{ten.1}) provides only a small perturbation. This assumption
is justified because in a third order nonlinear medium the
anharmonic constant $\lambda$, which is a function of $\chi_{3}$,
is very small (i.e $\lambda\ll1)$ {[}\ref{pathak1}{]}.}
\section{\textcolor{black}{\normalsize Higher order squeezing}}
\textcolor{black}{Higher order squeezing is defined in various
ways. The definition which we have used in this work is by Hillery
{[}\ref{hillery}{]}. This definition is different from that of
Hong and Mandel {[}\ref{hong1}{]}. Present definition is also
called amplitude squared squeezing. According to this definition
of higher order squeezing, higher order quadrature variables are
defined as \begin{equation}
Y_{1}=\frac{1}{\sqrt{2}}(a^{\dagger2}+a^{2})\label{eq:highersq1}\end{equation}
and \begin{equation}
Y_{2}=\frac{i}{\sqrt{2}}(a^{\dagger2}-a^{2})\label{eq:highersq2}\end{equation}
From the commutation relation $[Y_{1'}Y_{2}]=i(4N+2)$ it is easy
to conclude that a state is squeezed in $Y_{1}$ variable if
\begin{equation} (\bigtriangleup\, Y_{1})^{2}<\left\langle
2N+1\right\rangle \label{eq:cond}\end{equation} or if,
\begin{equation} f=(\bigtriangleup\, Y_{1})^{2}-\left\langle
2N+1\right\rangle <0\label{eq:cond2}\end{equation}
A strenuous but straight forward operator algebra yields} \begin{equation}
\begin{array}{lcl}
\Delta Y_{1}^{2}=\langle Y_{1}^{2}\rangle-\langle Y_{1}\rangle^{2} & = & \left[2|\alpha|^{2}+1-\frac{\lambda}{4}\left[4|\alpha|^{2}\left(2|\alpha|^{2}+3\right)\sin t\sin(2\theta-t)-12|\alpha|^{4}t\sin(4\theta)\right.\right.\\
& + & 3\left(2|\alpha|^{4}+4|\alpha|^{2}+1\right)\sin^{2}2t-12|\alpha|^{2}\left(2|\alpha|^{2}+3\right)\sin t\sin(2\theta+t)\\
& - & \left.\left.2|\alpha|^{4}\sin2t\sin\left(2(2\theta-2t)\right)\right]\right]\end{array}\label{eq:dely1}\end{equation}
\textcolor{black}{where $\alpha=|\alpha|\exp(i\theta)$ is used. Here we would like to note that since we are working
in interaction picture we have to take all the expectation values
with respect to the initial coherent state $|\alpha\rangle$ which
is defined as $a|\alpha\rangle=\alpha|\alpha\rangle$. By taking
the expectation value of $N(t)=a^{\dagger(t)}a(t)$ we obtain
\begin{equation} \langle
N(t)\rangle=|\alpha|^{2}-\frac{\lambda}{4}\left[2|\alpha|^{2}\left(2|\alpha|^{2}+3\right)\sin
t\sin(2\theta-t)-|\alpha|^{4}\sin2t\sin\left(2(2\theta-2t)\right)\right].\label{eq:<n(t)>}\end{equation}
Substituting (\ref{eq:dely1}) and (\ref{eq:<n(t)>}) in
(\ref{eq:cond2}) we obtain a closed form analytic expression for
$f$ as \begin{equation}
\begin{array}{lcl}
f & = & -\frac{3\lambda}{4}\left[-4|\alpha|^{2}(2|\alpha|^{2}+3)\sin t\sin(t+2\theta)-4|\alpha|^{4}t\sin(4\theta)\right.\\
& + & \left.(2|\alpha|^{4}+4|\alpha|^{2}+1)\sin^{2}(2t)\right]\end{array}\label{eq:result}\end{equation}
} From (\ref{eq:result}) we can observe that $f$ oscillates
between positive and negative values depending upon the phase of
the input coherent light $\theta$ and the interaction time $t$.
Both of these parameters can be tuned to produce higher order
squeezed state and to increase the depth of noclassicality by
increasing the negativity of $f$. If we consider
$\theta=\frac{\pi}{2}$ then \textcolor{black}{(\ref{eq:result})
reduces to \begin{equation}
f=-\frac{3\lambda}{4}\left[4|\alpha|^{2}(2|\alpha|^{2}+3)\sin^{2}t+(2|\alpha|^{4}+4|\alpha|^{2}+1)\sin^{2}(2t)\right]\label{eq:result1}\end{equation}
which is always negative and thus we always have higher order
squeezing. }
\section{\textcolor{black}{\normalsize Higher order antibunching of photons}}
\textcolor{black}{As we have already mentioned the higher order
antibunching {[}\ref{lee1}{]} is not yet studied rigorously. Using
the negativity of P function {[}\ref{elements of quantum
optics}{]}, Lee introduced the criterion for HOA as }
\textcolor{black}{\begin{equation} R(l,m)=\frac{\left\langle
N_{x}^{(l+1)}\right\rangle \left\langle N_{x}^{(m-1)}\right\rangle
}{\left\langle N_{x}^{(l)}\right\rangle \left\langle
N_{x}^{(m)}\right\rangle }-1<0,\label{eq:ho3}\end{equation} where
$N$ is the usual number operator, $N^{(i)}=N(N-1)...(N-i+1)$ is
the $ith$ factorial moment of number operator, $\left\langle
\right\rangle $ denotes the quantum average, $l$ and $m$ are
integers satisfying the conditions $l\leq m\leq1$ and the
subscript $x$ denotes a particular mode. Ba An {[}\ref{ba an}{]}
choose $m=1$ and reduced the criterion of $l$th order antibunching
to \begin{equation} A_{x,l}=\frac{\left\langle
N_{x}^{(l+1)}\right\rangle }{\left\langle N_{x}^{(l)}\right\rangle
\left\langle N_{x}\right\rangle
}-1<0\label{eq:bhuta1}\end{equation} or, \begin{equation}
\left\langle N_{x}^{(l+1)}\right\rangle <\left\langle
N_{x}^{(l)}\right\rangle \left\langle N_{x}\right\rangle
.\label{eq:ba an (cond)}\end{equation} Physically, a state which
is antibunched in $l$th order has to be antibunched in $(l-1)th$
order. Therefore, we can further simplify (\ref{eq:ba an (cond)})
as \begin{equation} \left\langle N_{x}^{(l+1)}\right\rangle
<\left\langle N_{x}^{(l)}\right\rangle \left\langle
N_{x}\right\rangle <\left\langle N_{x}^{(l-1)}\right\rangle
\left\langle N_{x}\right\rangle ^{2}<\left\langle
N_{x}^{(l-2)}\right\rangle \left\langle N_{x}\right\rangle
^{3}<...<\left\langle N_{x}\right\rangle
^{l+1}\label{eq:ineq}\end{equation} and obtain the condition for
$l-th$ order antibunching as \begin{equation} d(l)=\left\langle
N_{x}^{(l+1)}\right\rangle -\left\langle N_{x}\right\rangle
^{l+1}<0.\label{eq:ho21}\end{equation} This simplified criterion
(\ref{eq:ho21}) coincides exactly with the physical criterion of
HOA introduced by Pathak and Garica {[}\ref{the:martin}{]}. }
\textcolor{black}{By using (\ref{ev6}) and the condition for
higher order antibunching (\ref{eq:ho21}), it is easy to derive
that for a third order non-linear medium having inversion
symmetry, we have
\begin{equation}
d(1)=\frac{3\lambda|\alpha|^{2}}{4}\left[2\left(2|\alpha|^{2}+1\right)\sin(t-2\theta)\sin(t)+|\alpha|^{2}\sin\left(2(t-2\theta)\right)\sin(2t)\right],\label{app1}\end{equation}
\begin{equation}
d(2)=\frac{3\lambda|\alpha|^{4}}{2}\left[\sin(t-2\theta)\sin(t)+\sin\left(2(t-2\theta)\right)\sin(2t)\right]\label{app2}\end{equation}
and\begin{equation}
d(3)=\frac{3\lambda|\alpha|^{4}}{4}\sin\left(2(t-2\theta)\right)\sin(2t).\label{app3}\end{equation}
Antibunching of fourth or higher order can not be observed with a
first order solution of the model Hamiltonian. In order to study
the possibilities of their occurrence we have to use higher order
operator solutions of (\ref{eqm}). }
\textcolor{black}{Equations (\ref{app1}-\ref{app3}) coincides
exactly with our recent result {[}\ref{the:martin}{]} which was
reported as a special case of a generalized Hamiltonian. Here we
can observe that if we choose interaction time $t=2\theta$ then
$d=0$. Therefore, we can observe higher order coherence. But for
$\theta=0$ or $\theta=n\pi$, (i.e. when input is real) $d$ is a
sum of square terms only. So $d$ is always positive and we have
higher order bunching of photons. For other values of phase
$(\theta)$ of input radiation field, value of $d$ oscillates from
positive to negative, so we can observe higher order bunching,
anti-bunching or coherence in the output depending upon the
interaction time $t$.}
\section{\textcolor{black}{\normalsize Summary and concluding remarks}}
\textcolor{black}{From (\ref{eq:result1}) we know that for}
$\theta=\frac{\pi}{2}$ we always obtain higher order squeezed
state. But for $\theta=\frac{\pi}{2}$, (\ref{app2}) and
(\ref{app3}) reduces to \textcolor{black}{\begin{equation}
d(2)=\frac{3\lambda|\alpha|^{4}}{2}\left[-\sin^{2}(t)+\sin^{2}(2t)\right]\label{app2.1}\end{equation}
and\begin{equation}
d(3)=\frac{3\lambda|\alpha|^{4}}{4}\sin^{2}(2t).\label{app3.1}\end{equation}
}respectively. Here we can see that $d(3)$ is always positive for
this particular choice of $\theta$, and therefore, third order
antibunching will not appear simultaneously with the amplitude
squared squeezing. On the other hand $d(2)$ oscillates between
positive (higher order bunching) and negative (higher order
antibunching) values. Thus we can conclude that neither second and
third order antibunching appears simultaneously nor the higher
order squeezing appears simultaneously with the higher order
antibunching. Alternatively, we can state that, in general higher
order nonclassical effects may appear separately. The possibility
of their appearance may be tuned by tuning the phase ($\theta$) of
the input coherent state and interaction time $(t)$.
\textbf{Acknowledgement:} Author is thankful to his student Mr.
Prakash Gupta for his interest in this work and help in correcting
the manuscript.
\end{document} |
\begin{document}
\title[On the asymptotic Dirichlet problem]{On the asymptotic Dirichlet problem for a class of mean curvature type partial differential equations}
\maketitle
\date{\{\}}
\begin{abstract}
We study the Dirichlet problem for the following prescribed mean curvature PDE
$$
\begin{cases}
-\operatorname{div}\dfrac{\nabla v}{\sqrt{1+|\nabla v|^{2}}}=f(x,v)
\text{ in }\Omega\\
v=\varphi \text{ on }\partial\Omega.
\end{cases}
$$
where $\Omega$ is a domain contained in a complete Riemannian manifold $M,$
$f:\Omega\times\mathbb{R\rightarrow R}$ is a fixed function and $\varphi$ is a given continuous function on $\partial\Omega$. This is done in three parts. In the first one we consider this problem in the most general form, proving the existence of solutions when $\Omega$ is a bounded $C^{2,\alpha}$ domain, under suitable conditions on $f$, with no restrictions on $M$ besides completeness. In the second part we study the asymptotic Dirichlet problem when $M$ is the hyperbolic space $\mathbb{H}^n$ and $\Omega$ is the whole space. This part uses in an essential way the geometric structure of $\mathbb{H}^n$ to construct special barriers which resemble the Scherk type solutions of the minimal surface PDE. In the third part one uses these Scherk type graphs to prove the non existence of isolated asymptotic boundary singularities for global solutions of this Dirichlet problem.
\end{abstract}
\
\noindent {\it Keywords:} Dirichlet problem with prescribed mean curvature, prescribed data on the asymptotic boundary, hyperbolic space, Scherk surfaces
\
\noindent {\it Mathematics Subject Classification:} 35J93, 58J05, 58J32
\
\section{Introduction}
A natural way of finding bounded entire solutions to a partial differential equation on a Cartan-Hadamard manifold (complete, simply connected Riemannian manifold with nonpositive sectional curvature) is by solving the asymptotic Dirichlet problem with a prescribed boundary data given at infinity. This problem has been extensively studied for the Laplace equation mostly motivated by the Green-Wu conjecture which asserts the existence of bounded non constant harmonic functions on a Cartan-Hadamard manifold under certain growth and decay conditions on the sectional curvature (see \cite{GW}, \cite{SY}). In the last years the asymptotic Dirichlet problem has been studied for other partial
differential equations such as the $p-$Laplace (\cite{Ho}) and the minimal
surface equation (\cite{GR}, \cite{CHR}, \cite{CHR2}, \cite{RT}).
In our paper we study the Dirichlet problem for the following prescribed mean curvature PDE
\begin{equation}
\begin{cases}
-\operatorname{div}\dfrac{\nabla v}{\sqrt{1+|\nabla v|^{2}}}=f(x,v)
\text{ in }\Omega\\
v=\varphi \text{ on }\partial\Omega.
\end{cases}
\label{eq-fgraph}
\end{equation}
where $\Omega$ is a domain contained in a complete Riemannian manifold $M,$
$f:\Omega\times\mathbb{R\rightarrow R}$ is a fixed function satisfying some
conditions and $\varphi$ is a given continuous function on $\partial\Omega$.
The objective of this paper is threefold: first, to investigate the existence of solutions of \eqref{eq-fgraph} when $\Omega$ is bounded; second, to study the asymptotic Dirichlet problem in the case where $M$ is the hyperbolic space $\mathbb{H}^n$ and, third, to study the existence or not of isolated asymptotic boundary
singularities for the solutions to the problem discussed in the second step. We make some comments on each of such problems.
Problem \eqref{eq-fgraph} in the case of bounded domains and where $f$ is a constant or a function depending only on $x$ is a classical topic of study in the Euclidean geometry which, more recently, has been studied in the Riemannian setting. The theory has now reached a well developped stage and problem \eqref{eq-fgraph} is completely
solved for a large class of PDE's on bounded domains of complete Riemannian manifolds (see \cite{RT} and references therein for the case where $f=0$ and \cite{dajczer2008killing} for $f$ depending on $x$ and for the mean curvature PDE). Concerning the case where $f$ depends on $x$ and $u,$ a uniqueness result for \eqref{eq-fgraph} has been obtained in \cite{AMP} provided that $\Omega$ is bounded and $f(x,\cdot)$, for a fixed $x\in \Omega$, is nonincreasing. Here, we will provide an existence result for the problem \eqref{eq-fgraph} under the same assumption on $f$ and some geometric assumptions on the domain which are well know to be necessary when dealing with the mean curvature operator. We shall use the method of a priori estimates for proving the existence of classical solutions. In the case where $f$ depends only on $x$ and for the mean curvature PDE our existence results recover the ones mentioned above.
It is natural to investigate the asymptotic Dirichlet problem once the solvability of \eqref{eq-fgraph} has been established in bounded domains for continuous boundary data. Despite the vast literature on this problem most of the results deal only with homogeneous PDEs of the form $\operatorname{div}(a(|\nabla u|)\nabla u)=0.$ We study here the existence of solutions to the inhomogeneous asymptotic problem \eqref{eq-fgraph} but only in the hyperbolic space. The reason for confining ourselves to this space comes from the construction of barriers at infinity, which are fundamental to prove the continuous extension to the asymptotic boundary of a prospective entire bounded solution. As it is well known from several works, the construction of such barriers is closely related to the existence of Scherk type super and sub solutions to \eqref{eq-fgraph} (see definition \ref{def-Scherkprob}). Due the inhomogeneous part of the PDE \eqref{eq-fgraph}, the geometric structure of the background manifold seems to be fundamental for the construction of such sub (super) solutions. Indeed, one uses here strongly the symmetries of the hyperbolic geometry to construct such barriers in a quite explicit way (see Secion \ref{sec-STSolutions}). This is the largest part of the paper which, despite being elementary, is more involved. The construction of such barriers requires a decay on $|f(x,u)|$ as $x$ goes to the asymptotic boundary as well as some global assumption on $f(x,u)$ (see \ref{propphi1} and \ref{propphi2} for a precise statement).
The fact that some sort of decay of $|f|$ at infinity is necessary follows from the geometric nature of the mean curvature operator (see \cite{PigolaRigoliSetti}). Indeed, an application of the tangency principle gives an obstruction to the mean curvature of a solution by comparing the mean curvature of its graph with the mean curvature of geodesic spheres. Let us point out that existence results have been obtained in \cite{CHHfmin} for a related equation on more general Cartan-Hadamard manifolds with a different method not using Scherk type solutions.
The existence or not of interior or boundary singularities for the minimal surface equation in the Euclidean space is a classical topic of study. In \cite{BR} the authors extended this study to the asymptotic Dirichlet problem in a Riemanian manifold and to a large class or partial differential equations which includes the $p-$Laplace and the minimal surface equation. In particular, they prove that isolated singularities at the asymptotic boundary of solutions of the minimal surface equation are removable. We here obtain the same result to the inhomogeneous PDE \eqref{eq-fgraph} in the hyperbolic space.
We state precisely our main results. Let us begin with our existence results on bounded domains, proved in Section \ref{sec-genExist}.
\begin{thm}
\label{thm-existence}
Let $\Omega \subset M$ be a bounded $C^{2,\alpha}$ domain in a complete Riemannian manifold $M$ and let $f\in C^{1}\bigl(\overline{\Omega}\times \mathbb{R}\bigr)$ be given.
Suppose there is a constant $F$ such that
\begin{equation}
|f(x,t)|\leq F, \text{ and } f_t(x,t)\le 0 \text{ for all }\left(x,t\right) \in\overline{\Omega}\times\mathbb{R}
\label{eq-f-form}
\end{equation}
and
\begin{equation}
\operatorname*{Ric}\nolimits_{\Omega}\geq-\dfrac{F^{2}}{n-1},\quad
H_{\partial\Omega}\geq F, \label{curvboundary}
\end{equation}
where
$\operatorname*{Ric}\nolimits_{\Omega}$ stands for the Ricci curvature of
$\Omega$ and $H_{\partial\Omega}$ for the inward mean curvature of
$\partial\Omega$.
Then the Dirichlet problem \eqref{eq-fgraph}
is solvable
for all $\varphi\in C^{0}(\partial\Omega)$. If $\varphi\in C^{2,\alpha}(\overline
{\Omega}),$ then the solution is also in $C^{2,\alpha}(\overline{\Omega}).$
\end{thm}
We observe that Theorem \ref{thm-existence} extends \cite[Theorem 1]{dajczer2008killing} to the case where the function $f$ depends also on $u$.
In Section \ref{sec-STSolutions}, we construct Scherk type sub and super solutions (see Definition \ref{def-Scherkprob} and Theorem \ref{s}) which are used to prove the following results:
\begin{thm}\label{thm-Scherk}
Suppose that $f\in C^{1}(\mathbb{H}^n\times\mathbb{R})$ satisfies
$f_t(x,t)\le 0$ in $\mathbb{H}^n\times\mathbb{R}$ and condition \ref{propphi1} for a function $\phi(r)\leq (n-1)\coth(r).$ Then the asymptotic
Dirichlet problem
\begin{equation}
\begin{cases}\label{eq-asymDP}
-\operatorname{div}\dfrac{\nabla v}{\sqrt{1+|\nabla v|^{2}}}=f(x,v)
\text{ in }\mathbb{H}^n\\
v=\varphi \text{ on }\partial_{\infty}\mathbb{H}^n
\end{cases}
\end{equation}
is solvable for any $\varphi\in C^{0}(\partial_{\infty}\mathbb{H}^n)$. Moreover,
the assumption $\phi(r)\leq (n-1)\coth(r)$ can be replaced by condition \ref{propphi2} on $f$.
\end{thm}
Let us observe that, even using Perron's method, we are not able to prove Theorem \ref{thm-Scherk} only assuming some asymptotic decay condition on $f$ i.e. condition \ref{propphi1}. Beyond $f_t \leq 0$, we also need some global assumptions on $f$, precisely, we require either that $|f(x,t)|\leq (n-1)\coth (r(x))$, for all $t\in \mathbb{R}$ and $x\in \mathbb{H}^n$, or condition \ref{propphi2}. Notice that these assumptions guarantee the solvability of some Dirichlet problem on balls $B_R (o)$.
To see their importance, first observe that for $H > n-1$ there are hemispheres of mean curvature $H$ which are graphs of functions $u: B_R(o)\to \mathbb{R}$ with infinite gradient at the boundary. Consider the case $f(x,t) = H$ for $r(x) < R$ and $t \in \mathbb{R}$. Then $f(x,t)$ does not satisfy either $|f(x,t)|\leq (n-1)\coth (r(x))$ or condition \ref{propphi2}, even if condition \ref{propphi1} and $f_t \le 0$ hold. We can prove using a comparison argument with these hemispheres that equation $Q(v)=f(x,v)$ has no solution in any domain containing $B_R(o)$.
\
We conclude this paper by generalizing partially Theorem 1.3 of \cite{BR}:
\begin{thm}
Suppose that $f\in C^{1}(\mathbb{H}^n\times\mathbb{R})$ satisfies $f_t(x,t)\le 0$ in $\mathbb{H}^n\times\mathbb{R}$, \ref{propphi1} and \ref{propphi2}. Let $p_1, \dots, p_k \in\partial_{\infty}\mathbb{H}^n$ and $\varphi \in C^0(\partial_\infty \mathbb{H}^n)$ be given. If $u \in C^2(\mathbb{H}^n)\cap C^0(\overline{\mathbb{H}^n} \backslash \{p_1, \dots, p_k \})$ is a solution to the Dirichlet problem
\begin{equation}
\begin{cases}
-\operatorname{div}\dfrac{\nabla v}{\sqrt{1+|\nabla v|^{2}}}=f(x,v)
\text{ in }\mathbb{H}^n\\
v=\varphi \text{ on }\partial_{\infty}\mathbb{H}^n\backslash\{p_1,\,\dots,\, p_k\},
\end{cases}
\end{equation}
then $u \in C^0\bigl(\overline{\mathbb{H}^n}\bigr)$.
\label{ContinuityOfSolutionWithSing}
\end{thm}
\section{A general existence theorem in bounded domains of Riemannian manifolds}\label{sec-genExist}
In this section we prove Theorem \ref{thm-existence}. As it is well-known from the theory of second order quasi linear elliptic PDE (see \cite{GilTru}) Theorem \ref{thm-existence} will follow once we get a priori height and gradient estimates for solutions of \eqref{eq-fgraph}.
From \cite[Theorem 1]{dajczer2008killing} given $\varphi \in C^{2,\alpha}(\partial\Omega),$ under the hypothesis of Theorem \ref{thm-existence}, there are functions $w^+, w^-\in C^{2,\alpha}(\overline{\Omega})$ such that $Q(w^+)=F,$ $Q(w^-)=-F$ in $\Omega$ and $w^+, w^-$ are equal to $\varphi$ on $\partial\Omega$, where \begin{equation}\label{def-opQ}
Q(v)=-\operatorname{div}\left(\dfrac{\nabla v}{\sqrt{1+|\nabla v|^{2}}}\right).
\end{equation}
Hence if $u\in C^1(\overline\Omega)$ is a solution to \eqref{eq-fgraph}, it holds that $w^-\leq u \leq w^+$ in $\Omega$ and these functions coincide on $\partial \Omega.$ Therefore,
taking $$C=\max\{\sup_\Omega |w^+|,\sup_\Omega |w^-|, \sup_{\partial\Omega} |\nabla w^+|, \sup_{\partial\Omega} |w^-|\}= C(\varphi, \Omega, F),$$ we have
\begin{lem}
\label{lem-altEgradbordo} Suppose that $f\in C^{1}\bigl(\overline{\Omega}\times \mathbb{R}\bigr)$ satisfies \eqref{eq-f-form} and that
\eqref{curvboundary} holds. Let $u \in C^{2}(\Omega)\cap C^1(\overline{\Omega})$ be a
solution of \eqref{eq-fgraph}. Then, there exists a constant
\[
C= C(\varphi, \Omega, F)\]
such that
\[\sup_{\Omega} |u| \le C \text{ and }
\sup_{\partial\Omega} |\nabla u| \le C.
\]
\end{lem}
We also need local and global gradient estimates as stated below.
\begin{proposition}
Consider problem \eqref{eq-fgraph} in $\Omega \subset M$, a bounded $C^{2,\alpha}$ domain. Suppose that $f\in C^1(\overline{\Omega} \times \mathbb{R})$ is a function for which there is a constant $A,$ such that
\begin{equation}
|f(x,t)| \le A, \quad |f_x(x,t)| \le A \text{ and } f_t(x,t) \le 0
\label{boundOf-f-form}
\end{equation}
for any $(x,t) \in \overline{\Omega} \times \mathbb{R}$. Let $u\in C^3(\Omega)$ be a solution of \eqref{eq-fgraph}.
\begin{enumerate}
\item[(a)] If $u \in C^1(\overline{\Omega})$, then there is $L=L(\max_\Omega u, \max_{\partial \Omega}|\nabla u|, A,\mathbb{R}ic_{\Omega} ) > 0$ such that, for any $x \in \Omega,$
$$ |\nabla u(x)| \le L.$$
\item[(b)] For any normal geodesic ball $B_R(x_0) \subset \subset \Omega$, there exists \\ $L=L(\max_\Omega u, A, \mathbb{R}ic_{\Omega}, R) > 0$ such that
$$ |\nabla u(x_0)| \le L.$$
\label{lem-gradint}
\end{enumerate}
\end{proposition}
First, observe that if $u$ is a classical solution of \eqref{eq-fgraph}, then $u$ satisfies
\begin{equation}
|\nabla u|^2 \Delta u + b(|\nabla u|) \nabla^2u(\nabla u, \nabla u) = -f(x,u) \, c( |\nabla u| ),
\label{mingrapheqOpen}
\end{equation} where $b(s)= -s^2/(s^2+1)$ and $c(s) = s^2\sqrt{1+s^2}$.
The next lemma is very close to Lemma 6 of \cite{RT} and therefore we omit its proof.
\begin{lem}
If $u$ solves \eqref{mingrapheqOpen}, then in an orthonormal frame $E_1, \dots, E_n$ with $E_1=|\nabla u|^{-1}\nabla u$, the following equality holds
\begin{align*} (b+1) |\nabla u| \nabla^2 |\nabla u|(E_1,E_1) &+ |\nabla u| \sum_{i=2}^n \nabla^2 |\nabla u|(E_i,E_i) \\[5pt]
+ b' |\nabla u| \nabla^2 u(E_1,E_1)^2 &+ b\sum_{i=2}^n \nabla^2 u (E_1,E_i)^2 \\[5pt]
- \sum_{i=1,j=2}^n \nabla^2 u(E_i,E_j)^2 - \mathbb{R}ic(\nabla u,\nabla u) &= \frac{2fc}{|\nabla u|^4} \nabla^2 u(\nabla u, \nabla u) \\[5pt]
&- \frac{f}{|\nabla u|^2}\langle \nabla c, \nabla u\rangle - \frac{c}{|\nabla u|^2} \langle \tilde{\nabla} f , \nabla u \rangle,
\end{align*}
where $\tilde{\nabla}f = \nabla_x f(x,u) + f_t(x,u) \nabla u.$
\label{lemmaAuxiliar1}
\end{lem}
As in \cite{RT} we obtain an estimate for $|\nabla u|$ by considering a function of the form \begin{equation}\label{eq-funG}
G(x):= g(x)h(u(x))F(|\nabla u(x)|)
\end{equation}
and assuming that this function attains its maximum at an interior point $y_0$ of $\Omega$.
Then the matrix $(\nabla^2 \ln G (E_i, E_j))_{i,j}$ is nonpositive at $y_0$ and it holds
$$ 0 \ge \theta:= (b+1) \nabla^2 \ln G (E_1, E_1)(y_0) + \sum_{i \ge 2} \nabla^2 \ln G (E_i, E_i)(y_0).$$
At the end, with appropriate choices of $g$, $h$ and $F$, the inequality above
gives an upper bound for $|\nabla u|.$
The next lemma is the version of Lemma 7 of \cite{RT} to our setting and its proof follows the same steps as the ones presented there.
\begin{lem}
If $y_0 \in \Omega$ is a local maximum of $G$ and $\nabla u(y_0) \ne 0$, then at $y_0$
\begin{equation}
\frac{F'}{F}\nabla^2u(E_1,E_i) = -\frac{1}{g}\langle \nabla g, E_i\rangle - \frac{h'}{h} \langle \nabla u, E_i \rangle \text{ for }i\in \{1, \dots, n\}
\label{criticalRelation}
\end{equation}
and
\begin{align}
\theta &= \left[ -\frac{F'b'}{F} + (b+1) \left( \frac{F''}{F} - \frac{(F')^2}{F^2} \right) \right] \nabla^2 u(E_1,E_1)^2 \nonumber \\[5pt]
&+ \frac{F'}{F|\nabla u|}\sum_{i\ge 2, j=1}^n \nabla^2 u(E_i,E_j)^2 \nonumber \\[5pt]
&+ \left[-\frac{F'b}{F|\nabla u|} + \frac{F''}{F} - \frac{(F')^2}{F^2} \right] \sum_{i \ge 2}^n \nabla^2 u(E_1,E_i)^2 \nonumber \\[5pt]
&+ (b+1)\left( \frac{h''}{h} - \frac{(h')^2}{h^2}\right)|\nabla u|^2 +\frac{|\nabla u| F'}{F} \mathbb{R}ic(E_1,E_1) \nonumber \\[5pt]
&+ \frac{1}{g} \left[ (b+1) \nabla^2g (E_1,E_1) + \sum_{i \ge 2} \nabla^2 g(E_i,E_i) \right] \nonumber \\[5pt]
&-\frac{1}{g^2} \left[ (b+1)\langle \nabla g, E_1 \rangle^2 + \sum_{i \ge 2} \langle \nabla g, E_i\rangle^2 \right]
+\Bigg\{ \frac{2fc}{|\nabla u|^4} \nabla^2 u (\nabla u, \nabla u) \nonumber \\[5pt]
&- \frac{f}{|\nabla u|^2} \langle \nabla c, \nabla u \rangle - \frac{c}{|\nabla u|^2} \langle \tilde{\nabla} f, \nabla u \rangle \Bigg\} \frac{F'}{F|\nabla u|} + \frac{h'}{h} \left[ \frac{-fc}{|\nabla u|^2} \right] \le 0.
\label{negativityOfTheMaximum}
\end{align}
\end{lem}
We are now in position to prove Proposition \ref{lem-gradint}.
\begin{proof}
Choose $h(t) = e^{kt}$ in \eqref{eq-funG}, where $k$ is a positive constant.
Then, from \eqref{criticalRelation}, at the maximum point $y_0$, we get
\begin{align}
\nabla^2u(E_1,E_i) &= -\frac{F \langle \nabla g, E_i\rangle}{F'g} - k \frac{F}{F'} \langle \nabla u, E_i \rangle \nonumber \\[5pt]
&= -\frac{F\, E_i(g)}{F'g} - k \frac{F}{F'} \langle |\nabla u| E_1, E_i \rangle = -\frac{F\, E_i(g)}{F'g} -k \frac{F}{F'}|\nabla u| \delta_{1i},
\label{criticalRelationGlobal}
\end{align}
where $\delta_{1i}$ is the Kronecker delta.
Hence, we have
$$ \nabla^2u (\nabla u, \nabla u) = |\nabla u|^2 \nabla^2 u ( E_1 , E_1) = -k \frac{F}{F'}|\nabla u|^3 - \frac{F \, E_1(g)}{F' g}|\nabla u|^2 $$
and
\begin{align*}
\langle \nabla c(|\nabla u|), \nabla u \rangle &= c'(|\nabla u|) \nabla^2 u \left(\frac{\nabla u}{|\nabla u|} , \nabla u \right) \\[5pt]
&= -c'(|\nabla u|)\left( k \frac{F}{F'}|\nabla u|^2 + \frac{F \, E_1(g)}{F' g}|\nabla u|\right) \\[5pt]
&= \! -k \frac{F}{F'} \! \left(\frac{2|\nabla u|^3 + 3 |\nabla u|^5}{\sqrt{1+|\nabla u|^2}}\right) \! - \frac{F E_1(g)}{F' g} \! \left(\frac{2|\nabla u|^2 + 3 |\nabla u|^4}{\sqrt{1+|\nabla u|^2}}\right)\! .
\end{align*}
Therefore, we can obtain an expression for the last four terms of \eqref{negativityOfTheMaximum}:
\begin{align*}
&\left\{ \frac{2fc}{|\nabla u|^4} \nabla^2 u (\nabla u, \nabla u) - \frac{f}{|\nabla u|^2} \langle \nabla c, \nabla u \rangle
- \frac{c}{|\nabla u|^2} \langle \tilde{\nabla} f, \nabla u \rangle \right\}\frac{F'}{F|\nabla u|} - \frac{kfc}{|\nabla u|^2} \\[5pt]
&= -\frac{2fc}{|\nabla u|^5} \left( k \frac{F}{F'}|\nabla u|^3 + \frac{F \, E_1(g)}{F' g}|\nabla u|^2 \right)\frac{F'}{F} - \frac{c}{|\nabla u|^3} \langle \tilde{\nabla} f, \nabla u \rangle \frac{F'}{F} - \frac{kfc}{|\nabla u|^2} \\[5pt]
&+ \frac{f}{|\nabla u|^3} \left[k \frac{F}{F'} \, \left(\frac{2|\nabla u|^3 + 3 |\nabla u|^5}{\sqrt{1+|\nabla u|^2}}\right) +\frac{F\, E_1(g)}{F' g} \, \left(\frac{2|\nabla u|^2 + 3 |\nabla u|^4}{\sqrt{1+|\nabla u|^2}}\right)\right] \frac{F'}{F} \\[5pt]
&= kf\left[ -3 \sqrt{1+|\nabla u|^2} + \frac{2 + 3 |\nabla u|^2}{\sqrt{1+|\nabla u|^2}} \right] - \frac{c}{|\nabla u|^3} \langle \tilde{\nabla} f, \nabla u \rangle \frac{F'}{F} \\[5pt]
& -\frac{f E_1(g)}{|\nabla u| g} \left[2 \sqrt{1+|\nabla u|^2} - \frac{2 + 3 |\nabla u|^2}{\sqrt{1+|\nabla u|^2}} \right] \\[5pt]
&= - \frac{kf}{ \sqrt{1+|\nabla u|^2} } - \frac{F'\sqrt{1+|\nabla u|^2}}{F |\nabla u|}\langle \nabla_x f(x,u) + f_t(x,u) \nabla u, \nabla u \rangle \\[5pt]
& + \frac{f E_1(g)}{ g} \left[ \frac{|\nabla u|}{\sqrt{1+|\nabla u|^2}} \right].
\end{align*}
\
\noindent Hence, since $f$ satisfies \eqref{boundOf-f-form} (observe that $f_t \le 0$), we have
\begin{align*}
&\left\{ \frac{2fc}{|\nabla u|^4} \nabla^2 u (\nabla u, \nabla u) - \frac{f}{|\nabla u|^2} \langle \nabla c, \nabla u \rangle
- \frac{c}{|\nabla u|^2} \langle \tilde{\nabla} f, \nabla u \rangle \right\}\frac{F'}{F|\nabla u|} - \frac{kfc}{|\nabla u|^2} \\[5pt]
&\ge - \frac{k A}{ \sqrt{1+|\nabla u|^2} } - \frac{A F' \sqrt{1+|\nabla u|^2}}{F |\nabla u|} \\[5pt]
&\quad - f_t(x,u) \frac{F' |\nabla u| \sqrt{1+|\nabla u|^2}}{F} - \frac{A| E_1(g)| |\nabla u|}{g \sqrt{1+|\nabla u|^2}} \\[5pt]
& \ge - \frac{k A}{ \sqrt{1+|\nabla u|^2} } - \frac{A F' \sqrt{1+|\nabla u|^2}}{F |\nabla u|} - \frac{A| E_1(g)| |\nabla u|}{g \sqrt{1+|\nabla u|^2}}.
\end{align*}
assuming that $g, F >0$ and $F' \ge 0$.
\noindent Then, inequality \eqref{negativityOfTheMaximum} implies that
\begin{align}
0 \ge \theta &\ge \left[ -\frac{F'b'}{F} + (b+1) \left( \frac{F''}{F} - \frac{(F')^2}{F^2} \right) \right] \nabla^2 u(E_1,E_1)^2 \nonumber \\[5pt]
&+ \frac{F'}{F|\nabla u|}\sum_{i\ge 2, j=1}^n \nabla^2 u(E_i,E_j)^2 \nonumber \\[5pt]
&+ \left[-\frac{F'b}{F|\nabla u|} + \frac{F''}{F} - \frac{(F')^2}{F^2} \right] \sum_{i \ge 2}^n \nabla^2 u(E_1,E_i)^2 \nonumber \\[5pt]
&+ \frac{|\nabla u| F'}{F} \mathbb{R}ic(E_1,E_1) + \frac{1}{g} \left[ (b+1) \nabla^2g (E_1,E_1) + \sum_{i \ge 2} \nabla^2 g(E_i,E_i) \right] \nonumber \\[5pt]
&-\frac{1}{g^2} \left[ (b+1)\langle \nabla g, E_1 \rangle^2 + \sum_{i \ge 2} \langle \nabla g, E_i\rangle^2 \right] - \frac{k A}{ \sqrt{1+|\nabla u|^2} } \nonumber \\[5pt]
&- \frac{A F' \sqrt{1+|\nabla u|^2}}{F |\nabla u|} - \frac{A| E_1(g)| |\nabla u|}{g \sqrt{1+|\nabla u|^2}} .
\label{negativityOfTheMaximumNEW}
\end{align}
Now we can prove (a) and (b).
\
\noindent Proof of (a): Choose $F(s)=s$ and $g(x)\equiv 1$. Then, from \eqref{negativityOfTheMaximumNEW}, we have
\begin{align*}
0 \ge \theta &\ge \left[ -\frac{b'}{|\nabla u|} + (b+1) \left(-\frac{1}{|\nabla u|^2} \right) \right] \nabla^2 u(E_1,E_1)^2 \\[5pt]
&+ \frac{1}{|\nabla u|^2}\sum_{i\ge 2, j=1}^n \nabla^2 u(E_i,E_j)^2 + \left[-\frac{b}{|\nabla u|^2} - \frac{1}{|\nabla u|^2} \right] \sum_{i \ge 2}^n \nabla^2 u(E_1,E_i)^2 \\[5pt]
&+ \mathbb{R}ic(E_1,E_1) - \frac{k A}{ \sqrt{1+|\nabla u|^2} } - \frac{A \sqrt{1+|\nabla u|^2}}{ |\nabla u|^2}. \\[5pt]
\end{align*}
Observe that from \eqref{criticalRelationGlobal}, it follows that $\nabla^2 u(E_1,E_i) = - k |\nabla u|^2 \delta_{1i}$.
Hence, using that $b(s)=-s^2/(1+s^2)$, we conclude that
\begin{align}
0 \ge \theta &\ge \left[ \frac{|\nabla u|^2 -1}{(1+ |\nabla u|^2)^2 |\nabla u|^2} \right] \left(-k|\nabla u|^2 \right)^2 + \mathbb{R}ic(E_1,E_1) \nonumber \\[5pt]
&- \frac{k A}{ \sqrt{1+|\nabla u|^2} } - \frac{A \sqrt{1+|\nabla u|^2}}{ |\nabla u|^2}.
\label{negativityOfTheMaximumGlobal}
\end{align}
If $|\nabla u(y_0)| \ge 2$, this inequality and Young's inequality imply
\begin{align*}
0 \ge \theta &\ge \left[ \frac{3}{16 |\nabla u|^4} \right] k^2|\nabla u|^4 + \mathbb{R}ic(E_1,E_1) - k A - A \\[5pt]
&\ge \frac{3 k^2}{16} + \mathbb{R}ic(E_1,E_1) - \frac{k^2}{8}- 2 A^2 - A ,
\end{align*}
that is,
$$k \le 4\sqrt{2A^2 + A - \mathbb{R}ic(E_1,E_1)}. $$
Therefore, if we take $k=5 \sqrt{2A^2 + A - \mathbb{R}ic(E_1,E_1)}$, this inequality is not satisfied and, then, the maximum of $G$ cannot happen in some interior point $y_0$ such that $|\nabla u(y_0)| \ge 2$. Thus, either
$$ \max_{\Omega} G(y) \le G(y_0) = e^{ku(y_0)} |\nabla u(y_0)| \le 2 e^{kM}$$
or
$$ \max_{\Omega} G(y) \le \max_{\partial \Omega} G(x) = \max_{\partial \Omega} e^{ku(x)} |\nabla u(x)| \le e^{k M} \max_{\partial \Omega} |\nabla u(x)|,$$
that is,
$$ |\nabla u(y)| \le e^{k M - ku(y)} (2 + \max_{\partial \Omega} |\nabla u(x)|) \le e^{2k M}(2+ \max_{\partial \Omega} |\nabla u(x)|),$$
for any $y \in \Omega$, proving (a).
\
\noindent Proof of (b): Choose $F(s)=\ln s$ and $g(x)= 1 - r(x)^2/R^2$, where $r(x)$ is the distance from $x$ to $x_0$.
First observe that from \eqref{criticalRelationGlobal} we have, at $y_0,$
$$ \frac{ \langle \nabla g, E_i\rangle}{g} = -\frac{F'}{F} \nabla^2u(E_1,E_i) - k |\nabla u| \delta_{1i}. $$
Then
$$ - \frac{ \langle \nabla g, E_1\rangle^2}{g^2} \ge -2 \frac{(F')^2}{F^2} \nabla^2u(E_1,E_1)^2 - 2 k^2 |\nabla u|^2 $$
and
$$ - \frac{ \langle \nabla g, E_i\rangle^2}{g^2} = - \frac{(F')^2}{F^2} \nabla^2u(E_1,E_i)^2 \quad {\rm for} \quad i\ge 2.$$
Therefore, from inequality \eqref{negativityOfTheMaximumNEW}, we get
\begin{align*}
0 \ge \theta &\ge \left[ -\frac{F'b'}{F} + (b+1) \left( \frac{F''}{F} - 3\frac{(F')^2}{F^2} \right) \right] \nabla^2 u(E_1,E_1)^2 \nonumber \\[5pt]
&- 2 k^2 |\nabla u|^2 (b+1) + \frac{F'}{F|\nabla u|}\sum_{i\ge 2, j=1}^n \nabla^2 u(E_i,E_j)^2 \nonumber \\[5pt]
&+ \left[-\frac{F'b}{F|\nabla u|} + \frac{F''}{F} - 2 \frac{(F')^2}{F^2} \right] \sum_{i \ge 2}^n \nabla^2 u(E_1,E_i)^2 \nonumber \\[5pt]
&+ \frac{|\nabla u| F'}{F} \mathbb{R}ic(E_1,E_1) + \frac{1}{g} \left[ (b+1) \nabla^2g (E_1,E_1) + \sum_{i \ge 2} \nabla^2 g(E_i,E_i) \right] \nonumber \\[5pt]
& - \frac{k A}{ \sqrt{1+|\nabla u|^2} } - \frac{A F' \sqrt{1+|\nabla u|^2}}{F |\nabla u|} - \frac{A| E_1(g)| |\nabla u|}{g \sqrt{1+|\nabla u|^2}} .
\end{align*}
Since
$$ \frac{F'}{F|\nabla u|}\sum_{i\ge 2, j=1}^n \nabla^2 u(E_i,E_j)^2 \ge \frac{F'}{F|\nabla u|}\sum_{i\ge 2}^n \nabla^2 u(E_1,E_i)^2,$$
it follows that
\begin{align}
0 \ge \theta &\ge \left[ -\frac{F'b'}{F} + (b+1) \left( \frac{F''}{F} - 3\frac{(F')^2}{F^2} \right) \right] \nabla^2 u(E_1,E_1)^2 \nonumber \\[5pt]
&- 2 k^2 |\nabla u|^2 (b+1) + \left[\frac{F'(1-b)}{F|\nabla u|} + \frac{F''}{F} - 2 \frac{(F')^2}{F^2} \right] \sum_{i \ge 2}^n \nabla^2 u(E_1,E_i)^2 \nonumber \\[5pt]
&+ \frac{|\nabla u| F'}{F} \mathbb{R}ic(E_1,E_1) + \frac{1}{g} \left[ (b+1) \nabla^2g (E_1,E_1) + \sum_{i \ge 2} \nabla^2 g(E_i,E_i) \right] \nonumber \\[5pt]
& - \frac{k A}{ \sqrt{1+|\nabla u|^2} } - \frac{A F' \sqrt{1+|\nabla u|^2}}{F |\nabla u|} - \frac{A| E_1(g)| |\nabla u|}{g \sqrt{1+|\nabla u|^2}} .
\label{negativityOfTheMaximumNEWFor-b}
\end{align}
\
\
\noindent {\bf Claim:} Choosing $k=1,$ for $\mathbb{R}ic^-= - \displaystyle \min_{|\eta|=1} \min \{ \mathbb{R}ic(\eta, \eta), 0\}$, it holds that
\begin{align}
g(y_0) \ln |\nabla u(y_0)| &\le 8 \left( 2 + 2A + \frac{1+ 2A}{R} \right. \nonumber \\[5pt]
&+ \left. \max_{B_R(x_0)}|\mathbb{R}ic^-| + \frac{n}{r^2} \max_{B_R(x_0)}| \nabla^2 r^2| \right).
\label{eq-claim}
\end{align}
If $|\nabla u(y_0)| < e^{12},$ the inequality follows from $g\leq 1.$
Therefore we prove the claim assuming $|\nabla u(y_0)| \ge e^{12}.$ Observe that
\begin{align*}
-\frac{F'b'}{F} + (b+1) \left( \frac{F''}{F} - 3\frac{(F')^2}{F^2} \right) &= \frac{(|\nabla u|^2-1)\ln |\nabla u| - 3(|\nabla u|^2 + 1)}{ |\nabla u|^2 (1+|\nabla u|^2)^2 (\ln |\nabla u|)^2}\\[5pt]
&\ge \frac{1}{4}\,\left( \frac{(|\nabla u|^2+1)\ln |\nabla u|}{ |\nabla u|^2 (1+|\nabla u|^2)^2 (\ln |\nabla u|)^2}\right) \\[5pt]
&\ge \frac{1}{4}\left( \frac{1}{ |\nabla u|^2 (1+|\nabla u|^2) (\ln |\nabla u|)}\right).
\end{align*}
We have also that
\begin{align*}
\frac{F'(1-b)}{F|\nabla u|} + \frac{F''}{F} - 2 \frac{(F')^2}{F^2} &= \frac{|\nabla u|^2 \ln |\nabla u| -2 (1+|\nabla u|^2)}{|\nabla u|^2 (1+|\nabla u|^2) (\ln |\nabla u|)^2} > 0, \\[5pt]
\end{align*}
These two inequalities and \eqref{negativityOfTheMaximumNEWFor-b} yield
\begin{align}
0 \ge \theta &\ge \left[ \frac{1}{4 |\nabla u|^2 (1+|\nabla u|^2) (\ln |\nabla u|)} \right] \nabla^2 u(E_1,E_1)^2 - \frac{2 k^2 |\nabla u|^2}{1+ |\nabla u|^2}\nonumber \\[5pt]
&+ \frac{1}{\ln |\nabla u|} \mathbb{R}ic(E_1,E_1) + \frac{1}{g} \left[ (b+1) \nabla^2g (E_1,E_1) + \sum_{i \ge 2} \nabla^2 g(E_i,E_i) \right] \nonumber \\[5pt]
& - \frac{k A}{ \sqrt{1+|\nabla u|^2} } - \frac{A \sqrt{1+|\nabla u|^2}}{ |\nabla u|^2 \ln |\nabla u|} - \frac{A| E_1(g)| |\nabla u|}{g \sqrt{1+|\nabla u|^2}} .
\label{InqualityForGradient-1}
\end{align}
Using that
\begin{align*}
\nabla^2u(E_1,E_1) &= - \frac{F}{F'} \frac{ \langle \nabla g, E_1\rangle}{g} - k \frac{F}{F'} |\nabla u| \\[5pt]
&= -\frac{ |\nabla u| \ln |\nabla u| E_1(g)}{g} - k |\nabla u|^2 \ln |\nabla u|,
\end{align*}
we get
$$ \nabla^2u(E_1,E_1)^2 \ge -\frac{2 k |\nabla u|^3 (\ln |\nabla u|)^2 |E_1(g)|}{g} + k^2 |\nabla u|^4 (\ln |\nabla u|)^2.$$
Therefore, if $|\nabla u(y_0)| \ge e^{12}$,
\begin{align}
0 \ge \theta &\ge \frac{k^2 |\nabla u|^2 (\ln |\nabla u|)}{4 (1+|\nabla u|^2)} - \frac{k |\nabla u| (\ln |\nabla u|) |E_1(g)|}{2 g (1+|\nabla u|^2) } - 2 k^2 \nonumber \\[5pt]
&+ \frac{1}{\ln |\nabla u|} \mathbb{R}ic(E_1,E_1) + \frac{1}{g} \left[ (b+1) \nabla^2g (E_1,E_1) + \sum_{i \ge 2} \nabla^2 g(E_i,E_i) \right] \nonumber \\[5pt]
& - \frac{k A}{ \sqrt{1+|\nabla u|^2} } - \frac{A \sqrt{1+|\nabla u|^2}}{ |\nabla u|^2 \ln |\nabla u|} - \frac{A| E_1(g)| |\nabla u|}{g \sqrt{1+|\nabla u|^2}} .
\label{InqualityForGradient-2}
\end{align}
Since $$\nabla^2 g(E_i,E_i) \ge -\frac{| \nabla^2 r^2|}{R^2}, $$
$b+1 < 1$, $|E_1(g)| \le |\nabla g| \le \frac{2r}{R^2} \le \frac{2}{R}$ and $0 < g \le 1$, we conclude that
$$ 0 \ge \frac{k^2 (\ln |\nabla u|)}{8} - \frac{k }{R g} - \frac{2 k^2}{g} - \frac{|\mathbb{R}ic^-|}{g} - \frac{n| \nabla^2 r^2|}{R^2g} -\frac{k A}{g} - \frac{A}{g} - \frac{2A}{Rg}. $$
Hence, for $k=1$, we obtain \eqref{eq-claim}.
$$g(y_0) \ln |\nabla u(y_0)| \le 8 \left( 2 + 2A + \frac{1+ 2A}{R} + \max_{B_R(x_0)}|\mathbb{R}ic^-| + \frac{n}{R^2} \max_{B_R(x_0)}| \nabla^2 r^2| \right)$$
Therefore,
\begin{align*}
e^{u(x_0)} &\ln |\nabla u(x_0)| = G(x_0) \le G(y_0) \le g(y_0)e^{u(y_0)} \ln |\nabla u(y_0)| \\[5pt]
&\le 8 e^M \left( 2 + 2A + \frac{1+ 2A}{R} + \max_{B_R(x_0)}|\mathbb{R}ic^-| + \frac{n}{R^2} \max_{B_R(x_0)}| \nabla^2 r^2| \right),
\end{align*}
that is,
\begin{align}
|\nabla u(x_0)| &\le \exp\left[8 e^{2M} \left( 2 + 2A + \frac{1+ 2A}{R} \right. \right. \nonumber \\[5pt]
&+ \left. \left. \max_{B_R(x_0)}|\mathbb{R}ic^-| + \frac{n}{R^2} \max_{B_R(x_0)}| \nabla^2 r^2| \right)\right].
\label{IneqFinalForGrad}
\end{align}
Since $\max_{B_R(x_0)}| \nabla^2 r^2|$ is bounded by a constant depending on the curvature and on $R,$ the result follows.
\end{proof}
\
We are now in position to prove Theorem \ref{thm-existence}.
\begin{proof}
[Proof of Theorem \ref{thm-existence}]
We begin by assuming that $\varphi\in
C^{2,\alpha}\left(\overline{\Omega}\right).$ Consider the following family
of Dirichlet problems
\begin{equation}
\begin{cases}
\dv\dfrac{\nabla v}{\sqrt{1+|\nabla v|^{2}}}+\tau f(x,v)=0\text{ in }\Omega,\\
v=\tau\varphi\text{ in }\partial\Omega,\ 0\leq\tau\leq1.
\end{cases}
\label{eqfamily}
\end{equation}
Observe that from Lemma \ref{lem-altEgradbordo}, any solution $v_{\tau}$ to \eqref{eqfamily} is bounded by a constant that does not depend on $\tau$. So Proposition \ref{lem-gradint} applies. Hence, there exists a constant $C,$ not depending on $\tau,$ such that for any solution $v_{\tau}$ to \eqref{eqfamily},
\[
\lVert v_{\tau}\rVert_{C^{1}(\overline{\Omega})}\leq C.
\]
Thanks to this estimate, we obtain a solution $v\in C^{2,\alpha}\left(
\overline{\Omega}\right) $ to \eqref{eqfamily} by using the Leray-Schauder
method \cite[Theorem 13.8]{GilTru}.
If $\varphi\in C^{0}\left( \partial
\Omega\right) $ we take an approximating sequence of $\varphi$ by
$C^{2,\alpha}$ functions. Using then the previous case, the comparison
principle, Lemma \ref{lem-altEgradbordo} and Proposition \ref{lem-gradint} we obtain the existence of a solution
$v\in C^{2}\left( \Omega\right) \cap C^{0}\left( \overline{\Omega}\right)
$ to \eqref{eq-fgraph}.
This concludes the proof of Theorem \ref{thm-existence}.
\end{proof}
\section{Scherk type solutions in the hyperbolic space} \label{sec-STSolutions}
From now on we concentrate in the hyperbolic space $\mathbb{H}^n.$ In order to prove Theorems \ref{thm-Scherk} and \ref{ContinuityOfSolutionWithSing}, we construct barriers that take value $+\infty$ in a totally geodesic hypersphere of $\mathbb{H}^n.$
\begin{defin}
Let $S$ be a totally geodesic hypersphere (or a geodesic if $n=2$) of
$\mathbb{H}^{n}$ and $B$ be one connected component of $\mathbb{H}^{n}\backslash S$.
Given $f\in C^{1}(\mathbb{H}^n\times\mathbb{R})$ and a constant $c$, if $u\in C^2(B)\cap C^0(\overline{B})$ is a solution of
\begin{equation}
\begin{cases}
\label{ScherkProblem}
Q(v)= f(x,v)\text{ in } B\\
v = c\text{ on }\partial_{\infty}B\\
v= +\infty\text{ on } S,
\end{cases}
\end{equation}
where $\partial_{\infty}B$ is the asymptotic boundary of $B,$ we call $u$ a Scherk type solution to problem \eqref{ScherkProblem}. \label{def-Scherkprob}
Analogously we define Scherk type sub and supersolutions.
\end{defin}
Our next result is about the existence of Scherk type solutions and for that we assume that $f$ satisfies:
\begin{enumerate}[label=(\text{\mathbb{R}oman*}), ref=\text{\mathbb{R}oman*}]
\item \label{propphi1} for some fixed $o\in\mathbb{H}^{n}$, there exists a continuous
decreasing function $\phi:[0,+\infty)\rightarrow\mathbb{R}$ such that
\[
|f(x,t)|\leq\phi(r(x)),
\]
where $r(x)=\mathrm{dist}(x,o)$,
and $\displaystyle\int_{0}^{+\infty}\sqrt{\phi(r)}\,dr<\infty$;
\item \label{propphi2} there is a continuous function $h:\mathbb{R}\rightarrow\mathbb{R},$ such that $h(t)\rightarrow0$ as $t\rightarrow\pm\infty,$ for which
\[
|f(x,t)|\leq h(t), \, \forall x\in \mathbb{H}^n.
\]
\end{enumerate}
We also call $B$ from definition \ref{def-Scherkprob} a totally geodesic hyperball and we denote by $d=d_S$ the signed distance function to $S,$ positive in $B.$
Conditions \ref{propphi1} and \ref{propphi2} guarantee the existence of a nice function $\psi=\psi_S$ as stated below:
\begin{proposition}\label{prop-Psi}
Given $f\in C^{1}(\mathbb{H}^n\times\mathbb{R})$ satisfying conditions \ref{propphi1} and \ref{propphi2}, there exists a nonnegative $C^{1}$ function $\psi:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ such that
\begin{enumerate}[label=(\roman{enumi})]
\item \label{psiDefinitiona} $\psi
(d(x),t)\geq|f(x,t)|$, for any $(x,t)\in
\mathbb{H}^{n}\times\mathbb{R}$;
\item \label{psiDefinitionb} $\psi(d,t)$, $\frac{\partial\psi
}{\partial d}(d,t)$ and $\frac{\partial\psi}{\partial t}(d,t)$ are bounded
functions;
\item \label{psiDefinitionc} for $d\in\mathbb{R}$, the map $t\mapsto\psi(d,t)$ is decreasing for $t\geq0$ and constant for $t\leq 0$;
\item \label{psiDefinitiond} for $t\in\mathbb{R}$, the map $d\mapsto\psi(d,t)$ is increasing in $(-\infty,\tilde{d})$ and
decreasing in $(\tilde{d},+\infty)$, where $\tilde{d}$ is some real number that does not depend on $t$;
\item \label{psiDefinitionef} $\psi(d,t)$ converges to zero uniformly in
$t\in\mathbb{R}$ as $d\rightarrow\pm\infty$ and uniformly in $d\in\mathbb{R}$ as $t\rightarrow+\infty$;
\item \label{psiDefinitiong} for
any $t\in\mathbb{R}$, $\displaystyle\int_{0}^{+\infty}\psi(s,t)\;ds<+\infty$.
\end{enumerate}
\end{proposition}
\begin{proof}
First, observe that we can assume w.l.g. that $\phi$ and $h$ are $C^1$ functions, $\phi'(0)=0$, $h$ is even and decreasing on $[0,+\infty)$.
The proof follows by considering
$$\psi(d,t):= \left\{ \begin{array}{rr} \sqrt{ \phi(|d - d(o)|) h(t)} & {\rm if} \quad t \ge 0 \\[5pt]
\sqrt{ \phi(|d - d(o)|) h(0)} & {\rm if} \quad t < 0.
\end{array} \right.
$$
The conditions (ii)-(vi) can be verified directly from the definition. To prove condition $(i)$,
let $x \in \mathbb{H}^n$. By the triangle inequality
$$ r(x) + d(x) \ge d(o) \text{ and }
r(x) + d(o) \ge d(x),$$
that is, $r(x)\ge |d(x) - d(o)|$. (This holds even if $d(x) < 0$ or $d(o) < 0$.) Hence, using that $\phi$ is decreasing and hypothesis (1), we have that
$$ |f(x,t)| \le \phi(r(x))\le \phi(|d(x) - d(o)|).$$
From this and \ref{propphi2}, we have $|f(x,t)|^2\le \phi(|d(x) - d(o)|) h(t) \le \psi^2(d(x),t)$.
\end{proof}
The main result of this section is the following:
\begin{thm}
\label{s}
$\mathbb{H}^{n}\backslash S$.
Let $B\subset \mathbb{H}^n$ be a totally geodesic hyperball. Suppose that $f\in C^{1}(\mathbb{H}^n\times\mathbb{R})$ is nonnegative and satisfies $f_t(x,t)\le 0$ in $\mathbb{H}^n\times\mathbb{R},$ \ref{propphi1} and \ref{propphi2}. Then, for any constant $c\in \mathbb{R}$, there exists a solution $u$ to
the problem \eqref{ScherkProblem}.
Besides, if $f$ is not necessarily nonnegative and satisfies only conditions \ref{propphi1} and \ref{propphi2}, this Dirichlet problem has a supersolution. If we replace $v = +\infty$ on $S$ by $v = -\infty$ on $S$ and assume \ref{propphi1} and \ref{propphi2}, the problem has a subsolution.
\end{thm}
To prove this theorem, our main task is to construct supersolutions for this equation. The
existence of Scherk type solutions will then be an immediate consequence of Perron's method, which due to Theorem \ref{thm-existence} applies in our setting.
Now let us explain our strategy to construct supersolutions for \eqref{ScherkProblem}. We will look for a solution $w=w(d(x))$, where $d(x)=dist(x,S)$, $S=\partial B$, to the following problem
\begin{equation}
\begin{cases}
Q(v\circ d)= \psi(d ,v \circ d) \text{ in } B\\
v = c \text{ on } \partial_{\infty} B\\
v = + \infty\text{ on }S.
\end{cases} \label{superScherkEDP}
\end{equation}
Since $\Delta d= (n-1)\tanh d$, we can rewrite \eqref{superScherkEDP} as
\begin{equation}
\begin{cases}
\displaystyle \frac{w^{\prime\prime}}{(1 + w^{\prime2})^{3/2}} + (n-1)
\tanh(d) \left(\frac{w^{\prime}}{\sqrt{1 + w^{\prime2}}}\right) = -\psi(d,w)
\text{ for } d > 0\\
w(+\infty) = c\\
w(0) = + \infty. \label{superScherkEDO}
\end{cases}
\end{equation}
\
Next, we set $g=\frac{ w^{\prime}}{\sqrt{1+ w^{\prime2}}}$ on $[0,+\infty)$.
Observe that $-1 < g < 1$ and that the ODE in \eqref{superScherkEDO} rewrites as the following system for $(w,g)$
\begin{equation}
\begin{cases}
w^{\prime}(d) = \displaystyle \frac{g(d)}{\sqrt{1-g^{2}(d)}}\\[10pt]
g^{\prime}(d) = -(n-1)\tanh(d) \; g(d) - \psi(d, w(d)).
\end{cases} \label{systemForWG}
\end{equation}
or
\[
\left[
\begin{array}
[c]{c}
w(d)\\
g(d)
\end{array}
\right] ^{\prime}= F(d,w(d),g(d)),
\]
where $F: \mathbb{R} \times\mathbb{R} \times(-1,1) \to\mathbb{R}^{2}$
corresponds to the right-hand side of \eqref{systemForWG}. Given $d_{0} > 0$,
$h \in\mathbb{R}$ and $\gamma\in(-1,1)$, we consider the initial condition
\begin{equation}
\begin{cases}
w(d_{0}) = h\\
g(d_{0}) = \gamma.
\end{cases} \label{InitialConditionsForWG}
\end{equation}
Note that, from \eqref{systemForWG} and \eqref{InitialConditionsForWG}, we
get
\begin{equation}
w_{\gamma}(d) = h + \int_{d_{0}}^{d} \frac{g_{\gamma}(t)}{\sqrt{1-g_{\gamma
}^{2}(t)}}\; dt \label{wExpressionByG}
\end{equation}
and
\begin{equation}
g_{\gamma}(d) = \frac{1}{(\cosh d)^{n-1}} \left( \gamma\cosh^{n-1}(d_{0}) -
\int_{d_{0}}^{d} \psi(s,w_{\gamma}(s)) (\cosh s)^{n-1} ds \right) .
\label{gExpressionByW}
\end{equation}
Since $F$ is $C^{1}$, from the classical theory, there exists only one maximal
solution $(w_{d_{0},h,\gamma},g_{d_{0},h,\gamma})$ to the system
\eqref{systemForWG} with initial condition \eqref{InitialConditionsForWG}.
Let $I_{d_{0},h,\gamma}$ be the domain of this solution. Observe that
$w_{d_{0},h,\gamma}$ is the solution of the second order ODE in
\eqref{superScherkEDO} with the initial conditions
\[
w(d_{0}) = h \quad\mathrm{and} \quad w^{\prime}(d_{0}) = \frac{\gamma}
{\sqrt{1-\gamma^{2}}}.
\]
To solve \eqref{superScherkEDO}, we have to prove that there exist $d_{0} >
0$, $h \in\mathbb{R}$ and $\gamma\in(-1,1)$ such that $w_{d_{0},h,\gamma
}(0)=+\infty$ and $w_{d_{0},h,\gamma}(+\infty)=c$.
To do so, we first fix $h$ and $d_{0} >0$, and then study the behavior of
$I_{\gamma}:=I_{d_{0},h,\gamma}$, $w_{\gamma}:=w_{d_{0},h,\gamma}$ and
$g_{\gamma}:=g_{d_{0},h,\gamma}$ as $\gamma$ varies. We will prove, in Proposition \ref{uniquenessOfGamma-0}, that there exists a unique $\gamma_0=\gamma_0(d_0,h)$ such that $I_{\gamma_0} = (0,\infty)$. In a second moment, we will show in Lemma \ref{lemsurjective} that the application $h\mapsto \lim_{d\to \infty} w_{d_0,h,\gamma_0(d_0,h)}(d)$ is well-defined and surjective on $\mathbb{R}.$ This will establish the existence of solution for the problem \eqref{superScherkEDP}.
Let us now put into practice the strategy described above.
\
First, we establish several properties of $g_{\gamma}$ that we will use extensively along the proof. We begin by proving some bounds for $g^\prime_\gamma$ and $g^{\prime \prime}_\gamma$.
\begin{lem}
Let $d_{0} >0$ and $\gamma\in(-1,1)$. Then we have
\begin{enumerate}
\item[-] $|g^{\prime}_{\gamma}| \le n-1 + $ $\max \psi$;
\item[-] $g^{\prime\prime}_{\gamma}$ is bounded from
above (resp. from below) in the set $$\{ d \in I_{\gamma} \; | \; w^{\prime}_{\gamma}(d) \le0\ (\text{resp. }\ge0) \};$$
\item[-] if $g_{\gamma}(d_{1}) > 0$ for some $d_{1} \in I_{\gamma}$, then $g_{\gamma
}(d) > 0$ for any $d \le d_{1}$, $d\in I_{\gamma}$. (If $g_{\gamma}(d_{1}) \ge 0$ for some $d_{1} \in I_{\gamma}$, then $g_{\gamma
}(d) \ge 0$ for any $d \le d_{1}$, $d\in I_{\gamma}$.)
\end{enumerate}
\label{boundnessOfTheDerivativesOfg}
\end{lem}
\begin{proof}
Remind that $-1 <g_{\gamma}(d) < 1$ for any $d\in
I_{\gamma}$ and
$\psi$ is bounded from \ref{psiDefinitionc}, \ref{psiDefinitiond} and \ref{psiDefinitiond}. Then the bound on $|g_{\gamma}^{\prime}|$ follows directly from \eqref{systemForWG}.
Next, we prove that $g^{\prime\prime}_{\gamma}$ is bounded from above in the set $\{
d \in I_{\gamma} \; | \; w^{\prime}_{\gamma}(d) \le0 \}$ (the other statement follows in the same way). Observe that,
differentiating the second equation of \eqref{systemForWG}, we obtain
\begin{align*}
g_{\gamma}^{\prime\prime}(d) & =-(n-1)\;\mathrm{sech}^{2}(d)\,g_{\gamma
}(d)-(n-1)\tanh(d)\,g_{\gamma}^{\prime}(d)\\[5pt]
& \quad-\frac{\partial\psi}{\partial d}(d,w_{\gamma}(d))-\frac{\partial\psi
}{\partial t}(d,w_{\gamma}(d))\,w_{\gamma}^{\prime}(d).
\end{align*}
Using that $w_{\gamma}^{\prime}(d)\leq0$ and $\dfrac{\partial \psi}{\partial t}\leq0$ (from \ref{psiDefinitionc}), we get
\[
g_{\gamma}^{\prime\prime}(d)\leq-(n-1)\;\mathrm{sech}^{2}(d)\,g_{\gamma
}(d)-(n-1)\tanh(d)\,g_{\gamma}^{\prime}(d)-\frac{\partial\psi}{\partial
d}(d,w_{\gamma}(d)).
\]
Since $|g_{\gamma}^{\prime}|$ and $\dfrac{\partial \psi}{\partial d}$ are bounded (see \ref{psiDefinitionb}), it follows that $g_{\gamma
}^{\prime\prime}$ is bounded from above.
To prove the last statement, observe that, multiplying \eqref{systemForWG} by
$\cosh^{n-1} (d)$, we obtain
\begin{equation}
( (\cosh d)^{n-1} g_{\gamma}(d))^{\prime}=-(\cosh d)^{n-1} \psi(d,w_{\gamma}(d)) \le0,
\label{detivativeOfProductGammaCosh}
\end{equation}
that is, $(\cosh d)^{n-1} g_{\gamma}(d)$ is non increasing. Hence, if
$g_{\gamma}(d_{1}) > 0$, then
\[
(\cosh d)^{n-1} g_{\gamma}(d) \ge(\cosh d_{1})^{n-1} g_{\gamma}(d_{1}) >0
\quad\mathrm{for} \quad d \le d_{1}, \,d\in I_{\gamma}.
\]
\end{proof}
\begin{lem}
Let $d_{0} > \tilde{d} > 0$ ($\tilde{d}$ is defined in \ref{psiDefinitiond}) be such that
\begin{equation}\frac{1}{2} (n-1) \tanh d_{0} -
\psi(d_{0},0) > 0. \label{d0Condition1}
\end{equation}
Then, for any $\gamma\in(-1,1)$, for any $ d >
d_{0}$ ($d \in I_{\gamma}$), there holds
\begin{equation}
\min\{-\frac{1}{2} , \gamma\}< g_{\gamma}(d) \le \max\{0,\gamma\}
\label{gIsBoundedByAboveForDLarge1}
\end{equation}
Moreover, if $\gamma\ge0$, we have
\begin{equation}
\label{gIsBoundedByAboveForDLarge2}
g_{\gamma}(d) \ge0 \text{ for } d \le d_{0}\, (d \in I_{\gamma
}).
\end{equation}
\label{gIsBoundedByBelowForDLarge}
\end{lem}
\begin{comment}
\begin{lem}
Let $d_{0} >0$ be such that
\begin{equation}\frac{1}{2} (n-1) \tanh d_{0} -
\psi(d_{0},0) > 0. \label{d0Condition1}
\end{equation}
Then, for any $\gamma\in(-1,1)$, there holds
\[
g_{\gamma}(d) > \min\{-\frac{1}{2} , \gamma\} \quad\mathrm{for } \quad d >
d_{0} \quad(d \in I_{\gamma}),
\]
and
\begin{equation}
\label{gIsBoundedByAboveForDLarge1}
\max\{0,\gamma\} \ge g_{\gamma}(d) \quad\mathrm{for } \quad d > d_{0} \quad(d
\in I_{\gamma}).
\end{equation}
Moreover, if $\gamma\ge0$, we have
\begin{equation}
\label{gIsBoundedByAboveForDLarge2}
g_{\gamma}(d) \ge0 \quad\mathrm{for } \quad d \le d_{0} \quad(d \in I_{\gamma
}).
\end{equation}
\label{gIsBoundedByBelowForDLarge}
\end{lem}
\end{comment}
\begin{proof}
First observe that, since $\psi(d,0) \to0$ and $\tanh d \to1$ as $d
\to+\infty$, one can find $d_{0} >0$ satisfying \eqref{d0Condition1}.
Let $a_{\gamma}:[d_{0},+\infty)\rightarrow \mathbb{R}$ be defined by
\[
a_{\gamma}(d) = \frac{1}{(\cosh d)^{n-1}} \left( \gamma\cosh^{n-1}(d_{0}) -
\int_{d_{0}}^{d} \psi(s,0) (\cosh s)^{n-1} ds \right) .
\]
Using that $\psi(s,0) \ge\psi(s,w_{\gamma
}(s))$, for any $s$, (which follows from \ref{psiDefinitionc}) and \eqref{gExpressionByW}, we get that
\begin{equation}
g_{\gamma}(d) \ge a_{\tilde{\gamma}}(d) \quad\mathrm{for} \quad d \ge d_{0},
\label{relationBetweenGandA}
\end{equation}
where
$\tilde{\gamma} = \min\{ -\frac{1}{2}, \gamma\}$.
Now we prove that $a_{\tilde{\gamma}}$ is nondecreasing. First notice that
\begin{equation}
a_{\tilde{\gamma}}^{\prime}(d) = -\psi(d,0) -(n-1)\tanh(d) \; a_{\tilde
{\gamma}}(d). \label{EquationFor-a-gamma-tilde}
\end{equation}
Hence, from $-\tilde{\gamma} \ge1/2$ and \eqref{d0Condition1}, we deduce that
\begin{align}
a_{\tilde{\gamma}}^{\prime}(d_{0}) & = -\psi(d_{0},0) -(n-1)\tanh(d_{0}) \;
\tilde{\gamma}\nonumber\\[2pt]
& \ge-\psi(d_{0},0) + \frac{1}{2} (n-1)\tanh(d_{0}) > 0.
\label{a-gammaTildeLowerBound}
\end{align}
Suppose by contradiction that there exists some $d_{1} > d_{0}$ such that $a_{\tilde{\gamma}
}^{\prime}(d_{1}) < 0$. Since $a_{\tilde{\gamma}}^{\prime}$ is continuous and
$a_{\tilde{\gamma}}^{\prime}(d_{0}) > 0$, there is some $d_{2} \in(d_{0}
,d_{1})$ such that $a_{\tilde{\gamma}}^{\prime}(d_{2}) =0 $ and $a_{\tilde
{\gamma}}^{\prime}(d) < 0$ for $d \in(d_{2},d_{1}]$. By the Mean Value
Theorem, there is some $d_{3} \in(d_{2},d_{1})$ such that $a_{\tilde{\gamma}
}^{\prime\prime}(d_{3}) < 0$. Then, from \eqref{EquationFor-a-gamma-tilde}, we get that
\[
0> a_{\tilde{\gamma}}^{\prime\prime}(d_{3}) = -\psi^{\prime}(d_{3},0)
-(n-1)(\mathrm{sech}\, (d_{3}))^{2} \; a_{\tilde{\gamma}}(d_{3}) - (n-1)
\tanh(d_{3}) \; a_{\tilde{\gamma}}^{\prime}(d_{3}).
\]
We get a contradiction from the fact that $\psi(d,0)$ is decreasing in ($\tilde{d},+\infty)$, $a_{\tilde{\gamma}}$ is negative, and $a_{\tilde{\gamma}}^{\prime}(d_{3}) < 0$. Therefore we have proved that $a_{\tilde{\gamma}}(d)$ is nondecreasing for $d>d_0$. Since $a_{\tilde{\gamma}
}^{\prime}(d_{0}) >0$, using \eqref{relationBetweenGandA}, we conclude that
\[
g_{\gamma}(d) \ge a_{\tilde{\gamma}}(d) > a_{\tilde{\gamma}}(d_{0})= \tilde{\gamma} \quad\mathrm{for }
\quad d > d_{0}.
\]
Finally the upper bound of \eqref{gIsBoundedByAboveForDLarge1} and \eqref{gIsBoundedByAboveForDLarge2} are direct consequences of \eqref{gExpressionByW} and
$\psi\ge0$, and they hold for any $d_0>0$ i.e. not necessarily satisfying \eqref{d0Condition1}.
\end{proof}
\begin{rem}
Let $d_0$ be such that \eqref{d0Condition1} holds and $-1 < \gamma
\le-\frac{1}{2}$. Noticing that $\tilde \gamma =\gamma$ and $a_{\tilde \gamma}(d_0)=\gamma = g_{\gamma}(d_0)$, using \eqref{relationBetweenGandA} and \eqref{a-gammaTildeLowerBound}, we obtain
\begin{equation}
\label{lowerBoundForGgammaPrimeAtD0}
g_{\gamma}^{\prime}(d_{0})\ge a^\prime_{\tilde \gamma}(d_0) \ge\frac{1}{2} (n-1) \tanh d_{0} - \psi(d_{0},0).
\end{equation}
\end{rem}
\
Thanks to the two previous lemmas, we are able to prove that the domain of our maximal solution is of the form $(d_\gamma ,+\infty)$, for some $d_\gamma < d_0$. Moreover, we charaterize the behavior of our solution when $d$ goes to $d_\gamma$.
\begin{cor}\label{LimitOfg-gamma}
Let $d_{0} >0$ as in Lemma \ref{gIsBoundedByBelowForDLarge} and let $\gamma
\in(-1,1)$. Then the maximal interval $I_\gamma$ of $(w_\gamma, g_\gamma)$ has the form $I_{\gamma}=(d_{\gamma},
+\infty)$, where $d_{\gamma} \in[-\infty,d_{0})$. Furthermore:
\begin{enumerate}[label=(\alph{enumi})]
\item \label{cor-a}$\displaystyle \lim_{d\to+\infty}g_{\gamma}(d) = 0;$
\item \label{cor-b}if $d_{\gamma} \ne-\infty$, then $\displaystyle \lim_{d\to d_{\gamma}}g_{\gamma
}(d) = 1$ or $\displaystyle \lim_{d\to d_{\gamma}}g_{\gamma}(d) = -1$;
\item \label{cor-c}if $d_{\gamma} \ne-\infty$ and $\gamma\ge0$, then
$\displaystyle \lim_{d\to d_{\gamma}}g_{\gamma}(d) = 1$;
\item \label{cor-d}if
$\displaystyle \lim_{d\to d_{\gamma}}g_{\gamma}(d) = -1$, then $g_{\gamma} <
0$ in $I_{\gamma}$.
\end{enumerate}
\end{cor}
\
\begin{proof}
Suppose by contradiction that $I_{\gamma}=(d_{\gamma}, b)$,
where $b < +\infty$. Thanks to Lemma \ref{boundnessOfTheDerivativesOfg} and Lemma \ref{gIsBoundedByBelowForDLarge}, there exists $(\bar{w}, \bar{g})\in \mathbb{R} \times (-1,1)$ such that
\[
\lim_{d \to b} (d,w_{\gamma}(d), g_{\gamma}(d)) = (b, \bar{w}, \bar{g})
\in\Omega,
\]
where $\Omega= \mathbb{R} \times\mathbb{R} \times(-1,1)$ is the domain of $F$.
However by classical ODE theory, this contradicts the fact that $I_{\gamma}$
is maximal proving that $I_{\gamma}=(d_{\gamma},+\infty)$.
If $d_{\gamma} \ne-\infty$, using the same argument as before, it is clear that
$$\lim_{d \to d_{\gamma}} g_{\gamma}(d)= \pm 1$$
proving \ref{cor-b}.
The proof of $\ref{cor-c}$ is a consequence of \ref{cor-b} and \eqref{gIsBoundedByAboveForDLarge2} whereas
$\ref{cor-d}$ is a direct consequence of the last statement of Lemma
\ref{boundnessOfTheDerivativesOfg}.
Finally we prove $\ref{cor-a}$. Using \eqref{gExpressionByW}, we get, for $d\geq d_0$,
\begin{align*}
|g_{\gamma}(d)|& \le\frac{|\gamma|\cosh^{n-1}(d_{0})}{\cosh^{n-1}(d)} +
\frac{\displaystyle \left| \int_{d_{0}}^{d} \psi(s,w_{\gamma}(s)) \cosh
^{n-1}(s) ds \; \right| }{\cosh^{n-1}(d)}\\
&\leq \frac{ \cosh^{n-1}(d_{0})}{\cosh^{n-1}(d)} + \rho(d),
\end{align*}
where
\begin{equation}\label{eq-defrho}
\rho(d) = \frac{ \displaystyle \int_{d_{0}}^{d} \psi(s,0) \cosh^{n-1}(s) \; ds
}{ \cosh^{n-1}(d)}.
\end{equation}
It is easy to see that the proof boils down to show that $\rho (d)\rightarrow 0$ as $d\rightarrow 0$. Let us prove this last point.\\
Let $\varepsilon> 0$. Since $\psi(d,0) \to0$ as $d
\to+\infty$, there exists $d_{1} > d_{0}$ such that $\psi(d,0) <
\varepsilon/2^{n}$ for $d \ge d_{1}$ and there exists $d_2>d_1$ such that, for all $d\geq d_2$,
\[
\frac{\displaystyle \int_{d_{0}}^{d_{1}} \psi(s,0) \cosh^{n-1}(s) ds }
{\cosh^{n-1}(d)} \leq \dfrac{\varepsilon}{2}.
\]
Therefore, for $d \ge d_{2}$, we have
\begin{align*}
\rho(d) & = \frac{\displaystyle \int_{d_{0}}^{d_{1}} \psi(s,0) \cosh
^{n-1}(s) ds }{\cosh^{n-1}(d)} + \frac{\displaystyle \int_{d_{1}}^{d}
\psi(s,0) \cosh^{n-1}(s) ds }{\cosh^{n-1}(d)}\\[10pt]
& \le\frac{\varepsilon}{2} + \frac{\displaystyle \int_{d_{1}}^{d}
\varepsilon/2^{n} \cosh^{n-1}(s) ds }{\cosh^{n-1}(d)}<\varepsilon ,
\end{align*}
proving that $\displaystyle \lim_{d\to+\infty}\rho(d)= 0$. This concludes the proof.
\end{proof}
\begin{rem}\label{rmk-rhoint}
In the following, we will need some more refined estimate on $\rho$. More precisely, one can show that $|\rho|$ is integrable in
$[d_{0},+\infty)$ and
\begin{equation}
\displaystyle \int_{d_{1}}^{d_{2}} \rho(t) \; dt \leq \frac{2^{n-1}}{n-1} \left(
\rho(d_{1}) + \int_{d_{1}}^{d_{2}} \psi(s,0) \, ds \right) \quad\mathrm{for}
\quad d_{2} > d_{1} \ge d_{0}. \label{boundForIntegralOfRho}
\end{equation}
Indeed, let $d_{2} > d_{1} \ge d_{0}$. Using the
definition of $\rho$, we have
\begin{align*}
\int_{d_{1}}^{d_{2}} \rho(t) \, dt
& = \int_{d_{0}}^{d_{1}} \psi(s,0) \cosh^{n-1}(s) \int_{d_{1}}^{d_{2}}
\frac{1}{\cosh^{n-1}(t)} \, dt \,ds\nonumber\\[5pt]
& \quad+ \int_{d_{1}}^{d_{2}} \psi(s,0) \cosh^{n-1}(s) \int_{s}^{d_{2}}
\frac{1}{\cosh^{n-1}(t)} \, dt \,ds\nonumber\\[5pt]
& < \frac{2^{n-1}}{n-1} \left(
\rho(d_{1}) + \int_{d_{1}}^{d_{2}} \psi(s,0) \, ds \right) \quad\mathrm{for}
\quad d_{2} > d_{1} \ge d_{0}, \label{integralOfRho1}
\end{align*}
proving \eqref{boundForIntegralOfRho}. The integrability of $|\rho|$ in $[d_0, \infty )$ is then a direct consequence of \ref{psiDefinitiong}.
\end{rem}
\
For $d_{0} >0$ as in Lemma \ref{gIsBoundedByBelowForDLarge}, we set
\[
A = \{ \gamma\in(-1,1) \; \; | \; \displaystyle \inf_{d > 0 , d \in I_{\gamma
}} g_{\gamma}(d) > -1 \}.
\]
Observe that $A$ is nonempty since $0 \in A$ due to the fact $\inf_{d \in I_{\gamma}} g_{0}(d) \ge-1/2$
according to Lemma \ref{gIsBoundedByBelowForDLarge}. We define
\begin{equation}
\gamma_{0} = \inf A . \label{gammaoDefinition}
\end{equation}
We will show in the following that $\gamma_0$ is the unique initial data such that $I_{\gamma_0}=\mathbb{R}^+$. Before proceeding, let us show some preliminary properties of the set $A$ and of $\gamma_0$.
\begin{lem}
$\gamma\in A$ if and only if $d_{\gamma} < 0$ or $\displaystyle{\lim_{d \to d_{\gamma}
}g_{\gamma}(d) =1}$. \label{characterizationOFA}
\end{lem}
\begin{proof} Suppose that $\gamma\in A$. If $d_{\gamma} \ge0$, the fact that $\gamma \in A$ and {\it (b)} of Corollary \ref{LimitOfg-gamma} imply directly that $\lim_{d \to d_{\gamma}
}g_{\gamma}(d) = 1$.
Next, we prove the reverse implication. Using that
$g_{\gamma}$ is continuous, $g_{\gamma}(d) > -1$ for any $d \in I_{\gamma}$
and, using Corollary \ref{LimitOfg-gamma}, $\lim_{d \to+\infty}g_{\gamma}(d) = 0$, we conclude that
\[
\inf_{d > 0} g_{\gamma}(d) > -1,\ \text{if } d_\gamma <0,
\]
or
\[
\inf_{d \in I_{\gamma}} g_{\gamma}(d) > -1,\ \text{if } \lim_{d \to d_{\gamma_{0}}}
g_{\gamma_{0}}(d) = 1,
\]
proving that $\gamma\in A$.
\end{proof}
\begin{prop}
There exists $\delta>0$, that depends only on
$d_{0}$ and $\psi$, such that $\gamma_{0} \ge-1+\delta$.
\label{gamma-zero-biggerThan-1}
\end{prop}
\begin{proof}
First, using \eqref{lowerBoundForGgammaPrimeAtD0}, observe that there exists a constant $L>0$ such that $g_{\gamma}^{\prime}(d_{0}) \ge L$
for $\gamma\in(-1,-\frac{1}{2}]$.
Furthermore, from Lemma \ref{boundnessOfTheDerivativesOfg}, there exists $M >L
/ d_{0}$ such that $g^{\prime\prime}_{\gamma}(d) \le M$ for any $d \in
I_{\gamma}$ satisfying $w_{\gamma}^{\prime}(d) \le0$. Next, we set
\[
\delta= \min\left\{ \frac{1}{2}, \frac{L^{2}}{2M} \right\} .
\]
We will show that if $\gamma\in (-1, -1 + \delta )$, then $\gamma\not \in A$
proving that $\gamma_{0} > -1$.
First, we prove that, for $\gamma
\in(-1,-1+\delta)$,
\begin{equation}
g_{\gamma}^{\prime}(d) \ge0 \quad\mathrm{for} \quad d \in I_{\gamma}
\cap[d_{0}-L/M, d_{0}]. \label{g-gamma-primeIsNonnegativeInSomeInterval}
\end{equation}
Suppose by contradiction that $g_{\gamma}^{\prime}(d_{1}) < 0$ for some $d_{1} \in
I_{\gamma} \cap[d_{0}-L/M, d_{0}]$. Noticing that $g_{\gamma}^{\prime}(d_{0}) \ge L >0$, since $\gamma \leq -1/2$, and using the continuity of $g_{\gamma}^{\prime}$, there
exists $d_{2} \in(d_{1},d_{0})$ such that $g_{\gamma}^{\prime}(d_{2}) = 0$ and
$g_{\gamma}^{\prime}> 0$ in $(d_{2},d_{0}]$. This implies that $g_{\gamma}(d)\le
g_{\gamma}(d_{0})=\gamma< 0$ for $d \in[d_{2},d_{0}]$ and, from
\eqref{systemForWG},
\[
w_{\gamma}^{\prime}(d) = \frac{g_{\gamma}}{\sqrt{1- g_{\gamma}^{2}}} \le0
\text{ for } \quad d \in[d_{2},d_{0}].
\]
Thus, we deduce from the definition of $M$ that $g_{\gamma}^{\prime\prime}(d) \le M$ for $d \in[d_{2},d_{0}]$. However, using the Mean Value Theorem, we
get, for some $\xi\in(d_{2},d_{0})$,
\[
L \le g_{\gamma}^{\prime}(d_{0}) - g_{\gamma}^{\prime}(d_{2}) = g^{\prime
\prime}(\xi)(d_{0}-d_{2}) < M \left(\frac{L}{M}\right)= L,
\]
proving \eqref{g-gamma-primeIsNonnegativeInSomeInterval}. Now we consider
two possibilities: either $I_{\gamma} \supseteq[d_{0}-L/M, d_{0}]$ or $I_{\gamma}
\not \supseteq [d_{0}-L/M, d_{0}]$.
Next, we rule out the first one. Suppose by contradiction that $I_{\gamma} \supseteq[d_{0}-L/M, d_{0}]$. Then, we deduce from \eqref{g-gamma-primeIsNonnegativeInSomeInterval} that $g_{\gamma}$ is increasing in
$[d_{0}-L/M, d_{0}]$ and, therefore, $g_{\gamma} \le g_{\gamma}(d_{0})=\gamma<0$ in this interval.
From \eqref{systemForWG}, we get that $w^{\prime}_{\gamma} \le0$,
which implies that $g_{\gamma}^{\prime\prime}\le M$ in $[d_{0}-L/M, d_{0}]$,
according to the definition of $M$. Then, using Taylor's expansion, we obtain that, for some $\xi\in(d_{0} -L/M,d_{0})$,
\begin{align*}
g_{\gamma}\left( d_{0}-\frac{L}{M} \right) & = g_{\gamma}(d_{0}) +
g_{\gamma}^{\prime}(d_{0})\left( -\frac{L}{M}\right) + \frac{g_{\gamma
}^{\prime\prime}(\xi)}{2}\left( -\frac{L}{M} \right) ^{2}\\[5pt]
& \le\gamma+ L \left( -\frac{L}{M}\right) + \frac{M}{2}\left( \frac{L}
{M}\right) ^{2}\leq -1 ,
\end{align*}
which contradicts $g_{\gamma} > -1$.
Hence the second possibility must occur, that is, $d_{\gamma} \ge d_{0} -
L/M$. From $\ref{cor-b}$ of Corollary \ref{LimitOfg-gamma} and
\eqref{g-gamma-primeIsNonnegativeInSomeInterval}, we deduce that $\lim_{d
\to d_{\gamma}} g_{\gamma}(d) =-1$. Observe also that $d_{\gamma} \ge d_{0} -
L/M > 0$, since $M > L/d_{0}$. Therefore, $\inf_{d > 0, d \in I_{\gamma}}
g_{\gamma}(d)=-1$, that is, $\gamma\not \in A$. This completes the proof.
\end{proof}
\begin{prop}
We have $\gamma_{0} \not \in A$. \label{gamma-zero-notInA}
\end{prop}
\begin{proof} Suppose by contradiction that $\gamma_{0} \in A$. If $d_{\gamma_{0}} <
0$, then $(w_{\gamma_{0}}, g_{\gamma_{0}})$ is defined in $[0,d_{0}]$. For
$\gamma< \gamma_{0}$ sufficiently close to $\gamma_{0}$, due to the continuous
dependence of solutions with respect to the initial conditions, it follows that
\[
| g_{\gamma}(d) - g_{\gamma_{0}}(d) | < \varepsilon\quad\mathrm{for} \quad d
\in[0,d_{0}],
\]
for $$\varepsilon= \displaystyle \frac{ 1 + \inf_{d > 0} g_{\gamma_{0}}(d)}{
2} > 0. $$
Thanks to our choice of $\varepsilon$, we get that $g_{\gamma}(d) > -1 + \varepsilon$ for $d \in[0,d_{0}]$.
Combining this fact with Lemma \ref{gIsBoundedByBelowForDLarge}, we deduce that
\[
\inf_{d > 0} g_{\gamma}(d) \ge\min\left\{ - \frac{1}{2}, \gamma, -1 +
\varepsilon\right\} > -1.
\]
Hence, $\gamma\in A$ contradicting that $\gamma_{0} = \inf A$.
Next, let us consider the case $d_{\gamma_{0}} \ge0$. Using Lemma
\ref{characterizationOFA}, we have in this case that $\lim_{d \to d_{\gamma_{0}}}
g_{\gamma_{0}}(d) =1$. Therefore, for some $d_{1} \in(d_{\gamma_{0}}, d_{0})$,
we have
\begin{equation}
g_{\gamma_{0}}(d) > 1/2 \quad \text{ for } d \in(d_{\gamma_{0}},d_{1}].
\label{gamma-0-d1BiggerThan1-2}
\end{equation}
Again, using the continuous dependence of solutions on initial conditions, we
get, for $\gamma <\gamma_0$ sufficiently close to $\gamma_0$,
\[
| g_{\gamma}(d) - g_{\gamma_{0}}(d) | < \min \left\{ 1/4, \varepsilon\right\} , \text{ for } d \in[d_{1},d_{0}].
\]
As before, this implies that $g_{\gamma}(d) > -1+\varepsilon$ for $d \in[d_{1},d_{0}]$ and, together with \eqref{gamma-0-d1BiggerThan1-2}, we conclude that $g_{\gamma}(d_1) > 0$. Hence, from Lemma \ref{boundnessOfTheDerivativesOfg}, we have
that $g_{\gamma}(d) > 0$ for $d \le d_{1}$, $d \in I_{\gamma}$. On the other hand, using
Lemma \ref{gIsBoundedByBelowForDLarge}, we see that $g_{\gamma} \ge\min\{-1/2, \gamma\}$ in
$[d_{0},+\infty)$. Therefore, we obtain that $\inf_{d \in I_{\gamma}} g_{\gamma} > -1$.
Thus, $\gamma\in A$, which contradicts $\gamma_{0}=\inf A$.
\end{proof}
We are now in position to prove that $\gamma_0$ is such that $I_{\gamma_0}=\mathbb{R}^+$.
\begin{thm}
Let $d_{0} >0$ be as in Lemma \ref{gIsBoundedByBelowForDLarge} and
$\gamma_{0}$ defined by \eqref{gammaoDefinition}. Then, $\gamma_{0} < 0$,
$I_{\gamma_{0}} = (0,+\infty)$ (that is, $d_{\gamma_{0}}=0$) and
\[
\lim_{d \to0} g_{\gamma_{0}}(d) = -1.
\]
\label{g-Gamma0IsSol}
\end{thm}
\begin{proof} Recalling that $0\in A$, we deduce, from Propositions \ref{gamma-zero-notInA} and \ref{gamma-zero-biggerThan-1}, that $-1<\gamma_0<0$, and, from Lemma \ref{characterizationOFA} and Corollary \ref{LimitOfg-gamma}, that $d_{\gamma_{0}} \ge0$ and $\lim_{d \to d_{\gamma_{0}}}
g_{\gamma_{0}}(d) = -1.$ Thus, it only remains to prove that $d_{\gamma_{0}}=0$. Suppose by contradiction that $d_{\gamma_{0}} >
0$. Let $\gamma_{k} \in A$ such that $\gamma_{k} < 0$ and $\gamma_{k}
\to\gamma_{0}$. Since the proof is quite lenghty, we split it into three claims.
\
\noindent\textsl{Claim 1:} There exists $\delta> 0$ such that, for all $k$ large enough, $(w_{\gamma_{k}
}, g_{\gamma_{k}})$ is defined in $(d_{\gamma_{0}}-\delta,+\infty)$. Furthermore, $g_{\gamma_{k}} < 0$ in $(d_{\gamma_{0}}-\delta,+\infty)$.
\
\noindent\textsl{Proof of Claim 1:}
Let $D:=n-1+ \max \psi$. We recall from Lemma \ref{boundnessOfTheDerivativesOfg} that $|g_{\gamma_k}^\prime|\leq D$. Next, we set
\[
\delta= \min\{ 1/(4D), d_{\gamma_{0}} \}.
\]
Since $\lim_{d \to d_{\gamma_{0}}} g_{\gamma_{0}}(d) = -1$, there exists $d_{\gamma_0}<d^{*} <
d_{\gamma_{0}}+\delta$ such that $g_{\gamma
_{0}}(d^{*}) < -3/4$. Observe now that from the continuous dependence of
solutions on initial conditions, for $\gamma$ close to $\gamma_0$, it follows that $ [d^{*},d_{0}] \subseteq I_\gamma $ and
\[
| g_{\gamma}(d) - g_{\gamma_{0}}(d) | < 1/4 \text{ for } d
\in[d^{*},d_{0}].
\]
Hence, using that $\gamma_{k} \to
\gamma_{0}$, we have that $I_{\gamma_{k}}=(d_{\gamma_{k}},+\infty)
\supseteq[d^{*},d_{0}]$ and $-1< g_{\gamma_{k}}(d^{*}) < -1/2$ for $k$ large enough.
Let $d \in(d_{\gamma_{0}} - \delta, d^{*}] \cap I_{\gamma_{k}}$ and notice that $0<
d^{*}-d <(d_{\gamma_{0}} +\delta) - (d_{\gamma_{0}} - \delta)= 2 \delta$. So doing a Taylor's expansion and recalling the definition of $D$, we get that
\[
g_{\gamma_{k}}(d) \le g_{\gamma_{k}}(d^{*}) + D |d^{*} - d| < -\frac{1}{2} + 2
\delta\; D \le-\frac{1}{2} + 2 \frac{1}{4D} D \le0.
\]
Then, for $k$ large enough, we have shown that
\begin{equation}
g_{\gamma_{k}}(d) < 0 \quad\mathrm{for } \quad d \in(d_{\gamma_{0}} - \delta,
d^{*}] \cap I_{\gamma_{k}}. \label{negativityOfG-gamma-k}
\end{equation}
Suppose now that, for $k$ large enough, $ (d_{\gamma_{0}}-\delta,
+\infty) \not \subseteq I_{\gamma_{k}}$, i.e. $d_{\gamma_{0}}-\delta< d_{\gamma_{k}} < d_{0}$. Using that $\delta\le d_{\gamma_{0}}$, we have
$d_{\gamma_{k}} > 0$. From this and $\gamma_{k} \in A$, Lemma
\ref{characterizationOFA} implies that $\lim_{d \to d_{\gamma_{k}}}
g_{\gamma_{k}}(d) = 1.$ However, this contradicts
\eqref{negativityOfG-gamma-k}. Then $(d_{\gamma_{0}
}-\delta, +\infty) \subseteq I_{\gamma_{k}}$ and $g_{\gamma_{k}} < 0$ in $(d_{\gamma_{0}} - \delta,
d^{*}]$. Hence, the last statement of Lemma \ref{boundnessOfTheDerivativesOfg}
implies that $g_{\gamma_{k}} < 0$ in $(d_{\gamma_{0}} - \delta, +\infty)$
proving the claim.
\
From now on, we only consider $k$ for which this claim holds. Observe also
that $|g_{\gamma_{k}}| < 1$ and from Lemma
\ref{boundnessOfTheDerivativesOfg}, we have that $|g_{\gamma_{k}}^{\prime}| <
D$ in $I_{\gamma_{k}} \supset(d_{\gamma_{0}}-\delta, +\infty)$. Then, applying Arzel\`a-Ascoli Theorem, there exists a subsequence,
which we denote by $g_{\gamma_{k}}$, that converges uniformly, in $[d_{\gamma_{0}},d_{0}]$, to some
continuous function $g$. On the other hand, from the continuous
dependence of solutions on initial conditions and $\gamma_{k} \to\gamma_{0}$,
we get that $g_{\gamma_{k}} \to g_{\gamma_{0}}$ uniformly in compact subsets
of $(d_{\gamma_{0}}, d_{0}]$. Therefore, we deduce that $g=g_{\gamma_{0}}$ in $(d_{\gamma_{0}
},d_{0}]$ and $g(d_{\gamma_{0}})= \lim_{d \to d_{\gamma_{0}}}g_{\gamma_{0}}(d)
= -1$ which implies that
\begin{equation}
\lim_{k \to+\infty} g_{\gamma_{k}}(d_{\gamma_{0}}) = -1.
\label{limitOfGkAtD-gamma-0}
\end{equation}
Moreover, since $g_{\gamma_{k}}^{\prime}(d_{\gamma_{0}})$ is bounded from
Lemma \ref{boundnessOfTheDerivativesOfg}, there exists $\alpha \in \mathbb{R}$ such that, up to a subsequence, we have
\begin{equation}
\lim_{k \to+\infty} g_{\gamma_{k}}^{\prime}(d_{\gamma_{0}}) = \alpha .
\label{limitOfgkPrimeAtD-gamma-0}
\end{equation}
\
\noindent\textsl{Claim 2:} We have $\alpha= 0$.
\
\noindent \textsl{Proof of Claim 2:} Suppose that $\alpha> 0$, the case $\alpha<0$ follows in the same way. We follow the same idea as in Proposition
\ref{gamma-zero-biggerThan-1}. First, observe that $g_{\gamma_{k}} < 0$ in
$(d_{\gamma_{0}} -\delta, +\infty)$ from Claim 1. Hence $w_{\gamma_{k}
}^{\prime}< 0$ in this interval from \eqref{systemForWG} and, therefore, Lemma
\ref{boundnessOfTheDerivativesOfg} implies that there exists $M > 0$ such
that
\[
g_{\gamma_{k}}^{\prime\prime}\le M \text{ in }(d_{\gamma_{0}}
-\delta, +\infty).
\]
Now let $d \in(d_{\gamma_{0}}-\alpha /2M, d_{\gamma_{0}})$ and define $\rho=d_{\gamma_{0}} - d >0$. From
\eqref{limitOfGkAtD-gamma-0} and \eqref{limitOfgkPrimeAtD-gamma-0}, there
exists $k$ such that
\[
g_{\gamma_{k}}(d_{\gamma_{0}}) < -1 + \frac{\alpha\rho}{4} \text{ and }
g_{\gamma_{k}}^{\prime}(d_{\gamma_{0}}) > \frac{\alpha}{2} .
\]
Hence, using a Taylor's expansion, we have, for some
$\xi\in(d,d_{\gamma_{0}})$,
\begin{align*}
g_{\gamma_{k}}\left( d \right) & = g_{\gamma_{k}}(d_{\gamma_{0}}) +
g_{\gamma_{k}}^{\prime}(d_{\gamma_{0}})\left( -\rho\right) + \frac
{g_{\gamma_{k}}^{\prime\prime}(\xi)}{2}\left( -\rho\right) ^{2}\\[5pt]
& \le-1 + \frac{\alpha\rho}{4} - \frac{\alpha}{2} \rho+ \frac{M}{2} \rho
^{2}<-1
\end{align*}
contradicting $g_{\gamma_{k}} > -1$. This proves the claim.
\
\noindent\textsl{Claim 3:} There holds $\displaystyle \lim_{k \to\infty} w_{\gamma_{k}
}(d_{\gamma_{0}}) = +\infty$.
\
\noindent \textsl{Proof of Claim 3:} Let $M$ as in Claim 2 and $d^{*} \in(d_{\gamma_{0}}, d_{0})$ such
that $(d^{*}-d_{\gamma_{0}})^{2} < \min\{1,1/2M\}$. Given $0 < \varepsilon<
1/4$, we deduce from \eqref{limitOfGkAtD-gamma-0}, \eqref{limitOfgkPrimeAtD-gamma-0} and
$\alpha=0$, that, for $k$ large enough,
\[
g_{\gamma_{k}}(d_{\gamma_{0}}) < -1 + \frac{\varepsilon}{2} \text{ and }
g_{\gamma_{k}}^{\prime}(d_{\gamma_{0}}) < \frac{\varepsilon}{2}.
\]
Using a Taylor's expansion again for $d \in[d_{\gamma_{0}}, d^{*}]$, there exists
$\xi\in(d_{\gamma_{0}},d)$ such that
\begin{align*}
g_{\gamma_{k}}\left( d \right) & = g_{\gamma_{k}}(d_{\gamma_{0}}) +
g_{\gamma_{k}}^{\prime}(d_{\gamma_{0}})\left( d-d_{\gamma_{0}} \right) +
\frac{g_{\gamma_{k}}^{\prime\prime}(\xi)}{2}\left( d-d_{\gamma_{0}} \right)
^{2}\\[5pt]
& < -1 + \varepsilon + \frac{M}{2}
\left( d-d_{\gamma_{0}} \right) ^{2}\leq - 1/2,
\end{align*}
and, therefore,
$$1-(g_{\gamma_{k}}(d))^{2}
\le 4[\varepsilon+ M(d-d_{\gamma_{0}})^{2}].$$
Then, using that $g_{\gamma_{k}} < 0$ in
$(d_{\gamma_{0}}-\delta, +\infty)$ and $d^{*} < d_{0}$, we get
\begin{align*}
\int_{d_{0}}^{d_{\gamma_{0}}} \!\!\frac{g_{\gamma_{k}}(t)}{\sqrt
{1-(g_{\gamma_{k}}(t))^{2}}}\, dt & = \int_{d_{\gamma_{0}}}^{d_{0}}
\!\!\frac{- g_{\gamma_{k}}(t)}{\sqrt{1-(g_{\gamma_{k}}(t))^{2}}} \, dt\\[5pt]
& \ge\int_{d_{\gamma_{0}}}^{d^{*}} \! \!\frac{- g_{\gamma_{k}}(t)}
{\sqrt{1-(g_{\gamma_{k}}(t))^{2}}} \, dt\\[5pt]
& \ge\frac{1}{4} \int_{d_{\gamma_{0}}}^{d^{*}}\!\! \frac{1}{\sqrt
{\varepsilon+ M(t-d_{\gamma_{0}})^{2}}} \, dt.
\end{align*}
Since the last expression goes to infinity as $\varepsilon\to0$, we conclude
that
\[
\lim_{k \to\infty} \int_{d_{0}}^{d_{\gamma_{0}}} \!\!\frac{g_{\gamma_{k}}
(t)}{\sqrt{1-(g_{\gamma_{k}}(t))^{2}}}\, dt =+\infty.
\]
Then, from \eqref{wExpressionByG}, it follows that $w_{\gamma_{k}}
(d_{\gamma_{0}}) \to+\infty$ as $k \to\infty$, proving the claim.
\
We are now in position to complete the proof of the theorem.\\
\noindent \textsl{End of the proof of Theorem \ref{g-Gamma0IsSol}:} Remind that, from
\eqref{systemForWG}, we have
\[
g_{\gamma_{k}}^{\prime}(d_{\gamma_{0}}) = -(n-1)\tanh(d_{\gamma_{0}}) \;
g_{\gamma_{k}}(d_{\gamma_{0}}) - \psi(d_{\gamma_{0}}, w_{\gamma_{k}}(d_{\gamma_{0}})).
\]
Now, observe that Claim $3$ and \ref{psiDefinitionef} implies that $\lim_{k \to\infty} \psi(d_{\gamma_{0}}, w_{\gamma_{k}}(d_{\gamma_{0}})) =0$. Hence, letting $k \to\infty$ in this equation and using \eqref{limitOfGkAtD-gamma-0},
\eqref{limitOfgkPrimeAtD-gamma-0} and Claim 2, we conclude that
\[
(n-1)\tanh(d_{\gamma_{0}}) =0.
\]
However, this contradicts the assumption $d_{\gamma_{0}} > 0$. Therefore,
$d_{\gamma_{0}}=0$, completing the proof.
\end{proof}
Next, we prove the uniqueness of such $\gamma_0$ in the following sense:
\begin{prop}
If $\gamma\in(-1,1)$ is such that $d_{\gamma}=0$ and $\lim_{d \to0}g_{\gamma
}(d)=-1$, then $\gamma=\gamma_{0}$. In other words, $\gamma_{0}$ is the unique
$\gamma$ such that $I_{\gamma}=(0,+\infty)$ and $g_{\gamma}(0^{+})=-1$.
\label{uniquenessOfGamma-0}
\end{prop}
\begin{proof}
The proof follows by contradiction. Assume that there
exists $\gamma_{1} \ne\gamma_{0}$ such that $d_{\gamma_{1}} =0$. Suppose without loss of generality
that $\gamma_{1} > \gamma_{0}$. By definition, we have that $g_{\gamma_{1}}(d_{0}) =
\gamma_{1} > \gamma_{0} = g_{\gamma_{0}}(d_{0})$. From the continuity of
$g_{\gamma_{i}}$, it follows that $g_{\gamma_{1}} > g_{\gamma_{0}}$ in some
neighborhood of $d_{0}$. Let $J$ be the largest open interval contained in
$(0,+\infty)$ such that $d_{0} \in J$ and $g_{\gamma_{1}} > g_{\gamma_{0}}$ in
$J$. Using that the application $z \mapsto z/\sqrt{1-z^{2}}$ is
increasing on $(-1,1)$, we obtain
\[
\frac{g_{\gamma_{1}}(t)}{\sqrt{1-g_{\gamma_{1}}^{2}(t)}} > \frac{g_{\gamma
_{0}}(t)}{\sqrt{1-g_{\gamma_{0}}^{2}(t)}} \text{ for } t \in J.
\]
Hence, from \eqref{wExpressionByG}, we conclude that $w_{\gamma_{1}}(d) <
w_{\gamma_{0}}(d)$ if $d < d_{0}$ and $d \in J$. Then, using
\eqref{systemForWG} and \ref{psiDefinitionc}, we have
\begin{align}
g_{\gamma_{1}}^{\prime}(d) & = -(n-1)\tanh(d) \; g_{\gamma_{1}}(d) - \psi(d,
w_{\gamma_{1}}(d))\nonumber\\[5pt]
& < -(n-1)\tanh(d) \; g_{\gamma_{0}}(d) - \psi(d, w_{\gamma_{0}}(d)) =
g_{\gamma_{0}}^{\prime}(d) , \label{g-gamma-1SmallerThanG-gamma-0}
\end{align}
for $d < d_{0}$ ($d \in J$).
Let $d^{*}$ be the left endpoint of the interval of $J$. Thus $0 \le d^{*} <
d_{0}$. If $d^{*} > 0$, then $g_{\gamma_{0}}$ and $g_{\gamma_{1}}$ are defined
at $d^{*}$ and, therefore, $g_{\gamma_{1}}(d^{*})=g_{\gamma_{0}}(d^{*})$ due
to the definition of $J$. Since $g_{\gamma_{1}}(d^{*})=g_{\gamma_{0}}(d^{*})$
and $g_{\gamma_{1}} > g_{\gamma_{0}}$ in $J$, there exists some $\xi\in
(d^{*},d_{0})$ such that $g_{\gamma_{1}}^{\prime}(\xi) > g_{\gamma_{0}
}^{\prime}(\xi)$. But, this contradicts \eqref{g-gamma-1SmallerThanG-gamma-0}.
Then $d^{*}=0$. Hence, we have $g_{\gamma_{1}}(0^{+})=-1=g_{\gamma_{0}}
(0^{+})$ and $g_{\gamma_{1}} > g_{\gamma_{0}}$ in $(0,d_{0}]$. As before, we
get a contradiction with \eqref{g-gamma-1SmallerThanG-gamma-0}.
\end{proof}
We are also able to prove that $w_{\gamma_0}(d)$ blows up when $d\rightarrow 0^+$ and therefore it satisfies the boundary condition on $S$ of \eqref{superScherkEDP}.
\begin{lem}
Let $\gamma_0$ be defined as in \eqref{gammaoDefinition}. Then, we have
\[
\lim_{d\rightarrow0}w_{\gamma_{0}}(d)=+\infty.
\]
\label{w-omega-goes-to-infinity-at-0}
\end{lem}
\begin{proof} This lemma will follow by comparing our solution to an ordinary Scherk graph namely a solution of
\begin{equation}
\begin{cases}
W^{\prime}(d) = \displaystyle \frac{G(d)}{\sqrt{1-G^{2}(d)}}\\[10pt]
G^{\prime}(d) = -(n-1)\tanh(d) \; G(d),
\end{cases} \label{systemForSomeScherkInB}
\end{equation}
with initial conditions
\[
W(d_{0})=h \text{ and } G(d_{0})= \frac{-1}{\cosh^{n-1}(d_{0})}.
\]
Observe that the previous system can be solved explicitly by
\begin{equation}
G(d) = \frac{-1}{\cosh^{n-1}(d)}\text{ and } W(d) = h -
\int_{d_{0}}^{d} \frac{1}{\sqrt{\cosh^{2n-2}(t) - 1}}\, dt.
\label{ExpressionForScherkInB}
\end{equation}
Also notice that
\[
(G (d)\cosh^{n-1}(d))^{\prime}= 0 \quad\mathrm{in} \quad(0,+\infty).
\]
On the other hand, from
\eqref{detivativeOfProductGammaCosh}, we have
\[
(g_{\gamma_{0}} (d)\cosh^{n-1}(d))^{\prime}\le0 \quad\mathrm{in} \quad
I_{\gamma_{0}} =(0,+\infty).
\]
Therefore, defining $H(d)= g_{\gamma_{0}} (d)\cosh^{n-1}(d) - G
(d)\cosh^{n-1}(d)$, it follows that $H$ is nonincreasing in $(0,+\infty),$ and
\[
\lim_{d \to0} H(d)= \lim_{d \to0} g_{\gamma_{0}} (d)\cosh^{n-1}(d) - G
(d)\cosh^{n-1}(d) =0.
\]
Hence, we deduce that $H(d) \le0$ for any $d \in(0,+\infty)$ which implies that
\begin{equation}
g_{\gamma_{0}} \le G \quad\mathrm{in} \quad(0,+\infty).
\label{g0BiggerThanG-gamma-0}
\end{equation}
Then, since the map $z \mapsto z/\sqrt{1-z^{2}}$ is increasing in $(-1,1)$, we have
\[
w_{\gamma_{0}}(d) = h + \int_{d_{0}}^{d} \frac{g_{\gamma_{0}}(t)}
{\sqrt{1-g_{\gamma_{0}}^{2}(t)}}\; dt \ge h + \int_{d_{0}}^{d} \frac{G
(t)}{\sqrt{1-G^{2}(t)}}\; dt = W(d)
\]
for $d \le d_{0}$. Since $\lim_{d\to0} W (d) = +\infty$, we
conclude that $\lim_{d\to0} w_{\gamma_{0}}(d) = +\infty$.
\end{proof}
Until now, we have proved that if $d_{0}>0$ satisfies \eqref{d0Condition1} and
$h \in\mathbb{R}$, then there exist $\gamma_{0}=\gamma_{0}(d_{0}, h)$ and a
solution $w_{\gamma_{0}}=w_{d_{0},h,\gamma_{0}}$ to the equation
\eqref{superScherkEDO} in $(0,+\infty)$ with initial condition $w_{\gamma_{0}}(d_0)=h.$
Besides $w_{\gamma_{0}}(0)=+\infty$.
Hence, it still remains to show that $w_{\gamma_{0}}(+\infty)=c$ for some such $d_{0}$ and $h \in\mathbb{R}$. Choose and fix such $d_{0}$. We analyze the relation between $h$ and
$w_{d_{0},h,\gamma_{0}}$ for this $d_{0}$. Since $d_{0}$ is fixed and
$\gamma_{0}$ depends on $h$, we use the notation $\gamma_{0}(h)=\gamma
_{0}(d_{0},h).$ Our goal now is to show that the application $\ell(h)=\lim_{d\to \infty} w_{d_{0},h,\gamma_{0}(h)}(d)$ is well-defined, continuous in $\mathbb{R}$ and surjective on $\mathbb{R}$.
\
\begin{lem}
The map $h\mapsto\gamma_{0}(h)$ is continuous. \label{continuityOfH}
\end{lem}
\begin{proof}
Suppose by contradiction that $\gamma_{0}(h)$ is not continuous at
some $h^{*} \in\mathbb{R}$. Then, there exists a sequence $(h_{k})$ and $\varepsilon >0$ such that
\begin{equation}
h_{k} \to h^{*} \quad\mathrm{and} \quad|\gamma_{0}(h_{k}) - \gamma_{0}(h^{*})|
> \varepsilon. \label{differenceBetweenGammaHkAndGammaHstar}
\end{equation}
Observe that, from Proposition \ref{gamma-zero-biggerThan-1} and Theorem
\ref{g-Gamma0IsSol}, $-1+\delta\le \gamma_{0}(h_{k}) < 0$. Hence, up to a subsequence, there exists $\gamma^{*}\in[-1+\delta,0]$ such that $\lim_{k\rightarrow \infty}\gamma_{0}(h_{k})=\gamma^{*}$.
Consider the solution ($w_{k},g_{k}$) of \eqref{systemForWG} associated to
$(\gamma_{0}(h_{k}))$, that is, $w_{k}(d_{0})=h_{k}$ and $g_{k}(d_{0}
)=\gamma_{0}(h_{k})$. Observe that the domain of ($w_{k},g_{k}$) is $(0,+\infty)$
and $g_{k}(0^{+})=-1$, thanks to Theorem \ref{g-Gamma0IsSol}. Moreover, $|g_{k}| < 1$ and $|g_{k}^{\prime}|$ is
bounded by Lemma \ref{boundnessOfTheDerivativesOfg}. Hence, extending
$g_{k}$ continuously at $0$ by $g_{k}(0)=-1$, we can apply Arzel\`a-Ascoli
Theorem to conclude that some subsequence of $(g_{k})$ converges uniformly to
some continuous function $\tilde{g}$ in $[0,d_{0}]$. Observe that
\[
\tilde{g}(0)=\lim_{k \to\infty} g_{k}(0)=-1 \text{ and }\tilde
{g}(d_{0})=\lim_{k \to\infty} g_{k}(d_{0})=\gamma^{*}.
\]
On the other hand, from the continuous dependence of solutions on initial
conditions, $h_{k} \to h^{*}$ and $\gamma_{0}(h_{k}) \to\gamma^{*}$, it
follows that $w_{k} \to w^{*}$ and $g_{k} \to g^{*}$ uniformly in compacts of
$I_{h^{*},\gamma^{*}}$, where ($w^{*},g^{*}$) is the solution of
\eqref{systemForWG} in $I_{h^{*},\gamma^{*}}$ such that $w^{*}(d_{0})=h^{*}$
and $g^{*}(d_{0})=\gamma^{*}$. Therefore, $\tilde{g}=g^{*}$ in $[0,d_{0}] \cap
I_{h^{*},\gamma^{*}}$. If $0 \in I_{h^{*},\gamma^{*}}$, then $g^{*}
(0)=\tilde{g}(0)=-1$, contradicting that $|g^{*}| < 1$ in $I_{h^{*},\gamma
^{*}}$. Hence, the left endpoint of $I_{h^{*},\gamma^{*}}$, denoted by $d^{*}
$, satisfies $0\le d^{*} < d_{0}$.
Next, notice that
\[
\lim_{d\to d^{*}}g^{*}(d)=-1,
\]
otherwise, according to Corollary \ref{LimitOfg-gamma}, $\lim_{d\to d^{*}
}g^{*}(d)=1$, which contradicts $g_{k} \to g^{*}$ and $g_{k} < 0$ from $\ref{cor-d}$
of Corollary \ref{LimitOfg-gamma}. Hence, the solution $g_{k}:(0,+\infty)
\to\mathbb{R}$ converges uniformly to the solution $g^{*}:(d^{*},+\infty)
\to\mathbb{R}$ in $(d^{*},d_{0}]$, and $\lim_{k\rightarrow \infty} g_{k}(d^{*})=\lim_{d\to
d^{*}}g^{*}(d)=-1$. Following the same argument as in Claims 2 and 3 of Theorem
\ref{g-Gamma0IsSol}, one can show that $d^{*}=0$
and $g^{*}$ is a solution of \eqref{systemForWG} in $(0,+\infty)$ satisfying
$g^{*}(d_{0})=\gamma^{*}$ and $w^{*}(d_{0})=h^{*}$. Moreover, $\lim_{d\to0}
g^{*}(d)=-1$, since $g_{k} \to g^{*}$ and $g_{k} < 0$. Hence, from Proposition
\ref{uniquenessOfGamma-0}, $\gamma_{0}(h^{*})=\gamma^{*}$. But, this
contradicts $\gamma_{0}(h_{k}) \to\gamma^{*}$ and
\eqref{differenceBetweenGammaHkAndGammaHstar}, proving that $\gamma_{0}$ is
continuous.
\end{proof}
\begin{lem}
For any $h\in\mathbb{R}$, there exists $\displaystyle{\lim_{d\rightarrow+\infty}w_{d_{0},h,\gamma_{0}}(d)}.$ We denote this limit by $\ell(h).$
Moreover, the convergence is uniform in $h$.
\label{estimateBetweenW-h-dAndTheLimit}
\end{lem}
\begin{proof} To simplify notation, we denote $g_{\gamma_{0}}(d)=g_{\gamma_{0}(h)}(d)$ by
$g_{h}(d)$. First recall that $g_{h} <0$ from $\ref{cor-d}$ of Corollary
\ref{LimitOfg-gamma}.
Moreover, from Lemma \ref{gIsBoundedByBelowForDLarge} and Proposition
\ref{gamma-zero-biggerThan-1}, we deduce that
there exists $\beta\in(0,1)$ that does not depend on $h$ such that $g_{h}(d) \ge-\beta$ for $d
\ge d_{0}$.
Therefore, we have
\begin{equation}
\frac{1}{\sqrt{1-g_{h}^{2}(d)}} \le C_{1}:=\frac{1}{\sqrt{1 -\beta^{2}}}
\text{ for } d \ge d_{0}. \label{g-hSupBoundForInverse}
\end{equation}
We also recall that, according to \eqref{eq-defrho}, there holds
\begin{equation}
|g_{h}(d)| \le\frac{\cosh^{n-1}(d_{0})}{\cosh^{n-1}(d)} + \rho(d)
\text{ for } d \ge d_{0}. \label{g-hSupBound}
\end{equation}
Since $d_{0}$ is fixed and
$\gamma_{0}$ depends on $h$, we use the notation $w_{h}(d) = w_{d_{0},h,\gamma_{0}}(d)$.
Using \eqref{wExpressionByG}, the
inequalities \eqref{g-hSupBound} and \eqref{g-hSupBoundForInverse} imply
\[
|w_{h}(d_{2})-w_{h}(d_{1})| = \left| \int_{d_{1}}^{d_{2}} \frac{g_{h}
(t)}{\sqrt{1-g_{h}^{2}(t)}} \; dt \right| \le C_{1} \int_{d_{1}}^{d_{2}}
\frac{\cosh^{n-1}(d_{0})}{\cosh^{n-1}(t)} + \rho(t) \; dt,
\]
for $d_{0} \le d_{1} < d_{2}$.
So using \eqref{boundForIntegralOfRho}, we obtain that
\begin{equation}
|w_{h}(d_{2})-w_{h}(d_{1})| \le C_{0} \left( \frac{1}{\cosh^{n-1}(d_{1}) } +
\rho(d_{1}) + \int_{d_{1}}^{d_{2}} \psi(s,0) \, ds \right) ,
\label{differenceEstimateBetweenWh1-Wh2}
\end{equation}
for some constant $C_0$ depending only on $n$, $d_{0}$ and $\psi$.
From this, we get that $w_{h}$ is bounded in $[d_{1},+\infty)$.
Furthermore, since $g_{h} < 0$ and $w_{h}$ satisfies \eqref{wExpressionByG}, we deduce that $w_{h}$ is decreasing.
Thus, $w_h(d)$ converges as $d \to \infty$. The uniform convergence is due to the fact that the right-hand side of \eqref{differenceEstimateBetweenWh1-Wh2} does not depend on $h$.
This establishes the proof.
\end{proof}
We are now in position to prove the following theorem:
\begin{thm}
\label{lemsurjective}
The application $h\mapsto\ell(h)$ is continuous in $\mathbb{R}$ and
surjective on $\mathbb{R}$. \label{continuityOfOmega-h}
\end{thm}
\begin{proof} Let $h_{0} \in\mathbb{R}$ and $\varepsilon> 0$. By Lemma
\ref{estimateBetweenW-h-dAndTheLimit}, we have, for $d_{1} > d_{0}$ large enough,
\[
|w_{h}(d_{1}) -\ell(h)| < \varepsilon/3 \text{ for any } h
\in\mathbb{R}.
\]
On the other hand, from Lemma \ref{continuityOfH}, if $h$ is close to $h_{0}$, then
$\gamma_{0}(h)$ is close to $\gamma_{0}(h_{0})$. Hence, using the continuous
dependence of solutions on initial conditions, there exists $\delta_{1} >0$
such that if $|h-h_{0}| < \delta_{1}$, then
\[
|w_{h}(d) - w_{h_{0}}(d)| = |w_{d_{0},h,\gamma_{0}(h)}(d) - w_{d_{0}
,h_{0},\gamma_{0}(h_{0})}(d)| < \varepsilon/3 \text{ for }d
\in[d_{0},d_{1}].
\]
In particular,
\[
|w_{h}(d_{1}) - w_{h_{0}}(d_{1})| < \varepsilon/3.
\]
Therefore, if $|h-h_{0}| < \delta_{1}$, we have
\[
|\ell(h) - \ell(h_{0})| \le|\ell(h) - w_{h}(d_{1})| + |w_{h}(d_{1}) -
w_{h_{0}}(d_{1})| + |w_{h_{0}}(d_{1}) -\ell(h_{0})| < \varepsilon.
\]
Thus, the application $h\mapsto\ell (h)$ is continuous. To prove
that $\ell$ is surjective, let
\[
\sigma:= C_{0} \left( \frac{1}{\cosh^{n-1}(d_{0}) } + \rho(d_{0}) +
\int_{d_{0}}^{+\infty} \psi(s,0) \, ds \right) .
\]
Then, from Lemma \ref{estimateBetweenW-h-dAndTheLimit} (see \eqref{differenceEstimateBetweenWh1-Wh2}), we have
\[
|w_{h}(d_{0}) - \ell(h) | \le\sigma \text{ for any } h
\in\mathbb{R}.
\]
Recalling that $w_{h}(d_{0})=h$, we deduce that $h - \sigma\le\ell(h) \le h + \sigma$
and, therefore,
\[
\lim_{h \to+\infty} \ell(h) = +\infty\text{ and }\lim_{h
\to-\infty} \ell(h) = -\infty.
\]
The continuity of $\ell$ implies that $\ell$ is an onto
application.
\end{proof}
\begin{cor}
For any $c\in\mathbb{R}$, problem \eqref{superScherkEDO} and, therefore,
problem \eqref{superScherkEDP} have a solution. The solution $w$ of problem \eqref{superScherkEDO} is a decreasing function in $(0,\infty),$ satisfying $w>c.$
Even if $f$ changes sign, this solution is a supersolution of \eqref{ScherkProblem}. \label{MainTheorem}
\end{cor}
\begin{proof}
According to Theorem \ref{continuityOfOmega-h},
$\ell(h)$ is surjective on $\mathbb{R}$. Then, there exists $h_{c}$ such
that $\ell(h_{c})=c$. Therefore, the solution $w_{h_{c}}(d)=w_{d_{0},h_{c},\gamma_{0}(h_{c})}(d)$ associated to $h_{c}$ is the one that we are
looking for.
Remind that $g_{\gamma_0}=g_{d_0,h_c,\gamma_0(h_c)}$ that corresponds to $w_{h_{c}}=w_{d_{0}
,h_{c},\gamma_{0}(h_{c})}$ satisfies
$$ \lim_{d \to0} g_{\gamma_{0}}(d) = -1 ,$$
according to Theorem \ref{g-Gamma0IsSol}. Hence, from the last item of Lemma \ref{boundnessOfTheDerivativesOfg}, $g_{\gamma_{0}} < 0$ in $[0,+\infty)$.
Therefore, $w_{h_{c}}'(d)<0$ for any $d >0$, that is, the solution of \eqref{superScherkEDO}, $w_{h_{c}}$ is a decreasing function. Then, using that $w_{h_c}(+\infty)=c$, we conclude that $ w_{h_c} > c$.
\end{proof}
By a standard comparison argument, we also have:
\begin{proposition}
There exists a subsolution $W:[0,+\infty) \to \mathbb{R}$ for \eqref{superScherkEDO}, given by \eqref{ExpressionForScherkInB}, satisfying $W(+\infty)=c$ and $W(d)\leq w_{h_{c}}(d)$, where $w_{h_{c}}$ is the solution from previous corollary.
\label{kindOfScherkBelowSol}
\end{proposition}
\begin{proof}
[Proof of Theorem \ref{s}]Let $W$ and $w_{h_c}$ be as stated in Proposition \ref{kindOfScherkBelowSol}.
Then they are sub and supersolutions of $Q(v)=f(x,v)$ in $B$ such that $W,w_{h_c}=+\infty$ on $S$ and
$W,w_{h_c}=c$ on $\partial_{\infty}B.$ Set
\[
\mathcal{S}_{B}=\left\{ \sigma\in C^{0}\left( B\right) \, |\,
\sigma\text{ is a subsolution of }Q\text{ and }W\leq\sigma\leq w_{h_c}\right\} .
\]
From the results from Section \ref{sec-genExist}, it is clear that Perron's method can be
applied to the equation $Q(v)=f(x,v)$ and we conclude that the function $u$ defined in
$B$ by
\[
u(x)=\sup_{\sigma\in\mathcal{S}_{B}}\sigma(x),\text{ }x\in B,
\]
is $C^{2}$ and satisfies $Q(u)=f(x,u).$ Clearly $u|_{S}=+\infty,$ $u$ extends
continuously to $\partial_{\infty}B$ and $u|_{\partial_{\infty}B}=c$, since $W \le u \le w_{h_c}$.
Observe that $w_{h_c}$ is a supersolution of \eqref{ScherkProblem} even if $f$ changes sign and satisfies only conditions \ref{propphi1} and \ref{propphi2}. If $\tilde{w}$ is a supersolution of $Q(v)=-f(x,v)$ that satisfies $v=-c$ on $\partial_{\infty} B$ and $v=+\infty$ on $S$, then $-\tilde{w}$ is a subsolution of \eqref{ScherkProblem} that satisfies $v=-\infty$ on $S$.
\end{proof}
\section{The asymptotic Dirichlet problem}
\label{The_asymptotic_Dirichlet_problem_subsection}
In this section, we solve the asymptotic Dirichlet problem for our prescribed mean curvature type equation by making use of the Scherk type solutions obtained in the previous section. We refer to \cite{CHHfmin} for related results using another method.
\begin{proof}[Proof of Theorem \ref{thm-Scherk}]
First consider the case where $f$ satisfies condition \ref{propphi2}. We use Perron's method by setting
\[
\mathcal{S}_{\varphi}=\left\{ \sigma\in C^{0}\left( \mathbb{H}^{n}\right)\,
\vert\, \sigma\text{ is a subsol. and }\limsup_{x\rightarrow x_{0}}\sigma
(x)\leq\varphi(x_{0}),\,x_{0}\in\partial_{\infty}\mathbb{H}^{n}\right\}
\]
and defining
\[
u(x)=\sup_{\sigma\in\mathcal{S}_{\varphi}}\sigma(x),\text{ }x\in\mathbb{H}
^{n}.
\]
We first prove that $u$ is well defined. Set
\[
C_{1}=\min_{\partial_{\infty}\mathbb{H}^{n}} \varphi
\text{ and }C_{2}=\max_{\partial_{\infty}\mathbb{H}^{n}} \varphi
.
\]
Let $S_{1}\subset B_2$ and $S_{2}\subset B_1$ be two totally geodesic hypersurfaces of
$\mathbb{H}^{n}$ where each $B_i$, $i=1,2,$ is a connected component of
$\mathbb{H}^{n}\backslash S_{i}$. By Theorem \ref{s}, let $w_{i}$ be two Scherk type supersolutions of \eqref{ScherkProblem} in $B_{i}$ such that $w_{i}|_{\partial_{\infty}B_{i}}=C_{2}$ and $w_{i}|_{S_{i}}=+\infty.$ Let $v_{i}$ be two Scherk type subsolutions of \eqref{ScherkProblem} in $B_{i}$ such that $v_{i}
|_{\partial_{\infty}B_{i}}=C_{1}$ and $v_{i}|_{S_{i}}=-\infty.$ Define super and
subsolutions $w,v$ in $\mathbb{H}^{n}$ by
\[w(x)=\begin{cases}
w_{1}(x)\text{ if }x\in B_{1}\backslash B_{2}\\
\inf\left\{ w_{1}(x),w_{2}(x)\right\} \text{ if }x\in B_{1}\cap B_{2}\\
w_{2}(x)\text{ if }x\in B_{2}\backslash B_{1}
\end{cases}\]
and
\[v(x)=\begin{cases}
v(x)\text{ if }x\in B_{1}\backslash B_{2}\\
\sup\left\{ v_{1}(x),v_{2}(x)\right\} \text{ if }x\in B_{1}\cap B_{2}\\
v_{2}(x)\text{ if }x\in B_{2}\backslash B_{1}.\end{cases}\]
Since $v\in\mathcal{S}_{\varphi}$ by the comparison principle we have that
$\sigma\leq w$ for all $\sigma\in\mathcal{S}_{\varphi}.$ It follows that $u$
is well defined. Perron's method then guarantees that $u\in C^{2}\left(
\mathbb{H}^{n}\right) $ and that $Q(u)(x)=f(x,u)$ for all $x\in\mathbb{H}
^{n}.$ We now prove that $u$ extends continuously to $\partial_{\infty}
\mathbb{H}^{n}$ and that $u|_{\partial_{\infty}\mathbb{H}^{n}}=\varphi.$
Choose $x_{0}\in\partial_{\infty}\mathbb{H}^{n}$ and let $\varepsilon>0$ be given. Since $\varphi$ is continuous, there exists an open neighborhood
$W\subset\partial_{\infty}\mathbb{H}^{n}$ of $x_{0}$ such that $\varphi
(x)<\varphi(x_{0})+\varepsilon$ for all $x\in W$. We may take a totally
geodesic hypersuface $S$ of $\mathbb{H}^{n}$ such that one of the connected
components of $\mathbb{H}^{n}\backslash S,$ say $B,$ is such that $x_{0}
\in\partial_{\infty}B\subset W.$ Define a Scherk type solution $\tilde{w}$
on $B$ such that $\tilde{w}|_S=+\infty$ and $\tilde{w}|_{\partial
_{\infty}B}=C,$ where $C=\varphi(x_{0})+\varepsilon.$ Note that given
$\sigma\in\mathcal{S}_{\varphi},$ and denoting by $\sigma_{B}=\sigma|_B,$ the
comparison principle implies that $\tilde{w}|_B\leq\sigma_{B}.$ It follows
that $u\leq\tilde{w}$ in $B$ and then
\[
\limsup_{x\rightarrow x_{0}}u(x)\leq\varphi(x_{0})+\varepsilon.
\]
We may also construct a subsolution $\tilde{v}\in\mathcal{S}_{\varphi}$
such that $\tilde{v}|_{\partial_{\infty}B}=C,$ where now $C=\varphi
(x_{0})-\varepsilon$ so that
\[
\liminf_{x\rightarrow x_{0}}u(x)\geq\varphi(x_{0})-\varepsilon.
\]
Since $\varepsilon>0$ is arbitrary we obtain that $\lim_{x\rightarrow x_{0}
}u(x)=\varphi(x_{0}),$ concluding the proof when condition \ref{propphi2} is satisfied.
If, instead of condition \ref{propphi2}, we assume $\phi(r)\leq (n-1)\coth(r),$ we obtain the following a priori bound for a solution $u$ to the \eqref{eq-asymDP}. Let $o\in \mathbb{H}^n$ be the point in condition \ref{propphi1}. Let $v:[0,\infty) \to \mathbb{R}$ be the solution of the following ODE
\begin{equation}
\begin{cases}
Q(v\circ r)= \phi(r)\\
v'(0)=0\\
v(\infty) = M,
\end{cases}
\end{equation}
where $M=\displaystyle{\sup_{\partial_\infty \mathbb{H}^n}\varphi}.$ Then
$$v(r)=\int_r^{+\infty} \frac{\tilde{\rho}(t)}{\sqrt{1-(\tilde{\rho}(t))^2}}dt+M$$ for $$\tilde{\rho}(t)=\frac{1}{\sinh^{n-1}(r)}\int_0^r\phi(s)\sinh^{n-1}(s)ds.$$
Observe that condition \ref{propphi1} and $\phi(r) \le (n-1)\coth r$ imply that $\sup |\tilde{\rho}(t)| < 1$. Hence
the fact that $v$ is well defined follows from the integrability of $\tilde{\rho}$, that is proved as Remark \ref{rmk-rhoint}. Therefore we have an upper barrier for the Dirichlet problem \eqref{eq-asymDP}.
We conclude the proof by modifying our function $f$ in \eqref{eq-asymDP} to a new function $\hat{f}\in C^2(\mathbb{H}^n\times \mathbb{R})$ satisfying:
$$\hat{f}(x,t)=\begin{cases} f(x,t) \text{ if }t\leq v(0)\\
g(x,t)\text{ if }t\geq v(0)+1\end{cases}$$
where $g$ satisfies conditions \ref{propphi1} and \ref{propphi2}. Hence, from the previous case, there is $\hat{u}$ solution to the \eqref{eq-asymDP} with $\hat{f}.$ Nevertheless, $\hat{u}$ also satisfies \eqref{eq-asymDP} for $f$ since $\hat{u}$ is bounded by $v\circ r$ and therefore by $v(0).$
\end{proof}
\section{Removable asymptotic singularities}
In this section, we show that there is no isolated singularity on the asymptotic boundary for the solution to \eqref{eq-asymDP}. For that, we study a Dirichlet problem similar to the one studied in section \ref{The_asymptotic_Dirichlet_problem_subsection}, in which we relax the boundary condition:
\begin{equation}
\begin{cases}
Q(v) = f(x,v)\text{ in } \mathbb{H}^n \\
v =\varphi\text{ on }\partial_{\infty} \mathbb{H}^n \backslash \{p_1, \dots, p_k\},
\end{cases}
\label{mingrapheqWithSing}
\end{equation}
where $\varphi\in
C^{0}(\partial_{\infty}\mathbb{H}^n)$ is a given function, $p_i \in \partial_{\infty} \mathbb{H}^n$ and $f \in C^1(\mathbb{H}^n \times \mathbb{R})$ satisfies conditions \ref{propphi1}, \ref{propphi2} and $f_t(x,t)\le 0$ in $\mathbb{H}^n\times\mathbb{R}$. Using the Scherk type solutions and following the same idea as in \cite{BR}, we prove that a solution to this problem can be extended continuously to
the points $p_i$, that is, such a solution satisfies $v =\varphi$ on $\partial_{\infty} \mathbb{H}^n$.
However, since our Scherk type solutions are not isometric, in contrast with the Scherk solutions used in \cite{BR}, we need some auxiliary results to prove that the solutions are bounded. For that, remind that $\psi=\psi_S$, defined in Proposition \ref{prop-Psi}, satisfies
$$ \psi_S(d,t) = \Psi ( d - d_S(o),t).$$
where
$$\Psi(z,t) = \left\{ \begin{array}{rr} \sqrt{ \phi(|z|) h(t)} & {\rm if} \quad t \ge 0 \\[5pt]
\sqrt{ \phi(|z|) h(0)} & {\rm if} \quad t < 0
\end{array} \right. $$
and $d(x)=d_S(x)$ is the signed distance function to $S$. Observe that for any totally geodesic hypersphere $S_0$ that contains the point $o$, we have $d_{S_0}(o)=0$ and, therefore, $\Psi(d,t)=\psi_{S_0}(d,t)$. Then, from Proposition \ref{prop-Psi}, $\Psi$ satisfies the conditions (i)-(vi). Indeed, we have:
\begin{lem}
Let $o \in \mathbb{H}^n$ as stated in \ref{propphi1}. Then there exists a nonnegative $C^1$ function $\Psi: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ that satisfies \ref{psiDefinitiona}
-\ref{psiDefinitiong} from Proposition \ref{prop-Psi} and such that
for any totally geodesic hypersphere $S$,
\begin{equation}
|f(x,t)| \le \Psi(d_S(x)-d_S(o), t) \text{ for }x\in \mathbb{H}^n \text{ and } t \ge 0.
\label{PsiBoundsfxt}
\end{equation}
\end{lem}
\noindent
\begin{rem}
Remind that for any totally geodesic hypersphere $S$ and any $c\in \mathbb{R}$, according to Corollary \ref{MainTheorem}, there exists a solution $w_{S,c}(d)$ of \eqref{superScherkEDO} with $\psi$ replaced by $\psi_S$.
Defining
$$g_{S,c}=\frac{w_{S,c}'}{\sqrt{1+(w_{S,c}')^2}},$$
we have that $(w_{S,c},g_{S,c})$ satisfies \eqref{systemForWG} with $\psi$ replaced by $\psi_S$.
Now we present a result of some uniform bound of $w_{S.c}$.
\label{SolutionOfScherkWithPsiS}
\end{rem}
\begin{lem}
There exist $c_0 >0$ and $d_1>0$ with the following property: if $c \ge c_0$, there exists a constant $M >c$ that depends only on $c$, $n$, and $\Psi$, such that, for any totally geodesic hypersphere $S$,
$$ w_{S,c}(d) \le M \quad {\rm for} \quad d \ge d_1. $$
\label{AuxiliarLemmaForUniformBoundedness2}
\end{lem}
\begin{proof}
Let $S$ be a totally geodesic hypersphere and $c\ge 0$. Remind, from Theorem \ref{g-Gamma0IsSol}, that
$$ \lim_{d \to 0}g_{S,c}(d) = -1.$$
Using this, \eqref{systemForWG} and $w_S(+\infty)=c$, we conclude that
\begin{equation}
w_{S,c}(d) = c - \int_{d}^{+\infty} \frac{g_{S,c}(t)}{\sqrt{1-g_{S,c
}^{2}(t)}}\; dt \label{wExpressionByGFinal}
\end{equation}
and
\begin{equation}
g_{S,c}(d) = \frac{1}{\cosh^{n-1} (d)} \left( -1 -
\int_{0}^{d} \psi_S(s,w_{S,c}(s)) \cosh^{n-1} (s) ds \right) .
\label{gExpressionByWFinal}
\end{equation}
Observe now that, from Corollary \ref{MainTheorem}, $w_{S,c}$ is decreasing and, therefore, $w_{S,c}(d) \ge c$. Hence, using \eqref{gExpressionByWFinal} and that $\psi_{S}(d,t)$ is nonincreasing in the variable $t$, we get
\begin{equation}
g_{S,c}(d) \ge \frac{1}{\cosh^{n-1} (d)} \left( -1 -
\int_{0}^{d} \psi_S(s,c) \cosh^{n-1} (s) ds \right).
\label{LowerBoundForGSc}
\end{equation}
From \ref{psiDefinitionef}, there exists $c_0 > 0$ such that
$$ \psi_{S}(d,t) = \Psi(d-d_S(o),t) \le \Psi(0,t) \le \frac{1}{2^{n+1}} $$
for any $d \in \mathbb{R}$ and $t \ge c_0$.
This $c_0$ does not depend on $S$, but only on $\phi(0)$ and $h$. Hence, if $c \ge c_0$, inequality \eqref{LowerBoundForGSc} implies that
$$ g_{S,c}(d) \ge \frac{1}{\cosh^{n-1}(d)} \left( -1 -
\int_{0}^{d} \frac{1}{2^{n+1}} \cosh^{n-1}(s) ds \right). $$
Therefore, using that $\cosh^{n-1} (d) > 4$ for $d \ge 4$ and
\[
\frac{\displaystyle \int_{0}^{d} \cosh ^{n-1}(s) ds}{\cosh ^{n-1}(d)} <
2^{n-1} \text{ for any } d \ge 0,
\]
we conclude that
$$ g_{S,c}(d) \ge - \frac{1}{2} \text{ for } d \ge 4 \text{ and } c \ge c_0.$$
Moreover, $ g_{S,c}(d) < 0$ according to the proof of Corollary \ref{MainTheorem}. Then $|g_{S,c}(d) | \le 1/2$ for $d \ge 4$ and $c \ge c_0$.
From this and \eqref{wExpressionByGFinal},
\begin{equation}
w_{S,c}(d) \le c + \frac{2}{\sqrt{3}} \int_{d}^{+\infty} |g_{S,c}(t)|\; dt \text{ for } d \ge 4 \text{ and } c \ge c_0.
\label{UpperBoundForWscForDBiggerThan4}
\end{equation}
Now using again \eqref{gExpressionByWFinal} and that $\psi_S(d,0) \ge \psi_S(d,t)$ for any $d$, we get
$$ |g_{S,c}(d)| < \frac{1}{\cosh^{n-1}( d)} + \rho(d),$$
where $\rho$ is defined in \eqref{eq-defrho} with $d_0=0$ and $\psi$ replaced by $\psi_S$. Then, from \eqref{UpperBoundForWscForDBiggerThan4} and \eqref{boundForIntegralOfRho}, we have
\begin{align*}
w_{S,c}(d) &\le c + \frac{2}{\sqrt{3}} \int_{d}^{+\infty} \left(\frac{1}{\cosh^{n-1}(s)} + \rho(s)\right) \; ds \\[5pt]
&\le c + \frac{2}{\sqrt{3}} \int_{0}^{+\infty} \frac{1}{\cosh^{n-1}(s)} \; ds + \frac{2^n}{\sqrt{3}(n-1)} \int_0^{+\infty} \psi_S(s,0) \; ds,
\end{align*}
for $d \ge 4$ and $c \ge c_0$. Since, by the definition of $\Psi$ and \ref{psiDefinitiong},
$$\int_0^{+\infty} \psi_S(s,0) \; ds =\int_{-d_S (o)}^\infty \Psi (s,0) ds \le \int_{-\infty}^{\infty} \Psi (s,0) ds < \infty,$$
the proof follows.
\end{proof}
Now we state a comparison principle for some unbounded domains.
\begin{lem}
Let $U$ be a domain in $\mathbb{H}^n$, possibly unbounded. Suppose that $w_1$ and $w_2$ are respectively a sub and a supersolution of $Q(w(x)) = \psi(d(x),w(x))$ in $U$ such that
$$ \limsup_{x \to p} w_1(x) \le \liminf_{x \to p} w_2(x) \text{ for any } p \in \partial U \cup \partial_{\infty} U$$
and, for some $A < \infty$,
we have $\limsup_{x \to q} w_1(x) \le A$ for any $q \in \partial U \cup \partial_{\infty} U$.
Then, $w_1 \le w_2$ in $U$.
\label{comparison-principle-for-unbounded}
\end{lem}
Now making some adjustments in the proof of Theorem 1.1 of \cite{BR}, we obtain:
\begin{prop}
If $u \in C^2(\mathbb{H}^n) \cap C^0(\overline{\mathbb{H}^n} \backslash \{p_1, \dots, p_k \})$ is a solution of \eqref{mingrapheqWithSing}, then $u$ is bounded.
\label{boundednessOfSolutionWithSing}
\end{prop}
\begin{proof}
For each $p_i$ let $B_i$ be a totally geodesic hyperball such that $p_j\in \partial_\infty B_i$ if and only if $i=j.$ We can suppose $p_i \in \text{int}\; \partial_{\infty} B_i$ and we denote by $H_i$ the hypersphere that bounds $B_i.$
Since $p_j \not\in \partial_{\infty}H_i$ for any $j \in \{1, \dots, k\}$, $u$ is continuous in $H_i \cup \partial_{\infty} H_i$ and, therefore, is bounded on $H_i$.
Then set
$$ c_1 =\max\{c_0, \sup_{\partial_{\infty} \mathbb{H}^n} \varphi, \sup_{H_1} u, \dots, \sup_{H_k} u\}, $$
where $c_0$ is given in Lemma \ref{AuxiliarLemmaForUniformBoundedness2}. Since $c_1 \ge c_0$, we deduce from Lemma \ref{AuxiliarLemmaForUniformBoundedness2} that there exist $M > c_1$ and $d_1>0$ such that
\begin{equation}
w_{H,c_1}(d) \le M \text{ for } d \ge d_1,
\label{UniformBoundedNessForSolutions}
\end{equation}
for any totally geodesic hypersphere $H$. Let $H^*$ be some totally geodesic hypersphere contained in $B_1$ such that $dist(H^*,\partial B_1) > d_1$. Hence,\\ $dist(H^*,\mathbb{H}^n \backslash B_1 \cup \dots \cup B_k) > d_1$ and, therefore \eqref{UniformBoundedNessForSolutions} implies that
$$ w_{H^*,c_1}(d_{H^*}(x)) \le M \text{ for } x \in \mathbb{H}^n \backslash B_1 \cup \dots \cup B_k. $$
Using that $u \le c_1$ on the boundary and asymptotic boundary of $\mathbb{H}^n \backslash B_1 \cup \dots \cup B_k$, the comparison principle (Lemma \ref{comparison-principle-for-unbounded}) and $w_{H^*,c_1} \ge c_1$, we conclude that
\begin{equation}
u \le w_{H^*,c_1} \le M \text{ in } \mathbb{H}^n \backslash B_1 \cup \dots \cup B_k.
\label{uIsSmallerThanC1}
\end{equation}
Now we prove that $u \le M$ in $B_i$. For that, take a sequence of totally geodesic hyperspheres $S_m$ that converges to $p_i$.
Let $Y_m$ be the connected component of $\mathbb{H}^n \backslash S_m$ whose asymptotic boundary does not contain $p_i$. Observe that $\mathbb{H}^n \backslash B_i \subset Y_m$ for $m$ large and $\cup_m Y_m = \mathbb{H}^n$. Consider the problem
$$\begin{cases}
Q(v\circ d_{S_m})= \psi_{S_m}(d_{S_m},v\circ d_{S_m}) \text{ in } Y_m\\[5pt]
v = c_1 \text{ on }\partial_{\infty} Y_m\\[5pt]
v = + \infty\text{ on } S_m,
\end{cases}
$$
where $\psi_{S_m}$ is defined in Proposition \ref{prop-Psi} and in the beginning of this section. According to Remark \ref{SolutionOfScherkWithPsiS}, this problem has a solution $w_{S_m,c_1}(d_{S_m}(x))$. Moreover, from \eqref{UniformBoundedNessForSolutions}, we get
\begin{equation}
w_{S_m,c_1}(d) \le M \quad {\rm for } \quad d \ge d_1.
\label{UniformBoundedNessForSolutionsWSm2}
\end{equation}
From Corollary \ref{MainTheorem}, $w_{S_m,c_1}(d) > c_1$ for any $d$. Hence, $w_{S_m,c_1} > c_1 \ge u$ on $H_i = \partial B_i$, $w_{S_m,c_1} \ge u$ on $\partial_{\infty} ( Y_m \cap B_i)$ and $w_{S_m,c_1} = +\infty > u$ on $\partial Y_m$. Therefore, Lemma \ref{comparison-principle-for-unbounded} implies that
$$u(x)\le w_{S_m,c_1}(d_{S_m}(x)) \text{ in } Y_m \cap B_i \text{ for }m \text{ large.}$$
Let $z \in B_i$. Then $z \in Y_m \cap B_i$ for $m$ sufficiently large, since $\cup Y_m = \mathbb{H}^n$.
Moreover, using that $S_m$ converges to $p_i$, we have $d_{S_m}(z) > d_1$ for $m$ large. Therefore, from \eqref{UniformBoundedNessForSolutionsWSm2}, we conclude that
$$ w_{S_m,c_1}(d_{S_m}(z)) \le M \text{ for } m \text{ large.}$$
Hence
$u \le M$ in $B_i$. From this and \eqref{uIsSmallerThanC1}, it follows that $u \le M$ in $\mathbb{H}^n$. Analogously, one can show that $u$ is bounded from below. This establishes the proof.
\end{proof}
The proof of the next result follows the same idea as in Theorem 1.1 of \cite{BR}.
\begin{proof}[Proof of Theorem \ref{ContinuityOfSolutionWithSing}]
For $p \in \{p_1, \dots, p_k\}$, we have to prove that
$$\lim_{x\to p} u(x) =\varphi(p).$$
Observe that $v:=u-\varphi(p)$ is a solution of $Q(v)=\tilde{f}(x,v)$ in $\mathbb{H}^n$ and $v=\varphi-\varphi(p)$ on $\mathbb{H}^n \backslash \{p_1, \dots, p_k \}$, where $\tilde{f}(x,v):=f(x,v+\varphi(p))$ satisfies \ref{propphi1} and \ref{propphi2}. Then, we can suppose w.l.g. that $\varphi(p)=0$.
That is, we need to show that
$$\lim_{x\to p} u(x) = 0.$$
Let
$$ K = \limsup_{x \to p} u(x) .$$
According to Proposition \ref{boundednessOfSolutionWithSing}, $u$ is bounded from above by some $M,$ so $K\leq M.$ We show now that, for any $\delta > 0$, we have $K \le \delta$. Suppose by contradiction that $K > \delta$, for some $\delta >0$.
Now consider a decreasing sequence $(V_j)$ of totally geodesic hyperballs ``concentric" at $p$
(that is, $p$ is one of the ending point of a geodesic that cross each $\partial V_j$ orthogonally),
such that $$\bigcap_j \overline{V}_j = \{p\},\quad \sup_{V_j} u < K + 1/j \quad \text{ and } \quad \sup_{\partial_\infty V_j}\varphi \le \frac{\delta}{2}. $$
For each $j$, let $\tilde{V}_j \subset V_j$ be a totally geodesic hyperball concentric with $V_j$ at $p$ such that
$$ dist( \partial \tilde{V}_j, \partial V_j ) \ge j \text{ and } \sup_{x \in \tilde{V}_j} u(x) > K -1/j.$$
Then there exists a sequence $(x_j)$ that satisfies $x_j \in \tilde{V}_j$ and
$$ K - 1/j < u(x_j) < K + 1/j .$$
Denote $A = V_1$ and let $T_j:\mathbb{H}^n \to \mathbb{H}^n$ be an isometry that preserves $p$, $T_j(\tilde{V}_j) \supset A$ and $y_j:=T_j(x_j) \in \partial A.$ Since $T_j(V_j)$ and $T_j(\tilde{V}_j)$ are totally geodesic hyperballs and $T_j(V_j) \supsetneq T_j(\tilde{V}_j) \supset A$, we have that $\partial_{\infty}A \subset {\rm int} \; \partial_{\infty} T_j(V_j)$ for any $j$.
Observe that $$u_j= u \circ T^{-1}_j$$
satisfies
\begin{equation}
\sup_{T_j(V_j)}u_j < K + 1/j \text{ and } u_j(y_j) > K - 1/j,
\label{limiteSuperiorWm}
\end{equation}
and is a solution of
$$Q(v(y)) = f(T^{-1}_j(y),v(y)),$$
since $Q$ is invariant under isometries. We have also that
$T_j(V_j)$ is a totally geodesic hyperball and
$u_j \le \delta/2$ on $ \partial_{\infty} (T_j(V_j)) \backslash \{p\}$
since $u=\varphi \le \delta/2$ on $\partial_{\infty} V_j \backslash \{p\}$.
Moreover, using that $A \subset T_j(V_j)$ and $p \not\in \partial_{\infty}( \mathbb{H}^n \backslash A )$, we have that $\partial_{\infty}A \cap \partial_{\infty}( \mathbb{H}^n \backslash A )\subset \partial_{\infty}T_j(V_j)\backslash \{p\}$. Therefore, $u_j \le \delta/2$ on $\partial_{\infty}A \cap \partial_{\infty}( \mathbb{H}^n \backslash A)$.
For $q \in \partial_{\infty}A \cap \partial_{\infty}( \mathbb{H}^n \backslash A)$, let $B_q$ be a totally geodesic hyperball disjoint with $V_2$ such that $q \in {\rm int} \partial_{\infty}B_q$ and $B_q \subset T_j(V_j)$ for any $j$. (This is possible since $(V_j)$ is a decreasing sequence, $\partial T_j(V_j)$ is a totally geodesic hypersphere, $dist(\partial T_j(V_j),A) \ge j$ as in \cite{BR} and, then some neighborhood of $\partial_{\infty}A \subset {\rm int} \, \partial_{\infty} T_j(V_j)$ for any $j$). As in Theorem \ref{thm-Scherk},
consider the supersolutions $w_q$ of
$$\begin{cases}
Q(v)(y) = f(T^{-1}_j(y) , v(y))\text{ in } B_q \\
v = +\infty \text{ on } \partial B_q \\
v = \delta/2 \text{ on } {\rm int} \; \partial_{\infty} B_q.\end{cases} $$
Such a problem is solvable according to Theorem \ref{s}, since $f(T^{-1}_j(y) , t)$ satisfies \ref{propphi1} and \ref{propphi2}.
Since $u_j \le w_q=\delta/2$ on ${\rm int} \; \partial_{\infty} B_q$, Lemma \ref{comparison-principle-for-unbounded} implies that $u_j \le w_q$ in $B_q$. Let $\tilde{B}_q \subset B_q$ be the hyperball with boundary equidistant to $\partial B_q$, for which $w_q < \delta$ in $\tilde{B}_q$. Hence $u_j < \delta$ in $\tilde{B}_q$ and, therefore,
\begin{equation}
u_j < \delta \text{ in } \tilde{B}
\label{u-jIsSmallerThanDelta}
\end{equation} for any $j$, where
$$ \tilde{B} = \bigcup_{q \in \partial_{\infty}A \cap \partial_{\infty}( \mathbb{H}^n \backslash A)} \tilde{B}_q .$$
Observe that $\tilde{B}$ is a neighborhood of $\partial_{\infty}A \cap \partial_{\infty}( \mathbb{H}^n \backslash A)$ and $\partial A \backslash \tilde{B}$ is compact.
\
Now we prove that there exists some $w$ defined in $\mathbb{H}^n$ that is the limit of some subsequence $(u_j)$ and is a solution of the minimal surface equation. This function is also not constant and satisfies $\max w = K$ contradicting the maximum principle.
First, remind that we have already noted that $T_j(V_j) \supset A$ and $$dist(\partial T_j(V_j),A) \ge j,$$ which implies that ``$T_j(V_j) \to \mathbb{H}^n$".
This means that any compact set $F \subset \mathbb{H}^n$ is contained in $T_j(V_j)$ for large $j$. Since $|u|$ is bounded by $M$, we have $\sup_F |u_j| \le M$, for large $j$.
In fact, this estimative holds in any neighborhood of $F$.
Hence, from Proposition \ref{lem-gradint},
\begin{equation}
\sup_F |\nabla u_j| \le L,
\label{uniform-boundedness-of-u_j}
\end{equation}
where $L$ is some positive constant that does not depend on $j$.
From Arzel\`a-Ascoli, there is some subsequence of $(u_j)$ that converges uniformly in $F$.
Taking an increasing sequence of compacts sets $F_m$ such that $\bigcup_m F_m =\mathbb{H}^n$ and applying a diagonal argument, we conclude that there exists some subsequence, that we rename by $u_j$, such that $u_j$ converges uniformly in any compact subset of $\mathbb{H}^n$. Let
$$w(x) = \lim_{j\to \infty} u_j(x).$$
From \eqref{uniform-boundedness-of-u_j}, for any bounded open set $U$, we have also that $u_j$ is uniformly bounded in $C^{1}(\overline{U})$. From the linear eliptic PDE theory, $(u_j)$ admits a converging subsequence in $C^{2,\alpha}$ for some $\alpha\in (0,1).$ Let $w$ denote the limit of this subsequence. Again, using a diagonal argument, we have that some subsequence converges to $w$ in $C^{2,\alpha}(\mathbb{H}^n)$. We can denote this subsequence by $u_j$.
Since ``$T^{-1}_j(F) \to \partial_{\infty}\mathbb{H}^n$" as $j \to \infty$ for any compact $F$, condition \ref{propphi1} implies that $f(T^{-1}_j(y), u_j(y)) \to 0$ uniformly in $F$ as $j \to \infty$. Hence, using \eqref{uniform-boundedness-of-u_j} and that $u_j$ converges
to $w$ in $C^2,$
we have that $Q(w)=0$. From the classical theory, the graph of $w$ is a minimal surface. Moreover, from $T_j(V_j) \supset F$ for large $j$ and \eqref{limiteSuperiorWm}, it follows that
$$\sup_{\mathbb{H}^n} w \le K.$$
Now, remind that $y_j=T_j(x_j) \in \partial A$. From \eqref{limiteSuperiorWm} and \eqref{u-jIsSmallerThanDelta}, we conclude that $y_j \in \partial A \backslash \tilde{B}$ if $1/j < K - \delta$. Since $\partial A \backslash \tilde{B}$ is compact, upon passing to a subsequence, $y_j$ converges to some $y \in \partial A \backslash \tilde{B}$. Hence, from \eqref{limiteSuperiorWm} and the fact that $u_j$ converges to $w$ uniformly in compact sets, we have that $w(y) = K$.
Then $y$ is a maximum point of $w$ and, therefore, by the maximum principle, $w$ is constant. However this contradicts that $w(y)=K$ and $w < \delta < K$ in $\tilde{B}$. From this, we conclude that $K \le \delta$ for any $\delta >0$ and, therefore, $K=0$.
By a similar argument, we can prove that $\liminf_{x \to p} u(x) =0$, proving that $\lim_{x \to p}w(x)=0=\varphi(p)$.
\end{proof}
\end{document} |
\begin{document}
\title[Attractors for non-autonomous reaction-diffusion equations]
{Global and cocycle attractors for non-autonomous reaction-diffusion equations. The case of null upper Lyapunov exponent}
\author[T. Caraballo]{Tom\'{a}s Caraballo}
\author[J.A. Langa]{Jos\'{e} A. Langa}
\author[R. Obaya]{Rafael Obaya}
\author[A.M. Sanz]{Ana M. Sanz}
\address[T. Caraballo and J.A. Langa]{Departamento de Ecuaciones Diferenciales y An\'{a}lisis Num\'{e}rico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain} \email{caraball@us.es} \email{langa@us.es}
\address[R. Obaya]{Departamento de Matem\'{a}tica
Aplicada, E. Ingenier\'{\i}as Industriales, Universidad de Valladolid,
47011 Valladolid, Spain, and member of IMUVA, Instituto de Investigaci\'{o}n en
Matem\'{a}ticas, Universidad de Valladolid.}
\email{rafoba@wmatem.eis.uva.es}
\address[A.M. Sanz]{Departamento de Did\'{a}ctica de las Ciencias Experimentales, Sociales y de la Matem\'{a}tica,
Facultad de Educaci\'{o}n, Universidad de Valladolid, 34004 Palencia, Spain,
and member of IMUVA, Instituto de Investigaci\'{o}n en Mate\-m\'{a}\-ti\-cas, Universidad de
Valladolid.} \email{anasan@wmatem.eis.uva.es}
\thanks{R. Obaya and A.M. Sanz were partly supported by MINECO/FEDER
under project MTM2015-66330, and the European Commission under project H2020-MSCA-ITN-2014 643073 CRITICS}
\thanks{T. Caraballo and J.A. Langa were partially supported by Junta de Andaluc\'{\i}a under Proyecto de Excelencia FQM-1492 and FEDER Ministerio de Econom\'{\i}a y Competitividad grant MTM2015-63723-P}
\date{}
\begin{abstract}
In this paper we obtain a detailed description of the global and cocycle attractors for the skew-product semiflows induced by the mild solutions of a family of scalar linear-dissipative parabolic problems over a minimal and uniquely ergodic flow. We consider the case of null upper Lyapunov exponent for the linear part of the problem. Then, two different types of attractors can appear, depending on whether the linear equations have a bounded or an unbounded associated real cocycle. In the first case (e.g.~in periodic equations), the structure of the attractor is simple, whereas in the second case (which occurs in aperiodic equations), the attractor is a pinched set with a complicated structure. We describe situations when the attractor is chaotic in measure in the sense of Li-Yorke. Besides, we obtain a non-autonomous discontinuous pitchfork bifurcation scenario for concave equations, applicable for instance to a linear-dissipative version of the Chafee-Infante equation.
\end{abstract}
\keywords{Non-autonomous dynamical systems; global and cocycle attractors; linear-dissipative PDEs; Li-Yorke chaos in measure; non-autonomous bifurcation theory}
\renewcommand{\mathbb{S}ubjclassname}{\thetaxtup{2010} Mathematics Subject Classification}
\maketitle
\mathbb{S}ection{Introduction}\label{sec-intro}\noindent
In this paper we investigate the dynamical structure of the global and cocycle attractors of the skew-product semiflow generated by a family of scalar linear-dissipative reaction-diffusion equations over a minimal and uniquely ergodic flow, with Neumann or Robin boundary conditions. We assume that the terms involved in the equations satisfy standard regularity assumptions which provide the existence, uniqueness, global definition and continuous dependence of mild solutions with respect to initial conditions.
\par
If $P$ denotes the hull of the time-dependent coefficients of a particular equation, then often the flow defined by time-translation on $P$ is minimal and uniquely ergodic, with a unique ergodic measure $\nu$. These are the hypotheses assumed in this
paper, which in particular includes the case of
almost periodic equations. If $U\mathbb{S}ubset \mathbb{R}^m$ denotes the spatial domain of the equation, the coefficients of the differential equations are continuous functions from $P \times \bar U \times \mathbb{R}$ to $\mathbb{R}$ that can be identified with continuous functions from $P \times C(\bar U)$ to $C(\bar U)$. In this formalism the solutions of the linear-dissipative equations generate a continuous global skew-product semiflow $\tau$ on $P \times C(\bar U)$.
\par
In this work we analyze the structure of the attractors when $\lambda_P$, the upper Lyapunov exponent of the linear part of the linear-dissipative equations, is null and the flow on $P$ is not periodic. We prove that generically the global attractor $\mathbb{A}= \cup_{p\in P} \{p\}\times A(p)$ is a pinched compact set with ingredients of dynamical complexity like sensitive dependence in relevant subsets of this compact set (see Glasner and Weiss~\cite{glwe}). We establish conditions on the coefficient of the linear equations that provide nontrivial sections $A(p)$ for the elements $p$ in an invariant subset $P_f$ of $P$ with complete measure and prove that, in this case, the restriction of the flow on the global attractor $(\mathbb{A}, \tau)$ is chaotic in the sense of Li-Yorke. We can understand these results in the framework of non-autonomous bifurcation theory as a discontinuous pitchfork bifurcation of minimal sets.
\par
We next describe the structure and main results of the paper. Section~\ref{sec-preli} contains some basic facts in non-autonomous dynamical systems which will be required in the rest of the paper.
\par
In Section \ref{sec-mild solutions} we review the construction of the skew-product semiflow induced by the mild solutions of a very general family of parabolic partial differential equations (PDEs for short) over a minimal flow $(P,\theta,\mathbb{R})$ just denoted by $\theta_tp=p{\cdot}t$. We also state a result on comparison of solutions and the strong monotonicity of the semiflow.
\par
Section \ref{sec-linear} is devoted to the study of families of scalar parabolic linear PDEs $\partial y /\partial t = \Delta \, y+h(p{\cdot}t,x)\,y$, $t>0$, $x\in U$ for each $p\in P$ ($P$ a minimal and uniquely ergodic flow) with Neumann or Robin boundary conditions. Mild solutions generate a linear skew-product semiflow $\tau_L$ on $P \times C(\bar U)$ which is strongly monotone and hence admits a continuous separation $C(\bar U)= X_1(p) \oplus X_2(p)$, for every $p \in P$, in the terms stated in Pol\'{a}\v{c}ik and Tere\v{s}\v{c}\'{a}k~\cite{pote} and Shen and Yi~\cite{shyi}. The restriction of $\tau_L$ to the principal bundle $\cup_{p\in P}\{p\}\times X_1(p)$ generates a continuous 1-dim linear cocycle $c(t,p)$ whose Lyapunov exponents match the upper Lyapunov exponent $\lambda_P$. We prove the continuous dependence of $\lambda_P(h)$ on $h$, the coefficient in the equations. Consequently, the set $C_0(P \times\bar U)= \{h \in C(P \times \bar U) \mid \lambda_P(h)=0\}$ is closed. We denote by $B(P\times \bar U)$ the subset of $C_0(P \times \bar U)$ formed by the functions $h$ with an associated coboundary cocycle $\ln c(t,p)$, that is, there is a $k \in C(P)$ such that $\ln c(t,p)= k(p{\cdot}t)-k(p)$ for any $p\in P$ and $t\in\mathbb{R}$.
\par
We show, on the one hand, that if the coefficient $h$ is in $B(P\times \bar U)$, then $P \times \Int C_+(\bar U)$ contains minimal sets that are copies of the base $P$, all the nontrivial positive solutions are strongly positive, remain uniformly away from 0 and bounded above and eventually approximate solutions with the same recurrence in time as that of the initial problem; for instance, they are asymptotically almost periodic if the base flow is almost periodic. On the other hand, if the linear coefficient $h$ is in $\mathcal{U}(P \times \bar U)=C_0(P \times \bar U)\mathbb{S}etminus B(P \times \bar U)$, then $P \times C_+(\bar U)$ contains pinched compact invariant sets, all the nontrivial positive solutions are strongly positive and for all the equations given by $p$ in a residual subset of P their modulus oscillates from $0$ to $\infty$ as time goes to $\infty$. In addition, when the base flow on $P$ is aperiodic, we deduce that $\mathcal{U}(P \times \bar U)$ is a residual subset of $C_0(P \times \bar U)$ as its complementary set $B(P \times\bar U)$ is a dense subset of first category. The above arguments allow us to show that $\lambda_P(h)$ has a strictly convex variation on $h$, which becomes linear in the trivial case when $h_2-h_1$, the difference of the coefficients involved in the convex combination, only depends on $p \in P$.
\par
In Section \ref{sec-nonlinear} we study the behaviour of the solutions of a family of scalar linear-dissipative reaction-diffusion equations $\partial y/\partial t = \Delta \, y+h(p{\cdot}t,x)\,y+g(p{\cdot}t,x,y)$, $t>0$, $x\in U$, for each $p\in P$, and the dynamical properties of the induced skew-product semiflow $\tau$ on $P \times C(\bar U)$. Standard arguments taken from Caraballo and Han~\cite{caha} or Carvalho et al.~\cite{calaro} allow us to deduce the existence of a global attractor $\mathbb{A}= \cup_{p\in P} \{p\}\times A(p)$, and thus $\{A(p)\}_{p \in P} $ defines the cocycle attractor of the continuous skew-product semiflow. In addition, for each $p \in P$ the invariant family of compact sets $\{A(p{\cdot}t)\}_{t \in \mathbb{R}}$ provides the pullback atractor of the process generated by the solutions of the parabolic equation obtained by evaluation of the coefficients along the trajectory of $p$.
\par
The structure of the global and cocycle attractors in the case that $\lambda_P$, the upper Lyapunov exponent of the linear part of the reaction-diffusion equations, is different from zero has been investigated in Cardoso et al.~\cite{cardoso}. In this work we study the same problem when $\lambda_P=0$ to show that these attractors exhibit a rich dynamics that frequently contains ingredients of high complexity. The global attractor has upper and lower boundaries given by the graphs of two semicontinuous functions $a$ and $b$. For simplicity we asume that the coefficients of the equation are odd with respect to the dependent variable $y$, which implies that $a=-b$.
\par
More precisely, if $h$, the coefficient of the linear part, is in $B(P \times \bar U)$, then $b$ is continuous and strongly positive and the global atractor is included in the principal bundle, whereas if $h$ is in $\mathcal{U} (P \times \bar U)$, then there is a residual invariant subset $P_{\rm s} \mathbb{S}ubset P$ such that $b(p)=0$ for every $p \in P_{\rm s}$ and $P_{\rm f}=P\mathbb{S}etminus P_{\rm s}$ is a dense invariant subset of first category with $b(p)\gg 0$ for every $p \in P_{\rm f}$. We prove that $p \in P_{\rm s}$ if and only if $\mathbb{S}up_{t\leq 0} c(t,p)= \infty$ and conversely $p \in P_{\rm f}$ if and only if $\mathbb{S}up_{ t\leq 0} c(t,p) < \infty$. Later we describe precise examples of functions $h$ such that $\nu (P_{\rm f})=1$ and prove that in this case the restriction of the equations on the section $A(p)$ of the attractor is linear for almost every $p \in P$ and the flow $(\mathbb{A}, \tau)$ is fiber-chaotic in measure in the sense of Li-Yorke. In consequence, the main results on the structure and properties of the attractors obtained in Caraballo et al.~\cite{caloNonl} for scalar almost periodic linear-dissipative ordinary differential equations (ODEs for short) remain valid for the class of reaction-diffusion models here considered.
\par
Finally, we introduce a parameter in the equations and analyze the evolution of the structure of the global attractor when the upper Lyapunov exponent of the linear part crosses through zero. Assuming that the nonlinear term is concave and using results by N\'{u}\~{n}ez et al.~\cite{nuos4} we show that this transition provides a discontinuous bifurcation of attractors and describes a discontinuous pitchfork bifurcation diagram for the minimal sets. The results in this work offer a dynamical description of the often complicated structure of the global attractor at the bifurcation point.
\mathbb{S}ection{Basic notions}\label{sec-preli}\noindent
In this section we include some preliminaries about
topological dynamics for non-autonomous dynamical systems.
\par
Let $(P,d)$ be a compact
metric space. A real {\em continuous flow\/} $(P,\theta,\mathbb{R})$ is
defined by a continuous map $\theta: \mathbb{R}\times P \to P,\;
(t,p)\mapsto \theta(t,p)=\theta_t(p)=p{\cdot}t$ satisfying
\begin{enumerate}
\renewcommand{(\roman{enumi})}{(\roman{enumi})}
\item $\theta_0=\thetaxt{Id},$
\item $\theta_{t+s}=\theta_t\circ\theta_s$ for each $s$, $t\in\mathbb{R}$\,.
\end{enumerate}
The set $\{ \theta_t(p) \mid t\in\mathbb{R}\}$ is called the {\em orbit\/}
of the point $p$. We say that a subset $P_1\mathbb{S}ubset P$ is {\em
$\theta$-invariant\/} if $\theta_t(P_1)=P_1$ for every $t\in\mathbb{R}$.
The flow $(P,\theta,\mathbb{R})$ is called {\em minimal\/} if it does not contain properly any other
compact $\theta$-invariant set, or equivalently, if every
orbit is dense. The flow is {\em distal\/} if the orbits of any two distinct
points $p_1,\,p_2\in P$ keep at a positive distance,
that is, $\inf_{t\in \mathbb{R}}d(\theta(t,p_1),\theta(t,p_2))>0$; and it is {\em almost periodic\/} if the family of maps $\{\theta_t\}_{t\in \mathbb{R}}:P\to P$ is uniformly equicontinuous. An almost periodic flow is always distal.
\par
A finite regular measure defined on the Borel sets of $P$ is called
a Borel measure on $P$. Given $\mu$ a normalized Borel measure on
$P$, it is {\em $\theta$-invariant\/} if $\mu(\theta_t(P_1))=\mu(P_1)$ for every Borel subset
$P_1\mathbb{S}ubset P$ and every $t\in \mathbb{R}$. It is {\em ergodic\/} if, in
addition, $\mu(P_1)=0$ or $\mu(P_1)=1$ for every
$\theta$-invariant subset $P_1\mathbb{S}ubset P$.
We denote by $\mathcal{M}(P)$ the set of
all positive and normalized $\theta$-invariant measures on $P$. This set is nonempty by the
Krylov\nobreakdash-Bogoliubov theorem when $P$ is a
compact metric space. We
say that $(P,\theta,\mathbb{R})$ is {\em uniquely ergodic\/} if it has a
unique normalized invariant measure, which is then necessarily
ergodic. A minimal and almost periodic flow $(P,\theta,\mathbb{R})$ is uniquely ergodic.
\par
A standard method to, roughly speaking, get rid of the time variation in a non-autonomous equation and build a non-autonomous dynamical system, is the so-called {\em hull\/} construction. More precisely, a function $f\in C(\mathbb{R}\times\mathbb{R}^m)$ is said to be {\em admissible\/} if for any
compact set $K\mathbb{S}ubset \mathbb{R}^m$, $f$ is bounded and uniformly continuous
on $\mathbb{R}\times K$. Provided that $f$ is admissible, its {\em hull\/} $P$ is the closure for the compact-open topology of the set of $t$-translates of $f$, $\{ f_t \mid t\in\mathbb{R}\}$ with $f_t(s,x)=f(t+s,x)$
for $s\in \mathbb{R}$ and $x\in\mathbb{R}^m$. The translation map $\mathbb{R}\times P\to P$,
$(t,p)\mapsto p{\cdot}t$ given by $p{\cdot}t(s,x)= p(s+t,x)$ ($s\in \mathbb{R}$ and $x\in\mathbb{R}^m$) defines a
continuous flow on the compact metric space $P$. This flow is minimal as far as the map $f$ has certain recurrent behaviour in time, such as periodicity, almost periodicity, or other weaker properties of recurrence. If the map $f(t,x)$ is uniformly almost periodic (that is, it is admissible and almost periodic in $t$ for any fixed $x$), then the flow on the hull is minimal and almost periodic. It is relevant to note that any minimal and uniquely ergodic flow which is not almost periodic is sensitive with respect to initial conditions (see Glasner and Weiss~\cite{glwe}).
\par
Let $\mathbb{R}_+=\{t\in\mathbb{R}\,|\,t\geq 0\}$. Given a continuous compact flow $(P,\theta,\mathbb{R})$ and a
complete metric space $(X,\textsf{d})$, a continuous {\em skew-product semiflow\/} $(P\times
X,\tau,\,\mathbb{R}_+)$ on the product space $P\times X$ is determined by a continuous map
\begin{equation*}
\begin{array}{cccl}
\tau \colon &\mathbb{R}_+\times P\times X& \longrightarrow & P\times X \\
& (t,p,x) & \mapsto &(p{\cdot}t,u(t,p,x))
\end{array}
\end{equation*}
which preserves the flow on $P$, referred to as the {\em base flow\/}.
The semiflow property means that
\begin{enumerate}
\renewcommand{(\roman{enumi})}{(\roman{enumi})}
\item $\tau_0=\thetaxt{Id},$
\item $\tau_{t+s}=\tau_t \circ \tau_s\;$ for all $\; t$, $s\geq 0\,,$
\end{enumerate}
where again $\tau_t(p,x)=\tau(t,p,x)$ for each $(p,x) \in P\times X$ and $t\in \mathbb{R}_+$.
This leads to the so-called semicocycle property,
\begin{equation*}
u(t+s,p,x)=u(t,p{\cdot}s,u(s,p,x))\quad\mbox{for $s,t\ge 0$ and $(p,x)\in P\times X$}\,.
\end{equation*}
\par
The set $\{ \tau(t,p,x)\mid t\geq 0\}$ is the {\em semiorbit\/} of
the point $(p,x)$. A subset $K$ of $P\times X$ is {\em positively
invariant\/} if $\tau_t(K)\mathbb{S}ubseteq K$
for all $t\geq 0$ and it is $\tau$-{\em invariant\/} if $\tau_t(K)= K$
for all $t\geq 0$. A compact $\tau$-invariant set $K$ for the
semiflow is {\em minimal\/} if it does not contain any nonempty
compact $\tau$-invariant set other than itself.
\par
A compact $\tau$-invariant set $K\mathbb{S}ubset P\times X$ is called a {\em pinched\/} set if there exists a residual set $P_0\mathbb{S}ubsetneq P$ such that for every $p\in P_0$ there is a unique element in $K$ with $p$ in the first component, whereas there are more than one if $p\notin P_0$.
\par
The reader can find in Ellis~\cite{elli}, Sacker and
Sell~\cite{sase}, Shen and Yi~\cite{shyi} and references therein, a
more in-depth survey on topological dynamics.
\par
We now state the definitions of global attractor and cocycle attractor for skew-product semiflows. The books by Caraballo and Han~\cite{caha}, Carvalho et al.~\cite{calaro} and Kloeden and Rasmussen~\cite{klra} are good references for this topic.
\par
We say that the skew-product semiflow $\tau$ has a {\em global attractor\/} if there exists an invariant compact set attracting bounded sets forwards in time; more precisely, if there is a compact set $\mathbb{A}\mathbb{S}ubset P\times X$ such that $\tau_t(\mathbb{A}) = \mathbb{A}$ for any $t\geq 0$ and $\lim_{t\to\infty} {\rm dist}(\tau_t(\mathbb{B}),\mathbb{A})=0$ for any bounded set $\mathbb{B}\mathbb{S}ubset P\times X$, for the semi-Hausdorff distance.
\par
A {\em non-autonomous set\/} is a family $\{A(p)\}_{p\in P}$ of subsets of $X$ indexed by $p\in P$. It is said to be {\em compact\/} provided that $A(p)$ is a compact set in $X$ for every $p\in P$; and it is said to be {\em invariant\/} if for every $p\in P$, $u(t,p,A(p))=A(p{\cdot}t)$ for any $t\geq 0$. A compact invariant non-autonomous set $\{A(p)\}_{p\in P}$ is called a {\em cocycle attractor\/} for the skew-product semiflow $\tau$ if it pullback attracts all bounded subsets $B\mathbb{S}ubset X$, that is, for any $p\in P$,
\[
\lim_{t\to\infty} {\rm dist}(u(t,p{\cdot}(-t),B),A(p))=0\,.
\]
It is well-known (see \cite{klra}) that, with $P$ compact, if $\mathbb{A}$ is a global attractor for $\tau$, then $\{A(p)\}_{p\in P}$, with $A(p)=\{x\in X\mid (p,x)\in \mathbb{A}\}$ for each $p\in P$, is a cocycle attractor.
\par
To finish, we include some basic notions on monotone skew-product semiflows. When the state space $X$ is a strongly ordered Banach space, that is, there is a closed convex solid cone of nonnegative vectors $X_+$ with a nonempty interior, then, a (partial) {\em strong order relation\/} on $X$ is
defined by
\begin{equation}\label{order}
\begin{split}
x\le y \quad &\Longleftrightarrow \quad y-x\in X_+\,;\\
x< y \quad &\Longleftrightarrow \quad y-x\in X_+\;\thetaxt{ and }\;x\ne y\,;
\\ x\ll y \quad &\Longleftrightarrow \quad y-x\in \Int X_+\,.\qquad\quad\quad~
\end{split}
\end{equation}
In this situation, the skew-product semiflow $\tau$
is {\em monotone\/} if
\begin{equation*}
u(t,p,x)\le u(t,p,y)\,\quad \thetaxt{for\, $t\ge 0$\,,\, $p\in P$ \,and\,
$x,y\in X$ \,with\, $x\le y$}\,.
\end{equation*}
\par
A Borel map $a:P\to X$ such that $u(t,p,a(p))$ exists for any $t\geq 0$ is said to be
\begin{itemize}
\item[(a)] an {\em equilibrium\/} if $a(p{\cdot}t)=u(t,p,a(p))$ for any $p\in P$ and $t\geq 0$;
\item[(b)] a {\em sub-equilibrium\/} if $a(p{\cdot}t)\leq u(t,p,a(p))$ for any $p\in P$ and $t\geq 0$;
\item[(c)] a {\em super-equilibrium\/} if $a(p{\cdot}t)\geq u(t,p,a(p))$ for any $p\in P$ and $t\geq 0$.
\end{itemize}
A super-equilibrium (resp.~sub-equilibrium) $a:P\to X$ is {\em strong\/} if for some $t_*>0$, $a(p{\cdot}t_*)\gg u(t_*,p,a(p))$ (resp.~$\ll$) for every $p\in P$. The study of semicontinuity properties of these maps and other related issues can be found in Novo et al.~\cite{nono2}.
\mathbb{S}ection{Skew-product semiflow induced by scalar parabolic PDEs}\label{sec-mild solutions}\noindent
Let us consider a family of scalar parabolic PDEs over a minimal flow $(P,\theta,\mathbb{R})$, with Neumann or Robin boundary
conditions
\begin{equation}\label{family}
\left\{\begin{array}{l} \displaystyle\frac{\partial y}{\partial t} =
\Delta \, y+f(p{\cdot}t,x,y)\,,\quad t>0\,,\;\,x\in U, \;\, \thetaxt{for each}\; p\in P,\\[.2cm]
By:=\alpha(x)\,y+\displaystyle\frac{\partial y}{\partial n} =0\,,\quad t>0\,,\;\,x\in \partial U,\,
\end{array}\right.
\end{equation}
where $p{\cdot}t$ denotes the flow on $P$; $U$, the spatial domain, is a bounded, open and
connected subset of $\mathbb{R}^m$ ($m\geq 1$) with a sufficiently smooth boundary
$\partial U$; $\Delta$ is the Laplacian operator on $\mathbb{R}^m$;
$f$ satisfies the following hypothesis:
\begin{itemize}
\item[(H)] $f\colon P\times \bar U\times \mathbb{R}\to\mathbb{R}$ is continuous and is Lipschitz in $y$ in bounded sets, uniformly for $p\in P$ and $x\in\bar U$, that is, given any $R>0$ there exists an $L_R>0$ such that
\[
|f(p,x,y_2)-f(p,x,y_1)|\leq L_R\,|y_2-y_1|
\]
for any $p\in P$, $x\in\bar U$ and $y_1,\,y_2\in\mathbb{R}$ with $|y_1|,\,|y_2|\leq R$\,;
\end{itemize}
$\partial/\partial n$ denotes the
outward normal
derivative at the boundary; and
$\alpha:\partial U\to \mathbb{R}$ is a nonnegative sufficiently regular function.
\par
In order to immerse the initial boundary value problem (IBV problem for short) associated with the parabolic problem~\eqref{family} into an abstract Cauchy problem (ACP for short), we consider the strongly ordered Banach space $X=C(\bar U)$ of the continuous functions on $\bar U$ with the sup-norm $\|\,{\cdot}\,\|$, and positive cone $X_+=\{z\in X\mid z(x)\geq 0 \;\forall x\in \bar U\}$ with nonempty interior $\Int X_+=\{z\in X\mid z(x)> 0 \;\forall x\in \bar U\}$, which induces a (partial) strong ordering in $X$ as in~\eqref{order}. Note that $X$ is also a Banach algebra for the usual product $(z_1\,z_2)(x)=z_1(x)\,z_2(x)$ for $z_1,\,z_2\in X$ and $x\in \bar U$.
\par
Now, following Smith~\cite{smit}, let $A$ be the closure of the differential operator
$A_0\colon D(A_0)\mathbb{S}ubset X\to X$, $A_0z=\Delta\,z$, defined on
\[
D(A_0)=\{z \in C^2(U)\cap C^1(\bar U)\;\mid\;A_0z\in C(\bar U),\;
Bz=0 \,\hbox{ on }\partial U\}\,.
\]
The operator $A$ is sectorial and it generates
an analytic compact semigroup of operators $\{T(t)\}_{t\geq 0}$ on
$X$ which is strongly continuous (that is, $A$ is densely defined).
\par
If we define $\tilde f:P\times X\to X$, $(p,z)\mapsto \tilde f(p,z)$, $\tilde f(p,z)(x)=f(p,x,z(x))$, $x\in \bar U$, the regularity conditions (H) on $f$ are transferred to $\tilde f$. This leads to the continuity of $\tilde f$, and Lipschitz continuity with respect to $z$ on any bounded set of $X$ with Lipschitz constant independent of $p$, that is, given any bounded set $B\mathbb{S}ubset X$, there exists an $L_B>0$ such that
\[
\|\tilde f(p,z_2)-\tilde f(p,z_1)\|\leq L_B\,\|z_2-z_1\|\quad\thetaxt{for any}\,\; p\in P,\; z_1,\,z_2\in B\,.
\]
With the former conditions on $A$ and these conditions on $\tilde f$, when
we consider the ACP given
for each fixed $p\in P$ and $z\in X$ by
\begin{equation}\label{acpnl}
\left\{\begin{array}{l} u'(t) =
A\, u(t)+\tilde f(p{\cdot}t,u(t))\,,\quad t>0\,,\\
u(0)=z\,,
\end{array}\right.
\end{equation}
this problem has a unique {\it mild solution\/}, that is, there exists a unique continuous map $u(t)=u(t,p,z)$ defined on a maximal interval $[0,\beta)$ for some $\beta=\beta(p,z)>0$ (possibly $\infty$) which satisfies the integral equation
\begin{equation*}
u(t)=T(t)\,z +\int_0^t T(t-s)\,
\tilde f(p{\cdot}s,u(s))\,ds\,,\quad t\in [0,\beta)\,.
\end{equation*}
(For instance, see Travis and Webb~\cite{trwe} or Hino et al.~\cite{hnms}.)
Mild solutions allow us to locally define a continuous
skew-product semiflow
\begin{equation*}
\begin{array}{cccl}
\tau: &\mathcal{U} \mathbb{S}ubseteq\mathbb{R}_+\times P\times X& \longrightarrow & \hspace{0.3cm}P\times X\\
& (t,p,z) & \mapsto
&(p{\cdot}t,u(t,p,z))\,,
\end{array}
\end{equation*}
for an appropriate open set $\mathcal{U}$. Besides, if a solution $u(t,p,z)$ remains bounded, then it is defined on the whole positive real line and the semiorbit of $(p,z)$ is relatively compact (see Proposition~2.4 in~\cite{trwe}, where the compactness of the operators $T(t)$ for $t>0$ is crucial).
\par
Note that the linear family
\begin{equation}\label{pdefamily}
\left\{\begin{array}{l} \displaystyle\frac{\partial y}{\partial t} =
\Delta \, y+h(p{\cdot}t,x)\,y\,,\quad t>0\,,\;\,x\in U, \;\, \thetaxt{for each}\; p\in P,\\[.2cm]
By:=\alpha(x)\,y+\displaystyle\frac{\partial y}{\partial n} =0\,,\quad t>0\,,\;\,x\in \partial U,\,
\end{array}\right.
\end{equation}
with $h:P\times\bar U\to\mathbb{R}$ a continuous map, is included in the general setting of \eqref{family}. In this case, $\tilde h:P\to X$, $p\mapsto \tilde h(p)$, $\tilde h(p)(x)=h(p,x)$, $x\in \bar U$ is continuous and bounded. In the associated linear ACP given for each $p\in P$ and $z\in X$ by
\begin{equation}\label{acplineal}
\left\{\begin{array}{l} v'(t) =
A\, v(t)+\tilde h(p{\cdot}t)\,v(t)\,,\quad t>0\,,\\
v(0)=z\,,
\end{array}\right.
\end{equation}
there appears the term $\tilde f(p,z)=\tilde h(p)\,z$, for $p\in P$, $z\in X$ which is globally Lipschitz continuous with respect to $z$, uniformly for $p\in P$. This implies that the mild solutions $v(t)=v(t,p,z)$, which in this linear case are solutions of the integral equations
\begin{equation*}
v(t)=T(t)\,z +\int_0^t T(t-s)\,
\tilde h(p{\cdot}s)\,v(s)\,ds\,,\quad t\geq 0\,,
\end{equation*}
allow us to define a globally defined continuous linear
skew-product semiflow
\begin{equation*}
\begin{array}{cccl}
\tau_L: & \mathbb{R}_+\times P\times X& \longrightarrow & \hspace{0.3cm}P\times X\\
& (t,p,z) & \mapsto
&(p{\cdot}t,\phi(t,p)\,z)\,,
\end{array}
\end{equation*}
where $\phi(t,p)\,z=v(t,p,z)$. In particular $\phi(t,p)$ are bounded operators on $X$ which are compact for $t>0$ and satisfy the linear semicocycle property $\phi(t+s,p)=\phi(t,p{\cdot}s)\,\phi(s,p)$, $p\in P$, $t,s\geq 0$. As before, bounded trajectories are relatively compact.
\par
Under additional regularity conditions in the nonlinear term $f(p{\cdot}t,x,y)$, such as a Lipschitz condition with respect to $t$ and H\"{o}lder-continuity with respect to $x$, mild solutions are known to generate classical solutions; namely, $y(t,x)=u(t,p,z)(x)$, $t\in [0,\beta(p,z))$, $x\in \bar U$ is a classical solution of the IBV problem given by~\eqref{family} for $p\in P$ with initial condition at time $t=0$, $y(0,x)=z(x)$, $x\in \bar U$, meaning that the corresponding partial derivatives exist, are continuous
and satisfy the corresponding equation in~\eqref{family} as well as the boundary conditions (see Smith~\cite{smit} and
Friedman~\cite{frie}).
\par
We next state a result of comparison of solutions which will be used through the paper, and the strong monotonicity of the semiflow.
\begin{teor}\label{teor-comparacion}
Let $f_1$ and $f_2$ satisfy hypothesis $\rm{(H)}$ and be such that $f_1\leq f_2$. For each $p\in P$ and $z\in X$, denote by $u_1(t,p,z)$ and $u_2(t,p,z)$ the mild solutions of the associated ACPs~\eqref{acpnl}, respectively. Then, $u_1(t,p,z)\leq u_2(t,p,z)$ for any $t\geq 0$ where both solutions are defined.
\end{teor}
\begin{proof}
Let us fix a $p\in P$ and a $z\in X$ and let $t_0>0$ be such that both $u_1(t_0,p,z)$ and $u_2(t_0,p,z)$ exist. The idea is to approximate the equations given by $f_1$ and $f_2$ by a sequence of equations to which the standard comparison of solutions result applies, and whose solutions approximate the mild solutions of the initial problems. \par
Let $R=\mathbb{S}up\{\|u_1(t,p,z)\|,\,\|u_2(t,p,z)\|\mid t\in [0,t_0]\}<\infty$. First, we apply Tietze's extension theorem to the continuous map
\[
g_i:[0,t_0]\times\bar U\times [-R,R]\to \mathbb{R}\,,\quad (t,x,y)\mapsto f_i(p{\cdot}t,x,y)
\]
for $i=1,2$. Thus there exist continuous maps $F_i:\mathbb{R}\times\mathbb{R}^m\times \mathbb{R}\to \mathbb{R}$ ($i=1,2$) with compact support such that the restriction $F_i |_{[0,t_0]\times\bar U\times [-R,R]}\equiv g_i$ and $\|F_i\| = \|g_i\|_{[0,t_0]\times\bar U\times [-R,R]}$. Now, for $i=1,2$, we apply to $F_i$ the regularization process used in the construction of solutions of the heat equation, using the convolution with the so-called {\em Gauss kernel\/}; namely, the maps defined on $\mathbb{R}\times \mathbb{R}^m\times \mathbb{R}$,
\[
F_{i,n}(t,x,y)=\left(\frac{n}{4\pi}\right)^{\!\frac{m+2}{2}}\int_{\mathbb{R}\times\mathbb{R}^m\times\mathbb{R}} e^{\frac{-n\,\|(t,x,y)-(\omegait t,\omegait x,\omegait y)\|^2}{4}}F_i(\omegait t,\omegait x,\omegait y)\,d\omegait t\,d\omegait x\,d\omegait y\,,\quad \;n\geq 1
\]
satisfy:
\begin{itemize}
\item[(i)] $F_{i,n}(t,x,y)$ is of class $C^\infty$ with respect to $t$, $x$ and $y$;
\item[(ii)] $\displaystyle\lim_{n\to\infty} F_{i,n}(t,x,y)=F_i(t,x,y)$ uniformly;
\item[(iii)] $|F_{i,n}(t,x,y)|\leq \|F_i\|$ for any $(t,x,y)\in \mathbb{R}\times\mathbb{R}^m\times\mathbb{R}$.
\end{itemize}
\par
At this point, for $i=1,2$ and for each $n\geq 1$ we denote by $u_{i,n}(t,p,z)$ the mild solution of the ACP for $p$ and $z$ given by
\begin{equation*}
\left\{\begin{array}{l} u'(t) =
A\, u(t)+\omegaidetilde F_{i,n}(t,u(t))\,,\quad t>0\,,\\
u(0)=z\,,
\end{array}\right.
\end{equation*}
where $\omegaidetilde F_{i,n}(t,u)(x)=F_{i,n}(t,x,u(x))$, $x\in \bar U$. Thanks to (iii), $u_{i,n}(t,p,z)$ is defined on $[0,t_0]$ for $n\geq 1$, and we affirm that $u_{i,n}(t,p,z)\to u_i(t,p,z)$ uniformly for $t\in [0,t_0]$ as $n\to\infty$. To see it, follow the argumentation in the proof of Proposition~3.2 in Novo et al.~\cite{nonuobsa} (inspired in the proof of Proposition~2.4 in Travis and Webb~\cite{trwe}).
\par
To finish, note that we can also assume that $F_{1,n}\leq F_1 \leq F_2\leq F_{2,n}$ for any $n\geq 1$ and besides, with the regularity conditions we have on $F_{i,n}$, the mild solutions of the ACPs give rise to classical solutions of the associated IBV problems. Therefore, the standard result of comparison of solutions says that $u_{1,n}(t,p,z)\leq u_{2,n}(t,p,z)$ for $t\in [0,t_0]$ for every $n\geq 1$. Therefore, taking limits, we finally obtain that $u_1(t,p,z)\leq u_2(t,p,z)$ for $t\in [0,t_0]$, as desired.
\end{proof}
\begin{prop}\label{prop-strong monot lineal}
Consider the linear problem \eqref{pdefamily} with $h:P\times\bar U\to\mathbb{R}$ continuous. Then, the induced linear skew-product semiflow $\tau_L$ is strongly monotone, that is, for any $p\in P$, $\phi(t,p)\,z\gg 0$ whenever $z>0$, for any $t>0$.
\end{prop}
\begin{proof}
We just consider a regular map $h_1:P\times\bar U\to\mathbb{R}$ with $h_1\leq h$. The linear skew-product semiflow associated to the regular map $h_1$ is well-known to be strongly monotone, so that the result follows by comparison applying Theorem~\ref{teor-comparacion}.
\end{proof}
To finish this section, in the nonlinear case we deduce the strong monotonicity of the induced skew-product semiflow $\tau$ by linearizing, provided that the nonlinear term is of class $C^1$ in the $y$ variable.
\begin{prop}\label{prop-strong monot}
Consider the nonlinear problem \eqref{family} with $f:P\times\bar U\times \mathbb{R}\to\mathbb{R}$ continuous and of class $C^1$ in the $y$ variable. Then, the induced skew-product semiflow is strongly monotone, that is, for any $p\in P$, $u(t,p,z_2)\gg u(t,p,z_1)$ whenever $z_2>z_1$, for any $t>0$ where both terms are defined.
\end{prop}
\begin{proof}
First of all, we define the map $\frac{\omegaidetilde {\partial f}}{\partial y}:P\times X \to X$, $(p,z)\mapsto \frac{\omegaidetilde {\partial f}}{\partial y}(p,z)$, $\frac{\omegaidetilde {\partial f}}{\partial y}(p,z)(x)=\frac{\partial f}{\partial y}(p,x,z(x))$, $x\in\bar U$, which is continuous. Then, given a pair $(p,z)\in P\times X$, we consider the associated variational ACP along the trajectory of $(p,z)$ with initial value $z_0\in X$, for $t>0$ as long as $\tau(t,p,z)$ exists:
\begin{equation*}
\left\{\begin{array}{l} v'(t) =
A\, v(t)+\displaystyle\frac{\omegaidetilde {\partial f}}{\partial y}(\tau(t,p,z))\,v(t)\,,\\
v(0)=z_0\,.
\end{array}\right.
\end{equation*}
Denoting by $v(t,p,z,z_0)$ the mild solution to this problem, we follow the argumentation in the proof of Theorem~3.5 in Novo et al.~\cite{nonuobsa} to see that $D_zu(t,p,z)\,z_0$ exists and $D_zu(t,p,z)\,z_0=v(t,p,z,z_0)$. Besides, the map $P\times X\to \mathcal{L}(X)$, $(p,z)\mapsto D_zu(t,p,z)$ is continuous for any $t>0$ in the interval of definition of $u(t,p,z)$.
\par
To finish the proof, given a $T>0$ such that $u(t,p,z_1)$ and $u(t,p,z_2)$ are defined on $[0,T]$, we can assume without loss of generality that also
$u(t,p,\lambda\,z_2+(1-\lambda)\,z_1)$ is defined on $[0,T]$ for any $\lambda\in(0,1)$, and then just write for $z_1< z_2$,
\begin{equation*}
u(t,p,z_2)-u(t,p,z_1)=\int_0^1 D_z u(t,p,\lambda\,z_2+(1-\lambda)\,z_1)(z_2-z_1)\,d\lambda\,.
\end{equation*}
\par
By applying Proposition~\ref{prop-strong monot lineal} to the mild solutions of the variational linear ACPs, we get the nonnegativity of the integrand. Since $z_1<z_2$, at $\lambda=0$ (for instance) apply the strong monotonicity, so that $D_z u(t,p,z_1)(z_2-z_1)\gg 0$, and this, together with the continuity of the integrand, is enough to conclude the proof.
\end{proof}
\mathbb{S}ection{Scalar linear parabolic PDEs with null upper Lyapunov exponent}\label{sec-linear}\noindent
In this section we concentrate on the linear case. Let us consider a family \eqref{pdefamily} of scalar linear parabolic PDEs over a minimal flow $(P,\theta,\mathbb{R})$, with Neumann or Robin boundary
conditions:
\begin{equation*}
\left\{\begin{array}{l} \displaystyle\frac{\partial y}{\partial t} =
\Delta \, y+h(p{\cdot}t,x)\,y\,,\quad t>0\,,\;\,x\in U, \;\, \thetaxt{for each}\; p\in P,\\[.2cm]
By:=\alpha(x)\,y+\displaystyle\frac{\partial y}{\partial n} =0\,,\quad t>0\,,\;\,x\in \partial U,\,
\end{array}\right.
\end{equation*}
with $h\in C(P\times \bar U)$, the Banach space of the continuous real maps defined on $P\times \bar U$. We keep the notation introduced in the previous section; in particular, $\tau_L$ is the globally defined linear skew-product semiflow given by the mild solutions of the associated ACPs, determined by the compact (for $t>0$) linear operators $\phi(t,p)\in \mathcal{L}(X)$. Recall also that $\tau_L$ is strongly monotone: see Proposition~\ref{prop-strong monot lineal}.
\par
The {\em Sacker-Sell spectrum\/} (or {\em continuous spectrum\/}; see Sacker and Sell~\cite{sase94}) of $\tau_L$ is the set
\[
\Sigma=\{\lambda\in \mathbb{R}\mid \tau_L^\lambda \thetaxt{ has no exponential dichotomy} \},
\]
where $\tau_L^\lambda$ denotes the linear skew-product semiflow $\tau_L^\lambda(t,p,z)=(p{\cdot}t,e^{-\lambda t}\phi(t,p)\,z)$ on $P\times X$. The {\em upper Lyapunov exponent\/} of $\tau_L$ is defined as $\lambda_P=\mathbb{S}up_{p\in P} \lambda(p)$, where $\lambda(p)$ is the Lyapunov exponent given by
\begin{equation}\label{lyap exp}
\lambda(p)=\limsup_{t\to \infty} \frac{\ln\|\phi(t,p)\|}{t}=\limsup_{t\to \infty} \frac{\ln\|\phi(t,p)\,z\|}{t}\,
\end{equation}
for any $z\gg 0$; since for a given $z\gg 0$ there exists an $l=l(z)>0$ such that
$\|\phi(t,p)\|\leq l\,\|\phi(t,p)\,z\|$ for any $t>0$ and
$p\in P$.
It is well-known that $\lambda_P=\mathbb{S}up \Sigma <\infty$ (see Shen and Yi~\cite{shyi} and Chow and Leiva~\cite{chle}
for further details).
\par
To emphasize the dependance of $\lambda_P$ on the coefficient $h$, we will write $\lambda_P(h)$. In particular, for $h=0$, the problem is autonomous and the solution semiflow is given by the semigroup $\{T(t)\}_{t\geq 0}$. Since $\|T(t)\|\leq 1$ for any $t\geq 0$ (see Smith~\cite{smit}), it follows that $\lambda_P(0)\leq 0$. One can arrive at this same conclusion by considering the strongly positive solution of problem~\eqref{pdefamily} with $h=0$ given by $y(t,x)=e^{-\gamma_0 t}e_0(x)$, where $\gamma_0\geq 0$ is the first eigenvalue and $e_0\in X$, with $e_0\gg 0$ and $\|e_0\|=1$ is the associated eigenfunction, of the boundary value problem
\begin{equation}\label{bvp}
\left\{\begin{array}{l}
\Delta \, u +\lambda\,u = 0\,,\quad x\in U,\\
Bu:=\alpha(x)\,u+\displaystyle\frac{\partial u}{\partial n} =0\,,\quad x\in \partial U.
\end{array}\right.
\end{equation}
More precisely, it turns out that $\lambda_P(0)=-\gamma_0\leq 0$.
\par
As proved by Pol\'{a}\v{c}ik and Tere\v{s}\v{c}\'{a}k~\cite{pote} in the discrete case, and then extended by Shen and Yi~\cite{shyi} to the continuous case, the operators $\phi(t,p)$ being compact and strongly positive make the linear skew-product semiflow $\tau_L$ admit a continuous separation. This means that there are two families of subspaces $\{X_1(p)\}_{p\in P}$ and $\{X_2(p)\}_{p\in P}$ of $X$ which satisfy:
\begin{itemize}
\item[(1)] $X=X_1(p)\oplus X_2(p)$ and $X_1(p)$, $X_2(p)$ vary
continuously in $P$;
\item[(2)] $X_1(p)=\langle e(p)\rangle$, with $e(p)\gg 0$ and
$\|e(p)\|=1$ for any $p\in P$;
\item[(3)] $X_2(p)\cap X_+=\{0\}$ for any $p\in P$;
\item[(4)] for any $t>0$, $p\in P$,
\begin{align*}
\phi(t,p)X_1(p)&= X_1(p{\cdot}t)\,,\\
\phi(t,p)X_2(p)&\mathbb{S}ubset X_2(p{\cdot}t)\,;
\end{align*}
\item[(5)] there are $M>0$, $\delta>0$ such that for any $p\in P$, $z\in
X_2(p)$ with $\|z\|=1$ and $t>0$,
\begin{equation*}
\|\phi(t,p)\,z\|\leq M \,e^{-\delta t}\|\phi(t,p)\,e(p)\|\,.
\end{equation*}
\end{itemize}
\par
In this situation, the 1-dim invariant subbundle
\begin{equation*}
\displaystyle\bigcup_{p\in P} \{p\} \times X_1(p)\,
\end{equation*}
is called the {\em principal bundle\/} and the Sacker-Sell spectrum of the restriction of $\tau_L$ to this invariant subbundle is called the {\em principal spectrum\/} of $\tau_L$, and is denoted by $\Sigma_{\thetaxt{pr}}(\tau_L)$ (see Mierczy{\'n}ski and Shen~\cite{mish}). It is well-known that $\Sigma_{\thetaxt{pr}}(\tau_L)$ is a possibly degenerate compact interval of the real line. Actually, if $c(t,p)$ is the real linear semicocycle associated with the continuous separation, that is, if for any $t\geq 0$ and $p\in P$, $c(t,p)$ is the positive number such that
\begin{equation}\label{c}
\phi(t,p)\,e(p)=c(t,p)\,e(p{\cdot}t)\,,
\end{equation}
then the Lyapunov exponents~\eqref{lyap exp} can be calculated by $\lambda(p)=\displaystyle\limsup_{t\to\infty}\frac{\ln c(t,p)}{t}$ for each $p\in P$ and besides $\Sigma_{\thetaxt{pr}}(L)=[\alpha_P,\lambda_P]$ with $\alpha_P\leq\lambda_P$, and
there are two ergodic measures $\mu_1$, $\mu_2\in\mathcal{M}(P)$ such that
\begin{equation}\label{rep int exponente}
\alpha_{P}=\int_P \ln c(1,p)\,d\mu_1\,\quad \thetaxt{and}\quad
\lambda_{P}=\int_P \ln c(1,p)\,d\mu_2\,.
\end{equation}
The reader is referred to Novo et al.~\cite{noos7} for all the details in an abstract setting.
\par
As a consequence, when the flow on $P$ is uniquely ergodic, the principal spectrum is a singleton determined by the upper Lyapunov exponent: $\Sigma_{\thetaxt{pr}}(\tau_L)=\{\lambda_P\}$. Furthermore, in the uniquely ergodic setting an application of Birkhoff's ergodic theorem permits to conclude that $\lambda_P=\lambda(p)$ for any $p\in P$ and besides, the superior limit in the definition of $\lambda(p)$ is an existing limit.
\par
Note that the linear semicocycle $c(t,p)$ can be extended to a linear cocycle just by taking $c(-t,p)=1/c(t,p{\cdot}(-t))$ for any $t>0$ and $p\in P$. Since this 1-dim linear cocycle is going to be a fundamental tool in this section, we give a definition.
\begin{defi}\label{defi-cociclo c}
For each $h\in C(P\times\bar U)$, $c(t,p)$ ($t\in\mathbb{R}$, $p\in P$) is the 1-dim linear cocycle driving the dynamics of $\tau_L$ when restricted to the principal bundle determined by the continuous separation (see~\eqref{c}).
\end{defi}
The kind of results that we are going to present in the linear case are in line with those in Caraballo et al.~\cite{caloNonl} given for families of scalar linear ODEs $x'=h(p{\cdot}t)\,x$, $p\in P$, with $P$ a minimal and almost periodic flow and with null upper Lyapunov exponent $\lambda_P=\lambda_P(h)=0$. A significant difference is that in the case of scalar ODEs $\lambda_P(h)$ is a linear map with respect to $h$,
\[
\lambda_P(h)=\int_P h\,d\nu\,,
\]
for the Haar measure $\nu$ in $P$, whereas in the present case of scalar parabolic PDEs we show that the dependance of the upper Lyapunov exponent on $h$ is continuous and convex but not linear any more: just note that $\lambda_P(0)=-\gamma_0<0$ for Robin boundary conditions (also, see Theorem~\ref{prop-convex estricta}).
\par
From now on, we assume that the flow on $P$ is minimal and uniquely ergodic and $\nu$ denotes the unique ergodic measure.
\begin{prop}\label{prop-continua}
The map $\lambda_P:C(P\times\bar U)\to \mathbb{R}$, $h\mapsto \lambda_P(h)$ is continuous. As a consequence,
\begin{equation*}
C_0(P\times\bar U)=\{h\in C(P\times\bar U)\mid \lambda_P(h)=0\}
\end{equation*}
is a closed complete set in $C(P\times\bar U)$.
\end{prop}
\begin{proof}
Let $h\in C(P\times\bar U)$ and let $(h_n)_n\mathbb{S}ubset C(P\times\bar U)$ be such that $h_n\to h$ as $n\to\infty$. Then, in particular, fixed an $\varepsilon>0$ there exists an $n_0$ such that $h-\varepsilon\leq h_n\leq h+\varepsilon$ for any $n\geq n_0$. Applying Theorem~\ref{teor-comparacion} we deduce that $\lambda_P(h-\varepsilon)\leq \lambda_P(h_n)\leq \lambda_P(h+\varepsilon)$ for any $n\geq n_0$. Now, since the linear cocycle for $h\pm\varepsilon$ is just given by $\exp(\pm\varepsilon\,t)\,\phi(t,p)$, it is straightforward that $\lambda_P(h\pm\varepsilon)=\lambda_P(h)\pm\varepsilon$, so that $\lambda_P(h_n)\to \lambda_P(h)$ as $n\to\infty$. The proof is finished.
\end{proof}
Let us now deal with the convexity of $\lambda_P(h)$.
\begin{prop}\label{prop-convex}
For any $h_1,\,h_2\in C(P\times\bar U)$ and any $0\leq r\leq 1$,
\[
\lambda_P(r h_1+(1-r)h_2)\leq r\lambda_P(h_1)+(1-r)\lambda_P(h_2)\,.
\]
\end{prop}
\begin{proof}
First, let us assume that $h_1$ and $h_2$ are regular enough so that the mild solutions of the associated IBV problems become classical solutions. For a fixed $p\in P$, and any fixed $z_0\in X$, $z_0\gg 0$, on the one hand, let $y_1(t,x)$ and $y_2(t,x)$ denote respectively the solution of the IBV problem for $i=1,2$:
\begin{equation*}
\left\{\begin{array}{l} \displaystyle\frac{\partial y}{\partial t} =
\Delta \, y+h_i(p{\cdot}t,x)\,y\,,\quad t>0\,,\;\, x\in U,\,\\[.2cm]
By:=\alpha(x)\,y+\displaystyle\frac{\partial y}{\partial n} =0\,,\quad t>0\,,\;\, x\in \partial U,\,\\[.2cm]
y(0,x)=z_0(x)\,,\quad x\in \bar U;\,
\end{array}\right.
\end{equation*}
and let $\phi_1(t,p)$ and $\phi_2(t,p)$ be the associated linear cocycles, so that $y_i(t,x)=(\phi_i(t,p)\,z_0)(x)$, $t\geq 0,\,x\in \bar U$, for $i=1,2$.
On the other hand, let $y(t,x)$ be the solution of the IBV problem
\begin{equation*}
\left\{\begin{array}{l} \displaystyle\frac{\partial y}{\partial t} =
\Delta \, y+(r h_1(p{\cdot}t,x)+(1-r) h_2(p{\cdot}t,x))\,y\,,\quad t>0\,,\;\, x\in U,\,\\[.2cm]
By:=\alpha(x)\,y+\displaystyle\frac{\partial y}{\partial n} =0\,,\quad t>0\,,\;\, x\in \partial U,\,\\[.2cm]
y(0,x)=z_0(x)\,,\quad x\in \bar U;\,
\end{array}\right.
\end{equation*}
with associated linear cocycle $\Phi(t,p)$, so that $y(t,x)=(\Phi(t,p)\,z_0)(x)$, $t\geq 0,\,x\in \bar U$. By the strong monotonicity of these problems, $y_1(t,x)$, $y_2(t,x)$, $y(t,x)>0$ for any $t\geq 0$, $x\in \bar U$, and we can consider $z(t,x)=\exp(r\ln y_1(t,x)+(1-r)\ln y_2(t,x))$.
\par
We do some routine calculations for $z(t,x)$:
\begin{align*}
\displaystyle\frac{\partial z}{\partial t} = & z\left(\frac{r}{y_1}\,\frac{\partial y_1}{\partial t}+\frac{1-r}{y_2}\,\frac{\partial y_2}{\partial t} \right)\\ = & z\left(\frac{r}{y_1}\,\Delta\,y_1+\frac{1-r}{y_2}\,\Delta\,y_2 +r h_1(p{\cdot}t,x)+(1-r) h_2(p{\cdot}t,x) \right);
\\ \displaystyle\frac{\partial z}{\partial x_i} = & z\left(\frac{r}{y_1}\,\frac{\partial y_1}{\partial x_i}+\frac{1-r}{y_2}\,\frac{\partial y_2}{\partial x_i} \right)\mathbb{R}ightarrow \nabla z = z\left(\frac{r}{y_1}\,\nabla y_1 + \frac{1-r}{y_2}\,\nabla y_2 \right);\\
\Delta\,z =& z \mathbb{S}um_{i=1}^m \left( \left( \frac{r}{y_1}\,\frac{\partial y_1}{\partial x_i}+\frac{1-r}{y_2}\,\frac{\partial y_2}{\partial x_i}\right)^{\!2} - \frac{r}{y_1^2}\,\left(\frac{\partial y_1}{\partial x_i}\right)^{\!2}- \frac{1-r}{y_2^2}\,\left(\frac{\partial y_2}{\partial x_i}\right)^{\!2} \right)\\ & + z\left( \frac{r}{y_1}\,\Delta\, y_1+\frac{1-r}{y_2}\,\Delta \,y_2 \right) \leq z\left( \frac{r}{y_1}\,\Delta\, y_1+\frac{1-r}{y_2}\,\Delta \,y_2 \right),
\end{align*}
where the convexity of the map $\mathbb{R}\to\mathbb{R}$, $s\mapsto s^2$ has been applied in the inequality. Therefore, $z(t,x)$ is a solution of the problem
\begin{equation*}
\left\{\begin{array}{l} \displaystyle\frac{\partial z}{\partial t} \geq
\Delta \, z+(r h_1(p{\cdot}t,x)+(1-r) h_2(p{\cdot}t,x))\,z\,,\quad t>0\,,\;\, x\in U,\,\\[.2cm]
Bz:=\alpha(x)\,z+\displaystyle\frac{\partial z}{\partial n} =0\,,\quad t>0\,,\;\, x\in \partial U,\,\\[.2cm]
z(0,x)=z_0(x)\,,\quad x\in \bar U.
\end{array}\right.
\end{equation*}
\par
Then, a standard argument of comparison of solutions (see Smith~\cite{smit}) says that $z(t,x)\geq y(t,x)$, that is, $y_1(t,x)^ry_2(t,x)^{1-r}\geq y(t,x)$ for $t\geq 0$, $x\in \bar U$. In other words, we have proved in $X$ that $(\phi_1(t,p)\,z_0)^r(\phi_2(t,p)\,z_0)^{1-r}\geq \Phi(t,p)\,z_0$. Applying monotonicity of the norm and the fact that $X$ is a Banach algebra,
\[
\|\phi_1(t,p)\,z_0\|^r\|\phi_2(t,p)\,z_0\|^{1-r}\geq \|(\phi_1(t,p)\,z_0)^r(\phi_2(t,p)\,z_0)^{1-r}\|\geq \|\Phi(t,p)\,z_0\|\,,
\]
and taking logarithm,
\begin{equation*}
r\ln \|\phi_1(t,p)\,z_0\|+ (1-r)\ln \|\phi_2(t,p)\,z_0\|\geq \ln \|\Phi(t,p)\,z_0\|\,,\quad t\geq 0\,.
\end{equation*}
\par
As it has been remarked before, in the uniquely ergodic case the upper Lyapunov exponent equals the value of any of the Lyapunov exponents, and in particular that of $p$, so that having~\eqref{lyap exp} in mind, it suffices to divide by $t$ and take limits as $t\to\infty$ to get the convexity relation.
\par
To finish the proof, consider any $h_1,\,h_2\in C(P\times\bar U)$. Using a result by Schwartzman~\cite{schw} we can approximate these maps by respective sequences $(h_{1,n})_n,\,(h_{2,n})_n$ of sufficiently regular maps. More precisely, maps of class $C^1$ in $U$ and of class $C^{1\!}$ along the orbits in $P$, that is, for any $p\in P$ and $x\in\bar U$ the maps $h_{i,n}(p{\cdot}t,x)$ are continuously differentiable in $t\in\mathbb{R}$ ($i=1,2$, $n\geq 1$). Since the convexity relation applies to the pairs $h_{1,n},\,h_{2,n}$ for any $n\geq 1$, with the continuity result in Proposition~\ref{prop-continua} we are done.
\end{proof}
As a corollary, since $\lambda_P(0)\leq 0$, we get the superlinear character of $\lambda_P$, that is,
$\lambda_P(r h)\leq r\lambda_P(h)$ for any $h\in C(P\times\bar U)$ and any $0\leq r\leq 1$.
\par
Once we have studied some basic properties of the map $\lambda_P(h)$, our aim is to give a description of the dynamics of the linear semiflow $\tau_L$ when $\lambda_P(h)=0$, depending on the map $h$. As it was also done in Caraballo et al.~\cite{caloNonl}, from now on we assume that the minimal and uniquely ergodic flow on $P$ is not periodic. In $C(P)$, the space of continuous functions on $P$, we consider the Banach space $C_0(P)=\left\{a\in C(P)\mid \int_P a\,d\nu=0\right\}$, its vector subspace
\[
B(P)=\left\{ a\in C_0(P) \;\mathbb{B}ig|\; \mathbb{S}up_{t\in\mathbb{R}} \left| \int_0^t a(p{\cdot}s)\,ds\right|<\infty \;\thetaxt{ for any}\;p\in P \right\}
\]
of the continuous functions on $P$ with zero mean and bounded primitive, and its complement $\mathcal{U}(P)=C_0(P)\mathbb{S}etminus B(P)$ of the continuous functions on $P$ with zero mean and unbounded primitive. As a consequence of Lemma 5.1 in Campos et al.~\cite{caot}, $B(P)$ is a dense set of first category in $C_0(P)$ and thus $\mathcal{U}(P)$ is a residual set (see Gottschalk and Hedlund~\cite{gohe} and Johnson~\cite{john} for the result in the almost periodic and aperiodic case).
\par
Now, in the complete metric space $C_0(P\times\bar U)=\{h\in C(P\times\bar U)\mid \lambda_P(h)=0\}$
we introduce the sets
\begin{align*}
B(P\times \bar U) &= \{ h\in C_0(P\times\bar U)\mid \mathbb{S}up_{t\in\mathbb{R}} | \ln c(t,p)|<\infty \;\thetaxt{ for any}\;p\in P \}\,\; \thetaxt{and} \\%\ln c(1,p)\in B(P)
\mathcal{U}(P\times \bar U) &=C_0(P\times\bar U)\mathbb{S}etminus B(P\times \bar U) \,,
\end{align*}
for the associated 1-dim linear cocycle $c(t,p)$ given in Definition~\ref{defi-cociclo c}.
Note that the condition determining $B(P\times \bar U)$ is equivalent to saying that for any $p\in P$ the linear positive cocycle $c(t,p)$ is both bounded away from $0$ and bounded above.
\par
Next, we state without proof two technical results given for general positive 1-dim linear cocycles $c(t,p)$, which are in correspondance with two classical results for maps in $C_0(P)$. The first one is the adaptation of Proposition~12 in~\cite{caloNonl} (proved in~\cite{gohe}), whereas the second one has the spirit of the oscillation result stated in Theorem~13 in~\cite{caloNonl} (proved in~\cite{john}). In fact, the proofs can be adapted respectively from the proofs of Proposition~A.1 and Theorem~A.2 in Jorba et al.~\cite{jnot}.
\begin{prop}\label{prop-gott}
Let $c(t,p)$ be a continuous positive 1-dim linear cocycle. Then, the following conditions are equivalent:
\begin{itemize}
\item[(i)] There exists a function $k\in C(P)$ such that
\[
k(p{\cdot}t)-k(p)=\ln c(t,p) \;\;\thetaxt{for all}\;\; p\in P,\;t\in\mathbb{R}\,.
\]
\item[(ii)] For any $p\in P$, $\mathbb{S}up_{t\in\mathbb{R}} | \ln c(t,p)|<\infty $.
\item[(iii)] There exists a $p_0\in P$ such that $\mathbb{S}up_{t\in\mathbb{R}} | \ln c(t,p_0)|<\infty $.
\item[(iv)] There exists a $p_0\in P$ such that
\[
\thetaxt{either}\;\; \mathbb{S}up_{t\geq 0} | \ln c(t,p_0)|<\infty \quad\thetaxt{or}\;\; \mathbb{S}up_{t\leq 0} | \ln c(t,p_0)|<\infty\,.
\]
\end{itemize}
\end{prop}
\begin{teor}\label{teor-johnson}
Let $c(t,p)$ be a continuous positive 1-dim linear cocycle and assume that it does not satisfy the conditions in Proposition~$\ref{prop-gott}$, and the associated real linear skew-product flow $\mathbb{R}\times P\times\mathbb{R}\to P\times\mathbb{R}$, $(t,p,y)\mapsto (p{\cdot}t,c(t,p)\,y)$ does not have an exponential dichotomy. Then, there exists an invariant and residual set $P_{\rm{o}}\mathbb{S}ubset P$ such that for any $p\in P_{\rm{o}}$ there exist sequences (depending on $p$) $(t_n^i)_n$, $i=1,2,3,4$ with $t_n^i\uparrow \infty$ for $i=1,2$ and $t_n^i\downarrow -\infty$ for $i=3,4$ such that
\[
\lim_{n\to\infty}c(t_n^i,p) =0 \;\;\thetaxt{for}\;\;i=1,3 \quad\thetaxt{and}\quad
\lim_{n\to\infty} c(t_n^i, p)=\infty\;\;\thetaxt{for}\;\;i=2,4\,.
\]
\end{teor}
We will sometimes refer to $P_{\rm{o}}$ as the oscillation set of $c(t,p)$.
\begin{nota}\label{remark-DE}
Note that if $h\in C_0(P\times\bar U)$, then $\Sigma_{\rm{pr}}(\tau_L)=\{0\}$, that is, $\tau_L$ restricted to the principal bundle does not have an exponential dichotomy. In other words, the associated real linear skew-product flow $\mathbb{R}\times P\times\mathbb{R}\to P\times\mathbb{R}$, $(t,p,y)\mapsto (p{\cdot}t,c(t,p)\,y)$ does not have an exponential dichotomy. This means that given any $h\in C_0(P\times\bar U)$, either the associated 1-dim cocycle $c(t,p)$ satisfies the equivalent conditions in Proposition~\ref{prop-gott} if $h\in B(P\times\bar U)$, or Theorem~\ref{teor-johnson} applies if $h\in \mathcal{U}(P\times \bar U)$. Note also that if the flow on $P$ is periodic, then $C_0(P\times\bar U)=B(P\times\bar U)$.
\end{nota}
For the sake of completeness, and because it will be used later on, we include here a result for 1-dim linear cocycles in line with the Corollary of Theorem~1 in Shneiberg~\cite{shne} given for integrable maps $f:P\to\mathbb{R}$ with zero mean, which says that for almost all $p\in P$ there exists a sequence $(t_n)_n\uparrow \infty$ such that $\int_0^{t_n} f(p{\cdot}s)\,ds = 0$ for any $n\geq 1$. The corresponding adaptation for cocycles reads as follows.
\begin{teor}\label{teor-shneiberg}
Let $c(t,p)$ be a continuous positive 1-dim linear cocycle and assume that the associated real linear skew-product flow $\mathbb{R}\times P\times\mathbb{R}\to P\times\mathbb{R}$, $(t,p,y)\mapsto (p{\cdot}t,c(t,p)\,y)$ does not have an exponential dichotomy. Then, for almost all $p\in P$ there exists a sequence $(t_n)_n\uparrow \infty$ such that $
c(t_n,p)=1$ for any $n\geq 1$.
\end{teor}
In the following result, the dynamics of the linear semiflow $\tau_L$ is described when $h\in B(P\times \bar U)$. Basically, it means bounded orbits, both away from $0$ and above, for strongly positive initial data.
\begin{teor}\label{teor-equivalencias}
Let $h\in C_0(P\times \bar U)$ and let us fix a reference vector $z_0\gg 0$ in $X$. The following statements are equivalent:
\begin{itemize}
\item[(i)] There exist a $p_0\in P$ and constants $c_0, C_0>0$ such that $c_0\,z_0\leq \phi(t,p_0)\,z_0\leq C_0\,z_0$ for any $t\geq 0$.
\item[(ii)] For any $p\in P$ and $z\in X$, $z\gg 0$, there exist constants $c(p,z), C(p,z)>0$ such that $c(p,z)\,z_0\leq \phi(t,p)\,z\leq C(p,z)\,z_0$ for any $t\geq 0$.
\item[(iii)] $h\in B(P\times \bar U)$.
\item[(iv)] For any $p\in P$ there exists a $C(p)>0$ such that $\phi(t,p)\,z_0\leq C(p)\,z_0$ for any $t\geq 0$.
\item[(v)] For any $p\in P$ there exists a $c(p)>0$ such that $c(p)\,z_0\leq \phi(t,p)\,z_0$ for any $t\geq 0$.
\end{itemize}
\end{teor}
\begin{proof}
(i)$\mathbb{R}ightarrow$(ii): Since the trajectory of $(p_0,z_0)$ under $\tau_L$ lies in the order-interval $[c_0\,z_0,C_0\,z_0]$ and the cone is normal, it is bounded, and we can consider the omega-limit set $K=\mathcal{O}(p_0,z_0)$ which is a compact $\tau_L$-invariant set which projects over the whole $P$. Besides, for any $(p,z)\in K$, $c_0\,z_0\leq z\leq C_0\,z_0$. Now, take a $p\in P$ and a $z\in X$ with $z\gg 0$. For $p$ there is a pair $(p,z^*)\in K$ and we can take constants $c_1(p,z), C_1(p,z)>0$ such that $c_1(p,z)\,z^*\leq z\leq C_1(p,z)\,z^*$. Then, for any $t\geq 0$, $c_0\,c_1(p,z)\,z_0\leq c_1(p,z)\, \phi(t,p)\,z^*\leq \phi(t,p)\,z\leq C_1(p,z)\, \phi(t,p)\,z^*\leq C_0\,C_1(p,z)\,z_0$ and it suffices to take $c(p,z)=c_0\,c_1(p,z)$ and $C(p,z)=C_0\,C_1(p,z)$.
\par
(ii)$\mathbb{R}ightarrow$(iii): Let us see that $\mathbb{S}up_{t\geq 0} | \ln c(t,p)|<\infty$ for any $p\in P$. First of all, from the continuity and strong positivity on the compact set $P$ of the map $e$ giving the leading direction in the continuous separation, and the fact that $z_0\gg 0$, one deduces that there exist constants $c_1,\,C_1>0$ such that $c_1\,z_0\leq e(p)\leq C_1\,z_0$ for any $p\in P$. Then, for $p\in P$ and $e(p)\gg 0$, take $c(p), C(p)>0$ given in (ii) such that $c(p)\,z_0\leq \phi(t,p)\,e(p)= c(t,p)\,e(p{\cdot}t)\leq C(p)\,z_0$ for any $t\geq 0$. We can then deduce that the values of $c(t,p)$ for $t\geq 0$ move between two positive constants. By Proposition~\ref{prop-gott} we can conclude that $h\in B(P\times\bar U)$.
\par
(iii)$\mathbb{R}ightarrow$(i), (iii)$\mathbb{R}ightarrow$(iv) and (iii)$\mathbb{R}ightarrow$(v): Using the previous relation $c_1\,z_0\leq e(p{\cdot}t)\leq C_1\,z_0$ for any $p\in P$ and $t\geq 0$ and~\eqref{c}, it is easy to deduce that
\begin{equation}\label{bounds}
\frac{c_1}{C_1} \, c(t,p)\,z_0\leq \phi(t,p)\,z_0\leq \frac{C_1}{c_1} \, c(t,p)\,z_0\,,\quad p\in P,\;t\geq 0\,.
\end{equation}
Since in particular $\mathbb{S}up_{t\geq 0} | \ln c(t,p)|<\infty$ for any $p\in P$, this means that for each $p\in P$, $c(t,p)$ is bounded away from $0$ and bounded above for any $t\geq 0$. From this, it is immediate to conclude.
\par
(iv)$\mathbb{R}ightarrow$(iii) and (v)$\mathbb{R}ightarrow$(iii): Once more, from \eqref{bounds}, for any $p\in P$ the semicocycle $c(t,p)$ is bounded above by a constant if (iv) holds, and is bounded below by a positive constant if (v) holds. According to Theorem~\ref{teor-johnson} this can only happen if $h\in B(P\times \bar U)$. The proof is finished.
\end{proof}
As for the dynamics when $h\in \mathcal{U}(P\times \bar U)$, we state an oscillation result for $\tau_L$ .
\begin{teor}\label{teor-oscilacion}
Let $h\in \mathcal{U}(P\times \bar U)$. Then, there exists an invariant and residual set $P_{\rm{o}}\mathbb{S}ubset P$ such that for any $p\in P_{\rm{o}}$ there exist sequences $(t_n^1)_n, (t_n^2)_n\uparrow \infty$ depending on $p$, such that for any $z\in X$ with $z\gg 0$ it holds:
\[ \lim_{n\to \infty} \| \phi(t_n^1,p)\,z \|=0\quad\thetaxt{ and }\quad \lim_{n\to \infty} \displaystyle\left\|\frac{1}{ \phi(t_n^2,p)\,z} \right\|=0\,.
\]
\end{teor}
\begin{proof}
Let $P_{\rm{o}}\mathbb{S}ubset P$ be the invariant and residual set determined in Theorem~\ref{teor-johnson} for the associated real cocycle $c(t,p)$.
Then, for each $p\in P_{\rm{o}}$ there exist sequences $(t_n^1)_n, (t_n^2)_n\uparrow \infty$ depending on $p$ such that
\[
\lim_{n\to\infty} c(t_n^1,p) =0 \quad\thetaxt{and}\quad
\lim_{n\to\infty} c(t_n^2,p) =\infty\,.
\]
\par
By the properties of $e(p)$, given $z\gg 0$, we can find constants $c_1, C_1>0$ such that $c_1\, e(p)\leq z\leq C_1\, e(p)$ for any $p\in P$.
Then, by relation~\eqref{c}, monotonicity of $\tau_L$ and monotonicity of the norm we get that $\| \phi(t_n^1,p)\,z \|\leq C_1 \,\| \phi(t_n^1,p)\,e(p) \|= C_1 \, c(t_n^1,p)\to 0$ as $n\to\infty$, and
\[
\displaystyle\left\|\frac{1}{\phi(t_n^2,p)\,z} \right\|\leq \frac{1}{c_1} \displaystyle\left\|\frac{1}{ \phi(t_n^2,p)\,e(p)} \right\|= \frac{1}{c_1 \,c(t_n^2,p)} \to 0 \quad\thetaxt{as } n\to\infty\,,
\]
as we wanted to see.
\end{proof}
\par
Now, for $h\in C_0(P\times\bar U)$ we prove the existence of an invariant compact set in $P\times X$, with a precise dynamical description depending on whether $h\in B(P\times \bar U)$ or $h\in \mathcal{U}(P\times \bar U)$. First, we give a definition. The operator $A$ below is the one defined in Section~\ref{sec-mild solutions}.
\begin{defi}\label{def solucion entera}
A solution $v:I\to X$ of the abstract equation
\begin{equation}\label{abstract eq}
v'(t) =
A\, v(t)+\tilde h(p{\cdot}t)\,v(t)\,,\quad t\in I\,,
\end{equation}
along the orbit of $p\in P$ is said to be an {\em entire solution} provided that $I=(-\infty,\infty)$. In that case, $v(t+s)=\phi(t,p{\cdot}s)\,v(s)$ for any $t\geq 0$ and $s\in \mathbb{R}$. An entire solution $v:(-\infty,\infty)\to X$ is said to be {\em negatively bounded} if $\{v(t)\mid t\leq 0\}\mathbb{S}ubset X$ is a bounded set.
\end{defi}
\begin{prop}\label{prop-compacto invariante}
Let $h\in C_0(P\times\bar U)$. Then,
the following items hold:
\begin{itemize}
\item[(i)] If $v:\mathbb{R}\to X$ is a negatively bounded solution of the abstract equation \eqref{abstract eq} along the orbit of $p_0\in P$, then $v(t)\in X_1(p_0{\cdot} t)$ for any $t\in \mathbb{R}$.
\item[(ii)] If $h\in B(P\times \bar U)$, then there exists a continuous map $\omegaidehat e: P \to \Int X_+$ such that $\omegaidehat e(p)\in X_1(p)$ for any $p\in P$ and $\omegaidehat e(p{\cdot} t)=\phi(t,p)\,\omegaidehat e(p)$ for any $p\in P$ and $t\geq 0$. Besides, $K=\{(p,\omegaidehat e(p))\mid p\in P\}$ is a minimal set which is a copy of the base $P$ and it is contained in $P\times \Int X_+$.
\item[(iii)] If $h\in \mathcal{U}(P\times \bar U)$, there exist a $p_0\in P$ and a bounded entire solution $v:\mathbb{R}\to X_+\mathbb{S}etminus \{0\}$ of~\eqref{abstract eq} along the orbit of $p_0$ such that $K_0=\cls \{(p_0{\cdot}t, v(t))\mid t\in \mathbb{R}\}$ is a pinched $\tau_L$-invariant compact set in $P\times (\Int X_+\cup \{0\})$.
\end{itemize}
\end{prop}
\begin{proof}
By the invariance of the 1-dim principal bundle, to prove (i) it suffices to check that $v(0)\in X_1(p_0)$. The continuous variation with respect to $p\in P$ of the projections $\Pi_{1,p}:X\to X_1(p)$, $\Pi_{2,p}:X\to X_2(p)$ implies that there exist $\rho_1,\, \rho_2>0$ such that $\|\Pi_{1,p}\|\leq \rho_1$ and $\|\Pi_{2,p}\|\leq \rho_2$ for any $p\in P$. Let $r=\mathbb{S}up\{\|v(t)\|\mid t\leq 0\}$. If we write $v(t)=z_1(t)+z_2(t)\in X_1(p_0{\cdot}t)\oplus X_2(p_0{\cdot}t)$ for any $t\in \mathbb{R}$, then $\|z_1(t)\|\leq \rho_1 r$ and $\|z_2(t)\|\leq \rho_2 r$ for any $t\leq 0$. Besides, $\phi(t,p_0{\cdot}(-t))\,z_1(-t)=z_1(0)$ and $\phi(t,p_0{\cdot}(-t))\,z_2(-t)=z_2(0)$ for any $t\geq 0$. Then, applying property~(5) in the definition of continuous separation,
\[
\|z_2(0)\|=\|\phi(t,p_0{\cdot}(-t))\,z_2(-t)\|\leq \|z_2(-t)\|\,M\,e^{-\delta\,t} \|\phi(t,p_0{\cdot}(-t))\,e(p_0{\cdot}(-t))\|\,.
\]
Since $\Sigma_{\thetaxt{pr}}(L)=\{\lambda_P(h)\}=\{0\}$, given $0<\lambda< \delta$, there is an exponential dichotomy with full stable subspace for the 1-dim semiflow $e^{-\lambda t}\,\phi(t,p) |_{X_1}$, that is, given $\varepsilon>0$ there exists a $t_0$ such that $
\|\phi(t,p_0{\cdot}(-t))\,e(p_0{\cdot}(-t))\|\leq \varepsilon\, e^{\lambda t}$ for any $t\geq t_0$. Therefore, we can easily deduce that $\|z_2(0)\|=0$, so that $v(0)\in X_1(p_0)$.
\par
Now assume that $h\in B(P\times \bar U)$. Then, by Proposition~\ref{prop-gott} there exists a function $k\in C(P)$ such that $k(p{\cdot}t)-k(p)=\ln c(t,p)$ for any $p\in P$, $t\in\mathbb{R}$,
whose exponential $\kappa:P\to \mathbb{R}_+$, $\kappa(p)=\exp k(p)$ is positive and satisfies
\begin{equation*}
\kappa(p{\cdot}t)=\kappa(p)\,c(t,p)\;\; \thetaxt{for any}\;\;p\in P ,\; t\in \mathbb{R}\,.
\end{equation*}
Now define the continuous map $\omegaidehat e: P \to \Int X_+$, $p\mapsto \kappa(p)\,e(p)$.
According with~\eqref{c} and the previous formula, for any $t\geq 0$ and any $p\in P$ this map satisfies
\[
\phi(t,p)\,\omegaidehat e(p)=\kappa(p)\,\phi(t,p)\,e(p) =\kappa(p)\,c(t,p)\,e(p{\cdot}t)=\kappa(p{\cdot}t)\,e(p{\cdot}t)=\omegaidehat e(p{\cdot}t)\,.
\]
In other words, it defines a continuous equilibrium for the linear skew-product semiflow $\tau_L$. As a consequence, the set $K=\{(p,\omegaidehat e(p))\mid p\in P\}$ is a minimal set in $P\times\Int X_+$ with the simplest possible structure, and (ii) is proved.
\par
Finally, let us assume that $h\in \mathcal{U}(P\times \bar U)$. Then, as explained in Remark~\ref{remark-DE}, the associated real linear skew-product flow $\pi:\mathbb{R}\times P\times\mathbb{R}\to P\times\mathbb{R}$, $\pi(t,p,y)=(p{\cdot}t,c(t,p)\,y)$ does not have an exponential dichotomy. In this case, a result by Selgrade~\cite{selg} says that there exists a $p_0\in P$ for which there is a nonzero bounded orbit, that is, $\{c(t,p_0)\mid t\in\mathbb{R}\}$ is bounded in $\mathbb{R}_+$. In this situation we claim that the set $K=\cls\{(p_0{\cdot}t, c(t,p_0))\mid t\in \mathbb{R}\}$ is an invariant compact set in $P\times \mathbb{R}$ for $\pi$ with a pinched structure (this is a generalization of Lemma~14 in Caraballo et al.~\cite{caloNonl}). The fact that $K$ is invariant and compact is clear. In particular, for any $p\in P$ there is at least one pair $(p,y)\in K$. Now, let $P_{\rm{o}}$ be the oscillation set of $c(t,p)$ given in Theorem~\ref{teor-johnson}. Then, for every $p\in P_{\rm{o}}$ the only pair in $K$ is $(p,0)$, as if there were a pair $(p,y)$ with $y\not=0$, the orbit of $(p,y)$ would remain in $K$ but this cannot happen because of its oscillating behaviour. Since $P$ is minimal and $0$ determines an orbit, in fact $(p,0)\in K$ for any $p\in P$. Finally $(p_0,1)\in K$, so that we have proved that $K$ has a pinched structure.
\par
At this point, the entire solution along the trajectory of $p_0$ defined by $v:\mathbb{R}\to X_+$, $t\mapsto v(t)= c(t,p_0)\,e(p_0{\cdot}t)$ is bounded and the set $K_0=\cls\{(p_0{\cdot}t, v(t))\mid t\in \mathbb{R}\}$ is an invariant compact set in $P\times (\Int X_+\cup \{0\})$ which is homeomorphic to $K$ and thus has a pinched structure. The proof is finished.
\end{proof}
For the sake of completeness, we collect the fundamental properties of the set $K_0$ in the third item of the previous proposition.
\begin{coro}
Let $h \in \mathcal{U}(P\times\bar U)$. Then, the pinched invariant compact set $K_0\mathbb{S}ubset P\times (\Int X_+\cup \{0\})$ given in Proposition~$\ref{prop-compacto invariante}~\rm{(iii)}$ satisfies:
\begin{itemize}
\item[(a)] $(p_0,e(p_0))\in K_0$.
\item[(b)] $(p,0)\in K_0$ for any $p\in P$.
\item[(c)] $(p,0)$ is the only element in $K_0$ if $p\in P_{\rm{o}}$, the oscillation set of the associated $1$-dim linear cocycle $c(t,p)$.
\end{itemize}
\end{coro}
After the dynamical description of $\tau_L$, depending on whether the coefficient map $h$ is in $B(P\times \bar U)$ or in $\mathcal U(P\times \bar U)$, we wonder which the topological size of these sets is. Before we move on, we make a remark which will let us play with an additional term in the equations, providing a technical tool in this paper. Note that, if $h\in C(P\times\bar U)$ and $k\in C(P)$, then $h+k\in C(P\times\bar U)$ and the linear parabolic problem for $h+k$ given by
\begin{equation}\label{pdefamily h+k}
\left\{\begin{array}{l} \displaystyle\frac{\partial y}{\partial t} =
\Delta \, y+h(p{\cdot}t,x)\,y+k(p{\cdot}t)\,y\,,\quad t>0\,,\;\,x\in U, \;\, \thetaxt{for each}\; p\in P,\\[.2cm]
By:=\alpha(x)\,y+\displaystyle\frac{\partial y}{\partial n} =0\,,\quad t>0\,,\;\,x\in \partial U,\,
\end{array}\right.
\end{equation}
admits the same treatment as the one developed for~\eqref{pdefamily}. Actually, the linear skew-product semiflow $\omegait \tau_L$ associated with~\eqref{pdefamily h+k}, which is given by
\begin{equation*}
\begin{array}{cccl}
\omegait \tau_L: & \mathbb{R}_+\times P\times X& \longrightarrow & \hspace{0.3cm}P\times X\\
& (t,p,z) & \mapsto
&(p{\cdot}t,\omegait \phi(t,p)\,z)=(p{\cdot}t,e^{\int_0^t k(p{\cdot}s)\,ds}\phi(t,p)\,z)\,,
\end{array}
\end{equation*}
admits a continuous separation sharing the principal bundle of $\tau_L$. Clearly, the associated 1-dim linear cocycle $\omegait c(t,p)$ satisfying $\omegait \phi(t,p)\,e(p) = \omegait c(t,p)\,e(p{\cdot}t)$ for $t\geq 0$, $p\in P$ is given by
\begin{equation}\label{ctilde}
\omegait c(t,p)= c(t,p)\,\exp \int_0^t k(p{\cdot}s)\,ds\,.
\end{equation}
\par
At this point, note that given $k\in C(P)$, if we consider the 1-dim linear cocycle
\[
\omegait c(t,p)=\exp \int_0^t k(p{\cdot}s)\,ds\,,\quad t\geq 0\,, \;p\in P,
\]
we can easily build a problem \eqref{pdefamily h+k} for which $ \omegait c(t,p)$ is the associated 1-dim cocycle. One just has to consider $h\equiv \gamma_0$ ($\gamma_0\geq 0$) the first eigenvalue of the boundary value problem~\eqref{bvp} with associated eigenfunction $e_0\in X$, with $e_0\gg 0$ and $\|e_0\|=1$. In this case, the principal bundle in the induced linear skew-product semiflow is just given by $e(p)=e_0$ for any $p\in P$.
\par
We next collect some easy facts for the terms of the type $h+k$. Recall that relation \eqref{rep int exponente} gives an integral representation of the upper Lyapunov exponent.
\begin{prop}\label{prop-h+k}
The following items hold:
\begin{itemize}
\item[(i)] If $h\in C(P\times\bar U)$ and $k\in C(P)$, then $h+k\in C(P\times\bar U)$ and
\[
\lambda_P(h+k)=\lambda_P(h)+\int_P k\,d\nu\,.
\]
\item[(ii)] If $h\in C_0(P\times\bar U)$ and $k\in C_0(P)$, then $h+k\in C_0(P\times\bar U)$.
\item[(iii)] If $h\in B(P\times \bar U)$ and $k\in B(P)$, then $h+k\in B(P\times \bar U)$.
\item[(iv)] If $h\in B(P\times \bar U)$ and $ k\in \mathcal{U}(P)$, or if $h\in \mathcal{U}(P\times \bar U)$ and $k\in B(P)$, then $h+k\in \mathcal{U}(P\times \bar U)$.
\end{itemize}
\end{prop}
\begin{proof}
To prove the formula in (i) we use relations \eqref{rep int exponente} and \eqref{ctilde} to get
\begin{align*}
\lambda_P(h+k)&=\int_P\ln \omegait c(1,p)\,d\nu=\int_P\ln c(1,p)\,d\nu+\int_P\int_0^1 k(p{\cdot}s)\,ds\,d\nu \\&= \lambda_P(h)+ \int_0^1 \int_P k(p{\cdot}s)\,d\nu\,ds=\lambda_P(h)+ \int_P k\,d\nu\,,
\end{align*}
where we have applied Fubini's theorem, and the invariance of the measure $\nu$.
\par
Clearly, (ii) follows from (i). For (iii) and (iv) we once more argue from \eqref{ctilde} which means that for any $t\in\mathbb{R}$ and any $p\in P$,
\[
\ln \omegait c(t,p)= \ln c(t,p) +\int_0^t k(p{\cdot}s)\,ds\,.
\]
The proof is finished.
\end{proof}
The properties of the decomposition $C_0(P)=B(P)\cup\mathcal{U}(P)$ can be transferred to the space $C_0(P\times\bar U)$ in the following sense.
\begin{teor}\label{teor-categoria}
Consider the complete metric space $C_0(P\times\bar U)$. Then $C_0(P\times\bar U)=B(P\times \bar U)\,\cup\, \mathcal{U}(P\times \bar U)$ where the union is disjoint and:
\begin{itemize}
\item[(i)] $B(P\times \bar U)$ is a dense set of first category in $C_0(P\times\bar U)$.
\item[(ii)] $\mathcal{U}(P\times \bar U)$ is a residual set in $C_0(P\times\bar U)$.
\end{itemize}
\end{teor}
\begin{proof}
The union is disjoint by the definition, and (ii) follows from (i). To see that $B(P\times \bar U)$ is of first category, let us fix a $p\in P$ and a vector $z_0\gg 0$ in $X$ and define the sets
\[
B_n=\left\{ h\in B(P\times \bar U) \;\mathbb{B}ig|\;\frac{1}{n}\,z_0 \leq \phi(t,p)\,z_0\leq n\,z_0 \;\,\thetaxt{for any}\,\;t\geq 0 \right\},\quad n\geq 1\,.
\]
By Theorem~\ref{teor-equivalencias} (iv)-(v), given $h\in B(P\times \bar U)$ it is clear that $h\in B_n$ for $n$ large enough. Therefore, $B(P\times \bar U)=\cup_{n=1}^\infty B_n$. Next we check that each $B_n$ is a closed set with an empty interior, so that $B_n$ is a nowhere dense set and we are done.
\par
Let us fix an $n\geq 1$ and consider a sequence $(h_j)_j\mathbb{S}ubset B_n$ with $h_j\to h_0\in C_0(P\times\bar U)$ as $j\to \infty$. For each $j\geq 0$, let $\phi_j(t,p)$ be the associated linear cocycle for $h_j$. Since for every $j\geq 1$, $\frac{1}{n}\,z_0 \leq \phi_j(t,p)\,z_0\leq n\,z_0$ for any $t\geq 0$, it follows that also $\frac{1}{n}\,z_0 \leq \phi_0(t,p)\,z_0\leq n\,z_0$ for any $t\geq 0$ (for this convergence result, see the proof of Theorem~\ref{teor-comparacion}). Then Theorem~\ref{teor-equivalencias} asserts that $h_0\in B_n$, and it is closed.
\par
About the empty interior, let us argue by contradiction and let us assume that for some $n_0\geq 1$ there exists an $h_0\in \Int B_{n_0}$. Since $\mathcal{U}(P)$ is dense in $C_0(P)$, there is a sequence $(k_j)_j\mathbb{S}ubset \mathcal{U}(P)$ with $\|k_j\|\leq 1/j$ for any $j\geq 1$. Note that by Proposition~\ref{prop-h+k}, $h_0+k_j\in \mathcal{U}(P\times\bar U)$ for any $j\geq 1$ and $\lim_{j\to\infty} h_0+k_j=h_0$, which is a contradiction.
\par
Finally, to see that $B(P\times \bar U)$ is dense in $C_0(P\times\bar U)$, once more using a result by Schwartzman~\cite{schw} it suffices to see that any map in $C_0(P\times\bar U)$ which is of class $C^1$ in $U$ and of class $C^{1\!}$ along the orbits in $P$ can be approximated by a sequence of maps in $B(P\times \bar U)$. So take such a regular map $h$ in $\mathcal{U}(P\times\bar U)$. The advantage is that one can associate a 1-dim cocycle $c_1(t,p)$ to this $h$ with the same behaviour, referring to boundedness, as that of $c(t,p)$, which is further differentiable. More precisely, note that in principle $c(t,p)=\|\phi(t,p)\,e(p)\|$ might not be always differentiable. However, by fixing a point $x_0\in U$ and taking
\begin{equation*}
z_1(p)=\frac{e(p)}{e(p)(x_0)}\in X\,,\quad p\in P,
\end{equation*}
it is not difficult to check that $\phi(t,p)\,z_1(p)=c_1(t,p)\,z_1(p{\cdot}t)$ for the positive coefficient
\[
c_1(t,p)=v(t,p,z_1(p))(x_0)\quad \thetaxt{for any } p\in P,\; t\geq 0\,,
\]
which defines a 1-dim differentiable linear cocycle: with the regularity conditions on $h$,
$y(t,x)= v(t,p,z_1(p))(x)$ is a classical solution of the IBV problem given by~\eqref{pdefamily} for $p\in P$ with $y(0,x)=z_1(p)(x)$, $x\in \bar U$. Therefore, the map $a(p):=\left.\frac{d}{dt} \ln c_1(t,p)\right|_{t=0}$ is well defined and continuous on $P$ and
\[
c_1(t,p)=\exp\int_0^t a(p{\cdot}s)\,ds\,,\quad p\in P,\; t\geq 0\,.
\]
\par
Note that the relation between $c(t,p)$ and $c_1(t,p)$ is given by
\[
c_1(t,p)=\frac{e(p{\cdot}t)(x_0)}{e(p)(x_0)}\,c(t,p)\,,\quad p\in P,\; t\geq 0\,,
\]
and there exist constants $c_0,\,C_0>0$ such that $c_0\leq e(p)(x_0)\leq C_0$ for any $p\in P$.
Besides, as commented in the second to last paragraph in the proof of Proposition~\ref{prop-convex}, for any $p\in P$ there exists the limit
\[
\lambda_P=\lim_{t\to\infty} \frac{\ln\|\phi(t,p)\,z_1(p)\|}{t} =\lim_{t\to\infty} \frac{\ln c_1(t,p)}{t}= \lim_{t\to\infty} \frac{1}{t}\,\int_0^t a(p{\cdot}s)\,ds=\int_P a\,d\nu\,,
\]
where the ergodic theorem of Birkhoff has been applied in the last equality. That is, $a\in C_0(P)$. Now the density of $B(P)$ in $C_0(P)$ permits us to find a sequence of maps $(k_n)_n\mathbb{S}ubset C_0(P)$ with $k_n\to 0$ as $n\to\infty$ and $a+k_n\in B(P)$ for every $n\geq 1$. Now, for each $n\geq 1$, we take $\omegait c_n(t,p)$ the associated 1-dim cocycle for $h+k_n$, which satisfies
\begin{align*}
\ln \omegait c_n(t,p)&= \ln c(t,p) +\int_0^t k_n(p{\cdot}s)\,ds = \ln \frac{e(p)(x_0)}{e(p{\cdot}t)(x_0)}+\ln c_1(t,p) +\int_0^t k_n(p{\cdot}s)\,ds\\
&=\ln \frac{e(p)(x_0)}{e(p{\cdot}t)(x_0)}+ \int_0^t a(p{\cdot}s)\,ds +\int_0^t k_n(p{\cdot}s)\,ds \,.
\end{align*}
Since $c_0\leq e(p)(x_0)\leq C_0$ for any $p\in P$ and $a+k_n\in B(P)$, from this relation it follows that for any $p\in P$, $\mathbb{S}up_{t\in\mathbb{R}}|\ln \omegait c_n(t,p)|<\infty$, meaning that $h+k_n\in B(P\times\bar U)$ for every $n\geq 1$. Since $h+k_n\to h$ as $n\to\infty$, we are done. The proof is finished.
\end{proof}
To finish this section, recall that in Proposition~\ref{prop-convex} we have proved the convexity of the upper Lyapunov exponent $\lambda_P(h)$. We now prove that it is strictly convex except for the case when the two maps $h_1$ and $h_2$ differ in a map $k(p)$.
\begin{teor}\label{prop-convex estricta}
Let $h_1,\,h_2 \in C(P\times\bar U)$ be of class $C^1$ in $x\in U$ and of class $C^1$ along the trajectories of $P$. Then, for any $0< r< 1$, $\lambda_P(r h_1+(1-r)h_2)=r\lambda_P( h_1)+(1-r)\lambda_P(h_2) $ if and only if $\nabla_x h_1= \nabla_x h_2$.
\end{teor}
\begin{proof}
First of all, we know that $\lambda_P(r h_1+(1-r)h_2)\leq r\lambda_P(h_1)+(1-r)\lambda_P(h_2)$ for any $h_1,\,h_2\in C(P\times\bar U)$. Now, if $\nabla_x h_1= \nabla_x h_2$, then $h_1=h_2 + k$ for some $k\in C(P)$. Then, using Proposition~\ref{prop-h+k}~(i) it is easy to check that $\lambda_P(r h_1+(1-r)h_2)= r\lambda_P(h_1)+(1-r)\lambda_P(h_2)$ for any $0< r< 1$. Note that in particular for maps $h_1,\,h_2\in C_0(P\times\bar U)$ this means that $r h_1+(1-r)h_2\in C_0(P\times\bar U)$ for any $0\leq r\leq 1$.
\par
For the converse, we first prove the result for maps in $B(P\times\bar U)$, then in $C_0(P\times\bar U)$ and finally in the general case.
\par
Assume that $h_1, h_2\in B(P\times\bar U)$ and $\displaystyle\frac{\partial h_1}{\partial x_i}\not=\displaystyle\frac{\partial h_2}{\partial x_i}$ for some $i\in\{1,\ldots,m\}$ and let us see that $\lambda_P(r h_1+(1-r)h_2)<0$ for any $0<r<1$. By Proposition~\ref{prop-compacto invariante} (ii) let $b_1,\,b_2: P\to \Int X_+ $ be continuous equilibria respectively for the linear skew-product semiflows
induced by the problems~\eqref{pdefamily} given by $h_1$ and $h_2$, and recall that they lie in the corresponding principle bundle.
\par
Then, as in the proof of Proposition~\ref{prop-convex}, we consider $b:P\to X$, $p\mapsto b(p)=\exp(r\ln b_1(p)+(1-r)\ln b_2(p))$, which satisfies: it is continuous, $C^{1\!}$ along the orbits in $P$, and for every $p\in P$
the map $\bar b:\mathbb{R}_+\times\bar U\to\mathbb{R},\;(t,x)\mapsto\bar b(t,x)= b(p{\cdot}t)(x)$
is continuously differentiable,
twice continuously differentiable in $x\in U$ and besides, denoting
\begin{equation*}
b'(p)(x)=\frac{\partial}{\partial t}b(p{\cdot}t)(x)|_{t=0}\,,\quad p\in P,\;\,x\in\bar U,
\end{equation*}
it holds
\begin{equation*}
\left\{\begin{array}{l} b'(p)(x) \geq
\Delta \, b(p)(x)+(r h_1(p,x)+(1-r) h_2(p,x))\,b(p)(x)\,,\quad p\in P,\;\,x\in\bar U,\\[.2cm]
B\bar b:=\alpha(x)\,\bar b+\displaystyle\frac{\partial \bar b}{\partial n} =0\,,\quad t>0\,,\;\, x\in \partial U.
\end{array}\right.
\end{equation*}
By Lemma~2.11 in N\'{u}\~{n}ez et al.~\cite{nuos3}, $b(p)$ defines a continuous super-equilibrium for the semiflow associated with the linear family~\eqref{pdefamily} with the term $r h_1+(1-r) h_2$, with associated linear cocycle $\Phi(t,p)$.
Let us now see that $ b(p)$ is a strong super-equilibrium. Once more according to Lemma~2.11 in~\cite{nuos3}, it suffices to find some $p_0\in P$ and $x_0\in U$ for which
\begin{equation}\label{strict}
b'(p_0)(x_0)>
\Delta \, b(p_0)(x_0)+(r h_1(p_0,x_0)+(1-r) h_2(p_0,x_0))\,b(p_0)(x_0)\,.
\end{equation}
\par
Now, it is not difficult to check that
\begin{align*}
\nabla_x h_1= \nabla_x h_2 &\Longleftrightarrow h_1= h_2 + k \;\thetaxt{ for some }k\in C(P) \\ & \Longleftrightarrow
b_1=\lambda\, b_2 \;\thetaxt{ for some positive map }\lambda\in C(P)\\&\Longleftrightarrow \nabla_x \ln b_1= \nabla_x \ln b_2\,.
\end{align*}
Since we are assuming that this is not the case, we deduce that there exists a $p_0\in P$ such that $\nabla_x \ln b_1(p_0) \not= \nabla_x \ln b_2(p_0)$, which means that there exists an $x_0\in U$ such that $\nabla_x \ln b_1(p_0)(x_0) \not= \nabla_x \ln b_2(p_0)(x_0$). That is, for some $i\in\{1,\ldots,m\}$,
\[
\frac{1}{ b_1(p_0)(x_0)} \frac{\partial b_1(p_0)(x_0)}{\partial x_i} \not= \frac{1}{ b_2(p_0)(x_0)} \frac{\partial b_2(p_0)(x_0)}{\partial x_i} \,,
\]
and going back to the calculations made in the proof of Proposition~\ref{prop-convex} for $z$, and recalling that $\mathbb{R}\to\mathbb{R}$, $s\mapsto s^2$ is a strictly convex map, we deduce that~\eqref{strict} holds, so that $b(p)$ is a strong super-equilibrium.
\par
Therefore, by considering any $\beta>0$, by linearity $\beta\, b(p)$ is a strong super-equilibrium too. This forces $\lim_{t\to\infty} \Phi(t,p)\,z=0$ for any $z\gg 0$. By Theorems~\ref{teor-equivalencias} and~\ref{teor-oscilacion} it cannot be $\lambda_P(r h_1+(1-r)h_2)=0$ and consequently $\lambda_P(r h_1+(1-r)h_2)<0$ (see also Sacker and Sell~\cite{sase94}), as we wanted to see.
\par
Next, we consider the case $h_1,h_2\in C_0(P\times\bar U)$. If some of them is not in $B(P\times\bar U)$, for instance $h_1\in \mathcal{U}(P\times\bar U)$, then, as seen in the proof of Theorem~\ref{teor-categoria} one can find a map $k\in C_0(P)$ such that $h_1+k \in B(P\times\bar U)$. But since $\lambda_P(h_1+k)=\lambda_P(h_1)$ and $\nabla_x (h_1+k) = \nabla_x h_1$, we can just replace $h_1$ by $h_1+k$ and apply the previous argument for maps in $B(P\times\bar U)$.
\par
Finally, it remains to deal with regular $h_1,h_2\in C(P\times\bar U)$. In this case we just need to note that $\lambda_P(h-\lambda_P(h))=0$ for any $h\in C(P\times\bar U)$, so that we fall into the previous case considered. The proof is finished.
\end{proof}
\mathbb{S}ection{Attractors for non-autonomous parabolic PDEs. The case $\lambda_P=0$}\label{sec-nonlinear}\noindent
In this section we consider a family of scalar linear-dissipative parabolic PDEs over a minimal, uniquely ergodic and aperiodic flow $(P,\theta,\mathbb{R})$, with Neumann or Robin boundary conditions, given for each $p\in P$ by
\begin{equation}\label{pdefamilynl}
\left\{\begin{array}{l} \displaystyle\frac{\partial y}{\partial t} =
\Delta \, y+h(p{\cdot}t,x)\,y+g(p{\cdot}t,x,y)\, \\
\;\;\;\;\;\;= \Delta \, y+ G(p{\cdot}t,x,y)\,,\quad t>0\,,\;\,x\in U,
\\[.2cm]
By:=\alpha(x)\,y+\displaystyle\frac{\partial y}{\partial n} =0\,,\quad t>0\,,\;\,x\in \partial U,\,
\end{array}\right.
\end{equation}
where $h\in C_0(P\times\bar U)=\{h\in C(P\times\bar U)\mid \lambda_P(h)=0\}$, and the nonlinear term $g:P\times \bar U\times \mathbb{R}\to \mathbb{R}$ is continuous and of class $C^1$ with respect to $y$ and satisfies the following conditions which render the equations dissipative:
\begin{itemize}
\item[(c1)] $g(p,x,0)=\displaystyle\frac{\partial g}{\partial y}(p,x,0)=0$ for any $p\in P$ and $x\in \bar U$;
\item[(c2)] $y\,g(p,x,y)\leq 0$ for any $p\in P$, $x\in \bar U$, and $y\in \mathbb{R}$;
\item[(c3)] $g(p,x,-y)=-g(p,x,y)$ for any $p\in P$, $x\in \bar U$, and $y\in \mathbb{R}$;
\item[(c4)] there exists an $r_0>0$ such that $g(p,x,y)=0$ if and only if $|y|\leq r_0$;
\item[(c5)] $\displaystyle\lim_{|y|\to\infty}\frac{g(p,x,y)}{y}=-\infty$ uniformly on $P\times\bar U$.
\end{itemize}
\par
Under these hypotheses, the {\it a priori\/} only locally defined skew-product semiflow
\begin{equation*}
\begin{array}{cccl}
\tau: & \mathbb{R}_+\times P\times X& \longrightarrow & \hspace{0.3cm}P\times X\\
& (t,p,z) & \mapsto
&(p{\cdot}t,u(t,p,z))\,,
\end{array}
\end{equation*}
induced by the mild solutions of the associated ACPs (see Section~\ref{sec-mild solutions}) is globally defined because of the boundedness of solutions, and it is strongly monotone as stated in Proposition~\ref{prop-strong monot}. Recall also that the section semiflow $\tau_t$ is compact for every $t>0$ (once more, see Travis and Webb~\cite{trwe}).
\par
Section 3.1 in Cardoso et al.~\cite{cardoso} is devoted to the existence of attractors for linear-dissipative parabolic PDEs of type~\eqref{pdefamilynl} with a general $h\in C(P\times \bar U)$ and conditions (c1), (c2), (c4) and (c5) for the nonlinear term (condition (c3) has been added here for the sake of simplicity). They prove that there exists an absorbing compact set for the semiflow, thanks to the presence of the nonlinear dissipative term $g(p,x,y)$ (see Proposition~2 in~\cite{cardoso}), so that $\tau$ has a global attractor $\mathbb{A}=\cup_{p\in P} \{p\}\times A(p)$ for the sets $A(p)=\{z\in X\mid (p,z)\in \mathbb{A}\}\mathbb{S}ubset X$, which is formed by bounded entire trajectories. Besides, in~\cite{cardoso} the structure of the attractor is studied in the cases $\lambda_P<0$ and $\lambda_P>0$, where $\lambda_P$ is the upper Lyapunov exponent of the linearized family along the null solution, which is of type~\eqref{pdefamily}:
\begin{equation*}
\left\{\begin{array}{l} \displaystyle\frac{\partial y}{\partial t} =
\Delta \, y+h(p{\cdot}t,x)\,y\,,\quad t>0\,,\;\,x\in U, \;\, \thetaxt{for each}\; p\in P,\\[.2cm]
By:=\alpha(x)\,y+\displaystyle\frac{\partial y}{\partial n} =0\,,\quad t>0\,,\;\,x\in \partial U.
\end{array}\right.
\end{equation*}
\par
In this section we concentrate on the unresolved case $\lambda_P=0$, by assuming that $h\in C_0(P\times\bar U)$; just the case that has been studied in detail in Section~\ref{sec-linear}. We keep the notation used up to now for the linear problem. In particular, $\tau_L$ is the induced linear skew-product semiflow.
\par
Note that, on the one hand, for each fixed $p\in P$, the family of compact sets $\{A(p{\cdot}t)\}_{t\in \mathbb{R}}$ is the pullback attractor for the process on $X$ defined, for each fixed $p\in P$, by $S_p(t,s)\,z=u(t-s,p{\cdot}s,z)$ for any $z\in X$ and $t\geq s$,
meaning that:
\begin{itemize}
\item[(i)] it is invariant, i.e., $S_p(t,s)\,A(p{\cdot}s)=A(p{\cdot}t)$ for any $t\geq s$;
\item[(ii)] it pullback attracts bounded subsets of $X$, i.e., for any bounded set $B\mathbb{S}ubset X$,
\[
\lim_{s\to -\infty}{\rm dist}(S_p(t,s)\,B,A(p{\cdot}t))=0 \quad \hbox{for any}\; t\in \mathbb{R}\,;
\]
\item[(iii)] it is the minimal family of closed sets with property (ii).
\end{itemize}
A nice reference for processes and pullback attractors is Carvalho et al.~\cite{calaro}.
\par
On the other hand, the non-autonomous set $\{A(p)\}_{p\in P}$ is a cocycle attractor for the non-autonomous dynamical system (see Section~\ref{sec-preli}). Besides, taking
\[
a(p)=\inf A(p)\quad\thetaxt{ and } \quad b(p)=\mathbb{S}up A(p) \quad\thetaxt{for any}\;\,p\in P,
\]
these are semicontinuous equilibria for $\tau$ and
\[
\mathbb{A}\mathbb{S}ubseteq \bigcup_{p\in P} \{p\}\times [a(p),b(p)]\,.
\]
\par
In our case $a(p)=-b(p)$, because of the odd character of the nonlinear term $g(p,x,y)$ in the $y$ variable assumed in (c3). Therefore, we are only going to concentrate on the properties of $b(p)$, but note that in the general case the properties of $b(p)$ can also be immediately transferred to $a(p)$. Finally, as stated in Proposition~3 in~\cite{cardoso}, the pullback attraction property of the cocycle attractor implies that fixed any $z_0\gg 0$, for $r>0$ large enough,
\begin{equation}\label{b(p)}
b(p)=\lim_{T\to\infty} u(T,p{\cdot}(-T),r z_0)\quad\thetaxt{for any}\;\,p\in P.
\end{equation}
\par
The aim of the next two results is to describe the structure of the global attractor depending on whether the map $h\in C_0(P\times\bar U)$ in the linear part lies in $B(P\times\bar U)$ or in $\mathcal{U}(P\times\bar U)$: in the first case, roughly speaking, $\mathbb{A}$ is a wide set, whereas in the second case it is a pinched set with a complex dynamical structure. In any case, the sections $A(p)$ of $\mathbb{A}$ are very thin sets in the infinite dimensional space $X$.
\begin{teor}\label{teor-estr atractor caso b}
Let $h \in B(P\times\bar U)$ and let $\omegaidehat e: P \to \Int X_+$ be the continuous map given in Proposition~$\ref{prop-compacto invariante}$ {\rm (ii)}.
Then, there exists an $r_*>0$ such that
\[
A(p)=\{r\,\omegaidehat e(p)\mid |r|\leq r_*\}\mathbb{S}ubset X_1(p)\quad\thetaxt{for any}\;\,p\in P.
\]
\end{teor}
\begin{proof}
The map $\omegaidehat e$ given in Proposition~\ref{prop-compacto invariante} (ii) defines a continuous equilibrium for the linear skew-product semiflow $\tau_L$, so that we have a family of continuous equilibria for the linear problem given by $\omegaidehat e_r(p)=r\,\omegaidehat e(p)$, $p\in P$ for each $r>0$. Due to condition (c4) in the nonlinear term, clearly for $r>0$ small enough $\omegaidehat e_r$ is also a continuous equilibrium for the nonlinear semiflow $\tau$. At this point we define
\[
r_*=\mathbb{S}up\{r>0\mid r\,\omegaidehat e(p) \leq \bar r_0 \,\hbox{ for any }\,p\in P\}\,,
\]
for the map $\bar r_0$ in $X$ identically equal to $r_0$, the constant given in (c4).
\par
Clearly, if $|r|\leq r_*$, $\omegaidehat e_r$ is a continuous equilibrium for $\tau$. Therefore, the set $\{r\,\omegaidehat e(p)\mid |r|\leq r_*\}\mathbb{S}ubseteq A(p)$ for any $p\in P$.
\par
By condition (c2) and Theorem~\ref{teor-comparacion}, we can compare solutions of the linear and the nonlinear problems. In particular, for $r>r_*$, $\omegaidehat e_r$ is a super-equilibrium for $\tau$, that is,
$\omegaidehat e_r(p{\cdot}t)\geq u(t,p,\omegaidehat e_r(p))$ for any $p\in P$ and $t\geq 0$, and since it is no longer an equilibrium, there are a $p_0\in P$ and a time $t_0>0$ such that $\omegaidehat e_r(p_0{\cdot}t_0)> u(t_0,p_0,\omegaidehat e_r(p_0))$. Let us compare solutions and apply the strong monotonicity of the nonlinear semiflow $\tau$ to get, for $t>0$, $
\omegaidehat e_r(p_0{\cdot}(t+t_0))=\phi(t,p_0{\cdot}t_0)\,\omegaidehat e_r(p_0{\cdot}t_0)\geq u(t,p_0{\cdot}t_0,\omegaidehat e_r(p_0{\cdot}t_0))\gg u(t,p_0{\cdot}t_0,u(t_0,p_0,\omegaidehat e_r(p_0))$,
that is, $\omegaidehat e_r(p_0{\cdot}(t+t_0))\gg u(t+t_0,p_0,\omegaidehat e_r(p_0))$. This implies that $\omegaidehat e_r$ is a strong super-equilibrium (see Novo et al.~\cite{nono2}).
Similar arguments to the ones used in the proof of Proposition~2 in~\cite{cardoso} lead to the fact that, fixed a $z_0\gg 0$, for $r$ large enough,
\[
\lim_{T\to\infty} u(T,p{\cdot}(-T),rz_0)=r_*\,\omegaidehat e(p)\quad\thetaxt{for any}\;\,p\in P,
\]
and thus, $b(p)=r_*\,\omegaidehat e(p)$ for any $p\in P$. Then, $\mathbb{A}$ is an invariant compact set also for the linear semiflow $\tau_L$. Since $\mathbb{A}$ is composed of bounded entire trajectories, by Proposition~\ref{prop-compacto invariante} (i) these entire trajectories lie in the principal bundle, that is, $A(p)\mathbb{S}ubset X_1(p)$, and consequently $ A(p)=\{r\,\omegaidehat e(p)\mid |r|\leq r_*\}$ for any $p\in P$, as we wanted to prove.
\end{proof}
We remark that in the previous situation, $b(p)$ is a continuous equilibrium.
\par
In the following theorem the subindexes s and f respectively stand for second and first category sets in the Baire sense. The result can be rephrased by saying that the presence of a pinched global attractor is generic in $C_0(P\times\bar U)$ (see Theorem~\ref{teor-categoria}).
\begin{teor}\label{teor-estr atractor caso u}
Let $h \in \mathcal{U}(P\times\bar U)$. Then, the global attractor $\mathbb{A}$ is a pinched set. More precisely:
\begin{itemize}
\item[(i)] There exists an invariant residual set $P_{\rm{s}}\mathbb{S}ubsetneq P$ such that $b(p)=0$ for any $p\in P_{\rm{s}}$. In fact $P_{\rm{s}}$ is the set of continuity points of $b$.
\item[(ii)] The set $P_{\rm{f}}=P\mathbb{S}etminus P_{\rm{s}}$ is an invariant dense set of first category and $b(p)\gg 0$ for any $p\in P_{\rm{f}}$.
\end{itemize}
\end{teor}
\begin{proof}
According to Proposition~\ref{prop-compacto invariante} (iii) there exist a $p_0\in P$ and a bounded entire solution $v:\mathbb{R}\to X_+\mathbb{S}etminus \{0\}$ of~\eqref{abstract eq} along the orbit of $p_0$ such that $K_0=\cls \{(p_0{\cdot}t, v(t))\mid t\in \mathbb{R}\}\mathbb{S}ubset P\times (\Int X_+\cup \{0\})$ is a pinched invariant compact set for the linear semiflow $\tau_L$. Taking $\delta>0$, the set $K_\delta=\{(p,\delta z)\mid (p,z)\in K_0\}\mathbb{S}ubset P\times (\Int X_+\cup \{0\})$ is still a pinched $\tau_L$-invariant compact set, and if $\delta$ is small enough so that $\|z\|\leq r_0$ for any $(p,z)\in K_\delta$ ($r_0$ the one given in condition (c4) for $g$), then $K_\delta$ is also a pinched invariant compact set for the nonlinear semiflow $\tau$. Therefore, $K_\delta\mathbb{S}ubset \mathbb{A}$ and for every $p\in P$, either $b(p)=0$ or $b(p)\gg 0$.
\par
Let $P_{\rm{s}}$ be the set of continuity points of $b$, which is a residual set. Theorem~7 in~\cite{cardoso} asserts that either there exists a $\lambda_0>0$ such that $b(p)\geq \lambda_0\, e_0$ for every $p\in P$ (where $e_0$ has been taken to be the first eigenfunction for~\eqref{bvp} but it might be any $e_0\gg 0$) or $b(p)=0$ for any $p\in P_{\rm{s}}$. So, assume by contradiction that $b(p)\geq \lambda_0\, e_0$ for every $p\in P$ and take a $p_1\in P_{\rm{o}}$, the set given in Theorem~\ref{teor-oscilacion}. Then, there exists a sequence $(t_n)_n\uparrow \infty$ such that $\lim_{n\to\infty}\|\phi(t_n,p_1)\,r e_0\|=0$ for any $r>0$. But if we take $r>0$ big enough so that $r e_0\geq b(p_1)$, then using Theorem~\ref{teor-comparacion} to compare solutions of the linear and nonlinear problems, $\phi(t_n,p_1)\,r e_0\geq u(t_n,p_1,r e_0)\geq u(t_n,p_1,b(p_1)) = b(p_1{\cdot}t_n)\geq \lambda_0\, e_0$ for every $n\geq 1$, which is a contradiction. Therefore, $b(p)=0$ for any $p\in P_{\rm{s}}$, and since the pinched set $K_\delta\mathbb{S}ubset \mathbb{A}$, also $\mathbb{A}$ is pinched.
\par
Note that if $b(p)=0$ for some $p\in P$, $p$ is a continuity point for $b$, so that $P_{\rm{s}}=\{p\in P\mid b(p)=0\}$. From here it follows that the set $P_{\rm{s}}$ is invariant, since $b(p)=0$ implies $b(p{\cdot}t)=u(t,p,b(p))=0$ for any $t\geq 0$, because the null map is a solution of the nonlinear problem; and if for a $t>0$ it were $b(p{\cdot}(-t))\gg 0$, it would be $u(t,p{\cdot}(-t),b(p{\cdot}(-t)))=b(p)\gg 0$ by the strong monotonicity. Therefore, it is straightforward that $P_{\rm{f}}=P\mathbb{S}etminus P_{\rm{s}}$ is an invariant dense set of first category and $b(p)\gg 0$ for any $p\in P_{\rm{f}}$. The proof is finished.
\end{proof}
From now on, we restrict attention to maps $h\in\mathcal{U}(P\times\bar U)$. For a further study of the dynamics of the global attractor $\mathbb{A}$, the sets $P_{\rm{s}}$ and $P_{\rm{f}}$ given in Theorem~\ref{teor-estr atractor caso u} play a fundamental role. Note that, roughly speaking, if $\nu(P_{\rm{s}})=1$, the boundary maps $a(p)$ and $b(p)$ of $\mathbb{A}$ touch each other over a set of full measure, whereas if $\nu(P_{\rm{f}})=1$ these maps only coincide over a set of null measure. \par
We state a technical result, which characterizes the points in the sets $P_{\rm{f}}$ and $P_{\rm{s}}$ in terms of the behaviour for negative times of the 1-dim linear cocycle $c(t,p)$ given in Definition~\ref{defi-cociclo c}. Recall that $e(p)\gg 0$, $p\in P$ are the generators of the principal bundle in the continuous separation of $\tau_L$.
\begin{prop}\label{prop-pasado c}
Let $h \in \mathcal{U}(P\times\bar U)$. Then, for $p\in P$:
\begin{itemize}
\item[(i)] $p\in P_{\rm{f}}$ if and only if $\displaystyle\mathbb{S}up_{t\leq 0} c(t,p)<\infty$;
\item[(ii)] $p\in P_{\rm{s}}$ if and only if $\displaystyle\mathbb{S}up_{t\leq 0} c(t,p)=\infty$.
\end{itemize}
\end{prop}
\begin{proof}
It clearly suffices to prove (i). Assume that $\mathbb{S}up_{t\leq 0} c(t,p)<\infty$ for a certain $p\in P$. Given $r_0$ in condition (c4) for $g$, $\delta\,c(t,p)\,e(p{\cdot}t)(x)\leq r_0$ for any $t\leq 0$ and $x\in \bar U$ provided that $\delta>0$ is small enough. Then, the solution $u(t,p,\delta\,e(p))$ coincides with the solution $\phi(t,p)\,\delta\,e(p)=\delta\,c(t,p)\,e(p{\cdot}t)$ of the linear abstract problem for negative time, and it is also bounded in forwards time. That is, it is an entire bounded solution which then lies in the global attractor, i.e., $u(t,p,\delta\,e(p))\in A(p{\cdot}t)$ for any $t\in \mathbb{R}$. In particular, $0\ll \delta\,e(p)\in A(p)$ which implies $b(p)\gg 0$, i.e., $p\in P_{\rm{f}}$.
\par
Now assume by contradiction that $\mathbb{S}up_{t\leq 0} c(t,p)=\infty$ for a certain $p\in P_{\rm{f}}$. Since $\{b(p{\cdot}t)\mid t\in \mathbb{R}\}$ is bounded in $X$, there exists a $C>0$ large enough such that $b(p{\cdot}t)\leq C\,e(p{\cdot}t)$ for any $t\in \mathbb{R}$. Then, by monotonicity, Theorem~\ref{teor-comparacion} and~\eqref{c},
\begin{align*}
0\ll b(p)&=u(t,p{\cdot}(-t),b(p{\cdot}(-t)))\leq u(t,p{\cdot}(-t),C\,e(p{\cdot}(-t)))\\&\leq C\,\phi(t,p{\cdot}(-t))\,e(p{\cdot}(-t))=C\, c(t,p{\cdot}(-t))\,e(p)\,\quad \hbox{for any}\;t>0\,.
\end{align*}
\par
By the linear cocycle property $c(t,p{\cdot}(-t))=1/c(-t,p)$,
and from the hypothesis we can take a sequence $(t_n)_n$ of positive times such that $\lim_{n\to\infty} c(-t_n,p)=\infty$. Then it follows that $\lim_{n\to\infty}c(t_n,p{\cdot}(-t_n))=0$ and consequently $b(p)=0$, which is a contradiction. The proof is finished.
\end{proof}
\begin{coro}\label{coro-1}
Let $h \in \mathcal{U}(P\times\bar U)$. Then the oscillation set $P_{\rm{o}}$ of the cocycle $c(t,p)$ satisfies $P_{\rm{o}}\mathbb{S}ubseteq P_{\rm{s}}$.
\end{coro}
\par
When looking at the forwards cocycle $c(t,p)$ for $t\geq 0$, the result is the following.
\begin{prop}\label{prop-Rf o Rs}
Let $h\in\mathcal{U}(P\times\bar U)$ and fix a $z_0\gg 0$ in $X$. Then:
\begin{itemize}
\item[(i)] $\nu(P_{\rm{s}})=1$ $\Leftrightarrow$ $\mathbb{S}up_{t\geq 0} \|\phi(t,p)\,z_0\|=\infty$ for a.e.~$p\in P$ $\Leftrightarrow$ $\displaystyle\mathbb{S}up_{t\geq 0} c(t,p)=\infty$ for a.e.~$p\in P$;
\item[(ii)] $\nu(P_{\rm{f}})=1$ $\Leftrightarrow$ $\mathbb{S}up_{t\geq 0} \|\phi(t,p)\,z_0\|<\infty$ for a.e.~$p\in P$ $\Leftrightarrow$ $\displaystyle\mathbb{S}up_{t\geq 0} c(t,p)<\infty$ for a.e.~$p\in P$.
\end{itemize}
\end{prop}
\begin{proof}
First of all, note that $\mathbb{S}up_{t\geq 0} \|\phi(t,p)\,z_0\|=\infty$ if and only if $\mathbb{S}up_{t\geq 0} c(t,p)=\infty$: it suffices to recall relation~\eqref{c}, take constants $\lambda_1,\lambda_2>0$ such that $\lambda_1\,e(p)\leq z_0\leq \lambda_2\,e(p)$ and apply the monotonicity of both the semiflow and the norm. Second, note that by the cocycle property the set $\{p\in P\mid \mathbb{S}up_{t\geq 0} c(t,p)=\infty\}$ is invariant, so that its measure is either full or null. Thus, we just need to prove (i).
\par
So, assume first that $\nu(P_{\rm{s}})=1$. By Theorem~\ref{teor-shneiberg} for almost every $p\in P$ there exists a sequence $(t_n)_n\uparrow \infty$ such that $c(t_n,p)=1$ for any $n\geq 1$. As a consequence, the set of the so-called {\it recurrent points at $\infty$\/},
\[
\{p\in P\mid \thetaxt{there exists a sequence }(t_n)_n\uparrow \infty \thetaxt{ such that } \lim_{n\to\infty}c(t_n,p)=1\},
\]
has full measure. An application of Fubini's theorem permits to see that for almost every recurrent point, its orbit is made of recurrent points too, so that we can consider the invariant set of full measure
\begin{equation*}
P_{\rm{r}}^+=\{p\in P \mid p{\cdot}t \;\thetaxt{is recurrent at $\infty$ for every}\;t\in\mathbb{R} \}\,.
\end{equation*}
\par
Now, if we take a $p\in P_{\rm{s}}\cap P_{\rm{r}}^+$, on the one hand, by Proposition~\ref{prop-pasado c} we can take a sequence $(t_n^1)_n\downarrow -\infty$ such that $c(t_n^1,p)\to \infty$ as $n\to\infty$. On the other hand, since for any $n\geq 1$, $p{\cdot}t_n^1$ is recurrent at $\infty$, we can find a sequence $(t_n^2)_n\uparrow \infty$ such that $c(t_n^2-t_n^1,p{\cdot}t_n^1)\to 1$ as $n\to\infty$. Then, by the cocycle property,
\[
c(t_n^2,p)=c(t_n^2-t_n^1,p{\cdot}t_n^1)\,c(t_n^1,p)\to \infty \;\thetaxt{ as }\; n\to \infty\,,
\]
so that $\mathbb{S}up_{t\geq 0} c(t,p)=\infty$ for almost every $p$ in $P$.
\par
Conversely, assume that $\mathbb{S}up_{t\geq 0} c(t,p)=\infty$ for almost every $p\in P$ and consider the cocycle $\omegaidehat c(t,p)=c(-t,p)$ ($t\in\mathbb{R}$, $p\in P$) for the time-reversed flow on $P$ given by $\omegaidehat \theta:\mathbb{R}\times P\to P$, $(t,p)\mapsto p{\cdot}(-t)$. Theorem~\ref{teor-shneiberg} applied to this cocycle ensures that for almost every $p\in P$ there exists a sequence $(t_n)_n\uparrow \infty$ such that $c(-t_n,p)=1$ for any $n\geq 1$. As a consequence, the set of the so-called {\it recurrent points at $-\infty$\/}
\[
\{p\in P\mid \thetaxt{there exists a sequence }(t_n)_n\uparrow \infty \thetaxt{ such that } \lim_{n\to\infty}c(-t_n,p)=1\}
\]
has full measure. At this point, a parallel argument to the former one permits to conclude that for almost every $p\in P$, $\mathbb{S}up_{t\leq 0} c(t,p)=\infty$. In other words, by Proposition~\ref{prop-pasado c}, $\nu(P_{\rm{s}})=1$, as we wanted to see.
\end{proof}
\mathbb{S}ubsection{The case $\nu(P_{\rm{f}})=1$: chaotic dynamics in the attractor}
In this section we show the presence of chaos, in a very precise sense, inside the attractor, when $h\in \mathcal{U}(P\times\bar U)$ is such that $\nu(P_{\rm{f}})=1$.
\par
We remark that this case often occurs. The references Johnson~\cite{john81} and Ortega and Tarallo~\cite{orta} provide examples (based on a previous construction by Anosov~\cite{anos}) of quasi-periodic flows $(P,\theta,\mathbb{R})$ and maps $k\in \mathcal{U}(P)$ with $\mathbb{S}up_{t\in\mathbb{R}}\int_0^t k(p{\cdot}s)\,ds<\infty$ for almost every $p\in P$. In fact these results are expected to be true in a more general setting. As a consequence, using the methods in Proposition~\ref{prop-h+k} and Proposition~\ref{prop-Rf o Rs}, we can assert that for any regular map $h\in C_0(P\times\bar U)$ there exists a map $k_1\in C_0(P)$ such that the map $h+k_1\in\mathcal{U}(P\times\bar U)$ and it satisfies $\nu(P_{\rm{f}})=1$. To see it, recall that for every regular $h\in C_0(P\times\bar U)$ we can find a $k_2\in C_0(P)$ such that $h+k_2\in B(P\times\bar U)$ (see the proof of Theorem~\ref{teor-categoria}) and just take $k_1=k_2+k$.
\par
For the reader not familiar with the notion of chaos in the sense of Li and Yorke~\cite{liyo}, we include the definition.
\begin{defi}\label{defi-li yorke}
Let $(K,\mathbb{S}igma,\mathbb{R})$ be a continuous flow on a compact metric space $(K,d)$. (i) A pair $\{x,y\}\mathbb{S}ubset K$ is called a {\it Li-Yorke pair\/} if
\[
\liminf_{t\to \infty} d(\mathbb{S}igma_tx,\mathbb{S}igma_ty)=0\;\quad\thetaxt{and}\;\quad \limsup_{t\to \infty} d(\mathbb{S}igma_tx,\mathbb{S}igma_ty)>0\,.
\]
\par
(ii) A set $D\mathbb{S}ubseteq K$ is said to be {\it scrambled\/} if every pair $\{x,y\}\mathbb{S}ubset D$ with $x\not=y$ is a Li-Yorke pair.
\par
(iii) The flow on $K$ is said to be {\it chaotic in the Li-Yorke sense\/} if there exists an uncountable scrambled set in $K$.
\end{defi}
Some dynamical properties associated to the Li-Yorke chaos and its relation with other notions of chaotic dynamics can be found in Blanchard et al.~\cite{bgkm}.
\par
Note that the restriction of the skew-product semiflow $\tau$ to the global attractor $\mathbb{A}$ is a continuous flow on a compact metric space. For almost periodic equations the base flow $(P,\theta,\mathbb{R})$ is almost periodic, and thus it is also distal. Consequently, in that case, if $\{(p_1,z_1),(p_2,z_2)\}\mathbb{S}ubset P\times X$ is a Li-Yorke pair, necessarily $p_1=p_2$. This motivates the following definition. The subindex ch stands for chaos.
\begin{defi}\label{defi-li yorke in measure}
The global attractor $\mathbb{A}$ is said to be {\it fiber-chaotic in measure in the sense of Li-Yorke\/} if there exists a set $P_{\rm{ch}}\mathbb{S}ubset P$ of full measure such that for every $p\in P_{\rm{ch}}$, $\{p\}\times A(p)$ is an uncountable scrambled set.
\end{defi}
Note that, if pairs in $\{p\}\times A(p)$ are to exist, it must be $p\in P_{\rm{f}}$. In other words, $P_{\rm{ch}}\mathbb{S}ubseteq P_{\rm{f}}$. Recall that we are assuming that $\nu(P_{\rm{f}})=1$ for the first category set $P_{\rm{f}}$, so that there is a chance for chaos in measure. Note also that this notion is different and complements in some way the notion of residually Li-Yorke chaotic sets analyzed by some authors in the context of skew-product flows; for instance, see Bjerklov and Johnson~\cite{bjjo} and Huang and Yi~\cite{huyi}.
\par
We now give a technical and fundamental result for our purposes, which is a nontrivial generalization of Theorem~35 in~\cite{caloNonl} to this infinite-dimensional setting. Basically, it says that with full measure the attractor consists of entire bounded trajectories of the linear semiflow $\tau_L$. The subindex l stands for linear.
\begin{teor}\label{teor-casi siempre zona lineal}
Let $h\in\mathcal{U}(P\times\bar U)$ be such that $\nu(P_{\rm{f}})=1$. For the constant $r_0>0$ given in condition {\rm (c4)}, let $\bar r_0\in X$ be the identically equal to $r_0$ map defined on $\bar U$.
Then, there exists an invariant set of full measure $P_{\rm{l}}\mathbb{S}ubset P_{\rm{f}}$ such that $0\ll b(p)\leq \bar r_0$ for every $p\in P_{\rm{l}}$.
\end{teor}
\begin{proof}
To see that the invariant set $\{p\in P\mid b(p{\cdot}t)\leq \bar r_0 \;\forall\;t\in \mathbb{R}\}$ has full measure, let us assume by contradiction that its complementary set
\[
D=\{p\in P\mid \thetaxt{there exist a } t\in \mathbb{R} \thetaxt{ and an } x\in \bar U \thetaxt{ such that } b(p{\cdot}t)(x) > r_0\}
\]
has measure one. Note that by Theorem~\ref{teor-estr atractor caso u}, $D\mathbb{S}ubset P_{\rm{f}}$ and $b(p)\gg 0$ for any $p\in D$.
\par
Recall that the metric space $X=C(\bar U)$ is separable (in particular it is a second-countable topological space) and the measure $\nu$ is a regular Borel measure. In these conditions, we can apply the general form of the classical Lusin's theorem (for instance, see Feldman~\cite{feld}) to the semicontinuous (thus measurable) function $b:P\to X$ to affirm that, fixed an $\varepsilon>0$, there exists a continuous map $\omegait b:P\to X$ such that $\nu(\{p\in P\mid b(p)=\omegait b(p)\})>1-\varepsilon$. Since $\nu$ is regular, we can take a compact set $E_0\mathbb{S}ubset D\cap \{p\in P\mid b(p)=\omegait b(p)\}$ with $\nu(E_0)>0$.
\par
A standard application of Birkhoff's ergodic theorem to the characteristic function of $E_0$ implies that for almost every $p\in P$ there exists a real sequence $(t_n)_n\uparrow \infty$
such that $p{\cdot}t_n\in E_0$ for every $n\geq 1$. Once more, since $\nu$ is a regular measure, we can take a compact set $E_1$ with $\nu(E_1)>0$ such that
\[
E_1\mathbb{S}ubset \{p\in E_0\mid \thetaxt{there exists a sequence }(t_n)_n\uparrow \infty \thetaxt{ with } p{\cdot}t_n\in E_0 \;\forall \,n\geq 1 \}\,.
\]
Finally, consider
\[
E_2= \{p\in E_1\mid \thetaxt{there exists a sequence }(s_n)_n\uparrow \infty \thetaxt{ with } p{\cdot}s_n\in E_1 \;\forall \,n\geq 1 \}
\]
which, again by Birkhoff's ergodic theorem, has $\nu(E_2)=\nu(E_1)>0$. Since the proof is rather technical, we continue with a series of statements to make it easier to read.
\par
{\it Statement 1\/}: $E_1\mathbb{S}ubset D_+$, for the set
\[
D_+= \{p\in P\mid \thetaxt{there exist a } t>0 \thetaxt{ and an } x\in \bar U \thetaxt{ such that } b(p{\cdot}t)(x) > r_0\}\,.
\]
\par
{\it Proof\/}. As a first step, let us prove that there exists a $T_0>0$ such that for any $p\in E_0$ there exist a $t=t(p)$ with $|t|\leq T_0$ and an $x=x(p)\in\bar U$ with $b(p{\cdot}t)(x) > r_0$. This follows from a compactness argument: note that for a fixed $p\in E_0$ there exist a $t=t(p)\in \mathbb{R}$ and an $x=x(p)\in \bar U$ such that $b(p{\cdot}t)(x) > r_0$. Since $b(p{\cdot}t)(x)=u(t,p,b(p))(x)=u(t,p,\omegait b(p))(x)$, by continuity, there exists a ball $B(p,\delta(p))$ for an appropriate $\delta(p)>0$ such that for any $\omegait p\in B(p,\delta(p))\cap E_0$, also $u(t,\omegait p,\omegait b(\omegait p))(x)=b(\omegait p{\cdot}t)(x) > r_0$. Then, since $E_0\mathbb{S}ubset \cup_{p\in E_0} B(p,\delta(p))\cap E_0$, there is a finite covering, say $E_0\mathbb{S}ubset \cup_{i=1}^N B(p_i,\delta(p_i))\cap E_0$ and it suffices to take $T_0=\max\{|t(p_1)|,\ldots,|t(p_N)|\}$.
\par
Now, to finish, take $p\in E_1$ and let us check that $p\in D_+$. Take $s>T_0$ with $p{\cdot}s\in E_0$ and apply the first step: then, there exist a $t=t(p{\cdot}s)$ with $|t|\leq T_0$ and an $x=x(p{\cdot}s)\in\bar U$ with $b(p{\cdot}(t+s))(x) > r_0$. Since $t+s>0$, $p\in D_+$ and we are done.
\par
{\it Statement 2\/}: If $p_2\in E_2$ and $p_2{\cdot}s_n\in E_1$, $n\geq 1$ for a sequence $(s_n)_n\uparrow\infty$, then, $\lim_{n\to\infty} \|\phi(s_n-1,p_2)\,b(p_2)\|=\infty$.
\par
{\it Proof\/}. Argue by contradiction and assume without loss of generality that $\{\phi(s_n-1,p_2)\,b(p_2)\mid n\geq 1\}$ is a bounded set in $X$. Once more arguing as in Proposition~2.4 in Travis and Webb~\cite{trwe}, we obtain that the set $\{\phi(s_n,p_2)\,b(p_2)\mid n\geq 1\}$ is relatively compact: just write $\phi(s_n,p_2)\,b(p_2)=\phi(1,p_2{\cdot}(s_n-1))\,\phi(s_n-1,p_2)\,b(p_2)$.
Thus, the set $\{\phi(s_n,p_2)\,b(p_2)\mid n\geq 1\}$ has at least a limit point. Taking a subsequence if necessary, we can assume that $p_2{\cdot}s_n\to p_1\in E_1$ and $\phi(s_n,p_2)\,b(p_2)\to z$ as $n\to\infty$. Since $p_2{\cdot}s_n,\,p_1\in E_1\mathbb{S}ubset E_0$ for any $n\geq 1$, then $b(p_2{\cdot}s_n)\to b(p_1)\gg 0$. Comparing solutions of the nonlinear and the linear problems, $b(p_2{\cdot}s_n)\leq \phi(s_n,p_2)\,b(p_2)$ for any $n\geq 1$, so that in the limit $0\ll b(p_1)\leq z$.
\par
By Statement~1, for $p_1\in E_1\mathbb{S}ubset D_+$, there exist a $t_1>0$ and an $x_1\in\bar U$ such that $b(p_1{\cdot}t_1)(x_1) > r_0$. If we look at the solution $b(p_1{\cdot}t)$, $t\geq 0$ of the nonlinear problem for $p_1$, it lies below the solution $\phi(t,p_1)\,b(p_1)$ of the linear problem with the same initial condition $b(p_1)$. Since at time $t_1$, $b(p_1{\cdot}t_1)(x_1) > r_0$, the zone where the problem is strictly nonlinear, there must exist a time $t_2>t_1$ such that $b(p_1{\cdot}t_2)< \phi(t_2,p_1)\,b(p_1)$. Now, once more comparing solutions and applying the strong monotonicity of the semiflow, for any $t> t_2$, $b(p_1{\cdot}t)\ll \phi(t,p_1)\,b(p_1)$. Let us fix a time $t_3>0$ such that $b(p_1{\cdot}t_3)\ll \phi(t_3,p_1)\,b(p_1)$.
\par
Let $\gamma_1\geq 1$ be the biggest possible such that $0\ll \gamma_1\,b(p_1)\leq z$. We can then take a sufficiently close $\gamma_2>\gamma_1$ such that
\[
\gamma_2\,b(p_1{\cdot}t_3)\ll \phi(t_3,p_1)\,\gamma_1\,b(p_1)\,.
\]
\par
Now, since $\lim_{n\to\infty} \phi(s_n+t_3,p_2)\,b(p_2)=\lim_{n\to\infty} \phi(t_3,p_2{\cdot}s_n)\,\phi(s_n,p_2)\,b(p_2)=\phi(t_3,p_1)\,z \geq \phi(t_3,p_1)\,\gamma_1\,b(p_1)\gg \gamma_2\,b(p_1{\cdot}t_3)= \gamma_2\,\lim_{n\to\infty} b(p_2{\cdot}(s_n+t_3))$ (recall that $b$ is an equilibrium for the nonlinear problem), we deduce that there exists an $n_0\in\mathbb{N}$ such that for any $n\geq n_0$, $\phi(s_n+t_3,p_2)\,b(p_2)\geq \gamma_2\,b(p_2{\cdot}(s_n+t_3))$. For any $n\geq n_0$ such that $s_n>s_{n_0}+t_3$ we write $s_n=r_n+s_{n_0}+t_3$ for a positive $r_n$. Then, $\phi(s_n,p_2)\,b(p_2)=\phi(r_n,p_2{\cdot}(s_{n_0}+t_3))\,\phi(s_{n_0}+t_3,p_2)\,b(p_2)\geq \phi(r_n,p_2{\cdot}(s_{n_0}+t_3))\,\gamma_2\,b(p_2{\cdot}(s_{n_0}+t_3))\geq \gamma_2\,b(p_2{\cdot}(r_n+s_{n_0}+t_3))=\gamma_2\,b(p_2{\cdot}s_n)$. Taking limits as $n\to \infty$, we deduce that $z\geq \gamma_2\,b(p_1)$ with $\gamma_2>\gamma_1$, in contradiction with the definition of $\gamma_1$. We are done.
\par
{\it Statement 3\/}: For any $p_2\in E_2$, $\mathbb{S}up_{t\geq 0} c(t,p_2)=\infty$.
\par
{\it Proof\/}. Since for $p_2\in E_2$, $b(p_2)\gg 0$, there exists a $\gamma>0$ such that $e(p_2)\geq \gamma\,b(p_2)$, so that by monotonicity $c(t,p_2)\,e(p_2{\cdot}t)=\phi(t,p_2)\,e(p_2)\geq \gamma\,\phi(t,p_2)\,b(p_2)$ for any $t\geq 0$. The boundedness of $e(p_2{\cdot}t)$ for $t\geq 0$ and Statement~2 then imply that $\mathbb{S}up_{t\geq 0} c(t,p_2)=\infty$, as wanted.
\par
To finish the proof, note that, since $E_2 \mathbb{S}ubset P_{\rm{f}}$, Statement 3 falls into contradiction with Proposition~\ref{prop-Rf o Rs}. Therefore, the invariant set $\{p\in P\mid b(p{\cdot}t)\leq \bar r_0 \;\forall\;t\in \mathbb{R}\}$ has full measure and it suffices to take the intersection of this set with $P_{\rm{f}}$ to obtain the set $P_{\rm{l}}$ in the statement of the theorem. The proof is finished.
\end{proof}
We can now prove that there is chaos in the global attractor.
\begin{teor}\label{teor-caos}
Let $h\in\mathcal{U}(P\times\bar U)$ be such that $\nu(P_{\rm{f}})=1$. Then, the global attractor $\mathbb{A}$ is fiber-chaotic in measure in the sense of Li-Yorke.
\end{teor}
\begin{proof}
For the set $P_{\rm{l}}\mathbb{S}ubset P_{\rm{f}}$ given in Theorem~\ref{teor-casi siempre zona lineal}, let us take a compact set $E_0\mathbb{S}ubset P_{\rm{l}}$ with $\nu(E_0)>0$ such that the restriction $b|_{E_0}$ is continuous (which exists by the generalized Lusin's theorem) and consider the set of full measure
\[
P_{\rm{ch}}= \{p\in P_{\rm{l}} \mid \thetaxt{there exists a sequence }(s_n)_n\uparrow \infty \thetaxt{ with } p{\cdot}s_n\in E_0 \;\forall \,n\geq 1 \}\,.
\]
\par
Now, take a $p\in P_{\rm{ch}}$, and let us see that any pair of distinct points $(p,z_1),\,(p,z_2)\in \{p\}\times A(p)\mathbb{S}ubset \{p\}\times [-b(p),b(p)]$ is a Li-Yorke pair. Since $p\in P_{\rm{l}}$, $b(p)\gg 0$ and the orbits $b(p{\cdot}t)$ and $u(t,p,z_i)$ for $i=1,2$ lie, roughly speaking, in the linear zone of the problem, so that they are entire bounded trajectories for the linear skew-product semiflow, and by Proposition~\ref{prop-compacto invariante} (i) they lie inside the principal bundle. As a consequence $z_1, z_2, b(p) \in X_1(p)$ and there exist distinct $\lambda_1,\,\lambda_2\in\mathbb{R}$ such that
\[
\|u(t,p,z_2)-u(t,p,z_1)\|=|\lambda_2-\lambda_1|\,\|b(p{\cdot}t)\| \quad\hbox{for any}\; t\geq 0\,.
\]
\par
Then, first take a sequence $(s_n)_n\uparrow \infty$ with $p{\cdot}s_n\in E_0$ for every $n\geq 1$ to obtain that $\limsup_{t\to\infty}\|u(t,p,z_2)-u(t,p,z_1)\|>0$, since $b\gg 0$ over the compact set $E_0$ where $b$ is continuous. Second, we now take a $p_0\in P_{\rm{s}}$ and a sequence $(t_n)_n\uparrow \infty$ with $p{\cdot}t_n\to p_0$ as $n\to\infty$. By Theorem~\ref{teor-estr atractor caso u}, $\lim_{n\to\infty} b(p{\cdot}t_n)=b(p_0)=0$, so that we can conclude that $\liminf_{t\to\infty}\|u(t,p,z_2)-u(t,p,z_1)\|=0$. The proof is finished.
\end{proof}
\begin{nota} Since the set $P_{\rm{ch}}$ is also invariant, the previous dynamical behaviour can be interpreted in the formulation of processes, by saying that for every $p\in P_{\rm{ch}}$ the pullback attractor $\{A(p{\cdot}t)\}_{t\in \mathbb{R}}$ for the process in $X$, $S_p(t,s)(\,{\cdot}\,)=u(t-s,p{\cdot}s,\,{\cdot}\,)$ ($t\geq s$), is Li-Yorke chaotic.
\end{nota}
As a consequence of the next result, we can affirm that the closed set
\[
\mathbb{F}=\bigcup_{p\in P} \{p\}\times[-b(p),b(p)]\mathbb{S}ubset P\times X
\]
is also fiber-chaotic in measure in the sense of Li-Yorke, meaning that for a subset of $P$ with full measure its sections contain a big uncountable scrambled set. Recall that we have denoted by $\Pi_{1,p}:X\to X_1(p)$, and $\Pi_{2,p}:X\to X_2(p)$ ($p\in P$) the projections over the subspaces of the continuous separation for $\tau_L$.
\begin{prop}
Let $h\in\mathcal{U}(P\times\bar U)$ be such that $\nu(P_{\rm{f}})=1$. Consider the complete metric space $\mathbb{F}$ which is positively invariant for $\tau$. Given a pair of distinct points $(p,z_1), (p,z_2)\in \mathbb{F}$ with $p\in P_{\rm{ch}}$, two things can happen:
\begin{itemize}
\item[(i)] either $\Pi_{1,p}(z_1)=\Pi_{1,p}(z_2)$, and then it is an asymptotic pair, meaning that
\[
\lim_{t\to\infty}\|u(t,p,z_2)-u(t,p,z_1)\|=0\,;
\]
\item[(ii)] or $\Pi_{1,p}(z_1)\not=\Pi_{1,p}(z_2)$, and then it is a Li-Yorke pair.
\end{itemize}
\end{prop}
\begin{proof}
First note that since $P_{\rm{ch}}\mathbb{S}ubset P_{\rm{l}}$ for the set $P_{\rm{l}}$ given in Theorem~\ref{teor-casi siempre zona lineal}, $b(p)\gg 0$ and the semiorbits of the pairs $(p,z_1), (p,z_2)\in \mathbb{F}$ for $\tau$ are actually semiorbits for the linear semiflow $\tau_L$, so that $\|u(t,p,z_2)-u(t,p,z_1)\|=\|\phi(t,p)\,z_2-\phi(t,p)\,z_1\|$, and besides, $b(p)\in X_1(p)$ by Proposition~\ref{prop-compacto invariante} (i).
\par
So, if $\Pi_{1,p}(z_1)=\Pi_{1,p}(z_2)$, then $\|u(t,p,z_2)-u(t,p,z_1)\|=\|\phi(t,p)\,(\Pi_{2,p}(z_2)-\Pi_{2,p}(z_1))\|\leq M\,e^{-\delta t}\,c(t,p)\,\|\Pi_{2,p}(z_2)-\Pi_{2,p}(z_1)\|$ by using property (5) in the description of the continuous separation of $\tau_L$ and relation~\eqref{c}. Since for $p\in P_{\rm{l}}$ the semiorbit of $r e(p)$ for $r>0$ small enough remains in the bounded zone $\mathbb{F}$, $\mathbb{S}up_{t\geq 0}c(t,p)<\infty$ and consequently the pair is asymptotic.
\par
Finally, if $\Pi_{1,p}(z_1)\not=\Pi_{1,p}(z_2)$, $\lim_{t\to\infty}\|\phi(t,p)\,(\Pi_{2,p}(z_2)-\Pi_{2,p}(z_1))\|=0$ as before; and for $\Pi_{1,p}(z_1),\,\Pi_{1,p}(z_2),\, b(p)\in X_1(p)$ we just argue as in the proof of Theorem~\ref{teor-caos} to conclude that the pair is Li-Yorke chaotic.
\end{proof}
\mathbb{S}ubsection{A non-autonomous discontinuous pitchfork bifurcation diagram}
The purpose of this final section is to present the conclusions of the
previous sections of the paper in terms of non-autonomous bifurcation theory.
We want to emphasize that the classical bifurcation patterns can exhibit, in this
non-autonomous framework, ingredients of dynamical complexity which are not
possible in the autonomous models.
\par
Precisely, we look at the one-parametric family ($\gamma\in\mathbb{R}$) of scalar reaction-diffusion problems over a minimal, uniquely ergodic and aperiodic flow $(P,\theta,\mathbb{R})$, with Neumann or Robin boundary conditions, given for each $p\in P$ by
\begin{equation}\label{bifurcation}
\left\{\begin{array}{l} \displaystyle\frac{\partial y}{\partial t} =
\Delta \, y+(\gamma+h(p{\cdot}t,x))\,y+g(p{\cdot}t,x,y)\,,\quad t>0\,,\;\,x\in U,
\\[.2cm]
By:=\alpha(x)\,y+\displaystyle\frac{\partial y}{\partial n} =0\,,\quad t>0\,,\;\,x\in \partial U,\,
\end{array}\right.
\end{equation}
where we assume that $h\in \mathcal{U}(P\times\bar U)$ and $g:P\times \bar U\times \mathbb{R}\to \mathbb{R}$ is continuous, of class $C^1$ with respect to $y$, and it satisfies conditions (c1)-(c5) and also
\begin{itemize}
\item[(c6)] $g(p,x,y)$ is convex in $y$ for $y\leq 0$ and concave in $y$ for $y\geq 0$.
\end{itemize}
\par
For instance, $g$ might just be the map given by
\[
g(p,x,y)=\left\{\begin{array}{lr} k(p,x)\,(y+r_0)^3\,, & y\leq -r_0\\
0\,, & -r_0\leq y \leq r_0\\
-k(p,x)\,(y-r_0)^3\,, & y\geq r_0
\end{array}\right.
\]
for a certain positive map $k\in C(P\times\bar U)$ and the constant $r_0$ in (c4), which provides a non-autonomous version of the classical Chafee-Infante equation. The autonomous equation was studied by Chafee and Infante~\cite{chin} and some non-autonomous versions of this equation together with bifurcation problems have also been treated in the literature; for instance, see Carvalho et al.~\cite{calaro12}.
\par
The main result reads as follows. Let us denote by $\mathbb{A}_\gamma$ the global attractor of the corresponding skew-product semiflow $\tau_\gamma$ for the value $\gamma$ of the parameter. Let us also denote by $b_\gamma$ its upper boundary map.
\begin{teor}
The following assertions hold:
\begin{itemize}
\item[(i)] If $\gamma<0$, then $\mathbb{A}_\gamma=P\times \{0\}$ is the global attractor and it is globally exponentially stable.
\item[(ii)] If $\gamma=0$, then the global attractor $\mathbb{A}_0\mathbb{S}ubseteq \bigcup_{p\in P} \{p\}\times [-b_0(p),b_0(p)]$ is a pinched set which contains a unique minimal set $P\times\{0\}$. Its structure has been described in detail in Theorem~$\ref{teor-estr atractor caso u}$. In particular, if $\nu(P_{\rm{f}})=1$, then $\mathbb{A}_0$ is fiber-chaotic in measure in the sense of Li-Yorke.
\item[(iii)] If $\gamma>0$, then the global attractor $\mathbb{A}_\gamma\mathbb{S}ubseteq \bigcup_{p\in P} \{p\}\times [-b_\gamma(p),b_\gamma(p)]$ with $b_\gamma(p)\gg 0$ for every $p\in P$ and the maps $\pm b_\gamma$ define continuous equilibria. The copies of the base $K_\gamma^\pm=\{(p,\pm b_\gamma(p))\mid p\in P\}$ are globally exponentially stable minimal sets in $P\times \Int X_\pm$, whereas the trivial minimal set $P\times\{0\}$ is unstable. In addition, $b_0(p)=\lim_{\gamma\to 0^+} b_\gamma(p)$.
\end{itemize}
\end{teor}
\begin{proof}
First of all note that by Proposition~\ref{prop-h+k} (i) the upper Lyapunov exponent of the linearized problem of~\eqref{bifurcation} is $\lambda_P(\gamma+h)=\gamma+\lambda_P(h)=\gamma$, since $\lambda_P(h)=0$.
\par
With no need of condition (c6), (i) has been proved in Proposition~5 in Cardoso et al.~\cite{cardoso} and (ii) has been proved in Theorems~\ref{teor-estr atractor caso u} and~\ref{teor-caos}. Also it has been mentioned in~\cite{cardoso} that if $\lambda_P(\gamma+h)=\gamma>0$, then the semiflow is uniformly persistent in the interior of both the negative and the positive cones (this follows from Mierczy{\'n}ski and Shen~\cite{mish} or from the general theory developed in Novo et al.~\cite{noos7}), so that $b_\gamma(p)\gg 0$ for every $p\in P$, there exists a global attractor for the restriction of the semiflow to both of these cones, and the trivial minimal set $P\times\{0\}$ is unstable.
\par
The fact that $K_\gamma^\pm=\{(p,\pm b_\gamma(p))\mid p\in P\}$ are globally exponentially stable minimal sets follows, once (c6) is brought into play, from the general theory for monotone and concave skew-product semiflows written by N\'{u}\~{n}ez et al.~\cite{nuos4}. More precisely, the uniform persistence of the semiflow precludes the existence of infinitely many strongly positive minimal sets as the ones in Case A2 of Theorem~3.8 in~\cite{nuos4}, so that only Case A1 of this theorem can hold: there exists exactly one strongly positive minimal set, which is a globally exponentially stable copy of the base (the same for the negative cone). Note that, in particular, the maps $\pm b_\gamma$ define continuous equilibria.
\par
Finally, let us see that $b_0(p)=\lim_{\gamma\to 0^+} b_\gamma(p)$. Fix $z_0\gg 0$ and $r>0$ and note that if $0\leq\gamma_1\leq \gamma_2$, then by Theorem~\ref{teor-comparacion}, $u_{\gamma_1}(T,p{\cdot}(-T),r z_0)\leq u_{\gamma_2}(T,p{\cdot}(-T),r z_0)$ for any $p\in P$ and $T\geq 0$. Since relation~\eqref{b(p)} holds for $r>0$ large enough, it follows that $b_{\gamma_1}(p)\leq b_{\gamma_2}(p)$ for any $p\in P$. By monotonicity and compactness of the semiflow, there exists the limit $\lim_{\gamma\to 0^+}b_{\gamma}(p)= b_*(p)$ for each $p\in P$. Since $b_0\leq b_\gamma$ for any $\gamma>0$, it is $b_0(p)\leq b_*(p)$ for each $p\in P$. On the other hand, $b_*(p)$ defines an equilibrium for the problem with $\gamma=0$, and therefore it must be contained in the global attractor $\mathbb{A}_0$, so that $b_*(p)\leq b_0(p)$ for any $p\in P$. Therefore, $b_*(p)= b_0(p)$ for any $p\in P$ and the proof is finished.
\end{proof}
\mathbb{S}ubsection*{Acknowledgements} The authors would like to thank Prof.~Anthony Quas for providing them with a short proof of Theorem~\ref{teor-shneiberg}.
\end{document} |
\begin{document}
\title{Transformations on the product of Grassmann spaces}
\author{Hans Havlicek \and Mark Pankov}
\maketitle
\section{Introduction}\label{sect:intro}
Let ${\mathcal G}_k$ denote the set of all $k$-dimensional subspaces of an
$n$-dimensional vector space. We recall that two elements of ${\mathcal G}_k$ are called
\emph{adjacent} if their intersection has dimension $k-1$. The set ${\mathcal G}_k$ is
point set of a partial linear space, namely a \emph{Grassmann space} for $1< k<
n-1$ (see Section \ref{sect:thm:GxG}) and a projective space for
$k\in\{1,n-1\}$. Two adjacent subspaces are---in the language of partial linear
spaces---two distinct collinear points.
W.L.~Chow \cite{Chow} determined all bijections of ${\mathcal G}_k$ that preserve
adjacency in both directions in the year 1949. In this paper we call such a
mapping, for short, an \emph{{A-{\hspace{0pt}}}transformation}. Disregarding the trivial
cases $k=1$ and $k=n-1$, every {A-{\hspace{0pt}}}transformation of ${\mathcal G}_k$ is induced by a
semilinear transformation $V\to V$ or (only when $k=2n$) by a semilinear
transformation of $V$ onto its dual space $V^*$. There is a wealth of related
results, and we refer to \cite{benz-92}, \cite{huang-98}, and \cite{wan-96} for
further references.
In the present note, we aim at generalizing Chow's result to products of
Grassmann spaces. However, we consider only products of the form ${\mathcal G}_k\times
{\mathcal G}_{n-k}$, where ${\mathcal G}_k$ and ${\mathcal G}_{n-k}$ stem from the same vector space $V$.
Furthermore, for a fixed $k$ we restrict our attention to a certain subset of
${\mathcal G}_k\times {\mathcal G}_{n-k}$. This subset, say ${\mathcal G}$, is formed by all pairs of
\emph{complementary\/} subspaces. Our definition of an adjacency on ${\mathcal G}$ in
formula (\ref{eq:G.adjacent}) is motivated by the definition of lines in a
product of partial linear spaces; cf.\ e.g.\ \cite{NP}.
One of our main results (Theorem~\ref{thm:A}) states that Chow's theorem
remains true, mutatis mutandis, for the {A-{\hspace{0pt}}}transformations of ${\mathcal G}$. However,
in Theorem~\ref{thm:C} we can show even more: Let us say that two elements
$(S,U)$ and $(S',U')$ of ${\mathcal G}$ are \emph{close\/} to each other, if their
Hamming distance is $1$ or, said differently, if they coincide in precisely one
of their components. Then the bijections of ${\mathcal G}$ onto itself which preserve
this closeness relation in both directions---we call them
\emph{{C-{\hspace{0pt}}}transformations\/} of ${\mathcal G}$---are precisely the {A-{\hspace{0pt}}}transformations
of ${\mathcal G}$. In this way, we obtain for $1<k<n-1$ two characterizations of the
semilinear bijections $V\to V$ and $V\to V^*$ via their action on the set
${\mathcal G}$.
Finally, we turn to the following question: What happens to our results if we
replace the set ${\mathcal G}$ with the entire cartesian product ${\mathcal G}_k\times
{\mathcal G}_{n-k}$? Clearly, the basic notions of adjacency and closeness remain
meaningful. We describe all {C-{\hspace{0pt}}}transformations of ${\mathcal G}_k\times {\mathcal G}_{n-k}$ in
Theorem~\ref{thm:CGxG}. However, in sharp contrast to Theorem~\ref{thm:C}, this
is a rather trivial task, and the transformations of this kind do not deserve
any interest. Then, using a result of A.~Naumowicz and K.~Pra\.zmowski
\cite{NP}, we also determine all {A-{\hspace{0pt}}}transformations of ${\mathcal G}_k\times {\mathcal G}_{n-k}$
in Theorem~\ref{thm:AGxG}. Such mappings are closely related with collineations
of the underlying partial linear space, and in general they can be described in
terms of \emph{two\/} semilinear bijections, but not in terms of a
\emph{single\/} semilinear bijection.
Before we close this section, it is worthwhile to mention that the results from
\cite{NP} could be used to describe the {A-{\hspace{0pt}}}transformations of arbitrary finite
products of Grassmann spaces, but this is not the topic of the present article.
\section{{A-{\hspace{0pt}}}transformations and {C-{\hspace{0pt}}}transformations}\label{sect:trafos}
First, we collect our basic assumptions and definitions. Throughout this paper,
let $V$ be a $n$-dimensional left vector space over a division ring, $2\leq
n<\infty$. Suppose that $P,T\subset V$ are subspaces. They are said to be
\emph{incident} (in symbols: $P\mathrel{\mathrm{I}} T$) if $P\subset T$ or if $T\subset P $.
Note that according to this definition every subspace of $V$ is incident with
$0$ (the zero subspace) and with $V$. Furthermore, we define
\begin{equation}\label{eq:adjacent}
P\sim T \;:\Leftrightarrow\; \dim P = \dim T = \dim (P\cap T) +1,
\end{equation}
where ``$\sim$'' is to be read as \emph{adjacent}.
We put ${\mathcal G}_{i}$, for the set $i$-dimensional subspaces of $V$,
$i=0,1,\dots, n$. In what follows \emph{we fix a natural number\/}
$k\in\{1,2,\ldots,n-1\}$ and adopt the notation
\begin{equation}\label{eq:G}
{\mathcal G}:=\{(S,U)\in{\mathcal G}_{k}\times {\mathcal G}_{n-k}\mid S+U=V\}.
\end{equation}
Hence $(S,U)\in{\mathcal G}$ means that $S$ and $U$ are \emph{complementary\/}
subspaces. On the set ${\mathcal G}$ we define two binary relations: Elements $(S,U)$
and $(S',U')$ of ${\mathcal G}$ are said to be \emph{adjacent\/} if
\begin{equation}\label{eq:G.adjacent}
( S=S'\mbox{ and } U\sim U')\mbox{ or }( S\sim S'\mbox{ and } U=U').
\end{equation}
By abuse of notation, this relation on ${\mathcal G}$ will also be denoted by the symbol
``$\sim$''. Our elements are said to be \emph{close\/} to each other (in
symbols: $(S,U)\approx (S',U')$) if
\begin{equation}\label{eq:G.close}
( S=S'\mbox{ and } U\neq U')\mbox{ or }( S\neq S'\mbox{ and } U=U').
\end{equation}
According to this definition, any two adjacent elements of ${\mathcal G}$ are
close; the converse holds only for $k=1$ and $k=n-1$.
We shall establish in Lemma~\ref{lemma:connect} that any two elements $(S,U)$
and $(S',U')$ of ${\mathcal G}$ can be connected by a finite sequence
\begin{equation}\label{eq:A.connect}
(S,U)=(S_{0},U_{0})\sim (S_{1},U_{1})\sim\cdots\sim (S_{i},U_{i})=(S',U').
\end{equation}
Consequently, we also have
\begin{equation}\label{eq:C.connect}
(S,U)=(S_{0},U_{0})\approx (S_{1},U_{1})\approx\cdots\approx
(S_{i},U_{i})=(S',U').
\end{equation}
We refer to the definition of a \emph{Pl\"ucker space\/} in
\cite[p.~199]{benz-92}, and we point out the (inessential) difference that our
relations $\sim$ and $\approx$ are anti-reflexive.
A bijection $f:{\mathcal G}\to {\mathcal G}$ is said to be an \emph{adjacency
preserving transformation} (shortly: an \emph{{A-{\hspace{0pt}}}transformation}) if $f$ and
$f^{-1}$ transfer adjacent elements of ${\mathcal G}$ to adjacent elements; if $f$ and
$f^{-1}$ map close elements of ${\mathcal G}$ to close elements then we say that $f$ is
a \emph{closeness preserving transformation\/} (shortly: a
\emph{{C-{\hspace{0pt}}}transformation}).
\begin{exmp}\label{exmp:1}{\rm
For any two mappings $f':{\mathcal G}_{k}\to {\mathcal G}_{k}$ and
$f'':{\mathcal G}_{n-k}\to {\mathcal G}_{n-k}$ we put
\begin{equation}
f'\times f'': {\mathcal G}_{k}\times {\mathcal G}_{n-k}\to
{\mathcal G}_{k}\times {\mathcal G}_{n-k} :
(S,U)\mapsto \big(f'(S),f''(U)\big).
\end{equation}
Each semilinear isomorphism $l:V\to V$ induces, for $i=1,2,\dots,n-1$,
bijections
\begin{equation}
G_{i}(l):{\mathcal G}_{i}\to {\mathcal G}_{i} : S\mapsto l(S).
\end{equation}
Obviously, the restriction of
\begin{equation}
G_{k}(l)\times G_{n-k}(l)
\end{equation}
to ${\mathcal G}$ is an {A-{\hspace{0pt}}}transformation and a {C-{\hspace{0pt}}}transformation.
}\end{exmp}
\begin{exmp}\label{exmp:2}{\rm
For any two mappings $g':{\mathcal G}_{k}\to {\mathcal G}_{n-k}$ and
$g'':{\mathcal G}_{n-k}\to {\mathcal G}_{k}$ we put
\begin{equation}
g'\mathbin{\dot\times} g'':{\mathcal G}_{k}\times {\mathcal G}_{n-k}\to
{\mathcal G}_{k}\times {\mathcal G}_{n-k} :
(S,U)\mapsto \big(g''(U),g'(S)\big).
\end{equation}
Let $V^*$ denote the dual space of $V$. Each semilinear isomorphism $s:V\to
V^{*}$ induces, for $i=1,2,\dots,n-1$, the bijections
\begin{equation}
D_{i}(s):{\mathcal G}_{i}\to{\mathcal G}_{n-i} :
S\mapsto \big(s(S)\big)^{\circ},
\end{equation}
where $\big(s(S)\big)^{\circ}$ denotes the annihilator of $s(S)$. The
restriction of
\begin{equation}
D_{k}(s)\mathbin{\dot\times} D_{n-k}(s)
\end{equation}
to ${\mathcal G}$ is an {A-{\hspace{0pt}}}transformation and a {C-{\hspace{0pt}}}transformation. Observe
that a necessary and sufficient condition for the existence of such an
isomorphism $s$ is that the underlying division ring admits an
anti-auto\-mor\-phism.}
\end{exmp}
\begin{exmp}\label{exmp:3}{\rm
Now suppose that $n=2k$. We assume that $l:V\to V$ and $s:V\to V^{*}$ are
semilinear isomorphisms. The restrictions of
\begin{equation}
G_{k}(l)\mathbin{\dot\times} G_{k}(l)\;\mbox{ and }\;D_{k}(s)\times D_{k}(s)
\end{equation}
to ${\mathcal G}$ both are {A-{\hspace{0pt}}}transformations and {C-{\hspace{0pt}}}transformations.
}\end{exmp}
\begin{exmp}\label{exmp:n=2}{\rm
Let $n=2$ and $k=1$. Choose an arbitrary bijection $f:{\mathcal G}_1\to{\mathcal G}_1$. Then the
restrictions of $f\times f$ and $f\mathbin{\dot\times} f$ to ${\mathcal G}$ both are
{A-{\hspace{0pt}}}transformations and {C-{\hspace{0pt}}}transformations.}
\end{exmp}
We are now in a position to state our main results:
\begin{theorem}\label{thm:C}
Every closeness preserving transformation of ${\mathcal G}$ is one of the
mappings considered in Examples~{\rm\ref{exmp:1}--\ref{exmp:n=2}}. Hence it is
an adjacency preserving transformation.
\end{theorem}
It is trivial that each {A-{\hspace{0pt}}}transformation is a {C-{\hspace{0pt}}}transformation if $k=1$ or
if $k=n-1$. In Section~\ref{sect:thm:A} we shall prove this statement for the
general case. Thus the following statement holds true.
\begin{theorem}\label{thm:A}
Every adjacency preserving transformation of ${\mathcal G}$ is one of the
mappings considered in Examples~{\rm\ref{exmp:1}--\ref{exmp:n=2}}. Hence it is
a closeness preserving transformation.
\end{theorem}
It is clear that our definitions of adjacency and closeness remain meaningful
on the entire cartesian product ${\mathcal G}_{k}\times{\mathcal G}_{n-k}$. Also the notions of
{C-{\hspace{0pt}}} and {A-{\hspace{0pt}}}transformation and Examples~\ref{exmp:1}--\ref{exmp:n=2} can be
carried over accordingly. However, Theorems~\ref{thm:C} and \ref{thm:A} do not
remain unaltered when ${\mathcal G}$ is replaced with ${\mathcal G}_{k}\times{\mathcal G}_{n-k}$:
\begin{exmp}\label{exmp:CGxG}{\rm
Let $f':{\mathcal G}_{k}\to{\mathcal G}_{k}$ and $f'':{\mathcal G}_{n-k}\to{\mathcal G}_{n-k}$ be bijections. Then
$f'\times f''$ is a {C-{\hspace{0pt}}}transformation. Also, if $g':{\mathcal G}_{k}\to{\mathcal G}_{n-k}$ and
$g'':{\mathcal G}_{n-k}\to{\mathcal G}_{k}$ are bijections then $g'\mathbin{\dot\times} g''$ is a
{C-{\hspace{0pt}}}transformation.}
\end{exmp}
For the sake of completeness, let us state the following rather trivial result:
\begin{theorem}\label{thm:CGxG}
Every closeness preserving transformation of ${\mathcal G}_{k}\times{\mathcal G}_{n-k}$ is one of
the mappings considered in Example~{\rm\ref{exmp:CGxG}}.
\end{theorem}
\begin{exmp}\label{exmp:AGxG}{\rm
If $f':{\mathcal G}_{k}\to{\mathcal G}_{k}$ and $f'':{\mathcal G}_{n-k}\to{\mathcal G}_{n-k}$ are bijections which
preserve adjacency in both directions then $f'\times f''$ is an
{A-{\hspace{0pt}}}transformation. Also, if $g':{\mathcal G}_{k}\to{\mathcal G}_{n-k}$ and
$g'':{\mathcal G}_{n-k}\to{\mathcal G}_{k}$ are bijections which preserve adjacency in both
directions then $g'\mathbin{\dot\times} g''$ is an {A-{\hspace{0pt}}}transformation.
Suppose that $k=1$ or $k=n-1$. Then it suffices to require that the mappings
$f'$, $f''$, $g'$ and $g''$ from above are bijections in order to obtain an
{A-{\hspace{0pt}}}transformation of ${\mathcal G}_{k}\times{\mathcal G}_{n-k}$.
Provided that $1<k<n-1$, we can apply Chow's theorem (\cite[p.~38]{Chow},
\cite[p.~81]{dieu-71}) to describe explicitly the mappings from above.
In the first case we have $f'=G_{k}(l')$ or $f'=D_{k}(s')$ (only when $n=2k$),
and $f''=G_{n-k}(l'')$ or $f''=D_{k}(s'')$ (only when $n=2k$).
In the second case we have $g'=D_{k}(s')$ or $g'=G_{k}(l')$ (only when $n=2k$),
and $g''=D_{n-k}(s'')$ or $g''=G_{k}(l'')$ (only when $n=2k$).
Here $l',l'':V\to V$ and $s',s'':V\to V^*$ denote semilinear isomorphisms.
}\end{exmp}
We shall see that the following result is a consequence of
\cite[Theorem~1.14]{NP}:
\begin{theorem}\label{thm:AGxG}
Every adjacency preserving transformation of ${\mathcal G}_{k}\times{\mathcal G}_{n-k}$ is one of
the mappings considered in Example~{\rm\ref{exmp:AGxG}}.
\end{theorem}
\begin{rem}{\rm
Suppose that the underlying division ring of $V$ is not of characteristic $2$.
Let $u\in \GL(V)$ be an involution. Then there exist two invariant subspaces
$U_{+}(u)$ and $U_{-}(u)$ with $V=U_{+}(u)\oplus U_{-}(u)$ such that $u(x)=\pm
x$ for each $x\in U_{\pm}(u)$. If $\dim U_{+}(u)= r$ then $\dim U_{-}(u)=n-r$, and
$u$ is called an \emph{$(r,n-r)$-involution}.
For our fixed $k$ let $J$ be the set of all $(k,n-k)$-involutions. There exists
a bijection
\begin{equation}\label{}
\gamma: J\to {\mathcal G} : u \mapsto \big(U_{+}(u),U_{-}(u)\big).
\end{equation}
Two $(k,n-k)$-involutions $u$ and $v$ are said to be \emph{adjacent\/} if the
corresponding elements of ${\mathcal G}$ are adjacent. This holds if, and only
if, the product of $u$ and $v$ (in any order) is a transvection $\neq 1_V$.
Now let $f:J\to J$ be a bijection which preserves adjacency in both directions.
We apply Theorem~\ref{thm:A} to the {A-{\hspace{0pt}}}transformation $\gamma f\gamma^{-1}:
{\mathcal G}\to{\mathcal G}$. If $n>2$ and $n\ne 2k$ then this last mapping is given as
in Example~\ref{exmp:1} or \ref{exmp:2}. This means that $f$ can be extended
to an automorphism of the group $\GL(V)$ as follows: To each $u\in \GL(V)$ we
assign $lul^{-1}$ or the contragredient of $sus^{-1}$, respectively.}
\end{rem}
\section{Proof of Theorem~\ref{thm:C}}\label{sect:thm:C}
Our proof of Theorem~\ref{thm:C} will be based on several lemmas and the
subsequent characterization. In the case $n=2k$ this statement is a particular
case of a result in \cite{BlunckHavlicek}. The direct analogue of
Theorem~\ref{thm:3} for buildings can be found in \cite[Proposition~4.2]{AVM}.
\begin{theorem}\label{thm:3}
Let $1\leq k\leq n-1$. Then for any two distinct $S_{1},S_{2}\in {\mathcal
G}_{k}$ the following two conditions are equivalent:
\begin{enumerate}
\item[(a)] $S_{1}$ and $S_{2}$ are adjacent,
\item[(b)]
There exists an $S\in {\mathcal G}_{k}-\{S_{1},S_{2}\}$ such that for all
$U\in {\mathcal G}_{n-k}$ the condition $(S,U)\in {\mathcal G}$ implies that
$(S_{1},U)$ or $(S_{2},U)$ belongs to ${\mathcal G}$.
\end{enumerate}
\end{theorem}
\begin{proof}
(a) $\Rightarrow$ (b). If $S_{1}$ and $S_{2}$ are adjacent then $S_{1}\cap
S_{2}\in{\mathcal G}_{k-1}$ and $S_{1}+S_{2}\in{\mathcal G}_{k+1}$. Every
$S\in {\mathcal G}_{k}-\{S_{1},S_{2}\}$ satisfying the condition
\begin{equation}
S_{1}\cap S_{2}\subset S\subset S_{1}+S_{2}
\end{equation}
has the required property, and at least one such $S$ exists.
(b) $\Rightarrow$ (a). The proof of this implication will be given in several
steps. First we show that
\begin{equation}\label{eq:W-S}
0\neq W_{1}\subset S_1\mbox{ and } 0\neq W_{2}\subset S_{2} \Rightarrow
(W_{1}+W_{2})\cap S\neq 0.
\end{equation}
Assume, contrary to (\ref{eq:W-S}), that $(W_{1}+W_{2})\cap S=0$. Then there
exists a complement $U\in {\mathcal G}_{n-k}$ of $S$ containing
$W_{1}+W_{2}$. By our hypothesis, $U$ is a complement of $S_{1}$ or $S_{2}$.
This contradicts $W_{1}\subset S_{1}$ and $W_{2}\subset S_{2}$.
Our second assertion is
\begin{equation}\label{eq:S1-S2-S}
S_{1}\cap S_{2}\subset S.
\end{equation}
This inclusion is trivial if $S_{1}\cap S_{2}$ is zero. Otherwise, let
$P\subset S_{1}\cap S_{2}$ be an arbitrarily chosen $1$-dimensional subspace.
We apply (\ref{eq:W-S}) to $W_{1}=W_{2}=P$. This shows that $P\cap S\ne 0$.
Hence $P\subset S$, as required.
The third step is to show that
\begin{equation}\label{eq:dim.Si-S}
\dim (S\cap S_{1}) =\dim (S\cap S_{2}) = k-1.
\end{equation}
By symmetry, it suffices to establish that
\begin{equation}
W_1\cap(S\cap S_{1})\ne 0
\end{equation}
for all $2$-dimensional subspaces $W_1\subset S_{1}$: Let us take a
$1$-dimensional subspace $P_{2}\subset S_{2}$ such that $P_{2}\cap S=0$. Then
(\ref{eq:S1-S2-S}) implies that $P_{2}$ is not contained in $S_{1}$, and for
every $2$-dimensional subspace $W_1\subset S_{1}$ the subspace $W_1+P_{2}$ is
$3$-dimensional. Let $P_{1}$ and $Q_{1}$ be distinct $1$-dimensional
subspaces contained in $W_1$. It follows from (\ref{eq:W-S}) that
$P_{1}+P_{2}$ and $Q_{1}+P_{2}$ meet $S$ in $1$-dimensional subspaces ($\neq
P_{2}$) which will be denoted by $P$ and $Q$, respectively. As $P_{1}$ and
$Q_{1}$ are distinct, so are $P$ and $Q$. Therefore $P+Q$ is a
$2$-dimensional subspace of $S$. Since $W_1$ and $P+Q$ lie in the
$3$-dimensional subspace $W_1+P_{2}$, they have a common $1$-dimensional
subspace contained in $W_1\cap S = W_1\cap(S\cap S_{1})$. This proves
(\ref{eq:dim.Si-S}).
Finally, we read off from (\ref{eq:S1-S2-S}) that
\begin{equation}\label{eq:S1-S2}
S_{1}\cap S_{2}=(S\cap S_{1})\cap (S\cap S_{2}),
\end{equation}
and we shall finish the proof by showing that this subspace has dimension
$k-1$. By (\ref{eq:dim.Si-S}) and because of $S_1\neq S_2$, the dimension of
$S_{1}\cap S_{2}$ is either $k-2$ or $k-1$. Suppose, to the contrary, that
\begin{equation}\label{eq:k-2}
\dim S_{1}\cap S_{2}=k-2.
\end{equation}
Then $S\cap S_{1}$ and $S\cap S_{2}$ are distinct $(k-1)$-dimensional
subspaces spanning $S$.
There exist $1$-dimensional subspaces
$P_{1},P_{2}$ such that
\begin{equation}
S_{i}=(S\cap S_{i})+P_{i}
\end{equation}
for $i=1,2$. We have $P_{1}\ne P_{2}$ (otherwise (\ref{eq:S1-S2-S}) would
give $P_{1}=P_{2}\subset S_{1}\cap S_{2}\subset S$ which is impossible), and
(\ref{eq:W-S}) guarantees that $(P_{1}+P_{2})\cap S$ is a $1$-dimensional
subspace. Then $S_{1}+S_{2}$ is contained in the $(k+1)$-dimensional subspace
$S+P_1$ which, by the dimension formula for subspaces, contradicts
(\ref{eq:k-2}).
\end{proof}
\begin{lemma}\label{lemma:1}
If $l:V\to V$ is a semilinear isomorphism such that $G_{j}(l)$ is the identity
for at least one $j\in\{1,2,\ldots,n-1\}$ then the same holds for all
$i=1,2\ldots, n-1$.
\end{lemma}
\begin{proof}
This is well known.
\end{proof}
\begin{lemma}\label{lemma:2}
Let $l_{i}:V\to V$ and $s_{i}:V\to V^{*}$ be semilinear isomorphisms, $i=1,2$.
Then the following assertions hold.
\begin{enumerate}
\item[(a)] If one of the mappings $G_{k}(l_{1})\times G_{n-k}(l_{2})$ or
$G_{k}(l_{1})\mathbin{\dot\times} G_{k}(l_{2})$, when restricted to ${\mathcal G}$, is a
{C-{\hspace{0pt}}}transformation then $G_{i}(l_{1})=G_{i}(l_{2})$ for all $i=1,2,\dots, n-1$.
\item[(b)] If one of the mappings $D_{k}(s_{1})\mathbin{\dot\times} D_{n-k}(s_{2})$ or
$D_{k}(s_{1})\times D_{k}(s_{2})$, when restricted to ${\mathcal G}$, is a
{C-{\hspace{0pt}}}transformation then $D_{i}(s_{1})=D_{i}(s_{2})$ for all $i=1,2,\dots, n-1$.
\item[(c)] If $n=2k>2$ then none of the mappings
$G_{k}(l_{1})\times D_{k}(s_{2})$,
$D_{k}(s_{1})\times G_{k}(l_{2})$,
$G_{k}(l_{1})\mathbin{\dot\times} D_{k}(s_{2})$, and
$D_{k}(s_{1})\mathbin{\dot\times} G_{k}(l_{2})$
is a {C-{\hspace{0pt}}}transformation, when it is restricted to ${\mathcal G}$.
\end{enumerate}
\end{lemma}
\begin{proof}
(a) Let the restriction of $G_{k}(l_{1})\times G_{n-k}(l_{2})$ to ${\mathcal G}$ be a
{C-{\hspace{0pt}}}transformation. Then $G_{k}(1_{V})\times G_{n-k}(l^{-1}_{1}l_{2})$ gives
also a {C-{\hspace{0pt}}}transformation. This means that for each $U\in {\mathcal G}_{n-k}$
the mapping $G_{k}(1_{V})$ transfers the set of all $k$-dimensional subspaces
having a non-zero intersection with $U$ onto the set of all $k$-dimensional
subspaces having a non-zero intersection with $l^{-1}_{1}l_{2}(U)$. However,
$G_{k}(1_{V})$ is the identity. Thus
\begin{equation}
l^{-1}_{1}l_{2}(U)=U,
\end{equation}
and $G_{n-k}(l_{2}l^{-1}_{1})$ is the identity. Hence we can apply
Lemma~\ref{lemma:1} to show the assertion in this particular case.
Next, let the restriction of $G_{k}(l_{1})\mathbin{\dot\times} G_{k}(l_{2})$ to ${\mathcal G}$ be a
{C-{\hspace{0pt}}}transformation. Thus $n=2k$ and the assertion follows from the previous
case and
\begin{equation}
G_{k}(l_{1})\mathbin{\dot\times} G_{k}(l_{2}) = \big(G_{k}(1_V)\mathbin{\dot\times} G_{k}(1_V)\big)
\big(G_{k}(l_{1})\times G_{k}(l_{2})\big).
\end{equation}
(b) can be verified similarly to (a).
(c) Assume, contrary to our hypothesis, that $G_{k}(l_{1})\times D_{k}(s_{2})$
gives a {C-{\hspace{0pt}}}transformation. Hence $G_{k}(1_V)\times D_{k}(s_{2}l_1^{-1})$ is
also a {C-{\hspace{0pt}}}transformation and, as above, we infer that
\begin{equation}
D_{k}(s_{2}l_1^{-1})(U)=\big((s_{2}l_1^{-1})(U)\big)^\circ = U
\end{equation}
for all $U\in{\mathcal G}_{k}$. Let $W\in{\mathcal G}_{k-1}$. Then there are subspaces
$U_1,U_2,\ldots U_{k+1}\in{\mathcal G}_{k}$ such that $V = \sum_{i=1}^{k+1}U_i$ and $W
= \bigcap_{i=1}^{k+1}U_i$. Consequently,
\begin{equation}
0=\big(s_{2}l_1^{-1}(V)\big)^\circ
=\bigcap_{i=1}^{k+1}\big((s_{2}l_1^{-1})(U_i)\big)^\circ
=\bigcap_{i=1}^{k+1} U_i
= W
\end{equation}
which implies $k=1$, an absurdity.
The remaining cases can be shown in the same way.
\end{proof}
Let us remark that in general the assumption $n>2$ in part (c) of this lemma
cannot be dropped. Indeed, if $n=2k=2$ and if $K$ is a commutative field then
there exists a non-degenerate alternating bilinear form $b:V\times V\to K$.
Hence $s:V\to V^*: v\mapsto b(v,\cdot)$ is a linear bijection, and
$G_1(1_V)\times D_1(s)$ is the identity on ${\mathcal G}_1\times{\mathcal G}_1$.
\begin{lemma}\label{lemma:n=2}
Let $n=2$, whence $k=1$. Suppose that $g':{\mathcal G}_1\to {\mathcal G}_1$ and
$g'':{\mathcal G}_1\to{\mathcal G}_1$ are bijections such that one of the mappings $g'\times g''$
or $g'\mathbin{\dot\times} g''$, when restricted to ${\mathcal G}$, is a {C-{\hspace{0pt}}}transformation. Then
$g'=g''$.
\end{lemma}
\begin{proof}
It suffices to discuss the first case, since $1_{\mathcal G}\mathbin{\dot\times} 1_{\mathcal G}$ yields a
{C-{\hspace{0pt}}}transformation. Now we can proceed as in the proof of Lemma~\ref{lemma:2}
(a) in order to establish that the restriction of $g'{}^{-1}g''$ to ${\mathcal G}$
equals $1_{\mathcal G}$.
\end{proof}
We say that ${\mathcal X}\subset{\mathcal G}$ is a \emph{{C-{\hspace{0pt}}}subset} if any
two distinct elements of ${\mathcal X}$ are close. (If we consider the graph of
the closeness relation on $\mathcal G$ then a {C-{\hspace{0pt}}}subset is just a clique, i.e.
a complete subgraph.) A {C-{\hspace{0pt}}}subset is said to be \emph{maximal\/} if it is not
properly contained in any {C-{\hspace{0pt}}}subset. In order to describe the maximal
{C-{\hspace{0pt}}}subsets the following notation will be useful. If $P$ and $T$ are subspaces
of $V$ then we put
\begin{equation}\label{}
{\mathcal G}(P,T):=\{(S,U)\in{\mathcal G}\mid S\mathrel{\mathrm{I}} P \mbox{ and } U\mathrel{\mathrm{I}} T\};
\end{equation}
here we use the incidence relation from the beginning of
Section~\ref{sect:trafos}.
\begin{lemma}\label{lemma:3}
The maximal {C-{\hspace{0pt}}}subsets of ${\mathcal G}$ are precisely the sets ${\mathcal G}(S,V)$
with $S\in{\mathcal G}_{k}$, and ${\mathcal G}(V,U)$ with $U\in{\mathcal G}_{n-k}$.
\end{lemma}
\begin{proof}
Easy verification.
\end{proof}
We refer to the sets described in the lemma as maximal {C-{\hspace{0pt}}}subsets of
\emph{first kind} and \emph{second kind}, respectively.
\begin{myproof}{\ref{thm:C}}
(a) Let $f$ be a {C-{\hspace{0pt}}}transformation of ${\mathcal G}$. Then $f$ and $f^{-1}$
map maximal {C-{\hspace{0pt}}}subsets to maximal {C-{\hspace{0pt}}}subsets. Observe that two maximal
{C-{\hspace{0pt}}}subsets have a unique common element if, and only if, one of them is of
first kind, say ${\mathcal G}(S,V)$, the other is of second kind, say ${\mathcal G}(V,U)$, and
$(S,U)\in{\mathcal G}$.
Given $S,S'\in{\mathcal G}_{k}$ there exists a subspace $U\in{\mathcal G}_{n-k}$ such that
$S+U=S'+U=V$. We conclude from
\begin{equation}
f\big({\mathcal G}(S,V)\big)\cap f\big({\mathcal G}(V,U)\big)=\{f\big((S,U)\big)\}
\end{equation}
that $f\big({\mathcal G}(S,V)\big)$ and $f\big({\mathcal G}(V,U)\big)$ are maximal {C-{\hspace{0pt}}}subsets of
different kind. Likewise, $f\big({\mathcal G}(S',V)\big)$ and $f\big({\mathcal G}(V,U)\big)$ are
of different kind, so that $f\big({\mathcal G}(S,V)\big)$ and $f\big({\mathcal G}(S',V)\big)$ are
of the same kind.
A similar argument holds for maximal {C-{\hspace{0pt}}}subsets of second kind; altogether the
action of the {C-{\hspace{0pt}}}transformation $f$ on the set of maximal {C-{\hspace{0pt}}}subsets is
either \emph{type preserving\/} or \emph{type interchanging\/}.
(b) Suppose that $f$ is type preserving. Then there exist bijections
\begin{eqnarray*}\label{}
&g':{\mathcal G}_{k} \to {\mathcal G}_{k} \mbox{ such that } f\big({\mathcal G}(S,V)\big) = {\mathcal G}\big(g'(S),V\big)
\mbox{ for all }S\in{\mathcal G}_{k},&\\
&g'':{\mathcal G}_{n-k} \to {\mathcal G}_{n-k} \mbox{ such that }
f\big({\mathcal G}(V,U)\big) = {\mathcal G}\big(V,g''(U)\big)\mbox{ for all }U\in{\mathcal G}_{n-k};&
\end{eqnarray*}
thus $f$ equals the restriction of $g'\times g''$ to ${\mathcal G}$. We distinguish four
cases:
Case 1: $n=2$. Hence $k=1$; we deduce from Lemma~\ref{lemma:n=2} (a) that
$g'=g''$, whence $f$ is given as in Example~\ref{exmp:n=2}.
Case 2: $n>2$ and $k=1$. Then for each $U\in {\mathcal G}_{n-1}$ the mapping
$g'$ transfers the set of all $1$-dimensional subspaces contained in $U$ to the
set of all $1$-dimensional subspaces contained in $g''(U)$. This means, by the
fundamental theorem of projective geometry, that there exists a semilinear
isomorphism $l':V\to V$ with $g'=G_1(l')$. Similarly, $g''$ is induced by a
semilinear isomorphism $l'':V\to V$.
Case 3: $n>2$ and $k=n-1$. By symmetry, this coincides with the previous case.
Case 4: $n>2$ and $1<k<n-1$. Then Theorem~\ref{thm:3} guarantees that $g'$ and
$g''$ are adjacency preserving in both directions; Chow's theorem
(\cite[p.~38]{Chow}, \cite[p.~81]{dieu-71}) says that $g'$ and $g''$ are
induced by semilinear isomorphisms. More precisely, we have $g'=G_{k}(l')$ with
a semilinear bijection $l':V\to V$, or $g'=D_{k}(s')$ with a semilinear
bijection $s':V\to V^*$ (only when $n=2k$). A similar description holds for
$g''$.
In cases 2--4 we infer from Lemma~\ref{lemma:2} (c) that there are only two
possibilities:
Case A. $g'=G_{k}(l')$ and $g''=G_{n-k}(l'')$. Now Lemma~\ref{lemma:2} (a)
yields that $G_i(l')=G_i(l'')$ for all $i=1,2\ldots,n-1$, whence $f$ is the
restriction to ${\mathcal G}$ of $G_{k}(l')\times G_{n-k}(l')$; cf.\
Example~\ref{exmp:1}.
Case B. $n=2k$, $g'=D_{k}(s')$, and $g''=D_{k}(s'')$. Now Lemma~\ref{lemma:2}
(b) yields that $D_i(s')=D_i(s'')$ for all $i=1,2\ldots,n-1$, whence $f$ is the
restriction to ${\mathcal G}$ of $D_{k}(s')\times D_{k}(s')$; cf.\ Example~\ref{exmp:3}.
(c) If $f$ is type interchanging then there exist bijections
\begin{eqnarray*}\label{}
&g':{\mathcal G}_{k} \to {\mathcal G}_{n-k} \mbox{ such that } f\big({\mathcal G}(S,V)\big) = {\mathcal G}\big(V,g'(S)\big)
\mbox{ for all }S\in{\mathcal G}_{k},&\\
&g'':{\mathcal G}_{n-k} \to {\mathcal G}_{k} \mbox{ such that }
f\big({\mathcal G}(V,U)\big) = {\mathcal G}(g''(U),V)\mbox{ for all }U\in{\mathcal G}_{n-k};&
\end{eqnarray*}
thus $f$ is the restriction to ${\mathcal G}$ of $g'\mathbin{\dot\times} g''$. Now we can proceed,
mutatis mutandis, as in (b). So $f$ is given as in Example~\ref{exmp:n=2},
\ref{exmp:2}, or \ref{exmp:3}.
This completes the proof.
\end{myproof}
\section{Proof of Theorem~\ref{thm:A}}\label{sect:thm:A}
First, let us introduce the following notion: We say that ${\mathcal
X}\subset{\mathcal G}$ is an \emph{{A-{\hspace{0pt}}}subset} if any two distinct elements of
${\mathcal X}$ are adjacent. (As before, such a set is just a clique of the
graph given by the adjacency relation on $\mathcal G$.) An {A-{\hspace{0pt}}}subset is said
to be \emph{maximal} if it is not properly contained in any {A-{\hspace{0pt}}}subset.
If $k=1$ or if $k=n-1$ then an {A-{\hspace{0pt}}}subset is the same as a {C-{\hspace{0pt}}}subset, and
Lemma~\ref{lemma:3} can be applied.
\begin{lemma}\label{lemma:5}
Let $1< k < n-1$. Then the maximal {A-{\hspace{0pt}}}subsets of ${\mathcal G}$ are precisely
the following sets:
{\arraycolsep0pt\begin{eqnarray}\label{eq:typeA1}
{\mathcal G}(S,T)
\mbox{ with }&S\in{\mathcal G}_{k}\mbox{, }T\in{\mathcal G}_{n-k+1}\mbox{, and }&S+T =V.\\
\label{eq:typeA2}
{\mathcal G}(S,T)
\mbox{ with }&S\in{\mathcal G}_{k}\mbox{, }T\in{\mathcal G}_{n-k-1}\mbox{, and }&S\cap T =0.\\
\label{eq:typeA3}
{\mathcal G}(T,U)
\mbox{ with }&T\in{\mathcal G}_{k+1}\mbox{, }U\in{\mathcal G}_{n-k}\mbox{, and }&T+U =V.\\
\label{eq:typeA4}
{\mathcal G}(T,U)
\mbox{ with }&T\in{\mathcal G}_{k-1}\mbox{, }U\in{\mathcal G}_{n-k}\mbox{, and }&T\cap U =0.
\end{eqnarray}}
\end{lemma}
\begin{proof}
From \cite[p.~36]{Chow} we recall the following: Let ${\mathcal Y}\subset
{\mathcal G}_{i}$, $1<i<n-1$, be a maximal set of mutually adjacent
$i$-dimensional subspaces of $V$. Then there exists a subspace $T\in {\mathcal
G}_{i\pm 1}$ such that ${\mathcal Y}=\{Y\in{\mathcal G}_i\mid Y\mathrel{\mathrm{I}} T\}$.
Suppose now that ${\mathcal X}\subset{\mathcal G}$ is a maximal {A-{\hspace{0pt}}}subset. Clearly, there exists
an element $(S,U)\in{\mathcal X}$. Since ${\mathcal X}$ is also a {C-{\hspace{0pt}}}subset, we obtain that
${\mathcal X}\subset{\mathcal G}(S,V)$ or that ${\mathcal X}\subset{\mathcal G}(V,U)$.
Let ${\mathcal X}\subset{\mathcal G}(S,V)$. Then the second components of the elements of ${\mathcal X}$
are mutually adjacent elements of ${\mathcal G}_{n-k}$. Hence, by the above, they all
are incident with a subset $T\in{\mathcal G}_{n-k\pm 1}$. So, due to its maximality,
the set ${\mathcal X}$ is given as in (\ref{eq:typeA1}) or (\ref{eq:typeA2}).
Similarly, if ${\mathcal X}\subset{\mathcal G}(V,U)$ then ${\mathcal X}$ can be written as in
(\ref{eq:typeA3}) or (\ref{eq:typeA4}).
Conversely, it is obvious that (\ref{eq:typeA1})--(\ref{eq:typeA4}) define
maximal {A-{\hspace{0pt}}}subsets.
\end{proof}
We shall also make use of the following result:
\begin{lemma}\label{lemma:connect}
Any two elements $(S,U)$ and $(S',U')$ of ${\mathcal G}$ can be connected by a
finite sequence which is given as in formula {\rm(\ref{eq:A.connect})}. In
particular, if $S=S'$ {\rm(}or $U=U'${\rm)} then this sequence can be chosen in
such a way that $S=S_0=S_1=\cdots=S_i$ {\rm(}or $U=U_0=U_1=\cdots=U_i${\rm)}.
\end{lemma}
\begin{proof}
(a) First, we show the particular case when $(S,U),(S,U')\in{\mathcal G}(S,V)$ with
$S\in{\mathcal G}_k$. We proceed by induction on $d:=(n-k)-\dim(U\cap U')$, the case
$d=0$ being trivial.
Let $d>0$. There exists an $(n-k-1)$-dimensional subspace $W$ such that $U\cap
U'\subset W\subset U$. So $H:=W\oplus S$ is a hyperplane of $V$. It cannot
contain $U'$ because of $(S,U')\in{\mathcal G}$. Thus $W' := H\cap U'$ has dimension
$n-k-1$, and there exists a $1$-dimensional subspace $P'\subset U'$ with
$U'=P'\oplus W'$. Consequently, $P'\not\subset H$ and we obtain
\begin{equation}\label{}
V = P'\oplus H = P'\oplus W\oplus S.
\end{equation}
This means that $U'':=P'\oplus W$ is a complement of $S$. We have $(S,U)\sim
(S,U'')$ and $(n-k)-\dim(U''\cap U')=d-1$. So the assertion follows from the
induction hypothesis, applied to $(S,U'')$ and $(S,U')$.
Similarly, any two elements of ${\mathcal G}(V,U)$ with $U\in {\mathcal
G}_{n-k}$ can be connected.
(b) Now we consider the general case. Let $(S,U)$ and $(S',U')$ be elements of
${\mathcal G}$. There exists $U''\in {\mathcal G}_{n-k}$ which is complementary
to both $S$ and $S'$. Then, by (a), there exists a sequence
\begin{equation}\label{}
(S,U)\sim\cdots\sim (S,U'')\sim\cdots\sim(S',U'')\sim\cdots\sim(S',U')
\end{equation}
which completes the proof.
\end{proof}
The statement in (a) from the above is just a particular case of a more general
result on the connectedness of a \emph{spine space}; cf.\
\cite[Proposition~2.9]{praz+z-02}.
\begin{myproof}{\ref{thm:A}} (a) We shall accomplish our task by showing that every {A-{\hspace{0pt}}}transformation is a
{C-{\hspace{0pt}}}transformation. As has been noticed in Section~\ref{sect:trafos}, this is
trivial if $k=1$ or if $k=n-1$. So let $f$ be an {A-{\hspace{0pt}}}transformation of
${\mathcal G}$ and assume that $1<k<n-1$.
(b) We claim that
\begin{equation}\label{eq:Cinvariant}
f\big({\mathcal G}(S,V)) \mbox{ is a maximal {C-{\hspace{0pt}}}subset for all }S\in{\mathcal G}_{k}.
\end{equation}
Let us take $T\in {\mathcal G}_{n-k+1}$ such that ${\mathcal G}(S,T)$ is a
maximal {A-{\hspace{0pt}}}subset. Then $f\big({\mathcal G}(S,T)\big)$ is also a maximal
{A-{\hspace{0pt}}}subset. According to Lemma~\ref{lemma:5} there are four possible cases.
Case 1: $f\big({\mathcal G}(S,T)\big)$ is given according to (\ref{eq:typeA1}).
This means $f\big({\mathcal G}(S,T)\big)={\mathcal G}(W,Z)$ with $W\in{\mathcal G}_{k}$,
$Z\in{\mathcal G}_{n-k+1}$, and $W+Z=V$. We assert that in this case
\begin{equation}\label{eq:Ctrafo}
f\big((S,U')\big)\in{\mathcal G}(W,V) \mbox{ for all } (S,U')\in{\mathcal G}(S,V).
\end{equation}
In order to show this we choose an element $(S,U)\in{\mathcal G}(S,T)$. Clearly,
$f\big((S,U)\big)\in{\mathcal G}(W,Z)\subset{\mathcal G}(W,V)$.
First, we suppose that $(S,U)$ and $(S,U')$ are adjacent. Then $P:=U\cap
U'\in{\mathcal G}_{n-k-1}$. We consider the \emph{pencil\/} given by $P$ and $T$, i.e.
the set
\begin{equation}
\{X\in{\mathcal G}_{n-k}\mid P\subset X\subset T\}.
\end{equation}
It contains at least three elements; precisely one them is not complementary to
$S$. Consequently, the intersection of the maximal {A-{\hspace{0pt}}}subsets ${\mathcal
G}(S,T)$ and ${\mathcal G}(S,P)$ contains more than one element. The same
property holds for the intersection of the maximal {A-{\hspace{0pt}}}subsets $f\big({\mathcal
G}(S,T)\big)={\mathcal G}(W,Z)$ and $f\big({\mathcal G}(S,P)\big)$. But this means that
$W$ is the first component of every element of $f\big({\mathcal G}(S,P)\big)$
so that $ f\big((S,U')\big)\in{\mathcal G}(W,V)$.
Next, we suppose that $(S,U)$ and $(S,U')$ are arbitrary. By
Lemma~\ref{lemma:connect}, $(S,U)$ and $(S,U')$ can be connected by a finite
sequence
\begin{equation} (S,U)=(S,U_{0})\sim (S,U_{1})\sim\dots\sim (S,U_{i})=(S,U'),
\end{equation}
and the arguments considered above yield that (\ref{eq:Ctrafo}) holds.
Since $f^{-1}$ is adjacency preserving, we can repeat our previous proof, with
${\mathcal G}(W,Z)$ taking over the role of ${\mathcal G}(S,T)$. Altogether, this proves
\begin{equation}\label{eq:}
f\big({\mathcal G}(S,V)\big)={\mathcal G}(W,V).
\end{equation}
The remaining cases, i.e., when $f\big({\mathcal G}(S,T)\big)$ is given
according to (\ref{eq:typeA2}), (\ref{eq:typeA3}), or (\ref{eq:typeA4}), can be
treated similarly, whence (\ref{eq:Cinvariant}) holds true.
(c) Dual to (b), it can be shown that $f\big({\mathcal G}(V,U)\big)$ is a maximal
{C-{\hspace{0pt}}}subset for all $U\in{\mathcal G}_{n-k}$. Thus $f$ is a {C-{\hspace{0pt}}}transformation.
\end{myproof}
\section{Proofs of Theorem~\ref{thm:CGxG} and Theorem~\ref{thm:AGxG}}
\label{sect:thm:GxG}
In the following proof we use the term \emph{maximal {C-{\hspace{0pt}}}subset} just like in
Section~\ref{sect:thm:C}.
\begin{myproof}{\ref{thm:CGxG}}
Obviously, each maximal {C-{\hspace{0pt}}}subset of ${\mathcal G}_{k}\times{\mathcal G}_{n-k}$ has either the
form $\{S\}\times{\mathcal G}_{n-k}$ with $S\in{\mathcal G}_k$ (\emph{first kind}) or
${\mathcal G}_k\times\{U\}$ with $U\in{\mathcal G}_{n-k}$ (\emph{second kind}). Distinct maximal
{C-{\hspace{0pt}}}subsets of the same kind have empty intersection, whereas maximal
{C-{\hspace{0pt}}}subsets of different kind have a unique common element. So every
{C-{\hspace{0pt}}}transformation is either type preserving, whence it can be written as
$f'\times f''$, or type interchanging, whence it can be written as $g'\mathbin{\dot\times}
g''$.
\end{myproof}
Let $1<k<n-1$. We shall consider below the following well known \emph{partial
linear spaces}: For each $i=2,3,\ldots,n-2$ the set ${\mathcal G}_i$ is the point set of
the \emph{Grassmann space\/} $({\mathcal G}_i,{\mathcal L}_i)$; the elements of its line set
${\mathcal L}_i$ are the pencils
\begin{equation}
{\mathcal G}_i[P,T]:=\{X\in{\mathcal G}_i\mid P\subset X\subset T\},
\end{equation}
where $P\in{\mathcal G}_{i-1}$, $T\in{\mathcal G}_{i+1}$, and $P\subset T$. The \emph{Segre
product\/} (or \emph{product space\/}) of $({\mathcal G}_{k},{\mathcal L}_{k})$ and
$({\mathcal G}_{n-k},{\mathcal L}_{n-k})$ is the partial linear space with point set
\begin{equation}\label{}
{\mathcal P}:={\mathcal G}_{k}\times{\mathcal G}_{n-k}
\end{equation}
and line set
\begin{equation}\label{}
{\mathcal L}:=\big\{\{S\}\times l \mid S\in{\mathcal G}_{k},\, l\in{\mathcal L}_{n-k} \big\}
\cup
\big\{m\times \{U\} \mid m\in{\mathcal L}_{k},\, U\in{\mathcal G}_{n-k} \big\}.
\end{equation}
See \cite{NP} for further details and references.
\begin{myproof}{\ref{thm:AGxG}}
(a) If $k=1$ or if $k=n-1$ then the assertion follows from
Theorem~\ref{thm:CGxG}.
(b) Let $1<k<n-1$. Given a subset ${\mathcal M}\subset {\mathcal P}$ we put
\begin{equation}\label{}
{\mathcal M}^\perp:=\{(S,U)\in{\mathcal P} \mid (S,U)\perp (X,Y) \mbox{ for
all }(X,Y)\in{\mathcal M}\},
\end{equation}
where the sign ``$\perp$'' on the right hand side means ``adjacent or equal''.
Now let $(S,U)$ and $(S,U')$ be adjacent elements of ${\mathcal P} $. Then
\begin{equation}
\{(S,U),(S,U')\}^\perp =
\{(S,Y)\in{\mathcal P} \mid U\cap U'\subset Y
\mbox{ or } Y\subset U+U' \}
\end{equation}
and
\begin{equation}\label{eq:line1}
\{(S,U),(S,U')\}^\perp{}^\perp =
\{(S,Y)\in{\mathcal P} \mid U\cap U'\subset Y\subset U+U' \}.
\end{equation}
Similarly, if $(S,U)$ and $(S',U)$ are adjacent elements of ${\mathcal P} $ then
\begin{equation}\label{eq:line2}
\{(S,U),(S',U)\}^\perp{}^\perp =
\{(X,U)\in {\mathcal P} \mid S\cap S'\subset X\subset S+S' \}.
\end{equation}
Next, suppose that $g:{\mathcal P}\to{\mathcal P}$ is an {A-{\hspace{0pt}}}transformation. Every line of
$({\mathcal P},{\mathcal L})$ can be written in the form (\ref{eq:line1}) or (\ref{eq:line2}),
since it contains at least two distinct collinear points or, said differently,
two adjacent elements of ${\mathcal P}$. Thus $g$ is a collineation of the product space
$({\mathcal P},{\mathcal L})$. By \cite[Theorem~1.14]{NP}, there are two possibilities:
Case 1. There exist collineations of Grassmann spaces $f':{\mathcal G}_{k}\to{\mathcal G}_{k}$
and $f'':{\mathcal G}_{n-k}\to{\mathcal G}_{n-k}$ such that $g=f'\times f''$. Clearly, $f'$ and
$f''$ are adjacency preserving in both directions.
Case 2. There exist collineations of Grassmann spaces $g':{\mathcal G}_{k}\to{\mathcal G}_{n-k}$
and $g'':{\mathcal G}_{n-k}\to{\mathcal G}_{k}$ such that $g=g'\mathbin{\dot\times} g''$. As above, $g'$ and
$g''$ are adjacency preserving in both directions.
So $g$ is given as in Example~\ref{exmp:AGxG}.
\end{myproof}
\tiny
\ \\
Hans Havlicek\\
INSTITUT F\"UR DISKRETE MATHEMATIK UND GEOMETRIE\\
TECHNISCHE UNIVERSIT\"AT WIEN\\
WIEDNER HAUPTSTRASSE 8--10\\
A-1040 WIEN, AUSTRIA\\
havlicek@geometrie.tuwien.ac.at\\
\ \\
Mark Pankov\\
INSTITUTE OF MATHEMATICS\\
NATIONAL ACADEMY OF SCIENCE OF UKRAINE\\
TERESHCHENKIVSKA 3\\
01601 KIEV, UKRAINE\\
pankov@imath.kiev.ua
\end{document} |
\begin{document}
\title[Graphical $C(6)$ and $C(7)$ small cancellation groups]{Groups with graphical $C(6)$ and $C(7)$ small cancellation presentations}
\subjclass[2010]{Primary: 20F06; Secondary: 20F65, 20F67.}
\keywords{Small cancellation theory}
\author{Dominik Gruber}
\thanks{The work is supported by the ERC grant of Prof. Goulnara Arzhantseva ``ANALYTIC" no. 259527.}
\address{University of Vienna, Department of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria}
\email{dominik.gruber@univie.ac.at}
\begin{abstract}
We extend fundamental results of small cancellation theory to groups whose presentations satisfy the generalizations of the classical $C(6)$ and $C(7)$ conditions in \emph{graphical} small cancellation theory. Using these graphical small cancellation conditions, we construct lacunary hyperbolic groups and groups that coarsely contain prescribed infinite sequences of finite graphs. We prove that groups given by (possibly infinite) graphical $C(7)$ presentations contain non-abelian free subgroups.
\end{abstract}
\maketitle
\section{Introduction}
In his influential paper ``Random walk in random groups" \cite{Gr}, Gromov introduced small cancellation theory for labelled graphs as a far-reaching generalization of classical small cancellation theory.
A main feature of this theory is that it allows constructions of finitely generated groups that contain prescribed subgraphs in an appropriate metric sense. Gromov applied this observation to construct \emph{Gromov's monster}, a group that coarsely contains an expander graph. (For details, see \cite{AD}.)
Subsequently, Ollivier gave details on another theorem of Gromov that extends results of classical $C'(\frac{1}{6})$ small cancellation theory to its analogue in graphical small cancellation theory \cite{Oll}.
In the present paper, we generalize Ollivier's proof to the graphical analogues of the classical $C(6)$ and $C(7)$ conditions. The method of proof we present is purely combinatorial; most of our statements rely solely on Lyndon's curvature formulas \cite{LS}, which are consequences of the formula for the Euler characteristic of planar 2-complexes, and their applications to suitable van Kampen diagrams.
We first use our method to prove generalizations of fundamental results about classical $C(6)$ and $C(7)$ groups concerning Dehn functions and asphericity. These results have been stated before \cite{AD,Gr,Oll}, but proofs have never been published.
We then establish new results about groups given by non-metric graphical small cancellation presentations: Any group given by a (possibly infinite) graphical $C(7)$ presentation, see Definition \ref{defi:graphical1}, contains a non-abelian free subgroup, see Theorem \ref{thm:free}. This in particular implies non-amenability and exponential growth of the group.
We moreover prove that the $Gr(6)$ condition, see Definition \ref{defi:graphical2}, can be used to construct groups that coarsely contain prescribed infinite sequences of finite graphs, see Theorem \ref{thm:coarse_embedding}. In the view of applications to the Baum-Connes conjecture \cite{HLS}, this is of particular interest when considering an expander graph: If an expander graph $X:=(X_n)_{n\in\N}$ admits a labelling such that the disjoint union $\sqcup_{n\in\N}X_n$ satisfies the $Gr(6)$ condition, then the group defined by the labelled graph $X$ coarsely contains $X$. Thus, such a labelling would yield a new construction of Gromov's monster.
Continuing our investigations of groups defined by infinite sequences of finite graphs, we show that if the $Gr(7)$ condition, see Definition \ref{defi:graphical2}, is satisfied, lacunary hyperbolic groups can be constructed by passing to appropriate subsequences of defining graphs, see Proposition \ref{prop:7_lacunary}. If, moreover, the metric $Gr'(\frac{1}{6})$ condition is satisfied, we are able to give combinatorial conditions on the graphs that yield lacunary hyperbolicity, see Proposition \ref{prop:1/6_lacunary}.
The construction of Gromov's monster inspired further constructions of groups with extreme properties through graphical small cancellation theory \cite{AD, OW, Sil}. Common features of all these constructions are their use of \emph{metric} graphical small cancellation conditions and of \emph{random} labellings of graphs. Therefore, the group presentations obtained in this manner are non-explicit. Turning them into explicit constructions has been unattainable so far.
We believe that the weaker \emph{non-metric} small cancellation conditions
we present are more accessible to making explicit constructions. Our
theorems show that, in some respects, they yield results as good
as the metric ones.
\subsection{Graphical small cancellation conditions}
Before stating the graphical small cancellation conditions, we explain how a group is constructed from a labelled graph. This idea first appeared in \cite{RS}. Throughout this paper, $S$ denotes a finite set (our \emph{alphabet}).
Let $\Gamma$ be a graph. A \emph{labelling} of $\Gamma$ with letters from $S$ is a choice of orientation on each edge of $\Gamma$ and a map assigning to each edge of $\Gamma$ an element of $S$ (the \emph{label}). A map of labelled graphs is a simplicial map that preserves the labelling. A labelling is called \emph{reduced} if at every vertex $v$, any two oriented edges starting at $v$ have distinct labels and any two oriented edges ending at $v$ have distinct labels.
Let $M(S)$ denote the free monoid on $S\sqcup S^{-1}$. The labelling of $\Gamma$ induces a map
$$\ell:\{\text{paths on }\Gamma\}\to M(S),$$
given as follows: For a path $p$, $\ell(p)$ is obtained by reading the labels of the edges in $p$ starting from the initial vertex of $p$,
and each letter is given exponent 1 if the corresponding edge is traversed in its direction and exponent $-1$ if it is traversed in the opposite direction.
Assume $\Gamma$ is connected and fix a base vertex $v$ in $\Gamma$. Then $\ell$ induces a homomorphism:
$$\ell_*:\pi_1(\Gamma,v)\to F(S),$$
where $F(S)$ denotes the free group on $S$. Similarly, if $\Gamma$ is the disjoint union of a set of connected components with base points $\{(\Gamma_i,v_i)|i\in I\}$, we obtain a homomorphism $\ell_*:*_{i\in I}\pi_1(\Gamma_i,v_i)\to F(S)$.
\begin{defi}\label{defi:groupdefinedbygraph} For a labelled graph $\Gamma$, we define
$$G(\Gamma):=F(S)/\langle\langle \operatorname {im} \ell_* \rangle\rangle,$$
the \emph{group defined by $\Gamma$}. (Here $\langle\langle - \rangle\rangle$ denotes the normal subgroup generated by $-$.) Thus $G(\Gamma)$ is given by the presentation $\langle S|R\rangle$, where $R$ is the set of words read on closed paths in $\Gamma$.
\end{defi}
Let $\Gamma$ be the disjoint union of its connected components $\Gamma_i$ for $i\in I$. Given vertices $v_i\in \Gamma_i$ and group elements $g_i\in G(\Gamma)$, there is a unique map of labelled graphs $\Gamma\to\Cay(G(\Gamma),S)$ that maps each $v_i$ to $g_i$. Any normal subgroup $N$ of $F(S)$ with the property that there exists a map of labelled graphs $\Gamma\to \Cay(F(S)/N,S)$ contains $\langle\langle \operatorname {im} \ell_* \rangle\rangle$, since any closed path in $\Gamma$ has to be mapped to a closed path in $\Cay(F(S)/N,S)$. Thus, we can think of $G(\Gamma)$ as the largest quotient $G$ of $F(S)$ such that $\Gamma$ maps to $\Cay(G,S)$.
We now state the graphical small cancellation conditions. The graphical $C(n)$ and $C'(\lambda)$ conditions are based on \cite{Oll} and \cite[Section 2.2]{AD}, who both slightly modified Gromov's original definition \cite[Section 2]{Gr}. Gromov's graphical small cancellation conditions will be denoted by $Gr(n)$ and $Gr'(\lambda)$. We call a closed path on a graph \emph{nontrivial} if it is not 0-homotopic.
\begin{defi}\label{defi:graphical1}
Let $\Gamma$ be a labelled graph. A \emph{piece} on $\Gamma$ is a labelled path $p$ (considered as a labelled graph) for which there are two distinct maps of labelled graphs $p\to\Gamma$. Let $n\in\N$ and $\lambda>0$.
\begin{itemize}
\item $\Gamma$ satisfies the $C(n)$ \emph{condition} if the labelling is reduced and no nontrivial closed path is the concatenation of fewer than $n$ pieces.
\item $\Gamma$ satisfies the $C'(\lambda)$ \emph{condition} if the labelling is reduced and for any simple closed path $\gamma$, any piece $p$ that is a subpath of $\gamma$ satisfies $|p|<\lambda |\gamma|$.
\end{itemize}
\end{defi}
\begin{defi}\label{defi:graphical2}
Two maps of labelled graphs $\phi_1,\phi_2:\Gamma'\to \Gamma$ are \emph{essentially equal} if there is an automorphism of labelled graphs $\psi:\Gamma\to\Gamma$ with $\phi_2=\psi\circ\phi_1$. Otherwise they are \emph{essentially distinct}. An \emph{essential piece} on $\Gamma$ is a labelled path $p$ for with there exist two essentially distinct maps of labelled graphs $p\to\Gamma$.
\begin{itemize}
\item $\Gamma$ satisfies the $Gr(n)$ \emph{condition} if the labelling is reduced and no nontrivial closed path is the concatenation of fewer than $n$ essential pieces.
\item $\Gamma$ satisfies the $Gr'(\lambda)$ \emph{condition} if the labelling is reduced and for any simple closed path $\gamma$, any essential piece $p$ that is a subpath of $\gamma$ satisfies $|p|<\lambda |\gamma|$.
\end{itemize}
\end{defi}
Observe that the graphical small cancellation conditions are generalizations of the classical small cancellation conditions: For any classical $C(n)$ or $C'(\lambda)$ presentation $\langle S|R\rangle$, we can construct a graph as the disjoint union of cycle graphs, each labelled by (the class of cyclic conjugates and their inverses) of a relator of the presentation. If $\langle S|R\rangle$ satisfies the classical $C(n)$ or $C'(\lambda)$ condition, the constructed graph satisfies the $Gr(n)$ or $Gr'(\lambda)$ condition, respectively. If no relator is a proper power, the graph satisfies the $C(n)$ or $C'(\lambda)$ condition, respectively.
\subsection{Statement of results}
We first state those results which have been expected based on \cite{Gr}, \cite{Oll} and \cite{AD}. They are generalizations of fundamental facts about classical $C(6)$ and $C(7)$ presentations. Let $S$ be a finite set.
\begin{thm*}[cf.\ Theorem \ref{thm:linear_isoperimetry}]
Let $\Gamma$ be a $Gr(7)$-labelled graph. Let $R$ be the set of words read on all simple cycles of $\Gamma$. Then the presentation $\langle S|R\rangle$ satisfies the linear isoperimetric inequality:
$$\Area_R(w)\leq 8|w|.$$
In particular, if $\Gamma$ is finite, then the group $G(\Gamma)$ is Gromov-hyperbolic.
\end{thm*}
\begin{thm*}[cf.\ Theorem \ref{thm:quadratic_isoperimetry}]
Let $\Gamma$ be a $Gr(6)$-labelled graph. Let $R$ be the set of words read on all simple cycles of $\Gamma$. Then the presentation $\langle S|R\rangle$ satisfies the quadratic isoperimetric inequality:
$$\Area_R(w)\leq 3|w|^2.$$
\end{thm*}
\begin{thm*}[cf. Theorem \ref{thm:asphericity}] Let $\Gamma$ be a $C(6)$-labelled graph. Let $R$ be the set of cyclic reductions of words read on free generating sets of the fundamental groups of the connected components of $\Gamma$. Then the presentation complex associated to $\langle S|R\rangle$ is aspherical. The group $G(\Gamma)$ defined by $\Gamma$ has an at most 2-dimensional $K(G(\Gamma),1)$ space and hence cohomological dimension at most 2. Therefore, $G(\Gamma)$ is torsion-free.
\end{thm*}
We next state our original results. The following theorem in particular shows that only non-amenable groups arise from graphical $C(7)$ presentations. While for finitely presented graphical $C(7)$ groups the theorem follows from the fact that non-elementary hyperbolic groups have non-abelian free subgroups \cite[$5.3.\mathrm{C}_1$]{Grhyp}, it is new for infinitely presented graphical $C(7)$ groups.
\begin{thm*}[cf. Theorem \ref{thm:free}]
Let $\Gamma$ be a $C(7)$-labelled graph. Then $G(\Gamma)$ contains a non-abelian free subgroup unless it is trivial or infinite cyclic.
\end{thm*}
In fact, we explain in Remark \ref{rem:obvious} that $G(\Gamma)$ is trivial or infinite cyclic if and only if some easily checkable conditions on the graph hold. We also show a version of this theorem applicable to classical small cancellation presentations:
\begin{thm*}[cf. Theorem \ref{thm:ess-free}]
Let $\Gamma$ be a $Gr(7)$-labelled graph with infinitely many pairwise non-isomorphic connected components with nontrivial fundamental groups. Then $G(\Gamma)$ contains a non-abelian free subgroup.
\end{thm*}
\begin{cor*}[cf. Corollary \ref{cor:classical-free}]
Let $G$ be a group with an infinite classical $C(7)$ presentation. Then $G$ contains a non-abelian free subgroup.
\end{cor*}
We next consider how the defining graph $\Gamma$ of a group embeds into the Cayley graph $\Cay(G(\Gamma),S)$ as a metric space. Let $(\Gamma_n)_{n\in\N}$ be a sequence of connected finite graphs and let $\Gamma:=\bigsqcup_{n\in\N}\Gamma_n$ be their disjoint union. We endow $\Gamma$ with a metric that coincides with the graph metric on each connected component such that $d(\Gamma_ {a_n},\Gamma_{b_n})\to\infty$ as $a_n+b_n\to \infty$ assuming $a_n\neq b_n$ for almost all $n$. We call the resulting metric space the \emph{coarse union} of the $\Gamma_n$. A \emph{coarse embedding} is a map $f:X\rightarrow Y$, where $X$ and $Y$ are metric spaces, such that for all sequences $(x_n,y_n)_{n\in\N}$ in $X\times X$ we have $d(x_n,y_n)\to\infty\Leftrightarrow d(f(x_n),f(y_n))\to\infty$.
The following theorem shows that whenever we find a labelling of an infinite sequence of finite graphs such that their disjoint union satisfies the $Gr(6)$ condition, we can construct a finitely generated group that coarsely contains the coarse union of these graphs:
\begin{thm*}[cf.\ Theorem \ref{thm:coarse_embedding}] Let $(\Gamma_n)_{n\in\N}$ be a sequence of finite, connected graphs such that $\Gamma:=\bigsqcup_{n\in\N}\Gamma_n$ is $Gr(6)$-labelled and such that $|\Gamma_n|$ is unbounded. Then the coarse union $\Gamma$ embeds coarsely into $\Cay(G(\Gamma),S)$.
\end{thm*}
This map is also injective as stated by Gromov \cite{Gr} and proven in Lemma \ref{lem:embedding_injective}.
A finitely generated group is called \emph{lacunary hyperbolic} if one of its asymptotic cones is an $\R$-tree. The following proposition shows that the $Gr(7)$ condition can be used to construct lacunary hyperbolic groups:
\begin{prop*}[cf.\ Proposition \ref{prop:7_lacunary}]
Let $(\Gamma_n)_{n\in\N}$ be a sequence of finite, connected graphs such that their disjoint union is $Gr(7)$-labelled. Then there exists an infinite subsequence of graphs $(\Gamma_{k_n})_{n\in\N}$ such that $G(\bigsqcup_{n\in\N}\Gamma_{k_n})$ is lacunary hyperbolic.
\end{prop*}
Moreover, we apply results of \cite{Oll} and an argument given in
\cite[Proposition 3.12]{OOS}, where the question when a group given by an infinite classical $C'(\frac{1}{6})$ presentation is lacunary hyperbolic is answered, to show:
\begin{prop*}[cf.\ Proposition \ref{prop:1/6_lacunary}] Let $(\Gamma_n)_{n\in\N}$ be a sequence of finite, connected graphs such that
\begin{itemize}
\item $\Gamma:=\sqcup_{n\in\N}\Gamma_n$ is $Gr'(\frac{1}{6})$-labelled and
\item $\Delta(\Gamma_n)=O(g(\Gamma_n))$,
\end{itemize}
where $\Delta(\Gamma_n)$ denotes the diameter and $g(\Gamma_n)$ denotes the girth of each graph. (If $\Gamma_n$ is a tree, set $g(\Gamma_n)=0$.)
Then $G(\Gamma)$ is lacunary hyperbolic if and only if the set of girths $L:=\{g(\Gamma_n)|n\in\N\}$ is sparse, i.e. for all $K\in \R^+$ there exists $a \in \R^+$ such that $[a,aK]\cap L=\emptyset$.
\end{prop*}
\section{Extending classical small cancellation theory}\label{section:preliminaries}
\subsection{Examples and first observations}\label{subsection:p1}
Following the definitions of the graphical small cancellation conditions given in the introduction, we provide two examples and make some basic observations about pieces and the graphical small cancellation conditions.
\begin{example}
Let $S=\{a,b,c\}$, and let $\Gamma$ be as in Figure \ref{figure:C(6)-example}. The group $G(\Gamma)$ is given by the presentation $$\langle a,b,c|a^2c^{-1}b^{-2}a^{-1}b^{-1},a^2b^{-1}c^{-2}a^{-1}c^{-1}\rangle.$$ The graph $\Gamma$ satisfies the $C'(\frac{1}{6})$ condition: The labelling is reduced, any piece that is subpath of a simple cycle has length at most 1, and any cycle has length at least 7. Note that the presentation does not satisfy the classical $C'(\frac{1}{6})$ condition. Also note that this is a maximal graphical $C'(\frac{1}{6})$ presentation in the alphabet $S$, i.e.\ one cannot add more $S$-labelled cycles to the graph retaining the $C'(\frac{1}{6})$ condition, since any such cycle would produce new pieces of length 2 on $\Gamma$ that would lead to a violation of the $C'(\frac{1}{6})$ condition.
\end{example}
\begin{figure*}
\caption{The $C'(\frac{1}
\label{figure:C(6)-example}
\end{figure*}
\begin{example}\label{ex:cayley}
Let $G$ be a group generated by a finite set $S$. Then the Cayley graph $\Cay(G,S)=:\Gamma$ has a natural reduced labelling, and we have $G=G(\Gamma)$. Note that a path on a graph with a reduced labelling is uniquely determined by its starting vertex and its label. The group of labelled-graph-automorphisms of $\Gamma$ acts transitively on the vertex set of $\Gamma$. Therefore, whenever a labelled path has two maps to $\Gamma$, they are essentially equal. Thus $\Gamma$ has no essential pieces and satisfies any $Gr(n)$ or $Gr(\lambda)$ condition. Unless the group is free on $S$, or trivial, it will not satisfy $C(2)$ or $C'(1)$, i.e.\ no nontrivial $Gr(n)$ or $Gr'(\lambda)$ condition, since there will be two distinct cycles with the same label and thus a cycle made up of one piece.
\end{example}
Observe that any essential piece is a piece. Thus the $C(n)$ condition implies the $Gr(n)$ condition, and the $C'(\lambda)$ condition implies the $Gr'(\lambda)$ condition. The treatment of the conditions from Definition \ref{defi:graphical1} and those from Definition \ref{defi:graphical2} is very similar; in fact all results of Subsections \ref{subsection:p1}-\ref{subsection:p3} hold for both conditions and for pieces and essential pieces alike. We will therefore mostly restrict ourselves to stating them for the $C(n)$ or $C'(\lambda)$ conditions and for pieces only. The only difference in statements and proofs is that ``piece" has to be replaced by ``essential piece", ``unique" by ``essentially unique", ``equal" by ``essentially equal", etc.
\emph{Backtracking} in a path on a graph $\Gamma$ means that an edge of $\Gamma$ is traversed two consecutive times in opposite directions. A path in a graph is \emph{reduced} if there is no backtracking. The reduction of a path is obtained by removing any backtracking. We have the following observations for a piece $p$:
\begin{itemize}
\item The reduction of $p$ is a piece.
\item Every subpath of $p$ is a piece.
\item The inverse path of $p$ is a piece.
\end{itemize}
\begin{lem} Let $\Gamma$ be a labelled graph, $n\in\N$ and $\lambda>0$. Then the following are equivalent:
\begin{itemize}
\item[i)] $\Gamma$ satisfies $C(n)$.
\item[ii)] No reduced cycle is the concatenation of fewer than $n$ pieces.
\item [iii)] No simple cycle is the concatenation of fewer than $n$ pieces.
\end{itemize}
Moreover, if $\Gamma$ satisfies $C'(\lambda)$, then $\Gamma$ satisfies $C(\lfloor \frac{1}{\lambda} \rfloor +1)$.
\end{lem}
\begin{proof}
The implications $\text{i)}\Rightarrow \text{ii)}\Rightarrow \text{iii)}$ are obvious. Now suppose iii) holds. Let $\gamma$ be a nontrivial cycle on $\Gamma$, and suppose $\gamma=p_1p_2\ldots p_k$ where $p_i$ is a piece for each $i$. Then the reduction of $\gamma$ may be written as $q_1q_2\ldots q_l$ where each $q_i$ is a subpath of the reduction of a $p_j$, and $l\leq k$. Let $\gamma'$ be a subpath of $\gamma$ that is a simple cycle. Then $\gamma'=r_1r_2\ldots r_m$, where each $r_i$ is a subpath of some $q_j$, and $m\leq l$. By assumption, we have $m\geq n$ and hence $k\geq l\geq m\geq n$.
The last statement follows from the implication $\text{iii)}\Rightarrow \text{i)}$.
\end{proof}
\subsection{Diagrams}\label{subsection:p2}
We briefly recall tools and results of classical small cancellation theory, which we will use to present graphical small cancellation theory. For an introduction to classical small cancellation theory see \cite[Chapter V]{LS}.
A \emph{singular disk diagram} $D$ in the alphabet $S$ is a finite, simply connected, 2-dimensional CW-complex embedded into the plane with the following additional data:
\begin{itemize}
\item Each 1-cell is oriented and labelled by an element of $S$.
\item A base vertex $v$ in the topological boundary of $\R^2\setminus D$ is specified. (We also denote $(D,v)$. We omit mentioning the base vertex if this is not necessary.)
\end{itemize}
A \emph{simple} disk diagram is a singular disk diagram that is homeomorphic to a disk. Equivalently, a simple disk diagram is singular disk diagram without cut-points.
The \emph{boundary path} $\partial D$ of a singular disk diagram $D$ is the closed simplicial path along the topological boundary of $\R^2\setminus D$ from the base vertex in counterclockwise direction. The \emph{label} of $\partial D$ (or \emph{boundary label} of $D$) is the element of the free monoid on $S\sqcup S^{-1}$ obtained by reading the labels of the edges of $\partial D$,
where each letter is given exponent 1 if the corresponding edge is traversed in its direction and exponent $-1$ if it is traversed in the opposite direction.
We also introduce spherical diagrams following \cite{CH}. A spherical complex is a set of 2-spheres embedded into $\R^3$ connected by simple curves such that no sphere contains the other, such that the intersection of two spheres consists of at most one point and such that the whole complex is simply connected. (Recall that a curve is simple if it has no self-intersections.)
A \emph{spherical diagram} in the alphabet $S$ is finite 2-complex tessellating a spherical complex, where each 1-cell is oriented and labelled by an element of $S$. A \emph{simple} spherical diagram is a spherical diagram tessellating a single 2-sphere. It is our convention that a spherical diagram has empty boundary.
The word \emph{diagram} will refer to a singular disk diagram or a spherical diagram. The \emph{area} of a diagram is the number of its 2-cells.
A \emph{face} $f$ of a diagram is the image of a closed 2-cell $b$ under its characteristic map. The \emph{boundary path} $\partial f$ of $f$ is the simplicial path obtained as the image of the simple closed path along the boundary of $b$. The \emph{label} of $\partial f$ (or \emph{boundary label} of $f$) is obtained by reading the labels of the edges of $\partial f$. (Here choices of basepoints and orientations are necessary. If these are not specified, the boundary path and boundary label are still unique up to inversion and cyclic shifts, which will often be sufficient for our considerations.)
For the finite set $S$, let $M(S)$ denote the free monoid on $S\sqcup S^{-1}$. For a set of words $R\subset M(S)$, a \emph{diagram over $R$} is a diagram $D$ such that for each face $f$ of $D$ there exist an orientation and a basepoint for $\partial f$ such that the label of $\partial f$ lies in $R$. Given a labelled graph $\Gamma$, a \emph{diagram over $\Gamma$} is a diagram over the set of words read on nontrivial cycles on $\Gamma$.
Let $w\in M(S)$. A \emph{diagram for $w$ over $R$} (respectively \emph{over $\Gamma$}) is a diagram over $R$ (respectively $\Gamma$) whose boundary label is $w$. A \emph{minimal} diagram for $w$ over $R$ (respectively $\Gamma$) is a diagram $D$ for $w$ over $R$ (respectively $\Gamma$) such that
\begin{itemize}
\item the area of $D$ is minimal among all diagrams for $w$ over $R$ (respectively $\Gamma$), and
\item the number of edges of $D$ is minimal among all such diagrams of minimal area.
\end{itemize}
The following theorem is a version of the so-called ``van Kampen Lemma" \cite[Section V.1]{LS}, which is central in small cancellation theory.
\begin{thm}
Let $G$ be a group given by the presentation $\langle S|R\rangle$. An element $w$ of the free monoid $M(S)$ over $S$ satisfies $w=1$ in $G$ if and only if there exists a singular disk diagram over $R$ such that the boundary label of $D$ is $w$.
\end{thm}
An \emph{arc} in a diagram is an embedded path all interior vertices of which have degree 2 and whose initial and terminal vertices have degrees different from 2. A \emph{spur} is an arc which has a vertex of degree $1$. A face $f$ of a diagram $D$ is called \emph{interior} if the intersection $\partial f\cap \partial D$ contains no edge. All other faces are called \emph{exterior} or \emph{boundary faces}.
Let $p,q$ be positive integers. A diagram $D$ is a \emph{$(p,q)$-diagram} if every interior vertex has degree at least $p$ and if the boundary of every interior face consists of at least $q$ edges. If $D$ is a diagram in which the boundary path of each interior face is the concatenation of at least $q$ arcs, we can consider $D$ as a $(3,q)$-diagram if we ``forget" vertices of degree two. Forgetting a vertex $v$ of degree two means removing the vertex and replacing the two edges incident at $v$ by a single edge (cf. Figure \ref{figure:forget}). In this context, labels and orientations of edges will play no role and will therefore be ignored.
A diagram is a \emph{$[p,q]$-diagram} if every interior vertex has degree at least $p$ and if the boundary of every face consists of at least $q$ edges. Note that a spherical $(p,q)$-diagram is a $[p,q]$-diagram.
\begin{figure}
\caption{Forgetting a vertex of degree two.}
\label{figure:forget}
\end{figure}
The following are standard results from classical small cancellation theory which we will use later on:
\begin{thm}[Curvature formula I, cf. {\cite[Corollary V.3.4]{LS}}]
Let $D$ be a $(3,6)$-singular disk diagram with at least two faces. For a face $f$, let $i(f)$ denote the number of its interior edges. Then
$$\sum_{f\mathrm{\ a\ boundary\ face\ of\ } D} (4-i(f))\geq 6.$$
\end{thm}
In fact, more refined versions of this result can be found in \cite{MW} and \cite{Str}.
\begin{thm}[Curvature formula II, cf. {\cite[Corollary V.3.3]{LS}}] Let $D$ be a $[3,6]$-singular disk diagram. For a vertex $v$, let $d(v)$ denote its degree. Then
$$\sum_{v\in\partial D}(2+\frac{1}{2}-d(v))\geq 3,$$
where the sum is taken over all vertices in the boundary.
\end{thm}
The following theorem is an immediate conclusion of \cite[Theorem 6.2]{LS} and the first curvature formula.
\begin{thm}[Area theorem I]
Let $D$ be a $(3,6)$-singular disk diagram. Then
$$ |D|\leq 3|\partial D|^2,$$
where $|D|$ is the number of faces of $D$ and $|\partial D|$ is the number of its boundary edges.
\end{thm}
A stronger statement holds for $(3,7)$-diagrams. It is proved in \cite[Proposition 2.7]{Str}.
\begin{thm}[Area theorem II]
Let $D$ be a $(3,7)$-singular disk diagram. Then
$$ |D|\leq 8 |\partial D|.$$
\end{thm}
\subsection{Basic tools and results}\label{subsection:p3}
We will now explain how to apply the aforementioned results from classical small cancellation theory to graphical small cancellation presentations. The following definition is given in \cite{Oll}.
\begin{defi}[To originate from $\Gamma$] Let $\Gamma$ be a $C(2)$-labelled graph, and let $f$ be a face in a diagram $D$ over $\Gamma$. Choose a basepoint and an orientation for $\partial f$. Then $\partial f$ maps (we say \emph{lifts}) to a cycle in $\Gamma$, and since $C(2)$ is satisfied, this cycle is unique. Let $p$ be a path in an interior arc of $D$ that lies in the intersection of faces $f_1$ and $f_2$. Then we can choose basepoints and orientations for $\partial f_1$ and $\partial f_2$, such that $p$ is an initial subpath of $\partial f_1$ and $\partial f_2$. If the images of $p$ via the lifts of $\partial f_1$ and $\partial f_2$ to $\Gamma$ coincide, we say $p$ \emph{originates from} $\Gamma$.
\end{defi}
Observe that any interior arc of a diagram that does not originate from $\Gamma$ is a piece.
The above definition can be extended to $Gr(2)$-labelled graphs. In that case, lifts are considered up to composition with an automorphism of $\Gamma$. Any interior arc of a diagram that does not essentially originate from $\Gamma$ is an essential piece. As stated before, all proofs and conclusions of Subsections \ref{subsection:p1}-\ref{subsection:p3} for $C(n)$- (respectively $C'(n)$-) labelled graphs are true for $Gr(n)$- (respectively $Gr'(\lambda)$-) labelled graphs as well. We will not state them doubly to avoid confusing notation.
The following lemma is our main tool in extending classical small cancellation theory to graphical small cancellation theory. It is based on \cite{Oll}, in which the $C'(\frac{1}{6})$ condition is investigated.
\begin{lem}\label{lem:graphical_basic} Let $D$ be a singular disk diagram over a $C(n)$-labelled graph $\Gamma$, where $n\geq 6$. Then one of the following holds:
\begin{itemize}
\item Removing all edges of $D$ originating from $\Gamma$ and merging the corresponding 2-cells yields a $(3,n)$-diagram $D'$ over $\Gamma$, all faces of which are simply connected.
\item $D$ has a subdiagram that is a simple disk diagram with at least one face, with freely trivial boundary word, all interior edges of which originate from $\Gamma$ and no boundary edge of which is an interior edge of $D$ that originates from $\Gamma$.
\end{itemize}
\end{lem}
\begin{proof} Note that removing an arc $a$ from a diagram $D$ may yield a resulting space $D'$ that is not a diagram. This can be the case if $a$ is the image of two distinct, non-consecutive subpaths of $\partial f$ for a single face $f$. In this case, after removing the arc $a$, $f$ becomes the image of a closed annulus. If multiple arcs are removed, this can happen multiple times. Then $f$ becomes the image of a closed disk with holes, i.e. the image of a space $b'=\overline b\setminus \sqcup_{i=1}^nb_i$, where $b=D^2\subset \R^2$, $n\geq 1$ and each $b_i$ is an open disk of radius $\epsilon$ contained in $b$ such that $\overline b_i\cap \partial b=\emptyset$ and for $i\neq j: \overline b_i\cap \overline b_j=\emptyset$. We call such a space $f$ a \emph{face with holes}.
Let $D$ be as in the statement and assume the second claim does not hold. Let $D'$ be obtained from $D$ by deleting all edges originating from $\Gamma$ and merging the corresponding faces. We will show that there are no faces with holes, and hence $D'$ is a diagram. Moreover, there may be faces in $D'$ that are not simply connected. We rule out this case as well. (See Figure \ref{figure:holey} for illustrations.)
There are no spurs in $D'$ whose endpoints lie in the interior of $D'$, as such a spur would lie entirely in one face and, by the reducedness of the labelling, consist of edges originating from $\Gamma$. Assume there is a face with holes or a non-simply connected face $f$. By choosing $f$ to be innermost, we can assume that there is a subdiagram $\Delta$ which is enclosed by $f$ (i.e. the interior of $\Delta$ lies in the union of bounded connected components of $\R^2\setminus f$), such that $\Delta$ is in fact a diagram, $\Delta$ has at least one face, all faces of $\Delta$ are simply connected and $\Delta$ has no spur.
Since $\Delta$ is enclosed by $f$, $\Delta$ has at most one vertex (say $v$) that meets an edge that is not in $\Delta$. By assumption, all faces of $\Delta$ bear the labels of nontrivial cycles in $\Gamma$. Note that all interior arcs of $D'$ are pieces. Now we can forget vertices of degree two in $D'$. This turns the subdiagram $\Delta$ into a $[3,6]$-diagram, due to our small cancellation assumption. There is at most one vertex in $\Delta$ (namely $v$) that has degree (in $\Delta$) less than $3$. This contradicts the second curvature formula for $\Delta$.
Hence $D'$ is a diagram and all faces are simply connected. Since the second claim does not hold, every face bears a freely nontrivial boundary word, i.e. a word read on a nontrivial cycle of $\Gamma$. The small cancellation assumption and the observation that interior arcs are pieces yield that $D'$ is a $(3,n)$-diagram.
\end{proof}
\begin{figure}
\caption{On the left: A face $f$ with one hole enclosing the subdiagram $\Delta$. The dashed line represents an arc in the original diagram $D$ that was removed when passing to $D'$. On the right: A non-simply connected face $f$ enclosing the subdiagram $\Delta$.}
\label{figure:holey}
\end{figure}
Suppose in a singular disk diagram $D$, there is a simply connected face $f$ that has freely trivial boundary word and such that there are no spurs whose endpoints lie inside $f$. Then we can fold the boundary of $f$, thus removing $f$ without altering anything else in the diagram. This is done as follows:
Since the boundary cycle $\partial f$ embeds into $D$, any two consecutive edges with inverse labels in $\partial f$ can folded together (i.e. identified) in $D$ such that the resulting edge only intersects the resulting face $f'$ in a vertex. If $f$ had only two boundary edges, we let $f'$ be a vertex. The recursive application of this folding turns $f$ into a labelled tree, which, considered as a diagram, has the same boundary word as $f$. See Figure \ref{figure:fold} for an example.
\begin{figure}
\caption{Folding a face $f$ with freely trivial boundary word.}
\label{figure:fold}
\end{figure}
After folding away $f$, the resulting diagram has smaller area than the original one. This observation also applies to any subdiagram $\Delta$ of $D$ with at least one face and with freely trivial boundary word, since we can simply replace $\Delta$ by a face $f$ with the same boundary word and then fold away $f$. Hence removing all edges originating from $\Gamma$ on a diagram $D$ of minimal area for a word $w$ yields a diagram as in the first case of Lemma \ref{lem:graphical_basic}. We therefore obtain:
\begin{cor}\label{cor:graphical_basic} Let $w\in M(S)$ satisfying $w=1$ in $G(\Gamma)$, and let $D$ be a minimal diagram for $w$ over $\Gamma$. Then no interior edge of $D$ originates from $\Gamma$, and therefore, $D$ is a $(3,n)$-diagram, all faces of which are simply connected.
\end{cor}
The following lemma is proved just as Lemma \ref{lem:graphical_basic} with the additional observation that there is no spherical $[3,6]$-diagram with more than one face. This follows from the second curvature formula.
\begin{lem}\label{lem:graphical_basic_spherical} Let $D$ be a simple spherical diagram over a $C(6)$-labelled graph $\Gamma$. Then one of the following holds:
\begin{itemize}
\item All edges of $D$ originate from $\Gamma$.
\item $D$ has a subdiagram that is a simple disk diagram with at least one face, with freely trivial boundary word, all interior edges of which originate from $\Gamma$ and no boundary edge of which originates from $\Gamma$.
\end{itemize}
\end{lem}
The set of all words read on nontrivial cycles of a labelled graph $\Gamma$ is generally infinite. The set of all words read on all \emph{simple} cycles of $\Gamma$ generates the same normal subgroup of $F(S)$. If $\Gamma$ has finitely generated fundamental group, this set is finite. The following lemma improves the conclusions of Lemma \ref{lem:graphical_basic} to show, that considering the presentation given by the words read on all simple cycles is sufficient to obtain $(3,n)$-diagrams from $C(n)$ labelled graphs if $n\geq 6$.
\begin{lem}\label{lem:graphical_simple} Let $\Gamma$ be a $C(n)$ labelled graph for $n\geq 6$. Let $w\in M(S)$ satisfying $w=1$ in $G(\Gamma)$. Then there exists a diagram $D$ for $w$ such that no interior edge of $D$ originates from $\Gamma$ and such that every face bears the label of a simple cycle in $\Gamma$.
\end{lem}
Note that by Lemma \ref{lem:graphical_basic}, $D$ is a $(3,n)$ diagram, all faces of which are simply connected.
\begin{proof}[Proof of the lemma] Let $D$ be a diagram for $w$ over $\Gamma$ with a minimal number of edges whose number of vertices is minimal among all diagrams for $w$ over $\Gamma$ with a minimal number of edges.
Suppose $D$ has a subdiagram $\Delta$ as in the second case of Lemma \ref{lem:graphical_basic}. Then we can remove the interior edges of $\Delta$ and fold its boundary thus removing $\Delta$, leaving a diagram over $\Gamma$. This diagram has fewer edges than $D$ while having the same boundary word, which contradicts the assumptions.
Thus we are in the first case of Lemma \ref{lem:graphical_basic}. Since deleting all interior edges originating from $\Gamma$ gives a diagram over $\Gamma$, the minimality assumptions yield that no interior edge originates from $\Gamma$. Hence there is no spur whose endpoint lies in the interior of $D$, and, by Lemma \ref{lem:graphical_basic}, every face is simply connected.
Suppose $D$ has a face $f$ bearing a boundary word that cannot be read on a simple cycle. Since $f$ is simply connected and since there are no spurs inside $f$, $\partial f$ embeds into $D$. Thus there are two distinct vertices $v_1$ and $v_2$ in $\partial f$ mapping to the same vertex in $\Gamma$ via the lift of $\partial f$ to $\Gamma$. We can identify $v_1$ and $v_2$, i.e.\ ``pinch" $v_1$ and $v_2$ together to a single vertex (cf. Figure \ref{figure:pinching}).
If faces with freely trivial boundary words arise, they can be folded and removed to leave a diagram over $\Gamma$. This contradicts the minimality of number of edges of $D$. If no face with freely trivial boundary arises, the diagram obtained by pinching vertices is a diagram over $\Gamma$ with the same number of edges as $D$, but with fewer vertices, again contradicting the assumptions.
\end{proof}
\begin{figure}
\caption{Pinching together two vertices lying on a face that has a boundary path not lifting to a simple cycle.}
\label{figure:pinching}
\end{figure}
\begin{defi}[Gromov-hyperbolic group]\label{defi:hyperbolic_group}
Let $X$ be a geodesic metric space. A \emph{geodesic triangle} in $X$ is a triple of geodesics $(g_1,g_2,g_3)$ such that the endpoint $\tau(g_1)$ of $g_1$ equals the starting point $\iota(g_2)$ of $g_2$, $\tau(g_2)=\iota(g_3)$ and $\tau(g_3)=\iota(g_1)$. Let $\delta\geq 0$. Then $X$ is called \emph{$\delta$-hyperbolic} if and only if for all geodesic triangles $(g_1,g_2,g_3)$, $g_3$ is contained in the $\delta$-neighborhood of $g_1\cup g_2$. The space $X$ is called \emph{hyperbolic} if it is $\delta$-hyperbolic for some $\delta\geq0$.
Let $G$ be a group generated by a finite set $S$. The Cayley graph $\Cay(G,S)$ of $G$ with respect to $S$ is a connected graph endowed with the graph metric on its vertex set. If we consider each edge of $\Cay(G,S)$ isometric to the unit interval $[0,1]$, $\Cay(G,S)$ becomes a geodesic metric space. We say $G$ is $\delta$-hyperbolic with respect to $S$ if $\Cay(G,S)$ is $\delta$-hyperbolic. The group $G$ is called \emph{Gromov-hyperbolic} if it is $\delta$-hyperbolic with respect to $S$ for some $\delta\geq 0$ and some finite generating set $S$.
\end{defi}
\begin{defi}[Isoperimetric inequality] Let $G$ be a group given by the presentation $\langle S|R\rangle$. For $w\in M(S)$ satisfying $w=1$ in $G$, let $\operatorname{Area}_R(w)$ denote the minimal area of a singular disk diagram for $w$ over $R$.
The \emph{Dehn function} of $G$ with respect to $\langle S|R\rangle$ is the map $f:\N\to \N$ given by
$$f(l)=\max\{\operatorname{Area}_R(w)| |w|=l\}.$$
We say a group presentation satisfies a \emph{linear} (resp. \emph{quadratic}) \emph{isoperimetric inequality} if the corresponding Dehn function is bounded from above by a linear (resp. quadratic) map $\R\to\R$.
\end{defi}
We state two immediate consequences of Lemma \ref{lem:graphical_simple} using the area theorems. For classical $C(6)$ and $C(7)$ small cancellation presentations, these are well-known. It is shown in \cite[Theorem 2.5 and Proposition 2.10]{Al} that a group is Gromov-hyperbolic if and only if it has a finite presentation satisfying a linear isoperimetric inequality.
We choose to state the theorems for the $Gr(7)$ and $Gr(6)$ conditions (instead of the graphical $C(7)$ and $C(6)$ conditions) as they are more general.
\begin{thm}\label{thm:linear_isoperimetry}
Let $\Gamma$ be a $Gr(7)$-labelled graph. Let $R$ be the set of words read on all simple cycles of $\Gamma$. Then the presentation $\langle S|R\rangle$ satisfies the linear isoperimetric inequality:
$$\operatorname{Area}_R(w)\leq 8|w|.$$
In particular, if $\Gamma$ is finite, then the group $G(\Gamma)$ is Gromov-hyperbolic.
\end{thm}
\begin{thm}\label{thm:quadratic_isoperimetry}
Let $\Gamma$ be a $Gr(6)$-labelled graph. Let $R$ be the set of words read on all simple cycles of $\Gamma$. Then the presentation $\langle S|R\rangle$ satisfies the quadratic isoperimetric inequality:
$$\operatorname{Area}_R(w)\leq 3|w|^2.$$
\end{thm}
\subsection{Asphericity}
For a group presentation $\langle S|R \rangle$, the \emph{presentation complex} is a CW-complex constructed as follows: The 0-skeleton is a single point $v$. For each generator in $S$, a labelled, oriented loop is glued to $v$, and for each relator $r$ in $R$, a 2-cell whose boundary label is $r$ is glued along its boundary label onto the 1-skeleton. A CW-complex is called \emph{aspherical} if its universal cover is contractible.
We show that a group given by a graphical $C(6)$ presentation has an aspherical presentation complex. This is a generalization of the corresponding well-known result for classical $C(6)$ presentations where no relator is a proper power. (For a proof, see \cite[Theorem 13.3]{Ols} and \cite{CCH}.) It can easily be seen from Example \ref{ex:cayley} that this result does not hold for $Gr(6)$ presentations in general.
\begin{thm}\label{thm:asphericity} Let $\Gamma$ be a $C(6)$-labelled graph. Let $R$ be the set of cyclic reductions of words read on free generating sets of the fundamental groups of the connected components of $\Gamma$. Then the presentation complex associated to $\langle S|R\rangle$ is aspherical.
\end{thm}
Together with \cite[§VIII.2]{Br} this yields:
\begin{cor} The group $G(\Gamma)$ defined by $\Gamma$ has an at most 2-dimensional $K(G(\Gamma),1)$ space and hence cohomological dimension at most 2. Therefore, $G(\Gamma)$ is torsion-free.
\end{cor}
\begin{lem}\label{lem:infinite_presentation} Let $\Gamma$ and $R$ be as in Theorem \ref{thm:asphericity}. Let $R'$ be a proper subset of $R$. Then the homomorphism $\langle S|R'\rangle\to\langle S|R\rangle$ induced by the identity on $S$ is not injective. In particular, if $R$ is infinite, then $G(\Gamma)$ is not finitely presented.
\end{lem}
We postpone the proof of this lemma as it will be derived from our proof of Theorem \ref{thm:asphericity}. To prove asphericity, we will show that spherical diagrams over a $C(6)$ graph are reducible in an appropriate sense. We use definitions and results from \cite{CH} that allow an algebraic treatment of diagrams:
Let $R\subset F(S)$, and set $R^S:=\{gr^\epsilon g^{-1}|r\in R, g\in F(S),\epsilon\in\{\pm1\}\}\subset F(S)$. A \emph{sequence over $R$} is a finite sequence of elements of $R^S$. On all sequences over $R$, we consider the following operations:
\begin{itemize}
\item Exchange: Replace a pair $(x,y)$ by $(xyx^{-1},x)$ or by $(y,y^{-1}xy)$.
\item Deletion: Delete a pair $(x,x^{-1})$.
\item Insertion: Insert at any position a pair $(x,x^{-1})$ for any $x\in R^S$.
\end{itemize}
We call two sequences over $R$ \emph{equivalent} if one can be transformed into the other by a finite sequence of these operations. We call any finite sequence over $R$ an \emph{identity sequence} if the product of its elements (taken in the order they appear in the sequence) is trivial in $F(S)$. We call a sequence \emph{trivial} if it is equivalent to the empty sequence.
To a diagram $(D,v)$ over $R$, we can associate a sequence $\Sigma$ over $R$ as follows: From $D$, we construct a singular disk diagram $D'$ by ``ungluing'' faces of $D$ along edges, such that $D'$ is a bouquet of faces, each connected to the basepoint $v$ by an arc, such that the boundary word of $D'$ is freely equal to that of $D$. If $D$ is spherical, we choose some embedding in the plane after ungluing the first edge. The boundary word of $D'$ is freely trivial in that case.
The sequence $\Sigma$ is then the sequence of the labels of faces of $D'$, each conjugated by the label of the path that connects it to $v$, read in the order in which they appear in the boundary path. The resulting sequence is called a \emph{derived sequence} for $(D,v)$. Its length is equal to the number of faces of $D$, and the product over all elements of $\Sigma$ is freely equal to the boundary word of $D$.
Whilst a derived sequence for a diagram $(D,v)$ is not unique, \cite[Proposition 8]{CH} yields that any two derived sequences for a singular disk diagram $(D,v)$ are equivalent. Moreover, \cite[Corollary 1 of Proposition 8]{CH} implies that if $\Sigma$ and $\Sigma'$ are derived sequences for a spherical diagram $D$, $\Sigma$ is trivial if and only if $\Sigma'$ is.
For any sequence $\Sigma$ over $R$, we can construct a singular disk diagram $D$ over $R$ which has $\Sigma$ as a derived sequence. (This is done by reversing the procedure described above.) If $\Sigma$ is an identity sequence, the diagram has freely trivial boundary word. This freely trivial boundary word can be ``sewn up'' to obtain a spherical diagram. Sewing up means that, whenever there are two edges bearing inverse labels in the boundary, we fold them together until eventually we are left with a spherical diagram. (See \cite[Section 1.5]{CH} for details.) We call a diagram constructed from a sequence $\Sigma$ an \emph{associated diagram} for $\Sigma$
The following theorem is deduced from \cite[Proposition 1.3 and Proposition 1.5]{CCH}. A presentation $\langle S|R\rangle$ is \emph{concise} if for any relator $r\in R$, no $r'\in R$ that is distinct from $r$ is conjugate to $r$ or $r^{-1}$.
\begin{thm} Let $\langle S|R\rangle$ be a presentation where each relator is nontrivial and freely reduced. Then the associated presentation complex is aspherical if and only if:
\begin{itemize}
\item The presentation is concise,
\item no relator is a proper power,
\item and any identity sequence over $R$ is trivial.
\end{itemize}
\end{thm}
We will show that the presentation in our theorem satisfies these three conditions.
\begin{lem}\label{lem:concise} Let $\Gamma$ be a $C(2)$-labelled graph. Let $R$ be the set of cyclic reductions of words read on free generating sets of the fundamental groups of the connected components of $\Gamma$ (with respect to some base points).
Then $R$ is concise and contains no proper powers.
\end{lem}
\begin{proof}
For any cycle $\gamma$ on $\Gamma$, there exists a maximal initial subpath $p$ such that $\gamma=p\gamma'p^{-1}$ for some subpath $\gamma'$. Then $\gamma'$ is a cycle and, since the labelling of $\Gamma$ is reduced, its label $\ell(\gamma')$ equals the cyclic reduction of $\ell(\gamma)$. We call $\gamma'$ the \emph{cyclic reduction} of $\gamma$.
Let $\Gamma_i$ for $i\in I$, where $I$ is some index set, denote the connected components of $\Gamma$, and let $\nu_i$ denote a base vertex in $\Gamma_i$ for each $i$. Let $\gamma$ be the reduced representative of an element of a free generating set of some $\pi_1(\Gamma_i,\nu_i)$ and $\gamma'$ its cyclic reduction such that $\gamma=p\gamma' p^{-1}$ for some path $p$. Suppose $\ell(\gamma')=w^n$ for $w\neq 1$ and $n>1$. Let $\gamma'=\gamma_0...\gamma_{n-1}$ such that $\ell(\gamma_j)=w$ for $j\in \Z/n\Z$.
Then the map of labelled graphs $\gamma'\to\gamma'$ induced by $\gamma_j\to\gamma_{j+1}$ must be the identity, for otherwise $\gamma'$ would have two distinct maps of labelled graphs to $\Gamma$ and thus would be a single piece. Therefore, $\gamma_j=\gamma_{j+1}$ for all $j$, and each $\gamma_j$ is a cycle. Thus $[\gamma]=[p\gamma_0 p^{-1}]^n$ in $\pi_1(\Gamma_i,\nu_i)$. But it is easy to see that an element of a free generating set of $\pi_1(\Gamma_i,\nu_i)$ cannot be a proper power.
Now assume that there are two distinct relators $r$ and $r'$ in $R$ coming from cycles $[\gamma]$ and $[\delta]$ in the free generating sets, where $\gamma$ and $\delta$ are reduced, such that $r'$ is conjugate to $r$ (or $r^{-1}$). Since $r$ and $r'$ are cyclically reduced, they coincide up to a cyclic permutation (and possibly inversion). Again, consider the cyclic reductions $\gamma'$ and $\delta'$. Then we can perform a cyclic shift on $\delta'$ such that the resulting cycle $\tilde \delta '$ has the same label as $\gamma'$ (or $(\gamma')^{-1}$). But then $\tilde \delta'$ must equal $\gamma'$ (or $(\gamma')^{-1})$) for otherwise, it would be a single piece. This implies that $[\gamma]$ and $[\delta]$ lie on the same connected component of $\Gamma$, say $\Gamma_i$, and that they are conjugate in $\pi_1(\Gamma_i,\nu_i)$ (or conjugate up to inversion). This cannot hold for two elements of a free generating set.
\end{proof}
\begin{lem}\label{lem:asphericity}
Let $\Gamma$ be a connected $C(2)$-labelled graph. Let $R$ be the set of cyclic reductions of words read a set of free generators of $\pi_1(\Gamma,\nu)$ for some $\nu\in\Gamma$. Let $(D,v)$ be a simple disk diagram over $R$ with freely trivial boundary word such that every interior edge originates from $\Gamma$. Then any derived sequence is a trivial identity sequence.
\end{lem}
\begin{proof} Let $\{\phi_i|i\in I\}$ be a set of free generators of $\pi_1(\Gamma,\nu)$ such that
$$\phi_i=[\gamma_i \rho_i\gamma_i^{-1}],$$
where $\ell(\rho_i)$ is equal to the cyclic reduction of $\ell_*(\phi_i)$ and $\gamma_i$ is a reduced path. Denote the terminal vertex of $\gamma_i$ by $\nu_i$. Then $\{\ell(\rho_i)|i\in I\}=R$, and $\{\ell_*(\phi_i)|i\in I\}\subset R^S$.
The diagram $D$ globally lifts to $\Gamma$ via a map $\lambda:D^{1}\to \Gamma$, where $D^1$ denotes the 1-skeleton of $D$. Let $v\in\partial D$. Since $\Gamma$ is connected, we may assume that $\nu$ was chosen such that $\nu=\lambda(v)$. The path $\lambda(\partial D)$ is a cycle on $\Gamma$ based at $\nu$. Since the labelling of $\Gamma$ is reduced, the assumption that the boundary label of $D$ is freely trivial implies that $\lambda(D)$ is 0-homotopic, i.e.\ $[\lambda(\partial D)]=1\in\pi_1(\Gamma,\nu)$.
We number the faces of $D$ as $f_1,...,f_n$ and choose an orientation on each face. For a face $f_j$ with boundary label $\ell(\rho_i)^{\epsilon_j}$, where $i\in I$ and $\epsilon_j\in\{\pm1\}$, let $v_j$ denote the vertex that lifts to $\nu_i$, and set $t_j:=i$. Then there are paths $p_j$ in $D^1$ from $v$ to $v_j$ such that in $\pi_1(\Gamma,\nu)$:
\begin{equation*}
1=[\lambda(\partial D)]=[\lambda(p_1)\rho_{t_1}^{\epsilon_1}\lambda(p_1)^{-1}]\ldots [\lambda(p_n)\rho_{t_n}^{\epsilon_n}\lambda(p_n)^{-1}].
\end{equation*}
For each $j$, the path $\lambda(p_j)\gamma_{t_j}^{-1}$ is a cycle based at $\nu$; we denote $\eta_j:=[\lambda(p_j)\gamma_{t_j}^{-1}]$. The sequence
$$\Sigma:=(\ell_*(\eta_1\phi_{t_1}^{\epsilon_1}\eta_1^{-1}),\ldots)=(\ell_*(\eta_1)\ell_*(\phi_{t_1}^{\epsilon_1})\ell_*(\eta_1^{-1}),\ldots)$$
is a derived sequence for $(D,v)$. For each $j$, we can express $\eta_j$ in the free generators $\{\phi_i|i\in I\}$ as a reduced word $W_j$ and write:
$$\Sigma=(\ell_*(W_1)\ell_*(\phi_{t_1}^{\epsilon_1})\ell_*(W_1^{-1}),\ldots ,\ell_*(W_n)\ell_*(\phi_{t_n}^{\epsilon_n})\ell_*(W_n^{-1})).$$
Now suppose $W_1\neq1$, and let the first letter of $W_1$ be $\phi_{f_1}^{e_1}$. Then we can insert the pair $(\ell_*(\phi_{f_1}^{e_1}),\ell_*(\phi_{f_1}^{-e_1}))$ into $\Sigma$ at the first position and perform an exchange operation to obtain
$$(\ell_*(\phi_{f_1}^{e_1}),\ell_*(W_1')\ell_*(\phi_{t_1}^{\epsilon_1})\ell_*(W_1'^{-1}),\ell_*(\phi_{f_1}^{-e_1}),\ell_*(W_2)\ell_*(\phi_{t_2}^{\epsilon_2})\ell_*(W_2^{-1}),\ldots ),$$
where $\phi_{f_1}^{e_1}W_1'=W_1$, and $W_1'$ is shorter than $W_1$. Iterating this procedure and applying it to all $W_i$ yields a sequence $\Sigma'$ of the form
$$\Sigma'=(\ell_*(\phi_{g_1}^{h_1}),\ldots ,\ell_*(\phi_{g_N}^{h_N})),$$
where each $h_i\in\{\pm 1\}$. Since $1=\phi_{g_1}^{h_1}\ldots \phi_{g_N}^{h_N}$ in $\pi^1(\Gamma,\nu)$ and since a free reduction on the right hand side of this equation corresponds to a deletion operation in $\Sigma'$, we see that $\Sigma'$ can be transformed into the empty sequence by deletion operations.
\end{proof}
\begin{lem}
Let $\Gamma$ be a $C(6)$-labelled graph. Let $R$ be the set of cyclic reductions of words read on free generating sets of the fundamental groups of the connected components of $\Gamma$. Then any identity sequence over $R$ is trivial.
\end{lem}
\begin{proof} Let $\Sigma$ be a nontrivial identity sequence over $R$ of minimal length. Then there is an associated spherical diagram $D$. We may restrict ourselves to the case that $D$ is a simple spherical diagram, as the general case can be constructed from this. Lemma \ref{lem:graphical_basic_spherical} implies that all edges of $D$ originate from $\Gamma$, or that $D$ has a subdiagram $f$ that is a simple disk diagram with at least one face and with freely trivial boundary word, and all interior edges of $f$ originate from $\Gamma$. Assume the latter. Note that the 1-skeleton of $f$ maps to a connected component of $\Gamma$.
Changing the base point of a diagram corresponds to a conjugation of all elements of the derived sequence with an element $g\in F(S)$. A sequence is equivalent to a shorter sequence if and only if its conjugate is. Hence we can assume that the base point $v$ lies in the boundary of $f$.
We now associate to $(D,v)$ a derived sequence $\Sigma'$ that has an initial subsequence $\sigma$ that is a derived sequence for $(f,v)$: We cut $f$ out of $D$ and glue $f$ and $D\setminus f$ together at the point $v$ such that the boundary labels of $f$ and $D\setminus f$ read from $v$ are inverse. Let $\Sigma'$ be the concatenation of a sequence $\sigma$ for $f$ and one for $D\setminus f$. Since $f$ has at least one face, $\sigma$ is nonempty. Note that $\Sigma$ and $\Sigma'$ have the same length. By Lemma \ref{lem:asphericity}, $\sigma$ reduces to the trivial sequence, and therefore $\Sigma'$ is equivalent to a shorter sequence. Thus the minimality assumption on $\Sigma$ yields that $\Sigma'$ is trivial. Now, as mentioned before, \cite[Corollary 1 to Proposition 8]{CH} implies that $\Sigma$ is trivial, a contradiction.
If all edges of $D$ originate from $\Gamma$, we can unglue two faces along any edge of $D$ to obtain a simple disk diagram with freely trivial boundary word that globally lifts to $\Gamma$. Lemma \ref{lem:asphericity} yields that any derived sequence is trivial, and therefore $\Sigma$ is trivial.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:infinite_presentation}] Let $R$ be as in Theorem \ref{thm:asphericity}. Assume there is $r\in R$ such that $r\in\langle\langle R\setminus \{r\}\rangle\rangle$. Then there exists a diagram for $r$ over $R\setminus \{r\}$ and hence a spherical diagram $D$ over $R$ with exactly one face labelled by $r$. There exists a derived sequence for $D$ containing a conjugate of $r$, but (using Lemma \ref{lem:concise}) no conjugate of $r^{-1}$. Therefore the sequence is not trivial, which is a contradiction.
\end{proof}
\section{Free subgroups}
We prove that graphical $C(7)$ groups are either cyclic, or they have non-abelian free subgroups.
\begin{thm}\label{thm:free}
Let $\Gamma$ be a $C(7)$-labelled graph. Then $G(\Gamma)$ contains a non-abelian free subgroup unless it is trivial or infinite cyclic.
\end{thm}
\begin{cor}
Graphical $C(7)$ groups have exponential growth and are non-amenable, unless they are trivial or infinite cyclic.
\end{cor}
\begin{remark}\label{rem:obvious} We show that it is easy to check if $G(\Gamma)$ is trivial or infinite cyclic. First, either one of the curvature formulas implies that there is no diagram over $\Gamma$ whose boundary has length 1 and which has more than one face. Hence $G(\Gamma)$ is trivial if and only if it contains a loop of length 1 labelled with $s$ for each $s\in S$.
Second, we check when an infinite cyclic group can arise. From the labelled graph $\Gamma$, we construct a graph $\Gamma'$ by iteratively removing every edge that is contained in a simple cycle and that is not a piece. We also remove the letters labelling those edges from the alphabet $S$ to obtain the alphabet $S'$. Then the group defined by $\Gamma'$ over the alphabet $S'$ is isomorphic to $G(\Gamma)$ since the changes we made correspond to Tietze transformations. If $\Gamma'$ is a forest, then the group defined by $\Gamma$ is the free group on $S'$. Assume this is not the case. Then $\Gamma'$ has girth at least $7$.
Assume $G(\Gamma)$ is cyclic and hence abelian, and let $a,b\in S'$ with $a\neq b$. Then any diagram for the word $aba^{-1}b^{-1}$ over $\Gamma'$ has more than one face. Since there are only 4 boundary edges and the girth of $\Gamma'$ is at least 7, any face has at least 3 interior arcs. Therefore, the first curvature formula is violated.
We deduce that $G(\Gamma)$ is infinite cyclic if and only if $|S'|= 1$. Note that in this case, $\Gamma'$ is a forest, and the free rank of the free product of the fundamental groups of the connected components of $\Gamma$ is $|S|-1$.
\end{remark}
While, by Example \ref{ex:cayley}, the above theorem cannot hold for arbitrary $Gr(7)$ groups, our method of proof yields the following special case:
\begin{thm}\label{thm:ess-free}
Let $\Gamma$ be a $Gr(7)$-labelled graph with infinitely many pairwise non-isomorphic connected components with non-trivial fundamental groups. Then $G(\Gamma)$ contains a non-abelian free subgroup.
\end{thm}
\begin{cor}\label{cor:classical-free}
Let $G$ be a group with an infinite classical $C(7)$ presentation. Then $G$ contains a non-abelian free subgroup.
\end{cor}
\begin{proof}[Proof of Theorem \ref{thm:free} in the finitely presented case]
If $\Gamma$ has finitely \linebreak many distinct simple cycles, Theorem \ref{thm:linear_isoperimetry} and Theorem \ref{thm:asphericity} imply that the group is hyperbolic and torsion-free. It is well-known (cf. \cite[$5.3.\mathrm{C}_1$]{Grhyp}) that non-elementary hyperbolic groups contain non-abelian free subgroups. Any torsion-free elementary hyperbolic group is either trivial or infinite cyclic. Therefore, the claim holds in this case.
\end{proof}
We now make preliminary observations and definitions for the proof of Theorem \ref{thm:free} in the remaining case. The proof of Theorem \ref{thm:ess-free} will be deduced from this proof at the end of the section.
Let $\Gamma$ be a labelled graph. Suppose there is an edge $e$ that is not a piece. Then the label $s$ of $e$ appears only once on the graph. If $e$ lies on a simple cycle, then removing $e$ from $\Gamma$ as well as $s$ from $S$ corresponds to a Tietze transformation and yields an isomorphic group. If $e$ does not lie on a simple cycle, consider the presentation $\langle S|R\rangle$ for $G(\Gamma)$, where $R$ is the set of words read on simple cycles in $\Gamma$. Then $s$ appears in no word of $R$ and therefore generates a free factor. In this case, the claim of Theorem \ref{thm:free} holds. Therefore, it is no restriction to assume that all edges in $\Gamma$ are pieces, and we therefore do so until our proof is concluded.
Assume that all edges on the labelled graph $\Gamma$ are pieces. Let $x,y$ be vertices lying in a connected component of $\Gamma$. The \emph{piece distance} $d_p(x,y)$ is the least number of pieces whose concatenation is a path from $x$ to $y$. We observe that the restriction of $d_p$ to a connected component of $\Gamma$ is a metric.
\begin{lem}\label{lem:free1} Let $\gamma$ be a simple cycle on a $C(n)$-labelled graph $\Gamma$. Let $x$ be a vertex on $\gamma$. If $n$ is even, there exists a vertex $y$ on $\gamma$ such that $d_p(x,y)=\frac{n}{2}$. If $n$ is odd, one of the following holds:
\begin{itemize}
\item There is a vertex $y$ on $\gamma$ such that $d_p(x,y)=\frac{n+1}{2}$, or
\item there are vertices $y$ and $z$ on $\gamma$ such that $d_p(x,y)=d_p(x,z)=\frac{n-1}{2}$ and $d(y,z)=1$. Any path starting at $x$ and traversing both $y$ and $z$ consists of at least $\frac{n+1}{2}$ pieces. We call the edge connecting $y$ and $z$ the \emph{opposite} edge to $x$.
\end{itemize}
\end{lem}
\begin{proof}
We observe: For any $x,y$ in a connected component of $\Gamma$ with $d_p(x,y)=d<\frac{n}{2}$, there is a unique simple path from $x$ to $y$ consisting of $d$ pieces.
Now assume that the simple cycle $\gamma$ is based at $x$. Let $\gamma^+$ be the longest initial subpath that can be made up of less than $\frac{n}{2}$ pieces, and let $\gamma^-$ be the longest terminal subpath with this property. Then, by our observation, $\gamma^+$ and $\gamma^-$ have only the vertex $x$ in common. We may write $\gamma=\gamma^+\delta\gamma^-$ for some nonempty simple path $\delta.$
Assume that $n$ is even. Then $\delta$ consists of at least 2 pieces and hence at least two edges. Any interior vertex $y$ of $\delta$ satisfies $d_p(x,y)\geq \frac{n}{2}$.
Assume that $n$ is odd. If $\delta$ consists of more than one edge, the argument above applies, and the first claim holds. If $\delta$ consists of exactly one edge, the second claim holds.
\end{proof}
We now recall a basic fact about fundamental groups of graphs: Let $\Gamma$ be a connected graph and $T$ a spanning tree of $\Gamma$. Let $E$ denote the set of edges in $\Gamma$ but not in $T$. Then there is a bijective map from $E$ to a free generating set of $\pi_1(\Gamma,\nu)$ ($\nu$ arbitrary in $\Gamma$). In particular, $\pi_1(\Gamma,\nu)$ is not finitely generated if and only if $E$ is infinite. (For a proof, see \cite[Section 6]{KM}.)
\begin{lem}\label{lem:free2} Let $\Gamma$ be a graph of bounded vertex-degree containing infinitely many pairwise distinct simple cycles. Then $\Gamma$ contains infinitely many pairwise vertex-disjoint simple cycles.
\end{lem}
\begin{proof} First assume that $\Gamma$ has infinitely many connected components that have nontrivial fundamental group. Then the claim is obvious. Therefore, we will assume that $\Gamma$ is connected.
We prove by contradiction. Let $C$ be a maximal set of pairwise disjoint simple cycles on $\Gamma$ and assume that $C$ is finite. Consider a spanning tree $T$ of $\Gamma$. Then there are infinitely many edges in $\Gamma$ that do not lie in $T$, and therefore, since $\Gamma$ is of bounded degree, $T$ is infinite. The forest $T-C$ (i. e. $T$ with all vertices and edges of elements of $C$ removed) has finitely many connected components. Moreover, there are infinitely many edges in $\Gamma$ that do not meet a vertex in $C$ and that do not lie in $T$. Thus one of the following must occur:
\begin{itemize}
\item There exists a component $K$ of $T-C$ such that there is an edge in $\Gamma$ but not in $T$ and disjoint from $C$ that connects two vertices of $K$, or
\item there exist two components $K$, $K'$ of $T-C$ such that there are two edges in $\Gamma$ but not in $T$ and disjoint from $C$ connecting vertices of $K$ and $K'$.
\end{itemize}
In both cases, we obtain a simple cycle that is vertex-disjoint from all elements of $C$. This contradicts the maximality of $C$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:free} in the infinitely presented case]
We assume that the \\ free product of the fundamental groups of the components of $\Gamma$ is not finitely generated. The finitely generated case has been remarked upon.
First assume there are four pairwise vertex-disjoint simple cycles $\gamma_i$, $i\in\{1,2,3,4\}$, such that for each $i$, there are vertices $x_i$ and $y_i$ in $\gamma_i$ with $d_p(x_i,y_i)=4$ . For each $i$, let $w_i$ be the label of a path from $x_i$ to $y_i$. The method of proof below indicates how to show that $\alpha:=w_1w_2$ and $\beta:=w_3w_4$ generate a rank 2 free subgroup of $G(\Gamma)$.
Now assume the first assumption does not hold. We deduce from Lemmas \ref{lem:free1} and \ref{lem:free2} that there are infinitely many pairwise vertex-disjoint simple cycles on $\Gamma$ on each of which the maximal piece distance between two points is 3. Moreover, for any vertex in one of these cycles $\gamma$, there is an opposite edge on $\gamma$. Let $x_a,x_a',x_b,x_b'$ be vertices with opposite edges $a,a',b,b'$ in pairwise disjoint cycles. Since we are considering an infinite set of cycles labelled with a finite set of letters, we may assume that $\ell(a)=\ell(a')$ and $\ell(b)=\ell(b')$.
We add $\alpha,\beta$ to the alphabet $S$ and add the following relations to the presentation:
\begin{itemize}
\item $R_\alpha:=\{\alpha \red(\ell(p)^{-1}\ell(p'))|p:\iota(a)\to x_a,p':\iota(a') \to x_a'\},$
\item $R_\beta:=\{\beta \red(\ell(p)^{-1}\ell(p'))|p:\iota(b)\to x_b,p':\iota(b') \to x_b'\},$
\end{itemize}
where $\red(-)$ denotes the free reduction, $p:w\to x$ denotes a path from $w$ to $x$, and $\iota(e)$ denotes the initial vertex of an oriented edge $e$. We call a face in a diagram bearing a relator in $R_\alpha\cup R_\beta$ \emph{special}. Let $R$ denote the set of labels of all nontrivial cycles on $\Gamma$. Then
$$G(\Gamma)=\langle S|R\rangle\cong \langle S\cup\{\alpha,\beta\}|R\cup R_\alpha\cup R_\beta\rangle,$$
since we only applied the Tietze transformations of adding generators. (Note that $\ell(p_1)=\ell(p_2)$ in $G$ for any two paths $p_1,p_2$ from vertices $x$ to $y$.) We show that $\alpha$ and $\beta$ generate a free subgroup of rank 2 in $G(\Gamma)$.
Let $D$ be a nontrivial diagram for a cyclically reduced word $\omega$ in $\alpha,\beta$ over the extended presentation. First, we can assume that no edge labelled $\alpha$ or $\beta$ is an interior edge: If there is such an interior edge, it can only occur when two special faces are glued together in such a way that the boundary read around these two faces can be obtained by gluing together two cycles lifting to $\Gamma$. Thus we can replace the two faces containing $\alpha$ (or $\beta$) by two that do not.
Second, we choose $D$ to be of minimal area with this property and without interior spurs. Then the proofs of Lemma \ref{lem:graphical_basic} and Corollary \ref{cor:graphical_basic} yield that no interior face of $D$ contains an edge originating from $\Gamma$. (Here we associate to the interior edges of special faces the lifts that arise from the fact that they are labels of paths in $\Gamma$.) Hence any arc in the intersection of two interior faces is a piece.
We first cover the case that no interior edge originates from $\Gamma$. We construct from $D$ a diagram $D'$ that violates the first curvature formula by replacing each special face by two faces over the original presentation and performing some foldings. For the sake of brevity we describe the process for one special face $F$ whose label lies in $R_\alpha$:
The subpath of $\partial F$ that lies in the interior of $D$ comes from two reduced paths $p:\iota(a)\to x_a$ and $p':\iota(a')\to x_a'$, where a free reduction may have occurred corresponding to cutting off initial subpaths of $p$ and $p'$ that bear the same labels.
We replace $F$ by two faces $f$ and $f'$ constructed as follows: $\partial f=\ell (p) \ell (q)$ and $\partial f'=\ell (p') \ell (q')$, where $q$ (resp. $q'$) is a path in $\Gamma$ such that the concatenation $pq$ (resp. $p'q'$) is a closed reduced path in $\Gamma$. The two faces are glued together at the vertex lifting to the initial vertex of $p$ resp. $p'$, and edges of $f$ and $f'$ are folded together starting at this vertex in correspondence to the free reduction of $\ell(p)^{-1}\ell(p')$. Then we glue these two faces into the diagram instead of $F$. Note that interior edges arising from gluing $f$ and $f'$ together do not originate from $\Gamma$. Hence any interior arc in $\partial f$ or $\partial f'$ is a piece. Now there are three cases to consider:
\begin{itemize}
\item[(A)] Suppose neither $\ell(p)$ nor $\ell(p')$ start with $\ell(a)$. Then we can choose $q$ and $q'$ such that $\ell(q)$ and $\ell(q')$ terminate with $\ell(a)^{-1}$. Thus we can fold $f$ and $f'$ together along the corresponding edge as well. This edge does not lift to $\Gamma$. Then the interior paths of $f$ and $f'$ consist of at least 4 pieces by Lemma \ref{lem:free1}.
\item[(B)] Suppose both start with $a$. Then $p$ and $p'$ consist of at least 4 pieces by Lemma \ref{lem:free1}.
\item[(C)] Suppose $\ell(p)$ starts with $\ell(a)$, and $\ell(p')$ does not (or the symmetric case). Then $f$ has interior degree at least 4, $f'$ has interior degree at least 3, and there is no folding between $f$ and $f'$. We can, however just fold together the two exterior edges of $f$ and $f'$ incident at the vertex where the two faces intersect (disregarding the label) to increase the interior degrees of $f$ and $f'$.
\end{itemize}
For all cases we obtain that after these choices and additional foldings, the constructed boundary faces $f$ and $f'$ have interior degree at least 4. Any arc in the intersection of $f$ or $f'$ with an interior face is a piece. Hence, after we performing this procedure for all special faces, we obtain a $(3,7)$-diagram violating the first curvature formula, which is a contradiction.
We are left with the case that $D$ does contain interior edges originating from $\Gamma$. As above, we can construct from $D$ a diagram $D'$ over $\Gamma$ (in particular making the choices in case (A)), but without performing the foldings in case (A) or case (C). By our assumptions, an interior edge $e$ originating from $\Gamma$ lies in the the boundary of one or two special faces, and therefore it lies in the intersection of boundary faces $f_1$ and $f_2$ of $D'$, where possibly $f_1=f_2$.
If $f_1=f_2$, then, after forgetting vertices of degree 2 in $D'$, $f_1$ encloses some nontrivial $[3,6]$-subdiagram $\delta$ with at most one boundary vertex of degree (in $\delta$) less than 3, contradicting the second curvature formula. The second curvature formula also yields that any $[3,6]$-diagram with at most two boundary vertices of degree less than 3 is an arc, whence $f_1\cap f_2$ is an arc.
Since the boundary word of $D$ is cyclically reduced, there can be no vertex in $\partial D'\cap f_1\cap f_2$ whose lifts via $f_1$ and $f_2$ coincide.
Therefore, $\partial D'\cap f_1\cap f_2$ is empty, and there are nontrivial subdiagrams on both sides of $f_1\cup f_2$. We call two faces of $D'$ containing an edge originating from $\Gamma$ a \emph{bottle neck}.
If there exists a bottle neck, there also exists an \emph{extremal} bottle neck $f_1\cup f_2$, i.e. one such that in one component $C$ of $D'\setminus(f_1\cup f_2)$ there are no more bottle necks. This means in the diagram $\Delta:=C\cup f_1\cup f_2$, all interior edges originating from $\Gamma$ lie in $f_1\cap f_2$.
Now consider the diagram $\Delta$. We can remove all edges originating from $\Gamma$ to merge $f_1$ and $f_2$ to one face $f$. The resulting diagram has no more interior edges originating from $\Gamma$. Thus we can apply our technique from above: Performing the foldings indicated in case (A) and case (C) above yields a $(3,7)$-diagram where all but one boundary face (namely $f$) have interior degree at least 4. This contradicts the first curvature formula.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:ess-free}]
This proof is obtained precisely as above by making the following obervations and adjustments:
\begin{itemize}
\item Since the alphabet $S$ is finite and $\Gamma$ has infinitely many components, we can assume that on all but finitely many components, all edges are essential pieces.
\item On these infinitely many components, Lemma \ref{lem:free1} can be directly adapted to the $Gr(7)$ case.
\item Instead of choosing vertex-disjoint cycles, we choose cycles $\gamma_i$ such that that are essentially vertex-disjoint, i. e. $\gamma_i$ and $\phi(\gamma_j)$ are vertex-disjoint for all labelled automorphisms $\phi$ of $\Gamma$ if $i\neq j$. This can be be done due to our assumptions on $\Gamma$.
\item In the proof, we replace all occurrences of piece, originating from, coincide, etc. by {essential} piece, {essentially} originating from, {essentially} coincide, etc.
\end{itemize}
\end{proof}
\section{Coarse embedding of the graph}
We now consider the question of how $\Gamma$ embeds into $G(\Gamma)$. Similar arguments have been used by Ollivier \cite{Oll} to show that a connected $C'(\frac{1}{6})$-labelled graph embeds isometrically into $G(\Gamma)$.
Let $\Gamma$ be a connected, labelled graph. Let $v$ be a vertex in $\Gamma$ and $g\in G(\Gamma)$. There is a unique map of labelled graphs $\iota:\Gamma\to\Cay(G(\Gamma),S)$ satisfying $\iota(v)=g$. We prove a claim of Gromov \cite[Theorem 2.3]{Gr}:
\begin{lem}\label{lem:embedding_injective}
Let $\Gamma$ be a connected component of a $Gr(6)$-labelled graph. Then the map $\iota:\Gamma\to\Cay(G(\Gamma),S)$ is injective.
\end{lem}
\begin{proof}
Let $x\neq y\in \Gamma$, and assume $\iota(x)=\iota(y)$. Let $p$ be a path from $x$ to $y$ on $\Gamma$ such that the area of $w=\ell(p)$ is minimal among all paths from $x$ to $y$. We may assume that $w$ is reduced. Since $x\neq y$, this area is greater than zero. Let $D$ be a minimal diagram for $w$ over $\Gamma$. For each boundary edge of $D$ bounding a face, there is a lift to $\Gamma$ via $p$ and one via the face it lies on. Suppose there were an edge $e$ bounding a face $f$ for which these two lifts were essentially equal. Then we could remove $e$ and the interior of $f$. The resulting diagram would have a boundary word read on a path connecting $x$ and $y$ of lesser area than $D$, which is a contradiction. Thus every arc of $D$ bounding a face is a piece. By construction, $D$ has at most one spur (connecting the base vertex to the rest of the diagram) and hence at most one vertex of degree one. Now forgetting vertices of degree two leaves a $[3,6]$-diagram violating the second curvature formula, which
is a contradiction.
\end{proof}
\begin{cor}
For any $C(6)$-labelled graph $\Gamma$ with a connected component having more than one vertex, the group $G(\Gamma)$ is infinite.
\end{cor}
\begin{proof}
This follows since $G(\Gamma)$ is nontrivial and, by Theorem \ref{thm:asphericity}, torsion-free.
\end{proof}
Let $(\Gamma_n)_{n\in\N}$ be a sequence of connected finite graphs and let $\Gamma:=\bigsqcup_{n\in\N}\Gamma_n$ be their disjoint union. We endow $\Gamma$ with a metric that coincides with the graph metric on each connected component such that $d(\Gamma_ {a_n},\Gamma_{b_n})\to\infty$ as $a_n+b_n\to \infty$ assuming $a_n\neq b_n$ for almost all $n$. (For example, set $d(x,y)=\diam(X_m)+\diam(X_n)+m+n$ if $x\in X_m$, $y \in X_n$ and $m\neq n$.) We call the resulting metric space the \emph{coarse union} of the $\Gamma_n$.
If $\Cay(G(\Gamma),S)$ is infinite, it contains an infinite geodesic ray. Then we can map $\Gamma$ into $\Cay(G(\Gamma),S)$ via a map of labelled graphs $\iota$ by lining the $\Gamma_n$'s up on this geodesic ray such that, for all sequences $a_n,b_n$ we have $d(\iota(\Gamma_{a_n}),\iota(\Gamma_{b_n}))\to \infty$ if and only if $d(\Gamma_{a_n},\Gamma_{b_n})\to \infty$. We claim that such a map $\iota$ is a \emph{coarse embedding}, i.e.\ it satisfies for every sequence of pairs of points $(x_n,y_n)_{n\in\N}$ in $\Gamma\times \Gamma$:
$$d(x_n,y_n)\to \infty \Leftrightarrow d(\iota(x_n),\iota(y_n))\to\infty.$$
\begin{thm}\label{thm:coarse_embedding} Let $(\Gamma_n)_{n\in\N}$ be a sequence of connected finite graphs such that $\Gamma:=\bigsqcup_{n\in\N}\Gamma_n$ is $Gr(6)$-labelled and such that $|\Gamma_n|$ is unbounded. Then the coarse union $\Gamma$ embeds coarsely into $\Cay(G(\Gamma),S)$.
\end{thm}
\begin{proof}
First note that the assumption implies that $\Cay(G(\Gamma),S)$ is infinite, since by Lemma \ref{lem:embedding_injective}, each $\Gamma_n$ maps injectively into $\Cay(G,S)$.
Let $(x_n,y_n)_{n\in\N}$ be a sequence of pairs of points such that $d(x_n,y_n)\to\infty$. We claim: $d(\iota(x_n),\iota(y_n))\to\infty$. By the above construction of $\iota$, it is sufficient to consider the case where for all $n$, $x_n$ and $y_n$ lie in the same connected component. Suppose our claim is false. Then $(x_n,y_n)$ has a subsequence $(x_n',y_n')$ such that $d(\iota(x_n'),\iota(y_n'))$ is bounded. Hence there is a subsequence $(x_n'',y_n'')$ such that for all $n$, the labels of paths from $x_n''$ to $y_n''$ define the same element $w$ of $G$. Since all $\Gamma_n$ are bounded, we also assume that for $n\neq m$, the graphs containing $\{x_n'',y_n''\}$ and $\{x_m'',y_m''\}$ are distinct and non-isomorphic.
Let $n\in\N$ and choose paths $p_n,q_n$ from $x_1''$ to $y_1''$, respectively from $x_n''$ to $y_n''$, such that the minimal area of a diagram for the word $\ell(p_n)\ell(q_n)^{-1}$ over $\Gamma$ is minimal. We claim that the area of such a minimal diagram $D_n$ is zero. We may assume that the paths are reduced. We can argue as in Lemma \ref{lem:embedding_injective} that every arc bounding a face in $D_n$ is a piece. Now forgetting vertices of degree two yields a $[3,6]$-diagram with at most two vertices of degree one. By the second curvature formula, such a diagram has exactly two vertices and hence no faces. Thus $D_n$ has no face, $\ell(p_n)=\ell(q_n)$, and $p_n$ and $q_n$ are essential pieces and hence simple paths.
Let $X$ be the connected component of $\Gamma$ containing $x_1''$. Then $|q_n|=|p_n|$ is bounded from above by the maximal length of a simple path on the finite graph $X$. Since $n$ was arbitrary, this contradicts the assumption $d(x_n'',y_n'')\to\infty$.
Obviously, $d(\iota(x),\iota(y))\leq d(x,y)$ for any $x,y$ in a connected component of $\Gamma$. Hence $d(\iota(x_n),\iota(y_n))\to\infty$ implies $d(x_n,y_n)\to\infty$.
\end{proof}
\begin{example}
Let $p\in\N$. We construct a sequence of finite, connected labelled graphs $(\Gamma_n)_{n\in\N}$ such that their disjoint union $\Gamma$ has a $C(p)$-labelling and such that any map of labelled graphs $\iota:\Gamma\to\Cay(G(\Gamma),S)$ is \emph{not} a quasi-isometric embedding. Let $S=S_1\cup S_2\cup\{a,b\}$, where $S_1,S_2$ and $\{a,b\}$ are pairwise disjoint and $|S_1|>1$. Let $(w_n)_{n\in\N}$ be a sequence of pairwise distinct, reduced words in the free monoid $M(S_1)$ on $S_1\cup S_1^{-1}$ such that $|w_n|=O(\log n)$, and let $(v_n)_{n\in\N}$ a sequence of pairwise distinct, reduced words in $M(S_2)$. For each $n$, let $\Gamma_n$ be the graph given in Figure \ref{figure:example_embedding}:
\begin{figure}
\caption{The graph $\Gamma_n$.}
\label{figure:example_embedding}
\end{figure}
\noindent Here $p_n,x_n,y_n$ are words in $S$ given as follows:
\begin{itemize}
\item $p_n=b^{f(n)}$, where $f(n)$ is some function $\N\to\N$ such that $\log n=o(f(n))$,
\item $x_n=w_{(p-1)n+1}a w_{(p-1)n+2}a\ldots aw_{pn}a$,
\item $y_n=v_{(p-1)n+1}a v_{(p-1)n+2}a\ldots av_{pn}a^{f(n)}$.
\end{itemize}
The labels $\eta^n$ and $\nu^n$ are vertex-labels for reference. The reader can easily check that the resulting graph $\Gamma$ has a $C(p)$-labelling by considering how many non-consecutive instances of $a$ can occur in a piece. By construction $|p_n|\leq |y_n|$ and hence $d(\eta^n,\nu^n)=|p_n|$. Since $p_n$ and $x_n$ are equal in $G(\Gamma)$, we have $d(\iota(\eta^n),\iota(\nu^n))\leq |x_n|=o(d(\eta^n,\nu^n))$. Therefore, $\iota$ cannot be a quasi-isometric embedding.
\end{example}
\section{Lacunary hyperbolicity}
We now consider an infinite sequence of finite graphs $(\Gamma_n)_{n\in\N}$ such that their disjoint union is $Gr(7)$-labelled or $Gr'(\frac{1}{6})$-labelled. The resulting group is a limit of hyperbolic groups. In the view of \cite{OOS} it is natural to ask whether these groups are \emph{lacunary hyperbolic}. We give some preliminary definitions:
\begin{defi}[Ultrafilter] An \emph{ultrafilter} is a finitely additive map $\omega:2^\N\to\{0,1\}$ such that $\omega(\N)=1$. An ultrafilter $\omega$ is called \emph{non-principal} if $\omega(F)=0$ for all finite subsets $F$ of $\N$. Let $f:\N\to \R$ be a sequence. Then for $x\in \R$ we say $x=\lim_n^{\omega}f(n)$ if $\forall \epsilon >0: \omega(f^{-1}([x-\epsilon,x+\epsilon]))=1$.
\end{defi}
It is a fact that given an ultrafilter $\omega$, any bounded sequence $f:\N\to\R$ has a limit with respect to $\omega$.
\begin{defi}[Asymptotic cone] Let $\omega$ be a non-principal ultrafilter and $(d_n)_{n\in\N}$ a sequence of real numbers such that $d_n\to\infty$ as $n\to\infty$. The sequence $d_n$ is called \emph{scaling sequence}. Let $G$ be a group generated by a finite set $S$. Let $G^\N$ denote the space of sequences of elements of $G$, and $X:=\{(x_n)\in G^\N|d_S(1,x_n)=O(d_n)\}$, where $d_S$ denotes the distance in $\Cay(G,S)$. We define a pseudo-metric $d$ on $X$ by setting $d((x_n),(y_n))=\lim_n^\omega \frac{d(x_n,y_n)}{d_n}$. An equivalence relation on $X$ is induced by $(x_n)\sim (y_n):\Leftrightarrow d(x_n,y_n)=0$. The \emph{asymptotic cone} of $G$ with respect to $S$, $\omega$ and $(d_n)$ is defined as $X/\sim$ with the metric induced by $d$. We denote it by $\Con^w(G,d_n)$.
\end{defi}
Let $G$ be a group generated by a finite set $S$, and let $\pi:G\to H$ be a group homomorphism. Then the \emph{injectivity radius} of $\pi$, denoted $r_S(\pi)$, is the largest $r\in\R$ such that $\pi$ restricted to the open ball of radius $r$ at $1$ in $\Cay(G,S)$ is injective. An $\R$-tree is a 0-hyperbolic space (see Definition \ref{defi:hyperbolic_group}). In particular, an $\R$-tree contains no nontrivial embedded cycles. We recall the definition and a characterization of lacunary hyperbolicity from \cite[Section 3.1]{OOS}:
\begin{defi}[Lacunary hyperbolic group]
A finitely generated group $G$ is called \emph{lacunary hyperbolic} if the following equivalent conditions hold:
\begin{itemize}
\item Some asymptotic cone of $G$ is an $\R$-tree.
\item $G$ is the direct limit of finitely generated groups and epimorphisms
$$G_1\xrightarrow{\alpha_1}G_2\xrightarrow{\alpha_2}\ldots$$
such that $G_i$ is generated by a finite set $S_i$, $\alpha_i(S_i)=S_{i+1}$ and each $G_i$ is $\delta_i$-hyperbolic, where $\delta_i=o(r_{S_i}(\alpha_i))$.
\end{itemize}
\end{defi}
\subsection{The $Gr(7)$ case}
Since the defining condition for lacunary hyperbolicity has a metric nature, the $Gr(7)$ condition does not yield optimal results. We are able to prove, however:
\begin{prop}\label{prop:7_lacunary}
Let $(\Gamma_n)_{n\in\N}$ be a sequence of finite, connected labelled graphs such that their disjoint union is $Gr(7)$-labelled. Then there exists an infinite subsequence of graphs $(\Gamma_{k_n})_{n\in\N}$ such that $G(\bigsqcup_{n\in\N}\Gamma_{k_n})$ is lacunary hyperbolic.
\end{prop}
\begin{lem}\label{lem:c7_injectivity_infinity}
Let $\Gamma$ be a finite labelled graph and $(\Gamma_n)_{n\in\N}$ a sequence of connected, finite labelled graphs such that $\Gamma':=\Gamma\sqcup\bigsqcup_{n\in\N}\Gamma_n$ is $Gr(7)$-labelled and such that for $n\neq n'$, the labelled graphs $\Gamma_n$ and $\Gamma_{n'}$ are non-isomorphic. Then the injectivity radii $\rho_n$ of the projections $\pi_n:G(\Gamma)\to G(\Gamma\sqcup\Gamma_n)$ induced by the identity on $S$ tend to infinity.
\end{lem}
\begin{proof}
Suppose this is false. Then there exists an infinite sequence $(k_n)_{n\in\N}$ and a word $w$ such that $w$ is nontrivial in $G(\Gamma)$ and such that for all $n\in\N$, $w$ is trivial in $G(\Gamma\sqcup \Gamma_{k_n})$ and $|w|=\rho_{k_n}$. For each $n\in\N$, let $D_n$ be a diagram for $w$ over $\Gamma\sqcup\Gamma_{k_n}$ such that the labels of all faces are read on simple cycles and such that no interior edge originates from the graph. By construction, each $D_n$ contains at least one face from $\Gamma_{k_n}$. For each $n\in\N$, let $D_n'$ be an inclusion-minimal subdiagram of $D_n$ containing all faces that lift to $\Gamma_{k_n}$. Then all boundary faces of $D_n'$ lift to $\Gamma_{k_n}$.
The first curvature formula implies that for each $n$, there is at least one boundary face $f_n$ of $D_n'$ such that one arc $p_n$ of $f_n\cap\partial D_n'$ is not an essential piece in $\Gamma'$. This implies that all $p_n$ bear distinct labels and hence $|p_n|\to \infty$. Note that $p_n$ is a subpath of $\partial D_n$ and/or the boundaries of some faces of $D_n$ that lift to $\Gamma$. By Theorem \ref{thm:linear_isoperimetry}, the length of any such path is bounded by $|w|+8|w| (V(\Gamma)+1)$, where $V(\Gamma)$ is the number of vertices of $\Gamma$ (Hence $V(\Gamma)+1$ is an upper bound for the length of a simple closed path on $\Gamma$). This is a contradiction.
\end{proof}
\begin{proof}[Proof of the proposition]
If all but finitely many $\Gamma_n$ are isomorphic, we can consider $G(\Gamma)$ as given by a finite $Gr(7)$-labelled graph. In that case, $G(\Gamma)$ is hyperbolic and hence lacunary hyperbolic.
Therefore, we can assume that for $n\neq n'$, $\Gamma_n$ and $\Gamma_{n'}$ are non-isomorphic. We choose the subsequence recursively: Let $k_0=1$, and let $k_1,\ldots,k_N$ be chosen. Set $\Gamma^N:=\sqcup_{i=1}^N \Gamma_{k_i}$. Then, by Theorem \ref{thm:linear_isoperimetry}, $G(\Gamma^N)$ is $\delta_N$-hyperbolic for some $\delta_N>0$. By Lemma \ref{lem:c7_injectivity_infinity}, the injectivity radii $\rho_n$ of the maps $G(\Gamma^N)\to G(\Gamma^N\sqcup \Gamma_n)$ induced by the identity on $S$ tend to infinity. Hence we may choose $k_{N+1}$ such that $k_{N+1}>\max\{k_1,\ldots,k_N\}$ and such that $\rho_{k_{N+1}}>N\delta_N$. The resulting limit group $G(\sqcup_{i\in\N}\Gamma_{k_i})$ is lacunary hyperbolic.
\end{proof}
\subsection{The $Gr'(\frac{1}{6})$ case}
Finitely presented $C'(\frac{1}{6})$ groups have been investigated in [Oll]. (In fact, Ollivier uses a stronger variant of the $C'(\frac{1}{6})$ condition.) His proofs of the facts we will use generalize to our $Gr'(\frac{1}{6})$ condition. The question, when classical $C'(\frac{1}{6})$ groups are lacunary hyperbolic has been solved in \cite[Proposition 3.12]{OOS}. We follow their arguments to extend this result:
\begin{prop}\label{prop:1/6_lacunary} Let $(\Gamma_n)_{n\in\N}$ be a sequence of finite, connected graphs such that
\begin{itemize}
\item $\Gamma:=\sqcup_{n\in\N}\Gamma_n$ has a $Gr'(\frac{1}{6})$-labelling and
\item $\Delta(\Gamma_n)=O(g(\Gamma_n))$,
\end{itemize}
where $\Delta(\Gamma_n)$ denotes the diameter and $g(\Gamma_n)$ denotes the girth of each graph. (If $\Gamma_n$ is a tree, set $g(\Gamma_n)=0$.)
Then $G(\Gamma)$ is lacunary hyperbolic if and only if the set of girths $L:=\{g(\Gamma_n)|n\in\N\}$ is sparse, i.e.\ for all $K\in \R^+$ there exists $a \in \R^+$ such that $[a,aK]\cap L=\emptyset$.
\end{prop}
We first give a precise estimate for the hyperbolicity constant of a group given by a finite $Gr'(\frac{1}{6})$-labelled graph. We observe as in Section \ref{section:preliminaries}:
\begin{lem}\label{lem:metric_diagrams} Let $\Gamma$ be a $Gr'(\frac{1}{6})$-labelled graph. Let $w\in M(S)$ satisfying $w=1$ in $G(\Gamma)$ and let $D$ be a minimal diagram over $\Gamma$ for $w$. Let $f$ be a face of $D$. Then $f$ is simply connected. Any interior arc of $\partial f$ has length less than $\frac{1}{6}|\partial f|$.
\end{lem}
Note that for a singular disk diagram $D$ over a group presentation $G=\langle S|R\rangle$, the 1-skeleton of $D$ maps into $\Cay(G,S)$. Singular disk diagrams with the property of Lemma \ref{lem:metric_diagrams} whose boundaries are geodesic triangles in the Cayley graph have been classified in \cite[Section 3.4]{Str}. This classification yields:
\begin{lem}\label{lem:metric_delta} Let $\Gamma$ be a finite $Gr'(\frac{1}{6})$-labelled graph, and let $\Delta$ be the maximum of the diameters of its connected components. Then $G(\Gamma)$ is $2\Delta$-hyperbolic.
\end{lem}
We will use the following two facts \cite[Lemma 13 and Theorem 1]{Oll} proven by Ollivier:
\begin{lem}\label{lem:metric_injectivity}
Let $\Gamma$ be a $Gr'(\frac{1}{6})$-labelled graph. Let $w\in M(S)$ satisfying $w=1$ in $G(\Gamma)$, and let $D$ be a minimal diagram for $w$ over $\Gamma$. Then for any face $f$, we have $|\partial f|\leq |\partial D|$.
\end{lem}
\begin{thm}\label{thm:metric_embedding} Let $\Gamma$ be a $Gr'(\frac{1}{6})$-labelled graph. Then each connected component of $\Gamma$ embeds isometrically into $\Cay(G(\Gamma),S)$.
\end{thm}
\begin{proof}[Proof of the proposition] Note that given the previous lemmas and theorem, a proof can be deduced from
\cite[Proof of Proposition 3.12]{OOS}. In fact, the first part of our proof uses the same arguments.
By identifying isomorphic connected components of $\Gamma$, we can assume that for $n\neq n'$, $\Gamma_n$ and $\Gamma_{n'}$ are non-isomorphic. Assume that $L$ is sparse. Then there exists a sequence $(\alpha_n)_{n\in\N}$ of positive real numbers such that for all $n\in \N$, we have $L\cap [\alpha_n,n \alpha_n]=\emptyset$. We may assume $n\alpha _n<\alpha_{n+1}$ for all $n$. Fix $C\in\R^+$ such that for all $n: \Delta(\Gamma_n)\leq C g(\Gamma_n)$. Let
$$\Gamma^k:=\sqcup_{g(\Gamma_n)<\alpha_k}\Gamma_n.$$
Then for all $k$, the graph $\Gamma^k$ is a finite and $Gr'(\frac{1}{6})$-labelled. For any connected component of $\Gamma^k$, the girth is bounded from above by $\alpha_k$, and hence the diameter is bounded by $C\alpha_k$. Lemma \ref{lem:metric_delta} implies that $G_k:=G(\Gamma^k)$ is $2C\alpha_k$-hyperbolic. Set $\delta_k:=2C\alpha_k$.
The injectivity radius $r_S(\pi_k)$ of the map $G_k\to G_{k+1}$ induced by the identity on $S$ is the length of the shortest word $w$ in $S$ that is trivial in $G_{k+1}$ but not trivial in $G_k$. Hence a minimal diagram $D$ for $w$ over $\Gamma^{k+1}$ has a face $f$ from $\Gamma^{k+1}\setminus \Gamma^k$. Such a face satisfies $|\partial f|\geq k\alpha_k$ by construction. Hence Lemma \ref{lem:metric_injectivity} implies $r_S(\pi_k)=|w|\geq k\alpha_k$. Therefore $\frac{\delta_k}{r_S(\pi_k)}\to 0$ as $k\to \infty$, and $G(\Gamma)$ is lacunary hyperbolic.
We now prove the converse. Assume that the set of girths $L$ is not sparse. Choose any non-decreasing scaling sequence $(d_n)_{n\in\N}$ tending to infinity and any non-principal ultrafilter $\omega$ on $\N$. We show that $Y:=\Con^\omega(G(\Gamma),d_n)$ is not an $\R$-tree. Theorem \ref{thm:metric_embedding} implies that from each graph $\Gamma_n$, there is a cycle $p_n$ of length $g(\Gamma_n)$ isometrically embedded into $X:=\Cay(G(\Gamma),S)$. Since $L$ is not sparse, there exists $K>0$ such that for all $a >0$ we have $[a,aK]\cap L\neq\emptyset$. Hence for all $n\in\N$ there is $k(n)\in \N$ such that the inequality
$$d_n\leq g(\Gamma_{k(n)})\leq K d_n $$
holds, or in other words $1\leq |p_{k(n)}|/d_n\leq K$. Since the interval $[1,K]$ is bounded, the sequence $(|p_{k(n)}|/d_n)_{n\in\N}$ converges to some $R\in[1,K]$ with respect to the ultrafilter $\omega$. We may view each cycle $p_{k(n)}$ as a continuous map $p_{k(n)}:\R/\Z\to X$, and we may assume that for all $t$ and for all $\epsilon\in(-\frac{1}{2},\frac{1}{2}]$ we have $d(p_{k(n)}(t),p_{k(n)}(t+\epsilon))=|\epsilon| |p_{k(n)}|$.
Consider the cycle $\gamma:\R/\Z\to Y, t\mapsto [(p_{k(n)}(t))_{n\in \N}]$. Let $t\neq t'\in \R/\Z$ and $\epsilon\in(-\frac{1}{2},\frac{1}{2}]$ such that $t'=t+\epsilon$. Then for any $n$,
$$d(\gamma(t),\gamma(t'))=\lim_n^\omega \frac{d(p_{k(n)}(t),p_{k(n)}(t+\epsilon))}{d_n}=\lim_n^\omega|\epsilon|\frac{|p_{k(n)}|}{ d_n}=|\epsilon|R>0.$$ Hence $\gamma$ is injective, and thus $Y$ is not an $\R$-tree.
\end{proof}
\end{document} |
\begin{document}
\begin{center}
\LARGE
Distance-preserving graph contractions
\renewcommand{\fnsymbol{footnote}}{\fnsymbol{footnote}}
\footnote{An extended abstract of this work has appeared in the Proceedings of the 9th Innovations in Theoretical Computer Science Conference (ITCS) 2018 \cite{DBLP:conf/innovations/BernsteinDDKMS18}.}
\par
\normalsize
\renewcommand\fnsymbol{footnote}{\@fnsymbol\c@footnote}
Aaron Bernstein\textsuperscript{1}\footnote{E-Mail: \texttt{bernstei@gmail.com}},
Karl D\"aubel\textsuperscript{1}\footnote{\label{mail}E-Mail: \texttt{\{daeubel,muetze,smolny\}@math.tu-berlin.de}},
Yann Disser\textsuperscript{2}\footnote{Supported by the `Excellence Initiative' of the German Federal and State Governments and the Graduate School~CE at TU~Darmstadt. E-Mail: \texttt{disser@mathematik.tu-darmstadt.de}}, \\
Max Klimm\textsuperscript{3}\footnote{E-Mail: \texttt{max.klimm@hu-berlin.de}},
Torsten M\"utze\textsuperscript{1}\footref{mail} and
Frieder Smolny\textsuperscript{1}\footref{mail} \par
\textsuperscript{1}Institut f\"{u}r Mathematik, TU Berlin \par
\textsuperscript{2}Department of Mathematics, Graduate School CE, TU Darmstadt \par
\textsuperscript{3}Wirtschaftswissenschaftliche Fakult\"{a}t, HU Berlin \par
\par
\end{center}
\begin{abstract}
Compression and sparsification algorithms are frequently applied in a preprocessing step before analyzing or optimizing large networks/graphs.
In this paper we propose and study a new framework contracting edges of a graph (merging vertices into super-vertices) with the goal of preserving pairwise distances as accurately as possible.
Formally, given an edge-weighted graph, the contraction should guarantee that for any two vertices at distance $d$, the corresponding super-vertices remain at distance at least $\varphi(d)$ in the contracted graph, where $\varphi$ is a tolerance function bounding the permitted distance distortion.
We present a comprehensive picture of the algorithmic complexity of the contraction problem for affine tolerance functions $\varphi(x)=x/\alpha-\beta$, where $\alpha\geq 1$ and $\beta\geq 0$ are arbitrary real-valued parameters.
Specifically, we present polynomial-time algorithms for trees as well as hardness and inapproximability results for different graph classes, precisely separating easy and hard cases.
Further we analyze the asymptotic behavior of contractions, and find efficient algorithms to compute (non-optimal) contractions despite our hardness results.
\end{abstract}
\section{Introduction}
\label{sec:intro}
When dealing with large networks, it is often beneficial to compress or sparsify the data to manageable size before analyzing or optimizing the network directly.
To be useful, a meaningful compression should represent salient features of the original network with good approximation, while being much smaller in size.
In this paper, we focus on a compression of undirected edge-weighted graphs that approximately maintains all distances between vertices in the graph.
In this context, an extensively studied concept are \emph{spanners} (e.g.\ \cite{MR982872,MR1184695,MR2298319,MR3536579}).
Given an undirected graph $G=(V,E)$ and real numbers $\alpha \geq 1$ and $\beta \geq 0$, a subgraph $H=(V,E')$, $E'\subseteq E$, is an \emph{$(\alpha,\beta)$-spanner of~$G$} if $\dist_{H}(u,v) \leq \alpha\cdot \dist_G(u,v) + \beta$ holds for all $u,v \in V$.
While the number of edges in a spanner may be much smaller than that of the original graph, the number of vertices is the same for both, leaving further potential for compression untapped.
For illustration, consider the road network of Europe with about 50~million vertices \cite{MR3077319}, any spanner of which must again have about 50 million vertices and edges.
However, to approximately represent distances in Europe's road network one may also merge nearby vertices into super-vertices, thus achieving a much better compression of the network.
This is akin to the visual process of zooming out of a graphical representation of the map, where neighbored vertices fade into each other and edges between merged vertices vanish.
At a large enough zoom level, the entire network merges into a single vertex.
In this paper we propose and study a new framework for contracting networks that formalizes this intuitive idea and makes it applicable to general graphs.
Specifically, we study a contraction problem on graphs where a subset of edges $C \subseteq E$ is contracted.
We denote by $G/C$ the resulting simple graph obtained from $G$ by contracting the edges in $C$ and by deleting resulting loops and multiple edges, keeping only the minimum length edge between any two vertices.
For any two vertices in $G$, we compare their distance in $G$ with the distance of the corresponding super-vertices in $G/C$.
It is interesting to contrast this concept with graph spanners.
When constructing a spanner, the length of the removed edges is implicitly set to~$\infty$, resulting in an overall increase of distances.
On the other hand, a contraction implicitly sets the length of the contracted edges to zero, leading to an overall decrease of distances.
For both problems, the ultimate goal is to reduce the complexity of the network while maintaining an approximation guarantee on the distances.
The following example shows that contractions may be better suited than spanners to achieve this goal.
In a subgraph with small radius, a spanner can at best result in a spanning tree of the same order, while a contraction can reduce the whole subgraph to a single vertex, while entailing a multiplicative distance distortion of similar magnitude.
In addition, the contraction may also merge many edges entering the contracted subgraph.
Clearly, the objective here is to maximize the total number of contracted and deleted edges, as this minimizes the memory required to represent the resulting network in a computer (using e.g.\ adjacency lists).
\begin{figure}
\caption{Top: two iterations of \Contraction{}
\label{fig:contraction}
\end{figure}
Given the results presented in this paper and the known results for spanners (discussed in detail below), we further believe that the combination of spanners and contractions is very powerful, promising and flexible.
As the former only increases and the latter only decreases the distances, the respective distortion guarantees provably also hold for the overall distortion.
In fact, both effects may even compensate each other.
This is true \emph{regardless} of the order in which both compression operations are applied, even when they are applied repeatedly.
In order to measure the distance distortion of the contraction, we assume a non-decreasing tolerance function $\varphi\colon \RR \to \RR$, similar to the corresponding function for spanners, see e.g.~\cite{MR2298319}.
We are interested in computing contractions that preserve distances in the following sense:
For any two vertices~$u$ and~$v$ at distance~$d$ in~$G$, the distance of the corresponding vertices in the contracted graph~$G/C$ must be at least~$\varphi(d)$.
If this condition is satisfied, we call~$C$ a \emph{$\varphi$-distance preserving contraction}, or \emph{$\varphi$-contraction} for short.
Formally, the algorithmic problem \Contraction{} considered in this paper is to compute for a given graph $G = (V,E)$ with edge lengths $\ell\colon E\to \RR_{>0}$ and a given tolerance function~$\varphi$, a $\varphi$-contraction $C \subseteq E$ such that the number of contracted and deleted edges is maximized.
We are specifically interested in the case where the tolerance function~$\varphi$ is an affine function $\varphi(x)=x/\alpha-\beta$ for real-valued parameters $\alpha \geq 1$ and $\beta \geq 0$.
We then simply write \emph{$(\alpha,\beta)$-contraction} instead of $\varphi$-contraction.
See Figure~\ref{fig:contraction} for some example instances of the problem \Contraction{}.
When considering the case of a purely multiplicative error ($\beta=0$), a slight subtlety has to be taken into account.
Specifically, for a graph with positive edge lengths it is not feasible to contract a single edge.
Therefore, we propose a slight modification of our original model:
We say that a set $C\subseteq E$ of edges of $G$ is a \emph{weak $\varphi$-distance preserving contraction}, or \emph{weak $\varphi$-contraction} for short, if it does not contract the entire graph and, for any two vertices $u$ and $v$ at distance $d$ in $G$, the distance of the corresponding vertices in $G/C$ is either zero or at least $\varphi(d)$.
We will refer to the corresponding algorithmic problem as \WContraction{}.
Put differently, in a weak contraction, the distances between different super-vertices satisfy the given distortion guarantee, but for vertices belonging to the same super-vertex, no guarantee is given.
\subsection{Our results}
\label{sec:results}
In this paper, we present a comprehensive picture of the algorithmic complexity of the described contraction problems.
Recall that we are given an input graph with edge lengths and tolerance function $\varphi$, and our goal is to compute a (weak) contraction that maximizes the total number of contracted and deleted edges.
Our main results concern affine tolerance functions $\varphi(x)=x/\alpha-\beta$ with parameters $\alpha\geq 1$ and $\beta\geq 0$.
For the reader's convenience, our results are summarized in Tables~\ref{tab:results} and~\ref{tab:results-asymp}.
Within the tables and throughout this paper, $n$ and $m$ denote the number of vertices and edges, respectively, of the input graph under consideration.
\newcommand{\myline}[1]{\raisebox{1.5mm}{\underline{\hspace{#1}}}}
\newcolumntype{P}[1]{>{\centering\arraybackslash}p{#1}}
\begin{table}[tb]
\caption{Overview of algorithmic and hardness results presented in this paper.
\label{tab:results}
}
\footnotesize
\begin{minipage}{\textwidth}
\begin{tabular*}{\linewidth}{
@{}l
@{\extracolsep{\fill}} P{2.0cm}
@{\extracolsep{0.6ex}} P{2.0cm}
@{\extracolsep{0.6ex}} P{2.35cm}
@{\extracolsep{0.6ex}} P{2.35cm}
}
\toprule
Problem &
\multicolumn{4}{c}{\myline{3.9cm}\,Graph classes\,\myline{3.9cm}} \\
&
\multicolumn{1}{c}{Path} &
\multicolumn{1}{c}{Tree} &
\multicolumn{1}{c}{Cycle} &
\multicolumn{1}{c}{General} \\
\Contraction{} &\\[1mm]
addit.\ {\scriptsize($\alpha\!\!=\!\!1$)}, unit lg. \rule{0pt}{1.1em}&
\multicolumn{1}{c}{\cellcolor{green!50}} &
\multicolumn{1}{c}{\cellcolor{green!50}$\cO(n)$~\scriptsize{[Th.~\ref{thm:unit-tree}]}} &
\multicolumn{1}{c}{\cellcolor{green!50}} & \multicolumn{1}{c}{\cellcolor{red!40}$m^{\frac{1}{2}\!-\!\varepsilon}$-inapx.\footnote{even for bipartite graphs and $\beta=1$} \scriptsize{[Th.~\ref{thm:inapx-bip}]}} \\
&
\multicolumn{1}{c}{\cellcolor{green!50}}&
&
\multicolumn{1}{c}{\cellcolor{green!50}}&
\multicolumn{1}{c}{\cellcolor{red!40}}\\[-2ex]
affine {\scriptsize($\alpha,\beta$)}, unit lg. \rule{0pt}{1.1em}&
\multicolumn{1}{c}{\cellcolor{green!50}$\cO(n)$ \scriptsize{[Th.~\ref{thm:path}]}} &
\multicolumn{1}{c}{\cellcolor{green!40}}&
\multicolumn{1}{c}{\cellcolor{green!50}$\cO(n)$~\scriptsize{[Th.~\ref{thm:cycle}]}} &
\multicolumn{1}{c}{\cellcolor{red!40}} \\
&
&
\multicolumn{1}{c}{\cellcolor{green!40}}&
&
\\[-2ex]
addit.\ {\scriptsize($\alpha\!\!=\!\!1$)} \rule{0pt}{1.1em}&
\multicolumn{2}{c}{\cellcolor{green!40}} &
\multicolumn{1}{c}{\cellcolor{red!30}NP-hard~\scriptsize{[Th.~\ref{thm:cycle-hard-fix}]}} &
\multicolumn{1}{c}{\cellcolor{red!50}$n^{1-\varepsilon}$-inapx.~\scriptsize{[Th.~\ref{thm:inapx-contr}]}} \\
&
\multicolumn{2}{c}{\cellcolor{green!40}} &
\multicolumn{1}{c}{\cellcolor{red!30}} &
\multicolumn{1}{c}{\cellcolor{red!50}}\\[-2ex]
affine {\scriptsize($\alpha,\beta$)} \rule{0pt}{1.1em}&
\multicolumn{2}{c}{\cellcolor{green!40}$\cO(n^3)$~\scriptsize{[Th.~\ref{thm:dp}]}} &
\multicolumn{1}{c}{\cellcolor{red!30}} &
\multicolumn{1}{c}{\cellcolor{red!50}} \\
\midrule
\WContraction{} &\\[1mm]
additive {\scriptsize($\alpha\!\!=\!\!1$)} \rule{0pt}{1.1em}&
\multicolumn{2}{c}{\cellcolor{green!30}} &
\multicolumn{2}{l}{\cellcolor{red!30}NP-hard\footnote{also NP-hard for planar graphs with arb.\ large girth, $(\alpha,\beta)=(2,0)$, and unit lg.\ ($\ell=1$) [Th.~\ref{thm:weak-planar}].} \scriptsize{[Th.~\ref{thm:cycle-hard-fix}]}} \\
&\multicolumn{2}{c}{\cellcolor{green!30}}& \multicolumn{1}{c}{\cellcolor{red!30}}&\\[-2ex]
affine {\scriptsize($\alpha,\beta$)} \rule{0pt}{1.1em}&
\multicolumn{2}{c}{\cellcolor{green!30}$\cO(n^5)$~\scriptsize{[Th.~\ref{thm:weak-dp}]}} &
\multicolumn{1}{c}{\cellcolor{red!30}} &
\multicolumn{1}{c}{\cellcolor{red!50}$n^{1-\varepsilon}$-inapx.\footnote{even if $(\alpha,\beta)=(3/2,0)$.} \scriptsize{[Th.~\ref{thm:inapx-weak}]}} \\
\bottomrule
\end{tabular*}
\end{minipage}
\end{table}
\subsubsection*{Algorithmic results}
We develop linear time greedy algorithms for \Contraction{} with unit lengths on paths and cycles for general $\alpha$ and $\beta$, as well as on trees with $\alpha = 1$ (Theorems~\ref{thm:path},~\ref{thm:cycle} and~\ref{thm:unit-tree}).
The first two algorithms are inspired by LP rounding techniques, the latter algorithm relies on a structural characterization of optimal solutions.
We present dynamic programming algorithms solving \Contraction{} and \WContraction{} on trees in time $\cO(n^3)$ or $\cO(n^5)$, respectively (Theorems~\ref{thm:dp} and~\ref{thm:weak-dp}).
These dynamic programs compute optimal solutions on subtrees, in the latter case combining several Pareto optimal solutions in a two-dimensional parameter space (hence the larger running time).
Note that instead of maximizing the number of contracted and deleted edges, we could optimize for $\alpha$ or $\beta$ while fixing the other parameters.
The resulting problems are polynomially equivalent to our setting, via binary search over one of the parameters.
\subsubsection*{Hardness results}
We complement these algorithms by several hardness results.
First we consider the purely additive case where $\alpha=1$.
We show that here both \Contraction{} and \WContraction{} are NP-hard on cycles for any fixed $\beta>0$, by a reduction of a variant of \Partition{} (Theorem~\ref{thm:cycle-hard-fix}).
As mentioned before, both problems can be solved efficiently on graphs without cycles, and there is a linear time algorithm for \Contraction{} on cycles with unit lengths.
By reductions from \Clique{} we show that both the general as well as the unit lengths case of \Contraction{} with $\alpha=1$ are hard to approximate within factors of $n^{1-\varepsilon}$ or $m^{1/2-\varepsilon}$, respectively (Theorem~\ref{thm:inapx-contr} and Theorem~\ref{thm:inapx-bip}).
Further we consider the purely multiplicative case where $\beta=0$ (here \Contraction{} is trivial).
We show that in this case \WContraction{} is NP-hard on planar graphs with arbitrarily large girth and unit length edges by a reduction from a special case of \textsc{Planar 3SAT} (Theorem~\ref{thm:weak-planar}).
Since these graphs are locally tree-like, this result constitutes another rather sharp separation from the polynomially solvable tree case.
Furthermore, we show that the problem is hard to approximate within a factor of $n^{1-\varepsilon}$ by a reduction from \IndSet{} (Theorem~\ref{thm:inapx-weak}).
\begin{table}[tb]
\caption{Overview of asympotic bounds presented in this paper.
\label{tab:results-asymp}
}
\footnotesize
\begin{minipage}{\textwidth}
\centering
\begin{tabular}{
@{}l
@{\extracolsep{1ex}} P{3.5cm}
@{\extracolsep{1ex}} P{2.5cm}
@{\extracolsep{1ex}} P{2.5cm}
}
\toprule
\Contraction{} with unit lg.\ {\scriptsize($\ell\!\!=\!\!1$)} &
\# of edges in $G/C$ &
Time &
Reference \\[1mm]
\rowcolor{green!50}
\multicolumn{1}{@{}l}{\cellcolor{white}$(\alpha, \beta)=(2k -1, 1)$}&
\rule{0pt}{1.1em}
$n^{1+1/k}$ &
$\cO(m)$ &
[Th.~\ref{thm:asymp-mult}] \\
&
&
&\\[-2ex]
\rowcolor{green!50}
\multicolumn{1}{@{}l}{\cellcolor{white}$(\alpha, \beta)=(2\log_2 n -1, 1)$}&
\rule{0pt}{1.1em}
$2n$ &
$\cO(m)$ &
[Cor.~\ref{cor:asymp-mult}] \\
&
&
&\\[-2ex]
\rowcolor{red!50}
\multicolumn{1}{@{}l}{\cellcolor{white}$(\alpha, \beta)=(k -1, 1)$}&
\rule{0pt}{1.1em}
$\Omega(n^{1+1/k})$ &
--- &
[Th.~\ref{thm:asymp-girth}] \\
&
&
&\\[-2ex]
\rowcolor{green!50}
\multicolumn{1}{@{}l}{\cellcolor{white}$(\alpha, \beta)=(1, k)$}&
\rule{0pt}{1.1em}
$m - km/(2n)$ &
$\cO(m)$ &
[Th.~\hyperref[thm:asymp-add-first]{\ref*{thm:asymp-add}~\ref*{thm:asymp-add-first}}]\\
&
&
&\\[-2ex]
\rowcolor{green!50}
\multicolumn{1}{@{}l}{\cellcolor{white}$(\alpha, \beta)=(1, k)$}&
\rule{0pt}{1.1em}
$\cO(n^2/k)$ &
$\cO(m)$ &
[Th.~\hyperref[thm:asymp-add-high]{\ref*{thm:asymp-add}~\ref*{thm:asymp-add-high}}] \\
&
&
&\\[-2ex]
\rowcolor{red!50}
\multicolumn{1}{@{}l}{\cellcolor{white}$(\alpha, \beta)=(1, \cO(1))$}&
\rule{0pt}{1.1em}
$\Omega(n^{4/3 - o(1)})$ &
--- &
\cite{MR3536579} \\
\midrule
\Contraction{} with unit lg.\ {\scriptsize($\ell\!\!=\!\!1$)}\\ \ and min.~degree $D$&
\# of vertices in $G/C$ &
Time &
Reference \\
\rowcolor{green!50}
\multicolumn{1}{@{}l}{\cellcolor{white}$(\alpha, \beta)=(5, 1)$}&
\rule{0pt}{1.1em}
$n/D$ &
$\cO(m)$ &
[Th.~\ref{thm:min-degree}] \\
&
&
&\\[-2ex]
\rowcolor{red!50}
\multicolumn{1}{@{}l}{\cellcolor{white}$(\alpha, \beta)=(k, 1)$}&
\rule{0pt}{1.1em}
$n/((k+1) D)$ &
--- &
[Th.~\ref{thm:asymp-degree}] \\
\bottomrule
\end{tabular}
\end{minipage}
\end{table}
\subsubsection*{Asymptotic bounds}
We now discuss our asymptotic bounds for contractions.
In this setting, we are interested in (non-optimal) contractions for graphs with unit lengths that can be computed efficiently despite the above-mentioned hardness results.
We prove that for any $k\geq 1$ any graph $G$ has a $(2k-1,1)$-contraction $C$ such that $G/C$ has at most $n^{1+1/k}$ edges, and such a contraction can be computed in time $\cO(m)$ (Theorem~\ref{thm:asymp-mult}) by successively growing clusters around center vertices.
Assuming Erd\H{o}s' girth conjecture, we show a corresponding (not tight) lower bound (Theorem~\ref{thm:asymp-girth}).
For a purely additive error, we observe two simple $(1,k)$-contractions that can be computed in $\cO(m)$ time (Theorem~\ref{thm:asymp-add}).
We show that for any even integer $0 \leq k \leq n$, the edges incident to the $k/2$ vertices of highest degrees form a $(1,k)$-contraction with objective value at least $km/(2n)$, which is asymptotically best possible for paths.
Another $(1, k)$-contraction $C$ is implicitly used by Bernstein and Chechik in their faster deterministic algorithm for dynamic shortest paths in dense graphs \cite{MR3536582}.
For any number $0 < k \leq n$, it consists of the edges incident to two vertices of degree at least $n/k$, and $G/C$ has $\cO(n^2/k)$ edges.
Both of these contractions can be computed in $\cO(m)$ time.
Further we note that the main result in~\cite{MR3536579} implies that for all $\varepsilon > 0$, any contraction $C$ such that $G/C$ has $\cO(n^{4/3 - \varepsilon})$ edges does not admit a constant additive error.
One possible advantage of contraction compared to spanners is the potentially significant reduction of \emph{vertices} as well as edges, e.g. reducing the complexity of performing algorithmic tasks in the smaller graph.
To ground this intuition, we exhibit a contraction that significantly reduces the number of vertices in any graph with minimum degree $D$ to $\cO(n/D)$ (Theorem~\ref{thm:min-degree}). We also present a lower bound (Theorem~\ref{thm:asymp-degree}) showing that we cannot guarantee $o(n/D)$ vertices, even if we allow larger approximation error.
\subsection{Comparison with previous results}
There are several models aiming to compress graphs while preserving distances.
They differ by their choice of compression operation, such as replacing the graph by a subgraph or minor, and by whether the aim is to preserve all or only certain distances.
As discussed before, graph spanners are a concept closely related to contractions, where the length of removed edges is set to $\infty$ rather than to $0$.
Our results highlight further intrinsic similarities of the two models.
Like contractions, spanners are NP-hard to compute optimally (see \cite{MR982872,MR1232326}).
While the spanner literature considers the problem of minimizing the number of remaining edges, we analyze the objective of maximizing the number of contracted edges, prohibiting a direct comparison of the respective inapproximability results.
We note however that approximation algorithms for spanner problems have been studied extensively, even though strong lower bounds are known.
For instance, computing $(2,0)$-spanners in unweighted graphs is $\Theta(\log n)$-hard to approximate (\cite{MR1291540,MR1822929}); for further references see e.g.~\cite{MR3627763}.
Despite these negative results, it is still possible to obtain powerful asymptotic guarantees in both models.
In particular, our $(2k-1, 1)$-contraction with $\cO(n^{1+1/k})$ edges for unweighted graphs has a clear analogy to the classic $(2k-1,0)$-spanner with the same number of edges~\cite{MR1184695} (note that the additive error of 1 in our result is strictly necessary, as discussed above).
There is, however, a major difference between the two results: whereas the $(2k-1,0)$-spanner can trivially be shown to be optimal assuming Erd\H{o}s' girth conjecture, applying this conjecture to the contraction model only yields a lower bound of $n^{1+1/(2k)}$ edges for a $(2k-1,1)$-contraction. Closing this gap thus remains as an interesting open problem in the contraction model, whose solution would likely yield further insight into the relationship to spanners.
Halperin and Zwick showed how an optimal $(2k-1,0)$-spanner can be constructed in linear time (see~\cite{MR2080715}). We achieve the same running time for our $(2k-1,1)$-contraction.
It is interesting to note that the clustering yielding our $(2k-1,1)$-contraction was previously used in~\cite{MR982872} to obtain a $(4k+1,0)$-spanner of the same asymptotic density.
There are also spanner results that significantly sparsify unweighted graphs at the cost of a purely additive error, as a (1,2)-spanner with $\cO(n^{3/2})$ edges~\cite{MR1681058}, or a (1,6)-spanner with $\cO(n^{4/3})$ edges~\cite{MR2298319}.
We do not know if analogous results are possible in the contraction model.
The incompressibility result in~\cite{MR3536579} mentioned above implies the same lower bound for spanners as for contractions and every other distance oracle with additive error: For every $\varepsilon > 0$ any spanner of size $\cO(n^{4/3 - \varepsilon})$ does not admit a constant additive error.
Finally, for spanners there are results that combine multiplicative and additive error, such as the $(k,k-1)$-spanner of~\cite{MR2298319}.
Gupta~\cite{MR1958411} considered the problem of approximating a tree metric on a subset of the vertices by another tree, and gave a linear time algorithm computing an $8$-approximation.
As Chan et al.~\cite{MR2304999} observed later, on complete binary trees a solution of minimum distortion is always achieved by a minor (with possibly different edge lengths) of the input tree, so this seems to be the first investigation of contractions that approximate graph distances.
Krauthgamer et al.~\cite{MR3158782} considered an extension to general graphs, studying the size of minors preserving all distances between a given terminal set of fixed size.
Cheung et al.~\cite{DBLP:conf/icalp/CheungGH16} introduced a multiplicative distortion to this model.
As here no two terminals may be merged, these approaches cannot compress a graph at all if every vertex is a terminal.
The \emph{pairwise preservers} due to Coppersmith et al.~\cite{MR2257273} combine spanners with the aim of preserving only terminal distances.
Given a graph $G$ and a set of $k$ terminal pairs, a pairwise preserver is a spanning subgraph inducing exactly the same terminal distances as~$G$. Coppersmith et al.~\cite{MR2257273} proved that for every undirected weighted graph there exists a pairwise preserver of size $\cO(n + n^{1/2}k)$. Furthermore, they showed that every directed weighted graph has a pairwise preserver of size $\cO(nk^{1/2})$.
For the special case of undirected unweighted graphs, Bodwin et al.~\cite{MR3478437} showed the existence of a pairwise preserver with $\cO(n^{2/3}k^{2/3} + nk^{1/3})$ edges.
Recently, Bodwin~\cite{MR3627768} proved that any directed weighted graph has a pairwise preserver of size $\cO(n + n^{2/3}k)$.
\subsection{Further related work}
The preservation of graph properties other than distances has been studied as well.
Biedl et al.~\cite{MR1844744} considered contractions in capacitated networks with the goal of maintaining the maximum flow in the network.
Here an edge $e$ is called \emph{useless}, if for every capacity function there is a maximum flow not using $e$.
Biedl et al.\ showed that finding all useless edges is NP-complete, but solvable in $\cO(n^2)$ time on certain planar graphs.
For undirected networks, Misio{\l}ek et al.~\cite{MR2190897} gave an algorithm finding all useless edges in $\cO(n + m)$ time.
Toivonen et al.~\cite{DBLP:conf/icdm/ZhouMT10} considered a more general model aiming to maintain the quality of paths with respect to any given function, e.g., distance or capacity.
They investigated strategies of removing edges, without decreasing the quality of the best path between any pair of vertices.
Graph simplification problems have also been studied in several other contexts, and we conclude this section by mentioning two such examples:
H{\"u}bler et al.~\cite{DBLP:conf/icdm/HublerKBG08} studied a problem related to graph mining, examining how to choose an induced subgraph with a given number of vertices and with similar topological properties as the input graph.
Numerous papers investigate, directly or as a tool, sparsifiers that preserve the effective resistance between certain or all pairs of vertices, see e.g.~\cite{MR3017573,MR3441994,MR3631020,MR3678224,DBLP:conf/focs/ChuGPSSW18}.
\subsection{Outline of this paper}
In Section~\ref{sec:prelim} we introduce important definitions and notations that will be used throughout this paper.
In Section~\ref{sec:greedy} we present our three greedy algorithms for solving \Contraction{} with unit lengths on paths, cycles and trees (the latter result requires $\alpha=1$).
In Section~\ref{sec:trees} we discuss efficient dynamic programming algorithms for \Contraction{} and \WContraction{} on trees.
Sections~\ref{sec:hard-add} and \ref{sec:hard-mult} are devoted to our hardness results, focussing on the cases of purely additive and multiplicative error, respectively.
In Section~\ref{sec:asymp} we present our asymptotic results on contractions.
\section{Preliminaries}
\label{sec:prelim}
Throughout this paper we consider simple undirected graphs~$G$ (without parallel edges or loops).
We let~$V(G)$ and~$E(G)$ denote the vertex and edge set of~$G$, respectively, and we define $n(G):=|V(G)|$ and $m(G):=|E(G)|$.
If the context is clear, we simply write $V$, $E$, $n$ and~$m$.
We also use the notation $[n]:=\{1,2,\ldots,n\}$.
We assume that~$G$ is connected, otherwise the contraction problem can be solved independently for each connected component.
Edge lengths are given by a function $\ell\colon E\to\RR_{>0}$.
The \emph{distance} $\dist_\ell(u,v)$ between two vertices~$u$ and~$v$ is the length of a shortest path between~$u$ and~$v$ in~$G$ with respect to~$\ell$.
Given a subset of edges $C\subseteq E$, we denote the resulting simple graph obtained from $G$ by contracting the edges in $C$, deleting resulting loops and keeping only the minimum length edge between any two vertices by $G/C$.
We denote the number of deleted loops and multi-edges by $\Delta(C)$ (thus $m(G/C)=m(G)-|C|-\Delta(C)$).
Instead of contracting a set $C\subseteq E$ of edges in $G$, setting their edge lengths to zero has the same effect on the distances in the resulting graph.
This is somewhat cleaner conceptually, so we will often adopt this viewpoint.
Specifically, we let $\ell_C$ be the new length function that assigns 0 to every edge in $C$, and that is equal to the original edge lengths $\ell$ on the edges $E\setminus C$.
A \emph{tolerance function} is a non-decreasing function $\varphi\colon\RR\to \RR$.
Roughly speaking, this function describes by how much the distance between two vertices may drop when contracting edges (i.e., setting edge lengths to zero).
Formally, given a graph $G$ with edge lengths $\ell$ and a tolerance function $\varphi$, we say that a subset of edges $C\subseteq E$ is a \emph{$\varphi$-distance preserving contraction} or \emph{$\varphi$-contraction} for short, if
\begin{equation}
\label{eq:contr-cond}
\dist_{\ell_C}(u,v)\geq \varphi(\dist_\ell(u,v))
\end{equation}
holds for any two vertices~$u$ and~$v$ in~$G$.
Similarly, we say that $C$ is a \emph{weak $\varphi$-distance preserving contraction} or \emph{weak $\varphi$-contraction} for short, if any two vertices $u$ and $v$ satisfy relation \eqref{eq:contr-cond} or the relation $\dist_{\ell_C}(u,v)=0$, and if the graph $(V,C)$ is disconnected (equivalently, if $G/C$ is not a single vertex).
The last condition prevents solutions $C\subseteq E$ for which the graph is contracted to a single vertex.
If $\varphi(x)=x/\alpha-\beta$, then we simply write (weak) $(\alpha,\beta)$-contraction instead of (weak) $\varphi$-contraction.
An \emph{instance} of the problem \Contraction{} or \WContraction{} is a triple $(G,\ell,\varphi)$, where~$G$ is the underlying graph, $\ell$ the length function and~$\varphi$ the tolerance function, and the objective is to find a (weak) $\varphi$-distance preserving contraction $C\subseteq E$, such that
\begin{equation}
\label{eq:objective}
\Phi(C) := |C|+\Delta(C)=m(G)-m(G/C)
\end{equation}
is maximized.
This quantity equals the number of edges we save when going from $G$ to $G/C$.
Note that on trees we have $\Phi(C)=|C|$ for any (weak) contraction $C$, whereas on general graphs we have $\Phi(C)\geq |C|$.
\begin{problem}{\wContraction{}}
Input: & A graph $G = (V,E)$ with edge lengths $\ell\colon E \to \RR_{>0}$ and a non-decreasing function $\varphi\colon \RR \to \RR$. \\
Output: & A (weak) $\varphi$-distance preserving contraction $C \subseteq E$ maximizing $\Phi(C)$. \\
\end{problem}
In this context we sometimes refer to a set of edges that forms a (weak) contraction as a \emph{feasible} solution, and to a (weak) contraction of maximum value $\Phi(C)$ as an \emph{optimal} solution.
We begin by proving that our contraction model behaves nicely when contracting edges in phases, i.e., the total error is simply the error accumulated over the contraction phases (but not more).
To state this result we denote the composition of tolerance functions~$\varphi$ and~$\psi$ as $(\psi \circ \varphi)(x):=\psi(\varphi(x))$.
\begin{thm}
\label{thm:contract-comp}
Let~$C$ be a (weak) $\varphi$-contraction for~$G$, and let~$C'$ be a (weak) $\psi$-contraction for~$G/C$.
Then $C\cup C'$ is a (weak) $(\psi \circ \varphi)$-contraction for~$G$.
\end{thm}
\begin{proof}
We only prove the statement for contractions~$\varphi$ and~$\psi$.
The proof for weak contractions works analogously.
Let~$\ell$ denote the edge lengths of~$G$ and consider a pair of vertices $u,v\in V(G)$.
Then we have $\dist_{\ell_{C\cup C'}}(u,v) \geq \psi(\dist_{\ell_C}(u,v))$ by the definition of~$C'$ and $\dist_{\ell_C}(u,v)\geq \varphi(\dist_\ell(u,v))$ by the definition of~$C$.
Combining these inequalities and using that~$\psi$ is non-decreasing we obtain $\dist_{\ell_{C\cup C'}}(u,v)\geq \psi(\varphi(\dist_\ell(u,v)))$, as desired.
\end{proof}
Note that Theorem~\ref{thm:contract-comp} only concerns the \emph{feasibility} of repeated contractions, but not about their \emph{optimality} when searching for contractions of maximum cardinality.
With respect to solution quality, contracting in phases may be arbitrarily bad:
Consider a star with~$k$ unit length edges and additive tolerance functions $\varphi(x)=\psi(x)=x-1$.
An optimum $(\psi \circ \varphi)$-contraction contains all~$k$ edges, whereas finding an optimal $\varphi$-contraction~$C$ and then an optimal $\psi$-contraction of~$G/C$ allows contracting only one edge in each phase, leading to a $(\psi \circ \varphi)$-contraction of value 2.
\section{Greedy algorithms}
\label{sec:greedy}
In this section we consider three special cases of the problem \Contraction{} with affine tolerance function $\varphi(x)=x/\alpha-\beta$.
We obtain simple greedy algorithms computing maximum size $\varphi$-contractions in $\cO(n)$ time on paths and cycles with unit lengths, and on trees with unit lengths and $\alpha=1$.
\subsection{Paths with unit length edges}
\label{sec:paths}
In this section we consider the special case of contracting a path~$P_n$ with~$n-1$ unit length edges~$\ell=1$ and the tolerance function $\varphi(x)=x/\alpha-\beta$.
In this case optimal solutions have a very special structure, which leads to a straightforward greedy algorithm running in linear time.
Recall that as a path is a tree, our objective functions satisfies $\Phi(C)=|C|$ for any contraction $C$.
Observe that a solution $C\subseteq E(P_n)$ for the instance $(P_n,\ell,\varphi)$ of the problem \Contraction{} is feasible, if and only if every subpath $P'\subseteq P_n$ satisfies the condition
\begin{equation}
\label{eq:path-cond}
|E(P')\cap C| \leq (1-1/\alpha)|E(P')|+\beta .
\end{equation}
This observation leads to the following natural greedy algorithm $\Greedy(P_n,\alpha,\beta)$:
The algorithm considers the edges $e_1,e_2,\ldots,e_{n-1}$ of~$P_n$ as they are encountered when starting from one of the two end vertices of~$P_n$.
It iteratively constructs a solution~$C$ for the subpath on the first~$i$ edges $e_1,e_2,\ldots,e_i$ for $i=1,2,\ldots,n-1$, by initializing $C:=\emptyset$, and by adding the edge~$e_i$ to~$C$ if and only if the condition $|C|+1\leq (1-1/\alpha)i+\beta$ is satisfied (so after adding~$e_i$ to~$C$, \eqref{eq:path-cond} is still satisfied).
\begin{thm}
\label{thm:path}
Let~$P_n$ be a path with unit length edges~$\ell=1$ and consider the tolerance function $\varphi(x)=x/\alpha-\beta$, $\alpha,\beta\geq 1$.
The set of edges computed by the algorithm $\Greedy(P_n,\alpha,\beta)$ is an optimal solution for the instance $(P_n,\ell,\varphi)$ of the problem \Contraction{}, and it is computed in time $\cO(n)$.
\end{thm}
\begin{proof}
Let $C\subseteq E(P_n)$ be the set of edges computed by the algorithm $\Greedy(P_n,\alpha,\beta)$.
Clearly, we have $|C|=\lfloor(1-1/\alpha)|E(P_n)|+\beta\rfloor$, and this is optimal according to \eqref{eq:path-cond}.
However, it remains to show that~$C$ is feasible.
For $1\leq i\leq j\leq n-1=|E(P_n)|$ we let~$P_{i,j}$ denote the subpath of~$P_n$ formed by the edges $e_i,e_{i+1},\ldots,e_j$.
By the definition of our algorithm we know that $|E(P_{1,i})\cap C|=\lfloor (1-1/\alpha)i+\beta\rfloor$, from which we obtain that
\begin{align*}
|E(P_{i,j})\cap C| &= |E(P_{1,j})\cap C| - |E(P_{1,i-1})\cap C| \\
&= \lfloor (1-1/\alpha)j+\beta \rfloor - \lfloor (1-1/\alpha)(i-1)+\beta \rfloor \\
&\leq (1-1/\alpha)j+\beta - ((1-1/\alpha)(i-1)+\beta - 1) \\
&= (1-1/\alpha)(j-i+1)+1 \\
&\leq (1-1/\alpha)|E(P_{i,j})|+\beta ,
\end{align*}
where we used the assumption $\beta\geq 1$ in the last step.
Using \eqref{eq:path-cond} it thus follows that~$C$ is feasible.
\end{proof}
\subsection{Cycles with unit length edges}
\label{sec:cycles}
In this section we consider the special case of contracting a cycle~$C_n$ with~$n$ vertices and unit length edges~$\ell=1$ and the tolerance function $\varphi(x)=x/\alpha-\beta$, $\alpha\geq 1$, $\beta\geq 0$.
For this case we present a greedy algorithm running in linear time.
The main purpose of this result is to clearly separate the polynomially solvable cases of \Contraction{} from the NP-hard cases, and the case of a cycle with unit length edges precisely forms this boundary on the polynomially solvable side.
Recall in this context that we can solve \Contraction{} in polynomial time on any tree (this will be proved in Section~\ref{sec:dp} below), and that \Contraction{} is NP-hard already on a cycle for $\alpha=1$ (with arbitrary edge lengths; we will show this in Section~\ref{sec:cycles-hard} below).
We first argue that on a cycle it is equivalent to maximize the number of contracted edges $|C|$ or to maximize our objective function $\Phi(C)$ defined in \eqref{eq:objective}.
This is because the set of pairs $(|C|,\Phi(C))$ for all feasible contractions $C$ in a cycle $G=C_n$ is given by $\{(1,1),(2,2),\ldots,(n-3,n-3),(n-2,n-1),(n-1,n),(n,n)\}$, so it forms a monotone function, implying that maximizing either one of the two quantities is equivalent.
Based on this argument, for the rest of this section we consider maximizing the number $|C|$ of contracted edges.
Observe that a solution $C\subseteq E(C_n)$ ($C_n$ is the cycle we want to contract, and~$C$ is the set of edges to be contracted) for the instance $(C_n,\ell,\varphi)$ of the problem \Contraction{} is feasible, if and only if every subpath $P\subseteq C_n$ of length $d:=|E(P)|\in\{1,2,\ldots,n-1\}$ satisfies the condition
\begin{equation}
\label{eq:cycle-cond}
|E(P)\cap C| \leq \lfloor d - \min\{d,n-d\}/\alpha + \beta \rfloor .
\end{equation}
Rounding down on the right-hand side of \eqref{eq:cycle-cond} is justified because $|E(P)\cap C|$ is always an integer.
Defining
\begin{subequations}
\label{eq:uniform-solution}
\begin{align}
\lambda' &:= \min_{d\in\{1,2,\ldots,n-1\}} \frac{\lfloor d - \min\{d,n-d\}/\alpha + \beta \rfloor}{d} , \label{eq:uniform-sol1} \\
\lambda &:= \min\{1,\lambda'\} , \label{eq:uniform-sol2}
\end{align}
\end{subequations}
we obtain from \eqref{eq:cycle-cond} that $\lambda\in[0,1]$ is the maximal amount by which we can contract each edge in a uniform \emph{fractional} solution.
Inspired by the rounding technique from \cite{MR589671}, we turn this fractional solution into an integer optimal solution, yielding the following greedy algorithm $\Greedy(C_n,\alpha,\beta)$:
The algorithm considers the edges $e_1,e_2,\ldots,e_n$ of~$C_n$ as they are encountered when walking around the cycle.
It iteratively constructs a solution~$C$ by initializing $C:=\emptyset$ and by adding the edge~$e_i$ to~$C$ if and only if $\lfloor \lambda i\rfloor - \lfloor \lambda (i-1)\rfloor = 1$ for all $i=1,2,\ldots,n$ (since $\lambda\in[0,1]$, this difference is always either 0 or 1).
Note that we contract all edges of~$C_n$ if and only if~$\lambda=1$.
\begin{thm}
\label{thm:cycle}
Let~$C_n$ be a cycle with unit length edges~$\ell=1$ and consider the tolerance function $\varphi(x) = x/\alpha - \beta$, $\alpha \geq 1$, $\beta \geq 0$.
The set of edges computed by the algorithm $\Greedy(C_n,\alpha,\beta)$ is an optimal solution for the instance $(C_n,\ell,\varphi)$ of the problem \Contraction{}, and it is computed in time $\cO(n)$.
\end{thm}
The next lemma shows that the contraction computed by our algorithm has the maximum size.
\begin{lem}
\label{lem:lambda-prop}
For any feasible solution $C \subseteq E(C_n)$ we have $|C| \leq \lfloor \lambda n \rfloor$ with~$\lambda$ defined in \eqref{eq:uniform-solution}.
\end{lem}
\begin{proof}
If $\lambda=1$ this inequality is trivial.
So let us assume that $\lambda=\lambda'<1$ and that the minimum in \eqref{eq:uniform-sol1} is attained for some $d\in\{1,2,\ldots,n-1\}$.
Starting at some vertex~$u$ of the cycle, we walk along the cycle and cover it with~$n$ consecutive paths $P_1,P_2,\ldots,P_n$ of length~$d$ each ($P_{i+1}$ starts where~$P_i$ ends).
The sum of the lengths of the paths is~$n d$, so this process ends at the starting vertex~$u$, and each edge of the cycle and each edge of~$C$ is covered exactly~$d$ times.
We therefore obtain
\begin{equation*}
|C| = \frac{1}{d} \sum_{i=1}^n |E(P_i)\cap C|
\leBy{eq:cycle-cond} \sum_{i=1}^n \frac{\lfloor d - \min\{d,n-d\}/\alpha + \beta\rfloor}{d} \eqBy{eq:uniform-solution} \lambda n .
\end{equation*}
As~$|C|$ must be integral this inequality yields the desired bound $|C|\leq \lfloor\lambda n\rfloor$.
\end{proof}
With Lemma~\ref{lem:lambda-prop} in hand, we are now ready to prove Theorem~\ref{thm:cycle}.
\begin{proof}[Proof of Theorem~\ref{thm:cycle}]
In this proof we will use that for any two real numbers~$x$ and~$y$ we have
\begin{subequations}
\begin{align}
\lfloor x\rfloor + \lfloor y\rfloor &\leq \lfloor x+y\rfloor , \label{eq:floor-sum} \\
\lfloor x \rfloor - \lfloor y \rfloor &\leq \lceil x-y \rceil . \label{eq:floor-diff}
\end{align}
\end{subequations}
Let $C\subseteq E(C_n)$ be the set of edges computed by the algorithm $\Greedy(C_n,\alpha,\beta)$.
Clearly, we have $|C|=\sum_{i=1}^n (\lfloor \lambda i\rfloor - \lfloor \lambda (i-1)\rfloor) = \lfloor \lambda n \rfloor$, which is optimal by Lemma~\ref{lem:lambda-prop}.
However, it remains to show that~$C$ is feasible.
We consider a path~$P$ of length $d:=|E(P)|\in\{1,2,\ldots,n-1\}$ on the edges $e_k,e_{k+1},\ldots,e_{k+d-1}$ (indices are considered cyclically modulo~$n$, so $e_{n+i}=e_i$).
We distinguish two cases:
If $k+d-1\leq n$, we have
\begin{equation}
\label{eq:PcapC1}
|E(P)\cap C| = \sum_{i=k}^{k+d-1} (\lfloor \lambda i \rfloor - \lfloor \lambda(i-1)\rfloor)
= \lfloor \lambda (k+d-1) \rfloor - \lfloor \lambda(k-1) \rfloor \leBy{eq:floor-diff} \lceil \lambda d\rceil .
\end{equation}
If $k+d-1>n$, we obtain
\begin{align}
|E(P)\cap C| &= \sum_{i=k}^n (\lfloor \lambda i\rfloor - \lfloor \lambda(i-1) \rfloor)
+ \sum_{i=1}^{d-n+k-1} (\lfloor \lambda i\rfloor - \lfloor \lambda(i-1) \rfloor) \notag \\
&= \lfloor \lambda n \rfloor - \lfloor \lambda(k-1)\rfloor + \lfloor \lambda (d-n+k-1) \rfloor \notag \\
&\leBy{eq:floor-sum} \lfloor \lambda (d+k-1) \rfloor - \lfloor \lambda(k-1)\rfloor
\leBy{eq:floor-diff} \lceil \lambda d\rceil . \label{eq:PcapC2}
\end{align}
Applying \eqref{eq:uniform-solution} and using that $\lceil \lfloor x\rfloor\rceil=\lfloor x\rfloor$ shows that the right-hand sides of \eqref{eq:PcapC1} and \eqref{eq:PcapC2} can both be bounded from above by $\lfloor d - \min\{d,n-d\}/\alpha + \beta \rfloor$, proving that~$C$ is indeed feasible by \eqref{eq:cycle-cond}.
\end{proof}
\subsection{Trees with unit length edges and additive error}
\label{sec:trees-unit-length}
In this section we consider the special case of contracting a tree~$T$ with unit length edges $\ell=1$ and the tolerance function $\varphi(x)=x-\beta$ (purely additive error; we can assume w.l.o.g.\ that~$\beta$ is an integer).
Note that in this setting the objective function defined in \eqref{eq:objective} satisfies $\Phi(C)=|C|$ for any contraction $C$.
It turns out that in this case, optimal solutions have a very special structure that can be exploited to compute them in linear time.
Specifically, an optimal solution is obtained by taking all edges of~$T$ which have the property that only short paths start from one of its end vertices.
Formally, for the tree~$T$ and $d\in\NN_{\geq 0}$, we let $L(T,d)$ denote the set of all edges~$e$ of~$T$ which have one end vertex~$v$ such that all paths that start at~$v$ and do not contain~$e$ have length at most~$d-1$ (together with~$e$ these paths have length at most~$d$).
E.g., we have $L(T,0)=\emptyset$, and the set $L(T,1)$ are all the edges incident to a leaf (see Figure~\ref{fig:L}).
\begin{figure}
\caption{Illustration of the sets of edges $L(T,d)$, $d=1,2,3$.
The set $L(T,3)$ is an optimal solution of the problem \Contraction{}
\label{fig:L}
\end{figure}
Clearly, the set $L(T,d)$ can be computed in linear time by repeatedly removing all leaves of~$T$ in~$d$ rounds.
This is a variant of the well-known linear time algorithm to compute the so-called center of a tree (see \cite[Section~15.11]{DBLP:books/daglib/0022194}).
\begin{thm}
\label{thm:unit-tree}
Let~$T$ be a tree with unit length edges~$\ell=1$ and consider the tolerance function $\varphi(x)=x-\beta$, $\beta\in\NN_{\geq 0}$.
If~$\beta$ is even, the set of edges $L(T,d)$ with $d:=\lfloor\beta/2\rfloor$ is an optimal solution for the instance $(T,\ell,\varphi)$ of the problem \Contraction{}.
If~$\beta$ is odd, $L(T,d)\cup \{e\}$, $e\in E\setminus L(T,d)$, is an optimal solution.
These solutions can be computed in time $\cO(n)$.
\end{thm}
\begin{proof}
We define $C:=L(T,d)$ if~$\beta$ is even and $C:=L(T,d)\cup\{e\}$, for some $e\in E\setminus L(T,d)$, if~$\beta$ is odd.
We first argue that~$C$ is a feasible solution.
To see this note that for the given tolerance function we only need to verify that the path~$P$ between any two \emph{leaves} $u,v$ of~$T$ contains at most~$\beta$ edges.
Consider all the edges of~$P$ for which both end vertices have distance at least~$d$ from both~$u$ and~$v$.
None of those edges is in $L(T,d)$ by its definition.
It follows that $|P\cap L(T,d)|\leq 2d=2\lfloor\beta/2\rfloor$ and therefore $|P\cap C|\leq \beta$.
To prove that~$C$ is a solution of maximum size we argue by induction over~$\beta$.
The claim is trivially true for $\beta=0$ and $\beta=1$ (in these cases $|C|=0$ and $|C|=1$, respectively).
So let~$D$ be an arbitrary feasible solution of the instance $(T,\ell,\beta)$ of the problem \Contraction{} for some $\beta\geq 2$.
We need to show that $|C|\geq |D|$.
To this end we let $V^*\subseteq V(T)$ denote the set of leaves of~$T$ and we define $E^*:=L(T,1)$.
Moreover, we define $T^*:=T\setminus V^*$ and $C^*:=C\setminus E^*$.
By induction, $C^*=L(T^*,d-1)$ is an optimal solution for the instance $(T^*,\ell,\beta-2)$.
We first consider the case that $E^*\setminus D=\emptyset$ or $D\setminus E^*=\emptyset$ (this is equivalent to $E^*\subseteq D$ or $D\subseteq E^*$).
In this case we define $D^*:=D\setminus E^*$, and observe that~$D^*$ is a feasible solution for the instance $(T^*,\ell,\beta-2)$.
It follows that $|C^*|\geq |D^*|$, implying that $|C|=|C^*|+|E^*|\geq |D^*|+|E^*|\geq |D|$, as claimed.
\begin{figure}
\caption{Illustration of the proof of Theorem~\ref{thm:unit-tree}
\label{fig:exchange}
\end{figure}
It remains to consider the case that both sets $E^*\setminus D$ and $D\setminus E^*$ are nonempty, so there is an edge $e'\in E^*\setminus D$ and an edge $f\in D\setminus E^*$.
We denote the leaf incident to~$e'$ by~$v$.
We will now remove an edge $e\in D\setminus E^*$ from~$D$ and add~$e'$ instead to obtain another feasible solution~$D'$ satisfying $|D|=|D'|$.
Repeating this exchange argument and applying the reasoning from the first case then proves the lemma.
The edge $e\in D\setminus E^*$ to be removed from~$D$ is obtained by considering the path that connects~$v$ and~$f$ in~$T$ and that contains~$f$, and by choosing the first edge from~$D$ (or equivalently, from $D\setminus E^*$) that is encountered when following this path from~$v$ to~$f$.
It may happen that $e=f$ is the first such edge we encounter.
To complete the proof of the lemma it remains to show that $D'=D\setminus\{e\}\cup\{e'\}$ is feasible.
To prove this we only need to check paths which start in~$v$ and contain~$e'$ but not~$e$.
Let~$P'$ be such a path, let~$Q$ be any path that also starts in~$v$ but does contain~$e$, and consider the path $P:=(P'\setminus Q)\cup (Q\setminus P')$ (see Figure~\ref{fig:exchange}).
Here and in the following we slightly abuse notation and interpret these set unions/differences/intersections in terms of the edge sets of the graphs.
As~$D$ is feasible and as $P\cap Q$ contains~$e$, the number of edges in~$D$ or~$D'$ on $P'\setminus Q=P\setminus Q$ is at most $\beta-1$.
By the choice of~$e$, the number of edges of~$D'$ on $P'\cap Q$ is 1 (the only edge of~$D'$ on this path is~$e'$).
As $P'=(P'\setminus Q)\cup (P'\cap Q)$, we obtain that the number of edges from~$D'$ on~$P'$ is at most $\beta-1+1=\beta$, as desired.
This completes the proof.
\end{proof}
\section{Dynamic programs for general trees}
\label{sec:trees}
In this section we describe dynamic programming algorithms for the problems \Contraction{} and \WContraction{} on trees with general edge lengths and affine tolerance functions.
Recall that on trees our objective function satisfies $\Phi(C)=|C|$ for any contraction~$C$.
\subsection{\Contraction{} on trees}
\label{sec:dp}
In this section we describe a dynamic programming algorithm for the problem of computing an optimal contraction of a tree~$T$ with arbitrary edge lengths $\ell\colon E\to \RR_{>0}$ and an affine tolerance function $\varphi(x)=x/\alpha-\beta$, $\alpha\geq 1$, $\beta\geq 0$, generalizing the solution for the special case presented at the beginning of the previous section.
The goal is to prove the following result.
\begin{thm}
\label{thm:dp}
Let~$T$ be a tree with edge lengths $\ell\colon E\to\RR_{>0}$ and consider the tolerance function $\varphi(x)=x/\alpha-\beta$, $\alpha\geq 1$, $\beta\geq 0$.
An optimal solution for the instance $(T,\ell,\varphi)$ of the problem \Contraction{} can be computed by dynamic programming in time $\cO(n^3)$.
\end{thm}
Observe that a solution $C\subseteq E$ is feasible if and only if for any two vertices~$u$ and~$v$ of~$T$ we have $\load_{C,\alpha}(u,v)\leq \beta$, where the \emph{load between~$u$ and~$v$} is defined as
\begin{subequations}
\label{eq:load}
\begin{equation}
\label{eq:load-path}
\load_{C,\alpha}(u,v):=\dist_\ell(u,v)/\alpha-\dist_{\ell_C}(u,v)
\end{equation}
(recall \eqref{eq:contr-cond}).
For any vertex~$v$ of~$T$ we also define the \emph{load of~$T$ at~$v$} as
\begin{equation}
\label{eq:load-tree}
\load_{C,\alpha}(T,v):=\max\{\load_{C,\alpha}(u,v): u \in V(T) \} .
\end{equation}
\end{subequations}
Note that $\load_{C,\alpha}(T,v)\geq 0$, as we have $\load_{C,\alpha}(v,v)=0$.
The next lemma states a criterion when feasible solutions of subtrees can be combined to a feasible solution of the entire tree.
The definitions \eqref{eq:load-path}, \eqref{eq:load-tree} and the lemma are illustrated in Figure~\ref{fig:load}.
\begin{lem}
\label{lem:tree-partition}
Consider a partition of~$T$ into two subtrees~$T_1$ and~$T_2$ that only have a vertex~$v\in V$ in common.
Then $C\subseteq E$ is a feasible solution for the instance $(T,\ell,\varphi)$ of the problem \Contraction{} if and only if the following two conditions hold: $C\cap E(T_1)$ and $C\cap E(T_2)$ are feasible solutions for the instances $(T_1,\ell,\varphi)$ and $(T_2,\ell,\varphi)$ respectively; and we have $\load_{C,\alpha}(T_1,v)+\load_{C,\alpha}(T_2,v)\leq \beta$.
\end{lem}
\begin{proof}
Observe that the path between two vertices $u\in T_1$ and $w\in T_2$ contains the vertex~$v$, so we obtain $\load_{C,\alpha}(u,w)=\load_{C,\alpha}(u,v)+\load_{C,\alpha}(v,w)$ from \eqref{eq:load-path}.
Using \eqref{eq:load-tree} it follows that the condition $\load_{C,\alpha}(u,w)\leq \beta$ holding for all such pairs of vertices $u,w$ is equivalent to $\load_{C,\alpha}(T_1,v)+\load_{C,\alpha}(T_2,v)\leq \beta$.
\end{proof}
We will use this lemma to formulate our dynamic programming algorithm.
The idea is to compute optimal solution for subtrees and combining them to an optimal solution for the entire tree.
To describe the algorithm we introduce a few definitions.
An \emph{ordered rooted tree} is a rooted tree with a specified left-to-right ordering for the children of each vertex.
Given the tree~$T$, we can pick an arbitrary vertex as the root, and for each descendant of the root an arbitrary left-to-right ordering of its children, yielding an ordered rooted tree (different roots and orderings yield different ordered rooted trees, but any one of them is good for our purposes).
We slightly abuse notation in the following and use~$T$ to denote this ordered rooted tree.
All trees considered in the rest of this section are ordered and rooted.
For any vertex~$v$ of~$T$, we let~$T_v$ denote the subtree of~$T$ rooted at~$v$, and we use~$c(v)$ to denote the number of children of~$v$.
If $u_1,u_2,\ldots,u_{c(v)}$ are the children of~$v$ (in the specified ordering), we write $T_{v,i}$, $i\in\{1,\ldots, c(v)\}$, for the subtree of~$T$ that contains~$v$,~$u_i$ and all the descendants of~$u_i$.
We also define $T_{v,0}:=\{v\}$.
Furthermore, we define $T_{v,i}^+:=\bigcup_{0\leq j\leq i} T_{v,i}$, so we have $T_v=T_{v,c(v)}^+$.
These definitions are illustrated in Figure~\ref{fig:load}.
\begin{figure}
\caption{Illustration of the definitions of Section~\ref{sec:dp}
\label{fig:load}
\end{figure}
Using these definitions it follows straightforwardly from \eqref{eq:load-path} and \eqref{eq:load-tree} that for any set of edges $C\subseteq E(T_{u_i})$ we have
\begin{subequations}
\label{eq:load-add-edge}
\begin{align}
\load_{C\cup\{\{v,u_i\}\},\alpha}(T_{v,i},v) &= \load_{C,\alpha}(T_{u_i},u_i)+\ell(v,u_i)/\alpha , \label{eq:load-add-edge-yes} \\
\load_{C,\alpha}(T_{v,i},v) &= \max\{ \load_{C,\alpha}(T_{u_i},u_i)-(1-1/\alpha)\ell(v,u_i), 0 \} . \label{eq:load-add-edge-no}
\end{align}
\end{subequations}
Note that the load increases if the edge $\{v,u_i\}$ is added to $C$ (see \eqref{eq:load-add-edge-yes}), and it decreases otherwise (see \eqref{eq:load-add-edge-no}).
Moreover, for any set of edges $C\subseteq T_{v,i}^+$ and any $i=1,2,\ldots,c(v)$ we obtain from those definitions that
\begin{equation}
\label{eq:load-join-trees}
\load_{C,\alpha}(T_{v,i}^+,v) = \max \{\load_{C,\alpha}(T_{v,i-1}^+,v), \load_{C,\alpha}(T_{v,i},v) \} .
\end{equation}
These rules allow us to compute the load of all subtrees of~$T$ in a bottom-up fashion.
Our dynamic program maintains the minimum load of all subtrees of~$T$ in three-dimensional matrices~$L$ and~$L^+$.
We begin defining these matrices in an abstract way, and then establish several recursive relations which directly translate into a dynamic program.
Specifically, for $v\in V$, $i\in\{0,1,\ldots,c(v)\}$ and $s\in\{0,1,\ldots,m\}$ (recall that $m=|E|$) we define
\begin{equation}
\label{eq:def-L}
L(v,i,s) := \min \{\load_{C,\alpha}(T_{v,i},v): \text{$C$ is a feasible solution of $(T_{v,i},\ell,\varphi)$ of size $|C|=s$}\} .
\end{equation}
If there is no feasible solution of the required size, we have $L(v,i,s)=\infty$.
The entries of $L^+(v,i,s)$ are defined analogously to \eqref{eq:def-L} by considering the load of~$T_{v,i}^+$ instead of~$T_{v,i}$.
In words, the entries~$L(v,i,s)$ and~$L^+(v,i,s)$ describe feasible solutions~$C$ of size~$s$ of the instances $(T_{v,i},\ell,\varphi)$ or $(T_{v,i}^+,\ell,\varphi)$, respectively, of the problem \Contraction{} for which the load at the vertex~$v$ is as small as possible (the matrices contain the minimum achievable load, not the corresponding set of edges).
\begin{lem}
\label{lem:L-Lp-rec}
Let~$v$ be a vertex of~$T$ and let $u_1,u_2,\ldots,u_{c(v)}$ be the children of~$v$.
Then the matrices~$L$ and~$L^+$ defined in and directly after \eqref{eq:def-L} satisfy the relations
\begin{subequations}
\begin{align}
L(v,i,0) &= L^+(v,i,0) = 0 \quad \text{for all } i\in\{0,1,\ldots,c(v)\} , \label{eq:dp-init-L1} \\
L(v,0,s) &= L^+(v,0,s) = \infty \quad \text{for all } s\in\{1,2,\ldots,m\} , \label{eq:dp-init-L2} \\
L(v,i,s) &= \begin{cases}
\mu & \text{if } \mu \leq \beta , \\
\infty & \text{otherwise},
\end{cases} \label{eq:dp-update-L}
\end{align}
where $\mu:=\min \big\{ L^+(u_i,c(u_i),s-1)+\ell(v,u_i)/\alpha, \, \max\{L^+(u_i,c(u_i),s)-(1-1/\alpha)\ell(v,u_i), 0\}\big\}$ for all $i\in\{1,2,\ldots,c(v)\}$ and $s\in\{1,2,\ldots,m\}$.
Moreover, we have
\begin{align}
L^+(v,i,s) &= \min \big\{\max \{L^+(v,i-1,t),L(v,i,s-t)\} : t\in\{0,1,\ldots,s\} \text{ and } \notag \\
& \hspace{3cm} L^+(v,i-1,t)+L(v,i,s-t)\leq \beta \big\} , \label{eq:dp-update-Lp1}
\end{align}
\end{subequations}
for all $i\in\{1,2,\ldots,c(v)\}$ and $s\in\{1,2,\ldots,m\}$.
\end{lem}
The most interesting of these recursive relations are of course \eqref{eq:dp-update-L} and \eqref{eq:dp-update-Lp1}.
The relation \eqref{eq:dp-update-L} captures the two possibilities of either adding the edge $\{v,u_i\}$ or not adding it to a partial solution in the tree~$T_{u_i,c(u_i)}^+=T_{u_i}$ to obtain a solution for the tree~$T_{v,i}$ (recall \eqref{eq:load-add-edge}).
The relation \eqref{eq:dp-update-Lp1}, on the other hand, describes how to distribute~$s$ contraction edges in~$T_{v,i}^+$ among the two subtrees~$T_{v,i-1}^+$ and~$T_{v,i}$ ($t$ is the number of edges contracted in the first tree, and~$s-t$ the number of edges in the second tree, respectively).
\begin{proof}
The relations \eqref{eq:dp-init-L1} and \eqref{eq:dp-init-L2} follow immediately from the definitions of the trees $T_{v,i}$ and~$T_{v,i}^+$ and from \eqref{eq:def-L}.
The relation \eqref{eq:dp-update-L} follows from \eqref{eq:load-add-edge} and \eqref{eq:def-L}.
The relation \eqref{eq:dp-update-Lp1} follows from \eqref{eq:load-join-trees} and \eqref{eq:def-L} with the help of Lemma~\ref{lem:tree-partition}.
\end{proof}
We are now ready to prove Theorem~\ref{thm:dp}.
\begin{proof}[Proof of Theorem~\ref{thm:dp}]
Given the instance $(T,\ell,\varphi)$, we fix an arbitrary root~$r$ of~$T$ and an arbitrary ordering of the children of each vertex, making~$T$ an ordered rooted tree.
We then compute the entries of the matrices~$L$ and~$L^+$ using Lemma~\ref{lem:L-Lp-rec}.
We first initialize various entries using \eqref{eq:dp-init-L1} and \eqref{eq:dp-init-L2}, and compute the remaining entries in a bottom-up fashion moving upwards from the leaves to the root.
Specifically, at a vertex~$v$ with children $u_1,u_2,\ldots,u_{c(v)}$ for which all the entries of~$L$ and~$L^+$ have already been computed, we first compute $L(v,i,s)$ for all $i\in\{1,2,\ldots,c(v)\}$ and $s\in\{1,2,\ldots,m\}$ using \eqref{eq:dp-update-L}, and then $L^+(v,i,s)$ for all $i\in\{1,2,\ldots,c(v)\}$ and $s\in\{1,2,\ldots,m\}$ using \eqref{eq:dp-update-Lp1}.
Let~$s^*$ be the largest~$s$ such that $L^+(r,c(r),s)\leq \beta$.
From \eqref{eq:def-L} we obtain that~$s^*$ is the size of an optimal solution of the instance $(T,\ell,\varphi)$.
The corresponding set of edges $C\subseteq E$ can be obtained by keeping track of the arguments for which the minima and maxima in \eqref{eq:dp-update-L} and \eqref{eq:dp-update-Lp1} are attained in each step.
Clearly,~$L$ and~$L^+$ both have $\cO(n^2)$ entries, and computing each entry takes time $\cO(n)$, so the running time of our dynamic program is $\cO(n^3)$.
\end{proof}
\subsection{\WContraction{} on trees}
\label{sec:weak-dp}
In this section we consider the problem of computing weak contractions for a tree~$T$ with affine tolerance function $\varphi(x)=x/\alpha-\beta$.
Here, our main result is a dynamic programming algorithm that builds on the algorithmic ideas presented in Section~\ref{sec:dp}.
\begin{thm}
\label{thm:weak-dp}
Let~$T$ be a tree with edge lengths $\ell\colon E\to\RR_{>0}$ and consider the tolerance function $\varphi(x)=x/\alpha-\beta$, $\alpha\geq 1$, $\beta\geq 0$.
An optimal solution for the instance $(T,\ell,\varphi)$ of the problem \WContraction{} can be computed by dynamic programming in time $\cO(n^5)$.
\end{thm}
In this setting we need to specifically keep track of pairs of vertices whose distance remains positive when contracting a set of edges $C\subseteq E$ (i.e., not all edges in between these vertices are contracted).
To this end we extend the definitions \eqref{eq:load} as follows:
For any vertex~$v$ of~$T$ we define the \emph{weak load of~$T$ at~$v$} as
\begin{equation}
\label{eq:wload}
\wload_{C,\alpha}(T,v) := \max \{\load_{C,\alpha}(u,v): u\in V(T) \text{ and } \dist_{\ell_C}(u,v)>0\} .
\end{equation}
Note that in the maximization we have to consider all vertices~$u$ such that at least one edge on the path from~$u$ to~$v$ is not in~$C$.
This definition together with \eqref{eq:load-tree} yields $\wload_{C,\alpha}(T,v)\leq \load_{C,\alpha}(T,v)$.
In contrast to the load, the weak load may be negative.
In particular, $\wload_{C,\alpha}(T,v)=-\infty$ if and only if $C=E$.
The following lemma is the counterpart to Lemma~\ref{lem:tree-partition} for weak contractions.
It describes how to combine feasible solutions on subtrees to a feasible solution of the entire tree.
There is one important subtlety here:
While the notion of a weak contraction forbids contracting all edges of~$T$, we clearly have to allow this for partial solutions on subtrees of~$T$ (as long as some other edge not in the subtree is is not contracted, this might still yield a feasible solution).
\begin{lem}
\label{lem:weak-tree-partition}
Consider a partition of~$T$ into two subtrees~$T_1$ and~$T_2$ that only have a vertex~$v\in V$ in common.
Then $C\subsetneq E$ is a feasible solution for the instance $(T,\ell,\varphi)$ of the problem \WContraction{} if and only if the following two conditions hold: For~$i=1,2$, either~$C$ contains every edge of~$T_i$ or $C\cap E(T_i)$ is a feasible solution for the instance $(T_i,\ell,\varphi)$ of \WContraction{}; and we have
\begin{equation}
\label{eq:weak-tree-cond}
\load_{C,\alpha}(T_1,v)+\wload_{C,\alpha}(T_2,v)\leq \beta \quad \text{and} \quad \wload_{C,\alpha}(T_1,v)+\load_{C,\alpha}(T_2,v)\leq \beta .
\end{equation}
\end{lem}
\begin{proof}
Let $C\subsetneq E$.
For the rest of the proof we omit the subscripts~$C$ and~$\alpha$ and simply write $\load_{C,\alpha}=\load$ and $\wload_{C,\alpha}=\wload$.
We first assume that~$C$ is a feasible solution for the instance $(T,\ell,\varphi)$ of the problem \WContraction{}.
I.e., any two vertices $u,w$ of~$T$ with $\dist_{\ell_C}(u,w)>0$ satisfy the condition $\load(u,w)\leq \beta$.
This is true in particular for all pairs of vertices $u,w\in T_i$, $i=1,2$, implying that either $C\supseteq T_i$ or $C\cap T_i\subsetneq T_i$ is a feasible solution for the instance $(T_i,\ell,\varphi)$.
If $\wload(T_2,v)=-\infty$, the claimed inequality $\load(T_1,v)+\wload(T_2,v)\leq \beta$ is trivially satisfied.
So suppose that $\wload(T_2,v)$ is a finite number, and let $u\in T_1$ and $w\in T_2$ be such that $\load(u,v)=\load(T_1,v)$, and $\dist_{\ell_C}(v,w)>0$ as well as $\load(v,w)=\wload(T_2,v)$.
Then we also have $\dist_{\ell_C}(u,w)>0$, so we know that $\load(u,w)\leq \beta$ by the assumption that~$C$ is feasible for $(T,\ell,\varphi)$.
Combining this last inequality with the relation $\load(u,w)=\load(u,v)+\load(v,w)=\load(T_1,v)+\wload(T_2,v)$ proves that the right hand side of the equation is at most~$\beta$, as claimed.
The proof of the second inequality $\wload(T_1,v)+\load(T_2,v)\leq \beta$ works symmetrically.
This proves one direction of the equivalence.
To prove the reverse direction, we now assume that either $C\supseteq T_i$ or $C\cap T_i\subsetneq T_i$ is a feasible solution for the instance $(T_i,\ell,\varphi)$ for $i=1,2$, and that $\load(T_1,v)+\wload(T_2,v)\leq \beta$ and $\wload(T_1,v)+\load(T_2,v)\leq \beta$.
To show that~$C$ is a feasible solution for the instance $(T,\ell,\varphi)$, let $u\in T_1$ and $w\in T_2$ be such that $\dist_{\ell_C}(u,w)>0$.
It follows that $\dist_{\ell_C}(u,v)>0$ or $\dist_{\ell_C}(v,w)>0$.
We first consider the case that $\dist_{\ell_C}(u,v)>0$.
By the definitions \eqref{eq:load} and \eqref{eq:wload} we have $\load(u,v)\leq \wload(T_1,v)$, and also $\load(v,w)\leq \load(T_2,v)$, yielding $\load(u,w)=\load(u,v)+\load(v,w)\leq \wload(T_1,v)+\load(T_2,v)\leq \beta$ (the last inequality holds by assumption).
This proves that $\load(u,w)\leq \beta$, as desired.
The proof of the other case $\dist_{\ell_C}(v,w)>0$ works symmetrically.
This completes the proof of the lemma.
\end{proof}
As in Section~\ref{sec:dp}, we view~$T$ as an ordered rooted tree, and consider its subtrees $T_v$, $T_{v,i}$ and~$T_{v,i}^+$ for all $v\in V$ and $i\in\{0,1,\ldots,c(v)\}$ (recall the definitions given after Lemma~\ref{lem:tree-partition}).
Let us briefly highlight the differences between Lemmas~\ref{lem:tree-partition} and \ref{lem:weak-tree-partition}.
The dynamic programming algorithm presented in Section~\ref{sec:dp} exploits the fact that the optimal way to contract exactly $|C|=s$ edges in a subtree~$T_v$ of~$T$ rooted at a particular vertex~$v$ is to contract a set of edges that minimizes $\load_{C,\alpha}(T_v,v)$.
This is possible as the optimality condition in Lemma~\ref{lem:tree-partition} only depends on this parameter.
Here the situation is more complicated, as Lemma~\ref{lem:weak-tree-partition} also considers $\wload_{C,\alpha}(T_v,v)$.
Figure~\ref{fig:wload} illustrates that it is not sufficient to minimize only one of these parameters.
\begin{figure}
\caption{Example of the behavior of the parameters~$\load()$ and~$\wload()$ for $(\alpha,\beta)=(2,1/2)$.
Consider the tree~$T$ in (c) and (d) with length functions that differ only in the value they assign to the topmost edge.
Parts (a) and (b) of the figure show the subtree~$T_v$ of~$T$ and two different subsets of edges~$C$ and~$C'$ of~$T_v$, respectively, drawn by dashed edges.
We have $\load_{C,\alpha}
\label{fig:wload}
\end{figure}
Consequently, we keep track of an entire Pareto front of non-dominated partial solutions (see Figure~\ref{fig:pareto}).
Formally, we define the \emph{set $F(T_v,s)$ of feasible partial solutions of size~$s$} as the family of all sets $C\subseteq E(T_v)$ with $|C|=s$ such that either $C=E(T_v)$ or~$C$ is a feasible solution for the instance $(T_v,\ell,\varphi)$ of \WContraction{}.
For two sets $C,C'\in F(T_v,s)$ we say that \emph{$C$ dominates~$C'$ at~$v$} if $\load_{C,\alpha}(T_v,v) \leq \load_{C',\alpha}(T_v,v)$ and $\wload_{C,\alpha}(T_v,v) \leq \wload_{C',\alpha}(T_v,v)$, and we define the \emph{Pareto front} $P(T_v,v,s)$ as a minimal family of sets $C\in F(T_v,s)$ such that no set $C'\in F(T_v,s)$ dominates~$C$ at~$v$.
Note that the domination relation is reflexive, so there may be several different such minimal families, all with the same pairs of load and weak load values, and any choice among them is equally good for us.
This definition is illustrated in Figure~\ref{fig:pareto}.
The following crucial lemma asserts that the number of points on the Pareto front, i.e., the size of the family $P(T_v,v,s)$ is at most~$n+1$.
This property is essential for our dynamic programming approach, and it does not follow immediately from the definition of $P(T_v,v,s)$, as the set of feasible solutions $F(T_v,s)$ is typically of exponential size.
\begin{lem}
\label{lem:pareto}
For any $C\subseteq E(T_v)$, we have $\load_{C,\alpha}(T_v,v) \in \Lambda(T_v,v):=\{\dist_\ell(u,v)/\alpha : u\in V(T_v)\}$ or $\load_{C,\alpha}(T_v,v)=\wload_{C,\alpha}(T_v,v)$.
Consequently, the Pareto front $P(T_v,v,s)$ has size at most~$n+1$.
\end{lem}
\begin{proof}
By the definitions \eqref{eq:load} and \eqref{eq:wload} we have $\wload_{C,\alpha}(T_v,v)\leq \load_{C,\alpha}(T_v,v)$ for all $C\subseteq E(T_v)$.
Now let $C\subseteq E(T_v)$ be such that $\wload_{C,\alpha}(T_v,v)<\load_{C,\alpha}(T_v,v)$.
Again by the previously mentioned definitions this implies that $\load_{C,\alpha}(T_v,v)=\load_{C,\alpha}(u,v)=\dist_\ell(u,v)/\alpha$ for some $u\in V(T_v)$, which is indeed an element of the set $\Lambda(T_v,v)$.
Consequently, the Pareto front $P(T_v,v,s)$ consists of at most one set $C\in F(T_v,s)$ with $\wload_{C,\alpha}(T_v,v)=\load_{C,\alpha}(T_v,v)$ and at most one set $C\in F(T_v,s)$ with $\wload_{C,\alpha}(T_v,v)<\load_{C,\alpha}(T_v,v)$ for each number in $\Lambda(T_v,v)$.
Using that $|\Lambda(T_v,v)|\leq |V(T_v)|\leq n$ it follows that $|P(T_v,v,s)|\leq n+1$.
\end{proof}
\begin{figure}
\caption{The right hand side of the figure shows the set of all pairs of $(\load,\wload)$ values for all feasible solutions of size $s=3$ for the rooted tree $(T_v,v)$ shown on the left for the parameter values $(\alpha,\beta)=(2,13/2)$.
The points in $P(T_v,v,s)$ are highlighted with small circles.
The vertical dashed lines represent all values in the set $\Lambda(T_v,v)=\{0,1/2,3/2,2,3,7/2,4\}
\label{fig:pareto}
\end{figure}
By Lemma~\ref{lem:pareto} the load values of all points on the Pareto front with $\wload()<\load()$ are in the set $\Lambda(T_v,v)$.
There might also be one point with $\wload()=\load()$ on the Pareto front (as in the example shown in Figure~\ref{fig:pareto}), and this load value might not be an element of $\Lambda(T_v,v)$.
We extend the set $\Lambda(T_v,v)$ accordingly by defining for $s\in\{0,1,\ldots,m\}$ (recall that $m=|E|$)
\begin{subequations}
\label{eq:lambda-Lambda*}
\begin{align}
\lambda^*(T_v,v,s) &:= \min \{\load_{C,\alpha}(T_v,v) : C \in F(T_v,s)\} , \label{eq:lambda*} \\
\Lambda^*(T_v,v,s) &:= \Lambda(T_v,v) \cup \{\lambda^*(T_v,v,s)\} . \label{eq:Lambda*}
\end{align}
\end{subequations}
If the set $F(T_v,s)$ is empty, we have $\lambda^*(T_v,v,s)=\infty$.
We now describe recursive relations for the weak load that are analogous to \eqref{eq:load-add-edge} and \eqref{eq:load-join-trees} for the load.
It follows straightforwardly from \eqref{eq:load} and \eqref{eq:wload} that for any vertex~$v$ of~$T$ and its children $u_i$, $i=1,2,\ldots,c(v)$, and for any set of edges $C\subseteq E(T_{u_i})$ we have
\begin{subequations}
\label{eq:wload-add-edge}
\begin{align}
\wload_{C\cup\{\{v,u_i\}\},\alpha}(T_{v,i},v) &= \wload_{C,\alpha}(T_{u_i},u_i)+\ell(v,u_i)/\alpha , \label{eq:wload-add-edge-yes} \\
\wload_{C,\alpha}(T_{v,i},v) &= \load_{C,\alpha}(T_{u_i},u_i) - (1-1/\alpha)\ell(v,u_i) . \label{eq:wload-add-edge-no}
\end{align}
\end{subequations}
Note that the weak load increases if the edge $\{v,u_i\}$ is added (see \eqref{eq:wload-add-edge-yes}).
On the other hand, if the edge $\{v,u_i\}$ is not added, it may decrease or increase (the right hand side of \eqref{eq:wload-add-edge-no} refers to the load, \emph{not} to the weak load).
Moreover, for any set of edges $C\subseteq T_{v,i}^+$ and any $i=1,2,\ldots,c(v)$ the definition \eqref{eq:wload} readily implies
\begin{equation}
\label{eq:wload-join-trees}
\wload_{C,\alpha}(T_{v,i}^+,v) = \max \{\wload_{C,\alpha}(T_{v,i-1}^+,v), \wload_{C,\alpha}(T_{v,i},v) \} .
\end{equation}
These rules together with the corresponding relations \eqref{eq:load-add-edge} and \eqref{eq:load-join-trees} allow us to compute the weak load and the load of all Pareto optimal partial solutions in a bottom-up fashion, similar to the approach taken in Section~\ref{sec:dp}.
Before it was sufficient to compute one optimal partial solution for every subtree~$T_{v,i}$ and~$T_{v,i}^+$, $i\in\{1,2,\ldots,c(v)\}$, and every possible size~$s$ of the contracted set of edges, but now our dynamic program keeps track of the entire Pareto fronts $P(T_{v,i},v,s)$ and $P(T_{v,i}^+,v,s)$.
We store the corresponding pairs of load and weak load values on the Pareto front in separate four-dimensional matrices $W$, $W^+$, $L$ and~$L^+$ (the entries of~$W$ and~$W^+$ are certain weak load values, and the entries of~$L$ and~$L^+$ are the corresponding load values).
We begin defining these matrices in an abstract way, and then establish several recursive relations which directly translate into a dynamic programming algorithm.
Specifically, for $v\in V$, $i\in\{0,1,\ldots,c(v)\}$, $s\in\{0,1,\ldots,m\}$ and $\lambda\in \Lambda(T_{v,i},v)$ with $\Lambda(T_{v,i},v)$ as in Lemma~\ref{lem:pareto} we define
\begin{subequations}
\label{eq:weak-def-LW}
\begin{align}
W(v,i,s,\lambda) &:= \min \{\wload_{C,\alpha}(T_{v,i},v): C \in F(T_{v,i},s) \text{ with } \load_{C,\alpha}(T_{v,i},v) \leq \lambda\} , \label{eq:weak-def-W} \\
L(v,i,s,\lambda) &:= \min \{\load_{C,\alpha}(T_{v,i},v) : C \in F(T_{v,i},s), \wload_{C,\alpha}(T_{v,i},v) = W(v,i,s,\lambda)\} . \label{eq:weak-def-L}
\end{align}
\end{subequations}
If there is no set~$C$ satisfying these requirements, we have $W(v,i,s,\lambda)=L(v,i,s,\lambda)=\infty$.
The entries of $W^+(v,i,s,\lambda)$ and $L^+(v,i,s,\lambda)$ are defined analogously to \eqref{eq:weak-def-LW} by considering the tree~$T_{v,i}^+$ instead of~$T_{v,i}$ (in particular, in this case we have $\lambda\in\Lambda(T_{v,i}^+,v)$).
The definitions of $W(v,i,s,\lambda)$ and $L(v,i,s,\lambda)$ given in \eqref{eq:weak-def-LW} extend straightforwardly to the value $\lambda=\lambda^*(T_{v,i},v,s)$ defined in \eqref{eq:lambda*}.
Similarly, the definitions of $W^+(v,i,s,\lambda)$ and $L^+(v,i,s,\lambda)$ from before extend to the value $\lambda=\lambda^*(T_{v,i}^+,v,s)$.
It is easy to see that we have in fact
\begin{equation}
\label{eq:L-lambda*}
L(v,i,s,\lambda^*(T_{v,i},v,s)) = \lambda^*(T_{v,i},v,s)
\end{equation}
(an analogous relation holds for the entries of~$L^+$).
The recursive relations satisfied by the matrices $W$, $L$, $W^+$ and~$L^+$ defined before are captured by the following two lemmas.
The initialization steps and the recursive computation of~$W$ and~$L$ are treated in Lemma~\ref{lem:W-L-rec1}.
The recursive computation of~$W^+$ and~$L^+$ is somewhat more technical, and is treated separately in Lemma~\ref{lem:W-L-rec2}.
\begin{lem}
\label{lem:W-L-rec1}
Let~$v$ be a vertex of~$T$ and let $u_1,u_2,\ldots,u_{c(v)}$ be the children of~$v$.
Then the matrices $W$, $W^+$, $L$ and~$L^+$ defined in and directly after \eqref{eq:weak-def-LW} satisfy the relations
\begin{subequations}
\begin{align}
W(v,0,0,0) &= W^+(v,0,0,0) = -\infty , \label{eq:weak-dp-init-W1} \\
W(v,i,0,\lambda) &= -(1-1/\alpha)\ell(v,u_i) \quad \text{for all } i\in\{1,2,\ldots,c(v)\} \text{ and } \lambda\in\Lambda^*(T_{v,i},v,0) , \label{eq:weak-dp-init-W2} \\
L(v,i,0,\lambda) &= 0 \quad \text{for all } i\in\{0,1,\ldots,c(v)\} \text{ and } \lambda\in\Lambda^*(T_{v,i},v,0) , \label{eq:weak-dp-init-L1} \\
L^+(v,i,0,\lambda) &= 0 \quad \text{for all } i\in\{0,1,\ldots,c(v)\} \text{ and } \lambda\in\Lambda^*(T_{v,i}^+,v,0) , \label{eq:weak-dp-init-L2} \\
W(v,0,s,\lambda) &= L(v,0,s,\lambda) = \infty \quad \text{for all } s\in\{1,2,\ldots,m\} \text{ and } \lambda\in\Lambda^*(T_{v,0},v,s) , \label{eq:weak-dp-init-WL1} \\
W^+(v,0,s,\lambda) &= L^+(v,0,s,\lambda) = \infty \quad \text{for all } s\in\{1,2,\ldots,m\} \text{ and } \lambda\in\Lambda^*(T_{v,0}^+,v,s) . \label{eq:weak-dp-init-WL2}
\end{align}
Furthermore, we have
\begin{align}
W(v,i,s,\lambda) &= \begin{cases}
\mu & \text{if } \lambda = 0 , \\
\min\{\mu,\nu\} & \text{if } \lambda > 0 \text{ and } \mu\leq \min\{\beta,\lambda\} , \\
\nu & \text{if } \lambda > 0 \text{ and } \mu > \min\{\beta,\lambda\} \text{ and } \nu \leq \beta , \\
\infty & \text{otherwise} ,
\end{cases} \label{eq:weak-dp-update-W1} \\
L(v,i,s,\lambda) &= \begin{cases}
\max\{\mu,0\} & \text{if } W(v,i,s,\lambda) = \mu , \\
L^+(u_i,c(u_i),s-1,\lambda-\ell(v,u_i)/\alpha)+\ell(v,u_i)/\alpha & \text{otherwise} ,
\end{cases} \label{eq:weak-dp-update-L1}
\end{align}
with $\mu:=\lambda^*(T_{u_i},u_i,s) - (1-1/\alpha)\ell(v,u_i)$ and $\nu:=W^+(u_i,c(u_i),s-1,\lambda-\ell(v,u_i)/\alpha)+\ell(v,u_i)/\alpha$ for all $i\in\{1,2,\ldots,c(v)\}$, $s\in\{1,2,\ldots,m\}$ and $\lambda\in \Lambda(T_{v,i},v)$.
Finally, we have
\begin{align}
L(v,i,s,\lambda^*) &= \lambda^* = \begin{cases}
\min\{\lambda+\ell(v,u_i)/\alpha,\max \{0, \mu\}\} & \text{if } \mu \leq \beta , \\
\lambda+\ell(v,u_i)/\alpha & \text{otherwise} ,
\end{cases} \label{eq:weak-dp-update-L2} \\
W(v,i,s,\lambda^*) &= \begin{cases}
\rho & \text{if } L(v,i,s,\lambda^*) = \lambda+\ell(v,u_i)/\alpha , \\
\mu & \text{otherwise} ,
\end{cases} \label{eq:weak-dp-update-W2}
\end{align}
\end{subequations}
where $\lambda \in \Lambda^*(T_{u_i},u_i,s-1)$ is minimal such that $\rho := W^+(u_i,c(u_i),s-1,\lambda)+\ell(v,u_i)/\alpha \leq \beta$, if such a value~$\lambda$ exists, and $\lambda := \rho := \infty$ otherwise, for all $i\in\{1,2,\ldots,c(v)\}$, $s\in\{1,2,\ldots,m\}$ and $\lambda^*=\lambda^*(T_{v,i},v,s)$.
\end{lem}
Note that the relations \eqref{eq:weak-dp-init-W1}--\eqref{eq:weak-dp-init-WL2} are the initialization steps, and the relations \eqref{eq:weak-dp-update-W1}--\eqref{eq:weak-dp-update-W2} capture the two possibilities of either adding or not adding the edge $\{v,u_i\}$ to a partial solution in the tree $T_{u_i,c(u_i)}^+=T_{u_i}$ to obtain a solution for the tree~$T_{v,i}$ (recall \eqref{eq:load-add-edge} and \eqref{eq:wload-add-edge}).
We only refer to well-defined entries of~$W^+$ and~$L^+$ in~\eqref{eq:weak-dp-update-L1} and in the definition of~$\nu$, as $\lambda-\ell(v,u_i)/\alpha \in \Lambda(T_{u_i}, u_i)$ holds for every $\lambda \in \Lambda(T_{v,i},v) \setminus \{0\}$.
Note that we either have $\nu \leq \lambda$ or $\nu = \infty$, while~$\mu$ may also take a value in the open interval $(\lambda,\infty)$.
\begin{proof}
The relations~\eqref{eq:weak-dp-init-W1}--\eqref{eq:weak-dp-init-WL2} follow immediately from the definitions of the trees~$T_{v,i}$ and~$T_{v,i}^+$ and the definitions of the respective matrices given in~\eqref{eq:weak-def-LW} and afterwards.
The relations~\eqref{eq:weak-dp-update-W1} and~\eqref{eq:weak-dp-update-L2} follow from~\eqref{eq:wload-add-edge} and the definitions of~$W$ and~$L$, respectively:
Consider a partial solution $C\in F(T_{v_i},s)$.
If $\load_{C,\alpha}(T_{v,i},v) = 0$, then~$C$ does not contain the edge $\{v,u_i\}$, so we have $W(v,i,s,0)=\mu$.
The other cases of~\eqref{eq:weak-dp-update-W1} as well as \eqref{eq:weak-dp-update-L2} are implied by the following observation:
If $\wload_{C,\alpha}(T_{v,i},v) \leq \lambda$ and $\{v, u_i\} \notin C$, then by~\eqref{eq:wload-add-edge} we have $\mu \leq \beta$ and $\mu \leq \lambda$.
The relation~\eqref{eq:weak-dp-update-L1} is closely related to~\eqref{eq:weak-dp-update-W1}.
If $\mu \neq \nu$, then~\eqref{eq:weak-dp-update-L1} follows immediately from~\eqref{eq:weak-dp-update-W1} and the definitions of~$W$ and~$L$.
If $\mu = \nu \leq \min\{\beta,\lambda\}$, then both a partial solution containing the edge $\{v,u_i\}$ as well as one missing this edge minimize the weak load.
As the weak load is bounded from above by the load, we get $L(v,i,s,\lambda) = W(v,i,s,\lambda) = \mu$ in this case.
This implies~\eqref{eq:weak-dp-update-L1}.
An analogous argument yields~\eqref{eq:weak-dp-update-W2}.
\end{proof}
The following lemma describes the recursive relations satisfied by the entries of~$W^+$ and~$L^+$.
Specifically, the lemma describes how to distribute~$s$ contraction edges in~$T_{v,i}^+$ among the two subtrees~$T_{v,i-1}^+$ and~$T_{v,i}$ ($t$ is the number of edges contracted in the first tree, and~$s-t$ the number of edges in the second tree, respectively).
To compute a single point on the Pareto front $P(T_{v,i}^+,v,s)$, we need to consider all points on the Pareto fronts $P(T_{v,i-1}^+,v,t)$ and $P(T_{v,i},v,s-t)$.
\begin{lem}
\label{lem:W-L-rec2}
Let~$v$ be a vertex of~$T$, and let $s \in\{0,1,\dots,m\}$ and $i\in\{1,2,\dots,c(v)\}$ be fixed throughout this lemma.
For $t\in \{0,1,\ldots,s\}$ we let $\Pi(t)$ denote the set of all pairs $(\lambda_1,\lambda_2)$ with $\lambda_1\in \Lambda^*(T_{v,i-1}^+,v,t)$ and $\lambda_2\in \Lambda^*(T_{v,i},v,s-t)$ such that $W^+(v,i-1,t,\lambda_1)+L(v,i,s-t,\lambda_2)\leq \beta$ and $L^+(v,i-1,t,\lambda_1)+W(v,i,s-t,\lambda_2)\leq \beta$.
For $t\in \{0,1,\ldots,s\}$ and $\lambda\in \Lambda(T_{v,i}^+,v)$ we let $\Pi(t,\lambda)\subseteq\Pi(t)$ denote the set of all pairs $(\lambda_1,\lambda_2)\in\Pi(t)$ satisfying $\max\{L^+(v,i-1,t,\lambda_1),L(v,i,s-t,\lambda_2)\}\leq \lambda$. \\
For all $\lambda\in\Lambda(T_{v,i}^+,v)$, defining
\begin{subequations}
\begin{align}
W(t) &:= \min\big\{\max\{W^+(v,i-1,t,\lambda_1),W(v,i,s-t,\lambda_2)\} : (\lambda_1,\lambda_2)\in \Pi(t,\lambda) \big\} , \label{eq:Wt} \\
L(t) &:= \min\big\{\max\{L^+(v,i-1,t,\lambda_1),L(v,i,s-t,\lambda_2)\} : (\lambda_1,\lambda_2) \text{ minimizes } \eqref{eq:Wt} \big\} , \label{eq:Lt}
\end{align}
we have
\begin{align}
W^+(v,i,s,\lambda) &= \min\{W(t):t\in\{0,1,\ldots,s\}\} , \label{eq:weak-dp-update-W3} \\
L^+(v,i,s,\lambda) &= \min\{L(t):t \text{ minimizes } \eqref{eq:weak-dp-update-W3} \} . \label{eq:weak-dp-update-L3}
\end{align}
For $\lambda^*=\lambda^*(T_{v,i}^+,v,s)$, defining
\begin{align}
L^*(t) &:= \min\big\{\max\{L^+(v,i-1,t,\lambda_1),L(v,i,s-t,\lambda_2)\} : (\lambda_1,\lambda_2)\in \Pi(t) \big\} , \label{eq:Lt*} \\
W^*(t) &:= \min\big\{\max\{W^+(v,i-1,t,\lambda_1),W(v,i,s-t,\lambda_2)\} : (\lambda_1,\lambda_2) \text{ minimizes } \eqref{eq:Lt*} \big\} , \label{eq:Wt*}
\end{align}
we have
\begin{align}
L^+(v,i,s,\lambda^*) &= \lambda^* = \min\{L^*(t):t\in\{0,1,\ldots,s\}\} , \label{eq:weak-dp-update-L4} \\
W^+(v,i,s,\lambda^*) &= \min\{W^*(t):t \text{ minimizes } \eqref{eq:weak-dp-update-L4} \} . \label{eq:weak-dp-update-W4}
\end{align}
\end{subequations}
\end{lem}
\begin{proof}
The relation \eqref{eq:weak-dp-update-W3} follows by combining the definitions \eqref{eq:weak-def-W} and \eqref{eq:Wt} with the relations \eqref{eq:load-join-trees}, \eqref{eq:wload-join-trees} and the condition \eqref{eq:weak-tree-cond} from Lemma~\ref{lem:weak-tree-partition}.
The argument for \eqref{eq:weak-dp-update-L3} is analogous, using the definitions \eqref{eq:weak-def-L} and \eqref{eq:Lt} instead of \eqref{eq:weak-def-W} and \eqref{eq:Wt}.
The relation \eqref{eq:weak-dp-update-L4} follows by combining the definitions \eqref{eq:lambda*} and \eqref{eq:Lt*} (recall also \eqref{eq:L-lambda*}) with the relations \eqref{eq:load-join-trees}, \eqref{eq:wload-join-trees} and the condition \eqref{eq:weak-tree-cond} from Lemma~\ref{lem:weak-tree-partition}.
The argument for \eqref{eq:weak-dp-update-W4} is analogous, using the definitions \eqref{eq:weak-def-W} and \eqref{eq:Wt*} instead of \eqref{eq:lambda*} and \eqref{eq:Lt*}.
\end{proof}
We can trivially compute the quantities $W(t)$, $L(t)$, $W^*(t)$ and~$L^*(t)$ as defined in Lemma~\ref{lem:W-L-rec2} in time $\cO(n^2)$ (using that $|\Pi(t)|=\cO(n^2)$ and $|\Pi(t,\lambda)|=\cO(n^2)$ by Lemma~\ref{lem:pareto}).
The following lemma shows how to do the same computation in time $\cO(n)$, so that the entries $W^+(v,i,s,\lambda)$ and $L^+(v,i,s,\lambda)$ can be computed via \eqref{eq:weak-dp-update-W3}, \eqref{eq:weak-dp-update-L3}, \eqref{eq:weak-dp-update-L4} and \eqref{eq:weak-dp-update-W4} in time $\cO(n^2)$ (instead of the trivial bound $\cO(n^3)$).
\begin{lem}
\label{lem:wt-lt-comp}
If the numbers in the sets $\Lambda^*(T_{v,i-1}^+,v,t)$ and $\Lambda^*(T_{v,i},v,s-t)$ are sorted increasingly, the quantities $W(t)$, $L(t)$, $W^*(t)$ and~$L^*(t)$ defined in Lemma~\ref{lem:W-L-rec2} can be computed in time $\cO(n)$.
Consequently, $W^+(v,i,s,\lambda)$ and $L^+(v,i,s,\lambda)$ can be computed for all $s\in\{0,1,\ldots,m\}$ and all $\lambda\in\Lambda^*(T_{v,i}^+,v,s)$ in time $\cO(n^2)$.
\end{lem}
\begin{proof}
We define the sequence~$P_1$ of all pairs of finite numbers $(L^+(v,i-1,t,\lambda),W^+(v,i-1,t,\lambda))$ for all $\lambda\in\Lambda^*(T_{v,i-1}^+,v,t)$ in increasing order of $\lambda$-values.
Similarly, we define the sequence~$P_2$ of all pairs of finite numbers $(L(v,i,s-t,\lambda),W(v,i,s-t,\lambda))$ for all $\lambda\in\Lambda^*(T_{v,i},v,s-t)$ in increasing order of $\lambda$-values.
By Lemma~\ref{lem:pareto} each of these lists has size $\cO(n)$.
Note that these sequences correspond to the Pareto fronts $P(T_{v,i-1}^+,v,t)$ and $P(T_{v,i},v,s-t)$, respectively.
Some pairs of points may appear multiple times consecutively in~$P_1$ and~$P_2$, and in a preprocessing step we eliminate these duplicates in time $\cO(n)$.
We know that after this preprocessing step, the first entries in the simplified lists~$P_1$ and~$P_2$ are strictly increasing, and the second entries are strictly decreasing (recall Figure~\ref{fig:pareto}).
We first argue how to compute~$W(t)$ and~$L(t)$.
We begin discarding all pairs from each list whose first entry ($L^+$ or~$L$, respectively) is strictly greater than~$\lambda$ in time $\cO(n)$.
We then process the remaining lists~$P_1$ and~$P_2$ beginning at the last entries $(L_j^+,W_j^+)$ and $(L_k,W_k)$ (with smallest~$W^+$ or~$W$-values, respectively) in two phases.
In the first phase we compute~$W(t)$ as follows:
If $L_j^+ +W_k>\beta$, we discard the last element of~$P_1$ by decreasing~$j$ by 1 (by our sorting of the lists we know that $L_j^+ +W_{k'}>\beta$ for all $k'\leq k$).
If $W_j^+ +L_k>\beta$, we discard the last element of~$P_2$ by decreasing~$k$ by 1 (by our sorting of the lists we know that $W_{j'}^++L_k>\beta$ for all $j'\leq j$).
Once $L_j^+ +W_k\leq \beta$ and $W_j^+ +L_k\leq \beta$ for the first time, we have found $W(t)=\max\{W_j^+,W_k\}$. If this never happens we know that $W(t)=\infty$.
This computation is correct by the definition of $\Pi(t,\lambda)$ in Lemma~\ref{lem:W-L-rec2} and by \eqref{eq:Wt}, and it takes time $\cO(n)$.
In the second phase we compute~$L(t)$ as follows:
If $W(t)=\infty$, we know that $L(t)=\infty$, too.
Otherwise we distinguish two cases:
If $W_j^+\geq W_k$, we decrease~$k$ further as long as both inequalities $W_j^+\geq W_k$ and $L_j^+ +W_k\leq \beta$ are still satisfied (so that they still hold for the final~$k$).
If $W_j^+\leq W_k$, we decrease~$j$ further as long as both inequalities $W_j^+\leq W_k$ and $W_j^+ +L_k\leq \beta$ are still satisfied (so that they still hold for the final~$j$).
In the end we set $L(t)=\max\{L_j^+,L_k\}$.
Note that in the first case, the third constraint $W_j^+ +L_k\leq \beta$ remains valid by the monotonicity $L_{k'}\leq L_k$ for all $k'\leq k$, and in the second case, the third constraint $L_j^+ +W_k\leq \beta$ remains valid by the monotonicity $L_{j'}^+\leq L_j^+$ for all $j'\leq j$.
Therefore, the correctness of the computation of~$L(t)$ follows from \eqref{eq:Lt}.
The procedure to compute~$W^*(t)$ and~$L^*(t)$ processes~$P_1$ and~$P_2$ (as obtained from the preprocessing step explained in the beginning) starting at the first entries $(L_j^+,W_j^+)$,~$j=1$, and $(L_k,W_k)$,~$k=1$, in two phases very similarly to before.
We omit the details here.
\end{proof}
We are now ready to prove Theorem~\ref{thm:weak-dp}.
\begin{proof}[Proof of Theorem~\ref{thm:weak-dp}]
Given the instance $(T,\ell,\varphi)$, we fix an arbitrary root~$r$ of~$T$ and an arbitrary ordering of the children of each vertex, making~$T$ an ordered rooted tree.
We begin precomputing and sorting all of the sets $\Lambda(T_{v,i},v)$ and $\Lambda(T_{v,i}^+,v)$, $v\in V$, $i\in\{0,1,\ldots,c(v)\}$, and we maintain them as sorted lists throughout the algorithm.
This takes time $\cO(n^2\log n)$ in total (recall Lemma~\ref{lem:pareto}).
We then compute the entries of the matrices $W$, $L$, $W^+$ and~$L^+$ using Lemmas~\ref{lem:W-L-rec1} and \ref{lem:wt-lt-comp}.
We first initialize various entries using \eqref{eq:weak-dp-init-W1}--\eqref{eq:weak-dp-init-WL2}, and compute the remaining entries in a bottom-up fashion moving upwards from the leaves to the root.
Specifically, at a vertex~$v$ with children $u_1,u_2,\ldots,u_{c(v)}$ for which all the entries of $W$, $L$, $W^+$ and~$L^+$ have already been computed, we first compute $W(v,i,s,\lambda)$ and then $L(v,i,s,\lambda)$ for all $i\in\{1,2,\ldots,c(v)\}$, $s\in\{1,2,\ldots,m\}$ and $\lambda \in \Lambda(T_{v,i},v)$ using \eqref{eq:weak-dp-update-W1} and \eqref{eq:weak-dp-update-L1}, then we compute $L(v,i,s,\lambda^*(T_{v,i},v,s))=\lambda^*(T_{v,i},v,s)$ and $W(v,i,s,\lambda^*(T_{v,i},v,s))$ for all $i\in\{1,2,\ldots,c(v)\}$ and $s\in\{1,2,\ldots,m\}$ using \eqref{eq:weak-dp-update-L2} and \eqref{eq:weak-dp-update-W2}.
We obtain sorted lists containing the numbers in $\Lambda^*(T_{v,i},v,s)$ by inserting $\lambda^*(T_{v,i},v,s)$ at the correct position into the precomputed list $\Lambda(T_{v,i},v)$.
Next, we compute $W^+(v,i,s,\lambda)$ and then $L^+(v,i,s,\lambda)$ for all $i\in\{1,2,\ldots,c(v)\}$, $s\in\{1,2,\ldots,m\}$ and $\lambda\in \Lambda(T_{v,i}^+,v)$ using \eqref{eq:weak-dp-update-W3} and \eqref{eq:weak-dp-update-L3}, and then we compute $L^+(v,i,s,\lambda^*(T_{v,i}^+,v,s))=\lambda^*(T_{v,i}^+,v,s)$ and $W^+(v,i,s,\lambda^*(T_{v,i}^+,v,s))$ for all $i\in\{1,2,\ldots,c(v)\}$ and $s\in\{1,2,\ldots,m\}$ using \eqref{eq:weak-dp-update-L4} and \eqref{eq:weak-dp-update-W4}.
We obtain sorted lists containing the numbers in $\Lambda^*(T_{v,i}^+,v,s)$ by inserting $\lambda^*(T_{v,i}^+,v,s)$ at the correct position into the precomputed list $\Lambda(T_{v,i}^+,v)$.
Let~$s^*$ be the largest~$s$ such that $W^+(r,c(r),s,\lambda^*(T,r,s))$ is finite.
From \eqref{eq:weak-def-LW} we obtain that~$s^*$ is the size of an optimal solution of the instance $(T,\ell,\varphi)$.
The corresponding set of edges $C\subseteq E$ can be obtained by keeping track of the arguments for which the minima and maxima in \eqref{eq:weak-dp-update-W1}--\eqref{eq:weak-dp-update-W2} and \eqref{eq:Wt}--\eqref{eq:weak-dp-update-W4} are attained in each step.
Each of the matrices $W$, $L$, $W^+$ and~$L^+$ has $\cO(n^3)$ entries (recall Lemma~\ref{lem:pareto}).
Computing an entry of~$W$ or~$L$ takes $\cO(n)$ time by Lemma~\ref{lem:W-L-rec1}, while computing an entry of~$W^+$ or~$L^+$ can be achieved in time $\cO(n^2)$ by Lemma~\ref{lem:wt-lt-comp}, so the runnning time of our dynamic program is $\cO(n^5)$.
\end{proof}
\section{Hardness for additive tolerance functions}
\label{sec:hard-add}
In this section we prove that the problems \Contraction{} and \WContraction{} for the tolerance function $\varphi(x)=x-\beta$ (purely additive error) are hard already on cycles (Section~\ref{sec:cycles-hard} below).
We then prove that \Contraction{} with the same tolerance function is hard to approximate for general graphs and for bipartite graphs (Section~\ref{sec:inapx-contraction}).
\subsection{Hardness of \Contraction{} and \WContraction{}}
\label{sec:cycles-hard}
Recall that we can compute optimal (weak) $(\alpha,\beta)$-contractions in polynomial time on trees (this was shown in Section~\ref{sec:dp}), and have a linear time algorithm for \Contraction{} on cycles with unit length edges (this was shown in Section~\ref{sec:cycles}).
We now show that the problem with $\alpha=1$ is NP-hard on cycles with arbitrary edge lengths.
\begin{thm}
\label{thm:cycle-hard-fix}
For any fixed $\beta>0$, the problems \Contraction{} and \WContraction{} with tolerance function $\varphi(x)=x-\beta$, $\beta\geq 0$, are NP-hard on cycles.
\end{thm}
Theorem~\ref{thm:cycle-hard-fix} (where~$\beta$ is not part of the input) follows immediately from Theorem~\ref{thm:cycle-hard} below (where~$\beta$ is part of the input).
The reason is that an instance with $\alpha=1$ does not change when multiplying all edge lengths and $\beta$ by some constant.
\begin{thm}
\label{thm:cycle-hard}
The problems \Contraction{} and \WContraction{} with tolerance function $\varphi(x)=x-\beta$, $\beta\geq 0$, are NP-hard on cycles.
\end{thm}
The rest of this section is devoted to proving Theorem~\ref{thm:cycle-hard}.
For our proof we will use the following variant of the well-known problem \Partition{}, referred to as \CPartition{}.
To state the problem we say that a set of positive rational numbers $\{a_1,a_2,\ldots,a_n\}$ is \emph{close to 1}, if $\sum_{i=1}^n a_i = n$ and $\varepsilon:=\sum_{i=1}^n |a_i-1|<1/5$.
\begin{problem}{\CPartition{}}
Input: & A set of positive rational numbers $\{a_1,a_2,\ldots,a_n\}$ that is close to 1. \\
Output: & `Yes' if there is a subset $I \subseteq [n]$ such that $\sum_{i \in I} a_i = \sum_{i\in [n]\setminus I} a_i$, `No' otherwise. \\
\end{problem}
Note that for a `Yes'-instance of this problem, the solution $I\subseteq [n]$ must have size~$n/2$, so $|I|=|[n]\setminus I|=\sum_{i \in I} a_i=\sum_{i\in [n]\setminus I} a_i=n/2$.
In particular, this implies that~$n$ is even.
In the classical problem \Partition{}, the input set is not constrained to be close to 1.
\Partition{} was shown to be NP-complete already in Karp's seminal paper~\cite{MR0378476}.
The fact that \CPartition{} is also NP-complete follows from a straightforward rescaling argument.
\begin{lem}
\label{lem:cpartition}
\CPartition{} is NP-complete.
\end{lem}
\begin{proof}
Given an instance $\{a_1,a_2,\ldots,a_n\}$ of \Partition{}, we first add~$n$ additional zeroes $a_{n+1}=a_{n+2}=\cdots=a_{2n}=0$ to the instance (by this we ensure that a partition with equal sums is transformed into one where both partition classes have the same number~$n$ of summands).
We then linearly transform all the~$a_i$ according to $a_i':=(a_i+C)/D$, where~$C$ and~$D$ are sufficiently large constants so that the transformed values~$a_i'$ are close to 1.
The transformed set of numbers has even cardinality~$2n$, is close to 1, and it admits a partition into two sets of size~$n$ with equal sum if and only if the original instance allows a partition into two sets with equal sum.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:cycle-hard}]
We first focus on the problem \Contraction{}.
We reduce \CPartition{}, which is NP-complete by Lemma~\ref{lem:cpartition}, to the problem \Contraction{} on a cycle with tolerance function $\varphi(x)=x-\beta$, $\beta\geq 0$.
Let $\cI=\{a_1,a_2,\ldots,a_n\}$ be an instance of \CPartition{} such that $a_1\geq a_2 \geq \cdots \geq a_n$.
This ensures that all~$a_i$ that are bigger than 1 appear before all~$a_i$ that are smaller than 1, which is the only property of the ordering that we exploit in the proof later on.
The instance of \Contraction{} we construct is on the cycle $C_{2n+4}$ with~$2n+4$ edges.
We label the vertices of the cycle by walking around the cycle as follows:
The first~$n+1$ vertices are labelled $u_0,u_1,\ldots,u_n$, then there are two special vertices $v_1$, $v_2$, and the remaining~$n+1$ vertices are labelled $w_0,w_1,\ldots,w_n$, see Figure~\ref{fig:cycle-hard}.
We denote the subpath $(u_0,\ldots,u_n)$ as~$P_u$, and the subpath $(w_0,\ldots,w_n)$ by~$P_w$.
We now define $\varepsilon:=\sum_{i=1}^n |1-a_i|<1/5$, $\beta:=n/2+2\varepsilon$ and $\beta':=\beta+1 > \beta$, and the length function~$\ell$ on the cycle edges by setting $\ell(u_{i-1},u_i):=a_i$ and $\ell(w_{i-1},w_i)=2-a_i$ for all $i\in [n]$, and by $\ell(u_n,v_1)=\ell(v_2,w_1):=\varepsilon$, $\ell(v_1,v_2):=\beta'$, and $\ell(w_n,u_0):=\beta'+2\varepsilon$ (see Figure~\ref{fig:cycle-hard}).
\begin{figure}
\caption{Reduction from \CPartition{}
\label{fig:cycle-hard}
\end{figure}
Now consider the instance $\cJ := (C_{2n+4}, \ell, \varphi)$ with $\varphi(x) = x-\beta$ of the problem \Contraction{}.
Observe that no $\varphi$-contraction may contain an edge $\{u,v\}$ of length greater than~$\beta$ (in particular, no feasible solution may contain one of the edges of length~$\beta'$ or $\beta'+2\varepsilon$).
Furthermore any (weak) $\varphi$-contraction $C$ on this graph satisfies $\Phi(C) = |C|$.
We will show that~$\cJ$ has an optimal solution of cardinality (and thus of value)~$n+2$ if and only if~$\cI$ is a `Yes'-instance.
In particular, we will see that any feasible solution of~$\cJ$ of size~$n+2$ contains the two edges of length~$\varepsilon$ and exactly~$n/2$ edges with length~$a_i$,~$i\in I$, from~$P_u$ and the corresponding edges with length~$2-a_i$,~$i\in I$, from~$P_w$.
Such solutions correspond to subsets of~$[n]$ in the following natural way:
For any subset $I\subseteq [n]$ of size~$n/2$ we let~$C(I)$ be the subset of edges of the cycle~$C_{2n+4}$ consisting of the two edges of length~$\varepsilon$ and of all edges $\{u_{i-1},u_i\}$ and $\{w_{i-1},w_i\}$ (of length~$a_i$ or~$2-a_i$, respectively) for all $i\in I$.
Thus we will show that~$C(I)$ is an optimal solution of the instance~$\cJ$ of \Contraction{} if and only if $\sum_{i \in I} a_i = \sum_{i\in[n]\setminus I} a_i = n/2$, i.e.,~$\cI$ is a `Yes'-instance of \CPartition{}.
Both directions of this equivalence are captured and proved as Claim~2 and 4 below.
Claims~1 and 3 are auxiliary statements used in the proofs of these two main claims.
For any path~$P$ on the cycle we let~$\ell(P)$ denote the sum of~$\ell(e)$ over all edges~$e$ of~$P$.
For all $i\in [n]$ we denote by $P_i^\sqsupset$ and $P_i^\sqsubset$ the path on the cycle between the vertices~$u_i$ and~$w_i$ that contains and that does not contain the edge $\{v_1,v_2\}$, respectively (in Figure~\ref{fig:cycle-hard}, these are the right and left segment of the cycle).
\underline{Claim~1:}
For all~$i\in [n]$, the number $\ell(P_i^\sqsupset)$ lies in the interval $[n+\beta'+\varepsilon,n+\beta'+2\varepsilon]$ and the number $\ell(P_i^\sqsubset)$ lies in the interval $[n+\beta'+2\varepsilon,n+\beta'+3\varepsilon]$.
In particular, we have $\dist_\ell(u_i,w_i)=\min\{\ell(P_i^\sqsupset),\ell(P_i^\sqsubset)\}=\ell(P_i^\sqsupset)$ and the difference $\ell(P_i^\sqsubset)-\ell(P_i^\sqsupset)$ lies in the interval $[0,2\varepsilon]$.
\underline{Proof of Claim~1:}
Note that the condition $\sum_{i=1}^n a_i=n$ implies that
\begin{equation}
\label{eq:ai-split}
\varepsilon=2\sum_{a_i:a_i\geq 1} (a_i-1)=2\sum_{a_i:a_i<1}(1-a_i) .
\end{equation}
By our assumption $a_1\geq a_2\geq\cdots \geq a_n$, the numbers $\ell(P_i^\sqsupset)$ form a unimodal sequence for $i=0,1,\ldots,n$ that is maximized for $i=0$ and $i=n$, proving that $\ell(P_i^\sqsupset)\leq n+\beta'+2\varepsilon$ (note that $\ell(P_u)=\ell(P_w)=n$).
By \eqref{eq:ai-split} the minimum of this unimodal sequence is at most~$\varepsilon$ smaller than the maximum.
This proves the first part of the claim.
As $\ell(P_i^\sqsupset)+\ell(P_i^\sqsubset)=2(n+\beta'+2\varepsilon)$, we obtain the second part of the claim.
The last part of the claim is an immediate consequence of the first two.
\qedclaim
\underline{Claim~2:}
If $I \subseteq [n]$ is a solution of the instance~$\cI$ of \CPartition{} such that $\sum_{i \in I} a_i = \sum_{i\in[n]\setminus I} a_i = n/2$, then~$C(I)$ is a $(1,\beta)$-contraction.
\underline{Proof of Claim~2:}
It suffices to prove that there is no pair of vertices whose distance decreases by more than~$\beta$ when contracting the edges in~$C(I)$.
We start by verifying this for the pairs $u_i,w_i$ for $i \in [n]$.
We first consider the path $P_i^\sqsupset$ between~$u_i$ and~$w_i$.
Observe that $\sum_{e\in C(I)\cap P_i^\sqsupset}\ell(e)$ lies in the interval $[n/2+\varepsilon,n/2+2\varepsilon]=[\beta-\varepsilon,\beta]$.
Similarly to before, this follows from the observation that by the assumption $a_1\geq a_2\geq\cdots \geq a_n$ those sums form a unimodal sequence for $i=0,1,\ldots,n$ that is maximized for~$i=0$ and~$i=n$, and by using \eqref{eq:ai-split} (recall also that $|I|=n/2$).
Consequently, we have
\begin{equation}
\label{eq:ell-Piright}
\ell_{C(I)}(P_i^\sqsupset)\geq \ell(P_i^\sqsupset)-\beta .
\end{equation}
Since $\sum_{e\in C(I)} \ell(e)=n+2\varepsilon=2\beta-2\varepsilon$, we obtain that $\sum_{e\in C(I)\cap P_i^\sqsubset}\ell(e)$ lies in the interval $[\beta-2\varepsilon,\beta-\varepsilon]$, yielding
\begin{equation}
\label{eq:ell-Pileft}
\ell_{C(I)}(P_i^\sqsubset)\geq \ell(P_i^\sqsubset)-(\beta-\varepsilon)\geq \ell(P_i^\sqsubset)-\beta .
\end{equation}
Combining \eqref{eq:ell-Piright} and \eqref{eq:ell-Pileft} proves that
\begin{equation}
\label{eq:dist-ui-wi}
\dist_{\ell_{C(I)}}(u_i,w_i) \geq \dist_\ell(u_i,w_i)-\beta .
\end{equation}
Now consider two vertices~$u_i$ and~$w_j$,~$j<i$ (the case~$j>i$ can be treated analogously).
Let $P_{i,j}^\sqsupset$ and $P_{i,j}^\sqsubset$ be the path on the cycle between the vertices~$u_i$ and~$w_j$ that contains and that does not contain the edge $\{v_1,v_2\}$, respectively.
Using that $P_{i,j}^\sqsupset\subseteq P_i^\sqsupset$ we obtain
\begin{equation}
\label{eq:ell-Pijsmile}
\ell_{C(I)}(P_{i,j}^\sqsupset)\geq \ell(P_{i,j}^\sqsupset)-\beta
\end{equation}
from \eqref{eq:ell-Piright}.
We know that $a_i\leq 1+1/5\leq 8/5$ and consequently
\begin{equation}
\label{eq:2mai}
2-a_i\geq 2/5\geq 2\varepsilon
\end{equation}
by the assumption that the input $\{a_1,a_2,\ldots,a_n\}$ of the instance~$\cI$ is close to 1 (there is plenty of leeway in all those inequalities).
Furthermore, we have
\begin{equation}
\label{eq:ell-Pij}
\dist_\ell(u_i,w_j)\leq \ell(P_i^\sqsupset)-(2-a_i)\leBy{eq:2mai} \ell(P_i^\sqsupset)-2\varepsilon \leq \min\{\ell(P_i^\sqsupset),\ell(P_i^\sqsubset)\}=\dist_\ell(u_i,w_i) ,
\end{equation}
where the second-to-last inequality follows from Claim~1.
Combining those observations yields
\begin{equation}
\label{eq:ell-Pijfrown}
\ell_{C(I)}(P_{i,j}^\sqsubset) \geq \dist_{\ell_{C(I)}}(u_i,w_i) \geBy{eq:dist-ui-wi} \dist_\ell(u_i,w_i)-\beta \geBy{eq:ell-Pij} \dist_\ell(u_i,w_j)-\beta .
\end{equation}
Combining \eqref{eq:ell-Pijsmile} and \eqref{eq:ell-Pijfrown} proves that
\begin{equation}
\label{eq:dist-ui-wj}
\dist_{\ell_{C(I)}}(u_i,w_j) \geq \dist_\ell(u_i,w_j)-\beta .
\end{equation}
From \eqref{eq:ell-Pijsmile} and \eqref{eq:ell-Pijfrown} we can derive analogous relations for the remaining cases where we need to consider the distance between a vertex~$u_i$,~$i\in[n]$, and a vertex $w\in\{v_1,v_2,u_0,u_1,\allowbreak \ldots, u_{i-1},u_{i+1},\ldots,u_n\}$, between a vertex $w_i$, $i\in[n]$, and a vertex $u\in\{v_1,v_2,w_0,w_1,\ldots,\allowbreak w_{i-1},w_{i+1},\ldots,w_n\}$, and between the vertices~$v_1$ and~$v_2$.
This completes the proof of Claim~2.
\qedclaim
\underline{Claim~3:}
Every $(1,\beta)$-contraction~$C$ contains at most~$n/2$ edges in $(P_u \cup P_w) \cap P_i^\sqsupset$ for all $i\in [n]$ and at most~$n/2$ edges in $(P_u \cup P_w) \cap P_i^\sqsubset$ for all $i\in [n]$.
\underline{Proof of Claim~3:}
Note that for any $I\subseteq [n]$ and $k\in\{0,1,\ldots,n\}$ we have $\sum_{i\in I:i>k} a_i+\sum_{i\in I:i\leq k}(2-a_i)\geq |I|-\varepsilon$ by the definition of $\varepsilon$.
Consequently, assuming for the sake of contradiction that~$C$ contains strictly more than~$n/2$ edges in $(P_u \cup P_w) \cap P_i^\sqsupset$, we have $\ell_C(P_i^\sqsupset)-\ell(P_i^\sqsupset)\geq n/2+1-\varepsilon$.
Similarly, assuming that~$C$ contains strictly more than~$n/2$ edges in $(P_u \cup P_w) \cap P_i^\sqsubset$ yields $\ell_C(P_i^\sqsubset)-\ell(P_i^\sqsubset)\geq n/2+1-\varepsilon$.
By Claim~1 the difference $\ell(P_i^\sqsubset)-\ell(P_i^\sqsupset)$ lies in the interval $[0,2\varepsilon]$, so in both cases we obtain
\begin{equation*}
\dist_\ell(u_i,w_i) - \dist_{\ell_C}(u_i,w_i) \geq \frac{n}{2}+1-\varepsilon -2\varepsilon > \frac{n}{2}+2\varepsilon = \beta ,
\end{equation*}
where we used that $\varepsilon<1/5$ in the second-to-last step.
This contradicts the fact that~$C$ is a $(1,\beta)$-contraction, proving Claim~3.
\qedclaim
\underline{Claim~4:}
Let~$C$ be a feasible solution of the instance~$\cJ$ of \Contraction{}.
Then we have $|C|\leq n+2$, and if $|C|=n+2$, we have $C=C(I)$ for some set $I \subseteq [n]$ with $\sum_{i \in I} a_i = \sum_{i\in[n]\setminus I} a_i = n/2$.
\underline{Proof of Claim~4:}
As~$C$ does not contain any of the edges of length~$\beta'$ or $\beta'+2\varepsilon$, we have $|C|\leq n+2$ by Claim~3 (the +2 comes from the two edges of length~$\varepsilon$ that may be contained in~$C$).
Suppose now that $|C|=n+2$.
Applying Claim~3 again shows that~$C$ must contain both edges of length~$\varepsilon$, and that it contains the edge $\{u_{i-1},u_i\}$ if and only if it contains the edge $\{w_{i-1},w_i\}$, for all $i\in[n]$.
Defining $I:=\{i \in [n]: \{u_{i-1},u_i\} \in C\}$ we have $|I|=n/2$ and $C=C(I)$.
By Claim~1 we have $\dist_\ell(u_0,w_0)=\ell(P_0^\sqsupset)$ and $\dist_\ell(u_n,w_n)=\ell(P_n^\sqsupset)$.
As~$C$ is a $(1,\beta)$-contraction containing the two edges of length~$\varepsilon$ we thus obtain $\sum_{i\in I} a_i = \sum_{e\in C\cap P_u} \ell(e) \leq \beta-2\varepsilon = n/2$.
Similarly, we have $\sum_{i\in [n]\setminus I} a_i = \sum_{i\in I} (2-a_i)\allowbreak = \sum_{e\in C\cap P_w} \ell(e)\allowbreak \leq \beta-2\varepsilon = n/2$.
As $\sum_{i\in[n]}a_i=n$, these two inequalities must be tight, yielding $\sum_{i\in I} a_i = \sum_{i\in[n]\setminus I} a_i = n/2$.
\qedclaim
Combining Claims~2 and 4 proves the statement of the theorem for the problem \Contraction{}.
We now focus on the problem \WContraction{}.
The hardness result follows immediately from the following claim.
\underline{Claim~5:}
For $n \geq 5$, any feasible weak $(1,\beta)$-contraction $C$ on the instance $\cJ$ is also a feasible $(1,\beta)$-contraction.
\underline{Proof of Claim~5:}
Suppose for the sake of contradiction that $C$ is not a feasible $(1,\beta)$-contraction.
This means there are vertices $a,b$ such that $\dist_{\ell_C}(a,b) = 0$ and $\dist_\ell(a,b)>\beta$, i.e., $a$ and $b$ lie on a (maximal) subpath $Q$ formed by edges from $C$ on the cycle.
Let $u$ be one end vertex of $Q$, and let $x$ be the neighbour of $u$ not on $Q$.
Let $v$ be the last vertex on $Q$ when traversed starting at $u$, such that the length of the $x$-$v$-path $P$ containing $u$ is at most $\beta+\ell(x, u)$, and let $y$ be the next vertex on $Q$ when traversed starting at $u$.
Such a vertex $y$ exists as $\ell(Q) > \beta$, and the $x$-$y$-path $P'$ containing $u$ has length strictly greater than $\beta+\ell(x, u)$.
We have $\dist_{\ell_C}(x, y) > 0$, as $C$ does not contract the entire cycle.
By~\eqref{eq:contr-cond}, we have $\dist_{\ell_C}(x, y) \geq \dist_\ell(x, y)-\beta$.
As $\dist_{\ell_C}(x, y) \leq \ell(x, u)$, we get $\dist_\ell(x, y) \leq \beta+\ell(x, u)$.
As we saw before, the $x$-$y$-path $P'$ has length strictly greater than $\beta+\ell(x, u)$, thus the $x$-$y$-path $P''$ not containing $u$ must have length at most $\beta+\ell(x, u)$.
As the entire cycle has length $2n+2\beta'+4\varepsilon = 3n+2+8\varepsilon$ and can be partitioned into $P, P''$ and the edge $\{v,y\}$, we get
\begin{align*}
3n+2+8\varepsilon &= \ell(P)+\ell(P'') + \ell(v, y) \\
&\leq 2(\beta + \ell(x, u)) + \ell(v, y)
\leq 5\beta+3+4\varepsilon = 5n/2+3+14\varepsilon,
\end{align*}
where the second inequality holds as the two longest edges of the cycle have length $\beta'+2\varepsilon=\beta+1+2\varepsilon$ and $\beta'=\beta+1$, respectively.
From this chain of inequalities we obtain $n \leq 2+12\varepsilon < 4+2/5$, contradicting the assumption $n\geq 5$.
\end{proof}
The reader might be tempted to `simplify' the previous reduction proof by omitting the four special edges of length $\varepsilon$, $\beta'$ and $\beta'+2\varepsilon$ and by setting $\beta:=n/2$ instead.
However, this would invalidate Claim~2 (specifically, the estimate \eqref{eq:ell-Pileft} would not always hold).
\subsection{Inapproximability of \Contraction{}}
\label{sec:inapx-contraction}
We are able to extend the before-mentioned hardness result for \Contraction{} as follows:
\begin{thm}
\label{thm:inapx-contr}
For any fixed $\beta>0$ and $\varepsilon > 0$, it is NP-hard to approximate the problem \Contraction{} with tolerance function $\varphi(x)=x-\beta$, $\beta\geq 0$, to within a factor of $n^{1-\varepsilon}$.
\end{thm}
For the following theorem the additive error is fixed to $\beta=1$.
\begin{thm}
\label{thm:inapx-bip}
For any $\varepsilon > 0$, it is NP-hard to approximate the problem \Contraction{} with tolerance function $\varphi(x)=x-1$ on bipartite graphs with unit length edges $\ell=1$ to within a factor of $m^{1/2-\varepsilon}$.
\end{thm}
Our reductions are based on the inapproximability of the well-known \Clique{} problem.
Recall that a \emph{clique} in a graph $G$ is a complete subgraph of $G$.
\begin{problem}{\Clique{}}
Input: & A graph $G$. \\
Output: & A clique in $G$ of maximum size. \\
\end{problem}
It was shown in \cite{MR2403018} that for any $\varepsilon > 0$, it is NP-hard to approximate \Clique{} to within a factor of $n^{1-\varepsilon}$.
The following lemma will be used in our proofs.
It shows that for $(1,\beta)$-contractions the feasibility condition \eqref{eq:contr-cond} needs not be checked for all pairs of vertices $u$ and $v$, but only for those satisfying certain extra conditions.
\begin{lem}
\label{lem:add-contr-cond}
A set of edges $C\subseteq E$ is a $(1,\beta)$-contraction if and only if all pairs of vertices $u,v\in V$ with the property that every shortest path with respect to $\ell_C$ between $u$ and $v$ starts and ends with an edge from $C$ satisfy condition \eqref{eq:contr-cond}.
\end{lem}
\begin{proof}
Suppose for the sake of contradiction that all pairs of vertices $u,v\in V$ as in the lemma satisfy condition \eqref{eq:contr-cond} and that $C$ is \emph{not} a $(1,\beta)$-contraction.
Then there is a pair of vertices $u,v \in V$ violating \eqref{eq:contr-cond} and a shortest path $P$ with respect to $\ell_C$ between $u$ and $v$ that does not start or end with an edge from $C$.
We choose $u$ and $v$ such that $\dist_{\ell_C}(u,v)$ is minimal, and we may assume that the first edge $\{u,w\}$ of $P$ is not contained in $C$, so $\dist_\ell(u,v) - \dist_{\ell_C}(u,v) = \dist_\ell(w,v) - \dist_{\ell_C}(w,v)$.
By our choice of $u$ and $v$, the vertices $w$ and $v$ satisfy \eqref{eq:contr-cond}, i.e., the right-hand side of this equation is bounded by $\beta$, a contradiction.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:inapx-contr}]
Let $\beta,\varepsilon>0$ be fixed and let $G=(V,E)$ be an instance of \Clique{}.
We define a graph $H=H(G)$ as follows, see Figure~\ref{fig:inapx-reduc1}:
The vertex set of $H$ is given by $(V\times\{1,2\})\cup \{s\}$, i.e., we create two copies of each original vertex and add a special vertex $s$.
The edge set of $H$ is given by $\{\{(u,1),(v,1)\}:\{u,v\}\in E\}$ plus the edges $\{(v,1),(v,2)\}$ and $\{s,(v,2)\}$ for all $v\in V$.
The first set of edges are simply the original edges of $G$ on the first copies of the vertices, the second set is a perfect matching between the two copies of the vertex set, and the third set of edges connects the special vertex~$s$ to all vertices of the second copy of the vertex set.
The length function $\ell$ on the edges of $H$ is set to $2\beta+2$, $\beta$ or $\beta+1$ for those three sets of edges, respectively.
\begin{figure}
\caption{An instance $G$ of \Clique{}
\label{fig:inapx-reduc1}
\end{figure}
Now consider the instance $\cI:=(H,\ell,\varphi)$ of the problem \Contraction{} with the tolerance function $\varphi(x)=x-\beta$.
Clearly, any $(1,\beta)$-contraction $C$ in $H$ can contain only edges of the form $\{(u,1),(u,2)\}$ for some $u\in V$.
As $H$ does not contain two edges between two different connected components of $(V,C)$, our objective function defined in \eqref{eq:objective} satisfies $\Phi(C)=|C|$ for any feasible solution $C$ of $\cI$.
We will show that it allows a feasible solution with $k$ edges (and thus of value $k$) if and only if $G$ has a clique with $k$ vertices.
Formally, for $U \subseteq V$ we define $C(U) := \{\{(u,1),(u,2)\} : u \in U\}$ (see Figure~\ref{fig:inapx-reduc1}).
We proceed to show that $U$ induces a clique in $G$ if and only if $C(U)$ is a $(1,\beta)$-contraction in $H=H(G)$.
Note that for any two vertices $u,v \in U$ we have
\begin{align*}
\dist_{\ell_{C(U)}}((u,1),(v,1)) &= 2\beta+2 =
\begin{cases}
\dist_\ell((u,1),(v,1)) & \text{if } \{u,v\} \in E,\\
\dist_\ell((u,1),(v,1))-2\beta & \text{otherwise},
\end{cases} \\
\dist_{\ell_{C(U)}}((u,1),(v,2)) &= 2\beta+2 = \dist_\ell((u,1),(v,2)) - \beta, \\
\dist_{\ell_{C(U)}}((u,2),(v,2)) &= 2\beta+2 = \dist_\ell((u,2),(v,2)) , \\
\dist_{\ell_{C(U)}}((u,1),(u,2)) &= 0 = \dist_\ell((u,1),(u,2)) - \beta.
\end{align*}
These relations together with Lemma~\ref{lem:add-contr-cond} show that $C(U)$ is a $(1,\beta)$-contraction in $H$ if and only if $U$ is a clique in $G$.
As $n(H)$ differs from $n(G)$ only by a constant factor, an $n^{1-\varepsilon}$-approximation algorithm for \Contraction{} would yield an $n^{1-\varepsilon'}$-approximation algorithm for \Clique{} via this reduction.
Together with the before-mentioned inapproximability of \Clique{} \cite{MR2403018} this proves the theorem.
\end{proof}
The rest of this section is devoted to proving Theorem~\ref{thm:inapx-bip}, so we now focus on $(1,1)$-contractions in bipartite graphs with unit length edges $\ell=1$.
The next lemma characterizes the structure of contractions in this setting.
\begin{lem}
\label{lem:bip-contr}
Let $G=(V, E)$ be a bipartite graph with unit edge lengths $\ell=1$ and let $C \subseteq E$ be a set of edges.
\begin{enumerate}[label=(\roman*),leftmargin=7mm]
\item
If $C$ is a $(1,1)$-contraction, then $C$ is a matching.
\item
If $C = \{e,f\}$ with edges $e = \{u_1,u_2\}, f = \{v_1,v_2\} \in E$, then $C$ is a $(1,1)$-contraction if and only if $\dist_\ell(u_1,v_1) = \dist_\ell(u_2,v_2)$ and $\dist_\ell(u_1,v_2) = \dist_\ell(u_2,v_1)$.
\item
$C$ is a $(1,1)$-contraction if and only if all two-element subsets of $C$ are.
\end{enumerate}
\end{lem}
\begin{proof}
\begin{enumerate}[label=(\roman*),leftmargin=7mm]
\item
Suppose for the sake of contradiction that $C$ contains a path $(u,v,w)$ on two edges.
As $G$ is bipartite, it has no triangles, so $\dist_\ell(u,w)=2$ and $\dist_{\ell_C}(u,w)=0$, a contradiction to the assumption that $C$ is a $(1,1)$-contraction.
\item
For the edges $e=\{u_1,u_2\}$ and $f=\{v_1,v_2\}$ we define $d_{i,j}:=\dist_\ell(u_i,v_j)$ for $i,j\in\{1,2\}$.
Let $C=\{e,f\}$ be a $(1,1)$-contraction.
Both $d_{1,1}$ and $d_{2,2}$ must have the same parity (as $G$ is bipartite), so if $d_{1,1}<d_{2,2}$, the difference between them is exactly~2.
However, this would mean that $\dist_{\ell_C}(u_2,v_2)=d_{1,1}=d_{2,2}-2=\dist_\ell(u_2,v_2)-2$, a contradiction to the assumption that $C$ is a $(1,1)$-contraction.
Repeating the same argument with $d_{1,1}$ and $d_{2,2}$ interchanged shows that $d_{1,1}=d_{2,2}$.
An analogous argument shows that $d_{1,2}=d_{2,1}$.
Now suppose that $d_{1,1}=d_{2,2}$ and $d_{1,2}=d_{2,1}$.
From these conditions it follows that for all $i,j\in\{1,2\}$ every path between $u_i$ and $v_j$ that contains both edges $e$ and $f$ has length at least $d_{i,j}+2$ with respect to $\ell$.
Consequently, we have $\dist_{\ell_C}(u_i,v_j)\geq \dist_\ell(u_i,v_j)-1$ for $C=\{e,f\}$.
By Lemma~\ref{lem:add-contr-cond}, $C$ is a $(1,1)$-contraction.
\item
One direction of the equivalence is obvious, so we only need to prove the other direction.
So we assume that all two-element subsets of $C$ are $(1,1)$-contractions, and we need to prove that $C$ is a $(1,1)$-contraction.
The argument is a straightforward generalization of the argument for (ii) from before.
Let $P$ be a path that contains exactly $k$ edges from $C$, and that starts and ends with an edge from $C$.
Let $e_1,e_2,\ldots,e_k$ be those edges and $u_{1,1},u_{1,2},u_{2,1},u_{2,2},\ldots,u_{k,1},u_{k,2}$ their end vertices as they are encountered when traversing $P$ (so $u_{1,1}$ and $u_{k,2}$ are the end vertices of $P$).
For all $i=1,2,\ldots,\lfloor k/2\rfloor$ the pair of edges $e_{2i-1}$ and $e_{2i}$ and their end vertices satisfy the distance conditions from~(ii).
From these conditions it follows that the subpath of $P$ between $u_{2i-1,1}$ and $u_{2i,2}$ has length at least $\dist_\ell(u_{2i-1,1},u_{2i,2})+2$.
So overall the length of $P$ is at least $\dist_\ell(u_{1,1},u_{k,2})+2\lfloor k/2\rfloor\geq \dist_\ell(u_{1,1},u_{k,2})+(k-1)$.
Consequently, we have $\dist_{\ell_C}(u_{1,1},u_{k,2})\geq \dist_\ell(u_{1,1},u_{k,2})-1$.
By Lemma~\ref{lem:add-contr-cond}, $C$ is a $(1,1)$-contraction.
\end{enumerate}
\end{proof}
With Lemma~\ref{lem:bip-contr} in hand, we are now ready to prove Theorem~\ref{thm:inapx-bip}.
\begin{proof}[Proof of Theorem~\ref{thm:inapx-bip}]
Let $\varepsilon>0$ be fixed and let $G=(V,E)$ be an instance of \Clique{}.
We construct a bipartite graph $H=H(G)$ as follows, see Figure~\ref{fig:inapx-reduc-bip}:
For every vertex $v \in V$, the graph $H$ contains two vertices $(v,1)$ and $(v,2)$ and the edge $f_v:=\{(v,1),(v,2)\}$.
For every edge $e=\{u,v\} \in E$, we add a vertex $x_e$ and the edges $f_{e,u}:=\{x_e,(u,1)\}$ and $f_{e,v}:=\{x_e,(v,1)\}$ to $H$.
Furthermore, we add a new special vertex~$s$ to $H$ and all the edges $\{s,(v,2)\}$, $v\in V$, and $\{s,x_e\}$, $e\in E$.
It is easy to check that the graph~$H$ defined in this way is bipartite.
All edges of $H$ receive unit lengths ($\ell = 1$) and we consider the instance $\cI=(H,\ell,\varphi)$ of the problem \Contraction{} with the tolerance function $\varphi(x)=x-1$.
\begin{figure}
\caption{An instance $G$ of \Clique{}
\label{fig:inapx-reduc-bip}
\end{figure}
For any set of vertices $U \subseteq V$ we define $C(U) := \{f_u : u \in U\}$ (see Figure~\ref{fig:inapx-reduc-bip}).
\underline{Claim~1:}
If $U \subseteq V$ is a clique in $G$, then $C(U)$ is a $(1,1)$-contraction in $H$ and $\Phi(C(U)) = |U|$.
\underline{Proof of Claim~1:}
Let $U$ be a a set of vertices in $G$ that form a clique, and let $u,v \in U$ be two vertices from this clique.
Then we have $\dist_\ell((u,1),(v,1))=\dist_\ell((u,2),(v,2))=2$ and $\dist_\ell((u,1),(v,2))=\dist_\ell((u,2),(v,1))=3$, so Lemma~\ref{lem:bip-contr}~(ii) implies that $C(\{u,v\})$ is a $(1,1)$-contraction in $H$.
Repeating this argument for every pair of vertices from $U$ and applying Lemma~\ref{lem:bip-contr}~(iii) yields that $C(U)$ is a $(1,1)$-contraction in $H$.
As there are never two edges in $H$ between any two connected components of the graph $(V,C(U))$, we have $\Phi(C(U)) = |C(U)| = |U|$.
\qedclaim
For any set of edges $C \subseteq E(H)$, we let $U(C)$ be the set of vertices $v \in V$ for which $(v,1)$ is incident to an edge in $C$.
\underline{Claim~2:}
If $C\subseteq E(H)$ is a $(1,1)$-contraction, then $C$ is a matching in $H$ and $U(C)$ is a clique in $G$ of size at least $\Phi(C)-3$.
\underline{Proof of Claim~2:}
$C$ is a matching by Lemma~\ref{lem:bip-contr}~(i).
Let $u,v \in U(C)$.
We will show that $e=\{u,v\}\in E$ by applying Lemma~\ref{lem:bip-contr}~(ii) to the two edges in $C$ incident to $(u,1)$ and $(v,1)$.
To prove that $e\in E$ it suffices to show that $\dist_\ell((u,1),(v,1))=2$.
Let us first consider the case that $f_u,f_v \in C$.
As $\dist_\ell((u,2),(v,2)) = 2$ (the shortest path between those vertices goes via~$s$), Lemma~\ref{lem:bip-contr}~(ii) implies that $\dist_\ell((u,1),(v,1))=2$.
We now consider the case that there is an edge $e'\in E\setminus\{e\}$ with $f_u, f_{e',v} \in C$.
We then have $\dist_\ell((u,2),x_{e'}) = 2$ (via~$s$), so Lemma~\ref{lem:bip-contr}~(ii) yields $\dist_\ell((u,1),(v,1))=2$.
Finally, we consider the case that there are two edges $e',e''\in E\setminus \{e\}$ with $f_{e',u}, f_{e'',v} \in C$.
We then have $\dist_\ell(x_{e'},x_{e''}) = 2$ (via~$s$), again implying that $\dist_\ell((u,1),(v,1))=2$.
This proves that indeed $e\in E$, so $U(C)$ forms a clique in $G$.
Every edge in $H$ is either incident to $s$ or to a vertex of the form $(v,1)$, $v\in V$.
Since at most one of the edges incident to $s$ can be in $C$, the definition of $U(C)$ shows that the size of $U(C)$ is either $|C|-1$ or $|C|$.
Therefore, to finish the proof of Claim 2, it suffices to show that $\Phi(C) \leq |C|+2$.
If $C$ contains no two edges that are connected by more than one edge in $H$, then we have $\Phi(C)=|C|$.
Otherwise we consider two such edges $f$ and $g$ from $C$.
It is easy to check that either $f$ or $g$ must be incident to $s$, so suppose that the edge $f$ contains $s$.
We first consider the case that $f=\{s,x_e\}$ for some edge $e=\{u,v\}\in E$.
In this case it follows that $g=\{(u,1),(u,2)\}$ or $g=\{(v,1),(v,2)\}$, so we have $\Phi(C)=|C|+2$.
Now consider the case that $f=\{s,(u,2)\}$ for some vertex $u \in V$.
In this case it follows that $g=\{(u,1),x_e\}$ for exactly one edge $e \in E$ incident to $u$ in $G$, showing that $\Phi(C)=|C|+2$.
In all three cases we have $\Phi(C)\leq |C|+2$, as claimed.
\qedclaim
Combining Claims~1 and 2 will allow us to prove the following claim:
\underline{Claim~3:}
If there is an $n^{1/2-\varepsilon}$-approximation algorithm for \Contraction{}, then there is an $n^{1-\varepsilon/2}$-approximation algorithm for \Clique{}.
\underline{Proof of Claim~3:}
Suppose for the sake of contradiction that such an approximation algorithm for \Contraction{} exists.
We use it to compute a clique in a given instance $G$ of \Clique{} as follows:
We construct $\cI=(H(G),\ell,\varphi)$ and compute a solution $C$ of \Contraction{} for this instance, and we define the clique $U(C)$ as before (recall Claim~2).
If $U(C)\neq \emptyset$, we return $U(C)$, otherwise we return any vertex from $G$.
We denote the clique computed in this fashion by $U$.
We may assume that $n(G) \geq 16^{1/\varepsilon}$, in particular $n(H) \geq 16^{1/\varepsilon}$.
It follows that
\begin{equation}
\label{eq:nHnG}
n(H) = 1+2n(G)+m(G) \leq 1+2n(G)+\binom{n(G)}{2} \leq n(G)^2.
\end{equation}
By assumption we know that
\begin{equation}
\label{eq:PhiC*}
\Phi(C)\cdot n(H)^{1/2-\varepsilon} \geq \Phi(C^*),
\end{equation}
where $C^*$ is an optimal solution of $\cI$.
In particular, $\Phi(C)$ is positive.
Combining these observations we get
\begin{align*}
|U|\cdot n(G)^{1-\varepsilon/2} &\geBy{eq:nHnG} |U|\cdot n(H)^{1/2-\varepsilon/2} \\
&\geq \max\{\Phi(C)-3,1\} \cdot n(H)^{1/2-\varepsilon/2} \\
&= \big(\max\{\Phi(C)-3,1\}\cdot n(H)^{\varepsilon/2}\big) \cdot n(H)^{1/2-\varepsilon} \\
&\geq \Phi(C) \cdot n(H)^{1/2-\varepsilon} \\
&\geBy{eq:PhiC*} \Phi(C^*)\\
&\geq \omega(G),
\end{align*}
where the second inequality holds because of Claim~2, and the last inequality involving the clique number $\omega(G)$ holds because of Claim~1.
\qedclaim
As $m(H)=\Theta(n(H))$, Claim~3 implies the theorem (using the inapproximability of \Clique{} proved in \cite{MR2403018}).
\end{proof}
\section{Hardness for multiplicative tolerance function}
\label{sec:hard-mult}
By Theorem~\ref{thm:cycle-hard-fix}, the problem \WContraction{} with purely additive tolerance function $\varphi(x)=x-\beta$ is NP-hard on cycles.
In this section we prove the hardness and inapproximability of this problem also in the case of a purely multiplicative tolerance function $\varphi(x)=x/\alpha$, $\alpha\geq 1$.
Recall that the problem \Contraction{} is trivial for this tolerance function (we may not contract any edges).
\subsection{Hardness of planar \WContraction{}}
\label{sec:planar}
To state the main result of this section recall that the \emph{girth} of a graph $G$ is defined as the minimum length of a cycle in $G$.
\begin{thm}
\label{thm:weak-planar}
For any $g\geq 2$, the problem \WContraction{} with tolerance function $\varphi(x)=x/2$, is NP-hard for planar graphs with girth at least $3g$ and unit length edges $\ell=1$.
\end{thm}
Theorem~\ref{thm:weak-planar} implies that \WContraction{} is hard for a general multiplicative tolerance function $\varphi(x)=x/\alpha$, $\alpha\geq 1$, but it leaves open the question whether this is true also for other fixed values of $\alpha$ other than 2 (when $\alpha$ is not part of the input).
The arguments given in this section for $\alpha=2$ carry over straightforwardly to any fixed value $2\leq \alpha<3$, but not to 3 or larger values (for $\alpha<2$ and unit length edges the problem is trivial).
We first characterize the set of feasible solutions in this special case.
\begin{lem}
\label{lem:weak-planar}
Let $G=(V,E)$ be a graph with girth at least 6 and unit length edges $\ell=1$, and consider the tolerance function $\varphi(x)=x/2$.
Furthermore, let $C\subseteq E$ be a set of edges such that $(V,C)$ is disconnected.
Then $C$ is a weak $(2,0)$-contraction if and only if for any two edges $e,f\in C$ either $e$ and $f$ are incident and both contain a degree-1 vertex, or any path containing $e$ and $f$ also contains at least two edges not in $C$.
\end{lem}
Recall that the assumption that $(V,C)$ is disconnected prevents solutions $C\subseteq E$ for which the contracted graph $G/C$ is a single vertex.
Note that Lemma~\ref{lem:weak-planar} does not require $G$ to be planar.
\begin{proof}
To prove the equivalence, we need the following auxiliary claim:
\underline{Claim:}
If $C$ is a weak $(2,0)$-contraction, then every component of $(V,C)$ that is not a single edge is a star with the property that each of its vertices except the center of the star has degree~1 in $G$.
\underline{Proof of Claim:}
Let $M$ be a component of $(V,C)$ with more than one edge.
Clearly, there must be an edge $\{u,v\}$ with vertices $u\notin V(M)$ and $v\in V(M)$.
If $M$ contains a path $P$ on two edges starting at $v$ and ending at some vertex $w$, then $\dist_\ell(u,w)=3$ and $\dist_{\ell_C}(u,w)=1$, a contradiction to the assumption that $C$ is a weak $(2,0)$-contraction (note that $P\cup \{u,v\}$ is the shortest path between $u$ and $w$, as the girth of $G$ is at least 6).
Thus the edges of $M$ must form a star centered at $v$.
By the same argument, no vertex outside $M$ can be connected to any vertex of $M$ other than $v$.
This proves the claim.
\qedclaim
We first assume that $C$ is a weak $(2,0)$-contraction, and we need to show that any two edges $e,f\in C$ satisfy the conditions of the lemma.
If $e$ and $f$ are incident, the statement follows from the auxiliary claim from before.
If $e$ and $f$ are not incident, we consider an inclusion-minimal path $P$ containing both $e$ and $f$.
We let $u$ and $v$ be the end vertices of $P$, $u'$ the other end vertex of $e$, and $v'$ the other end vertex of $f$ ($u'$ and $v'$ are the vertices at distance 1 from the ends of the path).
If the distance between $u'$ and $v'$ was only 1, we have $\dist_\ell(u,v)=3$ and $\dist_{\ell_C}(u,v)=1$ (here we need again the assumption that the girth is at least 6), a contradiction to the assumption that $C$ is a weak $(2,0)$-contraction.
Therefore at least two edges lie between $u'$ and $v'$.
The auxiliary claim from before implies that no two incident edges on $P$ between $u$ and $v$ are contained in $C$, therefore $P$ must contain at least two edges not in $C$.
This proves one direction of the equivalence.
To prove the other direction, we now assume that any two edges $e,f$ satisfy the conditions of the lemma, and we need to show that $C$ is a weak $(2,0)$-contraction.
Consider any two vertices $u$ and $v$ with $\dist_{\ell_C}(u,v)>0$, and any path between $u$ and $v$.
As no inner vertex of $P$ is a leaf, we know that between any two consecutive edges from $C$ on $P$ there are at least 2 edges not in $C$.
This proves that $\dist_{\ell_C}(u,v)\geq \dist_\ell(u,v)/2$, as desired.
This completes the proof of the lemma.
\end{proof}
For a given propositional formula $F$ in conjunctive normal form (CNF) the bipartite \emph{variable-clause graph} $\Gamma(F)$ is defined as follows:
The two partition classes of $\Gamma(F)$ are given by the sets of variables and clauses of $F$, and there is an edge between a variable $x$ and a clause $c$ if $x$ appears in $c$.
If $c$ contains $x$ as a positive or negative literal, we call the corresponding edge of $\Gamma(F)$ a \emph{positive or negative edge}, respectively.
A planar drawing of $\Gamma(F)$, where positive and negative edges appear in cyclically contiguous intervals around every variable vertex, is called \emph{contiguous}.
We call a $k$-CNF formula \emph{regular}, if every clause contains exactly $k$ literals, no clause contains a literal twice, every variable appears at least once as a positive literal and at least once as a negative literal in the formula.
Consider now the following variant of \textsc{3SAT}.
\begin{problem}{\CPSAT{}}
Input: & A regular 3-CNF formula $F$ and a contiguous planar drawing of $\Gamma(F)$. \\
Output: & `Yes', if $F$ has a satisfying assignment, `No' otherwise. \\
\end{problem}
\begin{lem}
\label{lem:cpsat}
\CPSAT{} is NP-complete.
\end{lem}
\begin{proof}
The more general variant of \CPSAT{} not requiring $F$ to be regular was shown to be NP-complete in \cite{DBLP:journals/ijcga/BergK12}.
We now show how to reduce this generalization to \CPSAT{}, which will prove the lemma.
Given a (not necessarily regular) 3-CNF formula $F$ we first eliminate all variables appearing only as negative or only as positive literals and all clauses containing exactly one literal, as well as multiple appearances of literals in the same clause.
This yields a formula $F'$ in which all clauses have two or three literals, no clause contains a literal twice, and every variable appears at least once as a positive literal and at least once as a negative literal in $F'$.
Moreover, since $\Gamma(F')$ is a subgraph of $\Gamma(F)$, we also obtain a contiguous planar drawing of $\Gamma(F')$.
As a last step we eliminate clauses $c$ with two literals by introducing a new variable $x$ for each of them and replacing $c$ by the equivalent formula $(c \lor x) \land (c \lor \ol{x})$.
It is easy to check that the resulting formula $F''$ is regular and equisatisfiable to $F$, and to obtain a contiguous planar drawing of $\Gamma(F'')$, see Figure~\ref{fig:exact-3}.
\begin{figure}
\caption{The clause $c = (y \lor \ol{z}
\label{fig:exact-3}
\end{figure}
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:weak-planar}]
We first present the proof for the case $g=2$, and then sketch how to generalize it for larger values of $g$.
We reduce \CPSAT{} to \WContraction{}.
Consider an instance $F$ of \CPSAT{} with variables $x_1,x_2, \ldots, x_n$ and clauses $c_1,c_2,\linebreak \ldots, c_m$.
Given the formula $F$, we construct from it a graph $G=G(F)$ as follows, see Figures~\ref{fig:gadgets} and~\ref{fig:sat}.
For every variable $x_i$, $i\in[n]$, we add a variable gadget $H(x_i)$ as shown on the left hand side of Figure~\ref{fig:gadgets} to the graph $G$.
The vertices $u_i$ and $\ol{u}_i$ will be used later to connect this gadget to other parts of the graph.
The idea of the variable gadget is that an optimal solution of our instance of \WContraction{} should contain either the four edges $T_i:=\{t_i,\ol{t}_i,t_i',t_i''\}$ or the four edges $F_i:=\{f_i,\ol{f}_i,f_i',f_i''\}$, corresponding to setting $x_i$ to \ttrue{} or \tfalse{}, respectively.
For every clause $c_j$, $j\in [m]$, we add a clause gadget $H(c_j)$ (a star with three edges) as shown on the right hand side of Figure~\ref{fig:gadgets} to the graph $G$.
The vertices $v_j^1$, $v_j^2$ and $v_j^3$ will be used later to connect this gadget to other parts of the graph.
The idea of the clause gadget is that a feasible solution contains at most one of these three edges, and if it does contain one of them, this restricts the choice we have inside the respective neighbouring variable gadget.
\begin{figure}
\caption{Variable gadget (left) and clause gadget (right) used in the proof of Theorem~\ref{thm:weak-planar}
\label{fig:gadgets}
\end{figure}
We connect the variable and clause gadgets in $G$ as follows (see Figure~\ref{fig:sat}):
For every $j\in[m]$ and $k\in[3]$, if the $k$-th literal in the clause $c_j$ is $x_i$, we add an edge connecting $u_i$ to $v_j^k$, and if the $k$-th literal in the clause $c_j$ is $\ol{x_i}$, we add an edge connecting $\ol{u}_i$ to $v_j^k$.
We refer to the edges added to $G$ in this step as \emph{connection edges}.
\begin{figure}
\caption{The graphs $\Gamma(F)$ (top) and $G=G(F)$ (bottom) constructed in the proof of Theorem~\ref{thm:weak-planar}
\label{fig:sat}
\end{figure}
This completes the definition of the graph $G=G(F)$.
It is easy to see that this graph is planar.
Specifically, a planar embedding can be obtained from the given planar embedding of $\Gamma(F)$ by replacing variable vertices $x_i$ in $\Gamma(F)$ by the variable gadgets $H(x_i)$ in $G$, and by replacing clause vertices $c_j$ by the clause gadgets $H(c_j)$.
Using that for each variable vertex $x_i$ in $\Gamma(F)$ the positive and negative edges appear in cyclically contiguous intervals around $x_i$, the connection edges in $G$ (that connect the variable and clause gadgets) can also be drawn in a planar fashion.
Moreover, it is easy to check that $G$ has girth~6 and no degree-1 vertices.
Now consider the instance $\cI:=(G,\ell,\varphi)$ of the problem \WContraction{} with $\ell=1$ (unit length edges) and the tolerance function $\varphi(x)=x/2$.
Lemma~\ref{lem:weak-planar} implies that any feasible solution of $\cI$ is a matching, as $G$ has no vertices of degree 1. As $G$ contains no cycles of length 3 or 4, it cannot contain two edges between vertex sets of two different components of $(V, C)$ for any such feasible solution $C$.
This implies that our objective function satisfies $\Phi(C)=|C|$.
We proceed to show that $F$ is satisfiable if and only if $\cI$ has an optimal solution of cardinality (and thus of value) $4n+m$.
Specifically, a satisfying assignment of $F$ corresponds to a solution that contains exactly all edges of either $T_i$ or $F_i$ in $H(x_i)$ for every variable $i\in[n]$ (corresponding to the value \ttrue{} or \tfalse{} assigned to this variable, respectively) and exactly one edge in $H(c_j)$ for each clause $j\in[m]$ (corresponding to a literal that satisfies this clause).
Formally, for any variable assignment $\tau\colon \{x_1,x_2,\ldots,x_n\}\to\{\ttrue,\tfalse\}$, we define the set of edges $C(\tau) \subseteq E(G)$ as follows:
$C(\tau)$ contains all edges of $T_i$ for any variable $x_i$, $i\in[n]$, that $\tau$ sets to \ttrue{}, and it contains all edges of $F_i$ for any variable $x_i$ that $\tau$ sets to \tfalse{}.
Moreover, for every clause $c_j$, $j\in[m]$, that is satisfied by $\tau$, we choose an index $k\in [3]$ of a literal in $c_j$ that is satisfied by $\tau$ and add the edge $e_j^k$ to $C(\tau)$.
The following claim is an immediate consequence of Lemma~\ref{lem:weak-planar}.
\underline{Claim~1:}
Any subset $C\subseteq E(G)$ is a feasible solution if and only if every path containing two edges from $C$ also contains at least two edges not in $C$.
By Claim~1, for every variable assignment $\tau$ of $F$, the set $C(\tau)$ is a feasible solution of $\cI$.
In particular, if $\tau$ satisfies $F$, then $C(\tau)$ is a feasible solution of size $4n+m$.
The remainder of the proof is devoted to showing the converse, i.e., if $C\subseteq E(G)$ is a feasible solution of size $4n+m$, then $F$ is satisfiable.
For all $i\in[n]$ we let $H(x_i)^+$ denote the subgraph of $G$ induced by all edges of $H(x_i)$ and all connection edges incident to either $u_i$ or $\ol{u}_i$.
\underline{Claim~2:}
For any $j\in[m]$, $C$ contains at most one edge from $H(c_j)$.
For any $i\in[n]$, $C$ contains at most four edges from $H(x_i)^+$.
Moreover, if $C$ contains one of the connection edges incident to $u_i$ or $\ol{u}_i$ for some $i\in[n]$, it does not contain any edges from the gadget $H(c_j)$ that is connected to $H(x_i)$ via this edge.
\underline{Proof of Claim~2:}
The first and last statement are immediate consequences of Claim~1.
The argument for the second statement is as follows:
For all $i\in[n]$ we let $E_i$ denote the set of edges $\{h_i,t_i,f_i\}$ plus the connection edges incident to $u_i$, and we let $\ol{E}_i$ denote the set of edges $\{\ol{h}_i,\ol{t}_i,\ol{f}_i\}$ plus the connection edges incident to $\ol{u}_i$.
By Claim~1, $C$ contains at most two edges from $E_i$, and if the intersection size is two, then $C$ must contain the edge $h_i$.
Similarly, $C$ contains at most two edges from $\ol{E}_i$, and if the intersection size is two, then $C$ must contain the edge $\ol{h}_i$.
As $C$ cannot contain $h_i$ and $\ol{h}_i$ simultaneously, $C$ contains at most three edges from $E_i\cup\ol{E}_i$, and if the intersection size is three, then $C$ must contain either $h_i$ or $\ol{h}_i$.
Again by Claim~1, $C$ contains at most two edges from the 6-cycle $\{f_i',h_i',t_i',f_i'',h_i'',t_i''\}$.
However, if $C$ contains one of the edges $h_i$ or $\ol{h}_i$, it contains at most one edge from this 6-cycle.
This proves that $C$ indeed contains at most four edges from $H(x_i)^+$.
\qedclaim
Note that every edge of $G$ belongs to exactly one subgraph $H(x_i)^+$ or $H(c_j)$.
So if $|C|=4n+m$, we know by Claim~2 that $C$ contains exactly four edges from $H(x_i)$ for all $i\in[n]$ and exactly one edge from $H(c_j)$ for all $j\in[m]$, and none of the connection edges in $G$.
\underline{Claim~3:}
For any $i\in[n]$, if $C$ contains four edges from $H(x_i)$ and if $f_i$ is not among them, then those edges must be $T_i$.
On the other hand, if $\ol{t}_i$ is not among them, those edges must be $F_i$.
In particular, these two cases cannot occur simultaneously.
\underline{Proof of Claim~3:}
If $C$ contains four edges from $H(x_i)$ and $f_i$ is not among them, Claim~1 enforces taking first the edge $t_i$, then $t_i'$ and $t_i''$, and eventually $\ol{t}_i$.
This proves the first part of the statement.
The argument for the second part is symmetric.
The third part of the statement is a consequence of the first two.
\qedclaim
So given a solution $C$ of $\cI$ of size $4n+m$, we can derive from it a satisfying assignment $\tau$ of $F$ as follows:
For every clause $c_j$, $j\in[m]$, we consider the unique edge $e_j^k$ from $H(c_j)$ that belongs to $C$.
We follow the attachment edge incident to $e_j^k$, leading to the corresponding variable gadget $H(x_i)$, and connecting to either $u_i$ or $\ol{u}_i$.
If the attachment edge connects to $u_i$, then by Claim~1, $f_i\notin C$, so by Claim~3, the four edges of $H(x_i)$ contained in $C$ must be $T_i$, so we define $\tau(x_i):=\ttrue{}$.
If the attachment edge connects to $\ol{u}_i$, then by Claim~1, $\ol{t}_i\notin C$, so by Claim~3, the four edges of $H(x_i)$ contained in $C$ must be $F_i$, so we define $\tau(x_i):=\tfalse{}$.
This process does not lead to any contradicting variable assignments by the last statement of Claim~3.
However, this process may leave some variables $x_i$ undefined, and we can set them arbitrarily, e.g., $\tau(x_i):=\ttrue{}$.
By construction, each clause receives a satisfying literal, so the assignment $\tau$ is indeed a satisfying assignment of $F$.
This proves that $F$ is satisfiable if and only if $\cI$ has a feasible solution of size $4n+m$ (which must be optimal by Claim~2), completing the proof of the theorem in the case $g=2$.
For values $g\geq 2$, the construction of the gadgets $H(x_i)$ and $H(c_j)$ can be generalized as follows:
We subdivide each of the edges $h_i,\ol{h}_i$ and $h_i''$, and each of the edges $e_j^1$, $e_j^2$ and $e_j^3$ into $1+3(g-2)$ edges.
Then the resulting graph $G=G(F)$ clearly has girth $3g$, and the above arguments can be easily modified to show that any solution $C$ of $\cI$ contains at most $1+3(g-2)=3g-5$ edges from $H(c_j)$ for all $j\in[m]$, and at most $4+3(g-2)=3g-2$ edges from $H(x_i)^+$ for all $i\in[n]$, and that $F$ is satisfiable if and only if $\cI$ has an optimal solution of size $(3g-2)n+(3g-5)m$.
This completes the proof.
\end{proof}
\subsection{Inapproximability of \WContraction{}}
\label{sec:inapx-wcontraction}
We are able to further extend our hardness results for \WContraction{} as follows:
\begin{thm}
\label{thm:inapx-weak}
For any $\varepsilon > 0$, it is NP-hard to approximate the problem \WContraction{} with tolerance function $\varphi(x)=2x/3$ to within a factor of $n^{1-\varepsilon}$.
\end{thm}
Theorem~\ref{thm:inapx-weak} implies that \WContraction{} is hard to approximate for general multiplicative tolerance functions $\varphi(x)=x/\alpha$, $\alpha\geq 1$, but it leaves open the question whether this is true also for other fixed values of $\alpha$ other than $3/2$ (when $\alpha$ is not part of the input).
The arguments given in this section for $\alpha=3/2$ carry over straightforwardly to any fixed value $1<\alpha<2$, but not to 2 or larger values (for $\alpha=1$ the problem is trivial).
This time we reduce from the well-known \IndSet{} problem (which is equivalent to \Clique{} by considering the complement graph).
Recall that an \emph{independent set} in a graph $G$ is a subset of vertices of $G$ such that no two vertices in the subset are adjacent.
\begin{problem}{\IndSet{}}
Input: & A graph~$G$. \\
Output: & An independent set in $G$ of maximum size. \\
\end{problem}
We use again the fact that for any $\varepsilon > 0$, \IndSet{} is NP-hard to approximate to within a factor of $n^{1-\varepsilon}$ \cite{MR2403018}.
\begin{proof}[Proof of Theorem~\ref{thm:inapx-weak}]
Let $G=(V,E)$ be an instance of \IndSet{}.
We construct a graph $H=H(G)$ and a length function $\ell$ on the edges of $H$ as follows, see Figure~\ref{fig:indset}:
We start with a copy of $G$, and all edges of this copy receive length~2.
The vertices of this copy are also denoted by $V$.
We then add additional vertices and edges to $H$ as follows:
To every vertex $v\in V$ we attach two pending edges $\{v,(v,1)\}$ and $\{v,(v,2)\}$ of length~1 or 2, respectively.
We may assume $G$, and thus also $H$, to be connected.
\begin{figure}
\caption{An instance $G$ of \IndSet{}
\label{fig:indset}
\end{figure}
Now consider the instance $\cI=(H,\ell,\varphi)$ of the problem \WContraction{} with the tolerance function $\varphi(x)=2x/3$.
We proceed to show that $\cI$ has a feasible solution of value $k$ if and only if $G$ has an independent set of size $k$.
This is an immediate consequence of Claim~3 below.
To prove Claim~3 we need the following two auxiliary claims.
\underline{Claim~1:}
For any induced subgraph of $H$ that is a path on two edges, a feasible solution $C$ of $\cI$ does not contain only the longer of the two edges (either it contains none of the two, the shorter of the two if there is one, or both).
\underline{Proof of Claim~1:}
Consider a path on two edges $\{u,v\}$, $\{v,w\}$ of length~2 in $H$ such that $\{u,w\} \notin E(H)$, and suppose for the sake of contradiction that $\{u,v\}\in C$, but $\{v,w\}\notin C$.
Then we have $\dist_\ell(u,w)=4$ and $\dist_{\ell_C}(u,w)=2$, violating the condition \eqref{eq:contr-cond} for the given tolerance function.
A similar contradiction arises if one of the edges has length~1 and the other length~2, and only the edge of length~2 is contracted.
This proves the claim.
\qedclaim
\underline{Claim~2:}
No feasible solution of $\cI$ contains an edge of length~2.
\underline{Proof of Claim~2:}
Assume for the sake of contradiction that a feasible solution $C$ contains an edge $e$ of length~2. Note that any edge $f$ of $H$ may be reached from $e$ via a walk $e_1\dots e_k$ where $e_1 = e$ and $e_k = f$, and for all $i<k-1$ we have $\ell(e_i) = 2$ and the edges $e_i$ and $e_{i+1}$ induce a path in $H$. Now successively applying Claim~1 to the subgraphs induced by $e_i$ and $e_{i+1}$ for $i < k$ shows that $C$ contracts $f$. Thus $C$ violates the condition that a weak contraction must not contract every edge.
\qedclaim
Claim~2 implies that our objective functions satisfies $\Phi(C)=|C|$ for every feasible solution $C$ of $\cI$, because $H$ never contains two edges between two different connected components of $(V,C)$.
For any set of vertices $U \subseteq V(G)$ we define $C(U) := \{\{u,(u,1)\}: u \in U\}$.
\underline{Claim~3:}
A set of edges $C\subseteq E(H)$ is a feasible solution of $\cI$ if and only if $C=C(U)$ for an independent set $U$ in $G$.
\underline{Proof of Claim~3:}
Let $C$ be a feasible solution of $\cI$.
By Claim~2, $C$ contains only edges of length~1, so we have $C=C(U)$ for some set of vertices $U$ in $G$.
Suppose that two such vertices $u,v\in U$ are connected by an edge, then we would have $\dist_\ell(u,v)=4$ and $\dist_{\ell_C}(u,v)=2$, violating the condition \eqref{eq:contr-cond} for the given tolerance function.
It follows that $U$ is an independent set.
To prove the other direction of the equivalence, let $U$ be an independent set in $G$ and consider the set of edges $C(U)$ in $H$.
To verify that $C$ is a weak $(3/2,0)$-contraction, it suffices to check condition \eqref{eq:contr-cond} between the end vertices of paths on two edges, one of length~1 from $C$ and the other of length~2, and for paths on $k$ edges that start and end with an edge of length~1 from $C$.
In the first case the contraction $C(U)$ changes the distance from 3 to 2, which is compatible with \eqref{eq:contr-cond}.
In the second case the contraction $C(U)$ changes the distance from $2k-2$ to $2k-4$, which is also compatible with \eqref{eq:contr-cond}, where we use that $k\geq 4$ because of the assumption that $U$ is an independent set.
\qedclaim
Claim~3 implies that $\cI$ has a feasible solution with $k$ edges if and only if $G$ has an independent set of size $k$.
As $n(H)=3n(G)=\cO(n(G))$, the theorem follows from the \cite{MR2403018} result.
\end{proof}
\section{Asymptotic bounds}
\label{sec:asymp}
In this section we show how to compute contractions for graphs that are not optimal, but can be computed efficiently despite our hardness results from the previous section.
In this vein, the main results of this section are Theorem~\ref{thm:asymp-mult} and the corresponding (not tight) lower bound (Theorem~\ref{thm:asymp-girth}) for the case of tolerance functions of the form $\varphi(x)=x/\alpha-1$.
Further we consider purely additive tolerance functions (Section~\ref{sec:asymp-add}) and the factor by which a contraction can reduce the number of vertices (Section~\ref{sec:asymp-vert}).
Throughout this section, we assume all graphs to have unit length edges $\ell=1$.
\subsection{Almost multiplicative contractions}
\label{sec:asymp-mult}
As mentioned in the introduction, a purely multiplicative tolerance function ($\beta=0$) forbids decreasing any distances.
In this section we thus consider an `almost' purely multiplicative tolerance function of the form $\varphi(x)=x/\alpha-1$.
\begin{thm}
\label{thm:asymp-mult}
Let $k\geq 1$ be a real number.
Any graph $G$ has a $(2k-1,1)$-contraction $C$ such that the contracted graph $G/C$ has at most $n^{1+1/k}$ edges, and such a contraction can be computed in time $\cO(m)$.
\end{thm}
Recall that here and throughout, $n$ and $m$ denote the number of vertices and edges of the input graph $G$, not of the contracted graph $G/C$.
Setting $k:=\log_2 n$ in Theorem~\ref{thm:asymp-mult} yields the following corollary.
\begin{cor}
\label{cor:asymp-mult}
Any graph $G$ has a $(2\log_2 n-1,1)$-contraction $C$ such that the contracted graph $G/C$ has at most $2n$ edges, and such a contraction can be computed in time $\cO(m)$.
\end{cor}
To prove Theorem~\ref{thm:asymp-mult}, we use a clustering approach as presented in \cite{MR810338}, yielding the next lemma.
Specifically, the following crucial lemma appears in a slightly weaker form in that paper.
For any real number $r\geq 1$, we define an \emph{$r$-partition} of a graph $G=(V,E)$ as a set of \emph{clusters} $P_i \subseteq V$, $i\in[l]$, with corresponding \emph{cluster centers $p_i \in P_i$}, where the sets $P_i$ are required to form a partition of the vertex set $V$ and where $\dist_\ell(p_i,u)\leq r-1$ for all $u\in P_i$ and $i\in[l]$.
We denote the resulting $r$-partition by $P:=\{(p_i,P_i) : i\in[l]\}$.
We write $\rho(P)$ for the number of pairs $1\leq i<j\leq l$ for which $P_i$ and $P_j$ are connected by at least one edge, and we refer to this quantity as the \emph{density of $P$}.
\begin{lem}
\label{lem:awerbuch}
Let $r\geq 1$ be a real number.
Any graph $G$ with unit length edges has an $r$-partition $P$ with density $\rho(P)\leq n^{1+1/r}$, and such a partition can be computed in time $\cO(m)$.
\end{lem}
\begin{proof}
The idea of the algorithm is to build an $r$-partition $P$ of $G$ iteratively in rounds.
In each round, we build a new cluster and remove all vertices from that cluster from the graph, processing the subgraph on the remaining vertices in the next round.
The algorithm proceeds until all vertices are assigned to a cluster.
In round $i$, we choose an arbitrary vertex $p_i$ as a cluster center, and define layers $L_{i,0},L_{i,1},\ldots$ around the vertex $p_i$, where the layer $L_{i,j}$ consists of all vertices at distance exactly $j$ from $p_i$ (this distance is measured in the subgraph of $G$ under consideration in this round).
We continue computing these layers as long as the number of vertices in the new layer is at least the number of vertices in all previous layers times the factor $n^{1/r}$.
The cluster $P_i$ is defined as the union of all layers around $p_i$ satisfying this expansion condition.
We refer to the first layer violating this condition (which is \emph{not} added to $P_i$ anymore) as the \emph{rejected layer}.
We let $P$ denote the partition of the vertices of $G$ computed in this fashion.
To verify that $P$ is indeed an $r$-partition, we proceed to show that each vertex within a cluster has distance at most $r-1$ from the center vertex of that cluster, and that the density $\rho(P)$ of the partition is at most $n^{1+1/r}$.
Intuitively, the expansion condition in the definition of the layers ensures that a cluster has few layers and that the number of edges that go to unclustered vertices is small.
Consider a cluster $P_i$ with center vertex $p_i$ and the layers $L_{i,0},L_{i,1},\ldots,L_{i,d}$.
Suppose for the sake of contradiction that $d\geq r$.
By the definition of the layers in the algorithm we know that $|L_{i,j}|\geq n^{1/r}\sum_{k=0}^{j-1} |L_{i,k}|$ holds for all $j\in [d]$, implying that $|L_{i,j}|\geq n^{j/r}$.
Consequently, the size of the cluster satisfies $|P_i|=\sum_{j=0}^d |L_{i,j}|\geq 1+n^{r/r}=n+1$, a contradiction.
We now show that $\rho(P)\leq n^{1+1/r}$.
The key idea is that the number of vertices in the rejected layer of a cluster $P_i$ is at most $n^{1/r}|P_i|$.
Thus the number of edges from $P_i$ to clusters that are created later is at most $n^{1/r}|P_i|$.
For every edge between two clusters we let the cluster that is created first account for that edge.
Summing over all these edges between clusters yields the desired upper bound of $\rho(P)\leq n\cdot n^{1/r}=n^{1+1/r}$.
Using breadth-first search, the partitioning algorithm described above runs in time $\cO(m)$ (recall that $G$ is assumed to be connected).
This completes the proof of the lemma.
\end{proof}
With Lemma~\ref{lem:awerbuch} in hand, we are now ready to prove Theorem~\ref{thm:asymp-mult}.
\begin{proof}[Proof of Theorem~\ref{thm:asymp-mult}]
Given $G=(V,E)$, we first compute a $k$-partition $P$ into $l$ clusters as described by Lemma~\ref{lem:awerbuch}.
We define the set $C$ of contracted edges as the union of all edges within the clusters, $C := \{\{u,v\} \in E : u,v \in P_i \text{ for some } i \in [l]\}$.
We thus contract each cluster into a single vertex and remove from every set of resulting parallel edges all but a single edge.
We proceed to show that $C$ is a $(2k-1,1)$-contraction, i.e., we show that $\dist_{\ell_C}(u,v) \geq \dist_{\ell}(u,v)/(2k-1)-1$ for all $u,v\in V$.
Consider two vertices $u \in P_i$ and $v \in P_j$, where $i$ and $j$ might be equal.
Let $Q_{u,v}$ be the shortest path from $u$ to $v$ in $G$ with edge lengths $\ell_C$ (all edges from $C$ receive length zero).
The length $d$ of $Q_{u,v}$ is the number of edges on that path that connect different clusters.
Note that $Q_{u,v}$ enters and leaves each of the $d+1$ visited clusters at most once, using at most $2k-2$ edges in every cluster, so in $G$ (where all edges have unit lengths) we get $\dist_{\ell}(u,v) \leq d + (d+1)(2k-2)$.
Combining these observations we obtain
\begin{equation*}
\dist_{\ell_C}(u,v) = d \geq d - \frac{1}{2k -1} = \frac{d + (d+1)(2k-2)}{2k -1} -1 \geq \frac{\dist_{\ell}(u,v)}{2k -1} -1,
\end{equation*}
proving the claim.
It remains to show that the contracted graph $G/C$ has at most $n^{1+1/k}$ edges, which is an immediate consequence of the upper bound $m(G/C)=\rho(P)\leq n^{1+1/k}$ given by Lemma~\ref{lem:awerbuch}.
This completes the proof of the theorem.
\end{proof}
Erd\H{o}s' girth conjecture \cite{MR0180500} asserts that there exist graphs with $\Omega(n^{1+1/k})$ edges and girth $2k+1$.
It has been verified for $k=1,2,3,5$ \cite{MR1109426} and the strongest spanner lower bounds depend on it.
We derive from the conjecture the following (not tight) lower bound.
\begin{thm}
\label{thm:asymp-girth}
Assuming Erd\H{o}s' girth conjecture, there exists for any integer $k\geq 2$ a graph $G$ such that any $(k-1,1)$-contraction $C$ results in a graph $G/C$ with $\Omega(n^{1+1/k})$ edges.
\end{thm}
\begin{proof}
For a given integer $k\geq 2$ let $G$ be a graph that is guaranteed by Erd\H{o}s' girth conjecture, i.e., $G$ has girth $2k+1$ and $\Omega(n^{1+1/k})$ edges.
Consider any $(k-1,1)$-contraction $C$ on $G$, and consider a connected component of the graph $(V,C)$.
Applying \eqref{eq:contr-cond} shows that $\dist_\ell(u,v)\leq k-1$ holds for any two vertices $u$ and $v$ in that component.
Using that the girth of $G$ is $2k+1$, it follows that for any cycle in $G$, the connected component of $(V,C)$ does not contain a contiguous segment of cycle edges of length at least half of the cycle.
This implies that all connected components of the graph $(V,C)$ are trees with diameter at most $k-1$.
Therefore, the total number of edges within all connected components of $(V,C)$ is at most $n$.
We will further argue that there is at most one edge between any two connected components.
Suppose for the sake of contradiction that there are two components of $(V,C)$ with two different edges connecting them, say $\{u,v\}$ and $\{u',v'\}$, where $u$ and $u^\prime$ lie in the same connected component and $v$ and $v'$ in the other.
As the diameter of each component is at most $k-1$, it follows that in $G$ there is a path from $u$ to $u'$ of length at most $k-1$, and a path from $v$ to $v'$ of length at most $k-1$.
Together with the two edges connecting the components we obtain a cycle of length at most $2(k-1)+2=2k$, contradicting the assumption that $G$ has girth $2k+1$.
Therefore, the resulting graph after the contraction has $\Omega(m)=\Omega(n^{1+1/k})$ edges.
\end{proof}
\subsection{Additive contractions}
\label{sec:asymp-add}
Turning to the case of a purely additive error, we obtain the following two results.
\begin{thm}
\label{thm:asymp-add}
Let $G$ be a graph with unit length edges.
\begin{enumerate}[label=(\roman*),leftmargin=7mm]
\item For any even integer $0 \leq k \leq n$, the set of edges incident to the $k/2$ vertices of highest degrees is a $(1,k)$-contraction $C$ in $G$ with $\Phi(C) \geq km/(2n)$.\label{thm:asymp-add-first}
\item For any real number $0 < k \leq n$, the set of edges incident to two vertices of degree at least $n/k$ is a $(1,k)$-contraction $C$ in $G$ such that $G/C$ has $\cO(n^2/k)$ edges.\label{thm:asymp-add-high}
\end{enumerate}
These contractions can be computed in time $\cO(m)$.
\end{thm}
As mentioned in the introduction, Bernstein and Chechik analyzed the contraction of Theorem~\hyperref[thm:asymp-add-high]{\ref*{thm:asymp-add}~\ref*{thm:asymp-add-high}} in~\cite{MR3536582} and used it in their dynamic shortest paths algorithm, so this part is already proved.
\begin{proof}[Proof of Theorem~{\hyperref[thm:asymp-add-first]{\ref*{thm:asymp-add}~\ref*{thm:asymp-add-first}}}]
Let $U$ be the set of $k$ vertices in $G$ of highest degree.
Then we have
\begin{equation*}
\sum_{u \in U} \deg(u) \geq k/n \sum_{v \in V} \deg(v) = k/n\cdot 2m = 2km/n.
\end{equation*}
Let $C$ be the set of edges incident to any vertex in $U$.
As each edge is incident to at most two vertices in $U$, we get $|C| \geq 1/2 \sum_{u \in U} \deg(u) \geq km/n$ from the previous inequality.
As no shortest path visits a vertex in $U$ twice, $C$ is indeed a $(1,2k)$-contraction.
The set $C$ can be computed as follows: We first compute the degrees of all vertices in time $\cO(m)$, then find the $k$-th largest element in this list in time $\cO(n)$, and by another linear time sweep over this list we select $k$ vertices of highest degree.
Overall, the required time is $\cO(m)$.
\end{proof}
This result implies that the number of edges in $G/C$ is at most $m-k m/n$.
If $G$ is a path, no $(1,2k)$-contraction has an objective value greater than $2k$, and $k m/n=k(1-1/n)$, showing that the objective value in Theorem~\hyperref[thm:asymp-add-first]{\ref*{thm:asymp-add}~\ref*{thm:asymp-add-first}} can be improved by at most a factor of two.
The information theoretic lower bound in~\cite{MR3536579} implies that for all $\varepsilon > 0$, any contraction $C$ such that $G/C$ has $\cO(n^{4/3 - \varepsilon})$ edges does not admit a constant additive error.
\subsection{Vertex reduction}
\label{sec:asymp-vert}
All of the results above show that contractions can be effectively used to reduce the number of edges in a dense graph.
But one possible advantage of using a contraction instead of a spanner is that it also has the potential to reduce the number of \emph{vertices} in the graph.
Unfortunately, for constant approximation errors, it is not possible to guarantee more than a constant-factor reduction in general graphs: it is not hard to see that given a path on $n$ vertices, any $(k,1)$-contraction will still result in at least $n/(k+1)$ vertices.
The same problem applies to general dense graphs, since they could still contain a long path within them.
That being said, it seems likely that in practice contraction can lead to significant vertex reduction in many dense graphs.
We ground this practical intuition with the following theoretical result for the special case of graphs with large minimum degree.
\begin{thm}
\label{thm:min-degree}
Let $D$ be an integer.
Any graph $G$ with minimum degree at least $D$ has a $(5,1)$-contraction $C$ such that the contracted graph $G/C$ has at most $n/D$ vertices, and such a contraction can be computed in time $\cO(m)$.
\end{thm}
\begin{proof}
Recall the definition of an $r$-partition.
For a cluster $P_i$ with center vertex $p_i$ we refer to $r$ as the \emph{radius} of that cluster.
This is the maximum distance of all cluster vertices from $p_i$.
We will show how to construct a $3$-partition in which the number of clusters $P_i$ is at most $n/D$.
Using the exact same argument as in the proof of Theorem~\ref{thm:asymp-mult}, such a $3$-partition yields the desired $(5,1)$-contraction.
Our construction first builds clusters of radius~1, and then extends them to clusters of radius~2.
The clustering with radius~1 proceeds very similarly as in the proof of Lemma~\ref{lem:awerbuch} before with $r=1$.
The crucial difference is that we choose as center vertices only vertices with degree at least $D$.
If no such vertices are left, the clustering process terminates, and the remaining unclustered vertices have degree strictly less than $D$.
It is easy to see that since those vertices have degree at least $D$ in the original graph, they must be adjacent to a vertex in a radius~1 cluster.
We can thus assign each of those vertices to such a cluster arbitrarily, yielding a clustering of all vertices of $G$ with radius~2.
The number of clusters is at most $n/D$ because by construction every cluster contains at least $D$ vertices.
This shows that the number of vertices in the contracted graph is at most $n/D$.
This algorithm can be implemented in time $\cO(m)$ by using an adjacency list representation where we keep track of degree information after removing an edge from the graph.
\end{proof}
To see that we cannot guarantee less than $n/D$ vertices, even with larger approximation error, consider the graph $G$ that consists of $n/D$ isolated $D$-cliques.
We now show that even if $G$ is connected, we cannot guarantee $o(n/D)$ vertices in the contracted graph, even if we allow a larger (constant) approximation error.
\begin{thm}
\label{thm:asymp-degree}
Let $D$ and $k$ be integers.
There exists an infinite family of $n$-vertex graphs $G$ with minimum degree $D$ such that any $(k,1)$-contraction $C$ results in a graph $G/C$ with $n/((k+1)D)$ vertices.
\end{thm}
\begin{proof}
Assume for simplicity that $n$ is divisible by $D$.
We construct the graph $G$ as follows.
We partition the $n$ vertices into $n/D$ layers, with each layer containing exactly $D$ vertices.
For $1 \leq i < n/D$, all vertices in layer~$i$ receive an edge to all vertices in layer~$i+1$.
Clearly all vertices in the resulting graph have degree at least $D$.
Let $u$ and $v$ be two vertices in layers~$i$ and $j$, respectively.
Then clearly we have $\dist_{\ell}(u,v) \geq |j-i|$.
Now let $C$ be any $(k,1)$-contraction on $G$, and consider the connected components of the graph $(V,C)$.
Applying \eqref{eq:contr-cond} shows that $\dist_\ell(u,v)\leq k$ holds for any two vertices $u$ and $v$ in the same component.
Combining these two inequalities shows that every connected component contains vertices from at most $k+1$ layers.
As there are $n/D$ layers, the contracted graph has at least $n/((k+1)D)$ vertices.
\end{proof}
{}
\end{document} |
\begin{document}
\preprint{APS/123-QED}
\title{Quantum noise correlations of an optical parametric oscillator based on a non-degenerate four wave mixing process in hot alkali atoms}
\author{A. Monta\~na Guerrero$^1$}
\author{P. Nussenzveig $^1$}
\author{M. Martinelli $^1$}
\author{A. M. Marino $^2$}
\author{H. M. Florez $^1$}
\email{hans@if.usp.br}
\affiliation{$^1$ Instituto de F\'{\i}sica, Universidade de S\~ao Paulo, 05315-970 S\~ao Paulo, SP-Brazil}
\affiliation{$^2$ Center for Quantum Research and Technology and Homer L. Dodge Department of Physics and Astronomy, The University of Oklahoma, Norman, Oklahoma 73019, USA}
\date{\today}
\pacs{Valid PACS appear here}
\maketitle
\section{Data Analysis}
Fig. \ref{fig:datoBruto} shows an example of raw data in which line 1 corresponds to the electronic noise, line 2 to the shot noise associated with the sum of the power of the two beams, line 3 to the shot noise associated with a single beam, line 4 to the noise of the intensity difference between signal and idler beams without normalization, and line 5 to the intensity noise associated to a single beam (signal or idler).
\begin{figure}
\caption{Measured noise spectra used to evaluate the normalized curves on the main text. T=91$^{\circ}
\label{fig:datoBruto}
\end{figure}
For the evaluation of the normalized noise presented in the main paper, all the measured spectra had their electronic noise subtracted before normalization by their corresponding shot noise level. This reference level was obtained using a sample of the pump laser, strongly attenuated for obtaining the same intensity of the output fields generated by the OPO. We have tested the intensity noise of the laser, verifying the linear response of the noise spectra with the power of the reference beam within a precision of 1\%.
\begin{figure}
\caption{Normalized noise of the subtraction of the photocurrents. Curves 1 to 6 correspond to a input pump power of $\sigma$= 1.26, 1.44, 1.62, 1.73, 1.82 and 2.1mW respectively, all of them with $P_{Th}
\label{fig:RuidoVsFrecuenciaT}
\end{figure}
The evolution of the narrow peak in the noise spectra for different pump powers can be seen in Fig. \ref{fig:RuidoVsFrecuenciaT}. We present the normalized power spectrum of the intensity difference between signal and idler. Other parameters are the same as in Fig. \ref{fig:datoBruto}. Curves 1 to 6 correspond to an input pump power of 278, 318, 359, 383, 403 and 465 mW respectively.
The peak frequency was evaluated by subtraction of the background noise taken as a baseline, followed by a fit to a Lorenztian. Central frequency is shown in Fig. \ref{fig:my_label2}, as a function of the pump power, while the error bars are estimated from the width of the fitted curve.
The linear adjust corresponds to proportionality ratio of 38.4 (3) MHz/W.
\begin{figure}
\caption{Peak frequency as a function of the pump power}
\label{fig:my_label2}
\end{figure}
\end{document} |
\begin{document}
\title{Existence of families of Galois
representations and new cases of the Fontaine-Mazur conjecture}
\begin{abstract}
In a previous article, we have proved a result asserting the existence
of a compatible family of Galois representations containing a given
crystalline irreducible odd two-dimensional representation. We apply this
result to establish new cases of the Fontaine-Mazur conjecture, namely,
an irreducible Barsotti-Tate $\lambda$-adic $2$-dimensional Galois
representation unramified at $3$ and such that the traces $a_p$
of the images of Frobenii verify ${\mathbb{Q} }(\{ a_p^2 \}) = {\mathbb{Q} } $ always
comes from an abelian variety. We also show the non-existence of
irreducible Barsotti-Tate $2$-dimensional Galois representations of conductor $1$
and apply this to the irreducibility of Galois representations on
level 1 genus 2 Siegel cusp forms.
\end{abstract}
\section{Existence of families}
The following is a slight generalization of a result proved in [D3],
which follows from the results and techniques in [T1], [T2]:
\begin{teo}
\label{teo:Taylor} Let q be an odd prime, and $Q$ a place above $q$. Let
$\sigma_{Q}$ be a two dimensional odd irreducible $Q$-adic Galois
representation (of the absolute Galois group of ${\mathbb{Q} }$, continuous)
ramified only at $q$ and at a finite set of primes $S$. Assume that
$\sigma_Q$ is crystalline at $q$, with Hodge-Tate weights $\{0, w \}$ ($w$ odd).
Assume also that $q \geq 2w + 1$.
Then, there exists a compatible family of
Galois representations $\{ \sigma_{\lambda} \}$ containing $
\sigma_{Q}$, such that
for every $\ell \not\in
S$, $\lambda \mid \ell$, the representation $\sigma_{\lambda}$ is
unramified outside $\{ \ell \} \cup S$ and is crystalline at $\ell$ with
Hodge-Tate weights $\{ 0 , w \}$. Moreover, the family $\{ \sigma_{\lambda} \}$
is strictly compatible (see [T1] for the definition) and all its members are
irreducible.
\end{teo}
Remark: the proof given in [D3] applies in this generality, just
observe that for the case of $w =1$, in general when Taylor proves
that the restriction of $\sigma_Q$ to some totally real field $F$ will correspond
to a representation attached to a Hilbert modular form $h$, this is not
enough to conclude that the representation is motivic (the construction
of Blasius-Rogawski does not apply in some cases), thus is not
enough in general to prove the Fontaine-Mazur conjecture, but in any
case there is a strictly compatible family of Galois representation
attached to $h$ (by previous results of Taylor) and so the argument
in [D3] (descent of this family to a compatible family of $G_{\mathbb{Q} }$-representations)
can be applied also in this case. The family obtained will be
strictly compatible, as follows from the results in [T1]. All representations
in the family are irreducible because when restricted to the Galois group of
$F$ they agree with the modular Galois representations attached to
$h$, which are irreducible (because $h$ is cuspidal).\\
\section{Fontaine-Mazur for ``projectively rational" Barsotti-Tate representations}
Now suppose that we are given a representation $\sigma_Q$ as in
theorem \ref{teo:Taylor} with $w = 1$ and $\det (\sigma_Q) = \chi$.
Then, we will prove the following:
\begin{teo}
\label{teo:abel} Assume that $\sigma_Q$ verifies also the following two
conditions:\\
1) If $q \neq 3$, then $3 \not\in S$ ($\sigma_Q$ unramified at $3$).\\
2) The traces $\{ a_p \}$ of the images of Frobenii,
for every $p \neq q, p \not\in S$
verify: $a_p^2 \in \mathbb{Z}$.\\
Then $\sigma_Q$ can be attached to an
abelian variety $A$ (of course, $A$ is of ${\rm GL}_2$-type). \\
\end{teo}
Remark: In particular, in case all the traces are integers, this result
shows that a Galois representation that ``looks like" the one attached
to an elliptic curve with good reduction at $3$
does indeed come from such an elliptic curve.\\
Remark: A similar result is proved in [T1] without restriction on the traces
but (sticking to the case $w=1$) with the extra assumption
that there is
a prime $u \in
S$ such that the restriction of $\sigma_Q$ to the decomposition group $D_u$
is of a particular type
(corresponding to discrete
series under the local Langlands correspondence). Thus, this result
does not apply, for example, if $\sigma_Q$ has conductor 1 or
is semistable (i.e., unipotent) locally at every prime of $ S$. Therefore, in the
semistable case, the results of [T1] are not enough to prove the Fontaine-Mazur
conjecture for $\sigma_Q$.\\
Proof:
The proof follows from the combination of theorem
\ref{teo:Taylor} with modularity results … la Wiles. We know that
there exists a strictly compatible family $\{ \sigma_\lambda \}$
containing $\sigma_Q$. Take $t \mid 3$ and consider the Galois
representation $\sigma_t$ (if $q=3$, just take $t=Q$).
As $3 \not\in S$, we know that $\sigma_t$
is crystalline at $t$, and has Hodge-Tate weights $\{ 0, 1\}$.
Following the initial idea of Wiles (see also [D1]) we know, from
condition 2) in the theorem (via results of Langlands and Tunnell), that the
residual representation $\bar{\sigma}_t$ will be either modular or
reducible. The information we have on $\sigma_t$ is enough then to
conclude, via a combination of modularity results … la Taylor-Wiles
and Skinner-Wiles, that $\sigma_t$ is modular. This is a non-trivial
assertion, but this is done in [D1]
in exactly the same situation! So we conclude that the family $\{\sigma_\lambda\}$
is modular, and from $w=1$ we easily check that it will be attached
to a weight $2$ cusp form f. This proves that $\sigma_Q$ can be
attached to the abelian variety $A_f$. \\
Remark: It follows from condition 2) in the theorem that the
variety $A_f$ will have a large endomorphism algebra (cf. [R]).\\
\section{Non-existence of Barsotti-Tate representations of conductor 1}
In this section we will prove the following result:
\begin{teo}
\label{teo:nohay} There are no
irreducible Barsotti-Tate $2$-dimensional Galois representations of conductor $1$
and odd residual characteristic.
\end{teo}
Remark: By Barsotti-Tate we mean crystalline Galois representations as
$\sigma_Q$ in theorem \ref{teo:Taylor} with $w=1$.\\
Proof: As in the previous section, $q$ is odd and $\sigma_Q$ has $w=1$.
Now there is no restriction in the field of coefficients, but $S$ is
empty. If $q \neq 3$, once again we use the results in section 1 to
construct a strictly compatible family $\{\sigma_\lambda\}$ containing
$\sigma_Q$. We consider again $\sigma_t$ for $t \mid 3$ (just take $t=Q$ if
$q=3$). From strict compatibility, this $t$-adic Galois representation
will be unramified outside 3. But such a representation can not exist, as
was proved in [D2]. Let us briefly recall the argument for the convenience
of the reader: the residual representation $\bar{\sigma}_t$
has coefficient in a finite
field of characteristic $3$ and is unramified outside $3$. Then, a
result of Serre tells us that it must be reducible. An application of
results of Skinner-Wiles (that can be applied because $\sigma_t$ is Barsotti-Tate)
shows that $\sigma_t$ is modular, but this is a contradiction because
it must correspond to a level 1, weight 2, cuspidal modular form.\\
\begin{cor}
\label{teo:masqueirred} Let $f$ be a genus $2$, level $1$, cuspidal
Siegel modular form (Hecke eigenform) having multiplicity one and weight $k>3$. Suppose
that $f$ is not a Maass spezialform. Then, for every odd prime $\ell$,
$\lambda \mid
\ell$ in $E$= field generated by the eigenvalues of $f$, the Galois
representation $\rho_{f,\lambda}$ attached to $f$ is absolutely
irreducible. In particular, this representation can be defined over
$E_\lambda$.
\end{cor}
Proof: In [D2] it is shown that the only possible reducible case is:
$\rho_{f,\lambda} \cong \sigma_{1,\lambda} \oplus \sigma_{2,\lambda}$
where one of the two (necessarily irreducible) components, say
$\sigma_{2,\lambda}$, is crystalline, with Hodge-Tate weights
$\{k-2 , k-1\}$, and has conductor 1. But theorem \ref{teo:nohay} says
that such a representation (after twisting by $\chi^{2-k}$)
can not exist. This proves the
corollary.\\
Remark: This improves the main result of [D3], for the level
1 case: ``uniformity of reducibility" now does not make sense, because all
representations will be irreducible.\\
In the semistable case, the
main theorem of [D3] can be extended, just by applying theorem
\ref{teo:Taylor}, to be valid for every prime:
\begin{prop}
\label{teo:lomas} Let $\{ \rho_\lambda\}$ be a compatible family of
4-dimensional, pure, symplectic, Galois representations, with
finite ramification set $S$, semistable (at every place in $S$) in the sense of [D3]
and such that for every $\ell \not\in S$, $\lambda \mid \ell$,
$\rho_\lambda$ is crystalline with Hodge-Tate weights $\{0 , k-2, k-1,
2k-3\}$. Then, if for some $q >2, q\not\in S$, $Q \mid q$, the representation
$\rho_Q$ is reducible,
all the representations in the family are
reducible.
\end{prop}
Proof: It is known (cf. [D2], [D3]) that the only possible reducible case
is: $\rho_Q \cong \sigma_{1,Q} \oplus \sigma_{2,Q}$, where the two components
are irreducible odd two-dimensional Galois representations of the same
determinant, one of them having Hodge-Tate weights $\{0, 2k-3\}$, the
other having weights $\{ k-2 , k-1\}$.
If $q \geq 4k-5$, an application of theorem \ref{teo:Taylor} to both
components (for one of them, you may twist by a power of $\chi$ before applying
the theorem, and then untwist to obtain the desired family)
proves the result, because from compatibility and
Cebotarev density theorem it is clear that the families $\{\sigma_{1,\lambda}\}$
and $\{\sigma_{2,\lambda} \}$ verify: $\rho_{\lambda} \cong
\sigma_{1,\lambda} \oplus \sigma_{2,\lambda}$ (up to
semisemplification) for every $\lambda$.\\
If $2 <q < 4k-5$, then we can only apply (with the twist and untwist trick)
theorem \ref{teo:Taylor} to one of the components, say $\sigma_{2,
Q} \in \{ \sigma_{2,\lambda }\}$. The techniques of [D3]
apply precisely in this situation: it is enough (because we are
assuming semistability) to have the ``existence of a family" result
for one component, to conclude reducibility of $\{\rho_\lambda\}$
for almost every prime
But this implies that we can take now a
second prime $r$ as large as we want ($r \geq 4k-5, r \not\in S$), and $R \mid
r$ such that $\rho_R$ is reducible: $\rho_R \cong \sigma_{1,R} \oplus
\sigma_{2,R}$, and now because $r$ is sufficiently large we can
apply theorem \ref{teo:Taylor} also to the other component,
and conclude as before that the whole family $\{ \rho_\lambda\}$
is reducible.\\
Remark: When we assume that $\rho_Q$ is reducible, reducibility must
necessarily ``occur over the field of coefficients" (the field generated by the traces
of the images of Frobenii), i.e., the field of coefficients of
$\rho_Q$ must contain the fields of coefficients of its irreducible
components (this is required in the proof above, because we have used
the results of [D3]). That reducibility always occurs over the field of coefficients
was proved in [D2] for $q \geq 4k-5$ but the proof
holds for every odd prime $q$: if we assume that this is not the case, following
the arguments in [D2], we conclude that the field of coefficients
of $\sigma_{2,Q}$ is an infinite extension of that of $\rho_Q$ (to orient the
reader, let us point out that this result
strongly depends on the fact that $\rho_Q$ has four different Hodge-Tate
weights, and that when reducibility is not over the fields of coefficients this
forces the
images of the representations $\{ \rho_\lambda \}$ to be ``generically
large", cf. [D2], [D3])), but
this contradicts the fact that $\sigma_{2,Q}$ (after twisting) is
potentially modular (thus, the field of coefficients of $\rho_Q |_{G_F}$ is a number
field for some totally real number field $F$,
and a fortiori the field of coefficients of $\rho_Q$ is also a finite extension
of ${\mathbb{Q} }$).
\section{Bibliography}
[D1] Dieulefait, L., {\it Modularity of Abelian Surfaces with Quaternionic
Multiplication}, Math. Res. Letters {\bf 10} no. 2-3 (2003)\\
\newline
[D2] Dieulefait, L., {\it Irreducibility of Galois actions on level $1$
Siegel cusp forms}, preprint, (2002)\\
available at http://www.math.leidenuniv.nl/gtem/view.php (preprint
19)\\
\newline
[D3] Dieulefait, L., {\it Uniform behavior of families of Galois
representations on Siegel modular forms}, preprint, (2002)\\
available at http://www.math.leidenuniv.nl/gtem/view.php (preprint
68)\\
\newline
[R] Ribet, K., {\it Abelian varieties over ${\mathbb{Q} }$ and modular forms},
Algebra and Topology 1992, KAIST Math. Workshop, Taejon, Korea
(1992) 53-79\\
\newline
[T1] Taylor, R., {\it On the meromorphic continuation of degree two
L-functions}, preprint, (2001);
available at http://abel.math.harvard.edu/$\sim$rtaylor/ \\
\newline
[T2] Taylor, R., {\it Galois Representations}, proceedings of ICM, Beijing,
August 2002, longer version available at
http://abel.math.harvard.edu/$\sim$rtaylor/ \\
\newline
\end{document} |
\begin{document}
\title[Pinching estimates in higher dimensions]
{Pinching estimates for solutions of the linearized Ricci flow system\\
in higher dimensions}
\author{Jia-Yong Wu and Jian-Biao Chen}
\address{Department of Mathematics, Shanghai Maritime University,
1550 Haigang Avenue, Shanghai 201306, P. R. China}
\email{jywu81@yahoo.com}
\email{chenjb@shmtu.edu.cn}
\thanks{The first author is partially supported by the NSFC (11101267, 11271132)
and the Innovation Program of Shanghai Municipal Education Commission
(13YZ087). The second author is partially supported by the NSFC (51109128)}
\subjclass[2000]{Primary 53C44.}
\date{\today}
\dedicatory{}
\keywords{Ricci flow, the linearized Ricci flow,
Lichnerowicz Laplacian heat equation, pinching estimate.}
\begin{abstract}
We prove pinching estimates for solutions of the linearized Ricci flow
system on a closed manifold of dimension $n\geq 4$ with positive
scalar curvature and vanishing Weyl tensor. If the vanishing
Weyl tensor condition is removed, we only give a rough pinching
estimate controlled by some blow-up function in a short time. These results
extend the $3$-dimensional case due to Anderson and Chow (2005) \cite{[AC]}.
\end{abstract}
\maketitle
\section{Introduction and main results}
Given an $n$-dimensional closed Riemannian manifold $M^n$, a
smooth family of Riemannian metrics $g(t)$, $t\in [0, T)$, is
said to be evolving under the \emph{Ricci flow} if
\begin{equation}\label{Rf}
\frac{\partial}{\partial t}g_{ij}=-2R_{ij},
\end{equation}
where $R_{ij}$ is the Ricci curvature of the metric $g(t)$.
This geometric flow was introduced by R. Hamilton \cite{[Ham1]}.
In \cite{[Ham1]}, he proved that for any an initial metric $g_0$,
there is a unique solution to this flow over some short time
interval, with $g(0)=g_0$. This proof was later simplified by
D. DeTurck \cite{[De]} by linearizing the modified Ricci flow,
which leads to the following \emph{Lichnerowicz Laplacian heat
equation}
\begin{equation}\label{LLh}
\frac{\partial}{\partial t}h_{ij}=(\Delta_Lh)_{ij}
:=\Delta h_{ij}+2R_{ikjl}h_{kl}-R_{ik}h_{kj}-R_{jk}h_{ki},
\end{equation}
for a symmetric $2$-tensor $h$, where $R_{ikjl}$ denotes the
Riemannian curvature of the metric $g(t)$ moving under the
Ricci flow. Nowadays the Ricci flow \eqref{Rf} coupled with
equation \eqref{LLh} is often called the \emph{linearized
Ricci flow system}, which is useful for studying the singularities
of the Ricci flow. If we let $h_{ij}=R_{ij}$, then equation
\eqref{LLh} is exactly the evolution of the Ricci curvature
under the Ricci flow.
On the subject of differential Harnack inequalities for the
linearized Ricci flow system, there have been many important
contributions. In \cite{[ChHam]}, B. Chow and R. Hamilton
proved a linear trace differential Harnack inequality for
the coupled system \eqref{Rf}-\eqref{LLh} by adapting
similar techniques of Hamilton's trace inequality for
the Ricci flow \cite{[Ham2]} and Li-Yau's seminal inequality
for the heat equation \cite{[Li-Yau]}. Meanwhile in
\cite{[ChCh]}, B. Chow and S.-C. Chu gave a geometric
interpretation in terms of this linear trace differential
Harnack inequality. In \cite{[GIK]}, C. Guenther,
J. Isenberg and D. Knopf studied the linearized Ricci flow
system at flat solutions from the point of view of maximal
regularity theory. Besides, the related Li-Yau-Hamilton type
differential Harnack inequalities for the linearized Ricci flow
system were also appeared in \cite {[Chow]}, \cite{[ChKn]} and
\cite{[WuZh]}.
In another direction, in \cite{[AC]}, G. Anderson and B. Chow
considered the quantity $|h|^2/R^2$ for solutions to the linearized
Ricci flow system \eqref{Rf}-\eqref{LLh} on closed $3$-manifolds
with positive scalar curvature. They proved an interesting pinching
estimate for solutions of the linearized Ricci flow system on a closed
$3$-manifold:
\begin{thm}[see Anderson and Chow \cite{[AC]}]\label{ACthm}
Let $(M^3,g(t))$ be a solution to the Ricci flow on a closed $3$-manifold
on a time interval $[0,T)$ with $T<\infty$ and let $\rho\in[0,\infty)$ be
such that $R_{min}(0)>-\rho$. If the pair $(g,h)$ is any solution to the
linearized Ricci flow system \eqref{Rf}-\eqref{LLh}, then there exists a
constant $C<\infty$ depending only on $g(0)$, $h(0)$, $\rho$ and $T$
such that
\[
\frac{|h|}{R+\rho}\leq C
\]
on $M\times[0,T)$. Furthermore, when $\rho=0$, $C$ is independent of $T$.
\end{thm}
The inspiration for the above pinching estimate partly comes from the works of
R. Hamilton (see Section 10 of \cite{[Ham1]} and Section 24 of \cite{[Ham3]})
and M. Gursky \cite{[Gu]}. We refer the reader to the excellent introduction
of \cite{[AC]} for nice explanations on this subject. In Theorem \ref{ACthm},
if we take $h=Rc$ and $\rho=0$, we immediately obtain Hamilton's Ricci
pinching estimate:
\begin{thm}[see Hamilton \cite{[Ham1]}]\label{Hamth}
If $(M^3,g(t))$, $t\in[0,T)$, $T<\infty$ is a solution to the Ricci flow
\eqref{Rf} on a closed $3$-manifold with positive scalar curvature,
then there exists a constant $C<\infty$ depending only on $g(0)$
such that
\[
\frac{|Rc|}{R}\leq C
\]
on $M\times[0,T)$.
\end{thm}
Recently, S. Brendle \cite{[Br1]} has successfully applied Theorem
\ref{ACthm} (see Proposition 6.1 in \cite{[Br1]}) to give a complete
proof of the uniqueness of the $3$-dimensional Bryant soliton
which is $k$-noncollapsed and non-flat. This resolves a well-known
question mentioned in Perelman's paper \cite{[Per]}. Later in
\cite{[Br2]}, Brendle generalized his unique result to the higher
dimensional setting. That is, he obtained a similar uniqueness
theorem for the steady gradient Ricci soliton of dimension
$n\geq 4$ which has positive sectional curvature and is
asymptotically cylindrical. In the proof of the higher dimensions,
Brendle adapted the arguments in \cite{[Br1]} but needed some
extra technique work. For example, due to the lack of an analogue of
Theorem \ref{ACthm} in higher dimensions, he employed a new
argument to prove an estimate related the Lichnerowicz-type
equation (see Proposition 4.2 in \cite{[Br2]}), which is quite different
from the $3$-dimensional case.
In this paper, we mainly generalize Anderson-Chow's pinching estimate
to the higher dimensions under some curvature assumptions. Our
results could be viewed as a partial answer to a question implied in
the paper \cite{[Br2]}, which may be expected to study the singularities
of the Ricci flow. The results of this paper can be divided into two
parts.
On one hand, we will establish an Anderson-Chow's pinching estimate
for solutions of the linearized Ricci flow system on a closed manifold
of dimension $n\geq 4$ as long as scalar curvature preserves positive
and the Weyl tensor remains identically zero under the linearized
Ricci flow system. The Weyl tensor is an important geometric quantity
in understanding higher dimensional Riemannian manifolds. This tensor
is known to depend on the conformal structure, so that if
$\tilde{g}_{ij}=\phi g_{ij}$, then $\tilde{W}_{ijkl}=\phi W_{ijkl}$.
If $n\leq 3$, the Weyl tensor vanishes; if $n\geq 4$, in local coordinate,
the Weyl tensor can be written as
\begin{equation}
\begin{aligned}\label{Weyl}
W_{ikjl}&=R_{ikjl}-\frac{1}{n-2}(g_{ij}R_{kl}+g_{kl}R_{ij}-g_{il}R_{jk}-g_{jk}R_{il})\\
&\quad+\frac{R}{(n-1)(n-2)}(g_{ij}g_{kl}-g_{il}g_{jk}).
\end{aligned}
\end{equation}
There exist some examples of the Ricci flow with vanishing Weyl tensors
for all time. One distinct example is the 1-parameter family of conformally
equivalent metrics
\[
g(t)=\left(1-2(n-1)t\right)g_{S^n}
\]
on the unit round sphere $\mathbb{S}^n$ with the standard metric $g_{S^n}$.
This is an ancient solution of the Ricci flow with vanishing Weyl tensors for
$-\infty<t<\frac{1}{2(n-1)}$. Another interesting example is the time-parameter
family of metrics
\[
g(t)=\left(1-2(n-2)t\right)g_{S^{n-1}}+dr^2
\]
on the round cylinder $\mathbb{S}^{n-1}\times \mathbb{R}$
with the metric $g_0=g_{S^{n-1}}+dr^2$, $n\geq 3$. It is easy to check
that it is an ancient solution of the Ricci flow
with vanishing Weyl tensors for $-\infty<t<\frac{1}{2(n-2)}$.
Now we give the first main result, which describes a precise pointwise measure of the size
of $h$ relative to the scalar curvature in higher dimensions when the Weyl tensor remains
identically zero under the Ricci flow.
\begin{theorem}\label{Main}
Let $(M^n,g(t),h(t))$, $t\in[0,T)$, $T<\infty$ be a solution to the
linearized Ricci flow system \eqref{Rf}-\eqref{LLh} on a closed
$n$-dimensional ($n\geq 4$) manifold with positive scalar curvature.
Assume that the Weyl tensor remains identically zero under the
linearized Ricci flow system. Then there exists a constant
$c_0<\infty$ depending only on $g(0)$ and $h(0)$ such that
\[
\frac{|h|}{R}(x,t)\leq c_0
\]
on $M\times[0,T)$.
\end{theorem}
The proof of Theorem \ref{Main} essentially follows the arguments of \cite{[AC]}.
The main difference is more complicated curvature terms due to the higher
dimensions. We reduce this difficulty to a matrix problem, which is discussed in
Section \ref{NonP}. The assumption of the Weyl tensor in Theorem
\ref{Main} seems to be necessary because the Weyl tensor remains
identically zero in Theorem A. It is interesting to ask if there exists
a similar pinching estimate under different curvature assumptions.
\begin{remark}
In Theorem \ref{Main}, we assume that the scalar curvature is positive; however
in Theorem A, the scalar curvature only needs to be a lower bound. This is
because Anderson and Chow use a special property of curvature estimate in
dimension 3 due to Hamilton-Ivey estimate (see Theorem 24.4 in \cite{[Ham3]}
or \cite{[Ivey]}), whereas this property does not hold in higher dimensions.
Therefore the scalar curvature condition in Theorem \ref{Main} is a
little stronger than the case of Theorem A.
\end{remark}
\begin{remark}
If we let $h=Rc$, then Theorem \ref{Main} is just as a
special case of Knopf's Ricci curvature pinching
estimate \cite{[Kno]} (see also \cite{[Caox]}).
\end{remark}
\begin{remark}
Recently, X.-D. Cao and H. Tran \cite{[CaoxTr]} observed
that the Ricci flow solution with nonnegative isotropic
curvature implies a Riemannian curvature pinching result:
\[
\frac{|Rm|}{R}(x,t)\leq c
\]
for some constant $c=c(n,g(0))<\infty$.
\end{remark}
On the other hand, if the vanishing Weyl tensor condition is removed in
Theorem \ref{Main}, we can give another Anderson-Chow's type pinching estimate
for solutions of the linearized Ricci flow system on a closed manifold.
Though this pinching estimate is not uniformed by a constant, it could
be controlled by some blow-up function in a short time.
\begin{theorem}\label{Main2}
Let $(M^n,g(t),h(t))$, $t\in[0,T)$, $T<\infty$ be a solution to the
linearized Ricci flow system \eqref{Rf}-\eqref{LLh} on a closed
$n$-dimensional ($n\geq 4$) manifold with positive scalar curvature,
and let $K:=\max_{t=0}|Rm|$. Then there exist finite constants $c_0:=c_0(g(0),h(0))$
and $c:=c(n)$ such that
\[
\frac{|h|^2}{R^2}(t)\leq c_0\cdot(1-8Kt)^{-c}
\]
on $M\times[0,T')$, where $T':=\min\{T, \frac{1}{8K}\}$.
\end{theorem}
The rest of this paper organized as follows. In Section \ref{BaRe},
following the arguments of \cite{[AC]}, we will prove Theorem \ref{Main}
by the straightforward computation and the usage of parabolic maximum
principle. The main difference is that we need to estimate the
nonnegativity of some complicated terms in the evolution equation due
to the higher dimensions. In Section \ref{BaRe2}, we first give
an estimate on the norm of Riemannian curvature under the Ricci flow
on a closed manifold. Then we apply this estimate and parabolic
maximum principles to prove Theorem \ref{Main2}. In Section \ref{NonP},
we will use the basic matrix theory to justify the nonnegativity
of the above mentioned terms appeared in the proof of
Theorem \ref{Main}.
\textbf{Acknowledgement}
The author would like to thank the referee for many valuable
comments and suggestions on an earlier version of this paper.
\section{Proof of Theorem \ref{Main}}\label{BaRe}
Along the Ricci flow \eqref{Rf}, we have
\begin{equation}\label{evolubian}
\frac{\partial }{\partial t}R=\Delta R+2|Rc|^2.
\end{equation}
Combining equations \eqref{Rf}, \eqref{LLh}, \eqref{evolubian},
and the key estimate (see Claim \ref{Klem}), we complete
the proof of Theorem \ref{Main}.
\begin{proof}[Proof of Theorem \ref{Main}]
The proof involves a direct computation and the parabolic
maximum principle. Here we can borrow Anderson-Chow's
computation to simplify our calculation. From
the evolution equation (16) in \cite{[AC]}, we have
\begin{equation}
\begin{aligned}\label{evolu}
\frac{\partial}{\partial t}\left(\frac{|h|^2}{R^2}\right)
&=\Delta\left(\frac{|h|^2}{R^2}\right)+\frac{2}{R}\nabla R\cdot\nabla\left(\frac{|h|^2}{R^2}\right)
-\frac{2}{R^4}\left|R\nabla_ih_{jk}-\nabla_iRh_{jk}\right|^2\\
&\quad+\frac{4}{R^2}R_{ikjl}h_{ij}h_{kl}-\frac{4}{R^3}|h|^2|Rc|^2.
\end{aligned}
\end{equation}
The above formula holds for all dimensions. Since here $n\geq 4$
and the Weyl tensor $W=0$ by assumption, from \eqref{Weyl}
we have
\begin{equation*}
\begin{aligned}
R_{ikjl}&=\frac{1}{n-2}(g_{ij}R_{kl}+g_{kl}R_{ij}-g_{il}R_{jk}-g_{jk}R_{il})\\
&\quad-\frac{R}{(n-1)(n-2)}(g_{ij}g_{kl}-g_{il}g_{jk}).
\end{aligned}
\end{equation*}
Then substituting this into \eqref{evolu} yields
\begin{equation}
\begin{aligned}\label{evolu2}
\frac{\partial}{\partial t}\left(\frac{|h|^2}{R^2}\right)
&=\Delta\left(\frac{|h|^2}{R^2}\right)
+\frac{2}{R}\nabla R\cdot\nabla\left(\frac{|h|^2}{R^2}\right)\\
&\quad-\frac{2}{R^4}\left|R\nabla_ih_{jk}-\nabla_iRh_{jk}\right|^2
-\frac{4}{R^3}P,
\end{aligned}
\end{equation}
where
\begin{equation*}
\begin{aligned}
P:=&|h|^2|Rc|^2-\frac{Rh_{ij}h_{kl}}{n-2}(g_{ij}R_{kl}+g_{kl}R_{ij}-g_{il}R_{jk}-g_{jk}R_{il})\\
&+\frac{R^2h_{ij}h_{kl}}{(n-1)(n-2)}(g_{ij}g_{kl}-g_{il}g_{jk})\\
=&|h|^2|Rc|^2-\frac{2R}{n-2}(HR_{ij}h_{ij}-R_{jk}h_{ji}h_{ik})+\frac{R^2(H^2-|h|^2)}{(n-1)(n-2)}
\end{aligned}
\end{equation*}
and where $H:=g^{ij}h_{ij}$.
Now we need to deal with the troublesome term $P$. Fortunately,
we observe that $P$ is always nonnegative
without any assumption, which shall be confirmed
in the Section \ref{NonP}
(see Corollary \ref{cor}).
\begin{claim}\label{Klem}
If $n\geq 4$, for any metric $g$ and symmetric $2$-tensor $h$, we have
\[
|h|^2|Rc|^2-\frac{2R}{n-2}(HR_{ij}h_{ij}-R_{jk}h_{ji}h_{ik})+\frac{R^2(H^2-|h|^2)}{(n-1)(n-2)}\geq 0.
\]
\end{claim}
Proceeding our proof, by Claim \ref{Klem}
and $R>0$, from \eqref{evolu2} we derive that
\[
\frac{\partial}{\partial t}\left(\frac{|h|^2}{R^2}\right)
\leq\Delta\left(\frac{|h|^2}{R^2}\right)
+\frac{2}{R}\nabla R\cdot\nabla\left(\frac{|h|^2}{R^2}\right).
\]
Finally, applying the parabolic maximum principle to the above equation
yields
\[
\frac{|h|^2}{R^2}(x,t)\leq c_0
\]
on $M\times[0,T)$, where $c_0:=\max_{t=0}|h|^2/R^2$.
\end{proof}
\begin{remark}
The theorem is also true on complete noncompact Riemannian manifolds
when we can apply the maximum principle.
\end{remark}
\section{Proof of Theorem \ref{Main2}}\label{BaRe2}
In this section, we will prove Theorem \ref{Main2}. First,
we give a Riemannian curvature estimate under the Ricci flow
on a closed manifold, which will be useful in the proof of
Theorem \ref{Main2}.
\begin{lemma}\label{Riemcon}
Let $(M^n,g(t))$ be a solution to the Ricci flow on a closed $n$-dimensional
manifold on a time interval $[0,T)$ with $T<\infty$. Then
\begin{equation}\label{Riem}
|Rm(x,t)|\leq \frac{K}{1-8Kt}
\end{equation}
on $M\times[0,T')$, where $T':=\min\{T, \frac{1}{8K}\}$ and
$K:=\max_{t=0}|Rm|$.
\end{lemma}
\begin{proof}
Under the Ricci flow,
\[
\frac{\partial}{\partial t}|Rm|^2
\leq\Delta|Rm|^2-2|\nabla Rm|^2+16|Rm|^3.
\]
By the maximum principle, we have
\[
|Rm(g(t))|\leq \frac{K}{1-8Kt}
\]
on $M\times[0,T')$, where $T':=\min\{T, \frac{1}{8K}\}$ and
$K:=\max_{t=0}|Rm|$.
\end{proof}
Now we use Lemma \ref{Riemcon} to finish the proof of Theorem
\ref{Main2}.
\begin{proof}[Proof of Theorem \ref{Main2}]
As before, we still have the evolution equation \eqref{evolu}.
Since $(M^n,g(t))$, $t\in[0,T)$, $T<\infty$ has a solution of the
Ricci flow \eqref{Rf} on a closed $n$-dimensional ($n\geq 4$)
manifold, by Lemma \ref{Riemcon}, we have
\begin{equation}\label{Rmest}
|Rm(x,t)|\leq\frac{K}{1-8Kt}
\end{equation}
on $M\times[0,T')$, where $T':=\min\{T, \frac{1}{8K}\}$ and
$K:=\max_{t=0}|Rm|$. Substituting \eqref{Rmest}
into \eqref{evolu} and noticing $R>0$ by our assumption, we have
\[
\frac{\partial}{\partial t}\left(\frac{|h|^2}{R^2}\right)
\leq\Delta\left(\frac{|h|^2}{R^2}\right)+\frac{2}{R}\nabla R\cdot\nabla\left(\frac{|h|^2}{R^2}\right)
+\frac{C(n)K}{1-8Kt}\cdot\frac{|h|^2}{R^2}.
\]
Applying the parabolic maximum principle to the above
evolution equation (for example, see Proposition 4.3
in \cite{[ChKn2]}), we conclude that
\[
\frac{|h|^2}{R^2}(t)\leq c_0\cdot(1-8Kt)^{-c}
\]
on $M\times[0,T')$, where $c_0:=\max_{t=0}|h|^2/R^2$ and $c:=c(n)$.
This finishes the proof of the theorem.
\end{proof}
\section{Nonnegativity of a degree $4$ homogeneous polynomial\\
in $2n$ variables}\label{NonP}
In this section, we will prove Claim \ref{Klem} in Section \ref{BaRe}.
Our proof seems to be different from \cite{[AC]}. Adapting the
Anderson-Chow's symbols in \cite{[AC]}, since $h$ is a symmetric tensor,
we may assume $h$ is diagonal. Let $h_1$, $h_1$,$\ldots$, $h_n$ denote the
eigenvalues of $h$ and let $r_1=R_{11}$, $r_2=R_{22}$, $\ldots$, $r_n=R_{nn}$
denote the diagonal entries of $R_{ij}$. Then
\begin{equation}
\begin{aligned}\label{PQ}
P&=|Rc|^2|h|^2-\frac{2R}{n-2}(HR_{ij}h_{ij}-R_{jk}h_{ji}h_{ik})+\frac{R^2(H^2-|h|^2)}{(n-1)(n-2)}\\
&\geq Q:=\sum^n_{i=1}r^2_i\cdot\sum^n_{i=1}h^2_i\\
&\quad+\frac{2}{n-2}\sum^n_{i=1}r_i\cdot\left(-\sum^n_{i=1}h_i\sum^n_{i=1}r_ih_i+\sum^n_{i=1}r_ih^2_i\right)\\
&\quad+\frac{1}{(n-1)(n-2)}\left(\sum^n_{i=1}r_i\right)^2\cdot
\left[\left(\sum^n_{i=1}h_i\right)^2-\sum^n_{i=1}h^2_i\right],
\end{aligned}
\end{equation}
where we used the fact $|Rc|^2\geq\sum^n_{i=1}r^2_i$.
Now we rewrite $Q$ as a bilinear form in $\gamma=(r_1,r_2,\ldots,r_n)^T$ with
coefficients in $h=(h_1,h_2,\ldots,h_n)^T$:
\[
Q=\gamma^T(s_2I_n+\alpha_0\beta^T+\beta\alpha^T_0)\gamma,
\]
where $I_n$ is the identity matrix of order $n$,
\begin{equation}\label{def1}
\alpha_0:=(1,1,\ldots,1)^T\in \mathbb{R}^n,
\quad\quad
\alpha_k:=(h^k_1,h^k_2,\ldots,h^k_n)^T\in \mathbb{R}^n
\end{equation}
and
\begin{equation}\label{def2}
s_0=n,\quad\quad s_k:=\sum^n_{i=1}h^k_i
\end{equation}
for $k=1,2,3,4$; and where
\begin{equation}\label{def3}
\beta:=\frac{(s^2_1-s_2)\alpha_0}{2(n-1)(n-2)}-\frac{s_1\alpha_1}{n-2}+\frac{\alpha_2}{n-2}.
\end{equation}
In the rest of this section, we shall prove $Q\geq 0$. To achieve
it, we begin
with two technical lemmas.
\begin{lemma}\label{tech1}
If the matrix $\Lambda\in \mathbb{R}^{n\times n}$ is nonsingular, then for any
column vector $\xi_i,\eta_i\in\mathbb{R}^n$, $(i=1,2)$
\[
\det\left(\Lambda+\xi_1\eta_1^T+\xi_2\eta_2^T\right)
=\det(\Lambda)\cdot\det\left(I_2+(\eta_1,\eta_2)^T\Lambda^{-1}(\xi_1,\xi_2)\right).
\]
\end{lemma}
\begin{proof}
One can easily check that
\begin{align*}
\begin{bmatrix}
I_{2} & ~~~(\eta_1,\eta_2)^T \\
0 & ~~~\Lambda+\xi_1\eta_1^T+\xi_2\eta_2^T
\end{bmatrix}
=&\begin{bmatrix}
I_{2} & ~~~0 \\
(\xi_1,\xi_2) & ~~~I_n
\end{bmatrix}
\times
\begin{bmatrix}
I_{2} & ~~~(\eta_1,\eta_2)^T\Lambda^{-1}\\
0 & ~~~I_n
\end{bmatrix}\\
&\times\begin{bmatrix}
I_2+(\eta_1,\eta_2)^T\Lambda^{-1}(\xi_1,\xi_2) &~~~ 0\\
-(\xi_1,\xi_2) & ~~~\Lambda
\end{bmatrix}.
\end{align*}
Then the conclusion can be immediately obtained by the above
identity.
\end{proof}
\begin{lemma}\label{tech2}
Let $\alpha_0$, $\alpha_k$, $s_k$ and $\beta$ be defined by \eqref{def1},
\eqref{def2} and \eqref{def3}. Then
\begin{equation}\label{buden1}
s_2+\alpha_0^T\beta\geq0
\end{equation}
and
\begin{equation}\label{buden2}
(s_2+\alpha_0^T\beta)^2-n\beta^T\beta\geq0.
\end{equation}
\end{lemma}
\begin{proof}
We first check \eqref{buden1}. Clearly,
$\alpha^T_i\alpha_j=s_{i+j}$ for all $i,j=0,1,2$.
Using this, by the definition $\beta$, direct computation yields
\begin{equation*}
\begin{aligned}
s_2+\alpha_0^T\beta&=s_2+\frac{n(s_1^2-s_2)}{2(n-1)(n-2)}-\frac{s_1^2}{n-2}+\frac{s_2}{n-2}\\
&=\left(1+\frac{1}{2(n-1)}\right)s_2-\frac{s_1^2}{2(n-1)}\\
&=\alpha_1^T\left[\left(1+\frac{1}{2(n-1)}\right)I_n-\frac{\alpha_0\alpha_0^T}{2(n-1)}\right]\alpha_1.
\end{aligned}
\end{equation*}
The fact that $1>\sum^{n-1}_{i=1}\frac{1}{2(n-1)}=\frac 12$ tells us that the real
matrix
\[
A:=\left(1+\frac{1}{2(n-1)}\right)I_n-\frac{\alpha_0\alpha_0^T}{2(n-1)}
\]
is strictly diagonally dominant. Moreover all main diagonal entries of
$A$ are positive. Hence, by Theorem 6.1.10 of \cite{[HoJo]}, we conclude that
$A$ is positive definite and
\[
s_2+\alpha_0^T\beta=\alpha_1^TA\alpha_1\geq 0.
\]
Then, we shall prove \eqref{buden2}. According to the definitions
$\alpha_k$ and $s_k$, using the relation $\alpha^T_i\alpha_j=s_{i+j}$,
we expand the expression
$(s_2+\alpha_0^T\beta)^2-n\beta^T\beta$ as
\begin{equation*}
\begin{aligned}
&(s_2+\alpha_0^T\beta)^2-n\beta^T\beta
=\left[s_2+\frac{n(s_1^2-s_2)}{2(n-1)(n-2)}-\frac{s^2_1}{n-1}+\frac{s_2}{n-1}\right]^2\\
&-n\left[\frac{n(s_1^2-s_2)^2}{4(n-1)^2(n-2)^2}+\frac{s_1^2s_2+s_4-2s_1s_3}{(n-2)^2}
+\frac{(s_1^2-s_2)s_2-(s_1^2-s_2)s_1^2}{(n-1)(n-2)^2}\right].
\end{aligned}
\end{equation*}
From above, we easily judge that
\[
f(h_1,h_2,\ldots,h_n):=(s_2+\alpha_0^T\beta)^2-n\beta^T\beta
\]
is a real homogeneous symmetric polynomial of degree $4$
with respect to $n$ variables: $h_1,h_2,\ldots,h_n$. Now
we recall an interesting fact due to V. Timofte
(see Corollary 5.6 in \cite{[Tim]}).
\begin{thm}[see Timofte \cite{[Tim]}]\label{propTi}
If $f$ is a real homogeneous symmetric polynomial of degree $4$
on $\mathbb{R}^n$, then
\[
f(h_1,h_2,\ldots,h_n)\geq 0\quad \Longleftrightarrow
\quad
\varphi_k(t)\geq 0, t\in[-1,1], k=1,2,\ldots,n-1,
\]
where
\[
\varphi_k(t):=\left.f(h_1,h_2,\ldots,h_n)\right|_{h_1=h_2=\ldots=h_k=t,\,h_{k+1}=\ldots=h_n=1}
\]
for $k=1,2,\ldots,n-1$.
\end{thm}
In order to prove \eqref{buden2}, by Theorem \ref{propTi},
we only need to check $\varphi_k(t)\geq 0$ for all
$k=1,2,\ldots,n-1$. Indeed,
\[
\varphi_1(t)=(t-1)^2 t^2\geq 0,\quad\quad\varphi_{n-1}(t)=(t-1)^2\geq 0
\]
and
\[
\varphi_{k}(t)=\frac{k (n-k) (t-1)^2 }{(n-1)(n-2)^2}(a_1t^2+b_1t+c_1),\quad k=2,\ldots,n-2
\]
where
\begin{equation*}
\begin{aligned}
a_1:=&\left[(n-1)k-1\right](n-1-k)>0,\\
b_1:=& -2(k-1)(n-1-k)(n-1),\\
c_1:=& (k-1)\left[(n-k)(n-1)-1\right].
\end{aligned}
\end{equation*}
Since $a_1>0$ and
\[
b_1^2-4a_1c_1=-4(k-1)(n-1-k)n(n-2)^2<0,
\]
we conclude that
\[
a_1t^2+b_1t+c_1>0 \quad\mathrm{and}\quad
\varphi_{k}(t)\geq 0
\]
for $k=2,\ldots,n-2$. Therefore \eqref{buden2} follows.
\end{proof}
Using Lemmas \ref{tech1} and \ref{tech2},
we now finish the proof of the nonnegativity of $Q$.
\begin{theorem}\label{thm}
Let $\alpha_0$, $\alpha_k$, $s_k$ and $\beta$ be defined by \eqref{def1},
\eqref{def2} and \eqref{def3}. Then
\[
Q=\gamma^T(s_2I_n+\alpha_0\beta^T+\beta\alpha^T_0)\gamma\geq 0.
\]
\end{theorem}
\begin{proof}[Proof of Theorem \ref{thm}]
Let $B:=s_2I_n+\alpha_0\beta^T+\beta\alpha^T_0$. We only to show that
all eigenvalues of real matrix $B$ are real, nonnegative, and their
number is $n$. In fact, if $\lambda\neq s_2$, by Lemma
\ref{tech1}, we compute that
\begin{equation*}
\begin{aligned}
\det(\lambda I_n-B)&=\det((\lambda-s_2)I_n-\alpha_0\beta^T-\beta\alpha_0^T)\\
&=\det((\lambda-s_2)I_n)\cdot\det\left(I_2+\frac{1}{(\lambda-s_2)}(\beta,\alpha_0)^T(-\alpha_0,-\beta)\right)\\
&=(\lambda-s_2)^{n-2}\det\left((\lambda-s_2)I_2-(\beta,\alpha_0)^T(\alpha_0,\beta)\right)\\
&=(\lambda-s_2)^{n-2}(\lambda^2-p\lambda+q),
\end{aligned}
\end{equation*}
where
\[
p:=2(s_2+\alpha_0^T\beta)\quad\mathrm{and}
\quad
q:=(s_2+\alpha_0^T\beta)^2-n\beta^T\beta.
\]
In other words, when
$\lambda\neq s_2$, we have the identity
\begin{equation}\label{eigenequ}
\det(\lambda I_n-B)=(\lambda-s_2)^{n-2}(\lambda^2-p\lambda+q).
\end{equation}
Notice that two hand sides of the above identity are continuous
with respect to the parameter $\lambda$. Thus if $\lambda\to s_2$,
we know that \eqref{eigenequ} also holds for $\lambda=s_2$.
Therefore, \eqref{eigenequ} in fact holds \emph{without} any condition.
Now from \eqref{eigenequ}, we easily conclude that $s_2$ is
an nonnegative eigenvalue of multiplicity $n-2$ in $B$.
On the other hand, by Lemma \ref{tech2}, we know that $p,\,q\geq 0$.
Meanwhile,
\[
p^2-4q=n\beta^T\beta\geq 0.
\]
Hence, equation $\lambda^2-p\lambda+q=0$ has two real solutions,
which implies that $B$ has another two nonnegative real eigenvalues.
In summary, we prove that the number of
nonnegative real eigenvalues of matrix $B$ is $n$. Therefore
$Q=\gamma^TB\gamma\geq 0$.
\end{proof}
Combining Theorem \ref{thm} and \eqref{PQ} implies that
\begin{corollary}\label{cor}
If $n\geq 4$, for any Riemannian metric $g$ and symmetric $2$-tensor $h$, we have
\[
|h|^2|Rc|^2-\frac{2R}{n-2}(HR_{ij}h_{ij}-R_{jk}h_{ji}h_{ik})+\frac{R^2(H^2-|h|^2)}{(n-1)(n-2)}\geq 0.
\]
\end{corollary}
\begin{remark}
We would like to point out that our proof method is also
suitable for the case $n=3$, which has been proved by
G. Anderson and B. Chow \cite{[AC]}.
\end{remark}
\end{document} |
\begin{document}
\title{Long term Optimal Investment in Matrix Valued Factor Models}
\date{\today}
\author[]{Scott Robertson}
\address[Scott Robertson]{Department of Mathematical Sciences,
Wean Hall 6113,
Carnegie Mellon University,
Pittsburgh, PA 15213,
USA}
\email{scottrob@andrew.cmu.edu}
\author[]{Hao Xing}
\address[Hao Xing]{Department of Statistics,
London School of Economics and Political Science,
10 Houghton st,
London, WC2A 2AE,
UK}
\email{h.xing@lse.ac.uk}
\begin{abstract}
Long term optimal investment problems are studied in a factor model with matrix valued state variables. Explicit parameter restrictions are obtained under which, for an isoelastic investor, the finite horizon value function and optimal strategy converge to their long-run counterparts as the investment horizon approaches infinity. This convergence also yields portfolio turnpikes for general utilities. By using results on large time behavior of semi-linear partial differential equations, our analysis extends affine models, where the Wishart process drives investment opportunities, to a non-affine setting. Furthermore, in the affine setting, an example is constructed where the value function is not exponentially affine, in contrast to models with vector-valued state variables.
\end{abstract}
\keywords{Portfolio choice, Long-run, Risk sensitive control, Portfolio turnpike, Wishart process}
\maketitle
\section{Introduction}
When investment opportunities are stochastic and the market is incomplete, optimal strategies in portfolio choice problems rarely admit explicit forms. The main source of difficulty is that the hedging demand depends implicitly upon the investment horizon. This difficulty motivates approximating optimal policies, and one useful approximation occurs by considering the long run limit. This approximation enables tractability for optimal strategies and illuminates the relationship between investor preferences, underlying economic factors and dynamic asset demand. Long run approximations typically take two forms: first, the \emph{long run optimal investment} or \emph{risk sensitive control} problem seeks to identify growth optimal policies for isoelastic utilities; second, the \emph{portfolio turnpike} problem seeks to connect optimal policies for general utilities with those for a corresponding isoelastic utility.
In this article, long run optimal investment and portfolio turnpike problems are studied in a multi-asset factor model where the state variable takes values in the space of positive definite matrices. Such models generalize the Wishart model of \cite{buraschi2010correlation, Hata-Sekine} (amongst many others), which has been successfully employed in a wide-range of problems in Mathematical Finance. In addition to identifying optimal long run policies and proving turnpike theorems, we are particularly concerned with connecting the finite horizon and long run problems. Here, the goal is to provide conditions when optimal policies for finite horizons converge to their long-run counterparts. Positive results in this direction are necessary to validate long-run analysis. Though heuristics indicate convergence, from a technical standpoint it is not a priori clear that the long-run policy arises as the limit of finite horizon policies.
For isoelastic utilities, the risk sensitive control, or long run optimal investment, problem aims to maximize the expected utility growth rate. This problem has been addressed by many authors : see, for example, \cite{MR1675114, MR1790132, MR1882297, MR1802598, MR1910647, MR1932164, MR1995925, Nagai-03, MR2435642, Guasoni-Robertson, MR2477847}. In these studies, an ergodic Hamilton Jacobi Bellman (HJB) equation is analyzed. This ergodic equation is typically obtained via a heuristic argument, where one first derives the finite horizon HJB equation, and then conjectures that for long horizons the (reduced) value function decomposes into the sum of a spatial component and a temporal growth component. Thus, if $v(T,\cdot)$ denotes the finite horizon value function, the long-run value function takes the form $\hat{\lambda}T + \hat{v}(\cdot)$. Then ergodic HJB equation follows by substituting the latter function into the finite horizon HJB equation.
The above heuristic derivation indicates that finite and infinite horizon optimal investment problems are parallel in many aspects. Of primary importance is to connect these two class of problems. As the investment horizon $T$ approaches infinity, does the finite horizon value function $v(T, \cdot)$ converge to its long-run analogue $\hat{\lambda} T + \hat{v}(\cdot)$? If so, in what sense? Does the optimal strategy for the finite horizon problem converge to a long-run limit? As previously mentioned, affirmative answers to these questions verify the intuition underpinning the study of the risk sensitive controls, and provide consistency between the finite horizon and long-run problems.
Moving away from the isoelastic case, portfolio turnpikes provide another approximation for optimal policies of generic utility functions. Qualitatively, turnpike theorems state that in a growing market (i.e. one where the riskless asset tends to infinity), as the investment horizon becomes large, the optimal trading strategy of a generic utility converges, over any finite time window, to the optimal trading strategy of its isoelastic counterpart (see Assumption \ref{ass: ratio} for a precise formulation of ``counterpart''). Turnpike theorems were first investigated in \cite{mossin1968optimal} for utilities with affine risk tolerance, and have since been extensively studied: in particular we mention \cite{leland1972turnpike, ross1974portfolio, hakansson1974convergence, MR736053, MR1172445, MR1629559, MR1805320, dybvig1999portfolio, detemple2010dynamic} where turnpike theorems are proved in differing levels of generality.
For the risk-sensitive control and turnpike approximations, we summarize the relationship between the finite and long horizon problems in Statements \ref{stat: long hor} and \ref{stat: turnpike} respectively. Verification of these statements allows investors with a long horizon to replace their optimal, but implicit, strategies with explicit long-run approximations, which lead to minimal loss of their wealth and utility, while providing considerable tractability. Each of Statements \ref{stat: long hor} and \ref{stat: turnpike} have been proved in \cite{guasoni.al.11} in a factor model with univariate state variable and constant correlation of hedgeable and unhedgeable shocks. The present paper extends these results to a multivariate setting, which allows for stochastic interest rates, volatility, and correlation. Here, in our main results, Proposition \ref{prop: wishart} and Theorems \ref{thm: power}, \ref{thm: turnpike}, we provide explicit parameter assumptions upon the model coefficients under which both Statements \ref{stat: long hor} and \ref{stat: turnpike} hold.
As previously stated, we focus on a factor model where the state variable is matrix valued. This is motivated by consideration of the Wishart process (cf. \cite{Bru} and Example \ref{ex: wishart} below), which has been applied to option pricing (cf. \cite{Gourieroux06, Gourieroux09, DaFonseca08, DaFonseca10}). Its application to portfolio optimization was pioneered by \cite{buraschi2010correlation}, which highlighted the impact of the multivariate state variable on the hedging demand. In particular, using practical relevant parameters, the numerical example in Section B.3 therein showed that the hedging demand converges to a steady-state level when the investment horizon is longer than $5$ years. Our results confirm this observation. In \cite{Hata-Sekine}, the portfolio optimization problem is solved in the Wishart case via a matrix Riccati differential equation. In \cite{Bauerle-Li}, logarithmic utility is studied, and in \cite{Richter} the indifference pricing is discussed.
In contrast to the aforementioned results, which exploit the affine structure of the Wishart process, our results rely upon large time asymptotic analysis of partial differential equations with quadratic nonlinearities in the gradient. Using techniques developed in \cite{Robertson-Xing}, we are able to consider non-affine models, and hence discuss general matrix-valued state variables as in Section \ref{subsec: state var}. Moreover, stochastic correlation between the state variable and risky assets can be treated, whereas a special (constant) correlation structure is needed to ensure the affine structure. Furthermore, our analysis, when applied to affine models, yields new insight: we construct a counter-example (Example \ref{exa: counter-exa}) to the long-held belief that optimal policies are affine in affine models. Indeed, the model in this example is affine, but the associated value function is not exponentially affine, hence the optimal policy is not affine. This happens when the dimension of state variable is larger than the number of risky assets, and is due to the noncommuntative property of the matrix product.
The paper is organized as follows: after the model and Statements \ref{stat: long hor} and \ref{stat: turnpike} are introduced in Section \ref{sec: setup}, the main results are presented in Section \ref{sec: converge}. For ease of exposition, the general results are first specified to when the state variable follows a Wishart process in Section \ref{subsec: wishart}. Here, the investment model may or may not be affine depending upon the asset drifts and covariances. Proposition \ref{prop: wishart} provides simple, mild (especially in the case where the investor risk aversion exceeds that of a logarithmic investor) parameter restrictions under which the main results follow. Proposition \ref{prop: wishart_good_case} explicitly identifies the long-run limit policy when the model is further specified to the ``classical'' affine Wishart model considered in \cite{buraschi2010correlation, Hata-Sekine} and Example \ref{exa: counter-exa} constructs the non exponentially affine counter example. After considering the Wishart case, the main results for general matrix valued state variables are given in Section \ref{subsec: general_case} : see Theorem \ref{thm: power} for the long run limit results and Theorem \ref{thm: turnpike} for the turnpike results. All proofs are deferred to Appendices \ref{app: A}, \ref{app: C} and \ref{app: B}. Finally, we summarize several notations used throughout the paper:
\begin{itemize}
\item $\mathbb{M}^dim{d}{k}$ denotes the space of $d\times k$ matrices with $\mathbb{M}^d\, := \, \mathbb{M}^dim{d}{d}$. For $x\in \mathbb{M}^dim{d}{k}$, denote by $x'$ the transpose of $x$. For $x\in \mathbb{M}^d$, denote by $\trace{x}$ the trace of $x$ and $\norm{x} = \sqrt{\trace{x'x}}$. For $x,y\in \mathbb{M}^d$, the Kronecker product of $x$ and $y$ is denoted by $x\otimes y\in \mathbb{M}^{d^2}$. Denote by $\idmat{d}$ the identity matrix in $\mathbb{M}^d$ and $1_d$ the $d$-dimensional vector with each component $1$.
\item $\mathbb{S}^d$ denotes the space of $d\times d$ symmetric matrices, and $\mathbb{S}_{++}^d$ the cone of positive definite matrices. For $x\in\mathbb{S}_{++}^d$, denote by $\sqrt{x}$ the unique element $y\in\mathbb{S}_{++}^d$ such that $y^2 = x$. For $x, y\in \mathbb{S}_{++}^d$, $x\geq y$ when $x-y$ is positive semi-definite.
\item For $E\subset \mathbb{M}^dim{d}{k}$, $F\subset \mathbb{M}^dim{m}{n}$, and $\gamma\in(0,1]$, denote by $C^{\ell,\gamma}(E; F)$ the space of $\ell$ times continuously differentiable functions from $E$ to $F$ whose derivatives of order up to $\ell$ is locally H\"{o}lder continuous with exponent $\gamma$.
\end{itemize}
\section{Set up}\label{sec: setup}
Let $(\Omega, (\mathcal{F}_t)_{t\geq 0}, \mathcal{F}, \mathbb{P})$ be a filtered probability space with $(\mathcal{F}_t)_{t\geq 0}$ a right-continuous filtration. Following the treatment in \cite{guasoni.al.11}, all $N$-negligible sets (cf. \cite[Definition 1.3.23]{Bitchteler} and \cite{N-N}) are included into $\mathcal{F}_0$. Such a completion of $\mathcal{F}_0$ ensures, for all $T\geq 0$, that $(\Omega, (\mathcal{F}_t)_{0\leq t\leq T}, \mathcal{F}_T, \mathbb{P})$ satisfies the usual conditions.
Consider a financial model with one risk-free asset $S^0$ and $n$ risky assets $(S^1,...,S^n)$. Investment opportunities are driven by a $\mathbb{S}_{++}^d$ valued state variable $X$. Before writing down the dynamics for the assets, it is necessary to introduce the state variable $X$, as the dynamics for $X$ involve matrix notation.
\subsection{A $\mathbb{S}_{++}^d$-valued state variable}\label{subsec: state var}
Let $B= (B^{ij})_{i,j=1,...d}$ be a $\mathbb{M}^d$-valued Brownian motion on $(\Omega, (\mathcal{F}_t)_{t\geq 0}, \mathcal{F}, \mathbb{P})$. The state variable $X$ has dynamics
\begin{equation}\label{eq: state}
dX_t = b(X_t)dt + F(X_t)dB_t G(X_t) + G(X_t)'dB'_t F(X_t)',\qquad X_0\in\mathbb{S}_{++}^d.
\end{equation}
Here, $b\in C^{1,\gamma}(\mathbb{S}_{++}^d; \mathbb{S}^d)$ and $F, G \in C^{2,\gamma}(\mathbb{S}_{++}^d; \mathbb{M}^d)$ are given functions. We require $b,F,G$ to be such that $X$ possesses a unique strong solution which is non-explosive, i.e.,
\begin{equation*}\label{eq: state_no_expl}
\mathbb{P}^x\bra{X_t \in \mathbb{S}_{++}^d,\ \forall \ t\geq 0} =1, \qquad \text{ for all } x\in \mathbb{S}_{++}^d,
\end{equation*}
where $\mathbb{P}^x$ is the probability such that $X_0= x$ a.s..
To enforce this requirement through restrictions upon $b,F$ and $G$, the results as well as notation of \cite{Mayerhofer-Pfaffel-Stelzer} are used. Namely, define
\begin{equation}\label{eq: f_g_def}
f(x):= FF'(x) \quad \text{and} \quad g(x):= G'G(x),\qquad x\in \mathbb{S}_{++}^d.
\end{equation}
Next, given $b,f,g: \mathbb{S}_{++}^d\rightarrow \mathbb{S}^d$ and $\delta \in \mathbb R$, define $H_\delta: \mathbb{S}_{++}^d \rightarrow \mathbb R$ via
\begin{equation}\label{eq: Hdelta_def}
H_\delta(x; b) := \trace{b\, x^{-1}} - (1+
\delta)\,\trace{fx^{-1}g x^{-1}} - \trace{f\,x^{-1}}\,
\trace{g\, x^{-1}}, \qquad x\in\mathbb{S}_{++}^d.
\end{equation}
Here, we have omitted the function arguments from $b,f,g$ but have explicitly identified the drift function $b$ in $H_\delta$, since in the sequel $H_\delta$ will be used with various $b$.
To understand $H_\delta$, note that if $X$ from \eqref{eq: state} has a strong solution satisfying \eqref{eq: state_no_expl} then It\^{o}'s formula implies the drift in the dynamics for $\log(\det(X_t)))$ is $H_0(X_t;b)$. Thus, the following assumption ensures that $X$ from \eqref{eq: state} neither explodes in norm nor has degenerate determinate and hence possesses a unique global strong solution $(X_t)_{t\in \mathbb R_+}$ on $\mathbb{S}_{++}^d$, cf. \cite[Theorem 3.4]{Mayerhofer-Pfaffel-Stelzer}.
\begin{ass}\label{ass: wp}\text{}
\begin{enumerate}[i)]
\item $G'\otimes F$ and $b$ are locally Lipschitz and of linear growth.
\item $\inf_{x\in \mathbb{S}_{++}^d} H_0(x; b)>-\infty$.
\end{enumerate}
\end{ass}
\begin{rem}\label{rem: kron}
A direct calculation, using \cite[Section 4.2]{Horn-Johnson}, shows that
\begin{equation*}
\begin{split}
\|G'\otimes F(x) - G'\otimes F(y)\|^2 &\leq 2\left(\|G(x)\|^2\|F(x)-F(y)\|^2 + \|F(y)\|^2\|G(x)-G(y)\|^2\right),\\
\|G'\otimes F(x)\|^2 &= \|F(x)\|^2\|G(x)\|^2 = \trace{f}\trace{g},\qquad \text{ for } x,y\in\mathbb{S}_{++}^d.
\end{split}
\end{equation*}
Thus, $G'\otimes F$ will be locally Lipschitz and of linear growth once $F$ and $G$ are locally Lipschitz and $\|F(x)\| \|G(x)\|\leq C(1+\|x\|)$ or equivalently if $\trace{f}\trace{g}\leq C(1+\|x\|^2)$.
\end{rem}
Assumption \ref{ass: wp} establishes well-posedness of \eqref{eq: state}. The next assumption implies that the volatility of $X$ is non-degenerate in the interior of $\mathbb{S}_{++}^d$.
\begin{ass}\label{ass: loc_ellip}
For each $x\in\mathbb{S}_{++}^d$, $f(x) > 0$ and $g(x) > 0$.
\end{ass}
Indeed, note that \eqref{eq: state} is short-hand for the following system:
\begin{equation*}
dX^{ij}_t = b_{ij}(X_t)dt + \sum_{k,l=1}^{d} F(X_t)_{ik}dB^{kl}_tG(X_t)_{lj}+\sum_{k,l=1}^d F(X_t)_{jk}dB^{kl}_tG(X_t)_{li}, \qquad i,j = 1,...,d.
\end{equation*}
For $i,j=1,...,d$ define the matrix $a^{ij}:\mathbb{S}_{++}^d\rightarrow\mathbb{M}^d$ by
\begin{equation*}\label{eq: a_def}
a^{ij}_{kl}(x) \, := \, \left(F_{ik}G_{lj} + F_{jk}G_{li}\right)(x),\qquad k,l = 1,...,d,\,
x\in \mathbb{S}_{++}^d.
\end{equation*}
Then the above system takes the form
\begin{equation*}\label{eq: state_alt}
dX_t^{ij} = b_{ij}(X_t)dt + \trace{a^{ij}(X_t)dB'_t}.
\end{equation*}
Then \cite[Lemma 5.1]{Robertson-Xing} shows that under Assumption \ref{ass: loc_ellip}, for any $x\in \mathbb{S}_{++}^d$ and $\theta\in \mathbb{S}^d$,
\begin{equation}\label{eq: ellip}
\sum_{i,j,k,l=1}^d \theta_{ij}\trace{a^{ij}(a^{kl})'}(x) \theta_{kl} = 4 \trace{f(x) \theta g(x) \theta}\geq c(x)\norm{\theta}^2,
\end{equation}
for some constant $c(x)>0$.
\begin{exa}\label{ex: wishart}
The primary example to keep in mind is when $X$ is the Wishart process, cf. \cite{Bru}:
\begin{equation}\label{eq: wishart}
dX_t = \pare{L L' + K X_t + X_t K'} dt + \sqrt{X_t} dB_t \Lambda' + \Lambda
dB'_t \sqrt{X_t},
\end{equation}
where $K,L,\Lambda \in \mathbb{M}^d$. Then both Assumptions \ref{ass: wp} and \ref{ass: loc_ellip} are satisfied when
\begin{equation}\label{eq: wishart_cond}
LL' \geq (d+1)\Lambda\Lambda' > 0.
\end{equation}
Indeed, here $b(x) = LL' + Kx + xK'$, $f(x) = x$, and $g(x) = \Lambda\Lambda'$. Using Remark \ref{rem: kron} it follows that $b,G'\otimes F$ are locally Lipschitz and of linear growth. Furthermore, calculation shows that $H_0(x; b) = \trace{(LL' - (d+1)\Lambda\Lambda')x^{-1}} + 2\trace{K}$. Thus, the first inequality in \eqref{eq: wishart_cond} implies $H_0(x;b) \geq 2\trace{K}$ on $\mathbb{S}_{++}^d$ and Assumption \ref{ass: wp} holds. Assumption \ref{ass: loc_ellip} readily follows from the second inequality in \eqref{eq: wishart_cond}.
\end{exa}
\subsection{The financial model}\label{subsec: fin market}
Having fixed notation and established well-posedness for the state variable, we may now define the financial model. As mentioned above, there is one risk-free asset $S^0$ and $n$ risky assets $(S^1,...,S^n)$ whose dynamics are given by
\begin{align}
\frac{dS^0_t}{S^0_t} &= r(X_t)dt,\qquad S^0_0 = 1, \label{eq: sde safe}\\
\frac{dS^i_t}{S^i_t} &= \left(r(X_t) + \mu_i(X_t)\right) dt +
\sum_{j=1}^m \sigma_{ij}(X_t)dZ^j_t,\qquad S^i_0 > 0,\qquad i=1,...,n.\label{eq: sde S}
\end{align}
Here, $r\in C^\gamma(\mathbb{S}_{++}^d; \mathbb R)$, $\mu\in C^{1,\gamma}(\mathbb{S}_{++}^d; \mathbb R^n)$, $\sigma\in C^{2,\gamma}(\mathbb{S}_{++}^d; \mathbb{M}^dim{n}{m})$ and $Z = (Z^1,...,Z^m)$ is a $\mathbb R^m$ valued Brownian motion. That $\sigma$ is of full rank, as well as the existence of \emph{market price of risk}, i.e., $\nu: \mathbb{S}_{++}^d \rightarrow \mathbb R^n$ such that $\mu = \sigma \sigma' \nu$ on $\mathbb{S}_{++}^d$, are ensured by the following assumption:
\begin{ass}\label{ass: sig_ellip}\text{}
\begin{enumerate}[i)]
\item When $m > n$, $\Sigma(x)\, := \, \sigma\sigma'(x) > 0$ for $x\in \mathbb{S}_{++}^d$. Then $\nu:= \Sigma^{-1} \mu$.
\item When $m < n$, $\sigma'\sigma(x) > 0$ for $x\in\mathbb{S}_{++}^d$ and there exists $\nu\in C^{1,\gamma}(\mathbb{S}_{++}^d; \mathbb R^n)$ such that $\mu = \Sigma\nu$.
\item When $m=n$, $\Sigma(x)>0$ for $x\in\mathbb{S}_{++}^d$ and $\sigma = \sqrt{\Sigma}$. Here again, $\nu = \Sigma^{-1}\mu$.
\end{enumerate}
\end{ass}
To allow for potentially stochastic instantaneous correlations between asset returns and the state variable, we define $Z$ in terms of the Brownian motion $B$ which drives $X$ and an independent $\mathbb R^m$ valued Brownian motion $W$. Specifically, let $C\in C^{2,\gamma}(\mathbb{S}_{++}^d;\mathbb{M}^dim{m}{d})$ and $\rho\in C^{2,\gamma}(\mathbb{S}_{++}^d; \mathbb R^d)$ be such that
\begin{ass}\label{ass: rho}
$\rho'\rho(x) CC'(x) \leq \idmat{m}$ for each $x\in \mathbb{S}_{++}^d$.
\end{ass}
Set $D:=\sqrt{\idmat{m}- \rho'\rho C C'} \in C^{2,\gamma}(\mathbb{S}_{++}^d; \mathbb{S}^d)$. We then may define $Z$ by
\begin{equation}\label{eq: BM Z}
Z^j_t:= \sum_{k,l=1}^d\int_0^tC_{jk}(X_u)dB^{kl}_u\rho_l(X_u) + \sum_{k=1}^m\int_0^tD_{jk}(X_u)dW^k_u,\qquad t\geq 0, j=1,...,m.
\end{equation}
By construction, $Z$ is a $m$ dimensional Brownian motion. Furthermore, the instantaneous correlation between $Z$ and $B$ is $d\langle Z^j, B^{kl} \rangle_t= C_{jk}(X_t) \rho_l(X_t) dt$, for $1\leq j\leq m, 1\leq k,l\leq d$. In particular, when $m=d$, $C= \idmat{d}$ and $\rho\in\mathbb R^d$ is constant, $d\langle Z^i, B^{jl} \rangle_t= \delta_{ij} \rho_l dt$, where $\delta_{ij}=1$ for $i=j$ and $0$ otherwise. This particular correlation structure is assumed in \cite{buraschi2010correlation, Hata-Sekine, Bauerle-Li, Richter}. Here, the matrix $C$ introduces general correlation structure and allow its dependence upon the state variable $X$.
\subsection{The optimal investment problem}\label{subsec: inv prob}
Consider an investor whose preference is described by a utility function $U: \mathbb R_+ \rightarrow \mathbb R$ which is strictly increasing, strictly concave, continuously differentiable and satisfies the Inada conditions $U'(0) =\infty$ and $U'(\infty)=0$. In particular, we pay special attention to utilities with constant relative risk aversion (henceforth CRRA) $U(x)= x^p/p$ for $0\neq p<1$.
Starting from an initial capital, this investor trades in the market until a time horizon $T\in \mathbb R_+$. She puts a proportion of her wealth $(\pi_t)_{t\leq T}$ into the risky assets and the remaining into the risk free asset. Given her strategy $\pi$, the price dynamics in \eqref{eq: sde safe} and \eqref{eq: sde S} imply that the wealth process $\mathcal{W}^\pi$ has dynamics
\begin{equation}\label{eq: wealth_dyn}
\frac{d\mathcal{W}^\pi_t}{\mathcal{W}^\pi_t} = (r(X_t) + \pi_t'\Sigma(X_t)\nu(X_t))dt + \pi_t'\sigma(X_t) dZ_t.
\end{equation}
The set of \emph{admissible} strategies are those $\pi$ which are $\mathbb{F}$-adapted and such that $\mathbb{P}^x\bra{\mathcal{W}^\pi_t >0, \forall t\leq T} =1$ for all $x\in\mathbb{S}_{++}^d$. In \eqref{eq: M_eta_def} below, positive super-martingale $M$ are constructed such that $M \mathcal{W}^\pi$ is a super-martingale for any admissible strategy $\pi$. In the presence of such \emph{super-martingale deflators}, arbitrage is excluded from the model (cf. \cite{Karatzas-Kardaras}).
The investor seeks to maximize the expected utility of her terminal wealth at $T$ by choosing admissible strategies, i.e.,
\begin{equation}\label{eq: op}
\mathbb{E}\bra{U(\mathcal{W}^\pi_T)}\rightarrow \text{Max}.
\end{equation}
In the remainder of this section, we will focus on the optimal investment problem for CRRA utilities and derive the associated HJB equation via a heuristic argument. To this end, define the (reduced) value function $v$ via
\begin{equation}\label{eq: power op}
\sup_{\pi \text{ admissible}} \mathbb{E}\bra{\left. \frac{1}{p}\pare{\mathcal{W}^\pi_T}^p \right|\mathcal{W}_t=w, X_t=x} = \frac{1}{p} w^p e^{v(T-t, x)},\qquad 0\leq t\leq T, w>0, x\in\mathbb{S}_{++}^d.
\end{equation}
Set $L$ as the infinitesimal generator of \eqref{eq: state_alt}:
\begin{equation}\label{eq: state_gen}
L := \frac12 \sum_{i,j,k,l=1}^d \trace{a^{ij}(a^{kl})'} D^2_{(ij),(kl)} + \sum_{i,j=1}^d b_{ij} D_{(ij)},
\end{equation}
where $D_{(ij)}=\partial_{x^{ij}}$ and $D^2_{(ij),(kl)}=\partial^2_{x^{ij} x^{kl}}$. The standard dynamic programming argument yields the following HJB equation for $v$:
\begin{equation}\label{eq: v eqn}
\begin{split}
\partial_t v = &L v + \frac12 \sum_{i,j,k,l=1}^d D_{(ij)} v \trace{a^{ij}(a^{kl})'} D_{(kl)} v
+ p\,r
\\
&+ \sup_{\pi} \left\{p \pi' \pare{\Sigma \nu + \sum_{i,j=1}^d \sigma C a^{ij} \rho D_{(ij)} v} + \frac12 p(p-1) \pi' \Sigma \pi\right\}, \qquad t>0, x\in \mathbb{S}_{++}^d,\\
0= &v(0, x), \quad x\in \mathbb{S}_{++}^d.
\end{split}
\end{equation}
The optimizer $\pi$ in the previous equation can be obtained pointwise and is given by
\begin{equation}\label{eq: pi_v_map}
\pi(t,x; v) \, := \, \begin{cases} \frac{1}{1-p}\Sigma^{-1}\left(\Sigma\nu +
\sum_{i,j=1}^d\sigma C a^{ij}\rho D_{(ij)}v\right)(t,x), & m > n \\
\frac{1}{1-p}\sigma(\sigma'\sigma)^{-1}\left(\sigma'\nu + \sum_{i,j=1}^d
C a^{ij}\rho D_{(ij)}v\right)(t,x), & m \leq n\end{cases},\qquad t>0,x\in\mathbb{S}_{++}^d.
\end{equation}
Define $q:=p/(p-1)$ as the conjugate of $p$ and the function $\mathcal{T}heta: \mathbb{S}_{++}^d \rightarrow \mathbb{S}_{++}^d$ via
\begin{equation}\label{eq: Theta_def}
\mathcal{T}heta(x)\, := \, \begin{cases} \sigma'\Sigma^{-1}\sigma(x) & m> n\\ \idmat{m}
& m \leq n\end{cases},\qquad x\in\mathbb{S}_{++}^d.
\end{equation}
Plugging in the formula for $\pi$ in \eqref{eq: pi_v_map} into \eqref{eq: v eqn}, a lengthy calculation yields the following semi-linear Cauchy problem for $v$:
\begin{equation}\label{eq: v_pde}
\begin{split}
v_t(t,x) &= \mathfrak{F}[v](t,x),\qquad 0< t, x\in\mathbb{S}_{++}^d,\\
v(0,x)& = 0,\qquad x\in\mathbb{S}_{++}^d.
\end{split}
\end{equation}
Here, the differential operator $\mathfrak{F}$ is defined as
\begin{equation}\label{eq: x_ops}
\mathfrak{F} \, := \,
\frac{1}{2}\sum_{i,j,k,l=1}^{d}A_{(ij),(kl)}D^2_{(ij),(kl)}
+ \sum_{i,j=1}^{d} \bar{b}_{ij} D_{(ij)}+
\frac{1}{2}\sum_{i,j,k,l=1}^{d}D_{(ij)}\bar{A}_{(ij),(kl)}D_{(kl)}
+ V,
\end{equation}
with
\begin{equation}\label{eq: functions}
\begin{split}
A_{(ij),(kl)}(x) &:= \trace{a^{ij}(a^{kl})'}(x),\\
\bar{A}_{(ij),(kl)}(x) &:=\trace{a^{ij}(a^{kl})'}(x)-q\rho'(a^{ij})'C'\mathcal{T}heta
C a^{kl}\rho(x),\\
\bar{b}_{ij}(x)&:= b_{ij}(x) - q\nu'\sigma C a^{ij}\rho(x),\\
V(x)&:= pr(x) - \frac{1}{2}q\nu'\Sigma\nu(x), \qquad i,j,k,l = 1,...,d, x\in \mathbb{S}_{++}^d.
\end{split}
\end{equation}
Note that $\pi$ in \eqref{eq: pi_v_map} and $\mathfrak{F}$ in \eqref{eq: x_ops} take different forms depending on $m>n$ or $m\leq n$ (with the two forms coinciding at $m=n$), and that using the definition of $L$ from \eqref{eq: state_gen} we have
\begin{equation}\label{eq: x_ops_L}
\mathfrak{F} = L - q\sum_{i,j=1}^d \nu'\sigma C a^{ij} \rho D_{(ij)} + \frac{1}{2}\sum_{i,j,k,l=1}^{d}D_{(ij)}\bar{A}_{(ij),(kl)}D_{(kl)}
+ V.
\end{equation}
In Section \ref{sec: converge} well-posedness of \eqref{eq: v_pde} is proved under appropriate parameter assumptions, and it is shown that the solution $v$, with appropriate growth constraint, to \eqref{eq: v_pde} is the reduced value function in \eqref{eq: power op}. Moreover the optimal strategy for \eqref{eq: power op} is given by
\begin{equation}\label{eq: opt_strat_T}
\pi^T_t \, := \, \pi(T-t, X_t; v), \qquad 0\leq t\leq T,
\end{equation}
for $\pi(\cdot, \cdot; v)$ from \eqref{eq: pi_v_map}.
\subsection{Long Horizon Convergence}
As mentioned in the introduction, this article is concerned with the large time behavior of the optimal investment problem. Such behavior for a CRRA investor is closely related to the ergodic analog of \eqref{eq: v_pde}, given by
\begin{equation}\label{eq: v_ergodic}
\begin{split}
\lambda & = \mathfrak{F}[v](x), \qquad x\in \mathbb{S}_{++}^d.
\end{split}
\end{equation}
A solution to \eqref{eq: v_ergodic} is defined as a pair $(\lambda, v)$ where $\lambda\in\mathbb R$ and $v\in C^2(\mathbb{S}_{++}^d; \mathbb R)$ which satisfy \eqref{eq: v_ergodic}. Since $\mathfrak{F}[v]$ only depends on derivatives of $v$, $v$ in a solution is only determined up to an additive constant. In particular we are interested in the \emph{smallest} $\lambda$ such that \eqref{eq: v_ergodic} admits a solution.
In the study of long horizon optimal investment and risk sensitive control problems, when the state variable is in $E\subseteq \mathbb R^d$, under appropriate restrictions \cite{Ichihara, Guasoni-Robertson}, there does exist a smallest $\hat{\lambda}$ such that \eqref{eq: v_ergodic} has a solution $\hat{v}$, such that the candidate reduced long run value function, accounting for the growth rate, is $\hat{\lambda} T + \hat{v}(x)$. The candidate long run optimal strategy is
\begin{equation}\label{eq: opt_strat}
\hat{\pi}_t := \pi(X_t; \hat{v}), \qquad t\geq 0,
\end{equation}
where $\pi(\cdot; \hat{v})$ from \eqref{eq: pi_v_map} with $v$ replaced by $\hat{v}$ which does not have a time argument. Now when the state variable is matrix valued, Proposition \ref{prop: ergodic wellposed} below establishes the existence of such $(\hat{\lambda}, \hat{v})$.
Comparing the finite and long horizon problems, we are interested in proving the following claim:
\begin{stat}[Long Horizon Convergence]\label{stat: long hor}$\,$
\begin{enumerate}[i)]
\item Define $h(T, x) := v(T,x)-\hat{\lambda}T - \hat{v}(x)$, for $T\geq 0$ and $x\in \mathbb{S}_{++}^d$. Then
\[
h(T, \cdot) \rightarrow C \quad \text{ and } \quad \nabla h(T,\cdot) \rightarrow 0 \quad \text{ in } C(\mathbb{S}_{++}^d), \quad \text{ as } T\rightarrow \infty.
\]
Here $C$ is a constant, $\nabla=(D_{(ij)})_{1\leq i,j\leq d}$ is the gradient operator, and convergence in $C(\mathbb{S}_{++}^d)$ stands for locally uniformly convergence in $\mathbb{S}_{++}^d$.
\item As functions of $x\in\mathbb{S}_{++}^d$ the finite horizon strategies converge to the long-run counterpart, i.e.
\[
\lim_{T\rightarrow \infty} \pi(T, \cdot; v) = \pi(\cdot; \hat{v}) \quad \text{ in } C(\mathbb{S}_{++}^d).
\]
\item Let $\pi^T$ and $\hat{\pi}$ be as in \eqref{eq: opt_strat_T} and \eqref{eq: opt_strat}. Let $\mathcal{W}^T$ and $\hat{\mathcal{W}}$ be the wealth processes employing $\pi^T$ and $\hat{\pi}$ respectively starting with initial capital $w$. Then for all $x\in\mathbb{S}_{++}^d$ and all $t\geq 0$:
\begin{align}
&\mathbb{P}^x-\lim_{T\rightarrow \infty} \sup_{0\leq u\leq t} \left|\frac{\mathcal{W}^T_u}{\hat{\mathcal{W}}_u} -1\right| =0,\label{eq: ratio wealth power}\\
&\mathbb{P}^x-\lim_{T\rightarrow \infty} \int_0^t (\pi^T_u- \hat{\pi}_u)' \Sigma(X_u) (\pi^T_u - \hat{\pi}_u) \, du =0.\label{eq: dist strategy}
\end{align}
Here $\mathbb{P}^x-\lim$ stands for convergence in probability $\mathbb{P}^x$.
\end{enumerate}
\end{stat}
In Statement \ref{stat: long hor}, i) claims that the reduced value function for the finite horizon problem converges to its infinite horizon counterpart; moreover ii) indicates that the finite horizon optimal strategy also converges, in feedback form, to a myopic long run limit. In addition to these analytic results, iii) states convergence in probabilistic terms: that is, the ratio between optimal wealth processes and distance between optimal strategies, when measured in a finite time window $[0,t]$, converge to zero in probability. Therefore when Statement \ref{stat: long hor} holds, a CRRA investor with long horizon can slightly modify her optimal strategy $\pi^T$ to $\hat{\pi}$, at the beginning of investment period, and incur a minimal loss of wealth and utility. Indeed, under appropriate parameter assumptions, Statement \ref{stat: long hor} is proved in \cite{guasoni.al.11} when the state variable is $\mathbb R$ valued and has constant correlation with risky assets. In Section \ref{sec: converge} below, we will verify Statement \ref{stat: long hor} in the matrix setting.
\subsection{Turnpike Theorems}
To state turnpike results, we consider two investors: the first one has a general utility function $U$ which satisfies conditions at the beginning of Section \ref{subsec: inv prob}; the second investor has a CRRA utility $U(x) = x^p/p$ for $0\neq p < 1$ \footnote{The logarithmic utility case is excluded here, since \cite[Proposition 2.5]{guasoni.al.11} already shows that turnpike theorems hold in a general semimartingale setting including the current case.}. The two investors are connected through the ratio of their marginal utilities $U'(x)/x^{p-1}$ as in the following assumption:
\begin{ass}\label{ass: ratio}
With $\mathfrak{R}(x)\, := \, U'(x)/x^{p-1}$ it follows that
\begin{equation}\label{eq: conv}
\lim_{x\uparrow\infty} \mathfrak{R}(x) = 1.
\end{equation}
\end{ass}
Assumption \ref{ass: ratio} ensures that preferences of the two investors are similar for large wealths. The next assumption ensures that the market described in Section \ref{subsec: fin market} is growing over time.
\begin{ass}\label{ass: grow}
For $r(x)$ as in \eqref{eq: sde safe} there exits constants $0 < \underbar{r} < \bar{r}$ such that $\underbar{r}\leq r(x) \leq \bar{r}$ for all $x\in\mathbb{S}_{++}^d$.
\end{ass}
In order to present the turnpike results, for the investor with general utility $U$, set $\pi^{1,T}$ as the optimal strategy of \eqref{eq: op} and $\mathcal{W}^{1,T}$ as the associated optimal wealth process starting from initial wealth $w$. We are interested in proving the turnpike theorem:
\begin{stat}[Turnpike Theorem]\label{stat: turnpike}
For all $x\in\mathbb{S}_{++}^d$ and all $t\geq 0$,
\begin{align}
&\mathbb{P}^x-\lim_{T\rightarrow\infty} \ \sup_{u\leq t}\left|\frac{\mathcal{W}^{1,T}_u}{\hat{\mathcal{W}}_u} - 1\right| = 0,\label{eq: turnpike wealth}\\
&\mathbb{P}^x-\lim_{T\rightarrow\infty} \int_0^t \left(\pi^{1,T}_u - \hat{\pi}_u\right)'\Sigma(X_u)\left(\pi^{1,T}_u - \hat{\pi}_u\right) du = 0, \label{eq: turnpike strat}
\end{align}
where $\hat{\pi}$ from \eqref{eq: opt_strat} and $\hat{\mathcal{W}}$ is the wealth process starting from $w$ following $\hat{\pi}$.
\end{stat}
The first convergence above states that the ratio, when measured in an finite time window, of the optimal wealth process for the generic investor and the long run wealth process for the CRRA investor is uniformly close to one in probability as the horizon becomes large. The message behind the second convergence is that, as the horizon becomes long, the optimal investment strategy for the generic utility investor approaches the long-run limit strategy of the CRRA investor.
Such a result is called an ``explicit" turnpike using the terminology of \cite{guasoni.al.11}, where Statement \ref{stat: turnpike} is proved in a factor model with $\mathbb R$ valued state variable and constant correlation. In Section \ref{sec: converge} below, we will extend this result to when the state variable is matrix valued.
\begin{rem}\label{rem: general state space}
Statements \ref{stat: long hor} and \ref{stat: turnpike} are not specific to models with matrix valued state variables. As mentioned in introduction, the main technique to confirm these statements is the large time asymptotic analysis of \eqref{eq: v eqn} in \cite{Robertson-Xing}. In particular, a general framework is introduced in \cite[Section 2]{Robertson-Xing}, where convergence results (cf. Theorems 2.9 and 2.11 therein) are obtained for a general state space $E$. The main message therein is, when two ``Lyapunov'' functions $\phi$ and $\psi$ exist and satisfy appropriate assumptions, then the desired convergence results hold. When the state space is specified, assumptions on $\phi$ and $\psi$ are translated to explicit parameter restrictions. In particular, when the state space is $\mathbb R^d$, these parameter restrictions are given in \cite[Section 3.1]{Robertson-Xing}. Therefore, proof of Statements \ref{stat: long hor} and \ref{stat: turnpike} in this case follows from essentially the same line of reasoning as in the matrix case and is, in fact, much more straightforward.
\end{rem}
\section{Main results}\label{sec: converge}
\subsection{The (generalized) Wishart factor model}\label{subsec: wishart}
Before presenting results for the general matrix setting in Section \ref{subsec: state var}, let us highlight the case when $X$ is a Wishart process as in Example \ref{ex: wishart}. We specify the financial model in Section \ref{subsec: fin market} to the following:
\begin{align*}
& m=d, \quad C(x) = \idmat{d}, \quad D(x) = \sqrt{1 - \rho' \rho(x)}\idmat{d},\\
& r(x) = r_0+ \trace{r_1 x}, \quad \sigma(x) = \zeta(x) \sqrt{x}, \quad \mu(x) = \zeta(x)x\zeta'(x)\nu(x); \quad \text{ for } x\in \mathbb{S}_{++}^d,
\end{align*}
where $r_0\in \mathbb R$ and $r_1 \in \mathbb{M}^d$. We assume that $\nu\in C^{1,\gamma}(\mathbb{S}_{++}^d; \mathbb R^n)$, $\zeta \in C^{2,\gamma}(\mathbb{S}_{++}^d; \mathbb{M}^{n\times d})$, and $\rho \in C^{2,\gamma}(\mathbb{S}_{++}^d; \mathbb R^d)$ are all bounded functions and $\sup_{x\in \mathbb{S}_{++}^d} \rho'\rho(x) < 1$. When these functions are not constant, the previous model is not affine, in contrast to \cite{buraschi2010correlation, Hata-Sekine, Bauerle-Li, Richter}. For the given $\sigma$, Assumption \ref{ass: sig_ellip} takes the form
\begin{ass}\label{ass: sig_ellip_gW}\text{}
\begin{enumerate}[i)]
\item When $d> n$, $\zeta\zeta'(x) > 0$ for $x\in\mathbb{S}_{++}^d$.
\item When $d < n$, $\zeta'\zeta(x) > 0$ for $x\in\mathbb{S}_{++}^d$.
\item When $d=n$, $\zeta(x) = \zeta'(x) > 0$ for $x\in\mathbb{S}_{++}^d$.
\end{enumerate}
\end{ass}
The following proposition verifies Statements \ref{stat: long hor} and \ref{stat: turnpike} in the current model under explicit parameter restrictions. The proof of Proposition \ref{prop: wishart} is in Appendix \ref{app: B}.
\begin{prop}\label{prop: wishart}
Let Assumption \ref{ass: sig_ellip_gW} hold. Assume the following parameter restrictions:
\begin{enumerate}[i)]
\item $LL' > (d+1)\Lambda \Lambda'>0$.
\item When $p<0$, $r_1$ satisfies $r_1 + r_1 \geq 0$ and there exists $\varepsilonilon > 0$ such that either
\begin{equation*}
-p(r_1+r_1')+q \zeta' \nu\nu'\zeta(x)\geq \varepsilonilon \,\idmat{d},\qquad x\in\mathbb{S}_{++}^d;
\end{equation*}
or
\begin{equation*}
(K-q\Lambda \rho \nu' \zeta)(x) + (K-q\Lambda \rho \nu' \zeta)'(x) \leq -\varepsilonilon \,\idmat{d},\qquad x\in \mathbb{S}_{++}^d.
\end{equation*}
\item When $0<p<1$, there exists $\varepsilonilon>0$ such that
\begin{equation*}
(K-q \Lambda \rho \nu' \zeta)(x) + (K- q \Lambda \rho \nu' \zeta)'(x) \leq -\varepsilonilon \idmat{d},\qquad x\in \mathbb{S}_{++}^d;
\end{equation*}
and
\begin{equation}\label{eq: p>0 cond}
\varepsilonilon^2 > 8(1-q) \sqrt{d} \,\trace{\Lambda \Lambda'} \sup_{x\in \mathbb{S}_{++}^d}\norm{p(r_1 + r_1') - q\zeta' \nu \nu' \zeta(x)}.
\end{equation}
\end{enumerate}
Then, the long-horizon convergence results in Statement \ref{stat: long hor} hold. Additionally when $r_1 = 0$, the turnpike theorems in Statement \ref{stat: turnpike} hold for all utility functions $U$ satisfying Assumption \ref{ass: ratio}.
\end{prop}
In the previous parameter restrictions, part i) is slightly stronger than the well-posedness condition \eqref{eq: wishart_cond}. The restriction in the $p<0$ case is \emph{mild}.
When $r_1 + r_1'>0$, it follows that $-p(r_1 + r_1') + q \sigma' \nu \nu' \sigma (x) \geq \varepsilonilon \, \idmat{d}$ for some $\varepsilonilon>0$ since $q\zeta'\nu\nu'\zeta \geq 0$. Thus, part $ii)$ holds. When $r_1+r_1'$ is non-negative but may degenerate, consider a (generalized) Wishart process $\overline{X}$ with dynamics
\[
d\overline{X}_t = \pare{L L' + \overline{K}(\overline{X}_t) \overline{X_t} + X_t \overline{K}(\overline{X}_t)}' dt + \sqrt{\overline{X}_t} dB_t \Lambda' + \Lambda
dB'_t \sqrt{\overline{X}_t},
\]
where $\overline{K}(x):=K- q \Lambda \rho \nu' \zeta(x)$.\footnote{This SDE admits a unique global strong solution $\overline{X}$. This is because $H_0(x; b)\geq 2 \trace{\overline{K}(x)}$ which is uniformly bounded from below due to the boundedness assumption of $\rho, \nu$, and $\zeta$ on $\mathbb{S}_{++}^d$. Hence the existence follows from \cite[Theorem 3.4]{Mayerhofer-Pfaffel-Stelzer}.} Then we require $\overline{X}$ is mean-reverting to verify part $ii)$. When $0<p<1$, we require the force of mean-reversion to be sufficiently strong. In this case, \eqref{eq: p>0 cond} is necessary because the potential
\begin{equation*}
V(x) = pr_0 + p\trace{r_1 x} - \frac{1}{2}q\nu'\zeta(x) x \zeta(x)'\nu = pr_0 +\frac{1}{2}\trace{x(p(r_1+r_1') - q\zeta'\nu\nu'\zeta(x))},
\end{equation*}
may not be uniformly bounded from above on $\mathbb{S}_{++}^d$.
\subsubsection{An Explicit Long Run Optimal Strategy and a Counter-Example}\label{subsubsec: wishart_examples} We now focus on the ``classical'' Wishart model where $\rho,\nu$ and $\zeta$ in the previous section are constants taking values in $\mathbb R^d$, $\mathbb R^n$ and $\mathbb{M}^dim{n}{d}$ respectively. Here, it is shown that if the dimension $d$ of the Wishart process is \emph{less than or equal to} $n$, the number of risky assets, then the solution $\hat{v}$ to \eqref{eq: v_ergodic} with minimal $\hat{\lambda}$ is an affine function of $x$: i.e. up to an additive constant, $\hat{v}(x) = \textrm{Tr}(\hat{M}x)$ for a symmetric matrix $\hat{M}$ satisfying the Riccati equation given in \eqref{eq: d_leq_n_Ricatti} below. However, surprisingly, if $d > n$ then $\hat{v}$ may \emph{not} be affine, hence $\hat{\pi}$ in \eqref{eq: opt_strat} is not affine either. This is due to the non-commutative property of matrix product.
To streamline the presentation, we assume that $p<0$ and $r_1 + r_1' > 0$. Hence Proposition \ref{prop: wishart} follows if $LL' > (d+1)\Lambda\Lambda' > 0$ and the constant matrix $\zeta$ satisfies Assumption \ref{ass: sig_ellip_gW}. We consider candidate solutions to \eqref{eq: v_ergodic} given by
\begin{equation}\label{eq: cand_hatv}
v(x) = \trace{Mx},\qquad M=M'.\footnote{We can assume $M=M'$ without loss of generality since $x\in \mathbb{S}_{++}^d$ implies $\trace{Mx} = \trace{M'x} = (1/2)\trace{(M+M')x}$.}
\end{equation}
First we present the result when $d\leq n$:
\begin{prop}\label{prop: wishart_good_case}
Assume $d\leq n$ and $\rho,\nu,\zeta$ are constant. Let $\zeta$ satisfy Assumption \ref{ass: sig_ellip_gW} and assume $p<0$, $r_1+r_1' > 0$, $LL' > (d+1)\Lambda\Lambda' >0$. Consider the following matrix Riccati equation in $M$:
\begin{equation}\label{eq: d_leq_n_Ricatti}
0 = 2M\Lambda(1-q\rho\rho')\Lambda'M + (K-q\Lambda\rho\nu'\zeta)'M + M(K-q\Lambda\rho\nu'\zeta) + \frac{1}{2}\left(p(r_1+r_1') - q\zeta'\nu\nu'\zeta\right).
\end{equation}
There exists a unique $\hat{M}\in\mathbb{S}^d$ solving \eqref{eq: d_leq_n_Ricatti} such that $(\hat{\lambda}, \hat{v})$, with $\hat{\lambda}= \textrm{Tr}(LL'\hat{M})+ p r_0$ and $\hat{v}(x)=\textrm{Tr}(\hat{M}x)$, solves \eqref{eq: v_ergodic} and $\hat{\lambda}$ is the smallest $\lambda$ with accompanying $v$.
\end{prop}
We next present a counter-example in the $d > n$ case showing that solutions $(\hat{\lambda},\hat{v})$ to \eqref{eq: v_ergodic} \emph{cannot} be of the affine form in \eqref{eq: cand_hatv}. However the existence of solutions to \eqref{eq: v_ergodic} is still ensured by Proposition \ref{prop: wishart}.
\begin{exa}\label{exa: counter-exa}
Take $n=1, d=2$ and
\begin{equation}\label{eq: example_coeffs}
\begin{split}
\Lambda &= \idmat{2},\quad L = \ell \idmat{2}\textrm{ for }\ell > \sqrt{3}, \quad K = \idmat{2}, \quad C= \idmat{2},\\
\zeta &= \left(\begin{array}{c c} 1 & 0 \end{array}\right),\quad \nu = \nu\in\mathbb R,\quad \rho = \rho\left(\begin{array}{c c} 1 &1\end{array}\right)'\textrm{ for } 0<2\rho^2 < 1,\\
r_0 &> 0, \quad r_1 = r_1\idmat{2}\textrm{ for } r_1 > 0.
\end{split}
\end{equation}
Consider functions $v$ as in \eqref{eq: cand_hatv}. Writing the generic element $X\in \mathbb{S}_{++}^d$ and the matrix $M$ as
\begin{equation}\label{eq: wishart_x_v_ex}
X= \left(\begin{array}{c c} x & y \\ y & z\end{array}\right),\quad x,z > 0, y^2 < xz,\qquad M = \left(\begin{array}{c c} M_1 & M_2 \\ M_2 & M_3\end{array}\right),
\end{equation}
we have that $\Sigma(X) = \zeta X \zeta' = x>0$ so that Assumption \ref{ass: sig_ellip_gW} holds. Furthermore, $LL' - 3\Lambda\Lambda' = (\ell^2 - 3)\idmat{2} > 0$ and for $p<0$, $-p(r_1+r_1') + q\zeta'\nu\nu'\zeta (x) \geq -2pr_1 \idmat{2} > 0$. Thus, the assumptions of Proposition \ref{prop: wishart} hold for $p<0$. A lengthy calculation shows that (cf. Lemma \ref{lem: example_calc} in Appendix \ref{app: C})
\begin{equation}\label{eq: F_example}
\begin{split}
\mathfrak{F}[v] &= x\left(2(M_1^2 + M_2^2) - 2q\rho^2(M_1+M_2)^2 + 2M_1 - 2q\rho\nu(M_1+M_2) + pr_1 - \frac{1}{2}q\nu^2\right)\\
&\qquad + y\left(4M_2(M_2+M_3) - 4q\rho^2(M_1+M_2)(M_2+M_3) + 4M_2 - 2q\rho\nu(M_2+M_3)\right)\\
&\qquad + z\left(2(M_2^2+M_3^2) +2M_3 + pr_1\right)\\
&\qquad + \frac{y^2}{x}\left(-2q\rho^2(M_2+M_3)^2\right)\\
&\qquad + pr_0 + \ell^2(M_1 + M_3).
\end{split}
\end{equation}
As can be seen from \eqref{eq: wishart_bar_A} in Lemma \ref{lem: wishart_op_affine_v} below, the problem term $y^2/x$ arises when evaluating $\bar{A}$ from \eqref{eq: functions}, since for $d>n$:
\begin{equation}\label{eq: wishart_counter_ex_Theta}
\sqrt{X}\mathcal{T}heta(X)\sqrt{X} = X\zeta'(\zeta X\zeta')^{-1}\zeta X = \frac{1}{x}X\left(\begin{array}{c} 1\\ 0\end{array}\right)\left(\begin{array}{c c} 1 & 0\end{array}\right) X = \left(\begin{array}{c c} x & y \\ y & \frac{y^2}{x}\end{array}\right);
\end{equation}
whereas, for arbitrary model coefficients, if $d\leq n$ then $\sqrt{X}\mathcal{T}heta(X)\sqrt{X} = X$.
Thus, if $\mathfrak{F}[v] = \lambda$ for some constant $\lambda$ it must be that each coefficient of $x,y,z, y^2/x$ in \eqref{eq: F_example} is equal to zero. By considering $y^2/x$ it follows that $M_2 + M_3 = 0$. Plugging this into the coefficient of $y$
gives $M_2 = 0$ and hence $M_3=0$. Then the coefficient of $z$ being zero yields $0 = pr_1$ a contradiction since $r_1 > 0$. Thus, the function $\hat{v}$ cannot be affine.
\end{exa}
\subsection{General State Variables}\label{subsec: general_case}
We now consider the general case when $X$ has dynamics as in \eqref{eq: state} where, in addition to the aforementioned regularity restrictions, the model coefficients satisfy Assumptions \ref{ass: wp} and \ref{ass: loc_ellip}. As in the previous section, the goal is to provide conditions, based entirely upon the model coefficients, under which Statements \ref{stat: long hor} and \ref{stat: turnpike} hold.
To list the coefficient assumptions, let $f,g$ be as in \eqref{eq: f_g_def}, $\bar{b},V$ as in \eqref{eq: x_ops}, and recall $H_\delta(x;b)$ from \eqref{eq: Hdelta_def}. Assumption \ref{ass: coeff_master_list} below gives a number of restrictions under which the main convergence results hold. Though the list below is lengthy, it can be readily checked for particular models of interest.
\begin{ass}\label{ass: coeff_master_list}
There exists $n_0>0$ such that the following hold for $\norm{x}\geq n_0$:
\begin{enumerate}[1)]
\item $\bar{b}$ has at most linear growth.
\item There exists $\alpha_1 > 0$ so that $\trace{f(x)}\trace{g(x)} \leq
\alpha_1\norm{x}$.
\item There exits $\beta_1\in\mathbb R$, $C_1 > 0$ so that $\trace{\bar{b}(x)'x} \leq
-\beta_1\norm{x}^2 + C_1$.
\item There exists $\gamma_1,\gamma_2\in\mathbb R$ and $C_2 > 0$ so that
$-\gamma_2\norm{x} - C_2 \leq V(x) \leq -\gamma_1\norm{x} + C_2$. $V(x)$ is uniformly bounded from above for $\norm{x}\leq n_0$.
\item $\max\cbra{\gamma_1,\beta_1} > 0$. Furthermore
\begin{enumerate}[i)]
\item If $\gamma_1 > 0, \beta_1\leq 0$, then there exist $\alpha_2>0$,
$C_3\in\mathbb R$ so that $\trace{f(x)xg(x)x} \geq \alpha_2\norm{x}^3 - C_3$.
\item If $\gamma_1< 0, \beta_1 > 0$, then
$\beta_1^2 + 16\overline{\kappa}\alpha_1\gamma_1 > 0$, where $\alpha_1$ is from part $2)$, $\overline{\kappa}=1$ when $p<0$, and $\overline{\kappa}=1-q$ when $0<p<1$.
\item If $\gamma_1 \geq 0, \beta_1 > 0$ then no additional restrictions are
necessary.
\end{enumerate}
\end{enumerate}
There exists $\varepsilon, c_0, c_1 > 0$ such that
\begin{enumerate}[A)]
\item $\inf_{x\in\mathbb{S}_{++}^d} H_\varepsilon(x; \bar{b})
> -\infty$ (note : here we are using $\bar{b}$ instead of $b$ in \eqref{eq: Hdelta_def}).
\item $\liminf_{\det{x}\downarrow 0}\left(H_{\varepsilon}(x; \bar{b}) + c_0
\log(\det{x})\right) > -\infty$.
\item $\lim_{\det{x}\downarrow
0}\left(H_0(x;\bar{b}) + c_1 V(x)\right) = \infty$.
\end{enumerate}
\end{ass}
\begin{rem} When $p<0$ and the interest rate function $r(x)$ is bounded from below on $\mathbb{S}_{++}^d$ (e.g. $r(x)\geq 0$), then $\gamma_1 \geq 0$, hence the complicated part $5-ii)$ in Assumption \ref{ass: coeff_master_list} is never required.
\end{rem}
The parameter restrictions in Assumption \ref{ass: coeff_master_list} have a similar interpretation to those in Proposition \ref{prop: wishart}. Indeed, consider a $\mathbb{S}_{++}^d$-valued diffusion $X$ with dynamics:
\begin{equation}\label{eq: sde barX}
d\bar{X}^{ij}_t = \bar{b}_{ij}(\bar{X}_t) dt + \trace{a^{ij}(\bar{X}_t) dW'_t}, \qquad i,j=1,\cdots, d.
\end{equation}
Comparing to \eqref{eq: state_alt}, the drift is adjusted to $\bar{b}$. The given regularity assumptions and parts 1) and 2) imply that the coefficients of $\bar{X}$ are locally Lipschitz and have at most linear growth. On the other hand, due to the second inequality in \eqref{eq: ellip}, $H_\delta$ is decreasing in $\delta$. Hence part A) implies $H_0(x; \bar{b})$ is bounded from below on $\mathbb{S}_{++}^d$. As a result, Assumption \ref{ass: wp} specified to $\overline{X}$ from \eqref{eq: sde barX} holds and \cite[Theorem 3.4]{Mayerhofer-Pfaffel-Stelzer} ensures that \eqref{eq: sde barX} has a unique global strong solution.
In Assumption \ref{ass: coeff_master_list} parts 3) and 4), if $\beta_1 >0$ then $\bar{X}$ is mean-reverting and if $\gamma_1>0$, the potential $V$ decays to $-\infty$ uniformly as $\norm{x}\rightarrow\infty$. Thus, part 5) requires either mean reversion or a decaying potential. If both happen, then no additional parameter restrictions is necessary. However, if mean reversion fails we require uniform ellipticity for $A(x)$ in the direction of $x$. If $\gamma_1<0$, then a delicate relationship in $5-ii)$ between the growth and degeneracy of $A$, mean reversion of $\bar{b}$ and the growth of $V$ is needed.
Finally, Assumption \ref{ass: coeff_master_list} parts B) and C) are restrictions when the determinant of $\bar{X}$ is small. These two assumptions help to bound the value function $v$ from above and below, ensuring $v$ is finite close to the boundary $\{x\in \mathbb{S}_{++}^d\,:\, \det(x) =0\}$ of the state space.
From a technical point of view, Assumption \ref{ass: coeff_master_list} helps to construct an upper bound for solutions to \eqref{eq: v_pde}. It is shown in \cite[Section 3]{Robertson-Xing} that well-posedness of \eqref{eq: v_pde} is established among solutions which are bounded from above (up to an additive constant) by
\[
\phi_0(x) := -\underline{c} \log(\det(x)) + \overline{c} \norm{x} \eta(\norm{x}) + C,
\]
where $\underline{c}, \overline{c} > 0$ and $C>0$ is chosen so that $\phi_0$ is non-negative on $\mathbb{S}_{++}^d$. Here, $\eta\in C^{\infty}(0,\infty)$ is a cutoff function satisfying $0\leq \eta \leq 1$, $\eta(x) =1$ when $x>n_0+2$ and $\eta(x)=0$ for $x<n_0+1$, for the given $n_0$. Assumption \ref{ass: coeff_master_list} helps to verify the heuristic argument in Section \ref{subsec: inv prob}: \cite[Propositions 2.5, 2.7, and Theorem 3.9]{Robertson-Xing} prove that
\begin{prop}\label{prop: v wellposed}
Let Assumptions \ref{ass: loc_ellip}, \ref{ass: sig_ellip}, \ref{ass: rho} and \ref{ass: coeff_master_list} hold. Then there exists a unique solution $v\in C^{1,2}((0,\infty)\times \mathbb{S}_{++}^d) \cap C([0,\infty)\times \mathbb{S}_{++}^d)$ to \eqref{eq: v_pde} such that
\[
\sup_{(t,x)\in [0,T]\times \mathbb{S}_{++}^d} (v(t,x)-\phi_0(x))<\infty, \quad \text{ for each } T\geq 0.
\]
\end{prop}
Combining with the following verification result whose proof is deferred to Appendix \ref{app: A}, we obtain that the optimization problem in \eqref{eq: power op} is well-posed for any horizon $T>0$.
\begin{prop}\label{prop: verification}
Let Assumptions \ref{ass: loc_ellip}, \ref{ass: sig_ellip}, \ref{ass: rho} and \ref{ass: coeff_master_list} hold. Then for $v$ in Proposition \ref{prop: v wellposed} and any $T>0$, \eqref{eq: power op} holds and $\pi^T$ from \eqref{eq: opt_strat_T} is the optimal strategy for \eqref{eq: power op}.
\end{prop}
The aforementioned parameter assumptions also ensure the well-posedness of \eqref{eq: v_ergodic}: \cite[Proposition 2.3 and Lemma 5.3]{Robertson-Xing} prove that
\begin{prop}\label{prop: ergodic wellposed}
Let Assumptions \ref{ass: loc_ellip}, \ref{ass: sig_ellip}, \ref{ass: rho} and \ref{ass: coeff_master_list} hold. There exists $(\hat{\lambda}, \hat{v})$ solving \eqref{eq: v_ergodic} such that $\hat{v}$ is unique (up to an additive constant) and $\hat{\lambda}$ is the smallest $\lambda$ such that there exists a corresponding $v$ solving \eqref{eq: v_ergodic}.
\end{prop}
We are now ready to state our first main result, whose proof is presented in Appendix \ref{app: B}.
\begin{thm}\label{thm: power}
Let Assumptions \ref{ass: loc_ellip}, \ref{ass: sig_ellip}, \ref{ass: rho} and \ref{ass: coeff_master_list} hold. Then the long horizon results in Statement \ref{stat: long hor} hold.
\end{thm}
To state the portfolio turnpike result, we need to make an additional assumption which is a mild strengthening of Assumption \ref{ass: rho}:
\begin{ass}\label{ass: rho_strong}
For $\rho$ and $C$ in Assumption \ref{ass: rho}, $\rho'\rho CC'(x) < \idmat{m}$ for all $x\in\mathbb{S}_{++}^d$.
\end{ass}
Under the previous assumption, it is possible to construct not only super-martingale deflators (cf. \eqref{eq: M_eta_def} below), but also equivalent local martingale measures $\mathbb{Q}^T$, for all $T>0$; i.e. $\mathbb{Q}^T$ is equivalent to $\mathbb{P}$ on $\mathcal{F}_T$ and $e^{-\int_0^\cdot r(X_u)\,du}S$ is a $\mathbb{Q}^T$ local martingale on $[0,T]$. This is needed to utilize duality results in \cite{karatzas.zitkovic.03} to establish the existence of an optimal strategy to \eqref{eq: op} for the generic utility $U$.
We are now ready to state the following turnpike result:
\begin{thm}\label{thm: turnpike}
Let Assumptions \ref{ass: wp}, \ref{ass: loc_ellip}, \ref{ass: sig_ellip}, \ref{ass: coeff_master_list} and \ref{ass: rho_strong} hold. Then the turnpike theorems in Statement \ref{stat: turnpike} hold.
\end{thm}
\appendix
\section{Proof of Proposition \ref{prop: verification}}\label{app: A}
We first define a class of supermartingale deflators on $[0,T]$ for any $T>0$. Given a $\mathbb{M}^{d}$-valued process $\eta$ with $\int_0^T \norm{\eta_u}^2 du<\infty$ a.s., define $M^\eta$ via (note: for a function $g$ of $\mathbb{S}_{++}^d$ we will write $g_u$ for $g(X_u)$):
\begin{equation}\label{eq: M_eta_def}
\begin{split}
M^{\eta}_t &\, := \,
e^{-\int_0^t r_udu}\mathcal{E}\left(\int\left(-\nu_u'\sigma_u C_u dB_u \rho_u + \trace{\eta_u dB_u'} - \rho_u'\eta_u'C_u'\mathcal{T}heta_uC_u dB_u\rho_u\right)\right)_t\\
&\qquad\times\mathcal{E}\left(-\int\left(\nu_u'\sigma_u D_u +\rho_u'\eta_u'C_u'\mathcal{T}heta_uD_u\right)dW_u\right)_t,\\
&= e^{-\int_0^t r_u du}\mathcal{E}\left(\int \sum_{k,l=1}^d dB^{kl}_u\left(-(C'\sigma'\nu)_k\rho_l + \eta_{kl} - (C'\mathcal{T}heta C\eta \rho)_k\rho_l\right)_u\right)_t\\
&\qquad\times\mathcal{E}\left(-\int \sum_{k=1}^d dW^k_u\left((D'\sigma'\nu)_k + (D'\mathcal{T}heta C\eta\rho)_k\right)_u\right)_t,\qquad t\leq T.
\end{split}
\end{equation}
When $\eta=0$, $e^{\int_0^\cdot r_u du} M^\eta$ defines the \emph{minimal martingale measure}, provided the stochastic exponentials are indeed martingales, see \cite{MR1108430}. Hence we call $\eta$ a \emph{risk premia}. For any admissible strategy $\pi$, $M^\eta \mathcal{W}^\pi$ is a positive super-martingale. Indeed, using \eqref{eq: BM Z}, \eqref{eq: wealth_dyn}, and \eqref{eq: M_eta_def}, the stochastic integration by parts formula shows that the drift of $M^\eta \mathcal{W}^\pi$ has the following integrand (omitting function arguments and time subscripts):
\begin{equation*}
\begin{split}
M^{\eta}\mathcal{W}^{\pi}&\pi' \bra{\Sigma\nu + \sigma C\left(-C'\sigma'\nu\rho' + \eta - C'\mathcal{T}heta C\eta\rho\rho'\right)\rho - \sigma D\left(D'\sigma'\nu + D'\mathcal{T}heta C\eta\rho\right)}\\
= &M^{\eta}\mathcal{W}^{\pi}\pi'\bra{\Sigma\nu - \sigma\left(CC'\rho'\rho + DD'\right)\sigma'\nu + \sigma C \eta\rho - \sigma\left(CC'\rho'\rho + DD'\right)\mathcal{T}heta C\eta\rho},\\
= &M^{\eta}\mathcal{W}^{\pi}\pi'\bra{\sigma C\eta\rho - \sigma\mathcal{T}heta C\eta\rho},\\
= &0,
\end{split}
\end{equation*}
where the second identity follows from $(CC'\rho'\rho + DD')(x)= 1_m$ and the third identity holds due to $\sigma\mathcal{T}heta = \sigma$. Therefore $M^\eta \mathcal{W}^\pi$ is a positive local martingale hence a super-martingale.
Before proving Proposition \ref{prop: verification}, we must introduce some notation. For a fixed $\phi\in C^{(1,2),\gamma}((0,\infty)\times \mathbb{S}_{++}^d, \mathbb R)$, the regularity assumptions on the coefficients and ellipticity assumption in \eqref{eq: ellip} ensure that the \emph{generalized} martingale problem on $\mathbb{S}_{++}^d$ for
\begin{equation}\label{eq: cL_phi_time}
\mathcal{L}^{\phi,T-t} := \frac12 \sum_{i,j,k,l=1}^d A_{(ij),(kl)} D_{(ij),(kl)} + \sum_{i,j=1}^d \pare{\bar{b}_{ij} + \sum_{k,l=1}^d \bar{A}_{(ij),(kl)} D_{(kl)} \phi(T-t,\cdot)} D_{(ij)}, \quad t\leq T,
\end{equation}
has a unique solution $\left(\mathbb{P}^{\phi,T,x}\right)_{x\in\mathbb{S}_{++}^d}$ cf. \cite{Pinsky}. When $\phi$ does not depend upon $t$ we will write $\mathcal{L}^{\phi}$ and denote the solution as $\left(\mathbb{P}^{\phi,x}\right)_{x\in\mathbb{S}_{++}^d}$. The martingale problem for $\mathcal{L}^{\phi,T-\cdot}$ is \emph{well-posed} if the coordinate process $X$ does not hit the boundary $\mathbb{S}_{++}^d$, $\mathbb{P}^{\phi,T,x}$-a.s., before $T$ for any $x\in\mathbb{S}_{++}^d$. Similarly, if $\phi$ does not depend upon time, then well-posedness follows if the coordinate process does not hit the boundary in finite time $\mathbb{P}^{\phi, x}$-a.s. for any $x\in\mathbb{S}_{++}^d$.
For the given $\phi$, define the stochastic exponential
\begin{equation}\label{eq: Z_phi}
\begin{split}
Z^{\phi,T}_t \, := \, &\mathcal{E}\left(\int_0^\cdot \sum_{k,l=1}^d dB^{kl}_u \left(-q(C'\sigma'\nu)_k\rho_l + \sum_{i,j=1}^d \left(a^{ij}_{kl} - q(C'\mathcal{T}heta Ca^{ij}\rho)_k\rho_l\right)D_{(ij)}\phi\right)(T-u,X_u)\right)_t\\
&\times \mathcal{E}\left(\int_0^\cdot \sum_{k=1}^m dW^k_u\left(-q(D'\sigma'\nu)_k - q\sum_{i,j=1}^d(D'\mathcal{T}heta Ca^{ij}\rho)_kD_{(ij)}\phi\right)(T-u,X_u)\right)_t,\qquad t\leq T.
\end{split}
\end{equation}
For $\phi$ not depending upon time, write $Z^\phi$ for $Z^{\phi,T}$ and note that $Z^{\phi}$ is defined for all $t\geq 0$.
Recall from Section \ref{subsec: state var} that Assumption \ref{ass: wp} ensures the well-posedness of \eqref{eq: state}. Hence the martingale problem for $L$ in \eqref{eq: state_gen} is well-posed. Now if the martingale problem for $\mathcal{L}^{\phi,T-\cdot}$ is also well-posed, it follows from (\cite[Remark 2.6]{Cheridito-Filipovic-Yor}) that the first stochastic exponential on the right hand side of \eqref{eq: Z_phi} is a $\mathbb{P}^x$-martingale on $[0,T]$. On the other hand, since $X$ and $W$ are $\mathbb{P}^x$-independent, it follows from \cite[Lemma 4.8]{Karatzas-Kardaras} that $Z^{\phi, T}$ is also a $\mathbb{P}^x$-martingale on $[0,T]$. Therefore, we may define a new measure $\mathbb{P}^{\phi, T, x}$ on $\mathcal{F}_T$ via
$ d\mathbb{P}^{\phi,T,x}/d\mathbb{P}^x |_{\mathcal{F}_T} = Z^{\phi,T}_T$.
Moreover, Girsanov's theorem yields that $X$ has generator $\mathcal{L}^{\phi,T-\cdot}$ under $\mathbb{P}^{\phi,T,x}$. When $\phi$ does not have time argument and the martingale problem for $\mathcal{L}^\phi$ is well-posed, the same argument as above yields that $Z^\phi$ is a $\mathbb{P}^x$-martingale on $[0,\infty)$. Hence a new measure $\mathbb{P}^{\phi, x}$ is defined via
$ d\mathbb{P}^{\phi,x}/d\mathbb{P}^x |_{\mathcal{F}_T} = Z^{\phi}_T$, $T\geq 0$.
Note that $\mathbb{P}^{\phi,x}$ is consistently defined on $\vee_{T\geq 0} \mathcal{F}_T$. Lastly we recall that $\mathbb{P}^{\phi}$ is \emph{ergodic} if $X$ is recurrent under $\mathbb{P}^\phi$ and there exists an invariant probability measure.
\begin{rem}\label{rem: numeraire}
Set $\phi=\hat{v}$ from Proposition \ref{prop: ergodic wellposed}, if $\mathbb{P}^{\hat{v},x}$ is well defined, then Girsanov's theorem together with \eqref{eq: sde S} and \eqref{eq: Z_phi} yield the following dynamics of $S$ under $\mathbb{P}^{\hat{v},x}$:
\[
\frac{dS^i_t}{S^i_t} = \pare{r(X_t) + \frac{1}{1-p} \pare{\Sigma \nu + \sum_{k,l=1}^d \sigma C a^{kl} \rho D_{(kl)} \hat{v}}(T-t, X_t)} dt + \sum_{j=1}^m \sigma_{ij}(X_t) d\hat{Z}^j_t, \quad i=1, \dots, n,
\]
where $\hat{Z}$ is a $\mathbb{P}^{\hat{v},x}$ Brownian motion. Comparing the previous dynamics with $\hat{\pi}$ in \eqref{eq: opt_strat}, it follows that $\hat{\pi}$ is the optimal strategy for a logarithmic investor under $\mathbb{P}^{\hat{v},x}$. Hence its associated wealth process $\hat{\mathcal{W}}$ has the \emph{num\'{e}raire} property, i.e., $\mathcal{W}/\hat{\mathcal{W}}$ is a $\mathbb{P}^{\hat{v},x}$-supermartingale for any admissible wealth process $\mathcal{W}$.
\end{rem}
For the proof of Proposition \ref{prop: verification}, we prepare following two lemmas, whose proofs are postponed until after the proof of Proposition \ref{prop: verification}.
\begin{lem}\label{lem: barA}
Let Assumptions \ref{ass: loc_ellip}, \ref{ass: sig_ellip} and \ref{ass: rho} hold. Let $A$ and $\bar{A}$ be as in \eqref{eq: functions}. Set
\begin{equation}\label{eq: kappa}
\underline{\kappa} = \left\{\begin{array}{ll}1, & 0<p<1\\ 1-q, & p<0\end{array}\right. \quad \text{ and } \quad \overline{\kappa} = \left\{\begin{array}{ll}1-q, & 0<p<1\\ 1,& p<0\end{array}\right..
\end{equation}
Then, for all $x\in \mathbb{S}_{++}^d$ and $\theta\in\mathbb{S}^d$:
\begin{equation}\label{eq: A_barA_compare}
\underline{\kappa} \sum_{i,j,k,l=1}^d \theta_{ij} A_{(ij),(kl)}(x) \theta_{kl} \leq \sum_{i,j,k,l=1}^d \theta_{ij}\overline{A}_{(ij),(kl)}(x)\theta_{kl} \leq \overline{\kappa} \sum_{i,j,k,l=1}^d \theta_{ij} A_{(ij),(kl)}(x)\theta_{kl}.
\end{equation}
\end{lem}
For $\eta\in C^{(1,2), \gamma}((0,\infty)\times \mathbb{S}_{++}^d, \mathbb R)$, define function $\eta: \mathbb{S}_{++}^d \rightarrow \mathbb{M}^d$ via
\begin{equation}\label{eq: eta_v_map}
\eta_{kl}(t,x; \phi) \, := \, \left(\sum_{i,j=1}^d a^{ij}_{kl}D_{(ij)}\phi\right)(t,x),\qquad k,l = 1,...,d,\, t\geq 0,x\in\mathbb{S}_{++}^d.
\end{equation}
Define $\eta^T_t:= \eta(T-t, X_t; \phi)$, $t\in[0,T]$. When $\phi$ is $v$ from Proposition \ref{prop: v wellposed} (resp. $\hat{v}$ from Proposition \ref{prop: ergodic wellposed}), then $\eta(T-\cdot, X_\cdot; v)$ (resp. $\eta(X_\cdot; \hat{v})$) is expected to be the optimal risk premium for the dual problem of \eqref{eq: power op} (resp. its long run analogue).
The following result is the key to prove Proposition \ref{prop: verification}.
\begin{lem}\label{lem: identity}
Let $\phi \in C^{(1,2), \gamma}((0,\infty) \times \mathbb{S}_{++}^d, \mathbb R)$ satisfy $\phi_t = \mathfrak{F}[\phi]$ on $(0,\infty)\times \mathbb{S}_{++}^d$ where $\mathfrak{F}$ is defined in \eqref{eq: x_ops}. For any $T\geq 0$, let $\pi_t= \pi(T-t, X_t; \phi)$, $\eta_t = \eta(T-t, X_t; \phi)$, for $t\in[0,T]$, and let $\mathcal{W}^\pi$ and $M^\eta$ be the associated wealth process and super-martingale deflator respectively. Then, the following identities hold:
\begin{equation}\label{eq: prop_1_1}
\begin{split}
p\log\left(\mathcal{W}^\pi_T\right) - p\log\left(\mathcal{W}^\pi_t\right) + \phi(0,X_T) - \phi(T-t,X_t) &=
\log\pare{Z^{\phi,T}_T} - \log\pare{Z^{\phi,T}_t},\\
q\log\left(M^\eta_T\right) - q\log\left(M^\eta_t\right) + (1-q)(\phi(0,X_T) -
\phi(T-t,X_t)) &=\log\pare{Z^{\phi,T}_T} - \log\pare{Z^{\phi,T}_t},
\end{split}
\end{equation}
where $Z^{\phi, T}$ is given in \eqref{eq: Z_phi}.
\end{lem}
Using Lemmas \ref{lem: barA} and \ref{lem: identity}, the proof of Proposition \ref{prop: verification} is now given.
\begin{proof}[Proof of Proposition \ref{prop: verification}]
Note that in \eqref{eq: kappa}, $0<\underline{\kappa} <\overline{\kappa}$ holds for both $0<p<1$ and $p<0$. Thus, \cite[Assumption 3.4]{Robertson-Xing} is ensured by Assumption \ref{ass: loc_ellip} and Lemma \ref{lem: barA}. Additionally, \cite[Assumptions 3.5 and 3.6]{Robertson-Xing} are exactly Assumption \ref{ass: coeff_master_list} here. As the assumptions of \cite[Lemma 4.1]{Robertson-Xing} are verified, the well-posedness of the martingale problem for $\mathcal{L}^{v, T-\cdot}$ follows from \cite[Lemma 4.1]{Robertson-Xing}. Since the martingale problem for $L$ is also well-posed, it then follows from the discussion after \eqref{eq: Z_phi} that $Z^{v,T}$ is a $\mathbb{P}^x$-martingale. Applying Lemma \ref{lem: identity} to $v$, it then follows from \eqref{eq: prop_1_1} and $v(0,x)=0$ that
\begin{equation}\label{eq: duality}
\mathbb{E}\bra{\left.\left(\frac{\mathcal{W}^\pi_T}{\mathcal{W}^{\pi}_t}\right)^p\right| \mathcal{F}_t} =e^{v(T-t, X_t)}=\left(\mathbb{E}\bra{\left.\left(\frac{M^{\eta}_T}{M^{\eta}_t}\right)^q\right| \mathcal{F}_t}\right)^{1/(1-q)}, \quad \text{ for all } t\leq T.
\end{equation}
Therefore the optimality of $\pi$ follows from \cite[Lemma 5]{Guasoni-Robertson} and \eqref{eq: power op} is verified in the previous identity.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem: barA}]
From \eqref{eq: x_ops}:
\begin{equation*}\label{eq: A_barA_compare_2}
\sum_{i,j,k,l=1}^d \theta_{ij}\overline{A}_{(ij),(kl)}(x)\theta_{kl} = \sum_{i,j,k,l=1}^d \theta_{ij} \trace{a^{ij}(a^{kl})'}(x)\theta_{kl} - q\sum_{i,j,k,l=1}^d \theta_{ij}\rho'(a^{ij})'C'\mathcal{T}heta C a^{kl}\rho \theta_{kl}.
\end{equation*}
Define the matrix $Y$ via $Y_{kl} \, := \, \sum_{i,j=1}^d a^{ij}_{kl}\theta_{ij}$, for $k,l=1,...,d$. It then follows that
\begin{equation*}\label{eq: bar_A_to_A_compare_1}
\sum_{i,j,k,l=1}^d \theta_{ij}\rho'(a^{ij})'C'\mathcal{T}heta C a^{kl}\rho \theta_{kl} = \rho'Y'C'\mathcal{T}heta C Y \rho.
\end{equation*}
We claim that
\begin{equation}\label{eq: Y_mat_bounds}
0 \leq \rho'Y'C'\mathcal{T}heta C Y\rho \leq \trace{YY'}.
\end{equation}
Admitting this fact, and plugging back in for $Y$ yields
\begin{equation}\label{eq: bar_A_to_A_compare_5}
0 \leq \sum_{i,j,k,l=1}^d \theta_{ij}\rho'(a^{ij})'C'\mathcal{T}heta C a^{kl}\rho \theta_{kl} \leq \sum_{i,j,k,l=1}^d \theta_{ij} \trace{a^{ij}(a^{kl})'}(x)\theta_{kl}.
\end{equation}
If $p<0$ then $q>0$ and \eqref{eq: A_barA_compare} holds for $\underline{\kappa} = 1-q$ and $\overline{\kappa} = 1$. If $0<p<1$ then $q<0$ and hence \eqref{eq: A_barA_compare} holds for $\underline{\kappa} = 1$ and $\overline{\kappa} = 1-q$.
It remains to show \eqref{eq: Y_mat_bounds}. When $\rho(x) = 0_d$, the $d$-dimensional vector with all components $0$, it is clear that $\rho'Y'C'\mathcal{T}heta C Y \rho=0$ and \eqref{eq: Y_mat_bounds} holds. When $\rho(x)\neq 0_d$, it follows from $\mathcal{T}heta\geq 0$ that $\rho'Y'C'\mathcal{T}heta C Y \rho \geq 0$. On the other hand, since by construction $\mathcal{T}heta\leq 1$ (see \eqref{eq: Theta_def}), we have
\begin{equation*}
\rho'Y'C'\mathcal{T}heta C Y \rho \leq \rho'Y'C'C Y\rho \leq \frac{1}{\rho'\rho} \rho'Y'Y\rho = \frac{1}{\rho'\rho}\trace{Y\rho \rho'Y'},
\end{equation*}
where the second inequality holds by Assumption \ref{ass: rho} and the fact
that $C'C$ and $CC'$ have the same eigenvalues. Note that the eigenvalues of
$(1/\rho'\rho)\rho\rho'$ are $1$ and $0$,
and that $\trace{N M N'} \leq \lambda^{+,M}\trace{NN'}$ for any $n\in \mathbb{M}^d$ and $M\in \mathbb{S}^d$, where $\lambda^{+,M}$ is the maximal eigenvalue of $M$. Therefore, $(1/\rho'\rho)\trace{Y\rho\rho'Y} \leq \trace{YY'}$ and \eqref{eq: Y_mat_bounds} is confirmed, finishing the proof.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem: identity}]
The proof is similar that of \cite[Lemma B.3]{Guasoni-Robertson-fundsep}. However, since herein we work with a semi-linear equation and a matrix valued state variable, the notational differences in the calculations are such that, for clarity, we will present a detailed proof.
First of all, set
\begin{equation}\label{eq: A_B_def}
\begin{split}
\textbf{A} &:= p\log\left(\mathcal{W}^\pi_T\right) - p\log\left(\mathcal{W}^\pi_t\right) + \phi(0,X_T)
- \phi(T-t,X_t),\\
\textbf{B} &:= q\log\left(M^\eta_T\right) - q\log\left(M^\eta_t\right) + (1-q)(\phi(0,X_T) -
\phi(T-t,X_t)).
\end{split}
\end{equation}
The identities in \eqref{eq: prop_1_1} are verified in the following four steps.
\begin{enumerate}[1)]
\item Use the dynamics for $\mathcal{W}^\pi$ in \eqref{eq: wealth_dyn}, the definition of $M^{\eta}$ in \eqref{eq: M_eta_def}, and the
definitions of $\pi$, $\eta$ in \eqref{eq: pi_v_map} and \eqref{eq: eta_v_map} to write
\begin{equation}\label{eq: AB iden}
\begin{split}
\textbf{A} = \int_t^T \textbf{A1}_udu + \sum_{k,l=1}^d
\int_t^T\textbf{A2}^{kl}_udB^{kl}_u +
\sum_{k=1}^{m}\int_t^T\textbf{A3}^k_u dW^k_u,\\
\textbf{B} = \int_t^T \textbf{B1}_udu + \sum_{k,l=1}^d
\int_t^T\textbf{B2}^{kl}_udB^{kl}_u + \sum_{k=1}^{m}\int_t^T\textbf{B3}^k_u dW^k_u,\\
\end{split}
\end{equation}
where $\textbf{A1},\textbf{B1}:[0,T]\times\mathbb{S}_{++}^d\rightarrow\mathbb R$, $\textbf{A2},\textbf{B2}:[0,T]\times\mathbb{S}_{++}^d\rightarrow\mathbb{M}^d$, and $\textbf{A3},\textbf{B3}:[0,T]\times\mathbb{S}_{++}^d\rightarrow\mathbb R^{m}$. These functions with time subscripts represent, for example, $\textbf{A1}_u = \textbf{A1}(T-u, X_u)$.
\item Add and subtract
\begin{equation}\label{eq: addsub}
\begin{split}
&\frac{1}{2}\sum_{k,l=1}^d \int_t^T
\left(\textbf{A2}^{kl}_u\right)^2 du + \frac{1}{2}\sum_{k=1}^{m}\int_t^T
\left(\textbf{A3}^k_u\right)^2du,\\
&\frac{1}{2}\sum_{k,l=1}^d \int_t^T
\left(\textbf{B2}^{kl}_u\right)^2 du+ \frac{1}{2}
\sum_{k=1}^{m}\int_t^T\left(\textbf{B3}^k_u\right)^2du,
\end{split}
\end{equation}
to the right-hand-side of \textbf{A} and \textbf{B}, respectively, to obtain
\begin{equation*}
\begin{split}
\textbf{A} &= \int_t^T \left(\textbf{A1}_u+
\frac{1}{2}\sum_{k,l=1}^d\left(\textbf{A2}_u^{kl}\right)^2 +
\frac{1}{2}\sum_{k=1}^{m}\left(\textbf{A3}_u^k\right)^2\right)du
+ \log(\mathcal{Z}_T) - \log(\mathcal{Z}_t),\\
\textbf{B} &= \int_t^T \left(\textbf{B1}_u+
\frac{1}{2}\sum_{k,l=1}^d\left(\textbf{B2}_u^{kl}\right)^2 +
\frac{1}{2}\sum_{k=1}^{m}\left(\textbf{B3}^k_u\right)^2\right)du
+ \log(\tilde{\mathcal{Z}}_T) - \log(\tilde{\mathcal{Z}}_t),\\
\end{split}
\end{equation*}
where
\begin{equation}\label{eq: cZ def}
\begin{split}
\mathcal{Z} = \mathcal{E}\left(\int \sum_{k,l=1}^d
\textbf{A2}_u^{kl}dB^{kl}_u + \int
\sum_{k=1}^{m}\textbf{A3}^{k}_udW^l_u\right), \quad
\tilde{\mathcal{Z}} = \mathcal{E}\left(\int \sum_{k,l=1}^d
\textbf{B2}^{kl}_udB^{kl}_u + \int
\sum_{k=1}^{m}\textbf{B3}^k_udW^k_u\right).
\end{split}
\end{equation}
\item Show that for $u\leq T$ and $x\in\mathbb{S}_{++}^d$:
\begin{equation*}
\begin{split}
\left(\textbf{A1}+
\frac{1}{2}\sum_{k,l=1}^d\left(\textbf{A2}^{kl}\right)^2 +
\frac{1}{2}\sum_{k=1}^{m}\left(\textbf{A3}^k\right)^2\right)(T-u,x) =
\left(-\phi_t+ \mathfrak{F}[\phi]\right)(T-u,x) = 0,\\
\left(\textbf{B1}+
\frac{1}{2}\sum_{k,l=1}^d\left(\textbf{B2}^{kl}\right)^2 +
\frac{1}{2}\sum_{k=1}^{m}\left(\textbf{B3}^k\right)^2\right)(T-u,x) =
\left(-\phi_t+ \mathfrak{F}[\phi]\right)(T-u,x) = 0.\\
\end{split}
\end{equation*}
\item Show that $\mathcal{Z} = \tilde{\mathcal{Z}} = Z^{\phi,T}$.
\end{enumerate}
Combining the above four steps, \eqref{eq: prop_1_1} is then verified.
\begin{rem}
For notational ease the following conventions are used: 1) we will omit $\int_t^T$ and the integrator $du$ from all integrals; 2) we will suppress the argument $(T-u,X_u)$ from
all functions; 3) we will also drop all time subscripts. Thus, for
example, we will write
\begin{equation*}
f + g'dB\rho + h'dW =\int_t^T f(T-u,X_u)du + \int_t^T g(T-u,X_u)'dB_u\rho(X_u) + \int_t^T h(T-u,X_u)'dW_u.
\end{equation*}
\end{rem}
The first identity in \eqref{eq: prop_1_1} is now shown. Using $\rho'\rho CC' + DD' = \idmat{m}$ and the dynamics of $\mathcal{W}^\pi$ in \eqref{eq: wealth_dyn}, It\^{o}'s formula gives \eqref{eq: AB iden} where
\begin{equation}\label{eq: A def}
\begin{split}
\textbf{A1} &= pr + p\pi'\Sigma\nu - \frac{1}{2}p\pi'\Sigma\pi - \phi_t + L\phi,\\
\textbf{A2}^{kl} &= p(C'\sigma'\pi)_k\rho_l + \sum_{i,j=1}^d
a^{ij}_{kl}D_{(ij)}\phi,\\
\textbf{A3}^k &= p(D'\sigma'\pi)_k.
\end{split}
\end{equation}
While the second step follows from definitions of $Z$ and $\tilde{Z}$, we move onto the third step. For $u\leq T$ and $ x\in\mathbb{S}_{++}^d$, it follows that
\begin{equation}\label{eq: prop_1_2}
\begin{split}
\textbf{A1} +& \frac{1}{2}\sum_{k,l=1}^d (\textbf{A2}^{kl})^2 +
\sum_{k=1}^{m}(\textbf{A3}^k)^2\\
=& pr + p\pi'\Sigma\nu - \frac{1}{2}p\pi'\Sigma\pi - \phi_t +L\phi + \frac{1}{2}p^2\pi'\sigma CC'\sigma'\pi \rho'\rho + p\pi'\left(\sum_{i,j1}^d \sigma C a^{ij}\rho D_{(ij)}\phi \right)\\
& + \frac{1}{2}\sum_{i,j,k,l=1}^d D_{(ij)}\phi \trace{a^{ij}(a^{kl})'} D_{(kl)}\phi + \frac{1}{2}p^2\pi'\sigma DD'\sigma'\pi,\\
=&\frac{1}{2}p(p-1)\pi'\Sigma\pi
+ p\pi'\Sigma\nu + p\pi'\left(\sum_{ij=1}^d \sigma C a^{ij}\rho D_{(ij)}\phi \right)\\
& +pr - \phi_t + L\phi + \frac{1}{2}\sum_{i,j,k,l=1}^d D_{(ij)}\phi\
\trace{a^{ij}(a^{kl})'} D_{(kl)}\phi.
\end{split}
\end{equation}
The terms above containing $\pi$ are
\begin{equation*}
\frac{1}{2}p(p-1)\pi'\Sigma\pi + p\pi'\left(\Sigma\nu + \sum_{i,j=1}^d
\sigma C a^{ij}\rho D_{(ij)}\phi\right).
\end{equation*}
Using \eqref{eq: pi_v_map}, we obtain the following expression for the quadratic function in the previous line:
\[
-\frac{1}{2}q\nu'\Sigma\nu - q\sum_{i,j=1}^d\ \nu'\sigma C
a^{ij}\rho D_{(ij)}\phi - \frac{1}{2}q \sum_{i,j,k,l=1}^d D_{(ij)}\phi\ \rho'(a^{ij})'C'\mathcal{T}heta
C a^{kl}\rho D_{(kl)}\phi,
\]
for both cases $m\geq n$ or $m<n$. Thus, substituting the previous expression into \eqref{eq: prop_1_2}, using the expressions for $\bar{A}, V$ in \eqref{eq: functions} and $\mathfrak{F}$ in \eqref{eq: x_ops_L} gives
\begin{equation}\label{eq: prop_1_3}
\begin{split}
\textbf{A1} +& \frac{1}{2}\sum_{k,l=1}^d (\textbf{A2}^{kl})^2 +
\sum_{k=1}^{m}(\textbf{A3}^k)^2\\
=& pr-\frac{1}{2}q\nu'\Sigma\nu - q\sum_{i,j=1}^d \nu'\sigma C a^{ij}\rho D_{(ij)}\phi - \frac{1}{2}\sum_{i,j,k,l=1}^d D_{(ij)}\phi \rho'(a^{ij})'C'\mathcal{T}heta C a^{kl}\rho D_{(kl)}\phi \\
&- \phi_t + L\phi + \frac{1}{2}\sum_{i,j,k,l=1}^d D_{(ij)}\phi \trace{a^{ij}(a^{kl})'} D_{(kl)}\phi\\
=& -\phi_t + L\phi -
q\sum_{i,j=1}^d\nu'\sigma C a^{ij}\rho D_{(ij)}\phi + \frac{1}{2}\sum_{i,j,k,l=1}^d D_{(ij)}\phi \bar{A}_{(ij),(kl)} D_{(kl)}\phi + V\\
=& -\phi_t + \mathfrak{F}[\phi] \\
=&0,
\end{split}
\end{equation}
finishing the third step. For the last step, recall the definition of $Z^{\phi,T}$ from \eqref{eq: Z_phi}.
Comparing with the definition of $\mathcal{Z}$ in \eqref{eq: cZ def}, it suffices to show that
\begin{equation}\label{eq: A2_A3}
\begin{split}
\textbf{A2}^{kl} &= -q(C'\sigma'\nu)_k\rho_l + \sum_{i,j=1}^d \left(
a^{ij}_{kl} - q(C'\mathcal{T}heta C a^{ij}\rho)_k\rho_l\right)D_{(ij)}\phi, \\
\textbf{A3}^k & = -q(D'\sigma'\nu)_k - q\sum_{i,j=1}^d
\left(D'\mathcal{T}heta C a^{ij}\rho\right)_kD_{(ij)}\phi.
\end{split}
\end{equation}
Using \eqref{eq: pi_v_map} for $m\geq n$ it follows that (recall $\mathcal{T}heta = \sigma'\Sigma^{-1}\sigma$ when $m\geq n$)
\begin{equation*}
\begin{split}
p(\sigma'\pi)&= -q\sigma'\Sigma^{-1}\left(\Sigma\nu + \sum_{i,j=1}^d \sigma C a^{ij}\rho D_{(ij)}\phi\right) =-q\sigma'\nu - q\sum_{i,j=1}^d \mathcal{T}heta C a^{ij}\rho D_{(ij)}\phi.
\end{split}
\end{equation*}
Similarly, using \eqref{eq: pi_v_map} for $m< n$ gives (recall $\mathcal{T}heta = 1_m$ for $m<n$):
\begin{equation*}
\begin{split}
p(\sigma'\pi)&= -q\sigma'\sigma(\sigma'\sigma)^{-1}\left(\sigma'\nu + \sum_{i,j=1}^d C a^{ij}\rho D_{(ij)}\phi\right)=-q\sigma'\nu - q\sum_{i,j=1}^d \mathcal{T}heta C a^{ij}\rho D_{(ij)}\phi.
\end{split}
\end{equation*}
Therefore, in both cases $m\geq n$, $m< n$ we have, using the definition of $\textbf{A2},\textbf{A3}$ in \eqref{eq: A def} that
\begin{equation*}
\begin{split}
\textbf{A2}^{kl} &= p(C'\sigma'\pi)_k \rho_l + \sum_{i,j=1}^d a^{ij}_{kl} D_{(ij)}\phi =-q(C'\sigma'\nu)_k\rho_l + \sum_{i,j=1}^d \left(a^{ij}_{kl} - q(C'\mathcal{T}heta C a^{ij}\rho)_k\rho_l\right) D_{(ij)}\phi,\\
\textbf{A3}^k &= p(D'\sigma'\pi)_k = -q(D'\sigma'\nu)_k - q\sum_{i,j=1}^d (D'\mathcal{T}heta C a^{ij}\rho)_k D_{(ij)}\phi,
\end{split}
\end{equation*}
which verifies \eqref{eq: A2_A3}.
The proof for the second identity in \eqref{eq: prop_1_1} is similar. First, using the definition of $M^\eta$ in \eqref{eq: M_eta_def}, It\^{o}'s formula yields the second identity in \eqref{eq: AB iden}, where
\begin{equation}\label{eq: B def}
\begin{split}
\textbf{B1} = & -qr + (1-q)(-\phi_t + L\phi)\\ &-\frac{1}{2}q\left(\sum_{k,l=1}^{d}\left(-(C'\sigma'\nu)_k\rho_l + \eta_{kl}
- (C'\mathcal{T}heta C\eta\rho)_k\rho_l\right)^2 +
\sum_{k=1}^{m}\left((D'\sigma'\nu)_k + (D'\mathcal{T}heta
C\eta\rho)_k\right)^2\right),\\
\textbf{B2}^{kl} &= q\left(-(C'\sigma'\nu)_k\rho_l + \eta_{kl} - (C'\mathcal{T}heta
C\eta\rho)_k\rho_l\right) + (1-q)\sum_{i,j=1}^d a^{ij}_{kl}D_{(ij)}\phi,\\
\textbf{B3}^{k} &= -q\left((D'\sigma'\nu)_k + (D'\mathcal{T}heta C\eta\rho)_k\right).
\end{split}
\end{equation}
Using $(1-q)p=-q$ we obtain
\begin{equation}\label{eq: prop_1_45}
\begin{split}
&\textbf{B1} + \frac{1}{2}\sum_{k,l=1}^d (\textbf{B2}^{kl})^2 + \frac{1}{2}\sum_{k=1}^{m} (\textbf{B3}^k)^2\\
= & (1-q)pr + (1-q)(-\phi_t + L\phi)\\
& -\frac{1}{2}q(1-q)\left(\sum_{k,l=1}^{d}\left(-(C'\sigma'\nu)_k\rho_l + \eta_{kl}
- (C'\mathcal{T}heta C\eta\rho)_k\rho_l\right)^2 +
\sum_{k=1}^{m}\left((D'\sigma'\nu)_k + (D'\mathcal{T}heta
C\eta\rho)_k\right)^2\right)\\
&+ q(1-q)\sum_{i,j,k,l=1}^d \left(-(C'\sigma'\nu)_k\rho_l + \eta_{kl}
- (C'\mathcal{T}heta C\eta\rho)_k\rho_l\right) a^{ij}_{kl}D_{(ij)}\phi\\
&+ \frac{1}{2}(1-q)^2\sum_{k,l=1}^d\left(\sum_{i,j=1}^da^{ij}_{kl}D_{(ij)}\phi\right)^2.
\end{split}
\end{equation}
Now, using $\rho'\rho CC' + DD' = \idmat{m}$ gives
\begin{equation*}
\begin{split}
&\sum_{k,l=1}^{d}\left(-(C'\sigma'\nu)_k\rho_l + \eta_{kl}
- (C'\mathcal{T}heta C\eta\rho)_k\rho_l\right)^2 +
\sum_{k=1}^{m}\left((D'\sigma'\nu)_k + (D'\mathcal{T}heta
C\eta\rho)_k\right)^2\\
&=\nu'\sigma CC'\sigma' \nu \rho'\rho + \trace{\eta'\eta} + \rho'\eta'C'\mathcal{T}heta CC'\mathcal{T}heta C\eta\rho\rho'\rho - 2\nu'\sigma C\eta\rho + 2\nu'\sigma CC'\mathcal{T}heta C\eta\rho \rho'\rho -2\rho'\eta'C'\mathcal{T}heta C\eta \rho\\
&\qquad + \nu'\sigma DD'\sigma'\nu + \rho'\eta'C'\mathcal{T}heta DD'\mathcal{T}heta C\eta\rho + 2\nu'\sigma DD'\mathcal{T}heta C\eta\rho\\
&=\nu'\sigma(CC'\rho'\rho + DD')\sigma'\nu + \rho'\eta'C'\mathcal{T}heta(CC'\rho'\rho + DD')\mathcal{T}heta C\eta\rho + 2\nu'\sigma(CC'\rho'\rho + DD')\mathcal{T}heta C\eta\rho\\
&\qquad +\trace{\eta'\eta} - 2\nu'\sigma C\eta\rho - 2\rho'\eta'C'\mathcal{T}heta C\eta\rho\\
&=\nu'\Sigma\nu + \rho'\eta'C'\mathcal{T}heta\mathcal{T}heta C\eta\rho + 2\nu'\sigma\mathcal{T}heta C\eta\rho +\trace{\eta'\eta} - 2\nu'\sigma C\eta\rho - 2\rho'\eta'C'\mathcal{T}heta C\eta\rho\\
&=\nu'\Sigma\nu + \trace{\eta'\eta} - \rho'\eta'C'\mathcal{T}heta C\eta\rho,
\end{split}
\end{equation*}
where the last equality follows since the definition of $\mathcal{T}heta$ in \eqref{eq: Theta_def} implies both $\mathcal{T}heta\mathcal{T}heta = \mathcal{T}heta$ and $\sigma\mathcal{T}heta = \sigma$. We also have
\begin{equation*}
\begin{split}
&\sum_{i,j,k,l=1}^d \left(-(C'\sigma'\nu)_k\rho_l + \eta_{kl}
- (C'\mathcal{T}heta C\eta\rho)_k\rho_l\right) a^{ij}_{kl}D_{(ij)}\phi \\
&\hspace{2cm} = \sum_{i,j=1}^d \left(-\nu'\sigma C a^{ij}\rho + \trace{\eta' a^{ij}} - \rho'\eta'C'\mathcal{T}heta C a^{ij}\rho\right)D_{(ij)}\phi,\\
&\sum_{k,l=1}^d\left(\sum_{i,j=1}^da^{ij}_{kl}D_{(ij)}\phi\right)^2 = \sum_{i,j,k,l=1}^d D_{(ij)}\phi \trace{a^{ij}(a^{kl})'} D_{(kl)}\phi.
\end{split}
\end{equation*}
Plugging all of this into \eqref{eq: prop_1_45} yields
\begin{equation}\label{eq: prop_1_55}
\begin{split}
&\frac{1}{1-q}\pare{\textbf{B1} + \frac{1}{2}\sum_{k,l=1}^d (\textbf{B2}^{kl})^2 + \frac{1}{2}\sum_{k=1}^{m} (\textbf{B3}^k)^2}\\
=& pr - \phi_t + L\phi - \frac{1}{2}q\left(\nu'\Sigma\nu + \trace{\eta'\eta} - \rho'\eta'C'\mathcal{T}heta C\eta\rho\right) \\
&+ q\sum_{i,j=1}^d\left(-\nu'\sigma C a^{ij}\rho + \trace{\eta'a^{ij}} - \rho'\eta'C\mathcal{T}heta C a^{ij}\rho\right)D_{(ij)}\phi\\
&+ \frac{1}{2}(1-q)\sum_{i,j,k,l=1}^d D_{(ij)}\phi\trace{a^{ij}(a^{kl})'}D_{(kl)}\phi.
\end{split}
\end{equation}
On the right-hand-side, terms involving $\eta$ are
\begin{equation}\label{eq: prop_1_8}
\begin{split}
-\frac{1}{2}q\trace{\eta'\eta}& + \frac{1}{2}q\rho'\eta'C'\mathcal{T}heta C\eta\rho +
q\sum_{i,j=1}^d \trace{\eta'a^{ij}}D_{(ij)}\phi - q\sum_{i,j=1}^d
\rho'\eta' C'\mathcal{T}heta C a^{ij}\rho D_{(ij)}\phi.
\end{split}
\end{equation}
For $\eta$ in \eqref{eq: eta_v_map}, the following identities hold
\begin{equation*}
\begin{split}
&\trace{\eta'\eta} = \sum_{i,j,k,l=1}^d D_{(ij)}\phi\ \trace{a^{ij}(a^{kl})'} D_{(kl)}\phi,\\
&\rho'\eta'C'\mathcal{T}heta C\eta\rho = \sum_{i,j,k,l=1}^d D_{(ij)}\phi\ \rho'(a^{ij})'C'\mathcal{T}heta C a^{kl}\rho D_{(kl)} \phi,\\
&\sum_{i,j=1}^d \trace{\eta'a^{ij}}D_{(ij)}\phi = \sum_{i,j,k,l=1}^d D_{(ij)}\phi\ \trace{a^{ij}(a^{kl})'}
D_{(kl)}\phi,\\
&\sum_{i,j=1}^d \rho'\eta' C'\mathcal{T}heta C a^{ij}\rho D_{(ij)}\phi = \sum_{i,j,k,l=1}^d D_{(ij)}\phi\ \rho'(a^{ij})'C'\mathcal{T}heta C a^{kl}\rho D_{(kl)} \phi.
\end{split}
\end{equation*}
Using above identities in \eqref{eq: prop_1_8}, we obtain the following expression for \eqref{eq: prop_1_8}:
\[
\frac{1}{2}q\sum_{i,j,k,l=1}^dD_{(ij)}\phi\ \left(\trace{a^{ij}(a^{kl})'} -
\rho'(a^{ij})'C'\mathcal{T}heta C a^{kl}\rho\right)D_{(kl)}\phi.
\]
Inserting this into \eqref{eq: prop_1_55} gives
\begin{equation*}\label{eq: prop_1_95}
\begin{split}
&\frac{1}{1-q}\pare{\textbf{B1} + \frac{1}{2}\sum_{i,j=1}^d (\textbf{B2}^{ij})^2 + \frac{1}{2}\sum_{l=1}^{m} (\textbf{B3}^l)^2}\\
=& pr - \phi_t + L\phi - \frac{1}{2}q\nu'\Sigma\nu - q\sum_{i,j=1}^d \nu'\sigma C a^{ij}\rho D_{(ij)}\phi + \frac{1}{2}(1-q)\sum_{i,j,k,l=1}^d D_{(ij)}\phi \trace{a^{ij}(a^{kl})'} D_{(kl)}\phi\\
&+ \frac{1}{2}q\sum_{i,j,k,l=1}^d D_{(ij)}\phi\left(\trace{a^{ij}(a^{kl})'} - \rho'(a^{ij})'C'\mathcal{T}heta C a^{kl}\rho\right)D_{(kl)}\phi\\
=&-\phi_t + L\phi - q\sum_{i,j=1}^d \nu'\sigma C a^{ij}\rho D_{(ij)}\phi + \frac{1}{2}\sum_{i,j,k,l=1}^d D_{(ij)}\phi\left(\trace{a^{ij}(a^{kl})'} - q\rho'(a^{ij})'C'\mathcal{T}heta C a^{kl}\rho\right)D_{(kl)}\phi\\
& + pr - \frac{1}{2}q\nu'\Sigma\nu\\
=& -\phi_t + \mathfrak{F}[\phi]\\
=&0,
\end{split}
\end{equation*}
where the second to last equality uses \eqref{eq: functions} and \eqref{eq: x_ops_L}. Thus, the third step is complete.
Turning to the last step, comparing $Z^{\phi,T}$ in \eqref{eq: Z_phi} with $\tilde{\mathcal{Z}}$ in \eqref{eq: cZ def}, it suffices to show
\begin{equation*}
\begin{split}
\textbf{B2}^{kl} &= -q(C'\sigma'\nu)_k\rho_l + \sum_{i,j=1}^d\left(
a^{ij}_{kl} - q(C'\mathcal{T}heta C a^{ij}\rho)_k\rho_l\right)D_{(ij)}\phi,\\
\textbf{B3}^k & = -q(D'\sigma'\nu)_k - q\sum_{i,j=1}^d\left(D'\mathcal{T}heta C a^{ij}\rho\right)_k D_{(ij)}\phi.
\end{split}
\end{equation*}
Using the definitions of $\textbf{B2}$ and $\textbf{B3}$ in \eqref{eq: B def} it suffices to show that
\begin{equation*}
\begin{split}
q\eta_{kl} - q(C'\mathcal{T}heta C\eta\rho)_k\rho_l + (1-q)\sum_{i,j=1}^d a^{ij}_{kl} D_{(ij)}\phi &=\sum_{i,j=1}^d \left(a^{ij}_{kl} - q(C'\mathcal{T}heta C a^{ij}\rho)_k\rho_l\right)D_{(ij)}\phi,\\
(D'\mathcal{T}heta C \eta\rho)_k &=\sum_{i,j=1}^d (D'\mathcal{T}heta C a^{ij}\rho)_k D_{(ij)}\phi.
\end{split}
\end{equation*}
Since $\eta_{kl} = \sum_{i,j=1}^d a^{ij}_{kl}D_{(ij)}\phi$ from \eqref{eq: eta_v_map} the last two identities readily follow, finishing the proof.
\end{proof}
\section{Proofs for Subsection \ref{subsubsec: wishart_examples}}\label{app: C}
Throughout this section, the model is from Section \ref{subsec: wishart} with $\rho,\nu$ and $\zeta$ constant. Furthermore, $\zeta$ is assumed to satisfy Assumption \ref{ass: sig_ellip_gW}. We begin with the following lemma, which identifies $\mathfrak{F}[v]$ for $v$ as in \eqref{eq: cand_hatv}.
\begin{lem}\label{lem: wishart_op_affine_v} For $v = \trace{Mx}$ as in \eqref{eq: cand_hatv} it follows for $d\leq n$ that
\begin{equation}\label{eq: wishart_fv_affine_d_leq_n}
\begin{split}
&\mathfrak{F}[v](x) = \trace{x\left(2M\Lambda(1-q\rho\rho')\Lambda'M + K'M + MK - q\zeta'\nu\rho'\Lambda'M - qM\Lambda\rho\nu'\zeta + \frac{1}{2}\left(p(r_1+r_1') - q\zeta'\nu\nu'\zeta\right)\right)}\\
&\qquad\qquad + \trace{LL'M} + p r_0.
\end{split}
\end{equation}
For $d > n$
\begin{equation}\label{eq: wishart_fv_affine_d_ge_n}
\begin{split}
&\mathfrak{F}[v](x) = \trace{x\left(2M\Lambda\Lambda'M + K'M + MK - q\zeta'\nu\rho'\Lambda'M - qM\Lambda\rho\nu'\zeta + \frac{1}{2}\left(p(r_1+r_1') - q\zeta'\nu\nu'\zeta\right)\right)}\\
&\qquad - 2q\trace{x\zeta'\left(\zeta x\zeta'\right)^{-1}\zeta x M\Lambda\rho\rho'\Lambda'M} + \trace{LL'M} + p r_0.
\end{split}
\end{equation}
\end{lem}
\begin{proof}
Plugging in the model coefficients gives
\begin{equation*}
\begin{split}
b(x) &= LL' + Kx + xK',\qquad a^{ij}_{kl}(x) = \sqrt{x}_{ik}\Lambda_{jl} + \sqrt{x}_{jk}\Lambda_{il},\\
r(x) &=r_0 + \trace{r_1x},\qquad \sigma(x) = \zeta\sqrt{x},\qquad \nu(x) = \nu,\\
C(x) &=\idmat{d},\qquad \rho(x) = \rho.
\end{split}
\end{equation*}
Therefore, using the definitions in \eqref{eq: functions}, calculation shows that
\begin{equation}\label{eq: wishart_b_A_V}
\begin{split}
\bar{b}_{ij}(x) =& (LL'+Kx+xK')_{ij}-q(x\zeta'\nu\rho'\Lambda')_{ij} - q(x\zeta'\nu\rho'\Lambda')_{ji},\\
A_{(ij),(kl)}(x) =& x_{ik}(\Lambda\Lambda')_{jl} + x_{il}(\Lambda\Lambda')_{jk} +x_{jk}(\Lambda\Lambda')_{il} + x_{jl}(\Lambda\Lambda')_{ik},\\
V(x) =& pr_0 + \frac{1}{2}p\trace{x(r_1+r_1')} - \frac{1}{2}q\trace{x\zeta'\nu\nu'\zeta},
\end{split}
\end{equation}
and
\begin{equation}\label{eq: wishart_bar_A}
\begin{split}
\bar{A}_{(ij),(kl)}(x) =& x_{ik}(\Lambda\Lambda')_{jl} - q(\sqrt{x}\mathcal{T}heta(x)\sqrt{x})_{ik}(\Lambda\rho\rho'\Lambda')_{jl} + x_{il}(\Lambda\Lambda')_{jk} - q(\sqrt{x}\mathcal{T}heta(x)\sqrt{x})_{il}(\Lambda\rho\rho'\Lambda')_{jk}\\
&+ x_{jk}(\Lambda\Lambda')_{il} - q(\sqrt{x}\mathcal{T}heta(x)\sqrt{x})_{jk}(\Lambda\rho\rho'\Lambda')_{il} + x_{jl}(\Lambda\Lambda')_{ik} - q(\sqrt{x}\mathcal{T}heta(x)\sqrt{x})_{jl}(\Lambda\rho\rho'\Lambda')_{ik}.
\end{split}
\end{equation}
For the given $v$, $D_{(ij)}v = D_{(ji)}v = M_{ij}$ and $D_{(ij),(kl)}v = 0$. Therefore
\begin{equation}\label{eq: wishart_temp_op_calc}
\begin{split}
&\sum_{i,j,k,l=1}^d A_{(ij),(kl)}D_{(ij),(kl)}v = 0,\\
&\sum_{i,j=1}^d \bar{b}_{ij}D_{(ij)}v = \trace{x\left(K'M + MK - q\zeta'\nu\rho'\Lambda'M - qM\Lambda\rho\nu'\zeta\right)} + \trace{LL'M},
\end{split}
\end{equation}
where we have used repeatedly that $M,X$ are symmetric and that $\trace{ABC} = \trace{BCA} = \trace{CAB}$ for matrices $A,B,C$. When $d\leq n$, it follows that $\mathcal{T}heta(x) =\idmat{d}$ and $\bar{A}$ from \eqref{eq: wishart_bar_A} simplifies to
\begin{equation*}
\begin{split}
\bar{A}_{(ij),(kl)}(x) =& x_{ik}\left(\Lambda\Lambda'-q\Lambda\rho\rho'\Lambda'\right)_{jl}+ x_{il}\left(\Lambda\Lambda'-q\Lambda\rho\rho'\Lambda'\right)_{jk}\\
& + x_{jk}\left(\Lambda\Lambda'-q\Lambda\rho\rho'\Lambda'\right)_{il}+ x_{jl}\left(\Lambda\Lambda'-q\Lambda\rho\rho'\Lambda'\right)_{ik},\\
\end{split}
\end{equation*}
and hence using the symmetry for $\Lambda\Lambda'-q\Lambda\rho\rho'\Lambda'$:
\begin{equation}\label{eq: wishart_temp_op_calc_2}
\frac{1}{2}\sum_{i,j,k,l=1}^d \bar{A}_{(ij),(kl)}D_{(ij)}v D_{(kl)}v = 2\trace{x\left(M\Lambda(1-q\rho\rho'\Lambda'M)\right)}.
\end{equation}
Therefore, \eqref{eq: wishart_fv_affine_d_leq_n} follows using \eqref{eq: wishart_b_A_V}, \eqref{eq: wishart_temp_op_calc}, \eqref{eq: wishart_temp_op_calc_2} and the definition of $\mathfrak{F}$ in \eqref{eq: x_ops}. When $d>n$:
\begin{equation*}
\sqrt{x}\mathcal{T}heta(x)\sqrt{x} =\sqrt{x}\left(\sigma'\Sigma^{-1}\sigma\right)(x)\sqrt{x} = x\zeta'\left(\zeta x \zeta'\right)^{-1}\zeta x,
\end{equation*}
thus, using \eqref{eq: wishart_bar_A} it follows that
\begin{equation}\label{eq: wishart_temp_op_calc_3}
\begin{split}
\frac{1}{2}\sum_{i,j,k,l=1}^d \bar{A}_{(ij),(kl)}D_{(ij)}v D_{(kl)}v & = 2\trace{xM\Lambda\Lambda'M} - 2q\trace{x\zeta'\left(\zeta x\zeta'\right)^{-1}\zeta x M \Lambda\rho\rho'\Lambda'M}.
\end{split}
\end{equation}
\eqref{eq: wishart_fv_affine_d_ge_n} now follows from \eqref{eq: wishart_b_A_V}, \eqref{eq: wishart_temp_op_calc} and \eqref{eq: wishart_temp_op_calc_3}.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop: wishart_good_case}]
Using Lemma \ref{lem: wishart_op_affine_v} it follows for $d\leq n$ that if $M$ solves \eqref{eq: d_leq_n_Ricatti} then $\mathfrak{F}[v] = \lambda$ with $\lambda = \trace{LL'M} + pr_0$. Now, with $D=-M$, \eqref{eq: d_leq_n_Ricatti} takes the form
\begin{equation*}
D\left(2\Lambda(1-q\rho\rho')\Lambda'\right)D - D(K-q\Lambda\rho\nu'\zeta) - (K-q\Lambda\rho\nu'\zeta)'D - \frac{1}{2}\left(-p(r_1+r_1') + q\zeta'\nu\nu'\zeta\right) = 0.
\end{equation*}
Since the eigenvalues of $\rho\rho'$ are $\rho'\rho$ and $0$, then
\begin{equation*}
2\Lambda(1-q\rho\rho')\Lambda' \geq 2(1-q\rho'\rho)\Lambda\Lambda' > 0.
\end{equation*}
Furthermore, by assumption $-p(r_1+r_1') + q\zeta'\nu\nu'\zeta > 0$. Thus, the Riccati equation takes the form
\begin{equation}\label{eq: matrix riccati}
D\textbf{B}\textbf{B}'D - D\textbf{A} - \textbf{A}'D - \textbf{C}\textbf{C}' = 0,
\end{equation}
where $\textbf{B} = \sqrt{2\Lambda(1-q\rho\rho')\Lambda'}$, $\bold{A} = K-q\Lambda\rho\nu'\zeta$ and $\textbf{C} = (1/\sqrt{2})\sqrt{-p(r_1+r_1') + q\zeta'\nu\nu'\zeta}$. By \cite[Lemma 2.4.1]{MR1997753}, if there exists matrices $F_1$ and $F_2$ such that $\textbf{A}-\textbf{B}F_1 < 0$ \footnote{Here and in what follows, we write $M<0$ for a given matrix $M\in\mathbb{M}^d$ with $M+M'<0$.} and $\textbf{A}'-\textbf{C}F_2 < 0$ then there is a unique solution $\hat{M} = -\hat{D}$ to the above such that
\begin{equation}\label{eq: A_B_rel}
\begin{split}
\textbf{A} - \textbf{B}\textbf{B}'\hat{D} &= \textbf{A} + \textbf{B}\textbf{B}'\hat{M}= (K-q\Lambda\rho\nu'\zeta) + 2\Lambda(1-q\rho\rho')\Lambda' \hat{M} < 0.
\end{split}
\end{equation}
Note that $F_1 = \textbf{B}^{-1}\left(\idmat{d} - \textbf{A}\right)$ and $F_2 = \textbf{C}^{-1}\left(\idmat{d} - \textbf{A}'\right)$ are two such matrices. Hence \eqref{eq: matrix riccati} admits a unique solution $\hat{M}$ such that \eqref{eq: A_B_rel} holds.
For $\phi = \hat{v} = \textrm{Tr}(\hat{M}x)$, consider the generator $\mathcal{L}^{\hat{v}}$ from \eqref{eq: cL_phi_time}, which takes the form
\begin{equation*}
\mathcal{L}^{\hat{v}} = \frac{1}{2}\sum_{i,j,k,l=1}^d A_{(ij),(kl)}D_{(ij),(kl)} + \sum_{i,j=1}^d\left(\bar{b}_{ij} + \sum_{k,l=1}^d \bar{A}_{(ij),(kl)}\hat{M}_{kl}\right)D_{(ij)}.
\end{equation*}
The drift (i.e. the first order term) above takes the form
\begin{equation*}
\begin{split}
\bar{b}^{ij} &+ \sum_{k,l=1}^d \bar{A}_{(ij),(kl)}\hat{M}_{kl}\\
&= \left(LL' + \left(K-q\Lambda\rho\nu'\zeta + 2\Lambda(1-q\rho\rho')\Lambda'\hat{M}\right)x + x\left(K-q\Lambda\rho\nu'\zeta + 2\Lambda(1-q\rho\rho')\Lambda'\hat{M}\right)'\right)_{ij}\\
&=\left(LL' + (\bold{A}+\bold{B}\bold{B}'\hat{M})x + x(\bold{A}+\bold{B}\bold{B}\hat{M})'\right)_{ij}.
\end{split}
\end{equation*}
Thus, we see that the process $X$ with generator given by $\mathcal{L}^{\hat{v}}$ is a Wishart process of the form in \eqref{eq: wishart}. Moreover, \eqref{eq: A_B_rel} implies that $K:=\bold{A} + \bold{B}\bold{B}'\hat{M}<0$, hence $X$ is ergodic. Indeed, $LL' > (d+1)\Lambda\Lambda' > 0$ ensures $X$ does not explode to the boundary of $\mathbb{S}_{++}^d$. Furthermore, consider
\begin{equation*}
u(x) = -\underline{c}\log\left(\det{x}\right) + \overline{c} \norm{x}\eta(\norm{x}),
\end{equation*}
where $\underline{c}, \overline{c}$ are two constants to be determined later, and $\eta(y)$ is a smooth function satisfying $0\leq \eta(y)\leq 1$, $\eta(y) = 1$ for $y>1$ and $0$ for $y<1/2$. Observe that $\lim_{\norm{x}\rightarrow \infty}u(x) =\infty$ and $\lim_{\det(x)\rightarrow 0} u(x) =\infty$, where both limits are uniform as $x$ approaches the boundaries. On the other hand, a calculation similar to that in \cite[Lemmas 5.2 and 5.3]{Robertson-Xing} (with $\bar{\kappa}$ therein equal to $0$) shows the existence of $\underline{c}, \overline{c}, \varepsilonilon>0$ and a sufficiently large sub-domain $E\subset \mathbb{S}_{++}^d$ such that $\mathcal{L}^{\hat{v}}u(x)\leq -\varepsilonilon$ for all $x\in \mathbb{S}_{++}^d \setminus E$. Therefore \cite[Theorem 6.1.3]{Pinsky} shows that $\mathbb{P}^{\hat{v}}$ is ergodic. Hence $\hat{v}$ is equal to $\textrm{Tr}(\hat{M}x)$ and $\hat{\lambda} = \textrm{Tr}(LL'\hat{M}) + pr_0$. This fact follows from \cite[Proposition 2.3]{Robertson-Xing} and \cite[Theorems 2.1,2.2]{Ichihara} which shows the equivalency between $\mathcal{L}^{\hat{v}}$ being ergodic and $\hat{\lambda}$ being the smallest $\lambda$ with accompanying solution $v$ to $\mathfrak{F}[v]=\lambda$.
\end{proof}
\begin{lem}\label{lem: example_calc}
In the setting of Example \ref{exa: counter-exa}, for $v$ as in \eqref{eq: cand_hatv}, $\mathfrak{F}[v]$ takes the form in \eqref{eq: F_example}.
\end{lem}
\begin{proof}
$\mathfrak{F}[v]$ is given in \eqref{eq: wishart_fv_affine_d_ge_n} of Lemma \ref{lem: wishart_op_affine_v}. Specifying to the example coefficients and using the representation for $X$,$M$ from \eqref{eq: wishart_x_v_ex}:
\begin{equation*}
\begin{split}
&2M\Lambda\Lambda'M + K'M + MK -q\zeta'\nu\rho'\Lambda'M - qM\Lambda\rho\nu'\zeta + \frac{1}{2}\left(p(r_1+r_1') - q\zeta'\nu\nu'\zeta\right)\\
&\qquad = 2M^2 + 2M - q\rho\nu\left(\begin{array}{c c} 1 & 1\\ 0 & 0\end{array}\right)M - q\rho\nu M\left(\begin{array}{c c} 1 & 0\\ 1& 0\end{array}\right) + pr_1\left(\begin{array}{c c} 1&0\\0&1\end{array}\right) - \frac{1}{2}q\nu^2\left(\begin{array}{c c} 1 & 0 \\ 0 & 0\end{array}\right),\\
&\qquad = 2\left(\begin{array}{c c} M_1^2+M_2^2 & M_2(M_1+M_3)\\ M_2(M_1+M_3) & M_2^2 + M_3^2\end{array}\right) + 2\left(\begin{array}{c c} M_1 & M_2 \\ M_2 & M_3\end{array}\right) - q\rho\nu\left(\begin{array}{c c} M_1+M_2 & M_2 + M_3\\ 0 & 0 \end{array}\right),\\
&\qquad\qquad - q\rho\nu\left(\begin{array}{c c} M_1+M_2 & 0 \\ M_2 + M_3 & 0\end{array}\right) + pr_1\left(\begin{array}{c c} 1 & 0 \\ 0 & 1\end{array}\right) - \frac{1}{2}q\nu^2\left(\begin{array}{c c} 1 & 0 \\ 0 & 0\end{array}\right),\\
&\qquad = \left(\begin{array}{c c} 2(M_1^2+M_2^2) + 2M_1 - 2q\rho\nu(M_1+M_2) + pr_1 - \tfrac{1}{2}q\nu^2 & 2M_2(M_1+M_3) + 2M_2 - q\rho\nu(M_2+M_3)\\ 2M_2(M_1+M_3) + 2M_2 -q\rho\nu(M_2+M_3) & 2(M_2^2+M_3^2)+2M_3 + pr_1\end{array}\right).
\end{split}
\end{equation*}
Thus,
\begin{equation}\label{eq: wishart_counterex_good}
\begin{split}
&\trace{X\left(2M\Lambda'\Lambda'M + K'M + MK -q\zeta'\nu\rho'\Lambda'M - qM\Lambda\rho\nu'\zeta + \frac{1}{2}\left(p(r_1+r_1') - q\zeta'\nu\nu'\zeta\right)\right)}\\
&\qquad = x\left(2(M_1^2+M_2^2) + 2M_1 - 2q\rho\nu(M_1+M_2) + pr_1 - (1/2)q\nu^2\right)\\
&\qquad\qquad + y\left(4M_2(M_1+M_3) + 4M_2 -2q\rho\nu(M_2+M_3)\right)\\
&\qquad\qquad + z\left(2(M_2^2+M_3^2)+2M_3 + pr_1\right).
\end{split}
\end{equation}
Now, as for the non-constant term on the second line of \eqref{eq: wishart_fv_affine_d_ge_n}, from \eqref{eq: wishart_counter_ex_Theta} we have
\begin{equation}\label{eq: wishart_counterex_bad}
\begin{split}
&-2q\trace{X\zeta'(\zeta X \zeta')^{-1}\zeta X M \Lambda\rho\rho'\Lambda'M}\\
&\qquad = -2q\rho^2\trace{\left(\begin{array}{c c} x &y \\ y &y^2/x\end{array}\right)M\left(\begin{array}{c c} 1&1\\1&1\end{array}\right)M},\\
&\qquad = -2q\rho^2\trace{\left(\begin{array}{c c} x& y\\ y & y^2/x\end{array}\right)\left(\begin{array}{c c} (M_1+M_2)^2 & (M_1+M_2)(M_2+M_3)\\ (M_1+M_2)(M_2+M_3) & (M_2+M_3)^2\end{array}\right)},\\
&\qquad = x\left(-2q\rho^2(M_1+M_2)^2\right)+y\left(-4q\rho^2 (M_1+M_2)(M_2+M_3)\right)+\frac{y^2}{x}\left(-2q\rho^2(M_2+M_3)^2\right).
\end{split}
\end{equation}
Since $\trace{LL'M} + pr_0 = \ell^2(M_1+M_3) + pr_0$, \eqref{eq: F_example} follows from \eqref{eq: wishart_counterex_good} and \eqref{eq: wishart_counterex_bad}.
\end{proof}
\section{Remaining Proofs from Section \ref{sec: converge}}\label{app: B}
\begin{proof}[Proof of Theorem \ref{thm: power}]
Under Assumptions of Theorem \ref{thm: power}, Statement \ref{stat: long hor} part i) is proved in \cite[Theorems 2.11 and 3.9]{Robertson-Xing}. Note that $\nabla h = \nabla v- \nabla \hat{v}$, part ii) follows from $\nabla h(T, \cdot)\rightarrow 0$ in part i) and the form of $\pi$ in \eqref{eq: pi_v_map}.
To prove part iii), let us collect two facts from \cite{Robertson-Xing}. First \cite[Proposition 2.3 i)]{Robertson-Xing} implies that $\mathbb{P}^{\hat{v},x}$, as the solution to the martingale problem for $\mathcal{L}^{\hat{v}}$, is a well defined probability measure. Therefore discussion after \eqref{eq: Z_phi} proves that $\mathbb{P}^{\hat{v}, x}$ is equivalent to $\mathbb{P}^x$ on $\mathcal{F}_t$ for any $t\geq 0$. Second,
\begin{equation}\label{eq: hatv_quad_var_lim}
\lim_{T\rightarrow \infty} \mathbb{E}^{\mathbb{P}^{\hat{v},x}}\bra{\int_0^t
\sum_{i,j,k,l=1}^d D_{(ij)}h\bar{A}_{(ij),(kl)}D_{(kl)} h (T-u,X_u)\,du} = 0.
\end{equation}
Indeed,
since the integrand in \eqref{eq: hatv_quad_var_lim} is independent of the Brownian motion $W$, \eqref{eq: hatv_quad_var_lim} is proved in \cite[Theorems 2.9 and 3.9]{Robertson-Xing}.
Let us use the previous two facts to prove \eqref{eq: dist strategy} first. To this end, using \eqref{eq: pi_v_map}, we obtain in either cases $m\geq n$ or $m<n$,
\begin{equation*}
\begin{split}
&\left(\pi(T-t,x;v) - \pi(x;\hat{v})\right)'\Sigma(x)\left(\pi(T-t,x;v) -
\pi(x;\hat{v})\right),\\
&\qquad = \frac{1}{(1-p)^2}\left(\sum_{i,j,k,l=1}^d D_{(ij)}h
\rho'(a^{ij})'C'\mathcal{T}heta C a^{kl}\rho D_{(kl)}h\right)(T-t,x),\\
&\qquad \leq \frac{1}{(1-p)^2}\left(\sum_{i,j,k,l=1}^d D_{(ij)}h
\trace{a^{ij}(a^{kl})'} D_{(kl)}h\right)(T-t,x),\\
&\qquad \leq \frac{1}{\underline{\kappa}(1-p)^2}\left(\sum_{i,j,k,l=1}^d
D_{(ij)}h\bar{A}_{(ij),(kl)} D_{(kl)}h\right)(T-t,x),
\end{split}
\end{equation*}
where the first inequality follows from \eqref{eq: bar_A_to_A_compare_5} and the second inequality follows from the first inequality in \eqref{eq: A_barA_compare}. Then \eqref{eq: hatv_quad_var_lim} yields
\[
\lim_{T\rightarrow \infty} \mathbb{E}^{\mathbb{P}^{\hat{v},x}}\bra{\int_0^t\left(\pi^T_u - \hat{\pi}_u\right)'\Sigma(X_u)\left(\pi^T_u -
\hat{\pi}_u\right)\, du}=0.
\]
This implies the convergence in probability $\mathbb{P}^{\hat{v},x}$, hence in $\mathbb{P}^x$, since $\mathbb{P}^{\hat{v},x}$ is equivalent to $\mathbb{P}^x$ on $\mathcal{F}_t$.
To prove \eqref{eq: ratio wealth power},
apply the first identity of \eqref{eq: prop_1_1}, where we choose $\phi=v$ from Proposition \ref{prop: v wellposed} and $\pi= \pi^T$ from \eqref{eq: opt_strat_T}. Taking difference of this identity when $t=t$ and $t=0$ respectively yields
\[
\pare{\frac{\mathcal{W}^T_t}{w}}^p = Z^{v, T}_t e^{v(T, x) - v(T-t, X_t)}.
\]
On the other hand, apply the first identity of \eqref{eq: prop_1_1} again, but choose $\pi= \hat{\pi}$ from \eqref{eq: opt_strat} and $\phi(t,x)= \hat{\lambda} t + \hat{v}(x)$, where $(\hat{\lambda}, \hat{v})$ comes from Proposition \ref{prop: ergodic wellposed} and the current choice of $\phi$ satisfies $\phi_t = \mathfrak{F}[\phi]$ due to \eqref{eq: v_ergodic}. Taking difference of this identity when $t=t$ and $t=0$ respectively, we obtain
\[
\pare{\frac{\hat{\mathcal{W}}_t}{w}}^p = Z^{\hat{v}}_t e^{\hat{\lambda} T+ \hat{v}(x) - \hat{\lambda}(T-t) - \hat{v}(X_t)}.
\]
Therefore, the ratio between the previous two identities reads
\begin{equation}\label{eq: ratio wealth prowr resp}
\frac{\mathcal{W}^T_t}{\hat{\mathcal{W}}_t} = \pare{\frac{Z^{v,T}_t}{Z^{\hat{v}}_t} e^{h(T, x) - h(T-t, X_t)}}^{\frac1p},
\end{equation}
where $h$ is defined in Statement \ref{stat: long hor} part i).
It has been proved in part i) that $h(T, \cdot)\rightarrow C$ for some constant $C$. Therefore $e^{h(T,x) - h(T-t,X_t)} \rightarrow 1$ a.s. as $T\rightarrow \infty$.
In the next paragraph, we will show
\begin{equation}\label{eq: ratio Z}
\mathbb{P}^{\hat{v}, x}-\lim_{T\rightarrow \infty} \frac{Z^{v,T}_t}{Z^{\hat{v}}_t} =1.
\end{equation}
Plugging the previous two convergence back into \eqref{eq: ratio wealth prowr resp}, it follows \[\mathbb{P}^{\hat{v},x}-\lim_{T\rightarrow \infty} \frac{\mathcal{W}^T_t}{\hat{\mathcal{W}}_t} =1.\]
Recall from Remark \ref{rem: numeraire} that $\mathcal{W}^T/\hat{\mathcal{W}}$ is a $\mathbb{P}^{\hat{v},x}$-supermartingale. Combining the previous convergence with Scheff\'{e}'s lemma, we obtain
\[
\lim_{T\rightarrow \infty} \mathbb{E}^{\mathbb{P}^{\hat{v},x}}\bra{\left|\frac{\mathcal{W}^T_t}{\hat{\mathcal{W}}_t}-1\right|}=0,
\]
Applying \cite[Lemma 3.9]{guasoni.al.11} under $\mathbb{P}^{\hat{v},x}$, the previous convergence then yields
\[
\mathbb{P}^{\hat{v},x}-\lim_{T\rightarrow \infty} \sup_{0\leq u\leq t}\left|\frac{\mathcal{W}^T_u}{\hat{\mathcal{W}}_u}-1\right|=0.
\]
Hence \eqref{eq: ratio wealth power} is confirmed after utilizing the equivalence between $\mathbb{P}^{\hat{v},x}$ and $\mathbb{P}^x$.
It remains to prove \eqref{eq: ratio Z}. To this end, using \eqref{eq: Z_phi} for $v$ and $\hat{v}$, and the definition of $h$, it follows that $Z^{v,T}_t/Z^{\hat{v}}_t = \mathcal{E}(L^T)_t$, where the $\mathbb{P}^{\hat{v}, x}$-local martingale $L^T$ takes the form
\begin{equation*}
\begin{split}
L^T_t = &\int_0^t \sum_{k,l=1}^d d\hat{B}^{kl}_u\left(\sum_{i,j=1}^d \left(a^{ij}_{kl} - q(C'\mathcal{T}heta C a^{ij}\rho)_k\rho_l\right)D_{(ij)}h \right)(T-u,X_u)\\
& + \int_0^t \sum_{k=1}^m d\hat{W}^k_u\left(-q\sum_{i,j=1}^d (D'\mathcal{T}heta C a^{ij}\rho)_k D_{(ij)}h\right) (T-u,X_u),\qquad t\leq T,
\end{split}
\end{equation*}
where $\hat{B}$ and $\hat{W}$ are $\mathbb{P}^{\hat{v},x}$ independent $\mathbb{M}^d$ and $\mathbb R^m$ dimensional Brownian motions. Calculation using $\rho'\rho CC' + DD' = 1_m$ and $\mathcal{T}heta\mathcal{T}heta = \mathcal{T}heta$ shows that
\begin{equation*}
[L^T,L^T]_t = \int_0^t \left(\sum_{i,j,k,l=1}^d D_{(ij)}h\left(\bar{A}_{(ij),(kl)} - q(1-q)\rho'(a^{ij})'C'\mathcal{T}heta C a^{kl}\rho\right)D_{(kl)}h\right)(T-u,X_u)du.
\end{equation*}
Using \eqref{eq: bar_A_to_A_compare_5} at $\theta = D h\in\mathbb{S}^d$ it follows for $p < 0$ ($0<q<1$) that
\begin{equation*}
[L^T,L^T]_t \leq \int_0^t \left(\sum_{i,j,k,l=1}^d D_{(ij)}h \bar{A}_{(ij),(kl)}D_{(kl)}h\right)(T-u,X_u)du,
\end{equation*}
and for $0 < p < 1$ ($q<0$) that
\begin{equation*}
\begin{split}
[L^T,L^T]_t &\leq \int_0^t \left(\sum_{i,j,k,l=1}^d D_{(ij)}h\left(\bar{A}_{(ij),(kl)} - q(1-q)\trace{a^{ij}(a^{kl})'}\right)D_{(kl)}h\right)(T-u,X_u)du\\
&\leq \pare{1-\frac{q(1-q)}{\underline{\kappa}}}\int_0^t \left(\sum_{i,j,k,l=1}^d D_{(ij)}h\bar{A}_{(ij),(kl)}D_{(kl)}h\right)(T-u,X_u)du
\end{split}
\end{equation*}
where the last inequality uses Lemma \ref{lem: barA}. From \eqref{eq: hatv_quad_var_lim} it thus follows that
\begin{equation*}
\lim_{T\uparrow\infty} \mathbb{E}^{\mathbb{P}^{\hat{v},x}}\bra{[L^T,L^T]_t} = 0,
\end{equation*}
which implies $\mathbb{P}^{\hat{v},x}-\lim_{T\rightarrow \infty}[L^T,L^T]_t=0$. Combining the previous convergence and the fact that $L^T$ is continuous local martingales, it follows $\mathbb{P}^{\hat{v},x}-\lim_{T\rightarrow \infty} \mathcal{E}(L^T)_t=1$, hence \eqref{eq: ratio Z} holds.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm: turnpike}]
Given results in \cite[Theorems 2.9 and 3.9]{Robertson-Xing}, the statement follows from the same argument in \cite[Theorem 2.9]{guasoni.al.11}. We now check that the assumptions in \cite{guasoni.al.11} are satisfied in the current setting. First, for each $T>0$, there exists a probability measure $\mathbb{Q}^{T,x}$ such that $\mathbb{Q}^{T,x}$ is equivalent to $\mathbb{P}^x$ on $\mathcal{F}_T$ and such that $e^{-\int_0^\cdot r(X_u)du}S$ is a $\mathbb{Q}^{T,x}$-local martingale on $[0,T]$. Indeed, let $\theta : \mathbb{S}_{++}^d\mapsto \mathbb R^k$ be a continuous function and set
\begin{equation*}
Z_t = \mathcal{E}\left(-\int_0^\cdot \sum_{k=1}^d \theta_k (X_u)dW^k_u\right)_t,
\end{equation*}
The continuity of $\theta$ and the $\mathbb{P}$ independence of $X$ and $W$ ensure that $Z$ is also a $\mathbb{P}^x$-martingale, cf. \cite[Lemma 4.8]{Karatzas-Kardaras}. Under Assumption \ref{ass: rho_strong} we may choose
$\theta = D'(DD')^{-1}\sigma'\nu$,
and it follows that $\theta$ is continuous. Since $Z$ is a $\mathbb{P}^x$-martingale, for each $T$ we may define a probability $\mathbb{Q}^{T,x}$, which is equivalent to $\mathbb{P}^x$ on $\mathcal{F}_T$, via $d\mathbb{Q}^{T,x}/d\mathbb{P}^x |_{\mathcal{F}_T} = Z_T$. Using Girsanov's theorem, a direct calculation shows that $e^{-\int_0^\cdot r(X_u) du}S$ is $\mathbb{Q}^{T,x}$-local martingale. Therefore \cite[Assumption 2.3]{guasoni.al.11} is satisfied. On the other hand, Propositions \ref{prop: v wellposed} and \ref{prop: verification} combined implies that the value of the optimization problem in \eqref{eq: power op} is finite for all $T\geq 0$. Therefore \cite[Assumption 2.4]{guasoni.al.11} is satisfied as well. On the other hand, Assumptions \ref{ass: ratio} and \ref{ass: grow} are exactly \cite[Assumptions 2.1 and 2.2]{guasoni.al.11} respectively.
Therefore \cite[Proposition 2.5]{guasoni.al.11} proves that, for all $\varepsilon > 0$,
\begin{equation}\label{eq: PT_turnpikes}
\begin{split}
\lim_{T\uparrow\infty}& \mathbb{P}^{v,T,x}\bra{ \sup_{u\leq t}\left|\frac{\mathcal{W}^{1,T}_u}{\mathcal{W}^T_u} - 1\right| \geq \varepsilon} = 0,\\
\lim_{T\uparrow\infty}& \mathbb{P}^{v,T,x}\bra{ \int_0^t \left(\pi^{1,T}_u - \pi^{T}_u\right)'\Sigma(X_u)\left(\pi^{1,T}_u - \pi^{T}_u\right) du \geq \varepsilon} = 0.
\end{split}
\end{equation}
Here since the martingale problem for $\mathcal{L}^{v,T-\cdot}$ is well-posed, cf. \cite[Lemma 4.1]{Robertson-Xing}, $\mathbb{P}^{T,v,x}$ is defined via \eqref{eq: Z_phi} with $\phi=v$. From the definitions of $\mathbb{P}^{v,T,x}$ and $\mathbb{P}^{\hat{v},x}$, it follows
\[
\left.\frac{d\mathbb{P}^{v,T,x}}{d\mathbb{P}^{\hat{v},x}}\right|_{\mathcal{F}_t} = \frac{Z^{v,T}_t}{Z^{\hat{v}}_t}.
\]
Note that both events on the left-hand-side of \eqref{eq: PT_turnpikes} are $\mathcal{F}_t$-measurable. Therefore, \eqref{eq: ratio Z} implies \eqref{eq: PT_turnpikes} holds when $\mathbb{P}^{v,T,x}$ is replaced by $\mathbb{P}^{\hat{v},x}$, hence also by $\mathbb{P}^x$, since $\mathbb{P}^{\hat{v},x}$ and $\mathbb{P}^x$ are equivalent on $\mathcal{F}_t$. Lastly, the extension to Statement \ref{stat: turnpike} is immediate after utilizing Statement \ref{stat: long hor} part iii).
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop: wishart}]
Let us verify Assumption \ref{ass: coeff_master_list} is satisfied under the parameter restrictions of this proposition. Then the statements readily follow from Theorems \ref{thm: power} and \ref{thm: turnpike}. First, for the Wishart factor model described in Section \ref{subsec: wishart}:
\begin{align*}
& V(x) = pr_0 + \frac12 \trace{\pare{x(p(r_1+r_1') -q \zeta' \nu \nu' \zeta(x))}},\\
& \overline{b}(x) = LL' + \overline{K}(x) x+ x \overline{K}(x)',
\end{align*}
where $\overline{K} = K-q \Lambda \rho \nu' \zeta(x)$. Since $\rho, \nu, \zeta$ are bounded, it is clear that $\overline{b}$ has at most linear growth. We have seen from Example \ref{ex: wishart} that $f(x)=x$ and $g(x)= \Lambda \Lambda'$. Then $\trace{f(x)}\trace{g(x)} =\trace{x} \trace{\Lambda\Lambda'}\leq \sqrt{d} \trace{\Lambda \Lambda'} \norm{x}$. In particular, $\alpha_1$ in Assumption \ref{ass: coeff_master_list} part 2) can be chosen as $\sqrt{d} \trace{\Lambda \Lambda'}$. To see the previous inequality, let $(\lambda_i)_{i=1, \dots, d}$ be eigenvalues of $x$, then Cauchy-Schwarz inequality yields $\trace{x} = \sum_{i=1}^d \lambda_i \leq \sqrt{d} (\sum_{i=1}^d \lambda_i^2)^\frac12 = \sqrt{d} \norm{x}$. To verify Assumption \ref{ass: coeff_master_list} part 3), we choose $-\beta_1$ to be larger than any largest eigenvalue of $(\overline{K}+\overline{K}')(x)$ for $x\in \mathbb{S}_{++}^d$. Since $\overline{K}(x)$ is bounded on $\mathbb{S}_{++}^d$, its largest eigenvalue is uniformly bounded on $\mathbb{S}_{++}^d$. Move on to Assumption \ref{ass: coeff_master_list} part 4). When $0<p<1$, $q<0$, then
\[
-|p r_0| - \frac12 \norm{p(r_1+ r_1') - q \zeta' \nu \nu' \zeta(x)}\norm{x} \leq V(x) \leq |p r_0| + \frac12 \norm{p(r_1+ r_1') - q \zeta' \nu \nu' \zeta(x)}\norm{x}, \quad x\in \mathbb{S}_{++}^d.
\]
Hence we can choose $-\gamma_1 = \gamma_2 = (1/2)\sup_{x\in \mathbb{S}_{++}^d} \norm{p(r_1+ r_1') - q \zeta' \nu \nu' \zeta(x)}$. When $p<0$, $q>0$, then
\[
-|p r_0| - \frac12 \norm{p(r_1+ r_1') - q \zeta' \nu \nu' \zeta(x)}\norm{x} \leq V(x) \leq |p r_0| - \lambda_{min}(x) \norm{x},
\]
where $\lambda_{min}(x)$ is the smallest eigenvalue of $(1/2)(-p(r_1+ r_1') + q \zeta' \nu \nu' \zeta(x))$. Hence we can choose the same $\gamma_2$ as above, but $\inf_{x\in \mathbb{S}_{++}^d} \lambda_{min}(x)$ as $\gamma_1$. Therefore Assumption \ref{ass: coeff_master_list} part 4) is verified.
Let us now check part 5). When $p<0$, because $r_1+r_1'\geq 0$ and $\zeta' \nu \nu' \zeta\geq 0$, $\lambda_{min}(x)\geq 0$ for any $x\in \mathbb{S}_{++}^d$, then $\gamma_1 \geq 0$. When $(\overline{K} + \overline{K}')(x)\leq -\varepsilonilon \idmat{d}$ for any $x\in \mathbb{S}_{++}^d$, $\beta_1 >0$, hence part 5)-iii) is satisfied. When $-p(r_1 + r_1') + q(\zeta' \nu \nu' \zeta(x)) \geq \varepsilonilon \idmat{d}$ for any $x\in \mathbb{S}_{++}^d$, $\gamma_1>0$, hence we are in part 5)-i). In such a case, $\trace{f(x) x g(x) x} = \trace{x^3 \Lambda \Lambda'} \geq \alpha_2 \norm{x}^3$ for some $\alpha_2>0$, where the inequality holds due to $\Lambda \Lambda' >0$. When $0<p<1$, then $\gamma_1 = -(1/2)\sup_{x\in \mathbb{S}_{++}^d} \norm{p(r_1+ r_1') - q \zeta' \nu \nu' \zeta(x)}<0$. Recall $\overline{\kappa}=1-q$ from Lemma \ref{lem: barA} and $\alpha_1= \sqrt{d} \trace{\Lambda \Lambda'}$ from part 2), then \eqref{eq: p>0 cond} is equivalent to $\beta_1^2 + 16 \overline{\kappa} \alpha_1 \gamma_1>0$ from part 5)-ii). Therefore, Assumption \ref{ass: coeff_master_list} part 5) is satisfied as well.
Finally, let us verify part A)-C). For A), calculation shows that
\begin{equation*}
H_\varepsilonilon(x; \overline{b}) = \trace{(LL' - (1+d + \varepsilonilon) \Lambda \Lambda')x^{-1}} + 2 \trace{\overline{K}(x)}.
\end{equation*}
Then $LL' > (d+1)\Lambda \Lambda'$ ensures the existence of $\varepsilonilon>0$ such that $LL'-(1+d+\varepsilonilon)\Lambda \Lambda' > 0$. Hence the previous inequality and the assumption that $\overline{K}$ is bounded on $\mathbb{S}_{++}^d$ implies that $\inf_{x\in \mathbb{S}_{++}^d} H_\varepsilonilon(x; \overline{b})>-\infty$. As for B), part A) implies the existence of $\delta>0$ such that $H_\varepsilonilon(x; \overline{b}) \geq \delta \trace{x^{-1}} + 2\trace{\overline{K}(x)}$. Observe that, for any $c_0>0$, $\delta \trace{x^{-1}} + c_0 \log(\det{x})\rightarrow \infty$ as $\det{x}\downarrow 0$. Then part B) is confirmed. Lastly, for part C), there exist $\delta, C>0$ such that $H_0(x, \overline{b}) + c_1 V(x) \geq \delta \trace{x^{-1}} - \gamma_2 \norm{x} + C$, which goes to $\infty$ as $\det{x} \downarrow 0$. This concludes verification of all parameter restrictions in Assumption \ref{ass: coeff_master_list}.
\end{proof}
\end{document} |
\begin{document}
\textitle[Bijectivity of antipode]{\bf A note on the bijectivity of antipode of a Hopf algebra and its applications}
\author{Jiafeng L\"u}
\texthanks{The first author is supported by the National Natural Science Foundation of China: No. 11571316, No. 11001245 and the Natural Science Foundation of Zhejiang Province: No. LY16A010003.}
\address{Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China} \email{jiafenglv@zjnu.edu.cn}
\author{Sei-Qwon Oh}
\texthanks{The second author is supported by Chungnam National University Grant.}
\address{Department of Mathematics, Chungnam National University, 99 Daehak-ro, Yuseong-gu, Daejeon 34134, Korea} \email{sqoh@cnu.ac.kr}
\author{Xingting Wang}
\address{Department of Mathematics, Temple University, Philadelphia, 19122, USA} \email{xingting@temple.edu}
\texthanks{The third author is supported by AMS-Simons travel grant.}
\author{Xiaolan Yu}
\address{Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China}\email{xlyu@hznu.edu.cn}
\texthanks{The fourth author is supported by the National Natural Science Foundation of China: No. 11301126, No. 11571316, No. 11671351.}
\date{}
\begin{abstract}
Certain sufficient homological and ring-theoretical conditions are given for a Hopf algebra to have bijective antipode with applications to noetherian Hopf algebras regarding their homological behaviors.
\end{abstract}
\keywords{antipode; Hopf algebra; Calabi-Yau algebra; AS-Gorenstein; AS-regular}
\subjclass[2000]{16E65, 16W30, 16W35.}
\maketitle
\geqslantqslantction*{Introduction}\leftarrowbel{0}
A classical result due to Larson and Sweedler \cite{ls} states that any finite-dimensional Hopf algebra has bijective antipode. In general, the antipode of an infinite-dimensional Hopf algebra does not need to be bijective. For instance, Takeuchi \cite{take} constructed the free Hopf algebra generated by a coalgebra whose antipode is injective but not surjective. On the other hand, Schauenburg \cite{sch} gave examples of Hopf algebras whose antipode is surjective but not injective.
In recent development, the study of infinite-dimensional Hopf algebras seems to be of growing importance, which reveals that some well-known results about finite-dimensional Hopf algebras surprisingly have incarnations in the realm of noetherian Hopf algebras (see, e.g., survey papers \cite{Bro, good}). Among these progress, it is worthy to point out that the bijectivity of the antipode frequently plays an essential role in establishing these properties (see, e.g.,\cite{bz,hoz, rrz, wyz}). Therefore, one prompts to ask for criterions concerning the bijectivity of the antipode of a Hopf algebra.
In \cite{skr}, Skryabin gave two sufficient conditions for the bijectivity, which are purely ring-theoretic. As a corollary, he proved that the antipode of any noetherian Hopf algebra is always injective, and it is surjective if certain quotient ring exits \cite[Corollary 1]{skr}. Moreover, he proposed
\begin{conjecture}(Skryabin)\leftarrowbel{bna}
Every noetherian Hopf algebra has bijective antipode.
\end{conjecture}
Recently, Meur showed that, by imposing a purely homological restriction, any twisted Calabi-Yau Hopf algebra has bijective antipode \cite[Proposition 1]{Meur}. The next result proved in the present paper uses both homological and ring-theoretic restrictions on a Hopf algebra.
\begin{thm}\leftarrowbel{Bijection}
Let $H$ be a Hopf algebra such that the left or right trivial module ${}_\varepsilon \mathbbm{k}$ or $\mathbbm{k}_\varepsilon$ has a resolution by finitely generated projective modules. Suppose $H$ satisfies one of the following conditions.
\begin{itemize}
\textitem[(i)] $\operatorname {dim} \operatorname {Ext}_{H}^i({}_\varepsilon \mathbbm{k}, H)=1$ for some integer $i\ge 0$;
\textitem[(ii)] $\operatorname {dim} \operatorname {Ext}_{H^{\mathrm{op}}}^j(\mathbbm{k}_\varepsilon,H)=1$ for some integer $j\ge 0$.
\end{itemize}
Then $H$ has injective antipode. Moreover, if both (i) and (ii) hold for $H$ with $i=j$ and $H$ additionally has one of following properties
\begin{itemize}
\textitem[(iii)] every left invertible element is regular;
\textitem[(iv)] every right invertible element is regular.
\end{itemize}
Then $H$ has bijective antipode.
\end{thm}
The class of Hopf algebras satisfies the above assumptions is large. For instance, the homological restrictions (i) and (ii) are weaker versions of AS-Gorenstein condition (see, e.g., Definition \ref{defn as}), and the ring-theoretic restrictions (iii) and (iv) are held by any Hopf algebra that is weakly finite, which includes all noetherian Hopf algebras and Hopf domains. We are able therefore to obtain
\begin{cor}\leftarrowbel{App}
Any noetherian AS-Gorenstein Hopf algebra has bijective antipode.
\end{cor}
By a celebrated result of Wu and Zhang \cite{wz}, any noetherian affine PI Hopf algebra is AS-Gorenstein, which yields another proof of the following
\begin{cor}\cite[Corollary 2]{skr}\leftarrowbel{PI}
Any noetherian affine PI Hopf algebra has bijective antipode.
\end{cor}
Now it becomes clear that an affirmative answer to the following question \cite[Question E]{Bro} regarding the homological behaviors of noetherian Hopf algebras will help to answer Conjecture \ref{bna}.
\begin{question}(Brown)\leftarrowbel{Brown}
Is every noetherian Hopf algebra AS-Gorenstein?
\end{question}
The proof of our main theorem is based on analyzing the bimodule structures arising from the Hochschild cohomlogy of $H$ with coefficients in certain bimodule over $H$ (see Theorem \ref{FLemma}). With the help of Corollary \ref{App}, we apply the same idea to noetherian Hopf algebras. We are able to extend Radford $S^4$ formula to any noetherian AS-Gorenstein Hopf algebra (see Theorem \ref{RS4}) and establish equivalent conditions regarding the homological behaviors of noetherian Hopf algebras (see Theorem \ref{Gorenstein} and Theorem \ref{CY}).
\geqslantqslantction{Preliminaries}\leftarrowbel{1}
Throughout this paper, we work over a fixed field $\mathbbm{k}$. Unless stated otherwise all algebras and vector spaces are over $\mathbbm{k}$. The unadorned tensor $\otimes$ means $\otimes_\mathbbm{k}$.
Given an algebra $A$, we write $A^\mathrm{op}$ for the \textit{opposite algebra} of $A$ and $A^e$ for the \textit{enveloping algebra} $A\otimes A^{\mathrm{op}}$. The category of left (resp. right) $A$-modules is denoted by $\operatorname {Mod}(A)$ (resp. $\operatorname {Mod}(A^{\mathrm{op}})$). An $A$-bimodule $M$ can be identified with a left $A^e$-module, that is, an object $M$ in $\operatorname {Mod}(A^e)$ with action
$$(a\otimes b)\cdot m=amb$$
for all $a\otimes b\in A^e$ and $m\in M$.
Note that an $A$-bimodule $M$ can also be a right $A^e$-module with right $A^e$-action
$$m\cdot(a\otimes b)=b m a$$
for all $a\otimes b\in A^e$ and $m\in M$. Conversely, if $M$ is a right $A^e$-module then $M$ becomes an $H$-bimodule with bimodule action
$$b m a=m\cdot(a\otimes b)$$
for all $a,b\in A$ and $m\in M$.
For an $A$-bimodule $M$ and two algebra homomorphisms $\mu$ and $\nu$, we let $^\mu M^\nu$ denote the \textit{twisted $A$-bimodule} such that $^\mu M^\nu\cong M$ as vector spaces, and the bimodule structure is given by
$$a\cdot m \cdot b=\mu(a)m\nu(b),$$
for all $a,b\in A$ and $m\in M$. If one of the homomorphisms is the identity, we will omit it.
We preserve $H$ for a Hopf algebra, and as usual, we use the symbols $\Delta$, $\varepsilon$ and $S$ respectively for its comultiplication, counit, and antipode. We use Sweedler's (sumless) notation for the comultiplication of $H$. We write ${}_\varepsilon\mathbbm{k}$ (resp. $\mathbbm{k}_\varepsilon$) for the left (resp. right) trivial module defined by the counit of $H$.
\begin{defn}
Let $\xi:H\rightarrow \mathbbm{k}$ be an algebra homomorphism. The \textextit{left winding automorphism} $\Xi^\ell_{\xi}$ of $H$ given by $\xi$ is defined to be
$$\Xi_\xi^\ell (a)=\xi(a_1)a_2,$$
for any $a\in H$. Similarly, the \textextit{right winding automorphism} of $H$ given by $\xi$ is defined to be $$\Xi_{\xi}^r(a)=a_1\xi(a_2),$$
for any $a\in H$.
\end{defn}
We recall some well-known properties of winding automorphisms.
\begin{lem}(cf. \cite[Lemma 2.5]{bz})\leftarrowbel{windingaut}
\
\begin{enumerate}
\textitem[(i)] $(\Xi_\xi^\ell)^{-1}=\Xi_{\xi S}^\ell$.
\textitem[(ii)] $\xi S^2=\xi$, so $\Xi_{\xi}^\ell=\Xi_{\xi S^2}^\ell$.
\textitem[(iii)] $\Xi_\xi^\ell S^2=S^2 \Xi_\xi^\ell$.
\textitem[(iv)] The above are true for right winding automorphisms.
\textitem[(v)] Left and right winding automorphisms always commute with each other.
\end{enumerate}
\end{lem}
\begin{defn}(cf. \cite[definition 1.2]{bz})\leftarrowbel{defn as} Let $H$ be a noetherian Hopf algebra.
\begin{enumerate}
\textitem[(i)] We say $H$ has \textextit{finite injective dimension} if the injective dimensions of $\!_HH$ and $H_H$ are both finite. In this case these integers are equal by \cite{za}, and
we write $d$ for the common value. We say $H$ is \textextit{regular} if it has finite global dimension. Right global dimension always equals left global dimension for Hopf algebras \cite[Proposition 2.1.4]{wyz}; and, when finite, the global dimension equals the injective dimension.
\textitem[(ii)] The Hopf algebra $H$ is said to be \textit{Artin-Schelter Gorenstein}, which we usually abbreviate to AS-Gorenstein, if
\begin{enumerate}
\textitem[(AS1)] $\operatorname{injdim} {_HH}=d<\infty$,
\textitem[(AS2)] $\operatorname {Ext}_H^i({_\varepsilon\mathbbm{k}},{H})=0$ for $i\neq d$ and $\operatorname {dim}\operatorname {Ext}_H^d({_\varepsilon\mathbbm{k}},{H})=1$,
\textitem[(AS3)] the right $H$-module versions of (AS1,AS2) hold.
\end{enumerate}
\textitem[(iii)] If, in addition, the global dimension of $H$ is finite, then $H$ is called \textit{Artin-Schelter regular}, which is usually shorten to AS-regular.
\end{enumerate}
\end{defn}
Suppose $H$ is noetherian AS-Gorenstein of finite injective dimension $d$. Then $\operatorname {Ext}^d_H({}_\varepsilon\mathbbm{k}, H)$ is a one-dimensional right $H$-module. Any nonzero element in $\operatorname {Ext}^d_H({}_\varepsilon\mathbbm{k}, H)$ is called a \textit{left homological integral} of $H$. Usually, $\operatorname {Ext}^d_H({}_\varepsilon\mathbbm{k}, H)$ is denoted by $\int^\ell_H$. Similarly, any nonzero element in $\operatorname {Ext}^d_{H^{op}}(\mathbbm{k}_\varepsilon, H)$ is called a \textit{right homological integral}. And $\operatorname {Ext}^d_{A^{op}}(\mathbbm{k}_\varepsilon, A )$ is denoted by $\int^r_H$. Abusing the language slightly, $\int^\ell$ (resp. $\int^r$) is also called the left (resp. right) (homological) integral. Since the right $H$-module structure on $\int^\ell$ is given by some algebra homomorphism from $H$ to $\mathbbm{k}$, we can define left and right winding automorphisms given by $\int^\ell$. This also applies to $\int^r$ by using its left $H$-module structure. We say $H$ is \textit{unimodular} if $\int^\ell\cong \mathbbm{k}_\varepsilon$ as right $H$-modules. Clearly it is equivalent to the left or right winding automorphism given by $\int^\ell$ is identity.
In \cite{g2}, Ginzburg introduced Calabi-Yau algebras whose algebraic structures arises naturally in the geometry of Calabi-Yau manifolds and mirror symmetry. Calabi-Yau algebras are one of the examples satisfying the Van den Bergh duality, which was introduced by Van den Bergh \cite{vanh} in order to study Poincar\'e duality between Hochschild homology and cohomology. We adopt all these definitions to noetherian Hopf algebras.
\begin{defn}(cf. \cite{g2, vanh, bz})
Let $H$ be a noetherian Hopf algebra.
\begin{itemize}
\textitem[(i)] We say $H$ satisfies the \textit{Van den Bergh condition} if $H$ has finite injective dimension $d$ and
\[
\operatorname {Ext}_{H^e}^i(H,H^e)=
\begin{cases}
0 & i\neq d\\
U & i=d
\end{cases}
\]
where $U$ is an invertible $H$-bimodule. We usually call $U$ the \textit{Van den Bergh dualising module} for $H$.
\textitem[(ii)] We say $H$ has the \textit{Van den Bergh duality} if it satisfies the Van den Bergh condition and $H$ is \textit{homologically smooth}, that is, $H$ has a bounded resolution in $\operatorname {Mod}(H^e)$ by finitely generated projective modules.
\textitem[(iii)] We say $H$ is \textit{twisted Calabi-Yau} if $H$ has the Van den Bergh duality with the Van den Bergh dualising module given by $H^\nu$ for some algebra automorphism $\nu$ of $H$. Moreover, we say $H$ is \textit{Calabi-Yau} if $\nu$ can be chosen as an inner automorphism.
\end{itemize}
\end{defn}
\geqslantqslantction{An isomorphism lemma for Hopf bimodules}
In this section, we aim at investigating the bimodule structures arising from the Hochschild cohomology of $H$ with coefficients in the envelop algebra $H^e$. In particular, we do not require $H$ to be noetherian or have bijective antipode.
Note that the following map $$(1\otimes S)\Delta: H\texto H^e,\ \ a\mapsto a_1\otimes S(a_2)$$ is an algebra homomorphism.
\begin{defn}
We define the left adjoint functor $\mathscr L$ from the category of left $H^e$-modules into the category of left $H$-modules such that, for every left $H^e$-module $M$, $\mathscr L(M)=M$ as vector spaces with the left action
$$a\cdot m=(1\otimes S)\Delta(a)\cdot m=(a_1\otimes S(a_2))\cdot m $$
for $a\in H$ and $m\in M$.
Similarly, the right adjoint functor $\mathscr R$ from the category of right $H^e$-modules into the category of right $H$-modules such that, for every right $H^e$-module $M$, $\mathscr R(M)=M$ as vector spaces with the right action
$$ m\cdot a= m\cdot (1\otimes S)\Delta(a)=m\cdot(a_1\otimes S(a_2))$$
for $a\in H$ and $m\in M$.
\end{defn}
Here we introduce natural module actions and elementary properties which will be used.
Since the envelope algebra $H^e$ is an algebra, $H^e$ is equipped with a natural $H^e$-bimodule structure induced by the multiplication of $H^e$. That is, the left action is given by
\begin{equation}\leftarrowbel{HHlm}
(a\otimes b)\rightarrow (x\otimes y)=(a\otimes b)(x\otimes y)=a x\otimes y b,\end{equation}
called the \textit{outer action},
and the right action is given by
\begin{equation}\leftarrowbel{HHrm}(x\otimes y)\leftarrow(a\otimes b)=(x\otimes y)(a\otimes b) =x a\otimes b y,\end{equation}
called the \textit{inner action}.
As a consequence, $\mathscr L(H^e)$ can be viewed as an $H$-$H^e$-bimodule, where the left $H$-action is given by applying the left adjoint functor to the outer action
$$a\cdot (x\otimes y)=((1\otimes S)\Delta(a))(x\otimes y)=a_1 x\otimes y S(a_2)$$
and the inner action gives the right $H^e$-module structure. On the other hand, $\mathscr R(H^e)$ is an $H^e$-$H$-bimodule with the right action
$$(x\otimes y)\cdot a=(x\otimes y)((1\otimes S)\Delta(a))=xa_1\otimes S(a_2)y$$
together with the outer action for the left $H^e$-module structure.
Let $M$ and $N$ be two left $H$-modules. Then $M\otimes N$ is a left $H\otimes H$-module with a natural left $H\otimes H$-action
$$(a\otimes b)\rightarrow (x\otimes y)=(a\cdot x)\otimes (b\cdot y).$$
Since there are two natural algebra homomorphisms from $H$ into $H\otimes H$ such that
$$H\texto H\otimes H,\ \ \ a\mapsto a\otimes 1$$
and
$$H\texto H\otimes H,\ \ \ a\mapsto 1\otimes a,$$
there are two left $H$-module actions on $M\otimes N$ such that
$$a\cdot(x\otimes y)=(a\otimes 1)\rightarrow (x\otimes y)=(a\cdot x)\otimes y,\ \ \ (\textext{denoted by ${}_*M\otimes N$})$$
and
$$a\cdot(x\otimes y)=(1\otimes a)\rightarrow (x\otimes y)=x\otimes (a\cdot y). \ \ \ (\textext{denoted by $M\otimes {}_*N$})$$
Analogously, for any right $H$-modules $M$ and $N$, there are two right $H$-module actions $M_*\otimes N$ and $M\otimes N_*$.
Since the co-multiplication map $\Delta:H\texto H\otimes H$ is an algebra homomorphism, every left (respectively, right) $H\otimesimes H$-module becomes a left
(respectively, right) $H$-module with the action induced by $\Delta$, namely
$$a\cdot(x\otimes y)=\Delta(a)\rightarrow (x\otimes y)=(a_1\cdot x)\otimes (a_2\cdot y).$$
Let $R$ and $T$ be algebras. For a left $R$-module $_RN$ and an $R$-$T$-bimodule $_RM_T$, $\textext{Hom}_R(_RN,_RM_T)$ is a right $T$-module
with the right $T$-action
$$(ft)(n)=f(n)t$$
for $f\in \textext{Hom}_R(_RN,_RM_T)$, $t\in T, n\in N$. For a right $T$-module $N_T$ and an $R$-$T$-bimodule $_RM_T$, $\textext{Hom}_T(N_T,_RM_T)$ is a left $R$-module
with the left $R$-action
$$(rf)(n)=rf(n)$$
for $f\in \textext{Hom}_T(N_T,_RM_T)$, $r\in R, n\in N$.
We often write $\textext{Hom}_{T^{\textext{op}}}(N_T,_RM_T)$ for $\textext{Hom}_T(N_T,_RM_T)$.
For a $R$-$T$-bimodule $_RN_T$ and a left $R$-module $_RM$, $\textext{Hom}_R(_RN_T,_RM)$ is a left $T$-module
with the left $T$-action
$$(tf)(n)=f(nt)$$
for $f\in \textext{Hom}_R(_RN_T,_RM)$, $t\in T, n\in N$.
The following is parallel to Lemma 2.4 in \cite{bz} and Lemma 2.1.2 in \cite{wyz}. For the sake of completeness, we include a proof here.
\begin{lem}\leftarrowbel{adjointH} Let $A$ be an algebra. There are natural isomorphisms for all integers $i\ge 0$.
\begin{enumerate}
\textitem[(i)] Let $M$ be an $H^e$-$A$-bimodule. Then $\operatorname {Ext}^i_{H^e}(H,M)\cong \operatorname {Ext}^i_H({}_\varepsilon\mathbbm{k},\mathscr L(M))$ as right $A$-modules.
\textitem[(ii)] Let $M$ be an $A$-$H^e$-bimodule. Then $\operatorname {Ext}^i_{H^e}(H,M)\cong \operatorname {Ext}^i_{H^\mathrm{op}}(\mathbbm{k}_\varepsilon,\mathscr R(M))$ as left $A$-modules.
\end{enumerate}
\end{lem}
\rhoroof We only prove (i), the proof of (ii) is quite similar. Note that $H^e$-$A$-bimodule $N$ is canonically a left $H^e\otimesimes A^\textext{op}$-module and that $H^e\otimes A^\textext{op}$ is a right $H^e$-module with the right action induced by the
multiplication of $H^e\otimes A^\textext{op}$ since $H^e$ is considered as a subalgebra of $H^e\otimes A^\textext{op}$ by the inclusion map
$H^e\texto H^e\otimes A^\textext{op}$, $x\mapsto x\otimes1$.
First of all, one sees easily that any injective $H^e$-$A$-bimodule $N$ is still injective when viewed as a left $H^e$-module since
\begin{align*}
\operatorname {Hom}_{H^e}(-,N)&\, \cong \operatorname {Hom}_{H^e}(-, \operatorname {Hom}_{H^e\otimesimes A^{\mathrm{op}}}((H^e\otimesimes A^{\mathrm{op}})_{H^e},N))\\
&\, \cong \operatorname {Hom}_{H^e\otimesimes A^\textext{op}}((H^e\otimesimes A^{\mathrm{op}})_{H^e}\otimesimes -,N)
\end{align*}
by \cite[Theorem 2.11]{Rot}.
Next, we view $H^e$ as an $H^e$-$H$-bimodule, where the left $H^e$-action is given by \eqref{HHlm} and the right $H$-action is given by
$$(x\otimes y)\cdot a=xa_1\otimes S(a_2)y.$$
We simply denote it as ${}_{H^e}H^e_H$, which is free as a right module by the fundamental theorem of Hopf modules. Indeed,
there is an
$H^e$-$H$-bimodule isomorphism ${}_{H^e}H^e_H \rightarrow H_*\otimes H$ defined by $\;x\otimes y \mapsto x_1 \otimes x_2 y$ with inverse given by $x\otimes y \mapsto x_1 \otimes S(x_2) y$, where the left $H^e$-action on $H_*\otimes H$ is given by $(a\otimes b)\cdot(x\otimesimes y)=a_1x\otimes a_2yb$ and the right $H$-action on $H_*\otimes H$ is given by $(x\otimes y)\cdot a=(x\otimes y)(a\otimes 1)=xa\otimes y$. Since $\mathscr L\cong \operatorname {Hom}_{H^e}({}_{H^e}H^e_H,-)$ as functors, one gets that
\begin{align*}
\operatorname {Hom}_H(-,\mathscr L(M))&\cong \operatorname {Hom}_H(-,\operatorname {Hom}_{H^e}({}_{H^e}H^e_H,M))\\
&\cong \operatorname {Hom}_{H^e}({}_{H^e}H^e_H\otimes_H-,M).
\end{align*}
As a consequence, $\mathscr L$ is exact and preserves injectivity.
Since there is an isomorphism ${}_{H^e}H^e_H \rightarrow H_*\otimes H$ by the above paragraph, we have the canonical isomorphism
$${}_{H^e}H^e_H\otimes_H {}_\varepsilon\mathbbm{k}\cong H_*\otimes H\otimes_H {}_\varepsilon\mathbbm{k}\cong {}_{H^e}H.$$
Hence (i) holds for $i=0$.
It follows that (i) holds for all $i\ge 0$ by taking an injective resolution of $M$ as $H^e$-$A$-bimodules.\qed
\begin{lem}\leftarrowbel{HomProj}
\
\begin{itemize}
\textitem[(i)] Let $P$ be a finitely generated projective left $H$-module. Then
$$\operatorname {Hom}_H(P,\mathscr L (H^e))\cong \operatorname {Hom}_H(P,H)\otimesimes \!_*H^{S^2}$$
as $H$-bimodules, where the bimodule structure on $\operatorname {Hom}_H(P,H)\otimesimes \!_*H^{S^2}$ is given by $a(x\otimesimes y)b=xb_1\otimesimes ayS^2(b_2)$.
\textitem[(ii)] Let $Q$ be a finitely generated projective right $H$-module. Then
$$\operatorname {Hom}_{H^\mathrm{op}}(Q,\mathscr R(H^e))\cong \!^{S^2}H_*\otimesimes \operatorname {Hom}_H(P,H)$$
as $H$-bimodules, where the bimodule structure on $\!^{S^2}H_*\otimesimes \operatorname {Hom}_H(P,H)$ is given by $a(x\otimesimes y)b=S^2(a_1)xb\otimesimes a_2y$.
\end{itemize}
\end{lem}
\rhoroof
(i) Note that $\operatorname {Hom}_H(P,\mathscr L (H^e))$ is a left $H^e$-module and thus a $H$-bimodule since $\mathscr L (H^e)$ is a $H$-$H^e$-bimodule. First of all, we claim that $\mathscr L(H^e)\cong \!_*H\otimesimes H:=V$ as $H$-$H^e$-bimodules, where the left $H$-action on $V$ is defined by the left multiplication on the first factor $H$ of $V$ and the right $H^e$-action is given by $(x\otimesimes y)\leftarrow(a\otimesimes b)=xa_1\otimesimes byS^2(a_2)$. It can be proved via the explicit $H$-$H^e$-isomorphism $\mathscr L(H^e)\texto V$ defined by $x\otimesimes y\mapsto x_1\otimesimes yS^2(x_2)$ with inverse given by $x\otimesimes y\mapsto x_1\otimesimes yS(x_2)$.
Next for any left $H$-module $M$, there exists a natural $H$-bimodule map
$$\Phi_M: \operatorname {Hom}_H(M,H)\otimesimes \!_*H^{S^2}\texto \operatorname {Hom}_H(M,\mathscr L(H^e))\cong \operatorname {Hom}_H(M,V)$$
defined by $\Phi_M(f\otimesimes h)(m)=f(m)\otimesimes h$. One checks that $\Phi$ commutes with finite direct sum, that is, $\Phi_{\opluslus_{i\in I} M_i}=\bigoplus_{i\in I} \Phi_{M_i}$ since the following diagram
\[
\xymatrix{
\operatorname {Hom}_H(\bigoplus_{i\in I} M_i, H)\otimesimes \!_*H^{S^2} \ar[rr]^-{\Phi_{\opluslus_{i\in I}M_i}}\ar[d]^-{\cong} && \operatorname {Hom}_H(\bigoplus_{i\in I}M_i, V)\ar[d]^-{\cong}\\
\bigoplus_{i\in I} \geqslantft(\operatorname {Hom}_H(M_i,H)\otimesimes \!_*H^{S^2}\right) \ar[rr]^-{\bigoplus_{i\in I} \Phi_{M_i}} && \bigoplus_{i\in I} \operatorname {Hom}_H(M_i,V)
}
\]
commutes whenever $I$ is a finite index set. Suppose $P$ is finitely generated projective, then there exists another left $H$-module $Q$ such that $P\bigoplus Q=\bigoplus_{i\in I}H_i$ over a finite index set $I$, where each $H_i\cong H$ as left $H$-modules. Note that $\Phi_H$ is clearly an isomorphism. Hence $\Phi_P\bigoplus \Phi_Q=\Phi_{P\opluslus Q}=\bigoplus_{i\in I} \Phi_{H_i}$ is an isomorphism, which implies that $\Phi_P$ is an isomorphism.
Finally, denote by $W=H\otimesimes H_*$ the $H^e$-$H$-bimodule, where the right $H$-action is the right multiplication on the second factor $H$ of $W$ and the left $H^e$-action is given by $(a\otimesimes b)\rightarrow (x\otimesimes y)=S^2(a_1)xb\otimesimes a_2y$. Then (ii) can be proved in the same fashion by using the $H^e$-$H$-isomorphism $\mathscr R(H^e)\cong W$ via $x\otimesimes y\mapsto S^2(x_1)y\otimesimes x_2$ with inverse $x\otimesimes y\mapsto y_2\otimesimes S(y_1)x$.
\qed
\begin{lem}\leftarrowbel{Compact}
The following are equivalent.
\begin{itemize}
\textitem[(i)] $H$ has a resolution in $\operatorname {Mod}(H^e)$ by finitely generated projective modules.
\textitem[(ii)] ${}_\varepsilon\mathbbm{k}$ has a resolution in $\operatorname {Mod}(H)$ by finitely generated projective modules.
\textitem[(iii)] The right $H$-module version of (ii) holds.
\end{itemize}
\end{lem}
\rhoroof
(i) $\Rightarrow$ (ii), (iii)
Let $M$ be an $H$-bimodule and let $I=\ker\varepsilon$. Then it is easy to see that $\mathbbm{k}_\varepsilon\otimesimes_H M\cong M/IM$.
Let $\mathcal B^\bullet$ be a resolution of $H$ in $\operatorname {Mod}(H^e)$ by finitely generated projective modules. Then, using the above result, one can observe easily that $\mathbbm{k} \otimesimes_H \mathcal B^\bullet$ is a resolution of ${}_\varepsilon\mathbbm{k}_\varepsilon \otimesimes_H H\cong {}_\varepsilon\mathbbm{k}$ in $\operatorname {Mod}(H)$ by finitely generated projective modules. It is the same for (iii) when we tensor $\otimesimes_H{}_\varepsilon\mathbbm{k}_\varepsilon$ on the right side of $\mathcal B^\bullet$.
(iii), (ii) $\Rightarrow$ (i) In the proof of Lemma \ref{adjointH}, one sees that the left adjoint functor $\mathscr L: \operatorname {Mod}(H^e)\texto \operatorname {Mod}(H)$ is just a restriction functor, which certainly commutes with direct limits. Applying \cite[Corollary P. 130]{bro}, $\operatorname {Ext}_H^i({}_\varepsilon\mathbbm{k},\mathscr L(-))$ commutes with direct limits for all $i$ for ${}_\varepsilon\mathbbm{k}$ has a resolution in $\operatorname {Mod}(H)$ by finitely generated projective modules. This implies that $\operatorname {Ext}_{H^e}^i(H,-)$ commutes with direct limits in $\operatorname {Mod}(H^e)$ for all $i$ since $\operatorname {Ext}_{H^e}^i(H,-)\cong \operatorname {Ext}_H^i({}_\varepsilon\mathbbm{k},\mathscr L(-))$ by Lemma \ref{adjointH}. Then one concludes again by \cite[Corollary P. 130]{bro} that $H$ has a resolution in $\operatorname {Mod}(H^e)$ by finitely generated projective modules. The proof for (iii) is exactly the same.
\qed
\begin{thm}\leftarrowbel{FLemma}
Assume the conditions in Lemma \ref{Compact} hold. Then there are $H$-bimodule isomorphisms
$$\operatorname {Ext}_{H^e}^i(H,H^e)\cong \operatorname {Ext}_H^i\geqslantft({}_\varepsilon\mathbbm{k},H\right)\otimesimes \!_*H^{S^2}\cong\, \!{}^{S^2}H_* \otimesimes \operatorname {Ext}^i_{H^\mathrm{op}}\geqslantft(\mathbbm{k}_\varepsilon,H\right)$$
for all $i$, where the bimodule structures on the second and third ones are given by $a(x\otimesimes y)b=xb_1\otimesimes ayS^2(b_2)$ and $a(x\otimesimes y)b=S^2(a_1)xb\otimesimes a_2y$, respectively.
\end{thm}
\rhoroof
Since $(H^e)^\mathrm{op}\cong H^e$, there is an equivalence between the category of left $H^e$-modules and the category of right $H^e$-modules. As a consequence, $\operatorname {Ext}_{H^e}^i(H,H^e)$ can be computed by using both the outer action and the inner action of $H^e$ defined in \eqref{HHlm} and \eqref{HHrm}, respectively.
First of all, we use the outer action \eqref{HHlm} on $H^e$ to compute the Hochschild cohomology $\operatorname {Ext}_{H^e}^i(H,H^e)$. By Lemma \ref{Compact}, we can take $\mathcal P^\bullet$ to be a resolution of ${}_\varepsilon\mathbbm{k}$ in $\operatorname {Mod}(H)$ consisting of finitely generated projective modules. Then we have
\begin{align*}
\operatorname {Ext}_{H^e}^i(H,H^e)&\, =\operatorname {Ext}_{H}^i\geqslantft({}_\varepsilon \mathbbm{k}, \mathscr L(H^e)\right) \textag{Lemma \ref{adjointH}}\\
&\,=\mathrm{H}^i(\operatorname {Hom}_H(\mathcal P^\bullet, \mathscr L(H^e))\\
&\,=\mathrm{H}^i(\operatorname {Hom}_H(\mathcal P^\bullet, H)\otimesimes \!_* H^{S^2}) \textag{Lemma \ref{HomProj}}\\
&\,=\mathrm{H}^i(\operatorname {Hom}_H(\mathcal P^\bullet, H))\otimesimes \!_*H^{S^2} \textag{K\"unneth formula}\\
&\,=\operatorname {Ext}_H^i\geqslantft({}_\varepsilon\mathbbm{k}, H\right)\otimesimes \!_*H^{S^2}.
\end{align*}
On the other hand, we can apply the inner action \eqref{HHrm} on $H^e$ to compute the Hochschild cohomology $\operatorname {Ext}_{H^e}^i(H,H^e)$. We get $\operatorname {Ext}_{H^e}^i(H,H^e)\cong\!^{S^2}H_* \otimesimes \operatorname {Ext}^i_{H^\mathrm{op}}(\mathbbm{k}_\varepsilon,H)$ by the same argument. This proves the result.
\qed
\rhoroof[Proof of Theorem \ref{Bijection}]
For the injectivity of $S$, suppose (i) holds for $H$ and the proof for (ii) is analogous. Note that $\textext{Hom}_{H}(M,H)$
is a right $H$-module for any left $H$-module $M$. Hence we can write $\operatorname {Ext}_{H}^i({}_\varepsilon\mathbbm{k},H)=\mathbbm{k}^\xi$ for some $\xi\in \operatorname {Hom}_{\mathrm{Alg}}(H,\mathbbm{k})$. For simplicity, we denote the left winding automorphism $\Xi_\xi^\ell$ still by $\xi$. By Theorem \ref{FLemma}, we have the following isomorphisms
\begin{align}\leftarrowbel{BiExt}
\!^{S^2}H_*\otimesimes \operatorname {Ext}_{H^\mathrm{op}}^i(\mathbbm{k}_\varepsilon,H)\cong \operatorname {Ext}_H^i({}_\varepsilon\mathbbm{k}, H)\otimesimes \!_* H^{S^2}\cong \mathbbm{k}^\xi\otimesimes\!_*H^{S^2}\cong H^{S^2\xi}
\end{align}
as $H$-bimodules. Since the very left side of \eqref{BiExt} is a free right $H$-module, this implies that $H^{S^2\xi}$ is torsion free on the right side. Thus $S$ is injective.
Now assume that (i) and (ii) both hold for $H$ with $i=j$. Then we can further write $\operatorname {Ext}_{H^\mathrm{op}}^i(\mathbbm{k}_\varepsilon,H)=\!^\eta\mathbbm{k}$ for some $\eta\in \operatorname {Hom}_{\mathrm{Alg}}(H,\mathbbm{k})$. We still denote by $\eta$ the right winding automorphism $\Xi_\eta^r$. Then it is straightforward to check that \eqref{BiExt} implies that $\!^{S^2\eta}H$ and $H^{S^2\xi}$ are isomorphic as $H$-bimodules. Take $\Phi: \!^{S^2\eta}H\texto H^{S^2\xi}$ to be such an isomorphism with inverse $\Phi^{-1}$. Denote by $x=\Phi(1)$ and $y=\Phi^{-1}(1)$. One immediately, by the definition of the inverse $\Phi\Phi^{-1}=\mathrm{id}=\Phi^{-1}\Phi$, verifies that the following hold in $H$ for any $a,b\in H$.
\begin{align}\leftarrowbel{S4}
xS^4\xi\eta(a)S^2\xi(y)=a,\quad S^2\eta(x)S^4\xi\eta(b)y=b.
\end{align}
Here we use the fact that $\xi,\eta, S^2$ commute with each other by Lemma \ref{windingaut}. Let $a=b=1$. One gets $xS^2\xi(y)=S^2\eta(x)y=1$.
If (iii) holds. Applying $S^2\eta$ to $xS^2\xi(y)=1$, one sees that $S^2\eta(x)S^4\xi\eta(y)=S^2\eta(x)y=1$. So $S^2\eta(x)(y-S^4\xi\eta(y))=0$, which implies that $y=S^4\xi\eta(y)$ since $S^2\eta(x)$ is left invertible hence it is not a left zero divisor by (iii). Then one gets
\begin{align*}
S^2\xi(y)x=S^2\xi(y)\rightarrow\Phi(1)=\Phi(S^2\xi(y)\rightarrow 1)=\Phi(S^4\xi\eta(y))=\Phi(y)\\
=\Phi(1\leftarrow y)=\Phi(1)\leftarrow y=x S^2\xi(y)=1.
\end{align*}
Thus $x$ and $S^2\xi(y)$ are invertible to each other. One obtains from \eqref{S4} the following formula
\begin{align}\leftarrowbel{ProofS4}
S^4\xi\eta(a)=S^2\xi(y)ax.
\end{align}
As a consequence, $S^4\xi\eta$ is an inner automorphism given by the conjugation of the element $x$. Thus $S$ is bijective. Finally, the argument for (iv) is similar. This proves the result.
\qed
\geqslantqslantction{Applications to noetherian Hopf algebras}
In this section, we apply our result to noetherian Hopf algebras satisfying the AS-Gorenstein condition, which now we know have bijective antipodes by Corollary \ref{App}. We refine many results focusing on their homological behaviors, some of which were originally stated with the assumption of the bijectivity of the antipode (see, e.g., \cite{bz,hoz}). The first result is known to be the generalization of the famous Radford's $S^4$ formula \cite{rad} to the noetherian AS-Gorenstein Hopf algebra case by Brown and Zhang. We give another proof based on Theorem \ref{Bijection}.
\begin{thm}\cite[Corollary 4.6]{bz}\leftarrowbel{RS4}
Let $H$ be a noetherian AS-Gorenstein Hopf algebra. Then
$$S^4=\gamma \circ \rhohi \circ \xi^{-1}$$
where $\xi$ and $\rhohi$ are respectively the left and right winding automorphisms given by the left integral of $H$, and $\gamma$ is an inner automorphism.
\end{thm}
\rhoroof
The result basically can be derived from the proof of Theorem \ref{Bijection}. First of all, one checks that all the assumptions in Theorem \ref{Bijection} are satisfied when $H$ is noetherian AS-Gorenstein. Namely, noetherianness guarantees that ${}_\varepsilon\mathbbm{k}$ admits a resolution in $\operatorname {Mod}(H)$ by finitely generated projective modules. Conditions (i) and (ii) follow from AS-Gorenstein assumption with $i=j=d$. Note that in a noetherian ring, a left or right invertible element is always invertible and hence it is regular (cf. \cite[Exercise 5ZE]{gw}). So (iii) and (iv) hold.
Now we keep the same notations as in the proof of Theorem \ref{Bijection}. Denote by $\xi$ the left winding automorphism given by the left integral $\int^\ell$ and $\eta$ the right winding automorphism given by the right integral $\int^r$. We write $\int^r=\!^\rhoi \mathbbm{k}$ for some $\rhoi\in \operatorname {Hom}_{\mathrm{Alg}}(H,\mathbbm{k})$. By \cite[Lemma 2.1]{lwz} (note that $S$ is bijective), $\int^l=S(\int^r)=k^{\rhoi S}$. So by using Lemma \ref{windingaut}, one sees that $\eta^{-1}=(\Xi_\rhoi^r)^{-1}=\Xi_{\rhoi S}^r:=\rhohi$ is the right automorphism given by $\int^l$.
Finally, from \eqref{ProofS4} one gets that $S^4\eta\xi$ is an inner automorphism of $H$, which we now denote by $\gamma$. Note that $S^4, \eta, \xi$ and $\gamma$ commute with each other. Hence $S^4=\gamma \circ \eta^{-1} \circ \xi=\gamma \circ \rhohi \circ \xi^{-1}$.
\qed
\begin{question} (Brown-Zhang)
What is the inner automorphism $\gamma$ in Theorem \ref{RS4}?
\end{question}
The answer is known when $H$ is finite-dimensional, where $\gamma$ is the conjugation by the distinguish group-like element of $H$ given by $\int^\ell_{H^*}$. In view of Question \ref{Brown}, we expect Theorem \ref{RS4} should hold for any noetherian Hopf algebra.
Next, we establish several equivalent conditions regarding noetherian AS-Gorenstein and AS-regular Hopf algebras. Recall that the noncommutative version of the daulising complex was first introduced by Yekutieli in \cite{ye}, and rigid dualising complex was later introduced by Van den Bergh in \cite{vdb} in order to remedy its uniqueness.
\begin{thm}\leftarrowbel{Gorenstein}
Let $H$ be a noetherian Hopf algebra. Then the following are equivalent.
\begin{itemize}
\textitem[(i)] $H$ is AS-Gorenstein.
\textitem[(ii)] $H$ satisfies Van den Bergh condition.
\textitem[(iii)] $H$ has a rigid dualising complex $R=V[s]$, where $V$ is invertible and $s\in \mathbb Z$.
\end{itemize}
In these cases, the rigid dualising complex is $R=\!^{S^2\xi}H[d]$, where $\xi$ is the left winding automorphism given by the left integral of $H$ and $d$ is the injective dimension of $H$.
\end{thm}
\rhoroof
(ii) $\Leftrightarrow$ (iii) follows from \cite{vdb}, also see \cite[Proposition 4.3]{bz}.
(i) $\Rightarrow$ (iii) is \cite[Proposition 4.5]{bz}, where the assumption of the bijectivity of the antipode is automatically satisfied with the help of Corollary \ref{App}.
(ii) $\Rightarrow$ (i) Suppose $H$ satisfies Van den Bergh condition with injective dimension $d$. In view of Theorem \ref{FLemma}, one sees that $\operatorname {Ext}_H^i({}_\varepsilon\mathbbm{k},H)=0$ for $i\neq d$ and $\operatorname {Ext}_H^d({}_\varepsilon\mathbbm{k},H)\neq 0$. This holds for the right side versions of the Ext-groups as well. Moreover, the Van den Bergh dualising module $U$ is isomorphic to $\operatorname {Ext}_H^d({}_\varepsilon\mathbbm{k},H)\otimesimes \!_*H$ as $H$-bimodules, where the latter one is a free left $H$-module with basis given in $\operatorname {Ext}_H^d({}_\varepsilon\mathbbm{k},H)$. Since $U$ is invertible, it is finitely generated projective when viewed as a left $H$-module. It can be verified by considering the autoequivalence functor $U\otimesimes_A-: \operatorname {Mod}(H)\texto \operatorname {Mod}(H)$ with inverse functor given by $U^{-1}\otimesimes _A-$. Note that a left $H$-module $M$ is finitely generated if and only if $\operatorname {Hom}_H(M,-)$ commutes with inductive direct limits, which is certainly preserved under any autoequivalence functor. Hence $U=U\otimesimes_A A$ is finitely generated. As a consequence, this implies that $\operatorname {Ext}_H^d(\mathbbm{k},H)$ is finite-dimensional. By the same reason, $\operatorname {Ext}_{H^\mathrm{op}}^d(\mathbbm{k}_\varepsilon,H)$ is finite-dimensional. Then \cite[Lemma 3.2]{bz} shows that $H$ is AS-Gorenstein.
Finally, the formula of the rigid dualising complex is given in \cite[Proposition 4.5]{bz}.
\qed
\begin{thm}\leftarrowbel{CY}
Let $H$ be a noetherian Hopf algebra. Then the following are equivalent.
\begin{itemize}
\textitem[(i)] $H$ is twisted Calabi-Yau.
\textitem[(ii)] $H$ has Van den Bergh duality.
\textitem[(iii)] $H$ is AS-regular.
\textitem[(iv)] $H$ is regular and $\operatorname {Ext}_H^i({}_\varepsilon\mathbbm{k},H)$ are finite-dimensional for all $i$.
\textitem[(v)] $H$ is regular and $\operatorname {Ext}_{H^\mathrm{op}}^i(\mathbbm{k}_\varepsilon,H)$ are finite-dimensional for all $i$.
\end{itemize}
Moroever, $H$ is Calabi-Yau if and only if $H$ is unimodular and $S^2$ is inner.
\end{thm}
\rhoroof
(i) $\Leftrightarrow$ (ii) follows from \cite[Theorem 3.5.1]{Meur}.
(iii) $\Rightarrow$ (iv), (v) are clear.
(i) $\Leftrightarrow$ (iii) can be easily deduced from Theorem \ref{Gorenstein} ((i) $\Leftrightarrow$ (ii)). Since if $H$ is noetherian, then it is regular if and only if it is homologically smooth. One direction is clear. The other direction: suppose $H$ is regular, then ${}_\varepsilon\mathbbm{k}$ has a bounded resolution in $\operatorname {Mod}(H)$ by finitely generated projective modules. This implies that $H$ is homologically smooth by \cite[Proposition 2.1.5]{wyz}.
It remains to show that (iv), (v) $\Rightarrow$ (iii). Here we only prove (iv) $\Rightarrow$ (iii) and the other one is similar. We will use Ischebeck's spectral sequence such that
\[
\operatorname {Ext}_{H^\mathrm{op}}^p(\operatorname {Ext}_H^{-q}({}_\varepsilon\mathbbm{k},H), H)\Rightarrow \operatorname {Tor}^H_{-p-q}=\begin{cases} \mathbbm{k} & p+q=0 \\ 0 & \textext{elsewhere} \end{cases}.
\]
By \cite[Proposition 2.1.4]{wyz}, the right and left global dimension of $H$ are all equal to $d$. Since $H$ is noetherian, one sees that $\operatorname {Ext}_H^d({}_\varepsilon\mathbbm{k},H)\neq 0$ and $\operatorname {Ext}_{H^\mathrm{op}}^d(\mathbbm{k}_\varepsilon,H)\neq 0$ \cite[\S 1.12]{brg}. Applying \cite[Proposition 1.3]{brg}, we have
$$
\operatorname {Ext}_{H^\mathrm{op}}^d(\operatorname {Ext}_H^i({}_\varepsilon\mathbbm{k},H), H)\cong \operatorname {Ext}_{H^\mathrm{op}}^d(\mathbbm{k}_\varepsilon, H)^{\opluslus \operatorname {dim} \operatorname {Ext}_H^i({}_\varepsilon\mathbbm{k},H)},
$$
where we use the fact that $\operatorname {Ext}_H^i({}_\varepsilon\mathbbm{k},H)$ are finite-dimensional for all $i$. This implies that $\operatorname {Ext}_H^i({}_\varepsilon\mathbbm{k},H)=0$ for all $i \neq d$ since the lowest degree nonvanishing term $\operatorname {Ext}_H^i({}_\varepsilon\mathbbm{k},H)\neq 0$ if exists with $i<d$ would contribute to the final page of the spectral sequence at the highest degree $d-i\neq 0$. In view of Theorem \ref{FLemma}, one gets $\operatorname {Ext}_{H^\mathrm{op}}^i(\mathbbm{k}_\varepsilon,H)=0$ for all $i \neq d$. Finally, a dimension argument used in \cite[Lemma 3.2]{bz} yields that $\operatorname {dim} \operatorname {Ext}_H^d({}_\varepsilon\mathbbm{k},H)=\operatorname {dim} \operatorname {Ext}_{H^\mathrm{op}}^d(\mathbbm{k}_\varepsilon,H)=1$. This proves that $H$ is AS-Gorenstein of injective dimension $d$ and hence AS-regular.
Finally, the Calabi-Yau property is \cite[Theorem 2.3]{hoz}.
\qed
\noindent
{\bf Acknowledgments}
The second, third and fourth authors are grateful for the hospitality of the first author at Zhejiang Normal University summer 2016 during the time the project was started. And the third author wants to thank James Zhang and Martin Lorenz for some helpful conversations.
\end{document} |
\begin{document}
\subjclass[2010]{35A15; 35B06; 74G65; 35B65}
\keywords{Symmetry, non-convex problems, regularity of minimizing sequences.}
\begin{abstract}
Via a symmetric version of Ekeland's principle recently obtained by the author we improve,
in a ball or an annulus, a result of Boccardo-Ferone-Fusco-Orsina on the properties of minimizing
sequences of functionals of calculus of variations in the non-convex setting.
\end{abstract}
\title{On a result by Boccardo-Ferone-Fusco-Orsina}
\section{Introduction}
In the study of non-convex minimization problems \cite{ekeland2} of calculus of variations,
the idea of selecting minimizing sequences with nice properties to guarantee the convergence
towards a minimizer can be traced back to Hilbert and Lebesgue \cite{hilbert,lebesgue}.\
In \cite{squassinaJFPTA}, the author has recently obtained an abstract symmetric version of the celebrated
Ekeland's variational principle \cite{ekeland1} for lower semi-continuous
functionals, which is probably one of the main tools to perform the selection procedure
indicated above. More precisely, the new enhanced Ekeland type principle is able
to select points which are not only almost critical, in a suitable sense,
but also almost symmetric, provided that the functional does not increase under polarizations \cite{baernst}.
In turn, under rather mild assumptions, starting from a given minimizing sequence one can detect a new
minimizing sequence enriched with very nice features. The
additional symmetry characteristics play a r\v ole also in non-compact problems,
providing compactifying effects. In 1999, Boccardo-Ferone-Fusco-Orsina \cite{bffo}
considered functionals $J:W^{1,p}_0(\Omega)\to{\mathbb R}$ of calculus of variations,
$$
J(u)=\int_\Omega j(x,u,Du),\quad\,\, u\in W^{1,p}_0(\Omega),
$$
with no convexity assumption on $\xi\mapsto j(x,s,\xi)$ and showed that, by merely relying upon some classical \cite{ladura} growth
estimates on the integrand $j(x,s,\xi)$, the existence of minimizing sequences with enhanced
smoothness can be obtained by combining the application of the classical Ekeland's principle with
a priori estimates (cf.\ \cite[Lemmas 2.3 and 2.6]{bffo}) based upon suitable Gehring-type lemmas \cite{giaqu}.
We also refer the reader to \cite{marcsbord} for other results in the same spirit.
The main goal of the present note is to highlight that, if we restrict
the attention to the case where $\Omega$ is either a ball or an annulus of ${\mathbb R}^N$ and $J$
decreases upon polarizations, then arguing as in \cite{bffo} but using the Ekeland's principle from
\cite{squassinaJFPTA}, even more special minimizing sequences can be detected. More precisely,
consider $1<p<N$, let $p^*$ denote the critical Sobolev exponent and let
$j:\Omega\times{\mathbb R}\times{\mathbb R}^N\to{\mathbb R}$ be a Carath\'eodory function such that
\begin{equation}
\label{growth}
\alpha |\xi|^p-\varphi_2|s|^{\gamma_2}\leq j(x,s,\xi)\leq \beta |\xi|^p+\varphi_0+\varphi_1|s|^{\gamma_1},
\end{equation}
for a.e. $x\in \Omega$ and every $(s,\xi)\in {\mathbb R}\times{\mathbb R}^N$, for some $\alpha,\beta>0$,
$$
\varphi_0\in L^{r_0}(\Omega),\,\,\, r_0>1,
\quad\,\,
\varphi_1\in L^{r_1}(\Omega),\,\,\, r_1>N/p,
\quad\,\,
\varphi_2\in L^{r_2}(\Omega),\,\,\, r_2>N,
$$
and $\gamma_1,\gamma_2$ satisfying
$$
0\leq \gamma_1<p^*\frac{r_1-1}{r_1},\qquad
0\leq \gamma_2<\min\Big\{p,\frac{N}{N-1}\frac{r_2-1}{r_2}\Big\}.
$$
We consider the following classes of half-spaces in ${\mathbb R}^N$
\begin{align*}
& {\mathcal H}_*:=\{\text{$H\subset{\mathbb R}^N$ is a half-space with $0\in H$}\},\quad \text{if $\Omega$ is a ball}, \\
& {\mathcal H}_*:=\{\text{$H\subset{\mathbb R}^N$ is a half-space with ${\mathbb R}^+\times\{0\}\subset H$ and
$0\in \partial H$}\},\quad \text{if $\Omega$ is an annulus},
\end{align*}
For any nonnegative measurable function $u$ we define $u^H$ to be the polarization of $u$
with respect a half-space $H\in {\mathcal H}_*$. Moreover, we denote by $u^*$ the Schwarz
symmetrization (resp.\ the spherical cap symmetrization) if $\Omega$ is a ball (resp.\ if
$\Omega$ is an annulus). For definitions and properties of these notions, we refer
to \cite{baernst} and to the references therein.
\vskip3pt
\noindent
In this framework, merely under assumption \eqref{growth}, we have the following
\begin{theorem}
\label{main}
Assume that $\Omega$ is either a ball or an annulus in ${\mathbb R}^N$ with $N\geq 2$ and
\begin{equation}
\label{polar-ass}
J(u^H)\leq J(u)\qquad\text{for all $u\in W^{1,p}_{0+}(\Omega)$ and any $H\in{\mathcal H}_*$.}
\end{equation}
Then for an arbitrary minimizing sequence $(u_h)\subset W^{1,p}_{0+}(\Omega)$ for $J$ there exist $q>p$, a new minimizing sequence
$(v_h)\subset W^{1,p}_0(\Omega)$ for $J$ and continuous mappings ${\mathbb T}_h:W^{1,1}_{0+}(\Omega)\to W^{1,1}_{0+}(\Omega)$ such that
${\mathbb T}_h z$ is built from $z$ via iterated polarizations by half-spaces in ${\mathcal H}_*$, such that
$$
\sup_{h\geq 1}\|v_h\|_{W^{1,q}_0(\Omega)}<+\infty,\quad\text{if $r_0<\frac{N}{p}$},\qquad
\sup_{h\geq 1}\|v_h\|_{L^\infty(\Omega)}<+\infty,\quad\text{if $r_0>\frac{N}{p}$},
$$
and, in addition,
$$
\lim_h\|v_h-|v_h|^*\|_{L^{\frac{N}{N-1}}(\Omega)}=0,\quad\,\,\,
\limsup_h\|v_h-u_h\|_{W^{1,1}_0(\Omega)}\leq \limsup_h \|{\mathbb T}_h u_h-u_h\|_{W^{1,1}_0(\Omega)}.
$$
\end{theorem}
\noindent
We stress that, under \eqref{growth}, $J$ is bounded from below but,
since we are not assuming the convexity of $\xi\mapsto j(x,s,\xi)$,
we can by no means conclude that $J$ has a minimum point. Nevertheless, a smooth minimizing
sequence made by almost Schwarz symmetric (for the ball) or almost spherical cap symmetric (for the annulus) points can be constructed.
As it can be readily checked by direct computation, a class of integrands which
satisfy \eqref{polar-ass} (with the equality in place of the inequality) is, for instance, $j(x,s,\xi)=j_0(s,|\xi|)$, for some
continuous function $j_0:{\mathbb R}\times{\mathbb R}^+\to{\mathbb R}$.
Observe also that, of course, from the last conclusion of Theorem \ref{main}, the limit $v$ of
$(v_h)$ must be Schwarz symmetric, namely $v=v^*$.
\begin{remark}\rm
We conclude with an important remark, which is probably one of the main reasons why
the conclusion of Theorem~\ref{main} is rather powerful in the non-convex framework.
Should one {\em additionally} assume that $\xi\mapsto j(x,s,\xi)$ is convex, it is then often
the case that a functional which satisfies \eqref{polar-ass}, fulfills in turn the corresponding symmetrization inequality
$J(u^*)\leq J(u)$. In such a case, starting from a given minimizing sequence
$(u_h)\subset W^{1,p}_{0+}(\Omega)$ for $J$ one has that $(u_h^*)\subset W^{1,p}_{0+}(\Omega)$ is a minimizing sequence too
and it is then immediate from \cite{bffo} to find a further
almost symmetric regular minimizing sequence $(v_h)$.
On the other hand, without the convexity of $j(x,s,\xi)$ in the gradient, to the author knowledge, no symmetrization
inequality is available in the current literature. In some sense, while $J(u^H)\leq J(u)$ is often an algebraic fact,
$J(u^*)\leq J(u)$ is rather a more geometrical fact.
\end{remark}
\section{Symmetric Ekeland's principle}
Let $X$ and $V$ be two Banach spaces and $S\subseteq X$. We shall consider two maps $*:S\to V$, $u\mapsto u^*$,
the symmetrization map, and $h:S\times {\mathcal H}_*\to S$, $(u,H)\mapsto u^H$, the polarization map, ${\mathcal H}_*$
being a path-connected topological space. As in \cite{squassinaJFPTA}, we assume the following:
\begin{enumerate}
\item $X$ is continuously embedded in $V$;
\item $h$ is a continuous mapping;
\item for each $u\in S$ and $H\in {\mathcal H}_*$ it holds $(u^*)^H=(u^H)^*=u^*$ and $u^{HH}=u^H$;
\item there exists $(H_m)\subset {\mathcal H}_*$ such that, for $u\in S$, $u^{H_1\cdots H_m}$ converges
to $u^*$ in $V$;
\item for every $u,v\in S$ and $H\in {\mathcal H}_*$ it holds
$\|u^H-v^H\|_V\leq \|u-v\|_V$.
\end{enumerate}
Moreover, the mappings $*:S\to V$ and $h:S\times {\mathcal H}_*\to S$ can be extended to $*:X\to V$
and $h:X\times {\mathcal H}_*\to S$ respectively by setting
$u^*:=({\mathbb T}heta(u))^*$ and $u^H:=({\mathbb T}heta(u))^H$ for all $u\in X$, where ${\mathbb T}heta:(X,\|\cdot\|_V)\to (S,\|\cdot\|_V)$ is
Lipschitz of constant $C_{\mathbb T}heta$ and such that ${\mathbb T}heta|_{S}={\rm Id}|_{S}$.
In the above framework, we recall the result from \cite{squassinaJFPTA}.
\begin{theorem}
\label{mainthm}
Assume that $f:X\to{\mathbb R}\cup\{+\infty\}$ is a proper and lower
semi-continuous functional bounded from below such that
\begin{equation}
\label{assumptionpol}
\text{$f(u^H)\leq f(u)$\,\,\quad for all $u\in S$ and $H\in {\mathcal H}_*$}.
\end{equation}
Let $u\in S$, $\rho>0$ and $\sigma>0$ with
$$
f(u)\leq \inf_X f+\rho\sigma.
$$
Then there exist $v\in X$ and a continuous map ${\mathbb T}_\rho:S\to S$ such that ${\mathbb T}_\rho z$
is built via iterated polarizations of $z$ by half-spaces in ${\mathcal H}_*$ such that
\begin{enumerate}
\item[(a)] $\|v-v^*\|_V< C\rho$;
\item[(b)] $\|v-u\|_X\leq \rho+\|{\mathbb T}_\rho u-u\|_X;$
\item[(c)] $f(v)\leq f(u)$;
\item[(d)] $f(w)\geq f(v)-\sigma \|w-v\|_X,$\quad\text{for all $w\in X,$}
\end{enumerate}
for some positive constant $C$ depending only upon $V,X$ and ${\mathbb T}heta$.
\end{theorem}
\noindent
Let $\Omega$ be either a ball or an annulus of ${\mathbb R}^N$, $N\geq 2$. In particular, by choosing
$$
X=(W^{1,1}_{0}(\Omega),\|\cdot\|_{W^{1,1}_{0}(\Omega)}),\quad
\|u\|_{W^{1,1}_{0}(\Omega)}=\int_\Omega|Du|,\qquad
S=W^{1,1}_{0+}(\Omega),
$$
as well as
$$
V=(L^{\frac{N}{N-1}}(\Omega),\|\cdot\|_{L^{\frac{N}{N-1}}(\Omega)}),\quad\,\,\, {\mathbb T}heta(u)=|u|,
$$
then (1)-(5) hold true. The following by product, adapted to our purposes, holds true.
\begin{corollary}
\label{maincor}
Let $\Omega$ be either a ball or an annulus of ${\mathbb R}^N$ and
let $J:W^{1,1}_{0}(\Omega)\to{\mathbb R}\cup\{+\infty\}$ be a lower
semi-continuous functional bounded from below with
\begin{equation}
\label{polar-ass-concr}
J(u^H)\leq J(u)\qquad\text{for all $u\in W^{1,1}_{0+}(\Omega)$ and $H\in{\mathcal H}_*$.}
\end{equation}
Let $u\in W^{1,1}_{0+}(\Omega)$ and $\varepsilon>0$ be such that
$$
J(u)\leq \inf_{W^{1,1}_{0}(\Omega)} J+\varepsilon.
$$
Then there exist $v\in W^{1,1}_{0}(\Omega)$ and a continuous map ${\mathbb T}_\varepsilon:W^{1,1}_{0+}(\Omega)\to W^{1,1}_{0+}(\Omega)$
such that ${\mathbb T}_\varepsilon z$ is built via iterated polarizations of $z$ by half-spaces in ${\mathcal H}_*$ such that $J(v)\leq J(u)$,
\begin{equation}
\label{prima}
J(w)\geq J(v)-\sqrt{\varepsilon} \|w-v\|_{W^{1,1}_{0}(\Omega)}, \quad\text{for all $w\in W^{1,1}_{0}(\Omega),$}
\end{equation}
and
\begin{equation}
\label{seconda}
\|v-|v|^*\|_{L^{\frac{N}{N-1}}(\Omega)}\leq C\sqrt{\varepsilon},\quad\,\,
\|v-u\|_{W^{1,1}_{0}(\Omega)}\leq \sqrt{\varepsilon}+\|{\mathbb T}_{\varepsilon} u-u\|_{W^{1,1}_{0}(\Omega)},
\end{equation}
for some positive constant $C$.
\end{corollary}
\section{Proof of Theorem~\ref{main}}
\noindent
The argument closely follows \cite[proof of Theorem 3.1, p.128]{bffo}, aiming
to apply Corollary~\ref{maincor} in place of the standard Ekeland's variational principle \cite{ekeland1}.
We will denote by $C$ a generic positive constant which may vary from line to line.
Taking into account that $\gamma_2<p$ and $\gamma_2r_2'<N/(N-1)$,
it easily follows that $\tilde J:W^{1,1}_{0}(\Omega)\to {\mathbb R}\cup\{+\infty\}$,
$$
\tilde J(u)=
\begin{cases}
J(u) & \text{if $u\in W^{1,p}_{0}(\Omega)$,} \\
+\infty & \text{if $u\in W^{1,1}_{0}(\Omega)\setminus W^{1,p}_{0}(\Omega)$,}
\end{cases}
$$
is a lower semi-continuous functional bounded from below.
In light of assumption \eqref{polar-ass}, we have
\begin{equation*}
\tilde J(u^H)\leq \tilde J(u)\qquad\text{for all $u\in W^{1,1}_{0+}(\Omega)$ and $H\in{\mathcal H}_*$.}
\end{equation*}
Hence, we are in the framework of Corollary~\ref{maincor}.
Given a minimizing sequence $(u_h)\subset W^{1,p}_{0+}(\Omega)$ for $J$, let $(\varepsilon_h)\subset (0,1]$
be such that $\varepsilon_h\to 0$ as $h\to\infty$ and
\begin{equation}
\label{minimiseq}
J(u_h)\leq \inf_{W^{1,p}_{0}(\Omega)} J+\varepsilon_h,\quad\,\,\, \text{for any $h\geq 1$}.
\end{equation}
Since the infimum of $J$ over $W^{1,p}_{0}(\Omega)$ equals the infimum of $\tilde J$ over $W^{1,1}_{0}(\Omega)$, it holds
\begin{equation*}
\tilde J(u_h)\leq \inf_{W^{1,1}_{0}(\Omega)} \tilde J+\varepsilon_h,\quad\,\,\, \text{for any $h\geq 1$}.
\end{equation*}
Then, by applying Corollary~\ref{maincor} to $\tilde J$, $u_h$ and $\varepsilon_h$, for any $h\geq 1$, there exists $v_h\in W^{1,1}_{0}(\Omega)$
such that $J(v_h)=\tilde J(v_h)\leq \tilde J(u_h)=J(u_h)$ and, for any $h\geq 1$,
\begin{equation}
\label{addconclu}
\|v_h-|v_h|^*\|_{L^{\frac{N}{N-1}}(\Omega)}\leq C\sqrt{\varepsilon_h},\quad\,\,
\|v_h-u_h\|_{W^{1,1}_{0}(\Omega)}\leq \sqrt{\varepsilon_h}+\|{\mathbb T}_{\varepsilon_h} u_h-u_h\|_{W^{1,1}_{0}(\Omega)},
\end{equation}
for some continuous maps ${\mathbb T}_{\varepsilon_h}:W^{1,1}_{0+}(\Omega)\to W^{1,1}_{0+}(\Omega)$ as well as, for any $h\geq 1$,
\begin{equation*}
\tilde J(v_h)\leq \tilde J(w)+\sqrt{\varepsilon_h} \int_\Omega |Dw-Dv_h|, \quad\text{for all $w\in W^{1,1}_{0}(\Omega)$,}
\end{equation*}
that is, being $\tilde J(v_h)<+\infty$,
\begin{equation}
\label{conclquasi}
\int_\Omega j(x,v_h,Dv_h)\leq \int_\Omega j(x,w,Dw)+\sqrt{\varepsilon_h} \int_\Omega |Dw-Dv_h|, \quad\text{for all $w\in W^{1,p}_{0}(\Omega)$.}
\end{equation}
Observe that $(u_h)$ is bounded in $W^{1,p}_0(\Omega)$ since \eqref{minimiseq} and \eqref{growth} yield
\begin{equation}
\label{boundestimm}
\alpha\|Du_h\|^p_{L^p(\Omega)}\leq C+C\|\varphi_2\|_{L^{r_2}(\Omega)}\|Du_h\|_{L^p(\Omega)}^{\gamma_2}, \quad\,\, (\gamma_2<p).
\end{equation}
In turn, $(v_h)$ is bounded in $W^{1,1}_{0}(\Omega)$, since
by the second inequality of \eqref{addconclu}, it holds
\begin{align*}
\|v_h\|_{W^{1,1}_{0}(\Omega)} &\leq \|v_h-u_h\|_{W^{1,1}_{0}(\Omega)}+\|u_h\|_{W^{1,1}_{0}(\Omega)} \\
& \leq\sqrt{\varepsilon_h}+\|{\mathbb T}_{\varepsilon_h} u_h-u_h\|_{W^{1,1}_{0}(\Omega)}+\|u_h\|_{W^{1,1}_{0}(\Omega)} \\
& \leq\sqrt{\varepsilon_h}+3\|u_h\|_{W^{1,1}_{0}(\Omega)}\leq 1+C\|u_h\|_{W^{1,p}_{0}(\Omega)}\leq C.
\end{align*}
In the last line, we exploited the fact that, by construction of ${\mathbb T}_{\varepsilon_h}$, for any $h\geq 1$,
$$
\|{\mathbb T}_{\varepsilon_h} u_h\|_{W^{1,1}_{0}(\Omega)}=\int_\Omega \big|Du_h^{H_0\cdots H_{m_{\varepsilon_h}}}\big|
=\int_\Omega \big|Du_h^{H_0\cdots H_{m_{\varepsilon_h}-1}}\big|=\dots=\int_\Omega |Du_h|.
$$
In conclusion, $(v_h)$ is bounded in $W^{1,p}_0(\Omega)$ since by $J(v_h)\leq C$, \eqref{growth} and $\gamma_2r_2'<N/(N-1)$
\begin{equation}
\label{boundestimm-bis}
\alpha\|Dv_h\|^p_{L^p(\Omega)}\leq C+C\|\varphi_2\|_{L^{r_2}(\Omega)}\|v_h\|_{W^{1,1}_{0}(\Omega)}^{\gamma_2}\leq C,
\end{equation}
and the variational inequality \eqref{conclquasi}, choosing $w=v_h+\varphi$ for a $\varphi\in W^{1,p}_{0}(\Omega)$, yields
\begin{equation}
\label{finalsupt}
\int_{{\rm supt}(\varphi)} j(x,v_h,Dv_h)\leq \int_{{\rm supt}(\varphi)} j(x,v_h+\varphi,Dv_h+D\varphi)
+\sqrt{\varepsilon_h} \int_{{\rm supt}(\varphi)}|D\varphi|,
\end{equation}
for all $h\geq 1$ and any $\varphi\in W^{1,p}_{0}(\Omega)$.
Once these facts hold, the boundedness of $(v_h)$ in $W^{1,q}_0(\Omega)$ (case $r_0<N/p$) for some $q>p$ follows
as in \cite[proof of Theorem 3.4]{bffo} using \eqref{growth} in \eqref{finalsupt}.
The boundedness in $L^\infty(\Omega)$ (case $r_0>N/p$), follows by choosing
$w=\max(-k,\min(k,v_h))\in W^{1,p}_0(\Omega)$ in \eqref{conclquasi} and arguing as in
\cite[proof of Theorems 3.5]{bffo}. Recalling \eqref{addconclu}, the proof is complete.
\vskip25pt
\noindent
{\bf Acknowledgment.} The note is dedicated to the memory of my beloved mother Maria Grazia.
\vskip28pt
\end{document} |
\begin{document}
\flushbottom
\title{Simulating Anisotropic quantum Rabi model via frequency modulation}
\thispagestyle{empty}
\section*{Introduction}
The quantum Rabi model (QRM)~\cite{rabi1936,rabi1937,braak2011} is a fundamental model to describe the light-matter interaction, which has been at the heart of important discoveries of fundamental effects of quantum optics. When the ratio of coupling strength and mode frequency is much smaller than $1$, rotating wave approximation (RWA) is valid and the QRM in this regime can be reduced to the Jaynes-Cummings (JC) model~\cite{jc1963,shore1993}, which has been used to describe the basic interactions in various systems \cite{meekhof1996,leibfried2003,haffner2008,lv2017,blais2004,wallraff2004,blais2007,miller2005,walther2006}. Of particular interest is to implement the QRM in ultra-strong coupling (USC) regime (the coupling strength is comparable to the cavity frequency) \cite{lupascu2015,todorov2010,anappara2009,gunter2009,forndiaz2010,niemczyk2010,federov2010,muravev2011,schwartz2011,scalari2012,geiser2012,goryachev2014,zhang2016,chen2016,solano2003}, and even deep-strong coupling (DSC) regime (the coupling strength exceeds the cavity frequency) \cite{semba2016}, in which RWA is not suitable and the counter-rotating term (CRT) cannot be neglected. This is because various effects induced by CRT appear in these regimes \cite{solano2011,garz2016,wangx2017,Ridolfo2012,Ridolfo2013,Law2013,caox2010,aiq2010,lipb2012,wangx2014,reiter2013,hes2014}. Such tremendous advances in experiments have also motivated various potential applications to quantum information technologies \cite{felicettiprl2014,rossatto2016,kyawprb2015,romero2012,wangym2017}. Although great progresses have been achieved, it is also very challenging to implement such model in USC and DSC regimes experimentally. The quantum simulation proposal provides us with an experimental accessible approach to implement the QRM in USC and DSC regimes, respectively \cite{felicetti2015,lv2016,peder2015,chengxh2018,clerk2018,braum2017,lij2013,ballester2012,mezzacapo2014,langford2017,crespi2012,felicetti2017}.
Recently, a generalized QRM with distinct RT and CRT coupling constants, which has been referred to anisotropic quantum Rabi model (AQRM), is attracting interests \cite{xie2014,tomka2014,shenlt2014,zhangg2015,zhangyy2017,zhangyy2016,yuyx2013}. Due to such interesting characteristics, the AQRM has been utilized to study various theoretical issues, {\rm e.g.}, quantum phase transitions \cite{liu2017,shenlt2017}, quantum state engineering \cite{joshi2016}, quantum fisher information \cite{wangzh2017}, and so on. To date, people have proposed several methods to realize AQRM, which include the natural implementations of AQRM in quantum optics in a cross-electric and magnetic field \cite{xie2014}, electrons in semiconductors with spin-orbit coupling \cite{wangzh2017,schie2003}, and superconducting circuits systems \cite{baksic2014,yangwj2017}. Meanwhile, quantum simulation methods with superconducting circuits \cite{wangym2018} and trapped ions \cite{aedo2018} have also been proposed. The AQRM provides us with a paradigm to understand the light-matter interaction and solid-state system. However, these implementations of AQRM are limited on the tunabilities, which motivate us to develop a frequency modulated method to realize a tunable AQRM in USC or even DSC regimes.
\begin{figure*}
\caption{(a) The circuit QED architecture of the system: A transmon qubit is capacitively coupled to a LC resonator with frequency $\omega$ \cite{koch2007}
\label{fig1}
\end{figure*}
In this paper, we propose an effective method to simulate a tunable AQRM with a qubit coupled to a resonator in dispersive regime, and the transition frequency of the qubit is modulated by two periodic driving fields. The periodic driving have been widely used to modulate quantum systems \cite{strand2013,carlos2014,yanyy2015,liao2016,huangjf2017,silveri2017,basak2018,xuezy2015,xuezy2018_1,xuezy2018_2}. We show that all the parameters in the effective Hamiltonian depend on the external driving fields. The frequencies of qubit and resonator for the simulated system can be adjusted by controlling the frequencies of the driving fields, while the anisotropic coupling coefficients of the RT and CRT are decided by the amplitudes of the driving fields. Our proposal to implement the AQRM has three features: (i) The effective Hamiltonian is controllable, and all the parameters can be tuned by controlling the external driving fields. (ii) We can drive the system from weak-coupling regime to USC regime and even DSC regime by tuning the frequencies and amplitudes of the driving fields. (iii) The ratio of coupling constants of RT and CRT can be controlled in a wide range of parameter space, which makes it possible to study the transitions from JC regime to anti-JC regime.
\section*{The derivation of the effective Hamiltonian}
In this section, we consider a qubit coupled to a harmonic oscillator in dispersive regime, and the qubit is modulated by the periodic driving fields. Such setup can be realized in a variety of different physical contexts, such as trapped ions \cite{meekhof1996,leibfried2003,haffner2008,lv2017}, circuit QED \cite{blais2004,wallraff2004,blais2007}, cavity QED \cite{miller2005,walther2006}, and so on. Here, we adopt a circuit QED setup to illustrate our proposal (the architecture is depicted in Fig.~\ref{fig1}(a)). We consider a tunable transmon qubit, which is comprised of split junctions, is capacitively coupled to a LC resonator. Such split structure allows the qubit to be modulated by the magnetic flux through the pair junctions. The system is described by a time-dependent Hamiltonian as follows (we set $\hbar =1$)
\begin{equation}\label{Eq-II-1}
\hat{H}(t)=\hat{H}_{0}+\hat{H}_{\rm int} +\hat{H}_{d}(t),
\end{equation}
where $\hat{H}_{0}$, $\hat{H}_{\rm int}$ and $\hat{H}_{d}(t)$ are given as follows
\begin{subequations}
\label{eq-II-2}
\begin{align}
\hat{H}_{0} &=\omega \hat{a}^{\dag}\hat{a} + \frac{\varepsilon}{2}\hat{\sigma}_{z},\\
\hat{H}_{\rm int}&=g(\hat{a}+\hat{a}^{\dag})\hat{\sigma}_{x},\\
\hat{H}_{d}&=\sum_{j=1}^{n_{d}}\Omega_{j}\eta_{j}\cos(\Omega_{j}t+\varphi_{j})\hat{\sigma}_{z},\label{eq2c}
\end{align}
\end{subequations}
$\hat{H}(t)=\hat{H}_{0}+\hat{H}_{\mathrm{int}}+\hat{H}_{d}(t)$
where $\varepsilon$ is the transition frequency of the tranmon qubit. $\hat{\sigma}_{\alpha}$ is the $\alpha$-component of the Pauli matrices. $\omega$ is the frequency of the LC resonator. $\hat{a}$ ($\hat{a}^{\dag}$) is the annihilation (creation) operator. $g$ is the coupling constant between the qubit and the bosonic field, and $\hat{H}_{d}(t)$ describes $n_{d}$ periodic driving fields with frequencies $\Omega_{j}$ and normalized amplitudes $\eta_{i}$. In this work, we consider $n_{d}=2$ and the qubit coupled to the resonator in dispersive regime ({\it i.e.}, $|g|\ll |\Delta_{\pm}|$ with $\Delta_{\pm}=\omega \pm \varepsilon$). Without periodic driving, the RT and CRT terms can be ignored in dispersive regime. This is because all terms are fast oscillating terms in the rotating framework. If we choose proper modulation frequencies and amplitudes such that the near resonant physical transitions are remained and far off resonant transitions can be discarded. Moving to the rotating frame defined by the following unitary operator
\begin{eqnarray}
U_{1}(t) &=& \exp\left(-i\hat{H}_{0}t-i\sum_{j=1}^{2}\eta_{j}\sin(\Omega_{j}t+\varphi_{j})\hat{\sigma}_{z}\right),
\end{eqnarray}
we obtain the transformed Hamiltonian
\begin{small}
\begin{equation}
\begin{aligned}
\hat{H}'(t) =& U_{1}^{\dag}(t)\hat{H}(t)U_{1}-iU_{1}^{\dag}(t)\left(\partial_{t}U_{1}(t)\right) \\
=& g\hat{a}\left[\hat{\sigma}_{-}e^{-i\Delta_{+}t}\exp\left(-i\sum_{j=1}^{2}\eta_{j}\sin(\Omega_{j}t+\varphi_{j})\right)+ \hat{\sigma}_{+}e^{-i\Delta_{-}t}\exp\left(i\sum_{j=1}^{2}\eta_{j}\sin(\Omega_{j}t+\varphi_{j})\right)\right]+ {\rm H.c.},
\label{hp}
\end{aligned}
\end{equation}
\end{small}
where $\hat{\sigma}_{\pm}=(\hat{\sigma}_{x}\pm i\hat{\sigma}_{y})/2$. Using the following Jacobi-Anger expansion \cite{colton1998,krosch2006}
\begin{equation}\label{Iden}
\exp(2i\eta_{j}\sin(\Omega_{j}t+\varphi_{j}))=\sum_{n=-\infty}^{+\infty}J_{n}(2\eta_{j})\exp[in(\Omega_{j}t+\varphi_{j})],
\end{equation}
with $J_{n}(x)$ being the $n$-th order Bessel function of the first kind, we obtain
\begin{equation}
\label{eq-II-3}
H'(t)=g\left[\alpha(t)\hat{a}\hat{\sigma}_{+}+\beta(t)\hat{a}\hat{\sigma}_{-}\right]+{\rm H.c.},
\end{equation}
where
\begin{small}
\begin{equation}
\begin{aligned}
\alpha(t) & = \sum_{n_{1},n_{2}=-\infty}^{+\infty}J_{n_{1}}(2\eta_{1})J_{n_{2}}(2\eta_{2})e^{i(n_{1}\varphi_{1}+n_{2}\varphi_{2})}e^{i\Omega_{-}(n_{1},n_{2})t},\\
\beta(t) & = \sum_{n_{1},n_{2}=-\infty}^{+\infty}J_{n_{1}}(2\eta_{1})J_{n_{2}}(2\eta_{2})e^{-i(n_{1}\varphi_{1}+n_{2}\varphi_{2})}e^{-i\Omega_{+}(n_{1},n_{2})t}.
\label{eq-II-4}
\end{aligned}
\end{equation}
\end{small}
Here, $\Omega_{\pm}(n_{1},n_{2})=\Delta_{\pm}+ n_{1}\Omega_{1}+ n_{2}\Omega_{2}$. According to the RWA, only slowly varying terms appearing in $\alpha(t)$ and $\beta(t)$ will dominate the dynamics. We should choose the suitable driving frequencies to obtain the rotating and counter-rotating interaction terms. We assume there is a small detuning $\delta_{1}$ ($\delta_{2}$) between $\Omega_{1}$ ($\Omega_{2}$) and the red (blue) sideband, and the definition of the detunings read
\begin{equation}
\label{eq-II-4-1}
\delta_{1}=\Omega_{1}-\Delta_{-},\quad \delta_{2}=\Delta_{+}-\Omega_{2}.
\end{equation}
The energy levels of the modulated system are shown in Fig.~\ref{fig1}(b). Considering small detunings (\emph{i.e.} $|\delta_{i}|\ll |\Delta_{\pm}|$) and dispersive coupling regime (\emph{i.e.} $|g|\ll |\Delta_{\pm}|$), one can check that the RT and the CRT will contribute to the dynamics only for lowest oscillating frequencies $\Omega_{-}(-1,0)=-\delta_{1}$ and $\Omega_{+}(0,-1)=\delta_{2}$, respectively. When the oscillating frequencies are much larger than the effect couplings, {\rm i.e.}, $|\Omega_{+}(m_{1},m_{2})|\gg |g J_{m_{1}}(2\eta_{1})J_{m_{2}}(2\eta_{2})|$ with $(m_{1},m_{2})\neq (0,-1)$ and $|\Omega_{-}(q_{1},q_{2})|\gg |g J_{q_{1}}(2\eta_{1})J_{q_{2}}(2\eta_{2})|$ with $(q_{1},q_{2})\neq (-1,0)$, one may safely neglect these fast oscillating terms in Eq. (\ref{eq-II-3}). Then the dominant terms in Eq.~(\ref{eq-II-4}) are $\alpha(t) \approx -J_{1}(2\eta_{1})J_{0}(2\eta_{2})e^{-i\varphi_{1}}e^{-i\delta_{1}t}$ and $\beta(t) \approx -J_{0}(2\eta_{1})J_{1}(2\eta_{2})e^{i\varphi_{2}}e^{-i\delta_{2}t}$, where we have used the relation $J_{-n}(x)=(-1)^{n}J_{n}(x)$ for integer $n$. Then these approximations lead to the following near resonant time-dependent Hamiltonian
\begin{small}
\begin{equation}
\begin{aligned}
\hat{H}'(t) &\approx & \left(\tilde{g}_{r}\hat{a}\hat{\sigma}^{+}e^{-i(\delta_{1}t+\varphi_{1})} + \tilde{g}_{cr}\hat{a}^{\dag}\hat{\sigma}^{+}e^{i(\delta_{2}t-\varphi_{2})}\right)+{\rm H.c.},
\label{eq-II-5}
\end{aligned}
\end{equation}
\end{small}
where the effective coupling strengths of RT and CRT are
\begin{equation}
\label{eq-II-6}
\tilde{g}_{r}=-gJ_{1}(2\eta_{1})J_{0}(2\eta_{2}),\quad \tilde{g}_{cr}=-gJ_{0}(2\eta_{1})J_{1}(2\eta_{2}).
\end{equation}
The Hamiltonian in Eq. (\ref{eq-II-5}) is the so-called AQRM in interaction picture with effective resonator frequency $\tilde{\omega}=(\delta_{1}+\delta_{2})/2$ and qubit transition frequency $\tilde{\varepsilon}=(\delta_{2}-\delta_{1})/2$.
Defining the new rotating framework associated with the time-dependent unitary operator
\begin{eqnarray}
U_{2}(t)=\exp(i\tilde{\omega}\hat{a}^{\dag}\hat{a}t+i\frac{\tilde{\varepsilon}}{2}\hat{\sigma}_{z}t),
\end{eqnarray}
we obtain the effective Hamiltonian with anisotropic coupling strengths for RT and CRT
\begin{eqnarray}
\label{Heff}
\hat{H}_{\rm eff} &=& \tilde{\omega}\hat{a}^{\dag}\hat{a}+\frac{\tilde{\varepsilon}}{2}\hat{\sigma}_{z}+ \tilde{g}_{r}\left(\hat{a}\hat{\sigma}_{+}+\hat{a}^{\dag}\hat{\sigma}_{-}\right)+\tilde{g}_{cr}\left(\hat{a}\hat{\sigma}_{-}e^{i\theta}+\hat{a}^{\dag}\hat{\sigma}_{+}e^{-i\theta}\right),
\end{eqnarray}
where we have set $\varphi_{1} = 0$ and $\varphi_{2} = \theta$. The anisotropic parameter $\lambda$ is the ratio of RT and CRT coupling strengths ({\it i.e.}, $\lambda=\tilde{g}_{cr}/\tilde{g}_{r}$). Thus we obtain a controllable AQRM. Below we analyze the parameters in our scheme. In our circuit QED setup, we consider the following realistic parameters \cite{goerz2017,hofh2009}: the transition frequency of the transmon qubit is $\varepsilon = 2\pi\times 5.4~{\rm GHz}$ with the decay rate $\kappa = 2\pi\times 0.05~{\rm MHz}$, the resonator frequency is $\omega=2\pi\times 2.2~{\rm GHz}$ with the loss rate $\gamma = 2\pi\times 0.012~{\rm MHz}$, and the coupling strength of the resonator and qubit is $g = 2\pi\times 70~{\rm MHz}$. We can check that the dispersive condition ({\rm i.e.}, $|g|\ll |\Delta_{\pm}|$) is fulfilled. The frequency modulation can be implemented by applying proper biasing magnetic fluxes. The modulation parameters $\Omega_{i}$, $\eta_{i}$ and $\varphi_{i}$ can be chosen on demand by tuning the modulation fields. In circuit QED setups, the modulation frequency and modulation amplitude range from hundreds of megahertz to several gigahertz. It is reasonable to set the modulation amplitude $\eta_{i}\Omega_{i}$ ranges from $0$ to $2\pi\times10~{\rm GHz}$ \cite{lupascu2015}. The detunings $\delta_{i}$ can be tuned from $0$ to hundreds of megahertz to fulfill the condition $|\delta_{i}|\ll |\Delta_{\pm}|$.
\begin{table}
\centering
\caption{The system parameters are listed.}
\label{tab-I}
\begin{tabular}{ccccc}
\hline
\hline
$\varepsilon/2\pi$ & $\omega/2\pi$ & $g/2\pi$ & $\gamma/2\pi$ & $\kappa/2\pi$ \\
\hline
5.4~GHz & 2.2~GHz & 70~MHz & 12~KHz & 50~KHz\\
\hline
\hline
\end{tabular}
\end{table}
\section*{The simulation of QRM and AQRM in USC and DSC regimes}
\label{sec-III}
To assess the robustness of our proposal in circuit QED system, we should consider the dissipation effects in the following discussions~\cite{goerz2017}. Considering the zero-temperature Markovian environments and large driven frequencies $\Omega_{j}$, the master equation governing the evolution of the system can be derived as follows \cite{clerk2018}
\begin{equation}\label{master_eq}
\dot{\rho} = -i[\hat{H}(t),\rho]+\mathcal{L}_{q}[\rho]+\mathcal{L}_{r}[\rho],
\end{equation}
where $\mathcal{L}_{q}[\rho]=\frac{\kappa}{2}(2\hat{\sigma}_{-}\rho\hat{\sigma}_{+}-\rho\hat{\sigma}_{+}\hat{\sigma}_{-}-\hat{\sigma}_{+}\hat{\sigma}_{-}\rho)$ and $\mathcal{L}_{r}=\frac{\gamma}{2}(2\hat{a}\rho \hat{a}^{\dag}-\rho \hat{a}^{\dag}\hat{a}-\hat{a}^{\dag}\hat{a}\rho)$ are the standard Lindblad super-operators describing the losses of the system. To obtain the master equation in the framework of effective Hamiltonian, we set $U(t)=U_{2}(t)U_{1}(t)$. Let $\tilde{\rho}(t)$ be the density matrix in the same framework with effective Hamiltonian. Inserting $\rho(t)=U(t)\tilde{\rho}(t)U^{\dag}(t)$ to the master equation (\ref{master_eq}), we obtain the following master equation
\begin{equation}\label{master_eq_new}
\dot{\tilde{\rho}} = -i[\hat{\tilde{H}}(t),\tilde{\rho}]+\mathcal{L}_{q}[\tilde{\rho}]+\mathcal{L}_{r}[\tilde{\rho}],
\end{equation}
where $\hat{\tilde{H}}(t)=U^{\dag}(t)\hat{H}(t)U(t)-iU^{\dag}(t)\left(\partial_{t}U(t)\right)$ is the total system Hamiltonian in the new rotating framework. We show that the Hamiltonian $\hat{H}_{\rm eff}$ in Eq.~(\ref{Heff}) is the approximation of $\hat{\tilde{H}}(t)$ under RWA. Here we consider the initial phase difference of the driving fields is $\theta=0$. The parameters $\eta_{1,2}$ and the detuning of first sideband $\delta_{1,2}$ are tunable parameters. Such tunable parameters determine the parameters in the simulated system in Eq.~(\ref{Heff}). To verify the validity of the effective Hamiltonian in Eq.~(\ref{Heff}), we should study the fidelity of the evolution state. Let $\ket{\tilde{\psi}(0)}$ be an initial state in the new framework and the corresponding initial density matrix is $\tilde{\rho}(0)=\ket{\tilde{\psi}(0)}\bra{\tilde{\psi}(0)}$. Substituting $\tilde{\rho}(0)$ into Eq.~(\ref{master_eq_new}), we obtain the evolution density matrix $\tilde{\rho}(t)$. The ideal case can be obtained by solving the Sch\"{o}rdinger equation governed by the effective Hamiltonian (\ref{Heff}). We denote the ideal evolution state governed by the effective Hamiltonian (\ref{Heff}) with $\ket{\tilde{\psi}(t)}$. Then the fidelity of the evolution state reads $F(t)=\left|\bra{\tilde{\psi}(t)}\tilde{\rho}(t)\ket{\tilde{\psi}(t)}\right|$.
\subsection*{The simulation of QRM}
In this subsection, we will show the performance of the simulated QRM. To obtain equal effective RT and CRT coupling strengths ({\rm i.e.}, $\lambda=1$), we need to adjust the normalized amplitude $\eta_{i}$. A simple case is $\eta_{1}=\eta_{2}=\eta$. Then the simulated coupling strength $\tilde{g}_{r}=\tilde{g}_{cr}$ and we denote the simulated coupling strength with $\tilde{g}=-gJ_{0}(2\eta)J_{1}(2\eta)$. Assuming $\theta = 0$, we can obtain the following tunable QRM
\begin{eqnarray}
\label{QRM}
H_{\rm QRM} &=& \tilde{\omega}\hat{a}^{\dag}\hat{a}+\frac{\tilde{\varepsilon}}{2}\hat{\sigma}_{z}+\tilde{g}\left(\hat{a}^{\dag}+\hat{a}\right)\hat{\sigma}_{x}.
\end{eqnarray}
The effective frequencies of resonator and qubit are determined by the detunings $\delta_{i}$. One can tuning the ratios of modulation amplitudes and frequencies to obtain different relative coupling strength.
In Figs. \ref{fig2} and \ref{fig3}, we show the fidelity and dynamics under following four sets parameters: \ref{fig2}(a), \ref{fig2}(b) and \ref{fig2}(c) $\Omega_{2}= 2\pi\times 6.759~{\rm GHz}$, and $\eta_{2}\Omega_{2}=2\pi\times 4.849~{\rm GHz}$; \ref{fig2}(d), \ref{fig2}(e) and \ref{fig2}(f) $\Omega_{2}= 2\pi\times 7.516~{\rm GHz}$, and $\eta_{2}\Omega_{2}=2\pi\times 5.392~{\rm GHz}$; \ref{fig3}(a), \ref{fig3}(b) and \ref{fig3}(c) $\Omega_{2}= 2\pi\times 7.558~{\rm GHz}$, and $\eta_{2}\Omega_{2}=2\pi\times 5.422~{\rm GHz}$; \ref{fig3}(d), \ref{fig3}(e) and \ref{fig3}(f) $\Omega_{2}= 2\pi\times 7.565~{\rm GHz}$, and $\eta_{2}\Omega_{2}=2\pi\times 5.427~{\rm GHz}$. The red sideband modulation parameters are chosen as $\Omega_{1}=2\pi\times 3.2~{\rm GHz}$, and $\eta_{1}\Omega_{1}=2\pi\times 2.296~{\rm GHz}$. These sets parameters imply the normalized modulation amplitudes $\eta=0.7173$. One also can lead to resonant red sideband ({\rm i.e.}, $\delta_{1}=0$) and the detuned blue sideband, and the corresponding detunings read $\delta_{2}=2\pi\times 840.7~{\rm MHz}$, $2\pi\times 84.07~{\rm MHz}$, $2\pi\times 42.03~{\rm MHz}$ and $2\pi\times 35.03~{\rm MHz}$. These sets parameters correspond to the four relative coupling strengths $|\tilde{g}/\tilde{\omega}|=0.05$ (Figures~\ref{fig2}(a), \ref{fig2}(b), \ref{fig2}(c)), $|\tilde{g}/\tilde{\omega}|=0.5$ (Figures~\ref{fig2}(d), \ref{fig2}(e), \ref{fig2}(f)), $|\tilde{g}/\tilde{\omega}|=1$ (Figures~\ref{fig3}(a), \ref{fig3}(b), \ref{fig3}(c)) and $|\tilde{g}/\tilde{\omega}|=1.2$ (Figures~\ref{fig3}(d), \ref{fig3}(e), \ref{fig3}(f)). In the numerical simulation, we take $\ket{\tilde{\psi}(0)}=\ket{0}_{r}\otimes\ket{g}$ as initial state. Figures~\ref{fig2}(a) and \ref{fig2}(d) show the fidelity as a function of evolution time governed by the master equation in Eq.~(\ref{master_eq_new}) and the simulated Hamiltonian given in Eq.~(\ref{Heff}). Figures~\ref{fig2}(b) and \ref{fig2}(e) show the qubit excitation number $\langle \hat{\sigma}_{+}\hat{\sigma}_{-}\rangle$ as a function of evolution time. Figures~\ref{fig2}(c) and \ref{fig2}(f) show the excitation number of the resonator $\langle \hat{a}^{\dag}\hat{a}\rangle$ as a function of evolution time. The dynamics is governed by the master equation in Eq.~(\ref{master_eq_new}) (blue solid line) and the simulated Hamiltonian given in Eq.~(\ref{Heff}) (red dashed line with circles). In the case of $|\tilde{g}/\tilde{\omega}|=0.05$, the RWA is valid and the dynamics of qubit and the resonator are dominated by RT. The effects of CRT are very weak, and we can apply RWA safely. In the case of $|\tilde{g}/\tilde{\omega}|=0.5$, the RWA is not valid and the effects of CRT cannot be ignored. The qubit and resonator can be excited simultaneously. The Fig.~\ref{fig3} shows the fidelity and dynamics when $|\tilde{g}/\tilde{\omega}|=1$ (Figures~\ref{fig3}(a), \ref{fig3}(b), \ref{fig3}(c)) and $|\tilde{g}/\tilde{\omega}|=1.2$ (Figures~\ref{fig3}(d), \ref{fig3}(e), \ref{fig3}(f)). In these cases, the relative effective coupling strength reaches $1$ and even exceeds $1$. The CRT plays an important role in USC and DSC regimes. The exist of CRT makes the total excitation number operator $\hat{N}=\hat{a}^{\dag}\hat{a}+\hat{\sigma}_{+}\hat{\sigma}_{-}$ not a conserved quantity. The excitations of qubit and resonator can be excited from the vacuum. The Figures~\ref{fig3}(d), \ref{fig3}(e), \ref{fig3}(f) show the fidelity and dynamic when $|\tilde{g}/\tilde{\omega}|=1.2$. In this case, DSC regime is reached.
\begin{figure*}
\caption{The fidelity and dynamics of the simulated QRM with effective coupling ratio $|\tilde{g}
\label{fig2}
\end{figure*}
\begin{figure*}
\caption{The fidelity and dynamics of simulated QRM with effective coupling ratio $|\tilde{g}
\label{fig3}
\end{figure*}
\subsection*{The simulation of JC model and anti-JC model}
In this subsection, we will show how to obtain the JC model and anti-JC model by tuning the driving parameters to suppress the CRT or RT, respectively. To obtain the JC model, we chosen the modulation parameters as follows: $\Omega_{1}=2\pi\times 3.2~{\rm GHz}$, $\eta_{1}\Omega_{1}=2\pi\times 3.848~{\rm GHz}$, $\Omega_{2}=2\pi\times 7.565~{\rm GHz}$, and $\eta_{2}\Omega_{2}=2\pi\times 5.427~{\rm GHz}$, $\varphi_{1}=\varphi_{2}=0$. The other parameters are listed in Table \ref{tab-I}. These modulation parameters imply $\eta_{1}=1.2024$, $\eta_{2}=0.7173$, $\delta_{1}=0$, $\delta_{2}=2\pi\times 35.03$ {\rm MHz} and $\theta=0$. One can check that $\tilde{g}_{cr}=0$ and the relative coupling strength $|\tilde{g}_{r}/\tilde{\omega}|=1.137$. In this case, the rotating term is suppressed to zero and the effective Hamiltonian reduced to the following the JC model in DSC regime
\begin{eqnarray}
\label{JC}
H_{\rm JC} &=& \tilde{\omega}a^{\dag}a+\frac{\tilde{\varepsilon}}{2}\hat{\sigma}_{z}+\tilde{g}_{r}\left(a\hat{\sigma}_{+}+a^{\dag}\hat{\sigma}_{-}\right).
\end{eqnarray}
Taking the initial state $\ket{\tilde{\psi}(0)}=\ket{0}_{r}\otimes\ket{e}$, we obtain the fidelity and dynamics of the evolution state governed by the master equation in Eq.~(\ref{master_eq_new}) and the simulated Hamiltonian given in Eq.~(\ref{JC}), which are shown in figures \ref{fig4}(a), \ref{fig4}(b) and \ref{fig4}(c). The results show that the numerical simulation agrees well with the exact dynamics. It also shows that there exists the Rabi oscillation between states $\ket{0}_{r}\otimes\ket{e}$ and $\ket{1}_{r}\otimes\ket{g}$ with period $\pi/|\tilde{g}_{r}|$. For the case $\delta_{1}\neq 0$ and the initial state $\ket{\tilde{\psi}(0)}=\ket{0}_{r}\otimes\ket{e}$, the period of the Rabi oscillation is $2\pi/\sqrt{4\tilde{g}_{r}^{2}+\delta_{1}^{2}}$.
\begin{figure*}
\caption{The fidelity and dynamics of simulated QRM with effective coupling ratio $|\tilde{g}
\label{fig4}
\end{figure*}
To obtain the anti-JC model, we set $\Omega_{1}=2\pi\times 3.2~{\rm GHz}$, $\eta_{1}\Omega_{1}=2\pi\times 2.296~{\rm GHz}$, $\Omega_{2}=2\pi\times 7.565~{\rm GHz}$, and $\eta_{2}\Omega_{2}=2\pi\times 9.096~{\rm GHz}$, $\varphi_{1}=\varphi_{2}=0$. The other parameters are listed in Table \ref{tab-I}. These modulation parameters imply $\eta_{1}=0.7173$, $\eta_{2}=1.2024$, $\delta_{1}=0$, $\delta_{2}=2\pi\times 35.03$ {\rm MHz} and $\theta=0$. In this case, we can check that $\tilde{g}_{r}=0$ and the relative coupling strength $|\tilde{g}_{cr}/\tilde{\omega}|=1.137$. The effective Hamiltonian is reduced to the following anti-JC model in DSC regime
\begin{eqnarray}
\label{AJC}
H_{\rm AJC} &=& \tilde{\omega}\hat{a}^{\dag}\hat{a}+\frac{\tilde{\varepsilon}}{2}\hat{\sigma}_{z}+\tilde{g}_{cr}\left(\hat{a}\hat{\sigma}_{-}+\hat{a}^{\dag}\hat{\sigma}_{+}\right).
\end{eqnarray}
In the anti-JC model, the rotating term is suppressed to zero and only the CRT remains. We can check the validity and dynamics of the effective Hamiltonian. Let $\ket{\tilde{\psi}(0)}=\ket{0}_{r}\otimes\ket{g}$ be the initial state. The fidelity and dynamics of the evolution state governed by the master equation in Eq.~(\ref{master_eq_new}) and the simulated Hamiltonian given in Eq.~(\ref{AJC}) are shown in figures \ref{fig4}(d), \ref{fig4}(d), \ref{fig4}(f). The Fig.~\ref{fig4}(d) shows that the numerical simulation agrees well with the exact dynamics. The Figures~\ref{fig4}(e) and \ref{fig4}(f) show that excitation number of the resonator and qubit possesses the same behavior. It also shows that there exists the Rabi oscillation between states $\ket{0}_{r}\otimes\ket{g}$ and $\ket{1}_{r}\otimes\ket{e}$ with period $2\pi/\sqrt{4\tilde{g}_{cr}^{2}+\delta_{2}^{2}}$. For the case $\delta_{2}=0$, the period of the Rabi oscillation is $\pi/|\tilde{g}_{cr}|$. Such behavior is induced by pure effect of CRT and have been studied in Ref.~\cite{Garziano2015}.
\subsection*{The simulation of degenerate AQRM}
In this subsection, we will simulate the dynamics of the AQRM. For simplify, we choose the modulation parameters are as follows: $\Omega_{1}=2\pi\times 3.2~{\rm GHz}$, $\eta_{1}\Omega_{1}=2\pi\times 2.296~{\rm GHz}$, $\Omega_{2}=2\pi\times 7.6~{\rm GHz}$, $\varphi_{1}=\varphi_{2}=0$, and the blue sideband modulation amplitude ranges from $0$ to $2\pi\times 9.138~{\rm GHz}$. The other parameters are given in Table \ref{tab-I}. Then we can obtain $\delta_{1}=\delta_{2}=0$, $\eta_{1}=0.7173$ and $\theta=0$. The normalized amplitude of blue sideband ranges from $0$ to $1.2024$. In this case, only interaction terms remain and the effective Hamiltonian reduces to the following degenerate AQRM
\begin{eqnarray}
\label{H-DAQRM}
\hat{H}_{\rm DAQRM} &=& \tilde{g}_{r}\left(\hat{a}\hat{\sigma}_{+}+\hat{a}^{\dag}\hat{\sigma}_{-}\right)+\tilde{g}_{cr}\left(\hat{a}\hat{\sigma}_{-}+\hat{a}^{\dag}\hat{\sigma}_{+}\right).
\end{eqnarray}
We check that the simulated Hamiltonian varies from JC model to anti-JC model by tuning the normalized amplitude $\eta_{2}$. Let $\ket{\tilde{\psi}(0)}=\ket{0}_{r}\otimes\ket{g}$ be the initial state. We can obtain the dynamics of the evolution states governed by the master equation in Eq.~(\ref{master_eq}). The excitations of qubit and resonator as a function of evolution time and $\eta_2$ are shown in Fig.~\ref{fig5}. The Fig.~\ref{fig5}(a) shows the excitation of qubit $\langle\hat{\sigma}_{+}\hat{\sigma}_{-}\rangle$ as a function of evolution time and $\eta_2$. When $\eta_{2}\ll 1$, we can check that $\tilde{g}_{cr}$ approaches to zero and rotating term dominates the dynamics. The qubit and resonator are not excited in the evolution process. If we increase the normalized amplitude $\eta_{2}$, the effects of the CRT emerge. In this regime, the qubit and resonator are excited in the evolution process. When $\eta_{2} = 0.7173$ (red dashed line), the ration of the RT and CRT approaches to $1$. In this regime, the RT and CRT dominate the dynamics of the evolution. The Fig.~\ref{fig5}(b) shows the excitation number of resonator $\langle \hat{a}^{\dag}\hat{a}\rangle$. When $\eta_{2} = 0.7173$ (red dashed line), the excitation number reaches its maximum value in the evolution process, which originates from the competition of RT and CRT. When $\eta_{2}$ reaches 1.2024, we can check that when $\tilde{g}_{r}$ approaches to zero, the CRT dominates the evolution. The higher excitation number of the resonator can be excited. The dynamics of the qubit and resonator show the periodic oscillation behavior. The results show that we can drive the system from JC regime to anti-JC regime through quantum Rabi regime (indicated by red dashed line).
\begin{figure*}
\caption{The dynamics of simulated degenerate AQRM as a function of evolution time and $\eta_{2}
\label{fig5}
\end{figure*}
\section*{Some applications on the quantum information theory}
\label{sec-IV}
Our scheme could be utilized as a candidate platform to implement the quantum information and computation device. As an example, we show the generations of Schr\"{o}dinger cat states and quantum gate. For this purpose, we first generalize our scheme to the multi-qubit case \cite{blais2007}. Considering $N$ qubits coupled to a resonator, we can obtain the simulated anisotropic quantum Dicke model with the same treatment. We assume all the qubits possess the same energy split ({\it i.e.}, $\varepsilon_{i}=\varepsilon$) and the periodic driving fields described in Eq.~(\ref{eq2c}) act on all the qubits. By means of the same approach, we can obtain the simulated anisotropic quantum Dicke model. The simulated anisotropic quantum Dicke model in the interaction picture reads
\begin{eqnarray}
\label{DM}
\hat{H}_{\rm DM} &=& \tilde{g}_{r}\hat{a}\hat{J}_{+}e^{-i(\delta_{1}t+\varphi_{1})}+\tilde{g}_{cr}\hat{a}\hat{J}_{-}e^{-i(\delta_{2}t+\varphi_{2})}+{\rm H.c.},\nonumber\\
\end{eqnarray}
where $\hat{J}_{\pm}=\sum_{i}^{N}\hat{\sigma}_{i\pm}$ and $\hat{J}_{z}=\frac{1}{2}[\hat{J}_{+},\hat{J}_{-}]$. If we set the detunings of the blue and red sidebands $\delta_{1}=\delta_{2}=\delta$, we obtain the degenerate two-level system ({\it i.e.}, $\tilde{\varepsilon} = 0$), and the effective frequency of resonator is $\tilde{\omega}=\delta$. We also can adjust the normalized amplitudes of the driving fields to make $\tilde{g}_{r}=\tilde{g}_{cr}=\tilde{g}$. For simplicity, we set amplitudes $\eta_{1}=\eta_{2}$ and initial driving phases $\varphi_{i}=0$. In this case, the simulated Hamiltonian in the interaction picture reduces to the following form
\begin{eqnarray}
\label{H_dis}
\hat{H}_{\rm DM} &=& \tilde{g}\left(\hat{a}^{\dag}e^{i\tilde{\omega} t}+\hat{a}e^{-i\tilde{\omega} t}\right)\hat{J}_{x}.
\end{eqnarray}
The evolution operator for the Hamiltonian in Eq.~(\ref{H_dis}), which could be obtained by means of the Magnus expansion, reads \cite{blanes2009}
\begin{eqnarray}
\label{U_dis}
\mathcal{U}(t) &=& \exp[i\phi(t)\hat{J}_{x}^{2}]D[\xi(t)\hat{J}_{x}],
\end{eqnarray}
where $D(\xi)=\exp(\xi \hat{a}^{\dag}-\xi^{*}\hat{a})$, $\xi(t)=(\tilde{g}/\tilde{\omega})(1-e^{i\tilde{\omega}t})$ and $\phi(t)=(\tilde{g}/\tilde{\omega})^{2}(\tilde{\omega}t-\sin(\tilde{\omega}t))$.
Based on the dynamics of this effective Hamiltonian, the Schr\"{o}dinger cat states and quantum gate can be generated.
\subsection*{The generation of Schr\"{o}dinger cat states}
Superposition of coherent states plays an important role in quantum theory \cite{liao2016,huangjf2017,liu2005,liao2008,yin2013}. In this subsection, we consider how to generate superposition of coherent states for a single-qubit case. Assuming the initial state prepared on $\ket{\tilde{\psi}(0)}=\ket{0}_{r}\otimes\ket{g}$, we obtain the evolution state as follows
\begin{eqnarray}
\label{cat_dis1}
\ket{\tilde{\psi}(t)}=\frac{e^{i\phi(t)}}{\sqrt{2}}(\ket{\xi(t)}_{r}\otimes\ket{+}-\ket{-\xi(t)}_{r}\otimes\ket{-}),
\end{eqnarray}
where $\ket{\pm}=\frac{1}{\sqrt{2}}(\ket{e}\pm\ket{g})$ are the eigenstates of $\hat{\sigma}_{x}$ and $\ket{\pm\xi(t)}_{r}=D[\xi(t)]\ket{0}_{r}$ are the coherent states with amplitude $\pm \xi(t)$. In the basis $\ket{e}$ and $\ket{g}$, the above state can be rewritten as following form
\begin{equation}
\ket{\tilde{\psi}(t)}=\frac{e^{i\phi(t)}}{2}(\ket{\mathcal{C}_{-}(t)}_{r}\otimes\ket{e}+\ket{\mathcal{C}_{+}(t)}_{r}\otimes\ket{g}),
\label{cat_dis2}
\end{equation}
where $\ket{\mathcal{C}_{\pm}(t)}=\mathcal{N}_{\pm}(\ket{\xi(t)}\pm \ket{-\xi(t)})$ with normalization coefficients $\mathcal{N}_{\pm}=\sqrt{2(1\pm \exp(-2|\xi(t)|^{2}))}$. Performing a projection measurement on the states $\ket{e}$ and $\ket{g}$, we obtain the states $\ket{\mathcal{C}_{+}(t)}$ and $\ket{\mathcal{C}_{-}(t)}$, which correspond to the even and odd Schr\"{o}dinger cat states. The magnitude of the displacement for $\ket{\mathcal{C}_{\pm}(t)}$ is $|\xi(t)|=2|(\tilde{g}/\tilde{\omega})\sin(\tilde{\omega}t/2)|$. When the evolution time $t_{0}=\pi/\tilde{\omega}$, the magnitude of the displacement reaches its maximum value $2|\tilde{g}/\tilde{\omega}|$.
\subsection*{The implementation of quantum gate}
In this subsection, we consider two-qubit case. Assuming the evolution time $T=2\pi/\tilde{\omega}$, we obtain $\xi(T)=0$ and $\phi(T)=2\pi(\tilde{g}/\tilde{\omega})^{2}$. The evolution operator is reduced to the following form
\begin{eqnarray}
\label{U_gate}
\mathcal{U}(T) &=& \exp[i\phi(T)J_{x}^{2}].
\end{eqnarray}
where $J_{x}=\hat{\sigma}_{1x}+\hat{\sigma}_{2x}$ for two-qubit case. The Eq.~(\ref{U_gate}) can be rewritten as the form $\mathcal{U}(T)= \cos\vartheta \mathcal{I}+i \sin\vartheta \hat{\sigma}_{1x}\hat{\sigma}_{2x}$, where $\vartheta=2\phi(T)$ and $\mathcal{I}$ is identity operator for two-qubit. Here, we have omitted the total phase factor. To assess the capacity of the quantum gate, Zanardi {\it et. al.} introduced the entangling power \cite{zanardi1,zanardi2}. The entangling power for this unitary operator reads $e_{p}(\mathcal{U})=\frac{2}{9}\sin^{2}(2\vartheta)$. So the evolution operator can be viewed as a nontrival two-qubit quantum gate when $\theta\neq \frac{k}{2}\pi$ ($k$ is integer). When $\vartheta=\pi/4$ ({\it i.e.}, $\tilde{g}/\tilde{\omega}=0.25$), the quantum gate reads $\mathcal{U}(T)= \frac{1}{\sqrt{2}}\left( \mathcal{I}+i\hat{\sigma}_{1x}\hat{\sigma}_{2x}\right)$. Such quantum gate is local equivalent to the control-not (CNOT) gate \cite{Makhlin2002,Zhang2003}. The equivalent relation reads
\begin{eqnarray}
\label{U_equi}
{\rm CNOT}=(u_{1}\otimes u_{2})\mathcal{U}(T)(u_{3}\otimes u_{4}),
\end{eqnarray}
where local unitary operators are as follows
\begin{equation}
\label{eq_local}
\begin{array}{llll}
u_{1}=\frac{1}{\sqrt{2}}\left(
\begin{array}{cc}
-1 & 1 \\
1 & 1\\
\end{array}\right),~u_{2}=\left(
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}
\right),~u_{3}= \frac{1}{\sqrt{2}}\left(
\begin{array}{cc}
-1 & -i \\
1 & -i \\
\end{array}\right),~ u_{4}=\frac{1}{\sqrt{2}}\left(
\begin{array}{cc}
1 & i \\
i & 1 \\
\end{array}
\right).
\end{array}
\end{equation}
\section*{Discussion}
In conclusion, we have proposed a method to simulate a tunable AQRM, which is achieved by driving the qubit(s) with two-tone periodic driving fields. We have analyzed the parameter conditions under which this proposal works well. By choosing proper modulation frequencies and amplitudes, the coupling constants of RT or CRT can be suppressed to zero, respectively. Consequently, we study the dynamics induced by CRT or RT correspondingly. In addition, we have also discussed the applications of our scheme to the generations of quantum gate and Schr\"{o}dinger cat states. This proposal provides us with a reliable approach for studying the effects of RT and CRT in different regimes individually. Although we explore the scheme with the circuit QED system, which could be implemented in other systems, {\rm e.g.}, cavity QED and trapped ion systems. The presented proposal will pave a way to further study the stronger light-matter interaction in a system whose coupling strength is far away from the USC and DSC regimes in quantum optics.
Extensions of presented scheme to a variety of physically relevant systems, such as multi-qubit and multi-mode fields interaction system and the system coupling with the environments, deserve future investigations.
\begin{thebibliography}{10}
\expandafter\ifx\csname url\endcsname\relax
\def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\defURL {URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}
\bibitem{rabi1936} \bibinfo{author}{Rabi, I. I.}
\newblock \bibinfo{title}{On the Process of Space Quantization}.
\newblock\emph{\bibinfo{journal}{Phys. Rev.}} \textbf{\bibinfo{volume}{49}}, \bibinfo{pages}{324} (\bibinfo{year}{1936}).
\bibitem{rabi1937}\bibinfo{author}{Rabi, I. I.}
\newblock \bibinfo{title}{Space Quantization in a Gyrating Magnetic Field}.
\newblock\emph{\bibinfo{journal}{Phys. Rev.}} \textbf{\bibinfo{volume}{51}}, \bibinfo{pages}{652} (\bibinfo{year}{1937}).
\bibitem{braak2011}\bibinfo{author}{Braak, D.}
\newblock \bibinfo{title}{Integrability of the Rabi model}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}} \textbf{\bibinfo{volume}{107}}, \bibinfo{pages}{100401} (\bibinfo{year}{2011}).
\bibitem{jc1963}\bibinfo{author}{Jaynes, E. T.} \& \bibinfo{author}{Cummings, F. W.},
\newblock\bibinfo{title}{Comparison of quantum and semiclassical radiation theories with application to the beam maser}.
\newblock\emph{\bibinfo{journal}{Proc. IEEE}} \textbf{\bibinfo{volume}{51}}, \bibinfo{pages}{89} (\bibinfo{year}{1963}).
\bibitem{shore1993}\bibinfo{author}{Shore, B. W.} \& \bibinfo{author}{Knight, P. L.}
\newblock \bibinfo{title}{The Jaynes-Cummings model}.
\newblock\emph{\bibinfo{journal}{J. Mod. Opt.}} \textbf{\bibinfo{volume}{40}}, \bibinfo{pages}{1195} (\bibinfo{year}{1993}).
\bibitem{meekhof1996}\bibinfo{author}{Meekhof, D. M.}, \bibinfo{author}{Monroe, C.}, \bibinfo{author}{King, B. E.},
\bibinfo{author}{Itano, W. M.} \& \bibinfo{author}{Wineland, D. J.}
\newblock \bibinfo{title}{Generation of Nonclassical Motional States of a Trapped Atom}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}} \textbf{\bibinfo{volume}{76}}, \bibinfo{pages}{1796} (\bibinfo{year}{1996}).
\bibitem{leibfried2003}\bibinfo{author}{Leibfried, D.}, \bibinfo{author}{Blatt, R.}, \bibinfo{author}{Monroe, C.} \& \bibinfo{author}{Wineland, D. J.}
\newblock\bibinfo{title}{Quantum Dynamics of Single Trapped Ions}.
\newblock\emph{\bibinfo{journal}{Rev. Mod. Phys.}} \textbf{\bibinfo{volume}{75}}, \bibinfo{pages}{281} (\bibinfo{year}{2003}).
\bibitem{haffner2008}\bibinfo{author}{H\"{a}ffner, H. H.}, \bibinfo{author}{Roos, C. F.} \& \bibinfo{author}{Blatt, R.} \newblock \bibinfo{title}{Quantum Computing with Trapped Ions}.
\newblock\emph{\bibinfo{journal}{Phys. Rep.}} \textbf{\bibinfo{volume}{469}}, \bibinfo{pages}{155} (\bibinfo{year}{2008}).
\bibitem{lv2017}\bibinfo{author}{Lv, D.}, \bibinfo{author}{An, S.}, \bibinfo{author}{Um, M.}, \bibinfo{author}{Zhang, J.},
\bibinfo{author}{Zhang, J.-N.}, \bibinfo{author}{Kim, M. S.} \& \bibinfo{author}{Kim, K.}
\newblock \bibinfo{title}{Reconstruction of the Jaynes-Cummings Field State of Ionic Motion in a Harmonic Trap}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{95}}, \bibinfo{pages}{043813} (\bibinfo{year}{2017}).
\bibitem{blais2004}\bibinfo{author}{Blais, A.}, \bibinfo{author}{Huang, R.-S.}, \bibinfo{author}{Wallraff, A.}, \bibinfo{author}{Girvin, S. M.} \& \bibinfo{author}{Schoelkopf, R. J.}
\newblock \bibinfo{title}{Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{69}}, \bibinfo{pages}{062320} (\bibinfo{year}{2004}).
\bibitem{wallraff2004}\bibinfo{author}{Wallraff, A.}, \bibinfo{author}{Schuster, D. I.}, \bibinfo{author}{Blais, A.}, \bibinfo{author}{Frunzio, L.},
\bibinfo{author}{Huang, R.-S.}, \bibinfo{author}{Majer, J.}, \bibinfo{author}{Kumar, S.}, \bibinfo{author}{Girvin, S. M.} \& \bibinfo{author}{Schoelkopf, R. J.}
\newblock \bibinfo{title}{Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics}.
\newblock\emph{\bibinfo{journal}{Nature}} \textbf{\bibinfo{volume}{431}}, \bibinfo{pages}{062320} (\bibinfo{year}{2004}).
\bibitem{blais2007}\bibinfo{author}{Blais, A.}, \bibinfo{author}{Gambetta, J.}, \bibinfo{author}{Wallraff, A.}, \bibinfo{author}{Schuster, D. I.},
\bibinfo{author}{Girvin, S. M.}, \bibinfo{author}{Devoret, M. H.} \& \bibinfo{author}{Schoelkopf, R. J.}
\newblock \bibinfo{title}{Quantum information processing with circuit quantum electrodynamics}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{75}}, \bibinfo{pages}{032329} (\bibinfo{year}{2007}).
\bibitem{miller2005}\bibinfo{author}{Miller, R.}, \bibinfo{author}{Northup, T. E.}, \bibinfo{author}{Birnbaum, K. M.}, \bibinfo{author}{Boca, A.}, \bibinfo{author}{Boozer, A. D.}
\& \bibinfo{author}{Kimble, H. J.}
\newblock \bibinfo{title}{Trapped Atoms in Cavity QED: Coupling Quantized Light and Matter}.
\newblock\emph{\bibinfo{journal}{J. Phys. B}} \textbf{\bibinfo{volume}{38}}, \bibinfo{pages}{S551} (\bibinfo{year}{2005}).
\bibitem{walther2006}\bibinfo{author}{Walther, H.}, \bibinfo{author}{Varcoe, B. T. H.}, \bibinfo{author}{Englert, B. G. } \& \bibinfo{author}{Becker, T.}
\newblock \bibinfo{title}{Cavity Quantum Electrodynamics}.
\newblock\emph{\bibinfo{journal}{Rep. Prog. Phys.}} \textbf{\bibinfo{volume}{69}}, \bibinfo{pages}{1325} (\bibinfo{year}{2006}).
\bibitem{todorov2010}\bibinfo{author}{Todorov, Y.}, \bibinfo{author}{Andrews, A. M.}, \bibinfo{author}{Colombelli, R.}, \bibinfo{author}{Liberato, S. D.},
\bibinfo{author}{Ciuti, C.}, \bibinfo{author}{Klang, P.}, \bibinfo{author}{Strasser, G.} \& \bibinfo{author}{Sirtori, C.}
\newblock \bibinfo{title}{Ultrastrong Light-Matter Coupling Regime with Polariton Dots}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}} \textbf{\bibinfo{volume}{105}}, \bibinfo{pages}{196402} (\bibinfo{year}{2010}).
\bibitem{lupascu2015}\bibinfo{author}{Deng, C. Q.}, \bibinfo{author}{Orgiazzi, J.-L.}, \bibinfo{author}{Shen, F. R.}, \bibinfo{author}{Ashhab, S. } \& \bibinfo{author}{Lupascu, A.}
\newblock \bibinfo{title}{Observation of Floquet States in a Strongly Driven Artificial Atom}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}} \textbf{\bibinfo{volume}{115}}, \bibinfo{pages}{133601} (\bibinfo{year}{2015}).
\bibitem{forndiaz2010}\bibinfo{author}{Forn-D\'{i}az, P.}, \bibinfo{author}{Lisenfeld, J.}, \bibinfo{author}{Marcos, D.}, \bibinfo{author}{Garc\'{i}a-Ripoll, J. J.},
\bibinfo{author}{Solano, E.}, \bibinfo{author}{Harmans, C. J. P. M.} \& \bibinfo{author}{Mooij, J. E.}
\newblock \bibinfo{title}{Observation of the Bloch-Siegert Shift in a Qubit-Oscillator System in the Ultrastrong Coupling Regime}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}} \textbf{\bibinfo{volume}{105}}, \bibinfo{pages}{237001} (\bibinfo{year}{2010}).
\bibitem{niemczyk2010}\bibinfo{author}{Niemczyk, T.}, \bibinfo{author}{Deppe, F.}, \bibinfo{author}{Huebl, H.}, \bibinfo{author}{Menzel, E. P.},
\bibinfo{author}{Hocke, F.}, \bibinfo{author}{Schwarz, M. J.}, \bibinfo{author}{Garc\'{i}a-Ripoll, J. J.}, \bibinfo{author}{Zueco, D.}, \bibinfo{author}{H\"{u}mmer, T.}, \bibinfo{author}{Solano, E.},
\bibinfo{author}{Marx, A.} \& \bibinfo{author}{Gross, R.}
\newblock \bibinfo{title}{Circuit quantum electrodynamics in the ultrastrong-coupling regime}.
\newblock\emph{\bibinfo{journal}{Nat. Phys.}} \textbf{\bibinfo{volume}{6}}, \bibinfo{pages}{772} (\bibinfo{year}{2010}).
\bibitem{scalari2012}\bibinfo{author}{Scalari, G.}, \bibinfo{author}{Maissen, C.}, \bibinfo{author}{Turcinkova, D.}, \bibinfo{author}{Hagenm\"{u}ller, D.},
\bibinfo{author}{Liberato, S. D.}, \bibinfo{author}{Ciuti, C.}, \bibinfo{author}{Reichl, C.}, \bibinfo{author}{Schuh, D.}, \bibinfo{author}{Wegscheider, W.}, \bibinfo{author}{Beck, M.}
\& \bibinfo{author}{Faist, J.}
\newblock \bibinfo{title}{Ultrastrong Coupling of the Cyclotron Transition of a 2D Electron Gas to a THz Metamaterial}.
\newblock\emph{\bibinfo{journal}{Science}} \textbf{\bibinfo{volume}{335}}, \bibinfo{pages}{1323} (\bibinfo{year}{2012}).
\bibitem{anappara2009}\bibinfo{author}{Anappara, A. A.}, \bibinfo{author}{Liberato, S. D.}, \bibinfo{author}{Tredicucci, A.}, \bibinfo{author}{Ciuti, C.},
\bibinfo{author}{Biasiol, G.}, \bibinfo{author}{Sorba, L.} \& \bibinfo{author}{Beltram, F.}
\newblock \bibinfo{title}{Signatures of the ultrastrong light-matter coupling regime}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. B}} \textbf{\bibinfo{volume}{79}}, \bibinfo{pages}{201303(R)} (\bibinfo{year}{2009}).
\bibitem{gunter2009}\bibinfo{author}{G\"{u}nter, G.}, \bibinfo{author}{Anappara, A. A.}, \bibinfo{author}{Hees, J.}, \bibinfo{author}{Sell, A.}, \bibinfo{author}{Biasiol, G.},
\bibinfo{author}{Sorba, L.}, \bibinfo{author}{Liberato, S. D.}, \bibinfo{author}{Ciuti, C.}, \bibinfo{author}{Tredicucci, A.}, \bibinfo{author}{Leitenstorfer, A.}
\& \bibinfo{author}{Huber, R.}
\newblock \bibinfo{title}{Sub-cycle switch-on of ultrastrong light-matter interaction}.
\newblock\emph{\bibinfo{journal}{Nature}} \textbf{\bibinfo{volume}{458}}, \bibinfo{pages}{178} (\bibinfo{year}{2009}).
\bibitem{federov2010}\bibinfo{author}{Fedorov, A.}, \bibinfo{author}{Feofanov, A. K.}, \bibinfo{author}{Macha, P.}, \bibinfo{author}{Forn-D\'{i}az, P.},
\bibinfo{author}{Harmans, C. J. P. M.} \& \bibinfo{author}{Mooij, J. E.}
\newblock \bibinfo{title}{Strong Coupling of a Quantum Oscillator to a Flux Qubit at Its Symmetry Point}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}} \textbf{\bibinfo{volume}{105}}, \bibinfo{pages}{060503} (\bibinfo{year}{2010}).
\bibitem{muravev2011}\bibinfo{author}{Muravev, V. M.}, \bibinfo{author}{Andreev, I. V.}, \bibinfo{author}{Kukushkin, I. V.}, \bibinfo{author}{Schmult, S.}
\& \bibinfo{author}{Dietsche, W.}
\newblock \bibinfo{title}{Observation of hybrid plasmon-photon modes in microwave transmission of coplanar microresonators}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. B}} \textbf{\bibinfo{volume}{83}}, \bibinfo{pages}{075309} (\bibinfo{year}{2011}).
\bibitem{schwartz2011}\bibinfo{author}{Schwartz, T.}, \bibinfo{author}{Hutchison, J. A.}, \bibinfo{author}{Genet, C.} \& \bibinfo{author}{Ebbesen, T.W.}
\newblock \bibinfo{title}{Reversible Switching of Ultrastrong Light-Molecule Coupling}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}} \textbf{\bibinfo{volume}{106}}, \bibinfo{pages}{196405} (\bibinfo{year}{2011}).
\bibitem{geiser2012}\bibinfo{author}{Geiser, M.}, \bibinfo{author}{Castellano, F.}, \bibinfo{author}{Scalari, G.}, \bibinfo{author}{Beck, M.},
\bibinfo{author}{Nevou, L.} \& \bibinfo{author}{Faist, J.}
\newblock \bibinfo{title}{Ultrastrong Coupling Regime and Plasmon Polaritons in Parabolic Semiconductor Quantum Wells}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}} \textbf{\bibinfo{volume}{108}}, \bibinfo{pages}{106402} (\bibinfo{year}{2012}).
\bibitem{goryachev2014}\bibinfo{author}{Goryachev, M.}, \bibinfo{author}{Farr, W. G.}, \bibinfo{author}{Creedon, D. L.}, \bibinfo{author}{Fan, Y.},
\bibinfo{author}{Kostylev, M.} \& \bibinfo{author}{Tobar, M. E.}
\newblock \bibinfo{title}{High-Cooperativity Cavity QED with Magnons at Microwave Frequencies}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Applied}} \textbf{\bibinfo{volume}{2}}, \bibinfo{pages}{054002} (\bibinfo{year}{2014}).
\bibitem{zhang2016}\bibinfo{author}{Zhang, Q.}, \bibinfo{author}{Lou, M.}, \bibinfo{author}{Li, X.}, \bibinfo{author}{Reno, J. L.}, \bibinfo{author}{Pan, W.},
\bibinfo{author}{Watson, J. D.}, \bibinfo{author}{Manfra, M. J.} \& \bibinfo{author}{Kono, J.}
\newblock\bibinfo{title}{Collective, Coherent, and Ultrastrong Coupling of 2D Electrons with Terahertz Cavity Photons}.
\newblock\emph{\bibinfo{journal}{Nature Physics}} \textbf{\bibinfo{volume}{12}}, \bibinfo{pages}{1005} (\bibinfo{year}{2016}).
\bibitem{chen2016}\bibinfo{author}{Chen, Z.}, \bibinfo{author}{Wang, Y. M.}, \bibinfo{author}{Li, T. F.}, \bibinfo{author}{Tian, L.}, \bibinfo{author}{Qiu, Y. Y.},
\bibinfo{author}{Inomata, K.}, \bibinfo{author}{Yoshihara, F.}, \bibinfo{author}{Han, S. Y.}, \bibinfo{author}{Nori, F.}, \bibinfo{author}{Tsai, J. S.}
\& \bibinfo{author}{You, J. Q.}
\newblock \bibinfo{title}{Single-photon-driven high-order sideband transitions in an ultrastrongly coupled circuit-quantum-electrodynamics system}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{96}}, \bibinfo{pages}{012325} (\bibinfo{year}{2017}).
\bibitem{solano2003}\bibinfo{author}{Solano, E.}, \bibinfo{author}{Agarwal, G. S.} \& \bibinfo{author}{Walther, H.}
\newblock \bibinfo{title}{Strong-Driving-Assisted Multipartite Entanglement in Cavity QED}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}} \textbf{\bibinfo{volume}{90}}, \bibinfo{pages}{027903} (\bibinfo{year}{2003}).
\bibitem{semba2016}\bibinfo{author}{Yoshihara, F.}, \bibinfo{author}{Fuse, T.}, \bibinfo{author}{Ashhab, S.}, \bibinfo{author}{Kakuyanagi, K.},
\bibinfo{author}{Saito, S.} \& \bibinfo{author}{Semba, K.}
\newblock \bibinfo{title}{Superconducting qubit-oscillator circuit beyond the ultrastrong-coupling regime}.
\newblock\emph{\bibinfo{journal}{Nature Phys.}} \textbf{\bibinfo{volume}{13}}, \bibinfo{pages}{44} (\bibinfo{year}{2017}).
\bibitem{solano2011}\bibinfo{author}{Solano, E.}
\newblock \bibinfo{title}{The dialogue between quantum light and matter}.
\newblock\emph{\bibinfo{journal}{Physics}} \textbf{\bibinfo{volume}{4}}, \bibinfo{pages}{68} (\bibinfo{year}{2011}).
\bibitem{garz2016}\bibinfo{author}{Garziano, L.}, \bibinfo{author}{Macr\'i, V.}, \bibinfo{author}{Stassi, R.}, \bibinfo{author}{Stefano, O. D.},
\bibinfo{author}{Nori, F.} \& \bibinfo{author}{Savasta, S.}
\newblock \bibinfo{title}{One photon can simultaneously excite two or more atoms}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}} \textbf{\bibinfo{volume}{117}}, \bibinfo{pages}{043601} (\bibinfo{year}{2016}).
\bibitem{wangx2017}\bibinfo{author}{Wang, X.}, \bibinfo{author}{Miranowicz, A.}, \bibinfo{author}{Li, H.-R.} \& \bibinfo{author}{Nori, F.}
\newblock \bibinfo{title}{Observing pure effects of counter-rotating terms without ultrastrong coupling: A single photon can simultaneously excite two qubits}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{96}}, \bibinfo{pages}{063820} (\bibinfo{year}{2017}).
\bibitem{Ridolfo2012}\bibinfo{author}{Ridolfo, A.}, \bibinfo{author}{Leib, M.}, \bibinfo{author}{Savasta, S.} \& \bibinfo{author}{Hartmann, M. J.}
\newblock \bibinfo{title}{Photon Blockade in the Ultrastrong Coupling Regime}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}} \textbf{\bibinfo{volume}{109}}, \bibinfo{pages}{193602} (\bibinfo{year}{2012}).
\bibitem{Ridolfo2013}\bibinfo{author}{Ridolfo, A.}, \bibinfo{author}{Savasta, S.} \& \bibinfo{author}{Hartmann, M. J.}
\newblock\bibinfo{title}{Nonclassical Radiation from Thermal Cavities in the Ultrastrong Coupling Regime}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}} \textbf{\bibinfo{volume}{110}}, \bibinfo{pages}{163601} (\bibinfo{year}{2013}).
\bibitem{Law2013}\bibinfo{author}{Law, C. K.}
\newblock\bibinfo{title}{Vacuum Rabi oscillation induced by virtual photons in the ultrastrong-coupling regime}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{87}}, \bibinfo{pages}{045804} (\bibinfo{year}{2013}).
\bibitem{caox2010}\bibinfo{author}{Cao, X. F.}, \bibinfo{author}{You, J. Q.}, \bibinfo{author}{Zheng, H.}, \bibinfo{author}{Kofman, A. G.} \& \bibinfo{author}{Nori, F.}
\newblock\bibinfo{title}{Dynamics and quantum Zeno effect for a qubit in either a low- or high-frequency bath beyond the rotating-wave approximation}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{82}}, \bibinfo{pages}{022119} (\bibinfo{year}{2010}).
\bibitem{aiq2010}\bibinfo{author}{Ai, Q.}, \bibinfo{author}{Li, Y.}, \bibinfo{author}{Zheng, H.} \& \bibinfo{author}{Sun, C. P.}
\newblock\bibinfo{title}{Quantum anti-Zeno effect without rotating wave approximation}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{81}}, \bibinfo{pages}{042116} (\bibinfo{year}{2010}).
\bibitem{lipb2012}\bibinfo{author}{Li, P. B.}, \bibinfo{author}{Gao, S. Y.} \& \bibinfo{author}{Li, F. L.}
\newblock \bibinfo{title}{Engineering two-mode entangled states between two superconducting resonators by dissipation}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{86}}, \bibinfo{pages}{012318} (\bibinfo{year}{2012}).
\bibitem{wangx2014}\bibinfo{author}{Wang, X.}, \bibinfo{author}{Li, H. R.}, \bibinfo{author}{Li, P. B.}, \bibinfo{author}{Jiang, C. W.},
\bibinfo{author}{Gao, H.} \& \bibinfo{author}{Li, F. L.}
\newblock \bibinfo{title}{Preparing ground states and squeezed states of nanomechanical cantilevers by fast dissipation}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{90}}, \bibinfo{pages}{013838} (\bibinfo{year}{2014}).
\bibitem{reiter2013}\bibinfo{author}{Reiter, F.}, \bibinfo{author}{Tornberg, L.}, \bibinfo{author}{Johansson, G.} \& \bibinfo{author}{S{\o}rensen, A. S.}
\newblock \bibinfo{title}{Steady-state entanglement of two superconducting qubits engineered by dissipation}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{88}}, \bibinfo{pages}{032317} (\bibinfo{year}{2013}).
\bibitem{hes2014}\bibinfo{author}{He, S.}, \bibinfo{author}{Zhao, Y.} \& \bibinfo{author}{Chen, Q. H.}
\newblock \bibinfo{title}{Absence of collapse in quantum Rabi oscillations}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{90}}, \bibinfo{pages}{053848} (\bibinfo{year}{2014}).
\bibitem{rossatto2016}\bibinfo{author}{Rossatto, D. Z.}, \bibinfo{author}{Felicetti, S.}, \bibinfo{author}{Eneriz, H.}, \bibinfo{author}{Rico, E.},
\bibinfo{author}{Sanz, M.} \& \bibinfo{author}{Solano, E.}
\newblock \bibinfo{title}{Entangling polaritons via dynamical Casimir effect in circuit quantum electrodynamics}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. B}} \textbf{\bibinfo{volume}{93}}, \bibinfo{pages}{094514} (\bibinfo{year}{2016}).
\bibitem{felicettiprl2014}\bibinfo{author}{Felicetti, S.}, \bibinfo{author}{Sanz, M.}, \bibinfo{author}{Lamata, L.}, \bibinfo{author}{Romero, G.}, \bibinfo{author}{Johansson, G.},
\bibinfo{author}{Delsing, P.} \& \bibinfo{author}{Solano, E.}
\newblock \bibinfo{title}{Dynamical Casimir Effect Entangles Artificial Atoms}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}} \textbf{\bibinfo{volume}{113}}, \bibinfo{pages}{093602} (\bibinfo{year}{2014}).
\bibitem{kyawprb2015}\bibinfo{author}{Kyaw, T. H.}, \bibinfo{author}{Herrera-Mart\'i, D. A.}, \bibinfo{author}{Solano, E.},
\bibinfo{author}{Romero, G.} \& \bibinfo{author}{Kwek, L.-C.}
\newblock \bibinfo{title}{Creation of quantum error correcting codes in the ultrastrong coupling regime}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. B}} \textbf{\bibinfo{volume}{91}}, \bibinfo{pages}{064503} (\bibinfo{year}{2015}).
\bibitem{romero2012}\bibinfo{author}{Romero, G.}, \bibinfo{author}{Ballester, D.}, \bibinfo{author}{Wang, Y. M.}, \bibinfo{author}{Scarani, V.} \& \bibinfo{author}{Solano, E.}
\newblock\bibinfo{title}{Ultrafast Quantum Gates in Circuit QED}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}} \textbf{\bibinfo{volume}{108}}, \bibinfo{pages}{120501} (\bibinfo{year}{2012}).
\bibitem{wangym2017}\bibinfo{author}{Wang, Y. M.}, \bibinfo{author}{Guo, C.}, \bibinfo{author}{Zhang, G.-Q.}, \bibinfo{author}{Wang, G. C.} \& \bibinfo{author}{Wu, C. F.}
\newblock\bibinfo{title}{Ultrafast quantum computation in ultrastrongly coupled circuit QED systems}.
\newblock\emph{\bibinfo{journal}{Sci. Rep.}} \textbf{\bibinfo{volume}{7}}, \bibinfo{pages}{44251} (\bibinfo{year}{2017}).
\bibitem{felicetti2015}\bibinfo{author}{Felicetti, S.}, \bibinfo{author}{Pedernales, J. S.}, \bibinfo{author}{Egusquiza, I. L.}, \bibinfo{author}{Romero, G.}, \bibinfo{author}{Lamata, L.},
\bibinfo{author}{Braak, D.} \& \bibinfo{author}{Solano, E.}
\newblock \bibinfo{title}{Spectral collapse via two-phonon interactions in trapped ions}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{92}}, \bibinfo{pages}{033817} (\bibinfo{year}{2015}).
\bibitem{lv2016}\bibinfo{author}{Lv, D. S.}, \bibinfo{author}{An, S. M.}, \bibinfo{author}{Liu, Z. Y.}, \bibinfo{author}{Zhang, J.-N.}, \bibinfo{author}{Pedernales, J. S.},
\bibinfo{author}{Lamata, L.},\bibinfo{author}{Solano, E.} \& \bibinfo{author}{Kim, K.}
\newblock \bibinfo{title}{Quantum simulation of the quantum Rabi model in a trapped ion}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. X}} \textbf{\bibinfo{volume}{8}}, \bibinfo{pages}{021027} (\bibinfo{year}{2018}).
\bibitem{peder2015}\bibinfo{author}{Pedernales, J. S.}, \bibinfo{author}{Lizuain, I.}, \bibinfo{author}{Felicetti, S.}, \bibinfo{author}{Romero, G.},
\bibinfo{author}{Lamata, L.} \& \bibinfo{author}{Solano, E.}
\newblock \bibinfo{title}{Quantum Rabi Model with Trapped Ions}.
\newblock\emph{\bibinfo{journal}{Sci. Rep.}} \textbf{\bibinfo{volume}{5}}, \bibinfo{pages}{15472} (\bibinfo{year}{2015}).
\bibitem{chengxh2018}\bibinfo{author}{Cheng, X.-H.}, \bibinfo{author}{Arrazola, I.}, \bibinfo{author}{Pedernales, J. S.}, \bibinfo{author}{Lamata, L.},
\bibinfo{author}{Chen, X.} \& \bibinfo{author}{Solano, E.}
\newblock \bibinfo{title}{Nonlinear Quantum Rabi Model in Trapped Ions}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{97}}, \bibinfo{pages}{023624} (\bibinfo{year}{2018}).
\bibitem{clerk2018}\bibinfo{author}{Leroux, C.}, \bibinfo{author}{Govia, L. C. G.} \& \bibinfo{author}{Clerk, A. A.}
\newblock \bibinfo{title}{Enhancing Cavity Quantum Electrodynamics via Antisqueezing: Synthetic Ultrastrong Coupling}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}} \textbf{\bibinfo{volume}{120}}, \bibinfo{pages}{093602} (\bibinfo{year}{2018}).
\bibitem{braum2017}\bibinfo{author}{Braum\"{u}ller, J.}, \bibinfo{author}{Marthaler, M.}, \bibinfo{author}{Schneider, A.}, \bibinfo{author}{Stehli, A.},
\bibinfo{author}{Rotzinger, H.}, \bibinfo{author}{Weides, M.} \& \bibinfo{author}{Ustinov, A. V.}
\newblock \bibinfo{title}{Analog quantum simulation of the Rabi model in the ultra-strong coupling regime}.
\newblock\emph{\bibinfo{journal}{Nature Communications}} \textbf{\bibinfo{volume}{8}}, \bibinfo{pages}{779} (\bibinfo{year}{2017}).
\bibitem{lij2013}\bibinfo{author}{Li, J.}, \bibinfo{author}{Silveri, M. P.}, \bibinfo{author}{Kumar, K. S.}, \bibinfo{author}{Pirkkalainen, J.-M.}, \bibinfo{author}{Veps\"{a}l\"{a}inen, A.},
\bibinfo{author}{Chien, W. C.}, \bibinfo{author}{Tuorila, J.}, \bibinfo{author}{Sillanp\"{a}\"{a}, M. A.}, \bibinfo{author}{Hakonen, P. J.}, \bibinfo{author}{Thuneberg, E. V.}
\& \bibinfo{author}{Paraoanu, G. S.}
\newblock \bibinfo{title}{Motional Averaging in a Superconducting Qubit}.
\newblock\emph{\bibinfo{journal}{Nature Communications}} \textbf{\bibinfo{volume}{4}}, \bibinfo{pages}{1420} (\bibinfo{year}{2013}).
\bibitem{ballester2012}\bibinfo{author}{Ballester, D.}, \bibinfo{author}{Romero, G.}, \bibinfo{author}{Garc\'{a}-Ripoll, J. J.},
\bibinfo{author}{Deppe, F.} \& \bibinfo{author}{Solano, E.}
\newblock \bibinfo{title}{Quantum Simulation of the Ultrastrong-Coupling Dynamics in Circuit Quantum Electrodynamics}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. X}} \textbf{\bibinfo{volume}{2}}, \bibinfo{pages}{021007} (\bibinfo{year}{2012}).
\bibitem{mezzacapo2014}\bibinfo{author}{Mezzacapo, A.}, \bibinfo{author}{Las Heras, U.}, \bibinfo{author}{Pedernales, J. S.}, \bibinfo{author}{DiCarlo, L.},
\bibinfo{author}{Solano, E.} \& \bibinfo{author}{Lamata, L.}
\newblock \bibinfo{title}{Digital Quantum Rabi and Dicke Models in Superconducting Circuits}.
\newblock\emph{\bibinfo{journal}{Sci. Rep.}} \textbf{\bibinfo{volume}{4}}, \bibinfo{pages}{7482} (\bibinfo{year}{2014}).
\bibitem{langford2017}\bibinfo{author}{Langford, N. K.}, \bibinfo{author}{Sagastizabal, R.}, \bibinfo{author}{Kounalakis, M.},
\bibinfo{author}{Dickel, C.}, \bibinfo{author}{Bruno, A.}, \bibinfo{author}{Luthi, F.}, \bibinfo{author}{Thoen, D. J.}, \bibinfo{author}{Endo, A.}
\& \bibinfo{author}{DiCarlo, L.}
\newblock \bibinfo{title}{Experimentally simulating the dynamics of quantum light and matter at ultrastrong coupling}.
\newblock\emph{\bibinfo{journal}{Nature Communications}} \textbf{\bibinfo{volume}{8}}, \bibinfo{pages}{1715} (\bibinfo{year}{2017}).
\bibitem{crespi2012}\bibinfo{author}{Crespi, A.}, \bibinfo{author}{Longhi, S.} \& \bibinfo{author}{Osellame, R.}
\newblock \bibinfo{title}{Photonic Realization of the Quantum Rabi Model}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}} \textbf{\bibinfo{volume}{108}}, \bibinfo{pages}{163601} (\bibinfo{year}{2012}).
\bibitem{felicetti2017}\bibinfo{author}{Felicetti, S.}, \bibinfo{author}{Romero, G.}, \bibinfo{author}{Solano, E.} \& \bibinfo{author}{Sab\'{i}n, C.}
\newblock \bibinfo{title}{Quantum Rabi model in a superfluid Bose-Einstein condensate}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{96}}, \bibinfo{pages}{033839} (\bibinfo{year}{2017}).
\bibitem{tomka2014}\bibinfo{author}{Tomka, M.}, \bibinfo{author}{Araby, O. E.}, \bibinfo{author}{Pletyukhov, M.} \& \bibinfo{author}{Gritsev, V.}
\newblock \bibinfo{title}{Exceptional and regular spectra of a generalized Rabi model}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{90}}, \bibinfo{pages}{063839} (\bibinfo{year}{2014}).
\bibitem{shenlt2014}\bibinfo{author}{Shen, L. T.}, \bibinfo{author}{Yang, Z. B.}, \bibinfo{author}{Lu, M.}, \bibinfo{author}{Chen, R. X.} \& \bibinfo{author}{Wu, H. Z.}
\newblock \bibinfo{title}{Ground state of the asymmetric Rabi model in the ultrastrong coupling regime}.
\newblock\emph{\bibinfo{journal}{Appl. Phys. B}} \textbf{\bibinfo{volume}{117}}, \bibinfo{pages}{195} (\bibinfo{year}{2014}).
\bibitem{zhangg2015}\bibinfo{author}{Zhang, G. F.} \& \bibinfo{author}{Zhu, H. J.},
\newblock\bibinfo{title}{Analytical Solution for the Anisotropic Rabi Model: Effects of Counter-Rotating Terms}.
\newblock\emph{\bibinfo{journal}{Sci. Rep.}} \textbf{\bibinfo{volume}{5}}, \bibinfo{pages}{8756} (\bibinfo{year}{2015}).
\bibitem{zhangyy2017}\bibinfo{author}{Zhang, Y. Y.} \& \bibinfo{author}{Chen, X. Y.},
\newblock\bibinfo{title}{Analytical solutions by squeezing to the anisotropic Rabi model in the nonperturbative deep-strong-coupling regime}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{96}}, \bibinfo{pages}{063821} (\bibinfo{year}{2017}).
\bibitem{xie2014}\bibinfo{author}{Xie, Q. T.}, \bibinfo{author}{Cui, S.}, \bibinfo{author}{Cao, J. P.}, \bibinfo{author}{Amico, L.} \& \bibinfo{author}{Fan, H.}
\newblock \bibinfo{title}{Anisotropic Rabi model}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. X}} \textbf{\bibinfo{volume}{4}}, \bibinfo{pages}{021046} (\bibinfo{year}{2014}).
\bibitem{zhangyy2016}\bibinfo{author}{Zhang, Y. Y.}
\newblock \bibinfo{title}{Generalized squeezing rotating-wave approximation to the isotropic and anisotropic Rabi model in the ultrastrong-coupling regime}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{94}}, \bibinfo{pages}{063824} (\bibinfo{year}{2016}).
\bibitem{yuyx2013}\bibinfo{author}{Yu, Y.-X.}, \bibinfo{author}{Ye, J.} \& \bibinfo{author}{Liu, W.-M.},
\newblock\bibinfo{title}{Goldstone and Higgs modes of photons inside a cavity}.
\newblock\emph{\bibinfo{journal}{Sci. Rep.}} \textbf{\bibinfo{volume}{3}}, \bibinfo{pages}{3476} (\bibinfo{year}{2013}).
\bibitem{liu2017}\bibinfo{author}{Liu, M. X.}, \bibinfo{author}{Chesi, S.}, \bibinfo{author}{Ying, Z.-J.}, \bibinfo{author}{Chen, X. S.},
\bibinfo{author}{Luo, H.-G.} \& \bibinfo{author}{ Lin, H.-Q.}
\newblock \bibinfo{title}{Universal scaling and critical exponents of the anisotropic quantum Rabi model}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}} \textbf{\bibinfo{volume}{119}}, \bibinfo{pages}{220601} (\bibinfo{year}{2017}).
\bibitem{shenlt2017}\bibinfo{author}{Shen, L. T.}, \bibinfo{author}{Yang, Z. B.}, \bibinfo{author}{Wu, H. Z.} \& \bibinfo{author}{Zheng, S. B.}
\newblock \bibinfo{title}{Quantum phase transition and quench dynamics in the anisotropic Rabi model}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{95}}, \bibinfo{pages}{013819} (\bibinfo{year}{2017}).
\bibitem{joshi2016}\bibinfo{author}{Joshi, C.}, \bibinfo{author}{Larson, J.} \& \bibinfo{author}{Spiller, T. P.}
\newblock \bibinfo{title}{Quantum state engineering in hybrid open quantum systems}. \newblock\emph{\bibinfo{journal}{Phys. Rev. A}}
\textbf{\bibinfo{volume}{93}}, \bibinfo{pages}{043818} (\bibinfo{year}{2016}).
\bibitem{wangzh2017}\bibinfo{author}{Wang, Z. H.}, \bibinfo{author}{Zheng, Q.}, \bibinfo{author}{Wang, X. G.} \& \bibinfo{author}{Li, Y.}
\newblock\bibinfo{title}{The energy level crossing behavior and quantum Fisher information in a quantum well with spin-orbit coupling}.
\newblock\emph{\bibinfo{journal}{Sci. Rep.}} \textbf{\bibinfo{volume}{6}}, \bibinfo{pages}{22347} (\bibinfo{year}{2016}).
\bibitem{schie2003}\bibinfo{author}{Schiemann, J.}, \bibinfo{author}{Egues, J. C.} \& \bibinfo{author}{Loss, D.}
\newblock \bibinfo{title}{Variational study of the $\nu=1$ quantum Hall ferromagnet in the presence of spin-orbit interaction}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. B}} \textbf{\bibinfo{volume}{67}}, \bibinfo{pages}{085302} (\bibinfo{year}{2003}).
\bibitem{baksic2014}\bibinfo{author}{Baksic, A.} \& \bibinfo{author}{Ciuti, C.}
\newblock \bibinfo{title}{Controlling Discrete and Continuous Symmetries in Superradiant Phase Transitions with Circuit QED Systems}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}} \textbf{\bibinfo{volume}{112}}, \bibinfo{pages}{173601} (\bibinfo{year}{2014}).
\bibitem{yangwj2017}\bibinfo{author}{Yang, W. J.} \& \bibinfo{author}{Wang, X. B.},
\newblock\bibinfo{title}{Ultrastrong-coupling quantum-phase-transition phenomena in a few-qubit circuit QED system}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{95}}, \bibinfo{pages}{043823} (\bibinfo{year}{2017}).
\bibitem{wangym2018}\bibinfo{author}{Wang, Y. M.}, \bibinfo{author}{You, W. L.}, \bibinfo{author}{Liu, M. X.}, \bibinfo{author}{Dong, Y. L.}, \bibinfo{author}{Luo, H. G.},
\bibinfo{author}{Romero, G.} \& \bibinfo{author}{You, J. Q.}
\newblock \bibinfo{title}{Quantum criticality and state engineering in the simulated anisotropic quantum Rabi model}.
\newblock\emph{\bibinfo{journal}{New J. Phys.}} \textbf{\bibinfo{volume}{20}}, \bibinfo{pages}{053061} (\bibinfo{year}{2018}).
\bibitem{aedo2018}\bibinfo{author}{Aedo, I.} \& \bibinfo{author}{Lamata, L.}
\newblock \bibinfo{title}{Analog quantum simulation of generalized Dicke models in trapped ions}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{97}}, \bibinfo{pages}{042317} (\bibinfo{year}{2018}).
\bibitem{strand2013}\bibinfo{author}{Strand, J. D.}, \bibinfo{author}{Ware, M.}, \bibinfo{author}{Beaudoin, F.}, \bibinfo{author}{Ohki, T. A.}, \bibinfo{author}{Johnson, B. R.},
\bibinfo{author}{Blais, A.} \& \bibinfo{author}{Plourde, B. L. T.}
\newblock \bibinfo{title}{First-order sideband transitions with flux-driven asymmetric transmon qubits}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. B}} \textbf{\bibinfo{volume}{87}}, \bibinfo{pages}{220505(R)} (\bibinfo{year}{2013}).
\bibitem{carlos2014}\bibinfo{author}{Benlloch, C. N.}, \bibinfo{author}{Garc\'{i}a-Ripoll, J. J.} \& \bibinfo{author}{Porras, D.}
\newblock \bibinfo{title}{Inducing Nonclassical Lasing via Periodic Drivings in Circuit Quantum Electrodynamics}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}}
\textbf{\bibinfo{volume}{113}}, \bibinfo{pages}{193601} (\bibinfo{year}{2014}).
\bibitem{yanyy2015}\bibinfo{author}{Yan, Y. Y.}, \bibinfo{author}{L\"{u}, Z. G.} \& \bibinfo{author}{Zheng, H.}
\newblock \bibinfo{title}{Bloch-Siegert shift of the Rabi model}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}}
\textbf{\bibinfo{volume}{91}}, \bibinfo{pages}{053834} (\bibinfo{year}{2015}).
\bibitem{liao2016}\bibinfo{author}{Liao, J. Q.}, \bibinfo{author}{Huang, J. F.} \& \bibinfo{author}{Tian, L.}
\newblock \bibinfo{title}{Generation of macroscopic Schr\"{o}dinger-cat states in qubit-oscillator systems}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}}
\textbf{\bibinfo{volume}{93}}, \bibinfo{pages}{033853} (\bibinfo{year}{2016}).
\bibitem{huangjf2017}\bibinfo{author}{Huang, J. F.}, \bibinfo{author}{Liao, J. Q.}, \bibinfo{author}{Tian, L.} \& \bibinfo{author}{Kuang, L. M.}
\newblock \bibinfo{title}{Manipulating counter-rotating interactions in the quantum Rabi model via modulation of the transition frequency of the two-level system}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{96}}, \bibinfo{pages}{043849} (\bibinfo{year}{2017}).
\bibitem{silveri2017}\bibinfo{author}{Silveri, M. P.}, \bibinfo{author}{Tuorila, J. A.}, \bibinfo{author}{Thuneberg, E. V.} \& \bibinfo{author}{Paraoanu, G. S.}
\newblock \bibinfo{title}{Quantum systems under frequency modulation}.
\newblock\emph{\bibinfo{journal}{Rep. Prog. Phys.}} \textbf{\bibinfo{volume}{80}}, \bibinfo{pages}{056002} (\bibinfo{year}{2017}).
\bibitem{basak2018}\bibinfo{author}{Basak, S.}, \bibinfo{author}{Chougale, Y.} \& \bibinfo{author}{Nath, R.}
\newblock\bibinfo{title}{Periodically Driven Array of Single Rydberg Atoms}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Lett.}}
\textbf{\bibinfo{volume}{120}}, \bibinfo{pages}{123204} (\bibinfo{year}{2018}).
\bibitem{xuezy2015}\bibinfo{author}{Xue, Z. Y.}, \bibinfo{author}{Zhou, J.} \& \bibinfo{author}{Wang, Z. D.}
\newblock\bibinfo{title}{Universal holonomic quantum gates in decoherence-free subspace on superconducting circuits}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{92}}, \bibinfo{pages}{022320} (\bibinfo{year}{2015}).
\bibitem{xuezy2018_1}\bibinfo{author}{Chen, T.}, \bibinfo{author}{Zhang, J.} \& \bibinfo{author}{Xue, Z. Y.}
\newblock\bibinfo{title}{Nonadiabatic holonomic quantum computation on coupled transmons with ancillaries}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{98}}, \bibinfo{pages}{052314} (\bibinfo{year}{2018}).
\bibitem{xuezy2018_2}\bibinfo{author}{Chen, T.} \& \bibinfo{author}{Xue, Z. Y.}
\newblock\bibinfo{title}{Nonadiabatic Geometric Quantum Computation with Parametrically Tunable Coupling}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. Applied}} \textbf{\bibinfo{volume}{10}}, \bibinfo{pages}{054051} (\bibinfo{year}{2018}).
\bibitem{koch2007}\bibinfo{author}{Koch, J.}, \bibinfo{author}{Yu, T. M.}, \bibinfo{author}{Gambetta, J.}, \bibinfo{author}{Houck, A. A.},
\bibinfo{author}{Schuster, D. I.}, \bibinfo{author}{Majer, J.}, \bibinfo{author}{J, A.}, \bibinfo{author}{Devoret, M. H.}, \bibinfo{author}{Girvin, S. M.}
\& \bibinfo{author}{Schoelkopf, R. J.}
\newblock \bibinfo{title}{Charge-insensitive qubit design derived from the Cooper pair box}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{76}}, \bibinfo{pages}{042319} (\bibinfo{year}{2007}).
\bibitem{colton1998}\bibinfo{author}{Colton, D.} \& \bibinfo{author}{Kress, R.}
\newblock \emph{\bibinfo{title}{Inverse Acoustic and Electromagnetic Scattering Theory (Applied Mathematical Sciences).}}
(\bibinfo{publisher}{Springer, New York}, \bibinfo{year}{1998}).
\bibitem{krosch2006}\bibinfo{author}{Korsch, H. J.}, \bibinfo{author}{Klumpp, A.} \& \bibinfo{author}{Witthaut, D.}
\newblock \bibinfo{title}{On two-dimensional Bessel functions}.
\newblock\emph{\bibinfo{journal}{J. Phys. A: Math. Gen.}} \textbf{\bibinfo{volume}{39}}, \bibinfo{pages}{14947} (\bibinfo{year}{2006}).
\bibitem{goerz2017}\bibinfo{author}{Goerz, M. H.}, \bibinfo{author}{Motzoi, F.}, \bibinfo{author}{Whaley, K. B.} \& \bibinfo{author}{Koch, C. P.}
\newblock \bibinfo{title}{Charting the circuit QED design landscape using optimal control theory}.
\newblock\emph{\bibinfo{journal}{npj Quantum Information}} \textbf{\bibinfo{volume}{3}}, \bibinfo{pages}{37} (\bibinfo{year}{2017}).
\bibitem{hofh2009}\bibinfo{author}{Hofheinz, M.}, \bibinfo{author}{Wang, H.}, \bibinfo{author}{Ansmann, M.}, \bibinfo{author}{Bialczak, R. C.}, \bibinfo{author}{Lucero, E.},
\bibinfo{author}{Neeley, M.}, \bibinfo{author}{O'Connell, A. D.}, \bibinfo{author}{Sank, D.}, \bibinfo{author}{Wenner, J.}, \bibinfo{author}{Martinis, J. M.}
\& \bibinfo{author}{Cleland, A. N.}
\newblock \bibinfo{title}{Synthesizing arbitrary quantum states in a superconducting resonator}.
\newblock\emph{\bibinfo{journal}{Nature (London)}} \textbf{\bibinfo{volume}{459}}, \bibinfo{pages}{546} (\bibinfo{year}{2009}).
\bibitem{Garziano2015}\bibinfo{author}{Garziano, L.}, \bibinfo{author}{Stassi, R.}, \bibinfo{author}{Macr\`{\i}, V.}, \bibinfo{author}{Kockum, A. F.},
\bibinfo{author}{Savasta, S.} \& \bibinfo{author}{Nori, F.}
\newblock \bibinfo{title}{Multiphoton quantum Rabi oscillations in ultrastrong cavity QED}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{92}}, \bibinfo{pages}{063830} (\bibinfo{year}{2015}).
\bibitem{blanes2009}\bibinfo{author}{Blanes, S.}, \bibinfo{author}{Casas, F.}, \bibinfo{author}{Oteo, J. A.} \& \bibinfo{author}{Ros, J.}
\newblock\bibinfo{title}{The Magnus expansion and some of its applications}.
\newblock\emph{\bibinfo{journal}{Physics Reports}} \textbf{\bibinfo{volume}{470}}, \bibinfo{pages}{151} (\bibinfo{year}{2009}).
\bibitem{liu2005}\bibinfo{author}{Liu, Y.-X.}, \bibinfo{author}{Wei, L. F.} \& \bibinfo{author}{Nori, F.}
\newblock\bibinfo{title}{Preparation of macroscopic quantum superposition states of a cavity field via coupling to a superconducting charge qubit}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{71}}, \bibinfo{pages}{063820} (\bibinfo{year}{2005}).
\bibitem{liao2008}\bibinfo{author}{Liao, J. Q.} \& \bibinfo{author}{Kuang, L. M.}
\newblock\bibinfo{title}{Nanomechanical resonator coupling with a double quantum dot: quantum state engineering}.
\newblock\emph{\bibinfo{journal}{The European Physical Journal B}} \textbf{\bibinfo{volume}{63}}, \bibinfo{pages}{79} (\bibinfo{year}{2008}).
\bibitem{yin2013}\bibinfo{author}{Yin, Z.-Q.}, \bibinfo{author}{Li, T.}, \bibinfo{author}{Zhang, X.} \& \bibinfo{author}{Duan, L. M.}
\newblock\bibinfo{title}{Large quantum superpositions of a levitated nanodiamond through spin-optomechanical coupling}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{88}}, \bibinfo{pages}{033614} (\bibinfo{year}{2013}).
\bibitem{zanardi1}\bibinfo{author}{Zanardi, P.}, \bibinfo{author}{Zalka, C.} \& \bibinfo{author}{Faoro, L.}
\newblock \bibinfo{title}{On the entangling power of quantum evolutions}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{62}}, \bibinfo{pages}{030301(R)} (\bibinfo{year}{2000}).
\bibitem{zanardi2}\bibinfo{author}{Zanardi, P.}
\newblock \bibinfo{title}{Entanglement of quantum evolutions}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{63}}, \bibinfo{pages}{040304(R)} (\bibinfo{year}{2001}).
\bibitem{Makhlin2002}\bibinfo{author}{Makhlin, Y.}
\newblock \bibinfo{title}{Nonlocal Properties of Two-Qubit Gates and Mixed Statesand the Optimization of Quantum Computations}.
\newblock\emph{\bibinfo{journal}{Quantum Inf. Process.}} \textbf{\bibinfo{volume}{1}}, \bibinfo{pages}{243} (\bibinfo{year}{2002}).
\bibitem{Zhang2003}\bibinfo{author}{Zhang, J.}, \bibinfo{author}{Vala, J.}, \bibinfo{author}{Sastry, S.} \& \bibinfo{author}{Whaley, K. B.}
\newblock \bibinfo{title}{Geometric theory of nonlocal two-qubit operations}.
\newblock\emph{\bibinfo{journal}{Phys. Rev. A}} \textbf{\bibinfo{volume}{67}}, \bibinfo{pages}{042313} (\bibinfo{year}{2003}).
\end{thebibliography}
\section*{Author Contributions}
G.W., H.S., and K.X. initiated the idea. C.S. and R.X. developed the model and performed the calculations. G.W. provided numerical results. All authors developed the scheme and wrote the main manuscript text.
\section*{Additional Information}
\textbf{Competing Interests:} The authors declare no competing interests.
\end{document} |
\begin{document}
\title{The ordered Bell numbers as weighted sums of odd or even Stirling numbers of the second kind}
\begin{abstract}
For the Stirling numbers of the second kind $S(n,k)$ and the ordered Bell numbers $B(n)$, we prove the identity $\sum_{k=1}^{n/2} S(n,2k)(2k-1)! = B(n-1)$. An analogous identity holds for the sum over odd $k$'s.
\end{abstract}
\section{Introduction}
For integers $0 \leq k \leq n$, we denote by $S(n,k)$ the Stirling numbers of the second kind and by $B(n)$ the ordered Bell numbers. These sequences can be found in the On-Line Encyclopedia of Integer Sequences (OEIS) \cite{OEIS} as \seqnum{A008277} and \seqnum{A000670}. They are related by
\begin{align}
\label{eq:sum1}
B(n) = \sum_{k=0}^n k! S(n,k)
\text{.}
\end{align}
In this short note, we will give four different ways to express the ordered Bell numbers as a weighted sum of odd or even Stirling numbers of the second kind, where the mentioned parity refers to the second argument.
\begin{theorem}
\label{tm:4ways}
For $n \geq 1$, we have
\begin{align}
\label{eq:sumk0}
B(n)
&= (-1)^{n+1} + 2 \sum_{k=0}^{\floor{n/2}} (2k)! S(n,2k)
= (-1)^n + 2 \sum_{k=0}^{\floor{n/2}} (2k+1)! S(n,2k+1) \\
\label{eq:sumk1}
&=\sum_{k=1}^{\floor{(n+1)/2}} (2k-1)! S(n+1,2k)
=\sum_{k=0}^{\floor{(n+1)/2}} (2k)! S(n+1,2k+1)
\text{.}
\end{align}
\end{theorem}
We could not find the Identities \eqref{eq:sumk1} (respectively \eqref{eq:sum4} below) in the literature. The Identities \eqref{eq:sumk0} are direct consequences of known other equations, as explained below.
\section{Proof of the Theorem}
The formal (exponential) generating functions (gf's) of $B(n)$ and $S(k,n)$ (for fixed $k$) are given by
\begin{align}
\label{eq:gfB}
\sum_{n=0}^\infty \frac{x^n}{n!} B(n)
&= \frac{1}{2-e^x}
=: \mathcal{B}(x) \\
\label{eq:gfS}
\sum_{n=0}^\infty \frac{x^n}{n!} S(n,k)
&= \frac{(e^x-1)^k}{k!}
\text{,}
\end{align}
see Quaintance and Gould \cite{QGo16}. The following identities are known
\begin{align}
\label{eq:sum2}
\sum_{k=0}^n (-1)^k k! S(n,k) &= (-1)^n \\
\label{eq:sum3}
\sum_{k=1}^n (-1)^k (k-1)! S(n,k) &=
\begin{cases}
-1, & \text{if $n=1$;}\\
0, & \text{if $n\geq 2$, }
\end{cases}
\text{,}
\end{align}
see Boyadzhiev \cite[Appendix A]{Boy18}. Combining identities \eqref{eq:sum1} and \eqref{eq:sum2} gives
\begin{align*}
\sum_{k=0}^{\floor{n/2}} (2k)! S(n,2k)
&= \tfrac{1}{2} \left(B(n) +(-1)^n\right) \\
\sum_{k=0}^{\floor{n/2}} (2k+1)! S(n,2k+1)
&= \tfrac{1}{2} \left(B(n) -(-1)^n\right)
\text{.}
\end{align*}
This shows Equation \eqref{eq:sumk0} of Theoreom \ref{tm:4ways}.
We show below the following theorem.
\begin{theorem}
\label{tm:Hn}
Put $H(n):=\sum_{k=1}^n (k-1)! S(n,k)$ for $n \geq 1$. Then
\begin{align}
\label{eq:sum4}
H(n) &=
\begin{cases}
1, & \text{if $n=1$;}\\
2 B(n-1), & \text{if $n\geq 2$. }
\end{cases}
\end{align}
\end{theorem}
Combining identities \eqref{eq:sum3} and \eqref{eq:sum4} gives, for $n \geq 2$,
\begin{align}
\label{eq:sum5}
\sum_{k=1}^{\floor{n/2}} (2k-1)! S(n,2k)
= \sum_{k=0}^{\floor{n/2}} (2k)! S(n,2k+1)
= B(n-1)
\text{,}
\end{align}
and therefore \eqref{eq:sumk1} of Theoreom \ref{tm:4ways}.
It remains to prove \eqref{eq:sum4}.
\begin{proof}[Proof of Theorem \ref{tm:Hn}]
The gf $\mathcal{H}(x)$ of $H(n)$ is given by
\begin{align*}
\mathcal{H}(x)
&=\sum_{n=0}^\infty \frac{x^n}{n!} \sum_{k=1}^n (k-1)! S(n,k)
=\sum_{k=1}^\infty \frac{1}{k} (e^x-1)^k \\
&=-\log(2-e^x)
\text{,}
\end{align*}
where we used \eqref{eq:gfS} and the Taylor series
$\log(1-z)=-\sum_{k=1}^{\infty} \frac{1}{k} z^{k}$ for $\abs{z}<1$.
Further, defining $G(1)=2$ and $G(n)=H(n)$ for $n \geq 2$, we get
\begin{align*}
\mathcal{G}(x)
:= \sum_{n=0}^\infty \frac{x^n}{n!} G(n)
= x - \log(2-e^x)
\text{,}
\end{align*}
noticing that $H(1)=1$.
By elementary differentiation rules and \eqref{eq:gfB}, it follows that
$\mathcal{G}'(x)=\frac{2}{e^x-1}=2\mathcal{B}(x)$.
Hence, by Wilf \cite[Formula 2.3.1]{Wil90}, we get $H(n)=G(n) = 2 B(n-1)$ for $n \geq 2$.
\end{proof}
\begin{remark}
We put
$H_e(n):=\sum_{k=1}^{\floor{n/2}} (2k-1)! S(n,2k)$
and
$H_o(n):=\sum_{k=0}^{\floor{n/2}} (2k)! S(n,2k+1)$.
Using the same tools as in the proof above, we can also show that the corresponding gf's are given by
\begin{align*}
\sum_{n=0}^\infty \frac{x^n}{n!} H_e(n)
&=\artanh(e^x-1)
= \frac{1}{2} \log \left( \frac{e^x}{2-e^x} \right) \\
\sum_{n=0}^\infty \frac{x^n}{n!} H_o(n)
&=-\frac{1}{2} \log \left( e^x (2-e^x) \right)
\text{.}
\end{align*}
With these gf's, we can deduce \eqref{eq:sum5} directly, without using \eqref{eq:sum3}.
\end{remark}
\begin{remark}
The quantities $W(n,k):=k! S(n+1,k+1)$ are called the \textit{Worpitzky numbers}, see Vanderfelde \cite{Wil90} (OEIS \seqnum{A130850}). With this notation, \eqref{eq:sum5} can be written as ($n \geq 1$)
\begin{align*}
\sum_{k=1}^{\floor{n/2}} W(n,2k)
= \sum_{k=0}^{\floor{n/2}} W(n,2k+1)
= B(n)
\text{.}
\end{align*}
\end{remark}
\hrule
\noindent 2010 {\it Mathematics Subject Classification}: Primary 11A51;
Secondary 05A17.
\noindent \emph{Keywords:}
ordered Bell number, Stirling number of the second kind, Worpitzky number triangle.
\hrule
\end{document} |
\begin{document}
\title{A note on the high power Diophantine equations
}
\author{ Mehdi Baghalaghdam \and Farzali Izadi
}
\institute{Mehdi Baghalaghdam \at
Department of Mathematics \\ Faculty of Science \\ Azarbaijan Shahid Madani University \\Tabriz 53751-71379, Iran
\\
\email{mehdi.baghalaghdam@yahoo.com }
\\%\emph{Present address:} of F. Author
\and
Farzali Izadi \at
Department of Mathematics\\ Faculty of Science \\Urmia University \\Urmia 165-57153, Iran
\\
\email{f.izadi@urmia.ac.ir }
}
\date{Received: date / Accepted: date}
\maketitle
\begin{abstract}
In this paper, we solve the simultaneous Diophantine equations (SDE) $ x_{1}^\mu+ x_{2}^\mu +\cdots+ x_{n}^\mu=k \cdot (y_{1}^\mu+ y_{2}^\mu +\cdots+ y_{\frac{n}{k}}^\mu )$, $\mu=1,3$, where $ n \geq3$, and $k \neq n$, is a divisor of $n$ ($\frac{n}{k}\geq2$), and obtain nontrivial parametric solution for them. Furthermore we present a method for producing another solution for the above Diophantine equation (DE) for the case $\mu=3$, when a solution is given. We work out some examples and find nontrivial parametric solutions for each case in nonzero integers. \\ Also we prove that the other DE $\sum_{i=1}^n p_{i} \cdot x_{i}^{a_i}=\sum_{j=1}^m q_{j} \cdot y_{j}^{b_j}$, has parametric solution and infinitely many solutions in nonzero integers with the condition that: there is a $i$ such that $p_{i}=1$, and\\
$(a_{i},a_{1} \cdot a_{2} \cdots a_{i-1} \cdot a_{i+1} \cdots a_{n} \cdot b_{1} \cdot b_{2} \cdots b_{m})=1$, or there is a $j$ such that
$q_{j}=1$, and $(b_{j},a_{1} \cdots a_{n} \cdot b_{1} \cdots b_{j-1} \cdot b_{j+1} \cdots b_{m})=1$.
Finally we study the DE $x^a+y^b=z^c$.
\keywords{ Simultaneous Diophantine equations, Equal sums of the cubes, High power Diophantine equations.}
\subclass{Primary11D45, Secondary11D72, 11D25.}
\end{abstract}
\section{Introduction}\label{intro}
\noindent The cubic Diophantine equations has been studied by some mathematicians.\\ Gerardin gave partial solutions of the SDE
\begin{equation}\label{190}
x^k+y^k+z^k=u^k+v^k+w^k; k=1,3
\end{equation}
\noindent in 1915-16 (as quoted by Dickson, pp. 565, 713 of \cite{5})
and additional partial solutions were given by Bremner \cite{1}. Subsequently, complete solutions were given
in terms of cubic polynomials in four variables by Bremner and Brudno \cite{2},
as well as by Labarthe \cite{7}.
\noindent In \cite{4} Choudhry presented a complete four-parameter solution of \eqref{190}
in terms of quadratic polynomials in which each parameter occurs only in the first degree.\\
\noindent In this paper we are interested in the study of the SDE
\begin{equation}\label{301}
x_{1}^\mu+ x_{2}^\mu+ \cdots +x_{n}^\mu=k \cdot (y_{1}^\mu+ y_{2}^\mu + \cdots + y_{\frac{n}{k}}^\mu ), \mu=1,3,
\end{equation}
\noindent where $ n\geq3$, and $k \neq n$, is a divisor of $n$ ($\frac{n}{k}\geq2$).
\\
\noindent Also we study the other DE
\begin{equation}\label{80}
\sum_{i=1}^n p_{i}.x_{i}^{a_i}=\sum_{j=1}^m q_{j}.y_{j}^{b_j}
\end{equation}
where $m, n, a_{i}, b_{i} \in \mathbb{N}$ and $p_{i}, q_{i}\in \mathbb{Z}$.
\noindent This is a generalization of the Fermat's equation.\\
\noindent A common generalization of Fermat's equation is $Ax^a+By^b+Cz^c=0$, where $a$, $b$, $c$ $\in\mathbb{N}
_{\geq2}$ and $A, B, C\in\mathbb{Z}$ are given integers with $ABC\neq0$ and $x, y, z\in\mathbb{Z}$ are variables.\\
\noindent In $1995$, Darmond and Granville (See \cite{6}) proved:\\
If $A, B, C\in\mathbb{Z}$ , $ABC\neq0$ and $a, b, c \in\mathbb{N}_{\geq2}$ be such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}<1$, then the equation $Ax^a+By^b+Cz^c=0$ has only finitely many
solutions $x, y, z\in\mathbb{Z}$ with $(x,y,z)=1$.\\
\noindent The following theorem is due to Beukers (See \cite{3}):\\
Let $A, B, C\in\mathbb{Z}$ , $ABC\neq0$ and $a, b, c \in\mathbb{N}_{\geq2}$ such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}>1$. Then the equation $Ax^a+By^b+Cz^c=0$ has either zero or infinitely many solutions $x, y, z\in\mathbb{Z}$ with $(x,y,z)=1$.\\
\noindent To the best of our knowledge the SDE \eqref{301} and the DE \eqref{80} has not already been considered by any
other authors.\\
\noindent We prove the following main theorems:
\begin{theoremz}
Let $ n\geq3$, and $k \neq n$, be a divisor of $n$ ($\frac{n}{k}\geq2$).
Then the SDE \eqref{301}
have infinitely many nontrivial parametric solutions in nonzero integers.
\end{theoremz}
\begin{theoremz}
Let $m, n, a_{i}, b_{i} \in \mathbb{N}$ and $p_{i}, q_{i}\in \mathbb{Z}$. Suppose that there is a $i$ such that $p_{i}=1$, and
$(a_{i},a_{1} \cdot a_{2} \cdots a_{i-1} \cdot a_{i+1} \cdots a_{n} \cdot b_{1} \cdot b_{2} \cdots b_{m})=1$
or there is a $j$ such that
$q_{j}=1$, and $(b_{j},a_{1} \cdots a_{n} \cdot b_{1} \cdots b_{j-1} \cdot b_{j+1} \cdots b_{m})=1$.
Then the DE \eqref{80}
has parametric solution and infinitely many solutions in nonzero integers. This solves the DE $x^a+y^b=z^c$ in the special cases of $(a,b,c)=1$, or $(a,b,c)=2$.
\end{theoremz}
\section{The SDE ${\displaystyle \sum_{i=1}^n x_{i} ^\mu=k \cdot \sum_{j=1}^\frac{n}{k} y_{j}^\mu }$; $\mu=1,3$}
In this section, we proof the first main theorem.
\begin{proof}:
\noindent Firstly, it is clear that if\\
\noindent $X=(x_1, \cdots ,x_{n},y_1, \cdots ,y_{\frac{n}{k}})$, and
\noindent $ Y=(X_1, \cdots ,X_{n},Y_1, \cdots ,Y_{\frac{n}{k}}), $\\
\noindent be two solutions for the SDE \eqref{301}, then for any arbitrary rational number $t$, $X+tY$ is also a solution for $\mu=1$, i.e.,\\
\noindent $(x_1+tX_1)+(x_2+tX_2)+ \cdots +(x_{n}+tX_{n})=\\
k \cdot [(y_1+tY_1)+(y_2+tY_2)+ \cdots +(y_{\frac{n}{k}}+tY_{\frac{n}{k}})]$.\\
\noindent We say that $X$ is a trivial parametric solution of the SDE \eqref{301}, if it is nonzero and satisfies the system trivially.
Let $X$, and $Y$ be two proper trivial parametric solutions of the SDE \eqref{301}, (we will introduce them later).
\noindent We suppose that $ X+tY$, is also a solution for the case $\mu=3$, where $t$ is a parameter, we wish to find $t$.
\noindent By plugging\\
\noindent $X+tY=(x_1+tX_1,x_2+tX_2, \cdots ,x_{n}+tX_{n},y_1+tY_1, y_2+tY_2, \cdots ,y_{\frac{n}{k}}+tY_{\frac{n}{k}})$,\\
\noindent into the SDE \eqref{301}, we get:\\
\noindent $(x_1+tX_1)^3+(x_2+tX_2)^3+ \cdots +(x_{n}+tX_{n})^3=\\
k \cdot [(y_1+tY_1)^3+(y_2+tY_2)^3+ \cdots +(y_{\frac{n}{k}}+tY_{\frac{n}{k}})^3]$.
\\
\noindent Since $X$ and $Y$ are solutions for the SDE \eqref{301}, after some simplifications, we obtain:\\
\noindent $3t^2(x_1X_1^2+x_2X_2^2+ \cdots +x_{n}X_{n}^2-ky_1Y_1^2-ky_2Y_2^2- \cdots -ky_{\frac{n}{k}}Y_{\frac{n}{k}}^2)+$\\
\noindent $3t(x_1^2X_1+x_2^2X_2+ \cdots +x_{n}^2X_{n}-ky_1^2Y_1-ky_2^2Y_2- \cdots -ky_{\frac{n}{k}}^2Y_{\frac{n}{k}})=0.$\\
\noindent Therefore $t=0$ or\\
$$t=\frac{x_1^2X_1+ \cdots +x_{n}^2X_{n}-ky_1^2Y_1- \cdots -ky_{\frac{n}{k}}^2Y_{\frac{n}{k}}}{-x_1X_1^2- \cdots -x_{n}X_{n}^2+ky_1Y_1^2+ \cdots +ky_{\frac{n}{k}}Y_{\frac{n}{k}}^2}:=\frac{A}{B}.$$\\
\noindent By substituting $t$ in the above expressions, and clearing the denominator $B^3$, we get an integer solution for the SDE \eqref{301} as follows:\\
\noindent $(x'_1,x'_2, \cdots ,x'_{n},y'_1,y'_2, \cdots ,y'_{\frac{n}{k}})=$\\
\noindent $(x_1B+AX_1,x_2B+AX_2, \cdots ,x_{n}B+AX_{n},y_1B+AY_1,y_2B+AY_2, \cdots ,y_{\frac{n}{k}}B+AY_{\frac{n}{k}})$.\\
\noindent If we pick up trivial parametric solutions $X$ and $Y$ properly, we will get a nontrivial parametric solution for the SDE \eqref{301}.
\noindent We should mention that not every trivial parametric solutions of $X$ and $Y$ necessarily give rise to a nontrivial parametric solution for the SDE \eqref{301}. So the trivial parametric solutions must be chosen properly.\\
\noindent Now we introduce the proper trivial parametric solutions.\\
\noindent For the sake of simplicity, we only write down the trivial parametric solutions for the left hand side of the SDE \eqref{301}. It is clear that the trivial parametric solutions for the right hand side of the SDE \eqref{301} can be similarly found by the given trivial parametric solutions of the left hand side of the SDE \eqref{301}.
\noindent Let $p_i$, $q_i$, $s_i$, $r_i$$\in\mathbb{Z}$.\\
\noindent A) The case $n\geq3$ and $\frac{n}{k}\geq3$:\\
\noindent There are $4$ different possible cases for $n$:\\
\noindent $1$. $n=2\alpha+1$, $\alpha$ is even.\\
\noindent $X_{left}=(x_1,x_2, \cdots ,x_{n})=
(p_1,-p_1,p_2,-p_2, \cdots ,p_\alpha,-p_\alpha, 0)$, and\\
\noindent $Y_{left}=(X_1,X_2, \cdots ,X_{n}) =$
\\
\noindent $(r_1,r_2,-r_1,-r_2,r_3,r_4,-r_3,-r_4, \cdots ,r_{\alpha-1},r_\alpha,-r_{\alpha-1},0,-r_\alpha).$
\\
\noindent $2$. $n=2\alpha+1$, $\alpha$ is odd.\\
\noindent $X_{left}=(p_1,-p_1,p_2,-p_2, \cdots ,p_\alpha,-p_\alpha,0)$, and\\
\noindent $Y_{left}=(r_1,r_2,-r_1,-r_2, \cdots ,- r_{\alpha-2},-r_{\alpha-1},r_\alpha,0,-r_\alpha)$.\\
\noindent $3$. $n=2\alpha$, $\alpha$ is even.\\
\noindent $X_{left}=(p_1,-p_1,p_2,-p_2, \cdots ,p_\alpha,-p_\alpha)$, and\\
\noindent $Y_{left}=(r_1,r_2,-r_1,-r_2, \cdots , r_{\alpha-1},r_\alpha,-r_{\alpha-1},-r_\alpha).$\\
\noindent $4$. $n=2\alpha$, $\alpha$ is odd.\\
\noindent $X_{left}=(p_1,-p_1,p_2,-p_2, \cdots ,p_\alpha,-p_\alpha)$, and\\
\noindent $Y_{left}=(r_1,r_2,-r_1,-r_2, \cdots , r_{\alpha-2},r_{\alpha-1},-r_{\alpha-2},r_\alpha,-r_{\alpha-1},-r_\alpha).$\\
\noindent B) The case $n\geq3$ and $\frac{n}{k}=2$:\\
\noindent $5$. $n=2k\geq3$ and $\frac{n}{k}=2$.\\
\noindent $X=(p_1,-p_1,p_2,-p_2, \cdots ,p_{k},-p_{k},q_1
,-q_1)$,\\
\noindent $Y=(r_1,r_2,r_1,r_2, \cdots ,r_1,r_2)$.\\
\noindent (In the final case we present the trivial parametric solutions X, and Y, completely.)\\
\noindent Finally, it can be easily shown that for every $i \neq j$, we have:\\
\noindent $x_iB+AX_i$ $\neq\pm$ $(x_jB+AX_j)$, $y_iB+AY_i$ $\neq\pm$ $(y_jB+AY_j)$ and\\ $x_iB+AX_i$$\neq\pm$ $(y_jB+AY_j)$, \\
\noindent i.e., it is really a nontrivial parametric solution in terms of the parameters $p_i$, $q_i$, $r_i$ and $s_i$.\\
\noindent It is clear that by fixing all of the parameters $p_i$, $q_i$, $r_i$ and $s_i$, but one parameter, we can obtain a nontrivial one parameter parametric solution for the SDE \eqref{301} and by changing properly the fixed values, finally get infinitely many nontrivial one parameter parametric solutions for the SDE \eqref{301}.
\noindent Now, the proof of the first main theorem is completed. $\spadesuit$\\
\end{proof}
\noindent Now, we work out some examples.
\begin{exam} $a^\mu+b^\mu+c^\mu+d^\mu+e^\mu+f^\mu=3 \cdot (g^\mu+h^\mu)$; $\mu=1,3$.\\
\noindent Trivial parametric solutions (case $5$):\\
\noindent $X=(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8)=(p_1,-p_1,p_2,-p_2,p_3,-p_3,q_1,-q_1),$\\
\noindent $Y=(y_1,y_2,y_3,y_4,y_5,y_6,y_7,y_8)=(r_1,r_2,r_1,r_2,r_1,r_2,r_1,r_2)$,\\
\noindent $A=x_1^2y_1+x_2^2y_2+x_3^2y_3+x_4^2y_4+x_5^2y_5+x_6^2y_6-3x_7^2y_7-3x_8^2y_8$,\\
\noindent $B=-x_1y_1^2-x_2y_2^2-x_3y_3^2-x_4y_4^2-x_5y_5^2-x_6y_6^2+3x_7y_7^2+3x_8y_8^2$,\\
\noindent nontrivial parametric solution:\\
\noindent $a=x_1B+Ay_1=p_1B+r_1A$,\\
\noindent $b=x_2B+Ay_2=-p_1B+r_2A$,\\
\noindent $c=x_3B+Ay_3=p_2B+r_1A$,\\
\noindent $d=x_4B+Ay_4=-p_2B+r_2A$,\\
\noindent $e=x_5B+Ay_5=p_3B+r_1A$,\\
\noindent $f=x_6B+Ay_6=-p_3B+r_2A$,\\
\noindent $g=x_7B+Ay_7=q_1B+r_1A$,\\
\noindent $h=x_8B+Ay_8=-q_1B+r_2A$.\\
\noindent Example $1$:\\
\noindent $p_1=2$, $p_2=3$, $p_3=4$, $q_1=5$, $r_1=1$, $r_2=6$,\\
\noindent Solution:\\
\noindent $371^\mu+756^\mu+476^\mu+651^\mu+581^\mu+546^\mu=3 \cdot (686^\mu+441^\mu)$; $\mu=1,3$..\\
\noindent Example $2$:\\
\noindent $p_1=7$, $p_2=9$, $p_3=5$, $q_1=2$, $r_1=4$, $r_2=1$,\\
\noindent Solution:\\
\noindent $257^\mu+458^\mu+167^\mu+548^\mu+347^\mu+368^\mu=3 \cdot (482^\mu+233^\mu)$; $\mu=1,3$.
\end{exam}
\begin{exam}
$a^\mu+b^\mu+c^\mu+d^\mu+e^\mu+f^\mu+g^\mu+h^\mu=2 \cdot (i^\mu+j^\mu+k^\mu+l^\mu)$; $\mu=1,3$.\\
\noindent Trivial parametric solutions (case $3$):\\
\noindent $X=(x_1,x_2,\cdots,x_{12})=$
\noindent $(p_1,-p_1,p_2,-p_2,p_3,-p_3,p_4,-p_4,q_1,-q_1,q_2,-q_2)$,\\
\noindent $Y=(y_1,y_2,\cdots,y_{12})=$
\noindent $(r_1,r_2,-r_1,-r_2,r_3,r_4,-r_3,-r_4,s_1,s_2,-s_1,-s_2)$,
\\
\noindent Example:\\
$p_1=2$, $p_2=5$, $p_3=7$, $p_4=6$, $q_1=5$, $q_2=3$, $r_1=7$, $r_2=8$,
$r_3=9$, $r_4=9$, $s_1=10$, $s_2=6$, $A=-593$, $B=1129$,\\
\noindent Solution:\\
\noindent $(-1893)^\mu+(-7002)^\mu+9796^\mu+(-901)^\mu+2566^\mu+(-13240)^\mu+12111^\mu+(-1437)^\mu=
2 \cdot [(-285)^\mu+(-9203)^\mu+9317^\mu+171^\mu]$; $\mu=1,3$
\end{exam}
\begin{exam} $a^\mu+b^\mu+c^\mu+ \cdots +m^\mu+n^\mu+o^\mu=5 \cdot (p^\mu+q^\mu+r^\mu)$; $\mu=1,3$.\\
\noindent Trivial parametric solutions (case $2$):\\
\noindent $X=(x_1,x_2, \cdots,
x_{18})=$
\noindent $(p_1,-p_1,p_2,-p_2,p_3,-p_3,p_4,-p_4,p_5,-p_5,p_6,-p_6,p_7,-p_7,0,q_1,-q_1,0)$,\\
\noindent $Y=(y_1,y_2, \cdots,y_{18})=$\\
\noindent $(r_1,r_2,-r_1,-r_2,r_3,r_4,-r_3,-r_4,r_5,r_6,-r_5,-r_6,r_7,0,-r_7,s_1,0,-s_1)$,\\
\noindent Example:\\
$p_1=1$, $p_2=3$, $p_3=5$, $p_4=4$, $p_5=1$, $p_6=2$, $p_7=3$, $q_1=2$, $r_1=2$, $r_2=3$, $r_3=5$, $r_4=1$, $r_5=6$, $r_6=6$,
, $r_7=7$, $s_1=3$,
$A=-19$, $B=-253$,\\
\noindent Solution:\\
\noindent $(-291)^\mu+196^\mu+(-721)^\mu+816^\mu+(-1360)^\mu+1246^\mu+(-917)^\mu+1031^\mu
+(-367)^\mu+139^\mu+(-392)^\mu+
620^\mu+(-892)^\mu+759^\mu+133^\mu=
5 \cdot [(-563)^\mu+506^\mu+57^\mu]$; $\mu=1,3$
\end{exam}
\begin{exam} $a^\mu+b^\mu+c^\mu+d^\mu=2 \cdot (e^\mu+f^\mu)$; $\mu=1,3$.\\
\noindent Trivial parametric solutions(case $5$):\\
\noindent $X=(u,u,v,v,u,v)$,\\
\noindent $Y=(-v,1,-1,v,-u,u)$,\\
\noindent nontrivial parametric solution:\\
\noindent $a=-u^2-uv+uv^3+2u^4-vu^2+v^3-v^4$,\\
\noindent $b=-u^2v^2-uv-uv^3+2u^4+2u^3v-u^2v-v^2+v^3+2u^3-2uv^2$,\\
\noindent $c=-v^4+2u^2v^2-v^3-2u^3-u^2-uv-uv^3+2u^3v+u^2v+2uv^2$,\\
\noindent $d=u^2v^2-3uv^3+4u^3v-uv-v^2+vu^2-v^3$,\\
\noindent $e=u^2v^2-u^2-u^3-uv-2uv^3+3u^3v+uv^2$,\\
\noindent $f=-uv-v^2-v^4+u^3v+u^3-uv^2+2u^4$.\\
\noindent Example:\\
\noindent $u=2$, $ v=1$, or $u=10$, $v=7$, or $u=100$, $v=11$, or $u=100$, $v=1$,\\
\noindent Solutions:\\
\noindent $1^\mu+4^\mu+5^\mu+8^\mu=2 \cdot (2^\mu+7^\mu)$; $\mu=1,3$.\\
\noindent $201^\mu+257^\mu+168^\mu+224^\mu=2 \cdot (180^\mu+245^\mu)$; $\mu=1,3$.\\
\noindent $900895^\mu+1002355^\mu+100874^\mu+202334^\mu=2 \cdot (148400^\mu+954829^\mu)$; $\mu=1,3$.\\
\noindent $1^\mu+201^\mu+10000^\mu+10200^\mu=2 \cdot (100^\mu+10101^\mu)$; $\mu=1,3$.\\
\end{exam}
\begin{exam} $a^\mu+b^\mu+c^\mu+d^\mu+e^\mu+f^\mu=2 \cdot (g^\mu+h^\mu+i^\mu)$; $\mu=1,3$.\\
\noindent Trivial parametric solutions (case: $4$, $1$):\\
\noindent $X=(1,v,u,v,-u,-1,v,u,-u)$\\
\noindent $Y=(v,v,u,1,-1,-u,-u,v,u)$,\\
\noindent nontrivial parametric solution:\\
\noindent $a=-3u^3+v^4-v+u+u^2+vu^2-uv+2v^2u-u^3v+2v^3u-2v^2u^2$,\\
\noindent $b= -4u^3v+4v^3u$,\\
\noindent $c=-4u^3+4v^2u^2$,\\
\noindent $d=-v^4+v-u^3-u^2-u+uv-3u^3v+2u^2v^2+2v^3u-u^2v+2v^2u$,\\
\noindent $e=3u^4-v-v^3-v^2+u+uv-2v^2u^2+v^2u+v^3u-2vu^3-2v^2u+2u^2v$,\\
\noindent $f=4u^3+u^4+v-u+v^2+v^3-3v^2u-2vu^2-uv-uv^3-2u^2v^2+2u^3v$,\\
\noindent $g=u^4+u^3+u^2-v^3-v^4-v^2-u^3v+v^3u+u^2v-uv^2$,\\
\noindent $h=-3u^4+u^2+u^3+v^2+v^3+v^4-2uv+u^3v+v^3u-v^2u-u^2v$,\\
\noindent $i=2u^4-2u^3-2u^2+2uv+2v^3u+2v^2u-4vu^3$.\\
\noindent Example:\\
\noindent $u=2$, $v=1$, or $u=2$, $v=10$, or $u=7$, $v=100$,\\
\noindent Solutions:\\
\noindent $(-8)^\mu+(-8)^\mu+(-16)^\mu+(-8)^\mu+11^\mu+13^\mu=2 \cdot [7^\mu+(-11)^\mu+(-4)^\mu]$; $\mu=1,3$.
\\
\noindent $563^\mu+320^\mu+64^\mu+(-211)^\mu+(-5)^\mu+(-91)^\mu=2 \cdot [(-388)^\mu+536^\mu+172^\mu];$ $\mu=1,3$.
\\
\noindent $173777^\mu+42800^\mu+2996^\mu+(-130549)^\mu+7510^\mu+(-10934)^\mu=2 \cdot [(-144557)^\mu+165839^\mu+21518^\mu]$; $\mu=1,3$.
\end{exam}
\begin{exam} $a^\mu+b^\mu+c^\mu=d^\mu+e^\mu+f^\mu$; $\mu=1,3$.\\
\noindent Trivial parametric solutions (case $2$):\\
\noindent $X=(u,v,1,u,v,1)$,\\
\noindent $Y=(v,-1,-u,-1,-u,v)$,\\
\noindent nontrivial parametric solution:\\
\noindent $a=-u^3-v^3+vu^3-v^3u-2uv +v^2u-vu^2+u^2-v^2$,\\
\noindent $b=-uv^3-vu^2+uv+v^2u^2+v^3-u^2v+u+u^2+v^2u+v$,\\
\noindent $c=-uv^2-v+u+vu^2+v^2-u^3v+v^2u+u^3+v^2u^2+uv$,\\
\noindent $d=-u^2v^2-uv-u^3+u^2+vu^3+2uv^2-u^2v+v^2+u+u^2+v$,\\
\noindent $e=-uv^3-v^2-vu^2+2uv+2u^2v^2+v^3-u^3v+uv^2+u^2+u^3$,\\
\noindent $f=-uv^2-v-u^2+u+u^2v^2-v^3-uv-v^3u$.\\
\noindent Example:\\
\noindent $u=10$, $v=3$,\\
\noindent Solution:\\
\noindent $762^\mu+145^\mu+251^\mu=195^\mu+377^\mu+586^\mu$; $\mu=1,3$.\\
\end{exam}
\begin{exam} $a^\mu+b^\mu+c^\mu+d^\mu+e^\mu+f^\mu+g^\mu+h^\mu+i^\mu=3 \cdot (j^\mu+k^\mu+l^\mu)$; $\mu=1,3$.\\
\noindent Trivial parametric solutions (case: $1$, $2$):\\
\noindent $X=(x_1,x_2,\cdots,x_{12})=$
\noindent $(u,v,-u,t,-s,-t,0,s,-v,t,0,-t)$,\\
\noindent $Y=(y_1,y_2,\cdots,y_{12})=$
\noindent $(v,t,-s,u,-t,s,-v,-u,0,0,t,-t)$,\\
\noindent Example $1$:\\
$u=1$, $v=2$, $s=3$, $t=10$,\\
\noindent Solution:\\
\noindent $1168^\mu+7346^\mu+(-2003)^\mu+(-4185)^\mu+(-6844)^\mu+7525^\mu+(-1670)^\mu+(-2341)^\mu+
1004^\mu=
3 \cdot [(-5020)^\mu+8350^\mu+(-3330)^\mu]$; $\mu=1,3$.
\\
\noindent Example $2$:\\
\noindent $u=1$, $v=10$, $s=20$, $ t=30$,\\
\noindent Solution:\\
\noindent $83515^\mu+208720^\mu+(-173005)^\mu+(-170301)^\mu+
(-148970)^\mu+358230^\mu+(-89490)^\mu+(-128449)^\mu+59750^\mu=3 \cdot [(-179250)^\mu+268470^\mu+(-89220)^\mu]$; $\mu=1,3$.
\end{exam}
\section{Producing another solution of the DE ${\displaystyle \sum_{i=1}^n x_{i} ^3=k \cdot \sum_{j=1}^\frac{n}{k} y_{j}^3 }$, when a solution is given.}
Let $X=(x_1,x_2, \cdots ,x_n ,y_1,y_2, \cdots ,y_\frac{n}{k})$ be a primitive solution for the DE
\begin{equation}\label{97}
\sum_{i=1}^n x_{i} ^3=k \cdot \sum_{j=1}^\frac{n}{k} y_{j}^3,
\end{equation}
\noindent under conditions that :
\noindent $x_1+x_2+ \cdots +x_n -ky_1-ky_2- \cdots -ky_\frac{n}{k}\neq 0$, and\\
\noindent $-2x_1^{2}-2x_2^{2}- \cdots -2x_n^{2} +2ky_1^{2}+2ky_2^{2}+ \cdots +2ky_\frac{n}{k}^{2}\neq 0$.\\
\noindent (These conditions will be necessary because we wish the other solution not to be trivial.)
\\
\noindent We try to find another solution by using the first solution.\\
\noindent Define two variables function:
\noindent $F(x,y):=(2x_1x+y)^3+(2x_2x+y)^3+ \cdots +(2x_nx+y)^3$
\noindent $-k(2y_1x+y)^3-k(2y_2x+y)^3- \cdots -k(2y_\frac{n}{k}x+y)^3$.
\\
\noindent (Also, we may define: $F(x,y):=(tx_1x+y)^3+(tx_2x+y)^3+ \cdots +(tx_nx+y)^3$
\noindent $-k(ty_1x+y)^3-k(ty_2x+y)^3- \cdots -k(ty_\frac{n}{k}x+y)^3$, where $t$ is an arbitrary integer.)
\\
\noindent It is clear that if $F(x,y)=0$, then\\
\noindent $(2x_1x+y,2x_2x+y, \cdots ,2x_nx+y,2y_1x+y,2y_2x+y, \cdots ,2y_\frac{n}{k}x+y)$,\\
\noindent is another solution for the DE \eqref{97}.
\noindent We have
\noindent $F(x,y)=6xy[(2x_1^2+2x_2^2+ \cdots +2x_n^2-2ky_1^2- \cdots -2ky_\frac{n}{k}^2)x+
\\
(x_1+x_2+ \cdots +x_n-ky_1-ky_2- \cdots -ky_\frac{n}{k})y].$
\\
\noindent Therefore if we define\\
\noindent $x:=x_1+x_2+ \cdots +x_n-ky_1-ky_2- \cdots -ky_\frac{n}{k}\neq0$,
\noindent and
\noindent $y:=-2x_1^2-2x_2^2- \cdots -2x_n^2+2ky_1^2+ \cdots +2ky_\frac{n}{k}^2\neq0$,\\
\noindent then we get $F(x,y)=0$, which gives rise to another solution for the DE \eqref{97}:
\\
\noindent $x'_1=2x_1x+y=2x_1(x_1+x_2+ \cdots +x_n-ky_1-ky_2- \cdots -ky_\frac{n}{k})+$
\noindent $(-2x_1^2-2x_2^2- \cdots -2x_n^2+2ky_1^2+ \cdots +2ky_\frac{n}{k}^2),$
\\
\noindent $x'_2=2x_2x+y=2x_2(x_1+x_2+ \cdots +x_n-ky_1-ky_2- \cdots -ky_\frac{n}{k})+$
\noindent $(-2x_1^2-2x_2^2- \cdots -2x_n^2+2ky_1^2+ \cdots +2ky_\frac{n}{k}^2),$
\vdots
\noindent $x'_n=2x_nx+y=2x_2(x_1+x_2+ \cdots +x_n-ky_1-ky_2- \cdots -ky_\frac{n}{k})+$
\noindent $(-2x_1^2-2x_2^2- \cdots -2x_n^2+2ky_1^2+ \cdots +2ky_\frac{n}{k}^2),$
\\
\noindent $y'_1=2y_1x+y=2y_1(x_1+x_2+ \cdots +x_n-ky_1-ky_2- \cdots -ky_\frac{n}{k})+$
\noindent $(-2x_1^2-2x_2^2- \cdots -2x_n^2+2ky_1^2+ \cdots +2ky_\frac{n}{k}^2),$
\\
\noindent $y'_2=2y_2x+y=2y_2(x_1+x_2+ \cdots +x_n-ky_1-ky_2- \cdots -ky_\frac{n}{k})+$
\noindent$(-2x_1^2-2x_2^2- \cdots -2x_n^2+2ky_1^2+ \cdots +2ky_\frac{n}{k}^2),$
\\
\vdots
\noindent $y'_\frac{n}{k}=2y_\frac{n}{k}x+y=2y_2(x_1+x_2+ \cdots +x_n-ky_1-ky_2- \cdots -ky_\frac{n}{k})+$
\noindent $(-2x_1^2-2x_2^2- \cdots -2x_n^2+2ky_1^2+ \cdots +2ky_\frac{n}{k}^2)$.
\\
\noindent By continuing this method for the new obtained solution, we get infinitely many solutions for the DE \eqref{97}. This means that we may get infinitely many nontrivial solutions for the DE \eqref{97} by using a given solution.\\
If the primitive solution $X$ be a parametric solution, we get another parametric solution for the DE \eqref{97}.
\begin{exam} $a^3+b^3+c^3=A^3+B^3+C^3$\\
\noindent Let $(a,b,c, A, B,C)$ be a solution for the above DE.
\noindent We find another solution by using the first solution.\\
\noindent Example $1$: $(a,b,c,A,B,C)=(1,155,209,-41,227,107)$,\\
\noindent Another solution:\\
\noindent $125^3+169^3+250^3=62^3+277^3+97^3$.
\\
\noindent Example $2$: $(a,b,c,A,B,C)=(-62,169,250,-125,277,97),$\\
\noindent Another solution:\\
\noindent $81^3+12555^3+16929^3=(-3321)^3+18387^3+8667^3.$\\
\end{exam}
\section{The DE ${\displaystyle \sum_{i=1}^n p_{i} \cdot x_{i}^{a_i}=\sum_{j=1}^m q_{j} \cdot y_{j}^{b_j}}$;}
In this section, we prove the second main theorem. Let the above DE be in the form:\\
\noindent $ x_{1}^{a_1}+ p_{2}x_{2}^ {a_2}+ p_{3}x_{3}^{a_3}+ \cdots + p_{n}x_{n}^{a_n}=q_{1}y_{1}^{b_1}+ \cdots +q_{m}y_{m}^{b_m}$,\\
\noindent where $ (a_{1}, a_{2} \cdots a_{n} \cdot b_{1} \cdots b_{m})=1 $.\\
\noindent Then we have:
\begin{equation}\label{7}
x_{1}^ {a_1}=q_{1}y_{1}^{b_1}+ \cdots +q_{m}y_{m}^{b_m}- p_{2}x_{2}^{a_2}- p_{3}x_{3}^{a_3}- \cdots - p_{n}x_{n}^{a_n}.
\end{equation}
\noindent Define:\\ $m:=q_{1}s_{1}^{b_1}+ \cdots +q_{m}s_{m}^{b_m}- p_{2}t_{2}^{a_2}- p_{3}t_{3}^{a_3}- \cdots - p_{n}t_{n}^{a_n}$,\\
\noindent where $t_{i}$ and $ s_{i} $ are arbitrary integers. Now we introduce the our parametric solution:
\\
\noindent $y_{1}=s_{1} \cdot m^\frac{k}{b_{1}}$,
\\
$y_{2}=s_{2} \cdot m^\frac{k}{b_{2}}$,
\\
$\vdots$
\\
$y_{m}=s_{m} \cdot m^\frac{k}{b_{m}}$,
\\
$ x_{1}=m^\frac{k+1}{a_{1}}$,
\\
$ x_{2}=t_{1} \cdot m^\frac{k}{a_{2}}$,
\\
$\vdots$
\\
$ x_{n}=t_{n} \cdot m^\frac{k}{a_{n}}$,
\\
\noindent where $k$ is a natural number such that
\\
\noindent $k\equiv 0 \pmod {b_{1}}$,
\\
$k\equiv 0 \pmod {b_{2}}$,
\\
$\vdots$
\\
$k\equiv 0 \pmod {b_{m}}$,
\\
$k\equiv 0 \pmod {a_{2}}$,
\\
$\vdots$
\\
$k\equiv 0 \pmod {a_{n}}$,
\\
$k\equiv -1 \pmod {a_{1}}$.
\\
\noindent From the Chinese remainder theorem, we know that there exists a solution for $k$, since $(a_{1},a_{2} \cdots a_{n} \cdot b_{1} \cdots b_{m})=1$, that is, the all exponents used in $x_{i}$ and $y_{j}$
are natural numbers.
We claim that this is a parametric solution for the DE \eqref{7}:
\\
\noindent $q_{1}y_{1}^{b_1}+ \cdots +q_{m}y_{m}^{b_m}- p_{2}x_{2}^{a_2}- p_{3}x_{3}^{a_3}- \cdots - p_{n}x_{n}^{a_n}=$\\
\noindent $q_{1} \cdot (s_{1}m^\frac{k}{{b_1}})^{b_1}+q_{2} \cdot (s_{2}m^\frac{k}{b_{2}})^{b_2}+ \cdots +q_{m} \cdot (s_{m}m^\frac{k}{b_{m}})^{b_m} -p_{2}(t_{2} \cdot m^\frac{k}{a_{2}})^{a_2}-$\\
\noindent $ \cdots -p_{n}(t_{n} \cdot m^\frac{k}{a_{n}})^{a_n}=$\\
\noindent $q_{1} \cdot s_{1}^{b_1} \cdot m^k+q_{2} \cdot s_{2}^{b_2} \cdot m^k+ \cdots +q_{m} \cdot s_{m}^{b_m} \cdot m^k-p_{2} \cdot t_{2}^{a_2} \cdot m^k- \cdots -p_{n} \cdot t_{n}^{a_n} \cdot m^k=$\\
\noindent $m^k(q_{1} \cdot s_{1}^{b_1}+
q_{2} \cdot s_{2}^{b_2}+ \cdots +q_{m} \cdot s_{m}^{b_m}-p_{2} \cdot t_{2}^{a_2}- \cdots -p_{n} \cdot t_{n}^{a_n})=m^k \cdot m=$
\\
\noindent $m^{k+1}=(m^\frac{k+1}{a})^a=x_{1}^a.$
\\
\noindent Now the proof of the second main theorem is completed. Since $s_{i}$, and $ t_{i}$ were arbitrary, we also obtain infinitely many solutions for the above DE.
\begin{exam}
The DE $x_{1}^5+x_{2}^6=y_{1}^7+y_{2}^8+y_{3}^9$, has infinitely many solutions in integers.
\\
Since we have: $x_{1}^5=y_{1}^7+y_{2}^8+y_{3}^9-x_{2}^6 $ and $(5,6 \cdot 7 \cdot 8 \cdot 9)=1$, then by using the previous theorem, we conclude that the aforementioned DE has infinitely many solutions in integers.
Put: $m:=r^7+s^8+t^9-w^6$, where $r$, $s$, $t$ and $w$ are arbitrary integers. As an example, if we let: $(r,s,t,w)=(3,2,1,3)$ and $k=504$, then we get:
\\
$m=1715$,
\\
$y_{1}=r \cdot m^\frac{k}{7}=3 \cdot 1715^{72}$,
\\
$y_{2}=s \cdot m^\frac{k}{8}=2 \cdot 1715^{63}$,
\\
$y_{3}=t \cdot m^\frac{k}{9}=1715^{56}$,
\\
$x_{1}=m^\frac{k+1}{5}=1715^{101}$,
\\
$x_{2}=w \cdot m^\frac{k}{6}=3 \cdot 1715^{84}$.
\\
\noindent Namely, we have:\\
$(1715^{101})^5+(3 \cdot 1715^{84})^6=(3 \cdot 1715^{72})^7+(2 \cdot 1715^{63})^8+(1715^{56})^9.$
\\
\noindent By letting $(r,s,t,w)=(3,2,2,2)$ and $k=504$, we get:\\
$(2891^{101})^5+(2 \cdot 2891^{84})^6=(3 \cdot 2891^{72})^7+(2 \cdot 2891^{63})^8+(2 \cdot 2891^{56})^9.$\\
\noindent By changing $(r,s,t,w)$, we obtain infinitely many solutions.
\end{exam}
\begin{exam}
The DE $x^5=6(y_{1}^7+y_{2}^7+y_{3}^7)$ has infinitely many solutions in integers. If we put: $m:=6r^7+6s^7+6t^7$, where $(r,s,t)=$ $(1,1,1)$ and $k=14$, we get:
\\
$m=18$,
\\
$y_{1}=r \cdot m^\frac{k}{7}=18^2$,
\\
$y_{2}=s \cdot m^\frac{k}{7}=18^2$,
\\
$y_{3}=t \cdot m^\frac{k}{7}=18^2$,
\\
$x=m^\frac{k+1}{5}=18^3$.
\\
\noindent Namely, we have: $(18^3)^5=6((18^2)^7+(18^2)^7+(18^2)^7)$.
\\
\noindent By letting: $(r,s,t)=(3,2,5)$ and $k=14$, we obtain:\\
$(482640^3)^5=6((3 \cdot 482640^2)^7+(2 \cdot 482640^2)^7+(5 \cdot 482640^2)^7)$.
\end{exam}
\begin{exam}
The DE $x^7=13y_{1}^{11}+ 11y_{2}^{13}$
has infinitely many solutions in integers. If we put: $m:=13r^{11}+11s^{13}$, where $(r,s,t)=(1,1,1)$ and $k=286$, we get:
\\
$m=24$,
\\
$y_{1}=r \cdot m^\frac{k}{11}=24^{26}$,
\\
$y_{2}=s \cdot m^\frac{k}{13}=24^{22}$,
\\
$x=m^\frac{k+1}{7}=24^{41}$.
\\
\noindent Namely, we have: $(24^{41})^7=13 \cdot (24^{26})^{11}+11 \cdot (24^{22})^{13}$.
\end{exam}
\begin{exam}
Is the DE $x_{1}^{5}+x_{2}^{5}=y_{1}^{6}+y_{2}^{6}$
solvable? Yes. To do this, let $ y_{1}=y_{2}$. Then we have: $x_{1}^{5}+x_{2}^{5}=2y_{1}^{6}$. We may let $x_{1}=2u$, $x_{2}=2v$. So we get: $y_{1}^{6}=2^4(u^5+v^5)$, that is solvable.
\end{exam}
\section{The DE $x^a+y^b=z^c$}
In this part, we solve the above DE, where $a$, $b$, $c$ are fixed natural numbers and $x$, $y$, $z$ are variables. We have three cases: $(a,b,c)=1$ , $(a,b,c)\geq3$ or $(a,b,c)=2$.
If $(a,b,c)=1$, by using the second main theorem, we proved that the DE has infinitely many solutions in integers.
If $(a,b,c)\geq3$, the Fermat last theorem says that the DE dose not have any solutions in integers. Then it suffices to study the case $(a,b,c)=2$. However, we know that the Diophantine equations $x^4+y^4=z^2$ and $x^4-y^4=z^2$ have not any solutions in integers. So it suffices only to study the case where at most one of $a$, $b$, $c$ is divisible by $4$. In the sequel, we study this case by proving several theorems.
\begin{theorem}
The DE $x^2+y^{2B} =z^{2C}$, where $B$ and $C$ are both odd and $(B,C)=1$, has infinitely many solutions in integers.
\end{theorem}
\begin{proof}
: we have $x^2+(y^{B})^2 =(z^{C})^2$, then we get:\\
$x=m^2-n^2$,
\\
$ y^{B}=2mn$,
\\
$z^C=m^2+n^2$.
\\
We may suppose that $m=2^{B-1} \cdot t_{1}^{B}$, $ n=t_{2}^{B}$. By plugging these into the equations, we get :
\\
$y=2t_{1}t_{2}$,
\\
$x=2^{2(B-1)} \cdot t_{1}^{2B}-t_{2}^{2B}$,
\\
$z^{C}=2^{2(B-1)} \cdot t_{1}^{2B}+t_{2}^{2B}$.
\\
\noindent Since $(C,2B)=1$, the DE $z^{C}=2^{2(B-1)} \cdot t_{1}^{2B}+t_{2}^{2B}$ is solvable for $z$, $t_{1}$, $t_{2}$ from the second main theorem. Also $x$ and $y$ are computed from
$y=2t_{1}t_{2}$,
$x=2^{2(B-1)} \cdot t_{1}^{2B}-t_{2}^{2B}$, as well.
The proof is completed.
\end{proof}
\begin{theorem}
The DE $x^{2A}+y^{2B} =z^{2C}$, where $A$, $B$, $C$ are all odd and $(A,C)=1$ and $(AC,B)=1$ has infinitely many solutions in integers.
\end{theorem}
We note that in this case $(a,b,c)=(2A,2B,2C)=2$ and none of $a=2A$, $b=2B$ and $c=2C$ is divisible by $4$. We wish to solve $x^a+y^b=z^c$ in the case of $(a,b,c)=2$ and at most one of $a$, $b$, $c$ is divisible by $4$, this theorem is one of the desired cases where $(a,b,c)=2$ and none of $a$, $b$ and $c$ is divisable by $4$.
\begin{proof}
: We know that the DE $X^A+Y^B=Z^C$ with the condition $(AC,B)=1$ is solvable and its solution is:
\\
$m:=r^C-s^A$,
\\
$Z=m^{\frac{k}{C}} \cdot r$,
\\
$X=m^{\frac{k}{A}} \cdot s$,
\\
$Y=m^{\frac{k+1}{B}}$,
\\
and
\\
$k\equiv0 \pmod {AC}$,
\\
$k\equiv-1 \pmod{ B}$,
\\
where $r$ and $s$ are arbitrary integers.
Now, if we obtain a solution for the DE $X^A+Y^B=Z^C$ from the solution just introduced, where $X$, $Y$ and $Z$ are squares, then we get a solution for the DE $x^{2A}+y^{2B} =z^{2C}$ by putting $X=x^2$, $Y=y^2$ and $Z=z^2$.
From the above solution for $X$, $Y$ and $Z$ , we see that if $m$, $r$ and $s$ be squares, then $X$, $Y$ and $Z$ will be squares, as well.
Then since $m$, $r$ and $s$ are related together with $m:= r^C-s^A$, we see that it suffices to solve the DE $M^2=R^{2C}-S^{2A}$, where we set:
\\
$m=M^2$,
\\
$r=R^2$,
\\
$s=S^2$.
\\
(we wanted $m$, $r$ and $s$ to be squares).
Fortunately the DE $M^2=R^{2C}-S^{2A}$ is solvable from the previous theorem due to $(A,C)=1$. Then we obtain $m$, $r$, $s$, and in the end we get $x$, $y$, $z$, and the proof is completed.
\end{proof}
\begin{theorem}
The DE $x^{4A}+y^{2B} =z^{2C}$, where $B$ and $C$ are both odd and $(A,C)=1$ and $(AC,B)=1$ has infinitely many solutions in integers.
\end{theorem}
\begin{proof}
: We know that the DE $X^{2A}+Y^B=Z^C$ with the condition $(2AC,B)=1$ is solvable and its solution is:
\\
$m:=r^C-s^{2A}$,
\\
$Z=m^{\frac{k}{C}} \cdot r$,
\\
$X=m^{\frac{k}{2A}} \cdot s$,
\\
$Y=m^{\frac{k+1}{B}}$,
\\
and
\\
$k\equiv0 \pmod {C}$,
\\
$k\equiv0 \pmod {2A}$,
\\
$k\equiv-1 \pmod {B}$,
\\
where $r$ and $s$ are arbitrary integers.
Now, if we obtain a solution for the DE $X^{2A}+Y^B=Z^C$ from the solution just introduced, where $X$, $Y$ and $Z$ are squares, next we get a solution for the DE $x^{4A}+y^{2B} =z^{2C}$ by putting $x=X^2$, $y=Y^2$ and $z=Z^2 $.
From the above solution for $X$, $Y$, and $Z$, we see that if $m$, $r$ and $s$ be squares, then $X$, $Y$, and $Z$ will be squares, as well.
Then since m, r and s are related together with $m:=r^C-s^{2A}$, we see that it suffices to solve the DE $M^2=R^{2C}-S^{4A}$, where we set:
\\
$m=M^2$,
\\
$r=R^2$,
\\
$s=S^2$.
\\
(We wanted $m$, $r$ and $s$ to be squares).
Fortunately the DE $M^2=R^{2C}-S^{4A}$ is solvable from the previous theorems:
we have $M^2+(S^{2A})^2 =(R^{C})^2$, and then we get:
\\
$M=t_{1}^2-t_{2}^2$,
\\
$ S^{2A}=2t_{1}t_{2}$,
\\
$R^C=t_{1}^2+t_{2}^2$.
\\
We may suppose that $t_{1}=2^{2A-1} \cdot p^{2A}$, $ t_{2}=q^{2A}$. By substituting these in the above equations, we get :
\\
$S=2pq$,
\\
$M=2^{2(2A-1)} \cdot p^{4A}-q^{4A}$,
\\
$R^{C}=2^{2(2A-1)} \cdot p^{4A}+q^{4A}$.
\\
Since $(C,4A)=1$, the DE $R^{C}=2^{2(2A-1)} \cdot p^{4A}+q^{4A}$ is solvable for $R$, $p$ and $q$, from the second main theorem. And $M$, and $S$ are computed from
$S=2pq$,
$M=2^{2(2A-1)} \cdot p^{4A}-q^{4A}$, as well,
the proof is completed.
\end{proof}
We note that the previous theorem is the case that in the DE $x^a+y^b=z^c$, exactly one of $a$ or $b$ is divisible by $4$, and $(a,b,c)=2$.
\begin{exam}
We wish to solve the DE $x^{6}+y^{10}=z^{14}$.
We know that the DE $x^{3}+y^{5}=z^{7}$ is solvable and its solution is:
\\
$m:=r^7-s^3$,
\\
$z=m^{\frac{k}{7}} \cdot r$,
\\
$x=m^{\frac{k}{3}} \cdot s$,
\\
$y=m^{\frac{k+1}{5}}$,
\\
and
\\
$k\equiv0 \pmod {21}$,
\\
$k\equiv-1 \pmod {5}$,
\\
where $r$, and $s$ are arbitrary integers.\\
\noindent Now, if we obtain a solution for the DE $x^3+y^5=z^7$ from the solution just introduced, where $x$, $y$ and $z$ are squares, then we get a solution for the DE $x^6+y^{10} =z^{14}$.
From the above solution for $x$, $y$ and $z$, we see that if $m$, $r$ and $s$ be squares, then $x$, $y$ and $z$ will be squares, as well.
Since $m:=r^7-s^3$, we see that it suffices to solve the DE $M^2=R^{14}-S^{6}$, where:
\\
$m=M^2$,
\\
$r=R^2$,
\\
$s=S^2$.
Fortunately the DE $M^2=R^{14}-S^6$ is solvable from the previous theorems:
we have $M^2+S^6 =R^{14}$, and then we get:
\\
$M=t_{1}^2-t_{2}^2$,
\\
$ S^3=2t_{1}t_{2}$,
\\
$R^7=t_{1}^2+t_{2}^2$.
\\
We may suppose that $t_{1}=4p^3$,$ t_{2}=q^3$. By substituting these in the above equations, we get :
\\
$S=2pq$,
\\
$M=16p^6-q^6$,
\\
$R^7=16p^6+q^6$.
\\
Since $(7,6)=1$, the DE $R^7=16p^6+q^6$ is solvable for $R$, $p$ and $q$ from the previous theorems:
\\
$M':=16u^6+v^6$,
\\
$p=M'^{\frac{k}{6} } \cdot u$,
\\
$q=M'^{\frac{k}{6}} \cdot v$,
\\
$R=M'^{\frac{k+1}{7}}$,
\\
and
\\
$K\equiv0 \pmod {6}$ ,
\\
$k\equiv-1\pmod {7}$.
\\
\noindent By putting $u=v=1$, $k=6$, we get:\\
\noindent $(15^{28} \cdot 17^{170} \cdot 2)^{6}+(15^{17} \cdot 17^{102})^{10}=(15^{12} \cdot 17^{73})^{14}$.
\end{exam}
\begin{theorem}
The DE $x^2+y^2 =z^{2^n \cdot c}$, where $c$ is odd and $n\geq2$, is solvable.
\end {theorem}
\begin{proof}
: We try to solve the DE by using induction. If $n=2$, we have\\
$x^2+y^2=z^{4c}$,
then
\\
$x=m^2-n^2$,
\\
$y=2mn$,
\\
$z^{2c}=m^2+n^2$.
\\
Again we put:
\\
$n=2t_{1}t_{2}$,
\\
$m=2t_{1}^2-t_{2}^2$,
\\
$z^c=t_{1}^2+t_{2}^2$.
\\
Since $(2,c)=1$, the DE $z^c=t_{1}^2+t_{2}^2$ is solvable, then the main DE is solvable as well, and the proof for $n=2$ is complete.
Next, suppose the above DE is solvable for $n$, then we solve it for $n+1$. We have:
\\
$x^2+y^2 =z^{2^{n+1} \cdot c}$,
then
\\
$x=m^2-n^2$,
\\
$y=2mn$,
\\
$z^{2^{n} \cdot c}=m^2+n^2$.
\\
Since the DE $z^{2^{n} \cdot c}=m^2+n^2$, is solvable for $m$, $n$, and $z$ by our assumption, then the DE $x^2+y^2 =z^{2^{n+1} \cdot c}$, is solvable as well, and the proof is completed.
\end{proof}
\begin{rem}
If in the previous theorem, we take $c=1$, we can solve it by another beautiful method. We start from the identity
\\
$(m^2-n^2)^2+(2mn)^2=(m^2+n^2)^2$.\\
\noindent We see that: if in the above identity $m^2+n^2$, to be square, then we get a solution for the DE $x^2+y^2=z^4$. As an example, we can get $m$ and $n$ from the Pythagorean triples. For $m=4$, $n=3$, we obtain:\\
$7^2+24^2=(5^2)^{2}=(3^2+4^2)^{2}=5^4$.\\
\noindent By letting $n=7$, $m=24$, we obtain:
$572^2+336^2=(7^2+24^2)^{2}=(5^4)^{2}=5^8$.\\
\noindent By continuing
in this way, if we put $m=572$, $n=336$, we get a solution for the DE $x^2+y^2=z^{16}$.
If we change $m$, $n$, we obtain another identity. If $m=120$, $n=119$, we have :
$239^2+28560^2=13^8$.
It is clear that by using this method we can find infinitely many solutions for the DE
$x^2+y^2 =z^{2^n \cdot c}$ with $(x,y,z)=1$, as mentioned in the Beukers theorem (see \cite{3}) in the introduction.
\end{rem}
In the end, it is clear that at some variables parametric solutions obtained for the Diophantine equations, we may get one variable parametric solutions for each case of the above Diophantine equations by fixing the other variables.
\begin{center}\textbf{Acknowledgements}
\end{center}
The authors would like to express their hearty thanks to the
anonymous referee for a careful reading of the paper and for many
careful comments and remarks which improved its quality.
\end{document} |
\begin{document}
\title{Anti-Forging Quantum Data: Cryptographic Verification of Quantum Computational Power}
\vb{a}uthor{Man-Hong Yung}
\vb{e}mail{yung@sustech.edu.cn}
\vb{a}ffiliation{Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China.}
\vb{a}ffiliation{Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China.}
\vb{a}ffiliation{Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China.}
\vb{a}ffiliation{Shenzhen Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen, 518055, China.}
\vb{a}uthor{Bin Cheng}
\vb{e}mail{chengb@mail.sustech.edu.cn}
\vb{a}ffiliation{Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China.}
\vb{a}ffiliation{Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China.}
\vb{a}ffiliation{Centre for Quantum Software and Information, Faculty of Engineering and Information Technology, University of Technology Sydney, NSW 2007, Australia.}
\begin{abstract}
Quantum cloud computing is emerging as a popular model for users to experience the power of quantum computing through the internet, enabling quantum computing as a service. The question is, when the scale of the computational problems becomes out of reach of classical computers, how can users be sure that the output strings sent by the server are really from a quantum hardware?
In 2008, Shepherd and Bremner proposed a cryptographic verification protocol based on a simplified circuit model called IQP (instantaneous quantum polynomial-time), which can potentially be applied to most existing quantum cloud platforms. However, the Shepherd-Bremner protocol has recently been shown to be insecure by Kahanamoku-Meyer.
Here we present an extended model of IQP-based cryptographic verification protocol, where the Shepherd-Bremner construction can be regarded as a special case. This protocol not only can avoid the attack by Kahanamoku-Meyer but also provide several additional security measures for anti-forging quantum data. In particular, our protocol admits a simultaneous encoding of multiple secret strings, strengthening significantly the hardness for classical hacking. Furthermore, we provide methods for estimating the correlation functions associated with the secret strings, which are the key elements in our verification protocol.
\vb{e}nd{abstract}
\vb{m}aketitle
\vb{s}ection{Introduction}
Despite the fact that near-term quantum computers would be noisy and of intermediate scale~\vb{c}ite{Preskill2018}, they have the potential in performing specific tasks intractable for any classical computer, a status knows as quantum supremacy~\vb{c}ite{Preskill2012,Lund2017,Terhal2018,YungNSR,arute_quantum_2019}. A natural question arises: how could we tell if a remote device is truly quantum or not? This question is not only of fundamental interest in nature~\vb{c}ite{aharonov_is_2012}, but also relevant to many computing protocols involving two parties, namely a verifier and a prover.
This question is called verification of quantum computational power, or test of quantumness, and it is a simplified version of the verification of \vb{e}mph{any} quantum computation.
In the setting of the general problem, a quantum prover interacts with a classical verifier, in order to convince the verifier that the results are correct for the given problem in \textsf{BQP};
here, \textsf{BQP} is the class of decision problems that a quantum computer can efficiently solve.
Verification protocols for the general problem can be achieved if certain relaxation is allowed.
For example, one may assume that the verifier can use limited quantum computational power and quantum communication~\vb{c}ite{Broadbent2008,Broadbent2010-blind-original,Aharonov2008,Aharonov2017, Fitzsimons2017-verifiable-blind, Fitzsimons2018, Mills2018-it-secure}, or one may allow multiple spatially separated entangled provers ~\vb{c}ite{Reichardt2013-blind-vazirani, Fitzsimons2018}. These protocols can be made secure without any computational assumption.
However, the caveat is that they may not be applicable to the near-term quantum cloud computing model in practice.
On the other hand, with cryptographic assumptions of trapdoor claw-free functions (TCF), such as the quantum hardness of the learning-with-errors (LWE) problem~\vb{c}ite{regev2009lwe}, Mahadev devised a four-message verification protocol, involving a purely classical verifier and a single untrusted quantum prover~\vb{c}ite{Mahadev2018}.
Mahadev's protocol was later improved to a two-message protocol via the Fiat-Shamir transform~\vb{c}ite{alagic_two-message_2019}.
Back to the problem of verifying quantum computational power, several verification protocols were proposed based on similar cryptographic assumptions of TCFs as in Mahadev's protocol~\vb{c}ite{brakerski_cryptographic_2018, brakerski_simpler_2020, kahanamoku-meyer_classically-verifiable_2021}.
However, implementing these TCF-based protocols requires thousands of qubits to ensure security, far out of reach of current quantum technology.
Specifically, existing quantum cloud computing models involve purely classical clients who can only send out classical descriptions of the quantum circuits to the service provider and receive the resulting statistics of the output bit strings through the internet.
Again, from the practical point of view, it would be desirable to directly incorporate the verification process into the cloud computing process without much modifications.
In 2008, based on the model of IQP (instantaneous quantum polynomial-time) circuits, Shepherd and Bremner proposed~\vb{c}ite{IQP08} a cryptographic verification protocol which fits very well the scope of the near-term quantum cloud computing.
In their protocol, the verifier constructs an IQP circuit from the quadratic-residue code (QRC)~\vb{c}ite{macwilliams1977book}, which is encoded with a secret string kept by the verifier.
Then, he/she sends the classical description of the IQP circuit to the prover, who returns the output strings of the IQP circuit to the verifier. Finally, the verifiers checks if a certain measurement probability, associated with the secret string, is the same as a pre-determined value.
The Shepherd-Bremner protocol was previously believed to be secure for a long time, partly because of the classical intractability of simulating general IQP circuits~\vb{c}ite{IQP10, bremner_average-case_2016}; generally, IQP circuits cannot be sampled even approximately with efficient classical algorithms, unless the polynomial hierarchy collapses, which is a highly implausible complexity-theoretic consequence.
However, Kahanamoku-Meyer has recently found~\vb{c}ite{kahanamoku-meyer_forging_2019} a loophole in the Shepherd-Bremner protocol, and devised a classical algorithm to break the Shepherd-Bremner protocol. The loophole is originated from the special properties of QRC encoding the secret string from the IQP circuit. Once the secret is known, the prover can efficiently generate output strings that can pass the verification test, even without running the quantum circuit.
Here, we present a generalized model of IQP-based cryptographic verification protocol for near-term quantum cloud computing, which can be reduced to the Shepherd-Bremner construction as a special case.
Similar to the Shepherd-Bremner construction, our construction also involves only one round of interaction. In the context of quantum cloud computing, the verifier (Alice) does not need to inform the prover of her purpose, and she can perform the verification protocol at any time, as long as she has access to the quantum server.
The difference is that, we may choose to construct the secret-encoded IQP circuit without depending on any error-correcting code. Therefore, our approach is intrinsically immune to the attack of Kahanamoku-Meyer's approach.
Furthermore, this model allows us to encode an unspecified number of secret strings, instead of relying only on a single secret string as in the Shepherd-Bremner protocol.
Intuitively, this significantly increases the difficulty for the prover to cheat, i.e., hacking multiple secret strings simultaneously with the same set of output bit strings.
In practice, our protocol requires the quantum circuits to be in the supremacy regime, i.e. beyond the capability of classical simulation. Otherwise, the prover (Bob) can just use classical simulation to cheat, even though he is unable to find the secret strings.
In addition, we provide a general sampling method for an efficient estimation of any correlation function associated with a secret string, which gives the verifier a pre-determined value for checking.
Moreover, in the special case where all angles are $\vb{p}i/8$ for local IQP gates, we discuss the possibility of ``quantizing" the values of the correlation functions through a connection with a family of Clifford circuits.
This gives an evaluation method for the correlation functions in this case based on the Gottesman-Knill algorithm~\vb{c}ite{Gottesman98}.
The efficient evaluation of correlation functions allows Alice not to rely on any error correcting codes.
However, we further show that for random IQP circuits, the majority of these correlation functions can be exponentially small, supported by numerical simulations together with an example on the random 2-local IQP circuits.
Consequently, we propose a heuristic strategy in which correlation functions with sizable values can be constructed by starting with a small system (Hamming weight), followed by a scrambling technique to expand the Hamming weight if necessary.
\vb{s}ection{Results and Methods}
\vb{s}ubsection{A general framework}
In Ref.~\vb{c}ite{IQP08}, Shepherd and Bremner formulate an IQP-based verification protocol and give an explicit construction recipe for the IQP circuits used in the protocol.
An IQP circuit of $n$ qubits can be represented by an $m$-by-$n$ binary matrix $\vb{c}hi$, where each row represents a Pauli product. For example, a row vector $(1,1,0,0)$ represents $X_1 X_2$, where $X_i$ is the Pauli-$X$ acting on the $i$-th qubit. The Hamiltonian is the sum of $m$ Pauli operators, and the resulting IQP circuit is given by the time evolution of the Hamiltonian, i.e., $U_{\rm IQP} := e^{i\theta H}$. For example,
\begin{align}
\vb{c}hi = \begin{pmatrix}
1 & 1 & 0 & 0 \\
0 & 1 & 0 & 1
\vb{e}nd{pmatrix} & \quad \Longrightarrow \quad H = X_1 X_2 + X_2 X_4 \label{eq:matrix_to_hamiltonian}\\
& \quad \Longrightarrow \quad U_{\rm IQP} = e^{i\theta X_1 X_2} e^{i\theta X_2 X_4} \ .
\vb{e}nd{align}
Here, one may choose $\theta = \vb{p}i/8$ as in the QRC construction~\vb{c}ite{IQP08}. However, one may also consider the general cases where the angles of each term can be arbitrary, i.e., the angles may not necessarily be the same.
Our verification protocol is similar to the original Shepherd-Bremner construction, except that the encoding method is significantly extended.
Below, following the physical picture previously introduced by the authors in Ref.~\vb{c}ite{chen_experimental_2018}, we first present an overview of our protocol, and address several differences compared to the Shepherd-Bremner protocol.
An explicit construction recipe is given in Sec.~\ref{subsec:analysis}.
Let us consider a quantum circuit of $n$ qubits:
\begin{enumerate}
\item[] \textbf{The quantum cloud verification protocol:}
\item[\textbf{Step 1.}] Alice (the verifier) generates one, or multiple, $n$-bit random string(s) $\vb{s}:=(s_1,s_2, \vb{c}dots,s_n)^T \in\{0,1\}^n$ kept as a secret, where each string is associated with a Pauli product, $\mathcal{Z}_{\vb{s}} := Z^{s_1} \otimes \vb{c}dots \otimes Z^{s_n}$.
\item[\textbf{Step 2.}] Based on the secret string(s), Alice designs a Hamiltonian $H$ consisting of a linear combination of Pauli-$X$ products.
\item[\textbf{Step 3.}] Alice then sends the classical description about the Hamiltonian $H$ to Bob (the prover) and asks him to apply the time evolution of $H$ to the state $\ket{0^n}$, where the angles of each term (i.e. the evolution time) are also determined by Alice.
\item[\textbf{Step 4.}] Bob should perform the quantum computation $U_{\rm IQP} |0^n\rangle$ accordingly and measure in the computational basis multiple times. After that, he returns the output bit strings to Alice.
\item[\textbf{Step 5.}] Finally, Alice calculates the correlation function(s) $\vb{e}v{\mathcal{Z}_{\vb{s}}}:=\langle 0^n | U_{\rm IQP}^\vb{d}agger \mathcal{Z}_{\vb{s}} U_{\rm IQP} |0^n\rangle$ by classical means in Sec.~\ref{subsec:analysis}, and compares it with the results obtained from the bit strings given by Bob.
\vb{e}nd{enumerate}
Note that in the Shepherd-Bremner protocol~\vb{c}ite{IQP08}, Alice will instead check the probability bias $\mathcal{P}_{\vb{s} \vb{p}erp}$, the probability of receiving bit strings that are orthogonal to $\vb{s}$, which is defined as,
\begin{align}
\mathcal{P}_{\vb{s} \vb{p}erp} := \vb{s}um_{\vb{x} \vb{c}dot \vb{s} = 0 } p(\vb{x}) \ ,
\vb{e}nd{align}
where $p(\vb{x})$ is the output probability of the IQP circuit. However, as shown in Ref.~\vb{c}ite{chen_experimental_2018}, the probability bias can be related to the correlation function as follows,
\begin{align}\label{eq:bias_and_cor_func}
\mathcal{P}_{\vb{s} \vb{p}erp} = \frac{1}{2} (\vb{e}v{\mathcal{Z}_{\vb{s}}} + 1) \ .
\vb{e}nd{align}
Therefore, these two measures of success are equivalent, but we choose to work with the correlation function, as it fits better our framework.
Furthermore, in the Shepherd-Bremner construction, only one secret string is considered for each time, and the Hamiltonian is constructed from a specific error-correcting code, the quadratic-residue code (QRC)~\vb{c}ite{macwilliams1977book}, which can be regarded as a special instance in our framework. That is, if one constructs the Hamiltonian in Step~2 with QRC and chooses $\theta = \vb{p}i/8$, then our protocol reduces to the Shepherd-Bremner construction.
In order for the verification to work, Alice needs to know the value of the chosen correlation function in advance, which will be compared with the results from Bob's measurement data in Step~5. In the Shepherd-Bremner construction, $\vb{e}v{\mathcal{Z}_{\vb{s}}}$ is designed to always equal $1/\vb{s}qrt{2}$ (in terms of probability bias, 0.854) with only one specific $\vb{s}$, due to the properties of QRC.
In Sec.~\ref{subsec:analysis}, we present two methods for evaluating general correlation functions for IQP circuits, one corresponding to the most general case with arbitrary angles of each term, and the other corresponding to the case $\theta = \vb{p}i/8$.
The former method is based on random sampling, while the latter is based on Gottesman-Knill algorithm~\vb{c}ite{Gottesman98}.
With these two methods, Alice can calculate any $Z$-correlation functions as she wants, which allows her to test multiple secret strings.
We summarize the differences in Table.~\ref{tab:comparison}.
\begin{table}[t]
\begin{tabular}{|c|c|c|}
\hline
& Shepherd-Bremner & our protocol \\ \hline
circuit & \begin{tabular}[c]{@{}c@{}}based on\\ quadratic residue code\vb{e}nd{tabular} & more general IQP circuits \\ \hline
secret string & single & multiple \\ \hline
$\theta$ & $\vb{p}i/8$ & arbitrary \\ \hline
\vb{e}nd{tabular}
\vb{c}aption{Comparison between our protocol and the Shepherd-Bremner protocol. }
\label{tab:comparison}
\vb{e}nd{table}
We now turn to discuss one possible way to incorporate multiple secret strings into the Hamiltonian in Step~2.
Given secret strings $\vb{s}_1, \vb{s}_2, \vb{c}dots, \vb{s}_k$, Alice first generates the main part $H_M$ of the Hamiltonian, which has the property that every term in $H_M$ anti-commutes with $\mathcal{Z}_{\vb{s}_1}, \vb{c}dots\, \mathcal{Z}_{\vb{s}_k}$ simultaneously. This can be achieved in the following way.
For a vector $\vb{p} \in \{0,1\}^n$, the associated Hamiltonian term is $\vb{m}athcal{X}_{\vb{p}} := X^{p_1} \otimes \vb{c}dots \otimes X^{p_n}$. Then $\vb{m}athcal{X}_{\vb{p}}$ anti-commutes with every $\mathcal{Z}_{\vb{s}_i}$ if $\vb{p} \vb{c}dot \vb{s}_i = 1$ for $i = 1, \vb{c}dots, k$.
Therefore, the main part $H_M$ can be constructed from the solution space of this linear system.
To hide the secret strings, Alice would have to add a redundant part $H_R$, whose terms commute with every secret string $\vb{s}_i$.
The redundant part can be similarly constructed from the solution space of the linear system $\vb{p} \vb{c}dot \vb{s}_i = 0$ for $i = 1, \vb{c}dots, k$.
The whole Hamiltonian is $H = H_M + H_R$.
We remark that the values of the correlation functions depends only on the main part $H_M$ (see Sec.~\ref{subsec:analysis}). Therefore, Alice can add many redundant terms, to make the test harder.
Next, we make a further remark about the potential class of methods attacking our protocol, i.e., for Bob generating bit strings that can reproduce the value of the correlation function(s) without a quantum computer.
It would be possible if Alice's secret strings $\vb{s}$ are leaked to Bob; then, Bob could potentially evaluate the value of $\vb{e}v{\mathcal{Z}_{\vb{s}}}$, and output random bit strings according to the probability bias $\mathcal{P}_{\vb{s} \vb{p}erp}$.
Therefore, the security of the IQP-based protocols is based on the assumption that the secret string cannot be efficiently recovered from the Hamiltonian $H$. For the original Shepherd-Bremner protocol, such an attack has recently been found~\vb{c}ite{kahanamoku-meyer_forging_2019} based on the properties of QRC.
Here our framework extends the encoding method in the protocol, making it immune to such kind of attack.
Furthermore, the number of secret strings is not revealed to Bob, instead of relying on a single secret string as in the previous protocol~\vb{c}ite{IQP08}.
On the other hand, the IQP circuits constructed in our protocol is rather general. Although, a rigorous proof on the security of our protocol is missing, from a complexity-theoretic point of view, a general IQP circuit cannot be efficiently classically sampled, assuming some plausible conjectures~\vb{c}ite{IQP10,bremner_average-case_2016}.
In addition, the new features in our protocol should enhance the security compared to the original Shepherd-Bremner protocol.
\vb{s}ubsection{Detailed analysis}
\label{subsec:analysis}
\vb{p}aragraph{Evaluating the correlation function given the secret string}
For Alice to evaluate correlation function(s), recall that relative to each string, the Hamiltonian $H=H_M+H_R$ can be divided into two parts: main part and redundant part. (i) The main part $H_M$ anti-commutes with $\mathcal{Z}_{\vb{s}}$, i.e., $\{ \mathcal{Z}_{\vb{s}}, H_M \} = 0$. (ii) The redundant part $H_R$ commutes with $\mathcal{Z}_{\vb{s}}$, i.e., $[H_R, \mathcal{Z}_{\vb{s}}] = 0$.
Due to these commuting properties relative to the secret strings, the value of the correction function only depends on the main part, i.e. (see Appendix~\ref{app:properties_correlation}),
\begin{align}\label{eq:clifford}
\vb{e}v{\mathcal{Z}_{\vb{s}}} = \vb{m}el{0^n}{e^{i 2\theta H_M} }{0^n} \ .
\vb{e}nd{align}
Note that one can arrive at a similar expression if we further relax the condition where the weight of each term can be uneven, e.g. $ H = \vb{a}lpha X_1 X_2 + \beta X_2 X_4$.
On the other hand, the correlation function can be evaluated directly, if we confine the Hamming weight of the secret string to be sufficiently small, and the main part $H_M$ only acts on those qubits involved in the Pauli product $\mathcal{Z}_{\vb{s}}$.
\begin{figure}[t!]
\vb{c}entering
\includegraphics[width = 0.5\textwidth]{data.pdf}
\vb{c}aption{Properties of the correlation functions. \textbf{(a)} Fractions of correlation functions of specific values versus the number of qubits $n$ in the IQP circuits. Data is obtained by searching over 10000 random IQP circuits for each $n$. \textbf{(b)} Correlation functions from 500 randomly generated 6-qubit IQP circuits and Pauli operators $\mathcal{Z}_{\vb{s}}$. The data fluctuation is because the correlation functions are obtained via random sampling.}
\label{fig:data}
\vb{e}nd{figure}
For the general cases, the evaluation of the expression of $\vb{e}v{\mathcal{Z}_{\vb{s}}}$ can be achieved efficiently by sampling:
\begin{theorem}\label{thm:cor_func}
The correlation function of any Puali-Z product, $\vb{e}v{\mathcal{Z}_{\vb{s}}}:=\langle 0^n | U_{\rm IQP}^\vb{d}agger \mathcal{Z}_{\vb{s}} U_{\rm IQP} |0^n\rangle$ associated with an IQP circuit, can be classically estimated to $\vb{e}psilon$ precision with probability $1 - \vb{d}elta$, by taking $\order{ \frac{1}{\vb{e}psilon^2} \log{\frac{2}{\vb{d}elta}} }$ random samples.
\vb{e}nd{theorem}
To see why this theorem holds, one can apply local Hadamard gates to changing the main part $H_M$ to the $z$-basis, i.e.,
\begin{align}
\vb{e}v{\mathcal{Z}_{\vb{s}}} = \frac{1}{2^n} \vb{s}um_{\vb{x}} \vb{m}el{\vb{x}}{(U_M^{(z)})^2}{\vb{x}} \ ,
\vb{e}nd{align}
where $U_M^{(z)}$ is the main part of the circuit with Pauli-$X$ replaced by Pauli-$Z$ (see Appendix~\ref{app:properties_correlation}). Each term in this summation can be efficiently calculated, with an absolute value bounded by 1.
Then by the Chernoff bound argument, one can randomly sample bit strings $\vb{x}$, calculate $\vb{m}el{\vb{x}}{ (U_M^{(z)})^2 }{\vb{x}}$ for each bit string, and use the sample average to approximate $\vb{e}v{\mathcal{Z}_{\vb{s}}}$ to $\vb{e}psilon$ precision with probability $1 - \vb{d}elta$, using $\order{ \frac{1}{\vb{e}psilon^2} \log{\frac{2}{\vb{d}elta}} }$ samples.
Thus, for a polynomially small correlation function, the precision $\vb{e}psilon$ can also be polynomially small, and this method is efficient with classical means. Note that using the same sampling method, Bob can also evaluate any correlation function efficiently even if the redundant part of the Hamiltonian is included. However, the correlation function(s) associated with the secret string(s) is hidden from Bob.
The above method is based on random sampling and approximation error will be incurred.
In the special case of $\theta = \vb{p}i/8$, the correlation function can be evaluated exactly.
Observe that $ \vb{e}v{\mathcal{Z}_{\vb{s}}} = \vb{m}el{0^n}{e^{i (\vb{p}i/4) H_M} }{0^n}$ actually corresponds to a transition amplitude of a Clifford circuit $e^{i (\vb{p}i/4) H_M}$.
In this way, one can evaluate the correlation function efficiently using the Gottesman-Knill algorithm~\vb{c}ite{Gottesman98}.
Specifically, the absolute value of $\vb{e}v{\mathcal{Z}_{\vb{s}}}$ is either $0$ or $2^{-g/2}$, where $0 \leq g \leq n$ is an integer determined by the stabilizer groups of $\ket{0^n}$ and $e^{i (\vb{p}i/4) H_M} \ket{0^n}$~\vb{c}ite{aaronson_improved_2004}.
Note that in the Shepherd-Bremner construction, $g = 1$ for all QRC constructed IQP circuits.
We provide a example for the $g$-number in Appendix~\ref{app:properties_correlation}.
Also, see Fig.~\ref{fig:data}~(b) for an illustration of this `quantization' phenomenon with the simulation results of random 6-qubit instances; here, the randomness is from the random binary matrices $\vb{c}hi$ for the IQP circuits and random secret strings.
\vb{p}aragraph{Problem of random IQP circuits}
On the other hand, even though we have efficient classical algorithms for evaluating the correlation function to some additive error, it does not directly imply an effective solution to the problem.
For a random instance of $H_M$, the value of the resulting correlation function could be small from the experimental point of view; this makes it difficult to distinguish from the uniform distribution. For example, for random 2-local IQP circuits of the form, $U = e^{i\frac{\vb{p}i}{8} \left(\vb{s}um_{i<j} w_{ij} X_i \otimes X_j + \vb{s}um_i v_i X_i \right) }$ with $w_{ij}, v_i \in \{0, 1, \vb{c}dots, 7\}$, we have the following theorem, bounding the occurring probability of large correlation functions:
\begin{theorem}\label{thm:random_2-local}
For the class of random 2-local IQP circuits, the probability of finding a polynomial-sized correlation function $\vb{e}v{\mathcal{Z}_{\vb{s}}}$ is exponentially small, i.e.,
\begin{align}
\mathcal{P}r_{U, \vb{s}} \left( \vb{e}v{\mathcal{Z}_{\vb{s}}}^2 \geq \frac{1}{k} \right) \leq \frac{3k}{2^n} \ ,
\vb{e}nd{align}
where $k = \vb{p}oly(n)$ is a polynomial of $n$.
\vb{e}nd{theorem}
This is essentially due to the anti-concentration properties of IQP circuits~\vb{c}ite{bremner_average-case_2016}, an important ingredient for proving the quantum computational supremacy of IQP sampling.
For the proof, we refer to Appendix~\ref{app:anti-concentration}.
We performed numerical simulation in Fig.~\ref{fig:data}~(a), which shows that the fraction of correlation functions that are larger than certain value decays quickly with the number of qubits.
Theorem~\ref{thm:random_2-local} implies that for such kind of random Hamiltonians, it is not easy for Alice to keep secret strings associated with correlation functions with sizable values.
\begin{figure}[t!]
\vb{c}entering
\includegraphics[width = 1\vb{c}olumnwidth]{scrambling.pdf}
\vb{c}aption{The scrambling process in the matrix representation. Here, $H_M$ (the blue block) initially acts only on first few qubits. After the first scrambling, the third column of the matrix is added to the last column, and the last entry of $\vb{s}$ is added to the third one, correspondingly. Similarly, after the second scrambling, the first column of the matrix is added to the fifth one, and the fifth entry of $\vb{s}$ is added to the first one. The resulting matrix after 200 times of scrambling is shown in the lower left corner, and the acting range of the main part extends to the whole circuit. }
\label{fig:scrambling}
\vb{e}nd{figure}
\vb{p}aragraph{Construction from an initially small main part}
Therefore, practically, Alice needs to carefully design the main part $H_M$ such that the correlation function is sufficiently away from zero.
In the case of $\theta = \vb{p}i/8$, it would be more desirable to have $g$ being zero or one, so that the correlation function becomes $1$ or $0.707$.
However, as far as we are aware, there is no known efficient scheme of constructing an explicit Clifford circuit of the form ${e^{i (\vb{p}i/4) H_M} }$, for a fixed value of $g$.
We anticipate that an systematic scheme can be designed to generate the desired main part with a given $g$ by leveraging the relation between IQP circuits and stabilizer formalism.
We leave it for future exploration.
Here, for the general case, we propose a heuristic method to construct the main part, which utilizes the feature that the correlation function with respect to any given secret string can be efficiently evaluated.
In our heuristic construction, Alice may start from secret strings of relatively small Hamming weight.
Then, a brute-force search is taken to find an $H_M$ that can yield an IQP circuit with sizable $\vb{e}v{\mathcal{Z}_{\vb{s}}}$ for every $\vb{s}$.
From Fig.~\ref{fig:data}~(a), the initial Hamming weight of the secret string should be limited to about 10, so that the search can stop in a reasonable time.
Then, one can extend the Hamming weight of the secret strings by the method of scrambling as previously introduced in Ref.~\vb{c}ite{IQP08}.
Specifically, the scrambling process refers to a column manipulation of the matrix $\vb{c}hi$.
For example, we can add the first column to the third in \vb{e}qref{eq:matrix_to_hamiltonian}, resulting in a new Hamiltonian $H = X_1 X_2 X_3 + X_2 X_4$.
Moreover, when all angles for the Hamiltonian terms are the same, the following theorem (as a consequence of Theorem~1 of \vb{c}ite{IQP08}) implies that for IQP circuits, the scrambling process leaves the correlation function unchanged:
\begin{theorem}
\label{thm:scrambling_invariant}
Denote $\vb{m}athcal{C}_M$ as the linear subspace spanned by the column vectors of the main part, i.e., the matrix representation of $H_M$. When all $\theta$'s are identical, we can express the correlation functions in the following form:
\begin{align}\label{eq:scrambling_invariant}
\vb{e}v{\mathcal{Z}_{\vb{s}}} = \frac{1}{2^{d}} \vb{s}um_{\vb{c} \in \vb{m}athcal{C}_{M}} \vb{c}os[2\theta (q - 2 |\vb{c}| )] \ ,
\vb{e}nd{align}
where $d$ is the dimension of the linear subspace $\mathcal{C}_{M}$, $q$ is the number of terms in $H_M$ and $|\vb{c}|$ is the Hamming weight of $\vb{c}$.
\vb{e}nd{theorem}
For completeness, we give a proof in Appendix~\ref{app:invariance_CF}.
From this theorem, one can see that the value of $\vb{e}v{\mathcal{Z}_{\vb{s}}}$ depends only on the linear subspace $\mathcal{C}_M$ since each term in the expression depends only on $\vb{c} \in \mathcal{C}_M$.
Therefore, even if $H_M$ is scrambled, as long as the secret string is also scrambled accordingly, to preserve the inner product with rows in $\vb{c}hi$, then the linear subspace $\vb{m}athcal{C}_M$ will remain unchanged and so is the value of $\vb{e}v{\mathcal{Z}_{\vb{s}}}$.
We note that, this scrambling-invariance property is true even in the case where the angles are different; we discuss this point also in Appendix~\ref{app:invariance_CF}.
This scrambling-invariance property can be better explained with Fig.~\ref{fig:scrambling} as an example.
In the upper left corner of Fig.~\ref{fig:scrambling}, $\vb{s} = (0,1,1,0,0,0,0)$ and the main part is the first 3 rows (i.e. $q = 3$).
The linear subspace $\mathcal{C}_M$ is spanned by the column vectors in the blue block, namely $\vb{a}_1 = (0,0,1)^T$, $\vb{a}_2 = (0,1,0)^T$, and $\vb{a}_3 = (1,0,1)^T$ (i.e. $d = 3$). In this way, the vectors of the linear subspace takes the form $\vb{c} = \vb{s}um_{i = 1}^3 y_i \vb{a}_i$, with $y_i \in \{ 0,1 \}$. After the scrambling, one can check that each column in the new main part can be written as linear combination of the three vectors $\{ \vb{a}_1,\vb{a}_2,\vb{a}_3\}$ , which means the new linear subspace is the same as $\mathcal{C}_M$.
So, the correlation function after the scrambling remains the same too.
This scrambling process not only hides the secret, but also extends an initially small main part.
We remark that the scrambling invariance holds for the case of multiple secret strings as well, because those secrte strings all share with the same $\mathcal{C}_M$.
\vb{s}ection{Discussion}
Verification is important is various quantum computing models, such as blind quantum computation~\vb{c}ite{Fitzsimons2017-verifiable-blind,xu_parallel_2020}, distributed quantum computation~\vb{c}ite{buhrman2003distributed, Sheng_distributed_2017} and secure multi-party quantum computation~\vb{c}ite{crepeau_2002_secure, Qiang_2017}.
In this paper, we have discussed a generalized model of IQP-based cryptographic verification protocol, where several features are introduced to achieve anti-forging of the quantum data. This model can be reduced to the Shepherd-Bremner protocol as a special case, but one can construct the secret without relying on the quadratic residue code, avoiding a class of attacks proposed by Kahanamoku-Meyer.
We remark that the applicability of our framework is not limited to IQP circuits. For example, one may design a quantum circuit of the following form: $U = U_R U_M$, where $U_R$ commutes with a certain observable $O$.
In this way, we also have $\vb{m}el{0^n}{U^{\vb{d}agger} O U}{0^n} = \vb{m}el{0^n}{U_M^{\vb{d}agger} O U_M}{0^n}$ depends only on the main part.
In the case that $U_M$ acts on a small number of qubits (i.e., a small main part), the expectation can be calculated by classical simulation.
Furthermore, one may consider an extension to quantum circuits like $\vb{d}isplaystyle U=U_{\rm shallow} V^{\vb{d}agger } V$, where $U_{\rm shallow}$ is a shallow sub-circuit and $V$ contains random unitary gates; of course, the overall gate order should be scrambled to the prover. In this case, the verifier can check any observable based on shallow part only.
Returning to the IQP framework, the most imminent open question is a lack of a rigorous security proof of the general protocol. Practically, the protocol should be implemented in a scale beyond the capability of classical computing, i.e., in the regime of quantum advantage (supremacy). Furthermore, it is also necessary to take noises into account. These questions should be addressed with large-scale numerical simulations in future.
\textbf{Acknowledgement.---}
We thank Zhengfeng Ji and Michael Bremner for insightful discussions. This work is supported by the Natural Science Foundation of Guangdong Province (2017B030308003), the Key R\&D Program of Guangdong province (2018B030326001), the Science, Technology and Innovation Commission of Shenzhen Municipality (JCYJ20170412152620376 and JCYJ20170817105046702 and KYTDPT20181011104202253), National Natural Science Foundation of China (11875160 and U1801661), the Economy,Trade and Information Commission of Shenzhen Municipality (201901161512), Guangdong Provincial Key Laboratory(Grant No.2019B121203002).
BC thanks the support from the Sydney Quantum Academy, Sydney, NSW, Australia.
\vb{c}learpage
\begin{appendix}
\vb{s}ection{Properties of the correlation functions}
\label{app:properties_correlation}
Let the main part and the redundant part of the IQP circuit be $U_M$ and $U_R$. Since each gate in the IQP circuit commutes with each other, we can assume without loss of generality that $U_{\rm IQP} = U_R U_M$, which gives $\vb{e}v{\mathcal{Z}_{\vb{s}}} = \vb{m}el{0^n}{U_M^{\vb{d}agger} U_R^{\vb{d}agger} \mathcal{Z}_{\vb{s}} U_R U_M}{0^n}$. From the fact that the redundant part $U_R$ commutes with $\mathcal{Z}_{\vb{s}}$, we have,
\begin{align}
\vb{e}v{\mathcal{Z}_{\vb{s}}} = \vb{m}el{0^n}{U_M^{\vb{d}agger} \mathcal{Z}_{\vb{s}} U_M}{0^n} \ .
\vb{e}nd{align}
Furthermore, with the anti-commutation of $H_M$ and $\mathcal{Z}_{\vb{s}}$, we have $U_M^{\vb{d}agger} \mathcal{Z}_{\vb{s}} U_M = \mathcal{Z}_{\vb{s}} U_M^2$ and
\begin{align}\label{eq:corfunc_general}
\vb{e}v{\mathcal{Z}_{\vb{s}}} = \vb{m}el{0^n}{U_M^2}{0^n} \ .
\vb{e}nd{align}
Then we apply Hadamard gates to change the basis,
\begin{align}
\vb{e}v{\mathcal{Z}_{\vb{s}}} &= \frac{1}{2^n} \vb{s}um_{\vb{x}, \vb{y}} \vb{m}el{\vb{x}}{(U_M^{(z)})^2}{\vb{y}} \\
&= \frac{1}{2^n} \vb{s}um_{\vb{x}} \vb{m}el{\vb{x}}{(U_M^{(z)})^2}{\vb{x}} \ , \label{eq:corfunc_diag}
\vb{e}nd{align}
where $U_M^{(z)}$ is the main part of the circuit with Pauli-$X$ replaced by Pauli-$Z$ and we have used the fact that $\vb{m}el{\vb{x}}{(U_M^{(z)})^2}{\vb{y}} = 0$ if $\vb{x} \neq \vb{y}$. Each term in the summation can be efficiently calculated by tracking the phase. By the Chernoff bound argument, $\vb{e}v{\mathcal{Z}_{\vb{s}}}$ can be approximated to $\vb{e}psilon$ precision with probability $1 - \vb{d}elta$ using $\order{ \frac{1}{\vb{e}psilon^2} \log{\frac{2}{\vb{d}elta}} }$ samples of $\vb{x}$, as stated in the main text.
Specifically, if $\theta = \vb{p}i/8$, then $U_M^2 = e^{i2\theta H_M}$, and \vb{e}qref{eq:corfunc_general} becomes,
\begin{align}
\vb{e}v{\mathcal{Z}_{\vb{s}}} = \vb{m}el{0^n}{e^{i2\theta H_M}}{0^n} \ .
\vb{e}nd{align}
Here, $e^{i2\theta H_M}$ is a Clifford circuit since $2\theta = \vb{p}i/4$, and the correlation function can be exactly calculated by the Gottesman-Knill algorithm~\vb{c}ite{Gottesman98} in this case.
Moreover, the absolute value of $\vb{e}v{\mathcal{Z}_{\vb{s}}}$ is either $0$ or $2^{-g/2}$, where $0 \leq g \leq n$ is the minimum number of different generators of the stabilizer groups of two states: $\ket{0^n}$ and $e^{i2\theta H_M} \ket{0^n}$.
For example, if the generator associated to $\ket{0^n}$ is given by $\{ Z_1, Z_1 Z_2, Z_3, Z_3 Z_4 \}$, and that of $e^{i2\theta H_M} \ket{0^n}$ is given by $\{ Y_1 X_2, X_1 Y_2, Y_3 X_4, X_3 Y_4 \}$, then the number of different generators is 2; this can be seen by noting that the latter stabilizer group can be equivalently described by the following set of generators $\{ Y_1 X_2, Z_1 Z_2, Y_3 X_4, Z_3 Z_4 \}$.
The set of generators is not unique, and $g$ is the minimum over all possible generators associated to two states.
\vb{s}ection{The problem from anti-concentration}
\label{app:anti-concentration}
In Ref.~\vb{c}ite{bremner_average-case_2016}, it is proved that for random 2-local IQP circuits of the form
\begin{align}\label{eq:IQP_form}
U_{\rm IQP} = e^{i\frac{\vb{p}i}{8} \left(\vb{s}um_{i<j} w_{ij} X_i \otimes X_j + \vb{s}um_i v_i X_i \right) } \ ,
\vb{e}nd{align}
with $w_{ij}, v_i \in \{ 0, 1, \vb{c}dots, 7 \}$, the output probability is anti-concentrated. Specifically, the \vb{e}mph{anti-concentration theorem} states that,
\begin{align}
\bb{E}_{U} [p(\vb{x})^2] \leq \frac{3}{2^{2n}}
\vb{e}nd{align}
for all $\vb{x}$, where $\bb{E}_{U}$ denotes a uniform average over all IQP circuits of the form of \vb{e}qref{eq:IQP_form}, that is over uniform choices of $w_{ij}$ and $v_i$. Then we have,
\begin{align}\label{eq:anti-concentration}
\bb{E}_{U} \left[ \vb{s}um_{\vb{x}} p(\vb{x})^2 \right] \leq \frac{3}{2^{n}} \ .
\vb{e}nd{align}
By definition, the correlation function can be written as,
\begin{align}\label{eq:corfunc_def}
\vb{e}v{\mathcal{Z}_{\vb{s}}} = \vb{s}um_{\vb{x}} p(\vb{x}) (-1)^{\vb{s} \vb{c}dot \vb{x}} \ ,
\vb{e}nd{align}
which actually holds for a general quantum circuit. This means that $p(\vb{x})$ is the Fourier transform of $\vb{e}v{\mathcal{Z}_{\vb{s}}}$, and that
\begin{align}
p(\vb{x}) = \frac{1}{2^n} \vb{s}um_{\vb{s}} \vb{e}v{\mathcal{Z}_{\vb{s}}} (-1)^{\vb{s} \vb{c}dot \vb{x}} \ .
\vb{e}nd{align}
Then we can apply the Parseval's identity~\vb{c}ite{Boolean},
\begin{align}
\vb{s}um_{\vb{x}} p(\vb{x})^2 = \frac{1}{2^n} \vb{s}um_{\vb{s}} \vb{e}v{\mathcal{Z}_{\vb{s}}}^2 \ ,
\vb{e}nd{align}
which gives,
\begin{align}
\bb{E}_{U, \vb{s}} \left[ \vb{e}v{\mathcal{Z}_{\vb{s}}}^2 \right] &= \frac{1}{2^n} \vb{s}um_{\vb{s}} \bb{E}_U \left[ \vb{e}v{\mathcal{Z}_{\vb{s}}}^2 \right] \\
&= \bb{E} \left[ \vb{s}um_{\vb{x}} p(\vb{x})^2 \right] \\
& \leq \frac{3}{2^n} \ .
\vb{e}nd{align}
The Markov's inequality gives the following bound,
\begin{align}\label{eq:markov}
\mathcal{P}r_{U, \vb{s}}(\vb{e}v{\mathcal{Z}_{\vb{s}}}^2 \geq a) \leq \frac{\bb{E}_{U, \vb{s}} \left[ \vb{e}v{\mathcal{Z}_{\vb{s}}}^2 \right]}{a} \leq \frac{3}{a 2^n} \ ,
\vb{e}nd{align}
for $a > 0$. Setting $a = \order{1/\vb{p}oly(n)}$, we have,
\begin{align}
\mathcal{P}r_{U, s} \left( \vb{e}v{\mathcal{Z}_{\vb{s}}}^2 \geq \frac{1}{\vb{p}oly(n)} \right) \leq \frac{3\vb{p}oly(n)}{2^n} \ .
\vb{e}nd{align}
This means that for random 2-local IQP circuits, the probability that the correlation functions are polynomially small is exponentially small.
In practice, Alice will obtain the correlation function from Bob's data in the following way. Suppose Alice receives $T$ output strings from Bob, $\vb{x}_1, \vb{c}dots, \vb{x}_T$. For each string, Alice computes $(-1)^{\vb{s} \vb{c}dot \vb{x}_i}$, and the sample average
\begin{align}
s_T := \frac{1}{T} \vb{s}um_i (-1)^{\vb{s} \vb{c}dot \vb{x}_i}
\vb{e}nd{align}
gives an approximation of $\vb{e}v{\mathcal{Z}_{\vb{s}}}$. From the Chernoff bound argument, for $\vb{e}psilon$ precision and probability $1 - \vb{d}elta$, $T = \order{ \frac{1}{\vb{e}psilon^2} \log{\frac{2}{\vb{d}elta}} }$. In order for our protocol to be practical, $T$ should be a polynomial of $n$, which implies a polynomial precision $\vb{e}psilon = \order{1/\vb{p}oly(n)}$. However, if $\vb{e}v{\mathcal{Z}_{\vb{s}}}$ itself is exponentially small, then Alice will not be able to distinguish the correct data from data obtained from uniform distribution, since polynomial precision in this case is not sufficient.
\vb{s}ection{Scambling invariance of correlation function}
\label{app:invariance_CF}
Now we want to prove that $\vb{e}v{\mathcal{Z}_{\vb{s}}}$ depends only on the linear subspace spanned by the matrix representation of $H_M$. First, denote $M$ as the matrix representation of $H_M$. \vb{e}qref{eq:corfunc_diag} gives,
\begin{align}
\vb{e}v{\vb{m}athcal{Z}_{\vb{s}}} &= \frac{1}{2^n} \vb{s}um_{\vb{y} \in \{ 0,1 \}^n } \vb{m}el{\vb{y}}{ \vb{p}rod_{\vb{p}\in \row(M) } e^{i 2\theta_{\vb{p}} Z_{\vb{p}}} }{\vb{y}} \ ,
\vb{e}nd{align}
where $Z_{\vb{p}} := Z^{p_1} \otimes \vb{c}dots \otimes Z^{p_n}$. Then,
\begin{align}
\vb{e}v{\vb{m}athcal{Z}_{\vb{s}}} &= \frac{1}{2^n} \vb{s}um_{\vb{y} \in \{ 0,1 \}^n } \vb{p}rod_{\vb{p}\in \row(M) } \vb{e}xp(i 2\theta_{\vb{p}} (-1)^{\vb{p}\vb{c}dot \vb{y}} ) \\
&= \frac{1}{2^n} \vb{s}um_{\vb{y} \in \{ 0,1 \}^n } \vb{e}xp(i \vb{s}um_{\vb{p} \in \row(M) } 2 \theta_{\vb{p}} (-1)^{\vb{p}\vb{c}dot \vb{y}} ) \ .
\vb{e}nd{align}
Define $S$ as the scrambling matrix, which is product of elementary matrices in $\vb{m}athbb{F}_2$.
Then, after the scrambling, we have $\vb{c}hi \to \vb{c}hi \vb{c}dot S$, $M \to M \vb{c}dot S$ and $\vb{s} \to S^{-1} \vb{s}$, so that the inner-product relation between rows in $\vb{c}hi$ and $\vb{s}$ is preserved.
Moreover, we have $\vb{p} \vb{c}dot \vb{y} = \vb{p} \vb{c}dot S S^{-1} \vb{c}dot \vb{y}$.
Then, denoting the new secret string as $\vb{s}' = S^{-1} \vb{s}$, the associated correlation function is given by,
\begin{align}
\vb{e}v{\mathcal{Z}_{\vb{s}'}} &= \frac{1}{2^n} \vb{s}um_{\vb{y} \in \{ 0,1 \}^n } \vb{e}xp(i \vb{s}um_{\vb{p} \in \row(M \vb{c}dot S) } 2 \theta_{\vb{p}} (-1)^{\vb{p} \vb{c}dot S S^{-1} \vb{c}dot \vb{y}} ) \\
&= \frac{1}{2^n} \vb{s}um_{\vb{y} \in \{ 0,1 \}^n } \vb{e}xp(i \vb{s}um_{\vb{p} \in \row(M) } 2 \theta_{\vb{p}} (-1)^{\vb{p}\vb{c}dot \vb{y}} ) \\
&= \vb{e}v{\mathcal{Z}_{\vb{s}}} \ .
\vb{e}nd{align}
This proves the scrambling invariance of the correlation function.
If all angles are the same, then one can arrive at a similar theorem as Theorem~1 of Ref.~\vb{c}ite{IQP08}.
Define $\vb{c}_{\vb{y}} := M \vb{c}dot \vb{y} $ to be an encoding of $\vb{y}$ under $M$. Recall that $M$ contains $q$ rows, so we can write
\begin{align}
\vb{c}_{\vb{y}} = \begin{pmatrix} \vb{p}_1 \\ \vdots \\ \vb{p}_q \vb{e}nd{pmatrix} \vb{c}dot \vb{y} = \begin{pmatrix} \vb{p}_1 \vb{c}dot \vb{y} \\ \vdots \\ \vb{p}_q \vb{c}dot \vb{y} \vb{e}nd{pmatrix} \ ,
\vb{e}nd{align}
which means that each entry in $\vb{c}_{\vb{y}}$ equals $\vb{p} \vb{c}dot \vb{y}$ for $\vb{p} \in \row(M)$.
Then $\vb{s}um_{\vb{p} \in \row(M) } (-1)^{\vb{p}\vb{c}dot \vb{y}}$ equals the number of zeros in $\vb{c}_{\vb{y}}$ minus the number of ones (i.e., Hamming weight $|\vb{c}_{\vb{y}}| $), which gives,
\begin{align}
\vb{s}um_{\vb{p} \in \row(M) } (-1)^{\vb{p}\vb{c}dot \vb{y}} = q - 2 |\vb{c}_{\vb{y}}| \ .
\vb{e}nd{align}
So, we arrive at,
\begin{align}
\vb{e}v{\mathcal{Z}_{\vb{s}}} &= \frac{1}{2^n} \vb{s}um_{\vb{y} \in \{ 0,1 \}^n } \vb{c}os[ 2\theta (q - 2 |\vb{c}_{\vb{y}}| ) ] \ ,
\vb{e}nd{align}
where we used the fact that $\vb{e}v{\mathcal{Z}_{\vb{s}}}$ is real. Now, in the column picture, we can write $M = (\vb{a}_1, \vb{c}dots, \vb{a}_n)$, where $\vb{a}_i \in \vb{c}ol(M)$ is a column vector of length $q$. Thus,
\begin{align}
\vb{c}_{\vb{y}} = M \vb{c}dot \vb{y} = y_1 \vb{a}_1 + \vb{c}dots + y_n \vb{a}_n \ .
\vb{e}nd{align}
Suppose the dimension of $\mathcal{C}_{M}$ is $d$, and without loss of generality, assume $\{ \vb{a}_1, \vb{c}dots, \vb{a}_d \}$ forms a basis. Then,
\begin{align}
\vb{c} = y_1 \vb{a}_1 + \vb{c}dots + y_{d} \vb{a}_{d}
\vb{e}nd{align}
for any $\vb{c} \in \mathcal{C}_{M}$. So the expression of $\vb{e}v{\mathcal{Z}_{\vb{s}}}$ becomes,
\begin{align}
\vb{e}v{\mathcal{Z}_{\vb{s}}} &= \frac{1}{2^{n - d}} \vb{s}um_{y_{d+1}, \vb{c}dots, y_n} \left( \frac{1}{2^{d}} \vb{s}um_{\vb{c} \in \vb{m}athcal{C}_M} \vb{c}os[2\theta (q - 2 |\vb{c}| )] \right)
\vb{e}nd{align}
The first summation will give a factor of $2^{n - d}$, which cancels with $\frac{1}{2^{n -d}}$. So finally, it ends up giving,
\begin{align}
\vb{e}v{\mathcal{Z}_{\vb{s}}} = \frac{1}{2^{d}} \vb{s}um_{\vb{c} \in \vb{m}athcal{C}_M} \vb{c}os[2\theta (q - 2 |\vb{c}| )] \ .
\vb{e}nd{align}
Every term in the summation depends on the element $\vb{c}$, and therefore $\vb{e}v{Z_{\vb{s}}}$ depends only on the linear subspace $\vb{m}athcal{C}_{M}$.
\vb{e}nd{appendix}
\vb{e}nd{document} |
\begin{document}
\title{Non equilibrium dynamics of an Optomechanical Dicke Model}
\author{Kamanasish Debnath$^1$}
\author{Aranya B Bhattacherjee$^2$}
\address{$^1$Institute of Applied Sciences, Amity University, Noida - 201303 (U.P.), India \\
$^2$School of Physical Sciences, Jawaharlal Nehru University, New Delhi- 110067, India}
\begin{abstract}
Motivated by the experimental realization of Dicke model in optical cavities, we model an optomechanical system consisting of two level BEC atoms with transverse pumping. We investigate the transition from normal and inverted state to the superradiant phase through a detailed study of the phase portraits of the system. The rich phase portraits generated by analytical arguments display two types of superradiant phases, regions of coexistence and some portion determining the persistent oscillations. We study the time evolution of the system from any phase and discuss the role of mirror frequency in reaching their attractors. Further, we add an external mechanical pump to the mirror which is capable of changing the mirror frequency through radiation pressure and study the impact of the pump on the phase portraits and the dynamics of the system. We find the external mirror frequency changing the phase portraits and even shifting the critical transition point, thereby predicting a system with controllable phase transition.
\end{abstract}
\keywords{Dicke model, optomechanics, Bose Einstein Condensation}
\pacs{03.75.Gg,05.30.Rt,42.79.Gn}
\maketitle
\section{Introduction}
Cavity optomechanics \citep{7} has been playing an important role in the exploration of quantum mechanical systems, especially the coupling between the electromagnetic field of the cavity and the mechanical oscillator \citep{1,2,3}. The photons inside the ultrahigh finesse cavity are capable of displacing the mechanical mirror through radiation pressure and this has been a subject of early research in the context of nanomechanical cantilevers \citep{4,5,6}, vibrating microtoroids \citep{8}, membranes and Bose-Einstein condensates \citep{9}. Recent advancements in the field of laser cooling, high finesse nanomechanical mirrors have made it possible to study ultra cold atoms by combining the tools of cavity quantum electrodynamics. Experimental realisation of quantum entanglement, gravitational wave detection \citep{10,11} in the last few years has added new interest to the field of optomechanics. Such a system with an ensemble of N atoms with single optical mode has been an interesting theme in quantum optics after the work of Dicke \citep{12}, showing the effects of quantum phase transition and superradiant phases.The phase transition from a super fluid to a self organised phase, above a certain threshold frequency, when a laser driven BEC \cite{46, 47, 48, 49} is coupled to the vacuum field of the cavity refers to the basic Dicke model \citep{35, 36, 37}. The ultra cold atoms self organizes to form a checkerboard pattern trapped in the interference pattern of the pump and the cavity beams \citep{13,14,15}. This self organization initiates at the onset of the superradiance in an effective non equilibrium Dicke model. Since then many theoretical proposals for single mode, multi mode \citep{38} and optomechanical Dicke models has been made which are presumed to exhibit interesting physics \citep{39} and applications in the field of quantum simulation and quantum information \citep{40, 41, 42, 43}. In the present cold atom settings, the splitting of the two distinct momentum states of the BEC is controlled by the atomic recoil energy, and this enables the phase transition to be observed with optical frequencies with light. This is quite similar to the theoretical approach proposed by Dimer $\textit{et al.}$ \citep{17} for attaining Dicke phase transitions using Raman pumping schemes between the hyperfine levels \citep{44}.\\
In this paper, we propose an optomechanical system consisting of N, two level elongated cigar shaped BEC interacting with light in a high finesse optical cavity with a movable mirror. Such systems can be used to investigate the optomechanical effects on the second order phase transition to a superradiant regime. We study the dynamics of the system and bring out all the possible phases by analytical arguments and further propose a modification in the system that can be used to alter the phase portraits and transition point of the system.
\section{The Model}
We consider a Fabry- Perot optical cavity with one fixed and another movable high finesse mirror of mass $M$, capable of oscillating freely with frequency $\omega_m$. A two level, cigar shaped BEC is trapped within the cavity with transition frequency $\omega_a$. The optical cavity is subjected to a transverse pump beam with Rabi frequency $\Omega_p$, wave vector $k$ and frequency $\omega_p$. In order to avoid population inversion, the later is far detuned from the atomic transition $\omega_a$. Absorption and emission of cavity photons generates an effective two level spin system with spin down and spin up corresponding to the ground $\Ket{0,0}$ and excited states $\Ket{\pm k, \pm k}$ respectively. The effective Hamiltonian of such a system can be written as ($\hbar$= 1 throughout the paper) \citep{18, 21, 22}:-
\begin{eqnarray}
H&=& \omega_a S_z+ \omega a^{\dagger} a+ \omega_m b^{\dagger} b+ \delta_0 a^{\dagger} a (b+ b^{\dagger}) \nonumber\\
&+& g(a+ a^{\dagger})(S_{+}+ S_{-})+ US_z a^{\dagger}a,
\end{eqnarray}
\begin{figure}
\caption{The schematic representation of the model considered. One of the mirror is movable under the radiation pressure of the cavity beams. The optical cavity has a decay rate of $\kappa$ and the mechanical mirror has a damping rate $\Gamma_m$.}
\end{figure}
where $\omega= \omega_c- \omega_p- N(5/8)g_0^2/ (\omega_a- \omega_c)$ \citep{22}, $\omega_c$ being the cavity frequency. $U$ represents the back reaction of the cavity light on the BEC and is given by $U= -(1/4)g_0^2/ (\omega_a- \omega_c)$, which is generally negative, however both the signs are achievable experimentally and we shall deal with the both in the present paper. $\delta_0$ is the optomechanical single photon coupling strength which represents the optical frequency shift produced by a zero point displacement. $\delta_0$ can be identified as $\omega x_{ZPF}/{L}$, $L$ being the cavity length and $x_{ZPF}$ denoting the mechanical zero point fluctuations (width of the mechanical ground state wave function) \citep{50}. $\omega_m$ represents the frequency of the mechanical mirror, which generates phonons with $b (b^{\dagger})$ as the annihilation (creation) operator. In the experiments of \citep{14}, both the pump and cavity were red detuned from the atomic transition and hence $U$ was considered negative for the observed Dicke phase transition. $a (a^{\dagger})$ is the annihilation (creation) operator of the optical mode while $b (b^{\dagger})$ representing the same for the mechanical mode, following the commutation relation [$a (b), a^{\dagger} (b^{\dagger})]$= 1. $S_{+}, S_{-}$ and $S_z$ are the spin operators obeying the relation [$S_+, S_-$]= $2S_z$ and [$S_{\pm}, S_z$]= $\mp S_{\pm}$. $\textbf{S}= (S_x, S_y, S_z)$ is the effective collective spin of length $N/2$. The co and counter rotating matter light coupling has been taken equal throughout the paper and is denoted by $g$. The schematic representation of the model considered in this paper has been shown in Fig. (1). \\
In the thermodynamic limit, the semi classical equations for our system, takes the form: -
\begin{equation}
\dot{S_{-}}= -i(\omega_a+ U \mid a \mid ^2) S_{-}+ 2ig(a+ a^{\dagger})S_z,
\end{equation}
\begin{equation}
\dot{S_z}= -ig(a+ a^{\dagger})S_{+}+ ig(a+ a^{\dagger})S_{-},
\end{equation}
\begin{eqnarray}
\dot{a}&=& -[\kappa+ i(\omega+ US_z+ \delta_0 (b+ b^{\dagger})]a\nonumber\\
&-& ig(S_{+}+ S_{-}),
\end{eqnarray}
\begin{equation}
\dot{b}= -i\omega_m b- i\delta_0 \mid a \mid^2- \Gamma_m b,
\end{equation}
where $\kappa$ and $\Gamma_m$ are the cavity decay rate and damping rate of the mechanical mode respectively. We employ the steady state analysis $(\dot{S}_-= \dot{S_z}= \dot{a}= \dot{b}= 0)$ of the above equations to determine the critical atom- cavity coupling strength. We carry a numerical approach in this paper to determine the critical value $\lambda_c$ ($g\sqrt{N_c}$). The analytical process uses the c- number variables and quantum fluctuations, and one can refer \citep{21} for the complete process in the absence of back reaction term. $\lambda$ ($g\sqrt{N}$) $>$ $\lambda_c$ ($g\sqrt{N_c}$) marks the onset of the superradiance, which was first observed experimentally by Tilman Esslinger and his group \citep{14} for BEC atoms in 2010.
\section{Superradiant phases and Phase Portraits}
To study the dynamics of the present system, we employ the same mathematical technique as \citep{22, 27} and define $a= a_1+ ia_2$, $b= b_1+ ib_2$ and $S_{\pm}= S_x \pm iS_y$. Substituting the same in the above semi classical equations (Eq. (2)- (5)) and comparing the real and imaginary parts on both side yields: -
\begin{equation}
(\omega_a+ U\mid a \mid^2)S_y= 0,
\end{equation}
\begin{equation}
(\omega_a+ U\mid a \mid^2)S_x- 4ga_1S_z= 0,
\end{equation}
\begin{equation}
-\kappa a_1+ (\omega+ US_z+ 2\delta_0 b_1)a_2= 0,
\end{equation}
\begin{equation}
\kappa a_2+ (\omega+ US_z+ 2\delta_0 b_1)a_1+ 2gS_x= 0.
\end{equation}
\begin{equation}
b= -\Big(\frac{\delta_0 \mid a \mid ^2 \omega_m}{\Gamma_m^2+ \omega_m^2}\Big) - i\Big(\frac{\delta_0 \mid a \mid^2 \Gamma_m}{\Gamma_m^2+ \omega_m^2}\big)
\end{equation}
Clearly, from Eq. (6), either $S_y$= 0 or $(\omega_a+ U\mid a \mid^2)$= 0. We define the case arising from the first condition as the superradiant phase A (SRA) and the second condition as the superradiant phase B (SRB). SRA represents the quantum phase transition from normal (N) or inverted (I) states to a self organized states defined by $S_x$ and $S_z$ only. Similarly, the SRB represents the transition from the mixed states (N+ I) to a superradiant phase defined by all the components of $\textbf{S}$. The difference of transition from mixed states (N+ I) as in SRB phase compared to from normal (N) or inverted (I) as in SRA phase can be understood in the phase diagrams. Ofcourse, with increasing back action parameter, we expect a reduced phase transition region. Again, SRB phase condition limits $U$ to be only negative, since the phase is defined as $(\omega_a+ U\mid a \mid^2)$= 0. However, what might be the effect of the mechanical mirror motion on the phase transition of the system? In the absence of the back reaction parameter $U$, \citep{21, 27} suggests no change in the critical transition point, $\lambda_c$ for the SRA phase. However, in the presence of the back reaction term and in the SRB phase, what role can the mirror frequency play in defining the phase portraits, a question to be analyzed in this paper. In the next section, we shall analyze all the possible conditions and present the phase portraits of the system for both positive and negative back reaction parameter.
\subsection{SRA Phase}
As defined before, $S_y$= 0 marks the SRA phase, which is simply the transition from normal (N) or inverted (I) state to the regime of superradiance. The critical atom- cavity coupling point can be determined by setting $[S_x, S_y, S_z]= [0, 0, \pm N/2]$, which signifies the presence of either spin up (inverted) or spin down (normal) particles and no photons. The steady state equations (Eq. 6- 9) can be straightforwardly solved using matrix method for $S_z$, which yields a quadratic equation supporting two roots of $S_z$. The determinant representing the steady state equations, takes the form: -
\begin{equation}
\begin{vmatrix}
\omega_{a} + U\mid a \mid^2 & 0 & -4gS_{z} & 0 \\ 0 & \omega_{a} + U\mid a \mid^2 & 0 & 0 \\ 2g & 0 & \chi & \kappa \\ 0 & 0 & -\kappa & \chi \end{vmatrix}= 0,
\end{equation}
where $\chi=\omega+ US_z- \frac{2\delta_0^2\mid a \mid^2 \omega_m}{\Gamma_m^2+ \omega_m^2}$. The above determinant has been solved numerically and the results are too cumbersome to be reproduced here. The two supporting roots for $S_z$ when equated to $\pm N/2$, and solved for $\omega$, represents the dynamical phase portrait for SRA phase showing the transition from normal (N) and inverted (I) phase to regimes of superradiance. An important point to note here, is that the SRA phase exists for any value of the back action parameter, U. Although the two roots of $S_z$ must be independent, however, we shall find a small region in the phase portraits, where both the roots of $S_z$ are satisfied. Such regions has been defined as 2SRA phase, or more precisely as SRA (N)+ SRA (I) phase. The same also had its existence in \citep{22}, however, in this paper, we shall find the mirror frequency $\omega_m$ to determine the physics of such coexisting regime and we shall exploit such condition to alter the phase portraits.
\subsection{SRB phase}
We defined the condition $(\omega_a+ U\mid a \mid^2)$= 0 as the origin of the B type superradiance. The same condition when incorporated in Eq. (7), yields $4gS_z a_1$= 0. Evidently, this bounds $a$ to be purely imaginary. Correspondingly, the initial condition also yields:-
\begin{equation}
\mid a \mid ^2= -\frac{\omega_a}{U},
\end{equation}
which again suggests the same nature for $a$. Hence $a_1$= 0, which when plugged in Eq. (8) and Eq. (9) yields: -
\begin{equation}
S_x^2= -\frac{\kappa^2 \omega_a}{2gU}
\end{equation}
and
\begin{equation}
S_z^2= \Big(\frac{\omega+ 2\delta_0 b_1}{U}\Big)^2.
\end{equation}
As noted previously, $S_y$= 0 was defined in SRA phase and in SRB phase $S_y$ $\ne$ 0. Hence, it follows from the normalization condition that $S_x^2+ S_z^2 \le$ $N^2/4$, where the above expressions give the corresponding values, with Eq. (10) determining the expression for $b_1$ and $\mid b \mid^2$.
\begin{figure}
\caption{Dynamical phase portrait of the stable attractors as a function of cavity frequency $\omega_c$ and $g\sqrt{N}
\end{figure}
\subsection{The Phase portraits}
We finally summarize the phase portraits of the dynamical system, with chosen parameters that satisfy the Routh- Hurwitz criteria \citep{33, 34} for a stable optomechanical system. We plot the phase portraits as a function of $g\sqrt{N}$, where N is the number of atoms $\approx$ $10^6$. We consider all the cases possible through analytical treatment of the dynamical equations of the system and it is noteworthy to mention here that although all these phase regions can be investigated in various experimental conditions, however, not all will emerge in a single experiment. The designing of such a system to observe various phase regions discussed here is a matter of technological advancement in controlling the parameters of the system. Experiments reported by K. Baumann $\textit{et al.}$ \citep{14, 15} showed the system evolving from normal phase (N) with all spins pointing downwards and no photons.
The first panel of fig.2 shows the phase diagram for UN= 0 MHz. The purple line marks the onset of superradiance from both normal (N) and inverted (I) states with all spins pointing downwards and upwards respectively. For $\omega<$ 0, the normal state (N) becomes unstable and the inverted state (I) becomes stable instead. As the backaction parameter is reduced (UN= -40 MHz), the SRA phase boundaries (purple line) between the (N) and (I) state shift to higher and lower frequency respectively. Simultaneously, SRB phase (red line) emerges which coincides with SRA(N) and SRA(I) for negative U as discussed previously and few new regimes come to play as seen from the second panel of fig. 2. The (N) and (I) phase coexists due to the shift of the SRA boundary and also gives rise to (SRB+ N), (SRB+ I) and (SRB+ N+ I) regions. Due to the frequency shifts induced by negative U, there exists a small region where SRA(N) and SRA(I) coexists, where both the roots of $S_z$ are supported. These phases are represented as 2SRA (SRA(N)+ SRA(I)) in this paper and we shall deal with the same in next section. \\
Although we have portrayed all the possible cases (for $UN$= -40 MHz) in the middle panel of Fig. (2), not all can be simultaneously observed in any single experiment. As reported by Esslinger and his group \citep{14}, the first superradiant transition was observed from inverted state (I) to SRA (Inverted) which corresponds to the lower symmetrical half of the phase portrait. The realization of other transitions is purely dependent on the conditions of the system. Considering $S_y$= 0 and the initial state being the normal state (-N/2 and no photons), the phase transition would correspond to the SRA (N) denoted by the purple line on the positive Y axis of top panel of Fig. (2) and vice versa for system prepared with inverted state (N/2) and operated with negative effective cavity frequency ($\omega$). The purple line marks the phase transition from superfluid to a self organized state and as seen from the figure, the critical transition point increases as the effective cavity frequency ($\omega$) in increased. This also supports the analytical results in \citep{14, 17, 21, 22, 23, 29} which showed the critical point at $\frac{1}{2} \Big( \frac{\omega_a}{\omega} (\kappa^2+ \omega^2) \Big)^{1/2}$ for $U$= 0. \\
Interestingly, as we reduce the back reaction parameter $U$ (middle panel, Fig. (2)), the SRA phase boundary shifts towards each other by $\pm UN/2$ so as to offer an identical superradiant phase (area covered between the purple lines and black horizontal lines (at $\pm UN/2$) in the middle panel of Fig. 2). Although we can never witness both the transitions in a single experiment, however, theoretical study predicts an identical superradiant phase when operated with effective cavity frequency ranging between $\pm UN/2$ MHz and initial state being normal or inverted. In simple words, for $UN$= -40 MHz we can predict an identical phase transition when operated with $-20 \le \omega \le 20$ MHz without worrying whether we started from normal (N) or inverted (I) state. Thus we can start from any mixed state configuration ($i.e.$ a combination of spin up (I) and spin down (N)) and still expect to get a phase transition if $UN$ is negative. This is analogous to the case of preparing mixed atoms with 50$\%$ spin up and 50$\%$ spin down and still get an identical superradiance as in two atom Dicke model \citep{52}. The advantage lies in the fact that with negative back action parameter, we get a short window of selecting our effective cavity frequency ($- UN/2 \le \omega \le +UN/2$) and worry not about the initial condition (N or I) to observe a Type A superradiance. The comparison becomes evident when we see the top panel of Fig. (2), which showed phase transition only when the system is operated and prepared in a combination of either positive $\omega$ and Normal state (+$\omega$, N) or negative $\omega$ and Inverted state (-$\omega$, I). Thus a negative variation of $U$ gives us a freedom to choose our effective cavity frequency ($\omega$) and initial state. \\
As the backaction parameter is made positive (UN= 40 MHz), the SRB phase vanishes for obvious reasons discussed previously. The SRA (N) and SRA (I) shifts away from each other by $\pm UN/2$ as seen from the lower panel of Fig. (2). The separation of the boundaries in opposite direction leads to the formation of another region termed here as persistent oscillation regime. Evidently, no phase transition can be observed when the system is operated with effective cavity frequency ($\omega$) between $\pm UN/2$. As the name suggests, this regime describes persistent oscillation and no steady state is reached even for long duration experiments, thereby predicting the presence of limit cycle. The notion of persistent oscillation will become clear in time evolution section when we shall simulate the system with initial conditions described by point (c), which lies in the concerned region.\\
We observe the type B superradiance only when $(\omega_a + U\mid a \mid ^2)$=0 $i.e.$ when $U$ is negative since $\omega_a >$0. The critical line separating the superfluid and self organized state has been denoted with red colour in the middle panel of Fig. (2). The SRB imposing condition ($S_x^2+ S_z^2 \le N^2/4$) itself reveals the fact that it can take both $\pm N/2$ ($i.e.$ both Normal (N) and Inverted (I) state), which marks its appearance between $\pm UN/2$ in the phase portraits. Thus when $UN$= -40 MHz, and we have an initial mixed state configuration (N+ I) and effective cavity frequency $(\omega$) being operated between $\pm 20$ MHz, we can get either a Type A superradiance or a Type B superradiance depending on whether $S_y$= 0 or $(\omega_a+ U\mid a \mid^2)$= 0 respectively but never both simultaneously in a single experiment.
\section{2SRA phase}
\begin{figure}
\caption{Magnified view of the dynamical phase diagram for UN= -30 MHz for $\omega_m$= 0.2, 0.4 and 0.8 for upper, middle and lower panel respectively. Parameters chosen were same as in the previous plots.}
\end{figure}
In this section we aim to discuss the role of the mechanical mirror in defining the phase portraits of the system. As hinted previously, there are regions where both the roots of $S_z$ are supported and the SRA(N) and SRA(I) regions coincides to describe the new phase. Although evident from previous discussion that the mirror frequency plays no role in defining the SRA region, however, the SRB phase does have an explicit dependency on $\omega_m$ as seen from Eq. (13) and (14), together with Eq. (10) for the expression of $b_1$. We produce here a magnified view of the dynamical phase diagram for UN= -30 MHz and determine the variation in transition point for different values of $\omega_m$. Interestingly, the 2SRA phase is no more distinct as in the case of a fixed mirror \citep{22} and in the optomechanical case, the mirror frequency determines the physics of this tricritical point where all the phase boundaries cross each other. \\
For $\omega_m$= 0.2, the top panel of Fig. (3) shows no 2SRA region and the same starts becoming prominent as the mirror frequency $\omega_m$ is increased as seen from middle and the lowermost plots of Fig. (3). The mirror is therefore found to be altering the coexisting regime, for experimentally realizable values of the mirror frequency. These optical systems with a movable cantilever can therefore be efficiently used for controlling the crtitical point and also the coexisting regime. With these plots, the effect of the mirror frequency can be well established. However, we may demand to alter the phase portraits more since the change with the mechanical mirror is almost negligible for any use as in experimental phenomenon like quantum entanglement or manipulation etc. So can we devise and conceive any further modification to the system that can allow further manipulation of the critical transition point. We shall deal with the same in Sec VI, with an aim of modifying the phase diagrams by some easy controllable parameter.
\section{Time Evolution}
In order to get insight on the distinction between the described phases, we examine the time evolution of the system from various initial conditions lying in different phase regions. We mainly consider the points (a), (b) and (c) marked in the dynamical phase diagrams (fig 2) which lies in the (SRB+ N+ I), SRA and persistent oscillation regime respectively. We solve the semiclassical equations of the system numerically for $S_x$, $S_y$, and $S_z$ by fourth order Runge Kutta method and illustrate the relaxation time in reaching their corresponding stable attractors. The plots below shows the time evolution of the system from different initial conditions.
\begin{figure}
\caption{Time evolution of the system from different initial condition. The top panel describes the point (a) and point (b) of fig.1 and the lower panel shows the persistent oscillations predicting a limit cycle in persistent oscillation regime for positive back action parameters. Other parameters used are same as in previous plots.}
\end{figure}
The top panel of fig. 4 shows the time evolution for point (a) (UN= -40 MHz) and point (b) (UN= 40 MHz) in the superradiant regime that are close to the normal and inverted state. The plot well describes the relaxation time for reaching their stable attractors. For point (a), $S_z$ initially increases and finally attains a stable value in approximately 0.7 ms thereby prediciting a stable case for realistic experiments. Point (b) lies in the SRA (N) region just above the persistent oscillation region and the time evolution of $S_z$ (blue curve) shows the system reaching their stable attractors in approximately 0.7 ms. As the initial condition enters the oscillation regime (point(c)), all the system parameters ($S_y$, $S_z$) starts to oscillate periodically at long times and no stable points are reached even after long duration as shown in the lower panel of fig. 4. Since the motion is described in a two dimensional plane, the attractors represent a simple limit cycle \citep{24}, thereby tagging the entire bounded plane as persistent oscillation regime.
\section{Dicke model in the presence of an external pump}
We modify our previous model by adding an external mechanical pump, which can be any external object in physical contact with the mirror or an external laser that is capable of changing the mirror frequency via radiation pressure. The pump can excite the mirror by coupling with the mirror fluctuation quadratures. The Hamiltonian of the new system, takes the form ($\hbar$= 1 throughout the paper) \citep{19, 27}: -
\begin{eqnarray} \label{eqn1}
H&=& \omega_a S_z+ \omega a^{\dagger}a+ \omega_m b^{\dagger}b+ \delta_0 a^{\dagger}a(b+ b^{\dagger}) \nonumber \\
&+& g(a+ a^{\dagger})(S_{+}+ S_{-})+ US_za^{\dagger}a+ \eta_p(b+ b^{\dagger}),
\end{eqnarray}
\begin{figure}
\caption{The dynamical phase diagram in the presence of the external mechanical pump. UN= -20 MHz and $\eta_p$= 1. Other parameters are same as in previous plots.}
\end{figure}
where $\eta_p$ represents the mechanical pump frequency and the last additional term describes the energy due to it. The mechanical pump frequency will be considered to be small here, $\textit{i.e.}$ $ 0 \le \eta_p \le1$. To proceed further, we begin with the semiclassical equations, of which the following equations gets modified: -
\begin{equation}
\dot{b}= -i\omega_m b- i\delta_0\mid a \mid^2- i\eta_p- \Gamma_m b.
\end{equation}
We repeat the same analysis to determine the dynamical phase diagram of the system in the presence of the mechanical pump for both SRA and SRB phase. We produce here the dynamical phase portraits for SRA and SRB separately in fig. 5 to unveil the effect of the mechanical pump. The dotted lines in both the plots of fig. 5 marks the SRA and SRB phase boundaries in the absence of the mechanical pump and the bold curve represents the phase portrait when the external mechanical pump starts working. The blue shaded region named $\eta$- SRA and $\eta$- SRB represents the extra region created by the external mechanical pump. Clearly, the shaded region decreases the critical transition point both in positive and negative direction symmetrically. Although the external mechanical pump has changed the dynamical phase diagram to a large extend, the physics behind the time evolution remains almost same as in previous case with minor change in the relaxation time. The external mechanical pump frequency $\eta_p$ also enhances the 2SRA region to a large extend since both the SRA and SRB phase shows considerable increase in phase area. We don't produce these plots as these remains evident from the plots of fig. 4. Thus it is clear from the discussion in this section and section IV that the phase portraits can be altered and enhanced by a simple modification. Although the SRA phase region is unaltered by mirror frequency initially, the same can be modified when we add external force to the mirror. These systems can be used for altering the phase transition in Dicke model by simple controllable parameters like the external mechanical pump. Such can find use in experiments like detecting quantum entanglement, which tends to infinity at the critical point \citep{51}.
\section{Conclusion}
In this paper, we have explored the dynamics of an optomechanical system with ultracold atoms between the optical cavities. Within the framework of non equilibrium Dicke model, we presented the rich phase portrait of attractors, including regimes of coexistence and persistent oscillations. We conclude from the analytical methods that the optomechanical system remains handy over an optical system in terms of control over phase transition and dynamical phase regions. The cantilever was found to be enhancing the coexisting region to a large extend and the persistent oscillation regime predicted the existence of limit cycle that prohibits reaching any stable state even in very long duration experiments. To study the system further, we added an external mechanical pump and found the external pump enhancing both the SRA and SRB phases thereby predicting an enhancement even in coexisting regions. We thereby predict a system that alters the phase transition in a Dicke model through a simple and effective process. Such system can also be used to study the dynamical entanglement in different regimes in the presence and absence of mechanical pump which can be used as a tool to selectively modify and alter the entanglement \citep{45} between the modes.
\end{document} |
\begin{document}
\title{\bf Online Local Volatility Calibration by Convex Regularization}
author{Vinicius V.L. Albani\thanks{IMPA, Estr. D. Castorina
110, 22460-320 Rio de Janeiro, Brazil, \href{mailto:vvla@impa.br}{\tt
vvla@impa.br}} \, and \,
Jorge P. Zubelli\thanks{IMPA, Estr. D. Castorina
110, 22460-320 Rio de Janeiro, Brazil, \href{mailto:zubelli@impa.br}{\tt
zubelli@impa.br}}
}
\date{\today}
\maketitle
\begin{abstract}
We address the inverse problem of local volatility surface calibration from market given option prices. We integrate the ever-increasing flow of option price information into the well-accepted local volatility model of Dupire. This leads to considering both the local volatility surfaces and their corresponding prices as indexed by the observed underlying stock price as time goes by in appropriate function spaces. The resulting parameter to data map is defined in appropriate Bochner-Sobolev spaces. Under this framework, we prove key regularity properties. This enable us to build a calibration technique that combines online methods with convex Tikhonov regularization tools. Such procedure is used to solve the inverse problem of local volatility identification. As a result, we prove convergence rates with respect to noise and a corresponding discrepancy-based choice for the regularization parameter. We conclude by illustrating the theoretical results by means of numerical tests.
\end{abstract}
\noindent {\bf Keywords:} Local Volatility Calibration, Convex Regularization, Online Estimation, Morozov's Principle, Convergence Rates.
\section{Introduction}\label{sec:intro}
A number of interesting problems in nonlinear analysis are motivated by questions from mathematical finance.
Among those problems, the robust identification of the variable diffusion coefficient that appears in Dupire's local volatility model~\cite{dupire,volguide} presents substantial difficulties for its nonlinearity and ill-posedness. In previous works tools from Convex Analysis and Inverse Problem theory have been used to address this problem. See \cite{acpaper} and references therein.
In this work, we incorporate the fact that as time evolves more data is available for the identification of Dupire's volatility surface. Thus we develop an {\em online} approach to the ill-posed problem of the local volatility surface calibration. Such surface is characterized by a non-negative two-variable function $\sigma = \sigma({\tilde{a}}u,K)$ of the time to expiration ${\tilde{a}}u$ and the strike price $K$.
In what follows, we consider that the local volatility surfaces are indexed by the observed underlying asset price $S_0$.
The reason for that stems from the fact that if we try to use information of prices observed on different dates, there is no financial or economical reason for the volatility surface to stay exactly the same. Thus, in principle we may have different volatility surfaces, although such change may be small.
Let us quickly review the standard Black-Scholes setting and Dupire's local volatility model. Recall that an option or derivative is a contract whose value depends on the value of an underlying stock or index. Perhaps the most well known derivative is an European call option, where the holder has the right (but not the obligation) to buy the underlying at time $t = T$ for a strike value $K$. We shall denote the stochastic process defining such underlying $S(t) = S(t,\omega)$, where as usual we assume that it is an adapted stochastic process on a suitable filtered probability space $(\Omega,\mathscr{U},\mathbb{F},\widetilde{\mathbb{P}})$, where $\mathbb{F} = \{\mathbb{F}_t\}_{t \in \R}$ is a filtration \cite{korn}.
It is well known \cite{dupire,volguide,korn} that, by setting the current time as $t=0$, the value $C$ of an European call option with strike $K$ and expiration $T = {\tilde{a}}u$ satisfies:
\begin{equation}
\left\{
\begin{array}{rcll}
-\displaystyle\frac{\partial C}{\partial {\tilde{a}}u} +
\frac{1}{2}\sigma^2({\tilde{a}}u,K)K^2\frac{\partial^2 C}{\partial K^2} -
bK\frac{\partial C}{\partial K} &=& 0 & {\tilde{a}}u > 0, ~K \geq 0\\
C({\tilde{a}}u = 0,K) &=& (S_0 - K)^+, & \text{for}~ K>0,\\
\displaystyle\lim_{K\rightarrow +\infty}C({\tilde{a}}u,K) & = & 0,&\text{for }~ {\tilde{a}}u > 0,\\
\displaystyle\lim_{K\rightarrow 0^+}C({\tilde{a}}u,K) & = & S_0,&\text{for }~ {\tilde{a}}u > 0
\end{array}
\right.
\label{dup1}
\end{equation}
where $b$ is the difference between the continuously compounded interest and dividend rates of the underlying asset. In what follows, we assume that such quantities are constant.
Defining the diffusion parameter $a({\tilde{a}}u,K) = \sigma({\tilde{a}}u,K)^2/2$, Problem (\ref{dup1}) leads to the following parameter to solution map:
$$
\begin{array}{rcl}
F : D(F) \subset X &\longrightarrow & Y\\
a \in D(F) & \longmapsto & F(a) = C \in Y
\end{array}
$$
where $X$ and $Y$ are Hilbert spaces to be properly defined below. $D(F)$ is the domain of the parameter to solution map (not necessarily dense in $X$) and $C = C(a,{\tilde{a}}u,K)$ is the solution of Problem (\ref{dup1}) with diffusion parameter $a$.
The inverse problem of local volatility calibration, as it was tackled in previous works \cite{crepey,acthesis,acpaper,eggeng}, consists in given option prices $C$, find an element ${\tilde{a}}$ of $D(F)$ such that $F({\tilde{a}}) = C$ in the least-square sense below. Indeed, the operator $F$ is compact and weakly closed. Thus, this inverse problem is ill-posed. In \cite{crepey,acthesis,acpaper,eggeng} different aspects of the Tikhonov regularization were analyzed. In our case, it is characterized by the following: Find an element of
$$
argmin \left\{\|F(a) - C\|^2_Y + alpha f_{a_0}(a) \right\} ~~\text{subject to }~ a \in D(F) \subset X,
$$
where $f_{a_0}$ is a weak lower semi-continuous convex coercive functional. The analysis presented in \cite{crepey,acthesis,acpaper,eggeng} was based on an {\em a priori} choice of the regularization parameter with convex regularization tools.
In contrast, in the present work we explore the dependence of the local volatility surface on the observed asset price in order to incorporate different option price surfaces in the same procedure of Tikhonov regularization. More precisely, we consider the map
$$
\begin{array}{rcl}
{\mathcal{U}}: D({\mathcal{U}}) \subset \mathcal{X} & \longmapsto & \mathcal{Y}\\
{\mathcal{A}} \in D({\mathcal{U}}) & \longmapsto & {\mathcal{U}}({\mathcal{A}}): S\in [S_{\min},S_{\max}] \mapsto C(S,a(S))
\end{array}
$$
where $C(S,{\mathcal{A}}(S))$ is the solution of (\ref{dup1}) with $S_0 = S$ and $\sigma^2/2 = a(S)$. Moreover, ${\mathcal{A}}$ maps $S \in [S_{\min},S_{\max}]$ to $a(S) \in D(F)$ in a well-behaved way.
In this context the inverse problem becomes the following: Given a family of option prices $\mathcal{C} \in \mathcal{Y}$, find ${\tilde{a}}f \in D({\mathcal{U}})$ such that ${\mathcal{U}}({\tilde{a}}f) = \mathcal{C}$. We shall see that the operator ${\mathcal{U}}$ is also compact and weakly closed. Thus, this problem is also ill-posed. The corresponding regularized problem is defined by the following:
Find an element of
$$
argmin\left\{ \displaystyle\int_{S_{\min}}^{S_{\max}}\|F(a(S)) - C(S)\|^2_YdS + alpha f_{{\mathcal{A}}_0}({\mathcal{A}}) \right\} ~~\text{ subject to }~ {\mathcal{A}} \in D({\mathcal{U}}).
$$
The main contributions of the current work are the following:
Firstly, we extend the local volatility calibration problem to local volatility families. This new setting allows incorporating more data into the calibration problem, leading to an online Tikhonov regularization. We prove that the so-called direct problem is well-posed, i.e., the forward operator satisfies key regularity properties.
This framework generalizes in a nontrivial way the structure used in previous works \cite{crepey,acthesis,acpaper,eggeng} since it requires the introduction of more tools, in particular that of Bochner spaces.
Secondly, in this setting, we develop a convergence analysis in a general context, based on convex regularization tools. See \cite{schervar}.
Thirdly, we establish a relaxed version of Morozov's discrepancy principle with convergence rates. This allows us to find the regularization parameter appropriately for the present problem. See \cite{anram,moro}.
The article is divided as follows:
\noindent In Section~2, we present the setting of the direct problem. In Section~3, we define properly the forward operator and prove some key regularity properties that are important in the analysis of the inverse problem. This is done in Theorem~\ref{prop22} and Propositions~\ref{prop4}, \ref{prop6}, \ref{prop7} and \ref{prf1}. In Section~4, we tie up the inverse problem with convex Tikhonov regularization under an {\em a priori} choice of the regularization parameter. The convergence of the regularized solutions to the true one, with respect to $\delta\rightarrow 0$, is stated in Theorem~\ref{tc1}. In Section~5 we establish the Morozov discrepancy principle for the present problem with convergence rates. This is done in Theorems~\ref{tma} and \ref{mor:cr}. Illustrative numerical tests are presented in Section~6.
\section{Preliminaries}\label{sec:preliminar}\label{sec:dupsurv}
We start by setting the so-called direct problem. It is based on the pricing of European call options by a generalization of Black-Scholes-Merton model.
Performing the change
of variables $y := \text{log}(K/S_0)$ and ${\tilde{a}}u : = T$ on the Cauchy problem (\ref{dup1}) and defining
$u(S_0,{\tilde{a}}u,y) : = C(S_0,{\tilde{a}}u,S_0\text{e}^y)$ and $a(S_0,{\tilde{a}}u,y) :=
\frac{1}{2}\sigma^2(S_0,{\tilde{a}}u,S_0\text{e}^y)$,
it follows that $u(S_0,{\tilde{a}}u,y)$ satisfies
\begin{equation}
\left\{
\begin{array}{rcll}
-\displaystyle\frac{\partial u}{\partial {\tilde{a}}u} + a(S_0,{\tilde{a}}u,y)\left(\frac{\partial^2 u}{\partial y^2}
- \frac{\partial u}{\partial y}\right)
+ b\frac{\partial u}{\partial y} &=& 0 & {\tilde{a}}u > 0, ~y \in \R\\
u({\tilde{a}}u = 0,y) &=& S_0(1 - \text{e}^y)^+, &\text{for }~ y \in \R,\\
\displaystyle\lim_{y\rightarrow +\infty}u({\tilde{a}}u,y) & = & 0,&\text{for }~ {\tilde{a}}u > 0,\\
\displaystyle\lim_{y\rightarrow -\infty}u({\tilde{a}}u,y) & = & S_0,&\text{for }~ {\tilde{a}}u > 0.
\end{array}\right.
\label{dup2}
\end{equation}
Note that, $\sigma$ and $a$ are assumed strictly positive and are related by a smooth bijection (since $\sigma>0$).
Thus, in what follows we shall work only with the local variance $a$ instead of
volatility $\sigma$. This simplifies the analysis that follows.
Denote by $D:=(0,T)\times \R$ the set where problem \eqref{dup2} is
defined. From \cite{eggeng} we know that \eqref{dup2} has a
unique solution in $W^{1,2}_{2,loc}(D)$, the space of functions
$u : ({\tilde{a}}u,y) \in D \mapsto u({\tilde{a}}u,y) \in \mathbb{R}$ such that, it has locally
squared integrable weak derivatives up to order one in ${\tilde{a}}u$ and up to order two
in $y$.
We now define the set where the diffusion parameter $a$ lives. For fixed $\varepsilon > 0$, take
scalar constants $a_1,a_2 \in \mathbb{R}$ such that $0 < a_1 \leq a_2 < +\infty$
and a fixed function $a_0 \in \he$, with $a_0 < a < a_1$. Define
\begin{equation}
Q:= \{a \in a_0 + \he : a_1\leq a \leq a_2\}
\label{domopdi}
\end{equation}
Note that $Q$ is weakly closed and has nonempty interior under the standard topology of $\he$. See the first two chapters of \cite{acthesis,acpaper} and references therein.
\section{The Forward Operator}\label{sec:forward}
Since we assume that the local variance surface is dependent on the current price, we have to introduce proper spaces for the analysis of the problem. As it turns out, we have to make use of Bochner integral techniques. See \cite{evanspde,reedsimon1,yosida}. The main reference for this section is \cite{haschele}.
We start with some definitions. Given a time interval, say $[0,\overline{T}]$, the realized prices $S(t)$ vary within $[S_{\min},S_{\max}]$. After reordering $S(t)$ in ascending order, we perform the change of variables $s = S(t)-S_{\min}$, denote $S = S_{\max}-S_{\min}$. Thus $s \in [0,S]$. Hence, for each $s$, we denote $a(s) := a(s,{\tilde{a}}u,y)$ the local variance surface correspondent to $s$.
\begin{df}
Given ${\mathcal{A}} \in {L^2(0,S,H^{1+\varepsilon}(D))}$, with ${\mathcal{A}} : s\mapsto a(s)$ (see \cite{yosida}), we define its Fourier series $\hat{{\mathcal{A}}} = \{\hat{a}(k)\}_{k \in \Z}$ by
$$
\hat{a}(k) := \displaystyle\frac{1}{2S}\int^S_0 a(s)\exp(-iks\pi/S)ds +
\displaystyle\frac{1}{2S}\int^0_{-S} a(-s)\exp(-iks\pi/S)ds.
$$
\end{df}
\noindent It is well defined, since $\{s \mapsto a(s)\exp(-iks2\pi/S)\}$ is weakly measurable and ${L^2(0,S,H^{1+\varepsilon}(D))} \subset L^1(0,S,\he)$ by the Cauchy-Schwartz inequality.
We now define a class of Bochner-type Sobolev spaces:
\begin{df}
Let $\X$ be the space of ${\mathcal{A}} \in {L^2(0,S,H^{1+\varepsilon}(D))}$, such that
$$
\|{\mathcal{A}}\|_l := \displaystyle\sum_{k \in \Z} (1+|k|^l)^2\|\hat{a}(k)\|^2_{\he_\mathbb{C}} < \infty,
$$
where $\he_\mathbb{C} = \he \oplus i\he$ is the complexification of $\he$. Moreover, $\X$ is a Hilbert space with the inner product
$$
\langle {\mathcal{A}},{\tilde{a}}f\rangle_l := \displaystyle\sum_{k\in\Z}(1+|k|^l)^2\langle a(k),{\tilde{a}}(k)\rangle_{\he_\mathbb{C}}.
$$
\end{df}
\begin{pr}{\cite[Lemma~3.2]{haschele}}
For $l > 1/2$, each ${\mathcal{A}} \in \X$ has a continuous representative and the map $i_l : \X \hookrightarrow C(0,S,\he)$ is continuous (bounded).
Moreover, we have the estimate
\begin{equation}
\displaystyle\sup_{s\in[0,S]}\|u(s)\|_{\he} \leq \|{\mathcal{U}}\|_{l}\left(2\sum_{k = 0}^{\infty}\frac{1}{(1+k^l)^2}\right)^{1/2}.
\label{estimate1}
\end{equation}
Defining the application $\langle {\mathcal{A}} , x\rangle_{\he} : = \{s\mapsto \langle a(s), x \rangle\}$ for each $x$ in $\he$ and ${\mathcal{A}}$ in $\X$,
it follows that $\langle {\mathcal{A}} , x\rangle_{\he}$ is an element of $H^l[0,S]$ and the inequality
$\|\langle {\mathcal{A}} , x\rangle_{\he}\|_{ H^l[0,S]} \leq \|{\mathcal{A}}\|_l\|x\|_{\he}$ holds. Moreover, for every ${\mathcal{A}},\mathcal{B} \in {L^2(0,S,H^{1+\varepsilon}(D))}$, we have the identity
$$
\langle {\mathcal{A}}, \mathcal{B} \rangle_{L^2(0,S,H^{1+\varepsilon}(D))} = \sum_{k \in \Z}\langle \hat{a}(k),\hat{b}(k)\rangle_{\he_\mathbb{C}}.
$$
\label{p1}
\end{pr}
\begin{lem}
Assume that $l > 1/2$. If the sequence $\{{\mathcal{A}}_n\}_{n\in\N}$ converges weakly to ${\tilde{a}}f$ in $\X$, then, the sequence $\{a_k(s)\}_{k\in\N}$ weakly converges to ${\tilde{a}}(s)$ in $\he$ for every $s \in [0,S]$.
\label{lemw}
\end{lem}
\begin{proof}
Take a $\{{\mathcal{A}}_n\}_{n\in\N}$ and ${\tilde{a}}f$ as above. We want to show that, given a weak zero neighborhood $U$ of $\he$, then for a sufficiently large $n$, $a_n(s) - a(s) \in U$ for every $s \in [0,S]$. A weak zero neighborhood $U$ of $\he$ is defined by a set of $alpha_1,...,alpha_K \in \he$ and an $\epsilon > 0$ such that $g \in \he$ is an element of $U$ if $\max_{k = 1,...,K}|\langle g , alpha_n\rangle| < \epsilon$.
Since the immersion $H^l[0,S] \hookrightarrow C([0,S])$ is compact and $H^l[0,S]$ is reflexive, it follows that each weak zero neighborhood of $H^l[0,S]$ is a zero neighborhood of $C([0,S])$. Furthermore, from Proposition \ref{p1} we know that
$\langle {\mathcal{A}}, alpha \rangle_{\he} \in H^l[0,S]$
with its norm bounded by
$\|{\mathcal{A}}\|_l\|alpha\|_{\he}$, for every $n \in \N$ and $alpha \in \he$.
Thus, we take the smallest closed ball centered at zero, $B$, which contains
$\langle {\tilde{a}}f,alpha_k\rangle_{\he}$
with $k = 1,...,K$ and every $\langle {\mathcal{A}}_n,alpha_k\rangle_{\he}$ with $n\in \N$ and $k = 1,...,K$.
Therefore, choosing $\epsilon > 0$ as above, it is true that for each $k = 1,...,K$, there are
$f_{k,1}, ...,f_{k,M(k)} \in H^l[0,S]$
and
$\eta_k > 0$, such that $\|f\|_{C([0,S])} < \epsilon$
for every $f \in B$ with $\max_{m = 1,...,M(k)}|\langle f,f_{k,m}\rangle|<\eta_k$.
Hence, we define $\mathcal{C}_{k,m} := alpha_k \otimes f_{k,m} \in \X^*$ and the weak zero neighborhood $A = \cap^K_{k = 1}A_k$ of $\X$ with
$$
A_k := \{{\mathcal{A}} \in \X ~: ~|\langle {\mathcal{A}}, \mathcal{C}_{k,m}\rangle|\leq \eta_k, ~m=1,...,M(k) \}.
$$
As $A$ is a weak zero neighborhood of $\X$, it is true that for sufficiently large $n$,
${\mathcal{A}}_n - {\tilde{a}}f \in A$, which implies that $a_n(s) - {\tilde{a}}(s) \in U$ for every $s \in [0,S]$, i.e., $\{a_n(s)\}_{n\in \N}$
weakly converges to ${\tilde{a}}(s)$ for every $s \in [0,S]$.
\end{proof}
Define the set
${\mathfrak{Q}} := \{ {\mathcal{A}} \in \X : a(s) \in Q ,~\forall s \in [0,S]\}$,
i.e., each ${\mathcal{A}}$ in ${\mathfrak{Q}}$ is the map
${\mathcal{A}} : s \in [0,S] \mapsto a(s) \in Q$. Note that ${\mathfrak{Q}}$ is the space of $Q$-valued paths, with $Q$ defined in (\ref{domopdi}).
\begin{pr}
For $l > 1/2$, the set ${\mathfrak{Q}}$ is weakly closed and its interior is nonempty in $\X$.
\label{p2}
\end{pr}
\begin{proof}
By Lemma \ref{lemw} and the fact that $Q$ is weakly closed it follows that ${\mathfrak{Q}}$ is weakly closed. The interior of ${\mathfrak{Q}}$ is nonempty since the inclusion $\X \hookrightarrow C(0,S,\he)$
is continuous and bounded.
Note that, given $\epsilon > 0$, it follows that ${\tilde{a}}f = \{s \mapsto {\tilde{a}}(s)\}$ with
$\underline{a} + \epsilon \leq {\tilde{a}}(s) \leq \overline{a} + \epsilon$
for every $s \in [0,S]$ is in the interior of ${\mathfrak{Q}}$.
\end{proof}
We stress that, in what follows, we always assume that $l>1/2$, since it is enough to state our results concerning regularity aspects of the forward operator.
We define below the forward operator, that associates each family of local variance surfaces to the corresponding family of option price surfaces, determined by the Cauchy problem~\eqref{dup2}. Thus, for a given $a_0 \in Q$ we define:
$$
\begin{array}{rcl}
\mathcal{U}: {\mathfrak{Q}} &\longrightarrow& \Y,\\
{\mathcal{A}} & \longmapsto & {\mathcal{U}}({\mathcal{A}}) :s \in [0,S] \mapsto F(s,a(s)) \in \ya,
\end{array}
$$
where $[{\mathcal{U}}({\mathcal{A}})](s) = F(s,a(s)):=u(s,a(s))-u(s,a_0)$ and $u(s,a)$ is the solution of the Cauchy problem~\eqref{dup2} with local variance $a$. The following results state some regularity properties concerning the forward operator. See \cite{acpaper} and references therein.
\begin{pr}
The operator $F:[0,S]\times Q\longrightarrow \ya$ is continuous and compact. Moreover, it is sequentially weakly continuous and weakly closed.\label{prop21}
\end{pr}
We define below the concept of Frech\'et equi-differentiability for a family of operators.
\begin{df}
We call a family of operators
$\{\mathcal{F}_s:Q\longrightarrow\ya \left|~ s \in [0,S] \right.\}$
Frech\'et equi-differentiable, if for all $\tilde{a} \in Q$ and $\epsilon > 0$, there is a $\delta > 0$, such that
$$
\displaystyle\sup_{s \in [0,S]}\|\mathcal{F}_t(\tilde{a}+h) - \mathcal{F}_s(\tilde{a}) - \mathcal{F}^\prime_s(\tilde{a})h\| \leq \epsilon\|h\|,
$$
for $\|h\|_{\he}<\delta$ and $\mathcal{F}^\prime_s(\tilde{a})$ the Frech\'et derivative of $\mathcal{F}_s(\cdot)$ at $\tilde{a}$.
\end{df}
Using this concept, we have the following proposition.
\begin{pr}
The family of operators $\{F(s,\cdot) : Q \longrightarrow \ya \left|~s \in [0,S] \right.\}$ is Frech\'et equi-differentiable.
\label{prop4}
\end{pr}
\begin{proof}
Given ${\tilde{a}} \in Q$ and $\epsilon > 0$, define $w = F(s,{\tilde{a}}+h) - F(s,{\tilde{a}}) - \partial_a F(s,{\tilde{a}})h$, it is equivalent to
$w = u(s,{\tilde{a}}+h) - u(s,{\tilde{a}}) - \partial_a u(s,{\tilde{a}})h$. We denote $v := u(s,{\tilde{a}}+h) - u(s,{\tilde{a}})$. Thus, by linearity $w$ satisfies
$$-w_{\tilde{a}}u + {\tilde{a}}(w_{yy} - w_y) + bw_y = h(v_{yy}-v_y),$$
with homogeneous boundary condition. Such problem does not depend on $s$, as ${\tilde{a}}$ is independent of $s$. From the proof of Proposition \ref{prop21} (see also \cite{eggeng}), we have
$
\|w\|_{\ya} \leq C\|h\|_{L^2(D)}\|v\|_{\ya}
$.
By the continuity of the operator $F$, given $\epsilon > 0 $ we can chose $h \in \he$ with $\|h\|_{\he} \leq \delta$, such that $\|v\|_{\ya} \leq \epsilon /C$ and thus the assertion follows.
\end{proof}
The following theorem is the principal result of this section, since it states some properties that are at the core of the inverse problems analysis \cite{ern,schervar}. For its proof see Appendix \ref{app:results}.
\begin{theo}
The forward operator ${\mathcal{U}}: {\mathfrak{Q}} \longrightarrow \Y$ is well defined, continuous and compact. Moreover, it is sequentially weakly continuous and weakly closed.
\label{prop22}
\end{theo}
The next result states necessary conditions for the convergence analysis. See \cite{ern,schervar}. Its proof is in the Appendix \ref{app:results}.
\begin{pr}
The operator ${\mathcal{U}}(\cdot)$ admits a one sided derivative at ${\tilde{a}}f \in {\mathfrak{Q}}$ in the direction $\mathcal{H}$, such that ${\tilde{a}}f+\mathcal{H} \in {\mathfrak{Q}}$. The derivative ${\mathcal{U}}^\prime({\tilde{a}}f)$ satisfies
$$
\left\|\mathcal{U}^\prime({\tilde{a}}f)\mathcal{H}\right\|_{\Y} \leq c\|\mathcal{H}\|_{\X}.
$$
Moreover, ${\mathcal{U}}^\prime({\tilde{a}}f)$ satisfies the Lipschitz condition
$$
\left\|{\mathcal{U}}^\prime({\tilde{a}}f) - {\mathcal{U}}^\prime({\tilde{a}}f+\mathcal{H})\right\|_{\mathcal{L}\left(\X,\Y\right)} \leq \gamma\|\mathcal{H}\|_{\X}
$$
for all ${\tilde{a}}f,\mathcal{H}\in {\mathfrak{Q}}$ such that ${\tilde{a}}f,{\tilde{a}}f+\mathcal{H} \in {\mathfrak{Q}}$. \label{prop6}
\end{pr}
The following result is a consequence of the compactness of ${\mathcal{U}}(\cdot)$.
\begin{pr}
The Frech\'et derivative of the operator ${\mathcal{U}}(\cdot)$ is injective and compact.\label{prop7}
\end{pr}
\begin{proof}
Take $\mathcal{H} \in \ker\left({\mathcal{U}}^\prime({\tilde{a}}f)\right)$. Thus, from the proof of Proposition \ref{prop6}, we have
$
h(s)\cdot (u_{yy} - u_y) = 0.
$
However, for each $t$, $G = u_{yy}-u_y$ is the solution of
$$
\left\{\begin{array}{ll}
\partial_{\tilde{a}}u G = \displaystyle\frac{1}{2}\left(\partial^2_{yy} - \partial_y\right)\left(a(s)G + bG\right)\\
G\displaystyle\left|_{{\tilde{a}}u=0} = \delta(y)\right.,
\end{array}\right.
$$
i.e., $G$ is the Green's function of the Cauchy problem above. Thus, $G > 0$ for every $y$,${\tilde{a}}u > 0$ and $s \in [0,S]$. Therefore $h(t) = 0$. Since this holds for every $s \in [0,S]$, then the result follows.
\end{proof}
We now make use of the bounded embedding of the space
$
\Y
$
into the space
$
L^2(0,S,L^2(D)),
$
since it implies that ${\mathcal{U}}$ satisfies the same results presented above with $L^2(0,S,L^2(D))$ instead of $\Y$.
Thus, we characterize the range of ${\mathcal{U}}^\prime({\mathcal{A}})$ as a subset of $L^2(0,S,L^2(D))$ and the range of ${\mathcal{U}}^\prime({\mathcal{A}})^*$
as a subset of $\X$ in order to proceed in Section~4 the convergence analysis.
\begin{pr}
The operator $\mathcal{U}^\prime({\mathcal{A}}^\dagger)^*$ has a trivial kernel.
\label{prf1}
\end{pr}
\begin{proof}
For simplicity take $b = 0$. Denote by
$
\mathcal{L} := -\partial_{\tilde{a}}u+a(\partial{yy} - \partial_y)
$
the parabolic operator of Equation \eqref{dup2} with homogeneous boundary condition and
$\mathcal{G}_{u_{yy}-u_y}$ the multiplication operator by $u_{yy}-u_y$. Thus, for each
$s \in [0,S]$, we have $\partial_a u(s,{\tilde{a}}(s)) = \mathcal{L}^{-1}\mathcal{G}_{u_{yy}-u_y}$,
where $\mathcal{L}^{-1}$ is the left inverse of $\mathcal{L}$ with null boundary conditions. By definition of
$
{\mathcal{U}}^\prime({\tilde{a}}f)^*:L^2(0,S,L^2(D)) \rightarrow \X,
$
we have,
$$
\left\langle \mathcal{U}^\prime ({\tilde{a}}f)\mathcal{H},\mathcal{Z}\right\rangle_{L^2(0,S,L^2(D))} =
\langle \mathcal{H}, \Phi \rangle_{\X},$$
$\forall ~\mathcal{H} \in \X$ and $\forall ~\mathcal{Z} \in L^2(0,S,L^2(D))$,
with $\Phi = {\mathcal{U}}^\prime ({\tilde{a}}f)^*\mathcal{Z}$.
Thus, given any $\mathcal{Z} \in \ker\left({\mathcal{U}}^\prime ({\tilde{a}}f)^*\right)$, it follows that
$$
\begin{array}{rcl}
0 &=& \left\langle {\mathcal{U}}^\prime ({\tilde{a}}f)\mathcal{H},\mathcal{Z}\right\rangle_{L^2(0,S,L^2(D))}
= \displaystyle\int^S_0\left\langle\mathcal{L}^{-1}\mathcal{G}_{u_{yy}-u_y}h(s),z(s) \right\rangle_{L^2(D)}ds \\
&=& \displaystyle\int^S_0\left\langle \mathcal{G}_{u_{yy}-u_y}h(s),[\mathcal{L}^{-1}]^*z(s)\right \rangle_{L^2\left(D\right)} ds =\displaystyle\int^S_0\left\langle \mathcal{G}_{u_{yy}-u_y}h(s),g(s)\right\rangle_{L^2\left(D\right)}ds,
\end{array}
$$
where $g$ is a solution of the adjoint equation
$$
g_{\tilde{a}}u+(ag)_{yy} + (ag)_y = z
$$
for each $s \in [0,S]$, with homogeneous boundary conditions. Since $z(t) \in L^2(D)$, we have that $g(s) \in \he$ (see \cite{lady}) and $g \in L^2\left(0,S,\he\right)$.
Since $\mathcal{G}>0$, from the proof of Proposition \ref{prop7} and the fact that $h \in \X$ is arbitrary, it follows that $g = 0$. Therefore $\mathcal{Z} = 0$ almost everywhere in $s \in [0,S]$. It yields that $\ker\left(\mathcal{U}^\prime (a)^*\right) = \{0\}$. \end{proof}
\begin{rem}
From the last proposition it follows that
$$
\ker\{{\mathcal{U}}^\prime({\tilde{a}}f)\} = \{0\} \Rightarrow \overline{\mathcal{R}\left\{\left({\mathcal{U}}^\prime({\tilde{a}}f)\right)^*\right\}} =
\X.
$$
In other words, the range of the adjoint operator of the Frech\'et derivative of the forward operator ${\mathcal{U}}$ at ${\tilde{a}}f$ is dense in $\X$.
\end{rem}
To finish this section we shall present below the tangential cone condition for ${\mathcal{U}}$. It follows almost directly by the above results and Theorem 1.4.2 from \cite{acthesis}. See also \cite{acpaper2}.
\begin{pr}
The map ${\mathcal{U}}(\cdot)$ satisfies the local tangential cone condition
\begin{equation}
\left\|{\mathcal{U}}({\mathcal{A}}) - {\mathcal{U}}({\tilde{a}}f) - {\mathcal{U}}^\prime({\tilde{a}}f)({\mathcal{A}} - {\tilde{a}}f)\right\|_{\Y} \leq \gamma \left\|{\mathcal{U}}({\mathcal{A}})- {\mathcal{U}}({\tilde{a}}f)\right\|_{\Y}
\end{equation}
for all ${\mathcal{A}},{\tilde{a}}f$ in a ball $B({\mathcal{A}}^*,\rho) \subset {\mathfrak{Q}}$ with some $\rho >0$ and $\gamma < 1/2$.
\label{tang}
\end{pr}
As a corollary we have the following result:
\begin{cor}
The operator ${\mathcal{U}}$ is injective.
\end{cor}
\section{The Inverse Problem}\label{sec:tikho}
Following the notation of Section~3, we want to define a precise and robust way of relating each family of European option price surfaces to the corresponding family of local volatility surfaces, both parameterized by
the underlying stock price. We first present an analysis of existence and stability of regularized solutions, then we establish some convergence rates. We also prove Morozov's discrepancy principle for the present problem with the same convergence rates.
The inverse problem of local volatility calibration can be restated as:
\noindent{\it Given a family of European call option price surfaces
${\widetilde{\mathcal{U}}} = \{s \mapsto \tilde{u}(s)\}$ in the space $\Ya$, find the correspondent family of local variance surfaces
${\mathcal{A}}^\dagger = \{s \mapsto a^\dagger(s)\} \in {\mathfrak{Q}}$, satisfying
\begin{equation}
{\widetilde{\mathcal{U}}} = {\mathcal{U}}({\mathcal{A}}^\dagger).
\label{ip1a}
\end{equation}}
In what follows we assume that for a given data ${\widetilde{\mathcal{U}}}$, the inverse problem \eqref{ip1a} has always a unique solution ${\mathcal{A}}^\dagger$ in ${\mathfrak{Q}}$. Such uniqueness follows by the forward operator being injective.
Note that, ${\widetilde{\mathcal{U}}}$ is noiseless, i.e., is known without uncertainties. This is an idealized
situation, thus, to be more realistic, we assume that we can only observe corrupted data ${\mathcal{U}^\delta}$,
satisfying a perturbed version of (\ref{ip1a}),
\begin{equation}
{\mathcal{U}^\delta} = {\widetilde{\mathcal{U}}} + \mathcal{E} = {\mathcal{U}}({\mathcal{A}}^\dagger)+\mathcal{E}
\label{pi1}
\end{equation}
where $\mathcal{E} = \{s \mapsto E(s)\}$ compiles all the uncertainties associated to
this problem and ${\widetilde{\mathcal{U}}}$ is the unobservable noiseless data. We assume further that,
the norm of $\mathcal{E}$ is bounded by the noise level $\delta
> 0$. Moreover, for each $s \in [0,S]$, we assume that $\|E(s)\| \leq \delta/S$.
These hypotheses imply that
\begin{equation}
\|{\mathcal{U}^\delta} - {\widetilde{\mathcal{U}}}\|_{\Ya} \leq \delta ~\text{ and }~
\|u^\delta(s) - \tilde{u}(s)\|_{L^2(D)} \leq \delta/S \text{ for every } s \in [0,S].
\label{errorb}
\end{equation}
Proposition~\ref{prop22} gives that ${\mathcal{U}}(\cdot)$ is compact, implying that the
associated inverse problem is ill-posed. It means that such inverse problem cannot be solved directly in a
stable way. Hence, we must apply regularization techniques. This, roughly speaking,
relies on stating the original problem under a more robust setting.
More specifically, instead of looking for an ${\mathcal{A}}^\delta \in {\mathfrak{Q}}$ satisfying
(\ref{pi1}), we shall search for an ${\mathcal{A}}^\delta \in {\mathfrak{Q}}$
minimizing the Tikhonov functional
\begin{equation}
\mathcal{F}^{{\mathcal{U}}^\delta}_{{\mathcal{A}}_0,alpha}({\mathcal{A}}) = \|\mathcal{U}^\delta -
\mathcal{U}({\mathcal{A}})\|^2_{\Ya} + alpha f_{{\mathcal{A}}_0}({\mathcal{A}}).
\label{tik1}
\end{equation}
The functional $f_{{\mathcal{A}}_0}$ has the goal of stabilizing the inverse problem and allows us to incorporate {\em a priori} information through ${\mathcal{A}}_0$.
We shall see later that, the minimizers of \eqref{tik1} are approximations for the solution of (\ref{ip1a}).
In order to guarantee the existence of stable minimizers for the functional \eqref{tik1}, we assume that $f_{{\mathcal{A}}_0} :{\mathfrak{Q}} \rightarrow [0,\infty]$ is convex, coercive and weakly lower semi-continuous. A classical reference on convex analysis is \cite{ekte}. Note that, these assumptions are not too restrictive, since they are fulfilled by a large class of functionals on $\X$. A canonical example is
$$f_{{\mathcal{A}}_0}({\mathcal{A}}) = \|{\mathcal{A}} - {\mathcal{A}}_0\|^2_{\X},$$
which is leads us to the classical Tikhonov regularization.
Recall that ${\mathcal{U}}$ is weakly continuous and ${\mathfrak{Q}}$ is weakly closed. Combining that with the required properties of $f_{{\mathcal{A}}_0}$ we can apply \cite[Theorem 3.22]{schervar}, which gives for a fixed ${\mathcal{U}^\delta} \in \Ya$ the existence of at least one element of ${\mathfrak{Q}}$ minimizing $\mathcal{F}^{{\mathcal{U}}^\delta}_{{\mathcal{A}}_0,alpha}(\cdot)$,
the functional defined in \eqref{tik1}.
For the sake of completeness, we present the definition of stability of a minimizer:
\begin{df}[Stability]
If ${\tilde{a}}f$ is a minimizer of \eqref{tik1} with data ${\mathcal{U}}$, then it is called stable if for every sequence $\{{\mathcal{U}}_k\}_{k \in \N} \subset \Y$
converging strongly to ${\mathcal{U}}$, the sequence $\{{\mathcal{A}}_k\}_{k\in \N} \subset {\mathfrak{Q}}$ of
minimizers of $\mathcal{F}^{{\mathcal{U}}^k}_{{\mathcal{A}}_0,alpha}(\cdot)$ has a subsequence
converging weakly to ${\tilde{a}}f$.
\label{dfstab}
\end{df}
Then, by \cite[Theorem 3.23]{schervar}, it follows that the minimizers of \eqref{tik1} are stable in the sense of Definition~\ref{dfstab}.
\noindent By \cite[Theorem 3.26]{schervar}, when the noise level $\delta$ and the regularization parameter $alpha = alpha(\delta)$ vanish, we can find a sequence of minimizers of \eqref{tik1} converging weakly to the solution of (\ref{ip1a}). In other words, the minimizers of (\ref{ip1a}) are indeed approximations of the family of true local volatility surfaces.
In addition, as one interpretation of this theorem, we can say that the smaller the noise level $\delta$ is, if the regularization parameter $alpha$ is properly chosen, the less dependent on the regularization functional and the {\em a priori} information the Tikhonov minimizers are.
Making use of convex regularization tools, we provide some convergence rates with respect to the noise level. In order to do that, we need some abstract concepts, as the Bregman distance related to $f_{{\mathcal{A}}_0}$, $q$-coerciveness and the source condition related to operator ${\mathcal{U}}$. Such ideas were also used in \cite{crepey,acthesis,acpaper,eggeng}, but here they are extended to the context of online local volatility calibration. For the definitions of Bregman distance and $q$-coerciveness see Appendix~\ref{app:def}.
In what follows we always assume that (\ref{ip1a}) has a (unique) solution which is an element of the Bregman domain $\mathcal{D}_B(f_{{\mathcal{A}}_0})$.
Before stating the result about convergence rates, we need the following auxiliary lemma, which introduces the so-called source condition. For a review on Convex Regularization, see \cite[Chapter 3]{schervar}.
\begin{lem}
For every $\xi^\dagger \in \partial f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger)$, there exists $\omega^\dagger \in \Ya$
and $\mathscr{E} \in \X$ such that
$\xi^\dagger = \left[{\mathcal{U}}^\prime({\mathcal{A}}^\dagger)\right]^*\omega^\dagger + \mathscr{E}$ holds. Moreover, $\mathscr{E}$ can be chosen such that $\|\mathscr{E}\|_{\X}$ is arbitrarily small.
\label{lemmax}
\end{lem}
Lemma~\ref{lemmax} follows by $\mathcal{R}({\mathcal{U}}^\prime({\mathcal{A}}^\dagger)^*)$ being dense in
$\X$. See Proposition~\ref{prf1} in Section~3.
Observe also that, we identify $\Ya^*$ and $\X^*$ with $\Ya$ and $\X$, respectively,
since they are Hilbert spaces.
\begin{theo}[Convergence Rates]
Assume that (\ref{ip1a}) has a (unique) solution. Let the map
$alpha : (0,\infty) \rightarrow (0,\infty)$ be such that
$alpha(\delta) approx \delta$ as $\delta\searrow 0$. Furthermore, assume that the convex functional $f_{{\mathcal{A}}_0}(\cdot)$ is also $q$-coercive with constant $\zeta$, with respect to the norm of $\X$. Then under the source condition of Lemma~\ref{lemmax} it follows that
$$
D_{\xi^\dagger}({\mathcal{A}}^\delta_alpha,{\mathcal{A}}^\dagger) = \mathcal{O}(\delta) ~~~\text{ and }~~~
\|{\mathcal{U}}({\mathcal{A}}^\delta_alpha) - {\mathcal{U}^\delta} \| = \mathcal{O}(\delta).
$$
\label{tc1}
\end{theo}
\begin{proof}
Let ${\mathcal{A}}^\dagger$ and ${\mathcal{A}}^\delta_alpha$ denote the solution of (\ref{ip1a}) and the minimizer of \eqref{tik1}, respectively. It follows that,
$
\|{\mathcal{U}}({\mathcal{A}}^\delta_alpha) - {\mathcal{U}^\delta}\|^2 + alpha f_{{\mathcal{A}}_0}({\mathcal{A}}^\delta_alpha) \leq \|{\mathcal{U}}({\mathcal{A}}^\dagger) - {\mathcal{U}^\delta}\|^2 + alpha f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger) \leq \delta^2 + alpha f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger).
$
\noindent Since, $D_{\xi^\dagger}({\mathcal{A}}^\delta_alpha,{\mathcal{A}}^\dagger) = f_{{\mathcal{A}}_0}({\mathcal{A}}^\delta_alpha) - f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger) - \langle \xi^\dagger , {\mathcal{A}}^\delta_alpha - {\mathcal{A}}^\dagger\rangle$, it follows by Lemma~\ref{lemmax} and the above estimate that,
$$
\|{\mathcal{U}}({\mathcal{A}}^\delta_alpha) - {\mathcal{U}^\delta}\|^2 + alpha D_{\xi^\dagger}({\mathcal{A}}^\delta_alpha,{\mathcal{A}}^\dagger) \leq
\delta^2 - alpha(\langle \omega^\dagger , {\mathcal{U}}^\prime({\mathcal{A}}^\dagger)({\mathcal{A}}^\delta_alpha - {\mathcal{A}}^\dagger)\rangle + \langle \mathscr{E} , {\mathcal{A}}^\delta_alpha - {\mathcal{A}}^\dagger\rangle).
$$
\noindent By Proposition~\ref{tang}, it follows that
$
|\langle \omega^\dagger , {\mathcal{U}}^\prime({\mathcal{A}}^\dagger)({\mathcal{A}}^\delta_alpha - {\mathcal{A}}^\dagger)\rangle|
\leq (1+\gamma)\|\omega^\dagger\|\|{\mathcal{U}}({\mathcal{A}}^\delta_alpha) - {\mathcal{U}}({\mathcal{A}}^\dagger)\| \leq (1+\gamma)\|\omega^\dagger\|(\delta + \|{\mathcal{U}}({\mathcal{A}}^\delta_alpha) - {\mathcal{U}^\delta}\|).
$
Thus,
$
\|{\mathcal{U}}({\mathcal{A}}^\delta_alpha) - {\mathcal{U}^\delta}\|^2 + alpha D_{\xi^\dagger}({\mathcal{A}}^\delta_alpha,{\mathcal{A}}^\dagger) \leq \delta^2 + alpha (1+\gamma)\|\omega^\dagger\|(\delta + \|{\mathcal{U}}({\mathcal{A}}^\delta_alpha) - {\mathcal{U}^\delta}\|) + alpha \|\mathscr{E}\|\cdot\|{\mathcal{A}}^\delta_alpha - {\mathcal{A}}^\dagger\|.
$
\noindent Since $\|\mathscr{E}\|$ is arbitrarily small, it follows that, $(\zeta - \|\mathscr{E}\|)/\zeta > 0$.
Moreover, since $f_{{\mathcal{A}}_0}$ is $q$-coercive with constant $\zeta$ we divide the estimates in two cases, when $q=1$ and $q>1$.
For the case $q = 1$, the above inequalities imply that,
$$
(\|{\mathcal{U}}({\mathcal{A}}^\delta_alpha) - {\mathcal{U}^\delta}\| - alpha(1+\gamma)\|\omega^\dagger\|/2)^2 + alpha(1 - 1/\zeta\|\mathscr{E}\|)D_{\xi^\dagger}({\mathcal{A}}^\delta_alpha,{\mathcal{A}}^\dagger) \leq
(\delta + alpha(1+\gamma)\|\omega^\dagger\|)^2
$$
Hence, the assertions follow.
For the case $q > 1$, we denote $\beta_1 = \|\mathscr{E}\|/\zeta$ and we have that,
$$
\beta_1(D_{\xi^\dagger}({\mathcal{A}}^\delta_alpha,{\mathcal{A}}^\dagger))^{1/q} \leq \displaystyle\frac{\beta_1^q}{q} + \frac{1}{q}D_{\xi^\dagger}({\mathcal{A}}^\delta_alpha,{\mathcal{A}}^\dagger).
$$
Thus, assuming that $\beta_1 = \mathcal{O}(\delta^{1/q})$, we have the estimate:
\begin{multline}
\left(\|{\mathcal{U}}({\mathcal{A}}^\delta_alpha) - {\mathcal{U}^\delta}\| - alpha\frac{1+\gamma}{2}\|\omega^\dagger\|\right)^2 + \displaystylealpha\frac{q - 1}{q}D_{\xi^\dagger}({\mathcal{A}}^\delta_alpha,{\mathcal{A}}^\dagger) \leq\\
(\delta + alpha(1+\gamma)\|\omega^\dagger\|)^2 + alpha\displaystyle\frac{\beta_1^q}{q},
\end{multline}
and the assertions follow.
\end{proof}
Note that the rates obtained in Theorem~\ref{tc1} state that, in some sense, the distance between the true local variance and the Tikhonov solution is of order $\mathcal{O}(\delta)$. This can be seen as a measure of the reliability of Tikhonov minimizers for this specific example.
\section{Morozov's Principle}\label{sec:morozov}
We now establish a relaxed version of Morozov's discrepancy principle for the specific problem under consideration \cite{moro}. This is one of the most reliable ways of finding the regularization parameter $alpha$ as a function of the data ${\mathcal{U}^\delta}$ and the noise level $\delta$. Intuitively, the regularized solution should not fit the data more accurately than the noise level. We remark that this statement does not follow immediately because, the parameter now has to be chosen as a function of the noise level $\delta$ and the data ${\mathcal{U}^\delta}$. Thus, it is necessary to prove that such functional in fact satisfies the required criteria to achieve the desired convergence rates.
From Equation (\ref{errorb}), it follows that any ${\mathcal{A}} \in {\mathfrak{Q}}$ satisfying
\begin{equation}
\|{\mathcal{U}}({\mathcal{A}}) - {\mathcal{U}^\delta}\| \leq \delta
\end{equation}
could be an approximate solution for (\ref{ip1a}).
If ${\mathcal{A}}^\delta_alpha$ is a minimizer of \eqref{tik1}, then Morozov's discrepancy principle says that the regularization parameter $alpha$ should be chosen through the condition
\begin{equation}
\|{\mathcal{U}}({\mathcal{A}}^\delta_alpha) - {\mathcal{U}^\delta}\| = \delta
\label{m_init}
\end{equation}
whenever it is possible. In other words, the regularized solution should not satisfy the data more accurately than up to the noise level.
Since the identity \eqref{m_init} is restrictive, in what follows we combine two strategies. The first one is the relaxed Morozov's discrepancy principle studied in \cite{anram}. The second one is the sequential discrepancy principle studied in \cite{ahm}.
Note that, in the analysis that follows, we also require that if $f_{{\mathcal{A}}_0}({\mathcal{A}}) = 0$ then ${\mathcal{A}} = {\mathcal{A}}_0$.
\begin{df}{\cite{anram}}
Let the noise level $\delta > 0$ and the data ${\mathcal{U}^\delta}$ be fixed. Define the functionals
\begin{eqnarray}
L:{\mathcal{A}} \in {\mathfrak{Q}} &\longmapsto & L({\mathcal{A}}) = \|{\mathcal{U}}({\mathcal{A}}) - {\mathcal{U}^\delta}\|\in\mathbb{R}_+\cup \{+\infty\},\\
H:{\mathcal{A}} \in {\mathfrak{Q}} &\longmapsto & H({\mathcal{A}}) = f_{{\mathcal{A}}_0}({\mathcal{A}})\in\mathbb{R}_+\cup \{+\infty\},\\
I: alpha \in \mathbb{R}_+ &\longmapsto & I(alpha) = \mathcal{F}^{{\mathcal{U}}^\delta}_{{\mathcal{A}}_0,alpha}({\mathcal{A}}^\delta_alpha)\in\mathbb{R}_+\cup \{+\infty\}.
\label{func_moro}
\end{eqnarray}
We also define the set containing all minimizers of the functional \eqref{tik1} for each fixed $alpha \in (0,\infty)$ as
$$
M_alpha: = \left\{{\mathcal{A}}^\delta_alpha \in {\mathfrak{Q}} : L(a^\delta_alpha) \leq L({\mathcal{A}}) ,~\forall {\mathcal{A}} \in \X \right\}.
$$
Note that we have extended $L({\mathcal{A}})$ to be equal to $\|{\mathcal{U}}({\mathcal{A}}) - {\mathcal{U}^\delta}\|$ when ${\mathcal{A}} \in {\mathfrak{Q}}$ and to be equal to $+\infty$ otherwise.
\end{df}
The first strategy above mentioned is defined as follows:
\begin{df}[Morozov Criteria]
For prescribed $1< {\tilde{a}}u_1 \leq {\tilde{a}}u_2$, choose $alpha = alpha(\delta,{\mathcal{U}^\delta})$ such that $alpha>0$ and
\begin{equation}
{\tilde{a}}u_1\delta \leq \|{\mathcal{U}}({\mathcal{A}}^\delta_alpha) - {\mathcal{U}^\delta} \| \leq {\tilde{a}}u_2\delta
\label{morozov}
\end{equation}
holds for some ${\mathcal{A}}^\delta_alpha$ in $M_alpha$.
\end{df}
If the first is not possible, then we consider the following:
\begin{df}[Sequential Morozov Criteria]
For prescribed ${\tilde{{\tilde{a}}u}}>1$, $alpha_0 > 0$ and $0<q<1$, choose $alpha_n = q^nalpha_0$ such that the discrepancy
\begin{equation}
\|{\mathcal{U}}({\mathcal{A}}^\delta_{alpha_{n}}) - {\mathcal{U}^\delta} \| \leq {\tilde{{\tilde{a}}u}}\delta < \|{\mathcal{U}}({\mathcal{A}}^\delta_{alpha_{n-1}}) - {\mathcal{U}^\delta} \|
\label{seqmorozov}
\end{equation}
is satisfied for some $n \in \N$ and some ${\mathcal{A}}^\delta_{alpha_{n}} \in M_{alpha_n}$ and ${\mathcal{A}}^\delta_{alpha_{n-1}} \in M_{alpha_{n-1}}$.
\end{df}
It follows by \cite[Lemma 2.6.1]{tikar} that the functional $H(\cdot)$ is non-increasing and the functionals $L(\cdot)$ and $I(\cdot)$ are non-decreasing with respect to $alpha \in (0,\infty)$ in the following sense, if $0 < alpha < \beta$ then we have
$$
\sup_{{\mathcal{A}}^\delta_alpha \in M_alpha}L({\mathcal{A}}^\delta_alpha) \leq \inf_{{\mathcal{A}}^\delta_\beta \in M_\beta}L({\mathcal{A}}^\delta_\beta), \inf_{{\mathcal{A}}^\delta_alpha \in M_alpha}H({\mathcal{A}}^\delta_alpha) \geq \sup_{{\mathcal{A}}^\delta_\beta \in M_\beta}H({\mathcal{A}}^\delta_\beta) \text{ and } I(alpha) \leq I(\beta).
$$
By \cite[Lemma 2.6.3]{tikar}, the functional $I(\cdot)$ is continuous and the sets of discontinuities of $L(\cdot)$ and $H(\cdot)$ are at most countable and coincide. If we denote this set by $M$, then $L(\cdot)$ and $H(\cdot)$ are continuous in $(0,\infty) \backslash M$.
Since the set $M_alpha$ is weakly closed for each $alpha > 0$, we have the following:
\begin{lem}
For each $\overline{alpha}>0$, there exist ${\mathcal{A}}_1,{\mathcal{A}}_2 \in M_{\overline{alpha}}$ such that
$$
L({\mathcal{A}}_1) = \displaystyle\inf_{{\mathcal{A}} \in M_{\overline{alpha}}}L({\mathcal{A}}) ~~~\text{and} ~~~
L({\mathcal{A}}_2) = \displaystyle\sup_{{\mathcal{A}} \in M_{\overline{alpha}}}L({\mathcal{A}}).
$$
\end{lem}
\begin{pr}
Let $1<{\tilde{a}}u_1 \leq {\tilde{a}}u_2$ be fixed. Suppose that $\|{\mathcal{U}}({\mathcal{A}}_0) - {\mathcal{U}^\delta}\| > {\tilde{a}}u_2\delta$. Then, we can find $\underline{alpha},\overline{alpha}>0$, such that
$$
L({\mathcal{A}}_1) < {\tilde{a}}u_1\delta \leq {\tilde{a}}u_2 \delta < L({\mathcal{A}}_2),
$$
where ${\mathcal{A}}_1 := {\mathcal{A}}^\delta_{\underline{alpha}}$ and ${\mathcal{A}}_2 := {\mathcal{A}}^\delta_{\overline{alpha}}$.
\label{pr7}
\end{pr}
\begin{proof}
First, let the sequence $\{alpha_n\}_{n \in \N}$ converge to $0$. Then, we can find a sequence
$\{{\mathcal{A}}_n\}_{n \in \N}$ with ${\mathcal{A}}_n \in M_{alpha_n}$ for each $n \in \N$. Now, let ${\mathcal{A}}^\dagger$ be an
$f_{{\mathcal{A}}_0}$-minimizing solution of (\ref{pi1}). Hence, it follows that
$
L({\mathcal{A}}_n)^2 \leq I(alpha_n) \leq \mathcal{F}^{{\mathcal{U}}^\delta}_{{\mathcal{A}}_0,alpha_n}({\mathcal{A}}^\dagger) \leq \delta^2 + alpha_n f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger).
$
Thus, for a sufficiently large $n\in\N$, $L({\mathcal{A}}_n)^2 < ({\tilde{a}}u_1\delta)^2$, since $alpha_n f_{{\mathcal{A}}_0}(a^\dagger) \rightarrow 0$. Thus, we can set $\underline{alpha} := alpha_n$ for this same $n$ .
We now assume that $alpha_n \rightarrow \infty$. Taking ${\mathcal{A}}_n$ as before, we have the following estimates
$
H({\mathcal{A}}_n) \leq \displaystyle\frac{1}{alpha_n}I(alpha_n) \leq \displaystyle\frac{1}{alpha_n}\mathcal{F}^{{\mathcal{U}}^\delta}_{{\mathcal{A}}_0,alpha_n}({\mathcal{A}}_0) =
\displaystyle\frac{1}{alpha_n}\|{\mathcal{U}}({\mathcal{A}}_0) - {\mathcal{U}^\delta}\| \rightarrow 0$ whenever $n\rightarrow \infty$.
Thus, $\displaystyle\lim_{n\rightarrow \infty}f_{{\mathcal{A}}_0}({\mathcal{A}}_n) = 0$, which implies that $\{{\mathcal{A}}_n\}_{n \in \N}$ converges weakly to ${\mathcal{A}}_0$.
Then, by the weak continuity of ${\mathcal{U}}(\cdot)$ and the lower semi-continuity of the norm, it follows that
$$
\|{\mathcal{U}}({\mathcal{A}}_0) - {\mathcal{U}^\delta}\| \leq \displaystyle\liminf_{n\rightarrow \infty}\|{\mathcal{U}}({\mathcal{A}}_n) - {\mathcal{U}^\delta}\|,$$
which shows the existence of $\overline{alpha}$, such that
$$L({\mathcal{A}}^\delta_{\overline{alpha}}) > {\tilde{a}}u_2\delta.$$
\end{proof}
\begin{rem}
For prescribed $1<{\tilde{a}}u_1\leq {\tilde{a}}u_2$, the discrepancy principle \eqref{morozov} always works if we assume that there is no $alpha > 0$ such that the minimizers ${\mathcal{A}}_1,{\mathcal{A}}_2 \in M_{alpha}$ satisfy
\begin{equation}
\|{\mathcal{U}}({\mathcal{A}}_1) - {\mathcal{U}^\delta}\| < {\tilde{a}}u_1\delta \leq {\tilde{a}}u_2\delta < \|{\mathcal{U}}({\mathcal{A}}_2) - {\mathcal{U}^\delta}\|.
\label{moro_condition}
\end{equation}
In other words, only one of the inequalities of the discrepancy principle \eqref{morozov} could be violated by the minimizers associated to $alpha$. A sufficient condition for such assumption is the uniqueness of Tikhonov minimizers which we are not able to prove for this specific case. Thus, we have to introduce the sequential discrepancy principle \eqref{seqmorozov} whenever the condition \eqref{moro_condition} is violated. Note that the discrepancy principle \eqref{morozov} is always preferable since its lower inequality implies that the Tikhonov minimizers satisfying \eqref{morozov} do not reproduce noise. Whereas the same conclusion cannot be achieved with the sequential discrepancy principle \eqref{seqmorozov}.
See also \cite[Remark~4.7]{schu} for another discussion about the discrepancy principle~\eqref{morozov}.
\end{rem}
Under the condition \eqref{moro_condition} and Proposition~\ref{pr7}, by \cite[Theorem 3.10]{anram} we can always find $alpha := alpha(\delta)>0$ and a Tikhonov minimizer ${\mathcal{A}}^\delta_alpha \in M_alpha$, such that both the inequalities of the discrepancy principle \eqref{morozov} are satisfied. Proposition~\ref{pr7} also implies that the sequential discrepancy principle \eqref{seqmorozov} is well posed. See \cite[Lemma~2]{ahm}. For a convergence analysis under the sequential Morozov, see \cite{hm}.
\begin{theo}
Assume that the inverse problem \eqref{ip1a} has a (unique) solution. If condition \eqref{moro_condition} holds, then the regularizing parameter $alpha = alpha(\delta,{\mathcal{U}^\delta})$ obtained through Morozov's discrepancy principle (\ref{morozov}) satisfies the limits
$$
\displaystyle\lim_{\delta \rightarrow 0+}alpha(\delta,{\mathcal{U}^\delta}) = 0
~~~\text{ and }~~~
\displaystyle\lim_{\delta \rightarrow 0+}\frac{\delta^2}{alpha(\delta,{\mathcal{U}^\delta})} = 0.
$$
The same limits hold if $alpha$ is chosen through the sequential discrepancy principle \eqref{seqmorozov}.
\label{tma}
\end{theo}
\begin{proof}
Let $\{\delta_n\}_{n\in \N}$ be a sequence such that $\delta_n \downarrow 0$ and let ${\widetilde{\mathcal{U}}}$ be the noiseless data. Thus, $\|{\widetilde{\mathcal{U}}} - {\mathcal{U}}^{\delta_n}\|\leq \delta_n$. In addition, recall that the inverse problem \eqref{ip1a} has a unique solution ${\mathcal{A}}^\dagger$ and then ${\mathcal{U}}({\mathcal{A}}^\dagger) = {\widetilde{\mathcal{U}}}$. We only prove the case where the choice of the regularization parameter is based on the discrepancy principle \eqref{morozov}. Very similar arguments to the ones that follow show the theorem's claim when the choice is based on the sequential discrepancy principle \eqref{seqmorozov}. See \cite[Theorem 1]{ahm}. Thus, it is straightforward to build diagonal convergent subsequences with elements satisfying one of both strategies, in order to prove the limits above asserted.
Let $alpha_n := alpha(\delta_n,{\mathcal{U}}^{\delta_n})$ denote the regularizing parameter chosen through \eqref{morozov}. Thus, we denote by ${\mathcal{A}}_n:={\mathcal{A}}^{\delta_n}_{alpha_n}$ its associated minimizer of \eqref{tik1} with respect to $\delta_n$, $alpha_n$ and ${\mathcal{U}}^{\delta_n}$. This defines the sequence $\{{\mathcal{A}}_n\}_{n\in\N}$, which is pre-compact by the coerciveness of $f_{{\mathcal{A}}_0}$. Choose a convergent subsequence, denoting it by $\{{\mathcal{A}}_k\}_{k\in\N}$ and its weak limit by ${\tilde{a}}f$. We shall see that ${\tilde{a}}f = {\mathcal{A}}^\dagger$ and thus the original sequence is bounded and has the unique cluster point ${\mathcal{A}}^\dagger$.
The weakly lower semi-continuity of $\|{\mathcal{U}}(\cdot)-{\widetilde{\mathcal{U}}}\|$ and $f_{{\mathcal{A}}_0}$ implies that $\|{\mathcal{U}}({\tilde{a}}f) - {\widetilde{\mathcal{U}}}\| \leq \lim_{k\rightarrow\infty}({\tilde{a}}u_2+1)\delta_k = 0$. Thus, ${\tilde{a}}f$ is a solution of the inverse problem \eqref{ip1a}, which is unique, then ${\tilde{a}}f = {\mathcal{A}}^\dagger$.
Since, for each $k$, ${\mathcal{A}}_k$ is a Tikhonov minimizer satisfying the discrepancy principle \eqref{morozov}, it follows by the weakly lower semi-continuity of $f_{{\mathcal{A}}_0}$ that
\begin{equation}
f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger) \leq \displaystyle\liminf_{k\rightarrow\infty}f_{{\mathcal{A}}_0}({\mathcal{A}}_k) \leq
\displaystyle\limsup_{k\rightarrow\infty}f_{{\mathcal{A}}_0}({\mathcal{A}}_k) \leq f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger).
\label{moro4}
\end{equation}
In other words, $f_{{\mathcal{A}}_0}({\mathcal{A}}_k)\rightarrow f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger)$.
We now prove that $alpha(\delta,{\mathcal{U}}^\delta)\rightarrow 0$. Assume that with respect to the sequence of the beginning of the proof, there exist $\overline{alpha}>0$ and a subsequence $\{alpha_k\}_{k\in\N}$ such that $alpha_k \geq \overline{alpha}$ for every $k \in \N$. Denote also by $\{{\mathcal{A}}_k\}_{k\in\N}$ a sequence of minimizers of \eqref{tik1} with respect to $\delta_k$, $alpha_k$ and ${\mathcal{U}}^{\delta_k}$. Define further the sequence $\{{\overline{\mathcal{A}}}_k\}_{k\in\N}$ of minimizers of \eqref{tik1} with respect to $\delta_k$, $\overline{alpha}$ and ${\mathcal{U}}^{\delta_k}$.
Since $L$ in non-decreasing, by the discrepancy principle \eqref{morozov},
\begin{equation}
\|{\mathcal{U}}({\overline{\mathcal{A}}}_k) - {\mathcal{U}}^{\delta_k}\| \leq \|{\mathcal{U}}({\mathcal{A}}_k) - {\mathcal{U}}^{\delta_k}\| \leq {\tilde{a}}u_2\delta_k\rightarrow 0
\label{moro5}
\end{equation}
On the other hand, $\displaystyle\limsup_{k\rightarrow \infty} \overline{alpha}f_{{\mathcal{A}}_0}({\overline{\mathcal{A}}}_k) \leq \overline{alpha}f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger)$. By the coerciveness of $f_{{\mathcal{A}}_0}$, the sequence has a convergent subsequence, denoted also by $\{{\overline{\mathcal{A}}}_k\}_{k \in \N}$, with limit ${\overline{\mathcal{A}}} \in {\mathfrak{Q}}$. Thus, by the estimates (\ref{moro4}) and (\ref{moro5}), the weakly lower semi-continuity of $\|{\mathcal{U}}(\cdot) - {\widetilde{\mathcal{U}}}\|$ and $f_{{\mathcal{A}}_0}$, it follows that $\|{\mathcal{U}}({\overline{\mathcal{A}}}) - {\widetilde{\mathcal{U}}}\| =0$ and $f_{{\mathcal{A}}_0}({\overline{\mathcal{A}}}) \leq f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger)$.
Since the inverse problem \eqref{ip1a} has a unique solution, ${\overline{\mathcal{A}}} = {\mathcal{A}}^\dagger$ and thus
$ f_{{\mathcal{A}}_0}({\overline{\mathcal{A}}}_k)\rightarrow f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger)$. On the other hand, ${\overline{\mathcal{A}}}$ is a minimizer of \eqref{tik1} with regularization parameter $\overline{alpha}$ and the noiseless data ${\widetilde{\mathcal{U}}}$, since for each ${\mathcal{A}} \in {\mathfrak{Q}}$, the following estimate hold:
$$
\begin{array}{rcl}
\|{\mathcal{U}}({\overline{\mathcal{A}}}) - {\widetilde{\mathcal{U}}}\|^2 + \overline{alpha} f_{{\mathcal{A}}_0}({\overline{\mathcal{A}}})& \leq &
\displaystyle\liminf_{k\rightarrow \infty}\left(\|{\mathcal{U}}({\mathcal{A}}) - {\mathcal{U}}^{\delta_k}\|^2 + \overline{alpha}f_{{\mathcal{A}}_0}({\mathcal{A}})\right)\\
& = & \|{\mathcal{U}}({\mathcal{A}}) - {\mathcal{U}}^{\delta_k}\|^2 + \overline{alpha}f_{{\mathcal{A}}_0}({\mathcal{A}}).
\end{array}
$$
Since $f_{{\mathcal{A}}_0}$ is convex, it follows that for every $t \in [0,1)$
$$
f_{{\mathcal{A}}_0}((1-t){\overline{\mathcal{A}}} + t{\mathcal{A}}_0) \leq (1-t)f_{{\mathcal{A}}_0}({\overline{\mathcal{A}}}) + tf_{{\mathcal{A}}_0}({\mathcal{A}}_0) = (1-t)f_{{\mathcal{A}}_0}({\overline{\mathcal{A}}}).
$$
Thus,
$\overline{alpha}f_{{\mathcal{A}}_0}({\overline{\mathcal{A}}}) \leq \|{\mathcal{U}}((1-t){\overline{\mathcal{A}}} + t{\mathcal{A}}_0) - {\widetilde{\mathcal{U}}}\|^2 + \overline{alpha}(1-t)f_{{\mathcal{A}}_0}({\overline{\mathcal{A}}})$. This implies that
$\overline{alpha}tf_{{\mathcal{A}}_0}({\overline{\mathcal{A}}}) \leq \|{\mathcal{U}}((1-t){\overline{\mathcal{A}}} + t{\mathcal{A}}_0) - {\widetilde{\mathcal{U}}}\|^2$. Since ${\widetilde{\mathcal{U}}} = {\mathcal{U}}({\overline{\mathcal{A}}})$, by Proposition~\ref{prop6} with $\mathcal{H} = {\mathcal{A}}_0 - {\mathcal{A}}$,
$\overline{alpha}f_{{\mathcal{A}}_0}({\overline{\mathcal{A}}}) \leq \displaystyle\lim_{t\rightarrow 0^+}\frac{1}{t}\|{\mathcal{U}}((1-t){\overline{\mathcal{A}}} + t{\mathcal{A}}_0) - {\widetilde{\mathcal{U}}}\|^2 = 0$. Therefore, $f_{{\mathcal{A}}_0}({\overline{\mathcal{A}}}) = 0$.
But, by hypothesis, it could only hold if ${\overline{\mathcal{A}}} = {\mathcal{A}}_0$, i.e., ${\mathcal{A}}^\dagger = {\mathcal{A}}_0$. However, $\|{\mathcal{U}}({\mathcal{A}}_0) - {\mathcal{U}}^\delta\| \geq {\tilde{a}}u_2\delta$.
This is a contradiction. We conclude that $alpha(\delta,{\mathcal{U}}^\delta)\rightarrow 0$ when $\delta \rightarrow 0$.
In order to prove the second limit, consider again the subsequence $\{{\mathcal{A}}_k\}_{k\in\N}$ converging weakly to ${\mathcal{A}}^\dagger$, the solution of the inverse problem \eqref{ip1a}, when $\delta_k\downarrow 0$. Thus, since for each $k$ ${\mathcal{A}}_k$ satisfies the discrepancy principle (\ref{morozov}), it follows that ${\tilde{a}}u^2_1\delta^2_k + alpha_k f_{{\mathcal{A}}_0}({\mathcal{A}}_k) \leq \delta_k^2 + alpha_k f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger)$. This implies that
$({\tilde{a}}u_1^2 - 1)\displaystyle\frac{\delta_k^2}{alpha_k} \leq f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger) - f_{{\mathcal{A}}_0}({\mathcal{A}}_k) \rightarrow 0$.
\end{proof}
The following theorem states that, if the regularization parameter $alpha$ is chosen through the discrepancy principle \eqref{morozov}, we achieve the same convergence rates of the Theorem~\ref{tc1}.
\begin{theo}
Assume that the inverse problem \eqref{ip1a} has a (unique) solution.
Suppose that ${\mathcal{A}}^\delta_alpha$ is a minimizer of \eqref{tik1} and $alpha = alpha(\delta,{\mathcal{U}^\delta})$ is chosen through the discrepancy principle (\ref{morozov}) or the sequential discrepancy principle \eqref{seqmorozov}. Then, by the source condition of Lemma~\ref{lemmax}, we have the estimates
\begin{equation}
\|{\mathcal{U}}({\mathcal{A}}^\delta_alpha) - {\mathcal{U}}({\mathcal{A}}^\dagger)\| = \mathcal{O}(\delta) ~~~\text{ and }~~~
D_{\xi^\dagger}({\mathcal{A}}^\delta_alpha,{\mathcal{A}}^\dagger) = \mathcal{O}(\delta),
\label{rates_conv}
\end{equation}
with $\xi^\dagger \in \partial f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger)$. The estimates are achieved whenever \eqref{morozov} is used.
\label{mor:cr}
\end{theo}
\begin{proof}
Let ${\mathcal{A}}^\dagger$ be the solution of the inverse problem (\ref{ip1a}). If ${\mathcal{A}}^\delta_alpha \in M_alpha$, then, the first estimate is trivial since $
\|{\mathcal{U}}({\mathcal{A}}^\delta_alpha) - {\mathcal{U}}({\mathcal{A}}^\dagger)\| \leq ({\tilde{a}}u_2 + 1)\delta.$
\noindent If condition \eqref{moro_condition} holds, then by the first inequality of the discrepancy principle (\ref{morozov}), ${\tilde{a}}u_1\delta^2 +alpha f_{{\mathcal{A}}_0}({\mathcal{A}}^\delta_alpha) \leq \delta^2+alpha f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger)$, implying that $f_{{\mathcal{A}}_0}({\mathcal{A}}^\delta_alpha) \leq f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger)$, since ${\tilde{a}}u_1 - 1 > 0$.
Hence, for every $\xi^\dagger \in \partial f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger)$ satisfying the source condition of Lemma~\ref{lemmax} and assuming that $f_{{\mathcal{A}}_0}$ is $1$-coerciveness with constant $\zeta$, we have the estimates:
\begin{equation}
\begin{array}{rcl}
D_{\xi^\dagger}({\mathcal{A}}^\delta_alpha,{\mathcal{A}}^\dagger) &\leq&
|\langle \xi^\dagger , {\mathcal{A}}^\delta_alpha - {\mathcal{A}}^\dagger \rangle| =
|\langle {\mathcal{U}}^\prime({\mathcal{A}}^\dagger)^*\omega^\dagger + \mathcal{E}, {\mathcal{A}}^\delta_alpha - {\mathcal{A}}^\dagger\rangle|\\
&\leq&
\|\omega^\dagger\|\|{\mathcal{U}}^\prime({\mathcal{A}}^\dagger)({\mathcal{A}}^\delta_alpha - {\mathcal{A}}^\dagger)\| +
\|\mathcal{E}\|\|{\mathcal{A}}^\delta_alpha - {\mathcal{A}}^\dagger\|\\
&\leq&
(1+\gamma)\|\omega^\dagger\|\|{\mathcal{U}}({\mathcal{A}}^\delta_alpha) - {\mathcal{U}}({\mathcal{A}}^\dagger)\| +
\displaystyle\frac{1}{\zeta}\|\mathcal{E}\|D_{\xi^\dagger}({\mathcal{A}}^\delta_alpha,{\mathcal{A}}^\dagger)
\end{array}
\label{mo:eqcr}
\end{equation}
Since $\xi^\dagger$ can be chosen with $\|\mathcal{E}\|$ arbitrarily small, it follows that
$
1 - 1/\zeta\|\mathcal{E}\| > 0
$
and then, by \eqref{morozov},
$$
D_{\xi^\dagger}({\mathcal{A}}^\delta_alpha,{\mathcal{A}}^\dagger) \leq \displaystyle\frac{\zeta}{\zeta - \|\mathcal{E}\|}
(1+\gamma)\|\omega^\dagger\|\|{\mathcal{U}}({\mathcal{A}}^\delta_alpha) - {\mathcal{U}}({\mathcal{A}}^\dagger)\| \leq
{\tilde{a}}u_2 \frac{\zeta}{\zeta - \|\mathcal{E}\|}(1+\gamma)\|\omega^\dagger\|\cdot \delta.
$$
On the other hand, let $alpha$ be given by the sequential discrepancy principle \eqref{seqmorozov}. Since,
$ alpha D_{\xi^\dagger}({\mathcal{A}}^\delta_alpha,{\mathcal{A}}^\dagger) \leq \|{\mathcal{U}}({\mathcal{A}}^\delta_alpha)-{\mathcal{U}^\delta}\|^2 + alpha D_{\xi^\dagger}({\mathcal{A}}^\delta_alpha,{\mathcal{A}}^\dagger)$, then,
$$
D_{\xi^\dagger}({\mathcal{A}}^\delta_alpha,{\mathcal{A}}^\dagger) \leq \displaystyle\frac{\delta^2}{alpha} + |\langle \xi^\dagger , {\mathcal{A}}^\delta_alpha - {\mathcal{A}}^\dagger \rangle|.
$$
By the previous case, the second term in the right hand side of the above inequality has the order $\mathcal{O}(\delta)$. By Theorem~\ref{tma} the first term also vanishes. Since ${\tilde{{\tilde{a}}u}}\delta \leq \|{\mathcal{U}}({\mathcal{A}}^{\delta}_{alpha/q}) - {\mathcal{U}^\delta}\|$, it follows that the first term is of order $\mathcal{O}\left(|f_{{\mathcal{A}}_0}({\mathcal{A}}^\delta_{alpha/q}) - f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger)|\right)$ and $|f_{{\mathcal{A}}_0}({\mathcal{A}}^\delta_{alpha/q}) - f_{{\mathcal{A}}_0}({\mathcal{A}}^\dagger)| \leq |\langle \xi^\dagger , {\mathcal{A}}^\delta_{alpha/q} - {\mathcal{A}}^\dagger \rangle|$. See \cite[Proposition~10]{ahm}.
\end{proof}
As above mentioned, the above rates obtained in terms of Bregman distance state that, in some sense, the distance between the true local variance and the Tikhonov solution is of order $\mathcal{O}(\delta)$. Under a more practical perspective, consider $f_{{\mathcal{A}}_0}({\mathcal{A}}) = \|{\mathcal{A}} - {\mathcal{A}}_0\|^2_{\X}$. In this case, it follows that $\|{\mathcal{A}}^\delta_alpha - {\mathcal{A}}^\dagger\|_{\X} = \mathcal{O}(\delta^{1/2})$. In addition, if $l > 1/2$ in $\X$, it follows by the inequality \eqref{estimate1} that
$$\sup_{s\in [0,S]}\|a^\delta_alpha(s) - a^\dagger(s)\|_{\he} \leq C \|{\mathcal{A}}^\delta_alpha - {\mathcal{A}}^\dagger\|_{\X}.$$
Thus, the convergence rates also follows uniformly in $s$ and imply the convergence rates obtained in previous works, such as \cite{acpaper, eggeng, crepey}. This can be understood as the online solution is at least as good as the solution obtained in the standard case, i.e., the Tikhonov minimizers with only one price surface.
\begin{rem}
For $f_{{\mathcal{A}}_0}$ $q$-coercive with $q > 1$, a reasoning as the one used in Equation~\eqref{mo:eqcr}, gives that
$$\begin{array}{rcl}
D_{\xi^\dagger}({\mathcal{A}}^\delta_alpha,{\mathcal{A}}^\dagger)
&\leq& \beta_1 (D_{\xi^\dagger}({\mathcal{A}}^\delta_alpha,{\mathcal{A}}^\dagger))^{1/q} + \beta_2\|{\mathcal{U}}({\mathcal{A}}^\delta_alpha) - {\mathcal{U}}({\mathcal{A}}^\dagger)\|\\
&\leq& \displaystyle\frac{\beta_1^q}{q} + \frac{1}{q}D_{\xi^\dagger}({\mathcal{A}}^\delta_alpha,alpha) + \beta_2\|{\mathcal{U}}({\mathcal{A}}^\delta_alpha) - {\mathcal{U}}({\mathcal{A}}^\dagger)\|.
\end{array}$$
Assume further that $\beta_1 = \mathcal{O}(\delta^{\frac{1}{q}})$. Since
$
\|{\mathcal{U}}({\mathcal{A}}^\delta_alpha) - {\mathcal{U}}({\mathcal{A}}^\dagger)\| = \mathcal{O}(\delta)
$,
it follows that
$
\|{\mathcal{A}}^\delta_alpha - {\mathcal{A}}^\dagger\|^q \leq \displaystyle\frac{1}{\zeta}D_{\xi}({\mathcal{A}}^\delta_alpha,{\mathcal{A}}^\dagger) = \mathcal{O}(\delta).
$
\end{rem}
\section{Numerical Results}\label{sec:numerics}
We first perform tests with synthetic data for testing accuracy and advantages of the methods. Then, we present some examples with observed market prices.
We note that Problem~\eqref{dup2} is solved by a Crank-Nicolson scheme \cite[Chapter 5]{vvlathesis}. Since we shall use a gradient-based method to solve numerically the minimization of the Tikhonov functional \eqref{tik1}. Let $J^\delta({\mathcal{A}})$ and $\nabla J^\delta({\mathcal{A}})$ denote the quadratic residual and its gradient, respectively. More precisely, the residual is given by
$J^\delta({\mathcal{A}}) : = \|{\mathcal{U}}({\mathcal{A}}) - {\mathcal{U}^\delta}\|^2_{\Ya} = \int^S_0\|F(s,a(s)) - u^\delta(s) \|^2_{L^{2}(D)}ds$ and the gradient is given by
\begin{multline}
\langle \nabla J^\delta({\mathcal{A}}),\mathcal{H}\rangle_{\X} = 2\langle{\mathcal{U}}({\mathcal{A}}) - {\mathcal{U}^\delta} ,{\mathcal{U}}^\prime({\mathcal{A}})\mathcal{H}\rangle_{\Ya}\\
= 2\displaystyle\int^S_0\int_D\{[v(u_{yy}-u_y)h(t)](s,a(s))\}({\tilde{a}}u,y)d{\tilde{a}}u dyds,
\label{gradj}
\end{multline}
where, for each $s \in [0,S]$, $v$ is the solution of equation,
\begin{equation}
v_{\tilde{a}}u + (av)_{yy} + (av)_y +bv_y= u(t,a) - u^\delta(s)
\label{adj}
\end{equation}
with homogeneous boundary condition. Note that, $V = \{V: s \mapsto v(s)\}$ is an element of $\Y$. We also numerically solve Problem (\ref{adj}) by a Crank-Nicolson scheme. See \cite[Chapter 5]{vvlathesis}.
In the following examples we assume that $l=1$ in $\X$ and the regularization functional is
$
f_{{\mathcal{A}}_0}({\mathcal{A}}) = \displaystyle\|{\mathcal{A}} - {\mathcal{A}}_0\|^2_{\X}.
$
\subsection{Examples with Synthetic Data}
Consider the following local volatility surface:
$$
a(s,u,x) = \left\{
\begin{array}{ll}
\label{sig}
\displaystyle\frac{2}{5}\left(1 - \frac{2}{5}\text{e}^{-\frac{1}{2}( u - s)} \right)\cos(1.25\,\pi \,x),&(u,x) \in (0,1]\times \left[-\displaystyle\frac{2}{5},\displaystyle\frac{2}{5}\right],\\
\displaystyle\frac{2}{5}, & \text{otherwise.}
\end{array}
\right.
$$
We generate the data, i.e., evaluate the call prices with the above volatility, in a very fine mesh. Then we add a zero-mean Gaussian noise with standard deviation $\delta = 0.035,\, 0.01$. We interpolate the resulting prices in coarser grids. This avoids a so-called inverse crime \cite{somersalo}.
In the present test, we assume that, $r = 0.03$, $({\tilde{a}}u,y) \in [0,1]\times [-5,5]$. We generate the price data with step sizes $\Delta {\tilde{a}}u = 0.002$ and $\Delta y = 0.01$. Then, we solve the inverse problem with the step sizes $\Delta {\tilde{a}}u = 0.01, \,0.005$ and $\Delta y = 0.1$. We also assume that the asset price is given by $s \in [29.5, 32.5]$ with three different step sizes, $\Delta s = 0.25, 0.1, 0.01$.
In what follows, we refer to standard Tikhonov as the case when we consider a single price surface in the Tikhonov regularization. Whereas, we use the terminology ``online'' Tikhonov whenever we use more than one single price surface.
\begin{figure}
\caption{Left: Original local volatility. Center: Reconstruction with noise level $\delta = 0.035$. Right: Reconstruction with $\delta = 0.01$. When the noise level decreases, the reconstructions become more accurate.}
\label{test1}
\end{figure}
Figure~\ref{test1} shows reconstructions of the local volatility surface from price data with different noise levels. In addition, we can see that, when the noise level decreases, by refining the accuracy of the data, the resulting reconstructions become more similar to the original local volatility surface. This is an illustration of the Theorems \ref{tc1}, \ref{tma} and \ref{mor:cr}.
\begin{figure}
\caption{Comparison between standard and online Tikhonov. As the number of price surfaces increases, the reconstructions become more accurate.}
\label{test2}
\end{figure}
In Figure~\ref{test2}, we can see that the online Tikhonov presents better solutions than the standard one, as we increase the number of price surfaces in the calibration procedure. Here, the regularization parameter was obtained through the Morozov's discrepancy principle.
\begin{figure}
\caption{$L^2$ distance between original local variance and its reconstructions, as a function of the number of price surfaces. it is constant for standard Tikhonov and non-increasing for on line Tikhonov.}
\label{test3}
\end{figure}
Figure~\ref{test3} shows the evolution of the $L^2(D)$ distance between the reconstructions and the original local variance as a function of the number of surfaces of call prices: it is constant for standard Tikhonov and non-increasing for online Tikhonov.
\subsection{Examples with Market Data}
We now present some reconstructions of the local volatility by online Tikhonov regularization from market prices. We solve the inverse problem with the step sizes $\Delta {\tilde{a}}u = 0.01$ and $\Delta y = 0.1$. The regularizing functional is $f_{{\mathcal{A}}_0}({\mathcal{A}}) = \|{\mathcal{A}} - {\mathcal{A}}_0\|^2_{\X}$ and the regularization parameter is chosen through the discrepancy principle \eqref{morozov}. We estimate the noise level as half of the mean of the bid-ask spread in market prices. The market prices are interpolated linearly in the mesh where the inverse problem is solved. In the present example, we consider seven surfaces of call prices in each experiment. The data corresponds to vanilla option prices on futures of Light Sweet Crude Oil (WTI) and Henry Hub natural gas. For a survey on commodity markets, see the book \cite{geman}. For a study of of an application of Dupire's local volatility model on commodity markets, see \cite[Chapter~4]{vvlathesis}.
\begin{figure}
\caption{Local Volatility reconstruction from European vanilla options on futures of WTI oil. We used online Tikhonov regularization with the standard quadratic functional.}
\label{test4}
\end{figure}
Note that, in order to use the framework developed in the previous sections, we assumed that, the local volatility is indexed by the unobservable spot price, instead of the future price. For more details on such examples, see Chapters 4 and 5 of \cite{vvlathesis}.
\begin{figure}
\caption{Local Volatility reconstruction from European vanilla options on futures of Henry Hub natural gas. We used online Tikhonov regularization with the standard quadratic functional.}
\label{test5}
\end{figure}
Figures~\ref{test4} and~\ref{test5} present the best reconstructions of local volatility for WTI and HH data, respectively. We collected the data prices for Henry Hub natural gas and WTI oil between 2011/11/16 and 2011/11/25, i.e., seven consecutive commercial days.
\section{Conclusions}\label{sec:conclusion}
In this paper we have used convex regularization tools to solve the inverse problem associated to Dupire's local
volatility model when there is a steady flow of data. We first established results concerning existence, stability and convergence of the regularized solutions, making use of convex regularization tools and the regularity of the forward operator. We also proved some convergence rates. Furthermore, we established discrepancy-based choices of the regularization parameter, under a general framework, following \cite{anram,ahm}. Such analysis allowed us to implement the algorithms and perform numerical tests.
The main contribution, {\em vis a vis} previous works, and in particular~\cite{acpaper}, is that we extended the convex regularization techniques to incorporate the information and data stream that is constantly supplied by the market. Furthermore, we have proved discrepancy-based choices for the regularization parameter that are suitable to this context with regularizing properties.
A natural extension of the current work is the application of these techniques to the context of future markets, where the underlying asset is the future price of some financial instrument or commodity. In such markets, vanilla options represent a key instrument in hedging strategies of companies and in general they are far more liquid than in equity markets. The warning here is that, in general, we do not have an entire price surface. Actually in this case, we only have an option price curve for each future's maturity. Thus, in order to apply the techniques above to this context, it is necessary to assembly all option prices for futures on the same instrument (financial or commodity) in a unique surface in an appropriate way. This was discussed in \cite[Chapter 4]{vvlathesis} and will be published elsewhere.
\section{Acknowledgments}
V.A. acknowledges and thanks CNPq, Petroleo Brasileiro S.A. and Ag\^encia Nacional do Petr\'oleo for the financial support during the preparation of this work.
J.P.Z. acknowledges and thanks the financial support from CNPq through grants 302161/2003-1 and
474085/2003-1, and from FAPERJ through the programs {\em Cientistas do Nosso Estado} and {\em Pensa Rio}.
appendix
\section{Proofs, Technical Results and Definitions}
In this appendix we collect technical results and definitions that were used in the remaining parts of the article. We also present the proofs of some results of from Section~3.
\subsection{Bregman Distance and $q$-Coerciveness}\label{app:def}
\begin{df}{\cite[Definition 3.15]{schervar}}
Let $X$ denote a Banach space and $f: D(f) \subset X \rightarrow \R\cup {\infty}$ be a convex functional with sub-differential $\partial f(x)$ in $x \in D(f)$. The Bregman distance (or divergence) of $f$ at $x \in D(f)$ and $\xi \in \partial f(x) \subset X^*$ is defined by
$
D_{\xi}(\tilde{x},x) = f(\tilde{x}) - f(x) - \langle\xi,\tilde{x} - x\rangle,
$
for every $\tilde{x} \in X$, with $\langle\cdot,\cdot\cdot\rangle$ the dual product of $X^*$ and $X$. Moreover, the set
$
\mathcal{D}_B(f) = \{x \in D(f) ~:~ \partial f(x) \not= \emptyset\}
$
is called the Bregman domain of $f$.
\end{df}
We stress that the Bregman domain $\mathcal{D}_B(f)$ is dense in $D(f)$ and the interior of $D(f)$ is a subset of $\mathcal{D}_B(f)$. The map $\tilde{x}\mapsto D_{\xi}(\tilde{x},x)$ is convex, non-negative and satisfies $D_{\xi}(x,x) = 0$. In addition, if $f$ is strictly convex, then $D_{\xi}(\tilde{x},x) = 0$ if and only if $\tilde{x} = x$. For a survey in Bregman distances see \cite[Chapter I]{butiusem}.
\begin{df}
For $1\leq q <\infty$ and $x \in D(f)$, the Bregman distance $D_{\xi}(\cdot,x)$ is said to be $q$-coercive with constant $\zeta>0$ if
$
D_{\xi}(y,x) \geq \zeta \|y-x\|^q_X
$
for every $y \in D(f)$.
\end{df}
\subsection{Equicontinuity}
Let $X$ and $Y$ be locally convex spaces. Fix the sets $B_X \subset X$ and $M \subset C(B_X,Y)$. A set $M$ is called equicontinuous on $B_X$ if for every $x_0 \in B_X$ and every zero neighborhood, $V \subset Y$ there is a zero neighborhood $U \subset X$ such that $G(x_0) - G(x) \in V$ for all $G \in M$ and all $x \in B_X$ with $x-x_0 \in U$. Furthermore, $M$ is called uniformly equicontinuous if for every zero neighborhood $V \subset Y$ there exists a zero neighborhood $U \subset X$ such that $G(x) - G(x^\prime) \in V$ for all $G \in M$ and all $x,x^\prime \in B_X$ with $x-x^\prime \in U$.
From \cite{haschele} we have the technical result:
\begin{pr}
Let $F: [0,T]\times B_X \longrightarrow Y$ be a function, and $B_X$, $X$ and $Y$ be as above. If $M_1:= \{F(t,\cdot) : t \in [0,T]\} \subset C(B_X,Y)$, $M_2:=\{F(\cdot,x) : x \in B_X\} \subset C([0,T],Y)$ and $M_1$ (respectively $M_2$) is equicontinuous, then $F$ is continuous. Reciprocally, if $F$ is continuous, then $M_1$ is equicontinuous and if additionally $B_X$ is compact, then $M_2$ is equicontinuous, too.\label{prop11}
\end{pr}
\subsection{Proof of Results from Section~3}\label{app:results}
{\bf Proof of Theorem \ref{prop22}:}
{\it Well Posedness:} Take an arbitrary ${\tilde{a}}f \in {\mathfrak{Q}}$, by the continuity of ${\tilde{a}}f$ (see Proposition \ref{p1}) and $F$, it follows that $t \mapsto F(s,{\tilde{a}}(s))$ is continuous and then weakly measurable. Therefore,
$s \mapsto \|F(s,a(s))\|_{\ya}$ is bounded, then ${\mathcal{U}}({\tilde{a}}f) \in\Y$, which asserts the well-posedness of ${\mathcal{U}}(\cdot)$.
\noindent {\it Continuity:} As $F:[0,S]\times Q \longrightarrow \ya$ is continuous, it follows by Proposition \ref{p1} that the set $\{F(s,\cdot) \left|~ s \in [0,S]\right.\} \subset C(Q,\ya)$ is uniformly equicontinuous, i.e., given $\epsilon > 0$, there is a $\delta > 0$ such that, for all $a,{\tilde{a}} \in Q$ satisfying $\|a - {\tilde{a}}\| < \delta$, we have that
$\sup_{s \in [0,S]}\|F(s,a)-F(s,{\tilde{a}})\| < \epsilon.$
Thus, given $\epsilon > 0$ and ${\mathcal{A}}, {\tilde{a}}f \in {\mathfrak{Q}}$ such that $\sup_{s \in [0,S]}\|a(s) - {\tilde{a}}(s)\|_{\he}<\delta$, then, by the uniform equicontinuity of $\{F(s,\cdot), s \in [0,S]\}$, it follows that
$$
\displaystyle\|{\mathcal{U}}({\mathcal{A}}) - {\mathcal{U}}({\tilde{a}}f)\|^2_{\Y} = \displaystyle\int^S_0\|F(s,a(s)) - F(s,{\tilde{a}}(s))\|^2_{\ya}ds < \epsilon^2\cdot S,
$$
which asserts the continuity of ${\mathcal{U}}(\cdot)$.
\noindent {\it Compactness:} It is sufficient to prove that, given an $\epsilon > 0$ and a sequence $\{{\mathcal{A}}_n\}_{n \in \N}$in ${\mathfrak{Q}}$ converging weakly to ${\tilde{a}}f$, it follows that there exist an $n_0$ and a weak zero neighborhood $U$ of $\X$ such that for $n > n_0$, ${\mathcal{A}}_n-{\tilde{a}}f \in U$ and $\|{\mathcal{U}}({\mathcal{A}}_n) - {\mathcal{U}}({\tilde{a}}f)\|_{\Y}< \epsilon.$
Following the same arguments of the proof of Lemma \ref{lemw}, we can find a set of functionals $\mathcal{C}_{n,m} \in \X^*$, defining such zero neighborhood $U$. We first note that, since $F$ is weak continuous, it follows that, given an $\epsilon >0$, there are $alpha_1,...,alpha_N \in \he$ and $\delta > 0$, such that
$\sup_{s \in [0,S]}\|F(s,a) - F(s,{\tilde{a}})\| < \epsilon/S$ for all $a,{\tilde{a}} \in B$ with
\begin{equation}
\max\{|\langle a - {\tilde{a}}, alpha_n \rangle_{\he} |\,:\, n = 1,...,N\} < \delta.\label{p5:eq1}
\end{equation}
By Proposition \ref{p1}, the estimate $\langle {\mathcal{A}},alpha_n\rangle_{\he} \in H^l[0,S]$ holds with its norm bounded by $\|{\mathcal{A}}\|_l\|alpha_n\|_{\he}$. Then, there is a closed and bounded ball
$A \subset H^l[0,S]$ containing $\langle {\mathcal{A}},alpha_n\rangle_{\he}$, for all $n = 1,...,N,$
and ${\mathcal{A}} \in \mathbb{B}$.
For $n = 1,...,N$ and the same $\delta > 0$ of \eqref{p5:eq1}, there are $f_{n,1},...,f_{n,M(n)}$ in $H^l[0,S]$ and $\xi_n > 0$ such that, $\|f\|_{C([0,S])} < \delta$ for every $f \in A$ satisfying the estimate
$\max_{m = 1,...,M(n)}|\langle f,alpha_n\rangle_{\he}| < \xi_n.$
Define $\mathcal{C}_{n,m} : = alpha_n \otimes f_{n,m}$, with $n = 1,...,N$ and $ m=1,...,M(n)$. It is an element of $\X^*$, where, for each ${\mathcal{A}} \in \X$, we have that
$\langle{\mathcal{A}}, \mathcal{C}_{n,m}\rangle_l = \langle \langle {\mathcal{A}},alpha_n\rangle_{\he}, f_{n,m}\rangle_{H^l[0,S]}$ and thus
$$\langle{\mathcal{A}}, \mathcal{C}_{n,m}\rangle_l = \displaystyle\sum_{k\in \Z}(1 + |k|^l)^2\langle \hat{a}(k),alpha_n\rangle_{\he}\hat{f}_{n,m}(k).
$$
These functionals define a weak zero neighborhood $U := \cap^N_{n=1}U_n$ with
$$
U_n : = \{ {\mathcal{A}} \in \X : |\langle {\mathcal{A}}, \mathcal{C}_{n,m}\rangle_l| < \xi_n, ~m=1,...,M(n)\}.
$$
Therefore, if $\{{\mathcal{A}}_k\}_{k\in\N}\subset \mathbb{B}$ converges weakly to ${\tilde{a}}f \in \mathbb{B}$, then for a sufficient large $k$, ${\mathcal{A}}_k-{\tilde{a}}f \in U$ and by the definition of $U$, we have that for each $n = 1,...,N$,
$\xi_n > |\langle {\mathcal{A}}-{\tilde{a}}f, \mathcal{C}_{n,m}\rangle_l| = |\langle \langle {\mathcal{A}}-{\tilde{a}}f,alpha_n\rangle_{\he}, f_{n,m} \rangle_{H^l[0,S]}|
$
for all $m = 1,...,M(n)$.
By the choice of the $f_{n,m} \in H^l[0,S]$, it follows that
$\|\langle {\mathcal{A}}_k-{\tilde{a}}f,alpha_n\rangle_{\he}\|_{H^l[0,S]} < \delta$ for all $n = 1,...,N,$
which implies that $\|{\mathcal{U}}({\mathcal{A}}_k) - {\mathcal{U}}({\tilde{a}}f)\|_{\Y} \leq \epsilon\cdot T$.
\noindent {\it Weak Continuity:} The weak continuity follows directly from the proof of compactness, as we use the same framework, only changing the compactness of $F$, by the weakly equicontinuity of $\{F(s,\cdot) : ~s \in [0,S]\}$ on bounded subsets of $Q$.
\noindent {\it Weak Closedness:} Just note that the set ${\mathfrak{Q}}$ is weakly closed and the operator ${\mathcal{U}}(\cdot)$ is weakly continuous. \fim
{\bf Proof of Proposition \ref{prop6}}
By Proposition \ref{prop4}, the family of operators $\{F(s,\cdot) \,: \,s \in [0,S]\}$ is Frech\'et equi-differentiable. Take ${\tilde{a}}f,\mathcal{H} \in \X$, such that ${\tilde{a}}f,{\tilde{a}}f+\mathcal{H} \in {\mathfrak{Q}}$. Then, define the one sided derivative of ${\mathcal{U}}(\cdot)$ at ${\tilde{a}}f$ in the direction $\mathcal{H}$ as
${\mathcal{U}}^\prime({\tilde{a}}f)\mathcal{H} := \{s \mapsto \partial_a F(s,{\tilde{a}}(s))h(s)\}$,
where for each $s \in [0,S]$, dropping $t$ to easy the notation, $\partial_a F(s,{\tilde{a}})h$ is the solution of
$$
-v_{\tilde{a}}u + a(v_{yy}-v_y) + bv_y = h(u_{yy}-u_y)
$$
with homogeneous boundary conditions and $u = u(s,a(s))$. From Proposition \ref{prop21} we have the estimate
$\|\partial_a F(s,{\tilde{a}}(s))h(s)\|_{\ya} \leq C\|h(s)\|_{L^2(D)}\|u_{yy}(s,{\tilde{a}}(s))-u_{y}(s,{\tilde{a}}(s))\|_{L^2(D)}$.
Note that, $\|u_{yy}(s,a)-u_{y}(s,a)\|_{L^2(D)}$ is uniformly bounded in $[0,S]\times Q$. Thus, ${\mathcal{U}}^\prime({\tilde{a}}f)\mathcal{H}$ is well defined and
\begin{multline}
\left\| {\mathcal{U}}^\prime({\tilde{a}}f)\mathcal{H}\right\|^2_{\Y} = \displaystyle\int^S_0\|\partial_a F(s,{\tilde{a}}(s))h(s)\|^2_{\ya}ds \\ \leq C \displaystyle\int^S_0\|h(s)\|_{L^2(D)}\|u_{yy}(s,{\tilde{a}}(s))-u_{y}(s,{\tilde{a}}(s))\|_{L^2(D)}ds\\
\leq c\displaystyle\int^S_0\|h(s)\|^2_{L^2(D)}ds = c\|\mathcal{H}\|^2_{\X}
\end{multline}
Therefore, $\mathcal{U}^\prime({\tilde{a}}f)$ can be extended to a bounded linear operator from the space $\X$ into $\Y$.
Let ${\tilde{a}}f,\mathcal{H},\mathcal{G} \in \X$ be such that, ${\tilde{a}}f,{\tilde{a}}f+\mathcal{H},{\tilde{a}}f+\mathcal{G}, {\tilde{a}}f+\mathcal{H}+\mathcal{G}$ are in $Q$.
Define $v:=u(s,a(s)+h(s)) - u(s,a(s))$. Thus,
$$
w := \partial_a u(s,a(s)+h(s))g(s) - \partial_a u(s,a(s))g(s)
$$
satisfies
$$
-w_{\tilde{a}}u + a(w_{yy} - w_y) = -g[v_{yy} - v_{y}] - h[(\partial_a u(s,a+h)g)_{yy} - (\partial_a u(s,a+h)g)_{y}],
$$
with homogeneous boundary conditions (dropping the dependence on $s$). As above, we have
\begin{multline}
\left\|\mathcal{U}^\prime({\tilde{a}}f+\mathcal{H})\mathcal{G} - \mathcal{U}^\prime({\tilde{a}}f) \mathcal{G}\right\|^2_{\Y} = \displaystyle\int^S_0\|w\|^2_{\ya}ds\\
\leq c_1\displaystyle\int^S_0\|g(s)\|^2_{L^2(D)}\|v_{yy}(s,{\tilde{a}}(s)) - v_y(s,{\tilde{a}}(s))\|^2_{L^{2}(D)}ds\\
+ c_2 \displaystyle\int^S_0\|h(s)\|^2_{L^2(D)}\|\partial_a u(s,a(s)+h(s))g(s)\|^2_{\ya}ds \\
\leq C\|\mathcal{H}\|^2_{\X}\|\mathcal{G}\|^2_{\X},
\end{multline}
which yields the Lipschitz condition. \fim
addcontentsline{toc}{section}{Bibliography}
\noindent Instituto Nacional de Matem\'atica Pura e Aplicada\\
Estr. D. Castorina 110, 22460-320. Rio de Janeiro,\\
Brazil.
\noindent E-mail: \href{mailto:vvla@impa.br}{\tt vvla@impa.br} (Vinicius Albani) and \href{mailto:zubelli@impa.br}{\tt zubelli@impa.br} (Jorge Zubelli).
\end{document} |
\begin{document}
\markboth{GREGOR WEIHS}{PHOTONIC CRYSTAL WAVEGUIDES FOR PARAMETRIC
DOWN-CONVERSION}
\catchline{}{}{}{}{}
\title{PHOTONIC CRYSTAL WAVEGUIDES FOR PARAMETRIC DOWN-CONVERSION}
\author{GREGOR WEIHS}
\address{Institute for Quantum Computing and Department of Physics,
University of Waterloo, 200 University Ave W, Waterloo, Ontario, N2L
3G1, Canada \\ weihs@iqc.ca}
\maketitle
\begin{history}
\received{07 10 2005}
\end{history}
\begin{abstract}
Photonic crystals create dramatic new possibilities for nonlinear
optics. Line defects are shown to support modes suitable for the
production of pairs of photons by the material's second order
nonlinearity even if the phase-matching conditions cannot be
satisfied in the bulk. These structures offer the flexibility to
achieve specific dispersion characteristics and potentially very
high brightness. In this work, two phase matching schemes are
identified and analyzed regarding their dispersive properties.
\end{abstract}
\keywords{Nonlinear Optics, Parametric processes, Photonic
integrated circuits}
\section{INTRODUCTION}
For many applications there is a need for new sources of entangled
photon pairs that are brighter or show certain dispersion
characteristics. In quantum communication for example, it is
entanglement based quantum key distribution, quantum teleportation
and, most importantly, quantum repeaters that all depend on a good
supply of entangled photon pairs.
But photon pairs have more traditional applications as well. Phase
measurements and lithography may one day employ entanglement to beat
their classical limits. In these scenarios one would not only look
for entangled pairs, but possibly even for entangled states of
multiple photons. At present, though, the efforts to even create
enough pairs to begin with, far outweighs the possible gains in
precision. It is yet more difficult and costly in terms of resources
to interferometrically engineer higher entangled states from pairs.
This would not necessarily be the case if only there were sources
available that could be plugged into a wall outlet and had fiber
outputs providing the entangled pairs, or even higher dimensional
entangled photon states. To date the majority of experiments obtains
their pairs from parametric down-conversion in bulk
materials\cite{Kwiat95b}.
One standard way of achieving nonlinear interaction in materials
that are not a priori suitable for this purpose is to use periodic
poling. In periodic poling the sign of the nonlinear tensor
component is reversed periodically. However, the linear dispersion
characteristics all stay the same. Still, periodically poled
crystals and waveguides have been used as high-yield sources of
entangled photon pairs\cite{Sanaka01a,Tanzilli01a}.
Photonic crystals allow us to go beyond the properties of natural
materials in many ways\cite{Joannopoulos95a}. They have been shown
to exhibit super strong or very small dispersion, enhance nonlinear
interactions\cite{Cowan05a}, and to confine waves in guides and
resonators. Therefore it only seems natural to consider photonic
crystals for the production of entangled photon
pairs\cite{Vamivakas04a}. De Dood \textit{et al.}\cite{deDood04a}
recently suggested to exploit form birefringence in multilayer
stacks (one-dimensional photonic crystals) for achieving
phase-matching in GaAs.
\section{PARAMETRIC DOWN-CONVERSION IN PHOTONIC CRYSTAL DEFECT
WAVEGUIDES}
Spontaneous parametric down-conversion is difference-frequency
generation where a pump beam at angular frequency $\omega_p$ and
wavevector $\mbf{k}_p$ irradiates a material with some nonlinear
tensor $\chi^{(2)}$. Then there is a probability to create pairs of
photons at $(\omega_1, \mbf{k}_1)$ and $(\omega_2, \mbf{k}_2)$ so
that $\omega_1+\omega_2=\omega_p$ (\emph{energy conservation}) and
$\mbf{k}_1+\mbf{k}_2=\mbf{k}_p$ (\emph{phase-matching}). In some
cases phase-matching can even be arranged such that the photon pairs
are polarization entangled\cite{Kwiat95b}.
Obviously, $\chi^{(2)}$ should be large and one should choose the
largest component of the tensor by selecting the polarizations and
directions of the involved light fields to maximize the effective
nonlinearity. In bulk materials, however, it is not always possible
to work in the maximizing configuration because the natural
dispersion of the material requires one to choose certain directions
in order to satisfy the phase-matching condition.
Periodicity can come to the rescue. In a periodic structure we know
from Bloch's theorem that the solutions of Maxwell's equation will
be periodic functions multiplied by a plane wave. The periodic
functions are labeled by a wavevector $\mbf{k}$ and we only need to
consider wavevectors within the first Brillouin zone. This means
that the phase-matching condition now reads
$\mbf{k}_1+\mbf{k}_2=\mbf{k}_p+\mbf{G}$ where $\mbf{G}$ is any
wavevector of the reciprocal lattice. In other words, any wavevector
we consider can be mapped to one in the first Brillouin zone and
phase-matching has to be satisfied within this zone.
Periodicity has been used in the form of periodic poling. In certain
crystals such as LiNbO$_3$ and KTiOPO$_4$ it is possible to reverse
the sign of the nonlinear optic coefficient. This will make it
possible to achieve phase-matching in otherwise forbidden
interaction schemes and therefore to utilize the maximum
nonlinearity and a long interaction region. Other materials that are
promising because of their large nonlinearity, such as GaAs, cannot
easily be poled, even though some progress has been
made\cite{Skauli02a}.
Because the number of pairs created in parametric down-conversion is
in the low excitation regime linear with the applied pump power
there is usually no need to go to pulsed sources. However, for
applications that need more than one photon pair at a time, such as
entanglement swapping, one needs to down-convert from ultra-fast
pulses\cite{Zukowski93a}. In this case, in addition to the
phase-matching condition one should ideally achieve group matching
as well. Otherwise there will be some differential group delay
between the pump and converted waves, or even between the two
converted waves if they have different polarization. In the source
of entangled photon pairs first described by Kwiat \textit{et
al.}\cite{Kwiat95b} differential group delay severely limits the
achievable entanglement and/or brightness, when it is used with a
fast pulsed pump and not limited to a very short interaction
region\cite{Keller97a}.
While periodic poling solves the phase-matching problem, it does not
change the group velocities, because the effective refractive index
is not changed. A natural consequence is therefore to not only
modulate the nonlinearity but the dielectric as a whole, i.e. to
look at photonic crystal structures. In contrast to earlier
suggestions\cite{Vamivakas04a,deDood04a}, here we will concentrate
on waveguide structures that are formed by a line defect in a
slab-type photonic crystal as shown in Fig.~\ref{fig1}. The purpose
of having a waveguide is a long interaction region and good mode
overlap.
\begin{figure}
\caption{\textbf{(Left)}
\label{fig1}
\end{figure}
Waveguides in photonic crystal slabs (PCS) have first been studied
theoretically by Johnson \textit{et al.}\cite{Johnson00a}. There are
many possibilities to create waveguides in photonic crystals. The
most popular configuration is a line defect in a lattice of
air-holes produced by reactive ion etching with subsequent undercut
so as to create a free-standing membrane of materials such as GaAs,
InP or Si. Spectacular structures up to 1~cm long have been
demonstrated with losses as low as 0.76~dB/cm \citelow{Sugimoto04a}.
AlGaAs is a favorable material from many points of view. The fact
that its lattice constant is almost independent of the Aluminum
fraction makes it possible to grow arbitrary heterostructures. Thus,
the fundamental electronic band gap can be chosen freely. It has
very high refractive index varying from about 3.6 (for 15\%
Aluminium) at 775~nm to about 3.3 at 1550~nm. The reported nonlinear
susceptibilities scatter but in recent years a consensus seems to
have been reached\cite{Skauli02a} at values around 100~pm/V, which
is about 5 times higher than for LiNbO$_3$ and almost 50 times that
of $\beta$-BaB$_2$O$_4$. The maximum effective nonlinearity is
achieved for three identical polarizations parallel to the [111]
direction. Further, if any of the interacting waves is polarized
along [011] the effective nonlinearity is independent of the other
waves' polarization. Growth of AlGaAs is usually done on [100] cut
wafers. Light propagating in a membrane parallel to the surface
could therefore propagate in any direction orthogonal to [100].
Fully three-dimensional band structure calculations using the MIT
Photonic Bands package\cite{Johnson01a} helped identify possible
phase-matching schemes. In the literature there is very little data
on band gaps for PCS as a function of either hole radius, or slab
thickness or dielectric constant. Since for the problem at hand we
need to be able to choose the location of the band gap with some
care, I first calculated a series of bandstructures resulting in a
map of the gap locations as a function of the hole radius. The
shaded area in Fig.~\ref{fig1}(right) shows the bandgap for even
modes in an hexagonal array of holes patterned into a membrane as a
function of the hole radius.
A photonic crystal defect waveguide in which the defect has
increased refractive index with respect to the periodic structure
around it can support two types of guided modes. There are
index-guided modes lying below the lowest PCS bands and modes that
lie within the band gap of the PCS. Obviously, the confinement will
be better for the latter, but if there are no sharp bends, index
guided modes should not suffer from much higher losses. Further, in
a PCS that is symmetric about its central plane the bands split into
even and odd ones, closely corresponding to their TE and TM
polarized counterparts in two-dimensional photonic crystals.
The allowed regions for defect waveguide modes are determined by
projecting the bands of the perfect PCS onto the guiding
k-direction. This procedure masks all areas in the $(\omega,k_x)$
strip that are covered by states that are extended in the plane and
can thus not support modes that would be localized to a defect. This
is in addition to the restriction that all guided modes lie outside
the lightcone of the surrounding medium. In the cases investigated
here, the surrounding medium is air around the free-standing
Al$_{0.15}$Ga$_{0.85}$As membrane. The material choice is mainly
motivated by the desire to create photon pairs at telecommunications
wavelengths, i.e. 1550~nm. The pump would then have to be at 775~nm
for a symmetric source which requires a 10\% Aluminium fraction or
higher. At 15\% one should be safely outside all exciton and
impurity resonances and expect very little absorption for the pump.
From studying gap maps for hexagonal and square lattices of holes,
we can conclude that either may be suitable for waveguiding, with
the hexagonal ones yielding larger gaps and thus possibly more
flexibility. The most basic defect waveguide is a single missing row
of holes (see Fig.~\ref{fig1}), in which waves then propagate in the
$\Gamma$-K direction of the perfect PCS. It is well
known\cite{Johnson00a} that such a waveguide is not single-mode.
Single-mode behavior can be brought about by decreasing the width of
the defect\cite{Benisty96a}. It is clear that for a proper source,
suitable for the generation of entangled pairs single mode behavior
will be essential.
\begin{figure}
\caption{Dispersion diagram (frequency vs. $k_x$) for the z-even
modes of a structure like in Fig.~\protect\ref{fig1}
\label{eveneven}
\end{figure}
Figure~\ref{eveneven} shows a scenario in which photons from a
z-even gap-guided mode can down-convert into two photons of a z-even
index-guided band.\protect\footnote{Unfortunately the frequency
domain method used in the MIT Photonic Bands package cannot treat
dispersive materials. Therefore the calculation was split into two,
one for the low refractive index at long wavelengths, and one for
the high refractive index and shorter wavelength.}
Figure~\ref{oddeven} on the other hand shows a configuration in
which pump photons in a z-odd index-guided band down-convert into
photon pairs of a z-even index-guided band. The lattice is
responsible for producing such a vastly different dispersion for the
two different symmetries\cite{Joannopoulos95a}.
\begin{figure}
\caption{Similar to Fig.~\ref{eveneven}
\label{oddeven}
\end{figure}
In any case, the relative strength of the nonlinear interaction will
depend on the overlap of the three fields involved. The amplitude of
the one photon pair term in the output state will be proportional to
(in first order perturbation theory)
\[ \int_V d^3\mathbf{r} \chi^{(2)}_{ijk} E_i E_j E_k, \]
where $E_{i,j,k}$ are three components of the electrical field and
we'll have to take the sum over all indices. One of the field
components would be the pump (high frequency) and the two other ones
the down-conversion fields (low frequency). For the case where the
low-frequency modes are degenerate the integrand reduces to $E_i
E_j^2$. In the case of a periodic structure, we can take the
integral over a unit cell. Obviously, the field distribution within
the unit cell has to be similar between the three fields to achieve
substantial overlap. Also, the fields will only contribute within
the material, but not in the air space within the holes or above and
below the slab. Preliminary calculations show that at least the
phase-matching scheme shown in Fig.~\ref{eveneven} achieves
reasonable overlap for band number 9 (counted from 0 frequency) at
$k=0.44$ with band number 1 at $k=0.22$. It is part of our ongoing
work to calculate the effective nonlinearity in absolute units.
\section{DISCUSSION}
The goal for parametric down-conversion in photonic crystals is to
exploit the high nonlinearity of certain semiconductors and possibly
achieve group velocity matching. Group velocity matching is similar
to achieving phase-matching over an extended bandwidth, if the band
curvature is not too big.
It is obvious that the above schemes do not yet achieve
group-matching, where the second one is closer to the goal but still
too far away to be practically relevant for this purpose. Given that
the gap-guided modes exhibit very small group velocities, it would
be exciting if there was a way to phase-match between bands of
different gaps. However, since the gaps in a PCS are typically
shifted up in frequency by the vertical confinement it seems very
difficult to find gaps at one frequency and twice the frequency
simultaneously, where both are also below the light cone in the
waveguiding direction.
For photon pairs created by ultrafast pulses one has to compare the
differential group delay (DGD) with the pulse duration. If the group
velocities are $u_\mrm{pump}$ and $u_\mrm{dc}$ then the DGD per
length is just $|1/u_\mrm{pump}-1/u_\mrm{dc}|$, which means that for
small group velocities it will be more difficult to achieve a small
DGD.
Yet, for some applications, such as interfacing to
electromagnetically induced transparency (EIT) and stored light it
is desirable to have very narrow-band down-conversion sources. For
this purpose various groups have considered counter-propagating
solutions, which are typically phase-matched only in a point, or
equivalently, have an extremely high DGD, because one of the
velocities is negative. For this purpose the first even-even-even
phase-matching scheme (Fig.~\ref{eveneven}) appears to be a very
good solution.
\section{CONCLUSIONS}
So far, we have only considered degenerate cases, i.e. both
down-conversion photons have the same frequency. Asymmetric sources
have applications in the preparation of single-photon states and in
schemes where one photon needs to be detected with high efficiency,
whereas the other needs to propagate with very low loss through an
optical fiber. Clearly, there are many solutions for non-degenerate
phase-matching to be found in the above diagrams. Whether a certain
solution is interesting depends on the details of the intended
application.
I have shown that phase matching is in principle possible in
photonic crystal waveguides. The coupling of both the pump light and
the down-converted one in and out of the waveguide respectively is a
challenge. Cleaved edges can be used\cite{Kramper04a}, as well as
fiber tapers\cite{Barclay03a}, where the latter have achieved the
highest reported coupling efficiency to date.
The biggest interest in photon pairs is associated with
entanglement. With a single waveguide mode and polarization there
can be no entanglement a priori. However, the photons of a pair are
still created simultaneously and therefore it should always be
possible to construct a source of time-bin entangled photon pairs as
is customary for photon pairs from waveguides\cite{Tanzilli01a}.
With a solution that creates pairs in two different polarizations
one can, using two identical waveguides, even construct a source of
polarization entangled photon pairs.
In conclusion, we have seen that it is possible to exploit the high
optical nonlinearity of AlGaAs and to achieve phase-matching in a
slab-type photonic crystal defect waveguide. The next steps are to
perform fully dispersive calculations and to tune the waveguide
modes by changing the waveguide's width and edge shape so as to give
the involved modes the desired dispersion. While some details are
still missing in this picture an ultra bright source of photon pairs
at telecommunication wavelengths based on photonic crystals seems
within reach.
\end{document} |
\begin{document}
\title
{Non-critical dimensions for critical problems\\
involving fractional Laplacians}
\author{Roberta Musina\varphiootnote{Dipartimento di Matematica ed Informatica, Universit\`a di Udine,
via delle Scienze, 206 -- 33100 Udine, Italy. Email: {roberta.musina@uniud.it}.
{Partially supported by Miur-PRIN 2009WRJ3W7-001 ``Fenomeni di concentrazione e {pro\-ble\-mi} di analisi geometrica''.}}~ and
Alexander I. Nazarov\varphiootnote{
St.Petersburg Department of Steklov Institute, Fontanka 27, St.Petersburg, 191023, Russia,
and St.Petersburg State University,
Universitetskii pr. 28, St.Petersburg, 198504, Russia. E-mail: al.il.nazarov@gmail.com.
Supported by RFBR grant 11-01-00825 and by
St.Petersburg University grant 6.38.670.2013.}
}
\date{}
\maketitle
\begin{abstract}
We study the Brezis--Nirenberg effect in
two families of noncompact boundary value problems involving Dirichlet-Laplacian
of arbitrary real order $m>0$.
\varphiootnotesize
\noindent
\textbf{Keywords:} {Fractional Laplace operators, Sobolev inequality, Hardy inequality, critical dimensions.}
\noindent
\end{abstract}
\normalsize
\section{Introduction}
\label{S:Introduction}
Let $m,s$ be two given real numbers, with $0\le s<m<\varphirac{n}{2}$. Let
$\Omega\subset\mathbb{R}^n$ be a bounded and smooth domain in $\mathbb{R}^n$
and put
$$
{2_m^*}=\varphirac{2n}{n-2m}.
$$
We study equations
\begin{gather}
\label{eq:problem2}
(-\Delta)^mu=\lambda(-\Delta)^s u+|u|^{{2_m^*}-2}u\quad\textrm{in $\Omega$,}\\
\nonumber
~\\
\label{eq:problem1}
(-\Delta)^mu=\lambda|x|^{-2s} u+|u|^{{2_m^*}-2}u\quad\textrm{in $\Omega$,}
\end{gather}
under suitably defined
Dirichlet boundary conditions. In dealing with equation (\ref{eq:problem1}) we
always assume that $\Omega$ contains the origin. For the definition
of fractional Dirichlet--Laplace operators $(-\Delta)^m, (-\Delta)^s$ and
for the variational approach to (\ref{eq:problem2}), (\ref{eq:problem1}) we refer to the next section.
The celebrated paper \cite{BN} by Brezis and Nirenberg
was the inspiration for a fruitful line of research about the effect of lower
order perturbations in noncompact variational problems. They
took as model the case $n>2$, $m=1$, $s=0$, that is,
\begin{equation}
\label{eq:BN_problem}
-\Delta u=\lambda u+|u|^{\varphirac{4}{n-2}}u\quad\textrm{in $\Omega$,}\qquad
u=0\quad
\textrm{on $\partial\Omega$.}
\end{equation}
Brezis and Nirenberg pointed out a remarkable phenomenon that appears
for positive values of the parameter $\lambda$: they proved
existence of a nontrivial solution for any small $\lambda>0$ if $n\ge 4$; in contrast,
in the lowest dimension
$n=3$ non-existence phenomena for sufficiently small $\lambda>0$ can be observed.
For this reason, the dimension $n=3$ has been named {\em critical}\varphiootnote{\varphiootnotesize{
compare with \cite{PuSe}, \cite{GGS}.}}
for problem (\ref{eq:BN_problem}).
Clearly, as larger $s$ is, as stronger the effects of the lower order perturbations are
expected in equations (\ref{eq:problem2}), (\ref{eq:problem1}).
We are interested in the following question:
{\em Given $m<\varphirac{n}{2}$, how large must be $s$
in order to have the existence of a ground state solution, for any
arbitrarily small $\lambda>0$ ?} In case of an affirmative answer,
we say that $n$ is \underline{not} a critical dimension.
We present our main result, that holds for any dimension $n\ge 1$
(see Section \ref{S:proof_main} for a more precise statement).
\noindent{\bf THEOREM.}
{
\em If $s\ge 2m-\varphirac{n}{2}$ then $n$ is not a critical dimension for the Dirichlet boundary
value problems
associated to equations (\ref{eq:problem2})
and (\ref{eq:problem1}).}
We point out some particular cases that are included in this result.
\begin{description}
\item$\bullet$
If $m$ is an integer and $s=m-1$, then at most the lowest dimension
$n=2m+1$ is critical.
\item$\bullet$
For any $n>2m$ there always exist lower order perturbations
of the type $|x|^{-2s}u$ and of the type $\left(-\Delta\right)^{\!s}\! u$ such that $n$ is not a critical dimension.
\item$\bullet$
If $ m<1/4$ then no dimension is critical, for any choice of $s\in[0,m)$.
\end{description}
After \cite{BN}, a large number of papers have been focussed
on studying the effect of linear perturbations in noncompact variational problems
of the type (\ref{eq:problem2}). Most of these papers deal with $s=0$,
when the problems (\ref{eq:problem2}) and (\ref{eq:problem1}) coincide.
Moreover, as far as we know, all of them consider
either polyharmonic case $2\le m\in \mathbb N$, see for instance
\cite{PuSe}, \cite{EFJ}, \cite{BG1}, \cite{Gr}, \cite{Gaz98}, or the case $m\in(0,1)$,
see \cite{SV1}, \cite{SV2}. We cite also \cite{CM2}, where equation
(\ref{eq:problem2}) is studied in case $m=2$, $s=1$.
Thus, our Theorem \ref{T:main} covers all earlier existence results.
Finally, we mention \cite{BaCPS} (see also \cite{Tan})
where equation (\ref{eq:problem2}) for the so-called Navier-Laplacian is studied
in case $m\in(0,1)$, $s=0$. For a comparison between the Dirichlet
and Navier Laplacians we refer to \cite{FL}.
The paper is organized as follows. After introducing
some notation and preliminary facts in
Section \ref{S:Preliminaries}, we provide the main estimates in Section
\ref{S:estimates}. In Section \ref{S:proof_main} we prove Theorem 1 and
point out an existence result for the case $s< 2m-\varphirac{n}{2}$.
\section{Preliminaries}
\label{S:Preliminaries}
The fractional Laplacian $\left(-\Delta\right)^{\!m}\! u$ of a function $u\in {\cal C}^\infty_0(\mathbb{R}^n)$ is defined via the Fourier transform
$$
{\cal F}[u](\xi)=\varphirac{1}{(2\pi)^{n/2}}\int\limits_{\mathbb R^n} e^{-i~\!\!\xi\cdot x}u(x)~\!dx
$$
by the identity
\begin{equation}
\label{eq:classicalF}
{\cal F}\left[{\left(-\Delta\right)^{\!m}\! u}\right](\xi)=|\xi|^{2m}{\cal F}[u](\xi).
\end{equation}
In particular, Parseval's formula gives
$$
\int\limits_{\mathbb R^n}\left(-\Delta\right)^{\!m}\! u\cdot u~\!dx=
\int\limits_{\mathbb R^n}|\left(-\Delta\right)^{\!m}\!half u|^2~\!dx=
\int\limits_{\mathbb R^n}|\xi|^{2m}|{\cal F}[u]|^2~\!d\xi~\!.
$$
We recall the well known Sobolev inequality
\begin{equation}
\label{eq:Sobolev}
\int\limits_{\mathbb{R}^n}|\left(-\Delta\right)^{\!m}\!half u|^2~\!dx\ge {\cal S}_m\bigg(\,\int\limits_{\mathbb{R}^n}|u|^{2_m^*}~\!dx\bigg)^{2/{2_m^*}},
\end{equation}
that holds for any $u\in {\cal C}^\infty_0(\mathbb{R}^n)$ and $m<\varphirac n2$, see for example \cite[2.8.1/15]{Tr}.
Let $\left(-\Delta\right)^{\!s}\!pace$ be the Hilbert space obtained
by completing ${\cal C}^\infty_0(\mathbb{R}^n)$ with respect to the Gagliardo norm
\begin{equation}
\label{eq:Gagliardo}
\|u\|_m^2 =
\int\limits_{\mathbb{R}^n}|\left(-\Delta\right)^{\!m}\!half u|^2~\!dx.
\end{equation}
Thanks
to (\ref{eq:Sobolev}), the space
$\left(-\Delta\right)^{\!s}\!pace$ is continuously embedded into $L^{2_m^*}(\mathbb{R}^n)$. The
{\em best Sobolev constant} ${\cal S}_m$ was explicitly computed in \cite{CoTa}. Moreover,
it has been proved
in \cite{CoTa} that ${\cal S}_m$ is attained in $\left(-\Delta\right)^{\!s}\!pace$ by a unique family of functions, all of them
being obtained from
\begin{equation}
\label{eq:AT}
\phi(x)=(1+|x|^2)^{\varphirac{2m-n}{2}}
\end{equation}
by translations, dilations in $\mathbb{R}^n$ and multiplication by constants.
Dilations play a crucial role in the problems under consideration. Notice that
for any $\omega\in {\cal C}^\infty_0(\mathbb{R}^n)$, $R>0$ it turns out that
\begin{eqnarray}
\label{eq:dilation}
\int\limits_{\mathbb{R}^n}|\xi|^{2m}|{\cal F}[\omega](\xi)|^2~\!d\xi&=&
R^{n-2m}\int\limits_{\mathbb R^n}|\xi|^{2m}|{\cal F}[\omega(R\cdot)](\xi)|^2~\! d\xi\\
\int\limits_{\mathbb{R}^n}|\omega|^{2_m^*}~\!dx&=&
R^n\int\limits_{\mathbb R^n}|\omega(R\cdot)|^{2_m^*}~\! dx~\!.
\nonumber
\end{eqnarray}
Finally, we point out that the Hardy inequality
\begin{equation}
\label{eq:Hardy}
\int\limits_{\mathbb{R}^n}|\left(-\Delta\right)^{\!m}\!half u|^2~\!dx\ge {\cal H}_m\int\limits_{\mathbb{R}^n}|x|^{-2m}|u|^2~\!dx
\end{equation}
holds for any function $u\in {\left(-\Delta\right)^{\!s}\!pace}$. The
{\em best Hardy constant} ${\cal H}_m$ was explicitly computed in \cite{He}.
The natural ambient space to study the Dirichlet boundary value problems for (\ref{eq:problem2}), (\ref{eq:problem1}) is
$$\widetilde{H}^m(\Omega)=\{u\in \left(-\Delta\right)^{\!s}\!pace\,:\,{\rm supp}\, u\subset\overline{\Omega}\},
$$
endowed with the norm $\|u\|_m$.
By Theorem 4.3.2/1 \cite{Tr}, for $m+\varphirac{1}{2}\notin\mathbb{N}$ this space coincides with $H^m_0(\Omega)$ (that is the closure of
${\cal C}^{\infty}_0(\Omega)$ in $H^m(\Omega)$), while for $m+\varphirac{1}{2}\in\mathbb{N}$ one has
$\widetilde{H}^m(\Omega)\subsetneq H^m_0(\Omega)$. Moreover, ${\cal C}^{\infty}_0(\Omega)$ is dense in
$\widetilde{H}^m(\Omega)$.
Clearly, if $m$ is an integer then
$\widetilde{H}^m(\Omega)$ is the standard Sobolev space of
functions $u\in H^m(\Omega)$ such that $D^\alpha u=0$
for every multiindex $\alpha\in\mathbb N^n$ with $0\le|\alpha|<m$.
We agree that $(-\Delta)^0u=u$, $\widetilde H^0(\Omega)=L^2(\Omega)$, since
(\ref{eq:Gagliardo}) reduces to the standard $L^2$ norm in case $m=0$.
We define (weak) solutions of the Dirichlet problems for (\ref{eq:problem2}), (\ref{eq:problem1})
as suitably normalized critical points of the functionals
\begin{gather}
\label{eq:functional1}
{\cal R}^\Omega_{\lambda,m,s}[u]=
\varphirac
{\displaystyle \int\limits_\Omega|\left(-\Delta\right)^{\!m}\!half u|^2~\!dx-\lambda\int\limits_\Omega|\left(-\Delta\right)^{\!\frac{s}{2}}\! u|^2~\!dx}
{\bigg(\displaystyle\int\limits_{\Omega}|u|^{{2_m^*}} dx\bigg)
^{2/{2_m^*}\vphantom{\displaystyle 2^2}}}\\
\label{eq:functional2}
\widetilde {\cal R}^\Omega_{\lambda,m,s}[u]=
\varphirac
{\displaystyle \int\limits_\Omega|\left(-\Delta\right)^{\!m}\!half u|^2~\!dx-\lambda\int\limits_\Omega|x|^{-2s}| u|^2~\!dx}
{\bigg(\displaystyle\int\limits_{\Omega}|u|^{{2_m^*}} dx\bigg)
^{2/{2_m^*}\vphantom{\displaystyle 2^2}}}~\!,
\end{gather}
respectively. It is easy to see that both functionals (\ref{eq:functional1}),
(\ref{eq:functional2}) are well defined on $\widetilde{H}^m(\Omega)\setminus\{0\}$.
We conclude this preliminary section with some embedding results.
\begin{Proposition}
\label{P:Poincare}
Let $m,s$ be given, with $0\le s<m<n/2$.
\begin{description}
\item$i)$ The space $\widetilde{H}^m(\Omega)$ is compactly embedded into $\widetilde{H}^s(\Omega)$. In particular the infima
\begin{equation}
\label{eq:Poincare}
\Lambda_1(m,s):=\inf_{\scriptstyle u\in \widetilde{H}^m(\Omega)\atop\scriptstyle u\ne 0}
\varphirac{\|u\|_m^2}{\|u\|_s^2}~~,\qquad
\widetilde\Lambda_1(m,s):=\inf_{\scriptstyle u\in \widetilde{H}^m(\Omega)\atop\scriptstyle u\ne 0}
\varphirac{\|u\|_m^2}{\||x|^{-s}u\|_0^2}
\end{equation}
are positive and achieved.
\item$ii)$
$\displaystyle{\inf_{\scriptstyle u\in \widetilde{H}^m(\Omega)\atop\scriptstyle u\ne 0}
\varphirac{\|u\|_m^2}
{\|u\|^2_{L^{2^*_m}}}={\cal S}_m.}$
\end{description}
\end{Proposition}
Statement $i)$ is well known for $\Lambda_1(m,s)$
and follows from
(\ref{eq:Hardy}) for $\widetilde\Lambda_1(m,s)$. To check $ii)$, use the inclusion
$\widetilde{H}^m(\Omega)\hookrightarrow \left(-\Delta\right)^{\!s}\!pace$ and a rescaling argument. Clearly, the Sobolev
constant ${\cal S}_m$ is never achieved on $\widetilde{H}^m(\Omega)$.
\section{Main estimates}
\label{S:estimates}
Let $\phi$ be the extremal of the Sobolev inequality (\ref{eq:Sobolev}) given by
(\ref{eq:AT}). In particular, it holds that
\begin{equation}
\label{eq:M}
M:=\int\limits_{\mathbb{R}^n}|\left(-\Delta\right)^{\!m}\!half \phi|^2~\!dx={\cal S}_m
\Big(\int\limits_{\mathbb{R}^n}|\phi|^{{2_m^*}}~dx\Big)^{2/{2_m^*}}.
\end{equation}
Fix $\delta>0$ and a cutoff function $\varphi\in {\cal C}^\infty_0(\Omega)$,
such that $\varphi\equiv 1$ on the ball $\{|x|<\delta\}$ and $\varphi\equiv 0$ outside
$\{|x|<2\delta\}$. If $\delta$ is sufficiently small, the function
$$
u_\varepsilon(x):=\varepsilon^{2m-n}\varphi(x)\phi\left({\varphirac{x}{\varepsilon}}\right)=\varphi(x)\left(\varepsilon^2+|x|^2\right)^{\varphirac{2m-n}{2}}
$$
has compact support in $\Omega$.
Next we define
$$
\begin{array}{ll}
A^\varepsilon_m:=\displaystyle\int\limits_{\Omega}|\left(-\Delta\right)^{\!m}\!half u_\varepsilon|^2 dx~&\quad
A^\varepsilon_s:=\displaystyle\int\limits_{\Omega}|\left(-\Delta\right)^{\!\frac{s}{2}}\! u_\varepsilon|^2 dx~\\
&\\
\widetilde A^\varepsilon_s:=\displaystyle\int\limits_{\Omega}|x|^{-2s}|u_\varepsilon|^2 dx~&\quad
B^\varepsilon:=\displaystyle\int\limits_{\Omega}|u_\varepsilon|^{{2_m^*}} dx
\end{array}
$$
and we denote by $c$ any universal positive constant.
\begin{Lemma}
\label{L:estimates}
It holds that
\begin{subnumcases}
{\label{eq:estimate}}
\label{eq:Hm_estimate}
A^\varepsilon_m\le\varepsilon^{2m-n}\left(M+c\varepsilon^{n-2m}\right)&{}\\
\label{eq:Hs_estimate+}
A^\varepsilon_s, \widetilde A^\varepsilon_s\ge c\varepsilon^{4m-n-2s}&{}\text{if $s>2m-\varphirac{n}{2}$}\\
\label{eq:Hs_estimate0}
A^\varepsilon_s, \widetilde A^\varepsilon_s\ge c~\!|\log\varepsilon| &{}\text{if $s=2m-\varphirac{n}{2}$}\\
\label{eq:Lp_estimate}
B^\varepsilon\ge\varepsilon^{-n}\left(({M}{\cal S}_m^{-1})^{{2_m^*}/2}-c\varepsilon^n\right)~\!.&{}
\end{subnumcases}
\end{Lemma}
\noindent
{\bf Proof of (\ref{eq:Hm_estimate}).}
First of all, from (\ref{eq:dilation}) we get
\begin{equation}
\label{eq:Aalpha}
A_m^\varepsilon =\varepsilon^{2m-n}\int\limits_{\R^n}|\xi|^{2m}\left|{\cal F}\left[\varphi(\varepsilon~\!\cdot)\phi\right]\right|^2~\!d\xi.
\end{equation}
Thus
$$
\Gamma_{m}^\varepsilon:=\varepsilon^{n-2m}A^\varepsilon_m-M=
\int\limits_{\R^n}|\xi|^{2{m}}\left|{\cal F}\left[\varphi(\varepsilon~\!\cdot)\phi\right]\right|^2~\!d\xi-
\int\limits_{\R^n}|\xi|^{2{m}}\left|{\cal F}[\phi]\right|^2~\!d\xi.
$$
We need to prove that
\begin{equation}
\label{eq:Gamma}
|\Gamma^\varepsilon_m|\le c\varepsilon^{n-2m}.
\end{equation}
If $m\in\mathbb N$, the proof of (\ref{eq:Gamma})
has been carried out in \cite{BN}, \cite{Gaz98}. Here we limit ourselves to the more
difficult case, namely, when $m$ is not an integer. We denote by $k:=\lfloor m\rfloor\ge 0$ the integer part
of $m$, so that $m-k>0$. Then
\begin{multline*}
\Gamma_m^\varepsilon=
\int\limits_{\R^n}|\xi|^{2k}{\cal F}[U_-]\cdot |\xi|^{2(m-k)}\overline{{\cal F}[U_+]}~\!d\xi\\
=2^{2(m-k)+\varphirac n2}\,\varphirac {\Gamma(m-k+\varphirac{n}2)}{\Gamma(-(m-k))}\cdot
\int\limits_{\R^n} (-\Delta)^k U_-(x)\cdot
V.P.\int\limits_{\R^n}\underbrace{\varphirac {U_+(x)-U_+(y)}{|x-y|^{n+2(m-k)}}}_{\Psi(x,y)}~\!dy~\!dx,
\end{multline*}
where $U_{\pm}=\varphi(\varepsilon~\!\cdot~\!)\phi\pm\phi$ (the last equality follows from
\cite[Ch. 2, Sec. 3]{GSh}).
We split the interior integral as follows:
$$V.P.\int\limits_{\mathbb{R}^n}\Psi dy=
\underbrace{V.P.\!\!\!\int\limits_{|y-x|\le\varphirac {|x|}2\atop ~}\Psi dy}_{I_1}+
\underbrace{\int\limits_{|y-x|\ge\varphirac {|x|}2\atop |y|\le|x|}\Psi dy}_{I_2}+
\underbrace{\int\limits_{|y-x|\ge\varphirac {|x|}2\atop |y|\ge|x|}\Psi dy}_{I_3}.
$$
We claim that $|I_j|\le c|x|^{2k-n}$ for $j=1,2,3$. Indeed, the Lagrange formula gives
\begin{multline*}
|I_1|
\le \max\limits_{|y-x|\le\varphirac {|x|}2}|D^2U_+(y)|\cdot
\int\limits_{|z|\le\varphirac {|x|}2}\varphirac {dz}{|z|^{n+2(m-k)-2}}\\
\le c|x|^{-(n-2m+2)}\cdot |x|^{2-2(m-k)}=c|x|^{2k-n}.
\end{multline*}
As concerns the last two integrals we estimate
$$|I_2|\le\int\limits_{|y-x|\ge\varphirac {|x|}2\atop |y|\le|x|}\varphirac {c|y|^{-(n-2m)}}{|x-y|^{n+2(m-k)}}\,dy
\le |x|^{-(n+2(m-k))}\cdot c|x|^{2m}=c|x|^{2k-n}
$$
and finally
\begin{eqnarray*}
|I_3|\le\int\limits_{|y-x|\ge\varphirac {|x|}2\atop |y|\ge|x|}\varphirac {c|x|^{-(n-2m)}}{|x-y|^{n+2(m-k)}}\,dy
&\le& c|x|^{-(n-2m)}\cdot\int\limits_{|z|\ge\varphirac {|x|}2}\varphirac {dz}{|z|^{n+2(m-k)}}\\
&\le& c|x|^{-(n-2m)}\cdot |x|^{-2(m-k)}=c|x|^{2k-n},
\end{eqnarray*}
and the claim follows.
Now, since
$$|(-\Delta)^k U_-(x)|\le \varphirac {c}{|x|^{n-2(m-k)}}~\!\chi_{\{|x|\ge\delta/\varepsilon\}}+
\varphirac {c\varepsilon^{2k}}{|x|^{n-2m}}~\!\chi_{\{\delta/\varepsilon\le|x|\le2\delta/\varepsilon\}},
$$
we obtain
$$|\Gamma_m^\varepsilon|\le c\int\limits_{|x|\ge\delta/\varepsilon}\varphirac {dx}{|x|^{2n-2m}}+c\int\limits_{\delta/\varepsilon\le|x|\le2\delta/\varepsilon}
\varphirac {\varepsilon^{2k}\,dx}{|x|^{2n-2(m+k)}}\le c\varepsilon^{n-2m},
$$
that completes the proof of (\ref{eq:Gamma}) and of (\ref{eq:Hm_estimate}).
\noindent{\bf Proof of (\ref{eq:Hs_estimate+}) and (\ref{eq:Hs_estimate0}).}
We use the Hardy inequality (\ref{eq:Hardy}) to get
\begin{eqnarray*}
A^\varepsilon_s&\ge& c\widetilde A^\varepsilon_s
\ge
c\varepsilon^{4m-2s-n}\int\limits_{\mathbb{R}^n}|x|^{-2s}|\varphi(\varepsilon~\!\cdot)\phi|^2 dx\\
&\ge&c\varepsilon^{4m-2s-n}\int\limits_{|x|<{\delta}/{\varepsilon}}\varphirac{dx}{|x|^{2s}(1+|x|^2)^{n-2m}}~\!.
\end{eqnarray*}
The last integral converges as $\varepsilon\to 0$ if $s>2m-\varphirac{n}{2}$, and diverges
with speed $|\log\varepsilon|$ if $s=2m-\varphirac{n}{2}$.
\noindent{\bf Proof of (\ref{eq:Lp_estimate}).}
For $\varepsilon$ small enough we estimate by below
\begin{eqnarray*}
\int\limits_{\R^n}|u_\varepsilon|^{{2_m^*}}&=&\varepsilon^{-n}\int\limits_{\R^n}|\varphi(\varepsilon~\!\cdot)\phi|^{{2_m^*}}~\!dx
=\varepsilon^{-n}\Big(\int\limits_{\R^n}|\phi|^{{2_m^*}}~\!dx-\int\limits_{|x|>\delta/\varepsilon}|\varphi(\varepsilon~\!\cdot)\phi|^{{2_m^*}}~\!dx\Big)\\
&\ge&\varepsilon^{-n}\Big(({M}{\cal S}_m^{-1})^{{2_m^*}/2}
-c\int\limits_{|x|>{\delta}/{\varepsilon}}|x|^{-2n}~\!dx\Big)\\
&=& \varepsilon^{-n}(({M}{\cal S}_m^{-1})^{{2_m^*}/2}-c\varepsilon^{n})
\end{eqnarray*}
and the Lemma is completely proved.
{$\square$}\goodbreak
\section{Two noncompact minimization problems}
\label{S:proof_main}
In this section we deal with the minimization problems
\begin{equation*}
{\cal S}^\Omega_\lambda(m,s)=
\inf_{\scriptstyle u\in \widetilde{H}^m(\Omega)\atop\scriptstyle u\ne 0}{\cal R}^\Omega_{\lambda,m,s}[u];\qquad
\widetilde {\cal S}^\Omega_\lambda(m,s)=
\inf_{\scriptstyle u\in \widetilde{H}^m(\Omega)\atop\scriptstyle u\ne 0}\widetilde {\cal R}^\Omega_{\lambda,m,s}[u]~\!,
\end{equation*}
where the functionals ${\cal R}$ and $\widetilde {\cal R}$ are introduced in (\ref{eq:functional1})
and (\ref{eq:functional2}), respectively.
\begin{Lemma}
\label{L:standard1}
The following facts hold for any $\lambda\in\mathbb{R}$:
\begin{description}
\item$~~i)$ ${\cal S}^\Omega_\lambda(m,s) \le {\cal S}_m$;
\item$~ii)$ If $\lambda\le 0$ then ${\cal S}^\Omega_\lambda(m,s)= {\cal S}_m$ and it is not achieved;
\item$iii)$ If $0<{\cal S}^\Omega_\lambda(m,s)< {\cal S}_m$, then ${\cal S}^\Omega_\lambda(m,s)$ is achieved.
\end{description}
The same statements hold for $\widetilde {\cal S}^\Omega_\lambda(m,s)$ instead of ${\cal S}^\Omega_\lambda(m,s)$.
\end{Lemma}
\noindent{\textbf{Proof. }}
The proof is nowdays standard, and is essentially due to Brezis and Nirenberg \cite{BN}.
We sketch it for the infimum ${\cal S}^\Omega_\lambda(m,s)$, for the convenience of the reader.
Fix $\varepsilon>0$ and take $u\in {\cal C}^\infty_0(\mathbb{R}^n)\setminus\{0\}$ such that
\begin{equation}
\label{eq:Sob_eps}
({\cal S}_m+\varepsilon)\bigg(\,\displaystyle\int\limits_{\mathbb{R}^n}|u|^{{2_m^*}} dx\bigg)^{2/{2_m^*}}
\ge \int\limits_{\mathbb{R}^n}|\left(-\Delta\right)^{\!m}\!half u|^2~dx.
\end{equation}
Let $R>0$ be large enough, so that $u_R(\cdot):=u(R~\!\!\cdot)\in {\cal C}^\infty_0(\Omega)$.
Using (\ref{eq:dilation}) we get
\begin{eqnarray*}
{\cal S}^\Omega_\lambda(m,s)&\le&
\varphirac{\|u\|_m^2-
\lambda R^{2(s-m)}\|u\|_s^2}
{\|u\|^2_{L^{2^*_m}}}\le ({\cal S}_m+\varepsilon)\left(1+cR^{2(s-m)}\right)~\!,
\end{eqnarray*}
where $c$ depends only on $u$ and $\lambda$. Letting $R\to\infty$
we get ${\cal S}^\Omega_\lambda(m,s)\le ({\cal S}_m+\varepsilon)$ for any $\varepsilon>0$,
and $i)$ is proved.
Next, if $\lambda\le0$ then clearly ${\cal S}^\Omega_\lambda(m,s)={\cal S}_m$.
If $\lambda=0$ then ${\cal S}_m$ is not achieved. The more it is not achieved for $\lambda<0$,
and $ii)$ holds.
Finally, to prove $iii)$ take a minimizing sequence $u_h$. It is convenient
to normalize $u_h$ with respect to the $L^{{2_m^*}}$-norm, so that
$$
\displaystyle \int\limits_\Omega|\left(-\Delta\right)^{\!m}\!half u_h|^2~\!dx-\lambda\int\limits_\Omega|\left(-\Delta\right)^{\!\frac{s}{2}}\! u_h|^2~\!dx
={\cal S}^\Omega_\lambda(m,s)+o(1).
$$
We can assume that $u_h\to u$ weakly in $\widetilde{H}^m(\Omega)$ and strongly in $\widetilde{H}^s(\Omega)$ by Proposition~\ref{P:Poincare}.
Since
\begin{eqnarray*}
\lambda\int\limits_\Omega|\left(-\Delta\right)^{\!\frac{s}{2}}\! u|^2~\!dx&=&\lambda\int\limits_\Omega|\left(-\Delta\right)^{\!\frac{s}{2}}\! u_h|^2~\!dx+o(1)\\
&=& \int\limits_\Omega|\left(-\Delta\right)^{\!m}\!half u_h|^2~\!dx-{\cal S}^\Omega_\lambda(m,s)+o(1)\\
&\ge& ({\cal S}_m-{\cal S}^\Omega_\lambda(m,s))+o(1),
\end{eqnarray*}
then $u\neq 0$.
By the Brezis--Lieb lemma we get
$$
1=\|u_h\|^{2^*_m}_{L^{2^*_m}}=\|u_h-u\|^{2^*_m}_{L^{2^*_m}}+\|u\|^{2^*_m}_{L^{2^*_m}}+o(1).
$$
Thus
\begin{eqnarray*}
{\cal S}^\Omega_\lambda(m,s)&=&
{\|u_h\|_m^2-\lambda \|u_h\|^2_s}+o(1)\\
~&&\\
&=&
\Big(\|u_h-u\|_m^2+\|u\|_m^2\Big)-\lambda \Big(\|u_h-u\|^2_s+\|u\|_s^2\Big)+o(1)\\
~&&\\
&=&
\varphirac{\Big(\|u_h-u\|_m^2-\lambda \|u_h-u\|^2_s\Big)+
\Big(\|u\|^2_m- \lambda \|u\|_s^2\Big)}
{\Big(\|u_h-u\|^{2^*_m}_{L^{2^*_m}}+\|u\|^{2^*_m}_{L^{2^*_m}}\Big)^{2/2^*_m}}+o(1)\\
~&&\\
&\ge&
{\cal S}^\Omega_\lambda(m,s)\cdot\varphirac{\xi_h^{2}+1}{(\xi_h^{2^*_m}+1)^{2/2^*_m}}+o(1),
\end{eqnarray*}
where we have set
$$
\xi_h:=\varphirac{\|u_h-u\|_{L^{2^*_m}}}{\|u\|_{L^{2^*_m}}}.
$$
Since $2^*_m>2$, this implies that $\xi_h\to 0$, that is,
$u_h\to u$ in $L^{2^*_m}$ and hence
$u$ achieves
${\cal S}^\Omega_\lambda(m,s)$.
{$\square$}\goodbreak
We are in position to prove our existence result, that includes the
theorem already stated in the introduction.
\begin{Theorem}
\label{T:main}
Assume $s\ge 2m-\varphirac{n}{2}$.
\begin{description}
\item$~i)$~ If $0<\lambda<\Lambda_1(m,s)$ then ${\cal S}^\Omega_\lambda(m,s)$ is achieved and
(\ref{eq:problem2}) has a nontrivial solution in $\widetilde{H}^m(\Omega)$.
\item$ii)$~ If $0<\lambda<\widetilde\Lambda_1(m,s)$ then $\widetilde {\cal S}^\Omega_\lambda(m,s)$ is achieved and
(\ref{eq:problem1}) has a nontrivial solution in $\widetilde{H}^m(\Omega)$.
\end{description}
\end{Theorem}
\noindent{\textbf{Proof. }}
Since $0<\lambda<\Lambda_1(m,s)$ then ${\cal S}^\Omega_\lambda(m,s)$ is positive, by Proposition
\ref{P:Poincare}. The main estimates in Lemma \ref{L:estimates}
readily imply
${\cal S}^\Omega_\lambda(m,s)<{\cal S}_m$. By Lemma \ref{L:standard1}, ${\cal S}^\Omega_\lambda(m,s)$ is achieved
by a nontrivial $u\in \widetilde{H}^m(\Omega)$, that solves (\ref{eq:problem2}) after multiplication by a suitable constant.
Thus $i)$ is proved. For $ii)$ argue in the same way.
{$\square$}\goodbreak
In the case $s< 2m-\varphirac{n}{2}$ the situation is more complicated. We limit ourselves to point out
the next simple existence result.
\begin{Theorem}
\label{T:critical}
Assume $s< 2m-\varphirac{n}{2}$.
\begin{description}
\item$~i)$ There exists $\lambda^*\in[0,\Lambda_1(m,s))$ such that
the infimum ${\cal S}^\Omega_\lambda(m,s)$ is attained for any $\lambda\in(\lambda^*,\Lambda_1(m,s))$, and
hence (\ref{eq:problem2}) has a nontrivial solution.
\item$ii)$ There exists $\widetilde\lambda^*\in[0,\widetilde\Lambda_1(m,s))$ such that
the infimum $\widetilde{\cal S}^\Omega_\lambda(m,s)$ is attained for any $\lambda\in(\widetilde\lambda^*,\widetilde\Lambda_1(m,s))$, and
hence (\ref{eq:problem1}) has a nontrivial solution.
\end{description}
\end{Theorem}
\noindent{\textbf{Proof. }}
Use Proposition \ref{P:Poincare} to find $\varphi_1\in\widetilde{H}^m(\Omega)$, $\varphi_1\neq 0$, such that
$$
\int\limits_{\Omega}|\left(-\Delta\right)^{\!m}\!half \varphi_1|^2~\!dx=\Lambda_1(m,s)
\int\limits_{\Omega}|\left(-\Delta\right)^{\!\frac{s}{2}}\! \varphi_1|^2~\!dx~\!.
$$ Then test
${\cal S}^\Omega_\lambda(m,s)$ with $\varphi_1$ to get the strict inequality
${\cal S}^\Omega_\lambda(m,s)<{\cal S}_m$.
The first conclusion follows by Proposition \ref{P:Poincare} and Lemma \ref{L:standard1}. For
(\ref{eq:problem1}) argue similarily.
{$\square$}\goodbreak
\varphiootnotesize
\label{References}
\end{document} |
\begin{document}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{corollary}{Corollary}
\title{Unitary transformations can be distinguished locally}
\author{Xiang-Fa Zhou}
\email{xfzhou@mail.ustc.edu.cn}
\author{Yong-Sheng Zhang}
\email{yshzhang@ustc.edu.cn}
\author{Guang-Can Guo}
\affiliation{\textit{Key Laboratory of Quantum Information,
University of Science and Technology of China, Hefei, Anhui
230026, People's Republic of China}}
\begin{abstract}
We show that in principle, $N$-partite unitary transformations can
be perfectly discriminated under local measurement and classical
communication (LOCC) despite of their nonlocal properties. Based
on this result, some related topics, including the construction of
the appropriate quantum circuit together with the extension to
general completely positive trace preserving operations, are
discussed.
\end{abstract}
\pacs{03.65.Ud, 03.67.-a}
\maketitle
Superposition plays the central role in quantum mechanics. The
quantum nonorthogonality and entanglement due to superposition,
which show many counter-intuitive behaviors compared with those in
classical world, have drawn much attention in the past two
decades. Quantum nonorthogonality put many constraints on
physically accessible manipulations on input states. It is
well-known that two nonorthogonal pure state can not be perfectly
discriminated \cite{discrimination}. On the other hand, quantum
nonlocality due to entanglement, which was first brought into
attention by Einstein, Podolsky, and Rosen (EPR) in 1935
\cite{EPR}, is also one of the most interesting and important
parts in quantum information science. Today, quantum entanglement
has been viewed as a significant resource for quantum information
processing, and currently the behavior of entanglement in quantum
information science is still under investigation.
Although perfect identification of nonorthogonal quantum states
are impossible in quantum world, when we refer to quantum
operations, thing becomes very different. It was proved that two
unitary operations can be perfectly discriminated after applying
the unitary gate a finite number of times in parallel
\cite{unitary,unitary2}. On the other hand, the nonlocality of
unitary transformation has been extensively studied because of its
fundamental importance during the construction of universal
quantum circuit \cite{gates}. For example, it has been shown that
a sequence of a nonlocal gate (e.g., Control-not gate or
Control-phase gate) and single-qubit rotations can be used to
construct any desired transformations. Also nonlocal gate can be
classified and simulate each other under specific conditions
\cite{nonlocal1,nonlocal2}. Based on these results, one natural
problem arises - what is the influence of the nonlocality of
quantum operation on the discrimination.
In this work, we consider to discriminate two unitary
transformations with local methods. Compared with its counterpart,
i.e., local identification of quantum states, which is often
considered for orthogonal states \cite{locc1,locc2}, we find that
any two unitary transformations can be perfectly identified
locally despite of their nonlocal properties.
Before concentrating on the specific topics, let us make a few
remarks about the difference between the discrimination of quantum
states and of quantum operations. Generally to identify a quantum
state, one should make a measurement on the given state followed
by an estimation based on the measurement results. Such process
usually collapses the input states which thus cannot be used any
more. However, thing becomes different when we refer to quantum
operations. The reason lies in the fact that quantum operations
never collapse, and in principle it can be repeated any times if
we need. What's more, when unitary operations are considered, by
exchanging the input and output ports of the whole setup, we can
obtain the reverse transformations. Actually, these facts make the
discrimination of quantum operations very different from that of
quantum states.
Generally the strategy of operation identification is formulated
as this: we employ a quantum circuit $f(U)$ which is made up of
the selected operation $U$ on the suitable input state
$\rho_{s,a}$, where $s(a)$ denotes the circuit system (auxiliary
system). If only local methods are required, $\rho_{s,a}$ must
also be separable. To obtain the maximal distinguishability, the
overlap of the output states should be as small as possible for
different quantum operations. Fig. 1 shows the sketch of the
identification process under local operation and classical
communication (LOCC). When global operation are permitted, both of
the circuit and the input state can be constructed to realize a
perfect discrimination for unitary transformations
\cite{unitary,unitary2}. However, when only local operations and
resources are permitted, thing becomes not so obvious. To simplify
our consideration, in the following, we mainly focus on bipartite
system.
\begin{figure}
\caption{Illustration of
the identification of unitary transformations under local
operation and classical communication. Alice and Bob input a
locally implemented state $\rho_{AA',BB'}
\end{figure}
Let us begin with some simple observations. Here we mainly
concentrate on unitary operations, one can check that some of the
discussions are also suitable for general quantum operations. As
we have mentioned above, to realize perfect identification, one
need to find a suitable input state such that the corresponding
output states are orthogonal to each other for different selected
operations. Assume that we want to discriminate two unitary
operations $U$ and $V$. By inputting a locally implemented quantum
state $\rho_{AA',BB'}=\sum_i \lambda_i \rho^i_{AA'} \otimes
\rho^i_{BB'}$, we have that the two output states $\rho_U=(U
\otimes I_{A'B'}) \rho_{AA',BB'} (U^{\dag} \otimes I_{A'B'})$ and
$\rho_V=(V \otimes I_{A'B'}) \rho_{AA',BB'} (V^{\dag} \otimes
I_{A'B'})$ should be orthogonal to each other. Now consider the
spectral decompositions of $\rho^i_{AA'} = \sum_j
r_j|r^i_j\rangle_{(AA')}\langle r^i_j|$ and $\rho^i_{BB'}= \sum_k
s_k|s^i_k\rangle_{(BB')}\langle s^i_k|$. The requirement of
$\rho_U \perp \rho_v$ is equivalent to $(U \otimes I_{A'B'})
|r^i_j\rangle_{AA'}|s^i_k\rangle_{BB'} \perp (V \otimes I_{A'B'})
|r^{i'}_{j'}\rangle_{AA'}|s^{i'}_{k'}\rangle_{BB'}$ for any $i$,
$i'$, $j$, $j'$, $k$, $k'$. This observation shows in general, a
pure input state $|r\rangle_{AA'}|s\rangle_{BB'}$ is enough to
perfectly discriminate two unitary operations if they can.
Moreover, since two orthogonal pure states can be locally
identified \cite{locc1,locc2}, hence in this case $U$ and $V$ can
also be discriminated with local methods.
Consider two unitary transformations $U_{AB}$ and $V_{AB}$ with
zero overlap in trace norm, i.e.,
$\mbox{Tr}(V_{AB}^{\dag}U_{AB})=0$. Then by preparing the
following locally maximal entangled state as the input
\begin{eqnarray} \label{entanglement}
| \phi \rangle_{AB,A'B'}=|\phi\rangle_{AA'} \otimes
|\phi\rangle_{BB'},
\end{eqnarray}
where $|\phi\rangle_{AA'}=\sum_i|i\rangle_A|i'\rangle_{A'}$ (or
$|\phi\rangle_{BB'}=\sum_i|i\rangle_B|i'\rangle_{B'}$) is a
nonnormalized entangled state between the system $A$ and the
corresponding local environment $A'$ (or $B$ and $B'$). From the
following equation
\begin{eqnarray} \label{norm}
\langle \phi | V_{AB}^{\dag}U_{AB} \otimes I | \phi \rangle =
\mbox{Tr}(V_{AB}^{\dag}U_{AB})=0,
\end{eqnarray}
one immediately obtain that the two output states $U_{AB} \otimes
I |\phi\rangle_{AB,A'B'}$ and $V_{AB} \otimes I
|\phi\rangle_{AB,A'B'}$ are orthogonal to each other, hence can be
locally discriminated perfectly. Equations ($\ref{entanglement}$,
$\ref{norm}$) can be viewed as the extension of Jamiolkowski
isomorphism in local case \cite{nonlocal1}. The input state $|
\phi \rangle_{AB,A'B'}$ is universal for any two operations $U$
and $V$ satisfying $\mbox{Tr}(V^{\dag}U)=0$. Actually, given $U$
and $V$, if global input states are permitted, one can always
choose a suitable pure input state in the composite system of only
$A$ and $B$, namely, the auxiliary system can be neglected in this
case \cite{unitary,unitary2}. However, if only local resources are
required, in order to achieve the maximal distinguishability of
the output states, an entangled state between the system and the
environment seems to be required unless the global optimal pure
state is separable.
In the above case, perfect identification can be realized in a
single run for both global and local methods. In the more general
cases, one needs to run the selected gate $N$ times ($N$ is
finite). The optimal $N$ has been found for global discrimination
of $U$ and $V$, which asserts that if the minimal arclength
$\delta$ spread by the eigenvalue of $(U^{\dag}V)^{\otimes N}$ in
the circle $|z|=1$ is not less than $\pi$, then a perfect
discrimination scheme is allowed. Now assume $U^{\dag}V=U_1
\otimes U_2$ to be local operation, with $\delta_1$ and $\delta_2$
being the minimal arclengths of $U_1^{\otimes N}$ and
$U_2^{\otimes N}$ respectively. Then perfect global discrimination
can be implemented by inputting an entangled state if
$\delta_1+\delta_2 \ge \pi$. However, if only local input states
(e.g., $|r\rangle|s\rangle$) are allowed, since
\begin{eqnarray}
&&\langle r|U_1^{\otimes N}|r \rangle \langle s|U_2^{\otimes N}|s
\rangle =0 \nonumber \\
&\Leftrightarrow& \langle r|U_1^{\otimes N}|r \rangle =0 \mbox{ or
} \langle s|U_2^{\otimes N}|s \rangle =0,
\end{eqnarray}
this indicates that to distinguish $U$ and $V$ locally, at least
one of the two arclength $\delta_1$ and $\delta_2$ must be not
less than $\pi$. Therefore, generally in the local case the
optimal running times $N$ of the selected operation should be
greater than that of the global case.
As a special example, consider the following control unitary
transformation $U^{\dag}V=P_1 \otimes I + P_2 \otimes u$, where
$P_iP_j=\delta_{ij}P_i$ and $\sum_i P_i =I$, $I$ is the identity
operation, and $u$ is a local unitary manipulations. The
eigenvalues $r_i$ of $U^{\dag}V$ belong to the set $\{1, b_1, b_2,
...\}$ with $b_i$ and $|b_i\rangle$ being the eigenvalues and
eigenvectors of $u^{\otimes N}$ separately. If only local input
state $\rho_A \otimes \rho_B= \mbox{Tr} (
|\psi_{AA'}\rangle\langle \psi_{AA'}|\otimes |\psi_{BB'}\rangle
\langle \psi_{BB'}| )$ is permitted, then
\begin{eqnarray} \label{C-U2}
\mbox{Tr}[(U^{\dag}V)^{\otimes N}(\rho_A\otimes
\rho_B)]=x+(1-x)\sum_i b_i\langle b_i|\rho_B|b_i\rangle,
\end{eqnarray}
where $x=\mbox{Tr}(P_1\rho_A) \ge 0$ and can be chosen arbitrarily
by input appropriate $\rho_A$. In order to make the
right-hand-side of Eq. ($\ref{C-U2}$) to be zero, one can easily
obtain that the minimal angular spread of $\{1, b_1, b_2, ...\}$
should be not less than $\pi$. Therefore, in this case the minimal
$N$ required equals to that of global case. Similarly, suppose
$U^{\dag}V=(P_1 \otimes I + P_2 \otimes u) \cdot (U_1 \otimes
U_2)$ and $U2 \neq I$ or $U2 \neq u^{\dag}$. If $U2 \neq
u^{\dag}$, then by inputting appropriate state $|\psi\rangle_A
|\psi\rangle_B$ with $|\psi\rangle_A$ lying in the support of
$P_1$, $U^{\dag}V$ is equivalent to the local transformation
$(uU_2)|\psi\rangle_B$, hence can be perfectly identified.
In the above discussions, we have considered to discriminate
several special kinds of unitary transformations. They all can be
perfectly identified and the optimal quantum circuit and input
state can be easily obtained. In the following, we mainly focus on
the most general case. Although we cannot present the optimal
quantum circuit and input state, we prove that, in principle, any
two unitary operations $U$ and $V$ can be perfectly identified
locally.
Following \cite{exactuniverse}, we call a $2$-qudit gate $U_{AB}$
to be primitive if $U_{AB}$ maps a separable state to another
separable state; otherwise, $U_{AB}$ is imprimitive. Generally, a
primitive gate $U_{AB}$ can be expressed as the product of 1-qudit
gate up to a swap operation $P$, namely, $U_{AB}=U_A \otimes U_B$
or $U_{AB}=U_A \otimes U_B \cdot P$ with $P|\alpha \rangle_A
|\beta\rangle_B=|\beta \rangle_A |\alpha \rangle_B$. For
simplicity, in the following, we use $H$ to denote the set of all
2-qubit gates of the form $U_A \otimes U_B$. Under these
assumptions, we then introduce the following lemma.
\begin{lemma}
$H$ together with an imprimitive gate $Q$ can generate the unitary
group $U(d^2)$.
\end{lemma}
A detailed proof of this lemma can be found in
\cite{exactuniverse}, which is used to study the university of
quantum gate. This lemma indicate that if $Q$ and all local
unitary transformations are permitted, we can then construct
$H'=QHQ^{-1}$. By choosing suitable sequence of $H$ and $H'$, we
can obtain any desired elements in $U(d^2)$. The length of the
sequence is finite, therefore it is only need to run the
imprimitive gate a finite number of times.
Based on this lemma, we now prove the main theorem of this work.
\begin{theorem}
Any two unitary transformation $U_{AB}$ and $V_{AB}$ can be
perfectly identified with local methods.
\end{theorem}
Proof: Following our former discussions, we obtain that if both
$U_{AB}$ and $V_{AB}$ are primitive, then they can be perfectly
discriminated locally.
Now assume that only one of the two unitary gate is primitive.
Without loss of generality, we suppose $V_{AB}$ to be imprimitive.
According to the lemma, we obtain that there exists a quantum
circuit $f(V_{AB})$ made up of the elements in $H$ and
$H'=V_{AB}HV^{\dag}_{AB}$ such that $f(V_{AB}) \in (HH')^n$ is
some control unitary transformation. On the other hand, since
$U_{AB}$ is primitive, which means $H'=U_{AB}HU^{\dag}_{AB}=H$,
one immediately obtain that $f(U_{AB})$ is also primitive. Because
$f(U_{AB}) \neq f(V_{AB})$, we have that the two unitary
operations can be locally identified.
If $U_{AB}$ and $V_{AB}$ are both imprimitive, Following the
lemma, we obtain that there is a quantum circuit such that
$f(U_{AB})=e^{i L^A_{12} \otimes L^B_{12}}$ with
$(L^A_{12})_{ij}=\delta_{i1}\delta_{j2}+\delta_{i2}\delta_{j1}$
(or $(L^B_{12})_{ij}$). If $f(V_{AB})$ is primitive, then perfect
local discrimination can be realized. Otherwise, both $f(U_{AB})$
and $f(V_{AB})$ are imprimitive. Since
$f(U_{AB})^{\dag}=A.f(U_{AB}).A^{\dag}$ with
$A=\mbox{diag}\{\sigma_z,I_{(d-2)}\} \otimes I$, $I \otimes
\mbox{diag}\{\sigma_z,I_{(d-2)}\}$, $\mbox{diag}\{\sigma_y,
I_{(d-2)}\} \otimes I$, or $I \otimes
\mbox{diag}\{\sigma_y,I_{(d-2)}\}$. One can easily check that if
the similar result occurs for $V_{AB}$, then $f(V_{AB})$ can be
expressed as $f(V_{AB})=e^{i x L^A_{12} \otimes L^B_{12}}$ for
some $x \in \mathbb{R}$. Therefore the whole question can be
divided into the following two parts:
\emph{i}). If $f(V_{AB})\neq e^{i x L^A_{12} \otimes L^B_{12}}$
for any $x \in \mathbb{R}$, then by employing the transformation
$Af(\cdot)A^{\dag}f(\cdot)$, we can obtain an identity operation
for $U_{AB}$. Because $Af(V_{AB})A^{\dag }f(V_{AB}) \neq I$, the
two operations thus are locally distinguishable.
\emph{ii}). If $f(V_{AB})=e^{i x L^A_{12} \otimes L^B_{12}}$, then
when $x \neq 1$, $f(U_{AB})$ and $f(V_{AB})$ can be reduced to
$e^{i L^A_{12}} \otimes I$ and $e^{ix L^A_{12}} \otimes I$ by
inputting a product state $|\phi\rangle |\psi\rangle$ with
$|\psi\rangle$ being an eigenvector of $L^A_{12}$, which,
therefore, can be perfectly identified locally by running the
circuit a finite number of times in parallel. Otherwise we have
$f(U_{AB})=f(V_{AB})$. Since $e^{i L^A_{12} \otimes L^B_{12} }$ is
imprimitive, it can be used to construct the desired operator
$U^{\dag}_{AB}$. Thus the original problem is reduced to the
locally identification of the identity operation and
$U^{\dag}_{AB}V_{AB}$, which can be implemented perfectly.
This completes the proof.
The above theorem shows that in principle, to realize a perfect
local identification, we only need to run the selected unitary
operation a finite number of times. Although we have assumed that
the two subsystems $A$ and $B$ have equal dimensions, one can
easily obtain that the same result holds even if $A$ and $B$ have
different dimensions. For example, if $dim{{\cal H}_A} < dim{{\cal
H}_B}$, then by introducing another subsystem $A_1$ in Alice's
side such that $dim{{\cal H}_A} + dim{{\cal H}_{A_1}} = dim{{\cal
H}_B}$, we can obtain two extended unitary transformations $U
\oplus I_{A_1}$ and $V \oplus I_{A_1}$, which thus can be
identified with the methods described above.
It should be mentioned that the ancillary subsystem $A_1$ usually
plays nontrivial role during the discussion of operation
discrimination \cite{duanprivite}. In practice, given two
different operations $\{\xi_1, \xi_2 \}$ acting on the same
Hilbert space $A$, it is always possible to prepare a larger
system $A'$ such that $A'=A\oplus A_1$. Therefore, the original
problem can be reduced to the discrimination of the two newly
defined operations $\{\xi_1\oplus I_{A_1},\xi_2\oplus I_{A_1}\}$.
For instance, in the global discrimination of two unitary
operations $\{U, V \}$, the minimal running times usually reads
$N=\left[ \frac{\pi}{\delta} \right]$. However, when subsystem
$A_1$ is concerned, if $1$ is not one of the eigenvalues of
$(U^{\dag}V)^{\otimes N}$, and the two minimal arclengthes
$\{\delta, \delta' \}$, spread by the eigenvalues of
$(U^{\dag}V)^{\otimes N}$ and $(U^{\dag}V\oplus I_{A_1})^{\otimes
N}$ separately, are different, then we have $N' = \left[
\frac{\pi}{\delta'} \right] \ge N$. The subsystem $A_1$ can be
used to distinguish two unitary operations up to a phase factor.
For example, consider a three level system $\{ |0\rangle,
|1\rangle, |2\rangle \}$. Suppose the Hamiltonian of the whole
system is $H= \omega (|0\rangle\langle 0| + |1\rangle\langle 1|)$.
If we are restricted in the subspace $\{|0\rangle, |1\rangle \}$,
then when $T= \pi/\omega$, we obtain $U=-\mbox{diag}\{1,1\}$,
which cannot be discriminated from the identity operators $I$.
However, if the ancillary level $|2\rangle$ is concerned, then
perfect identification can be implemented by preparing suitable
pure input state in the total Hilbert space.
From the practical viewpoint, it will be valuable if one can
provide an optimal circuit to implement such kind of
identification operation \cite{exact-example}. Generally, it is
not easy to do this. Here, to simplify our consideration, we take
two-qubit gates as an example.
For any two-qubit unitary transformation $U$, it has the following
canonical decomposition \cite{two-qubit}
\begin{eqnarray} \label{decomposition}
U=(U_1 \otimes U_2) e^{i(h_x \sigma_x \otimes \sigma_x + h_y
\sigma_y \otimes \sigma_y + h_z \sigma_z \otimes \sigma_z )} (U_3
\otimes U_4),
\end{eqnarray}
where $\sigma_x$, $\sigma_y$, $\sigma_z$ are the usual Pauli
matrices, $U_i$ are local single-qubit gate and $\pi/4 \ge h_x \ge
h_y \ge |h_z|$. Benefitting from the nice decomposition
($\ref{decomposition}$), one need not to reverse the whole setup
because $U^{\dag}$ can be constructed from $U$ directly. Now
suppose we have two unitary operations $U$ and $V$. After applying
the selected gate at most 2 times, we can transform one of them,
e.g., $U$, into $f(U)=e^{i h^U_x \sigma_x \otimes \sigma_x}$. If
$f(V) \neq e^{i h^V_x \sigma_x \otimes \sigma_x}$ for some $h^V_x
\in \mathbb{R}$, we can employ the manipulation
$g(\cdot)=Af(\cdot)A^{\dag}f(\cdot)$($A=\sigma_y \otimes I,
\sigma_z \otimes I, I \otimes \sigma_y, \mbox{ or } I \otimes
\sigma_z$), where $A$ can be selected to meet the requirement,
i.e., to reduce the original $U$ and $V$ to $I$ and $g(V)$
respectively. Similarly, by running $g(V)$ at most 4 times, we can
then obtain two local unitary transformations $U'$ and $V'$. One
can easily check that by choosing suitable single-qubit gates,
$U'$ and $V'$ can always be different. Therefore, after repeating
the selected gate at most 20 times, we reduce the original problem
to the discrimination of two local gates, which can be perfectly
implemented with the method we described in the former context.
The same question can also be investigated in multi-partite case.
To answer this problem, we should introduce the generalized
version of the primitive gates. We call $U_{12\ldots N}$ is
$\{[i_s,\ldots,i_e],\ldots, [j_s,\ldots,j_e],\ldots\}$-primitive
if $U_{12\ldots N}$ together with all single-qudit gates can
generate the group $\mathcal{U}_{\beta}=U_{i_s\ldots i_e} \otimes
\ldots \otimes U_{i_s\ldots i_e}$. Similarly, if
$\mathcal{U}_{\beta}=U(d^N)$, then $U_{12\ldots N}$ is
imprimitive. Following the same routine in \cite{exactuniverse},
we can obtain that a $\{[i_s,\ldots,i_e],\ldots,
[j_s,\ldots,j_e],\ldots\}$-primitive gate can be expressed as
$U_{i_s\ldots i_e} \otimes \ldots \otimes U_{i_s\ldots i_e} \cdot
P_{\{[i_s,\ldots,i_e],\ldots, [j_s,\ldots,j_e],\ldots\}}$, where
$P_{\{[i_s,\ldots,i_e],\ldots, [j_s,\ldots,j_e],\ldots\}}$ is
permutation operator which preserves the structure of the
partition $\{[i_s,\ldots,i_e],\ldots, [j_s,\ldots,j_e],\ldots\}$.
For example, if $U_{12345}$ is $\{[1,2],[3,4],5 \}$-primitive,
then $P_{\{[1,2],[3,4],5 \}}=P_{12,34} \otimes I_5$ or
$I_{12345}$, where $P_{12,34}$ is the swap operation between
Hilbert spaces ${\cal H}_1 \otimes {\cal H}_2$ and ${\cal H}_3
\otimes {\cal H}_4$; if $U_{12345}$ is $\{1,2,3,4,5 \}$-primitive,
then $P_{\{1,2,3,4,5 \}}$ can be any element in the permutation
group $S_5$.
We take 3-partite unitary transformations as an instance.
According to the above discussion, if one of the two 3-partite
unitary transformations $U_{ABC}$ and $V_{ABC}$ is $\{A,B,C
\}$-primitive, then perfect local identification can be realized.
If both of the two selected transformations are imprimitive, then
there exists a sequence $f(U_{ABC})=(H'H)\ldots(H'H)$ with
$H'=U_{ABC}HU_{ABC}^{\dag}$, such that $f(U)=e^{i L_{12}^A \otimes
L_{12}^B \otimes L_{12}^C}$. Following the discussion of bipartite
case, we conclude that $U_{ABC}$ and $V_{ABC}$ can be locally
discriminated. Finally, if $U_{ABC}$ is $\{[A,B],C \}$-primitive
with $V_{ABC}$ being $\{A,[B,C] \}$-primitive, then there exists a
circuit such that $f(U_{ABC})=(U_{A_1}\otimes P_{B_1} +
U_{A_2}\otimes P_{B_2} )\otimes U_C$ and $f(V_{ABC})=V_A \otimes
V_{BC}$, where $(U_{A_1}\otimes P_{B_1} + U_{A_2}\otimes P_{B_2}
)$ is some control-unitary transformation. Since $U_{A_1} \neq
V_A$ or $U_{A_2} \neq V_A$, by choosing suitable input state, the
original problem cane be reduced to the discrimination of two
different local unitary manipulations, hence can be realized
perfectly.
The above discussion can be extended to $N$-partite case, and we
have that it is always possible to discriminate two unitary
operations locally, although in general, we need to run the
unknown operation many times. Interestingly, unlike the previous
results for quantum states, where ``the hidden entanglement" plays
a very important role, it seems that the nonlocality of unitary
transformations doesnot affect the distinguishability much (in
this work, it only changes the total run times $N$). We can also
generalize this result to the case of $M$ unitary transformations.
To discriminate the unknown operation from others, we should
perform $M-1$ tests; after each test, one of the $M$ operations
can be ruled out. Therefore perfect local identification can be
realized after a finite number of runs of the unknown gate.
One can also consider the same problem for nonunitary
transformations\cite{nonunitary}. For general completely positive
trace preserving operations $\xi_1$ and $\xi_2$, the reverse
transformations donot always exist unless they are unitary.
Moreover, the output states usually are mixed even if we employ a
pure input state, and $\xi_1$, $\xi_2$ may contain common Kraus
operators. To realize perfect identification operation, these
components should have no contribution to the output states. Thus
totally solve this problem seems to be quite complicated.
To summarize, we have shown that besides global operations,
multi-partite unitary transformations can also be discriminated
perfectly with local methods. Nonlocal schemes together with
entangled input states usually can improve the efficiency of the
identification, i.e., we can run the unknown operation less times
to realize perfect discrimination. However, it doesnot affect the
distinguishability of the whole problem. In principle, by running
the secretly chosen operations a finite number of times, we can
also realize perfect identification under LOCC. From the practical
viewpoint, one need to provide an optimal methods to implement the
discrimination operations. Our investigation indicates that this
question has a close relation to the exact universality of unitary
evolution and the optimal quantum circuit in $d$-level
system\cite{construction}.
The authors thank R. Duan for helpful comments and suggestions and
drawing our attention to their closely related works
\cite{duan}.This work was funded by the National Fundamental
Research Program (2006CB921900) , the National Natural Science
Foundation of China (Grant No. 10674127), the Innovation Funds
from the Chinese Academy of Sciences, and Program for New Century
Excellent Talents in University.
\renewcommand{A-\arabic{equation}}{A-\arabic{equation}}
\setcounter{equation}{0}
\section*{APPENDIX}
We now present a simple proof about the exact universality of
$N$-partite unitary transformations. The method used here are
mainly based on ref. \cite{exactuniverse}. First we introduce the
following lemma.
\begin{lemma}
Let $G$ be a compact Lie group. If $H_1$, $\ldots$, $H_k$ are
closed connected subgroups and they generate a dense subgroup of
$G$, then in fact they generate $G$.
\end{lemma}
Suppose $U$ is a $N$-partite unitary map, and we also use $H$ to
denote all $1$-qudit gates $V_1\otimes \ldots \otimes V_N$. We
introduce the subgroup $H_1 = U H U^{-1}$. Now consider the
$n$-fold products $\Sigma^n= \Sigma \ldots \Sigma $ with $\Sigma=
H_1 H$. One can find that when $n \rightarrow \infty$,
$\Sigma^{\infty}$ is a subgroup of all $N$-partite unitary
transformations $U(d^N)$, hence we have $H \subseteq
\Sigma^{\infty} \subseteq U(d^N)$.
Assume $h$, $r$, $g$ are the corresponding Lie algebras of the
group $H$, $\Sigma^{\infty}$, $U(d^N)$ separately. Consider the
representation of $K= SU(d)\otimes \ldots \otimes SU(d)$ on the
Lie algebra $g$
\[
\pi^{S_1,\ldots, S_N}(\xi)=(S_1\otimes \ldots
\otimes S_N) \xi (S_1\otimes \ldots \otimes S_N)^{\dag},
\hspace{5mm} \xi \in g .
\]
Since $K$ is a compact Lie group, $\pi$ can be decomposed as a
direct sum of irreducible representations of $K$. Therefore, we
obtain the following decomposation of $g$
\begin{eqnarray} \label{A1}
g=\bigoplus_{j=0}^N \bigoplus_{k=1}^{n_j} i^{N+1-j}
P_{[\alpha^{j,k}_1, \ldots \alpha^{j,k}_j]}
\end{eqnarray}
with
\begin{eqnarray}
P_0 &=& \mathbb{R} I \otimes \ldots \otimes I, \\
P_{[\alpha^{j}_1, \ldots \alpha^{j}_j]}&=& I_1 \otimes \ldots
\otimes su(d)_{\alpha^{j}_1} \otimes \ldots \otimes
su(d)_{\alpha^{j}_j} \otimes \ldots,
\end{eqnarray}
where $i^2=-1$, $su(d)$ is the Lie algebra of $SU(d)$, and
$\alpha^{j}_{j'}$ are indices selected from the set $\{ 1, \ldots,
N \}$.
Similarly, because $H \subseteq \Sigma^{\infty}$, $r$ can also be
decomposed into the direct sum of a finite number of terms on the
right-hand-side of Eq. ($\ref{A1}$)
\begin{eqnarray}
r=\bigoplus_{j=0}^{n\le N }\bigoplus_{k=1}^{n_j} c_{jk}
P_{[\alpha^{j,k}_1, \ldots \alpha^{j,k}_j]}, \hspace{5mm} and
\hspace{5mm} c_{jk} \in \{\pm 1, \pm i \}.
\end{eqnarray}
We call two indices $\alpha^{j,k}_{L}$ and $\alpha^{j',k'}_{L'}$
to be connected if there exists a subset $C=[\alpha^{j,k}_1,
\ldots \alpha^{j,k}_j]$ such that $\alpha^{j',k'}_{L'} \in C$ and
$\alpha^{j',k'}_{L'} \in C$. Thus the connectedness of indices
lead to the following decomposition of $\{ 1, \ldots, N \}$
\begin{eqnarray}
[\ldots,\alpha^{j_1,k_1}_{L_1}, \ldots ] \oplus
[\ldots,\alpha^{j_2,k_2}_{L_2}, \ldots ] \oplus \ldots.
\end{eqnarray}
On the other hand, since $r$ is a Lie algebra, one can immediately
obtained that $r$ is the Lie algebra of the compact Lie group
$\mathcal{U}_{\alpha}=U_{[\ldots,\alpha^{j_1,k_1}_{L_1}, \ldots
]}\otimes U_{[\ldots,\alpha^{j_2,k_2}_{L_2}, \ldots ]} \otimes
\ldots$. According to Lemma 2, we obtain that
$\Sigma^{\infty}=\mathcal{U}_{\alpha}$, hence there exist some $p$
such that $\Sigma^p=\mathcal{U}_{\alpha}$.
After we have obtained the group $\mathcal{U}_{\alpha}$, we can
now define the new $n$-fold products as $ \Sigma_1^n=\Sigma_1
\ldots \Sigma_1$ with $\Sigma_1 =
(U\mathcal{U}_{\alpha}U^{\dag})\cdot \mathcal{U}_{\alpha}$. Repeat
the above discussions, we have that $U$ together with all
$1$-qudit gates can generate the following unitary group
\begin{eqnarray}
\mathcal{U}_{\beta}= U_{[\ldots,\beta^{j_1,k_1}_{L_1}, \ldots
]}\otimes U_{[\ldots,\beta^{j_2,k_2}_{L_2}, \ldots ]} \otimes
\ldots
\end{eqnarray}
with $U \mathcal{U}_{\beta} U^{\dag}= \mathcal{U}_{\beta}$.
Therefore, $U$ is $\{[\ldots,\beta^{j_1,k_1}_{L_1}, \ldots ],
[\ldots,\beta^{j_2,k_2}_{L_2}, \ldots ], \ldots \}$-primitive.
Moreover, $U$ normalize $\mathcal{U}_{\beta}$. Following the
similar discussion in ref. \cite{exactuniverse}, we have that $U$
can be expressed as $U=U_{\beta} \cdot P_{\beta}$ for some
$U_{\beta} \in \mathcal{U}_{\beta}$, where $P_{\beta}$ is the
corresponding permutation operator of the Hilbert spaces ${\cal
H}_{[\ldots,\beta^{j_m,k_m}_{L_m}, \ldots ]}$ which have the same
dimension.
For example, if $U_{12345}$ is $\{[1,2],[3,4],5 \}$-primitive,
then $P_{\{[1,2],[3,4],5 \}}=P_{12,34} \otimes I_5$ or
$I_{12345}$, where $P_{12,34}$ is the swap operation between
Hilbert spaces ${\cal H}_1 \otimes {\cal H}_2$ and ${\cal H}_3
\otimes {\cal H}_4$; if $U_{12345}$ is $\{1,2,3,4,5 \}$-primitive,
then $P_{\{1,2,3,4,5 \}}$ can be any element in the permutation
group $S_5$.
\end{document} |
\begin{document}
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\allowdisplaybreaks
\title[]{Solutions of Fully Nonlinear Nonlocal
Systems}
\author[Pengyan Wang and Mei Yu]{}
\keywords{Fully nonlinear nonlocal operator, narrow region principle, decay at infinity, method of moving planes, radial symmetry, monotonicity, non-existence, positive solutions}
\subjclass{}
\email{wangpy119@126.com}
\email{yumei@nwpu.edu.cn}
\maketitle
\centerline{\scshape Pengyan Wang, Mei Yu}
{\footnotesize\centerline{Department of Applied Mathematics , Northwestern Polytechnical University, } \centerline{Xi'an, 710129,
Shaan xi, P.R. China} }
\begin{abstract}
In this paper we consider the system involving fully nonlinear nonlocal operators:
$$ \left\{
\begin{array}{ll}
F_{\alpha}(u(x)) = C_{n,\alpha} PV \int_{{R}^n} \frac{G(u(x)-u(y))}{|x-y|^{n+\alpha}} dy=f(v(x)),\\
F_{\beta}(v(x)) = C_{n,\beta} PV \int_{{R}^n} \frac{G(v(x)-v(y))}{|x-y|^{n+\beta}} dy=g(u(x)).
\end{array}
\right.
$$
A \textit{narrow region principle} and a \textit{decay at infinity} for the system for carrying on the method of moving planes are established. Then we
prove the radial symmetry and monotonicity for positive
solutions to the nonlinear system in the whole space. Non-existence of positive
solutions to the nonlinear system on a half space is proved.
\end{abstract}
\section{Introduction}
In this paper, we consider the nonlinear system involving fully nonlinear nonlocal operators: $$\left\{
\begin{array}{ll}
F_{\alpha}(u(x)) = f(v(x)),\\
F_{\beta}(v(x)) =g(u(x)),
\end{array}
\right.$$
with $$F_{\alpha}(u(x))
= C_{n,\alpha} PV \int_{{R}^n} \frac{G(u(x)-u(y))}{|x-y|^{n+\alpha}} dy,$$
where $PV$ stands for the Cauchy principal value, $G$ is at least local Lipschitz continuous, $G(0)=0$ and $0<\alpha, \beta<2$. The operators $F_\alpha$ was introduced by
Caffarelli and Silvestre in \cite{CS1}.
In order the integral to make sense, we require $$u \in C^{1,1}_{loc}\cap L_\alpha,~v\in C^{1,1}_{loc}\cap L_\beta$$
with $$ L_{\alpha }=\{u:R^{n}\rightarrow R \mid \int_{R^{n}} \frac{|u(x)|}{1+|x|^{n+\alpha}}dx <\infty \} , $$
and $L_\beta$ having a similar meaning.
In the special case when $G(\cdot)$ is an identity map, $F_\alpha$ becomes the usual fractional Laplacian $(-\Delta)^{\frac{\alpha}{2}}.$ The nonlocal nature of fractional operators makes them difficult to study. To circumvent this, Caffarelli and Silvestre \cite{CS} introduced the \textit{extension method} which turns the nonlocal problem involving the fractional Laplacian into a local one in higher dimensions. This method has been applied successfully to treat equations involving the fractional Laplacian
and a series of fruitful results has been obtained (see \cite{BCPS}, \cite{CZ}, etc.). One can
also use \textit{the integral equations method}, such as \textit{the method of moving planes in integral forms} (see \cite{CC}, \cite{CD}, \cite{ZCCY}, \cite{LZ}, \cite{LZr}) and \textit{regularity lifting }to investigate equations involving fractional Laplacian by showing that they are equivalent to corresponding integral equations (see \cite{CFY}, \cite{CLO}, \cite{CLO1} and the references therein). For more articles concerning the method of moving planes for nonlocal equations and
for integral equations, see \cite{FL}, \cite{HLZ}, \cite{HWY}, \cite{LLM}, \cite{LZ2}, \cite{MC}, \cite{MZ}, \cite{LZ} and the references therein.
For the fully nonlinear nonlocal equations, so far as we know, there is neither any corresponding {\em extension method} nor equivalent integral equations that one can work at. A probable reason is that very few results were obtained for fully nonlinear nonlocal operator. In \cite{CLL}, Chen, Li and Li developed a new method that can deal directly with these nonlocal operators. Inspired by the idea, we extend the method in \cite{CLL} to fully nonlinear nonlocal systems and consider the nonlinear systems involving fully nonlinear nonlocal operators
\begin{equation}\label{eq:a1}
\left\{
\begin{array}{ll}
F_{\alpha}(u(x)) = f(v(x)),\\
F_{\beta}(v(x)) =g(u(x)), &\mbox~ x\in R^n,\\
u(x)>0,v(x)>0,&\mbox~ x\in R^n,
\end{array}
\right.\end{equation}
and
\begin{equation}\label{eq:a2}
\left\{
\begin{array}{ll}
F_{\alpha}(u(x)) = f(v(x)),\\
F_{\beta}(v(x)) =g(u(x)), &\mbox~ x\in R^n_+,\\
u(x)\equiv 0,v(x)\equiv 0,&\mbox~ x\not\in R^n_+,
\end{array}
\right.
\end{equation}
where $f$ and $g$ are nonnegative continuous and nondecreasing functions.
We first establish the \textit{narrow region principle} and \textit{decay at infinity} for the system which play important roles in carrying out the method of moving planes.
To state them, denote by
$$T_\lambda=\{x\in R^n| x_1=\lambda \}$$
the moving plane, $$\Sigma_\lambda=\{x\in R^n|x_1<\lambda\}$$ the left region of the plane $T_\lambda$, $$x^\lambda=(2\lambda-x_1,x_2,\cdots, x_n)$$ the reflection
of $x$ about $T_\lambda$, and denote $$U_\lambda(x)=u_\lambda(x)-u(x)~ \mbox{and}~
V_\lambda(x)=v_\lambda(x)-v(x).$$ For simplicity of notations, we
stand for $U_\lambda(x)$ by $U(x)$ and $V_\lambda(x)$ by $V(x)$ in the sequel.
\begin{thm}\label{thma2}(Narrow Region Principle )
Let $\Omega$ be a bounded narrow region in $\Sigma_\lambda$ contained in
$$\{x | \lambda-l<x_1<\lambda \}$$
with small $l>0$. Suppose that $U(x)\in L_\alpha\cap C_{loc}^{1,1}(\Omega), V(x)\in L_\beta \cap C_{loc}^{1,1}(\Omega), $ and $U(x),V(x)$ are lower semi-continuous on $\bar{\Omega}$.
If
$c_i(x)\leq 0,i=1,2,$ are bounded from below in $\Omega$, $U(x)$ and $V(x)$ satisfy
\begin{equation}\label{eq:a3}
\left\{\begin{array}{ll}
F_\alpha (u_\lambda(x))-F_\alpha (u(x))+c_1(x)V(x)\geq0, \\
F_\beta (v_\lambda(x))-F_\beta (v(x))+c_2(x)U(x)\geq0, &\quad x \in \Omega,\\
U(x), V(x)\geq 0, &\quad x \in \Sigma_\lambda\backslash\Omega,\\
U(x^\lambda)=-U(x),
V(x^\lambda)=-V(x), &\quad x \in \Sigma_\lambda,
\end{array}
\right.
\end{equation}
then we have for sufficiently small $l$,
\begin{equation}\label{eq:a4}
U(x),V(x) \geq0 \mbox{ in } \Omega;\end{equation}
if $\Omega$ is unbounded, the conclusion still holds under the conditions
$$\underset{|x|\rightarrow \infty}{\underline{\lim}}U(x),\underset{|x|\rightarrow \infty}{\underline{\lim}}V(x)\geq0;$$
furthermore, if $U(x)$ or $V(x)$ attains 0 somewhere in $\Sigma_\lambda$, then
\begin{equation}\label{eq:a5}U(x)=V(x)\equiv 0,~ x\in R^n.\end{equation}
\end{thm}
We call \eqref{eq:a5} the strong maximum principle later. As we can see from the proof, to ensure \eqref{eq:a5}, $\Omega$ does not need to be narrow.
\begin{thm}\label{thma1}( Decay at Infinity)
Let $\Omega$ be a bounded or unbounded domain in $R^n$. Assume that $U(x)\in C^{1,1}_{loc}(\Omega)\cap L_\alpha(R^n), V(x)\in C^{1,1}_{loc}(\Omega)\cap L_\beta(R^n),$ $U(x)$ and $V(x) $ are lower semi-continuous on $ \bar {\Omega}$. If $U(x)$ and $V(x) $ satisfy
\begin{equation}\label{eq:a11}
\left\{
\begin{array}{ll}
F_{\alpha}(u_\lambda(x))-F_{\alpha}(u(x)) +c_1(x)V(x)\geq 0,\\
F_{\beta}(v_\lambda(x))-F_{\beta}(v(x)) +c_2(x)U(x)\geq 0, &\mbox x\in \Omega,\\
U(x),V(x)\geq0, &\mbox x \in {\Sigma_\lambda} \backslash{\Omega},\\
U(x^\lambda) =-U(x),\\
V(x^\lambda) =-V(x), &\mbox x \in \Sigma_\lambda ,
\end{array}
\right.\end{equation}
with \begin{equation}\label{eq:aa11}c_1(x)\sim o(\frac{1}{|x|^\alpha}),~c_2(x)\sim o(\frac{1}{|x|^\beta}),~~ \mbox{for} ~|x|~~\mbox{large}, \end{equation}
and $$c_i(x)\leq 0,~i=1,2,$$
then there exists a constant $R_0>0$ depending only on $c_i(x)$ such that if
$$U(\tilde{x})=\underset{\Omega} {min}U(x) <0~ \mbox{and} ~ V(\bar{x})=\underset{\Omega} {min}V(x) <0,$$
then \begin{equation}\label{eq:aaa11}|\tilde{x}|\leq R_0~\mbox{or} ~ |\bar x|\leq R_0.\end{equation}
\end{thm}
Based on Theorems \ref{thma2} and \ref{thma1}, we apply
the \textit{method of moving planes} to obtain symmetry and monotonicity of positive solutions to \eqref{eq:a1} in $R^n$, as well as nonexistence of positive solutions to \eqref{eq:a2} on the half space.
\begin{thm}\label{thma3}
Assume that $u(x)\in L_\alpha(R^n)\cap C_{loc}^{1,1}(R^n)~ \mbox{and}~v(x)\in L_\beta(R^n)\cap C_{loc}^{1,1}(R^n)$ are positive solutions of system \eqref{eq:a1}. Suppose that for some $ \gamma,\tau>0,$
\begin{equation}\label{eq:aa1} v(x)=o(\frac{1}{|x|^\gamma}),~u(x)=o(\frac{1}{|x|^\tau}),~ \mbox{as}~ |x|\rightarrow \infty,\end{equation} and \begin{equation}\label{eq:aa2}f'(s)\leq s^q, ~g'(t)\leq t^p,~ \mbox{with} ~~ q\gamma\geq\alpha,~ p\tau\geq\beta .\end{equation}
Then $u(x)$ and $v(x)$ must be radially symmetric and monotone decreasing about some point $x_0$ in $R^n$.
\end{thm}
\begin{thm}\label{thma4}Assume that $u(x)\in L_\alpha\cap C_{loc}^{1,1}(R^n_+),~v(x)\in L_\beta\cap C_{loc}^{1,1}(R^n_+)$ are nonnegative solutions of system \eqref{eq:a2}. Suppose \begin{equation}\label{eq:a28}\underset{|x|\rightarrow \infty}{\lim}u(x)=0,~\underset{|x|\rightarrow \infty}{\lim}v(x)=0,\end{equation}
$f(v),g(u)$ are Lipschitz continuous in the range of $ v(x),u(x)$ respectively, and $f(0)=0,~g(0)=0.$
Then $u(x)\equiv0,~v(x)\equiv0$.
\end{thm}
In section 2, we prove Theorems \ref{thma2} and \ref{thma1} with a key ingredient \eqref{eq:aa20} below. In section 3, the proofs of Theorems \ref{thma3} and \ref{thma4} are given by using the previous results and the method of moving planes.
\section{Proofs of Theorems \ref{thma2} and \ref{thma1} }
~~~~~Let $$F_{\alpha}(u(x))= C_{n,\alpha} PV \int_{\mathbb{R}^n} \frac{G(u(x)-u(y))}{|x-y|^{n+\alpha}} dy=C_{n,\alpha} \lim_{\epsilon \rightarrow 0} \int_{\mathbb{R}^n\setminus B_{\epsilon}(x)} \frac{G(u(x)-u(y))}{|x-y|^{n+\alpha}} dy
.$$
Throughout this and next section, we assume
\begin{equation}\label{eq:a33}G\in C^1(R),~G(0)=0, \mbox{and} ~G'(t)\geq c_0>0, ~\mbox{for} ~t \in R.\end{equation}
Using the simple maximum principle in \cite{CLLg}, we prove the following strong maximum principle.
\begin{lem}\label{lemaa1}
Let $\Omega$ be a bounded domain in $R^n$. Assume that $u(x)\in C^{1,1}_{loc}(\Omega)\cap L_\alpha(R^n),$ is lower semi-continuous on $\bar \Omega$,
and satisfies \begin{equation}\label{eq:a30}
\left\{
\begin{array}{ll}
F_{\alpha}(u(x)) \geq 0, &\mbox x\in \Omega,\\
u(x)\geq0,&\mbox x\in \Omega^c.
\end{array}
\right.\end{equation}
If $u(x)$ attains 0 somewhere in $\Sigma_\lambda$, then
$$u(x)\equiv 0, ~x\in R^n.$$
\end{lem}
{\bf Proof .}
If $u(x)$ is not identical to 0, there exists
an $x^0$ such that $u(x^0)=0$ and
$$\aligned F_{\alpha}(u(x))&=\int_{R^n} \frac{G(u(x^0)-u(z))}{|x^0-z|^{n+\alpha}}dz\\
&=\int_{R^n} \frac{G'(\Psi(z))[u(x^0)-u(z)]}{|x^0-z|^{n+\alpha}}dz\\
&\leq c_0 \int_{R^n} \frac{-u(z)}{|x^0-z|^{n+\alpha}}dz\\
&<0.
\endaligned$$
This contradicts \eqref{eq:a30} and the proof is ended.
{\bf Proof of Theorem \ref{thma2}.}
If \eqref{eq:a4} does not hold, without loss of generality, we assume $U(x)<0$ at some point in $\Omega$; then the lower semi-continuity of $U(x)$ on $\bar \Omega$ guarantees that there exists some $\tilde{x}\in \Omega $ such that
$$U(\tilde{x})=\underset{\Omega} {\min}U(x) <0.$$
And it deduces from the condition \eqref{eq:a3} that $\tilde{x}$ is in the interior of $\Omega.$
By the defining integral, we have
\begin{eqnarray}\label{eq:aa20}
F_\alpha (u_\lambda(\tilde x))-F_\alpha (u(\tilde x))&=&C_{n,\alpha} PV\int_{R^{n}}\frac{G(u_\lambda (\tilde x)-u_\lambda (y))-G(u (\tilde x)-u(y))}{|\tilde x -y|^{n+\alpha}}dy \nonumber\\
&=&C_{n,\alpha} PV\int_{\Sigma_\lambda}\frac{G(u_\lambda (\tilde x)-u_\lambda (y))-G(u (\tilde x)-u(y))}{|\tilde x -y|^{n+\alpha}}dy\nonumber\\&+&C_{n,\alpha} PV\int_{\Sigma_\lambda}\frac{G(u_\lambda (\tilde x)-u (y))-G(u (\tilde x)-u_\lambda(y))}{|\tilde x -y^\lambda|^{n+\alpha}}dy\nonumber\\
&\leq& C_{n,\alpha} PV\int_{\Sigma_\lambda}\frac{G(u_\lambda (\tilde x)-u_\lambda (y))-G(u (\tilde x)-u(y))}{|\tilde x -y^\lambda|^{n+\alpha}}dy\nonumber\\&+&C_{n,\alpha} PV\int_{\Sigma_\lambda}\frac{G(u_\lambda (\tilde x)-u (y))-G(u (\tilde x)-u_\lambda(y))}{|\tilde x -y^\lambda|^{n+\alpha}}dy\nonumber\\
&=&C_{n,\alpha} PV\int_{\Sigma_\lambda}\frac{2G'(\cdot)U(\tilde x)}{|\tilde x -y^\lambda|^{n+\alpha}}dy\nonumber\\
&\leq& 2C_{n,\alpha} c_0 U(\tilde x)\int_{\Sigma_\lambda}\frac{1}{|\tilde x -y^\lambda|^{n+\alpha}}dy.
\end{eqnarray}
Let $D=B_{2l}(\tilde x)\cap \tilde \Sigma_\lambda $, then
\begin{equation}\label{eq:a6}\aligned
\int_{\Sigma_\lambda}\frac{1}{|\tilde x -y^\lambda|^{n+\alpha}}dy
&\geq \int_{D}\frac{1}{|\tilde x -y|^{n+\alpha}}dy\\
&\geq\frac{1}{10}\int_{B_{2l}(\tilde x)}\frac{1}{|\tilde x -y|^{n+\alpha}}dy\\
&\geq\frac{1}{l^\alpha}.\endaligned
\end{equation}
Thus from \eqref{eq:aa20},
\begin{equation}\label{eq:a7}
F_\alpha (u_\lambda(\tilde x))-F_\alpha (u(\tilde x))\leq \frac{C U(\tilde x)}{l^\alpha}<0.
\end{equation}
Together \eqref{eq:a7} with \eqref{eq:a3}, it yields
\begin{equation}\label{eq:a8}
U(\tilde x)\geq -Cc_1(\tilde x)l^\alpha V(\tilde x)~\mbox{and}~V(\tilde x)\leq 0.
\end{equation}
We know from \eqref{eq:a8} that there exists $\bar x$ such that
$$V(\bar x)=\underset{\Omega}{\min} V(x)<0.$$
Similarly to \eqref{eq:a7}, it derives that
$$F_\beta (v_\lambda(\bar x))-F_\beta (v(\bar x))\leq \frac{C V(\bar x)}{l^\beta}<0.$$
Combining it with \eqref{eq:a8}, we have for $l$ sufficiently small,
$$\aligned
0&\leq F_\beta (v_\lambda(\bar x))-F_\beta (v(\bar x))+c_2(\bar x)U(\bar x)\\
&\leq\frac{CV(\bar x)}{l^\beta}+c_2(\bar x)U(\tilde x)\\
&\leq C(\frac{V(\bar x)}{l^\beta}-c_2(\bar x)c_1(\tilde x)l^\alpha V(\tilde x))\\
&\leq C(\frac{V(\bar x)}{l^\beta}-c_2(\bar x)c_1(\tilde x)l^\alpha V(\bar x))\\
&\leq C \frac{V(\bar x)}{l^\beta}(1-c_1(\tilde x)c_2(\bar x)l^{\alpha+\beta})\\
&<0.
\endaligned$$
This contradiction shows that \eqref{eq:a4} must be true.
Next we prove \eqref{eq:a5}. Without loss of generality, let us suppose that there exists $\eta\in \Omega$ such that
$$U(\eta)=0.$$
Then we use $\frac{1}{|x-y|}>\frac{1}{|x-y^\lambda|}$,~for $x,y\in \Sigma_\lambda$, to have
\begin{equation}\label{eq:aa10}\aligned
&F_\alpha (u_\lambda(\eta))-F_\alpha (u(\eta))\\&=C_{n,\alpha} PV\int_{R^{n}}\frac{G(u_\lambda (\eta)-u_\lambda (y))-G(u (\eta)-u(y))}{|\eta -y|^{n+\alpha}}dy\\
&=C_{n,\alpha} PV\int_{\Sigma_\lambda}\frac{G(u_\lambda (\eta)-u_\lambda (y))-G(u (\eta)-u(y))}{|\eta -y|^{n+\alpha}}dy\\&+C_{n,\alpha} PV\int_{\Sigma_\lambda}\frac{G(u_\lambda (\eta)-u (y))-G(u (\eta)-u_\lambda(y))}{|\eta -y^\lambda|^{n+\alpha}}dy \\
&=C_{n,\alpha} PV\int_{\Sigma_\lambda}[G(u_\lambda (\eta)-u_\lambda (y))-G(u (\eta)-u(y))](\frac{1}{|\eta -y|^{n+\alpha}}-\frac{1}{|\eta -y^\lambda|^{n+\alpha}})dy\\
&+C_{n,\alpha} PV\int_{\Sigma_\lambda}\frac{G(u_\lambda (\eta)-u (y))-G(u (\eta)-u_\lambda(y))+G(u_\lambda (\eta)-u_\lambda (y))-G(u (\eta)-u(y))}{|\eta -y^\lambda|^{n+\alpha}}dy\\
& =C_{n,\alpha} G'(\cdot)\int_{\Sigma_\lambda} (U(\eta)-U(y))(\frac{1}{|\eta -y|^{n+\alpha}}-\frac{1}{|\eta -y^\lambda|^{n+\alpha}})dy\\&+C_{n,\alpha} G'(\cdot)\int_{\Sigma_\lambda}\frac{2U(\eta)}{|\eta -y^\lambda|^{n+\alpha}}dy\\
&\leq -Cc_0 \int_{\Sigma_\lambda}U(y)(\frac{1}{|\eta -y|^{n+\alpha}}-\frac{1}{|\eta -y^\lambda|^{n+\alpha}})dy.
\endaligned
\end{equation}
If $U(x)\not \equiv0$, then \eqref{eq:aa10} implies $$F_\alpha (u_\lambda(\eta))-F_\alpha (u(\eta))<0.$$
Using it with \eqref{eq:a3}, it shows $V(\eta)<0.$
This is a contradiction with \eqref{eq:a4}. Hence $U(x)$ must be identically $0$ in $\Sigma_\lambda$.
Since $$U(x^\lambda)=-U(x),~x\in \Sigma_\lambda,$$ it gives $$U(x)\equiv0,~x\in R^n.$$
Again from \eqref{eq:a3}, we see $$V(x)\leq 0,~x\in \Sigma_\lambda.$$
Since we already know $$V(x)\geq 0,~x\in \Sigma_\lambda,$$
it must hold $$V(x)=0,~x\in \Sigma_\lambda.$$
Recalling $V(x^\lambda)=-V(x),$ we arrive at
$$V(x)\equiv0,~x\in R^n.$$
Similarly, one can show that if $U(x)$ or $V(x)$ attains 0 at one point in $\Sigma_\lambda$, then both $U(x)$ and $V(x)$ are identically 0 in $R^n$. This completes the proof.
{\bf Proof of Theorem \ref{thma1}.} There exists $\tilde x \in \Omega,$
such that $$U(\tilde x)=\underset{\Omega}{\min}U(x)<0.$$
Using \eqref{eq:aa20}, we have
$$
\aligned F_{\alpha}(u_\lambda(\tilde x)) -F_{\alpha}(u(\tilde x))
=&C_{n,\alpha}PV \int_{\Sigma_\lambda} \frac{G(u_\lambda(\tilde x)-u_\lambda(y))-G(u(\tilde x)-u(y))}{|\tilde x-y|^{n+\alpha}}\\+&C_{n,\alpha}PV \int_{ \Sigma_\lambda} \frac{G(u_\lambda(\tilde x)-u(y))-G(u(\tilde x)-u_\lambda(y))}{|\tilde x-y^\lambda|^{n+\alpha}}\\
\leq&C_{n,\alpha}PV \int_{\Sigma_\lambda}\frac{G'(\cdot)2U(\tilde x)}{|\tilde x-y^\lambda|^{n+\alpha}}dy\\
\leq&2C_{n,\alpha}c_0U(\tilde x)\int_{\Sigma_\lambda}\frac{1}{|\tilde x-y^\lambda|^{n+\alpha}}dy.
\endaligned
$$
For each fixed $\lambda$, there exists $C>0$ such that for $\tilde x \in \Sigma_\lambda$ and $|\tilde x|$ sufficiently large,
\begin{equation}\label{eq:a9}
\int_{\Sigma_\lambda}\frac{1}{|\tilde x-y^\lambda|^{n+\alpha}}dy\geq \int_{B_{3|\tilde x|}(\tilde x)\backslash B_{2|\tilde x|}(\tilde x)} \frac{1}{|\tilde x-y|^{n+\alpha}}dy \sim \frac{C}{|\tilde x|^\alpha}.
\end{equation}
Hence
\begin{equation}\label{eq:a10}
F_{\alpha}(u_\lambda(\tilde x)) -F_{\alpha}(u(\tilde x)) \leq \frac{CU(\tilde x)}{|\tilde x|^\alpha}<0.
\end{equation}
Together \eqref{eq:a10} with \eqref{eq:a11}, it is easy to deduce
\begin{equation}\label{eq:a12}
V(\tilde x)<0,
\end{equation}
and
\begin{equation}\label{eq:a13}
U(\tilde x)\geq -Cc_1(\tilde x)|\tilde x|^\alpha V(\tilde x).
\end{equation}
From \eqref{eq:a12}, there exists $\bar x$ such that
$$V(\bar x)=\underset{\Omega}{\min}V(x)<0.$$
Similarly to \eqref{eq:a9}, we can derive
\begin{equation}\label{eq:aa13}F_{\beta}(v_\lambda(\bar x)) -F_{\beta}(v(\bar x)) \leq \frac{CV(\bar x)}{|\bar x|^\beta}<0.\end{equation}
Combing \eqref{eq:a11} and \eqref{eq:a13}, we have for $\lambda$ sufficiently negative, $$
\aligned
0&\leq F_{\beta}(v_\lambda(\bar x)) -F_{\beta}(v(\bar x)) +c_2(\bar x)U(\bar x)\\
&\leq \frac{CV(\bar x)}{|\bar x|^\beta}+c_2(\bar x)U(\bar x)\\
&\leq C(\frac{V(\bar x)}{|\bar x|^\beta}-c_2(\bar x)c_1(\tilde x)|\tilde x|^\alpha V(\bar x))\\
&\leq\frac{CV(\bar x)}{|\bar x|^\beta}(1-c_1(\tilde x)|\tilde x|^\alpha c_2(\bar x)|\bar x|^\beta).
\endaligned
$$
It follows that $1\leq c_1(\tilde x)|\tilde x|^\alpha c_2(\bar x)|\bar x|^\beta.$
However, from \eqref{eq:aa11} we have $ c_1(\tilde x)|\tilde x|^\alpha c_2(\bar x)|\bar x|^\beta<1 $ for $|\tilde x|$ and $|\bar x|$ sufficiently large.
This contradiction shows that \eqref{eq:aaa11} must be true.
\section{Symmetry of solutions in the whole space $R^n$ }
{\bf Proof of Theorem \ref{thma3}.}
Choose an arbitrary direction as the $x_1$-axis. Let $T_\lambda =\{x\in R^n \mid x_1=\lambda\},~x^\lambda=(2\lambda-x_1,x'),~ u_\lambda(x)=u(x^\lambda),~\Sigma_\lambda=\{x\in R^n|x_1<\lambda\}$,
$$U_\lambda(x)=u_\lambda(x)-u(x),~V_\lambda(x)=v_\lambda(x)-v(x).$$
Step1. \textit{Start moving the plane $T_\lambda$ from $-\infty$ to the right in $x_1$-direction.}
We will show that for $\lambda$ sufficiently negative,
\begin{equation}\label{eq:a16}U_\lambda(x)\geq0,~V_\lambda(x)\geq0,~x\in \Sigma_\lambda.\end{equation}
For the fixed $\lambda$ and $x\in\Sigma_\lambda,$ by \eqref{eq:aa1},
$$u(x)\rightarrow0,~\mbox{as}~|x|\rightarrow+\infty.$$
As $|x|\rightarrow+\infty$, we have $|x^\lambda|\rightarrow+\infty$; it follows that
$$u_\lambda(x)=u(x^\lambda)\rightarrow0.$$
Thus for $x\in \Sigma_\lambda ,$ \begin{equation}\label{eq:aa17}U_\lambda(x)\rightarrow 0,~\mbox{as}~|x|\rightarrow+\infty.\end{equation}
Similarly, one can show that for $x\in \Sigma_\lambda ,$ $$V_\lambda(x)\rightarrow 0,~\mbox{as}~|x|\rightarrow+\infty.$$
By the mean value theorem it is easy to see that
$$F_\alpha (u_\lambda(x))-F_\alpha (u(x))=f(v_\lambda(x))-f(v(x))=f'(\xi_\lambda(x))V_\lambda(x),$$
and $$F_\beta (v_\lambda(x))-F_\beta (v(x))=g(u_\lambda(x))-g(u(x))=g'(\eta_\lambda (x))U_\lambda(x),$$
where $\xi_\lambda(x)$ is valued between $v_\lambda(x)$ and $v(x)$; $\eta_\lambda(x)$ is valued between $u_\lambda(x)$ and $u(x)$. By Theorem \ref{thma1}, it suffices to check the decay rate of $f'(\xi_\lambda(x))$ and $g'(\eta_\lambda(x))$,
at the points where $V_\lambda(x)$ and $U_\lambda(x)$ are negative respectively.
Since $u_\lambda( x)<u(x)$ and $v_\lambda(x)<v( x)$,
we have $$0\leq u_\lambda( x)\leq \eta_\lambda(x)\leq u( x),~0\leq v_\lambda( x)\leq \xi_\lambda(x)\leq v( x).$$
At those points for $|x|$ sufficiently large, the decay assumptions \eqref{eq:aa1} and \eqref{eq:aa2} instantly yields that
$$c_1(x)=f'(\xi_\lambda(x))\sim o(\frac{1}{| x|^\alpha}), ~ c_2(x)=g'(\eta_\lambda(x))\sim o(\frac{1}{| x|^\beta}).$$
Consequently, there exists $R_0>0,$ such that, if $\tilde x$ and $\bar x$ are negative minima of $U_\lambda(x)$ and $V_\lambda(x)$ in $\Sigma_\lambda$ respectively, then by Theorem \ref{thma1}, it holds that
\begin{equation}\label{eq:aa16}|\tilde x|\leq R_0 ~\mbox{or}~|\bar x|\leq R_0.\end{equation}
Without loss of generality, we may assume
\begin{equation}\label{eq:aa18}|\tilde x|\leq R_0 .\end{equation} For $\lambda$ sufficiently negative, combining \eqref{eq:aa17} with fact that $$U_\lambda(x)=0,~x\in T_\lambda,$$ we know if $U_\lambda(x)<0$ in $\Sigma_\lambda$, then $U_\lambda(x)$
must have a negative minimum in $\Sigma_\lambda$. This contradicts \eqref{eq:aa18}. Hence, for $\lambda$ sufficiently negative we have \begin{equation}\label{eq:aa108}U_\lambda(x)\geq0,\end{equation} it follows that $V_\lambda(x)\geq 0$ in $\Sigma_\lambda$. Otherwise, there exists $\bar x$ in $\Sigma_\lambda$ such that $$V_\lambda(\bar x)=\underset{\Sigma_\lambda}{\min}V_\lambda(x)<0,$$
from \eqref{eq:aa13}, we have \begin{equation}\label{eq:aaa13}F_\beta(v_\lambda(\bar x))-F_\beta(v(\bar x))< 0.\end{equation}
However, combining \eqref{eq:a11} with \eqref{eq:aa108}, we have $F_\beta(v_\lambda(\bar x))-F_\beta(v(\bar x))\geq 0.$ This is a contradiction with \eqref{eq:aaa13} and $V_\lambda(x)$ cannot attain its negative value in $\Sigma_\lambda.$
It follows that \eqref{eq:a16} must be true. This completes the preparation for the moving planes.
Step 2. \textit{Keep moving the planes to the right to the limiting position $T_{\lambda_0}$ as long as
\eqref{eq:a16} holds}.
Let $$\lambda_0=\sup\{\lambda \mid U_\mu(x),~ V_\mu(x) \geq 0, ~x\in \Sigma_\mu, ~\mu\leq \lambda\}.$$
Obviously, \begin{equation}\label{eq:a61}\lambda_0<\infty.\end{equation}
Otherwise, for any $\lambda>0,$
$$u(0^\lambda)>u(0)>0,~~v(0^\lambda)>v(0)>0.$$
Meanwhile, $$u(0^\lambda)\sim \frac{1}{|0^\lambda|^\beta},~~v(0^\lambda)\sim \frac{1}{|0^\lambda|^\alpha},~\lambda\rightarrow\infty.$$
This is a contradiction and \eqref{eq:a61} is proved.
Now, we point out that \begin{equation}\label{eq:a18} U_{\lambda_0}(x)\equiv0, ~ V_{\lambda_0}(x) \equiv 0,~ x\in \Sigma_{\lambda_0}.\end{equation}
Otherwise, we will show that the plane $T_\lambda$ can be moved further to the right. More rigorously, there exists some $\epsilon>0,$ such that for any
$\lambda\in [\lambda_0,\lambda_0+\epsilon)$ we have
\begin{equation}\label{eq:a17}U_\lambda(x)\geq 0,~V_\lambda(x)\geq 0, ~x\in \Sigma_\lambda.\end{equation}
This is a contradiction with the definition of $\lambda_0$. Hence we must have \eqref{eq:a18}.
Now we to prove \eqref{eq:a17} by using Theorem \ref{thma2} and Theorem \ref{thma1}.
Suppose \eqref{eq:a18} is false, then $U_{\lambda_0}(x)\geq 0$ and $V_{\lambda_0}(x)\geq 0$ are positive somewhere in $\Sigma_{\lambda_0}$, and Theorem \ref{thma2} gives $$U_{\lambda_0}(x)> 0,~V_{\lambda_0}(x)> 0, ~x\in \Sigma_{\lambda_0} .$$
Let $R_0$ be determined in Theorem \ref{thma1}. It follows that for any $\delta>\epsilon>0$,
$$U_{\lambda_0}(x)\geq c_0>0,~V_{\lambda_0}(x)\geq c_0>0,~ x\in \overline{\Sigma_{\lambda_0-\delta}\cap B_{R_0}(0)} .$$
From the continuity of $U_\lambda(x)$ and $ V_\lambda(x)$ with respect to $\lambda$, there exists $\epsilon>0,$ such that for all $\lambda\in [\lambda_0,\lambda_0+\epsilon),$ we have
\begin{equation}\label{eq:a19}
U_\lambda(x)\geq 0,~V_\lambda(x)\geq 0, ~x\in \overline{\Sigma_{\lambda_0-\delta}\cap B_{R_0}(0)} .
\end{equation}
Suppose that \eqref{eq:a17} is false, we have $U_\lambda(x)<0,~V_\lambda(x)<0,~x\in \Sigma_\lambda$. If $\tilde x$ and $\bar x$ are negative minima of $U_\lambda(x)$ and $ V_\lambda(x)$ in $\Sigma_\lambda$ respectively. Next we consider two possibilities.
\textbf{Case 1}. One of the negative minima of $U_\lambda(x)$ and $V_\lambda(x)$ lies in $B_{R_0}(0)$, i.e. it is in the narrow region $\Sigma_{\lambda_0+\epsilon}\backslash \Sigma_{\lambda_0-\delta}$. The other is outside of $B_{R_0}(0)$.
Without loss of generality, we may assume the negative minimum of $U_\lambda(x)$ lies in $B_{R_0}(0).$
from \eqref{eq:a8}, we have
\begin{equation}\label{eq:aa8}
U_\lambda(\tilde x)\geq -c_1(\tilde x)l^\alpha V_\lambda(\tilde x).
\end{equation}
Furthermore, we know
$$\aligned
0&\leq F_\beta (v_\lambda(\bar x))-F_\beta (v(\bar x))+c_2(\bar x)U_\lambda(\bar x)\\
&\leq\frac{CV_\lambda(\bar x)}{|\bar x|^\beta}+c_2(\bar x)U_\lambda(\tilde x)\\
&\leq C\{\frac{V_\lambda(\bar x)}{|\bar x|^\beta}-c_2(\bar x)c_1(\tilde x)l^\alpha V_\lambda(\tilde x)\}\\
&\leq C\{\frac{V_\lambda(\bar x)}{|\bar x|^\beta}-c_2(\bar x)c_1(\tilde x)l^\alpha V_\lambda(\bar x)\}\\
&\leq C \frac{V_\lambda(\bar x)}{|\bar x|^\beta}[1-c_1(\tilde x)l^{\alpha}c_2(\bar x)|\bar x|^\beta].
\endaligned$$
Hence \begin{equation}\label{eq:aaa20}1\leq c_1(\tilde x)l^{\alpha}c_2(\bar x)|\bar x|^\beta.\end{equation}
From \eqref{eq:aa11}, we know that $c_2(\bar x)|\bar x|^\beta$ is small for $|\bar x|$ sufficiently large. Since $l=\epsilon+\delta$ is very narrow and $c_1(\tilde x)$ is bounded from below in
$\Sigma_{\lambda_0+\epsilon}\backslash \Sigma_{\lambda_0-\delta}$, $c_1(\tilde x)l^{\alpha}$ can be small. Consequently, $c_1(\tilde x)l^{\alpha}c_2(\bar x)|\bar x|^\beta<1$. This is a contradiction with \eqref{eq:aaa20} and \eqref{eq:a17} is proved.
\textbf{Case 2}. The negative minima of $U_\lambda(x)$ and $V_\lambda(x)$ lie in $B_{R_0}(0)$, i.e. they are all in the narrow region $\Sigma_{\lambda_0+\epsilon}\backslash \Sigma_{\lambda_0-\delta}$.
By \eqref{eq:a7},
\begin{equation}\label{eq:a23} F_{\alpha}(u_\lambda(\tilde x)) -F_{\alpha}(u(\tilde x)) \leq \frac{CU_\lambda(\tilde x)}{l^\alpha}<0,
\end{equation} where $l=\delta+\epsilon.$
Together with \eqref{eq:a3}, it implies
\begin{equation}\label{eq:aa8}
U_\lambda(\tilde x)\geq -c_1(\tilde x)l^\alpha V_\lambda(\tilde x).
\end{equation}
Similarly to \eqref{eq:a23}, we derive
$$F_\beta (v_\lambda(\bar x))-F_\beta (v(\bar x))\leq \frac{C V_\lambda(\bar x)}{l^\beta}<0.$$
Noting \eqref{eq:aa8}, we have for $l$ sufficiently small,
$$\aligned
0&\leq F_\beta (v_\lambda(\bar x))-F_\beta (v(\bar x))+c_2(\bar x)U_\lambda(\bar x)\\
&\leq\frac{CV_\lambda(\bar x)}{l^\beta}+c_2(\bar x)U_\lambda(\tilde x)\\
&\leq C\{\frac{V_\lambda(\bar x)}{l^\beta}-c_2(\bar x)c_1(\tilde x)l^\alpha V_\lambda(\tilde x)\}\\
&\leq C\{\frac{V_\lambda(\bar x)}{l^\beta}-c_2(\bar x)c_1(\tilde x)l^\alpha V_\lambda(\bar x)\}\\
&\leq C \frac{V_\lambda(\bar x)}{l^\beta}[1-c_1(\tilde x)c_2(\bar x)l^{\alpha+\beta}]\\
&<0.
\endaligned$$
This contradiction shows that \eqref{eq:a17} must be true.
Now we have shown that $ U_{\lambda_0}(x)\equiv 0,~V_{\lambda_0}(x)\equiv0, ~ x\in \Sigma_{\lambda_0}.$
Since the $x_1$ direction can be chosen arbitrarily, we actually prove that $u(x)$ and $v(x)$ must be radially symmetric about some point $x^0.$
Also the monotonicity follows easily from the argument.
This completes the proof of Theorem \ref{thma3}.
\section{Non-existence of solutions on a half space $R^n_+$ }
We investigate the system \eqref{eq:a2}.
{\bf Proof of Theorem \ref{thma4}.} Based on \eqref{eq:a28} and $f(0)=0,~g(0)=0$, one can see
from the proof of Lemma \ref{lemaa1} that
$$\mbox{either} ~u(x)>0,~v(x)>0~ \mbox{or} ~u(x)\equiv 0,~v(x)\equiv 0 , ~\mbox{ for}~ x\in R^n_+.$$
In fact, without loss of generality, assume $u(x)\not \equiv 0,$ there exists $x^0$ such that $u(x^0)=0$, and
$$F_\alpha(u(x^0))=c_{n,\alpha} PV \int _{R^n}\frac{G(u(x^0)-u(y))}{|x^0-y|^{n+\alpha}}dy<0,$$ i.e. $0\leq f(v(x))= F_\alpha(u(x))<0 ,$ this is impossible.
Hence if $u(x)$ or $v(x)$
attains 0 somewhere in $R^n_+$, then $u(x)=v(x)\equiv 0, x\in R^n_+.$
Hence in the following, we assume that $u(x)>0$ and $v(x)>0$ in $R^n_+.$
Let us carry on the method of moving planes on the solution $u$ along $x_n$ direction.
Dote $T_\lambda=\{x\in R^n| x_n=\lambda\},~\lambda>0,$ ~$\Sigma_\lambda=\{x\in R^n| 0<x_n<\lambda\}.$
Let $x^\lambda=(x_1,\cdots,x_{n-1},2\lambda-x_n)$ be the reflection of $x$ about the plane $T_\lambda$, and $U_\lambda(x)=u_\lambda(x)-u(x),~V_\lambda(x)=v_\lambda(x)-v(x).$
The key ingredient \eqref{eq:aa20} is obtained in this proof of Theorem \ref{thma2}. To see that it still applies in this situation, we only need to take $\Sigma=\Sigma_\lambda\cup R^n_-,$
where $R^n_-=\{x\in R^n| x_n\leq 0\}.$
Step1. For $\lambda$ sufficiently small, we have immediately
\begin{equation}\label{eq:a26}
U_\lambda(x)\geq 0, V_\lambda(x)\geq 0,~x\in \Sigma_\lambda,\end{equation}
since $\Sigma_\lambda$ is a narrow region.
Step2. Since \eqref{eq:a26} provides a starting point, we move the plane $T_\lambda$ upward as long as \eqref{eq:a26} holds.
Define $$\lambda_0=\sup\{\lambda>0|U_\mu(x)\geq 0, V_\mu(x)\geq 0,~x\in \Sigma_\mu,~ \mu \leq \lambda \}.$$
We show that \begin{equation}\label{eq:a27}\lambda_0=\infty.\end{equation}
Otherwise, if $\lambda_0<\infty,$ then using \eqref{eq:a26}, Theorem \ref{thma2}, Theorem \ref{thma1} and going through
the similar arguments as in Section 3, we are able to show
$$U_{\lambda_0}\equiv 0,~ V_{\lambda_0}\equiv 0, ~x\in \Sigma_{\lambda_0}, $$
which implies
$$u(x_1,\cdots, x_{n-1},2\lambda_0)=u(x_1,\cdots, x_{n-1},0)=0,$$
$$v(x_1,\cdots, x_{n-1},2\lambda_0)=v(x_1,\cdots, x_{n-1},0)=0.$$
This is impossible, because we assume that $u(x),v(x)>0$ in $R^n_+.$
Therefore, \eqref{eq:a27} must be valid and the solutions $u(x),v(x)$
are increasing with respect to $x_n.$ This contradicts \eqref{eq:a28}.
This completes the proof of Theorem \ref{thma4}.
\end{document} |